LMFDB - Embedded newform 4012.2.b.b.237.10

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			237.10
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.37

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82354i q^{3} -0.284793i q^{5} -0.645277i q^{7} -0.325298 q^{9} +O(q^{10})$	$q-1.82354i q^{3} -0.284793i q^{5} -0.645277i q^{7} -0.325298 q^{9} +1.56411i q^{11} +1.12492 q^{13} -0.519331 q^{15} +(1.35405 - 3.89443i) q^{17} -0.845911 q^{19} -1.17669 q^{21} +2.49622i q^{23} +4.91889 q^{25} -4.87743i q^{27} +9.28760i q^{29} -5.94750i q^{31} +2.85222 q^{33} -0.183770 q^{35} -6.09474i q^{37} -2.05134i q^{39} -1.30564i q^{41} +3.23300 q^{43} +0.0926424i q^{45} +0.319650 q^{47} +6.58362 q^{49} +(-7.10164 - 2.46916i) q^{51} -3.07538 q^{53} +0.445448 q^{55} +1.54255i q^{57} +1.00000 q^{59} +4.55451i q^{61} +0.209907i q^{63} -0.320370i q^{65} +4.98126 q^{67} +4.55196 q^{69} -13.3681i q^{71} -13.2696i q^{73} -8.96980i q^{75} +1.00929 q^{77} +1.65938i q^{79} -9.87007 q^{81} +4.62138 q^{83} +(-1.10910 - 0.385622i) q^{85} +16.9363 q^{87} +1.32179 q^{89} -0.725887i q^{91} -10.8455 q^{93} +0.240909i q^{95} -6.36118i q^{97} -0.508802i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.82354i		−	1.05282i	−0.850230	−	0.526411i	$-0.823537\pi$
$3$		−	1.82354i		−	1.05282i	0.850230	−	0.526411i	$-0.176463\pi$
$4$		0			0
$5$		−	0.284793i		−	0.127363i	−0.997970	−	0.0636816i	$-0.979716\pi$
$5$		−	0.284793i		−	0.127363i	0.997970	−	0.0636816i	$-0.0202842\pi$
$6$		0			0
$7$		−	0.645277i		−	0.243892i	−0.992537	−	0.121946i	$-0.961087\pi$
$7$		−	0.645277i		−	0.243892i	0.992537	−	0.121946i	$-0.0389135\pi$
$8$		0			0
$9$	−0.325298			−0.108433
$10$		0			0
$11$			1.56411i			0.471598i	0.971802	+	0.235799i	$0.0757707\pi$
$11$			1.56411i			0.471598i	−0.971802	+	0.235799i	$0.924229\pi$
$12$		0			0
$13$	1.12492			0.311997			0.155999	−	0.987757i	$-0.450140\pi$
$13$	1.12492			0.311997			0.155999	+	0.987757i	$0.450140\pi$
$14$		0			0
$15$	−0.519331			−0.134091
$16$		0			0
$17$	1.35405	−	3.89443i	0.328404	−	0.944537i
$17$	1.35405	−	3.89443i	0.328404	−	0.944537i
$18$		0			0
$19$	−0.845911			−0.194065			−0.0970327	−	0.995281i	$-0.530935\pi$
$19$	−0.845911			−0.194065			−0.0970327	+	0.995281i	$0.530935\pi$
$20$		0			0
$21$	−1.17669			−0.256775
$22$		0			0
$23$			2.49622i			0.520498i	0.965542	+	0.260249i	$0.0838047\pi$
$23$			2.49622i			0.520498i	−0.965542	+	0.260249i	$0.916195\pi$
$24$		0			0
$25$	4.91889			0.983779
$26$		0			0
$27$		−	4.87743i		−	0.938661i
$28$		0			0
$29$			9.28760i			1.72466i	0.506344	+	0.862332i	$0.330997\pi$
$29$			9.28760i			1.72466i	−0.506344	+	0.862332i	$0.669003\pi$
$30$		0			0
$31$		−	5.94750i		−	1.06820i	−0.845420	−	0.534102i	$-0.820650\pi$
$31$		−	5.94750i		−	1.06820i	0.845420	−	0.534102i	$-0.179350\pi$
$32$		0			0
$33$	2.85222			0.496508
$34$		0			0
$35$	−0.183770			−0.0310629
$36$		0			0
$37$		−	6.09474i		−	1.00197i	−0.865456	−	0.500985i	$-0.832971\pi$
$37$		−	6.09474i		−	1.00197i	0.865456	−	0.500985i	$-0.167029\pi$
$38$		0			0
$39$		−	2.05134i		−	0.328477i
$40$		0			0
$41$		−	1.30564i		−	0.203907i	−0.994789	−	0.101954i	$-0.967491\pi$
$41$		−	1.30564i		−	0.203907i	0.994789	−	0.101954i	$-0.0325094\pi$
$42$		0			0
$43$	3.23300			0.493028			0.246514	−	0.969139i	$-0.420715\pi$
$43$	3.23300			0.493028			0.246514	+	0.969139i	$0.420715\pi$
$44$		0			0
$45$			0.0926424i			0.0138103i
$46$		0			0
$47$	0.319650			0.0466257			0.0233128	−	0.999728i	$-0.492579\pi$
$47$	0.319650			0.0466257			0.0233128	+	0.999728i	$0.492579\pi$
$48$		0			0
$49$	6.58362			0.940517
$50$		0			0
$51$	−7.10164	−	2.46916i	−0.994429	−	0.345751i
$52$		0			0
$53$	−3.07538			−0.422436			−0.211218	−	0.977439i	$-0.567743\pi$
$53$	−3.07538			−0.422436			−0.211218	+	0.977439i	$0.567743\pi$
$54$		0			0
$55$	0.445448			0.0600642
$56$		0			0
$57$			1.54255i			0.204316i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			4.55451i			0.583145i	0.956549	+	0.291572i	$0.0941785\pi$
$61$			4.55451i			0.583145i	−0.956549	+	0.291572i	$0.905822\pi$
$62$		0			0
$63$			0.209907i			0.0264458i
$64$		0			0
$65$		−	0.320370i		−	0.0397370i
$66$		0			0
$67$	4.98126			0.608558			0.304279	−	0.952583i	$-0.401585\pi$
$67$	4.98126			0.608558			0.304279	+	0.952583i	$0.401585\pi$
$68$		0			0
$69$	4.55196			0.547991
$70$		0			0
$71$		−	13.3681i		−	1.58650i	−0.608896	−	0.793250i	$-0.708387\pi$
$71$		−	13.3681i		−	1.58650i	0.608896	−	0.793250i	$-0.291613\pi$
$72$		0			0
$73$		−	13.2696i		−	1.55309i	−0.630065	−	0.776543i	$-0.716972\pi$
$73$		−	13.2696i		−	1.55309i	0.630065	−	0.776543i	$-0.283028\pi$
$74$		0			0
$75$		−	8.96980i		−	1.03574i
$76$		0			0
$77$	1.00929			0.115019
$78$		0			0
$79$			1.65938i			0.186695i	0.995634	+	0.0933476i	$0.0297568\pi$
$79$			1.65938i			0.186695i	−0.995634	+	0.0933476i	$0.970243\pi$
$80$		0			0
$81$	−9.87007			−1.09667
$82$		0			0
$83$	4.62138			0.507262			0.253631	−	0.967301i	$-0.418375\pi$
$83$	4.62138			0.507262			0.253631	+	0.967301i	$0.418375\pi$
$84$		0			0
$85$	−1.10910	−	0.385622i	−0.120299	−	0.0418266i
$86$		0			0
$87$	16.9363			1.81576
$88$		0			0
$89$	1.32179			0.140110			0.0700550	−	0.997543i	$-0.477683\pi$
$89$	1.32179			0.140110			0.0700550	+	0.997543i	$0.477683\pi$
$90$		0			0
$91$		−	0.725887i		−	0.0760936i
$92$		0			0
$93$	−10.8455			−1.12463
$94$		0			0
$95$			0.240909i			0.0247168i
$96$		0			0
$97$		−	6.36118i		−	0.645880i	−0.946419	−	0.322940i	$-0.895329\pi$
$97$		−	6.36118i		−	0.645880i	0.946419	−	0.322940i	$-0.104671\pi$
$98$		0			0
$99$		−	0.508802i		−	0.0511365i
$100$		0			0
$101$	1.83767			0.182855			0.0914275	−	0.995812i	$-0.470857\pi$
$101$	1.83767			0.182855			0.0914275	+	0.995812i	$0.470857\pi$
$102$		0			0
$103$	−9.66717			−0.952535			−0.476267	−	0.879301i	$-0.658011\pi$
$103$	−9.66717			−0.952535			−0.476267	+	0.879301i	$0.658011\pi$
$104$		0			0
$105$			0.335113i			0.0327036i
$106$		0			0
$107$			15.8474i			1.53202i	0.642828	+	0.766011i	$0.277761\pi$
$107$			15.8474i			1.53202i	−0.642828	+	0.766011i	$0.722239\pi$
$108$		0			0
$109$			3.38945i			0.324650i	0.986737	+	0.162325i	$0.0518994\pi$
$109$			3.38945i			0.324650i	−0.986737	+	0.162325i	$0.948101\pi$
$110$		0			0
$111$	−11.1140			−1.05489
$112$		0			0
$113$		−	15.2698i		−	1.43647i	−0.695803	−	0.718233i	$-0.744951\pi$
$113$		−	15.2698i		−	1.43647i	0.695803	−	0.718233i	$-0.255049\pi$
$114$		0			0
$115$	0.710906			0.0662923
$116$		0			0
$117$	−0.365934			−0.0338306
$118$		0			0
$119$	−2.51299	−	0.873735i	−0.230365	−	0.0800951i
$120$		0			0
$121$	8.55355			0.777595
$122$		0			0
$123$	−2.38089			−0.214678
$124$		0			0
$125$		−	2.82483i		−	0.252660i
$126$		0			0
$127$	1.76464			0.156586			0.0782931	−	0.996930i	$-0.475053\pi$
$127$	1.76464			0.156586			0.0782931	+	0.996930i	$0.475053\pi$
$128$		0			0
$129$		−	5.89550i		−	0.519070i
$130$		0			0
$131$			9.57745i			0.836785i	0.908266	+	0.418393i	$0.137406\pi$
$131$			9.57745i			0.836785i	−0.908266	+	0.418393i	$0.862594\pi$
$132$		0			0
$133$			0.545848i			0.0473310i
$134$		0			0
$135$	−1.38906			−0.119551
$136$		0			0
$137$	−6.59426			−0.563385			−0.281693	−	0.959505i	$-0.590896\pi$
$137$	−6.59426			−0.563385			−0.281693	+	0.959505i	$0.590896\pi$
$138$		0			0
$139$		−	22.3298i		−	1.89399i	−0.321247	−	0.946995i	$-0.604102\pi$
$139$		−	22.3298i		−	1.89399i	0.321247	−	0.946995i	$-0.395898\pi$
$140$		0			0
$141$		−	0.582894i		−	0.0490885i
$142$		0			0
$143$			1.75951i			0.147137i
$144$		0			0
$145$	2.64504			0.219659
$146$		0			0
$147$		−	12.0055i		−	0.990196i
$148$		0			0
$149$	18.9103			1.54919			0.774595	−	0.632458i	$-0.217954\pi$
$149$	18.9103			1.54919			0.774595	+	0.632458i	$0.217954\pi$
$150$		0			0
$151$	−9.21004			−0.749502			−0.374751	−	0.927125i	$-0.622272\pi$
$151$	−9.21004			−0.749502			−0.374751	+	0.927125i	$0.622272\pi$
$152$		0			0
$153$	−0.440467	+	1.26685i	−0.0356097	+	0.102419i
$154$		0			0
$155$	−1.69381			−0.136050
$156$		0			0
$157$	13.0359			1.04038			0.520190	−	0.854051i	$-0.325861\pi$
$157$	13.0359			1.04038			0.520190	+	0.854051i	$0.325861\pi$
$158$		0			0
$159$			5.60808i			0.444750i
$160$		0			0
$161$	1.61076			0.126945
$162$		0			0
$163$		−	9.99477i		−	0.782851i	−0.920210	−	0.391425i	$-0.871982\pi$
$163$		−	9.99477i		−	0.782851i	0.920210	−	0.391425i	$-0.128018\pi$
$164$		0			0
$165$		−	0.812292i		−	0.0632369i
$166$		0			0
$167$		−	14.8575i		−	1.14971i	−0.818257	−	0.574853i	$-0.805059\pi$
$167$		−	14.8575i		−	1.14971i	0.818257	−	0.574853i	$-0.194941\pi$
$168$		0			0
$169$	−11.7346			−0.902658
$170$		0			0
$171$	0.275173			0.0210430
$172$		0			0
$173$		−	6.62290i		−	0.503530i	−0.967788	−	0.251765i	$-0.918989\pi$
$173$		−	6.62290i		−	0.503530i	0.967788	−	0.251765i	$-0.0810110\pi$
$174$		0			0
$175$		−	3.17405i		−	0.239936i
$176$		0			0
$177$		−	1.82354i		−	0.137066i
$178$		0			0
$179$	−25.6065			−1.91392			−0.956960	−	0.290219i	$-0.906272\pi$
$179$	−25.6065			−1.91392			−0.956960	+	0.290219i	$0.906272\pi$
$180$		0			0
$181$			15.4252i			1.14655i	0.819363	+	0.573275i	$0.194327\pi$
$181$			15.4252i			1.14655i	−0.819363	+	0.573275i	$0.805673\pi$
$182$		0			0
$183$	8.30533			0.613947
$184$		0			0
$185$	−1.73574			−0.127614
$186$		0			0
$187$	6.09133	+	2.11788i	0.445442	+	0.154875i
$188$		0			0
$189$	−3.14729			−0.228932
$190$		0			0
$191$	−20.2292			−1.46373			−0.731867	−	0.681447i	$-0.761351\pi$
$191$	−20.2292			−1.46373			−0.731867	+	0.681447i	$0.761351\pi$
$192$		0			0
$193$			14.2173i			1.02338i	0.859169	+	0.511692i	$0.170981\pi$
$193$			14.2173i			1.02338i	−0.859169	+	0.511692i	$0.829019\pi$
$194$		0			0
$195$	−0.584207			−0.0418359
$196$		0			0
$197$		−	9.98969i		−	0.711736i	−0.934536	−	0.355868i	$-0.884185\pi$
$197$		−	9.98969i		−	0.711736i	0.934536	−	0.355868i	$-0.115815\pi$
$198$		0			0
$199$			1.47073i			0.104257i	0.998640	+	0.0521286i	$0.0166006\pi$
$199$			1.47073i			0.104257i	−0.998640	+	0.0521286i	$0.983399\pi$
$200$		0			0
$201$		−	9.08353i		−	0.640703i
$202$		0			0
$203$	5.99308			0.420631
$204$		0			0
$205$	−0.371838			−0.0259703
$206$		0			0
$207$		−	0.812015i		−	0.0564389i
$208$		0			0
$209$		−	1.32310i		−	0.0915208i
$210$		0			0
$211$			5.98113i			0.411758i	0.978577	+	0.205879i	$0.0660053\pi$
$211$			5.98113i			0.411758i	−0.978577	+	0.205879i	$0.933995\pi$
$212$		0			0
$213$	−24.3772			−1.67030
$214$		0			0
$215$		−	0.920735i		−	0.0627936i
$216$		0			0
$217$	−3.83779			−0.260526
$218$		0			0
$219$	−24.1976			−1.63512
$220$		0			0
$221$	1.52320	−	4.38093i	0.102461	−	0.294693i
$222$		0			0
$223$	−27.7405			−1.85764			−0.928822	−	0.370527i	$-0.879177\pi$
$223$	−27.7405			−1.85764			−0.928822	+	0.370527i	$0.879177\pi$
$224$		0			0
$225$	−1.60010			−0.106674
$226$		0			0
$227$		−	11.6734i		−	0.774793i	−0.921913	−	0.387396i	$-0.873375\pi$
$227$		−	11.6734i		−	0.774793i	0.921913	−	0.387396i	$-0.126625\pi$
$228$		0			0
$229$	−7.25514			−0.479433			−0.239717	−	0.970843i	$-0.577055\pi$
$229$	−7.25514			−0.479433			−0.239717	+	0.970843i	$0.577055\pi$
$230$		0			0
$231$		−	1.84048i		−	0.121094i
$232$		0			0
$233$			13.5248i			0.886038i	0.896512	+	0.443019i	$0.146092\pi$
$233$			13.5248i			0.886038i	−0.896512	+	0.443019i	$0.853908\pi$
$234$		0			0
$235$		−	0.0910339i		−	0.00593840i
$236$		0			0
$237$	3.02595			0.196557
$238$		0			0
$239$	2.89559			0.187300			0.0936501	−	0.995605i	$-0.470146\pi$
$239$	2.89559			0.187300			0.0936501	+	0.995605i	$0.470146\pi$
$240$		0			0
$241$			10.2515i			0.660356i	0.943919	+	0.330178i	$0.107109\pi$
$241$			10.2515i			0.660356i	−0.943919	+	0.330178i	$0.892891\pi$
$242$		0			0
$243$			3.36619i			0.215941i
$244$		0			0
$245$		−	1.87497i		−	0.119787i
$246$		0			0
$247$	−0.951584			−0.0605479
$248$		0			0
$249$		−	8.42727i		−	0.534057i
$250$		0			0
$251$	13.3824			0.844691			0.422345	−	0.906435i	$-0.361207\pi$
$251$	13.3824			0.844691			0.422345	+	0.906435i	$0.361207\pi$
$252$		0			0
$253$	−3.90437			−0.245466
$254$		0			0
$255$	−0.703197	+	2.02250i	−0.0440359	+	0.126654i
$256$		0			0
$257$	22.1868			1.38397			0.691987	−	0.721910i	$-0.256736\pi$
$257$	22.1868			1.38397			0.691987	+	0.721910i	$0.256736\pi$
$258$		0			0
$259$	−3.93280			−0.244372
$260$		0			0
$261$		−	3.02123i		−	0.187010i
$262$		0			0
$263$	−2.12333			−0.130930			−0.0654651	−	0.997855i	$-0.520853\pi$
$263$	−2.12333			−0.130930			−0.0654651	+	0.997855i	$0.520853\pi$
$264$		0			0
$265$			0.875847i			0.0538028i
$266$		0			0
$267$		−	2.41034i		−	0.147511i
$268$		0			0
$269$		−	12.8565i		−	0.783877i	−0.919991	−	0.391939i	$-0.871805\pi$
$269$		−	12.8565i		−	0.783877i	0.919991	−	0.391939i	$-0.128195\pi$
$270$		0			0
$271$	−25.5328			−1.55101			−0.775504	−	0.631342i	$-0.782504\pi$
$271$	−25.5328			−1.55101			−0.775504	+	0.631342i	$0.782504\pi$
$272$		0			0
$273$	−1.32368			−0.0801130
$274$		0			0
$275$			7.69371i			0.463948i
$276$		0			0
$277$		−	15.8770i		−	0.953954i	−0.878916	−	0.476977i	$-0.841733\pi$
$277$		−	15.8770i		−	0.953954i	0.878916	−	0.476977i	$-0.158267\pi$
$278$		0			0
$279$			1.93471i			0.115828i
$280$		0			0
$281$	27.1720			1.62095			0.810474	−	0.585775i	$-0.199210\pi$
$281$	27.1720			1.62095			0.810474	+	0.585775i	$0.199210\pi$
$282$		0			0
$283$		−	4.03060i		−	0.239594i	−0.992798	−	0.119797i	$-0.961776\pi$
$283$		−	4.03060i		−	0.239594i	0.992798	−	0.119797i	$-0.0382244\pi$
$284$		0			0
$285$	0.439308			0.0260224
$286$		0			0
$287$	−0.842503			−0.0497314
$288$		0			0
$289$	−13.3331	−	10.5465i	−0.784301	−	0.620380i
$290$		0			0
$291$	−11.5999			−0.679996
$292$		0			0
$293$	−1.89615			−0.110774			−0.0553870	−	0.998465i	$-0.517639\pi$
$293$	−1.89615			−0.110774			−0.0553870	+	0.998465i	$0.517639\pi$
$294$		0			0
$295$		−	0.284793i		−	0.0165813i
$296$		0			0
$297$	7.62885			0.442671
$298$		0			0
$299$			2.80805i			0.162394i
$300$		0			0
$301$		−	2.08618i		−	0.120245i
$302$		0			0
$303$		−	3.35107i		−	0.192514i
$304$		0			0
$305$	1.29709			0.0742712
$306$		0			0
$307$	4.01367			0.229072			0.114536	−	0.993419i	$-0.463462\pi$
$307$	4.01367			0.229072			0.114536	+	0.993419i	$0.463462\pi$
$308$		0			0
$309$			17.6285i			1.00285i
$310$		0			0
$311$			13.3227i			0.755464i	0.925915	+	0.377732i	$0.123296\pi$
$311$			13.3227i			0.755464i	−0.925915	+	0.377732i	$0.876704\pi$
$312$		0			0
$313$		−	22.7398i		−	1.28533i	−0.766147	−	0.642665i	$-0.777829\pi$
$313$		−	22.7398i		−	1.28533i	0.766147	−	0.642665i	$-0.222171\pi$
$314$		0			0
$315$	0.0597800			0.00336822
$316$		0			0
$317$			23.0591i			1.29513i	0.762010	+	0.647565i	$0.224213\pi$
$317$			23.0591i			1.29513i	−0.762010	+	0.647565i	$0.775787\pi$
$318$		0			0
$319$	−14.5269			−0.813348
$320$		0			0
$321$	28.8983			1.61294
$322$		0			0
$323$	−1.14540	+	3.29434i	−0.0637319	+	0.183302i
$324$		0			0
$325$	5.53337			0.306936
$326$		0			0
$327$	6.18080			0.341799
$328$		0			0
$329$		−	0.206263i		−	0.0113716i
$330$		0			0
$331$	−5.41739			−0.297767			−0.148883	−	0.988855i	$-0.547568\pi$
$331$	−5.41739			−0.297767			−0.148883	+	0.988855i	$0.547568\pi$
$332$		0			0
$333$			1.98261i			0.108646i
$334$		0			0
$335$		−	1.41863i		−	0.0775079i
$336$		0			0
$337$		−	0.00668662i		−	0.000364243i	−1.00000		0.000182122i	$-0.999942\pi$
$337$		−	0.00668662i		−	0.000364243i	1.00000		0.000182122i	$-5.79711e-5\pi$
$338$		0			0
$339$	−27.8452			−1.51234
$340$		0			0
$341$	9.30257			0.503763
$342$		0			0
$343$		−	8.76520i		−	0.473276i
$344$		0			0
$345$		−	1.29636i		−	0.0697939i
$346$		0			0
$347$		−	14.1057i		−	0.757236i	−0.925553	−	0.378618i	$-0.876399\pi$
$347$		−	14.1057i		−	0.757236i	0.925553	−	0.378618i	$-0.123601\pi$
$348$		0			0
$349$	23.9796			1.28360			0.641800	−	0.766872i	$-0.278188\pi$
$349$	23.9796			1.28360			0.641800	+	0.766872i	$0.278188\pi$
$350$		0			0
$351$		−	5.48672i		−	0.292860i
$352$		0			0
$353$	−14.8194			−0.788755			−0.394378	−	0.918948i	$-0.629040\pi$
$353$	−14.8194			−0.788755			−0.394378	+	0.918948i	$0.629040\pi$
$354$		0			0
$355$	−3.80713			−0.202062
$356$		0			0
$357$	−1.59329	+	4.58253i	−0.0843259	+	0.242533i
$358$		0			0
$359$	−14.1611			−0.747393			−0.373697	−	0.927551i	$-0.621910\pi$
$359$	−14.1611			−0.747393			−0.373697	+	0.927551i	$0.621910\pi$
$360$		0			0
$361$	−18.2844			−0.962339
$362$		0			0
$363$		−	15.5977i		−	0.818669i
$364$		0			0
$365$	−3.77908			−0.197806
$366$		0			0
$367$			13.8046i			0.720596i	0.932837	+	0.360298i	$0.117325\pi$
$367$			13.8046i			0.720596i	−0.932837	+	0.360298i	$0.882675\pi$
$368$		0			0
$369$			0.424723i			0.0221102i
$370$		0			0
$371$			1.98448i			0.103029i
$372$		0			0
$373$	34.7891			1.80131			0.900655	−	0.434534i	$-0.143087\pi$
$373$	34.7891			1.80131			0.900655	+	0.434534i	$0.143087\pi$
$374$		0			0
$375$	−5.15119			−0.266006
$376$		0			0
$377$			10.4478i			0.538090i
$378$		0			0
$379$		−	9.91747i		−	0.509426i	−0.967017	−	0.254713i	$-0.918019\pi$
$379$		−	9.91747i		−	0.509426i	0.967017	−	0.254713i	$-0.0819810\pi$
$380$		0			0
$381$		−	3.21788i		−	0.164857i
$382$		0			0
$383$	28.4705			1.45477			0.727386	−	0.686228i	$-0.240735\pi$
$383$	28.4705			1.45477			0.727386	+	0.686228i	$0.240735\pi$
$384$		0			0
$385$		−	0.287438i		−	0.0146492i
$386$		0			0
$387$	−1.05169			−0.0534602
$388$		0			0
$389$	−33.3610			−1.69147			−0.845734	−	0.533604i	$-0.820837\pi$
$389$	−33.3610			−1.69147			−0.845734	+	0.533604i	$0.820837\pi$
$390$		0			0
$391$	9.72135	+	3.38000i	0.491630	+	0.170934i
$392$		0			0
$393$	17.4649			0.880985
$394$		0			0
$395$	0.472580			0.0237781
$396$		0			0
$397$			13.4188i			0.673472i	0.941599	+	0.336736i	$0.109323\pi$
$397$			13.4188i			0.673472i	−0.941599	+	0.336736i	$0.890677\pi$
$398$		0			0
$399$	0.995375			0.0498311
$400$		0			0
$401$			35.3351i			1.76455i	0.470734	+	0.882275i	$0.343989\pi$
$401$			35.3351i			1.76455i	−0.470734	+	0.882275i	$0.656011\pi$
$402$		0			0
$403$		−	6.69048i		−	0.333277i
$404$		0			0
$405$			2.81093i			0.139676i
$406$		0			0
$407$	9.53287			0.472527
$408$		0			0
$409$	25.4471			1.25828			0.629139	−	0.777293i	$-0.283407\pi$
$409$	25.4471			1.25828			0.629139	+	0.777293i	$0.283407\pi$
$410$		0			0
$411$			12.0249i			0.593144i
$412$		0			0
$413$		−	0.645277i		−	0.0317520i
$414$		0			0
$415$		−	1.31614i		−	0.0646066i
$416$		0			0
$417$	−40.7193			−1.99403
$418$		0			0
$419$		−	25.0810i		−	1.22529i	−0.790360	−	0.612643i	$-0.790106\pi$
$419$		−	25.0810i		−	1.22529i	0.790360	−	0.612643i	$-0.209894\pi$
$420$		0			0
$421$	3.09469			0.150826			0.0754131	−	0.997152i	$-0.475972\pi$
$421$	3.09469			0.150826			0.0754131	+	0.997152i	$0.475972\pi$
$422$		0			0
$423$	−0.103981			−0.00505574
$424$		0			0
$425$	6.66040	−	19.1563i	0.323077	−	0.929216i
$426$		0			0
$427$	2.93892			0.142224
$428$		0			0
$429$	3.20853			0.154909
$430$		0			0
$431$		−	6.19055i		−	0.298189i	−0.988823	−	0.149094i	$-0.952364\pi$
$431$		−	6.19055i		−	0.298189i	0.988823	−	0.149094i	$-0.0476358\pi$
$432$		0			0
$433$	−2.71795			−0.130616			−0.0653081	−	0.997865i	$-0.520803\pi$
$433$	−2.71795			−0.130616			−0.0653081	+	0.997865i	$0.520803\pi$
$434$		0			0
$435$		−	4.82334i		−	0.231261i
$436$		0			0
$437$		−	2.11158i		−	0.101011i
$438$		0			0
$439$		−	19.8137i		−	0.945658i	−0.881154	−	0.472829i	$-0.843233\pi$
$439$		−	19.8137i		−	0.945658i	0.881154	−	0.472829i	$-0.156767\pi$
$440$		0			0
$441$	−2.14163			−0.101983
$442$		0			0
$443$	28.6289			1.36020			0.680101	−	0.733118i	$-0.261936\pi$
$443$	28.6289			1.36020			0.680101	+	0.733118i	$0.261936\pi$
$444$		0			0
$445$		−	0.376437i		−	0.0178448i
$446$		0			0
$447$		−	34.4836i		−	1.63102i
$448$		0			0
$449$			22.7766i			1.07489i	0.843298	+	0.537447i	$0.180611\pi$
$449$			22.7766i			1.07489i	−0.843298	+	0.537447i	$0.819389\pi$
$450$		0			0
$451$	2.04218			0.0961623
$452$		0			0
$453$			16.7949i			0.789092i
$454$		0			0
$455$	−0.206727			−0.00969153
$456$		0			0
$457$	−26.1381			−1.22269			−0.611345	−	0.791364i	$-0.709371\pi$
$457$	−26.1381			−1.22269			−0.611345	+	0.791364i	$0.709371\pi$
$458$		0			0
$459$	−18.9948	−	6.60426i	−0.886600	−	0.308260i
$460$		0			0
$461$	−26.1242			−1.21672			−0.608362	−	0.793660i	$-0.708173\pi$
$461$	−26.1242			−1.21672			−0.608362	+	0.793660i	$0.708173\pi$
$462$		0			0
$463$	−20.0320			−0.930964			−0.465482	−	0.885057i	$-0.654119\pi$
$463$	−20.0320			−0.930964			−0.465482	+	0.885057i	$0.654119\pi$
$464$		0			0
$465$			3.08872i			0.143236i
$466$		0			0
$467$	−21.7634			−1.00709			−0.503544	−	0.863970i	$-0.667971\pi$
$467$	−21.7634			−1.00709			−0.503544	+	0.863970i	$0.667971\pi$
$468$		0			0
$469$		−	3.21430i		−	0.148422i
$470$		0			0
$471$		−	23.7715i		−	1.09533i
$472$		0			0
$473$			5.05678i			0.232511i
$474$		0			0
$475$	−4.16095			−0.190917
$476$		0			0
$477$	1.00041			0.0458058
$478$		0			0
$479$			7.90053i			0.360984i	0.983576	+	0.180492i	$0.0577691\pi$
$479$			7.90053i			0.360984i	−0.983576	+	0.180492i	$0.942231\pi$
$480$		0			0
$481$		−	6.85611i		−	0.312612i
$482$		0			0
$483$		−	2.93728i		−	0.133651i
$484$		0			0
$485$	−1.81162			−0.0822613
$486$		0			0
$487$		−	12.6887i		−	0.574979i	−0.957784	−	0.287489i	$-0.907179\pi$
$487$		−	12.6887i		−	0.574979i	0.957784	−	0.287489i	$-0.0928206\pi$
$488$		0			0
$489$	−18.2259			−0.824202
$490$		0			0
$491$	27.8725			1.25787			0.628934	−	0.777459i	$-0.283492\pi$
$491$	27.8725			1.25787			0.628934	+	0.777459i	$0.283492\pi$
$492$		0			0
$493$	36.1699	+	12.5758i	1.62901	+	0.566387i
$494$		0			0
$495$	−0.144903			−0.00651291
$496$		0			0
$497$	−8.62613			−0.386935
$498$		0			0
$499$			18.1286i			0.811546i	0.913974	+	0.405773i	$0.132998\pi$
$499$			18.1286i			0.811546i	−0.913974	+	0.405773i	$0.867002\pi$
$500$		0			0
$501$	−27.0932			−1.21043
$502$		0			0
$503$			37.0824i			1.65342i	0.562628	+	0.826710i	$0.309790\pi$
$503$			37.0824i			1.65342i	−0.562628	+	0.826710i	$0.690210\pi$
$504$		0			0
$505$		−	0.523355i		−	0.0232890i
$506$		0			0
$507$			21.3984i			0.950337i
$508$		0			0
$509$	17.6485			0.782257			0.391129	−	0.920336i	$-0.372085\pi$
$509$	17.6485			0.782257			0.391129	+	0.920336i	$0.372085\pi$
$510$		0			0
$511$	−8.56255			−0.378785
$512$		0			0
$513$			4.12587i			0.182162i
$514$		0			0
$515$			2.75314i			0.121318i
$516$		0			0
$517$			0.499968i			0.0219886i
$518$		0			0
$519$	−12.0771			−0.530127
$520$		0			0
$521$			29.6418i			1.29863i	0.760520	+	0.649315i	$0.224944\pi$
$521$			29.6418i			1.29863i	−0.760520	+	0.649315i	$0.775056\pi$
$522$		0			0
$523$	31.1883			1.36377			0.681884	−	0.731460i	$-0.261161\pi$
$523$	31.1883			1.36377			0.681884	+	0.731460i	$0.261161\pi$
$524$		0			0
$525$	−5.78801			−0.252609
$526$		0			0
$527$	−23.1621	−	8.05319i	−1.00896	−	0.350802i
$528$		0			0
$529$	16.7689			0.729082
$530$		0			0
$531$	−0.325298			−0.0141167
$532$		0			0
$533$		−	1.46875i		−	0.0636185i
$534$		0			0
$535$	4.51321			0.195123
$536$		0			0
$537$			46.6945i			2.01502i
$538$		0			0
$539$			10.2975i			0.443546i
$540$		0			0
$541$		−	40.3276i		−	1.73382i	−0.498466	−	0.866909i	$-0.666103\pi$
$541$		−	40.3276i		−	1.73382i	0.498466	−	0.866909i	$-0.333897\pi$
$542$		0			0
$543$	28.1286			1.20711
$544$		0			0
$545$	0.965291			0.0413485
$546$		0			0
$547$		−	33.7138i		−	1.44150i	−0.693196	−	0.720749i	$-0.743798\pi$
$547$		−	33.7138i		−	1.44150i	0.693196	−	0.720749i	$-0.256202\pi$
$548$		0			0
$549$		−	1.48157i		−	0.0632319i
$550$		0			0
$551$		−	7.85648i		−	0.334697i
$552$		0			0
$553$	1.07076			0.0455334
$554$		0			0
$555$			3.16519i			0.134355i
$556$		0			0
$557$	−2.16460			−0.0917171			−0.0458586	−	0.998948i	$-0.514602\pi$
$557$	−2.16460			−0.0917171			−0.0458586	+	0.998948i	$0.514602\pi$
$558$		0			0
$559$	3.63687			0.153823
$560$		0			0
$561$	3.86204	−	11.1078i	0.163055	−	0.468971i
$562$		0			0
$563$	28.2368			1.19004			0.595020	−	0.803711i	$-0.297144\pi$
$563$	28.2368			1.19004			0.595020	+	0.803711i	$0.297144\pi$
$564$		0			0
$565$	−4.34874			−0.182953
$566$		0			0
$567$			6.36894i			0.267470i
$568$		0			0
$569$	−1.10346			−0.0462596			−0.0231298	−	0.999732i	$-0.507363\pi$
$569$	−1.10346			−0.0462596			−0.0231298	+	0.999732i	$0.507363\pi$
$570$		0			0
$571$			25.9942i			1.08782i	0.839143	+	0.543911i	$0.183057\pi$
$571$			25.9942i			1.08782i	−0.839143	+	0.543911i	$0.816943\pi$
$572$		0			0
$573$			36.8888i			1.54105i
$574$		0			0
$575$			12.2786i			0.512055i
$576$		0			0
$577$	13.6580			0.568588			0.284294	−	0.958737i	$-0.408241\pi$
$577$	13.6580			0.568588			0.284294	+	0.958737i	$0.408241\pi$
$578$		0			0
$579$	25.9258			1.07744
$580$		0			0
$581$		−	2.98207i		−	0.123717i
$582$		0			0
$583$		−	4.81025i		−	0.199220i
$584$		0			0
$585$			0.104215i			0.00430878i
$586$		0			0
$587$	23.2472			0.959514			0.479757	−	0.877401i	$-0.340725\pi$
$587$	23.2472			0.959514			0.479757	+	0.877401i	$0.340725\pi$
$588$		0			0
$589$			5.03106i			0.207301i
$590$		0			0
$591$	−18.2166			−0.749330
$592$		0			0
$593$	−39.9531			−1.64068			−0.820338	−	0.571879i	$-0.806215\pi$
$593$	−39.9531			−1.64068			−0.820338	+	0.571879i	$0.806215\pi$
$594$		0			0
$595$	−0.248833	+	0.715680i	−0.0102012	+	0.0293400i
$596$		0			0
$597$	2.68193			0.109764
$598$		0			0
$599$	46.7706			1.91100			0.955498	−	0.294998i	$-0.0953189\pi$
$599$	46.7706			1.91100			0.955498	+	0.294998i	$0.0953189\pi$
$600$		0			0
$601$			12.0432i			0.491252i	0.969365	+	0.245626i	$0.0789935\pi$
$601$			12.0432i			0.491252i	−0.969365	+	0.245626i	$0.921007\pi$
$602$		0			0
$603$	−1.62039			−0.0659875
$604$		0			0
$605$		−	2.43599i		−	0.0990370i
$606$		0			0
$607$			10.1246i			0.410945i	0.978663	+	0.205472i	$0.0658730\pi$
$607$			10.1246i			0.410945i	−0.978663	+	0.205472i	$0.934127\pi$
$608$		0			0
$609$		−	10.9286i		−	0.442850i
$610$		0			0
$611$	0.359581			0.0145471
$612$		0			0
$613$	3.63880			0.146970			0.0734848	−	0.997296i	$-0.476588\pi$
$613$	3.63880			0.146970			0.0734848	+	0.997296i	$0.476588\pi$
$614$		0			0
$615$			0.678061i			0.0273421i
$616$		0			0
$617$		−	17.4331i		−	0.701829i	−0.936407	−	0.350915i	$-0.885871\pi$
$617$		−	17.4331i		−	0.701829i	0.936407	−	0.350915i	$-0.114129\pi$
$618$		0			0
$619$		−	12.8289i		−	0.515638i	−0.966193	−	0.257819i	$-0.916996\pi$
$619$		−	12.8289i		−	0.515638i	0.966193	−	0.257819i	$-0.0830038\pi$
$620$		0			0
$621$	12.1751			0.488571
$622$		0			0
$623$		−	0.852924i		−	0.0341717i
$624$		0			0
$625$	23.7900			0.951599
$626$		0			0
$627$	−2.41273			−0.0963551
$628$		0			0
$629$	−23.7355	−	8.25256i	−0.946398	−	0.329051i
$630$		0			0
$631$	22.7859			0.907091			0.453546	−	0.891233i	$-0.350159\pi$
$631$	22.7859			0.907091			0.453546	+	0.891233i	$0.350159\pi$
$632$		0			0
$633$	10.9068			0.433507
$634$		0			0
$635$		−	0.502556i		−	0.0199433i
$636$		0			0
$637$	7.40606			0.293439
$638$		0			0
$639$			4.34861i			0.172028i
$640$		0			0
$641$			28.0217i			1.10679i	0.832919	+	0.553396i	$0.186668\pi$
$641$			28.0217i			1.10679i	−0.832919	+	0.553396i	$0.813332\pi$
$642$		0			0
$643$		−	19.2975i		−	0.761018i	−0.924777	−	0.380509i	$-0.875749\pi$
$643$		−	19.2975i		−	0.761018i	0.924777	−	0.380509i	$-0.124251\pi$
$644$		0			0
$645$	−1.67900			−0.0661104
$646$		0			0
$647$	−19.3721			−0.761597			−0.380798	−	0.924658i	$-0.624351\pi$
$647$	−19.3721			−0.761597			−0.380798	+	0.924658i	$0.624351\pi$
$648$		0			0
$649$			1.56411i			0.0613968i
$650$		0			0
$651$			6.99836i			0.274288i
$652$		0			0
$653$		−	11.8464i		−	0.463587i	−0.972765	−	0.231794i	$-0.925541\pi$
$653$		−	11.8464i		−	0.463587i	0.972765	−	0.231794i	$-0.0744594\pi$
$654$		0			0
$655$	2.72759			0.106576
$656$		0			0
$657$			4.31656i			0.168405i
$658$		0			0
$659$	−11.9559			−0.465737			−0.232869	−	0.972508i	$-0.574811\pi$
$659$	−11.9559			−0.465737			−0.232869	+	0.972508i	$0.574811\pi$
$660$		0			0
$661$	18.9669			0.737727			0.368863	−	0.929484i	$-0.379747\pi$
$661$	18.9669			0.737727			0.368863	+	0.929484i	$0.379747\pi$
$662$		0			0
$663$	−7.98880	−	2.77761i	−0.310259	−	0.107873i
$664$		0			0
$665$	0.155453			0.00602823
$666$		0			0
$667$	−23.1839			−0.897684
$668$		0			0
$669$			50.5860i			1.95577i
$670$		0			0
$671$	−7.12377			−0.275010
$672$		0			0
$673$			25.6384i			0.988287i	0.869380	+	0.494144i	$0.164518\pi$
$673$			25.6384i			0.988287i	−0.869380	+	0.494144i	$0.835482\pi$
$674$		0			0
$675$		−	23.9915i		−	0.923435i
$676$		0			0
$677$			5.12360i			0.196916i	0.995141	+	0.0984580i	$0.0313910\pi$
$677$			5.12360i			0.196916i	−0.995141	+	0.0984580i	$0.968609\pi$
$678$		0			0
$679$	−4.10473			−0.157525
$680$		0			0
$681$	−21.2870			−0.815718
$682$		0			0
$683$			23.8909i			0.914162i	0.889425	+	0.457081i	$0.151105\pi$
$683$			23.8909i			0.914162i	−0.889425	+	0.457081i	$0.848895\pi$
$684$		0			0
$685$			1.87800i			0.0717546i
$686$		0			0
$687$			13.2300i			0.504758i
$688$		0			0
$689$	−3.45957			−0.131799
$690$		0			0
$691$		−	4.91390i		−	0.186934i	−0.995622	−	0.0934668i	$-0.970205\pi$
$691$		−	4.91390i		−	0.186934i	0.995622	−	0.0934668i	$-0.0297949\pi$
$692$		0			0
$693$	−0.328319			−0.0124718
$694$		0			0
$695$	−6.35937			−0.241225
$696$		0			0
$697$	−5.08474	−	1.76790i	−0.192598	−	0.0669640i
$698$		0			0
$699$	24.6630			0.932839
$700$		0			0
$701$	35.7046			1.34855			0.674273	−	0.738482i	$-0.264457\pi$
$701$	35.7046			1.34855			0.674273	+	0.738482i	$0.264457\pi$
$702$		0			0
$703$			5.15561i			0.194448i
$704$		0			0
$705$	−0.166004			−0.00625207
$706$		0			0
$707$		−	1.18581i		−	0.0445969i
$708$		0			0
$709$		−	6.96839i		−	0.261703i	−0.991402	−	0.130852i	$-0.958229\pi$
$709$		−	6.96839i		−	0.261703i	0.991402	−	0.130852i	$-0.0417712\pi$
$710$		0			0
$711$		−	0.539793i		−	0.0202438i
$712$		0			0
$713$	14.8463			0.555998
$714$		0			0
$715$	0.501094			0.0187399
$716$		0			0
$717$		−	5.28023i		−	0.197194i
$718$		0			0
$719$		−	21.3544i		−	0.796383i	−0.917302	−	0.398192i	$-0.869638\pi$
$719$		−	21.3544i		−	0.796383i	0.917302	−	0.398192i	$-0.130362\pi$
$720$		0			0
$721$			6.23801i			0.232316i
$722$		0			0
$723$	18.6940			0.695237
$724$		0			0
$725$			45.6847i			1.69669i
$726$		0			0
$727$	17.0508			0.632378			0.316189	−	0.948696i	$-0.397597\pi$
$727$	17.0508			0.632378			0.316189	+	0.948696i	$0.397597\pi$
$728$		0			0
$729$	−23.4718			−0.869327
$730$		0			0
$731$	4.37763	−	12.5907i	0.161912	−	0.465683i
$732$		0			0
$733$	32.8717			1.21414			0.607072	−	0.794647i	$-0.292344\pi$
$733$	32.8717			1.21414			0.607072	+	0.794647i	$0.292344\pi$
$734$		0			0
$735$	−3.41908			−0.126115
$736$		0			0
$737$			7.79126i			0.286995i
$738$		0			0
$739$	9.33927			0.343551			0.171775	−	0.985136i	$-0.445050\pi$
$739$	9.33927			0.343551			0.171775	+	0.985136i	$0.445050\pi$
$740$		0			0
$741$			1.73525i			0.0637461i
$742$		0			0
$743$			13.7678i			0.505092i	0.967585	+	0.252546i	$0.0812679\pi$
$743$			13.7678i			0.505092i	−0.967585	+	0.252546i	$0.918732\pi$
$744$		0			0
$745$		−	5.38551i		−	0.197310i
$746$		0			0
$747$	−1.50332			−0.0550037
$748$		0			0
$749$	10.2259			0.373648
$750$		0			0
$751$			17.6064i			0.642467i	0.947000	+	0.321233i	$0.104097\pi$
$751$			17.6064i			0.642467i	−0.947000	+	0.321233i	$0.895903\pi$
$752$		0			0
$753$		−	24.4034i		−	0.889308i
$754$		0			0
$755$			2.62295i			0.0954590i
$756$		0			0
$757$	22.4997			0.817764			0.408882	−	0.912587i	$-0.365919\pi$
$757$	22.4997			0.817764			0.408882	+	0.912587i	$0.365919\pi$
$758$		0			0
$759$			7.11978i			0.258432i
$760$		0			0
$761$	24.3751			0.883597			0.441798	−	0.897114i	$-0.354341\pi$
$761$	24.3751			0.883597			0.441798	+	0.897114i	$0.354341\pi$
$762$		0			0
$763$	2.18714			0.0791796
$764$		0			0
$765$	0.360789	+	0.125442i	0.0130444	+	0.00453536i
$766$		0			0
$767$	1.12492			0.0406186
$768$		0			0
$769$	29.6146			1.06793			0.533965	−	0.845507i	$-0.320702\pi$
$769$	29.6146			1.06793			0.533965	+	0.845507i	$0.320702\pi$
$770$		0			0
$771$		−	40.4585i		−	1.45708i
$772$		0			0
$773$	48.6428			1.74956			0.874781	−	0.484519i	$-0.161005\pi$
$773$	48.6428			1.74956			0.874781	+	0.484519i	$0.161005\pi$
$774$		0			0
$775$		−	29.2551i		−	1.05088i
$776$		0			0
$777$			7.17162i			0.257280i
$778$		0			0
$779$			1.10446i			0.0395714i
$780$		0			0
$781$	20.9092			0.748190
$782$		0			0
$783$	45.2996			1.61887
$784$		0			0
$785$		−	3.71253i		−	0.132506i
$786$		0			0
$787$		−	13.3538i		−	0.476012i	−0.971264	−	0.238006i	$-0.923506\pi$
$787$		−	13.3538i		−	0.476012i	0.971264	−	0.238006i	$-0.0764938\pi$
$788$		0			0
$789$			3.87198i			0.137846i
$790$		0			0
$791$	−9.85328			−0.350342
$792$		0			0
$793$			5.12347i			0.181940i
$794$		0			0
$795$	1.59714			0.0566448
$796$		0			0
$797$	17.0532			0.604056			0.302028	−	0.953299i	$-0.402336\pi$
$797$	17.0532			0.604056			0.302028	+	0.953299i	$0.402336\pi$
$798$		0			0
$799$	0.432820	−	1.24485i	0.0153121	−	0.0440397i
$800$		0			0
$801$	−0.429976			−0.0151925
$802$		0			0
$803$	20.7551			0.732432
$804$		0			0
$805$		−	0.458731i		−	0.0161682i
$806$		0			0
$807$	−23.4444			−0.825283
$808$		0			0
$809$		−	20.5820i		−	0.723626i	−0.932251	−	0.361813i	$-0.882158\pi$
$809$		−	20.5820i		−	0.723626i	0.932251	−	0.361813i	$-0.117842\pi$
$810$		0			0
$811$			40.3759i			1.41779i	0.705315	+	0.708894i	$0.250806\pi$
$811$			40.3759i			1.41779i	−0.705315	+	0.708894i	$0.749194\pi$
$812$		0			0
$813$			46.5601i			1.63294i
$814$		0			0
$815$	−2.84644			−0.0997064
$816$		0			0
$817$	−2.73483			−0.0956796
$818$		0			0
$819$			0.236129i			0.00825102i
$820$		0			0
$821$			35.5729i			1.24150i	0.784007	+	0.620752i	$0.213173\pi$
$821$			35.5729i			1.24150i	−0.784007	+	0.620752i	$0.786827\pi$
$822$		0			0
$823$		−	28.4197i		−	0.990648i	−0.868708	−	0.495324i	$-0.835049\pi$
$823$		−	28.4197i		−	0.990648i	0.868708	−	0.495324i	$-0.164951\pi$
$824$		0			0
$825$	14.0298			0.488454
$826$		0			0
$827$			17.5000i			0.608534i	0.952587	+	0.304267i	$0.0984116\pi$
$827$			17.5000i			0.608534i	−0.952587	+	0.304267i	$0.901588\pi$
$828$		0			0
$829$	21.8924			0.760354			0.380177	−	0.924914i	$-0.375863\pi$
$829$	21.8924			0.760354			0.380177	+	0.924914i	$0.375863\pi$
$830$		0			0
$831$	−28.9523			−1.00434
$832$		0			0
$833$	8.91451	−	25.6394i	0.308870	−	0.888353i
$834$		0			0
$835$	−4.23130			−0.146430
$836$		0			0
$837$	−29.0085			−1.00268
$838$		0			0
$839$			40.6379i			1.40298i	0.712681	+	0.701489i	$0.247481\pi$
$839$			40.6379i			1.40298i	−0.712681	+	0.701489i	$0.752519\pi$
$840$		0			0
$841$	−57.2594			−1.97446
$842$		0			0
$843$		−	49.5493i		−	1.70657i
$844$		0			0
$845$			3.34191i			0.114965i
$846$		0			0
$847$		−	5.51941i		−	0.189649i
$848$		0			0
$849$	−7.34995			−0.252250
$850$		0			0
$851$	15.2138			0.521523
$852$		0			0
$853$			22.8928i			0.783835i	0.920000	+	0.391918i	$0.128188\pi$
$853$			22.8928i			0.783835i	−0.920000	+	0.391918i	$0.871812\pi$
$854$		0			0
$855$		−	0.0783672i		−	0.00268010i
$856$		0			0
$857$		−	8.50036i		−	0.290367i	−0.989405	−	0.145183i	$-0.953623\pi$
$857$		−	8.50036i		−	0.290367i	0.989405	−	0.145183i	$-0.0463772\pi$
$858$		0			0
$859$	−49.9059			−1.70277			−0.851383	−	0.524545i	$-0.824236\pi$
$859$	−49.9059			−1.70277			−0.851383	+	0.524545i	$0.824236\pi$
$860$		0			0
$861$			1.53634i			0.0523582i
$862$		0			0
$863$	−24.0287			−0.817945			−0.408972	−	0.912547i	$-0.634113\pi$
$863$	−24.0287			−0.817945			−0.408972	+	0.912547i	$0.634113\pi$
$864$		0			0
$865$	−1.88615			−0.0641312
$866$		0			0
$867$	−19.2319	+	24.3135i	−0.653149	+	0.825729i
$868$		0			0
$869$	−2.59546			−0.0880451
$870$		0			0
$871$	5.60353			0.189868
$872$		0			0
$873$			2.06928i			0.0700344i
$874$		0			0
$875$	−1.82280			−0.0616218
$876$		0			0
$877$		−	27.1739i		−	0.917598i	−0.888540	−	0.458799i	$-0.848280\pi$
$877$		−	27.1739i		−	0.917598i	0.888540	−	0.458799i	$-0.151720\pi$
$878$		0			0
$879$			3.45770i			0.116625i
$880$		0			0
$881$		−	6.53835i		−	0.220283i	−0.993916	−	0.110141i	$-0.964870\pi$
$881$		−	6.53835i		−	0.220283i	0.993916	−	0.110141i	$-0.0351303\pi$
$882$		0			0
$883$	−14.8594			−0.500060			−0.250030	−	0.968238i	$-0.580440\pi$
$883$	−14.8594			−0.500060			−0.250030	+	0.968238i	$0.580440\pi$
$884$		0			0
$885$	−0.519331			−0.0174571
$886$		0			0
$887$			29.2486i			0.982070i	0.871140	+	0.491035i	$0.163381\pi$
$887$			29.2486i			0.982070i	−0.871140	+	0.491035i	$0.836619\pi$
$888$		0			0
$889$		−	1.13868i		−	0.0381901i
$890$		0			0
$891$		−	15.4379i		−	0.517190i
$892$		0			0
$893$	−0.270395			−0.00904843
$894$		0			0
$895$			7.29255i			0.243763i
$896$		0			0
$897$	5.12060			0.170972
$898$		0			0
$899$	55.2380			1.84229
$900$		0			0
$901$	−4.16421	+	11.9769i	−0.138730	+	0.399007i
$902$		0			0
$903$	−3.80423			−0.126597
$904$		0			0
$905$	4.39300			0.146028
$906$		0			0
$907$			39.1311i			1.29933i	0.760222	+	0.649664i	$0.225090\pi$
$907$			39.1311i			1.29933i	−0.760222	+	0.649664i	$0.774910\pi$
$908$		0			0
$909$	−0.597790			−0.0198274
$910$		0			0
$911$			27.5613i			0.913146i	0.889686	+	0.456573i	$0.150923\pi$
$911$			27.5613i			0.913146i	−0.889686	+	0.456573i	$0.849077\pi$
$912$		0			0
$913$			7.22836i			0.239224i
$914$		0			0
$915$		−	2.36530i		−	0.0781943i
$916$		0			0
$917$	6.18011			0.204085
$918$		0			0
$919$	11.4456			0.377557			0.188778	−	0.982020i	$-0.439547\pi$
$919$	11.4456			0.377557			0.188778	+	0.982020i	$0.439547\pi$
$920$		0			0
$921$		−	7.31909i		−	0.241172i
$922$		0			0
$923$		−	15.0381i		−	0.494984i
$924$		0			0
$925$		−	29.9794i		−	0.985716i
$926$		0			0
$927$	3.14471			0.103286
$928$		0			0
$929$			22.6214i			0.742185i	0.928596	+	0.371092i	$0.121017\pi$
$929$			22.6214i			0.742185i	−0.928596	+	0.371092i	$0.878983\pi$
$930$		0			0
$931$	−5.56916			−0.182522
$932$		0			0
$933$	24.2946			0.795368
$934$		0			0
$935$	0.603157	−	1.73477i	0.0197253	−	0.0567329i
$936$		0			0
$937$	−16.4168			−0.536313			−0.268156	−	0.963375i	$-0.586414\pi$
$937$	−16.4168			−0.536313			−0.268156	+	0.963375i	$0.586414\pi$
$938$		0			0
$939$	−41.4669			−1.35322
$940$		0			0
$941$			25.1873i			0.821083i	0.911842	+	0.410541i	$0.134660\pi$
$941$			25.1873i			0.821083i	−0.911842	+	0.410541i	$0.865340\pi$
$942$		0			0
$943$	3.25918			0.106133
$944$		0			0
$945$			0.896326i			0.0291575i
$946$		0			0
$947$			44.7188i			1.45317i	0.687079	+	0.726583i	$0.258893\pi$
$947$			44.7188i			1.45317i	−0.687079	+	0.726583i	$0.741107\pi$
$948$		0			0
$949$		−	14.9272i		−	0.484558i
$950$		0			0
$951$	42.0493			1.36354
$952$		0			0
$953$	31.2106			1.01101			0.505505	−	0.862824i	$-0.331306\pi$
$953$	31.2106			1.01101			0.505505	+	0.862824i	$0.331306\pi$
$954$		0			0
$955$			5.76113i			0.186426i
$956$		0			0
$957$			26.4903i			0.856310i
$958$		0			0
$959$			4.25513i			0.137405i
$960$		0			0
$961$	−4.37281			−0.141059
$962$		0			0
$963$		−	5.15510i		−	0.166121i
$964$		0			0
$965$	4.04899			0.130341
$966$		0			0
$967$	−38.3363			−1.23281			−0.616405	−	0.787429i	$-0.711412\pi$
$967$	−38.3363			−1.23281			−0.616405	+	0.787429i	$0.711412\pi$
$968$		0			0
$969$	6.00736	+	2.08869i	0.192984	+	0.0670983i
$970$		0			0
$971$	2.73616			0.0878077			0.0439039	−	0.999036i	$-0.486020\pi$
$971$	2.73616			0.0878077			0.0439039	+	0.999036i	$0.486020\pi$
$972$		0			0
$973$	−14.4089			−0.461929
$974$		0			0
$975$		−	10.0903i		−	0.323149i
$976$		0			0
$977$	5.74239			0.183715			0.0918577	−	0.995772i	$-0.470720\pi$
$977$	5.74239			0.183715			0.0918577	+	0.995772i	$0.470720\pi$
$978$		0			0
$979$			2.06744i			0.0660756i
$980$		0			0
$981$		−	1.10258i		−	0.0352027i
$982$		0			0
$983$		−	3.59156i		−	0.114553i	−0.998358	−	0.0572764i	$-0.981758\pi$
$983$		−	3.59156i		−	0.114553i	0.998358	−	0.0572764i	$-0.0182416\pi$
$984$		0			0
$985$	−2.84499			−0.0906489
$986$		0			0
$987$	−0.376128			−0.0119723
$988$		0			0
$989$			8.07028i			0.256620i
$990$		0			0
$991$			19.4482i			0.617792i	0.951096	+	0.308896i	$0.0999595\pi$
$991$			19.4482i			0.617792i	−0.951096	+	0.308896i	$0.900040\pi$
$992$		0			0
$993$			9.87882i			0.313495i
$994$		0			0
$995$	0.418853			0.0132785
$996$		0			0
$997$			21.8618i			0.692369i	0.938167	+	0.346184i	$0.112523\pi$
$997$			21.8618i			0.692369i	−0.938167	+	0.346184i	$0.887477\pi$
$998$		0			0
$999$	−29.7267			−0.940510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.10	✓	46
17.16	even	2	inner	4012.2.b.b.237.37	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.10	✓	46	1.1	even	1	trivial
4012.2.b.b.237.37	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)

\(f(q)\)	\(=\)	\(q-1.82354i q^{3} -0.284793i q^{5} -0.645277i q^{7} -0.325298 q^{9} +O(q^{10})\)	\(q-1.82354i q^{3} -0.284793i q^{5} -0.645277i q^{7} -0.325298 q^{9} +1.56411i q^{11} +1.12492 q^{13} -0.519331 q^{15} +(1.35405 - 3.89443i) q^{17} -0.845911 q^{19} -1.17669 q^{21} +2.49622i q^{23} +4.91889 q^{25} -4.87743i q^{27} +9.28760i q^{29} -5.94750i q^{31} +2.85222 q^{33} -0.183770 q^{35} -6.09474i q^{37} -2.05134i q^{39} -1.30564i q^{41} +3.23300 q^{43} +0.0926424i q^{45} +0.319650 q^{47} +6.58362 q^{49} +(-7.10164 - 2.46916i) q^{51} -3.07538 q^{53} +0.445448 q^{55} +1.54255i q^{57} +1.00000 q^{59} +4.55451i q^{61} +0.209907i q^{63} -0.320370i q^{65} +4.98126 q^{67} +4.55196 q^{69} -13.3681i q^{71} -13.2696i q^{73} -8.96980i q^{75} +1.00929 q^{77} +1.65938i q^{79} -9.87007 q^{81} +4.62138 q^{83} +(-1.10910 - 0.385622i) q^{85} +16.9363 q^{87} +1.32179 q^{89} -0.725887i q^{91} -10.8455 q^{93} +0.240909i q^{95} -6.36118i q^{97} -0.508802i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 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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