LMFDB - Embedded newform 1840.2.e.d.369.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (of order $2$, degree $1$, not 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:	$14.6924739719$
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:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.4
Root			$0.790245 + 0.790245i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.d.369.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.580491i q^{3} +(-0.209755 + 2.22621i) q^{5} -0.315061i q^{7} +2.66303 q^{9} +O(q^{10})$	$q-0.580491i q^{3} +(-0.209755 + 2.22621i) q^{5} -0.315061i q^{7} +2.66303 q^{9} +4.34797 q^{11} -4.58049i q^{13} +(1.29229 + 0.121761i) q^{15} +0.917461i q^{17} +2.76748 q^{19} -0.182890 q^{21} -1.00000i q^{23} +(-4.91201 - 0.933914i) q^{25} -3.28734i q^{27} -7.03291 q^{29} +0.867829 q^{31} -2.52396i q^{33} +(0.701392 + 0.0660856i) q^{35} +4.68904i q^{37} -2.65893 q^{39} +4.69184 q^{41} +9.08944i q^{43} +(-0.558583 + 5.92846i) q^{45} -8.24762i q^{47} +6.90074 q^{49} +0.532578 q^{51} -10.9048i q^{53} +(-0.912006 + 9.67948i) q^{55} -1.60650i q^{57} +1.50205 q^{59} +11.4634 q^{61} -0.839018i q^{63} +(10.1971 + 0.960779i) q^{65} +1.68494i q^{67} -0.580491 q^{69} +5.36578 q^{71} +10.1484i q^{73} +(-0.542129 + 2.85138i) q^{75} -1.36988i q^{77} +7.39760 q^{79} +6.08082 q^{81} -15.3052i q^{83} +(-2.04246 - 0.192442i) q^{85} +4.08254i q^{87} +10.9326 q^{89} -1.44314 q^{91} -0.503767i q^{93} +(-0.580491 + 6.16098i) q^{95} +14.2158i q^{97} +11.5788 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q - 6 * q^5 - 8 * q^9	$ 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) $ 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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			0
$2$		0			0
$3$		−	0.580491i		−	0.335147i	−0.985860	−	0.167573i	$-0.946407\pi$
$3$		−	0.580491i		−	0.335147i	0.985860	−	0.167573i	$-0.0535931\pi$
$4$		0			0
$5$	−0.209755	+	2.22621i	−0.0938051	+	0.995591i
$5$	−0.209755	+	2.22621i	−0.0938051	+	0.995591i
$6$		0			0
$7$		−	0.315061i		−	0.119082i	−0.998226	−	0.0595410i	$-0.981036\pi$
$7$		−	0.315061i		−	0.119082i	0.998226	−	0.0595410i	$-0.0189637\pi$
$8$		0			0
$9$	2.66303			0.887677
$10$		0			0
$11$	4.34797			1.31096			0.655481	−	0.755212i	$-0.272466\pi$
$11$	4.34797			1.31096			0.655481	+	0.755212i	$0.272466\pi$
$12$		0			0
$13$		−	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			0
$15$	1.29229	+	0.121761i	0.333669	+	0.0314385i
$16$		0			0
$17$			0.917461i			0.222517i	0.993792	+	0.111258i	$0.0354881\pi$
$17$			0.917461i			0.222517i	−0.993792	+	0.111258i	$0.964512\pi$
$18$		0			0
$19$	2.76748			0.634903			0.317451	−	0.948275i	$-0.397173\pi$
$19$	2.76748			0.634903			0.317451	+	0.948275i	$0.397173\pi$
$20$		0			0
$21$	−0.182890			−0.0399099
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.91201	−	0.933914i	−0.982401	−	0.186783i
$26$		0			0
$27$		−	3.28734i		−	0.632648i
$28$		0			0
$29$	−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			0
$31$	0.867829			0.155867			0.0779333	−	0.996959i	$-0.475168\pi$
$31$	0.867829			0.155867			0.0779333	+	0.996959i	$0.475168\pi$
$32$		0			0
$33$		−	2.52396i		−	0.439364i
$34$		0			0
$35$	0.701392	+	0.0660856i	0.118557	+	0.0111705i
$36$		0			0
$37$			4.68904i			0.770873i	0.922734	+	0.385436i	$0.125949\pi$
$37$			4.68904i			0.770873i	−0.922734	+	0.385436i	$0.874051\pi$
$38$		0			0
$39$	−2.65893			−0.425770
$40$		0			0
$41$	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			0
$43$			9.08944i			1.38613i	0.720877	+	0.693063i	$0.243739\pi$
$43$			9.08944i			1.38613i	−0.720877	+	0.693063i	$0.756261\pi$
$44$		0			0
$45$	−0.558583	+	5.92846i	−0.0832686	+	0.883763i
$46$		0			0
$47$		−	8.24762i		−	1.20304i	−0.798858	−	0.601519i	$-0.794562\pi$
$47$		−	8.24762i		−	1.20304i	0.798858	−	0.601519i	$-0.205438\pi$
$48$		0			0
$49$	6.90074			0.985819
$50$		0			0
$51$	0.532578			0.0745758
$52$		0			0
$53$		−	10.9048i		−	1.49789i	−0.662630	−	0.748947i	$-0.730560\pi$
$53$		−	10.9048i		−	1.49789i	0.662630	−	0.748947i	$-0.269440\pi$
$54$		0			0
$55$	−0.912006	+	9.67948i	−0.122975	+	1.30518i
$56$		0			0
$57$		−	1.60650i		−	0.212786i
$58$		0			0
$59$	1.50205			0.195550			0.0977750	−	0.995209i	$-0.468827\pi$
$59$	1.50205			0.195550			0.0977750	+	0.995209i	$0.468827\pi$
$60$		0			0
$61$	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			0
$63$		−	0.839018i		−	0.105706i
$64$		0			0
$65$	10.1971	+	0.960779i	1.26480	+	0.119170i
$66$		0			0
$67$			1.68494i			0.205848i	0.994689	+	0.102924i	$0.0328198\pi$
$67$			1.68494i			0.205848i	−0.994689	+	0.102924i	$0.967180\pi$
$68$		0			0
$69$	−0.580491			−0.0698829
$70$		0			0
$71$	5.36578			0.636801			0.318401	−	0.947956i	$-0.396854\pi$
$71$	5.36578			0.636801			0.318401	+	0.947956i	$0.396854\pi$
$72$		0			0
$73$			10.1484i			1.18778i	0.804548	+	0.593888i	$0.202408\pi$
$73$			10.1484i			1.18778i	−0.804548	+	0.593888i	$0.797592\pi$
$74$		0			0
$75$	−0.542129	+	2.85138i	−0.0625997	+	0.329248i
$76$		0			0
$77$		−	1.36988i		−	0.156112i
$78$		0			0
$79$	7.39760			0.832295			0.416148	−	0.909297i	$-0.363380\pi$
$79$	7.39760			0.832295			0.416148	+	0.909297i	$0.363380\pi$
$80$		0			0
$81$	6.08082			0.675647
$82$		0			0
$83$		−	15.3052i		−	1.67997i	−0.542611	−	0.839984i	$-0.682564\pi$
$83$		−	15.3052i		−	1.67997i	0.542611	−	0.839984i	$-0.317436\pi$
$84$		0			0
$85$	−2.04246	−	0.192442i	−0.221536	−	0.0208732i
$86$		0			0
$87$			4.08254i			0.437694i
$88$		0			0
$89$	10.9326			1.15885			0.579424	−	0.815026i	$-0.303277\pi$
$89$	10.9326			1.15885			0.579424	+	0.815026i	$0.303277\pi$
$90$		0			0
$91$	−1.44314			−0.151282
$92$		0			0
$93$		−	0.503767i		−	0.0522382i
$94$		0			0
$95$	−0.580491	+	6.16098i	−0.0595571	+	0.632103i
$96$		0			0
$97$			14.2158i			1.44340i	0.692208	+	0.721698i	$0.256638\pi$
$97$			14.2158i			1.44340i	−0.692208	+	0.721698i	$0.743362\pi$
$98$		0			0
$99$	11.5788			1.16371
$100$		0			0
$101$	−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			0
$103$			0.287338i			0.0283122i	0.999900	+	0.0141561i	$0.00450618\pi$
$103$			0.287338i			0.0283122i	−0.999900	+	0.0141561i	$0.995494\pi$
$104$		0			0
$105$	0.0383621	−	0.407152i	0.00374375	−	0.0397340i
$106$		0			0
$107$			14.0935i			1.36247i	0.732063	+	0.681237i	$0.238558\pi$
$107$			14.0935i			1.36247i	−0.732063	+	0.681237i	$0.761442\pi$
$108$		0			0
$109$	−1.76338			−0.168901			−0.0844506	−	0.996428i	$-0.526914\pi$
$109$	−1.76338			−0.168901			−0.0844506	+	0.996428i	$0.526914\pi$
$110$		0			0
$111$	2.72194			0.258355
$112$		0			0
$113$			0.315061i			0.0296385i	0.999890	+	0.0148192i	$0.00471728\pi$
$113$			0.315061i			0.0296385i	−0.999890	+	0.0148192i	$0.995283\pi$
$114$		0			0
$115$	2.22621	+	0.209755i	0.207595	+	0.0195597i
$116$		0			0
$117$		−	12.1980i		−	1.12770i
$118$		0			0
$119$	0.289056			0.0264978
$120$		0			0
$121$	7.90483			0.718621
$122$		0			0
$123$		−	2.72357i		−	0.245576i
$124$		0			0
$125$	3.10940	−	10.7393i	0.278114	−	0.960548i
$126$		0			0
$127$		−	21.3284i		−	1.89259i	−0.323300	−	0.946296i	$-0.604792\pi$
$127$		−	21.3284i		−	1.89259i	0.323300	−	0.946296i	$-0.395208\pi$
$128$		0			0
$129$	5.27634			0.464556
$130$		0			0
$131$	−8.11373			−0.708900			−0.354450	−	0.935075i	$-0.615332\pi$
$131$	−8.11373			−0.708900			−0.354450	+	0.935075i	$0.615332\pi$
$132$		0			0
$133$		−	0.871925i		−	0.0756055i
$134$		0			0
$135$	7.31830	+	0.689534i	0.629859	+	0.0593456i
$136$		0			0
$137$			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			0
$139$	−10.8431			−0.919701			−0.459850	−	0.887996i	$-0.652097\pi$
$139$	−10.8431			−0.919701			−0.459850	+	0.887996i	$0.652097\pi$
$140$		0			0
$141$	−4.78767			−0.403194
$142$		0			0
$143$		−	19.9158i		−	1.66545i
$144$		0			0
$145$	1.47518	−	15.6567i	0.122507	−	1.30022i
$146$		0			0
$147$		−	4.00582i		−	0.330394i
$148$		0			0
$149$	7.90074			0.647254			0.323627	−	0.946185i	$-0.395098\pi$
$149$	7.90074			0.647254			0.323627	+	0.946185i	$0.395098\pi$
$150$		0			0
$151$	8.03700			0.654042			0.327021	−	0.945017i	$-0.393955\pi$
$151$	8.03700			0.654042			0.327021	+	0.945017i	$0.393955\pi$
$152$		0			0
$153$			2.44323i			0.197523i
$154$		0			0
$155$	−0.182031	+	1.93197i	−0.0146211	+	0.155179i
$156$		0			0
$157$			4.75928i			0.379832i	0.981800	+	0.189916i	$0.0608216\pi$
$157$			4.75928i			0.379832i	−0.981800	+	0.189916i	$0.939178\pi$
$158$		0			0
$159$	−6.33016			−0.502014
$160$		0			0
$161$	−0.315061			−0.0248303
$162$		0			0
$163$			2.11545i			0.165695i	0.996562	+	0.0828473i	$0.0264014\pi$
$163$			2.11545i			0.165695i	−0.996562	+	0.0828473i	$0.973599\pi$
$164$		0			0
$165$	5.61885	+	0.529411i	0.437427	+	0.0412146i
$166$		0			0
$167$			15.8760i			1.22852i	0.789102	+	0.614262i	$0.210546\pi$
$167$			15.8760i			1.22852i	−0.789102	+	0.614262i	$0.789454\pi$
$168$		0			0
$169$	−7.98090			−0.613915
$170$		0			0
$171$	7.36988			0.563589
$172$		0			0
$173$			0.0767241i			0.00583323i	0.999996	+	0.00291661i	$0.000928388\pi$
$173$			0.0767241i			0.00583323i	−0.999996	+	0.00291661i	$0.999072\pi$
$174$		0			0
$175$	−0.294240	+	1.54758i	−0.0222425	+	0.116986i
$176$		0			0
$177$		−	0.871925i		−	0.0655379i
$178$		0			0
$179$	−20.7247			−1.54904			−0.774520	−	0.632549i	$-0.782009\pi$
$179$	−20.7247			−1.54904			−0.774520	+	0.632549i	$0.782009\pi$
$180$		0			0
$181$	−11.5610			−0.859319			−0.429660	−	0.902991i	$-0.641366\pi$
$181$	−11.5610			−0.859319			−0.429660	+	0.902991i	$0.641366\pi$
$182$		0			0
$183$		−	6.65441i		−	0.491908i
$184$		0			0
$185$	−10.4388	−	0.983546i	−0.767474	−	0.0723118i
$186$		0			0
$187$			3.98909i			0.291711i
$188$		0			0
$189$	−1.03571			−0.0753371
$190$		0			0
$191$	−21.9683			−1.58957			−0.794784	−	0.606892i	$-0.792416\pi$
$191$	−21.9683			−1.58957			−0.794784	+	0.606892i	$0.792416\pi$
$192$		0			0
$193$		−	20.2914i		−	1.46061i	−0.683122	−	0.730305i	$-0.739378\pi$
$193$		−	20.2914i		−	1.46061i	0.683122	−	0.730305i	$-0.260622\pi$
$194$		0			0
$195$	0.557723	−	5.91934i	0.0399394	−	0.423893i
$196$		0			0
$197$			8.62159i			0.614263i	0.951667	+	0.307131i	$0.0993691\pi$
$197$			8.62159i			0.614263i	−0.951667	+	0.307131i	$0.900631\pi$
$198$		0			0
$199$	20.7796			1.47302			0.736512	−	0.676424i	$-0.236471\pi$
$199$	20.7796			1.47302			0.736512	+	0.676424i	$0.236471\pi$
$200$		0			0
$201$	0.978092			0.0689893
$202$		0			0
$203$			2.21580i			0.155519i
$204$		0			0
$205$	−0.984135	+	10.4450i	−0.0687350	+	0.729512i
$206$		0			0
$207$		−	2.66303i		−	0.185093i
$208$		0			0
$209$	12.0329			0.832334
$210$		0			0
$211$	4.16508			0.286736			0.143368	−	0.989669i	$-0.454207\pi$
$211$	4.16508			0.286736			0.143368	+	0.989669i	$0.454207\pi$
$212$		0			0
$213$		−	3.11479i		−	0.213422i
$214$		0			0
$215$	−20.2350	−	1.90655i	−1.38001	−	0.130026i
$216$		0			0
$217$		−	0.273419i		−	0.0185609i
$218$		0			0
$219$	5.89103			0.398079
$220$		0			0
$221$	4.20242			0.282685
$222$		0			0
$223$			18.7618i			1.25638i	0.778060	+	0.628190i	$0.216204\pi$
$223$			18.7618i			1.25638i	−0.778060	+	0.628190i	$0.783796\pi$
$224$		0			0
$225$	−13.0808	−	2.48704i	−0.872055	−	0.165803i
$226$		0			0
$227$			10.9637i			0.727689i	0.931460	+	0.363845i	$0.118536\pi$
$227$			10.9637i			0.727689i	−0.931460	+	0.363845i	$0.881464\pi$
$228$		0			0
$229$	−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			0
$231$	−0.795201			−0.0523204
$232$		0			0
$233$		−	8.23942i		−	0.539783i	−0.962891	−	0.269891i	$-0.913012\pi$
$233$		−	8.23942i		−	0.539783i	0.962891	−	0.269891i	$-0.0869878\pi$
$234$		0			0
$235$	18.3609	+	1.72998i	1.19773	+	0.112851i
$236$		0			0
$237$		−	4.29424i		−	0.278941i
$238$		0			0
$239$	16.0508			1.03824			0.519120	−	0.854701i	$-0.326260\pi$
$239$	16.0508			1.03824			0.519120	+	0.854701i	$0.326260\pi$
$240$		0			0
$241$	−2.09755			−0.135115			−0.0675574	−	0.997715i	$-0.521521\pi$
$241$	−2.09755			−0.135115			−0.0675574	+	0.997715i	$0.521521\pi$
$242$		0			0
$243$		−	13.3919i		−	0.859089i
$244$		0			0
$245$	−1.44746	+	15.3625i	−0.0924749	+	0.981473i
$246$		0			0
$247$		−	12.6764i		−	0.806580i
$248$		0			0
$249$	−8.88455			−0.563036
$250$		0			0
$251$	13.7018			0.864847			0.432423	−	0.901671i	$-0.357659\pi$
$251$	13.7018			0.864847			0.432423	+	0.901671i	$0.357659\pi$
$252$		0			0
$253$		−	4.34797i		−	0.273354i
$254$		0			0
$255$	−0.111711	+	1.18563i	−0.00699559	+	0.0742470i
$256$		0			0
$257$		−	0.998281i		−	0.0622711i	−0.999515	−	0.0311355i	$-0.990088\pi$
$257$		−	0.998281i		−	0.0622711i	0.999515	−	0.0311355i	$-0.00991235\pi$
$258$		0			0
$259$	1.47733			0.0917971
$260$		0			0
$261$	−18.7288			−1.15929
$262$		0			0
$263$			9.18527i			0.566388i	0.959063	+	0.283194i	$0.0913940\pi$
$263$			9.18527i			0.566388i	−0.959063	+	0.283194i	$0.908606\pi$
$264$		0			0
$265$	24.2764	+	2.28734i	1.49129	+	0.140510i
$266$		0			0
$267$		−	6.34625i		−	0.388384i
$268$		0			0
$269$	−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			0
$271$	25.2488			1.53376			0.766878	−	0.641793i	$-0.221809\pi$
$271$	25.2488			1.53376			0.766878	+	0.641793i	$0.221809\pi$
$272$		0			0
$273$			0.837727i			0.0507016i
$274$		0			0
$275$	−21.3572	−	4.06063i	−1.28789	−	0.244865i
$276$		0			0
$277$			30.8114i			1.85128i	0.378409	+	0.925638i	$0.376471\pi$
$277$			30.8114i			1.85128i	−0.378409	+	0.925638i	$0.623529\pi$
$278$		0			0
$279$	2.31105			0.138359
$280$		0			0
$281$	3.60821			0.215248			0.107624	−	0.994192i	$-0.465676\pi$
$281$	3.60821			0.215248			0.107624	+	0.994192i	$0.465676\pi$
$282$		0			0
$283$		−	5.23942i		−	0.311451i	−0.987800	−	0.155726i	$-0.950228\pi$
$283$		−	5.23942i		−	0.311451i	0.987800	−	0.155726i	$-0.0497716\pi$
$284$		0			0
$285$	3.57639	+	0.336970i	0.211847	+	0.0199604i
$286$		0			0
$287$		−	1.47822i		−	0.0872565i
$288$		0			0
$289$	16.1583			0.950486
$290$		0			0
$291$	8.25214			0.483749
$292$		0			0
$293$		−	1.90074i		−	0.111042i	−0.998458	−	0.0555211i	$-0.982318\pi$
$293$		−	1.90074i		−	0.111042i	0.998458	−	0.0555211i	$-0.0176820\pi$
$294$		0			0
$295$	−0.315061	+	3.34387i	−0.0183436	+	0.194688i
$296$		0			0
$297$		−	14.2932i		−	0.829378i
$298$		0			0
$299$	−4.58049			−0.264897
$300$		0			0
$301$	2.86373			0.165063
$302$		0			0
$303$			4.73975i			0.272292i
$304$		0			0
$305$	−2.40450	+	25.5199i	−0.137681	+	1.46127i
$306$		0			0
$307$		−	2.82467i		−	0.161213i	−0.996746	−	0.0806063i	$-0.974314\pi$
$307$		−	2.82467i		−	0.161213i	0.996746	−	0.0806063i	$-0.0256856\pi$
$308$		0			0
$309$	0.166797			0.00948875
$310$		0			0
$311$	−16.1028			−0.913107			−0.456554	−	0.889696i	$-0.650916\pi$
$311$	−16.1028			−0.913107			−0.456554	+	0.889696i	$0.650916\pi$
$312$		0			0
$313$		−	26.3740i		−	1.49075i	−0.666648	−	0.745373i	$-0.732272\pi$
$313$		−	26.3740i		−	1.49075i	0.666648	−	0.745373i	$-0.267728\pi$
$314$		0			0
$315$	1.86783	+	0.175988i	0.105240	+	0.00991579i
$316$		0			0
$317$		−	15.5020i		−	0.870682i	−0.900266	−	0.435341i	$-0.856628\pi$
$317$		−	15.5020i		−	0.870682i	0.900266	−	0.435341i	$-0.143372\pi$
$318$		0			0
$319$	−30.5789			−1.71209
$320$		0			0
$321$	8.18117			0.456628
$322$		0			0
$323$			2.53905i			0.141277i
$324$		0			0
$325$	−4.27779	+	22.4994i	−0.237289	+	1.24804i
$326$		0			0
$327$			1.02363i			0.0566067i
$328$		0			0
$329$	−2.59851			−0.143260
$330$		0			0
$331$	−19.0302			−1.04599			−0.522997	−	0.852335i	$-0.675186\pi$
$331$	−19.0302			−1.04599			−0.522997	+	0.852335i	$0.675186\pi$
$332$		0			0
$333$			12.4870i			0.684286i
$334$		0			0
$335$	−3.75102	−	0.353423i	−0.204940	−	0.0193096i
$336$		0			0
$337$		−	4.09354i		−	0.222989i	−0.993765	−	0.111495i	$-0.964436\pi$
$337$		−	4.09354i		−	0.222989i	0.993765	−	0.111495i	$-0.0355638\pi$
$338$		0			0
$339$	0.182890			0.00993324
$340$		0			0
$341$	3.77329			0.204335
$342$		0			0
$343$		−	4.37959i		−	0.236475i
$344$		0			0
$345$	0.121761	−	1.29229i	0.00655537	−	0.0695748i
$346$		0			0
$347$			17.4357i			0.935997i	0.883729	+	0.467998i	$0.155025\pi$
$347$			17.4357i			0.935997i	−0.883729	+	0.467998i	$0.844975\pi$
$348$		0			0
$349$	−8.48770			−0.454336			−0.227168	−	0.973856i	$-0.572947\pi$
$349$	−8.48770			−0.454336			−0.227168	+	0.973856i	$0.572947\pi$
$350$		0			0
$351$	−15.0576			−0.803716
$352$		0			0
$353$		−	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			0
$355$	−1.12550	+	11.9453i	−0.0597352	+	0.633993i
$356$		0			0
$357$		−	0.167795i		−	0.00888064i
$358$		0			0
$359$	−22.4121			−1.18286			−0.591432	−	0.806355i	$-0.701437\pi$
$359$	−22.4121			−1.18286			−0.591432	+	0.806355i	$0.701437\pi$
$360$		0			0
$361$	−11.3411			−0.596898
$362$		0			0
$363$		−	4.58868i		−	0.240843i
$364$		0			0
$365$	−22.5923	−	2.12866i	−1.18254	−	0.111419i
$366$		0			0
$367$			21.9401i			1.14526i	0.819812	+	0.572632i	$0.194078\pi$
$367$			21.9401i			1.14526i	−0.819812	+	0.572632i	$0.805922\pi$
$368$		0			0
$369$	12.4945			0.650438
$370$		0			0
$371$	−3.43569			−0.178372
$372$		0			0
$373$		−	0.853821i		−	0.0442092i	−0.999756	−	0.0221046i	$-0.992963\pi$
$373$		−	0.853821i		−	0.0442092i	0.999756	−	0.0221046i	$-0.00703668\pi$
$374$		0			0
$375$	−6.23404	−	1.80498i	−0.321924	−	0.0932088i
$376$		0			0
$377$			32.2142i			1.65911i
$378$		0			0
$379$	−2.52049			−0.129469			−0.0647345	−	0.997903i	$-0.520620\pi$
$379$	−2.52049			−0.129469			−0.0647345	+	0.997903i	$0.520620\pi$
$380$		0			0
$381$	−12.3810			−0.634296
$382$		0			0
$383$		−	13.5983i		−	0.694841i	−0.937709	−	0.347420i	$-0.887058\pi$
$383$		−	13.5983i		−	0.694841i	0.937709	−	0.347420i	$-0.112942\pi$
$384$		0			0
$385$	3.04963	+	0.287338i	0.155424	+	0.0146441i
$386$		0			0
$387$			24.2055i			1.23043i
$388$		0			0
$389$	−10.7398			−0.544527			−0.272264	−	0.962223i	$-0.587772\pi$
$389$	−10.7398			−0.544527			−0.272264	+	0.962223i	$0.587772\pi$
$390$		0			0
$391$	0.917461			0.0463980
$392$		0			0
$393$			4.70995i			0.237585i
$394$		0			0
$395$	−1.55168	+	16.4686i	−0.0780735	+	0.828625i
$396$		0			0
$397$			19.5868i			0.983031i	0.870869	+	0.491516i	$0.163557\pi$
$397$			19.5868i			0.983031i	−0.870869	+	0.491516i	$0.836443\pi$
$398$		0			0
$399$	−0.506145			−0.0253389
$400$		0			0
$401$	−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			0
$403$		−	3.97508i		−	0.198013i
$404$		0			0
$405$	−1.27548	+	13.5372i	−0.0633791	+	0.672668i
$406$		0			0
$407$			20.3878i			1.01058i
$408$		0			0
$409$	28.2350			1.39613			0.698065	−	0.716034i	$-0.254045\pi$
$409$	28.2350			1.39613			0.698065	+	0.716034i	$0.254045\pi$
$410$		0			0
$411$	4.19509			0.206929
$412$		0			0
$413$		−	0.473237i		−	0.0232865i
$414$		0			0
$415$	34.0726	+	3.21034i	1.67256	+	0.157590i
$416$		0			0
$417$			6.29433i			0.308235i
$418$		0			0
$419$	−21.9470			−1.07218			−0.536091	−	0.844160i	$-0.680100\pi$
$419$	−21.9470			−1.07218			−0.536091	+	0.844160i	$0.680100\pi$
$420$		0			0
$421$	0.747198			0.0364162			0.0182081	−	0.999834i	$-0.494204\pi$
$421$	0.747198			0.0364162			0.0182081	+	0.999834i	$0.494204\pi$
$422$		0			0
$423$		−	21.9637i		−	1.06791i
$424$		0			0
$425$	0.856830	−	4.50657i	0.0415624	−	0.218601i
$426$		0			0
$427$		−	3.61168i		−	0.174781i
$428$		0			0
$429$	−11.5610			−0.558168
$430$		0			0
$431$	−27.3642			−1.31808			−0.659042	−	0.752106i	$-0.729038\pi$
$431$	−27.3642			−1.31808			−0.659042	+	0.752106i	$0.729038\pi$
$432$		0			0
$433$		−	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			0
$435$	−9.08858	−	0.856331i	−0.435764	−	0.0410579i
$436$		0			0
$437$		−	2.76748i		−	0.132386i
$438$		0			0
$439$	−30.8097			−1.47047			−0.735233	−	0.677815i	$-0.762927\pi$
$439$	−30.8097			−1.47047			−0.735233	+	0.677815i	$0.762927\pi$
$440$		0			0
$441$	18.3769			0.875089
$442$		0			0
$443$		−	24.5778i		−	1.16773i	−0.811852	−	0.583863i	$-0.801541\pi$
$443$		−	24.5778i		−	1.16773i	0.811852	−	0.583863i	$-0.198459\pi$
$444$		0			0
$445$	−2.29315	+	24.3381i	−0.108706	+	1.15374i
$446$		0			0
$447$		−	4.58631i		−	0.216925i
$448$		0			0
$449$	−38.6816			−1.82550			−0.912749	−	0.408522i	$-0.866044\pi$
$449$	−38.6816			−1.82550			−0.912749	+	0.408522i	$0.866044\pi$
$450$		0			0
$451$	20.4000			0.960597
$452$		0			0
$453$		−	4.66541i		−	0.219200i
$454$		0			0
$455$	0.302704	−	3.21272i	0.0141910	−	0.150615i
$456$		0			0
$457$		−	6.40041i		−	0.299398i	−0.988732	−	0.149699i	$-0.952169\pi$
$457$		−	6.40041i		−	0.299398i	0.988732	−	0.149699i	$-0.0478305\pi$
$458$		0			0
$459$	3.01600			0.140775
$460$		0			0
$461$	−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			0
$463$		−	26.5529i		−	1.23402i	−0.786957	−	0.617008i	$-0.788345\pi$
$463$		−	26.5529i		−	1.23402i	0.786957	−	0.617008i	$-0.211655\pi$
$464$		0			0
$465$	1.12149	+	0.105667i	0.0520078	+	0.00490021i
$466$		0			0
$467$		−	18.6049i		−	0.860931i	−0.902607	−	0.430465i	$-0.858349\pi$
$467$		−	18.6049i		−	0.860931i	0.902607	−	0.430465i	$-0.141651\pi$
$468$		0			0
$469$	0.530859			0.0245128
$470$		0			0
$471$	2.76272			0.127300
$472$		0			0
$473$			39.5206i			1.81716i
$474$		0			0
$475$	−13.5939	−	2.58459i	−0.623729	−	0.118589i
$476$		0			0
$477$		−	29.0399i		−	1.32965i
$478$		0			0
$479$	29.9627			1.36903			0.684514	−	0.728999i	$-0.260014\pi$
$479$	29.9627			1.36903			0.684514	+	0.728999i	$0.260014\pi$
$480$		0			0
$481$	21.4781			0.979316
$482$		0			0
$483$			0.182890i			0.00832180i
$484$		0			0
$485$	−31.6473	−	2.98183i	−1.43703	−	0.135398i
$486$		0			0
$487$			37.2265i			1.68689i	0.537214	+	0.843446i	$0.319477\pi$
$487$			37.2265i			1.68689i	−0.537214	+	0.843446i	$0.680523\pi$
$488$		0			0
$489$	1.22800			0.0555320
$490$		0			0
$491$	0.691841			0.0312224			0.0156112	−	0.999878i	$-0.495031\pi$
$491$	0.691841			0.0312224			0.0156112	+	0.999878i	$0.495031\pi$
$492$		0			0
$493$		−	6.45242i		−	0.290602i
$494$		0			0
$495$	−2.42870	+	25.7768i	−0.109162	+	1.15858i
$496$		0			0
$497$		−	1.69055i		−	0.0758315i
$498$		0			0
$499$	−24.7988			−1.11014			−0.555072	−	0.831802i	$-0.687309\pi$
$499$	−24.7988			−1.11014			−0.555072	+	0.831802i	$0.687309\pi$
$500$		0			0
$501$	9.21589			0.411735
$502$		0			0
$503$			22.6049i			1.00790i	0.863732	+	0.503951i	$0.168121\pi$
$503$			22.6049i			1.00790i	−0.863732	+	0.503951i	$0.831879\pi$
$504$		0			0
$505$	1.71266	−	18.1772i	0.0762125	−	0.808873i
$506$		0			0
$507$			4.63284i			0.205752i
$508$		0			0
$509$	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			0
$511$	3.19735			0.141443
$512$		0			0
$513$		−	9.09763i		−	0.401670i
$514$		0			0
$515$	−0.639674	−	0.0602704i	−0.0281874	−	0.00265583i
$516$		0			0
$517$		−	35.8604i		−	1.57714i
$518$		0			0
$519$	0.0445377			0.00195499
$520$		0			0
$521$	12.8425			0.562638			0.281319	−	0.959614i	$-0.409228\pi$
$521$	12.8425			0.562638			0.281319	+	0.959614i	$0.409228\pi$
$522$		0			0
$523$		−	32.8289i		−	1.43551i	−0.696298	−	0.717753i	$-0.745171\pi$
$523$		−	32.8289i		−	1.43551i	0.696298	−	0.717753i	$-0.254829\pi$
$524$		0			0
$525$	0.898358	+	0.170804i	0.0392076	+	0.00745449i
$526$		0			0
$527$			0.796199i			0.0346830i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		−	21.4909i		−	0.930876i
$534$		0			0
$535$	−31.3752	−	2.95618i	−1.35647	−	0.127807i
$536$		0			0
$537$			12.0305i			0.519156i
$538$		0			0
$539$	30.0042			1.29237
$540$		0			0
$541$	−25.7261			−1.10605			−0.553026	−	0.833164i	$-0.686527\pi$
$541$	−25.7261			−1.10605			−0.553026	+	0.833164i	$0.686527\pi$
$542$		0			0
$543$			6.71103i			0.287998i
$544$		0			0
$545$	0.369877	−	3.92565i	0.0158438	−	0.168157i
$546$		0			0
$547$		−	36.0191i		−	1.54006i	−0.638005	−	0.770032i	$-0.720240\pi$
$547$		−	36.0191i		−	1.54006i	0.638005	−	0.770032i	$-0.279760\pi$
$548$		0			0
$549$	30.5274			1.30288
$550$		0			0
$551$	−19.4634			−0.829169
$552$		0			0
$553$		−	2.33070i		−	0.0991114i
$554$		0			0
$555$	−0.570940	+	6.05961i	−0.0242350	+	0.257216i
$556$		0			0
$557$		−	36.2148i		−	1.53447i	−0.641366	−	0.767235i	$-0.721632\pi$
$557$		−	36.2148i		−	1.53447i	0.641366	−	0.767235i	$-0.278368\pi$
$558$		0			0
$559$	41.6341			1.76094
$560$		0			0
$561$	2.31563			0.0977660
$562$		0			0
$563$		−	23.3308i		−	0.983277i	−0.870799	−	0.491638i	$-0.836398\pi$
$563$		−	23.3308i		−	0.983277i	0.870799	−	0.491638i	$-0.163602\pi$
$564$		0			0
$565$	−0.701392	−	0.0660856i	−0.0295078	−	0.00278024i
$566$		0			0
$567$		−	1.91583i		−	0.0804574i
$568$		0			0
$569$	−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			0
$571$	−38.7600			−1.62206			−0.811028	−	0.585007i	$-0.801092\pi$
$571$	−38.7600			−1.62206			−0.811028	+	0.585007i	$0.801092\pi$
$572$		0			0
$573$			12.7524i			0.532738i
$574$		0			0
$575$	−0.933914	+	4.91201i	−0.0389469	+	0.204845i
$576$		0			0
$577$			21.5902i			0.898812i	0.893328	+	0.449406i	$0.148364\pi$
$577$			21.5902i			0.898812i	−0.893328	+	0.449406i	$0.851636\pi$
$578$		0			0
$579$	−11.7790			−0.489518
$580$		0			0
$581$	−4.82209			−0.200054
$582$		0			0
$583$		−	47.4139i		−	1.96368i
$584$		0			0
$585$	27.1553	+	2.55858i	1.12273	+	0.105784i
$586$		0			0
$587$			24.3902i			1.00669i	0.864086	+	0.503345i	$0.167897\pi$
$587$			24.3902i			1.00669i	−0.864086	+	0.503345i	$0.832103\pi$
$588$		0			0
$589$	2.40170			0.0989602
$590$		0			0
$591$	5.00476			0.205868
$592$		0			0
$593$		−	13.9747i		−	0.573874i	−0.957949	−	0.286937i	$-0.907363\pi$
$593$		−	13.9747i		−	0.573874i	0.957949	−	0.286937i	$-0.0926370\pi$
$594$		0			0
$595$	−0.0606309	+	0.643500i	−0.00248562	+	0.0263809i
$596$		0			0
$597$		−	12.0623i		−	0.493679i
$598$		0			0
$599$	−20.9932			−0.857758			−0.428879	−	0.903362i	$-0.641091\pi$
$599$	−20.9932			−0.857758			−0.428879	+	0.903362i	$0.641091\pi$
$600$		0			0
$601$	−13.3007			−0.542546			−0.271273	−	0.962502i	$-0.587445\pi$
$601$	−13.3007			−0.542546			−0.271273	+	0.962502i	$0.587445\pi$
$602$		0			0
$603$			4.48704i			0.182726i
$604$		0			0
$605$	−1.65807	+	17.5978i	−0.0674103	+	0.715452i
$606$		0			0
$607$		−	12.3549i		−	0.501469i	−0.968056	−	0.250734i	$-0.919328\pi$
$607$		−	12.3549i		−	0.501469i	0.968056	−	0.250734i	$-0.0806721\pi$
$608$		0			0
$609$	1.28625			0.0521215
$610$		0			0
$611$	−37.7781			−1.52834
$612$		0			0
$613$		−	14.6291i		−	0.590865i	−0.955364	−	0.295432i	$-0.904536\pi$
$613$		−	14.6291i		−	0.590865i	0.955364	−	0.295432i	$-0.0954637\pi$
$614$		0			0
$615$	6.06324	+	0.571281i	0.244493	+	0.0230363i
$616$		0			0
$617$		−	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			0
$619$	40.0049			1.60793			0.803967	−	0.594674i	$-0.202719\pi$
$619$	40.0049			1.60793			0.803967	+	0.594674i	$0.202719\pi$
$620$		0			0
$621$	−3.28734			−0.131916
$622$		0			0
$623$		−	3.44443i		−	0.137998i
$624$		0			0
$625$	23.2556	+	9.17479i	0.930224	+	0.366991i
$626$		0			0
$627$		−	6.98499i		−	0.278954i
$628$		0			0
$629$	−4.30201			−0.171532
$630$		0			0
$631$	−15.7992			−0.628956			−0.314478	−	0.949265i	$-0.601829\pi$
$631$	−15.7992			−0.628956			−0.314478	+	0.949265i	$0.601829\pi$
$632$		0			0
$633$		−	2.41779i		−	0.0960985i
$634$		0			0
$635$	47.4815	+	4.47374i	1.88425	+	0.177535i
$636$		0			0
$637$		−	31.6088i		−	1.25238i
$638$		0			0
$639$	14.2892			0.565273
$640$		0			0
$641$	−37.3861			−1.47666			−0.738332	−	0.674437i	$-0.764386\pi$
$641$	−37.3861			−1.47666			−0.738332	+	0.674437i	$0.764386\pi$
$642$		0			0
$643$			44.1766i			1.74216i	0.491145	+	0.871078i	$0.336579\pi$
$643$			44.1766i			1.74216i	−0.491145	+	0.871078i	$0.663421\pi$
$644$		0			0
$645$	−1.10674	+	11.7462i	−0.0435777	+	0.462507i
$646$		0			0
$647$		−	4.88521i		−	0.192058i	−0.995379	−	0.0960288i	$-0.969386\pi$
$647$		−	4.88521i		−	0.192058i	0.995379	−	0.0960288i	$-0.0306141\pi$
$648$		0			0
$649$	6.53086			0.256359
$650$		0			0
$651$	−0.158717			−0.00622063
$652$		0			0
$653$			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			0
$655$	1.70189	−	18.0628i	0.0664984	−	0.705774i
$656$		0			0
$657$			27.0254i			1.05436i
$658$		0			0
$659$	−10.3388			−0.402743			−0.201371	−	0.979515i	$-0.564540\pi$
$659$	−10.3388			−0.402743			−0.201371	+	0.979515i	$0.564540\pi$
$660$		0			0
$661$	6.46980			0.251646			0.125823	−	0.992053i	$-0.459843\pi$
$661$	6.46980			0.251646			0.125823	+	0.992053i	$0.459843\pi$
$662$		0			0
$663$		−	2.43947i		−	0.0947411i
$664$		0			0
$665$	1.94109	+	0.182890i	0.0752722	+	0.00709218i
$666$		0			0
$667$			7.03291i			0.272315i
$668$		0			0
$669$	10.8910			0.421071
$670$		0			0
$671$	49.8426			1.92415
$672$		0			0
$673$			30.1889i			1.16370i	0.813297	+	0.581849i	$0.197670\pi$
$673$			30.1889i			1.16370i	−0.813297	+	0.581849i	$0.802330\pi$
$674$		0			0
$675$	−3.07009	+	16.1474i	−0.118168	+	0.621515i
$676$		0			0
$677$		−	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			0
$679$	4.47885			0.171882
$680$		0			0
$681$	6.36436			0.243883
$682$		0			0
$683$		−	25.7553i		−	0.985498i	−0.870171	−	0.492749i	$-0.835992\pi$
$683$		−	25.7553i		−	0.985498i	0.870171	−	0.492749i	$-0.164008\pi$
$684$		0			0
$685$	−16.0884	−	1.51585i	−0.614704	−	0.0579178i
$686$		0			0
$687$			15.0560i			0.574422i
$688$		0			0
$689$	−49.9495			−1.90292
$690$		0			0
$691$	−16.1707			−0.615162			−0.307581	−	0.951522i	$-0.599520\pi$
$691$	−16.1707			−0.615162			−0.307581	+	0.951522i	$0.599520\pi$
$692$		0			0
$693$		−	3.64802i		−	0.138577i
$694$		0			0
$695$	2.27439	−	24.1390i	0.0862726	−	0.915646i
$696$		0			0
$697$			4.30458i			0.163048i
$698$		0			0
$699$	−4.78291			−0.180906
$700$		0			0
$701$	−26.5049			−1.00107			−0.500537	−	0.865715i	$-0.666864\pi$
$701$	−26.5049			−1.00107			−0.500537	+	0.865715i	$0.666864\pi$
$702$		0			0
$703$			12.9768i			0.489429i
$704$		0			0
$705$	1.00423	−	10.6583i	0.0378217	−	0.401416i
$706$		0			0
$707$			2.57250i			0.0967489i
$708$		0			0
$709$	−38.6885			−1.45298			−0.726488	−	0.687179i	$-0.758849\pi$
$709$	−38.6885			−1.45298			−0.726488	+	0.687179i	$0.758849\pi$
$710$		0			0
$711$	19.7000			0.738809
$712$		0			0
$713$		−	0.867829i		−	0.0325004i
$714$		0			0
$715$	44.3368	+	4.17744i	1.65810	+	0.156227i
$716$		0			0
$717$		−	9.31735i		−	0.347963i
$718$		0			0
$719$	43.7242			1.63064			0.815319	−	0.579012i	$-0.196562\pi$
$719$	43.7242			1.63064			0.815319	+	0.579012i	$0.196562\pi$
$720$		0			0
$721$	0.0905291			0.00337148
$722$		0			0
$723$			1.21761i			0.0452833i
$724$		0			0
$725$	34.5457	+	6.56813i	1.28299	+	0.243934i
$726$		0			0
$727$			37.2280i			1.38071i	0.723471	+	0.690355i	$0.242545\pi$
$727$			37.2280i			1.38071i	−0.723471	+	0.690355i	$0.757455\pi$
$728$		0			0
$729$	10.4686			0.387726
$730$		0			0
$731$	−8.33921			−0.308437
$732$		0			0
$733$			27.9656i			1.03293i	0.856308	+	0.516466i	$0.172753\pi$
$733$			27.9656i			1.03293i	−0.856308	+	0.516466i	$0.827247\pi$
$734$		0			0
$735$	8.91778	+	0.840238i	0.328937	+	0.0309926i
$736$		0			0
$737$			7.32606i			0.269859i
$738$		0			0
$739$	−30.4555			−1.12032			−0.560162	−	0.828383i	$-0.689261\pi$
$739$	−30.4555			−1.12032			−0.560162	+	0.828383i	$0.689261\pi$
$740$		0			0
$741$	−7.35854			−0.270323
$742$		0			0
$743$		−	1.97918i		−	0.0726090i	−0.999341	−	0.0363045i	$-0.988441\pi$
$743$		−	1.97918i		−	0.0726090i	0.999341	−	0.0363045i	$-0.0115586\pi$
$744$		0			0
$745$	−1.65722	+	17.5887i	−0.0607157	+	0.644400i
$746$		0			0
$747$		−	40.7583i		−	1.49127i
$748$		0			0
$749$	4.44033			0.162246
$750$		0			0
$751$	−5.77719			−0.210813			−0.105406	−	0.994429i	$-0.533614\pi$
$751$	−5.77719			−0.210813			−0.105406	+	0.994429i	$0.533614\pi$
$752$		0			0
$753$		−	7.95374i		−	0.289851i
$754$		0			0
$755$	−1.68580	+	17.8920i	−0.0613525	+	0.651158i
$756$		0			0
$757$			26.9857i			0.980814i	0.871494	+	0.490407i	$0.163152\pi$
$757$			26.9857i			0.980814i	−0.871494	+	0.490407i	$0.836848\pi$
$758$		0			0
$759$	−2.52396			−0.0916138
$760$		0			0
$761$	5.95209			0.215763			0.107881	−	0.994164i	$-0.465593\pi$
$761$	5.95209			0.215763			0.107881	+	0.994164i	$0.465593\pi$
$762$		0			0
$763$			0.555573i			0.0201131i
$764$		0			0
$765$	−5.43913	−	0.512478i	−0.196652	−	0.0185287i
$766$		0			0
$767$		−	6.88012i		−	0.248427i
$768$		0			0
$769$	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			0
$771$	−0.579493			−0.0208699
$772$		0			0
$773$			2.24223i			0.0806474i	0.999187	+	0.0403237i	$0.0128389\pi$
$773$			2.24223i			0.0806474i	−0.999187	+	0.0403237i	$0.987161\pi$
$774$		0			0
$775$	−4.26278	−	0.810478i	−0.153124	−	0.0291132i
$776$		0			0
$777$		−	0.857579i		−	0.0307655i
$778$		0			0
$779$	12.9846			0.465220
$780$		0			0
$781$	23.3302			0.834822
$782$		0			0
$783$			23.1195i			0.826225i
$784$		0			0
$785$	−10.5952	−	0.998281i	−0.378157	−	0.0356302i
$786$		0			0
$787$		−	6.28634i		−	0.224084i	−0.993703	−	0.112042i	$-0.964261\pi$
$787$		−	6.28634i		−	0.224084i	0.993703	−	0.112042i	$-0.0357391\pi$
$788$		0			0
$789$	5.33197			0.189823
$790$		0			0
$791$	0.0992637			0.00352941
$792$		0			0
$793$		−	52.5081i		−	1.86462i
$794$		0			0
$795$	1.32778	−	14.0922i	0.0470915	−	0.499801i
$796$		0			0
$797$			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			0
$799$	7.56687			0.267696
$800$		0			0
$801$	29.1137			1.02868
$802$		0			0
$803$			44.1247i			1.55713i
$804$		0			0
$805$	0.0660856	−	0.701392i	0.00232921	−	0.0247208i
$806$		0			0
$807$			12.4456i			0.438106i
$808$		0			0
$809$	−7.48638			−0.263207			−0.131604	−	0.991302i	$-0.542013\pi$
$809$	−7.48638			−0.263207			−0.131604	+	0.991302i	$0.542013\pi$
$810$		0			0
$811$	39.7694			1.39649			0.698246	−	0.715858i	$-0.253964\pi$
$811$	39.7694			1.39649			0.698246	+	0.715858i	$0.253964\pi$
$812$		0			0
$813$		−	14.6567i		−	0.514033i
$814$		0			0
$815$	−4.70942	−	0.443724i	−0.164964	−	0.0155430i
$816$		0			0
$817$			25.1548i			0.880056i
$818$		0			0
$819$	−3.84311			−0.134289
$820$		0			0
$821$	30.2829			1.05688			0.528440	−	0.848970i	$-0.322777\pi$
$821$	30.2829			1.05688			0.528440	+	0.848970i	$0.322777\pi$
$822$		0			0
$823$		−	7.15992i		−	0.249579i	−0.992183	−	0.124790i	$-0.960174\pi$
$823$		−	7.15992i		−	0.249579i	0.992183	−	0.124790i	$-0.0398256\pi$
$824$		0			0
$825$	−2.35716	+	12.3977i	−0.0820658	+	0.431632i
$826$		0			0
$827$		−	29.1034i		−	1.01202i	−0.862527	−	0.506011i	$-0.831119\pi$
$827$		−	29.1034i		−	1.01202i	0.862527	−	0.506011i	$-0.168881\pi$
$828$		0			0
$829$	42.4480			1.47428			0.737140	−	0.675740i	$-0.236176\pi$
$829$	42.4480			1.47428			0.737140	+	0.675740i	$0.236176\pi$
$830$		0			0
$831$	17.8857			0.620449
$832$		0			0
$833$			6.33115i			0.219362i
$834$		0			0
$835$	−35.3433	−	3.33007i	−1.22311	−	0.115242i
$836$		0			0
$837$		−	2.85285i		−	0.0986088i
$838$		0			0
$839$	−32.2617			−1.11380			−0.556898	−	0.830581i	$-0.688009\pi$
$839$	−32.2617			−1.11380			−0.556898	+	0.830581i	$0.688009\pi$
$840$		0			0
$841$	20.4618			0.705579
$842$		0			0
$843$		−	2.09454i		−	0.0721396i
$844$		0			0
$845$	1.67403	−	17.7671i	0.0575884	−	0.611208i
$846$		0			0
$847$		−	2.49051i		−	0.0855749i
$848$		0			0
$849$	−3.04144			−0.104382
$850$		0			0
$851$	4.68904			0.160738
$852$		0			0
$853$		−	12.3883i		−	0.424168i	−0.977251	−	0.212084i	$-0.931975\pi$
$853$		−	12.3883i		−	0.424168i	0.977251	−	0.212084i	$-0.0680250\pi$
$854$		0			0
$855$	−1.54587	+	16.4069i	−0.0528675	+	0.561103i
$856$		0			0
$857$			11.8646i			0.405287i	0.979253	+	0.202643i	$0.0649532\pi$
$857$			11.8646i			0.405287i	−0.979253	+	0.202643i	$0.935047\pi$
$858$		0			0
$859$	6.57949			0.224489			0.112245	−	0.993681i	$-0.464196\pi$
$859$	6.57949			0.224489			0.112245	+	0.993681i	$0.464196\pi$
$860$		0			0
$861$	−0.858092			−0.0292437
$862$		0			0
$863$			41.6998i			1.41948i	0.704464	+	0.709739i	$0.251187\pi$
$863$			41.6998i			1.41948i	−0.704464	+	0.709739i	$0.748813\pi$
$864$		0			0
$865$	−0.170804	−	0.0160932i	−0.00580751	−	0.000547186i
$866$		0			0
$867$		−	9.37973i		−	0.318552i
$868$		0			0
$869$	32.1645			1.09111
$870$		0			0
$871$	7.71785			0.261509
$872$		0			0
$873$			37.8571i			1.28127i
$874$		0			0
$875$	−3.38352	−	0.979653i	−0.114384	−	0.0331183i
$876$		0			0
$877$		−	20.9988i		−	0.709079i	−0.935041	−	0.354540i	$-0.884638\pi$
$877$		−	20.9988i		−	0.709079i	0.935041	−	0.354540i	$-0.115362\pi$
$878$		0			0
$879$	−1.10336			−0.0372154
$880$		0			0
$881$	−12.3401			−0.415747			−0.207874	−	0.978156i	$-0.566654\pi$
$881$	−12.3401			−0.415747			−0.207874	+	0.978156i	$0.566654\pi$
$882$		0			0
$883$			35.5426i			1.19610i	0.801457	+	0.598052i	$0.204058\pi$
$883$			35.5426i			1.19610i	−0.801457	+	0.598052i	$0.795942\pi$
$884$		0			0
$885$	1.94109	+	0.182890i	0.0652489	+	0.00614779i
$886$		0			0
$887$		−	19.1997i		−	0.644663i	−0.946627	−	0.322331i	$-0.895533\pi$
$887$		−	19.1997i		−	0.644663i	0.946627	−	0.322331i	$-0.104467\pi$
$888$		0			0
$889$	−6.71977			−0.225374
$890$		0			0
$891$	26.4392			0.885747
$892$		0			0
$893$		−	22.8251i		−	0.763813i
$894$		0			0
$895$	4.34711	−	46.1376i	0.145308	−	1.54221i
$896$		0			0
$897$			2.65893i			0.0887792i
$898$		0			0
$899$	−6.10336			−0.203558
$900$		0			0
$901$	10.0048			0.333307
$902$		0			0
$903$		−	1.66237i		−	0.0553202i
$904$		0			0
$905$	2.42496	−	25.7371i	0.0806085	−	0.855530i
$906$		0			0
$907$			54.5806i			1.81232i	0.422937	+	0.906159i	$0.360999\pi$
$907$			54.5806i			1.81232i	−0.422937	+	0.906159i	$0.639001\pi$
$908$		0			0
$909$	−21.7439			−0.721198
$910$		0			0
$911$	−24.7619			−0.820397			−0.410199	−	0.911996i	$-0.634541\pi$
$911$	−24.7619			−0.820397			−0.410199	+	0.911996i	$0.634541\pi$
$912$		0			0
$913$		−	66.5467i		−	2.20237i
$914$		0			0
$915$	14.8141	+	1.39579i	0.489739	+	0.0461435i
$916$		0			0
$917$			2.55632i			0.0844172i
$918$		0			0
$919$	9.32187			0.307500			0.153750	−	0.988110i	$-0.450865\pi$
$919$	9.32187			0.307500			0.153750	+	0.988110i	$0.450865\pi$
$920$		0			0
$921$	−1.63970			−0.0540298
$922$		0			0
$923$		−	24.5779i		−	0.808992i
$924$		0			0
$925$	4.37916	−	23.0326i	0.143986	−	0.757306i
$926$		0			0
$927$			0.765190i			0.0251321i
$928$		0			0
$929$	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			0
$931$	19.0976			0.625900
$932$		0			0
$933$			9.34754i			0.306025i
$934$		0			0
$935$	−8.88055	−	0.836730i	−0.290425	−	0.0273640i
$936$		0			0
$937$			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			0
$939$	−15.3099			−0.499618
$940$		0			0
$941$	39.8390			1.29871			0.649357	−	0.760484i	$-0.275038\pi$
$941$	39.8390			1.29871			0.649357	+	0.760484i	$0.275038\pi$
$942$		0			0
$943$		−	4.69184i		−	0.152787i
$944$		0			0
$945$	0.217246	−	2.30571i	0.00706700	−	0.0750049i
$946$		0			0
$947$		−	12.7594i		−	0.414624i	−0.978275	−	0.207312i	$-0.933528\pi$
$947$		−	12.7594i		−	0.414624i	0.978275	−	0.207312i	$-0.0664715\pi$
$948$		0			0
$949$	46.4844			1.50895
$950$		0			0
$951$	−8.99880			−0.291806
$952$		0			0
$953$			7.04791i			0.228304i	0.993463	+	0.114152i	$0.0364151\pi$
$953$			7.04791i			0.228304i	−0.993463	+	0.114152i	$0.963585\pi$
$954$		0			0
$955$	4.60794	−	48.9059i	0.149110	−	1.58256i
$956$		0			0
$957$			17.7508i			0.573800i
$958$		0			0
$959$	2.27688			0.0735244
$960$		0			0
$961$	−30.2469			−0.975706
$962$		0			0
$963$			37.5315i			1.20944i
$964$		0			0
$965$	45.1730	+	4.25622i	1.45417	+	0.137013i
$966$		0			0
$967$			9.27901i			0.298393i	0.988808	+	0.149196i	$0.0476687\pi$
$967$			9.27901i			0.298393i	−0.988808	+	0.149196i	$0.952331\pi$
$968$		0			0
$969$	1.47390			0.0473484
$970$		0			0
$971$	−0.0274304			−0.000880284			−0.000440142	−	1.00000i	$-0.500140\pi$
$971$	−0.0274304			−0.000880284			−0.000440142		1.00000i	$0.500140\pi$
$972$		0			0
$973$			3.41625i			0.109520i
$974$		0			0
$975$	13.0607	+	2.48322i	0.418277	+	0.0795266i
$976$		0			0
$977$			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			0
$979$	47.5344			1.51921
$980$		0			0
$981$	−4.69594			−0.149930
$982$		0			0
$983$		−	5.80749i		−	0.185230i	−0.995702	−	0.0926151i	$-0.970477\pi$
$983$		−	5.80749i		−	0.185230i	0.995702	−	0.0926151i	$-0.0295226\pi$
$984$		0			0
$985$	−19.1935	−	1.80842i	−0.611554	−	0.0576210i
$986$		0			0
$987$			1.50841i			0.0480132i
$988$		0			0
$989$	9.08944			0.289027
$990$		0			0
$991$	−17.5926			−0.558847			−0.279423	−	0.960168i	$-0.590143\pi$
$991$	−17.5926			−0.558847			−0.279423	+	0.960168i	$0.590143\pi$
$992$		0			0
$993$			11.0469i			0.350561i
$994$		0			0
$995$	−4.35861	+	46.2596i	−0.138177	+	1.46653i
$996$		0			0
$997$		−	24.2200i		−	0.767055i	−0.923530	−	0.383527i	$-0.874709\pi$
$997$		−	24.2200i		−	0.767055i	0.923530	−	0.383527i	$-0.125291\pi$
$998$		0			0
$999$	15.4144			0.487691

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.3	✓	8	4.3	odd	2
115.2.b.b.24.6	yes	8	20.19	odd	2
575.2.a.i.1.3		4	20.7	even	4
575.2.a.j.1.2		4	20.3	even	4
1035.2.b.e.829.3		8	60.59	even	2
1035.2.b.e.829.6		8	12.11	even	2
1840.2.e.d.369.4		8	1.1	even	1	trivial
1840.2.e.d.369.5		8	5.4	even	2	inner
5175.2.a.bv.1.2		4	60.47	odd	4
5175.2.a.bw.1.3		4	60.23	odd	4
9200.2.a.ck.1.3		4	5.3	odd	4
9200.2.a.cq.1.2		4	5.2	odd	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.580491i q^{3} +(-0.209755 + 2.22621i) q^{5} -0.315061i q^{7} +2.66303 q^{9} +O(q^{10})\)	\(q-0.580491i q^{3} +(-0.209755 + 2.22621i) q^{5} -0.315061i q^{7} +2.66303 q^{9} +4.34797 q^{11} -4.58049i q^{13} +(1.29229 + 0.121761i) q^{15} +0.917461i q^{17} +2.76748 q^{19} -0.182890 q^{21} -1.00000i q^{23} +(-4.91201 - 0.933914i) q^{25} -3.28734i q^{27} -7.03291 q^{29} +0.867829 q^{31} -2.52396i q^{33} +(0.701392 + 0.0660856i) q^{35} +4.68904i q^{37} -2.65893 q^{39} +4.69184 q^{41} +9.08944i q^{43} +(-0.558583 + 5.92846i) q^{45} -8.24762i q^{47} +6.90074 q^{49} +0.532578 q^{51} -10.9048i q^{53} +(-0.912006 + 9.67948i) q^{55} -1.60650i q^{57} +1.50205 q^{59} +11.4634 q^{61} -0.839018i q^{63} +(10.1971 + 0.960779i) q^{65} +1.68494i q^{67} -0.580491 q^{69} +5.36578 q^{71} +10.1484i q^{73} +(-0.542129 + 2.85138i) q^{75} -1.36988i q^{77} +7.39760 q^{79} +6.08082 q^{81} -15.3052i q^{83} +(-2.04246 - 0.192442i) q^{85} +4.08254i q^{87} +10.9326 q^{89} -1.44314 q^{91} -0.503767i q^{93} +(-0.580491 + 6.16098i) q^{95} +14.2158i q^{97} +11.5788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q - 6 * q^5 - 8 * q^9	\( 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71} + 48 q^{79} + 16 q^{81} + 12 q^{85} + 16 q^{89} - 52 q^{91} + 4 q^{95} + 72 q^{99}+O(q^{100}) \) 8 * q - 6 * q^5 - 8 * q^9 - 4 * q^11 - 6 * q^15 - 8 * q^19 - 4 * q^21 - 16 * q^25 - 8 * q^29 - 28 * q^35 - 16 * q^39 - 16 * q^41 + 24 * q^45 - 20 * q^51 + 16 * q^55 - 16 * q^61 - 14 * q^65 + 4 * q^69 + 48 * q^71 + 48 * q^79 + 16 * q^81 + 12 * q^85 + 16 * q^89 - 52 * q^91 + 4 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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