LMFDB - Embedded newform 6012.2.h.a.3005.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(3005,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6012.3005");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.h (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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3005.18
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39469 q^{5} -4.57475 q^{7} +O(q^{10})$	$q-1.39469 q^{5} -4.57475 q^{7} +3.16668i q^{11} -0.678448i q^{13} +2.04770 q^{17} -3.83917 q^{19} +1.63709 q^{23} -3.05483 q^{25} -2.49162i q^{29} -4.99578 q^{31} +6.38038 q^{35} +10.1215i q^{37} -12.1835 q^{41} +4.71460i q^{43} -3.25120i q^{47} +13.9284 q^{49} -7.08207 q^{53} -4.41655i q^{55} +5.99637 q^{59} +9.50097 q^{61} +0.946227i q^{65} +9.54542i q^{67} -9.35726 q^{71} -10.4904i q^{73} -14.4868i q^{77} -17.2275i q^{79} -15.9720 q^{83} -2.85592 q^{85} +15.3957i q^{89} +3.10373i q^{91} +5.35446 q^{95} +7.44085 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) $ 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times$.

$n$	$3007$	$3341$	$4681$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.39469			−0.623726			−0.311863	−	0.950127i	$-0.600953\pi$
$5$	−1.39469			−0.623726			−0.311863	+	0.950127i	$0.600953\pi$
$6$		0			0
$7$	−4.57475			−1.72909			−0.864547	−	0.502552i	$-0.832394\pi$
$7$	−4.57475			−1.72909			−0.864547	+	0.502552i	$0.832394\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.16668i			0.954791i	0.878689	+	0.477395i	$0.158419\pi$
$11$			3.16668i			0.954791i	−0.878689	+	0.477395i	$0.841581\pi$
$12$		0			0
$13$		−	0.678448i		−	0.188168i	−0.995564	−	0.0940839i	$-0.970008\pi$
$13$		−	0.678448i		−	0.188168i	0.995564	−	0.0940839i	$-0.0299922\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.04770			0.496641			0.248320	−	0.968678i	$-0.420121\pi$
$17$	2.04770			0.496641			0.248320	+	0.968678i	$0.420121\pi$
$18$		0			0
$19$	−3.83917			−0.880765			−0.440383	−	0.897810i	$-0.645157\pi$
$19$	−3.83917			−0.880765			−0.440383	+	0.897810i	$0.645157\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.63709			0.341356			0.170678	−	0.985327i	$-0.445404\pi$
$23$	1.63709			0.341356			0.170678	+	0.985327i	$0.445404\pi$
$24$		0			0
$25$	−3.05483			−0.610966
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.49162i		−	0.462681i	−0.972873	−	0.231341i	$-0.925689\pi$
$29$		−	2.49162i		−	0.462681i	0.972873	−	0.231341i	$-0.0743112\pi$
$30$		0			0
$31$	−4.99578			−0.897269			−0.448635	−	0.893715i	$-0.648090\pi$
$31$	−4.99578			−0.897269			−0.448635	+	0.893715i	$0.648090\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.38038			1.07848
$36$		0			0
$37$			10.1215i			1.66396i	0.554802	+	0.831982i	$0.312794\pi$
$37$			10.1215i			1.66396i	−0.554802	+	0.831982i	$0.687206\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.1835			−1.90275			−0.951374	−	0.308037i	$-0.900328\pi$
$41$	−12.1835			−1.90275			−0.951374	+	0.308037i	$0.900328\pi$
$42$		0			0
$43$			4.71460i			0.718970i	0.933151	+	0.359485i	$0.117048\pi$
$43$			4.71460i			0.718970i	−0.933151	+	0.359485i	$0.882952\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.25120i		−	0.474237i	−0.971481	−	0.237118i	$-0.923797\pi$
$47$		−	3.25120i		−	0.474237i	0.971481	−	0.237118i	$-0.0762029\pi$
$48$		0			0
$49$	13.9284			1.98977
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.08207			−0.972797			−0.486398	−	0.873737i	$-0.661690\pi$
$53$	−7.08207			−0.972797			−0.486398	+	0.873737i	$0.661690\pi$
$54$		0			0
$55$		−	4.41655i		−	0.595528i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.99637			0.780661			0.390330	−	0.920675i	$-0.372361\pi$
$59$	5.99637			0.780661			0.390330	+	0.920675i	$0.372361\pi$
$60$		0			0
$61$	9.50097			1.21647			0.608237	−	0.793755i	$-0.291877\pi$
$61$	9.50097			1.21647			0.608237	+	0.793755i	$0.291877\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.946227i			0.117365i
$66$		0			0
$67$			9.54542i			1.16616i	0.812415	+	0.583079i	$0.198152\pi$
$67$			9.54542i			1.16616i	−0.812415	+	0.583079i	$0.801848\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.35726			−1.11050			−0.555251	−	0.831683i	$-0.687378\pi$
$71$	−9.35726			−1.11050			−0.555251	+	0.831683i	$0.687378\pi$
$72$		0			0
$73$		−	10.4904i		−	1.22781i	−0.789381	−	0.613903i	$-0.789598\pi$
$73$		−	10.4904i		−	1.22781i	0.789381	−	0.613903i	$-0.210402\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	14.4868i		−	1.65092i
$78$		0			0
$79$		−	17.2275i		−	1.93824i	−0.246590	−	0.969120i	$-0.579310\pi$
$79$		−	17.2275i		−	1.93824i	0.246590	−	0.969120i	$-0.420690\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.9720			−1.75316			−0.876580	−	0.481257i	$-0.840180\pi$
$83$	−15.9720			−1.75316			−0.876580	+	0.481257i	$0.840180\pi$
$84$		0			0
$85$	−2.85592			−0.309768
$86$		0			0
$87$		0			0
$88$		0			0
$89$			15.3957i			1.63194i	0.578094	+	0.815970i	$0.303797\pi$
$89$			15.3957i			1.63194i	−0.578094	+	0.815970i	$0.696203\pi$
$90$		0			0
$91$			3.10373i			0.325360i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.35446			0.549356
$96$		0			0
$97$	7.44085			0.755503			0.377752	−	0.925907i	$-0.376697\pi$
$97$	7.44085			0.755503			0.377752	+	0.925907i	$0.376697\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.07700			0.604684			0.302342	−	0.953200i	$-0.402232\pi$
$101$	6.07700			0.604684			0.302342	+	0.953200i	$0.402232\pi$
$102$		0			0
$103$			14.5730i			1.43592i	0.696083	+	0.717961i	$0.254924\pi$
$103$			14.5730i			1.43592i	−0.696083	+	0.717961i	$0.745076\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.24282i			0.410169i	0.978744	+	0.205084i	$0.0657469\pi$
$107$			4.24282i			0.410169i	−0.978744	+	0.205084i	$0.934253\pi$
$108$		0			0
$109$		−	5.51313i		−	0.528062i	−0.964514	−	0.264031i	$-0.914948\pi$
$109$		−	5.51313i		−	0.528062i	0.964514	−	0.264031i	$-0.0850521\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.6859			1.00524			0.502620	−	0.864507i	$-0.332369\pi$
$113$	10.6859			1.00524			0.502620	+	0.864507i	$0.332369\pi$
$114$		0			0
$115$	−2.28323			−0.212913
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.36773			−0.858738
$120$		0			0
$121$	0.972122			0.0883747
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.2340			1.00480
$126$		0			0
$127$	14.3908			1.27698			0.638490	−	0.769630i	$-0.279559\pi$
$127$	14.3908			1.27698			0.638490	+	0.769630i	$0.279559\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.21806			−0.368534			−0.184267	−	0.982876i	$-0.558991\pi$
$131$	−4.21806			−0.368534			−0.184267	+	0.982876i	$0.558991\pi$
$132$		0			0
$133$	17.5632			1.52293
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	15.7664i		−	1.34701i	−0.739182	−	0.673505i	$-0.764788\pi$
$137$		−	15.7664i		−	1.34701i	0.739182	−	0.673505i	$-0.235212\pi$
$138$		0			0
$139$			1.32836i			0.112670i	0.998412	+	0.0563350i	$0.0179415\pi$
$139$			1.32836i			0.112670i	−0.998412	+	0.0563350i	$0.982059\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.14843			0.179661
$144$		0			0
$145$			3.47504i			0.288586i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.7715			0.964362			0.482181	−	0.876072i	$-0.339845\pi$
$149$	11.7715			0.964362			0.482181	+	0.876072i	$0.339845\pi$
$150$		0			0
$151$		−	10.5889i		−	0.861709i	−0.902421	−	0.430854i	$-0.858212\pi$
$151$		−	10.5889i		−	0.861709i	0.902421	−	0.430854i	$-0.141788\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.96759			0.559650
$156$		0			0
$157$	15.4110			1.22993			0.614965	−	0.788555i	$-0.289170\pi$
$157$	15.4110			1.22993			0.614965	+	0.788555i	$0.289170\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.48927			−0.590237
$162$		0			0
$163$		−	12.6578i		−	0.991436i	−0.868484	−	0.495718i	$-0.834905\pi$
$163$		−	12.6578i		−	0.991436i	0.868484	−	0.495718i	$-0.165095\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.96454	−	12.7726i	0.152021	−	0.988377i
$167$	1.96454	−	12.7726i	0.152021	−	0.988377i
$168$		0			0
$169$	12.5397			0.964593
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	20.4501i		−	1.55479i	−0.629013	−	0.777395i	$-0.716541\pi$
$173$		−	20.4501i		−	1.55479i	0.629013	−	0.777395i	$-0.283459\pi$
$174$		0			0
$175$	13.9751			1.05642
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	0.305626i		−	0.0228436i	−0.999935	−	0.0114218i	$-0.996364\pi$
$179$		−	0.305626i		−	0.0228436i	0.999935	−	0.0114218i	$-0.00363574\pi$
$180$		0			0
$181$	10.4692			0.778168			0.389084	−	0.921202i	$-0.372792\pi$
$181$	10.4692			0.778168			0.389084	+	0.921202i	$0.372792\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	14.1164i		−	1.03786i
$186$		0			0
$187$			6.48442i			0.474188i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	14.0013i		−	1.01310i	−0.862212	−	0.506548i	$-0.830921\pi$
$191$		−	14.0013i		−	1.01310i	0.862212	−	0.506548i	$-0.169079\pi$
$192$		0			0
$193$		−	11.7565i		−	0.846255i	−0.906070	−	0.423127i	$-0.860932\pi$
$193$		−	11.7565i		−	0.846255i	0.906070	−	0.423127i	$-0.139068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−21.3280			−1.51956			−0.759780	−	0.650181i	$-0.774693\pi$
$197$	−21.3280			−1.51956			−0.759780	+	0.650181i	$0.774693\pi$
$198$		0			0
$199$	−10.0439			−0.711992			−0.355996	−	0.934488i	$-0.615858\pi$
$199$	−10.0439			−0.711992			−0.355996	+	0.934488i	$0.615858\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			11.3985i			0.800020i
$204$		0			0
$205$	16.9923			1.18679
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	12.1574i		−	0.840947i
$210$		0			0
$211$	−2.36608			−0.162888			−0.0814438	−	0.996678i	$-0.525953\pi$
$211$	−2.36608			−0.162888			−0.0814438	+	0.996678i	$0.525953\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	6.57542i		−	0.448440i
$216$		0			0
$217$	22.8545			1.55146
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	1.38926i		−	0.0934517i
$222$		0			0
$223$	5.63935			0.377639			0.188819	−	0.982012i	$-0.439534\pi$
$223$	5.63935			0.377639			0.188819	+	0.982012i	$0.439534\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.2355			0.944844			0.472422	−	0.881372i	$-0.343380\pi$
$227$	14.2355			0.944844			0.472422	+	0.881372i	$0.343380\pi$
$228$		0			0
$229$	5.42754			0.358662			0.179331	−	0.983789i	$-0.442607\pi$
$229$	5.42754			0.358662			0.179331	+	0.983789i	$0.442607\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	7.63054i		−	0.499894i	−0.968260	−	0.249947i	$-0.919587\pi$
$233$		−	7.63054i		−	0.499894i	0.968260	−	0.249947i	$-0.0804131\pi$
$234$		0			0
$235$			4.53443i			0.295794i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.38304i		−	0.412884i	−0.978459	−	0.206442i	$-0.933811\pi$
$239$		−	6.38304i		−	0.412884i	0.978459	−	0.206442i	$-0.0661885\pi$
$240$		0			0
$241$			17.0010i			1.09513i	0.836764	+	0.547564i	$0.184445\pi$
$241$			17.0010i			1.09513i	−0.836764	+	0.547564i	$0.815555\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−19.4258			−1.24107
$246$		0			0
$247$			2.60468i			0.165732i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			15.5787i			0.983316i	0.870788	+	0.491658i	$0.163609\pi$
$251$			15.5787i			0.983316i	−0.870788	+	0.491658i	$0.836391\pi$
$252$		0			0
$253$			5.18413i			0.325924i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.5886			−1.40904			−0.704520	−	0.709684i	$-0.748838\pi$
$257$	−22.5886			−1.40904			−0.704520	+	0.709684i	$0.748838\pi$
$258$		0			0
$259$		−	46.3034i		−	2.87715i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4.71783i		−	0.290914i	−0.989365	−	0.145457i	$-0.953535\pi$
$263$		−	4.71783i		−	0.290914i	0.989365	−	0.145457i	$-0.0464652\pi$
$264$		0			0
$265$	9.87731			0.606758
$266$		0			0
$267$		0			0
$268$		0			0
$269$	22.3567			1.36311			0.681556	−	0.731766i	$-0.261303\pi$
$269$	22.3567			1.36311			0.681556	+	0.731766i	$0.261303\pi$
$270$		0			0
$271$			6.60680i			0.401335i	0.979659	+	0.200667i	$0.0643110\pi$
$271$			6.60680i			0.401335i	−0.979659	+	0.200667i	$0.935689\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	9.67368i		−	0.583345i
$276$		0			0
$277$		−	26.6210i		−	1.59950i	−0.600334	−	0.799749i	$-0.704966\pi$
$277$		−	26.6210i		−	1.59950i	0.600334	−	0.799749i	$-0.295034\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	32.8224i		−	1.95802i	−0.203805	−	0.979011i	$-0.565331\pi$
$281$		−	32.8224i		−	1.95802i	0.203805	−	0.979011i	$-0.434669\pi$
$282$		0			0
$283$	1.71459			0.101922			0.0509609	−	0.998701i	$-0.483772\pi$
$283$	1.71459			0.101922			0.0509609	+	0.998701i	$0.483772\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	55.7367			3.29003
$288$		0			0
$289$	−12.8069			−0.753348
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.20416i		−	0.479292i	−0.970860	−	0.239646i	$-0.922969\pi$
$293$		−	8.20416i		−	0.479292i	0.970860	−	0.239646i	$-0.0770314\pi$
$294$		0			0
$295$	−8.36310			−0.486918
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	1.11068i		−	0.0642322i
$300$		0			0
$301$		−	21.5681i		−	1.24317i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.2509			−0.758747
$306$		0			0
$307$			4.84943i			0.276772i	0.990378	+	0.138386i	$0.0441914\pi$
$307$			4.84943i			0.276772i	−0.990378	+	0.138386i	$0.955809\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7.14797i		−	0.405324i	−0.979249	−	0.202662i	$-0.935041\pi$
$311$		−	7.14797i		−	0.405324i	0.979249	−	0.202662i	$-0.0649593\pi$
$312$		0			0
$313$			11.0218i			0.622988i	0.950248	+	0.311494i	$0.100829\pi$
$313$			11.0218i			0.622988i	−0.950248	+	0.311494i	$0.899171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5.26539i			0.295734i	0.989007	+	0.147867i	$0.0472408\pi$
$317$			5.26539i			0.295734i	−0.989007	+	0.147867i	$0.952759\pi$
$318$		0			0
$319$	7.89016			0.441764
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.86147			−0.437424
$324$		0			0
$325$			2.07254i			0.114964i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			14.8734i			0.820000i
$330$		0			0
$331$			8.27343i			0.454749i	0.973807	+	0.227374i	$0.0730141\pi$
$331$			8.27343i			0.454749i	−0.973807	+	0.227374i	$0.926986\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	13.3129i		−	0.727363i
$336$		0			0
$337$	34.1026			1.85769			0.928844	−	0.370471i	$-0.120804\pi$
$337$	34.1026			1.85769			0.928844	+	0.370471i	$0.120804\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	15.8201i		−	0.856704i
$342$		0			0
$343$	−31.6955			−1.71140
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.7405			−0.952359			−0.476179	−	0.879348i	$-0.657979\pi$
$347$	−17.7405			−0.952359			−0.476179	+	0.879348i	$0.657979\pi$
$348$		0			0
$349$			32.9216i			1.76226i	0.472879	+	0.881128i	$0.343215\pi$
$349$			32.9216i			1.76226i	−0.472879	+	0.881128i	$0.656785\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	23.4839i		−	1.24992i	−0.780656	−	0.624961i	$-0.785115\pi$
$353$		−	23.4839i		−	1.24992i	0.780656	−	0.624961i	$-0.214885\pi$
$354$		0			0
$355$	13.0505			0.692649
$356$		0			0
$357$		0			0
$358$		0			0
$359$			19.7233i			1.04096i	0.853875	+	0.520479i	$0.174246\pi$
$359$			19.7233i			1.04096i	−0.853875	+	0.520479i	$0.825754\pi$
$360$		0			0
$361$	−4.26079			−0.224252
$362$		0			0
$363$		0			0
$364$		0			0
$365$			14.6309i			0.765815i
$366$		0			0
$367$	24.6349			1.28593			0.642967	−	0.765894i	$-0.277703\pi$
$367$	24.6349			1.28593			0.642967	+	0.765894i	$0.277703\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	32.3987			1.68206
$372$		0			0
$373$			5.09861i			0.263996i	0.991250	+	0.131998i	$0.0421393\pi$
$373$			5.09861i			0.263996i	−0.991250	+	0.131998i	$0.957861\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.69043			−0.0870617
$378$		0			0
$379$			2.07234i			0.106449i	0.998583	+	0.0532244i	$0.0169498\pi$
$379$			2.07234i			0.106449i	−0.998583	+	0.0532244i	$0.983050\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			21.5206i			1.09965i	0.835280	+	0.549825i	$0.185305\pi$
$383$			21.5206i			1.09965i	−0.835280	+	0.549825i	$0.814695\pi$
$384$		0			0
$385$			20.2046i			1.02972i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.7818			0.597359			0.298679	−	0.954354i	$-0.403454\pi$
$389$	11.7818			0.597359			0.298679	+	0.954354i	$0.403454\pi$
$390$		0			0
$391$	3.35227			0.169531
$392$		0			0
$393$		0			0
$394$		0			0
$395$			24.0270i			1.20893i
$396$		0			0
$397$	−1.66291			−0.0834590			−0.0417295	−	0.999129i	$-0.513287\pi$
$397$	−1.66291			−0.0834590			−0.0417295	+	0.999129i	$0.513287\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.873382			0.0436146			0.0218073	−	0.999762i	$-0.493058\pi$
$401$	0.873382			0.0436146			0.0218073	+	0.999762i	$0.493058\pi$
$402$		0			0
$403$			3.38938i			0.168837i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−32.0516			−1.58874
$408$		0			0
$409$	−22.8880			−1.13174			−0.565869	−	0.824495i	$-0.691459\pi$
$409$	−22.8880			−1.13174			−0.565869	+	0.824495i	$0.691459\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−27.4319			−1.34984
$414$		0			0
$415$	22.2761			1.09349
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	0.989747i		−	0.0483523i	−0.999708	−	0.0241762i	$-0.992304\pi$
$419$		−	0.989747i		−	0.0483523i	0.999708	−	0.0241762i	$-0.00769626\pi$
$420$		0			0
$421$	−13.0798			−0.637468			−0.318734	−	0.947844i	$-0.603258\pi$
$421$	−13.0798			−0.637468			−0.318734	+	0.947844i	$0.603258\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.25538			−0.303431
$426$		0			0
$427$	−43.4646			−2.10340
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	25.2048i		−	1.21407i	−0.794674	−	0.607037i	$-0.792358\pi$
$431$		−	25.2048i		−	1.21407i	0.794674	−	0.607037i	$-0.207642\pi$
$432$		0			0
$433$	−7.31288			−0.351435			−0.175717	−	0.984441i	$-0.556224\pi$
$433$	−7.31288			−0.351435			−0.175717	+	0.984441i	$0.556224\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.28505			−0.300655
$438$		0			0
$439$			32.2996i			1.54158i	0.637090	+	0.770789i	$0.280138\pi$
$439$			32.2996i			1.54158i	−0.637090	+	0.770789i	$0.719862\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.0186			0.856092			0.428046	−	0.903757i	$-0.359202\pi$
$443$	18.0186			0.856092			0.428046	+	0.903757i	$0.359202\pi$
$444$		0			0
$445$		−	21.4723i		−	1.01788i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			40.4917i			1.91092i	0.295119	+	0.955460i	$0.404641\pi$
$449$			40.4917i			1.91092i	−0.295119	+	0.955460i	$0.595359\pi$
$450$		0			0
$451$		−	38.5814i		−	1.81673i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	4.32876i		−	0.202935i
$456$		0			0
$457$		−	29.5031i		−	1.38010i	−0.723763	−	0.690049i	$-0.757589\pi$
$457$		−	29.5031i		−	1.38010i	0.723763	−	0.690049i	$-0.242411\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			18.5480i			0.863865i	0.901906	+	0.431932i	$0.142168\pi$
$461$			18.5480i			0.863865i	−0.901906	+	0.431932i	$0.857832\pi$
$462$		0			0
$463$		−	34.3790i		−	1.59773i	−0.601513	−	0.798863i	$-0.705435\pi$
$463$		−	34.3790i		−	1.59773i	0.601513	−	0.798863i	$-0.294565\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			18.5733i			0.859470i	0.902955	+	0.429735i	$0.141393\pi$
$467$			18.5733i			0.859470i	−0.902955	+	0.429735i	$0.858607\pi$
$468$		0			0
$469$		−	43.6679i		−	2.01640i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.9296			−0.686466
$474$		0			0
$475$	11.7280			0.538118
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−20.6138			−0.941868			−0.470934	−	0.882169i	$-0.656083\pi$
$479$	−20.6138			−0.941868			−0.470934	+	0.882169i	$0.656083\pi$
$480$		0			0
$481$	6.86691			0.313104
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.3777			−0.471227
$486$		0			0
$487$			29.2837i			1.32697i	0.748190	+	0.663485i	$0.230923\pi$
$487$			29.2837i			1.32697i	−0.748190	+	0.663485i	$0.769077\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	32.2853i		−	1.45701i	−0.685038	−	0.728507i	$-0.740214\pi$
$491$		−	32.2853i		−	1.45701i	0.685038	−	0.728507i	$-0.259786\pi$
$492$		0			0
$493$		−	5.10209i		−	0.229786i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	42.8071			1.92016
$498$		0			0
$499$			14.7516i			0.660373i	0.943916	+	0.330187i	$0.107112\pi$
$499$			14.7516i			0.660373i	−0.943916	+	0.330187i	$0.892888\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.1022i			0.807137i	0.914949	+	0.403569i	$0.132230\pi$
$503$			18.1022i			0.807137i	−0.914949	+	0.403569i	$0.867770\pi$
$504$		0			0
$505$	−8.47555			−0.377157
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	38.7177i		−	1.71613i	−0.513541	−	0.858065i	$-0.671666\pi$
$509$		−	38.7177i		−	1.71613i	0.513541	−	0.858065i	$-0.328334\pi$
$510$		0			0
$511$			47.9909i			2.12299i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	20.3249i		−	0.895621i
$516$		0			0
$517$	10.2955			0.452797
$518$		0			0
$519$		0			0
$520$		0			0
$521$	25.1043			1.09984			0.549919	−	0.835218i	$-0.314659\pi$
$521$	25.1043			1.09984			0.549919	+	0.835218i	$0.314659\pi$
$522$		0			0
$523$	−30.4808			−1.33283			−0.666416	−	0.745580i	$-0.732173\pi$
$523$	−30.4808			−1.33283			−0.666416	+	0.745580i	$0.732173\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.2299			−0.445620
$528$		0			0
$529$	−20.3199			−0.883476
$530$		0			0
$531$		0			0
$532$		0			0
$533$			8.26590i			0.358036i
$534$		0			0
$535$		−	5.91743i		−	0.255833i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			44.1067i			1.89981i
$540$		0			0
$541$		−	9.04366i		−	0.388817i	−0.980921	−	0.194409i	$-0.937721\pi$
$541$		−	9.04366i		−	0.388817i	0.980921	−	0.194409i	$-0.0622788\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			7.68912i			0.329366i
$546$		0			0
$547$		−	39.1689i		−	1.67474i	−0.546635	−	0.837371i	$-0.684092\pi$
$547$		−	39.1689i		−	1.67474i	0.546635	−	0.837371i	$-0.315908\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			9.56573i			0.407514i
$552$		0			0
$553$			78.8113i			3.35140i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	7.61535i		−	0.322673i	−0.986899	−	0.161336i	$-0.948420\pi$
$557$		−	7.61535i		−	0.322673i	0.986899	−	0.161336i	$-0.0515804\pi$
$558$		0			0
$559$	3.19861			0.135287
$560$		0			0
$561$		0			0
$562$		0			0
$563$			41.6553i			1.75556i	0.479062	+	0.877781i	$0.340977\pi$
$563$			41.6553i			1.75556i	−0.479062	+	0.877781i	$0.659023\pi$
$564$		0			0
$565$	−14.9035			−0.626995
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.2979			−1.01862			−0.509311	−	0.860582i	$-0.670100\pi$
$569$	−24.2979			−1.01862			−0.509311	+	0.860582i	$0.670100\pi$
$570$		0			0
$571$			39.5904i			1.65681i	0.560133	+	0.828403i	$0.310750\pi$
$571$			39.5904i			1.65681i	−0.560133	+	0.828403i	$0.689250\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.00102			−0.208557
$576$		0			0
$577$	13.4711			0.560807			0.280404	−	0.959882i	$-0.409532\pi$
$577$	13.4711			0.560807			0.280404	+	0.959882i	$0.409532\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	73.0681			3.03138
$582$		0			0
$583$		−	22.4267i		−	0.928817i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.30680			−0.0539376			−0.0269688	−	0.999636i	$-0.508585\pi$
$587$	−1.30680			−0.0539376			−0.0269688	+	0.999636i	$0.508585\pi$
$588$		0			0
$589$	19.1797			0.790284
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.0491			0.535862			0.267931	−	0.963438i	$-0.413660\pi$
$593$	13.0491			0.535862			0.267931	+	0.963438i	$0.413660\pi$
$594$		0			0
$595$	13.0651			0.535617
$596$		0			0
$597$		0			0
$598$		0			0
$599$			15.3900i			0.628820i	0.949287	+	0.314410i	$0.101807\pi$
$599$			15.3900i			0.628820i	−0.949287	+	0.314410i	$0.898193\pi$
$600$		0			0
$601$	44.7420			1.82506			0.912532	−	0.409006i	$-0.134124\pi$
$601$	44.7420			1.82506			0.912532	+	0.409006i	$0.134124\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.35581			−0.0551216
$606$		0			0
$607$		−	7.47030i		−	0.303210i	−0.988441	−	0.151605i	$-0.951556\pi$
$607$		−	7.47030i		−	0.303210i	0.988441	−	0.151605i	$-0.0484442\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.20577			−0.0892360
$612$		0			0
$613$	10.4294			0.421238			0.210619	−	0.977568i	$-0.432452\pi$
$613$	10.4294			0.421238			0.210619	+	0.977568i	$0.432452\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			19.7728i			0.796021i	0.917381	+	0.398011i	$0.130299\pi$
$617$			19.7728i			0.796021i	−0.917381	+	0.398011i	$0.869701\pi$
$618$		0			0
$619$		−	13.2880i		−	0.534090i	−0.963684	−	0.267045i	$-0.913953\pi$
$619$		−	13.2880i		−	0.534090i	0.963684	−	0.267045i	$-0.0860473\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	70.4315i		−	2.82178i
$624$		0			0
$625$	−0.393856			−0.0157543
$626$		0			0
$627$		0			0
$628$		0			0
$629$			20.7258i			0.826392i
$630$		0			0
$631$	36.7439			1.46275			0.731375	−	0.681975i	$-0.238879\pi$
$631$	36.7439			1.46275			0.731375	+	0.681975i	$0.238879\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20.0708			−0.796486
$636$		0			0
$637$		−	9.44967i		−	0.374410i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−31.1486			−1.23029			−0.615147	−	0.788412i	$-0.710904\pi$
$641$	−31.1486			−1.23029			−0.615147	+	0.788412i	$0.710904\pi$
$642$		0			0
$643$			27.0760i			1.06777i	0.845556	+	0.533887i	$0.179269\pi$
$643$			27.0760i			1.06777i	−0.845556	+	0.533887i	$0.820731\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−25.5380			−1.00400			−0.502000	−	0.864867i	$-0.667402\pi$
$647$	−25.5380			−1.00400			−0.502000	+	0.864867i	$0.667402\pi$
$648$		0			0
$649$			18.9886i			0.745368i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	26.2589i		−	1.02759i	−0.857913	−	0.513794i	$-0.828239\pi$
$653$		−	26.2589i		−	1.02759i	0.857913	−	0.513794i	$-0.171761\pi$
$654$		0			0
$655$	5.88291			0.229864
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−29.7145			−1.15751			−0.578756	−	0.815501i	$-0.696462\pi$
$659$	−29.7145			−1.15751			−0.578756	+	0.815501i	$0.696462\pi$
$660$		0			0
$661$		−	2.40757i		−	0.0936436i	−0.998903	−	0.0468218i	$-0.985091\pi$
$661$		−	2.40757i		−	0.0936436i	0.998903	−	0.0468218i	$-0.0149093\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−24.4953			−0.949888
$666$		0			0
$667$		−	4.07899i		−	0.157939i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			30.0866i			1.16148i
$672$		0			0
$673$			12.5706i			0.484561i	0.970206	+	0.242280i	$0.0778954\pi$
$673$			12.5706i			0.484561i	−0.970206	+	0.242280i	$0.922105\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	51.1838i		−	1.96715i	−0.180495	−	0.983576i	$-0.557770\pi$
$677$		−	51.1838i		−	1.96715i	0.180495	−	0.983576i	$-0.442230\pi$
$678$		0			0
$679$	−34.0400			−1.30634
$680$		0			0
$681$		0			0
$682$		0			0
$683$	32.4862			1.24305			0.621524	−	0.783395i	$-0.286514\pi$
$683$	32.4862			1.24305			0.621524	+	0.783395i	$0.286514\pi$
$684$		0			0
$685$			21.9892i			0.840165i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.80482i			0.183049i
$690$		0			0
$691$		−	14.5875i		−	0.554936i	−0.960735	−	0.277468i	$-0.910505\pi$
$691$		−	14.5875i		−	0.554936i	0.960735	−	0.277468i	$-0.0894953\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	1.85265i		−	0.0702752i
$696$		0			0
$697$	−24.9482			−0.944982
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	34.6612i		−	1.30913i	−0.756004	−	0.654567i	$-0.772851\pi$
$701$		−	34.6612i		−	1.30913i	0.756004	−	0.654567i	$-0.227149\pi$
$702$		0			0
$703$		−	38.8581i		−	1.46556i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−27.8008			−1.04555
$708$		0			0
$709$			15.6451i			0.587563i	0.955873	+	0.293781i	$0.0949138\pi$
$709$			15.6451i			0.587563i	−0.955873	+	0.293781i	$0.905086\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.17853			−0.306288
$714$		0			0
$715$	−2.99640			−0.112059
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.1948			0.678550			0.339275	−	0.940687i	$-0.389818\pi$
$719$	18.1948			0.678550			0.339275	+	0.940687i	$0.389818\pi$
$720$		0			0
$721$		−	66.6679i		−	2.48284i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			7.61146i			0.282683i
$726$		0			0
$727$			2.38903i			0.0886041i	0.999018	+	0.0443021i	$0.0141064\pi$
$727$			2.38903i			0.0886041i	−0.999018	+	0.0443021i	$0.985894\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			9.65410i			0.357070i
$732$		0			0
$733$	10.8454			0.400585			0.200292	−	0.979736i	$-0.435811\pi$
$733$	10.8454			0.400585			0.200292	+	0.979736i	$0.435811\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−30.2273			−1.11344
$738$		0			0
$739$			48.9673i			1.80129i	0.434555	+	0.900645i	$0.356906\pi$
$739$			48.9673i			1.80129i	−0.434555	+	0.900645i	$0.643094\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3.09351i		−	0.113490i	−0.998389	−	0.0567448i	$-0.981928\pi$
$743$		−	3.09351i		−	0.113490i	0.998389	−	0.0567448i	$-0.0180722\pi$
$744$		0			0
$745$	−16.4177			−0.601497
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	19.4098i		−	0.709220i
$750$		0			0
$751$		−	1.99295i		−	0.0727238i	−0.999339	−	0.0363619i	$-0.988423\pi$
$751$		−	1.99295i		−	0.0727238i	0.999339	−	0.0363619i	$-0.0115769\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			14.7682i			0.537470i
$756$		0			0
$757$	14.4304			0.524483			0.262241	−	0.965002i	$-0.415538\pi$
$757$	14.4304			0.524483			0.262241	+	0.965002i	$0.415538\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	31.3336i		−	1.13584i	−0.823083	−	0.567921i	$-0.807748\pi$
$761$		−	31.3336i		−	1.13584i	0.823083	−	0.567921i	$-0.192252\pi$
$762$		0			0
$763$			25.2212i			0.913068i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	4.06823i		−	0.146895i
$768$		0			0
$769$			41.6037i			1.50027i	0.661287	+	0.750133i	$0.270011\pi$
$769$			41.6037i			1.50027i	−0.661287	+	0.750133i	$0.729989\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.95828			−0.286240			−0.143120	−	0.989705i	$-0.545713\pi$
$773$	−7.95828			−0.286240			−0.143120	+	0.989705i	$0.545713\pi$
$774$		0			0
$775$	15.2613			0.548201
$776$		0			0
$777$		0			0
$778$		0			0
$779$	46.7746			1.67588
$780$		0			0
$781$		−	29.6315i		−	1.06030i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−21.4936			−0.767139
$786$		0			0
$787$			53.0474i			1.89093i	0.325718	+	0.945467i	$0.394394\pi$
$787$			53.0474i			1.89093i	−0.325718	+	0.945467i	$0.605606\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−48.8851			−1.73816
$792$		0			0
$793$		−	6.44592i		−	0.228901i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.5774			−0.374671			−0.187336	−	0.982296i	$-0.559985\pi$
$797$	−10.5774			−0.374671			−0.187336	+	0.982296i	$0.559985\pi$
$798$		0			0
$799$		−	6.65749i		−	0.235525i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	33.2197			1.17230
$804$		0			0
$805$	10.4452			0.368146
$806$		0			0
$807$		0			0
$808$		0			0
$809$			21.2800i			0.748166i	0.927395	+	0.374083i	$0.122043\pi$
$809$			21.2800i			0.748166i	−0.927395	+	0.374083i	$0.877957\pi$
$810$		0			0
$811$		−	41.6959i		−	1.46414i	−0.681229	−	0.732070i	$-0.738554\pi$
$811$		−	41.6959i		−	1.46414i	0.681229	−	0.732070i	$-0.261446\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			17.6538i			0.618384i
$816$		0			0
$817$		−	18.1001i		−	0.633244i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.0559			0.455653			0.227826	−	0.973702i	$-0.426838\pi$
$821$	13.0559			0.455653			0.227826	+	0.973702i	$0.426838\pi$
$822$		0			0
$823$		−	39.1714i		−	1.36543i	−0.730686	−	0.682714i	$-0.760800\pi$
$823$		−	39.1714i		−	1.36543i	0.730686	−	0.682714i	$-0.239200\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.7523			0.443441			0.221720	−	0.975110i	$-0.428833\pi$
$827$	12.7523			0.443441			0.221720	+	0.975110i	$0.428833\pi$
$828$		0			0
$829$			11.5270i			0.400350i	0.979760	+	0.200175i	$0.0641511\pi$
$829$			11.5270i			0.400350i	−0.979760	+	0.200175i	$0.935849\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	28.5211			0.988198
$834$		0			0
$835$	−2.73993	+	17.8139i	−0.0948192	+	0.616476i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	33.9357i		−	1.17159i	−0.810459	−	0.585795i	$-0.800782\pi$
$839$		−	33.9357i		−	1.17159i	0.810459	−	0.585795i	$-0.199218\pi$
$840$		0			0
$841$	22.7918			0.785926
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.4890			−0.601641
$846$		0			0
$847$	−4.44722			−0.152808
$848$		0			0
$849$		0			0
$850$		0			0
$851$			16.5698i			0.568005i
$852$		0			0
$853$	−12.1652			−0.416527			−0.208264	−	0.978073i	$-0.566781\pi$
$853$	−12.1652			−0.416527			−0.208264	+	0.978073i	$0.566781\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	20.1719i		−	0.689061i	−0.938775	−	0.344530i	$-0.888038\pi$
$857$		−	20.1719i		−	0.689061i	0.938775	−	0.344530i	$-0.111962\pi$
$858$		0			0
$859$	−17.7396			−0.605268			−0.302634	−	0.953107i	$-0.597866\pi$
$859$	−17.7396			−0.605268			−0.302634	+	0.953107i	$0.597866\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	45.6561i		−	1.55415i	−0.629407	−	0.777076i	$-0.716702\pi$
$863$		−	45.6561i		−	1.55415i	0.629407	−	0.777076i	$-0.283298\pi$
$864$		0			0
$865$			28.5216i			0.969762i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	54.5539			1.85061
$870$		0			0
$871$	6.47607			0.219433
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−51.3928			−1.73740
$876$		0			0
$877$	41.2228			1.39200			0.695998	−	0.718044i	$-0.254962\pi$
$877$	41.2228			1.39200			0.695998	+	0.718044i	$0.254962\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	28.4484			0.958451			0.479226	−	0.877692i	$-0.340918\pi$
$881$	28.4484			0.958451			0.479226	+	0.877692i	$0.340918\pi$
$882$		0			0
$883$	3.99349			0.134392			0.0671958	−	0.997740i	$-0.478595\pi$
$883$	3.99349			0.134392			0.0671958	+	0.997740i	$0.478595\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.8899			0.802145			0.401073	−	0.916046i	$-0.368637\pi$
$887$	23.8899			0.802145			0.401073	+	0.916046i	$0.368637\pi$
$888$		0			0
$889$	−65.8345			−2.20802
$890$		0			0
$891$		0			0
$892$		0			0
$893$			12.4819i			0.417691i
$894$		0			0
$895$			0.426255i			0.0142481i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			12.4476i			0.415150i
$900$		0			0
$901$	−14.5020			−0.483130
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.6013			−0.485363
$906$		0			0
$907$	−4.31375			−0.143236			−0.0716178	−	0.997432i	$-0.522816\pi$
$907$	−4.31375			−0.143236			−0.0716178	+	0.997432i	$0.522816\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	31.8083i		−	1.05386i	−0.849910	−	0.526928i	$-0.823344\pi$
$911$		−	31.8083i		−	1.05386i	0.849910	−	0.526928i	$-0.176656\pi$
$912$		0			0
$913$		−	50.5784i		−	1.67390i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	19.2966			0.637230
$918$		0			0
$919$	18.2542			0.602152			0.301076	−	0.953600i	$-0.402654\pi$
$919$	18.2542			0.602152			0.301076	+	0.953600i	$0.402654\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			6.34842i			0.208961i
$924$		0			0
$925$		−	30.9195i		−	1.01663i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			34.2625i			1.12412i	0.827098	+	0.562058i	$0.189990\pi$
$929$			34.2625i			1.12412i	−0.827098	+	0.562058i	$0.810010\pi$
$930$		0			0
$931$	−53.4733			−1.75252
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	9.04378i		−	0.295763i
$936$		0			0
$937$		−	20.0049i		−	0.653531i	−0.945105	−	0.326766i	$-0.894041\pi$
$937$		−	20.0049i		−	0.653531i	0.945105	−	0.326766i	$-0.105959\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−58.9337			−1.92118			−0.960592	−	0.277961i	$-0.910341\pi$
$941$	−58.9337			−1.92118			−0.960592	+	0.277961i	$0.910341\pi$
$942$		0			0
$943$	−19.9455			−0.649515
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	55.1149i		−	1.79099i	−0.445069	−	0.895496i	$-0.646821\pi$
$947$		−	55.1149i		−	1.79099i	0.445069	−	0.895496i	$-0.353179\pi$
$948$		0			0
$949$	−7.11718			−0.231034
$950$		0			0
$951$		0			0
$952$		0			0
$953$	60.8692			1.97175			0.985873	−	0.167494i	$-0.0535674\pi$
$953$	60.8692			1.97175			0.985873	+	0.167494i	$0.0535674\pi$
$954$		0			0
$955$			19.5275i			0.631894i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			72.1272i			2.32911i
$960$		0			0
$961$	−6.04215			−0.194908
$962$		0			0
$963$		0			0
$964$		0			0
$965$			16.3968i			0.527831i
$966$		0			0
$967$	7.91549			0.254545			0.127273	−	0.991868i	$-0.459378\pi$
$967$	7.91549			0.254545			0.127273	+	0.991868i	$0.459378\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15.8769			0.509515			0.254757	−	0.967005i	$-0.418004\pi$
$971$	15.8769			0.509515			0.254757	+	0.967005i	$0.418004\pi$
$972$		0			0
$973$		−	6.07692i		−	0.194817i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−32.3902			−1.03625			−0.518127	−	0.855304i	$-0.673370\pi$
$977$	−32.3902			−1.03625			−0.518127	+	0.855304i	$0.673370\pi$
$978$		0			0
$979$	−48.7533			−1.55816
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−32.0878			−1.02344			−0.511720	−	0.859152i	$-0.670992\pi$
$983$	−32.0878			−1.02344			−0.511720	+	0.859152i	$0.670992\pi$
$984$		0			0
$985$	29.7461			0.947788
$986$		0			0
$987$		0			0
$988$		0			0
$989$			7.71821i			0.245425i
$990$		0			0
$991$			44.6089i			1.41705i	0.705687	+	0.708524i	$0.250639\pi$
$991$			44.6089i			1.41705i	−0.705687	+	0.708524i	$0.749361\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.0081			0.444087
$996$		0			0
$997$	30.1850			0.955969			0.477985	−	0.878368i	$-0.341367\pi$
$997$	30.1850			0.955969			0.477985	+	0.878368i	$0.341367\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.h.a.3005.18	yes	56
3.2	odd	2	inner	6012.2.h.a.3005.40	yes	56
167.166	odd	2	inner	6012.2.h.a.3005.39	yes	56
501.500	even	2	inner	6012.2.h.a.3005.17	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.17	✓	56	501.500	even	2	inner
6012.2.h.a.3005.18	yes	56	1.1	even	1	trivial
6012.2.h.a.3005.39	yes	56	167.166	odd	2	inner
6012.2.h.a.3005.40	yes	56	3.2	odd	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.39469 q^{5} -4.57475 q^{7} +O(q^{10})\)	\(q-1.39469 q^{5} -4.57475 q^{7} +3.16668i q^{11} -0.678448i q^{13} +2.04770 q^{17} -3.83917 q^{19} +1.63709 q^{23} -3.05483 q^{25} -2.49162i q^{29} -4.99578 q^{31} +6.38038 q^{35} +10.1215i q^{37} -12.1835 q^{41} +4.71460i q^{43} -3.25120i q^{47} +13.9284 q^{49} -7.08207 q^{53} -4.41655i q^{55} +5.99637 q^{59} +9.50097 q^{61} +0.946227i q^{65} +9.54542i q^{67} -9.35726 q^{71} -10.4904i q^{73} -14.4868i q^{77} -17.2275i q^{79} -15.9720 q^{83} -2.85592 q^{85} +15.3957i q^{89} +3.10373i q^{91} +5.35446 q^{95} +7.44085 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) \) 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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