LMFDB - Embedded newform 495.2.c.d.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(199,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("495.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	495.c (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:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	495.199
Dual form			495.2.c.d.199.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53919i q^{2} -0.369102 q^{4} +(-2.17009 + 0.539189i) q^{5} +0.290725i q^{7} -2.51026i q^{8} +O(q^{10})$	\(q-1.53919i q^{2} -0.369102 q^{4} +(-2.17009 + 0.539189i) q^{5} +0.290725i q^{7} -2.51026i q^{8} +(0.829914 + 3.34017i) q^{10} +1.00000 q^{11} -6.97107i q^{13} +0.447480 q^{14} -4.60197 q^{16} -4.78765i q^{17} -7.75872 q^{19} +(0.800984 - 0.199016i) q^{20} -1.53919i q^{22} +4.00000i q^{23} +(4.41855 - 2.34017i) q^{25} -10.7298 q^{26} -0.107307i q^{28} -7.41855 q^{29} +6.34017 q^{31} +2.06278i q^{32} -7.36910 q^{34} +(-0.156755 - 0.630898i) q^{35} -3.41855i q^{37} +11.9421i q^{38} +(1.35350 + 5.44748i) q^{40} +7.41855 q^{41} -0.290725i q^{43} -0.369102 q^{44} +6.15676 q^{46} -5.26180i q^{47} +6.91548 q^{49} +(-3.60197 - 6.80098i) q^{50} +2.57304i q^{52} +5.75872i q^{53} +(-2.17009 + 0.539189i) q^{55} +0.729794 q^{56} +11.4186i q^{58} +3.60197 q^{59} -6.68035 q^{61} -9.75872i q^{62} -6.02893 q^{64} +(3.75872 + 15.1278i) q^{65} +6.15676i q^{67} +1.76713i q^{68} +(-0.971071 + 0.241276i) q^{70} +5.07838 q^{71} +1.12783i q^{73} -5.26180 q^{74} +2.86376 q^{76} +0.290725i q^{77} +0.921622 q^{79} +(9.98667 - 2.48133i) q^{80} -11.4186i q^{82} +1.70928i q^{83} +(2.58145 + 10.3896i) q^{85} -0.447480 q^{86} -2.51026i q^{88} +4.34017 q^{89} +2.02666 q^{91} -1.47641i q^{92} -8.09890 q^{94} +(16.8371 - 4.18342i) q^{95} +4.68035i q^{97} -10.6442i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} - 2 q^{5}+O(q^{10}) $ 6 * q - 10 * q^4 - 2 * q^5	$ 6 q - 10 q^{4} - 2 q^{5} + 16 q^{10} + 6 q^{11} + 4 q^{14} + 10 q^{16} + 4 q^{19} - 14 q^{20} - 2 q^{25} + 16 q^{26} - 16 q^{29} + 16 q^{31} - 52 q^{34} + 12 q^{35} - 12 q^{40} + 16 q^{41} - 10 q^{44} + 24 q^{46} - 22 q^{49} + 16 q^{50} - 2 q^{55} - 76 q^{56} - 16 q^{59} + 4 q^{61} - 66 q^{64} - 28 q^{65} + 24 q^{70} + 24 q^{71} - 16 q^{74} - 36 q^{76} + 12 q^{79} + 58 q^{80} + 44 q^{85} - 4 q^{86} + 4 q^{89} + 16 q^{91} + 24 q^{94} + 44 q^{95}+O(q^{100}) $ 6 * q - 10 * q^4 - 2 * q^5 + 16 * q^10 + 6 * q^11 + 4 * q^14 + 10 * q^16 + 4 * q^19 - 14 * q^20 - 2 * q^25 + 16 * q^26 - 16 * q^29 + 16 * q^31 - 52 * q^34 + 12 * q^35 - 12 * q^40 + 16 * q^41 - 10 * q^44 + 24 * q^46 - 22 * q^49 + 16 * q^50 - 2 * q^55 - 76 * q^56 - 16 * q^59 + 4 * q^61 - 66 * q^64 - 28 * q^65 + 24 * q^70 + 24 * q^71 - 16 * q^74 - 36 * q^76 + 12 * q^79 + 58 * q^80 + 44 * q^85 - 4 * q^86 + 4 * q^89 + 16 * q^91 + 24 * q^94 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	1.53919i		−	1.08837i	−0.838965	−	0.544185i	$-0.816839\pi$
$2$		−	1.53919i		−	1.08837i	0.838965	−	0.544185i	$-0.183161\pi$
$3$		0			0
$3$		0			0
$4$	−0.369102			−0.184551
$5$	−2.17009	+	0.539189i	−0.970492	+	0.241133i
$5$	−2.17009	+	0.539189i	−0.970492	+	0.241133i
$6$		0			0
$7$			0.290725i			0.109884i	0.998490	+	0.0549418i	$0.0174973\pi$
$7$			0.290725i			0.109884i	−0.998490	+	0.0549418i	$0.982503\pi$
$8$		−	2.51026i		−	0.887511i
$9$		0			0
$10$	0.829914	+	3.34017i	0.262442	+	1.05626i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$		−	6.97107i		−	1.93343i	−0.255861	−	0.966714i	$-0.582359\pi$
$13$		−	6.97107i		−	1.93343i	0.255861	−	0.966714i	$-0.417641\pi$
$14$	0.447480			0.119594
$15$		0			0
$16$	−4.60197			−1.15049
$17$		−	4.78765i		−	1.16118i	−0.814197	−	0.580588i	$-0.802823\pi$
$17$		−	4.78765i		−	1.16118i	0.814197	−	0.580588i	$-0.197177\pi$
$18$		0			0
$19$	−7.75872			−1.77997			−0.889987	−	0.455987i	$-0.849286\pi$
$19$	−7.75872			−1.77997			−0.889987	+	0.455987i	$0.849286\pi$
$20$	0.800984	−	0.199016i	0.179105	−	0.0445013i
$21$		0			0
$22$		−	1.53919i		−	0.328156i
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	4.41855	−	2.34017i	0.883710	−	0.468035i
$26$	−10.7298			−2.10429
$27$		0			0
$28$		−	0.107307i		−	0.0202791i
$29$	−7.41855			−1.37759			−0.688795	−	0.724956i	$-0.741860\pi$
$29$	−7.41855			−1.37759			−0.688795	+	0.724956i	$0.741860\pi$
$30$		0			0
$31$	6.34017			1.13873			0.569364	−	0.822085i	$-0.307189\pi$
$31$	6.34017			1.13873			0.569364	+	0.822085i	$0.307189\pi$
$32$			2.06278i			0.364651i
$33$		0			0
$34$	−7.36910			−1.26379
$35$	−0.156755	−	0.630898i	−0.0264965	−	0.106641i
$36$		0			0
$37$		−	3.41855i		−	0.562006i	−0.959707	−	0.281003i	$-0.909333\pi$
$37$		−	3.41855i		−	0.562006i	0.959707	−	0.281003i	$-0.0906671\pi$
$38$			11.9421i			1.93727i
$39$		0			0
$40$	1.35350	+	5.44748i	0.214008	+	0.861322i
$41$	7.41855			1.15858			0.579291	−	0.815120i	$-0.303329\pi$
$41$	7.41855			1.15858			0.579291	+	0.815120i	$0.303329\pi$
$42$		0			0
$43$		−	0.290725i		−	0.0443351i	−0.999754	−	0.0221675i	$-0.992943\pi$
$43$		−	0.290725i		−	0.0443351i	0.999754	−	0.0221675i	$-0.00705673\pi$
$44$	−0.369102			−0.0556443
$45$		0			0
$46$	6.15676			0.907764
$47$		−	5.26180i		−	0.767512i	−0.923435	−	0.383756i	$-0.874630\pi$
$47$		−	5.26180i		−	0.767512i	0.923435	−	0.383756i	$-0.125370\pi$
$48$		0			0
$49$	6.91548			0.987926
$50$	−3.60197	−	6.80098i	−0.509395	−	0.961804i
$51$		0			0
$52$			2.57304i			0.356816i
$53$			5.75872i			0.791022i	0.918461	+	0.395511i	$0.129432\pi$
$53$			5.75872i			0.791022i	−0.918461	+	0.395511i	$0.870568\pi$
$54$		0			0
$55$	−2.17009	+	0.539189i	−0.292614	+	0.0727042i
$56$	0.729794			0.0975229
$57$		0			0
$58$			11.4186i			1.49933i
$59$	3.60197			0.468936			0.234468	−	0.972124i	$-0.424665\pi$
$59$	3.60197			0.468936			0.234468	+	0.972124i	$0.424665\pi$
$60$		0			0
$61$	−6.68035			−0.855331			−0.427665	−	0.903937i	$-0.640664\pi$
$61$	−6.68035			−0.855331			−0.427665	+	0.903937i	$0.640664\pi$
$62$		−	9.75872i		−	1.23936i
$63$		0			0
$64$	−6.02893			−0.753616
$65$	3.75872	+	15.1278i	0.466212	+	1.87638i
$66$		0			0
$67$			6.15676i			0.752167i	0.926586	+	0.376084i	$0.122729\pi$
$67$			6.15676i			0.752167i	−0.926586	+	0.376084i	$0.877271\pi$
$68$			1.76713i			0.214296i
$69$		0			0
$70$	−0.971071	+	0.241276i	−0.116065	+	0.0288380i
$71$	5.07838			0.602693			0.301346	−	0.953515i	$-0.402564\pi$
$71$	5.07838			0.602693			0.301346	+	0.953515i	$0.402564\pi$
$72$		0			0
$73$			1.12783i			0.132002i	0.997820	+	0.0660010i	$0.0210241\pi$
$73$			1.12783i			0.132002i	−0.997820	+	0.0660010i	$0.978976\pi$
$74$	−5.26180			−0.611671
$75$		0			0
$76$	2.86376			0.328496
$77$			0.290725i			0.0331311i
$78$		0			0
$79$	0.921622			0.103691			0.0518453	−	0.998655i	$-0.483490\pi$
$79$	0.921622			0.103691			0.0518453	+	0.998655i	$0.483490\pi$
$80$	9.98667	−	2.48133i	1.11654	−	0.277421i
$81$		0			0
$82$		−	11.4186i		−	1.26097i
$83$			1.70928i			0.187617i	0.995590	+	0.0938087i	$0.0299042\pi$
$83$			1.70928i			0.187617i	−0.995590	+	0.0938087i	$0.970096\pi$
$84$		0			0
$85$	2.58145	+	10.3896i	0.279997	+	1.12691i
$86$	−0.447480			−0.0482530
$87$		0			0
$88$		−	2.51026i		−	0.267595i
$89$	4.34017			0.460057			0.230029	−	0.973184i	$-0.426118\pi$
$89$	4.34017			0.460057			0.230029	+	0.973184i	$0.426118\pi$
$90$		0			0
$91$	2.02666			0.212452
$92$		−	1.47641i		−	0.153926i
$93$		0			0
$94$	−8.09890			−0.835337
$95$	16.8371	−	4.18342i	1.72745	−	0.429210i
$96$		0			0
$97$			4.68035i			0.475217i	0.971361	+	0.237609i	$0.0763635\pi$
$97$			4.68035i			0.475217i	−0.971361	+	0.237609i	$0.923636\pi$
$98$		−	10.6442i		−	1.07523i
$99$		0			0
$100$	−1.63090	+	0.863763i	−0.163090	+	0.0863763i
$101$	8.58145			0.853886			0.426943	−	0.904279i	$-0.359590\pi$
$101$	8.58145			0.853886			0.426943	+	0.904279i	$0.359590\pi$
$102$		0			0
$103$		−	6.73820i		−	0.663935i	−0.943291	−	0.331968i	$-0.892288\pi$
$103$		−	6.73820i		−	0.663935i	0.943291	−	0.331968i	$-0.107712\pi$
$104$	−17.4992			−1.71594
$105$		0			0
$106$	8.86376			0.860925
$107$		−	12.2329i		−	1.18260i	−0.806453	−	0.591298i	$-0.798616\pi$
$107$		−	12.2329i		−	1.18260i	0.806453	−	0.591298i	$-0.201384\pi$
$108$		0			0
$109$	6.31351			0.604725			0.302362	−	0.953193i	$-0.402225\pi$
$109$	6.31351			0.604725			0.302362	+	0.953193i	$0.402225\pi$
$110$	0.829914	+	3.34017i	0.0791291	+	0.318473i
$111$		0			0
$112$		−	1.33791i		−	0.126420i
$113$		−	16.4969i		−	1.55190i	−0.630794	−	0.775950i	$-0.717271\pi$
$113$		−	16.4969i		−	1.55190i	0.630794	−	0.775950i	$-0.282729\pi$
$114$		0			0
$115$	−2.15676	−	8.68035i	−0.201118	−	0.809446i
$116$	2.73820			0.254236
$117$		0			0
$118$		−	5.54411i		−	0.510377i
$119$	1.39189			0.127594
$120$		0			0
$121$	1.00000			0.0909091
$122$			10.2823i			0.930917i
$123$		0			0
$124$	−2.34017			−0.210154
$125$	−8.32684	+	7.46081i	−0.744775	+	0.667315i
$126$		0			0
$127$			4.87217i			0.432336i	0.976356	+	0.216168i	$0.0693558\pi$
$127$			4.87217i			0.432336i	−0.976356	+	0.216168i	$0.930644\pi$
$128$			13.4052i			1.18487i
$129$		0			0
$130$	23.2846	−	5.78539i	2.04219	−	0.507412i
$131$	8.68035			0.758405			0.379203	−	0.925314i	$-0.376198\pi$
$131$	8.68035			0.758405			0.379203	+	0.925314i	$0.376198\pi$
$132$		0			0
$133$		−	2.25565i		−	0.195590i
$134$	9.47641			0.818637
$135$		0			0
$136$	−12.0183			−1.03056
$137$			12.5958i			1.07613i	0.842902	+	0.538067i	$0.180845\pi$
$137$			12.5958i			1.07613i	−0.842902	+	0.538067i	$0.819155\pi$
$138$		0			0
$139$	9.91548			0.841020			0.420510	−	0.907288i	$-0.361851\pi$
$139$	9.91548			0.841020			0.420510	+	0.907288i	$0.361851\pi$
$140$	0.0578588	+	0.232866i	0.00488996	+	0.0196808i
$141$		0			0
$142$		−	7.81658i		−	0.655953i
$143$		−	6.97107i		−	0.582950i
$144$		0			0
$145$	16.0989	−	4.00000i	1.33694	−	0.332182i
$146$	1.73594			0.143667
$147$		0			0
$148$			1.26180i			0.103719i
$149$	−1.26180			−0.103370			−0.0516851	−	0.998663i	$-0.516459\pi$
$149$	−1.26180			−0.103370			−0.0516851	+	0.998663i	$0.516459\pi$
$150$		0			0
$151$	1.60197			0.130366			0.0651832	−	0.997873i	$-0.479237\pi$
$151$	1.60197			0.130366			0.0651832	+	0.997873i	$0.479237\pi$
$152$			19.4764i			1.57975i
$153$		0			0
$154$	0.447480			0.0360590
$155$	−13.7587	+	3.41855i	−1.10513	+	0.274585i
$156$		0			0
$157$		−	7.10504i		−	0.567044i	−0.958966	−	0.283522i	$-0.908497\pi$
$157$		−	7.10504i		−	0.567044i	0.958966	−	0.283522i	$-0.0915029\pi$
$158$		−	1.41855i		−	0.112854i
$159$		0			0
$160$	−1.11223	−	4.47641i	−0.0879293	−	0.353891i
$161$	−1.16290			−0.0916492
$162$		0			0
$163$			22.9360i			1.79649i	0.439499	+	0.898243i	$0.355156\pi$
$163$			22.9360i			1.79649i	−0.439499	+	0.898243i	$0.644844\pi$
$164$	−2.73820			−0.213818
$165$		0			0
$166$	2.63090			0.204197
$167$		−	4.81432i		−	0.372543i	−0.982498	−	0.186271i	$-0.940360\pi$
$167$		−	4.81432i		−	0.372543i	0.982498	−	0.186271i	$-0.0596404\pi$
$168$		0			0
$169$	−35.5958			−2.73814
$170$	15.9916	−	3.97334i	1.22650	−	0.304741i
$171$		0			0
$172$			0.107307i			0.00818209i
$173$		−	12.8865i		−	0.979746i	−0.871794	−	0.489873i	$-0.837043\pi$
$173$		−	12.8865i		−	0.979746i	0.871794	−	0.489873i	$-0.162957\pi$
$174$		0			0
$175$	0.680346	+	1.28458i	0.0514293	+	0.0971052i
$176$	−4.60197			−0.346886
$177$		0			0
$178$		−	6.68035i		−	0.500713i
$179$	1.84324			0.137771			0.0688853	−	0.997625i	$-0.478056\pi$
$179$	1.84324			0.137771			0.0688853	+	0.997625i	$0.478056\pi$
$180$		0			0
$181$	10.2823			0.764278			0.382139	−	0.924105i	$-0.375187\pi$
$181$	10.2823			0.764278			0.382139	+	0.924105i	$0.375187\pi$
$182$		−	3.11942i		−	0.231226i
$183$		0			0
$184$	10.0410			0.740235
$185$	1.84324	+	7.41855i	0.135518	+	0.545423i
$186$		0			0
$187$		−	4.78765i		−	0.350108i
$188$			1.94214i			0.141645i
$189$		0			0
$190$	−6.43907	−	25.9155i	−0.467139	−	1.88011i
$191$	11.5174			0.833373			0.416687	−	0.909050i	$-0.363191\pi$
$191$	11.5174			0.833373			0.416687	+	0.909050i	$0.363191\pi$
$192$		0			0
$193$		−	3.86603i		−	0.278283i	−0.990273	−	0.139141i	$-0.955566\pi$
$193$		−	3.86603i		−	0.278283i	0.990273	−	0.139141i	$-0.0444343\pi$
$194$	7.20394			0.517212
$195$		0			0
$196$	−2.55252			−0.182323
$197$			8.57304i			0.610804i	0.952224	+	0.305402i	$0.0987908\pi$
$197$			8.57304i			0.610804i	−0.952224	+	0.305402i	$0.901209\pi$
$198$		0			0
$199$	−8.31351			−0.589329			−0.294665	−	0.955601i	$-0.595208\pi$
$199$	−8.31351			−0.589329			−0.294665	+	0.955601i	$0.595208\pi$
$200$	−5.87444	−	11.0917i	−0.415386	−	0.784302i
$201$		0			0
$202$		−	13.2085i		−	0.929345i
$203$		−	2.15676i		−	0.151375i
$204$		0			0
$205$	−16.0989	+	4.00000i	−1.12440	+	0.279372i
$206$	−10.3714			−0.722608
$207$		0			0
$208$			32.0806i			2.22439i
$209$	−7.75872			−0.536682
$210$		0			0
$211$	25.9155			1.78410			0.892048	−	0.451941i	$-0.149268\pi$
$211$	25.9155			1.78410			0.892048	+	0.451941i	$0.149268\pi$
$212$		−	2.12556i		−	0.145984i
$213$		0			0
$214$	−18.8287			−1.28710
$215$	0.156755	+	0.630898i	0.0106906	+	0.0430269i
$216$		0			0
$217$			1.84324i			0.125128i
$218$		−	9.71769i		−	0.658165i
$219$		0			0
$220$	0.800984	−	0.199016i	0.0540023	−	0.0134176i
$221$	−33.3751			−2.24505
$222$		0			0
$223$			9.62863i			0.644781i	0.946607	+	0.322390i	$0.104486\pi$
$223$			9.62863i			0.644781i	−0.946607	+	0.322390i	$0.895514\pi$
$224$	−0.599701			−0.0400692
$225$		0			0
$226$	−25.3919			−1.68904
$227$			18.3896i			1.22056i	0.792185	+	0.610281i	$0.208943\pi$
$227$			18.3896i			1.22056i	−0.792185	+	0.610281i	$0.791057\pi$
$228$		0			0
$229$	−17.1506			−1.13334			−0.566672	−	0.823943i	$-0.691769\pi$
$229$	−17.1506			−1.13334			−0.566672	+	0.823943i	$0.691769\pi$
$230$	−13.3607	+	3.31965i	−0.880978	+	0.218892i
$231$		0			0
$232$			18.6225i			1.22263i
$233$		−	4.10731i		−	0.269079i	−0.990908	−	0.134539i	$-0.957045\pi$
$233$		−	4.10731i		−	0.269079i	0.990908	−	0.134539i	$-0.0429555\pi$
$234$		0			0
$235$	2.83710	+	11.4186i	0.185072	+	0.744864i
$236$	−1.32950			−0.0865428
$237$		0			0
$238$		−	2.14238i		−	0.138870i
$239$	−4.36683			−0.282467			−0.141234	−	0.989976i	$-0.545107\pi$
$239$	−4.36683			−0.282467			−0.141234	+	0.989976i	$0.545107\pi$
$240$		0			0
$241$	−5.20394			−0.335215			−0.167608	−	0.985854i	$-0.553604\pi$
$241$	−5.20394			−0.335215			−0.167608	+	0.985854i	$0.553604\pi$
$242$		−	1.53919i		−	0.0989428i
$243$		0			0
$244$	2.46573			0.157852
$245$	−15.0072	+	3.72875i	−0.958774	+	0.238221i
$246$		0			0
$247$			54.0866i			3.44145i
$248$		−	15.9155i		−	1.01063i
$249$		0			0
$250$	11.4836	+	12.8166i	0.726286	+	0.810592i
$251$	8.28231			0.522775			0.261388	−	0.965234i	$-0.415820\pi$
$251$	8.28231			0.522775			0.261388	+	0.965234i	$0.415820\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$	7.49920			0.470541
$255$		0			0
$256$	8.57531			0.535957
$257$		−	11.0205i		−	0.687441i	−0.939072	−	0.343721i	$-0.888313\pi$
$257$		−	11.0205i		−	0.687441i	0.939072	−	0.343721i	$-0.111687\pi$
$258$		0			0
$259$	0.993857			0.0617553
$260$	−1.38735	−	5.58372i	−0.0860400	−	0.346287i
$261$		0			0
$262$		−	13.3607i		−	0.825426i
$263$		−	2.97107i		−	0.183204i	−0.995796	−	0.0916020i	$-0.970801\pi$
$263$		−	2.97107i		−	0.183204i	0.995796	−	0.0916020i	$-0.0291988\pi$
$264$		0			0
$265$	−3.10504	−	12.4969i	−0.190741	−	0.767680i
$266$	−3.47187			−0.212874
$267$		0			0
$268$		−	2.27247i		−	0.138813i
$269$	−15.8432			−0.965980			−0.482990	−	0.875626i	$-0.660449\pi$
$269$	−15.8432			−0.965980			−0.482990	+	0.875626i	$0.660449\pi$
$270$		0			0
$271$	−6.28231			−0.381623			−0.190812	−	0.981627i	$-0.561112\pi$
$271$	−6.28231			−0.381623			−0.190812	+	0.981627i	$0.561112\pi$
$272$			22.0326i			1.33592i
$273$		0			0
$274$	19.3874			1.17123
$275$	4.41855	−	2.34017i	0.266449	−	0.141118i
$276$		0			0
$277$			5.12783i			0.308101i	0.988063	+	0.154051i	$0.0492319\pi$
$277$			5.12783i			0.308101i	−0.988063	+	0.154051i	$0.950768\pi$
$278$		−	15.2618i		−	0.915342i
$279$		0			0
$280$	−1.58372	+	0.393497i	−0.0946452	+	0.0235159i
$281$	−21.8888			−1.30578			−0.652889	−	0.757454i	$-0.726443\pi$
$281$	−21.8888			−1.30578			−0.652889	+	0.757454i	$0.726443\pi$
$282$		0			0
$283$		−	25.9649i		−	1.54345i	−0.635954	−	0.771727i	$-0.719393\pi$
$283$		−	25.9649i		−	1.54345i	0.635954	−	0.771727i	$-0.280607\pi$
$284$	−1.87444			−0.111228
$285$		0			0
$286$	−10.7298			−0.634466
$287$			2.15676i			0.127309i
$288$		0			0
$289$	−5.92162			−0.348331
$290$	−6.15676	−	24.7792i	−0.361537	−	1.45509i
$291$		0			0
$292$		−	0.416283i		−	0.0243611i
$293$			10.4163i			0.608526i	0.952588	+	0.304263i	$0.0984101\pi$
$293$			10.4163i			0.608526i	−0.952588	+	0.304263i	$0.901590\pi$
$294$		0			0
$295$	−7.81658	+	1.94214i	−0.455099	+	0.113076i
$296$	−8.58145			−0.498787
$297$		0			0
$298$			1.94214i			0.112505i
$299$	27.8843			1.61259
$300$		0			0
$301$	0.0845208			0.00487170
$302$		−	2.46573i		−	0.141887i
$303$		0			0
$304$	35.7054			2.04785
$305$	14.4969	−	3.60197i	0.830092	−	0.206248i
$306$		0			0
$307$		−	29.0700i		−	1.65911i	−0.558425	−	0.829555i	$-0.688594\pi$
$307$		−	29.0700i		−	1.65911i	0.558425	−	0.829555i	$-0.311406\pi$
$308$		−	0.107307i		−	0.00611439i
$309$		0			0
$310$	5.26180	+	21.1773i	0.298850	+	1.20279i
$311$	−5.44521			−0.308770			−0.154385	−	0.988011i	$-0.549340\pi$
$311$	−5.44521			−0.308770			−0.154385	+	0.988011i	$0.549340\pi$
$312$		0			0
$313$		−	25.0928i		−	1.41833i	−0.705044	−	0.709163i	$-0.749073\pi$
$313$		−	25.0928i		−	1.41833i	0.705044	−	0.709163i	$-0.250927\pi$
$314$	−10.9360			−0.617154
$315$		0			0
$316$	−0.340173			−0.0191362
$317$		−	21.1773i		−	1.18943i	−0.803935	−	0.594717i	$-0.797264\pi$
$317$		−	21.1773i		−	1.18943i	0.803935	−	0.594717i	$-0.202736\pi$
$318$		0			0
$319$	−7.41855			−0.415359
$320$	13.0833	−	3.25073i	0.731379	−	0.181721i
$321$		0			0
$322$			1.78992i			0.0997484i
$323$			37.1461i			2.06686i
$324$		0			0
$325$	−16.3135	−	30.8020i	−0.904911	−	1.70859i
$326$	35.3028			1.95524
$327$		0			0
$328$		−	18.6225i		−	1.02825i
$329$	1.52973			0.0843369
$330$		0			0
$331$	−6.70701			−0.368650			−0.184325	−	0.982865i	$-0.559010\pi$
$331$	−6.70701			−0.368650			−0.184325	+	0.982865i	$0.559010\pi$
$332$		−	0.630898i		−	0.0346250i
$333$		0			0
$334$	−7.41014			−0.405465
$335$	−3.31965	−	13.3607i	−0.181372	−	0.729973i
$336$		0			0
$337$			23.7503i			1.29376i	0.762591	+	0.646881i	$0.223927\pi$
$337$			23.7503i			1.29376i	−0.762591	+	0.646881i	$0.776073\pi$
$338$			54.7887i			2.98011i
$339$		0			0
$340$	−0.952819	−	3.83483i	−0.0516739	−	0.207973i
$341$	6.34017			0.343340
$342$		0			0
$343$			4.04557i			0.218440i
$344$	−0.729794			−0.0393479
$345$		0			0
$346$	−19.8348			−1.06633
$347$			31.1689i			1.67323i	0.547790	+	0.836616i	$0.315469\pi$
$347$			31.1689i			1.67323i	−0.547790	+	0.836616i	$0.684531\pi$
$348$		0			0
$349$	−11.0472			−0.591342			−0.295671	−	0.955290i	$-0.595543\pi$
$349$	−11.0472			−0.591342			−0.295671	+	0.955290i	$0.595543\pi$
$350$	1.97721	−	1.04718i	0.105687	−	0.0559742i
$351$		0			0
$352$			2.06278i			0.109947i
$353$		−	27.4329i		−	1.46011i	−0.683390	−	0.730054i	$-0.739495\pi$
$353$		−	27.4329i		−	1.46011i	0.683390	−	0.730054i	$-0.260505\pi$
$354$		0			0
$355$	−11.0205	+	2.73820i	−0.584908	+	0.145329i
$356$	−1.60197			−0.0849041
$357$		0			0
$358$		−	2.83710i		−	0.149945i
$359$	−19.1506			−1.01073			−0.505365	−	0.862905i	$-0.668642\pi$
$359$	−19.1506			−1.01073			−0.505365	+	0.862905i	$0.668642\pi$
$360$		0			0
$361$	41.1978			2.16830
$362$		−	15.8264i		−	0.831818i
$363$		0			0
$364$	−0.748046			−0.0392083
$365$	−0.608111	−	2.44748i	−0.0318300	−	0.128107i
$366$		0			0
$367$		−	14.5692i		−	0.760504i	−0.924883	−	0.380252i	$-0.875837\pi$
$367$		−	14.5692i		−	0.760504i	0.924883	−	0.380252i	$-0.124163\pi$
$368$		−	18.4079i		−	0.959577i
$369$		0			0
$370$	11.4186	−	2.83710i	0.593622	−	0.147494i
$371$	−1.67420			−0.0869203
$372$		0			0
$373$			8.81432i			0.456388i	0.973616	+	0.228194i	$0.0732820\pi$
$373$			8.81432i			0.456388i	−0.973616	+	0.228194i	$0.926718\pi$
$374$	−7.36910			−0.381047
$375$		0			0
$376$	−13.2085			−0.681175
$377$			51.7152i			2.66347i
$378$		0			0
$379$	23.5174			1.20801			0.604005	−	0.796980i	$-0.293571\pi$
$379$	23.5174			1.20801			0.604005	+	0.796980i	$0.293571\pi$
$380$	−6.21461	+	1.54411i	−0.318803	+	0.0792111i
$381$		0			0
$382$		−	17.7275i		−	0.907019i
$383$		−	19.3028i		−	0.986329i	−0.869936	−	0.493164i	$-0.835840\pi$
$383$		−	19.3028i		−	0.986329i	0.869936	−	0.493164i	$-0.164160\pi$
$384$		0			0
$385$	−0.156755	−	0.630898i	−0.00798900	−	0.0321535i
$386$	−5.95055			−0.302875
$387$		0			0
$388$		−	1.72753i		−	0.0877019i
$389$	−20.0410			−1.01612			−0.508060	−	0.861321i	$-0.669637\pi$
$389$	−20.0410			−1.01612			−0.508060	+	0.861321i	$0.669637\pi$
$390$		0			0
$391$	19.1506			0.968488
$392$		−	17.3596i		−	0.876795i
$393$		0			0
$394$	13.1955			0.664781
$395$	−2.00000	+	0.496928i	−0.100631	+	0.0250032i
$396$		0			0
$397$		−	33.3607i		−	1.67433i	−0.546954	−	0.837163i	$-0.684213\pi$
$397$		−	33.3607i		−	1.67433i	0.546954	−	0.837163i	$-0.315787\pi$
$398$			12.7961i			0.641409i
$399$		0			0
$400$	−20.3340	+	10.7694i	−1.01670	+	0.538470i
$401$	37.6475			1.88003			0.940014	−	0.341135i	$-0.110811\pi$
$401$	37.6475			1.88003			0.940014	+	0.341135i	$0.110811\pi$
$402$		0			0
$403$		−	44.1978i		−	2.20165i
$404$	−3.16743			−0.157586
$405$		0			0
$406$	−3.31965			−0.164752
$407$		−	3.41855i		−	0.169451i
$408$		0			0
$409$	25.2039			1.24625			0.623127	−	0.782120i	$-0.285862\pi$
$409$	25.2039			1.24625			0.623127	+	0.782120i	$0.285862\pi$
$410$	6.15676	+	24.7792i	0.304060	+	1.22376i
$411$		0			0
$412$			2.48709i			0.122530i
$413$			1.04718i			0.0515284i
$414$		0			0
$415$	−0.921622	−	3.70928i	−0.0452407	−	0.182081i
$416$	14.3798			0.705027
$417$		0			0
$418$			11.9421i			0.584109i
$419$	17.8432			0.871700			0.435850	−	0.900019i	$-0.356448\pi$
$419$	17.8432			0.871700			0.435850	+	0.900019i	$0.356448\pi$
$420$		0			0
$421$	−11.8120			−0.575684			−0.287842	−	0.957678i	$-0.592938\pi$
$421$	−11.8120			−0.575684			−0.287842	+	0.957678i	$0.592938\pi$
$422$		−	39.8888i		−	1.94176i
$423$		0			0
$424$	14.4559			0.702040
$425$	−11.2039	−	21.1545i	−0.543471	−	1.02614i
$426$		0			0
$427$		−	1.94214i		−	0.0939868i
$428$			4.51518i			0.218249i
$429$		0			0
$430$	0.971071	−	0.241276i	0.0468292	−	0.0116354i
$431$	−16.6803			−0.803464			−0.401732	−	0.915757i	$-0.631592\pi$
$431$	−16.6803			−0.803464			−0.401732	+	0.915757i	$0.631592\pi$
$432$		0			0
$433$			28.3135i			1.36066i	0.732906	+	0.680330i	$0.238164\pi$
$433$			28.3135i			1.36066i	−0.732906	+	0.680330i	$0.761836\pi$
$434$	2.83710			0.136185
$435$		0			0
$436$	−2.33033			−0.111603
$437$		−	31.0349i		−	1.48460i
$438$		0			0
$439$	27.4452			1.30989			0.654944	−	0.755677i	$-0.272692\pi$
$439$	27.4452			1.30989			0.654944	+	0.755677i	$0.272692\pi$
$440$	1.35350	+	5.44748i	0.0645258	+	0.259698i
$441$		0			0
$442$			51.3705i			2.44345i
$443$			0.412408i			0.0195941i	0.999952	+	0.00979704i	$0.00311854\pi$
$443$			0.412408i			0.0195941i	−0.999952	+	0.00979704i	$0.996881\pi$
$444$		0			0
$445$	−9.41855	+	2.34017i	−0.446482	+	0.110935i
$446$	14.8203			0.701761
$447$		0			0
$448$		−	1.75276i		−	0.0828100i
$449$	1.33403			0.0629568			0.0314784	−	0.999504i	$-0.489978\pi$
$449$	1.33403			0.0629568			0.0314784	+	0.999504i	$0.489978\pi$
$450$		0			0
$451$	7.41855			0.349326
$452$			6.08906i			0.286405i
$453$		0			0
$454$	28.3051			1.32842
$455$	−4.39803	+	1.09275i	−0.206183	+	0.0512291i
$456$		0			0
$457$			11.6514i			0.545030i	0.962151	+	0.272515i	$0.0878555\pi$
$457$			11.6514i			0.545030i	−0.962151	+	0.272515i	$0.912145\pi$
$458$			26.3980i			1.23350i
$459$		0			0
$460$	0.796064	+	3.20394i	0.0371167	+	0.149384i
$461$	2.05786			0.0958440			0.0479220	−	0.998851i	$-0.484740\pi$
$461$	2.05786			0.0958440			0.0479220	+	0.998851i	$0.484740\pi$
$462$		0			0
$463$		−	28.7792i		−	1.33748i	−0.743494	−	0.668742i	$-0.766833\pi$
$463$		−	28.7792i		−	1.33748i	0.743494	−	0.668742i	$-0.233167\pi$
$464$	34.1399			1.58491
$465$		0			0
$466$	−6.32192			−0.292857
$467$		−	1.84324i		−	0.0852952i	−0.999090	−	0.0426476i	$-0.986421\pi$
$467$		−	1.84324i		−	0.0852952i	0.999090	−	0.0426476i	$-0.0135793\pi$
$468$		0			0
$469$	−1.78992			−0.0826509
$470$	17.5753	−	4.36683i	0.810688	−	0.201427i
$471$		0			0
$472$		−	9.04187i		−	0.416186i
$473$		−	0.290725i		−	0.0133675i
$474$		0			0
$475$	−34.2823	+	18.1568i	−1.57298	+	0.833089i
$476$	−0.513749			−0.0235477
$477$		0			0
$478$			6.72138i			0.307429i
$479$	26.8371			1.22622			0.613109	−	0.789998i	$-0.289919\pi$
$479$	26.8371			1.22622			0.613109	+	0.789998i	$0.289919\pi$
$480$		0			0
$481$	−23.8310			−1.08660
$482$			8.00984i			0.364838i
$483$		0			0
$484$	−0.369102			−0.0167774
$485$	−2.52359	−	10.1568i	−0.114590	−	0.461195i
$486$		0			0
$487$			28.5646i			1.29439i	0.762326	+	0.647193i	$0.224057\pi$
$487$			28.5646i			1.29439i	−0.762326	+	0.647193i	$0.775943\pi$
$488$			16.7694i			0.759115i
$489$		0			0
$490$	5.73925	+	23.0989i	0.259273	+	1.04350i
$491$	−25.9877			−1.17281			−0.586405	−	0.810018i	$-0.699457\pi$
$491$	−25.9877			−1.17281			−0.586405	+	0.810018i	$0.699457\pi$
$492$		0			0
$493$			35.5174i			1.59963i
$494$	83.2495			3.74557
$495$		0			0
$496$	−29.1773			−1.31010
$497$			1.47641i			0.0662260i
$498$		0			0
$499$	27.5174			1.23185			0.615925	−	0.787805i	$-0.288782\pi$
$499$	27.5174			1.23185			0.615925	+	0.787805i	$0.288782\pi$
$500$	3.07346	−	2.75380i	0.137449	−	0.123154i
$501$		0			0
$502$		−	12.7480i		−	0.568973i
$503$			22.6576i			1.01025i	0.863046	+	0.505125i	$0.168554\pi$
$503$			22.6576i			1.01025i	−0.863046	+	0.505125i	$0.831446\pi$
$504$		0			0
$505$	−18.6225	+	4.62702i	−0.828690	+	0.205900i
$506$	6.15676			0.273701
$507$		0			0
$508$		−	1.79833i		−	0.0797880i
$509$	−27.8432			−1.23413			−0.617065	−	0.786912i	$-0.711678\pi$
$509$	−27.8432			−1.23413			−0.617065	+	0.786912i	$0.711678\pi$
$510$		0			0
$511$	−0.327887			−0.0145049
$512$			13.6114i			0.601546i
$513$		0			0
$514$	−16.9627			−0.748191
$515$	3.63317	+	14.6225i	0.160096	+	0.644344i
$516$		0			0
$517$		−	5.26180i		−	0.231413i
$518$		−	1.52973i		−	0.0672126i
$519$		0			0
$520$	37.9748	−	9.43537i	1.66530	−	0.413768i
$521$	−29.7009			−1.30122			−0.650609	−	0.759413i	$-0.725486\pi$
$521$	−29.7009			−1.30122			−0.650609	+	0.759413i	$0.725486\pi$
$522$		0			0
$523$		−	24.7565i		−	1.08252i	−0.840854	−	0.541262i	$-0.817947\pi$
$523$		−	24.7565i		−	1.08252i	0.840854	−	0.541262i	$-0.182053\pi$
$524$	−3.20394			−0.139965
$525$		0			0
$526$	−4.57304			−0.199394
$527$		−	30.3545i		−	1.32226i
$528$		0			0
$529$	7.00000			0.304348
$530$	−19.2351	+	4.77924i	−0.835521	+	0.207597i
$531$		0			0
$532$			0.832567i			0.0360963i
$533$		−	51.7152i		−	2.24004i
$534$		0			0
$535$	6.59583	+	26.5464i	0.285162	+	1.14770i
$536$	15.4551			0.667557
$537$		0			0
$538$			24.3857i			1.05134i
$539$	6.91548			0.297871
$540$		0			0
$541$	−12.5236			−0.538431			−0.269216	−	0.963080i	$-0.586764\pi$
$541$	−12.5236			−0.538431			−0.269216	+	0.963080i	$0.586764\pi$
$542$			9.66967i			0.415348i
$543$		0			0
$544$	9.87587			0.423425
$545$	−13.7009	+	3.40417i	−0.586881	+	0.145819i
$546$		0			0
$547$		−	14.2784i		−	0.610502i	−0.952272	−	0.305251i	$-0.901260\pi$
$547$		−	14.2784i		−	0.610502i	0.952272	−	0.305251i	$-0.0987404\pi$
$548$		−	4.64915i		−	0.198602i
$549$		0			0
$550$	−3.60197	−	6.80098i	−0.153588	−	0.289995i
$551$	57.5585			2.45207
$552$		0			0
$553$			0.267938i			0.0113939i
$554$	7.89269			0.335328
$555$		0			0
$556$	−3.65983			−0.155211
$557$			3.57918i			0.151655i	0.997121	+	0.0758274i	$0.0241598\pi$
$557$			3.57918i			0.151655i	−0.997121	+	0.0758274i	$0.975840\pi$
$558$		0			0
$559$	−2.02666			−0.0857187
$560$	0.721384	+	2.90337i	0.0304840	+	0.122690i
$561$		0			0
$562$			33.6910i			1.42117i
$563$			28.9588i			1.22047i	0.792222	+	0.610234i	$0.208924\pi$
$563$			28.9588i			1.22047i	−0.792222	+	0.610234i	$0.791076\pi$
$564$		0			0
$565$	8.89496	+	35.7998i	0.374214	+	1.50611i
$566$	−39.9649			−1.67985
$567$		0			0
$568$		−	12.7480i		−	0.534896i
$569$	−25.8264			−1.08270			−0.541350	−	0.840797i	$-0.682087\pi$
$569$	−25.8264			−1.08270			−0.541350	+	0.840797i	$0.682087\pi$
$570$		0			0
$571$	17.1194			0.716425			0.358213	−	0.933640i	$-0.383386\pi$
$571$	17.1194			0.716425			0.358213	+	0.933640i	$0.383386\pi$
$572$			2.57304i			0.107584i
$573$		0			0
$574$	3.31965			0.138560
$575$	9.36069	+	17.6742i	0.390368	+	0.737065i
$576$		0			0
$577$			22.5692i			0.939567i	0.882782	+	0.469783i	$0.155668\pi$
$577$			22.5692i			0.939567i	−0.882782	+	0.469783i	$0.844332\pi$
$578$			9.11450i			0.379113i
$579$		0			0
$580$	−5.94214	+	1.47641i	−0.246734	+	0.0613046i
$581$	−0.496928			−0.0206161
$582$		0			0
$583$			5.75872i			0.238502i
$584$	2.83114			0.117153
$585$		0			0
$586$	16.0326			0.662302
$587$		−	3.63317i		−	0.149957i	−0.997185	−	0.0749784i	$-0.976111\pi$
$587$		−	3.63317i		−	0.149957i	0.997185	−	0.0749784i	$-0.0238888\pi$
$588$		0			0
$589$	−49.1917			−2.02691
$590$	2.98932	+	12.0312i	0.123068	+	0.495317i
$591$		0			0
$592$			15.7321i			0.646584i
$593$			12.3051i			0.505310i	0.967556	+	0.252655i	$0.0813037\pi$
$593$			12.3051i			0.505310i	−0.967556	+	0.252655i	$0.918696\pi$
$594$		0			0
$595$	−3.02052	+	0.750491i	−0.123829	+	0.0307671i
$596$	0.465732			0.0190771
$597$		0			0
$598$		−	42.9192i		−	1.75510i
$599$	19.6865			0.804368			0.402184	−	0.915559i	$-0.368251\pi$
$599$	19.6865			0.804368			0.402184	+	0.915559i	$0.368251\pi$
$600$		0			0
$601$	25.8843			1.05584			0.527921	−	0.849294i	$-0.322972\pi$
$601$	25.8843			1.05584			0.527921	+	0.849294i	$0.322972\pi$
$602$		−	0.130094i		−	0.00530222i
$603$		0			0
$604$	−0.591290			−0.0240593
$605$	−2.17009	+	0.539189i	−0.0882266	+	0.0219211i
$606$		0			0
$607$		−	41.5357i		−	1.68588i	−0.538006	−	0.842941i	$-0.680822\pi$
$607$		−	41.5357i		−	1.68588i	0.538006	−	0.842941i	$-0.319178\pi$
$608$		−	16.0045i		−	0.649070i
$609$		0			0
$610$	−5.54411	−	22.3135i	−0.224474	−	0.903448i
$611$	−36.6803			−1.48393
$612$		0			0
$613$			20.6453i			0.833855i	0.908940	+	0.416927i	$0.136893\pi$
$613$			20.6453i			0.833855i	−0.908940	+	0.416927i	$0.863107\pi$
$614$	−44.7442			−1.80573
$615$		0			0
$616$	0.729794			0.0294042
$617$		−	8.69472i		−	0.350036i	−0.984565	−	0.175018i	$-0.944002\pi$
$617$		−	8.69472i		−	0.350036i	0.984565	−	0.175018i	$-0.0559984\pi$
$618$		0			0
$619$	−2.65368			−0.106661			−0.0533303	−	0.998577i	$-0.516984\pi$
$619$	−2.65368			−0.106661			−0.0533303	+	0.998577i	$0.516984\pi$
$620$	5.07838	−	1.26180i	0.203953	−	0.0506749i
$621$		0			0
$622$			8.38121i			0.336056i
$623$			1.26180i			0.0505528i
$624$		0			0
$625$	14.0472	−	20.6803i	0.561887	−	0.827214i
$626$	−38.6225			−1.54367
$627$		0			0
$628$			2.62249i			0.104649i
$629$	−16.3668			−0.652588
$630$		0			0
$631$	−10.6393			−0.423544			−0.211772	−	0.977319i	$-0.567923\pi$
$631$	−10.6393			−0.423544			−0.211772	+	0.977319i	$0.567923\pi$
$632$		−	2.31351i		−	0.0920265i
$633$		0			0
$634$	−32.5958			−1.29455
$635$	−2.62702	−	10.5730i	−0.104250	−	0.419578i
$636$		0			0
$637$		−	48.2083i		−	1.91008i
$638$			11.4186i			0.452065i
$639$		0			0
$640$	−7.22795	−	29.0905i	−0.285710	−	1.14990i
$641$	35.8576			1.41629			0.708145	−	0.706067i	$-0.249532\pi$
$641$	35.8576			1.41629			0.708145	+	0.706067i	$0.249532\pi$
$642$		0			0
$643$		−	12.2146i		−	0.481697i	−0.970563	−	0.240849i	$-0.922574\pi$
$643$		−	12.2146i		−	0.481697i	0.970563	−	0.240849i	$-0.0774258\pi$
$644$	0.429229			0.0169140
$645$		0			0
$646$	57.1748			2.24951
$647$		−	25.9421i		−	1.01989i	−0.860207	−	0.509945i	$-0.829666\pi$
$647$		−	25.9421i		−	1.01989i	0.860207	−	0.509945i	$-0.170334\pi$
$648$		0			0
$649$	3.60197			0.141390
$650$	−47.4101	+	25.1096i	−1.85958	+	0.984879i
$651$		0			0
$652$		−	8.46573i		−	0.331544i
$653$		−	14.3402i		−	0.561174i	−0.959829	−	0.280587i	$-0.909471\pi$
$653$		−	14.3402i		−	0.561174i	0.959829	−	0.280587i	$-0.0905292\pi$
$654$		0			0
$655$	−18.8371	+	4.68035i	−0.736026	+	0.182876i
$656$	−34.1399			−1.33294
$657$		0			0
$658$		−	2.35455i		−	0.0917899i
$659$	6.52359			0.254123			0.127062	−	0.991895i	$-0.459445\pi$
$659$	6.52359			0.254123			0.127062	+	0.991895i	$0.459445\pi$
$660$		0			0
$661$	3.16290			0.123022			0.0615112	−	0.998106i	$-0.480408\pi$
$661$	3.16290			0.123022			0.0615112	+	0.998106i	$0.480408\pi$
$662$			10.3234i			0.401228i
$663$		0			0
$664$	4.29072			0.166512
$665$	1.21622	+	4.89496i	0.0471631	+	0.189818i
$666$		0			0
$667$		−	29.6742i		−	1.14899i
$668$			1.77698i			0.0687532i
$669$		0			0
$670$	−20.5646	+	5.10957i	−0.794481	+	0.197400i
$671$	−6.68035			−0.257892
$672$		0			0
$673$			18.4885i			0.712680i	0.934356	+	0.356340i	$0.115976\pi$
$673$			18.4885i			0.712680i	−0.934356	+	0.356340i	$0.884024\pi$
$674$	36.5562			1.40809
$675$		0			0
$676$	13.1385			0.505327
$677$		−	15.0966i		−	0.580211i	−0.956995	−	0.290105i	$-0.906310\pi$
$677$		−	15.0966i		−	0.580211i	0.956995	−	0.290105i	$-0.0936903\pi$
$678$		0			0
$679$	−1.36069			−0.0522186
$680$	26.0806	−	6.48011i	1.00015	−	0.248501i
$681$		0			0
$682$		−	9.75872i		−	0.373681i
$683$		−	12.4657i		−	0.476988i	−0.971144	−	0.238494i	$-0.923346\pi$
$683$		−	12.4657i		−	0.476988i	0.971144	−	0.238494i	$-0.0766537\pi$
$684$		0			0
$685$	−6.79153	−	27.3340i	−0.259491	−	1.04438i
$686$	6.22690			0.237744
$687$		0			0
$688$			1.33791i			0.0510072i
$689$	40.1445			1.52938
$690$		0			0
$691$	−30.7214			−1.16870			−0.584348	−	0.811503i	$-0.698650\pi$
$691$	−30.7214			−1.16870			−0.584348	+	0.811503i	$0.698650\pi$
$692$			4.75646i			0.180813i
$693$		0			0
$694$	47.9748			1.82110
$695$	−21.5174	+	5.34632i	−0.816203	+	0.202797i
$696$		0			0
$697$		−	35.5174i		−	1.34532i
$698$			17.0037i			0.643599i
$699$		0			0
$700$	−0.251117	−	0.474142i	−0.00949134	−	0.0179209i
$701$	−17.9955			−0.679679			−0.339840	−	0.940483i	$-0.610373\pi$
$701$	−17.9955			−0.679679			−0.339840	+	0.940483i	$0.610373\pi$
$702$		0			0
$703$			26.5236i			1.00036i
$704$	−6.02893			−0.227224
$705$		0			0
$706$	−42.2245			−1.58914
$707$			2.49484i			0.0938281i
$708$		0			0
$709$	−23.5897			−0.885929			−0.442965	−	0.896539i	$-0.646073\pi$
$709$	−23.5897			−0.885929			−0.442965	+	0.896539i	$0.646073\pi$
$710$	4.21461	+	16.9627i	0.158172	+	0.636597i
$711$		0			0
$712$		−	10.8950i		−	0.408306i
$713$			25.3607i			0.949765i
$714$		0			0
$715$	3.75872	+	15.1278i	0.140568	+	0.565749i
$716$	−0.680346			−0.0254257
$717$		0			0
$718$			29.4764i			1.10005i
$719$	24.5646			0.916106			0.458053	−	0.888925i	$-0.348547\pi$
$719$	24.5646			0.916106			0.458053	+	0.888925i	$0.348547\pi$
$720$		0			0
$721$	1.95896			0.0729556
$722$		−	63.4112i		−	2.35992i
$723$		0			0
$724$	−3.79523			−0.141048
$725$	−32.7792	+	17.3607i	−1.21739	+	0.644760i
$726$		0			0
$727$			8.51130i			0.315667i	0.987466	+	0.157833i	$0.0504509\pi$
$727$			8.51130i			0.315667i	−0.987466	+	0.157833i	$0.949549\pi$
$728$		−	5.08745i		−	0.188553i
$729$		0			0
$730$	−3.76713	+	0.935998i	−0.139428	+	0.0346428i
$731$	−1.39189			−0.0514809
$732$		0			0
$733$			29.8615i			1.10296i	0.834188	+	0.551480i	$0.185937\pi$
$733$			29.8615i			1.10296i	−0.834188	+	0.551480i	$0.814063\pi$
$734$	−22.4247			−0.827711
$735$		0			0
$736$	−8.25112			−0.304140
$737$			6.15676i			0.226787i
$738$		0			0
$739$	36.7526			1.35197			0.675983	−	0.736917i	$-0.263719\pi$
$739$	36.7526			1.35197			0.675983	+	0.736917i	$0.263719\pi$
$740$	−0.680346	−	2.73820i	−0.0250100	−	0.100658i
$741$		0			0
$742$			2.57691i			0.0946015i
$743$		−	2.17501i		−	0.0797933i	−0.999204	−	0.0398966i	$-0.987297\pi$
$743$		−	2.17501i		−	0.0797933i	0.999204	−	0.0398966i	$-0.0127029\pi$
$744$		0			0
$745$	2.73820	−	0.680346i	0.100320	−	0.0249259i
$746$	13.5669			0.496719
$747$		0			0
$748$			1.76713i			0.0646128i
$749$	3.55640			0.129948
$750$		0			0
$751$	−44.4580			−1.62229			−0.811147	−	0.584842i	$-0.801157\pi$
$751$	−44.4580			−1.62229			−0.811147	+	0.584842i	$0.801157\pi$
$752$			24.2146i			0.883016i
$753$		0			0
$754$	79.5995			2.89884
$755$	−3.47641	+	0.863763i	−0.126519	+	0.0314356i
$756$		0			0
$757$		−	26.4247i		−	0.960422i	−0.877153	−	0.480211i	$-0.840560\pi$
$757$		−	26.4247i		−	0.960422i	0.877153	−	0.480211i	$-0.159440\pi$
$758$		−	36.1978i		−	1.31476i
$759$		0			0
$760$	−10.5015	−	42.2655i	−0.380928	−	1.53313i
$761$	27.6163			1.00109			0.500546	−	0.865710i	$-0.333133\pi$
$761$	27.6163			1.00109			0.500546	+	0.865710i	$0.333133\pi$
$762$		0			0
$763$			1.83549i			0.0664493i
$764$	−4.25112			−0.153800
$765$		0			0
$766$	−29.7107			−1.07349
$767$		−	25.1096i		−	0.906654i
$768$		0			0
$769$	−27.8432			−1.00405			−0.502027	−	0.864852i	$-0.667412\pi$
$769$	−27.8432			−1.00405			−0.502027	+	0.864852i	$0.667412\pi$
$770$	−0.971071	+	0.241276i	−0.0349950	+	0.00869499i
$771$		0			0
$772$			1.42696i			0.0513575i
$773$		−	18.0267i		−	0.648374i	−0.945993	−	0.324187i	$-0.894909\pi$
$773$		−	18.0267i		−	0.648374i	0.945993	−	0.324187i	$-0.105091\pi$
$774$		0			0
$775$	28.0144	−	14.8371i	1.00631	−	0.532964i
$776$	11.7489			0.421760
$777$		0			0
$778$			30.8469i			1.10592i
$779$	−57.5585			−2.06225
$780$		0			0
$781$	5.07838			0.181719
$782$		−	29.4764i		−	1.05407i
$783$		0			0
$784$	−31.8248			−1.13660
$785$	3.83096	+	15.4186i	0.136733	+	0.550312i
$786$		0			0
$787$			35.6391i			1.27040i	0.772349	+	0.635199i	$0.219082\pi$
$787$			35.6391i			1.27040i	−0.772349	+	0.635199i	$0.780918\pi$
$788$		−	3.16433i		−	0.112725i
$789$		0			0
$790$

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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.53919i q^{2} -0.369102 q^{4} +(-2.17009 + 0.539189i) q^{5} +0.290725i q^{7} -2.51026i q^{8} +O(q^{10})\)	\(q-1.53919i q^{2} -0.369102 q^{4} +(-2.17009 + 0.539189i) q^{5} +0.290725i q^{7} -2.51026i q^{8} +(0.829914 + 3.34017i) q^{10} +1.00000 q^{11} -6.97107i q^{13} +0.447480 q^{14} -4.60197 q^{16} -4.78765i q^{17} -7.75872 q^{19} +(0.800984 - 0.199016i) q^{20} -1.53919i q^{22} +4.00000i q^{23} +(4.41855 - 2.34017i) q^{25} -10.7298 q^{26} -0.107307i q^{28} -7.41855 q^{29} +6.34017 q^{31} +2.06278i q^{32} -7.36910 q^{34} +(-0.156755 - 0.630898i) q^{35} -3.41855i q^{37} +11.9421i q^{38} +(1.35350 + 5.44748i) q^{40} +7.41855 q^{41} -0.290725i q^{43} -0.369102 q^{44} +6.15676 q^{46} -5.26180i q^{47} +6.91548 q^{49} +(-3.60197 - 6.80098i) q^{50} +2.57304i q^{52} +5.75872i q^{53} +(-2.17009 + 0.539189i) q^{55} +0.729794 q^{56} +11.4186i q^{58} +3.60197 q^{59} -6.68035 q^{61} -9.75872i q^{62} -6.02893 q^{64} +(3.75872 + 15.1278i) q^{65} +6.15676i q^{67} +1.76713i q^{68} +(-0.971071 + 0.241276i) q^{70} +5.07838 q^{71} +1.12783i q^{73} -5.26180 q^{74} +2.86376 q^{76} +0.290725i q^{77} +0.921622 q^{79} +(9.98667 - 2.48133i) q^{80} -11.4186i q^{82} +1.70928i q^{83} +(2.58145 + 10.3896i) q^{85} -0.447480 q^{86} -2.51026i q^{88} +4.34017 q^{89} +2.02666 q^{91} -1.47641i q^{92} -8.09890 q^{94} +(16.8371 - 4.18342i) q^{95} +4.68035i q^{97} -10.6442i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} - 2 q^{5}+O(q^{10}) \) 6 * q - 10 * q^4 - 2 * q^5	\( 6 q - 10 q^{4} - 2 q^{5} + 16 q^{10} + 6 q^{11} + 4 q^{14} + 10 q^{16} + 4 q^{19} - 14 q^{20} - 2 q^{25} + 16 q^{26} - 16 q^{29} + 16 q^{31} - 52 q^{34} + 12 q^{35} - 12 q^{40} + 16 q^{41} - 10 q^{44} + 24 q^{46} - 22 q^{49} + 16 q^{50} - 2 q^{55} - 76 q^{56} - 16 q^{59} + 4 q^{61} - 66 q^{64} - 28 q^{65} + 24 q^{70} + 24 q^{71} - 16 q^{74} - 36 q^{76} + 12 q^{79} + 58 q^{80} + 44 q^{85} - 4 q^{86} + 4 q^{89} + 16 q^{91} + 24 q^{94} + 44 q^{95}+O(q^{100}) \) 6 * q - 10 * q^4 - 2 * q^5 + 16 * q^10 + 6 * q^11 + 4 * q^14 + 10 * q^16 + 4 * q^19 - 14 * q^20 - 2 * q^25 + 16 * q^26 - 16 * q^29 + 16 * q^31 - 52 * q^34 + 12 * q^35 - 12 * q^40 + 16 * q^41 - 10 * q^44 + 24 * q^46 - 22 * q^49 + 16 * q^50 - 2 * q^55 - 76 * q^56 - 16 * q^59 + 4 * q^61 - 66 * q^64 - 28 * q^65 + 24 * q^70 + 24 * q^71 - 16 * q^74 - 36 * q^76 + 12 * q^79 + 58 * q^80 + 44 * q^85 - 4 * q^86 + 4 * q^89 + 16 * q^91 + 24 * q^94 + 44 * q^95

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