LMFDB - Embedded newform 4012.2.b.b.237.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.18
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13278i q^{3} -2.09684i q^{5} +2.71039i q^{7} +1.71682 q^{9} +O(q^{10})$	$q-1.13278i q^{3} -2.09684i q^{5} +2.71039i q^{7} +1.71682 q^{9} -3.86380i q^{11} -2.19264 q^{13} -2.37525 q^{15} +(1.04117 - 3.98948i) q^{17} +7.57941 q^{19} +3.07027 q^{21} +3.94101i q^{23} +0.603254 q^{25} -5.34310i q^{27} -2.64568i q^{29} -4.90386i q^{31} -4.37683 q^{33} +5.68326 q^{35} +11.3346i q^{37} +2.48378i q^{39} +4.94646i q^{41} -5.42674 q^{43} -3.59989i q^{45} +6.03939 q^{47} -0.346228 q^{49} +(-4.51919 - 1.17941i) q^{51} -1.40329 q^{53} -8.10178 q^{55} -8.58578i q^{57} +1.00000 q^{59} -13.3980i q^{61} +4.65324i q^{63} +4.59763i q^{65} +1.98465 q^{67} +4.46428 q^{69} -4.42275i q^{71} +1.70785i q^{73} -0.683352i q^{75} +10.4724 q^{77} -4.77934i q^{79} -0.902097 q^{81} +9.20975 q^{83} +(-8.36531 - 2.18317i) q^{85} -2.99696 q^{87} -4.58193 q^{89} -5.94293i q^{91} -5.55498 q^{93} -15.8928i q^{95} -14.2966i q^{97} -6.63343i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.13278i		−	0.654009i	−0.945023	−	0.327005i	$-0.893961\pi$
$3$		−	1.13278i		−	0.654009i	0.945023	−	0.327005i	$-0.106039\pi$
$4$		0			0
$5$		−	2.09684i		−	0.937736i	−0.883268	−	0.468868i	$-0.844662\pi$
$5$		−	2.09684i		−	0.937736i	0.883268	−	0.468868i	$-0.155338\pi$
$6$		0			0
$7$			2.71039i			1.02443i	0.858857	+	0.512216i	$0.171175\pi$
$7$			2.71039i			1.02443i	−0.858857	+	0.512216i	$0.828825\pi$
$8$		0			0
$9$	1.71682			0.572272
$10$		0			0
$11$		−	3.86380i		−	1.16498i	−0.812838	−	0.582490i	$-0.802079\pi$
$11$		−	3.86380i		−	1.16498i	0.812838	−	0.582490i	$-0.197921\pi$
$12$		0			0
$13$	−2.19264			−0.608130			−0.304065	−	0.952651i	$-0.598344\pi$
$13$	−2.19264			−0.608130			−0.304065	+	0.952651i	$0.598344\pi$
$14$		0			0
$15$	−2.37525			−0.613288
$16$		0			0
$17$	1.04117	−	3.98948i	0.252521	−	0.967591i
$17$	1.04117	−	3.98948i	0.252521	−	0.967591i
$18$		0			0
$19$	7.57941			1.73884			0.869418	−	0.494078i	$-0.164494\pi$
$19$	7.57941			1.73884			0.869418	+	0.494078i	$0.164494\pi$
$20$		0			0
$21$	3.07027			0.669988
$22$		0			0
$23$			3.94101i			0.821757i	0.911690	+	0.410878i	$0.134778\pi$
$23$			3.94101i			0.821757i	−0.911690	+	0.410878i	$0.865222\pi$
$24$		0			0
$25$	0.603254			0.120651
$26$		0			0
$27$		−	5.34310i		−	1.02828i
$28$		0			0
$29$		−	2.64568i		−	0.491290i	−0.969360	−	0.245645i	$-0.921000\pi$
$29$		−	2.64568i		−	0.491290i	0.969360	−	0.245645i	$-0.0789998\pi$
$30$		0			0
$31$		−	4.90386i		−	0.880759i	−0.897812	−	0.440380i	$-0.854844\pi$
$31$		−	4.90386i		−	0.880759i	0.897812	−	0.440380i	$-0.145156\pi$
$32$		0			0
$33$	−4.37683			−0.761907
$34$		0			0
$35$	5.68326			0.960647
$36$		0			0
$37$			11.3346i			1.86339i	0.363242	+	0.931695i	$0.381670\pi$
$37$			11.3346i			1.86339i	−0.363242	+	0.931695i	$0.618330\pi$
$38$		0			0
$39$			2.48378i			0.397723i
$40$		0			0
$41$			4.94646i			0.772508i	0.922393	+	0.386254i	$0.126231\pi$
$41$			4.94646i			0.772508i	−0.922393	+	0.386254i	$0.873769\pi$
$42$		0			0
$43$	−5.42674			−0.827571			−0.413785	−	0.910374i	$-0.635794\pi$
$43$	−5.42674			−0.827571			−0.413785	+	0.910374i	$0.635794\pi$
$44$		0			0
$45$		−	3.59989i		−	0.536640i
$46$		0			0
$47$	6.03939			0.880936			0.440468	−	0.897768i	$-0.354812\pi$
$47$	6.03939			0.880936			0.440468	+	0.897768i	$0.354812\pi$
$48$		0			0
$49$	−0.346228			−0.0494611
$50$		0			0
$51$	−4.51919	−	1.17941i	−0.632814	−	0.165151i
$52$		0			0
$53$	−1.40329			−0.192756			−0.0963782	−	0.995345i	$-0.530726\pi$
$53$	−1.40329			−0.192756			−0.0963782	+	0.995345i	$0.530726\pi$
$54$		0			0
$55$	−8.10178			−1.09244
$56$		0			0
$57$		−	8.58578i		−	1.13721i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$		−	13.3980i		−	1.71544i	−0.514120	−	0.857718i	$-0.671881\pi$
$61$		−	13.3980i		−	1.71544i	0.514120	−	0.857718i	$-0.328119\pi$
$62$		0			0
$63$			4.65324i			0.586254i
$64$		0			0
$65$			4.59763i			0.570266i
$66$		0			0
$67$	1.98465			0.242464			0.121232	−	0.992624i	$-0.461316\pi$
$67$	1.98465			0.242464			0.121232	+	0.992624i	$0.461316\pi$
$68$		0			0
$69$	4.46428			0.537436
$70$		0			0
$71$		−	4.42275i		−	0.524883i	−0.964948	−	0.262442i	$-0.915472\pi$
$71$		−	4.42275i		−	0.524883i	0.964948	−	0.262442i	$-0.0845277\pi$
$72$		0			0
$73$			1.70785i			0.199889i	0.994993	+	0.0999445i	$0.0318665\pi$
$73$			1.70785i			0.199889i	−0.994993	+	0.0999445i	$0.968133\pi$
$74$		0			0
$75$		−	0.683352i		−	0.0789067i
$76$		0			0
$77$	10.4724			1.19344
$78$		0			0
$79$		−	4.77934i		−	0.537717i	−0.963180	−	0.268859i	$-0.913354\pi$
$79$		−	4.77934i		−	0.537717i	0.963180	−	0.268859i	$-0.0866465\pi$
$80$		0			0
$81$	−0.902097			−0.100233
$82$		0			0
$83$	9.20975			1.01090			0.505451	−	0.862856i	$-0.331326\pi$
$83$	9.20975			1.01090			0.505451	+	0.862856i	$0.331326\pi$
$84$		0			0
$85$	−8.36531	−	2.18317i	−0.907346	−	0.236798i
$86$		0			0
$87$	−2.99696			−0.321308
$88$		0			0
$89$	−4.58193			−0.485683			−0.242842	−	0.970066i	$-0.578080\pi$
$89$	−4.58193			−0.485683			−0.242842	+	0.970066i	$0.578080\pi$
$90$		0			0
$91$		−	5.94293i		−	0.622988i
$92$		0			0
$93$	−5.55498			−0.576025
$94$		0			0
$95$		−	15.8928i		−	1.63057i
$96$		0			0
$97$		−	14.2966i		−	1.45160i	−0.687906	−	0.725800i	$-0.741470\pi$
$97$		−	14.2966i		−	1.45160i	0.687906	−	0.725800i	$-0.258530\pi$
$98$		0			0
$99$		−	6.63343i		−	0.666685i
$100$		0			0
$101$	2.19359			0.218270			0.109135	−	0.994027i	$-0.465192\pi$
$101$	2.19359			0.218270			0.109135	+	0.994027i	$0.465192\pi$
$102$		0			0
$103$	−6.86304			−0.676236			−0.338118	−	0.941104i	$-0.609790\pi$
$103$	−6.86304			−0.676236			−0.338118	+	0.941104i	$0.609790\pi$
$104$		0			0
$105$		−	6.43787i		−	0.628272i
$106$		0			0
$107$		−	7.37881i		−	0.713337i	−0.934231	−	0.356668i	$-0.883913\pi$
$107$		−	7.37881i		−	0.713337i	0.934231	−	0.356668i	$-0.116087\pi$
$108$		0			0
$109$			8.73085i			0.836264i	0.908386	+	0.418132i	$0.137315\pi$
$109$			8.73085i			0.836264i	−0.908386	+	0.418132i	$0.862685\pi$
$110$		0			0
$111$	12.8395			1.21867
$112$		0			0
$113$			18.8264i			1.77103i	0.464607	+	0.885517i	$0.346196\pi$
$113$			18.8264i			1.77103i	−0.464607	+	0.885517i	$0.653804\pi$
$114$		0			0
$115$	8.26367			0.770591
$116$		0			0
$117$	−3.76437			−0.348016
$118$		0			0
$119$	10.8131	+	2.82198i	0.991232	+	0.258691i
$120$		0			0
$121$	−3.92895			−0.357178
$122$		0			0
$123$	5.60324			0.505227
$124$		0			0
$125$		−	11.7491i		−	1.05087i
$126$		0			0
$127$	0.105365			0.00934965			0.00467482	−	0.999989i	$-0.498512\pi$
$127$	0.105365			0.00934965			0.00467482	+	0.999989i	$0.498512\pi$
$128$		0			0
$129$			6.14729i			0.541239i
$130$		0			0
$131$		−	11.2515i		−	0.983050i	−0.870864	−	0.491525i	$-0.836440\pi$
$131$		−	11.2515i		−	0.983050i	0.870864	−	0.491525i	$-0.163560\pi$
$132$		0			0
$133$			20.5432i			1.78132i
$134$		0			0
$135$	−11.2036			−0.964256
$136$		0			0
$137$	12.3796			1.05766			0.528830	−	0.848728i	$-0.322631\pi$
$137$	12.3796			1.05766			0.528830	+	0.848728i	$0.322631\pi$
$138$		0			0
$139$		−	11.3513i		−	0.962806i	−0.876499	−	0.481403i	$-0.840127\pi$
$139$		−	11.3513i		−	0.962806i	0.876499	−	0.481403i	$-0.159873\pi$
$140$		0			0
$141$		−	6.84129i		−	0.576140i
$142$		0			0
$143$			8.47194i			0.708459i
$144$		0			0
$145$	−5.54757			−0.460701
$146$		0			0
$147$			0.392199i			0.0323480i
$148$		0			0
$149$	5.29922			0.434129			0.217064	−	0.976157i	$-0.430352\pi$
$149$	5.29922			0.434129			0.217064	+	0.976157i	$0.430352\pi$
$150$		0			0
$151$	−21.8368			−1.77705			−0.888527	−	0.458825i	$-0.848271\pi$
$151$	−21.8368			−1.77705			−0.888527	+	0.458825i	$0.848271\pi$
$152$		0			0
$153$	1.78750	−	6.84920i	0.144511	−	0.553725i
$154$		0			0
$155$	−10.2826			−0.825920
$156$		0			0
$157$	−24.9017			−1.98737			−0.993687	−	0.112188i	$-0.964214\pi$
$157$	−24.9017			−1.98737			−0.993687	+	0.112188i	$0.964214\pi$
$158$		0			0
$159$			1.58961i			0.126064i
$160$		0			0
$161$	−10.6817			−0.841834
$162$		0			0
$163$		−	24.0233i		−	1.88165i	−0.338893	−	0.940825i	$-0.610053\pi$
$163$		−	24.0233i		−	1.88165i	0.338893	−	0.940825i	$-0.389947\pi$
$164$		0			0
$165$			9.17751i			0.714468i
$166$		0			0
$167$		−	2.28288i		−	0.176654i	−0.996092	−	0.0883272i	$-0.971848\pi$
$167$		−	2.28288i		−	0.176654i	0.996092	−	0.0883272i	$-0.0281521\pi$
$168$		0			0
$169$	−8.19231			−0.630178
$170$		0			0
$171$	13.0124			0.995086
$172$		0			0
$173$		−	4.75068i		−	0.361187i	−0.983558	−	0.180594i	$-0.942198\pi$
$173$		−	4.75068i		−	0.361187i	0.983558	−	0.180594i	$-0.0578019\pi$
$174$		0			0
$175$			1.63505i			0.123598i
$176$		0			0
$177$		−	1.13278i		−	0.0851448i
$178$		0			0
$179$	20.3267			1.51929			0.759644	−	0.650339i	$-0.225373\pi$
$179$	20.3267			1.51929			0.759644	+	0.650339i	$0.225373\pi$
$180$		0			0
$181$			2.46150i			0.182962i	0.995807	+	0.0914810i	$0.0291601\pi$
$181$			2.46150i			0.182962i	−0.995807	+	0.0914810i	$0.970840\pi$
$182$		0			0
$183$	−15.1769			−1.12191
$184$		0			0
$185$	23.7668			1.74737
$186$		0			0
$187$	−15.4146	−	4.02288i	−1.12722	−	0.294182i
$188$		0			0
$189$	14.4819			1.05340
$190$		0			0
$191$	−6.81181			−0.492885			−0.246443	−	0.969157i	$-0.579262\pi$
$191$	−6.81181			−0.492885			−0.246443	+	0.969157i	$0.579262\pi$
$192$		0			0
$193$			1.68256i			0.121114i	0.998165	+	0.0605568i	$0.0192876\pi$
$193$			1.68256i			0.121114i	−0.998165	+	0.0605568i	$0.980712\pi$
$194$		0			0
$195$	5.20809			0.372959
$196$		0			0
$197$			24.2376i			1.72686i	0.504471	+	0.863429i	$0.331688\pi$
$197$			24.2376i			1.72686i	−0.504471	+	0.863429i	$0.668312\pi$
$198$		0			0
$199$			6.44922i			0.457173i	0.973524	+	0.228586i	$0.0734103\pi$
$199$			6.44922i			0.457173i	−0.973524	+	0.228586i	$0.926590\pi$
$200$		0			0
$201$		−	2.24817i		−	0.158574i
$202$		0			0
$203$	7.17083			0.503293
$204$		0			0
$205$	10.3720			0.724409
$206$		0			0
$207$			6.76598i			0.470268i
$208$		0			0
$209$		−	29.2853i		−	2.02571i
$210$		0			0
$211$			0.132264i			0.00910545i	0.999990	+	0.00455273i	$0.00144918\pi$
$211$			0.132264i			0.00910545i	−0.999990	+	0.00455273i	$0.998551\pi$
$212$		0			0
$213$	−5.00999			−0.343279
$214$		0			0
$215$			11.3790i			0.776043i
$216$		0			0
$217$	13.2914			0.902278
$218$		0			0
$219$	1.93462			0.130729
$220$		0			0
$221$	−2.28292	+	8.74752i	−0.153566	+	0.588422i
$222$		0			0
$223$	13.9089			0.931406			0.465703	−	0.884941i	$-0.345801\pi$
$223$	13.9089			0.931406			0.465703	+	0.884941i	$0.345801\pi$
$224$		0			0
$225$	1.03568			0.0690450
$226$		0			0
$227$			9.66541i			0.641516i	0.947161	+	0.320758i	$0.103938\pi$
$227$			9.66541i			0.641516i	−0.947161	+	0.320758i	$0.896062\pi$
$228$		0			0
$229$	−8.96547			−0.592455			−0.296228	−	0.955117i	$-0.595729\pi$
$229$	−8.96547			−0.592455			−0.296228	+	0.955117i	$0.595729\pi$
$230$		0			0
$231$		−	11.8629i		−	0.780522i
$232$		0			0
$233$		−	8.30445i		−	0.544042i	−0.962291	−	0.272021i	$-0.912308\pi$
$233$		−	8.30445i		−	0.544042i	0.962291	−	0.272021i	$-0.0876921\pi$
$234$		0			0
$235$		−	12.6637i		−	0.826086i
$236$		0			0
$237$	−5.41392			−0.351672
$238$		0			0
$239$	−20.9404			−1.35452			−0.677260	−	0.735744i	$-0.736833\pi$
$239$	−20.9404			−1.35452			−0.677260	+	0.735744i	$0.736833\pi$
$240$		0			0
$241$		−	17.1231i		−	1.10300i	−0.834175	−	0.551499i	$-0.814056\pi$
$241$		−	17.1231i		−	1.10300i	0.834175	−	0.551499i	$-0.185944\pi$
$242$		0			0
$243$		−	15.0074i		−	0.962727i
$244$		0			0
$245$			0.725985i			0.0463815i
$246$		0			0
$247$	−16.6189			−1.05744
$248$		0			0
$249$		−	10.4326i		−	0.661139i
$250$		0			0
$251$	5.61892			0.354663			0.177332	−	0.984151i	$-0.443253\pi$
$251$	5.61892			0.354663			0.177332	+	0.984151i	$0.443253\pi$
$252$		0			0
$253$	15.2273			0.957330
$254$		0			0
$255$	−2.47305	+	9.47604i	−0.154868	+	0.593412i
$256$		0			0
$257$	−25.3108			−1.57884			−0.789421	−	0.613852i	$-0.789619\pi$
$257$	−25.3108			−1.57884			−0.789421	+	0.613852i	$0.789619\pi$
$258$		0			0
$259$	−30.7211			−1.90892
$260$		0			0
$261$		−	4.54214i		−	0.281152i
$262$		0			0
$263$	−23.6724			−1.45970			−0.729851	−	0.683606i	$-0.760411\pi$
$263$	−23.6724			−1.45970			−0.729851	+	0.683606i	$0.760411\pi$
$264$		0			0
$265$			2.94247i			0.180755i
$266$		0			0
$267$			5.19030i			0.317641i
$268$		0			0
$269$		−	10.1941i		−	0.621547i	−0.950484	−	0.310774i	$-0.899412\pi$
$269$		−	10.1941i		−	0.621547i	0.950484	−	0.310774i	$-0.100588\pi$
$270$		0			0
$271$	−27.4239			−1.66588			−0.832940	−	0.553363i	$-0.813344\pi$
$271$	−27.4239			−1.66588			−0.832940	+	0.553363i	$0.813344\pi$
$272$		0			0
$273$	−6.73201			−0.407440
$274$		0			0
$275$		−	2.33085i		−	0.140556i
$276$		0			0
$277$		−	11.8920i		−	0.714522i	−0.934005	−	0.357261i	$-0.883711\pi$
$277$		−	11.8920i		−	0.714522i	0.934005	−	0.357261i	$-0.116289\pi$
$278$		0			0
$279$		−	8.41902i		−	0.504034i
$280$		0			0
$281$	10.7275			0.639947			0.319974	−	0.947426i	$-0.396326\pi$
$281$	10.7275			0.639947			0.319974	+	0.947426i	$0.396326\pi$
$282$		0			0
$283$			8.18183i			0.486360i	0.969981	+	0.243180i	$0.0781905\pi$
$283$			8.18183i			0.486360i	−0.969981	+	0.243180i	$0.921809\pi$
$284$		0			0
$285$	−18.0030			−1.06641
$286$		0			0
$287$	−13.4069			−0.791382
$288$		0			0
$289$	−14.8319	−	8.30746i	−0.872466	−	0.488674i
$290$		0			0
$291$	−16.1949			−0.949360
$292$		0			0
$293$	30.6434			1.79021			0.895103	−	0.445860i	$-0.147102\pi$
$293$	30.6434			1.79021			0.895103	+	0.445860i	$0.147102\pi$
$294$		0			0
$295$		−	2.09684i		−	0.122083i
$296$		0			0
$297$	−20.6447			−1.19793
$298$		0			0
$299$		−	8.64123i		−	0.499735i
$300$		0			0
$301$		−	14.7086i		−	0.847790i
$302$		0			0
$303$		−	2.48485i		−	0.142751i
$304$		0			0
$305$	−28.0935			−1.60863
$306$		0			0
$307$	24.9866			1.42606			0.713030	−	0.701134i	$-0.247322\pi$
$307$	24.9866			1.42606			0.713030	+	0.701134i	$0.247322\pi$
$308$		0			0
$309$			7.77430i			0.442264i
$310$		0			0
$311$		−	34.7067i		−	1.96804i	−0.178072	−	0.984018i	$-0.556986\pi$
$311$		−	34.7067i		−	1.96804i	0.178072	−	0.984018i	$-0.443014\pi$
$312$		0			0
$313$			20.8502i			1.17852i	0.807943	+	0.589261i	$0.200581\pi$
$313$			20.8502i			1.17852i	−0.807943	+	0.589261i	$0.799419\pi$
$314$		0			0
$315$	9.75712			0.549751
$316$		0			0
$317$		−	0.723762i		−	0.0406505i	−0.999793	−	0.0203253i	$-0.993530\pi$
$317$		−	0.723762i		−	0.0406505i	0.999793	−	0.0203253i	$-0.00647018\pi$
$318$		0			0
$319$	−10.2224			−0.572343
$320$		0			0
$321$	−8.35855			−0.466529
$322$		0			0
$323$	7.89145	−	30.2379i	0.439092	−	1.68248i
$324$		0			0
$325$	−1.32272			−0.0733714
$326$		0			0
$327$	9.89011			0.546925
$328$		0			0
$329$			16.3691i			0.902459i
$330$		0			0
$331$	−33.5150			−1.84215			−0.921076	−	0.389384i	$-0.872688\pi$
$331$	−33.5150			−1.84215			−0.921076	+	0.389384i	$0.872688\pi$
$332$		0			0
$333$			19.4593i			1.06637i
$334$		0			0
$335$		−	4.16150i		−	0.227367i
$336$		0			0
$337$			8.40561i			0.457883i	0.973440	+	0.228941i	$0.0735264\pi$
$337$			8.40561i			0.457883i	−0.973440	+	0.228941i	$0.926474\pi$
$338$		0			0
$339$	21.3261			1.15827
$340$		0			0
$341$	−18.9475			−1.02607
$342$		0			0
$343$			18.0343i			0.973763i
$344$		0			0
$345$		−	9.36090i		−	0.503974i
$346$		0			0
$347$			1.71718i			0.0921828i	0.998937	+	0.0460914i	$0.0146765\pi$
$347$			1.71718i			0.0921828i	−0.998937	+	0.0460914i	$0.985323\pi$
$348$		0			0
$349$	7.24134			0.387620			0.193810	−	0.981039i	$-0.437915\pi$
$349$	7.24134			0.387620			0.193810	+	0.981039i	$0.437915\pi$
$350$		0			0
$351$			11.7155i			0.625328i
$352$		0			0
$353$	−4.15382			−0.221086			−0.110543	−	0.993871i	$-0.535259\pi$
$353$	−4.15382			−0.221086			−0.110543	+	0.993871i	$0.535259\pi$
$354$		0			0
$355$	−9.27380			−0.492202
$356$		0			0
$357$	3.19668	−	12.2488i	0.169186	−	0.648275i
$358$		0			0
$359$	16.0839			0.848873			0.424437	−	0.905458i	$-0.360472\pi$
$359$	16.0839			0.848873			0.424437	+	0.905458i	$0.360472\pi$
$360$		0			0
$361$	38.4474			2.02355
$362$		0			0
$363$			4.45063i			0.233597i
$364$		0			0
$365$	3.58110			0.187443
$366$		0			0
$367$			11.1878i			0.583997i	0.956419	+	0.291998i	$0.0943202\pi$
$367$			11.1878i			0.583997i	−0.956419	+	0.291998i	$0.905680\pi$
$368$		0			0
$369$			8.49217i			0.442084i
$370$		0			0
$371$		−	3.80346i		−	0.197466i
$372$		0			0
$373$	−16.9616			−0.878236			−0.439118	−	0.898429i	$-0.644709\pi$
$373$	−16.9616			−0.878236			−0.439118	+	0.898429i	$0.644709\pi$
$374$		0			0
$375$	−13.3092			−0.687282
$376$		0			0
$377$			5.80103i			0.298768i
$378$		0			0
$379$		−	12.3024i		−	0.631930i	−0.948771	−	0.315965i	$-0.897672\pi$
$379$		−	12.3024i		−	0.631930i	0.948771	−	0.315965i	$-0.102328\pi$
$380$		0			0
$381$		−	0.119355i		−	0.00611476i
$382$		0			0
$383$	3.44810			0.176190			0.0880948	−	0.996112i	$-0.471922\pi$
$383$	3.44810			0.176190			0.0880948	+	0.996112i	$0.471922\pi$
$384$		0			0
$385$		−	21.9590i		−	1.11913i
$386$		0			0
$387$	−9.31672			−0.473595
$388$		0			0
$389$	33.5064			1.69884			0.849421	−	0.527716i	$-0.176951\pi$
$389$	33.5064			1.69884			0.849421	+	0.527716i	$0.176951\pi$
$390$		0			0
$391$	15.7226	+	4.10326i	0.795125	+	0.207511i
$392$		0			0
$393$	−12.7455			−0.642924
$394$		0			0
$395$	−10.0215			−0.504237
$396$		0			0
$397$			37.1344i			1.86372i	0.362816	+	0.931861i	$0.381815\pi$
$397$			37.1344i			1.86372i	−0.362816	+	0.931861i	$0.618185\pi$
$398$		0			0
$399$	23.2708			1.16500
$400$		0			0
$401$		−	7.63712i		−	0.381379i	−0.981650	−	0.190690i	$-0.938928\pi$
$401$		−	7.63712i		−	0.381379i	0.981650	−	0.190690i	$-0.0610724\pi$
$402$		0			0
$403$			10.7524i			0.535616i
$404$		0			0
$405$			1.89155i			0.0939921i
$406$		0			0
$407$	43.7945			2.17081
$408$		0			0
$409$	1.03726			0.0512892			0.0256446	−	0.999671i	$-0.491836\pi$
$409$	1.03726			0.0512892			0.0256446	+	0.999671i	$0.491836\pi$
$410$		0			0
$411$		−	14.0233i		−	0.691720i
$412$		0			0
$413$			2.71039i			0.133370i
$414$		0			0
$415$		−	19.3114i		−	0.947959i
$416$		0			0
$417$	−12.8585			−0.629684
$418$		0			0
$419$			7.64135i			0.373304i	0.982426	+	0.186652i	$0.0597637\pi$
$419$			7.64135i			0.373304i	−0.982426	+	0.186652i	$0.940236\pi$
$420$		0			0
$421$	9.05833			0.441476			0.220738	−	0.975333i	$-0.429153\pi$
$421$	9.05833			0.441476			0.220738	+	0.975333i	$0.429153\pi$
$422$		0			0
$423$	10.3685			0.504135
$424$		0			0
$425$	0.628090	−	2.40667i	0.0304668	−	0.116741i
$426$		0			0
$427$	36.3138			1.75735
$428$		0			0
$429$	9.59682			0.463339
$430$		0			0
$431$			9.22394i			0.444302i	0.975012	+	0.222151i	$0.0713077\pi$
$431$			9.22394i			0.444302i	−0.975012	+	0.222151i	$0.928692\pi$
$432$		0			0
$433$	17.3283			0.832745			0.416373	−	0.909194i	$-0.363301\pi$
$433$	17.3283			0.832745			0.416373	+	0.909194i	$0.363301\pi$
$434$		0			0
$435$			6.28416i			0.301302i
$436$		0			0
$437$			29.8705i			1.42890i
$438$		0			0
$439$		−	10.5581i		−	0.503912i	−0.967739	−	0.251956i	$-0.918926\pi$
$439$		−	10.5581i		−	0.503912i	0.967739	−	0.251956i	$-0.0810738\pi$
$440$		0			0
$441$	−0.594409			−0.0283052
$442$		0			0
$443$	20.7429			0.985525			0.492762	−	0.870164i	$-0.335987\pi$
$443$	20.7429			0.985525			0.492762	+	0.870164i	$0.335987\pi$
$444$		0			0
$445$			9.60757i			0.455443i
$446$		0			0
$447$		−	6.00283i		−	0.283924i
$448$		0			0
$449$			20.7979i			0.981514i	0.871296	+	0.490757i	$0.163280\pi$
$449$			20.7979i			0.981514i	−0.871296	+	0.490757i	$0.836720\pi$
$450$		0			0
$451$	19.1121			0.899956
$452$		0			0
$453$			24.7362i			1.16221i
$454$		0			0
$455$	−12.4614			−0.584199
$456$		0			0
$457$	22.8328			1.06807			0.534036	−	0.845462i	$-0.320675\pi$
$457$	22.8328			1.06807			0.534036	+	0.845462i	$0.320675\pi$
$458$		0			0
$459$	−21.3162	−	5.56308i	−0.994955	−	0.259662i
$460$		0			0
$461$	19.2864			0.898259			0.449129	−	0.893467i	$-0.351734\pi$
$461$	19.2864			0.898259			0.449129	+	0.893467i	$0.351734\pi$
$462$		0			0
$463$	−36.8200			−1.71117			−0.855584	−	0.517664i	$-0.826802\pi$
$463$	−36.8200			−1.71117			−0.855584	+	0.517664i	$0.826802\pi$
$464$		0			0
$465$			11.6479i			0.540159i
$466$		0			0
$467$	35.6659			1.65042			0.825210	−	0.564826i	$-0.191057\pi$
$467$	35.6659			1.65042			0.825210	+	0.564826i	$0.191057\pi$
$468$		0			0
$469$			5.37918i			0.248388i
$470$		0			0
$471$			28.2081i			1.29976i
$472$		0			0
$473$			20.9679i			0.964103i
$474$		0			0
$475$	4.57231			0.209792
$476$		0			0
$477$	−2.40919			−0.110309
$478$		0			0
$479$		−	12.6243i		−	0.576820i	−0.957507	−	0.288410i	$-0.906873\pi$
$479$		−	12.6243i		−	0.576820i	0.957507	−	0.288410i	$-0.0931266\pi$
$480$		0			0
$481$		−	24.8527i		−	1.13318i
$482$		0			0
$483$			12.1000i			0.550567i
$484$		0			0
$485$	−29.9777			−1.36122
$486$		0			0
$487$			4.63497i			0.210031i	0.994471	+	0.105015i	$0.0334892\pi$
$487$			4.63497i			0.210031i	−0.994471	+	0.105015i	$0.966511\pi$
$488$		0			0
$489$	−27.2130			−1.23062
$490$		0			0
$491$	−2.28619			−0.103174			−0.0515872	−	0.998668i	$-0.516428\pi$
$491$	−2.28619			−0.103174			−0.0515872	+	0.998668i	$0.516428\pi$
$492$		0			0
$493$	−10.5549	−	2.75460i	−0.475368	−	0.124061i
$494$		0			0
$495$	−13.9093			−0.625175
$496$		0			0
$497$	11.9874			0.537707
$498$		0			0
$499$			11.2080i			0.501739i	0.968021	+	0.250869i	$0.0807165\pi$
$499$			11.2080i			0.501739i	−0.968021	+	0.250869i	$0.919284\pi$
$500$		0			0
$501$	−2.58599			−0.115534
$502$		0			0
$503$		−	14.8013i		−	0.659959i	−0.943988	−	0.329980i	$-0.892958\pi$
$503$		−	14.8013i		−	0.659959i	0.943988	−	0.329980i	$-0.107042\pi$
$504$		0			0
$505$		−	4.59961i		−	0.204680i
$506$		0			0
$507$			9.28006i			0.412142i
$508$		0			0
$509$	38.9964			1.72848			0.864242	−	0.503076i	$-0.167798\pi$
$509$	38.9964			1.72848			0.864242	+	0.503076i	$0.167798\pi$
$510$		0			0
$511$	−4.62895			−0.204773
$512$		0			0
$513$		−	40.4975i		−	1.78801i
$514$		0			0
$515$			14.3907i			0.634131i
$516$		0			0
$517$		−	23.3350i		−	1.02627i
$518$		0			0
$519$	−5.38146			−0.236220
$520$		0			0
$521$			40.1427i			1.75868i	0.476191	+	0.879342i	$0.342017\pi$
$521$			40.1427i			1.75868i	−0.476191	+	0.879342i	$0.657983\pi$
$522$		0			0
$523$	−27.9817			−1.22355			−0.611777	−	0.791030i	$-0.709545\pi$
$523$	−27.9817			−1.22355			−0.611777	+	0.791030i	$0.709545\pi$
$524$		0			0
$525$	1.85215			0.0808346
$526$		0			0
$527$	−19.5639	−	5.10575i	−0.852215	−	0.222410i
$528$		0			0
$529$	7.46847			0.324716
$530$		0			0
$531$	1.71682			0.0745035
$532$		0			0
$533$		−	10.8458i		−	0.469785i
$534$		0			0
$535$	−15.4722			−0.668922
$536$		0			0
$537$		−	23.0256i		−	0.993629i
$538$		0			0
$539$			1.33775i			0.0576212i
$540$		0			0
$541$			42.9837i			1.84801i	0.382376	+	0.924007i	$0.375106\pi$
$541$			42.9837i			1.84801i	−0.382376	+	0.924007i	$0.624894\pi$
$542$		0			0
$543$	2.78833			0.119659
$544$		0			0
$545$	18.3072			0.784195
$546$		0			0
$547$			35.7790i			1.52980i	0.644150	+	0.764899i	$0.277211\pi$
$547$			35.7790i			1.52980i	−0.644150	+	0.764899i	$0.722789\pi$
$548$		0			0
$549$		−	23.0019i		−	0.981696i
$550$		0			0
$551$		−	20.0527i		−	0.854273i
$552$		0			0
$553$	12.9539			0.550855
$554$		0			0
$555$		−	26.9225i		−	1.14279i
$556$		0			0
$557$	21.8747			0.926862			0.463431	−	0.886133i	$-0.346618\pi$
$557$	21.8747			0.926862			0.463431	+	0.886133i	$0.346618\pi$
$558$		0			0
$559$	11.8989			0.503271
$560$		0			0
$561$	−4.55702	+	17.4613i	−0.192398	+	0.737215i
$562$		0			0
$563$	38.8742			1.63835			0.819176	−	0.573542i	$-0.194431\pi$
$563$	38.8742			1.63835			0.819176	+	0.573542i	$0.194431\pi$
$564$		0			0
$565$	39.4759			1.66076
$566$		0			0
$567$		−	2.44504i		−	0.102682i
$568$		0			0
$569$	−22.6908			−0.951246			−0.475623	−	0.879649i	$-0.657777\pi$
$569$	−22.6908			−0.951246			−0.475623	+	0.879649i	$0.657777\pi$
$570$		0			0
$571$		−	9.37266i		−	0.392233i	−0.980581	−	0.196117i	$-0.937167\pi$
$571$		−	9.37266i		−	0.392233i	0.980581	−	0.196117i	$-0.0628331\pi$
$572$		0			0
$573$			7.71627i			0.322352i
$574$		0			0
$575$			2.37743i			0.0991456i
$576$		0			0
$577$	−10.9027			−0.453886			−0.226943	−	0.973908i	$-0.572873\pi$
$577$	−10.9027			−0.453886			−0.226943	+	0.973908i	$0.572873\pi$
$578$		0			0
$579$	1.90597			0.0792094
$580$		0			0
$581$			24.9620i			1.03560i
$582$		0			0
$583$			5.42202i			0.224557i
$584$		0			0
$585$			7.89328i			0.326347i
$586$		0			0
$587$	−31.6476			−1.30623			−0.653117	−	0.757257i	$-0.726539\pi$
$587$	−31.6476			−1.30623			−0.653117	+	0.757257i	$0.726539\pi$
$588$		0			0
$589$		−	37.1683i		−	1.53149i
$590$		0			0
$591$	27.4558			1.12938
$592$		0			0
$593$	−16.2493			−0.667277			−0.333639	−	0.942701i	$-0.608276\pi$
$593$	−16.2493			−0.667277			−0.333639	+	0.942701i	$0.608276\pi$
$594$		0			0
$595$	5.91725	−	22.6733i	0.242584	−	0.929514i
$596$		0			0
$597$	7.30553			0.298995
$598$		0			0
$599$	−31.1316			−1.27200			−0.636000	−	0.771689i	$-0.719412\pi$
$599$	−31.1316			−1.27200			−0.636000	+	0.771689i	$0.719412\pi$
$600$		0			0
$601$			9.38086i			0.382653i	0.981526	+	0.191327i	$0.0612790\pi$
$601$			9.38086i			0.382653i	−0.981526	+	0.191327i	$0.938721\pi$
$602$		0			0
$603$	3.40728			0.138755
$604$		0			0
$605$			8.23839i			0.334938i
$606$		0			0
$607$		−	19.2786i		−	0.782493i	−0.920286	−	0.391246i	$-0.872044\pi$
$607$		−	19.2786i		−	0.782493i	0.920286	−	0.391246i	$-0.127956\pi$
$608$		0			0
$609$		−	8.12295i		−	0.329159i
$610$		0			0
$611$	−13.2422			−0.535724
$612$		0			0
$613$	37.5758			1.51767			0.758836	−	0.651282i	$-0.225768\pi$
$613$	37.5758			1.51767			0.758836	+	0.651282i	$0.225768\pi$
$614$		0			0
$615$		−	11.7491i		−	0.473770i
$616$		0			0
$617$		−	41.3023i		−	1.66277i	−0.555698	−	0.831384i	$-0.687549\pi$
$617$		−	41.3023i		−	1.66277i	0.555698	−	0.831384i	$-0.312451\pi$
$618$		0			0
$619$		−	7.57462i		−	0.304450i	−0.988346	−	0.152225i	$-0.951356\pi$
$619$		−	7.57462i		−	0.304450i	0.988346	−	0.152225i	$-0.0486438\pi$
$620$		0			0
$621$	21.0572			0.844996
$622$		0			0
$623$		−	12.4188i		−	0.497549i
$624$		0			0
$625$	−21.6198			−0.864793
$626$		0			0
$627$	−33.1737			−1.32483
$628$		0			0
$629$	45.2190	+	11.8012i	1.80300	+	0.470545i
$630$		0			0
$631$	14.8708			0.591997			0.295998	−	0.955188i	$-0.404348\pi$
$631$	14.8708			0.591997			0.295998	+	0.955188i	$0.404348\pi$
$632$		0			0
$633$	0.149826			0.00595505
$634$		0			0
$635$		−	0.220934i		−	0.00876750i
$636$		0			0
$637$	0.759154			0.0300788
$638$		0			0
$639$		−	7.59304i		−	0.300376i
$640$		0			0
$641$			37.6789i			1.48823i	0.668053	+	0.744114i	$0.267128\pi$
$641$			37.6789i			1.48823i	−0.668053	+	0.744114i	$0.732872\pi$
$642$		0			0
$643$		−	0.295144i		−	0.0116393i	−0.999983	−	0.00581966i	$-0.998148\pi$
$643$		−	0.295144i		−	0.0116393i	0.999983	−	0.00581966i	$-0.00185247\pi$
$644$		0			0
$645$	12.8899			0.507539
$646$		0			0
$647$	−29.3720			−1.15473			−0.577366	−	0.816486i	$-0.695919\pi$
$647$	−29.3720			−1.15473			−0.577366	+	0.816486i	$0.695919\pi$
$648$		0			0
$649$		−	3.86380i		−	0.151667i
$650$		0			0
$651$		−	15.0562i		−	0.590098i
$652$		0			0
$653$		−	16.1187i		−	0.630775i	−0.948963	−	0.315387i	$-0.897866\pi$
$653$		−	16.1187i		−	0.630775i	0.948963	−	0.315387i	$-0.102134\pi$
$654$		0			0
$655$	−23.5927			−0.921841
$656$		0			0
$657$			2.93207i			0.114391i
$658$		0			0
$659$	23.4823			0.914741			0.457370	−	0.889276i	$-0.348791\pi$
$659$	23.4823			0.914741			0.457370	+	0.889276i	$0.348791\pi$
$660$		0			0
$661$	8.39444			0.326506			0.163253	−	0.986584i	$-0.447801\pi$
$661$	8.39444			0.326506			0.163253	+	0.986584i	$0.447801\pi$
$662$		0			0
$663$	9.90899	+	2.58604i	0.384833	+	0.100433i
$664$		0			0
$665$	43.0758			1.67041
$666$		0			0
$667$	10.4266			0.403721
$668$		0			0
$669$		−	15.7556i		−	0.609148i
$670$		0			0
$671$	−51.7671			−1.99845
$672$		0			0
$673$		−	11.4909i		−	0.442943i	−0.975167	−	0.221472i	$-0.928914\pi$
$673$		−	11.4909i		−	0.442943i	0.975167	−	0.221472i	$-0.0710860\pi$
$674$		0			0
$675$		−	3.22325i		−	0.124063i
$676$		0			0
$677$			5.23834i			0.201326i	0.994921	+	0.100663i	$0.0320964\pi$
$677$			5.23834i			0.201326i	−0.994921	+	0.100663i	$0.967904\pi$
$678$		0			0
$679$	38.7494			1.48707
$680$		0			0
$681$	10.9488			0.419557
$682$		0			0
$683$			27.0797i			1.03618i	0.855327	+	0.518088i	$0.173356\pi$
$683$			27.0797i			1.03618i	−0.855327	+	0.518088i	$0.826644\pi$
$684$		0			0
$685$		−	25.9581i		−	0.991807i
$686$		0			0
$687$			10.1559i			0.387471i
$688$		0			0
$689$	3.07691			0.117221
$690$		0			0
$691$		−	4.40834i		−	0.167701i	−0.996478	−	0.0838506i	$-0.973278\pi$
$691$		−	4.40834i		−	0.167701i	0.996478	−	0.0838506i	$-0.0267218\pi$
$692$		0			0
$693$	17.9792			0.682974
$694$		0			0
$695$	−23.8019			−0.902858
$696$		0			0
$697$	19.7338	+	5.15011i	0.747472	+	0.195074i
$698$		0			0
$699$	−9.40709			−0.355809
$700$		0			0
$701$	13.5986			0.513613			0.256806	−	0.966463i	$-0.417330\pi$
$701$	13.5986			0.513613			0.256806	+	0.966463i	$0.417330\pi$
$702$		0			0
$703$			85.9092i			3.24013i
$704$		0			0
$705$	−14.3451			−0.540268
$706$		0			0
$707$			5.94549i			0.223603i
$708$		0			0
$709$			1.00228i			0.0376413i	0.999823	+	0.0188207i	$0.00599116\pi$
$709$			1.00228i			0.0376413i	−0.999823	+	0.0188207i	$0.994009\pi$
$710$		0			0
$711$		−	8.20524i		−	0.307721i
$712$		0			0
$713$	19.3261			0.723770
$714$		0			0
$715$	17.7643			0.664348
$716$		0			0
$717$			23.7208i			0.885868i
$718$		0			0
$719$			30.6299i			1.14230i	0.820845	+	0.571151i	$0.193503\pi$
$719$			30.6299i			1.14230i	−0.820845	+	0.571151i	$0.806497\pi$
$720$		0			0
$721$		−	18.6015i		−	0.692758i
$722$		0			0
$723$	−19.3967			−0.721371
$724$		0			0
$725$		−	1.59602i		−	0.0592745i
$726$		0			0
$727$	16.3145			0.605070			0.302535	−	0.953138i	$-0.402167\pi$
$727$	16.3145			0.605070			0.302535	+	0.953138i	$0.402167\pi$
$728$		0			0
$729$	−19.7064			−0.729865
$730$		0			0
$731$	−5.65017	+	21.6499i	−0.208979	+	0.800750i
$732$		0			0
$733$	42.5135			1.57027			0.785136	−	0.619324i	$-0.212593\pi$
$733$	42.5135			1.57027			0.785136	+	0.619324i	$0.212593\pi$
$734$		0			0
$735$	0.822379			0.0303339
$736$		0			0
$737$		−	7.66830i		−	0.282465i
$738$		0			0
$739$	19.9685			0.734554			0.367277	−	0.930112i	$-0.380290\pi$
$739$	19.9685			0.734554			0.367277	+	0.930112i	$0.380290\pi$
$740$		0			0
$741$			18.8256i			0.691574i
$742$		0			0
$743$		−	33.9693i		−	1.24621i	−0.782138	−	0.623106i	$-0.785871\pi$
$743$		−	33.9693i		−	1.24621i	0.782138	−	0.623106i	$-0.214129\pi$
$744$		0			0
$745$		−	11.1116i		−	0.407098i
$746$		0			0
$747$	15.8114			0.578510
$748$		0			0
$749$	19.9995			0.730765
$750$		0			0
$751$		−	46.5988i		−	1.70042i	−0.526447	−	0.850208i	$-0.676476\pi$
$751$		−	46.5988i		−	1.70042i	0.526447	−	0.850208i	$-0.323524\pi$
$752$		0			0
$753$		−	6.36499i		−	0.231953i
$754$		0			0
$755$			45.7883i			1.66641i
$756$		0			0
$757$	−52.1700			−1.89615			−0.948075	−	0.318047i	$-0.896973\pi$
$757$	−52.1700			−1.89615			−0.948075	+	0.318047i	$0.896973\pi$
$758$		0			0
$759$		−	17.2491i		−	0.626103i
$760$		0			0
$761$	0.909512			0.0329698			0.0164849	−	0.999864i	$-0.494752\pi$
$761$	0.909512			0.0329698			0.0164849	+	0.999864i	$0.494752\pi$
$762$		0			0
$763$	−23.6640			−0.856696
$764$		0			0
$765$	−14.3617	−	3.74810i	−0.519248	−	0.135513i
$766$		0			0
$767$	−2.19264			−0.0791718
$768$		0			0
$769$	−19.5541			−0.705137			−0.352569	−	0.935786i	$-0.614692\pi$
$769$	−19.5541			−0.705137			−0.352569	+	0.935786i	$0.614692\pi$
$770$		0			0
$771$			28.6715i			1.03258i
$772$		0			0
$773$	8.52160			0.306501			0.153250	−	0.988187i	$-0.451026\pi$
$773$	8.52160			0.306501			0.153250	+	0.988187i	$0.451026\pi$
$774$		0			0
$775$		−	2.95827i		−	0.106264i
$776$		0			0
$777$			34.8002i			1.24845i
$778$		0			0
$779$			37.4913i			1.34326i
$780$		0			0
$781$	−17.0886			−0.611478
$782$		0			0
$783$	−14.1361			−0.505184
$784$		0			0
$785$			52.2150i			1.86363i
$786$		0			0
$787$		−	1.83539i		−	0.0654248i	−0.999465	−	0.0327124i	$-0.989585\pi$
$787$		−	1.83539i		−	0.0654248i	0.999465	−	0.0327124i	$-0.0104145\pi$
$788$		0			0
$789$			26.8156i			0.954659i
$790$		0			0
$791$	−51.0268			−1.81430
$792$		0			0
$793$			29.3770i			1.04321i
$794$		0			0
$795$	3.33317			0.118215
$796$		0			0
$797$	−53.5743			−1.89770			−0.948850	−	0.315727i	$-0.897752\pi$
$797$	−53.5743			−1.89770			−0.948850	+	0.315727i	$0.897752\pi$
$798$		0			0
$799$	6.28804	−	24.0941i	0.222455	−	0.852386i
$800$		0			0
$801$	−7.86632			−0.277943
$802$		0			0
$803$	6.59880			0.232867
$804$		0			0
$805$			22.3978i			0.789418i
$806$		0			0
$807$	−11.5477			−0.406498
$808$		0			0
$809$			37.3629i			1.31361i	0.754061	+	0.656804i	$0.228092\pi$
$809$			37.3629i			1.31361i	−0.754061	+	0.656804i	$0.771908\pi$
$810$		0			0
$811$		−	1.33075i		−	0.0467290i	−0.999727	−	0.0233645i	$-0.992562\pi$
$811$		−	1.33075i		−	0.0467290i	0.999727	−	0.0233645i	$-0.00743782\pi$
$812$		0			0
$813$			31.0651i			1.08950i
$814$		0			0
$815$	−50.3731			−1.76449
$816$		0			0
$817$	−41.1315			−1.43901
$818$		0			0
$819$		−	10.2029i		−	0.356519i
$820$		0			0
$821$			37.9367i			1.32400i	0.749504	+	0.662000i	$0.230292\pi$
$821$			37.9367i			1.32400i	−0.749504	+	0.662000i	$0.769708\pi$
$822$		0			0
$823$		−	28.6307i		−	0.998004i	−0.866601	−	0.499002i	$-0.833700\pi$
$823$		−	28.6307i		−	0.998004i	0.866601	−	0.499002i	$-0.166300\pi$
$824$		0			0
$825$	−2.64034			−0.0919247
$826$		0			0
$827$			32.9677i			1.14640i	0.819416	+	0.573199i	$0.194298\pi$
$827$			32.9677i			1.14640i	−0.819416	+	0.573199i	$0.805702\pi$
$828$		0			0
$829$	−4.09265			−0.142144			−0.0710719	−	0.997471i	$-0.522642\pi$
$829$	−4.09265			−0.142144			−0.0710719	+	0.997471i	$0.522642\pi$
$830$		0			0
$831$	−13.4710			−0.467304
$832$		0			0
$833$	−0.360482	+	1.38127i	−0.0124900	+	0.0478581i
$834$		0			0
$835$	−4.78684			−0.165655
$836$		0			0
$837$	−26.2018			−0.905667
$838$		0			0
$839$			38.7903i			1.33919i	0.742727	+	0.669595i	$0.233532\pi$
$839$			38.7903i			1.33919i	−0.742727	+	0.669595i	$0.766468\pi$
$840$		0			0
$841$	22.0004			0.758634
$842$		0			0
$843$		−	12.1518i		−	0.418532i
$844$		0			0
$845$			17.1780i			0.590940i
$846$		0			0
$847$		−	10.6490i		−	0.365904i
$848$		0			0
$849$	9.26820			0.318084
$850$		0			0
$851$	−44.6696			−1.53125
$852$		0			0
$853$			17.5181i			0.599807i	0.953970	+	0.299904i	$0.0969545\pi$
$853$			17.5181i			0.599807i	−0.953970	+	0.299904i	$0.903046\pi$
$854$		0			0
$855$		−	27.2850i		−	0.933129i
$856$		0			0
$857$			35.9623i			1.22845i	0.789131	+	0.614224i	$0.210531\pi$
$857$			35.9623i			1.22845i	−0.789131	+	0.614224i	$0.789469\pi$
$858$		0			0
$859$	−17.6259			−0.601387			−0.300694	−	0.953721i	$-0.597218\pi$
$859$	−17.6259			−0.601387			−0.300694	+	0.953721i	$0.597218\pi$
$860$		0			0
$861$			15.1870i			0.517571i
$862$		0			0
$863$	45.8867			1.56200			0.781001	−	0.624529i	$-0.214709\pi$
$863$	45.8867			1.56200			0.781001	+	0.624529i	$0.214709\pi$
$864$		0			0
$865$	−9.96142			−0.338698
$866$		0			0
$867$	−9.41050	+	16.8013i	−0.319597	+	0.570601i
$868$		0			0
$869$	−18.4664			−0.626430
$870$		0			0
$871$	−4.35164			−0.147450
$872$		0			0
$873$		−	24.5446i		−	0.830710i
$874$		0			0
$875$	31.8448			1.07655
$876$		0			0
$877$			54.4778i			1.83958i	0.392406	+	0.919792i	$0.371643\pi$
$877$			54.4778i			1.83958i	−0.392406	+	0.919792i	$0.628357\pi$
$878$		0			0
$879$		−	34.7121i		−	1.17081i
$880$		0			0
$881$		−	8.78574i		−	0.295999i	−0.988987	−	0.148000i	$-0.952717\pi$
$881$		−	8.78574i		−	0.295999i	0.988987	−	0.148000i	$-0.0472834\pi$
$882$		0			0
$883$	22.8260			0.768156			0.384078	−	0.923301i	$-0.374519\pi$
$883$	22.8260			0.768156			0.384078	+	0.923301i	$0.374519\pi$
$884$		0			0
$885$	−2.37525			−0.0798433
$886$		0			0
$887$			0.205581i			0.00690273i	0.999994	+	0.00345136i	$0.00109861\pi$
$887$			0.205581i			0.00690273i	−0.999994	+	0.00345136i	$0.998901\pi$
$888$		0			0
$889$			0.285581i			0.00957808i
$890$		0			0
$891$			3.48552i			0.116769i
$892$		0			0
$893$	45.7750			1.53180
$894$		0			0
$895$		−	42.6219i		−	1.42469i
$896$		0			0
$897$	−9.78859			−0.326831
$898$		0			0
$899$	−12.9740			−0.432708
$900$		0			0
$901$	−1.46106	+	5.59839i	−0.0486750	+	0.186509i
$902$		0			0
$903$	−16.6616			−0.554462
$904$		0			0
$905$	5.16138			0.171570
$906$		0			0
$907$			0.0190783i			0.000633485i	1.00000		0.000316743i	$0.000100822\pi$
$907$			0.0190783i			0.000633485i	−1.00000		0.000316743i	$0.999899\pi$
$908$		0			0
$909$	3.76599			0.124910
$910$		0			0
$911$			27.0363i			0.895751i	0.894096	+	0.447876i	$0.147819\pi$
$911$			27.0363i			0.895751i	−0.894096	+	0.447876i	$0.852181\pi$
$912$		0			0
$913$		−	35.5846i		−	1.17768i
$914$		0			0
$915$			31.8236i			1.05206i
$916$		0			0
$917$	30.4960			1.00707
$918$		0			0
$919$	18.6036			0.613676			0.306838	−	0.951762i	$-0.400729\pi$
$919$	18.6036			0.613676			0.306838	+	0.951762i	$0.400729\pi$
$920$		0			0
$921$		−	28.3042i		−	0.932656i
$922$		0			0
$923$			9.69751i			0.319197i
$924$		0			0
$925$			6.83761i			0.224819i
$926$		0			0
$927$	−11.7826			−0.386991
$928$		0			0
$929$			41.5766i			1.36409i	0.731312	+	0.682043i	$0.238908\pi$
$929$			41.5766i			1.36409i	−0.731312	+	0.682043i	$0.761092\pi$
$930$		0			0
$931$	−2.62420			−0.0860047
$932$		0			0
$933$	−39.3149			−1.28711
$934$		0			0
$935$	−8.43533	+	32.3219i	−0.275865	+	1.05704i
$936$		0			0
$937$	15.9780			0.521978			0.260989	−	0.965342i	$-0.415951\pi$
$937$	15.9780			0.521978			0.260989	+	0.965342i	$0.415951\pi$
$938$		0			0
$939$	23.6186			0.770764
$940$		0			0
$941$		−	6.12647i		−	0.199717i	−0.995002	−	0.0998587i	$-0.968161\pi$
$941$		−	6.12647i		−	0.199717i	0.995002	−	0.0998587i	$-0.0318391\pi$
$942$		0			0
$943$	−19.4940			−0.634813
$944$		0			0
$945$		−	30.3663i		−	0.987815i
$946$		0			0
$947$		−	48.8422i		−	1.58716i	−0.608468	−	0.793579i	$-0.708215\pi$
$947$		−	48.8422i		−	1.58716i	0.608468	−	0.793579i	$-0.291785\pi$
$948$		0			0
$949$		−	3.74472i		−	0.121559i
$950$		0			0
$951$	−0.819861			−0.0265858
$952$		0			0
$953$	30.3537			0.983254			0.491627	−	0.870806i	$-0.336402\pi$
$953$	30.3537			0.983254			0.491627	+	0.870806i	$0.336402\pi$
$954$		0			0
$955$			14.2833i			0.462196i
$956$		0			0
$957$			11.5797i			0.374318i
$958$		0			0
$959$			33.5536i			1.08350i
$960$		0			0
$961$	6.95217			0.224264
$962$		0			0
$963$		−	12.6681i		−	0.408223i
$964$		0			0
$965$	3.52807			0.113573
$966$		0			0
$967$	42.2385			1.35830			0.679150	−	0.734000i	$-0.262349\pi$
$967$	42.2385			1.35830			0.679150	+	0.734000i	$0.262349\pi$
$968$		0			0
$969$	−34.2528	−	8.93926i	−1.10036	−	0.287170i
$970$		0			0
$971$	−37.4987			−1.20339			−0.601695	−	0.798726i	$-0.705508\pi$
$971$	−37.4987			−1.20339			−0.601695	+	0.798726i	$0.705508\pi$
$972$		0			0
$973$	30.7665			0.986330
$974$		0			0
$975$			1.49835i			0.0479856i
$976$		0			0
$977$	−3.83099			−0.122564			−0.0612821	−	0.998120i	$-0.519519\pi$
$977$	−3.83099			−0.122564			−0.0612821	+	0.998120i	$0.519519\pi$
$978$		0			0
$979$			17.7036i			0.565811i
$980$		0			0
$981$			14.9893i			0.478570i
$982$		0			0
$983$		−	53.9525i		−	1.72082i	−0.509605	−	0.860409i	$-0.670208\pi$
$983$		−	53.9525i		−	1.72082i	0.509605	−	0.860409i	$-0.329792\pi$
$984$		0			0
$985$	50.8224			1.61934
$986$		0			0
$987$	18.5426			0.590217
$988$		0			0
$989$		−	21.3868i		−	0.680062i
$990$		0			0
$991$			26.7390i			0.849393i	0.905336	+	0.424696i	$0.139619\pi$
$991$			26.7390i			0.849393i	−0.905336	+	0.424696i	$0.860381\pi$
$992$		0			0
$993$			37.9650i			1.20478i
$994$		0			0
$995$	13.5230			0.428707
$996$		0			0
$997$		−	33.1952i		−	1.05130i	−0.850700	−	0.525651i	$-0.823822\pi$
$997$		−	33.1952i		−	1.05130i	0.850700	−	0.525651i	$-0.176178\pi$
$998$		0			0
$999$	60.5617			1.91609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.18	✓	46
17.16	even	2	inner	4012.2.b.b.237.29	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.18	✓	46	1.1	even	1	trivial
4012.2.b.b.237.29	yes	46	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.13278i q^{3} -2.09684i q^{5} +2.71039i q^{7} +1.71682 q^{9} +O(q^{10})\)	\(q-1.13278i q^{3} -2.09684i q^{5} +2.71039i q^{7} +1.71682 q^{9} -3.86380i q^{11} -2.19264 q^{13} -2.37525 q^{15} +(1.04117 - 3.98948i) q^{17} +7.57941 q^{19} +3.07027 q^{21} +3.94101i q^{23} +0.603254 q^{25} -5.34310i q^{27} -2.64568i q^{29} -4.90386i q^{31} -4.37683 q^{33} +5.68326 q^{35} +11.3346i q^{37} +2.48378i q^{39} +4.94646i q^{41} -5.42674 q^{43} -3.59989i q^{45} +6.03939 q^{47} -0.346228 q^{49} +(-4.51919 - 1.17941i) q^{51} -1.40329 q^{53} -8.10178 q^{55} -8.58578i q^{57} +1.00000 q^{59} -13.3980i q^{61} +4.65324i q^{63} +4.59763i q^{65} +1.98465 q^{67} +4.46428 q^{69} -4.42275i q^{71} +1.70785i q^{73} -0.683352i q^{75} +10.4724 q^{77} -4.77934i q^{79} -0.902097 q^{81} +9.20975 q^{83} +(-8.36531 - 2.18317i) q^{85} -2.99696 q^{87} -4.58193 q^{89} -5.94293i q^{91} -5.55498 q^{93} -15.8928i q^{95} -14.2966i q^{97} -6.63343i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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