LMFDB - Embedded newform 4012.2.b.a.237.15

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			237.15
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.02370i q^{3} +3.54148i q^{5} +0.166941i q^{7} +1.95203 q^{9} +O(q^{10})$	$q-1.02370i q^{3} +3.54148i q^{5} +0.166941i q^{7} +1.95203 q^{9} -0.452734i q^{11} +3.43835 q^{13} +3.62542 q^{15} +(4.12255 + 0.0679254i) q^{17} +7.68707 q^{19} +0.170898 q^{21} -8.47857i q^{23} -7.54211 q^{25} -5.06941i q^{27} -1.93174i q^{29} -2.57764i q^{31} -0.463464 q^{33} -0.591220 q^{35} +1.31550i q^{37} -3.51984i q^{39} +4.56154i q^{41} -9.93091 q^{43} +6.91310i q^{45} +1.10859 q^{47} +6.97213 q^{49} +(0.0695354 - 4.22026i) q^{51} -3.35231 q^{53} +1.60335 q^{55} -7.86927i q^{57} -1.00000 q^{59} -8.42710i q^{61} +0.325875i q^{63} +12.1768i q^{65} -8.66678 q^{67} -8.67953 q^{69} -14.4683i q^{71} -4.18682i q^{73} +7.72087i q^{75} +0.0755800 q^{77} +12.9491i q^{79} +0.666545 q^{81} +10.1547 q^{83} +(-0.240557 + 14.5999i) q^{85} -1.97752 q^{87} -1.62855 q^{89} +0.574002i q^{91} -2.63873 q^{93} +27.2236i q^{95} +12.0497i q^{97} -0.883752i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.02370i		−	0.591034i	−0.955337	−	0.295517i	$-0.904508\pi$
$3$		−	1.02370i		−	0.591034i	0.955337	−	0.295517i	$-0.0954920\pi$
$4$		0			0
$5$			3.54148i			1.58380i	0.610651	+	0.791900i	$0.290908\pi$
$5$			3.54148i			1.58380i	−0.610651	+	0.791900i	$0.709092\pi$
$6$		0			0
$7$			0.166941i			0.0630979i	0.999502	+	0.0315490i	$0.0100440\pi$
$7$			0.166941i			0.0630979i	−0.999502	+	0.0315490i	$0.989956\pi$
$8$		0			0
$9$	1.95203			0.650678
$10$		0			0
$11$		−	0.452734i		−	0.136504i	−0.997668	−	0.0682522i	$-0.978258\pi$
$11$		−	0.452734i		−	0.136504i	0.997668	−	0.0682522i	$-0.0217423\pi$
$12$		0			0
$13$	3.43835			0.953625			0.476813	−	0.879005i	$-0.341792\pi$
$13$	3.43835			0.953625			0.476813	+	0.879005i	$0.341792\pi$
$14$		0			0
$15$	3.62542			0.936080
$16$		0			0
$17$	4.12255	+	0.0679254i	0.999864	+	0.0164743i
$17$	4.12255	+	0.0679254i	0.999864	+	0.0164743i
$18$		0			0
$19$	7.68707			1.76353			0.881767	−	0.471684i	$-0.156354\pi$
$19$	7.68707			1.76353			0.881767	+	0.471684i	$0.156354\pi$
$20$		0			0
$21$	0.170898			0.0372930
$22$		0			0
$23$		−	8.47857i		−	1.76790i	−0.467577	−	0.883952i	$-0.654873\pi$
$23$		−	8.47857i		−	1.76790i	0.467577	−	0.883952i	$-0.345127\pi$
$24$		0			0
$25$	−7.54211			−1.50842
$26$		0			0
$27$		−	5.06941i		−	0.975608i
$28$		0			0
$29$		−	1.93174i		−	0.358715i	−0.983784	−	0.179357i	$-0.942598\pi$
$29$		−	1.93174i		−	0.358715i	0.983784	−	0.179357i	$-0.0574019\pi$
$30$		0			0
$31$		−	2.57764i		−	0.462958i	−0.972840	−	0.231479i	$-0.925644\pi$
$31$		−	2.57764i		−	0.462958i	0.972840	−	0.231479i	$-0.0743564\pi$
$32$		0			0
$33$	−0.463464			−0.0806788
$34$		0			0
$35$	−0.591220			−0.0999345
$36$		0			0
$37$			1.31550i			0.216267i	0.994136	+	0.108133i	$0.0344873\pi$
$37$			1.31550i			0.216267i	−0.994136	+	0.108133i	$0.965513\pi$
$38$		0			0
$39$		−	3.51984i		−	0.563625i
$40$		0			0
$41$			4.56154i			0.712393i	0.934411	+	0.356197i	$0.115927\pi$
$41$			4.56154i			0.712393i	−0.934411	+	0.356197i	$0.884073\pi$
$42$		0			0
$43$	−9.93091			−1.51445			−0.757225	−	0.653155i	$-0.773445\pi$
$43$	−9.93091			−1.51445			−0.757225	+	0.653155i	$0.773445\pi$
$44$		0			0
$45$			6.91310i			1.03054i
$46$		0			0
$47$	1.10859			0.161704			0.0808521	−	0.996726i	$-0.474236\pi$
$47$	1.10859			0.161704			0.0808521	+	0.996726i	$0.474236\pi$
$48$		0			0
$49$	6.97213			0.996019
$50$		0			0
$51$	0.0695354	−	4.22026i	0.00973690	−	0.590954i
$52$		0			0
$53$	−3.35231			−0.460475			−0.230238	−	0.973134i	$-0.573950\pi$
$53$	−3.35231			−0.460475			−0.230238	+	0.973134i	$0.573950\pi$
$54$		0			0
$55$	1.60335			0.216196
$56$		0			0
$57$		−	7.86927i		−	1.04231i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$		−	8.42710i		−	1.07898i	−0.841992	−	0.539490i	$-0.818617\pi$
$61$		−	8.42710i		−	1.07898i	0.841992	−	0.539490i	$-0.181383\pi$
$62$		0			0
$63$			0.325875i			0.0410565i
$64$		0			0
$65$			12.1768i			1.51035i
$66$		0			0
$67$	−8.66678			−1.05882			−0.529408	−	0.848368i	$-0.677586\pi$
$67$	−8.66678			−1.05882			−0.529408	+	0.848368i	$0.677586\pi$
$68$		0			0
$69$	−8.67953			−1.04489
$70$		0			0
$71$		−	14.4683i		−	1.71707i	−0.512756	−	0.858534i	$-0.671376\pi$
$71$		−	14.4683i		−	1.71707i	0.512756	−	0.858534i	$-0.328624\pi$
$72$		0			0
$73$		−	4.18682i		−	0.490030i	−0.969519	−	0.245015i	$-0.921207\pi$
$73$		−	4.18682i		−	0.490030i	0.969519	−	0.245015i	$-0.0787929\pi$
$74$		0			0
$75$			7.72087i			0.891529i
$76$		0			0
$77$	0.0755800			0.00861314
$78$		0			0
$79$			12.9491i			1.45689i	0.685106	+	0.728443i	$0.259756\pi$
$79$			12.9491i			1.45689i	−0.685106	+	0.728443i	$0.740244\pi$
$80$		0			0
$81$	0.666545			0.0740606
$82$		0			0
$83$	10.1547			1.11462			0.557310	−	0.830305i	$-0.311834\pi$
$83$	10.1547			1.11462			0.557310	+	0.830305i	$0.311834\pi$
$84$		0			0
$85$	−0.240557	+	14.5999i	−0.0260921	+	1.58358i
$86$		0			0
$87$	−1.97752			−0.212013
$88$		0			0
$89$	−1.62855			−0.172626			−0.0863128	−	0.996268i	$-0.527508\pi$
$89$	−1.62855			−0.172626			−0.0863128	+	0.996268i	$0.527508\pi$
$90$		0			0
$91$			0.574002i			0.0601718i
$92$		0			0
$93$	−2.63873			−0.273624
$94$		0			0
$95$			27.2236i			2.79309i
$96$		0			0
$97$			12.0497i			1.22346i	0.791068	+	0.611728i	$0.209525\pi$
$97$			12.0497i			1.22346i	−0.791068	+	0.611728i	$0.790475\pi$
$98$		0			0
$99$		−	0.883752i		−	0.0888205i
$100$		0			0
$101$	13.0293			1.29647			0.648233	−	0.761442i	$-0.275508\pi$
$101$	13.0293			1.29647			0.648233	+	0.761442i	$0.275508\pi$
$102$		0			0
$103$	12.3351			1.21542			0.607709	−	0.794160i	$-0.292089\pi$
$103$	12.3351			1.21542			0.607709	+	0.794160i	$0.292089\pi$
$104$		0			0
$105$			0.605233i			0.0590647i
$106$		0			0
$107$		−	7.79701i		−	0.753765i	−0.926261	−	0.376883i	$-0.876996\pi$
$107$		−	7.79701i		−	0.753765i	0.926261	−	0.376883i	$-0.123004\pi$
$108$		0			0
$109$		−	14.5241i		−	1.39116i	−0.718449	−	0.695580i	$-0.755148\pi$
$109$		−	14.5241i		−	1.39116i	0.718449	−	0.695580i	$-0.244852\pi$
$110$		0			0
$111$	1.34668			0.127821
$112$		0			0
$113$			6.30644i			0.593260i	0.954992	+	0.296630i	$0.0958628\pi$
$113$			6.30644i			0.593260i	−0.954992	+	0.296630i	$0.904137\pi$
$114$		0			0
$115$	30.0267			2.80001
$116$		0			0
$117$	6.71177			0.620503
$118$		0			0
$119$	−0.0113396	+	0.688224i	−0.00103950	+	0.0630894i
$120$		0			0
$121$	10.7950			0.981367
$122$		0			0
$123$	4.66966			0.421049
$124$		0			0
$125$		−	9.00284i		−	0.805238i
$126$		0			0
$127$	0.942343			0.0836194			0.0418097	−	0.999126i	$-0.486688\pi$
$127$	0.942343			0.0836194			0.0418097	+	0.999126i	$0.486688\pi$
$128$		0			0
$129$			10.1663i			0.895092i
$130$		0			0
$131$		−	13.3539i		−	1.16674i	−0.812207	−	0.583369i	$-0.801734\pi$
$131$		−	13.3539i		−	1.16674i	0.812207	−	0.583369i	$-0.198266\pi$
$132$		0			0
$133$			1.28329i			0.111275i
$134$		0			0
$135$	17.9532			1.54517
$136$		0			0
$137$	−18.8082			−1.60690			−0.803448	−	0.595375i	$-0.797004\pi$
$137$	−18.8082			−1.60690			−0.803448	+	0.595375i	$0.797004\pi$
$138$		0			0
$139$		−	12.0802i		−	1.02463i	−0.858799	−	0.512313i	$-0.828789\pi$
$139$		−	12.0802i		−	1.02463i	0.858799	−	0.512313i	$-0.171211\pi$
$140$		0			0
$141$		−	1.13486i		−	0.0955727i
$142$		0			0
$143$		−	1.55666i		−	0.130174i
$144$		0			0
$145$	6.84122			0.568133
$146$		0			0
$147$		−	7.13738i		−	0.588681i
$148$		0			0
$149$	−9.82169			−0.804624			−0.402312	−	0.915503i	$-0.631793\pi$
$149$	−9.82169			−0.804624			−0.402312	+	0.915503i	$0.631793\pi$
$150$		0			0
$151$	11.2191			0.912995			0.456498	−	0.889725i	$-0.349104\pi$
$151$	11.2191			0.912995			0.456498	+	0.889725i	$0.349104\pi$
$152$		0			0
$153$	8.04735	+	0.132593i	0.650590	+	0.0107195i
$154$		0			0
$155$	9.12867			0.733232
$156$		0			0
$157$	7.37290			0.588422			0.294211	−	0.955741i	$-0.404943\pi$
$157$	7.37290			0.588422			0.294211	+	0.955741i	$0.404943\pi$
$158$		0			0
$159$			3.43177i			0.272157i
$160$		0			0
$161$	1.41542			0.111551
$162$		0			0
$163$			8.45331i			0.662115i	0.943611	+	0.331057i	$0.107405\pi$
$163$			8.45331i			0.662115i	−0.943611	+	0.331057i	$0.892595\pi$
$164$		0			0
$165$		−	1.64135i		−	0.127779i
$166$		0			0
$167$			7.93308i			0.613880i	0.951729	+	0.306940i	$0.0993051\pi$
$167$			7.93308i			0.613880i	−0.951729	+	0.306940i	$0.900695\pi$
$168$		0			0
$169$	−1.17778			−0.0905985
$170$		0			0
$171$	15.0054			1.14749
$172$		0			0
$173$			6.64183i			0.504969i	0.967601	+	0.252484i	$0.0812476\pi$
$173$			6.64183i			0.504969i	−0.967601	+	0.252484i	$0.918752\pi$
$174$		0			0
$175$		−	1.25909i		−	0.0951783i
$176$		0			0
$177$			1.02370i			0.0769461i
$178$		0			0
$179$	−14.5948			−1.09087			−0.545434	−	0.838154i	$-0.683635\pi$
$179$	−14.5948			−1.09087			−0.545434	+	0.838154i	$0.683635\pi$
$180$		0			0
$181$			2.87944i			0.214027i	0.994258	+	0.107013i	$0.0341288\pi$
$181$			2.87944i			0.214027i	−0.994258	+	0.107013i	$0.965871\pi$
$182$		0			0
$183$	−8.62684			−0.637714
$184$		0			0
$185$	−4.65882			−0.342523
$186$		0			0
$187$	0.0307522	−	1.86642i	0.00224882	−	0.136486i
$188$		0			0
$189$	0.846294			0.0615588
$190$		0			0
$191$	19.8966			1.43967			0.719835	−	0.694145i	$-0.244218\pi$
$191$	19.8966			1.43967			0.719835	+	0.694145i	$0.244218\pi$
$192$		0			0
$193$			25.7962i			1.85685i	0.371521	+	0.928425i	$0.378836\pi$
$193$			25.7962i			1.85685i	−0.371521	+	0.928425i	$0.621164\pi$
$194$		0			0
$195$	12.4655			0.892670
$196$		0			0
$197$			12.1242i			0.863814i	0.901918	+	0.431907i	$0.142159\pi$
$197$			12.1242i			0.863814i	−0.901918	+	0.431907i	$0.857841\pi$
$198$		0			0
$199$			6.78094i			0.480688i	0.970688	+	0.240344i	$0.0772603\pi$
$199$			6.78094i			0.480688i	−0.970688	+	0.240344i	$0.922740\pi$
$200$		0			0
$201$			8.87219i			0.625796i
$202$		0			0
$203$	0.322487			0.0226342
$204$		0			0
$205$	−16.1546			−1.12829
$206$		0			0
$207$		−	16.5505i		−	1.15034i
$208$		0			0
$209$		−	3.48020i		−	0.240730i
$210$		0			0
$211$			26.1549i			1.80057i	0.435296	+	0.900287i	$0.356644\pi$
$211$			26.1549i			1.80057i	−0.435296	+	0.900287i	$0.643356\pi$
$212$		0			0
$213$	−14.8112			−1.01485
$214$		0			0
$215$		−	35.1702i		−	2.39858i
$216$		0			0
$217$	0.430315			0.0292117
$218$		0			0
$219$	−4.28605			−0.289625
$220$		0			0
$221$	14.1747	+	0.233551i	0.953496	+	0.0157104i
$222$		0			0
$223$	−9.95510			−0.666643			−0.333321	−	0.942813i	$-0.608169\pi$
$223$	−9.95510			−0.666643			−0.333321	+	0.942813i	$0.608169\pi$
$224$		0			0
$225$	−14.7225			−0.981497
$226$		0			0
$227$			6.44011i			0.427445i	0.976894	+	0.213722i	$0.0685588\pi$
$227$			6.44011i			0.427445i	−0.976894	+	0.213722i	$0.931441\pi$
$228$		0			0
$229$	4.74189			0.313353			0.156677	−	0.987650i	$-0.449922\pi$
$229$	4.74189			0.313353			0.156677	+	0.987650i	$0.449922\pi$
$230$		0			0
$231$		−	0.0773714i		−	0.00509067i
$232$		0			0
$233$			13.4188i			0.879098i	0.898219	+	0.439549i	$0.144862\pi$
$233$			13.4188i			0.879098i	−0.898219	+	0.439549i	$0.855138\pi$
$234$		0			0
$235$			3.92604i			0.256107i
$236$		0			0
$237$	13.2560			0.861070
$238$		0			0
$239$	−18.4896			−1.19599			−0.597997	−	0.801498i	$-0.704036\pi$
$239$	−18.4896			−1.19599			−0.597997	+	0.801498i	$0.704036\pi$
$240$		0			0
$241$		−	3.13031i		−	0.201641i	−0.994905	−	0.100821i	$-0.967853\pi$
$241$		−	3.13031i		−	0.201641i	0.994905	−	0.100821i	$-0.0321468\pi$
$242$		0			0
$243$		−	15.8906i		−	1.01938i
$244$		0			0
$245$			24.6917i			1.57749i
$246$		0			0
$247$	26.4308			1.68175
$248$		0			0
$249$		−	10.3953i		−	0.658778i
$250$		0			0
$251$	10.0906			0.636915			0.318458	−	0.947937i	$-0.396835\pi$
$251$	10.0906			0.636915			0.318458	+	0.947937i	$0.396835\pi$
$252$		0			0
$253$	−3.83854			−0.241327
$254$		0			0
$255$	14.9460	+	0.246258i	0.935953	+	0.0154213i
$256$		0			0
$257$	0.0348730			0.00217532			0.00108766	−	0.999999i	$-0.499654\pi$
$257$	0.0348730			0.00217532			0.00108766	+	0.999999i	$0.499654\pi$
$258$		0			0
$259$	−0.219611			−0.0136460
$260$		0			0
$261$		−	3.77082i		−	0.233408i
$262$		0			0
$263$	−10.8829			−0.671066			−0.335533	−	0.942029i	$-0.608916\pi$
$263$	−10.8829			−0.671066			−0.335533	+	0.942029i	$0.608916\pi$
$264$		0			0
$265$		−	11.8722i		−	0.729300i
$266$		0			0
$267$			1.66715i			0.102028i
$268$		0			0
$269$		−	3.14016i		−	0.191459i	−0.995407	−	0.0957294i	$-0.969482\pi$
$269$		−	3.14016i		−	0.191459i	0.995407	−	0.0957294i	$-0.0305184\pi$
$270$		0			0
$271$	−30.2632			−1.83836			−0.919178	−	0.393842i	$-0.871146\pi$
$271$	−30.2632			−1.83836			−0.919178	+	0.393842i	$0.871146\pi$
$272$		0			0
$273$	0.587607			0.0355636
$274$		0			0
$275$			3.41457i			0.205906i
$276$		0			0
$277$		−	2.72167i		−	0.163529i	−0.996652	−	0.0817647i	$-0.973944\pi$
$277$		−	2.72167i		−	0.163529i	0.996652	−	0.0817647i	$-0.0260556\pi$
$278$		0			0
$279$		−	5.03164i		−	0.301237i
$280$		0			0
$281$	−16.2686			−0.970505			−0.485252	−	0.874374i	$-0.661272\pi$
$281$	−16.2686			−0.970505			−0.485252	+	0.874374i	$0.661272\pi$
$282$		0			0
$283$			18.1798i			1.08067i	0.841449	+	0.540337i	$0.181703\pi$
$283$			18.1798i			1.08067i	−0.841449	+	0.540337i	$0.818297\pi$
$284$		0			0
$285$	27.8689			1.65081
$286$		0			0
$287$	−0.761510			−0.0449505
$288$		0			0
$289$	16.9908	+	0.560052i	0.999457	+	0.0329442i
$290$		0			0
$291$	12.3352			0.723105
$292$		0			0
$293$	−24.4764			−1.42992			−0.714962	−	0.699163i	$-0.753556\pi$
$293$	−24.4764			−1.42992			−0.714962	+	0.699163i	$0.753556\pi$
$294$		0			0
$295$		−	3.54148i		−	0.206193i
$296$		0			0
$297$	−2.29509			−0.133175
$298$		0			0
$299$		−	29.1523i		−	1.68592i
$300$		0			0
$301$		−	1.65788i		−	0.0955586i
$302$		0			0
$303$		−	13.3381i		−	0.766256i
$304$		0			0
$305$	29.8444			1.70889
$306$		0			0
$307$	−12.7268			−0.726358			−0.363179	−	0.931719i	$-0.618309\pi$
$307$	−12.7268			−0.726358			−0.363179	+	0.931719i	$0.618309\pi$
$308$		0			0
$309$		−	12.6275i		−	0.718354i
$310$		0			0
$311$		−	9.28424i		−	0.526461i	−0.964733	−	0.263231i	$-0.915212\pi$
$311$		−	9.28424i		−	0.526461i	0.964733	−	0.263231i	$-0.0847880\pi$
$312$		0			0
$313$			11.3396i			0.640954i	0.947256	+	0.320477i	$0.103843\pi$
$313$			11.3396i			0.640954i	−0.947256	+	0.320477i	$0.896157\pi$
$314$		0			0
$315$	−1.15408			−0.0650252
$316$		0			0
$317$		−	27.9217i		−	1.56824i	−0.620610	−	0.784120i	$-0.713115\pi$
$317$		−	27.9217i		−	1.56824i	0.620610	−	0.784120i	$-0.286885\pi$
$318$		0			0
$319$	−0.874564			−0.0489662
$320$		0			0
$321$	−7.98181			−0.445501
$322$		0			0
$323$	31.6903	+	0.522148i	1.76330	+	0.0290531i
$324$		0			0
$325$	−25.9324			−1.43847
$326$		0			0
$327$	−14.8684			−0.822223
$328$		0			0
$329$			0.185069i			0.0102032i
$330$		0			0
$331$	−13.5486			−0.744696			−0.372348	−	0.928093i	$-0.621447\pi$
$331$	−13.5486			−0.744696			−0.372348	+	0.928093i	$0.621447\pi$
$332$		0			0
$333$			2.56790i			0.140720i
$334$		0			0
$335$		−	30.6932i		−	1.67695i
$336$		0			0
$337$			19.2389i			1.04801i	0.851715	+	0.524006i	$0.175563\pi$
$337$			19.2389i			1.04801i	−0.851715	+	0.524006i	$0.824437\pi$
$338$		0			0
$339$	6.45592			0.350637
$340$		0			0
$341$	−1.16698			−0.0631958
$342$		0			0
$343$			2.33253i			0.125945i
$344$		0			0
$345$		−	30.7384i		−	1.65490i
$346$		0			0
$347$			9.30984i			0.499779i	0.968274	+	0.249889i	$0.0803942\pi$
$347$			9.30984i			0.499779i	−0.968274	+	0.249889i	$0.919606\pi$
$348$		0			0
$349$	36.7239			1.96579			0.982894	−	0.184173i	$-0.0589607\pi$
$349$	36.7239			1.96579			0.982894	+	0.184173i	$0.0589607\pi$
$350$		0			0
$351$		−	17.4304i		−	0.930364i
$352$		0			0
$353$	21.2944			1.13339			0.566694	−	0.823929i	$-0.308222\pi$
$353$	21.2944			1.13339			0.566694	+	0.823929i	$0.308222\pi$
$354$		0			0
$355$	51.2392			2.71949
$356$		0			0
$357$	0.704536	+	0.0116083i	0.0372880	+	0.000614378i
$358$		0			0
$359$	−7.72841			−0.407890			−0.203945	−	0.978982i	$-0.565376\pi$
$359$	−7.72841			−0.407890			−0.203945	+	0.978982i	$0.565376\pi$
$360$		0			0
$361$	40.0910			2.11005
$362$		0			0
$363$		−	11.0509i		−	0.580021i
$364$		0			0
$365$	14.8276			0.776110
$366$		0			0
$367$		−	2.16874i		−	0.113208i	−0.998397	−	0.0566038i	$-0.981973\pi$
$367$		−	2.16874i		−	0.113208i	0.998397	−	0.0566038i	$-0.0180272\pi$
$368$		0			0
$369$			8.90429i			0.463539i
$370$		0			0
$371$		−	0.559639i		−	0.0290550i
$372$		0			0
$373$	27.0395			1.40005			0.700027	−	0.714116i	$-0.253171\pi$
$373$	27.0395			1.40005			0.700027	+	0.714116i	$0.253171\pi$
$374$		0			0
$375$	−9.21622			−0.475923
$376$		0			0
$377$		−	6.64199i		−	0.342080i
$378$		0			0
$379$			15.7156i			0.807254i	0.914924	+	0.403627i	$0.132251\pi$
$379$			15.7156i			0.807254i	−0.914924	+	0.403627i	$0.867749\pi$
$380$		0			0
$381$		−	0.964678i		−	0.0494219i
$382$		0			0
$383$	21.8835			1.11819			0.559097	−	0.829102i	$-0.311148\pi$
$383$	21.8835			1.11819			0.559097	+	0.829102i	$0.311148\pi$
$384$		0			0
$385$			0.267665i			0.0136415i
$386$		0			0
$387$	−19.3855			−0.985419
$388$		0			0
$389$	−15.3134			−0.776422			−0.388211	−	0.921571i	$-0.626907\pi$
$389$	−15.3134			−0.776422			−0.388211	+	0.921571i	$0.626907\pi$
$390$		0			0
$391$	0.575911	−	34.9533i	0.0291251	−	1.76766i
$392$		0			0
$393$	−13.6704			−0.689583
$394$		0			0
$395$	−45.8590			−2.30742
$396$		0			0
$397$		−	3.77544i		−	0.189484i	−0.995502	−	0.0947420i	$-0.969797\pi$
$397$		−	3.77544i		−	0.189484i	0.995502	−	0.0947420i	$-0.0302026\pi$
$398$		0			0
$399$	1.31371			0.0657676
$400$		0			0
$401$			16.9195i			0.844921i	0.906381	+	0.422460i	$0.138833\pi$
$401$			16.9195i			0.844921i	−0.906381	+	0.422460i	$0.861167\pi$
$402$		0			0
$403$		−	8.86282i		−	0.441488i
$404$		0			0
$405$			2.36056i			0.117297i
$406$		0			0
$407$	0.595571			0.0295213
$408$		0			0
$409$	19.4098			0.959755			0.479877	−	0.877336i	$-0.340681\pi$
$409$	19.4098			0.959755			0.479877	+	0.877336i	$0.340681\pi$
$410$		0			0
$411$			19.2540i			0.949731i
$412$		0			0
$413$		−	0.166941i		−	0.00821465i
$414$		0			0
$415$			35.9626i			1.76533i
$416$		0			0
$417$	−12.3665			−0.605589
$418$		0			0
$419$		−	25.9425i		−	1.26737i	−0.773590	−	0.633687i	$-0.781541\pi$
$419$		−	25.9425i		−	1.26737i	0.773590	−	0.633687i	$-0.218459\pi$
$420$		0			0
$421$	−33.2979			−1.62284			−0.811421	−	0.584462i	$-0.801306\pi$
$421$	−33.2979			−1.62284			−0.811421	+	0.584462i	$0.801306\pi$
$422$		0			0
$423$	2.16400			0.105217
$424$		0			0
$425$	−31.0927	−	0.512301i	−1.50822	−	0.0248503i
$426$		0			0
$427$	1.40683			0.0680814
$428$		0			0
$429$	−1.59355			−0.0769374
$430$		0			0
$431$		−	22.5681i		−	1.08707i	−0.839388	−	0.543533i	$-0.817086\pi$
$431$		−	22.5681i		−	1.08707i	0.839388	−	0.543533i	$-0.182914\pi$
$432$		0			0
$433$	−8.02995			−0.385895			−0.192947	−	0.981209i	$-0.561805\pi$
$433$	−8.02995			−0.385895			−0.192947	+	0.981209i	$0.561805\pi$
$434$		0			0
$435$		−	7.00337i		−	0.335786i
$436$		0			0
$437$		−	65.1754i		−	3.11776i
$438$		0			0
$439$			31.3823i			1.49779i	0.662686	+	0.748897i	$0.269416\pi$
$439$			31.3823i			1.49779i	−0.662686	+	0.748897i	$0.730584\pi$
$440$		0			0
$441$	13.6098			0.648088
$442$		0			0
$443$	12.7952			0.607920			0.303960	−	0.952685i	$-0.401691\pi$
$443$	12.7952			0.607920			0.303960	+	0.952685i	$0.401691\pi$
$444$		0			0
$445$		−	5.76747i		−	0.273404i
$446$		0			0
$447$			10.0545i			0.475561i
$448$		0			0
$449$		−	6.99517i		−	0.330122i	−0.986283	−	0.165061i	$-0.947218\pi$
$449$		−	6.99517i		−	0.330122i	0.986283	−	0.165061i	$-0.0527821\pi$
$450$		0			0
$451$	2.06516			0.0972448
$452$		0			0
$453$		−	11.4850i		−	0.539612i
$454$		0			0
$455$	−2.03282			−0.0953001
$456$		0			0
$457$	−9.80566			−0.458689			−0.229345	−	0.973345i	$-0.573658\pi$
$457$	−9.80566			−0.458689			−0.229345	+	0.973345i	$0.573658\pi$
$458$		0			0
$459$	0.344342	−	20.8989i	0.0160725	−	0.975475i
$460$		0			0
$461$	−40.4007			−1.88165			−0.940825	−	0.338893i	$-0.889948\pi$
$461$	−40.4007			−1.88165			−0.940825	+	0.338893i	$0.889948\pi$
$462$		0			0
$463$	19.3225			0.897991			0.448995	−	0.893534i	$-0.351782\pi$
$463$	19.3225			0.897991			0.448995	+	0.893534i	$0.351782\pi$
$464$		0			0
$465$		−	9.34503i		−	0.433366i
$466$		0			0
$467$	5.99232			0.277291			0.138646	−	0.990342i	$-0.455725\pi$
$467$	5.99232			0.277291			0.138646	+	0.990342i	$0.455725\pi$
$468$		0			0
$469$		−	1.44684i		−	0.0668090i
$470$		0			0
$471$		−	7.54765i		−	0.347777i
$472$		0			0
$473$			4.49606i			0.206729i
$474$		0			0
$475$	−57.9767			−2.66015
$476$		0			0
$477$	−6.54383			−0.299621
$478$		0			0
$479$			5.14717i			0.235180i	0.993062	+	0.117590i	$0.0375169\pi$
$479$			5.14717i			0.235180i	−0.993062	+	0.117590i	$0.962483\pi$
$480$		0			0
$481$			4.52314i			0.206237i
$482$		0			0
$483$		−	1.44897i		−	0.0659305i
$484$		0			0
$485$	−42.6736			−1.93771
$486$		0			0
$487$		−	5.30317i		−	0.240310i	−0.992755	−	0.120155i	$-0.961661\pi$
$487$		−	5.30317i		−	0.240310i	0.992755	−	0.120155i	$-0.0383391\pi$
$488$		0			0
$489$	8.65367			0.391333
$490$		0			0
$491$	−6.00075			−0.270810			−0.135405	−	0.990790i	$-0.543234\pi$
$491$	−6.00075			−0.270810			−0.135405	+	0.990790i	$0.543234\pi$
$492$		0			0
$493$	0.131214	−	7.96368i	0.00590959	−	0.358666i
$494$		0			0
$495$	3.12979			0.140674
$496$		0			0
$497$	2.41535			0.108343
$498$		0			0
$499$			34.7078i			1.55373i	0.629665	+	0.776867i	$0.283192\pi$
$499$			34.7078i			1.55373i	−0.629665	+	0.776867i	$0.716808\pi$
$500$		0			0
$501$	8.12110			0.362824
$502$		0			0
$503$			35.2610i			1.57221i	0.618091	+	0.786106i	$0.287906\pi$
$503$			35.2610i			1.57221i	−0.618091	+	0.786106i	$0.712094\pi$
$504$		0			0
$505$			46.1431i			2.05334i
$506$		0			0
$507$			1.20570i			0.0535468i
$508$		0			0
$509$	−13.0760			−0.579582			−0.289791	−	0.957090i	$-0.593586\pi$
$509$	−13.0760			−0.579582			−0.289791	+	0.957090i	$0.593586\pi$
$510$		0			0
$511$	0.698954			0.0309199
$512$		0			0
$513$		−	38.9689i		−	1.72052i
$514$		0			0
$515$			43.6847i			1.92498i
$516$		0			0
$517$		−	0.501895i		−	0.0220733i
$518$		0			0
$519$	6.79925			0.298454
$520$		0			0
$521$			8.02099i			0.351406i	0.984443	+	0.175703i	$0.0562199\pi$
$521$			8.02099i			0.351406i	−0.984443	+	0.175703i	$0.943780\pi$
$522$		0			0
$523$	5.32812			0.232982			0.116491	−	0.993192i	$-0.462835\pi$
$523$	5.32812			0.232982			0.116491	+	0.993192i	$0.462835\pi$
$524$		0			0
$525$	−1.28893			−0.0562536
$526$		0			0
$527$	0.175087	−	10.6264i	0.00762692	−	0.462895i
$528$		0			0
$529$	−48.8862			−2.12549
$530$		0			0
$531$	−1.95203			−0.0847111
$532$		0			0
$533$			15.6842i			0.679356i
$534$		0			0
$535$	27.6130			1.19381
$536$		0			0
$537$			14.9407i			0.644740i
$538$		0			0
$539$		−	3.15652i		−	0.135961i
$540$		0			0
$541$			9.67717i			0.416054i	0.978123	+	0.208027i	$0.0667042\pi$
$541$			9.67717i			0.416054i	−0.978123	+	0.208027i	$0.933296\pi$
$542$		0			0
$543$	2.94768			0.126497
$544$		0			0
$545$	51.4370			2.20332
$546$		0			0
$547$		−	15.6929i		−	0.670979i	−0.942044	−	0.335489i	$-0.891098\pi$
$547$		−	15.6929i		−	0.670979i	0.942044	−	0.335489i	$-0.108902\pi$
$548$		0			0
$549$		−	16.4500i		−	0.702069i
$550$		0			0
$551$		−	14.8494i		−	0.632606i
$552$		0			0
$553$	−2.16174			−0.0919265
$554$		0			0
$555$			4.76924i			0.202443i
$556$		0			0
$557$	−44.9506			−1.90462			−0.952309	−	0.305136i	$-0.901298\pi$
$557$	−44.9506			−1.90462			−0.952309	+	0.305136i	$0.901298\pi$
$558$		0			0
$559$	−34.1459			−1.44422
$560$		0			0
$561$	−1.91065	−	0.0314810i	−0.0806679	−	0.00132913i
$562$		0			0
$563$	11.2760			0.475226			0.237613	−	0.971360i	$-0.423635\pi$
$563$	11.2760			0.475226			0.237613	+	0.971360i	$0.423635\pi$
$564$		0			0
$565$	−22.3342			−0.939605
$566$		0			0
$567$			0.111274i			0.00467307i
$568$		0			0
$569$	26.4346			1.10819			0.554097	−	0.832452i	$-0.313064\pi$
$569$	26.4346			1.10819			0.554097	+	0.832452i	$0.313064\pi$
$570$		0			0
$571$		−	37.3794i		−	1.56428i	−0.623102	−	0.782141i	$-0.714128\pi$
$571$		−	37.3794i		−	1.56428i	0.623102	−	0.782141i	$-0.285872\pi$
$572$		0			0
$573$		−	20.3682i		−	0.850894i
$574$		0			0
$575$			63.9463i			2.66674i
$576$		0			0
$577$	−1.80909			−0.0753136			−0.0376568	−	0.999291i	$-0.511989\pi$
$577$	−1.80909			−0.0753136			−0.0376568	+	0.999291i	$0.511989\pi$
$578$		0			0
$579$	26.4076			1.09746
$580$		0			0
$581$			1.69523i			0.0703302i
$582$		0			0
$583$			1.51770i			0.0628569i
$584$		0			0
$585$			23.7696i			0.982753i
$586$		0			0
$587$	25.4265			1.04946			0.524731	−	0.851268i	$-0.324166\pi$
$587$	25.4265			1.04946			0.524731	+	0.851268i	$0.324166\pi$
$588$		0			0
$589$		−	19.8145i		−	0.816442i
$590$		0			0
$591$	12.4116			0.510544
$592$		0			0
$593$	−3.99697			−0.164136			−0.0820678	−	0.996627i	$-0.526152\pi$
$593$	−3.99697			−0.164136			−0.0820678	+	0.996627i	$0.526152\pi$
$594$		0			0
$595$	−2.43733	−	0.0401589i	−0.0999209	−	0.00164635i
$596$		0			0
$597$	6.94166			0.284103
$598$		0			0
$599$	27.0011			1.10323			0.551617	−	0.834097i	$-0.314011\pi$
$599$	27.0011			1.10323			0.551617	+	0.834097i	$0.314011\pi$
$600$		0			0
$601$		−	5.64762i		−	0.230371i	−0.993344	−	0.115186i	$-0.963254\pi$
$601$		−	5.64762i		−	0.230371i	0.993344	−	0.115186i	$-0.0367463\pi$
$602$		0			0
$603$	−16.9178			−0.688948
$604$		0			0
$605$			38.2304i			1.55429i
$606$		0			0
$607$		−	37.2751i		−	1.51295i	−0.654023	−	0.756475i	$-0.726920\pi$
$607$		−	37.2751i		−	1.51295i	0.654023	−	0.756475i	$-0.273080\pi$
$608$		0			0
$609$		−	0.330131i		−	0.0133776i
$610$		0			0
$611$	3.81171			0.154205
$612$		0			0
$613$	−31.4075			−1.26854			−0.634269	−	0.773112i	$-0.718699\pi$
$613$	−31.4075			−1.26854			−0.634269	+	0.773112i	$0.718699\pi$
$614$		0			0
$615$			16.5375i			0.666857i
$616$		0			0
$617$			27.7862i			1.11863i	0.828956	+	0.559314i	$0.188935\pi$
$617$			27.7862i			1.11863i	−0.828956	+	0.559314i	$0.811065\pi$
$618$		0			0
$619$			36.8215i			1.47998i	0.672616	+	0.739991i	$0.265170\pi$
$619$			36.8215i			1.47998i	−0.672616	+	0.739991i	$0.734830\pi$
$620$		0			0
$621$	−42.9813			−1.72478
$622$		0			0
$623$		−	0.271872i		−	0.0108923i
$624$		0			0
$625$	−5.82714			−0.233086
$626$		0			0
$627$	−3.56268			−0.142280
$628$		0			0
$629$	−0.0893558	+	5.42320i	−0.00356285	+	0.216237i
$630$		0			0
$631$	−12.8044			−0.509736			−0.254868	−	0.966976i	$-0.582032\pi$
$631$	−12.8044			−0.509736			−0.254868	+	0.966976i	$0.582032\pi$
$632$		0			0
$633$	26.7748			1.06420
$634$		0			0
$635$			3.33729i			0.132436i
$636$		0			0
$637$	23.9726			0.949829
$638$		0			0
$639$		−	28.2426i		−	1.11726i
$640$		0			0
$641$		−	45.2848i		−	1.78864i	−0.447426	−	0.894321i	$-0.647659\pi$
$641$		−	45.2848i		−	1.78864i	0.447426	−	0.894321i	$-0.352341\pi$
$642$		0			0
$643$			46.9370i			1.85101i	0.378731	+	0.925507i	$0.376361\pi$
$643$			46.9370i			1.85101i	−0.378731	+	0.925507i	$0.623639\pi$
$644$		0			0
$645$	−36.0037			−1.41765
$646$		0			0
$647$	10.3376			0.406412			0.203206	−	0.979136i	$-0.434864\pi$
$647$	10.3376			0.406412			0.203206	+	0.979136i	$0.434864\pi$
$648$		0			0
$649$			0.452734i			0.0177714i
$650$		0			0
$651$		−	0.440514i		−	0.0172651i
$652$		0			0
$653$		−	44.1028i		−	1.72588i	−0.505309	−	0.862938i	$-0.668622\pi$
$653$		−	44.1028i		−	1.72588i	0.505309	−	0.862938i	$-0.331378\pi$
$654$		0			0
$655$	47.2927			1.84788
$656$		0			0
$657$		−	8.17282i		−	0.318852i
$658$		0			0
$659$	−24.1383			−0.940294			−0.470147	−	0.882588i	$-0.655799\pi$
$659$	−24.1383			−0.940294			−0.470147	+	0.882588i	$0.655799\pi$
$660$		0			0
$661$	22.4443			0.872984			0.436492	−	0.899708i	$-0.356221\pi$
$661$	22.4443			0.872984			0.436492	+	0.899708i	$0.356221\pi$
$662$		0			0
$663$	0.239087	−	14.5107i	0.00928536	−	0.563549i
$664$		0			0
$665$	−4.54475			−0.176238
$666$		0			0
$667$	−16.3784			−0.634174
$668$		0			0
$669$			10.1911i			0.394009i
$670$		0			0
$671$	−3.81523			−0.147285
$672$		0			0
$673$			35.2869i			1.36021i	0.733115	+	0.680105i	$0.238066\pi$
$673$			35.2869i			1.36021i	−0.733115	+	0.680105i	$0.761934\pi$
$674$		0			0
$675$			38.2340i			1.47163i
$676$		0			0
$677$			15.6161i			0.600176i	0.953912	+	0.300088i	$0.0970160\pi$
$677$			15.6161i			0.600176i	−0.953912	+	0.300088i	$0.902984\pi$
$678$		0			0
$679$	−2.01159			−0.0771976
$680$		0			0
$681$	6.59275			0.252635
$682$		0			0
$683$		−	19.5712i		−	0.748872i	−0.927253	−	0.374436i	$-0.877836\pi$
$683$		−	19.5712i		−	0.748872i	0.927253	−	0.374436i	$-0.122164\pi$
$684$		0			0
$685$		−	66.6091i		−	2.54500i
$686$		0			0
$687$		−	4.85428i		−	0.185203i
$688$		0			0
$689$	−11.5264			−0.439121
$690$		0			0
$691$			39.9762i			1.52077i	0.649475	+	0.760383i	$0.274989\pi$
$691$			39.9762i			1.52077i	−0.649475	+	0.760383i	$0.725011\pi$
$692$		0			0
$693$	0.147535			0.00560439
$694$		0			0
$695$	42.7817			1.62280
$696$		0			0
$697$	−0.309845	+	18.8052i	−0.0117362	+	0.712297i
$698$		0			0
$699$	13.7369			0.519577
$700$		0			0
$701$	−27.4250			−1.03583			−0.517914	−	0.855433i	$-0.673291\pi$
$701$	−27.4250			−1.03583			−0.517914	+	0.855433i	$0.673291\pi$
$702$		0			0
$703$			10.1123i			0.381394i
$704$		0			0
$705$	4.01910			0.151368
$706$		0			0
$707$			2.17513i			0.0818043i
$708$		0			0
$709$			10.9104i			0.409749i	0.978788	+	0.204874i	$0.0656786\pi$
$709$			10.9104i			0.409749i	−0.978788	+	0.204874i	$0.934321\pi$
$710$		0			0
$711$			25.2771i			0.947964i
$712$		0			0
$713$	−21.8547			−0.818465
$714$		0			0
$715$	5.51287			0.206170
$716$		0			0
$717$			18.9278i			0.706873i
$718$		0			0
$719$		−	2.31372i		−	0.0862871i	−0.999069	−	0.0431436i	$-0.986263\pi$
$719$		−	2.31372i		−	0.0862871i	0.999069	−	0.0431436i	$-0.0137373\pi$
$720$		0			0
$721$			2.05925i			0.0766903i
$722$		0			0
$723$	−3.20450			−0.119177
$724$		0			0
$725$			14.5694i			0.541093i
$726$		0			0
$727$	1.25798			0.0466560			0.0233280	−	0.999728i	$-0.492574\pi$
$727$	1.25798			0.0466560			0.0233280	+	0.999728i	$0.492574\pi$
$728$		0			0
$729$	−14.2676			−0.528428
$730$		0			0
$731$	−40.9406	−	0.674561i	−1.51424	−	0.0249496i
$732$		0			0
$733$	−35.1793			−1.29938			−0.649689	−	0.760200i	$-0.725101\pi$
$733$	−35.1793			−1.29938			−0.649689	+	0.760200i	$0.725101\pi$
$734$		0			0
$735$	25.2769			0.932353
$736$		0			0
$737$			3.92374i			0.144533i
$738$		0			0
$739$	0.0244509			0.000899441			0.000449720	−	1.00000i	$-0.499857\pi$
$739$	0.0244509			0.000899441			0.000449720		1.00000i	$0.499857\pi$
$740$		0			0
$741$		−	27.0573i		−	0.993973i
$742$		0			0
$743$			31.3889i			1.15155i	0.817609	+	0.575774i	$0.195299\pi$
$743$			31.3889i			1.15155i	−0.817609	+	0.575774i	$0.804701\pi$
$744$		0			0
$745$		−	34.7834i		−	1.27436i
$746$		0			0
$747$	19.8223			0.725259
$748$		0			0
$749$	1.30164			0.0475610
$750$		0			0
$751$		−	20.3511i		−	0.742621i	−0.928509	−	0.371310i	$-0.878909\pi$
$751$		−	20.3511i		−	0.742621i	0.928509	−	0.371310i	$-0.121091\pi$
$752$		0			0
$753$		−	10.3298i		−	0.376439i
$754$		0			0
$755$			39.7322i			1.44600i
$756$		0			0
$757$	3.32526			0.120859			0.0604293	−	0.998172i	$-0.480753\pi$
$757$	3.32526			0.120859			0.0604293	+	0.998172i	$0.480753\pi$
$758$		0			0
$759$			3.92952i			0.142632i
$760$		0			0
$761$	−43.3994			−1.57323			−0.786614	−	0.617445i	$-0.788168\pi$
$761$	−43.3994			−1.57323			−0.786614	+	0.617445i	$0.788168\pi$
$762$		0			0
$763$	2.42468			0.0877793
$764$		0			0
$765$	−0.469575	+	28.4996i	−0.0169775	+	1.03040i
$766$		0			0
$767$	−3.43835			−0.124151
$768$		0			0
$769$	−0.917459			−0.0330844			−0.0165422	−	0.999863i	$-0.505266\pi$
$769$	−0.917459			−0.0330844			−0.0165422	+	0.999863i	$0.505266\pi$
$770$		0			0
$771$		−	0.0356996i		−	0.00128569i
$772$		0			0
$773$	−26.3136			−0.946436			−0.473218	−	0.880945i	$-0.656908\pi$
$773$	−26.3136			−0.946436			−0.473218	+	0.880945i	$0.656908\pi$
$774$		0			0
$775$			19.4408i			0.698335i
$776$		0			0
$777$			0.224816i			0.00806524i
$778$		0			0
$779$			35.0649i			1.25633i
$780$		0			0
$781$	−6.55028			−0.234387
$782$		0			0
$783$	−9.79277			−0.349965
$784$		0			0
$785$			26.1110i			0.931942i
$786$		0			0
$787$			25.5004i			0.908990i	0.890749	+	0.454495i	$0.150180\pi$
$787$			25.5004i			0.908990i	−0.890749	+	0.454495i	$0.849820\pi$
$788$		0			0
$789$			11.1408i			0.396623i
$790$		0			0
$791$	−1.05281			−0.0374335
$792$		0			0
$793$		−	28.9753i		−	1.02894i
$794$		0			0
$795$	−12.1535			−0.431042
$796$		0			0
$797$	−28.0359			−0.993084			−0.496542	−	0.868013i	$-0.665397\pi$
$797$	−28.0359			−0.993084			−0.496542	+	0.868013i	$0.665397\pi$
$798$		0			0
$799$	4.57020	+	0.0753013i	0.161682	+	0.00266397i
$800$		0			0
$801$	−3.17898			−0.112324
$802$		0			0
$803$	−1.89552			−0.0668913
$804$		0			0
$805$			5.01270i			0.176675i
$806$		0			0
$807$	−3.21459			−0.113159
$808$		0			0
$809$			17.0434i			0.599215i	0.954062	+	0.299608i	$0.0968558\pi$
$809$			17.0434i			0.599215i	−0.954062	+	0.299608i	$0.903144\pi$
$810$		0			0
$811$			14.9913i			0.526417i	0.964739	+	0.263208i	$0.0847807\pi$
$811$			14.9913i			0.526417i	−0.964739	+	0.263208i	$0.915219\pi$
$812$		0			0
$813$			30.9804i			1.08653i
$814$		0			0
$815$	−29.9373			−1.04866
$816$		0			0
$817$	−76.3396			−2.67078
$818$		0			0
$819$			1.12047i			0.0391525i
$820$		0			0
$821$		−	34.4623i		−	1.20274i	−0.798969	−	0.601372i	$-0.794621\pi$
$821$		−	34.4623i		−	1.20274i	0.798969	−	0.601372i	$-0.205379\pi$
$822$		0			0
$823$		−	22.7943i		−	0.794559i	−0.917698	−	0.397280i	$-0.869954\pi$
$823$		−	22.7943i		−	0.794559i	0.917698	−	0.397280i	$-0.130046\pi$
$824$		0			0
$825$	3.49550			0.121698
$826$		0			0
$827$			22.5911i			0.785568i	0.919631	+	0.392784i	$0.128488\pi$
$827$			22.5911i			0.785568i	−0.919631	+	0.392784i	$0.871512\pi$
$828$		0			0
$829$	41.4732			1.44043			0.720213	−	0.693753i	$-0.244044\pi$
$829$	41.4732			1.44043			0.720213	+	0.693753i	$0.244044\pi$
$830$		0			0
$831$	−2.78618			−0.0966515
$832$		0			0
$833$	28.7429	+	0.473585i	0.995883	+	0.0164088i
$834$		0			0
$835$	−28.0949			−0.972263
$836$		0			0
$837$	−13.0671			−0.451665
$838$		0			0
$839$			8.55106i			0.295216i	0.989046	+	0.147608i	$0.0471573\pi$
$839$			8.55106i			0.295216i	−0.989046	+	0.147608i	$0.952843\pi$
$840$		0			0
$841$	25.2684			0.871324
$842$		0			0
$843$			16.6542i			0.573602i
$844$		0			0
$845$		−	4.17109i		−	0.143490i
$846$		0			0
$847$			1.80214i			0.0619222i
$848$		0			0
$849$	18.6106			0.638716
$850$		0			0
$851$	11.1535			0.382339
$852$		0			0
$853$		−	43.8416i		−	1.50111i	−0.660809	−	0.750554i	$-0.729786\pi$
$853$		−	43.8416i		−	1.50111i	0.660809	−	0.750554i	$-0.270214\pi$
$854$		0			0
$855$			53.1415i			1.81740i
$856$		0			0
$857$			31.9231i			1.09047i	0.838282	+	0.545237i	$0.183560\pi$
$857$			31.9231i			1.09047i	−0.838282	+	0.545237i	$0.816440\pi$
$858$		0			0
$859$	43.7430			1.49249			0.746245	−	0.665671i	$-0.231855\pi$
$859$	43.7430			1.49249			0.746245	+	0.665671i	$0.231855\pi$
$860$		0			0
$861$			0.779559i			0.0265673i
$862$		0			0
$863$	−29.6490			−1.00926			−0.504631	−	0.863335i	$-0.668372\pi$
$863$	−29.6490			−1.00926			−0.504631	+	0.863335i	$0.668372\pi$
$864$		0			0
$865$	−23.5219			−0.799770
$866$		0			0
$867$	0.573326	−	17.3935i	0.0194712	−	0.590714i
$868$		0			0
$869$	5.86249			0.198871
$870$		0			0
$871$	−29.7994			−1.00971
$872$		0			0
$873$			23.5213i			0.796077i
$874$		0			0
$875$	1.50295			0.0508089
$876$		0			0
$877$		−	8.06461i		−	0.272323i	−0.990687	−	0.136161i	$-0.956523\pi$
$877$		−	8.06461i		−	0.272323i	0.990687	−	0.136161i	$-0.0434765\pi$
$878$		0			0
$879$			25.0565i			0.845134i
$880$		0			0
$881$			2.50249i			0.0843109i	0.999111	+	0.0421555i	$0.0134225\pi$
$881$			2.50249i			0.0843109i	−0.999111	+	0.0421555i	$0.986578\pi$
$882$		0			0
$883$	−48.9697			−1.64796			−0.823981	−	0.566617i	$-0.808252\pi$
$883$	−48.9697			−1.64796			−0.823981	+	0.566617i	$0.808252\pi$
$884$		0			0
$885$	−3.62542			−0.121867
$886$		0			0
$887$		−	21.0384i		−	0.706399i	−0.935548	−	0.353200i	$-0.885094\pi$
$887$		−	21.0384i		−	0.706399i	0.935548	−	0.353200i	$-0.114906\pi$
$888$		0			0
$889$			0.157316i			0.00527621i
$890$		0			0
$891$		−	0.301768i		−	0.0101096i
$892$		0			0
$893$	8.52179			0.285171
$894$		0			0
$895$		−	51.6873i		−	1.72772i
$896$		0			0
$897$	−29.8432			−0.996436
$898$		0			0
$899$	−4.97933			−0.166070
$900$		0			0
$901$	−13.8201	−	0.227707i	−0.460413	−	0.00758603i
$902$		0			0
$903$	−1.69717			−0.0564784
$904$		0			0
$905$	−10.1975			−0.338976
$906$		0			0
$907$		−	42.5635i		−	1.41330i	−0.707565	−	0.706648i	$-0.750206\pi$
$907$		−	42.5635i		−	1.41330i	0.707565	−	0.706648i	$-0.249794\pi$
$908$		0			0
$909$	25.4337			0.843582
$910$		0			0
$911$			10.2969i			0.341152i	0.985345	+	0.170576i	$0.0545629\pi$
$911$			10.2969i			0.341152i	−0.985345	+	0.170576i	$0.945437\pi$
$912$		0			0
$913$		−	4.59736i		−	0.152150i
$914$		0			0
$915$		−	30.5518i		−	1.01001i
$916$		0			0
$917$	2.22932			0.0736188
$918$		0			0
$919$	−16.9885			−0.560399			−0.280199	−	0.959942i	$-0.590401\pi$
$919$	−16.9885			−0.560399			−0.280199	+	0.959942i	$0.590401\pi$
$920$		0			0
$921$			13.0285i			0.429303i
$922$		0			0
$923$		−	49.7469i		−	1.63744i
$924$		0			0
$925$		−	9.92163i		−	0.326221i
$926$		0			0
$927$	24.0786			0.790846
$928$		0			0
$929$		−	30.0295i		−	0.985234i	−0.870246	−	0.492617i	$-0.836040\pi$
$929$		−	30.0295i		−	0.985234i	0.870246	−	0.492617i	$-0.163960\pi$
$930$		0			0
$931$	53.5953			1.75651
$932$		0			0
$933$	−9.50430			−0.311157
$934$		0			0
$935$	6.60988	+	0.108908i	0.216166	+	0.00356168i
$936$		0			0
$937$	21.7572			0.710776			0.355388	−	0.934719i	$-0.384349\pi$
$937$	21.7572			0.710776			0.355388	+	0.934719i	$0.384349\pi$
$938$		0			0
$939$	11.6084			0.378826
$940$		0			0
$941$		−	0.149812i		−	0.00488375i	−0.999997	−	0.00244187i	$-0.999223\pi$
$941$		−	0.149812i		−	0.00488375i	0.999997	−	0.00244187i	$-0.000777273\pi$
$942$		0			0
$943$	38.6754			1.25944
$944$		0			0
$945$			2.99714i			0.0974968i
$946$		0			0
$947$		−	2.76167i		−	0.0897422i	−0.998993	−	0.0448711i	$-0.985712\pi$
$947$		−	2.76167i		−	0.0897422i	0.998993	−	0.0448711i	$-0.0142877\pi$
$948$		0			0
$949$		−	14.3957i		−	0.467305i
$950$		0			0
$951$	−28.5835			−0.926884
$952$		0			0
$953$	−4.17245			−0.135159			−0.0675795	−	0.997714i	$-0.521528\pi$
$953$	−4.17245			−0.135159			−0.0675795	+	0.997714i	$0.521528\pi$
$954$		0			0
$955$			70.4636i			2.28015i
$956$		0			0
$957$			0.895292i			0.0289407i
$958$		0			0
$959$		−	3.13987i		−	0.101392i
$960$		0			0
$961$	24.3558			0.785670
$962$		0			0
$963$		−	15.2200i		−	0.490459i
$964$		0			0
$965$	−91.3567			−2.94088
$966$		0			0
$967$	25.7004			0.826469			0.413235	−	0.910625i	$-0.364399\pi$
$967$	25.7004			0.826469			0.413235	+	0.910625i	$0.364399\pi$
$968$		0			0
$969$	0.534523	−	32.4414i	0.0171714	−	1.04217i
$970$		0			0
$971$	−45.8981			−1.47294			−0.736470	−	0.676471i	$-0.763509\pi$
$971$	−45.8981			−1.47294			−0.736470	+	0.676471i	$0.763509\pi$
$972$		0			0
$973$	2.01668			0.0646517
$974$		0			0
$975$			26.5470i			0.850185i
$976$		0			0
$977$	−22.0722			−0.706153			−0.353077	−	0.935594i	$-0.614865\pi$
$977$	−22.0722			−0.706153			−0.353077	+	0.935594i	$0.614865\pi$
$978$		0			0
$979$			0.737298i			0.0235641i
$980$		0			0
$981$		−	28.3516i		−	0.905197i
$982$		0			0
$983$		−	32.6822i		−	1.04240i	−0.853435	−	0.521200i	$-0.825485\pi$
$983$		−	32.6822i		−	1.04240i	0.853435	−	0.521200i	$-0.174515\pi$
$984$		0			0
$985$	−42.9377			−1.36811
$986$		0			0
$987$	0.189456			0.00603044
$988$		0			0
$989$			84.1999i			2.67740i
$990$		0			0
$991$			53.2087i			1.69023i	0.534584	+	0.845115i	$0.320468\pi$
$991$			53.2087i			1.69023i	−0.534584	+	0.845115i	$0.679532\pi$
$992$		0			0
$993$			13.8697i			0.440141i
$994$		0			0
$995$	−24.0146			−0.761314
$996$		0			0
$997$			24.8766i			0.787851i	0.919142	+	0.393925i	$0.128883\pi$
$997$			24.8766i			0.787851i	−0.919142	+	0.393925i	$0.871117\pi$
$998$		0			0
$999$	6.66879			0.210991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.15	✓	40
17.16	even	2	inner	4012.2.b.a.237.26	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.15	✓	40	1.1	even	1	trivial
4012.2.b.a.237.26	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)

\(f(q)\)	\(=\)	\(q-1.02370i q^{3} +3.54148i q^{5} +0.166941i q^{7} +1.95203 q^{9} +O(q^{10})\)	\(q-1.02370i q^{3} +3.54148i q^{5} +0.166941i q^{7} +1.95203 q^{9} -0.452734i q^{11} +3.43835 q^{13} +3.62542 q^{15} +(4.12255 + 0.0679254i) q^{17} +7.68707 q^{19} +0.170898 q^{21} -8.47857i q^{23} -7.54211 q^{25} -5.06941i q^{27} -1.93174i q^{29} -2.57764i q^{31} -0.463464 q^{33} -0.591220 q^{35} +1.31550i q^{37} -3.51984i q^{39} +4.56154i q^{41} -9.93091 q^{43} +6.91310i q^{45} +1.10859 q^{47} +6.97213 q^{49} +(0.0695354 - 4.22026i) q^{51} -3.35231 q^{53} +1.60335 q^{55} -7.86927i q^{57} -1.00000 q^{59} -8.42710i q^{61} +0.325875i q^{63} +12.1768i q^{65} -8.66678 q^{67} -8.67953 q^{69} -14.4683i q^{71} -4.18682i q^{73} +7.72087i q^{75} +0.0755800 q^{77} +12.9491i q^{79} +0.666545 q^{81} +10.1547 q^{83} +(-0.240557 + 14.5999i) q^{85} -1.97752 q^{87} -1.62855 q^{89} +0.574002i q^{91} -2.63873 q^{93} +27.2236i q^{95} +12.0497i q^{97} -0.883752i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 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

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