LMFDB - Embedded newform 4012.2.b.a.237.12

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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.12
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.51601i q^{3} -1.74989i q^{5} -0.991299i q^{7} +0.701704 q^{9} +O(q^{10})$	$q-1.51601i q^{3} -1.74989i q^{5} -0.991299i q^{7} +0.701704 q^{9} +3.70237i q^{11} +3.69813 q^{13} -2.65285 q^{15} +(1.53378 + 3.82721i) q^{17} +0.316747 q^{19} -1.50282 q^{21} -5.72145i q^{23} +1.93789 q^{25} -5.61183i q^{27} +7.42574i q^{29} -6.07034i q^{31} +5.61284 q^{33} -1.73466 q^{35} +5.20586i q^{37} -5.60641i q^{39} +1.04527i q^{41} +0.358632 q^{43} -1.22790i q^{45} +5.00248 q^{47} +6.01733 q^{49} +(5.80210 - 2.32523i) q^{51} +8.46622 q^{53} +6.47873 q^{55} -0.480193i q^{57} -1.00000 q^{59} +7.21516i q^{61} -0.695599i q^{63} -6.47131i q^{65} +8.81340 q^{67} -8.67380 q^{69} +9.43765i q^{71} +2.28162i q^{73} -2.93787i q^{75} +3.67016 q^{77} +10.9138i q^{79} -6.40250 q^{81} -0.243518 q^{83} +(6.69719 - 2.68395i) q^{85} +11.2575 q^{87} -0.105838 q^{89} -3.66595i q^{91} -9.20272 q^{93} -0.554273i q^{95} -11.1656i q^{97} +2.59797i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * 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
$3$		−	1.51601i		−	0.875271i	−0.899153	−	0.437635i	$-0.855816\pi$
$3$		−	1.51601i		−	0.875271i	0.899153	−	0.437635i	$-0.144184\pi$
$4$		0			0
$5$		−	1.74989i		−	0.782574i	−0.920269	−	0.391287i	$-0.872030\pi$
$5$		−	1.74989i		−	0.782574i	0.920269	−	0.391287i	$-0.127970\pi$
$6$		0			0
$7$		−	0.991299i		−	0.374676i	−0.982296	−	0.187338i	$-0.940014\pi$
$7$		−	0.991299i		−	0.374676i	0.982296	−	0.187338i	$-0.0599860\pi$
$8$		0			0
$9$	0.701704			0.233901
$10$		0			0
$11$			3.70237i			1.11631i	0.829738	+	0.558153i	$0.188490\pi$
$11$			3.70237i			1.11631i	−0.829738	+	0.558153i	$0.811510\pi$
$12$		0			0
$13$	3.69813			1.02568			0.512838	−	0.858486i	$-0.328594\pi$
$13$	3.69813			1.02568			0.512838	+	0.858486i	$0.328594\pi$
$14$		0			0
$15$	−2.65285			−0.684964
$16$		0			0
$17$	1.53378	+	3.82721i	0.371997	+	0.928234i
$17$	1.53378	+	3.82721i	0.371997	+	0.928234i
$18$		0			0
$19$	0.316747			0.0726668			0.0363334	−	0.999340i	$-0.488432\pi$
$19$	0.316747			0.0726668			0.0363334	+	0.999340i	$0.488432\pi$
$20$		0			0
$21$	−1.50282			−0.327943
$22$		0			0
$23$		−	5.72145i		−	1.19301i	−0.802611	−	0.596503i	$-0.796557\pi$
$23$		−	5.72145i		−	1.19301i	0.802611	−	0.596503i	$-0.203443\pi$
$24$		0			0
$25$	1.93789			0.387578
$26$		0			0
$27$		−	5.61183i		−	1.08000i
$28$		0			0
$29$			7.42574i			1.37893i	0.724321	+	0.689463i	$0.242153\pi$
$29$			7.42574i			1.37893i	−0.724321	+	0.689463i	$0.757847\pi$
$30$		0			0
$31$		−	6.07034i		−	1.09027i	−0.838350	−	0.545133i	$-0.816479\pi$
$31$		−	6.07034i		−	1.09027i	0.838350	−	0.545133i	$-0.183521\pi$
$32$		0			0
$33$	5.61284			0.977070
$34$		0			0
$35$	−1.73466			−0.293212
$36$		0			0
$37$			5.20586i			0.855837i	0.903817	+	0.427919i	$0.140753\pi$
$37$			5.20586i			0.855837i	−0.903817	+	0.427919i	$0.859247\pi$
$38$		0			0
$39$		−	5.60641i		−	0.897744i
$40$		0			0
$41$			1.04527i			0.163243i	0.996663	+	0.0816215i	$0.0260099\pi$
$41$			1.04527i			0.163243i	−0.996663	+	0.0816215i	$0.973990\pi$
$42$		0			0
$43$	0.358632			0.0546909			0.0273455	−	0.999626i	$-0.491295\pi$
$43$	0.358632			0.0546909			0.0273455	+	0.999626i	$0.491295\pi$
$44$		0			0
$45$		−	1.22790i		−	0.183045i
$46$		0			0
$47$	5.00248			0.729687			0.364843	−	0.931069i	$-0.381123\pi$
$47$	5.00248			0.729687			0.364843	+	0.931069i	$0.381123\pi$
$48$		0			0
$49$	6.01733			0.859618
$50$		0			0
$51$	5.80210	−	2.32523i	0.812456	−	0.325598i
$52$		0			0
$53$	8.46622			1.16292			0.581462	−	0.813574i	$-0.302481\pi$
$53$	8.46622			1.16292			0.581462	+	0.813574i	$0.302481\pi$
$54$		0			0
$55$	6.47873			0.873592
$56$		0			0
$57$		−	0.480193i		−	0.0636031i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			7.21516i			0.923806i	0.886930	+	0.461903i	$0.152833\pi$
$61$			7.21516i			0.923806i	−0.886930	+	0.461903i	$0.847167\pi$
$62$		0			0
$63$		−	0.695599i		−	0.0876372i
$64$		0			0
$65$		−	6.47131i		−	0.802667i
$66$		0			0
$67$	8.81340			1.07673			0.538364	−	0.842712i	$-0.319042\pi$
$67$	8.81340			1.07673			0.538364	+	0.842712i	$0.319042\pi$
$68$		0			0
$69$	−8.67380			−1.04420
$70$		0			0
$71$			9.43765i			1.12004i	0.828478	+	0.560022i	$0.189207\pi$
$71$			9.43765i			1.12004i	−0.828478	+	0.560022i	$0.810793\pi$
$72$		0			0
$73$			2.28162i			0.267044i	0.991046	+	0.133522i	$0.0426287\pi$
$73$			2.28162i			0.267044i	−0.991046	+	0.133522i	$0.957371\pi$
$74$		0			0
$75$		−	2.93787i		−	0.339236i
$76$		0			0
$77$	3.67016			0.418253
$78$		0			0
$79$			10.9138i			1.22790i	0.789344	+	0.613951i	$0.210421\pi$
$79$			10.9138i			1.22790i	−0.789344	+	0.613951i	$0.789579\pi$
$80$		0			0
$81$	−6.40250			−0.711389
$82$		0			0
$83$	−0.243518			−0.0267295			−0.0133648	−	0.999911i	$-0.504254\pi$
$83$	−0.243518			−0.0267295			−0.0133648	+	0.999911i	$0.504254\pi$
$84$		0			0
$85$	6.69719	−	2.68395i	0.726412	−	0.291115i
$86$		0			0
$87$	11.2575			1.20693
$88$		0			0
$89$	−0.105838			−0.0112188			−0.00560939	−	0.999984i	$-0.501786\pi$
$89$	−0.105838			−0.0112188			−0.00560939	+	0.999984i	$0.501786\pi$
$90$		0			0
$91$		−	3.66595i		−	0.384296i
$92$		0			0
$93$	−9.20272			−0.954278
$94$		0			0
$95$		−	0.554273i		−	0.0568672i
$96$		0			0
$97$		−	11.1656i		−	1.13370i	−0.823822	−	0.566848i	$-0.808163\pi$
$97$		−	11.1656i		−	1.13370i	0.823822	−	0.566848i	$-0.191837\pi$
$98$		0			0
$99$			2.59797i			0.261106i
$100$		0			0
$101$	−7.41491			−0.737811			−0.368905	−	0.929467i	$-0.620267\pi$
$101$	−7.41491			−0.737811			−0.368905	+	0.929467i	$0.620267\pi$
$102$		0			0
$103$	17.3517			1.70972			0.854858	−	0.518861i	$-0.173644\pi$
$103$	17.3517			1.70972			0.854858	+	0.518861i	$0.173644\pi$
$104$		0			0
$105$			2.62977i			0.256639i
$106$		0			0
$107$		−	9.81743i		−	0.949087i	−0.880232	−	0.474543i	$-0.842613\pi$
$107$		−	9.81743i		−	0.949087i	0.880232	−	0.474543i	$-0.157387\pi$
$108$		0			0
$109$		−	10.7179i		−	1.02659i	−0.858214	−	0.513293i	$-0.828426\pi$
$109$		−	10.7179i		−	1.02659i	0.858214	−	0.513293i	$-0.171574\pi$
$110$		0			0
$111$	7.89215			0.749089
$112$		0			0
$113$		−	0.976230i		−	0.0918360i	−0.998945	−	0.0459180i	$-0.985379\pi$
$113$		−	0.976230i		−	0.0918360i	0.998945	−	0.0459180i	$-0.0146213\pi$
$114$		0			0
$115$	−10.0119			−0.933615
$116$		0			0
$117$	2.59499			0.239907
$118$		0			0
$119$	3.79391	−	1.52044i	0.347787	−	0.139378i
$120$		0			0
$121$	−2.70754			−0.246140
$122$		0			0
$123$	1.58464			0.142882
$124$		0			0
$125$		−	12.1405i		−	1.08588i
$126$		0			0
$127$	6.75624			0.599519			0.299760	−	0.954015i	$-0.403094\pi$
$127$	6.75624			0.599519			0.299760	+	0.954015i	$0.403094\pi$
$128$		0			0
$129$		−	0.543691i		−	0.0478693i
$130$		0			0
$131$		−	9.06972i		−	0.792425i	−0.918159	−	0.396213i	$-0.870324\pi$
$131$		−	9.06972i		−	0.792425i	0.918159	−	0.396213i	$-0.129676\pi$
$132$		0			0
$133$		−	0.313991i		−	0.0272265i
$134$		0			0
$135$	−9.82008			−0.845178
$136$		0			0
$137$	−19.9981			−1.70855			−0.854275	−	0.519821i	$-0.825999\pi$
$137$	−19.9981			−1.70855			−0.854275	+	0.519821i	$0.825999\pi$
$138$		0			0
$139$		−	1.69440i		−	0.143717i	−0.997415	−	0.0718587i	$-0.977107\pi$
$139$		−	1.69440i		−	0.143717i	0.997415	−	0.0718587i	$-0.0228931\pi$
$140$		0			0
$141$		−	7.58383i		−	0.638673i
$142$		0			0
$143$			13.6918i			1.14497i
$144$		0			0
$145$	12.9942			1.07911
$146$		0			0
$147$		−	9.12234i		−	0.752398i
$148$		0			0
$149$	−6.25451			−0.512389			−0.256195	−	0.966625i	$-0.582469\pi$
$149$	−6.25451			−0.512389			−0.256195	+	0.966625i	$0.582469\pi$
$150$		0			0
$151$	−12.0166			−0.977894			−0.488947	−	0.872313i	$-0.662619\pi$
$151$	−12.0166			−0.977894			−0.488947	+	0.872313i	$0.662619\pi$
$152$		0			0
$153$	1.07626	+	2.68557i	0.0870105	+	0.217115i
$154$		0			0
$155$	−10.6224			−0.853214
$156$		0			0
$157$	−19.7318			−1.57477			−0.787383	−	0.616465i	$-0.788564\pi$
$157$	−19.7318			−1.57477			−0.787383	+	0.616465i	$0.788564\pi$
$158$		0			0
$159$		−	12.8349i		−	1.01787i
$160$		0			0
$161$	−5.67167			−0.446990
$162$		0			0
$163$		−	16.3981i		−	1.28440i	−0.766538	−	0.642198i	$-0.778023\pi$
$163$		−	16.3981i		−	1.28440i	0.766538	−	0.642198i	$-0.221977\pi$
$164$		0			0
$165$		−	9.82185i		−	0.764630i
$166$		0			0
$167$		−	3.13777i		−	0.242808i	−0.992603	−	0.121404i	$-0.961260\pi$
$167$		−	3.13777i		−	0.242808i	0.992603	−	0.121404i	$-0.0387396\pi$
$168$		0			0
$169$	0.676138			0.0520106
$170$		0			0
$171$	0.222263			0.0169969
$172$		0			0
$173$			1.51723i			0.115353i	0.998335	+	0.0576764i	$0.0183691\pi$
$173$			1.51723i			0.115353i	−0.998335	+	0.0576764i	$0.981631\pi$
$174$		0			0
$175$		−	1.92103i		−	0.145216i
$176$		0			0
$177$			1.51601i			0.113951i
$178$		0			0
$179$	7.54467			0.563915			0.281958	−	0.959427i	$-0.409016\pi$
$179$	7.54467			0.563915			0.281958	+	0.959427i	$0.409016\pi$
$180$		0			0
$181$			11.8075i			0.877646i	0.898574	+	0.438823i	$0.144604\pi$
$181$			11.8075i			0.877646i	−0.898574	+	0.438823i	$0.855396\pi$
$182$		0			0
$183$	10.9383			0.808580
$184$		0			0
$185$	9.10967			0.669756
$186$		0			0
$187$	−14.1697	+	5.67863i	−1.03619	+	0.415262i
$188$		0			0
$189$	−5.56301			−0.404649
$190$		0			0
$191$	−10.1678			−0.735714			−0.367857	−	0.929882i	$-0.619908\pi$
$191$	−10.1678			−0.735714			−0.367857	+	0.929882i	$0.619908\pi$
$192$		0			0
$193$		−	26.8961i		−	1.93602i	−0.250905	−	0.968012i	$-0.580728\pi$
$193$		−	26.8961i		−	1.93602i	0.250905	−	0.968012i	$-0.419272\pi$
$194$		0			0
$195$	−9.81059			−0.702551
$196$		0			0
$197$			8.01401i			0.570975i	0.958382	+	0.285487i	$0.0921555\pi$
$197$			8.01401i			0.570975i	−0.958382	+	0.285487i	$0.907845\pi$
$198$		0			0
$199$		−	5.41772i		−	0.384052i	−0.981390	−	0.192026i	$-0.938494\pi$
$199$		−	5.41772i		−	0.384052i	0.981390	−	0.192026i	$-0.0615058\pi$
$200$		0			0
$201$		−	13.3612i		−	0.942429i
$202$		0			0
$203$	7.36113			0.516650
$204$		0			0
$205$	1.82910			0.127750
$206$		0			0
$207$		−	4.01477i		−	0.279046i
$208$		0			0
$209$			1.17272i			0.0811185i
$210$		0			0
$211$		−	17.1754i		−	1.18240i	−0.806525	−	0.591200i	$-0.798654\pi$
$211$		−	17.1754i		−	1.18240i	0.806525	−	0.591200i	$-0.201346\pi$
$212$		0			0
$213$	14.3076			0.980341
$214$		0			0
$215$		−	0.627566i		−	0.0427997i
$216$		0			0
$217$	−6.01753			−0.408496
$218$		0			0
$219$	3.45897			0.233736
$220$		0			0
$221$	5.67212	+	14.1535i	0.381548	+	0.952067i
$222$		0			0
$223$	−16.2133			−1.08572			−0.542862	−	0.839822i	$-0.682659\pi$
$223$	−16.2133			−1.08572			−0.542862	+	0.839822i	$0.682659\pi$
$224$		0			0
$225$	1.35983			0.0906551
$226$		0			0
$227$		−	6.30466i		−	0.418455i	−0.977867	−	0.209227i	$-0.932905\pi$
$227$		−	6.30466i		−	0.418455i	0.977867	−	0.209227i	$-0.0670949\pi$
$228$		0			0
$229$	13.9036			0.918775			0.459388	−	0.888236i	$-0.348069\pi$
$229$	13.9036			0.918775			0.459388	+	0.888236i	$0.348069\pi$
$230$		0			0
$231$		−	5.56401i		−	0.366085i
$232$		0			0
$233$		−	8.23385i		−	0.539418i	−0.962942	−	0.269709i	$-0.913073\pi$
$233$		−	8.23385i		−	0.539418i	0.962942	−	0.269709i	$-0.0869275\pi$
$234$		0			0
$235$		−	8.75378i		−	0.571034i
$236$		0			0
$237$	16.5455			1.07475
$238$		0			0
$239$	2.60780			0.168685			0.0843423	−	0.996437i	$-0.473121\pi$
$239$	2.60780			0.168685			0.0843423	+	0.996437i	$0.473121\pi$
$240$		0			0
$241$		−	4.47203i		−	0.288069i	−0.989573	−	0.144034i	$-0.953992\pi$
$241$		−	4.47203i		−	0.288069i	0.989573	−	0.144034i	$-0.0460076\pi$
$242$		0			0
$243$		−	7.12922i		−	0.457340i
$244$		0			0
$245$		−	10.5296i		−	0.672715i
$246$		0			0
$247$	1.17137			0.0745326
$248$		0			0
$249$			0.369176i			0.0233956i
$250$		0			0
$251$	−12.5552			−0.792475			−0.396237	−	0.918148i	$-0.629684\pi$
$251$	−12.5552			−0.792475			−0.396237	+	0.918148i	$0.629684\pi$
$252$		0			0
$253$	21.1829			1.33176
$254$		0			0
$255$	−4.06890	−	10.1530i	−0.254804	−	0.635807i
$256$		0			0
$257$	−17.3792			−1.08409			−0.542043	−	0.840351i	$-0.682349\pi$
$257$	−17.3792			−1.08409			−0.542043	+	0.840351i	$0.682349\pi$
$258$		0			0
$259$	5.16056			0.320662
$260$		0			0
$261$			5.21067i			0.322533i
$262$		0			0
$263$	−3.24920			−0.200355			−0.100177	−	0.994970i	$-0.531941\pi$
$263$	−3.24920			−0.200355			−0.100177	+	0.994970i	$0.531941\pi$
$264$		0			0
$265$		−	14.8149i		−	0.910074i
$266$		0			0
$267$			0.160451i			0.00981947i
$268$		0			0
$269$			20.7505i			1.26518i	0.774486	+	0.632591i	$0.218009\pi$
$269$			20.7505i			1.26518i	−0.774486	+	0.632591i	$0.781991\pi$
$270$		0			0
$271$	14.7364			0.895173			0.447586	−	0.894241i	$-0.352284\pi$
$271$	14.7364			0.895173			0.447586	+	0.894241i	$0.352284\pi$
$272$		0			0
$273$	−5.55763			−0.336363
$274$		0			0
$275$			7.17479i			0.432656i
$276$		0			0
$277$			25.5000i			1.53215i	0.642752	+	0.766074i	$0.277793\pi$
$277$			25.5000i			1.53215i	−0.642752	+	0.766074i	$0.722207\pi$
$278$		0			0
$279$		−	4.25959i		−	0.255015i
$280$		0			0
$281$	−15.5957			−0.930359			−0.465179	−	0.885216i	$-0.654010\pi$
$281$	−15.5957			−0.930359			−0.465179	+	0.885216i	$0.654010\pi$
$282$		0			0
$283$			20.8875i			1.24163i	0.783956	+	0.620817i	$0.213199\pi$
$283$			20.8875i			1.24163i	−0.783956	+	0.620817i	$0.786801\pi$
$284$		0			0
$285$	−0.840284			−0.0497742
$286$		0			0
$287$	1.03617			0.0611632
$288$		0			0
$289$	−12.2950	+	11.7402i	−0.723237	+	0.690600i
$290$		0			0
$291$	−16.9272			−0.992291
$292$		0			0
$293$	31.4672			1.83833			0.919167	−	0.393868i	$-0.128863\pi$
$293$	31.4672			1.83833			0.919167	+	0.393868i	$0.128863\pi$
$294$		0			0
$295$			1.74989i			0.101882i
$296$		0			0
$297$	20.7771			1.20561
$298$		0			0
$299$		−	21.1587i		−	1.22364i
$300$		0			0
$301$		−	0.355512i		−	0.0204914i
$302$		0			0
$303$			11.2411i			0.645784i
$304$		0			0
$305$	12.6257			0.722946
$306$		0			0
$307$	−20.1211			−1.14837			−0.574186	−	0.818725i	$-0.694681\pi$
$307$	−20.1211			−1.14837			−0.574186	+	0.818725i	$0.694681\pi$
$308$		0			0
$309$		−	26.3054i		−	1.49646i
$310$		0			0
$311$			10.3314i			0.585839i	0.956137	+	0.292919i	$0.0946267\pi$
$311$			10.3314i			0.585839i	−0.956137	+	0.292919i	$0.905373\pi$
$312$		0			0
$313$			4.93637i			0.279020i	0.990221	+	0.139510i	$0.0445528\pi$
$313$			4.93637i			0.279020i	−0.990221	+	0.139510i	$0.955447\pi$
$314$		0			0
$315$	−1.21722			−0.0685826
$316$		0			0
$317$		−	0.103657i		−	0.00582194i	−0.999996	−	0.00291097i	$-0.999073\pi$
$317$		−	0.103657i		−	0.00582194i	0.999996	−	0.00291097i	$-0.000926591\pi$
$318$		0			0
$319$	−27.4928			−1.53930
$320$		0			0
$321$	−14.8834			−0.830708
$322$		0			0
$323$	0.485821	+	1.21226i	0.0270318	+	0.0674518i
$324$		0			0
$325$	7.16656			0.397529
$326$		0			0
$327$	−16.2484			−0.898540
$328$		0			0
$329$		−	4.95895i		−	0.273396i
$330$		0			0
$331$	12.1205			0.666205			0.333103	−	0.942891i	$-0.391904\pi$
$331$	12.1205			0.666205			0.333103	+	0.942891i	$0.391904\pi$
$332$		0			0
$333$			3.65297i			0.200182i
$334$		0			0
$335$		−	15.4225i		−	0.842620i
$336$		0			0
$337$			23.4963i			1.27992i	0.768406	+	0.639962i	$0.221050\pi$
$337$			23.4963i			1.27992i	−0.768406	+	0.639962i	$0.778950\pi$
$338$		0			0
$339$	−1.47998			−0.0803814
$340$		0			0
$341$	22.4747			1.21707
$342$		0			0
$343$		−	12.9041i		−	0.696754i
$344$		0			0
$345$			15.1782i			0.817166i
$346$		0			0
$347$			29.3009i			1.57296i	0.617618	+	0.786478i	$0.288098\pi$
$347$			29.3009i			1.57296i	−0.617618	+	0.786478i	$0.711902\pi$
$348$		0			0
$349$	−0.648483			−0.0347125			−0.0173563	−	0.999849i	$-0.505525\pi$
$349$	−0.648483			−0.0347125			−0.0173563	+	0.999849i	$0.505525\pi$
$350$		0			0
$351$		−	20.7533i		−	1.10773i
$352$		0			0
$353$	22.6067			1.20323			0.601617	−	0.798785i	$-0.294523\pi$
$353$	22.6067			1.20323			0.601617	+	0.798785i	$0.294523\pi$
$354$		0			0
$355$	16.5148			0.876517
$356$		0			0
$357$	−2.30500	−	5.75161i	−0.121994	−	0.304408i
$358$		0			0
$359$	14.6426			0.772807			0.386404	−	0.922330i	$-0.373717\pi$
$359$	14.6426			0.772807			0.386404	+	0.922330i	$0.373717\pi$
$360$		0			0
$361$	−18.8997			−0.994720
$362$		0			0
$363$			4.10467i			0.215439i
$364$		0			0
$365$	3.99259			0.208982
$366$		0			0
$367$		−	31.5368i		−	1.64621i	−0.567890	−	0.823105i	$-0.692240\pi$
$367$		−	31.5368i		−	1.64621i	0.567890	−	0.823105i	$-0.307760\pi$
$368$		0			0
$369$			0.733467i			0.0381828i
$370$		0			0
$371$		−	8.39255i		−	0.435720i
$372$		0			0
$373$	−27.8202			−1.44047			−0.720237	−	0.693728i	$-0.755967\pi$
$373$	−27.8202			−1.44047			−0.720237	+	0.693728i	$0.755967\pi$
$374$		0			0
$375$	−18.4052			−0.950441
$376$		0			0
$377$			27.4613i			1.41433i
$378$		0			0
$379$		−	21.3575i		−	1.09706i	−0.836131	−	0.548530i	$-0.815188\pi$
$379$		−	21.3575i		−	1.09706i	0.836131	−	0.548530i	$-0.184812\pi$
$380$		0			0
$381$		−	10.2425i		−	0.524742i
$382$		0			0
$383$	−2.79820			−0.142981			−0.0714907	−	0.997441i	$-0.522776\pi$
$383$	−2.79820			−0.142981			−0.0714907	+	0.997441i	$0.522776\pi$
$384$		0			0
$385$		−	6.42236i		−	0.327314i
$386$		0			0
$387$	0.251654			0.0127923
$388$		0			0
$389$	−6.69722			−0.339563			−0.169781	−	0.985482i	$-0.554306\pi$
$389$	−6.69722			−0.339563			−0.169781	+	0.985482i	$0.554306\pi$
$390$		0			0
$391$	21.8972	−	8.77546i	1.10739	−	0.443794i
$392$		0			0
$393$	−13.7498			−0.693586
$394$		0			0
$395$	19.0980			0.960924
$396$		0			0
$397$		−	10.5186i		−	0.527915i	−0.964534	−	0.263958i	$-0.914972\pi$
$397$		−	10.5186i		−	0.527915i	0.964534	−	0.263958i	$-0.0850279\pi$
$398$		0			0
$399$	−0.476015			−0.0238306
$400$		0			0
$401$		−	25.4571i		−	1.27127i	−0.771992	−	0.635633i	$-0.780739\pi$
$401$		−	25.4571i		−	1.27127i	0.771992	−	0.635633i	$-0.219261\pi$
$402$		0			0
$403$		−	22.4489i		−	1.11826i
$404$		0			0
$405$			11.2037i			0.556714i
$406$		0			0
$407$	−19.2740			−0.955377
$408$		0			0
$409$	−10.5417			−0.521253			−0.260627	−	0.965440i	$-0.583929\pi$
$409$	−10.5417			−0.521253			−0.260627	+	0.965440i	$0.583929\pi$
$410$		0			0
$411$			30.3173i			1.49544i
$412$		0			0
$413$			0.991299i			0.0487787i
$414$		0			0
$415$			0.426129i			0.0209178i
$416$		0			0
$417$	−2.56874			−0.125792
$418$		0			0
$419$			18.4767i			0.902645i	0.892361	+	0.451323i	$0.149048\pi$
$419$			18.4767i			0.902645i	−0.892361	+	0.451323i	$0.850952\pi$
$420$		0			0
$421$	10.3993			0.506832			0.253416	−	0.967357i	$-0.418446\pi$
$421$	10.3993			0.506832			0.253416	+	0.967357i	$0.418446\pi$
$422$		0			0
$423$	3.51026			0.170675
$424$		0			0
$425$	2.97230	+	7.41671i	0.144178	+	0.359763i
$426$		0			0
$427$	7.15238			0.346128
$428$		0			0
$429$	20.7570			1.00216
$430$		0			0
$431$			20.2849i			0.977088i	0.872539	+	0.488544i	$0.162472\pi$
$431$			20.2849i			0.977088i	−0.872539	+	0.488544i	$0.837528\pi$
$432$		0			0
$433$	29.2733			1.40679			0.703393	−	0.710801i	$-0.251667\pi$
$433$	29.2733			1.40679			0.703393	+	0.710801i	$0.251667\pi$
$434$		0			0
$435$		−	19.6994i		−	0.944514i
$436$		0			0
$437$		−	1.81225i		−	0.0866919i
$438$		0			0
$439$			13.6931i			0.653535i	0.945105	+	0.326768i	$0.105959\pi$
$439$			13.6931i			0.653535i	−0.945105	+	0.326768i	$0.894041\pi$
$440$		0			0
$441$	4.22238			0.201066
$442$		0			0
$443$	−5.01441			−0.238242			−0.119121	−	0.992880i	$-0.538008\pi$
$443$	−5.01441			−0.238242			−0.119121	+	0.992880i	$0.538008\pi$
$444$		0			0
$445$			0.185204i			0.00877952i
$446$		0			0
$447$			9.48192i			0.448479i
$448$		0			0
$449$		−	6.59886i		−	0.311420i	−0.987803	−	0.155710i	$-0.950234\pi$
$449$		−	6.59886i		−	0.311420i	0.987803	−	0.155710i	$-0.0497665\pi$
$450$		0			0
$451$	−3.86996			−0.182229
$452$		0			0
$453$			18.2173i			0.855922i
$454$		0			0
$455$	−6.41500			−0.300740
$456$		0			0
$457$	29.7492			1.39161			0.695804	−	0.718232i	$-0.255048\pi$
$457$	29.7492			1.39161			0.695804	+	0.718232i	$0.255048\pi$
$458$		0			0
$459$	21.4776	−	8.60732i	1.00249	−	0.401755i
$460$		0			0
$461$	−5.76670			−0.268582			−0.134291	−	0.990942i	$-0.542876\pi$
$461$	−5.76670			−0.268582			−0.134291	+	0.990942i	$0.542876\pi$
$462$		0			0
$463$	−27.2234			−1.26518			−0.632589	−	0.774487i	$-0.718008\pi$
$463$	−27.2234			−1.26518			−0.632589	+	0.774487i	$0.718008\pi$
$464$		0			0
$465$			16.1037i			0.746793i
$466$		0			0
$467$	3.19668			0.147925			0.0739624	−	0.997261i	$-0.476436\pi$
$467$	3.19668			0.147925			0.0739624	+	0.997261i	$0.476436\pi$
$468$		0			0
$469$		−	8.73672i		−	0.403424i
$470$		0			0
$471$			29.9136i			1.37835i
$472$		0			0
$473$			1.32779i			0.0610518i
$474$		0			0
$475$	0.613822			0.0281641
$476$		0			0
$477$	5.94078			0.272010
$478$		0			0
$479$			13.9313i			0.636537i	0.948001	+	0.318268i	$0.103101\pi$
$479$			13.9313i			0.636537i	−0.948001	+	0.318268i	$0.896899\pi$
$480$		0			0
$481$			19.2519i			0.877812i
$482$		0			0
$483$			8.59833i			0.391238i
$484$		0			0
$485$	−19.5386			−0.887201
$486$		0			0
$487$		−	30.4947i		−	1.38185i	−0.722928	−	0.690923i	$-0.757204\pi$
$487$		−	30.4947i		−	1.38185i	0.722928	−	0.690923i	$-0.242796\pi$
$488$		0			0
$489$	−24.8597			−1.12419
$490$		0			0
$491$	2.40456			0.108516			0.0542582	−	0.998527i	$-0.482721\pi$
$491$	2.40456			0.108516			0.0542582	+	0.998527i	$0.482721\pi$
$492$		0			0
$493$	−28.4199	+	11.3895i	−1.27997	+	0.512956i
$494$		0			0
$495$	4.54615			0.204334
$496$		0			0
$497$	9.35554			0.419653
$498$		0			0
$499$		−	8.48589i		−	0.379881i	−0.981796	−	0.189940i	$-0.939171\pi$
$499$		−	8.48589i		−	0.379881i	0.981796	−	0.189940i	$-0.0608295\pi$
$500$		0			0
$501$	−4.75690			−0.212523
$502$		0			0
$503$		−	21.6852i		−	0.966894i	−0.875374	−	0.483447i	$-0.839385\pi$
$503$		−	21.6852i		−	0.966894i	0.875374	−	0.483447i	$-0.160615\pi$
$504$		0			0
$505$			12.9753i			0.577391i
$506$		0			0
$507$		−	1.02503i		−	0.0455233i
$508$		0			0
$509$	13.3378			0.591187			0.295593	−	0.955314i	$-0.404483\pi$
$509$	13.3378			0.591187			0.295593	+	0.955314i	$0.404483\pi$
$510$		0			0
$511$	2.26177			0.100055
$512$		0			0
$513$		−	1.77753i		−	0.0784800i
$514$		0			0
$515$		−	30.3636i		−	1.33798i
$516$		0			0
$517$			18.5210i			0.814554i
$518$		0			0
$519$	2.30014			0.100965
$520$		0			0
$521$		−	9.99019i		−	0.437678i	−0.975761	−	0.218839i	$-0.929773\pi$
$521$		−	9.99019i		−	0.437678i	0.975761	−	0.218839i	$-0.0702270\pi$
$522$		0			0
$523$	35.7931			1.56512			0.782562	−	0.622572i	$-0.213912\pi$
$523$	35.7931			1.56512			0.782562	+	0.622572i	$0.213912\pi$
$524$		0			0
$525$	−2.91231			−0.127103
$526$		0			0
$527$	23.2325	−	9.31058i	1.01202	−	0.405575i
$528$		0			0
$529$	−9.73501			−0.423262
$530$		0			0
$531$	−0.701704			−0.0304514
$532$		0			0
$533$			3.86552i			0.167434i
$534$		0			0
$535$	−17.1794			−0.742731
$536$		0			0
$537$		−	11.4378i		−	0.493578i
$538$		0			0
$539$			22.2784i			0.959597i
$540$		0			0
$541$			29.0265i			1.24795i	0.781446	+	0.623973i	$0.214482\pi$
$541$			29.0265i			1.24795i	−0.781446	+	0.623973i	$0.785518\pi$
$542$		0			0
$543$	17.9004			0.768178
$544$		0			0
$545$	−18.7551			−0.803379
$546$		0			0
$547$		−	2.25690i		−	0.0964980i	−0.998835	−	0.0482490i	$-0.984636\pi$
$547$		−	2.25690i		−	0.0964980i	0.998835	−	0.0482490i	$-0.0153641\pi$
$548$		0			0
$549$			5.06290i			0.216079i
$550$		0			0
$551$			2.35208i			0.100202i
$552$		0			0
$553$	10.8189			0.460065
$554$		0			0
$555$		−	13.8104i		−	0.586218i
$556$		0			0
$557$	6.05083			0.256382			0.128191	−	0.991750i	$-0.459083\pi$
$557$	6.05083			0.256382			0.128191	+	0.991750i	$0.459083\pi$
$558$		0			0
$559$	1.32627			0.0560951
$560$		0			0
$561$	8.60887	+	21.4815i	0.363467	+	0.906950i
$562$		0			0
$563$	7.57669			0.319319			0.159660	−	0.987172i	$-0.448960\pi$
$563$	7.57669			0.319319			0.159660	+	0.987172i	$0.448960\pi$
$564$		0			0
$565$	−1.70829			−0.0718685
$566$		0			0
$567$			6.34679i			0.266540i
$568$		0			0
$569$	−29.7526			−1.24730			−0.623648	−	0.781706i	$-0.714350\pi$
$569$	−29.7526			−1.24730			−0.623648	+	0.781706i	$0.714350\pi$
$570$		0			0
$571$			5.30711i			0.222096i	0.993815	+	0.111048i	$0.0354207\pi$
$571$			5.30711i			0.222096i	−0.993815	+	0.111048i	$0.964579\pi$
$572$		0			0
$573$			15.4145i			0.643949i
$574$		0			0
$575$		−	11.0875i		−	0.462383i
$576$		0			0
$577$	−23.6496			−0.984547			−0.492274	−	0.870441i	$-0.663834\pi$
$577$	−23.6496			−0.984547			−0.492274	+	0.870441i	$0.663834\pi$
$578$		0			0
$579$	−40.7748			−1.69454
$580$		0			0
$581$			0.241399i			0.0100149i
$582$		0			0
$583$			31.3451i			1.29818i
$584$		0			0
$585$		−	4.54094i		−	0.187745i
$586$		0			0
$587$	23.4801			0.969126			0.484563	−	0.874756i	$-0.338979\pi$
$587$	23.4801			0.969126			0.484563	+	0.874756i	$0.338979\pi$
$588$		0			0
$589$		−	1.92277i		−	0.0792262i
$590$		0			0
$591$	12.1493			0.499757
$592$		0			0
$593$	15.1879			0.623694			0.311847	−	0.950132i	$-0.399052\pi$
$593$	15.1879			0.623694			0.311847	+	0.950132i	$0.399052\pi$
$594$		0			0
$595$	−2.66059	−	6.63892i	−0.109074	−	0.272169i
$596$		0			0
$597$	−8.21334			−0.336150
$598$		0			0
$599$	−18.1977			−0.743538			−0.371769	−	0.928325i	$-0.621249\pi$
$599$	−18.1977			−0.743538			−0.371769	+	0.928325i	$0.621249\pi$
$600$		0			0
$601$		−	45.3384i		−	1.84939i	−0.380705	−	0.924696i	$-0.624319\pi$
$601$		−	45.3384i		−	1.84939i	0.380705	−	0.924696i	$-0.375681\pi$
$602$		0			0
$603$	6.18440			0.251848
$604$		0			0
$605$			4.73790i			0.192623i
$606$		0			0
$607$		−	31.1019i		−	1.26239i	−0.775626	−	0.631193i	$-0.782565\pi$
$607$		−	31.1019i		−	1.26239i	0.775626	−	0.631193i	$-0.217435\pi$
$608$		0			0
$609$		−	11.1596i		−	0.452209i
$610$		0			0
$611$	18.4998			0.748422
$612$		0			0
$613$	14.0559			0.567712			0.283856	−	0.958867i	$-0.408386\pi$
$613$	14.0559			0.567712			0.283856	+	0.958867i	$0.408386\pi$
$614$		0			0
$615$		−	2.77294i		−	0.111816i
$616$		0			0
$617$			43.6678i			1.75800i	0.476821	+	0.879000i	$0.341789\pi$
$617$			43.6678i			1.75800i	−0.476821	+	0.879000i	$0.658211\pi$
$618$		0			0
$619$		−	16.6051i		−	0.667415i	−0.942677	−	0.333708i	$-0.891700\pi$
$619$		−	16.6051i		−	0.667415i	0.942677	−	0.333708i	$-0.108300\pi$
$620$		0			0
$621$	−32.1078			−1.28844
$622$		0			0
$623$			0.104917i			0.00420341i
$624$		0			0
$625$	−11.5551			−0.462205
$626$		0			0
$627$	1.77785			0.0710006
$628$		0			0
$629$	−19.9239	+	7.98464i	−0.794417	+	0.318369i
$630$		0			0
$631$	12.9229			0.514452			0.257226	−	0.966351i	$-0.417192\pi$
$631$	12.9229			0.514452			0.257226	+	0.966351i	$0.417192\pi$
$632$		0			0
$633$	−26.0381			−1.03492
$634$		0			0
$635$		−	11.8227i		−	0.469168i
$636$		0			0
$637$	22.2528			0.881689
$638$		0			0
$639$			6.62244i			0.261980i
$640$		0			0
$641$		−	9.14595i		−	0.361243i	−0.983553	−	0.180622i	$-0.942189\pi$
$641$		−	9.14595i		−	0.361243i	0.983553	−	0.180622i	$-0.0578109\pi$
$642$		0			0
$643$			19.7105i			0.777307i	0.921384	+	0.388654i	$0.127060\pi$
$643$			19.7105i			0.777307i	−0.921384	+	0.388654i	$0.872940\pi$
$644$		0			0
$645$	−0.951399			−0.0374613
$646$		0			0
$647$	15.3967			0.605307			0.302653	−	0.953101i	$-0.402128\pi$
$647$	15.3967			0.605307			0.302653	+	0.953101i	$0.402128\pi$
$648$		0			0
$649$		−	3.70237i		−	0.145331i
$650$		0			0
$651$			9.12265i			0.357545i
$652$		0			0
$653$			22.1689i			0.867537i	0.901024	+	0.433769i	$0.142816\pi$
$653$			22.1689i			0.867537i	−0.901024	+	0.433769i	$0.857184\pi$
$654$		0			0
$655$	−15.8710			−0.620131
$656$		0			0
$657$			1.60102i			0.0624619i
$658$		0			0
$659$	−13.9129			−0.541969			−0.270985	−	0.962584i	$-0.587349\pi$
$659$	−13.9129			−0.541969			−0.270985	+	0.962584i	$0.587349\pi$
$660$		0			0
$661$	−13.2240			−0.514352			−0.257176	−	0.966365i	$-0.582792\pi$
$661$	−13.2240			−0.514352			−0.257176	+	0.966365i	$0.582792\pi$
$662$		0			0
$663$	21.4569	−	8.59900i	0.833316	−	0.333958i
$664$		0			0
$665$	−0.549450			−0.0213068
$666$		0			0
$667$	42.4860			1.64507
$668$		0			0
$669$			24.5796i			0.950303i
$670$		0			0
$671$	−26.7132			−1.03125
$672$		0			0
$673$			34.5954i			1.33356i	0.745256	+	0.666778i	$0.232327\pi$
$673$			34.5954i			1.33356i	−0.745256	+	0.666778i	$0.767673\pi$
$674$		0			0
$675$		−	10.8751i		−	0.418583i
$676$		0			0
$677$			8.32741i			0.320048i	0.987113	+	0.160024i	$0.0511572\pi$
$677$			8.32741i			0.320048i	−0.987113	+	0.160024i	$0.948843\pi$
$678$		0			0
$679$	−11.0685			−0.424769
$680$		0			0
$681$	−9.55794			−0.366261
$682$		0			0
$683$			41.9957i			1.60692i	0.595359	+	0.803460i	$0.297010\pi$
$683$			41.9957i			1.60692i	−0.595359	+	0.803460i	$0.702990\pi$
$684$		0			0
$685$			34.9944i			1.33707i
$686$		0			0
$687$		−	21.0780i		−	0.804177i
$688$		0			0
$689$	31.3091			1.19278
$690$		0			0
$691$		−	23.7264i		−	0.902595i	−0.892374	−	0.451298i	$-0.850961\pi$
$691$		−	23.7264i		−	0.902595i	0.892374	−	0.451298i	$-0.149039\pi$
$692$		0			0
$693$	2.57536			0.0978300
$694$		0			0
$695$	−2.96502			−0.112470
$696$		0			0
$697$	−4.00045	+	1.60321i	−0.151528	+	0.0607258i
$698$		0			0
$699$	−12.4826			−0.472136
$700$		0			0
$701$	−8.44424			−0.318935			−0.159467	−	0.987203i	$-0.550978\pi$
$701$	−8.44424			−0.318935			−0.159467	+	0.987203i	$0.550978\pi$
$702$		0			0
$703$			1.64894i			0.0621910i
$704$		0			0
$705$	−13.2708			−0.499809
$706$		0			0
$707$			7.35039i			0.276440i
$708$		0			0
$709$		−	4.87738i		−	0.183174i	−0.995797	−	0.0915870i	$-0.970806\pi$
$709$		−	4.87738i		−	0.183174i	0.995797	−	0.0915870i	$-0.0291940\pi$
$710$		0			0
$711$			7.65828i			0.287208i
$712$		0			0
$713$	−34.7312			−1.30069
$714$		0			0
$715$	23.9592			0.896022
$716$		0			0
$717$		−	3.95346i		−	0.147645i
$718$		0			0
$719$			27.2863i			1.01761i	0.860882	+	0.508804i	$0.169912\pi$
$719$			27.2863i			1.01761i	−0.860882	+	0.508804i	$0.830088\pi$
$720$		0			0
$721$		−	17.2008i		−	0.640590i
$722$		0			0
$723$	−6.77966			−0.252138
$724$		0			0
$725$			14.3903i			0.534441i
$726$		0			0
$727$	−48.1186			−1.78462			−0.892310	−	0.451424i	$-0.850916\pi$
$727$	−48.1186			−1.78462			−0.892310	+	0.451424i	$0.850916\pi$
$728$		0			0
$729$	−30.0155			−1.11168
$730$		0			0
$731$	0.550063	+	1.37256i	0.0203448	+	0.0507660i
$732$		0			0
$733$	15.9190			0.587983			0.293991	−	0.955808i	$-0.405016\pi$
$733$	15.9190			0.587983			0.293991	+	0.955808i	$0.405016\pi$
$734$		0			0
$735$	−15.9631			−0.588807
$736$		0			0
$737$			32.6305i			1.20196i
$738$		0			0
$739$	17.4362			0.641400			0.320700	−	0.947181i	$-0.396082\pi$
$739$	17.4362			0.641400			0.320700	+	0.947181i	$0.396082\pi$
$740$		0			0
$741$		−	1.77581i		−	0.0652362i
$742$		0			0
$743$		−	6.09751i		−	0.223696i	−0.993725	−	0.111848i	$-0.964323\pi$
$743$		−	6.09751i		−	0.223696i	0.993725	−	0.111848i	$-0.0356770\pi$
$744$		0			0
$745$			10.9447i			0.400982i
$746$		0			0
$747$	−0.170877			−0.00625207
$748$		0			0
$749$	−9.73201			−0.355600
$750$		0			0
$751$		−	5.89269i		−	0.215027i	−0.994204	−	0.107514i	$-0.965711\pi$
$751$		−	5.89269i		−	0.215027i	0.994204	−	0.107514i	$-0.0342889\pi$
$752$		0			0
$753$			19.0338i			0.693630i
$754$		0			0
$755$			21.0276i			0.765274i
$756$		0			0
$757$	−37.3178			−1.35634			−0.678169	−	0.734906i	$-0.737226\pi$
$757$	−37.3178			−1.35634			−0.678169	+	0.734906i	$0.737226\pi$
$758$		0			0
$759$		−	32.1136i		−	1.16565i
$760$		0			0
$761$	34.8054			1.26170			0.630848	−	0.775906i	$-0.282707\pi$
$761$	34.8054			1.26170			0.630848	+	0.775906i	$0.282707\pi$
$762$		0			0
$763$	−10.6246			−0.384637
$764$		0			0
$765$	4.69944	−	1.88334i	0.169909	−	0.0680922i
$766$		0			0
$767$	−3.69813			−0.133532
$768$		0			0
$769$	43.9868			1.58620			0.793102	−	0.609088i	$-0.208465\pi$
$769$	43.9868			1.58620			0.793102	+	0.609088i	$0.208465\pi$
$770$		0			0
$771$			26.3471i			0.948869i
$772$		0			0
$773$	49.4314			1.77792			0.888961	−	0.457983i	$-0.151428\pi$
$773$	49.4314			1.77792			0.888961	+	0.457983i	$0.151428\pi$
$774$		0			0
$775$		−	11.7637i		−	0.422563i
$776$		0			0
$777$		−	7.82348i		−	0.280666i
$778$		0			0
$779$			0.331085i			0.0118624i
$780$		0			0
$781$	−34.9417			−1.25031
$782$		0			0
$783$	41.6720			1.48924
$784$		0			0
$785$			34.5284i			1.23237i
$786$		0			0
$787$		−	38.7950i		−	1.38289i	−0.722428	−	0.691446i	$-0.756974\pi$
$787$		−	38.7950i		−	1.38289i	0.722428	−	0.691446i	$-0.243026\pi$
$788$		0			0
$789$			4.92584i			0.175364i
$790$		0			0
$791$	−0.967736			−0.0344087
$792$		0			0
$793$			26.6826i			0.947525i
$794$		0			0
$795$	−22.4596			−0.796561
$796$		0			0
$797$	24.4213			0.865047			0.432523	−	0.901623i	$-0.357623\pi$
$797$	24.4213			0.865047			0.432523	+	0.901623i	$0.357623\pi$
$798$		0			0
$799$	7.67271	+	19.1455i	0.271441	+	0.677320i
$800$		0			0
$801$	−0.0742668			−0.00262409
$802$		0			0
$803$	−8.44741			−0.298103
$804$		0			0
$805$			9.92479i			0.349803i
$806$		0			0
$807$	31.4581			1.10738
$808$		0			0
$809$		−	32.9021i		−	1.15678i	−0.815762	−	0.578388i	$-0.803682\pi$
$809$		−	32.9021i		−	1.15678i	0.815762	−	0.578388i	$-0.196318\pi$
$810$		0			0
$811$			44.6051i			1.56630i	0.621834	+	0.783149i	$0.286388\pi$
$811$			44.6051i			1.56630i	−0.621834	+	0.783149i	$0.713612\pi$
$812$		0			0
$813$		−	22.3406i		−	0.783518i
$814$		0			0
$815$	−28.6948			−1.00514
$816$		0			0
$817$	0.113596			0.00397421
$818$		0			0
$819$		−	2.57241i		−	0.0898874i
$820$		0			0
$821$			45.2478i			1.57916i	0.613648	+	0.789580i	$0.289701\pi$
$821$			45.2478i			1.57916i	−0.613648	+	0.789580i	$0.710299\pi$
$822$		0			0
$823$			2.83554i			0.0988409i	0.998778	+	0.0494204i	$0.0157374\pi$
$823$			2.83554i			0.0988409i	−0.998778	+	0.0494204i	$0.984263\pi$
$824$		0			0
$825$	10.8771			0.378691
$826$		0			0
$827$			8.63081i			0.300123i	0.988677	+	0.150061i	$0.0479471\pi$
$827$			8.63081i			0.300123i	−0.988677	+	0.150061i	$0.952053\pi$
$828$		0			0
$829$	41.9293			1.45626			0.728132	−	0.685437i	$-0.240389\pi$
$829$	41.9293			1.45626			0.728132	+	0.685437i	$0.240389\pi$
$830$		0			0
$831$	38.6584			1.34104
$832$		0			0
$833$	9.22926	+	23.0296i	0.319775	+	0.797927i
$834$		0			0
$835$	−5.49074			−0.190015
$836$		0			0
$837$	−34.0658			−1.17748
$838$		0			0
$839$		−	18.3115i		−	0.632184i	−0.948729	−	0.316092i	$-0.897629\pi$
$839$		−	18.3115i		−	0.632184i	0.948729	−	0.316092i	$-0.102371\pi$
$840$		0			0
$841$	−26.1416			−0.901436
$842$		0			0
$843$			23.6432i			0.814316i
$844$		0			0
$845$		−	1.18317i		−	0.0407021i
$846$		0			0
$847$			2.68398i			0.0922228i
$848$		0			0
$849$	31.6657			1.08676
$850$		0			0
$851$	29.7851			1.02102
$852$		0			0
$853$		−	1.47544i		−	0.0505179i	−0.999681	−	0.0252590i	$-0.991959\pi$
$853$		−	1.47544i		−	0.0505179i	0.999681	−	0.0252590i	$-0.00804104\pi$
$854$		0			0
$855$		−	0.388935i		−	0.0133013i
$856$		0			0
$857$		−	16.2911i		−	0.556492i	−0.960510	−	0.278246i	$-0.910247\pi$
$857$		−	16.2911i		−	0.556492i	0.960510	−	0.278246i	$-0.0897531\pi$
$858$		0			0
$859$	−28.1469			−0.960360			−0.480180	−	0.877170i	$-0.659429\pi$
$859$	−28.1469			−0.960360			−0.480180	+	0.877170i	$0.659429\pi$
$860$		0			0
$861$		−	1.57085i		−	0.0535344i
$862$		0			0
$863$	24.7065			0.841021			0.420510	−	0.907288i	$-0.361851\pi$
$863$	24.7065			0.841021			0.420510	+	0.907288i	$0.361851\pi$
$864$		0			0
$865$	2.65498			0.0902720
$866$		0			0
$867$	17.7983	+	18.6394i	0.604462	+	0.633028i
$868$		0			0
$869$	−40.4070			−1.37071
$870$		0			0
$871$	32.5931			1.10437
$872$		0			0
$873$		−	7.83495i		−	0.265173i
$874$		0			0
$875$	−12.0349			−0.406854
$876$		0			0
$877$		−	1.71685i		−	0.0579737i	−0.999580	−	0.0289869i	$-0.990772\pi$
$877$		−	1.71685i		−	0.0579737i	0.999580	−	0.0289869i	$-0.00922810\pi$
$878$		0			0
$879$		−	47.7047i		−	1.60904i
$880$		0			0
$881$		−	36.0275i		−	1.21380i	−0.794780	−	0.606898i	$-0.792414\pi$
$881$		−	36.0275i		−	1.21380i	0.794780	−	0.606898i	$-0.207586\pi$
$882$		0			0
$883$	−13.0401			−0.438836			−0.219418	−	0.975631i	$-0.570416\pi$
$883$	−13.0401			−0.438836			−0.219418	+	0.975631i	$0.570416\pi$
$884$		0			0
$885$	2.65285			0.0891747
$886$		0			0
$887$			20.1101i			0.675232i	0.941284	+	0.337616i	$0.109621\pi$
$887$			20.1101i			0.675232i	−0.941284	+	0.337616i	$0.890379\pi$
$888$		0			0
$889$		−	6.69745i		−	0.224625i
$890$		0			0
$891$		−	23.7044i		−	0.794128i
$892$		0			0
$893$	1.58452			0.0530240
$894$		0			0
$895$		−	13.2023i		−	0.441305i
$896$		0			0
$897$	−32.0768			−1.07101
$898$		0			0
$899$	45.0768			1.50340
$900$		0			0
$901$	12.9853	+	32.4020i	0.432604	+	1.07947i
$902$		0			0
$903$	−0.538961			−0.0179355
$904$		0			0
$905$	20.6618			0.686823
$906$		0			0
$907$		−	7.44825i		−	0.247315i	−0.992325	−	0.123657i	$-0.960538\pi$
$907$		−	7.44825i		−	0.247315i	0.992325	−	0.123657i	$-0.0394624\pi$
$908$		0			0
$909$	−5.20307			−0.172575
$910$		0			0
$911$			29.9948i			0.993773i	0.867815	+	0.496887i	$0.165524\pi$
$911$			29.9948i			0.993773i	−0.867815	+	0.496887i	$0.834476\pi$
$912$		0			0
$913$		−	0.901592i		−	0.0298383i
$914$		0			0
$915$		−	19.1408i		−	0.632774i
$916$		0			0
$917$	−8.99081			−0.296903
$918$		0			0
$919$	−46.1561			−1.52255			−0.761274	−	0.648430i	$-0.775426\pi$
$919$	−46.1561			−1.52255			−0.761274	+	0.648430i	$0.775426\pi$
$920$		0			0
$921$			30.5038i			1.00514i
$922$		0			0
$923$			34.9016i			1.14880i
$924$		0			0
$925$			10.0884i			0.331704i
$926$		0			0
$927$	12.1758			0.399905
$928$		0			0
$929$			3.80467i			0.124827i	0.998050	+	0.0624136i	$0.0198798\pi$
$929$			3.80467i			0.124827i	−0.998050	+	0.0624136i	$0.980120\pi$
$930$		0			0
$931$	1.90597			0.0624657
$932$		0			0
$933$	15.6625			0.512767
$934$		0			0
$935$	9.93696	+	24.7955i	0.324973	+	0.810898i
$936$		0			0
$937$	−38.2302			−1.24892			−0.624462	−	0.781055i	$-0.714682\pi$
$937$	−38.2302			−1.24892			−0.624462	+	0.781055i	$0.714682\pi$
$938$		0			0
$939$	7.48360			0.244218
$940$		0			0
$941$			40.8937i			1.33310i	0.745462	+	0.666548i	$0.232229\pi$
$941$			40.8937i			1.33310i	−0.745462	+	0.666548i	$0.767771\pi$
$942$		0			0
$943$	5.98044			0.194750
$944$		0			0
$945$			9.73464i			0.316668i
$946$		0			0
$947$			10.7951i			0.350795i	0.984498	+	0.175397i	$0.0561210\pi$
$947$			10.7951i			0.350795i	−0.984498	+	0.175397i	$0.943879\pi$
$948$		0			0
$949$			8.43773i			0.273900i
$950$		0			0
$951$	−0.157145			−0.00509577
$952$		0			0
$953$	−40.3470			−1.30697			−0.653484	−	0.756940i	$-0.726693\pi$
$953$	−40.3470			−1.30697			−0.653484	+	0.756940i	$0.726693\pi$
$954$		0			0
$955$			17.7925i			0.575750i
$956$		0			0
$957$			41.6795i			1.34731i
$958$		0			0
$959$			19.8241i			0.640153i
$960$		0			0
$961$	−5.84909			−0.188680
$962$		0			0
$963$		−	6.88893i		−	0.221993i
$964$		0			0
$965$	−47.0652			−1.51508
$966$		0			0
$967$	−48.5425			−1.56102			−0.780510	−	0.625143i	$-0.785041\pi$
$967$	−48.5425			−1.56102			−0.780510	+	0.625143i	$0.785041\pi$
$968$		0			0
$969$	1.83780	−	0.736511i	0.0590386	−	0.0236602i
$970$		0			0
$971$	−25.0568			−0.804111			−0.402056	−	0.915615i	$-0.631704\pi$
$971$	−25.0568			−0.804111			−0.402056	+	0.915615i	$0.631704\pi$
$972$		0			0
$973$	−1.67966			−0.0538475
$974$		0			0
$975$		−	10.8646i		−	0.347946i
$976$		0			0
$977$	59.7908			1.91288			0.956438	−	0.291936i	$-0.0942995\pi$
$977$	59.7908			1.91288			0.956438	+	0.291936i	$0.0942995\pi$
$978$		0			0
$979$		−	0.391850i		−	0.0125236i
$980$		0			0
$981$		−	7.52077i		−	0.240120i
$982$		0			0
$983$		−	14.1301i		−	0.450679i	−0.974280	−	0.225340i	$-0.927651\pi$
$983$		−	14.1301i		−	0.450679i	0.974280	−	0.225340i	$-0.0723492\pi$
$984$		0			0
$985$	14.0236			0.446830
$986$		0			0
$987$	−7.51784			−0.239296
$988$		0			0
$989$		−	2.05190i		−	0.0652465i
$990$		0			0
$991$			10.4022i			0.330437i	0.986257	+	0.165218i	$0.0528329\pi$
$991$			10.4022i			0.330437i	−0.986257	+	0.165218i	$0.947167\pi$
$992$		0			0
$993$		−	18.3749i		−	0.583110i
$994$		0			0
$995$	−9.48041			−0.300549
$996$		0			0
$997$		−	19.1383i		−	0.606115i	−0.952972	−	0.303058i	$-0.901992\pi$
$997$		−	19.1383i		−	0.606115i	0.952972	−	0.303058i	$-0.0980075\pi$
$998$		0			0
$999$	29.2144			0.924302

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.12	✓	40
17.16	even	2	inner	4012.2.b.a.237.29	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.12	✓	40	1.1	even	1	trivial
4012.2.b.a.237.29	yes	40	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)

\(f(q)\)	\(=\)	\(q-1.51601i q^{3} -1.74989i q^{5} -0.991299i q^{7} +0.701704 q^{9} +O(q^{10})\)	\(q-1.51601i q^{3} -1.74989i q^{5} -0.991299i q^{7} +0.701704 q^{9} +3.70237i q^{11} +3.69813 q^{13} -2.65285 q^{15} +(1.53378 + 3.82721i) q^{17} +0.316747 q^{19} -1.50282 q^{21} -5.72145i q^{23} +1.93789 q^{25} -5.61183i q^{27} +7.42574i q^{29} -6.07034i q^{31} +5.61284 q^{33} -1.73466 q^{35} +5.20586i q^{37} -5.60641i q^{39} +1.04527i q^{41} +0.358632 q^{43} -1.22790i q^{45} +5.00248 q^{47} +6.01733 q^{49} +(5.80210 - 2.32523i) q^{51} +8.46622 q^{53} +6.47873 q^{55} -0.480193i q^{57} -1.00000 q^{59} +7.21516i q^{61} -0.695599i q^{63} -6.47131i q^{65} +8.81340 q^{67} -8.67380 q^{69} +9.43765i q^{71} +2.28162i q^{73} -2.93787i q^{75} +3.67016 q^{77} +10.9138i q^{79} -6.40250 q^{81} -0.243518 q^{83} +(6.69719 - 2.68395i) q^{85} +11.2575 q^{87} -0.105838 q^{89} -3.66595i q^{91} -9.20272 q^{93} -0.554273i q^{95} -11.1656i q^{97} +2.59797i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more