LMFDB - Embedded newform 115.2.b.b.24.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(24,115)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("115.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	115.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.918279623245$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.527896576.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 2x^5 + 7x^4 - 10x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.3
Root			$0.790245 - 0.790245i$ of defining polynomial
Character	$\chi$	$=$	115.24
Dual form			115.2.b.b.24.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.751024i q^{2} +0.580491i q^{3} +1.43596 q^{4} +(-0.209755 + 2.22621i) q^{5} +0.435963 q^{6} +0.315061i q^{7} -2.58049i q^{8} +2.66303 q^{9} +O(q^{10})$	\(q-0.751024i q^{2} +0.580491i q^{3} +1.43596 q^{4} +(-0.209755 + 2.22621i) q^{5} +0.435963 q^{6} +0.315061i q^{7} -2.58049i q^{8} +2.66303 q^{9} +(1.67194 + 0.157531i) q^{10} -4.34797 q^{11} +0.833563i q^{12} -4.58049i q^{13} +0.236619 q^{14} +(-1.29229 - 0.121761i) q^{15} +0.933914 q^{16} +0.917461i q^{17} -2.00000i q^{18} -2.76748 q^{19} +(-0.301200 + 3.19675i) q^{20} -0.182890 q^{21} +3.26543i q^{22} +1.00000i q^{23} +1.49795 q^{24} +(-4.91201 - 0.933914i) q^{25} -3.44006 q^{26} +3.28734i q^{27} +0.452416i q^{28} -7.03291 q^{29} +(-0.0914452 + 0.970544i) q^{30} -0.867829 q^{31} -5.86237i q^{32} -2.52396i q^{33} +0.689035 q^{34} +(-0.701392 - 0.0660856i) q^{35} +3.82401 q^{36} +4.68904i q^{37} +2.07844i q^{38} +2.65893 q^{39} +(5.74471 + 0.541270i) q^{40} +4.69184 q^{41} +0.137355i q^{42} -9.08944i q^{43} -6.24352 q^{44} +(-0.558583 + 5.92846i) q^{45} +0.751024 q^{46} +8.24762i q^{47} +0.542129i q^{48} +6.90074 q^{49} +(-0.701392 + 3.68904i) q^{50} -0.532578 q^{51} -6.57741i q^{52} -10.9048i q^{53} +2.46887 q^{54} +(0.912006 - 9.67948i) q^{55} +0.813013 q^{56} -1.60650i q^{57} +5.28188i q^{58} -1.50205 q^{59} +(-1.85569 - 0.174844i) q^{60} +11.4634 q^{61} +0.651760i q^{62} +0.839018i q^{63} -2.53496 q^{64} +(10.1971 + 0.960779i) q^{65} -1.89555 q^{66} -1.68494i q^{67} +1.31744i q^{68} -0.580491 q^{69} +(-0.0496318 + 0.526763i) q^{70} -5.36578 q^{71} -6.87193i q^{72} +10.1484i q^{73} +3.52158 q^{74} +(0.542129 - 2.85138i) q^{75} -3.97400 q^{76} -1.36988i q^{77} -1.99692i q^{78} -7.39760 q^{79} +(-0.195893 + 2.07909i) q^{80} +6.08082 q^{81} -3.52369i q^{82} +15.3052i q^{83} -0.262624 q^{84} +(-2.04246 - 0.192442i) q^{85} -6.82639 q^{86} -4.08254i q^{87} +11.2199i q^{88} +10.9326 q^{89} +(4.45242 + 0.419509i) q^{90} +1.44314 q^{91} +1.43596i q^{92} -0.503767i q^{93} +6.19416 q^{94} +(0.580491 - 6.16098i) q^{95} +3.40306 q^{96} +14.2158i q^{97} -5.18262i q^{98} -11.5788 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9	$ 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9} + 6 q^{10} + 4 q^{11} + 8 q^{14} + 6 q^{15} + 4 q^{16} + 8 q^{19} - 8 q^{20} - 4 q^{21} + 24 q^{24} - 16 q^{25} + 12 q^{26} - 8 q^{29} - 2 q^{30} - 28 q^{34} + 28 q^{35} - 16 q^{36} + 16 q^{39} - 10 q^{40} - 16 q^{41} - 12 q^{44} + 24 q^{45} + 28 q^{50} + 20 q^{51} - 44 q^{54} - 16 q^{55} + 28 q^{56} - 16 q^{60} - 16 q^{61} + 40 q^{64} - 14 q^{65} - 16 q^{66} + 4 q^{69} - 28 q^{70} - 48 q^{71} + 72 q^{74} - 36 q^{76} - 48 q^{79} - 2 q^{80} + 16 q^{81} - 4 q^{84} + 12 q^{85} + 28 q^{86} + 16 q^{89} - 4 q^{90} + 52 q^{91} + 84 q^{94} - 4 q^{95} + 60 q^{96} - 72 q^{99}+O(q^{100}) $ 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9 + 6 * q^10 + 4 * q^11 + 8 * q^14 + 6 * q^15 + 4 * q^16 + 8 * q^19 - 8 * q^20 - 4 * q^21 + 24 * q^24 - 16 * q^25 + 12 * q^26 - 8 * q^29 - 2 * q^30 - 28 * q^34 + 28 * q^35 - 16 * q^36 + 16 * q^39 - 10 * q^40 - 16 * q^41 - 12 * q^44 + 24 * q^45 + 28 * q^50 + 20 * q^51 - 44 * q^54 - 16 * q^55 + 28 * q^56 - 16 * q^60 - 16 * q^61 + 40 * q^64 - 14 * q^65 - 16 * q^66 + 4 * q^69 - 28 * q^70 - 48 * q^71 + 72 * q^74 - 36 * q^76 - 48 * q^79 - 2 * q^80 + 16 * q^81 - 4 * q^84 + 12 * q^85 + 28 * q^86 + 16 * q^89 - 4 * q^90 + 52 * q^91 + 84 * q^94 - 4 * q^95 + 60 * q^96 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	0.751024i		−	0.531054i	−0.964103	−	0.265527i	$-0.914454\pi$
$2$		−	0.751024i		−	0.531054i	0.964103	−	0.265527i	$-0.0855460\pi$
$3$			0.580491i			0.335147i	0.985860	+	0.167573i	$0.0535931\pi$
$3$			0.580491i			0.335147i	−0.985860	+	0.167573i	$0.946407\pi$
$4$	1.43596			0.717981
$5$	−0.209755	+	2.22621i	−0.0938051	+	0.995591i
$5$	−0.209755	+	2.22621i	−0.0938051	+	0.995591i
$6$	0.435963			0.177981
$7$			0.315061i			0.119082i	0.998226	+	0.0595410i	$0.0189637\pi$
$7$			0.315061i			0.119082i	−0.998226	+	0.0595410i	$0.981036\pi$
$8$		−	2.58049i		−	0.912341i
$9$	2.66303			0.887677
$10$	1.67194	+	0.157531i	0.528713	+	0.0498156i
$11$	−4.34797			−1.31096			−0.655481	−	0.755212i	$-0.727534\pi$
$11$	−4.34797			−1.31096			−0.655481	+	0.755212i	$0.727534\pi$
$12$			0.833563i			0.240629i
$13$		−	4.58049i		−	1.27040i	−0.772348	−	0.635200i	$-0.780918\pi$
$13$		−	4.58049i		−	1.27040i	0.772348	−	0.635200i	$-0.219082\pi$
$14$	0.236619			0.0632390
$15$	−1.29229	−	0.121761i	−0.333669	−	0.0314385i
$16$	0.933914			0.233479
$17$			0.917461i			0.222517i	0.993792	+	0.111258i	$0.0354881\pi$
$17$			0.917461i			0.222517i	−0.993792	+	0.111258i	$0.964512\pi$
$18$		−	2.00000i		−	0.471405i
$19$	−2.76748			−0.634903			−0.317451	−	0.948275i	$-0.602827\pi$
$19$	−2.76748			−0.634903			−0.317451	+	0.948275i	$0.602827\pi$
$20$	−0.301200	+	3.19675i	−0.0673503	+	0.714815i
$21$	−0.182890			−0.0399099
$22$			3.26543i			0.696192i
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$	1.49795			0.305768
$25$	−4.91201	−	0.933914i	−0.982401	−	0.186783i
$26$	−3.44006			−0.674651
$27$			3.28734i			0.632648i
$28$			0.452416i			0.0854987i
$29$	−7.03291			−1.30598			−0.652989	−	0.757367i	$-0.726485\pi$
$29$	−7.03291			−1.30598			−0.652989	+	0.757367i	$0.726485\pi$
$30$	−0.0914452	+	0.970544i	−0.0166955	+	0.177196i
$31$	−0.867829			−0.155867			−0.0779333	−	0.996959i	$-0.524832\pi$
$31$	−0.867829			−0.155867			−0.0779333	+	0.996959i	$0.524832\pi$
$32$		−	5.86237i		−	1.03633i
$33$		−	2.52396i		−	0.439364i
$34$	0.689035			0.118169
$35$	−0.701392	−	0.0660856i	−0.118557	−	0.0111705i
$36$	3.82401			0.637335
$37$			4.68904i			0.770873i	0.922734	+	0.385436i	$0.125949\pi$
$37$			4.68904i			0.770873i	−0.922734	+	0.385436i	$0.874051\pi$
$38$			2.07844i			0.337168i
$39$	2.65893			0.425770
$40$	5.74471	+	0.541270i	0.908318	+	0.0855822i
$41$	4.69184			0.732742			0.366371	−	0.930469i	$-0.380600\pi$
$41$	4.69184			0.732742			0.366371	+	0.930469i	$0.380600\pi$
$42$			0.137355i			0.0211943i
$43$		−	9.08944i		−	1.38613i	−0.720877	−	0.693063i	$-0.756261\pi$
$43$		−	9.08944i		−	1.38613i	0.720877	−	0.693063i	$-0.243739\pi$
$44$	−6.24352			−0.941246
$45$	−0.558583	+	5.92846i	−0.0832686	+	0.883763i
$46$	0.751024			0.110732
$47$			8.24762i			1.20304i	0.798858	+	0.601519i	$0.205438\pi$
$47$			8.24762i			1.20304i	−0.798858	+	0.601519i	$0.794562\pi$
$48$			0.542129i			0.0782496i
$49$	6.90074			0.985819
$50$	−0.701392	+	3.68904i	−0.0991919	+	0.521708i
$51$	−0.532578			−0.0745758
$52$		−	6.57741i		−	0.912123i
$53$		−	10.9048i		−	1.49789i	−0.662630	−	0.748947i	$-0.730560\pi$
$53$		−	10.9048i		−	1.49789i	0.662630	−	0.748947i	$-0.269440\pi$
$54$	2.46887			0.335971
$55$	0.912006	−	9.67948i	0.122975	−	1.30518i
$56$	0.813013			0.108643
$57$		−	1.60650i		−	0.212786i
$58$			5.28188i			0.693545i
$59$	−1.50205			−0.195550			−0.0977750	−	0.995209i	$-0.531173\pi$
$59$	−1.50205			−0.195550			−0.0977750	+	0.995209i	$0.531173\pi$
$60$	−1.85569	−	0.174844i	−0.239568	−	0.0225722i
$61$	11.4634			1.46774			0.733870	−	0.679290i	$-0.237712\pi$
$61$	11.4634			1.46774			0.733870	+	0.679290i	$0.237712\pi$
$62$			0.651760i			0.0827737i
$63$			0.839018i			0.105706i
$64$	−2.53496			−0.316869
$65$	10.1971	+	0.960779i	1.26480	+	0.119170i
$66$	−1.89555			−0.233326
$67$		−	1.68494i		−	0.205848i	−0.994689	−	0.102924i	$-0.967180\pi$
$67$		−	1.68494i		−	0.205848i	0.994689	−	0.102924i	$-0.0328198\pi$
$68$			1.31744i			0.159763i
$69$	−0.580491			−0.0698829
$70$	−0.0496318	+	0.526763i	−0.00593214	+	0.0629602i
$71$	−5.36578			−0.636801			−0.318401	−	0.947956i	$-0.603146\pi$
$71$	−5.36578			−0.636801			−0.318401	+	0.947956i	$0.603146\pi$
$72$		−	6.87193i		−	0.809864i
$73$			10.1484i			1.18778i	0.804548	+	0.593888i	$0.202408\pi$
$73$			10.1484i			1.18778i	−0.804548	+	0.593888i	$0.797592\pi$
$74$	3.52158			0.409375
$75$	0.542129	−	2.85138i	0.0625997	−	0.329248i
$76$	−3.97400			−0.455848
$77$		−	1.36988i		−	0.156112i
$78$		−	1.99692i		−	0.226107i
$79$	−7.39760			−0.832295			−0.416148	−	0.909297i	$-0.636620\pi$
$79$	−7.39760			−0.832295			−0.416148	+	0.909297i	$0.636620\pi$
$80$	−0.195893	+	2.07909i	−0.0219015	+	0.232449i
$81$	6.08082			0.675647
$82$		−	3.52369i		−	0.389126i
$83$			15.3052i			1.67997i	0.542611	+	0.839984i	$0.317436\pi$
$83$			15.3052i			1.67997i	−0.542611	+	0.839984i	$0.682564\pi$
$84$	−0.262624			−0.0286546
$85$	−2.04246	−	0.192442i	−0.221536	−	0.0208732i
$86$	−6.82639			−0.736109
$87$		−	4.08254i		−	0.437694i
$88$			11.2199i			1.19604i
$89$	10.9326			1.15885			0.579424	−	0.815026i	$-0.303277\pi$
$89$	10.9326			1.15885			0.579424	+	0.815026i	$0.303277\pi$
$90$	4.45242	+	0.419509i	0.469326	+	0.0442201i
$91$	1.44314			0.151282
$92$			1.43596i			0.149709i
$93$		−	0.503767i		−	0.0522382i
$94$	6.19416			0.638879
$95$	0.580491	−	6.16098i	0.0595571	−	0.632103i
$96$	3.40306			0.347323
$97$			14.2158i			1.44340i	0.692208	+	0.721698i	$0.256638\pi$
$97$			14.2158i			1.44340i	−0.692208	+	0.721698i	$0.743362\pi$
$98$		−	5.18262i		−	0.523524i
$99$	−11.5788			−1.16371
$100$	−7.05346	−	1.34107i	−0.705346	−	0.134107i
$101$	−8.16508			−0.812456			−0.406228	−	0.913772i	$-0.633156\pi$
$101$	−8.16508			−0.812456			−0.406228	+	0.913772i	$0.633156\pi$
$102$			0.399979i			0.0396038i
$103$		−	0.287338i		−	0.0283122i	−0.999900	−	0.0141561i	$-0.995494\pi$
$103$		−	0.287338i		−	0.0283122i	0.999900	−	0.0141561i	$-0.00450618\pi$
$104$	−11.8199			−1.15904
$105$	0.0383621	−	0.407152i	0.00374375	−	0.0397340i
$106$	−8.18979			−0.795463
$107$		−	14.0935i		−	1.36247i	−0.732063	−	0.681237i	$-0.761442\pi$
$107$		−	14.0935i		−	1.36247i	0.732063	−	0.681237i	$-0.238558\pi$
$108$			4.72049i			0.454230i
$109$	−1.76338			−0.168901			−0.0844506	−	0.996428i	$-0.526914\pi$
$109$	−1.76338			−0.168901			−0.0844506	+	0.996428i	$0.526914\pi$
$110$	−7.26953	−	0.684939i	−0.693122	−	0.0653063i
$111$	−2.72194			−0.258355
$112$			0.294240i			0.0278031i
$113$			0.315061i			0.0296385i	0.999890	+	0.0148192i	$0.00471728\pi$
$113$			0.315061i			0.0296385i	−0.999890	+	0.0148192i	$0.995283\pi$
$114$	−1.20652			−0.113001
$115$	−2.22621	−	0.209755i	−0.207595	−	0.0195597i
$116$	−10.0990			−0.937668
$117$		−	12.1980i		−	1.12770i
$118$			1.12807i			0.103848i
$119$	−0.289056			−0.0264978
$120$	−0.314202	+	3.33475i	−0.0286826	+	0.304420i
$121$	7.90483			0.718621
$122$		−	8.60930i		−	0.779450i
$123$			2.72357i			0.245576i
$124$	−1.24617			−0.111909
$125$	3.10940	−	10.7393i	0.278114	−	0.960548i
$126$	0.630123			0.0561358
$127$			21.3284i			1.89259i	0.323300	+	0.946296i	$0.395208\pi$
$127$			21.3284i			1.89259i	−0.323300	+	0.946296i	$0.604792\pi$
$128$		−	9.82094i		−	0.868056i
$129$	5.27634			0.464556
$130$	0.721568	−	7.65829i	0.0632857	−	0.671676i
$131$	8.11373			0.708900			0.354450	−	0.935075i	$-0.384668\pi$
$131$	8.11373			0.708900			0.354450	+	0.935075i	$0.384668\pi$
$132$		−	3.62431i		−	0.315455i
$133$		−	0.871925i		−	0.0756055i
$134$	−1.26543			−0.109316
$135$	−7.31830	−	0.689534i	−0.629859	−	0.0593456i
$136$	2.36750			0.203011
$137$			7.22680i			0.617427i	0.951155	+	0.308713i	$0.0998984\pi$
$137$			7.22680i			0.617427i	−0.951155	+	0.308713i	$0.900102\pi$
$138$			0.435963i			0.0371116i
$139$	10.8431			0.919701			0.459850	−	0.887996i	$-0.347903\pi$
$139$	10.8431			0.919701			0.459850	+	0.887996i	$0.347903\pi$
$140$	−1.00717	−	0.0948964i	−0.0851217	−	0.00802021i
$141$	−4.78767			−0.403194
$142$			4.02983i			0.338176i
$143$			19.9158i			1.66545i
$144$	2.48704			0.207254
$145$	1.47518	−	15.6567i	0.122507	−	1.30022i
$146$	7.62166			0.630773
$147$			4.00582i			0.330394i
$148$			6.73328i			0.553472i
$149$	7.90074			0.647254			0.323627	−	0.946185i	$-0.395098\pi$
$149$	7.90074			0.647254			0.323627	+	0.946185i	$0.395098\pi$
$150$	−2.14145	−	0.407152i	−0.174849	−	0.0332438i
$151$	−8.03700			−0.654042			−0.327021	−	0.945017i	$-0.606045\pi$
$151$	−8.03700			−0.654042			−0.327021	+	0.945017i	$0.606045\pi$
$152$			7.14145i			0.579248i
$153$			2.44323i			0.197523i
$154$	−1.02881			−0.0829039
$155$	0.182031	−	1.93197i	0.0146211	−	0.155179i
$156$	3.81813			0.305695
$157$			4.75928i			0.379832i	0.981800	+	0.189916i	$0.0608216\pi$
$157$			4.75928i			0.379832i	−0.981800	+	0.189916i	$0.939178\pi$
$158$			5.55578i			0.441994i
$159$	6.33016			0.502014
$160$	13.0509	+	1.22966i	1.03176	+	0.0972131i
$161$	−0.315061			−0.0248303
$162$		−	4.56684i		−	0.358805i
$163$		−	2.11545i		−	0.165695i	−0.996562	−	0.0828473i	$-0.973599\pi$
$163$		−	2.11545i		−	0.165695i	0.996562	−	0.0828473i	$-0.0264014\pi$
$164$	6.73731			0.526095
$165$	5.61885	+	0.529411i	0.437427	+	0.0412146i
$166$	11.4946			0.892154
$167$		−	15.8760i		−	1.22852i	−0.789102	−	0.614262i	$-0.789454\pi$
$167$		−	15.8760i		−	1.22852i	0.789102	−	0.614262i	$-0.210546\pi$
$168$			0.471947i			0.0364115i
$169$	−7.98090			−0.613915
$170$	−0.144528	+	1.53394i	−0.0110848	+	0.117648i
$171$	−7.36988			−0.563589
$172$		−	13.0521i		−	0.995213i
$173$			0.0767241i			0.00583323i	0.999996	+	0.00291661i	$0.000928388\pi$
$173$			0.0767241i			0.00583323i	−0.999996	+	0.00291661i	$0.999072\pi$
$174$	−3.06609			−0.232439
$175$	0.294240	−	1.54758i	0.0222425	−	0.116986i
$176$	−4.06063			−0.306082
$177$		−	0.871925i		−	0.0655379i
$178$		−	8.21061i		−	0.615412i
$179$	20.7247			1.54904			0.774520	−	0.632549i	$-0.217991\pi$
$179$	20.7247			1.54904			0.774520	+	0.632549i	$0.217991\pi$
$180$	−0.802104	+	8.51305i	−0.0597853	+	0.634525i
$181$	−11.5610			−0.859319			−0.429660	−	0.902991i	$-0.641366\pi$
$181$	−11.5610			−0.859319			−0.429660	+	0.902991i	$0.641366\pi$
$182$		−	1.08383i		−	0.0803388i
$183$			6.65441i			0.491908i
$184$	2.58049			0.190236
$185$	−10.4388	−	0.983546i	−0.767474	−	0.0723118i
$186$	−0.378341			−0.0277413
$187$		−	3.98909i		−	0.291711i
$188$			11.8433i			0.863759i
$189$	−1.03571			−0.0753371
$190$	−4.62705	−	0.435963i	−0.335681	−	0.0316281i
$191$	21.9683			1.58957			0.794784	−	0.606892i	$-0.207584\pi$
$191$	21.9683			1.58957			0.794784	+	0.606892i	$0.207584\pi$
$192$		−	1.47152i		−	0.106198i
$193$		−	20.2914i		−	1.46061i	−0.683122	−	0.730305i	$-0.739378\pi$
$193$		−	20.2914i		−	1.46061i	0.683122	−	0.730305i	$-0.260622\pi$
$194$	10.6764			0.766521
$195$	−0.557723	+	5.91934i	−0.0399394	+	0.423893i
$196$	9.90920			0.707800
$197$			8.62159i			0.614263i	0.951667	+	0.307131i	$0.0993691\pi$
$197$			8.62159i			0.614263i	−0.951667	+	0.307131i	$0.900631\pi$
$198$			8.69594i			0.617993i
$199$	−20.7796			−1.47302			−0.736512	−	0.676424i	$-0.763529\pi$
$199$	−20.7796			−1.47302			−0.736512	+	0.676424i	$0.763529\pi$
$200$	−2.40996	+	12.6754i	−0.170410	+	0.896285i
$201$	0.978092			0.0689893
$202$			6.13217i			0.431458i
$203$		−	2.21580i		−	0.155519i
$204$	−0.764762			−0.0535440
$205$	−0.984135	+	10.4450i	−0.0687350	+	0.729512i
$206$	−0.215798			−0.0150353
$207$			2.66303i			0.185093i
$208$		−	4.27779i		−	0.296611i
$209$	12.0329			0.832334
$210$	−0.305781	−	0.0288108i	−0.0211009	−	0.00198814i
$211$	−4.16508			−0.286736			−0.143368	−	0.989669i	$-0.545793\pi$
$211$	−4.16508			−0.286736			−0.143368	+	0.989669i	$0.545793\pi$
$212$		−	15.6589i		−	1.07546i
$213$		−	3.11479i		−	0.213422i
$214$	−10.5846			−0.723548
$215$	20.2350	+	1.90655i	1.38001	+	0.130026i
$216$	8.48295			0.577191
$217$		−	0.273419i		−	0.0185609i
$218$			1.32434i			0.0896958i
$219$	−5.89103			−0.398079
$220$	1.30961	−	13.8994i	0.0882937	−	0.937096i
$221$	4.20242			0.282685
$222$			2.04424i			0.137201i
$223$		−	18.7618i		−	1.25638i	−0.778060	−	0.628190i	$-0.783796\pi$
$223$		−	18.7618i		−	1.25638i	0.778060	−	0.628190i	$-0.216204\pi$
$224$	1.84701			0.123408
$225$	−13.0808	−	2.48704i	−0.872055	−	0.165803i
$226$	0.236619			0.0157396
$227$		−	10.9637i		−	0.727689i	−0.931460	−	0.363845i	$-0.881464\pi$
$227$		−	10.9637i		−	0.727689i	0.931460	−	0.363845i	$-0.118536\pi$
$228$		−	2.30687i		−	0.152776i
$229$	−25.9367			−1.71394			−0.856971	−	0.515364i	$-0.827657\pi$
$229$	−25.9367			−1.71394			−0.856971	+	0.515364i	$0.827657\pi$
$230$	−0.157531	+	1.67194i	−0.0103873	+	0.110244i
$231$	0.795201			0.0523204
$232$			18.1484i			1.19150i
$233$		−	8.23942i		−	0.539783i	−0.962891	−	0.269891i	$-0.913012\pi$
$233$		−	8.23942i		−	0.539783i	0.962891	−	0.269891i	$-0.0869878\pi$
$234$	−9.16098			−0.598872
$235$	−18.3609	−	1.72998i	−1.19773	−	0.112851i
$236$	−2.15689			−0.140401
$237$		−	4.29424i		−	0.278941i
$238$			0.217088i			0.0140718i
$239$	−16.0508			−1.03824			−0.519120	−	0.854701i	$-0.673740\pi$
$239$	−16.0508			−1.03824			−0.519120	+	0.854701i	$0.673740\pi$
$240$	−1.20689	−	0.113714i	−0.0779045	−	0.00734021i
$241$	−2.09755			−0.135115			−0.0675574	−	0.997715i	$-0.521521\pi$
$241$	−2.09755			−0.135115			−0.0675574	+	0.997715i	$0.521521\pi$
$242$		−	5.93672i		−	0.381627i
$243$			13.3919i			0.859089i
$244$	16.4610			1.05381
$245$	−1.44746	+	15.3625i	−0.0924749	+	0.981473i
$246$	2.04547			0.130414
$247$			12.6764i			0.806580i
$248$			2.23942i			0.142204i
$249$	−8.88455			−0.563036
$250$	−8.06544	−	2.33524i	−0.510103	−	0.147693i
$251$	−13.7018			−0.864847			−0.432423	−	0.901671i	$-0.642341\pi$
$251$	−13.7018			−0.864847			−0.432423	+	0.901671i	$0.642341\pi$
$252$			1.20480i			0.0758952i
$253$		−	4.34797i		−	0.273354i
$254$	16.0182			1.00507
$255$	0.111711	−	1.18563i	0.00699559	−	0.0742470i
$256$	−12.4457			−0.777854
$257$		−	0.998281i		−	0.0622711i	−0.999515	−	0.0311355i	$-0.990088\pi$
$257$		−	0.998281i		−	0.0622711i	0.999515	−	0.0311355i	$-0.00991235\pi$
$258$		−	3.96266i		−	0.246704i
$259$	−1.47733			−0.0917971
$260$	14.6427	+	1.37964i	0.908101	+	0.0855618i
$261$	−18.7288			−1.15929
$262$		−	6.09361i		−	0.376464i
$263$		−	9.18527i		−	0.566388i	−0.959063	−	0.283194i	$-0.908606\pi$
$263$		−	9.18527i		−	0.566388i	0.959063	−	0.283194i	$-0.0913940\pi$
$264$	−6.51305			−0.400850
$265$	24.2764	+	2.28734i	1.49129	+	0.140510i
$266$	−0.654837			−0.0401506
$267$			6.34625i			0.388384i
$268$		−	2.41951i		−	0.147795i
$269$	−21.4398			−1.30721			−0.653603	−	0.756837i	$-0.726744\pi$
$269$	−21.4398			−1.30721			−0.653603	+	0.756837i	$0.726744\pi$
$270$	−0.517857	+	5.49622i	−0.0315158	+	0.334489i
$271$	−25.2488			−1.53376			−0.766878	−	0.641793i	$-0.778191\pi$
$271$	−25.2488			−1.53376			−0.766878	+	0.641793i	$0.778191\pi$
$272$			0.856830i			0.0519529i
$273$			0.837727i			0.0507016i
$274$	5.42750			0.327887
$275$	21.3572	+	4.06063i	1.28789	+	0.244865i
$276$	−0.833563			−0.0501746
$277$			30.8114i			1.85128i	0.378409	+	0.925638i	$0.376471\pi$
$277$			30.8114i			1.85128i	−0.378409	+	0.925638i	$0.623529\pi$
$278$		−	8.14344i		−	0.488411i
$279$	−2.31105			−0.138359
$280$	−0.170533	+	1.80994i	−0.0101913	+	0.108164i
$281$	3.60821			0.215248			0.107624	−	0.994192i	$-0.465676\pi$
$281$	3.60821			0.215248			0.107624	+	0.994192i	$0.465676\pi$
$282$			3.59565i			0.214118i
$283$			5.23942i			0.311451i	0.987800	+	0.155726i	$0.0497716\pi$
$283$			5.23942i			0.311451i	−0.987800	+	0.155726i	$0.950228\pi$
$284$	−7.70506			−0.457211
$285$	3.57639	+	0.336970i	0.211847	+	0.0199604i
$286$	14.9573			0.884442
$287$			1.47822i			0.0872565i
$288$		−	15.6117i		−	0.919927i
$289$	16.1583			0.950486
$290$	−11.7586	−	1.10790i	−0.690487	−	0.0650581i
$291$	−8.25214			−0.483749
$292$			14.5727i			0.852800i
$293$		−	1.90074i		−	0.111042i	−0.998458	−	0.0555211i	$-0.982318\pi$
$293$		−	1.90074i		−	0.111042i	0.998458	−	0.0555211i	$-0.0176820\pi$
$294$	3.00846			0.175457
$295$	0.315061	−	3.34387i	0.0183436	−	0.194688i
$296$	12.1000			0.703299
$297$		−	14.2932i		−	0.829378i
$298$		−	5.93364i		−	0.343727i
$299$	4.58049			0.264897
$300$	0.778477	−	4.09447i	0.0449454	−	0.236394i
$301$	2.86373			0.165063
$302$			6.03598i			0.347332i
$303$		−	4.73975i		−	0.272292i
$304$	−2.58459			−0.148236
$305$	−2.40450	+	25.5199i	−0.137681	+	1.46127i
$306$	1.83492			0.104895
$307$			2.82467i			0.161213i	0.996746	+	0.0806063i	$0.0256856\pi$
$307$			2.82467i			0.161213i	−0.996746	+	0.0806063i	$0.974314\pi$
$308$		−	1.96709i		−	0.112086i
$309$	0.166797			0.00948875
$310$	−1.45095	−	0.136710i	−0.0824087	−	0.00776459i
$311$	16.1028			0.913107			0.456554	−	0.889696i	$-0.349084\pi$
$311$	16.1028			0.913107			0.456554	+	0.889696i	$0.349084\pi$
$312$		−	6.86135i		−	0.388448i
$313$		−	26.3740i		−	1.49075i	−0.666648	−	0.745373i	$-0.732272\pi$
$313$		−	26.3740i		−	1.49075i	0.666648	−	0.745373i	$-0.267728\pi$
$314$	3.57434			0.201712
$315$	−1.86783	−	0.175988i	−0.105240	−	0.00991579i
$316$	−10.6227			−0.597572
$317$		−	15.5020i		−	0.870682i	−0.900266	−	0.435341i	$-0.856628\pi$
$317$		−	15.5020i		−	0.870682i	0.900266	−	0.435341i	$-0.143372\pi$
$318$		−	4.75410i		−	0.266597i
$319$	30.5789			1.71209
$320$	0.531718	−	5.64334i	0.0297240	−	0.315472i
$321$	8.18117			0.456628
$322$			0.236619i			0.0131862i
$323$		−	2.53905i		−	0.141277i
$324$	8.73183			0.485102
$325$	−4.27779	+	22.4994i	−0.237289	+	1.24804i
$326$	−1.58875			−0.0879928
$327$		−	1.02363i		−	0.0566067i
$328$		−	12.1073i		−	0.668511i
$329$	−2.59851			−0.143260
$330$	0.397601	−	4.21989i	0.0218872	−	0.232298i
$331$	19.0302			1.04599			0.522997	−	0.852335i	$-0.324814\pi$
$331$	19.0302			1.04599			0.522997	+	0.852335i	$0.324814\pi$
$332$			21.9778i			1.20619i
$333$			12.4870i			0.684286i
$334$	−11.9233			−0.652413
$335$	3.75102	+	0.353423i	0.204940	+	0.0193096i
$336$	−0.170804			−0.00931812
$337$		−	4.09354i		−	0.222989i	−0.993765	−	0.111495i	$-0.964436\pi$
$337$		−	4.09354i		−	0.222989i	0.993765	−	0.111495i	$-0.0355638\pi$
$338$			5.99385i			0.326022i
$339$	−0.182890			−0.00993324
$340$	−2.93289	−	0.276339i	−0.159059	−	0.0149866i
$341$	3.77329			0.204335
$342$			5.53496i			0.299296i
$343$			4.37959i			0.236475i
$344$	−23.4552			−1.26462
$345$	0.121761	−	1.29229i	0.00655537	−	0.0695748i
$346$	0.0576217			0.00309776
$347$		−	17.4357i		−	0.935997i	−0.883729	−	0.467998i	$-0.844975\pi$
$347$		−	17.4357i		−	0.935997i	0.883729	−	0.467998i	$-0.155025\pi$
$348$		−	5.86237i		−	0.314256i
$349$	−8.48770			−0.454336			−0.227168	−	0.973856i	$-0.572947\pi$
$349$	−8.48770			−0.454336			−0.227168	+	0.973856i	$0.572947\pi$
$350$	−1.16227	−	0.220982i	−0.0621261	−	0.0118120i
$351$	15.0576			0.803716
$352$			25.4894i			1.35859i
$353$		−	20.5696i		−	1.09481i	−0.836868	−	0.547404i	$-0.815616\pi$
$353$		−	20.5696i		−	1.09481i	0.836868	−	0.547404i	$-0.184384\pi$
$354$	−0.654837			−0.0348042
$355$	1.12550	−	11.9453i	0.0597352	−	0.633993i
$356$	15.6987			0.832032
$357$		−	0.167795i		−	0.00888064i
$358$		−	15.5648i		−	0.822625i
$359$	22.4121			1.18286			0.591432	−	0.806355i	$-0.298563\pi$
$359$	22.4121			1.18286			0.591432	+	0.806355i	$0.298563\pi$
$360$	15.2983	+	1.44142i	0.806293	+	0.0759694i
$361$	−11.3411			−0.596898
$362$			8.68256i			0.456345i
$363$			4.58868i			0.240843i
$364$	2.07229			0.108617
$365$	−22.5923	−	2.12866i	−1.18254	−	0.111419i
$366$	4.99762			0.261230
$367$		−	21.9401i		−	1.14526i	−0.819812	−	0.572632i	$-0.805922\pi$
$367$		−	21.9401i		−	1.14526i	0.819812	−	0.572632i	$-0.194078\pi$
$368$			0.933914i			0.0486837i
$369$	12.4945			0.650438
$370$	−0.738667	+	7.83977i	−0.0384015	+	0.407570i
$371$	3.43569			0.178372
$372$		−	0.723390i		−	0.0375060i
$373$		−	0.853821i		−	0.0442092i	−0.999756	−	0.0221046i	$-0.992963\pi$
$373$		−	0.853821i		−	0.0442092i	0.999756	−	0.0221046i	$-0.00703668\pi$
$374$	−2.99590			−0.154914
$375$	6.23404	+	1.80498i	0.321924	+	0.0932088i
$376$	21.2829			1.09758
$377$			32.2142i			1.65911i
$378$			0.777846i			0.0400081i
$379$	2.52049			0.129469			0.0647345	−	0.997903i	$-0.479380\pi$
$379$	2.52049			0.129469			0.0647345	+	0.997903i	$0.479380\pi$
$380$	0.833563	−	8.84694i	0.0427609	−	0.453838i
$381$	−12.3810			−0.634296
$382$		−	16.4987i		−	0.844147i
$383$			13.5983i			0.694841i	0.937709	+	0.347420i	$0.112942\pi$
$383$			13.5983i			0.694841i	−0.937709	+	0.347420i	$0.887058\pi$
$384$	5.70096			0.290926
$385$	3.04963	+	0.287338i	0.155424	+	0.0146441i
$386$	−15.2394			−0.775663
$387$		−	24.2055i		−	1.23043i
$388$			20.4134i			1.03633i
$389$	−10.7398			−0.544527			−0.272264	−	0.962223i	$-0.587772\pi$
$389$	−10.7398			−0.544527			−0.272264	+	0.962223i	$0.587772\pi$
$390$	4.44557	+	0.418864i	0.225110	+	0.0212100i
$391$	−0.917461			−0.0463980
$392$		−	17.8073i		−	0.899404i
$393$			4.70995i			0.237585i
$394$	6.47502			0.326207
$395$	1.55168	−	16.4686i	0.0780735	−	0.828625i
$396$	−16.6267			−0.835522
$397$			19.5868i			0.983031i	0.870869	+	0.491516i	$0.163557\pi$
$397$			19.5868i			0.983031i	−0.870869	+	0.491516i	$0.836443\pi$
$398$			15.6060i			0.782256i
$399$	0.506145			0.0253389
$400$	−4.58739	−	0.872196i	−0.229370	−	0.0436098i
$401$	−8.01209			−0.400105			−0.200052	−	0.979785i	$-0.564111\pi$
$401$	−8.01209			−0.400105			−0.200052	+	0.979785i	$0.564111\pi$
$402$		−	0.734570i		−	0.0366370i
$403$			3.97508i			0.198013i
$404$	−11.7247			−0.583328
$405$	−1.27548	+	13.5372i	−0.0633791	+	0.672668i
$406$	−1.66412			−0.0825888
$407$		−	20.3878i		−	1.01058i
$408$			1.37431i			0.0680386i
$409$	28.2350			1.39613			0.698065	−	0.716034i	$-0.254045\pi$
$409$	28.2350			1.39613			0.698065	+	0.716034i	$0.254045\pi$
$410$	7.84446	+	0.739109i	0.387410	+	0.0365020i
$411$	−4.19509			−0.206929
$412$		−	0.412607i		−	0.0203277i
$413$		−	0.473237i		−	0.0232865i
$414$	2.00000			0.0982946
$415$	−34.0726	−	3.21034i	−1.67256	−	0.157590i
$416$	−26.8526			−1.31655
$417$			6.29433i			0.308235i
$418$		−	9.03700i		−	0.442014i
$419$	21.9470			1.07218			0.536091	−	0.844160i	$-0.319900\pi$
$419$	21.9470			1.07218			0.536091	+	0.844160i	$0.319900\pi$
$420$	0.0550865	−	0.584655i	0.00268795	−	0.0285282i
$421$	0.747198			0.0364162			0.0182081	−	0.999834i	$-0.494204\pi$
$421$	0.747198			0.0364162			0.0182081	+	0.999834i	$0.494204\pi$
$422$			3.12807i			0.152272i
$423$			21.9637i			1.06791i
$424$	−28.1398			−1.36659
$425$	0.856830	−	4.50657i	0.0415624	−	0.218601i
$426$	−2.33928			−0.113339
$427$			3.61168i			0.174781i
$428$		−	20.2378i		−	0.978231i
$429$	−11.5610			−0.558168
$430$	1.43187	−	15.1970i	0.0690507	−	0.732863i
$431$	27.3642			1.31808			0.659042	−	0.752106i	$-0.270962\pi$
$431$	27.3642			1.31808			0.659042	+	0.752106i	$0.270962\pi$
$432$			3.07009i			0.147710i
$433$		−	25.4265i		−	1.22192i	−0.791662	−	0.610960i	$-0.790784\pi$
$433$		−	25.4265i		−	1.22192i	0.791662	−	0.610960i	$-0.209216\pi$
$434$	−0.205345			−0.00985685
$435$	9.08858	+	0.856331i	0.435764	+	0.0410579i
$436$	−2.53215			−0.121268
$437$		−	2.76748i		−	0.132386i
$438$			4.42430i			0.211401i
$439$	30.8097			1.47047			0.735233	−	0.677815i	$-0.237073\pi$
$439$	30.8097			1.47047			0.735233	+	0.677815i	$0.237073\pi$
$440$	−24.9778	−	2.35342i	−1.19077	−	0.112195i
$441$	18.3769			0.875089
$442$		−	3.15612i		−	0.150121i
$443$			24.5778i			1.16773i	0.811852	+	0.583863i	$0.198459\pi$
$443$			24.5778i			1.16773i	−0.811852	+	0.583863i	$0.801541\pi$
$444$	−3.90861			−0.185494
$445$	−2.29315	+	24.3381i	−0.108706	+	1.15374i
$446$	−14.0905			−0.667206
$447$			4.58631i			0.216925i
$448$		−	0.798667i		−	0.0377335i
$449$	−38.6816			−1.82550			−0.912749	−	0.408522i	$-0.866044\pi$
$449$	−38.6816			−1.82550			−0.912749	+	0.408522i	$0.866044\pi$
$450$	−1.86783	+	9.82401i	−0.0880503	+	0.463108i
$451$	−20.4000			−0.960597
$452$			0.452416i			0.0212799i
$453$		−	4.66541i		−	0.219200i
$454$	−8.23404			−0.386443
$455$	−0.302704	+	3.21272i	−0.0141910	+	0.150615i
$456$	−4.14555			−0.194133
$457$		−	6.40041i		−	0.299398i	−0.988732	−	0.149699i	$-0.952169\pi$
$457$		−	6.40041i		−	0.299398i	0.988732	−	0.149699i	$-0.0478305\pi$
$458$			19.4791i			0.910196i
$459$	−3.01600			−0.140775
$460$	−3.19675	−	0.301200i	−0.149049	−	0.0140435i
$461$	−23.8337			−1.11005			−0.555024	−	0.831835i	$-0.687291\pi$
$461$	−23.8337			−1.11005			−0.555024	+	0.831835i	$0.687291\pi$
$462$		−	0.597215i		−	0.0277850i
$463$			26.5529i			1.23402i	0.786957	+	0.617008i	$0.211655\pi$
$463$			26.5529i			1.23402i	−0.786957	+	0.617008i	$0.788345\pi$
$464$	−6.56813			−0.304918
$465$	1.12149	+	0.105667i	0.0520078	+	0.00490021i
$466$	−6.18801			−0.286654
$467$			18.6049i			0.860931i	0.902607	+	0.430465i	$0.141651\pi$
$467$			18.6049i			0.860931i	−0.902607	+	0.430465i	$0.858349\pi$
$468$		−	17.5159i		−	0.809671i
$469$	0.530859			0.0245128
$470$	−1.29925	+	13.7895i	−0.0599301	+	0.636062i
$471$	−2.76272			−0.127300
$472$			3.87602i			0.178408i
$473$			39.5206i			1.81716i
$474$	−3.22508			−0.148133
$475$	13.5939	+	2.58459i	0.623729	+	0.118589i
$476$	−0.415074			−0.0190249
$477$		−	29.0399i		−	1.32965i
$478$			12.0545i			0.551362i
$479$	−29.9627			−1.36903			−0.684514	−	0.728999i	$-0.739986\pi$
$479$	−29.9627			−1.36903			−0.684514	+	0.728999i	$0.739986\pi$
$480$	−0.713806	+	7.57591i	−0.0325806	+	0.345791i
$481$	21.4781			0.979316
$482$			1.57531i			0.0717533i
$483$		−	0.182890i		−	0.00832180i
$484$	11.3510			0.515957
$485$	−31.6473	−	2.98183i	−1.43703	−	0.135398i
$486$	10.0576			0.456223
$487$		−	37.2265i		−	1.68689i	−0.537214	−	0.843446i	$-0.680523\pi$
$487$		−	37.2265i		−	1.68689i	0.537214	−	0.843446i	$-0.319477\pi$
$488$		−	29.5812i		−	1.33908i
$489$	1.22800			0.0555320
$490$	11.5376	+	1.08708i	0.521215	+	0.0491092i
$491$	−0.691841			−0.0312224			−0.0156112	−	0.999878i	$-0.504969\pi$
$491$	−0.691841			−0.0312224			−0.0156112	+	0.999878i	$0.504969\pi$
$492$			3.91095i			0.176319i
$493$		−	6.45242i		−	0.290602i
$494$	9.52029			0.428338
$495$	2.42870	−	25.7768i	0.109162	−	1.15858i
$496$	−0.810478			−0.0363915
$497$		−	1.69055i		−	0.0758315i
$498$			6.67251i			0.299003i
$499$	24.7988			1.11014			0.555072	−	0.831802i	$-0.312691\pi$
$499$	24.7988			1.11014			0.555072	+	0.831802i	$0.312691\pi$
$500$	4.46499	−	15.4212i	0.199680	−	0.689656i
$501$	9.21589			0.411735
$502$			10.2903i			0.459281i
$503$		−	22.6049i		−	1.00790i	−0.863732	−	0.503951i	$-0.831879\pi$
$503$		−	22.6049i		−	1.00790i	0.863732	−	0.503951i	$-0.168121\pi$
$504$	2.16508			0.0964403
$505$	1.71266	−	18.1772i	0.0762125	−	0.808873i
$506$	−3.26543			−0.145166
$507$		−	4.63284i		−	0.205752i
$508$			30.6268i			1.35885i
$509$	19.1240			0.847655			0.423828	−	0.905743i	$-0.360686\pi$
$509$	19.1240			0.847655			0.423828	+	0.905743i	$0.360686\pi$
$510$	−0.890436	−	0.0838973i	−0.0394292	−	0.00371504i
$511$	−3.19735			−0.141443
$512$		−	10.2949i		−	0.454973i
$513$		−	9.09763i		−	0.401670i
$514$	−0.749733			−0.0330693
$515$	0.639674	+	0.0602704i	0.0281874	+	0.00265583i
$516$	7.57663			0.333542
$517$		−	35.8604i		−	1.57714i
$518$			1.10951i			0.0487492i
$519$	−0.0445377			−0.00195499
$520$	2.47928	−	26.3136i	0.108724	−	1.15393i
$521$	12.8425			0.562638			0.281319	−	0.959614i	$-0.409228\pi$
$521$	12.8425			0.562638			0.281319	+	0.959614i	$0.409228\pi$
$522$			14.0658i			0.615644i
$523$			32.8289i			1.43551i	0.696298	+	0.717753i	$0.254829\pi$
$523$			32.8289i			1.43551i	−0.696298	+	0.717753i	$0.745171\pi$
$524$	11.6510			0.508977
$525$	0.898358	+	0.170804i	0.0392076	+	0.00745449i
$526$	−6.89836			−0.300783
$527$		−	0.796199i		−	0.0346830i
$528$		−	2.35716i		−	0.102582i
$529$	−1.00000			−0.0434783
$530$	1.71785	−	18.2322i	0.0746185	−	0.791955i
$531$	−4.00000			−0.173585
$532$		−	1.25205i		−	0.0542834i
$533$		−	21.4909i		−	0.930876i
$534$	4.76619			0.206253
$535$	31.3752	+	2.95618i	1.35647	+	0.127807i
$536$	−4.34797			−0.187804
$537$			12.0305i			0.519156i
$538$			16.1018i			0.694198i
$539$	−30.0042			−1.29237
$540$	−10.5088	−	0.990145i	−0.452227	−	0.0426091i
$541$	−25.7261			−1.10605			−0.553026	−	0.833164i	$-0.686527\pi$
$541$	−25.7261			−1.10605			−0.553026	+	0.833164i	$0.686527\pi$
$542$			18.9625i			0.814507i
$543$		−	6.71103i		−	0.287998i
$544$	5.37850			0.230601
$545$	0.369877	−	3.92565i	0.0158438	−	0.168157i
$546$	0.629153			0.0269253
$547$			36.0191i			1.54006i	0.638005	+	0.770032i	$0.279760\pi$
$547$			36.0191i			1.54006i	−0.638005	+	0.770032i	$0.720240\pi$
$548$			10.3774i			0.443301i
$549$	30.5274			1.30288
$550$	3.04963	−	16.0398i	0.130037	−	0.683940i
$551$	19.4634			0.829169
$552$			1.49795i			0.0637571i
$553$		−	2.33070i		−	0.0991114i
$554$	23.1401			0.983128
$555$	0.570940	−	6.05961i	0.0242350	−	0.257216i
$556$	15.5703			0.660328
$557$		−	36.2148i		−	1.53447i	−0.641366	−	0.767235i	$-0.721632\pi$
$557$		−	36.2148i		−	1.53447i	0.641366	−	0.767235i	$-0.278368\pi$
$558$			1.73566i			0.0734763i
$559$	−41.6341			−1.76094
$560$	−0.655040	−	0.0617183i	−0.0276805	−	0.00260807i
$561$	2.31563			0.0977660
$562$		−	2.70986i		−	0.114308i
$563$			23.3308i			0.983277i	0.870799	+	0.491638i	$0.163602\pi$
$563$			23.3308i			0.983277i	−0.870799	+	0.491638i	$0.836398\pi$
$564$	−6.87491			−0.289486
$565$	−0.701392	−	0.0660856i	−0.0295078	−	0.00278024i
$566$	3.93493			0.165398
$567$			1.91583i			0.0804574i
$568$			13.8463i			0.580980i
$569$	−15.6845			−0.657529			−0.328764	−	0.944412i	$-0.606632\pi$
$569$	−15.6845			−0.657529			−0.328764	+	0.944412i	$0.606632\pi$
$570$	0.253072	−	2.68596i	0.0106000	−	0.112502i
$571$	38.7600			1.62206			0.811028	−	0.585007i	$-0.198908\pi$
$571$	38.7600			1.62206			0.811028	+	0.585007i	$0.198908\pi$
$572$			28.5984i			1.19576i
$573$			12.7524i			0.532738i
$574$	1.11018			0.0463379
$575$	0.933914	−	4.91201i	0.0389469	−	0.204845i
$576$	−6.75066			−0.281278
$577$			21.5902i			0.898812i	0.893328	+	0.449406i	$0.148364\pi$
$577$			21.5902i			0.898812i	−0.893328	+	0.449406i	$0.851636\pi$
$578$		−	12.1352i		−	0.504760i
$579$	11.7790			0.489518
$580$	2.11831	−	22.4825i	0.0879580	−	0.933533i
$581$	−4.82209			−0.200054
$582$			6.19756i			0.256897i
$583$			47.4139i			1.96368i
$584$	26.1877			1.08366
$585$	27.1553	+	2.55858i	1.12273	+	0.105784i
$586$	−1.42750			−0.0589694
$587$		−	24.3902i		−	1.00669i	−0.864086	−	0.503345i	$-0.832103\pi$
$587$		−	24.3902i		−	1.00669i	0.864086	−	0.503345i	$-0.167897\pi$
$588$			5.75220i			0.237217i
$589$	2.40170			0.0989602
$590$	−2.51133	−	0.236619i	−0.103390	−	0.00974144i
$591$	−5.00476			−0.205868
$592$			4.37916i			0.179982i
$593$		−	13.9747i		−	0.573874i	−0.957949	−	0.286937i	$-0.907363\pi$
$593$		−	13.9747i		−	0.573874i	0.957949	−	0.286937i	$-0.0926370\pi$
$594$	−10.7346			−0.440445
$595$	0.0606309	−	0.643500i	0.00248562	−	0.0263809i
$596$	11.3452			0.464716
$597$		−	12.0623i		−	0.493679i
$598$		−	3.44006i		−	0.140674i
$599$	20.9932			0.857758			0.428879	−	0.903362i	$-0.358909\pi$
$599$	20.9932			0.857758			0.428879	+	0.903362i	$0.358909\pi$
$600$	−7.35795	−	1.39896i	−0.300387	−	0.0571123i
$601$	−13.3007			−0.542546			−0.271273	−	0.962502i	$-0.587445\pi$
$601$	−13.3007			−0.542546			−0.271273	+	0.962502i	$0.587445\pi$
$602$		−	2.15073i		−	0.0876573i
$603$		−	4.48704i		−	0.182726i
$604$	−11.5408			−0.469590
$605$	−1.65807	+	17.5978i	−0.0674103	+	0.715452i
$606$	−3.55967			−0.144602
$607$			12.3549i			0.501469i	0.968056	+	0.250734i	$0.0806721\pi$
$607$			12.3549i			0.501469i	−0.968056	+	0.250734i	$0.919328\pi$
$608$			16.2240i			0.657970i
$609$	1.28625			0.0521215
$610$	19.1661	+	1.80584i	0.776013	+	0.0731163i
$611$	37.7781			1.52834
$612$			3.50838i			0.141818i
$613$		−	14.6291i		−	0.590865i	−0.955364	−	0.295432i	$-0.904536\pi$
$613$		−	14.6291i		−	0.590865i	0.955364	−	0.295432i	$-0.0954637\pi$
$614$	2.12140			0.0856126
$615$	−6.06324	−	0.571281i	−0.244493	−	0.0230363i
$616$	−3.53496			−0.142427
$617$		−	17.3723i		−	0.699384i	−0.936865	−	0.349692i	$-0.886286\pi$
$617$		−	17.3723i		−	0.699384i	0.936865	−	0.349692i	$-0.113714\pi$
$618$		−	0.125269i		−	0.00503904i
$619$	−40.0049			−1.60793			−0.803967	−	0.594674i	$-0.797281\pi$
$619$	−40.0049			−1.60793			−0.803967	+	0.594674i	$0.797281\pi$
$620$	0.261390	−	2.77423i	0.0104977	−	0.111416i
$621$	−3.28734			−0.131916
$622$		−	12.0936i		−	0.484909i
$623$			3.44443i			0.137998i
$624$	2.48322			0.0994082
$625$	23.2556	+	9.17479i	0.930224	+	0.366991i
$626$	−19.8075			−0.791667
$627$			6.98499i			0.278954i
$628$			6.83416i			0.272712i
$629$	−4.30201			−0.171532
$630$	−0.132171	+	1.40278i	−0.00526582	+	0.0558883i
$631$	15.7992			0.628956			0.314478	−	0.949265i	$-0.398171\pi$
$631$	15.7992			0.628956			0.314478	+	0.949265i	$0.398171\pi$
$632$			19.0894i			0.759337i
$633$		−	2.41779i		−	0.0960985i
$634$	−11.6424			−0.462379
$635$	−47.4815	−	4.47374i	−1.88425	−	0.177535i
$636$	9.08987			0.360437
$637$		−	31.6088i		−	1.25238i
$638$		−	22.9655i		−	0.909211i
$639$	−14.2892			−0.565273
$640$	21.8634	+	2.05999i	0.864229	+	0.0814281i
$641$	−37.3861			−1.47666			−0.738332	−	0.674437i	$-0.764386\pi$
$641$	−37.3861			−1.47666			−0.738332	+	0.674437i	$0.764386\pi$
$642$		−	6.14426i		−	0.242495i
$643$		−	44.1766i		−	1.74216i	−0.491145	−	0.871078i	$-0.663421\pi$
$643$		−	44.1766i		−	1.74216i	0.491145	−	0.871078i	$-0.336579\pi$
$644$	−0.452416			−0.0178277
$645$	−1.10674	+	11.7462i	−0.0435777	+	0.462507i
$646$	−1.90689			−0.0750256
$647$			4.88521i			0.192058i	0.995379	+	0.0960288i	$0.0306141\pi$
$647$			4.88521i			0.192058i	−0.995379	+	0.0960288i	$0.969386\pi$
$648$		−	15.6915i		−	0.616420i
$649$	6.53086			0.256359
$650$	16.8976	+	3.21272i	0.662778	+	0.126013i
$651$	0.158717			0.00622063
$652$		−	3.03770i		−	0.118966i
$653$			36.6211i			1.43309i	0.697540	+	0.716546i	$0.254278\pi$
$653$			36.6211i			1.43309i	−0.697540	+	0.716546i	$0.745722\pi$
$654$	−0.768769			−0.0300612
$655$	−1.70189	+	18.0628i	−0.0664984	+	0.705774i
$656$	4.38178			0.171080
$657$			27.0254i			1.05436i
$658$			1.95154i			0.0760790i
$659$	10.3388			0.402743			0.201371	−	0.979515i	$-0.435460\pi$
$659$	10.3388			0.402743			0.201371	+	0.979515i	$0.435460\pi$
$660$	8.06846	+	0.760215i	0.314065	+	0.0295913i
$661$	6.46980			0.251646			0.125823	−	0.992053i	$-0.459843\pi$
$661$	6.46980			0.251646			0.125823	+	0.992053i	$0.459843\pi$
$662$		−	14.2921i		−	0.555480i
$663$			2.43947i			0.0947411i
$664$	39.4950			1.53270
$665$	1.94109	+	0.182890i	0.0752722	+	0.00709218i
$666$	9.37807			0.363393
$667$		−	7.03291i		−	0.272315i
$668$		−	22.7974i		−	0.882057i
$669$	10.8910			0.421071
$670$	0.265430	−	2.81711i	0.0102544	−	0.108834i
$671$	−49.8426			−1.92415
$672$			1.07217i			0.0413599i
$673$			30.1889i			1.16370i	0.813297	+	0.581849i	$0.197670\pi$
$673$			30.1889i			1.16370i	−0.813297	+	0.581849i	$0.802330\pi$
$674$	−3.07435			−0.118419
$675$	3.07009	−	16.1474i	0.118168	−	0.621515i
$676$	−11.4603			−0.440780
$677$		−	0.443226i		−	0.0170345i	−0.999964	−	0.00851727i	$-0.997289\pi$
$677$		−	0.443226i		−	0.0170345i	0.999964	−	0.00851727i	$-0.00271117\pi$
$678$			0.137355i			0.00527509i
$679$	−4.47885			−0.171882
$680$	−0.496594	+	5.27055i	−0.0190435	+	0.202116i
$681$	6.36436			0.243883
$682$		−	2.83383i		−	0.108513i
$683$			25.7553i			0.985498i	0.870171	+	0.492749i	$0.164008\pi$
$683$			25.7553i			0.985498i	−0.870171	+	0.492749i	$0.835992\pi$
$684$	−10.5829			−0.404646
$685$	−16.0884	−	1.51585i	−0.614704	−	0.0579178i
$686$	3.28917			0.125581
$687$		−	15.0560i		−	0.574422i
$688$		−	8.48876i		−	0.323631i
$689$	−49.9495			−1.90292
$690$	−0.970544	−	0.0914452i	−0.0369480	−	0.00348126i
$691$	16.1707			0.615162			0.307581	−	0.951522i	$-0.400480\pi$
$691$	16.1707			0.615162			0.307581	+	0.951522i	$0.400480\pi$
$692$			0.110173i			0.00418815i
$693$		−	3.64802i		−	0.138577i
$694$	−13.0946			−0.497065
$695$	−2.27439	+	24.1390i	−0.0862726	+	0.915646i
$696$	−10.5350			−0.399326
$697$			4.30458i			0.163048i
$698$			6.37447i			0.241277i
$699$	4.78291			0.180906
$700$	0.422518	−	2.22227i	0.0159697	−	0.0839940i
$701$	−26.5049			−1.00107			−0.500537	−	0.865715i	$-0.666864\pi$
$701$	−26.5049			−1.00107			−0.500537	+	0.865715i	$0.666864\pi$
$702$		−	11.3086i		−	0.426817i
$703$		−	12.9768i		−	0.489429i
$704$	11.0219			0.415404
$705$	1.00423	−	10.6583i	0.0378217	−	0.401416i
$706$	−15.4483			−0.581403
$707$		−	2.57250i		−	0.0967489i
$708$		−	1.25205i		−	0.0470550i
$709$	−38.6885			−1.45298			−0.726488	−	0.687179i	$-0.758849\pi$
$709$	−38.6885			−1.45298			−0.726488	+	0.687179i	$0.758849\pi$
$710$	−8.97124	−	0.845275i	−0.336685	−	0.0317226i
$711$	−19.7000			−0.738809
$712$		−	28.2114i		−	1.05727i
$713$		−	0.867829i		−	0.0325004i
$714$	−0.126018			−0.00471610
$715$	−44.3368	−	4.17744i	−1.65810	−	0.156227i
$716$	29.7600			1.11218
$717$		−	9.31735i		−	0.347963i
$718$		−	16.8320i		−	0.628165i
$719$	−43.7242			−1.63064			−0.815319	−	0.579012i	$-0.803438\pi$
$719$	−43.7242			−1.63064			−0.815319	+	0.579012i	$0.803438\pi$
$720$	−0.521668	+	5.53667i	−0.0194414	+	0.206340i
$721$	0.0905291			0.00337148
$722$			8.51741i			0.316985i
$723$		−	1.21761i		−	0.0452833i
$724$	−16.6011			−0.616975
$725$	34.5457	+	6.56813i	1.28299	+	0.243934i
$726$	3.44621			0.127901
$727$		−	37.2280i		−	1.38071i	−0.723471	−	0.690355i	$-0.757455\pi$
$727$		−	37.2280i		−	1.38071i	0.723471	−	0.690355i	$-0.242545\pi$
$728$		−	3.72400i		−	0.138021i
$729$	10.4686			0.387726
$730$	−1.59868	+	16.9674i	−0.0591697	+	0.627992i
$731$	8.33921			0.308437
$732$			9.55548i			0.353181i
$733$			27.9656i			1.03293i	0.856308	+	0.516466i	$0.172753\pi$
$733$			27.9656i			1.03293i	−0.856308	+	0.516466i	$0.827247\pi$
$734$	−16.4776			−0.608198
$735$	−8.91778	−	0.840238i	−0.328937	−	0.0309926i
$736$	5.86237			0.216090
$737$			7.32606i			0.269859i
$738$		−	9.38368i		−	0.345418i
$739$	30.4555			1.12032			0.560162	−	0.828383i	$-0.310739\pi$
$739$	30.4555			1.12032			0.560162	+	0.828383i	$0.310739\pi$
$740$	−14.9897	−	1.41234i	−0.551032	−	0.0519185i
$741$	−7.35854			−0.270323
$742$		−	2.58029i		−	0.0947253i
$743$			1.97918i			0.0726090i	0.999341	+	0.0363045i	$0.0115586\pi$
$743$			1.97918i			0.0726090i	−0.999341	+	0.0363045i	$0.988441\pi$
$744$	−1.29997			−0.0476591
$745$	−1.65722	+	17.5887i	−0.0607157	+	0.644400i
$746$	−0.641240			−0.0234775
$747$			40.7583i			1.49127i
$748$		−	5.72819i		−	0.209443i
$749$	4.44033			0.162246
$750$	1.35558	−	4.68192i	0.0494989	−	0.170959i
$751$	5.77719			0.210813			0.105406	−	0.994429i	$-0.466386\pi$
$751$	5.77719			0.210813			0.105406	+	0.994429i	$0.466386\pi$
$752$			7.70257i			0.280884i
$753$		−	7.95374i		−	0.289851i
$754$	24.1936			0.881080
$755$	1.68580	−	17.8920i	0.0613525	−	0.651158i
$756$	−1.48725			−0.0540906
$757$			26.9857i			0.980814i	0.871494	+	0.490407i	$0.163152\pi$
$757$			26.9857i			0.980814i	−0.871494	+	0.490407i	$0.836848\pi$
$758$		−	1.89295i		−	0.0687550i
$759$	2.52396			0.0916138
$760$	−15.8984	−	1.49795i	−0.576694	−	0.0543364i
$761$	5.95209			0.215763			0.107881	−	0.994164i	$-0.465593\pi$
$761$	5.95209			0.215763			0.107881	+	0.994164i	$0.465593\pi$
$762$			9.29840i			0.336846i
$763$		−	0.555573i		−	0.0201131i
$764$	31.5456			1.14128
$765$	−5.43913	−	0.512478i	−0.196652	−	0.0185287i
$766$	10.2127			0.368998
$767$			6.88012i			0.248427i
$768$		−	7.22460i		−	0.260695i
$769$	29.5669			1.06621			0.533104	−	0.846050i	$-0.321025\pi$
$769$	29.5669			1.06621			0.533104	+	0.846050i	$0.321025\pi$
$770$	0.215798	−	2.29035i	0.00777681	−	0.0825384i
$771$	0.579493			0.0208699
$772$		−	29.1377i		−	1.04869i
$773$			2.24223i			0.0806474i	0.999187	+	0.0403237i	$0.0128389\pi$
$773$			2.24223i			0.0806474i	−0.999187	+	0.0403237i	$0.987161\pi$
$774$	−18.1789			−0.653426
$775$	4.26278	+	0.810478i	0.153124	+	0.0291132i
$776$	36.6837			1.31687
$777$		−	0.857579i		−	0.0307655i
$778$			8.06581i			0.289173i
$779$	−12.9846			−0.465220
$780$	−0.800870	+	8.49995i	−0.0286757	+	0.304347i
$781$	23.3302			0.834822
$782$			0.689035i			0.0246398i
$783$		−	23.1195i		−	0.826225i
$784$	6.44470			0.230168
$785$	−10.5952	−	0.998281i	−0.378157	−	0.0356302i
$786$	3.53728			0.126171
$787$			6.28634i			0.224084i	0.993703	+	0.112042i	$0.0357391\pi$
$787$			6.28634i			0.224084i	−0.993703	+	0.112042i	$0.964261\pi$
$788$			12.3803i			0.441029i
$789$	5.33197			0.189823
$790$	−12.3683	−	1.16535i	−0.440045	−	0.0414613i
$791$	−0.0992637			−0.00352941
$792$			29.8789i			1.06170i
$793$		−	52.5081i		−	1.86462i
$794$	14.7101			0.522043
$795$	−1.32778	+	14.0922i	−0.0470915	+	0.499801i
$796$	−29.8387			−1.05760
$797$			15.5627i			0.551258i	0.961264	+	0.275629i	$0.0888862\pi$
$797$			15.5627i			0.551258i	−0.961264	+	0.275629i	$0.911114\pi$
$798$		−	0.380127i		−	0.0134564i
$799$	−7.56687			−0.267696
$800$	−5.47496	+	28.7960i	−0.193569	+	1.01809i
$801$	29.1137			1.02868
$802$			6.01727i			0.212477i
$803$		−	44.1247i		−	1.55713i
$804$	1.40450			0.0495330
$805$	0.0660856	−	0.701392i	0.00232921	−	0.0247208i
$806$	2.98538			0.105156
$807$		−	12.4456i		−	0.438106i
$808$			21.0699i			0.741237i
$809$	−7.48638			−0.263207			−0.131604	−	0.991302i	$-0.542013\pi$
$809$	−7.48638			−0.263207			−0.131604	+	0.991302i	$0.542013\pi$
$810$	10.1667	+	0.957916i	0.357223	+	0.0336577i
$811$	−39.7694			−1.39649			−0.698246	−	0.715858i	$-0.746036\pi$
$811$	−39.7694			−1.39649			−0.698246	+	0.715858i	$0.746036\pi$
$812$		−	3.18180i		−	0.111659i
$813$		−	14.6567i		−	0.514033i
$814$	−15.3117			−0.536675
$815$	4.70942	+	0.443724i	0.164964	+	0.0155430i
$816$	−0.497382			−0.0174119
$817$			25.1548i			0.880056i
$818$		−	21.2052i		−	0.741421i
$819$	3.84311			0.134289
$820$	−1.41318	+	14.9987i	−0.0493504	+	0.523776i
$821$	30.2829			1.05688			0.528440	−	0.848970i	$-0.322777\pi$
$821$	30.2829			1.05688			0.528440	+	0.848970i	$0.322777\pi$
$822$			3.15061i			0.109890i
$823$			7.15992i			0.249579i	0.992183	+	0.124790i	$0.0398256\pi$
$823$			7.15992i			0.249579i	−0.992183	+	0.124790i	$0.960174\pi$
$824$	−0.741473			−0.0258304
$825$	−2.35716	+	12.3977i	−0.0820658	+	0.431632i
$826$	−0.355413			−0.0123664
$827$			29.1034i			1.01202i	0.862527	+	0.506011i	$0.168881\pi$
$827$			29.1034i			1.01202i	−0.862527	+	0.506011i	$0.831119\pi$
$828$			3.82401i			0.132894i
$829$	42.4480			1.47428			0.737140	−	0.675740i	$-0.236176\pi$
$829$	42.4480			1.47428			0.737140	+	0.675740i	$0.236176\pi$
$830$	−2.41105	+	25.5894i	−0.0836886	+	0.888220i
$831$	−17.8857			−0.620449
$832$			11.6113i			0.402551i
$833$			6.33115i			0.219362i
$834$	4.72719			0.163689
$835$	35.3433	+	3.33007i	1.22311	+	0.115242i
$836$	17.2788			0.597600
$837$		−	2.85285i		−	0.0986088i
$838$		−	16.4827i		−	0.569387i
$839$	32.2617			1.11380			0.556898	−	0.830581i	$-0.311991\pi$
$839$	32.2617			1.11380			0.556898	+	0.830581i	$0.311991\pi$
$840$	−1.05065	−	0.0989930i	−0.0362509	−	0.00341558i
$841$	20.4618			0.705579
$842$		−	0.561164i		−	0.0193390i
$843$			2.09454i			0.0721396i
$844$	−5.98090			−0.205871
$845$	1.67403	−	17.7671i	0.0575884	−	0.611208i
$846$	16.4952			0.567118
$847$			2.49051i			0.0855749i
$848$		−	10.1842i		−	0.349726i
$849$	−3.04144			−0.104382
$850$	−3.38454	−	0.643500i	−0.116089	−	0.0220719i
$851$	−4.68904			−0.160738
$852$		−	4.47272i		−	0.153233i
$853$		−	12.3883i		−	0.424168i	−0.977251	−	0.212084i	$-0.931975\pi$
$853$		−	12.3883i		−	0.424168i	0.977251	−	0.212084i	$-0.0680250\pi$
$854$	2.71246			0.0928184
$855$	1.54587	−	16.4069i	0.0528675	−	0.561103i
$856$	−36.3682			−1.24304
$857$			11.8646i			0.405287i	0.979253	+	0.202643i	$0.0649532\pi$
$857$			11.8646i			0.405287i	−0.979253	+	0.202643i	$0.935047\pi$
$858$			8.68256i			0.296418i
$859$	−6.57949			−0.224489			−0.112245	−	0.993681i	$-0.535804\pi$
$859$	−6.57949			−0.224489			−0.112245	+	0.993681i	$0.535804\pi$
$860$	29.0567	+	2.73774i	0.990825	+	0.0933561i
$861$	−0.858092			−0.0292437
$862$		−	20.5511i		−	0.699975i
$863$		−	41.6998i		−	1.41948i	−0.704464	−	0.709739i	$-0.748813\pi$
$863$		−	41.6998i		−	1.41948i	0.704464	−	0.709739i	$-0.251187\pi$
$864$	19.2716			0.655633
$865$	−0.170804	−	0.0160932i	−0.00580751	−	0.000547186i
$866$	−19.0959			−0.648906
$867$			9.37973i			0.318552i
$868$		−	0.392620i		−	0.0133264i
$869$	32.1645			1.09111
$870$	0.643125	−	6.82574i	0.0218040	−	0.231414i
$871$	−7.71785			−0.261509
$872$			4.55039i			0.154096i
$873$			37.8571i			1.28127i
$874$	−2.07844			−0.0703044
$875$	3.38352	+	0.979653i	0.114384	+	0.0331183i
$876$	−8.45930			−0.285813
$877$		−	20.9988i		−	0.709079i	−0.935041	−	0.354540i	$-0.884638\pi$
$877$		−	20.9988i		−	0.709079i	0.935041	−	0.354540i	$-0.115362\pi$
$878$		−	23.1388i		−	0.780897i
$879$	1.10336			0.0372154
$880$	0.851736	−	9.03981i	0.0287120	−	0.304732i
$881$	−12.3401			−0.415747			−0.207874	−	0.978156i	$-0.566654\pi$
$881$	−12.3401			−0.415747			−0.207874	+	0.978156i	$0.566654\pi$
$882$		−	13.8015i		−	0.464720i
$883$		−	35.5426i		−	1.19610i	−0.801457	−	0.598052i	$-0.795942\pi$
$883$		−	35.5426i		−	1.19610i	0.801457	−	0.598052i	$-0.204058\pi$
$884$	6.03452			0.202963
$885$	1.94109	+	0.182890i	0.0652489	+	0.00614779i
$886$	18.4585			0.620126
$887$			19.1997i			0.644663i	0.946627	+	0.322331i	$0.104467\pi$
$887$			19.1997i			0.644663i	−0.946627	+	0.322331i	$0.895533\pi$
$888$			7.02395i			0.235708i
$889$	−6.71977			−0.225374
$890$	18.2785	+	1.72221i	0.612698	+	0.0577287i
$891$	−26.4392			−0.885747
$892$		−	26.9412i		−	0.902057i
$893$		−	22.8251i		−	0.763813i
$894$	3.44443			0.115199
$895$	−4.34711	+	46.1376i	−0.145308	+	1.54221i
$896$	3.09420			0.103370
$897$			2.65893i			0.0887792i
$898$			29.0508i			0.969438i
$899$	6.10336			0.203558
$900$	−18.7836	−	3.57130i	−0.626119	−	0.119043i
$901$	10.0048			0.333307
$902$			15.3209i			0.510129i
$903$			1.66237i			0.0553202i
$904$	0.813013			0.0270404
$905$	2.42496	−	25.7371i	0.0806085	−	0.855530i
$906$	−3.50383			−0.116407
$907$		−	54.5806i		−	1.81232i	−0.422937	−	0.906159i	$-0.639001\pi$
$907$		−	54.5806i		−	1.81232i	0.422937	−	0.906159i	$-0.360999\pi$
$908$		−	15.7435i		−	0.522467i
$909$	−21.7439			−0.721198
$910$	2.41283	+	0.227338i	0.0799846	+	0.00753619i
$911$	24.7619			0.820397			0.410199	−	0.911996i	$-0.365459\pi$
$911$	24.7619			0.820397			0.410199	+	0.911996i	$0.365459\pi$
$912$		−	1.50033i		−	0.0496809i
$913$		−	66.5467i		−	2.20237i
$914$	−4.80686			−0.158997
$915$	−14.8141	−	1.39579i	−0.489739	−	0.0461435i
$916$	−37.2441			−1.23058
$917$			2.55632i			0.0844172i
$918$			2.26509i			0.0747592i
$919$	−9.32187			−0.307500			−0.153750	−	0.988110i	$-0.549135\pi$
$919$	−9.32187			−0.307500			−0.153750	+	0.988110i	$0.549135\pi$
$920$	−0.541270	+	5.74471i	−0.0178451	+	0.189397i
$921$	−1.63970			−0.0540298
$922$			17.8997i			0.589495i
$923$			24.5779i			0.808992i
$924$	1.14188			0.0375651
$925$	4.37916	−	23.0326i	0.143986	−	0.757306i
$926$	19.9418			0.655329
$927$		−	0.765190i		−	0.0251321i
$928$			41.2295i			1.35343i
$929$	16.9227			0.555217			0.277608	−	0.960694i	$-0.410458\pi$
$929$	16.9227			0.555217			0.277608	+	0.960694i	$0.410458\pi$
$930$	0.0793587	−	0.842266i	0.00260228	−	0.0276190i
$931$	−19.0976			−0.625900
$932$		−	11.8315i		−	0.387554i
$933$			9.34754i			0.306025i
$934$	13.9727			0.457201
$935$	8.88055	+	0.836730i	0.290425	+	0.0273640i
$936$	−31.4768			−1.02885
$937$			32.3925i			1.05822i	0.848554	+	0.529109i	$0.177474\pi$
$937$			32.3925i			1.05822i	−0.848554	+	0.529109i	$0.822526\pi$
$938$		−	0.398688i		−	0.0130176i
$939$	15.3099			0.499618
$940$	−26.3656	−	2.48418i	−0.859951	−	0.0810250i
$941$	39.8390			1.29871			0.649357	−	0.760484i	$-0.275038\pi$
$941$	39.8390			1.29871			0.649357	+	0.760484i	$0.275038\pi$
$942$			2.07487i			0.0676029i
$943$			4.69184i			0.152787i
$944$	−1.40278			−0.0456567
$945$	0.217246	−	2.30571i	0.00706700	−	0.0750049i
$946$	29.6809			0.965010
$947$			12.7594i			0.414624i	0.978275	+	0.207312i	$0.0664715\pi$
$947$			12.7594i			0.414624i	−0.978275	+	0.207312i	$0.933528\pi$
$948$		−	6.16637i		−	0.200274i
$949$	46.4844			1.50895
$950$	1.94109	−	10.2093i	0.0629772	−	0.331234i
$951$	8.99880			0.291806
$952$			0.745908i			0.0241750i
$953$			7.04791i			0.228304i	0.993463	+	0.114152i	$0.0364151\pi$
$953$			7.04791i			0.228304i	−0.993463	+	0.114152i	$0.963585\pi$
$954$	−21.8097			−0.706114
$955$	−4.60794	+	48.9059i	−0.149110	+	1.58256i
$956$	−23.0484			−0.745437
$957$			17.7508i			0.573800i
$958$			22.5027i			0.727029i
$959$	−2.27688			−0.0735244
$960$	3.27591	+	0.308658i	0.105729	+	0.00996189i
$961$	−30.2469			−0.975706
$962$		−	16.1306i		−	0.520070i
$963$		−	37.5315i		−	1.20944i
$964$	−3.01200			−0.0970099
$965$	45.1730	+	4.25622i	1.45417	+	0.137013i
$966$	−0.137355			−0.00441933
$967$		−	9.27901i		−	0.298393i	−0.988808	−	0.149196i	$-0.952331\pi$
$967$		−	9.27901i		−	0.298393i	0.988808	−	0.149196i	$-0.0476687\pi$
$968$		−	20.3983i		−	0.655628i
$969$	1.47390			0.0473484
$970$	−2.23942	+	23.7679i	−0.0719036	+	0.763141i
$971$	0.0274304			0.000880284			0.000440142	−	1.00000i	$-0.499860\pi$
$971$	0.0274304			0.000880284			0.000440142		1.00000i	$0.499860\pi$
$972$			19.2302i			0.616810i
$973$			3.41625i			0.109520i
$974$	−27.9580			−0.895831
$975$	−13.0607	−	2.48322i	−0.418277	−	0.0795266i
$976$	10.7058			0.342686
$977$			29.9665i			0.958714i	0.877620	+	0.479357i	$0.159130\pi$
$977$			29.9665i			0.958714i	−0.877620	+	0.479357i	$0.840870\pi$
$978$		−	0.922256i		−	0.0294905i
$979$	−47.5344			−1.51921
$980$	−2.07850	+	22.0599i	−0.0663952	+	0.704679i
$981$	−4.69594			−0.149930
$982$			0.519589i			0.0165808i
$983$			5.80749i			0.185230i	0.995702	+	0.0926151i	$0.0295226\pi$
$983$			5.80749i			0.185230i	−0.995702	+	0.0926151i	$0.970477\pi$
$984$	7.02815			0.224049
$985$	−19.1935	−	1.80842i	−0.611554	−	0.0576210i
$986$	−4.84592			−0.154326
$987$		−	1.50841i		−	0.0480132i
$988$			18.2028i			0.579110i
$989$	9.08944			0.289027
$990$	−19.3590	−	1.82401i	−0.615268	−	0.0579709i
$991$	17.5926			0.558847			0.279423	−	0.960168i	$-0.409857\pi$
$991$	17.5926			0.558847			0.279423	+	0.960168i	$0.409857\pi$
$992$			5.08754i			0.161529i
$993$			11.0469i			0.350561i
$994$	−1.26964			−0.0402707
$995$	4.35861	−	46.2596i	0.138177	−	1.46653i
$996$	−12.7579			−0.404249
$997$		−	24.2200i		−	0.767055i	−0.923530	−	0.383527i	$-0.874709\pi$
$997$		−	24.2200i		−	0.767055i	0.923530	−	0.383527i	$-0.125291\pi$
$998$		−	18.6245i		−	0.589547i
$999$	−15.4144			−0.487691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.b.b.24.3	✓	8
3.2	odd	2		1035.2.b.e.829.6		8
4.3	odd	2		1840.2.e.d.369.4		8
5.2	odd	4		575.2.a.i.1.3		4
5.3	odd	4		575.2.a.j.1.2		4
5.4	even	2	inner	115.2.b.b.24.6	yes	8
15.2	even	4		5175.2.a.bv.1.2		4
15.8	even	4		5175.2.a.bw.1.3		4
15.14	odd	2		1035.2.b.e.829.3		8
20.3	even	4		9200.2.a.ck.1.3		4
20.7	even	4		9200.2.a.cq.1.2		4
20.19	odd	2		1840.2.e.d.369.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.3	✓	8	1.1	even	1	trivial
115.2.b.b.24.6	yes	8	5.4	even	2	inner
575.2.a.i.1.3		4	5.2	odd	4
575.2.a.j.1.2		4	5.3	odd	4
1035.2.b.e.829.3		8	15.14	odd	2
1035.2.b.e.829.6		8	3.2	odd	2
1840.2.e.d.369.4		8	4.3	odd	2
1840.2.e.d.369.5		8	20.19	odd	2
5175.2.a.bv.1.2		4	15.2	even	4
5175.2.a.bw.1.3		4	15.8	even	4
9200.2.a.ck.1.3		4	20.3	even	4
9200.2.a.cq.1.2		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.751024i q^{2} +0.580491i q^{3} +1.43596 q^{4} +(-0.209755 + 2.22621i) q^{5} +0.435963 q^{6} +0.315061i q^{7} -2.58049i q^{8} +2.66303 q^{9} +O(q^{10})\)	\(q-0.751024i q^{2} +0.580491i q^{3} +1.43596 q^{4} +(-0.209755 + 2.22621i) q^{5} +0.435963 q^{6} +0.315061i q^{7} -2.58049i q^{8} +2.66303 q^{9} +(1.67194 + 0.157531i) q^{10} -4.34797 q^{11} +0.833563i q^{12} -4.58049i q^{13} +0.236619 q^{14} +(-1.29229 - 0.121761i) q^{15} +0.933914 q^{16} +0.917461i q^{17} -2.00000i q^{18} -2.76748 q^{19} +(-0.301200 + 3.19675i) q^{20} -0.182890 q^{21} +3.26543i q^{22} +1.00000i q^{23} +1.49795 q^{24} +(-4.91201 - 0.933914i) q^{25} -3.44006 q^{26} +3.28734i q^{27} +0.452416i q^{28} -7.03291 q^{29} +(-0.0914452 + 0.970544i) q^{30} -0.867829 q^{31} -5.86237i q^{32} -2.52396i q^{33} +0.689035 q^{34} +(-0.701392 - 0.0660856i) q^{35} +3.82401 q^{36} +4.68904i q^{37} +2.07844i q^{38} +2.65893 q^{39} +(5.74471 + 0.541270i) q^{40} +4.69184 q^{41} +0.137355i q^{42} -9.08944i q^{43} -6.24352 q^{44} +(-0.558583 + 5.92846i) q^{45} +0.751024 q^{46} +8.24762i q^{47} +0.542129i q^{48} +6.90074 q^{49} +(-0.701392 + 3.68904i) q^{50} -0.532578 q^{51} -6.57741i q^{52} -10.9048i q^{53} +2.46887 q^{54} +(0.912006 - 9.67948i) q^{55} +0.813013 q^{56} -1.60650i q^{57} +5.28188i q^{58} -1.50205 q^{59} +(-1.85569 - 0.174844i) q^{60} +11.4634 q^{61} +0.651760i q^{62} +0.839018i q^{63} -2.53496 q^{64} +(10.1971 + 0.960779i) q^{65} -1.89555 q^{66} -1.68494i q^{67} +1.31744i q^{68} -0.580491 q^{69} +(-0.0496318 + 0.526763i) q^{70} -5.36578 q^{71} -6.87193i q^{72} +10.1484i q^{73} +3.52158 q^{74} +(0.542129 - 2.85138i) q^{75} -3.97400 q^{76} -1.36988i q^{77} -1.99692i q^{78} -7.39760 q^{79} +(-0.195893 + 2.07909i) q^{80} +6.08082 q^{81} -3.52369i q^{82} +15.3052i q^{83} -0.262624 q^{84} +(-2.04246 - 0.192442i) q^{85} -6.82639 q^{86} -4.08254i q^{87} +11.2199i q^{88} +10.9326 q^{89} +(4.45242 + 0.419509i) q^{90} +1.44314 q^{91} +1.43596i q^{92} -0.503767i q^{93} +6.19416 q^{94} +(0.580491 - 6.16098i) q^{95} +3.40306 q^{96} +14.2158i q^{97} -5.18262i q^{98} -11.5788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9	\( 8 q - 4 q^{4} - 6 q^{5} - 12 q^{6} - 8 q^{9} + 6 q^{10} + 4 q^{11} + 8 q^{14} + 6 q^{15} + 4 q^{16} + 8 q^{19} - 8 q^{20} - 4 q^{21} + 24 q^{24} - 16 q^{25} + 12 q^{26} - 8 q^{29} - 2 q^{30} - 28 q^{34} + 28 q^{35} - 16 q^{36} + 16 q^{39} - 10 q^{40} - 16 q^{41} - 12 q^{44} + 24 q^{45} + 28 q^{50} + 20 q^{51} - 44 q^{54} - 16 q^{55} + 28 q^{56} - 16 q^{60} - 16 q^{61} + 40 q^{64} - 14 q^{65} - 16 q^{66} + 4 q^{69} - 28 q^{70} - 48 q^{71} + 72 q^{74} - 36 q^{76} - 48 q^{79} - 2 q^{80} + 16 q^{81} - 4 q^{84} + 12 q^{85} + 28 q^{86} + 16 q^{89} - 4 q^{90} + 52 q^{91} + 84 q^{94} - 4 q^{95} + 60 q^{96} - 72 q^{99}+O(q^{100}) \) 8 * q - 4 * q^4 - 6 * q^5 - 12 * q^6 - 8 * q^9 + 6 * q^10 + 4 * q^11 + 8 * q^14 + 6 * q^15 + 4 * q^16 + 8 * q^19 - 8 * q^20 - 4 * q^21 + 24 * q^24 - 16 * q^25 + 12 * q^26 - 8 * q^29 - 2 * q^30 - 28 * q^34 + 28 * q^35 - 16 * q^36 + 16 * q^39 - 10 * q^40 - 16 * q^41 - 12 * q^44 + 24 * q^45 + 28 * q^50 + 20 * q^51 - 44 * q^54 - 16 * q^55 + 28 * q^56 - 16 * q^60 - 16 * q^61 + 40 * q^64 - 14 * q^65 - 16 * q^66 + 4 * q^69 - 28 * q^70 - 48 * q^71 + 72 * q^74 - 36 * q^76 - 48 * q^79 - 2 * q^80 + 16 * q^81 - 4 * q^84 + 12 * q^85 + 28 * q^86 + 16 * q^89 - 4 * q^90 + 52 * q^91 + 84 * q^94 - 4 * q^95 + 60 * q^96 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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