LMFDB - Embedded newform 1176.2.k.b.881.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,2,Mod(881,1176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1176.881"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.19
Character	$\chi$	$=$	1176.881
Dual form			1176.2.k.b.881.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.48931 - 0.884275i) q^{3} +0.992103 q^{5} +(1.43612 - 2.63393i) q^{9} -2.50853i q^{11} -6.74633i q^{13} +(1.47755 - 0.877291i) q^{15} -3.75663 q^{17} +2.67036i q^{19} -1.74743i q^{23} -4.01573 q^{25} +(-0.190283 - 5.19267i) q^{27} +6.69055i q^{29} -2.82723i q^{31} +(-2.21823 - 3.73599i) q^{33} -3.28467 q^{37} +(-5.96561 - 10.0474i) q^{39} +9.02583 q^{41} +10.6558 q^{43} +(1.42478 - 2.61313i) q^{45} +10.5367 q^{47} +(-5.59480 + 3.32189i) q^{51} +10.2884i q^{53} -2.48872i q^{55} +(2.36133 + 3.97701i) q^{57} +4.64320 q^{59} -8.05303i q^{61} -6.69306i q^{65} -6.69770 q^{67} +(-1.54521 - 2.60247i) q^{69} -4.56245i q^{71} +3.23514i q^{73} +(-5.98069 + 3.55101i) q^{75} -14.4669 q^{79} +(-4.87514 - 7.56525i) q^{81} +16.8613 q^{83} -3.72696 q^{85} +(5.91628 + 9.96433i) q^{87} -3.17101 q^{89} +(-2.50005 - 4.21064i) q^{93} +2.64927i q^{95} -0.989407i q^{97} +(-6.60728 - 3.60254i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 16 q^{15} + 8 q^{25} + 16 q^{37} - 64 q^{39} + 16 q^{43} + 48 q^{51} + 48 q^{57} + 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} - 56 q^{99}+O(q^{100}) $ 24 * q - 16 * q^15 + 8 * q^25 + 16 * q^37 - 64 * q^39 + 16 * q^43 + 48 * q^51 + 48 * q^57 + 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 - 56 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.48931	−	0.884275i	0.859856	−	0.510536i
$3$	1.48931	−	0.884275i	0.859856	−	0.510536i
$4$		0			0
$5$	0.992103			0.443682			0.221841	−	0.975083i	$-0.428793\pi$
$5$	0.992103			0.443682			0.221841	+	0.975083i	$0.428793\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.43612	−	2.63393i	0.478706	−	0.877975i
$10$		0			0
$11$		−	2.50853i		−	0.756350i	−0.925734	−	0.378175i	$-0.876552\pi$
$11$		−	2.50853i		−	0.756350i	0.925734	−	0.378175i	$-0.123448\pi$
$12$		0			0
$13$		−	6.74633i		−	1.87110i	−0.353199	−	0.935548i	$-0.614906\pi$
$13$		−	6.74633i		−	1.87110i	0.353199	−	0.935548i	$-0.385094\pi$
$14$		0			0
$15$	1.47755	−	0.877291i	0.381503	−	0.226516i
$16$		0			0
$17$	−3.75663			−0.911115			−0.455558	−	0.890206i	$-0.650560\pi$
$17$	−3.75663			−0.911115			−0.455558	+	0.890206i	$0.650560\pi$
$18$		0			0
$19$			2.67036i			0.612623i	0.951931	+	0.306312i	$0.0990949\pi$
$19$			2.67036i			0.612623i	−0.951931	+	0.306312i	$0.900905\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.74743i		−	0.364365i	−0.983265	−	0.182182i	$-0.941684\pi$
$23$		−	1.74743i		−	0.364365i	0.983265	−	0.182182i	$-0.0583161\pi$
$24$		0			0
$25$	−4.01573			−0.803146
$26$		0			0
$27$	−0.190283	−	5.19267i	−0.0366201	−	0.999329i
$28$		0			0
$29$			6.69055i			1.24240i	0.783651	+	0.621202i	$0.213355\pi$
$29$			6.69055i			1.24240i	−0.783651	+	0.621202i	$0.786645\pi$
$30$		0			0
$31$		−	2.82723i		−	0.507786i	−0.967232	−	0.253893i	$-0.918289\pi$
$31$		−	2.82723i		−	0.507786i	0.967232	−	0.253893i	$-0.0817111\pi$
$32$		0			0
$33$	−2.21823	−	3.73599i	−0.386144	−	0.650352i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.28467			−0.539996			−0.269998	−	0.962861i	$-0.587023\pi$
$37$	−3.28467			−0.539996			−0.269998	+	0.962861i	$0.587023\pi$
$38$		0			0
$39$	−5.96561	−	10.0474i	−0.955262	−	1.60887i
$40$		0			0
$41$	9.02583			1.40960			0.704799	−	0.709407i	$-0.251037\pi$
$41$	9.02583			1.40960			0.704799	+	0.709407i	$0.251037\pi$
$42$		0			0
$43$	10.6558			1.62499			0.812495	−	0.582969i	$-0.198109\pi$
$43$	10.6558			1.62499			0.812495	+	0.582969i	$0.198109\pi$
$44$		0			0
$45$	1.42478	−	2.61313i	0.212393	−	0.389542i
$46$		0			0
$47$	10.5367			1.53694			0.768470	−	0.639886i	$-0.221018\pi$
$47$	10.5367			1.53694			0.768470	+	0.639886i	$0.221018\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−5.59480	+	3.32189i	−0.783428	+	0.465157i
$52$		0			0
$53$			10.2884i			1.41322i	0.707602	+	0.706611i	$0.249777\pi$
$53$			10.2884i			1.41322i	−0.707602	+	0.706611i	$0.750223\pi$
$54$		0			0
$55$		−	2.48872i		−	0.335579i
$56$		0			0
$57$	2.36133	+	3.97701i	0.312766	+	0.526768i
$58$		0			0
$59$	4.64320			0.604493			0.302246	−	0.953230i	$-0.402263\pi$
$59$	4.64320			0.604493			0.302246	+	0.953230i	$0.402263\pi$
$60$		0			0
$61$		−	8.05303i		−	1.03108i	−0.856864	−	0.515542i	$-0.827590\pi$
$61$		−	8.05303i		−	1.03108i	0.856864	−	0.515542i	$-0.172410\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.69306i		−	0.830172i
$66$		0			0
$67$	−6.69770			−0.818254			−0.409127	−	0.912477i	$-0.634167\pi$
$67$	−6.69770			−0.818254			−0.409127	+	0.912477i	$0.634167\pi$
$68$		0			0
$69$	−1.54521	−	2.60247i	−0.186021	−	0.313301i
$70$		0			0
$71$		−	4.56245i		−	0.541463i	−0.962655	−	0.270732i	$-0.912734\pi$
$71$		−	4.56245i		−	0.541463i	0.962655	−	0.270732i	$-0.0872656\pi$
$72$		0			0
$73$			3.23514i			0.378644i	0.981915	+	0.189322i	$0.0606290\pi$
$73$			3.23514i			0.378644i	−0.981915	+	0.189322i	$0.939371\pi$
$74$		0			0
$75$	−5.98069	+	3.55101i	−0.690590	+	0.410035i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−14.4669			−1.62766			−0.813828	−	0.581105i	$-0.802620\pi$
$79$	−14.4669			−1.62766			−0.813828	+	0.581105i	$0.802620\pi$
$80$		0			0
$81$	−4.87514	−	7.56525i	−0.541682	−	0.840584i
$82$		0			0
$83$	16.8613			1.85077			0.925386	−	0.379025i	$-0.123741\pi$
$83$	16.8613			1.85077			0.925386	+	0.379025i	$0.123741\pi$
$84$		0			0
$85$	−3.72696			−0.404245
$86$		0			0
$87$	5.91628	+	9.96433i	0.634292	+	1.06829i
$88$		0			0
$89$	−3.17101			−0.336126			−0.168063	−	0.985776i	$-0.553751\pi$
$89$	−3.17101			−0.336126			−0.168063	+	0.985776i	$0.553751\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.50005	−	4.21064i	−0.259243	−	0.436623i
$94$		0			0
$95$			2.64927i			0.271810i
$96$		0			0
$97$		−	0.989407i		−	0.100459i	−0.998738	−	0.0502295i	$-0.984005\pi$
$97$		−	0.989407i		−	0.100459i	0.998738	−	0.0502295i	$-0.0159953\pi$
$98$		0			0
$99$	−6.60728	−	3.60254i	−0.664057	−	0.362069i
$100$		0			0
$101$	−10.0416			−0.999173			−0.499586	−	0.866264i	$-0.666515\pi$
$101$	−10.0416			−0.999173			−0.499586	+	0.866264i	$0.666515\pi$
$102$		0			0
$103$		−	7.95653i		−	0.783980i	−0.919969	−	0.391990i	$-0.871787\pi$
$103$		−	7.95653i		−	0.783980i	0.919969	−	0.391990i	$-0.128213\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.3359i			1.57925i	0.613591	+	0.789624i	$0.289724\pi$
$107$			16.3359i			1.57925i	−0.613591	+	0.789624i	$0.710276\pi$
$108$		0			0
$109$	−9.43424			−0.903636			−0.451818	−	0.892110i	$-0.649224\pi$
$109$	−9.43424			−0.903636			−0.451818	+	0.892110i	$0.649224\pi$
$110$		0			0
$111$	−4.89191	+	2.90455i	−0.464319	+	0.275688i
$112$		0			0
$113$			18.5486i			1.74490i	0.488702	+	0.872451i	$0.337471\pi$
$113$			18.5486i			1.74490i	−0.488702	+	0.872451i	$0.662529\pi$
$114$		0			0
$115$		−	1.73363i		−	0.161662i
$116$		0			0
$117$	−17.7693	−	9.68853i	−1.64278	−	0.895705i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.70728			0.427934
$122$		0			0
$123$	13.4423	−	7.98132i	1.21205	−	0.719651i
$124$		0			0
$125$	−8.94453			−0.800023
$126$		0			0
$127$	10.2002			0.905118			0.452559	−	0.891734i	$-0.350511\pi$
$127$	10.2002			0.905118			0.452559	+	0.891734i	$0.350511\pi$
$128$		0			0
$129$	15.8698	−	9.42262i	1.39726	−	0.829616i
$130$		0			0
$131$	17.7607			1.55176			0.775880	−	0.630881i	$-0.217306\pi$
$131$	17.7607			1.55176			0.775880	+	0.630881i	$0.217306\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.188781	−	5.15166i	−0.0162477	−	0.443384i
$136$		0			0
$137$			0.588836i			0.0503076i	0.999684	+	0.0251538i	$0.00800755\pi$
$137$			0.588836i			0.0503076i	−0.999684	+	0.0251538i	$0.991992\pi$
$138$		0			0
$139$			2.88202i			0.244449i	0.992502	+	0.122225i	$0.0390029\pi$
$139$			2.88202i			0.244449i	−0.992502	+	0.122225i	$0.960997\pi$
$140$		0			0
$141$	15.6925	−	9.31736i	1.32155	−	0.784663i
$142$		0			0
$143$	−16.9234			−1.41520
$144$		0			0
$145$			6.63771i			0.551232i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			1.59071i			0.130316i	0.997875	+	0.0651581i	$0.0207552\pi$
$149$			1.59071i			0.130316i	−0.997875	+	0.0651581i	$0.979245\pi$
$150$		0			0
$151$	15.2164			1.23830			0.619148	−	0.785274i	$-0.287478\pi$
$151$	15.2164			1.23830			0.619148	+	0.785274i	$0.287478\pi$
$152$		0			0
$153$	−5.39495	+	9.89467i	−0.436156	+	0.799937i
$154$		0			0
$155$		−	2.80491i		−	0.225295i
$156$		0			0
$157$			2.04590i			0.163281i	0.996662	+	0.0816403i	$0.0260159\pi$
$157$			2.04590i			0.163281i	−0.996662	+	0.0816403i	$0.973984\pi$
$158$		0			0
$159$	9.09778	+	15.3227i	0.721501	+	1.21517i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	18.1622			1.42257			0.711285	−	0.702904i	$-0.248114\pi$
$163$	18.1622			1.42257			0.711285	+	0.702904i	$0.248114\pi$
$164$		0			0
$165$	−2.20071	−	3.70649i	−0.171325	−	0.288550i
$166$		0			0
$167$	−5.59063			−0.432616			−0.216308	−	0.976325i	$-0.569402\pi$
$167$	−5.59063			−0.432616			−0.216308	+	0.976325i	$0.569402\pi$
$168$		0			0
$169$	−32.5130			−2.50100
$170$		0			0
$171$	7.03354	+	3.83495i	0.537868	+	0.293266i
$172$		0			0
$173$	−4.49348			−0.341633			−0.170817	−	0.985303i	$-0.554641\pi$
$173$	−4.49348			−0.341633			−0.170817	+	0.985303i	$0.554641\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	6.91518	−	4.10586i	0.519777	−	0.308616i
$178$		0			0
$179$			19.9731i			1.49286i	0.665463	+	0.746431i	$0.268234\pi$
$179$			19.9731i			1.49286i	−0.665463	+	0.746431i	$0.731766\pi$
$180$		0			0
$181$		−	4.43963i		−	0.329995i	−0.986294	−	0.164998i	$-0.947238\pi$
$181$		−	4.43963i		−	0.329995i	0.986294	−	0.164998i	$-0.0527616\pi$
$182$		0			0
$183$	−7.12109	−	11.9935i	−0.526406	−	0.886585i
$184$		0			0
$185$	−3.25873			−0.239587
$186$		0			0
$187$			9.42361i			0.689122i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3.24209i		−	0.234590i	−0.993097	−	0.117295i	$-0.962578\pi$
$191$		−	3.24209i		−	0.234590i	0.993097	−	0.117295i	$-0.0374222\pi$
$192$		0			0
$193$	13.2677			0.955032			0.477516	−	0.878623i	$-0.341537\pi$
$193$	13.2677			0.955032			0.477516	+	0.878623i	$0.341537\pi$
$194$		0			0
$195$	−5.91850	−	9.96807i	−0.423833	−	0.713828i
$196$		0			0
$197$		−	18.7165i		−	1.33350i	−0.745282	−	0.666749i	$-0.767685\pi$
$197$		−	18.7165i		−	1.33350i	0.745282	−	0.666749i	$-0.232315\pi$
$198$		0			0
$199$		−	1.25058i		−	0.0886516i	−0.999017	−	0.0443258i	$-0.985886\pi$
$199$		−	1.25058i		−	0.0886516i	0.999017	−	0.0443258i	$-0.0141140\pi$
$200$		0			0
$201$	−9.97498	+	5.92260i	−0.703581	+	0.417748i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	8.95456			0.625413
$206$		0			0
$207$	−4.60260	−	2.50952i	−0.319903	−	0.174423i
$208$		0			0
$209$	6.69868			0.463358
$210$		0			0
$211$	−1.04084			−0.0716546			−0.0358273	−	0.999358i	$-0.511407\pi$
$211$	−1.04084			−0.0716546			−0.0358273	+	0.999358i	$0.511407\pi$
$212$		0			0
$213$	−4.03446	−	6.79493i	−0.276437	−	0.465581i
$214$		0			0
$215$	10.5716			0.720978
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.86075	+	4.81814i	0.193311	+	0.325579i
$220$		0			0
$221$			25.3435i			1.70478i
$222$		0			0
$223$			21.2360i			1.42207i	0.703156	+	0.711035i	$0.251773\pi$
$223$			21.2360i			1.42207i	−0.703156	+	0.711035i	$0.748227\pi$
$224$		0			0
$225$	−5.76706	+	10.5771i	−0.384471	+	0.705143i
$226$		0			0
$227$	−6.55372			−0.434985			−0.217493	−	0.976062i	$-0.569788\pi$
$227$	−6.55372			−0.434985			−0.217493	+	0.976062i	$0.569788\pi$
$228$		0			0
$229$			17.1926i			1.13612i	0.822988	+	0.568059i	$0.192306\pi$
$229$			17.1926i			1.13612i	−0.822988	+	0.568059i	$0.807694\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.4513i		−	0.815711i	−0.913047	−	0.407855i	$-0.866277\pi$
$233$		−	12.4513i		−	0.815711i	0.913047	−	0.407855i	$-0.133723\pi$
$234$		0			0
$235$	10.4535			0.681913
$236$		0			0
$237$	−21.5458	+	12.7927i	−1.39955	+	0.830978i
$238$		0			0
$239$			8.44730i			0.546410i	0.961956	+	0.273205i	$0.0880838\pi$
$239$			8.44730i			0.546410i	−0.961956	+	0.273205i	$0.911916\pi$
$240$		0			0
$241$			2.89976i			0.186790i	0.995629	+	0.0933951i	$0.0297720\pi$
$241$			2.89976i			0.186790i	−0.995629	+	0.0933951i	$0.970228\pi$
$242$		0			0
$243$	−13.9504	−	6.95609i	−0.894917	−	0.446233i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	18.0152			1.14628
$248$		0			0
$249$	25.1118	−	14.9101i	1.59140	−	0.944886i
$250$		0			0
$251$	−1.28400			−0.0810455			−0.0405227	−	0.999179i	$-0.512902\pi$
$251$	−1.28400			−0.0810455			−0.0405227	+	0.999179i	$0.512902\pi$
$252$		0			0
$253$	−4.38348			−0.275587
$254$		0			0
$255$	−5.55062	+	3.29565i	−0.347593	+	0.206382i
$256$		0			0
$257$	17.7919			1.10983			0.554913	−	0.831908i	$-0.312752\pi$
$257$	17.7919			1.10983			0.554913	+	0.831908i	$0.312752\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	17.6224	+	9.60841i	1.09080	+	0.594745i
$262$		0			0
$263$			13.9435i			0.859796i	0.902878	+	0.429898i	$0.141450\pi$
$263$			13.9435i			0.859796i	−0.902878	+	0.429898i	$0.858550\pi$
$264$		0			0
$265$			10.2072i			0.627021i
$266$		0			0
$267$	−4.72263	+	2.80404i	−0.289020	+	0.171604i
$268$		0			0
$269$	1.39664			0.0851546			0.0425773	−	0.999093i	$-0.486443\pi$
$269$	1.39664			0.0851546			0.0425773	+	0.999093i	$0.486443\pi$
$270$		0			0
$271$		−	10.6939i		−	0.649608i	−0.945781	−	0.324804i	$-0.894702\pi$
$271$		−	10.6939i		−	0.649608i	0.945781	−	0.324804i	$-0.105298\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			10.0736i			0.607460i
$276$		0			0
$277$	−3.52408			−0.211741			−0.105871	−	0.994380i	$-0.533763\pi$
$277$	−3.52408			−0.211741			−0.105871	+	0.994380i	$0.533763\pi$
$278$		0			0
$279$	−7.44672	−	4.06024i	−0.445824	−	0.243080i
$280$		0			0
$281$		−	7.84120i		−	0.467767i	−0.972265	−	0.233883i	$-0.924857\pi$
$281$		−	7.84120i		−	0.467767i	0.972265	−	0.233883i	$-0.0751434\pi$
$282$		0			0
$283$		−	22.1234i		−	1.31510i	−0.753411	−	0.657550i	$-0.771593\pi$
$283$		−	22.1234i		−	1.31510i	0.753411	−	0.657550i	$-0.228407\pi$
$284$		0			0
$285$	2.34269	+	3.94560i	0.138769	+	0.233717i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.88777			−0.169869
$290$		0			0
$291$	−0.874907	−	1.47354i	−0.0512880	−	0.0863803i
$292$		0			0
$293$	−20.5775			−1.20215			−0.601074	−	0.799193i	$-0.705260\pi$
$293$	−20.5775			−1.20215			−0.601074	+	0.799193i	$0.705260\pi$
$294$		0			0
$295$	4.60653			0.268203
$296$		0			0
$297$	−13.0260	+	0.477332i	−0.755843	+	0.0276976i
$298$		0			0
$299$	−11.7888			−0.681761
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−14.9550	+	8.87950i	−0.859145	+	0.510114i
$304$		0			0
$305$		−	7.98943i		−	0.457474i
$306$		0			0
$307$			29.4122i			1.67864i	0.543636	+	0.839321i	$0.317047\pi$
$307$			29.4122i			1.67864i	−0.543636	+	0.839321i	$0.682953\pi$
$308$		0			0
$309$	−7.03576	−	11.8498i	−0.400250	−	0.674110i
$310$		0			0
$311$	8.07689			0.457999			0.228999	−	0.973427i	$-0.426455\pi$
$311$	8.07689			0.457999			0.228999	+	0.973427i	$0.426455\pi$
$312$		0			0
$313$			1.21057i			0.0684254i	0.999415	+	0.0342127i	$0.0108924\pi$
$313$			1.21057i			0.0684254i	−0.999415	+	0.0342127i	$0.989108\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			29.7844i			1.67286i	0.548075	+	0.836429i	$0.315361\pi$
$317$			29.7844i			1.67286i	−0.548075	+	0.836429i	$0.684639\pi$
$318$		0			0
$319$	16.7834			0.939692
$320$		0			0
$321$	14.4454	+	24.3293i	0.806263	+	1.35793i
$322$		0			0
$323$		−	10.0316i		−	0.558171i
$324$		0			0
$325$			27.0915i			1.50276i
$326$		0			0
$327$	−14.0505	+	8.34246i	−0.776997	+	0.461339i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.4004			−0.681585			−0.340793	−	0.940138i	$-0.610695\pi$
$331$	−12.4004			−0.681585			−0.340793	+	0.940138i	$0.610695\pi$
$332$		0			0
$333$	−4.71717	+	8.65158i	−0.258499	+	0.474104i
$334$		0			0
$335$	−6.64481			−0.363044
$336$		0			0
$337$	1.60403			0.0873772			0.0436886	−	0.999045i	$-0.486089\pi$
$337$	1.60403			0.0873772			0.0436886	+	0.999045i	$0.486089\pi$
$338$		0			0
$339$	16.4020	+	27.6246i	0.890835	+	1.50036i
$340$		0			0
$341$	−7.09220			−0.384064
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.53301	−	2.58192i	−0.0825343	−	0.139006i
$346$		0			0
$347$		−	9.27425i		−	0.497868i	−0.968520	−	0.248934i	$-0.919920\pi$
$347$		−	9.27425i		−	0.497868i	0.968520	−	0.248934i	$-0.0800802\pi$
$348$		0			0
$349$		−	17.2999i		−	0.926043i	−0.886347	−	0.463021i	$-0.846765\pi$
$349$		−	17.2999i		−	0.926043i	0.886347	−	0.463021i	$-0.153235\pi$
$350$		0			0
$351$	−35.0315	+	1.28372i	−1.86984	+	0.0685197i
$352$		0			0
$353$	3.53395			0.188093			0.0940465	−	0.995568i	$-0.470020\pi$
$353$	3.53395			0.188093			0.0940465	+	0.995568i	$0.470020\pi$
$354$		0			0
$355$		−	4.52642i		−	0.240237i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.4164i		−	0.708091i	−0.935228	−	0.354046i	$-0.884806\pi$
$359$		−	13.4164i		−	0.708091i	0.935228	−	0.354046i	$-0.115194\pi$
$360$		0			0
$361$	11.8692			0.624693
$362$		0			0
$363$	7.01062	−	4.16253i	0.367962	−	0.218476i
$364$		0			0
$365$			3.20959i			0.167998i
$366$		0			0
$367$			9.10699i			0.475381i	0.971341	+	0.237691i	$0.0763904\pi$
$367$			9.10699i			0.475381i	−0.971341	+	0.237691i	$0.923610\pi$
$368$		0			0
$369$	12.9622	−	23.7734i	0.674783	−	1.23759i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−33.3259			−1.72555			−0.862775	−	0.505587i	$-0.831276\pi$
$373$	−33.3259			−1.72555			−0.862775	+	0.505587i	$0.831276\pi$
$374$		0			0
$375$	−13.3212	+	7.90942i	−0.687905	+	0.408441i
$376$		0			0
$377$	45.1367			2.32466
$378$		0			0
$379$	−11.8875			−0.610620			−0.305310	−	0.952253i	$-0.598760\pi$
$379$	−11.8875			−0.610620			−0.305310	+	0.952253i	$0.598760\pi$
$380$		0			0
$381$	15.1913	−	9.01974i	0.778271	−	0.462095i
$382$		0			0
$383$	−27.8159			−1.42133			−0.710663	−	0.703532i	$-0.751605\pi$
$383$	−27.8159			−1.42133			−0.710663	+	0.703532i	$0.751605\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	15.3029	−	28.0665i	0.777892	−	1.42670i
$388$		0			0
$389$			18.1795i			0.921736i	0.887469	+	0.460868i	$0.152462\pi$
$389$			18.1795i			0.921736i	−0.887469	+	0.460868i	$0.847538\pi$
$390$		0			0
$391$			6.56444i			0.331978i
$392$		0			0
$393$	26.4513	−	15.7053i	1.33429	−	0.792229i
$394$		0			0
$395$	−14.3527			−0.722162
$396$		0			0
$397$		−	2.14697i		−	0.107753i	−0.998548	−	0.0538767i	$-0.982842\pi$
$397$		−	2.14697i		−	0.107753i	0.998548	−	0.0538767i	$-0.0171578\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			11.7610i			0.587315i	0.955911	+	0.293658i	$0.0948725\pi$
$401$			11.7610i			0.587315i	−0.955911	+	0.293658i	$0.905127\pi$
$402$		0			0
$403$	−19.0735			−0.950117
$404$		0			0
$405$	−4.83664	−	7.50551i	−0.240334	−	0.372952i
$406$		0			0
$407$			8.23969i			0.408426i
$408$		0			0
$409$			19.0406i			0.941497i	0.882268	+	0.470748i	$0.156016\pi$
$409$			19.0406i			0.941497i	−0.882268	+	0.470748i	$0.843984\pi$
$410$		0			0
$411$	0.520692	+	0.876962i	0.0256839	+	0.0432573i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	16.7282			0.821154
$416$		0			0
$417$	2.54849	+	4.29223i	0.124800	+	0.210191i
$418$		0			0
$419$	−29.5570			−1.44395			−0.721976	−	0.691918i	$-0.756766\pi$
$419$	−29.5570			−1.44395			−0.721976	+	0.691918i	$0.756766\pi$
$420$		0			0
$421$	8.76696			0.427275			0.213638	−	0.976913i	$-0.431469\pi$
$421$	8.76696			0.427275			0.213638	+	0.976913i	$0.431469\pi$
$422$		0			0
$423$	15.1320	−	27.7530i	0.735742	−	1.34940i
$424$		0			0
$425$	15.0856			0.731759
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−25.2042	+	14.9649i	−1.21687	+	0.722513i
$430$		0			0
$431$		−	24.8468i		−	1.19683i	−0.801187	−	0.598414i	$-0.795798\pi$
$431$		−	24.8468i		−	1.19683i	0.801187	−	0.598414i	$-0.204202\pi$
$432$		0			0
$433$			14.9303i			0.717502i	0.933433	+	0.358751i	$0.116797\pi$
$433$			14.9303i			0.717502i	−0.933433	+	0.358751i	$0.883203\pi$
$434$		0			0
$435$	5.86956	+	9.88564i	0.281424	+	0.473980i
$436$		0			0
$437$	4.66627			0.223218
$438$		0			0
$439$		−	15.7716i		−	0.752735i	−0.926470	−	0.376368i	$-0.877173\pi$
$439$		−	15.7716i		−	0.752735i	0.926470	−	0.376368i	$-0.122827\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	13.2458i		−	0.629329i	−0.949203	−	0.314665i	$-0.898108\pi$
$443$		−	13.2458i		−	0.629329i	0.949203	−	0.314665i	$-0.101892\pi$
$444$		0			0
$445$	−3.14596			−0.149133
$446$		0			0
$447$	1.40663	+	2.36907i	0.0665311	+	0.112053i
$448$		0			0
$449$			14.0532i			0.663212i	0.943418	+	0.331606i	$0.107591\pi$
$449$			14.0532i			0.663212i	−0.943418	+	0.331606i	$0.892409\pi$
$450$		0			0
$451$		−	22.6416i		−	1.06615i
$452$		0			0
$453$	22.6621	−	13.4555i	1.06476	−	0.632195i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.8418			0.553938			0.276969	−	0.960879i	$-0.410670\pi$
$457$	11.8418			0.553938			0.276969	+	0.960879i	$0.410670\pi$
$458$		0			0
$459$	0.714824	+	19.5069i	0.0333651	+	0.910504i
$460$		0			0
$461$	−18.8583			−0.878318			−0.439159	−	0.898409i	$-0.644724\pi$
$461$	−18.8583			−0.878318			−0.439159	+	0.898409i	$0.644724\pi$
$462$		0			0
$463$	12.0554			0.560263			0.280131	−	0.959962i	$-0.409622\pi$
$463$	12.0554			0.560263			0.280131	+	0.959962i	$0.409622\pi$
$464$		0			0
$465$	−2.48031	−	4.17739i	−0.115021	−	0.193722i
$466$		0			0
$467$	−21.0877			−0.975824			−0.487912	−	0.872893i	$-0.662241\pi$
$467$	−21.0877			−0.975824			−0.487912	+	0.872893i	$0.662241\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.80914	+	3.04699i	0.0833606	+	0.140398i
$472$		0			0
$473$		−	26.7303i		−	1.22906i
$474$		0			0
$475$		−	10.7235i		−	0.492026i
$476$		0			0
$477$	27.0989	+	14.7754i	1.24077	+	0.676517i
$478$		0			0
$479$	1.48762			0.0679711			0.0339855	−	0.999422i	$-0.489180\pi$
$479$	1.48762			0.0679711			0.0339855	+	0.999422i	$0.489180\pi$
$480$		0			0
$481$			22.1595i			1.01039i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	0.981593i		−	0.0445718i
$486$		0			0
$487$	−31.2484			−1.41600			−0.708000	−	0.706213i	$-0.750402\pi$
$487$	−31.2484			−1.41600			−0.708000	+	0.706213i	$0.750402\pi$
$488$		0			0
$489$	27.0492	−	16.0603i	1.22321	−	0.726273i
$490$		0			0
$491$			28.3082i			1.27753i	0.769401	+	0.638766i	$0.220555\pi$
$491$			28.3082i			1.27753i	−0.769401	+	0.638766i	$0.779445\pi$
$492$		0			0
$493$		−	25.1339i		−	1.13197i
$494$		0			0
$495$	−6.55510	−	3.57409i	−0.294630	−	0.160644i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−29.2222			−1.30817			−0.654084	−	0.756422i	$-0.726946\pi$
$499$	−29.2222			−1.30817			−0.654084	+	0.756422i	$0.726946\pi$
$500$		0			0
$501$	−8.32621	+	4.94365i	−0.371987	+	0.220866i
$502$		0			0
$503$	37.0646			1.65263			0.826314	−	0.563210i	$-0.190434\pi$
$503$	37.0646			1.65263			0.826314	+	0.563210i	$0.190434\pi$
$504$		0			0
$505$	−9.96226			−0.443315
$506$		0			0
$507$	−48.4221	+	28.7504i	−2.15050	+	1.27685i
$508$		0			0
$509$	30.6883			1.36023			0.680117	−	0.733104i	$-0.261929\pi$
$509$	30.6883			1.36023			0.680117	+	0.733104i	$0.261929\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	13.8663	−	0.508126i	0.612212	−	0.0224343i
$514$		0			0
$515$		−	7.89369i		−	0.347838i
$516$		0			0
$517$		−	26.4317i		−	1.16247i
$518$		0			0
$519$	−6.69221	+	3.97347i	−0.293755	+	0.174416i
$520$		0			0
$521$	1.55801			0.0682577			0.0341288	−	0.999417i	$-0.489134\pi$
$521$	1.55801			0.0682577			0.0341288	+	0.999417i	$0.489134\pi$
$522$		0			0
$523$			2.45903i			0.107526i	0.998554	+	0.0537628i	$0.0171215\pi$
$523$			2.45903i			0.107526i	−0.998554	+	0.0537628i	$0.982879\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			10.6209i			0.462652i
$528$		0			0
$529$	19.9465			0.867238
$530$		0			0
$531$	6.66818	−	12.2298i	0.289374	−	0.530730i
$532$		0			0
$533$		−	60.8913i		−	2.63749i
$534$		0			0
$535$			16.2069i			0.700684i
$536$		0			0
$537$	17.6617	+	29.7463i	0.762160	+	1.28365i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.3080			−0.572155			−0.286077	−	0.958207i	$-0.592351\pi$
$541$	−13.3080			−0.572155			−0.286077	+	0.958207i	$0.592351\pi$
$542$		0			0
$543$	−3.92585	−	6.61201i	−0.168474	−	0.283748i
$544$		0			0
$545$	−9.35973			−0.400927
$546$		0			0
$547$	15.1745			0.648815			0.324408	−	0.945917i	$-0.394835\pi$
$547$	15.1745			0.648815			0.324408	+	0.945917i	$0.394835\pi$
$548$		0			0
$549$	−21.2111	−	11.5651i	−0.905267	−	0.493586i
$550$		0			0
$551$	−17.8662			−0.761125
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.85328	+	2.88161i	−0.206010	+	0.122318i
$556$		0			0
$557$		−	38.0016i		−	1.61018i	−0.593152	−	0.805091i	$-0.702117\pi$
$557$		−	38.0016i		−	1.61018i	0.593152	−	0.805091i	$-0.297883\pi$
$558$		0			0
$559$		−	71.8874i		−	3.04051i
$560$		0			0
$561$	8.33306	+	14.0347i	0.351822	+	0.592546i
$562$		0			0
$563$	38.6097			1.62721			0.813603	−	0.581421i	$-0.197503\pi$
$563$	38.6097			1.62721			0.813603	+	0.581421i	$0.197503\pi$
$564$		0			0
$565$			18.4021i			0.774181i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	24.2941i		−	1.01846i	−0.860630	−	0.509231i	$-0.829930\pi$
$569$		−	24.2941i		−	1.01846i	0.860630	−	0.509231i	$-0.170070\pi$
$570$		0			0
$571$	−30.3225			−1.26896			−0.634478	−	0.772941i	$-0.718785\pi$
$571$	−30.3225			−1.26896			−0.634478	+	0.772941i	$0.718785\pi$
$572$		0			0
$573$	−2.86690	−	4.82850i	−0.119766	−	0.201713i
$574$		0			0
$575$			7.01721i			0.292638i
$576$		0			0
$577$			17.0858i			0.711289i	0.934621	+	0.355645i	$0.115739\pi$
$577$			17.0858i			0.711289i	−0.934621	+	0.355645i	$0.884261\pi$
$578$		0			0
$579$	19.7598	−	11.7323i	0.821190	−	0.487578i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	25.8088			1.06889
$584$		0			0
$585$	−17.6290	−	9.61202i	−0.728870	−	0.397408i
$586$		0			0
$587$	−3.21402			−0.132657			−0.0663284	−	0.997798i	$-0.521129\pi$
$587$	−3.21402			−0.132657			−0.0663284	+	0.997798i	$0.521129\pi$
$588$		0			0
$589$	7.54974			0.311082
$590$		0			0
$591$	−16.5506	−	27.8748i	−0.680799	−	1.14662i
$592$		0			0
$593$	11.1298			0.457045			0.228522	−	0.973539i	$-0.426611\pi$
$593$	11.1298			0.457045			0.228522	+	0.973539i	$0.426611\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.10586	−	1.86251i	−0.0452598	−	0.0762276i
$598$		0			0
$599$			32.6688i			1.33481i	0.744695	+	0.667405i	$0.232595\pi$
$599$			32.6688i			1.33481i	−0.744695	+	0.667405i	$0.767405\pi$
$600$		0			0
$601$		−	27.2379i		−	1.11106i	−0.831497	−	0.555529i	$-0.812516\pi$
$601$		−	27.2379i		−	1.11106i	0.831497	−	0.555529i	$-0.187484\pi$
$602$		0			0
$603$	−9.61868	+	17.6412i	−0.391703	+	0.718407i
$604$		0			0
$605$	4.67010			0.189867
$606$		0			0
$607$		−	29.8529i		−	1.21169i	−0.795582	−	0.605846i	$-0.792835\pi$
$607$		−	29.8529i		−	1.21169i	0.795582	−	0.605846i	$-0.207165\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	71.0843i		−	2.87576i
$612$		0			0
$613$	−24.9907			−1.00937			−0.504683	−	0.863305i	$-0.668391\pi$
$613$	−24.9907			−1.00937			−0.504683	+	0.863305i	$0.668391\pi$
$614$		0			0
$615$	13.3362	−	7.91829i	0.537766	−	0.319296i
$616$		0			0
$617$			28.4800i			1.14656i	0.819359	+	0.573281i	$0.194330\pi$
$617$			28.4800i			1.14656i	−0.819359	+	0.573281i	$0.805670\pi$
$618$		0			0
$619$		−	45.7202i		−	1.83765i	−0.394667	−	0.918824i	$-0.629140\pi$
$619$		−	45.7202i		−	1.83765i	0.394667	−	0.918824i	$-0.370860\pi$
$620$		0			0
$621$	−9.07383	+	0.332507i	−0.364120	+	0.0133431i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11.2048			0.448190
$626$		0			0
$627$	9.97645	−	5.92348i	0.398421	−	0.236561i
$628$		0			0
$629$	12.3393			0.491999
$630$		0			0
$631$	37.3192			1.48565			0.742827	−	0.669483i	$-0.233484\pi$
$631$	37.3192			1.48565			0.742827	+	0.669483i	$0.233484\pi$
$632$		0			0
$633$	−1.55014	+	0.920391i	−0.0616127	+	0.0365823i
$634$		0			0
$635$	10.1196			0.401584
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−12.0172	−	6.55221i	−0.475392	−	0.259202i
$640$		0			0
$641$		−	37.9593i		−	1.49930i	−0.661832	−	0.749652i	$-0.730221\pi$
$641$		−	37.9593i		−	1.49930i	0.661832	−	0.749652i	$-0.269779\pi$
$642$		0			0
$643$			3.90637i			0.154052i	0.997029	+	0.0770262i	$0.0245425\pi$
$643$			3.90637i			0.154052i	−0.997029	+	0.0770262i	$0.975458\pi$
$644$		0			0
$645$	15.7445	−	9.34821i	0.619938	−	0.368086i
$646$		0			0
$647$	11.4573			0.450435			0.225217	−	0.974309i	$-0.427691\pi$
$647$	11.4573			0.450435			0.225217	+	0.974309i	$0.427691\pi$
$648$		0			0
$649$		−	11.6476i		−	0.457208i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	0.136439i		−	0.00533929i	−0.999996	−	0.00266964i	$-0.999150\pi$
$653$		−	0.136439i		−	0.00533929i	0.999996	−	0.00266964i	$-0.000849775\pi$
$654$		0			0
$655$	17.6204			0.688488
$656$		0			0
$657$	8.52111	+	4.64603i	0.332440	+	0.181259i
$658$		0			0
$659$		−	10.0047i		−	0.389726i	−0.980830	−	0.194863i	$-0.937574\pi$
$659$		−	10.0047i		−	0.389726i	0.980830	−	0.194863i	$-0.0624263\pi$
$660$		0			0
$661$		−	12.0093i		−	0.467109i	−0.972344	−	0.233554i	$-0.924964\pi$
$661$		−	12.0093i		−	0.467109i	0.972344	−	0.233554i	$-0.0750356\pi$
$662$		0			0
$663$	22.4106	+	37.7444i	0.870354	+	1.46587i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	11.6913			0.452688
$668$		0			0
$669$	18.7785	+	31.6271i	0.726018	+	1.22278i
$670$		0			0
$671$	−20.2013			−0.779861
$672$		0			0
$673$	25.4740			0.981950			0.490975	−	0.871174i	$-0.336641\pi$
$673$	25.4740			0.981950			0.490975	+	0.871174i	$0.336641\pi$
$674$		0			0
$675$	0.764127	+	20.8524i	0.0294113	+	0.802608i
$676$		0			0
$677$	−44.9146			−1.72621			−0.863105	−	0.505025i	$-0.831483\pi$
$677$	−44.9146			−1.72621			−0.863105	+	0.505025i	$0.831483\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−9.76054	+	5.79528i	−0.374025	+	0.222076i
$682$		0			0
$683$		−	23.5040i		−	0.899357i	−0.893191	−	0.449678i	$-0.851539\pi$
$683$		−	23.5040i		−	0.899357i	0.893191	−	0.449678i	$-0.148461\pi$
$684$		0			0
$685$			0.584186i			0.0223206i
$686$		0			0
$687$	15.2030	+	25.6052i	0.580030	+	0.976899i
$688$		0			0
$689$	69.4091			2.64427
$690$		0			0
$691$		−	20.5064i		−	0.780101i	−0.920793	−	0.390051i	$-0.872458\pi$
$691$		−	20.5064i		−	0.780101i	0.920793	−	0.390051i	$-0.127542\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2.85926i			0.108458i
$696$		0			0
$697$	−33.9067			−1.28431
$698$		0			0
$699$	−11.0104	−	18.5439i	−0.416450	−	0.701394i
$700$		0			0
$701$		−	20.4363i		−	0.771868i	−0.922526	−	0.385934i	$-0.873879\pi$
$701$		−	20.4363i		−	0.771868i	0.922526	−	0.385934i	$-0.126121\pi$
$702$		0			0
$703$		−	8.77126i		−	0.330814i
$704$		0			0
$705$	15.5686	−	9.24378i	0.586347	−	0.348141i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17.8764			−0.671362			−0.335681	−	0.941976i	$-0.608966\pi$
$709$	−17.8764			−0.671362			−0.335681	+	0.941976i	$0.608966\pi$
$710$		0			0
$711$	−20.7762	+	38.1048i	−0.779168	+	1.42904i
$712$		0			0
$713$	−4.94039			−0.185019
$714$		0			0
$715$	−16.7897			−0.627901
$716$		0			0
$717$	7.46973	+	12.5807i	0.278962	+	0.469834i
$718$		0			0
$719$	−33.7414			−1.25834			−0.629170	−	0.777267i	$-0.716605\pi$
$719$	−33.7414			−1.25834			−0.629170	+	0.777267i	$0.716605\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	2.56419	+	4.31866i	0.0953631	+	0.160613i
$724$		0			0
$725$		−	26.8674i		−	0.997832i
$726$		0			0
$727$			11.5388i			0.427950i	0.976839	+	0.213975i	$0.0686411\pi$
$727$			11.5388i			0.427950i	−0.976839	+	0.213975i	$0.931359\pi$
$728$		0			0
$729$	−26.9276	+	1.97616i	−0.997318	+	0.0731910i
$730$		0			0
$731$	−40.0297			−1.48055
$732$		0			0
$733$		−	22.0852i		−	0.815737i	−0.913041	−	0.407869i	$-0.866272\pi$
$733$		−	22.0852i		−	0.815737i	0.913041	−	0.407869i	$-0.133728\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			16.8014i			0.618887i
$738$		0			0
$739$	1.04873			0.0385782			0.0192891	−	0.999814i	$-0.493860\pi$
$739$	1.04873			0.0385782			0.0192891	+	0.999814i	$0.493860\pi$
$740$		0			0
$741$	26.8302	−	15.9303i	0.985634	−	0.585216i
$742$		0			0
$743$			22.3254i			0.819040i	0.912301	+	0.409520i	$0.134304\pi$
$743$			22.3254i			0.819040i	−0.912301	+	0.409520i	$0.865696\pi$
$744$		0			0
$745$			1.57815i			0.0578189i
$746$		0			0
$747$	24.2149	−	44.4115i	0.885975	−	1.62493i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−9.08649			−0.331571			−0.165785	−	0.986162i	$-0.553016\pi$
$751$	−9.08649			−0.331571			−0.165785	+	0.986162i	$0.553016\pi$
$752$		0			0
$753$	−1.91228	+	1.13541i	−0.0696875	+	0.0413767i
$754$		0			0
$755$	15.0963			0.549410
$756$		0			0
$757$	10.7231			0.389736			0.194868	−	0.980829i	$-0.437572\pi$
$757$	10.7231			0.389736			0.194868	+	0.980829i	$0.437572\pi$
$758$		0			0
$759$	−6.52839	+	3.87620i	−0.236965	+	0.140697i
$760$		0			0
$761$	−12.3093			−0.446211			−0.223105	−	0.974794i	$-0.571619\pi$
$761$	−12.3093			−0.446211			−0.223105	+	0.974794i	$0.571619\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−5.35235	+	9.81654i	−0.193515	+	0.354918i
$766$		0			0
$767$		−	31.3246i		−	1.13106i
$768$		0			0
$769$		−	3.10213i		−	0.111866i	−0.998435	−	0.0559328i	$-0.982187\pi$
$769$		−	3.10213i		−	0.111866i	0.998435	−	0.0559328i	$-0.0178133\pi$
$770$		0			0
$771$	26.4977	−	15.7329i	0.954291	−	0.566606i
$772$		0			0
$773$	−29.8457			−1.07348			−0.536738	−	0.843749i	$-0.680344\pi$
$773$	−29.8457			−1.07348			−0.536738	+	0.843749i	$0.680344\pi$
$774$		0			0
$775$			11.3534i			0.407827i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			24.1023i			0.863553i
$780$		0			0
$781$	−11.4450			−0.409536
$782$		0			0
$783$	34.7418	−	1.27310i	1.24157	−	0.0454969i
$784$		0			0
$785$			2.02974i			0.0724446i
$786$		0			0
$787$			40.7414i			1.45227i	0.687550	+	0.726137i	$0.258686\pi$
$787$			40.7414i			1.45227i	−0.687550	+	0.726137i	$0.741314\pi$
$788$		0			0
$789$	12.3299	+	20.7663i	0.438957	+	0.739301i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−54.3284			−1.92926
$794$		0			0
$795$	9.02593	+	15.2017i	0.320117	+	0.539148i
$796$		0			0
$797$	−34.2662			−1.21377			−0.606885	−	0.794789i	$-0.707581\pi$
$797$	−34.2662			−1.21377			−0.606885	+	0.794789i	$0.707581\pi$
$798$		0			0
$799$	−39.5826			−1.40033
$800$		0			0
$801$	−4.55394	+	8.35220i	−0.160905	+	0.295110i
$802$		0			0
$803$	8.11544			0.286387
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2.08004	−	1.23501i	0.0732208	−	0.0434745i
$808$		0			0
$809$		−	37.7362i		−	1.32673i	−0.748295	−	0.663367i	$-0.769127\pi$
$809$		−	37.7362i		−	1.32673i	0.748295	−	0.663367i	$-0.230873\pi$
$810$		0			0
$811$			33.3795i			1.17211i	0.810270	+	0.586057i	$0.199320\pi$
$811$			33.3795i			1.17211i	−0.810270	+	0.586057i	$0.800680\pi$
$812$		0			0
$813$	−9.45634	−	15.9266i	−0.331648	−	0.558570i
$814$		0			0
$815$	18.0187			0.631169
$816$		0			0
$817$			28.4548i			0.995506i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			8.11896i			0.283354i	0.989913	+	0.141677i	$0.0452494\pi$
$821$			8.11896i			0.283354i	−0.989913	+	0.141677i	$0.954751\pi$
$822$		0			0
$823$	47.6932			1.66248			0.831241	−	0.555913i	$-0.187631\pi$
$823$	47.6932			1.66248			0.831241	+	0.555913i	$0.187631\pi$
$824$		0			0
$825$	8.90781	+	15.0027i	0.310130	+	0.522328i
$826$		0			0
$827$		−	11.7724i		−	0.409367i	−0.978828	−	0.204684i	$-0.934383\pi$
$827$		−	11.7724i		−	0.409367i	0.978828	−	0.204684i	$-0.0656166\pi$
$828$		0			0
$829$		−	2.57828i		−	0.0895475i	−0.998997	−	0.0447738i	$-0.985743\pi$
$829$		−	2.57828i		−	0.0895475i	0.998997	−	0.0447738i	$-0.0142567\pi$
$830$		0			0
$831$	−5.24846	+	3.11625i	−0.182067	+	0.108102i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−5.54648			−0.191944
$836$		0			0
$837$	−14.6809	+	0.537976i	−0.507445	+	0.0185952i
$838$		0			0
$839$	15.1790			0.524036			0.262018	−	0.965063i	$-0.415612\pi$
$839$	15.1790			0.524036			0.262018	+	0.965063i	$0.415612\pi$
$840$		0			0
$841$	−15.7634			−0.543566
$842$		0			0
$843$	−6.93377	−	11.6780i	−0.238812	−	0.402212i
$844$		0			0
$845$	−32.2563			−1.10965
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.5632	−	32.9487i	−0.671406	−	1.13080i
$850$		0			0
$851$			5.73973i			0.196756i
$852$		0			0
$853$		−	5.00496i		−	0.171367i	−0.996322	−	0.0856833i	$-0.972693\pi$
$853$		−	5.00496i		−	0.171367i	0.996322	−	0.0856833i	$-0.0273073\pi$
$854$		0			0
$855$	6.97799	+	3.80467i	0.238642	+	0.130117i
$856$		0			0
$857$	−21.6219			−0.738591			−0.369296	−	0.929312i	$-0.620401\pi$
$857$	−21.6219			−0.738591			−0.369296	+	0.929312i	$0.620401\pi$
$858$		0			0
$859$			9.64857i			0.329205i	0.986360	+	0.164603i	$0.0526341\pi$
$859$			9.64857i			0.329205i	−0.986360	+	0.164603i	$0.947366\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			45.5846i			1.55172i	0.630907	+	0.775858i	$0.282683\pi$
$863$			45.5846i			1.55172i	−0.630907	+	0.775858i	$0.717317\pi$
$864$		0			0
$865$	−4.45800			−0.151576
$866$		0			0
$867$	−4.30079	+	2.55358i	−0.146063	+	0.0867240i
$868$		0			0
$869$			36.2907i			1.23108i
$870$		0			0
$871$			45.1849i			1.53103i
$872$		0			0
$873$	−2.60602	−	1.42090i	−0.0882006	−	0.0480903i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	51.5170			1.73960			0.869802	−	0.493401i	$-0.164247\pi$
$877$	51.5170			1.73960			0.869802	+	0.493401i	$0.164247\pi$
$878$		0			0
$879$	−30.6463	+	18.1961i	−1.03367	+	0.613740i
$880$		0			0
$881$	−10.4316			−0.351448			−0.175724	−	0.984439i	$-0.556227\pi$
$881$	−10.4316			−0.351448			−0.175724	+	0.984439i	$0.556227\pi$
$882$		0			0
$883$	−10.6345			−0.357880			−0.178940	−	0.983860i	$-0.557267\pi$
$883$	−10.6345			−0.357880			−0.178940	+	0.983860i	$0.557267\pi$
$884$		0			0
$885$	6.86057	−	4.07344i	0.230616	−	0.136927i
$886$		0			0
$887$	19.7177			0.662057			0.331028	−	0.943621i	$-0.392604\pi$
$887$	19.7177			0.662057			0.331028	+	0.943621i	$0.392604\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−18.9777	+	12.2294i	−0.635776	+	0.409701i
$892$		0			0
$893$			28.1369i			0.941565i
$894$		0			0
$895$			19.8154i			0.662356i
$896$		0			0
$897$	−17.5572	+	10.4245i	−0.586217	+	0.348064i
$898$		0			0
$899$	18.9157			0.630875
$900$		0			0
$901$		−	38.6497i		−	1.28761i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	4.40457i		−	0.146413i
$906$		0			0
$907$	−40.9516			−1.35978			−0.679888	−	0.733316i	$-0.737972\pi$
$907$	−40.9516			−1.35978			−0.679888	+	0.733316i	$0.737972\pi$
$908$		0			0
$909$	−14.4209	+	26.4487i	−0.478310	+	0.877249i
$910$		0			0
$911$		−	27.0641i		−	0.896672i	−0.893865	−	0.448336i	$-0.852017\pi$
$911$		−	27.0641i		−	0.896672i	0.893865	−	0.448336i	$-0.147983\pi$
$912$		0			0
$913$		−	42.2972i		−	1.39983i
$914$		0			0
$915$	−7.06485	−	11.8988i	−0.233557	−	0.393362i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−18.8158			−0.620675			−0.310337	−	0.950627i	$-0.600442\pi$
$919$	−18.8158			−0.620675			−0.310337	+	0.950627i	$0.600442\pi$
$920$		0			0
$921$	26.0084	+	43.8040i	0.857007	+	1.44339i
$922$		0			0
$923$	−30.7798			−1.01313
$924$		0			0
$925$	13.1904			0.433696
$926$		0			0
$927$	−20.9569	−	11.4265i	−0.688315	−	0.375296i
$928$		0			0
$929$	−27.3474			−0.897240			−0.448620	−	0.893723i	$-0.648084\pi$
$929$	−27.3474			−0.897240			−0.448620	+	0.893723i	$0.648084\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	12.0290	−	7.14219i	0.393813	−	0.233825i
$934$		0			0
$935$			9.34919i			0.305751i
$936$		0			0
$937$			49.4739i			1.61624i	0.589018	+	0.808120i	$0.299515\pi$
$937$			49.4739i			1.61624i	−0.589018	+	0.808120i	$0.700485\pi$
$938$		0			0
$939$	1.07047	+	1.80292i	0.0349336	+	0.0588360i
$940$		0			0
$941$	48.4859			1.58060			0.790298	−	0.612723i	$-0.209926\pi$
$941$	48.4859			1.58060			0.790298	+	0.612723i	$0.209926\pi$
$942$		0			0
$943$		−	15.7720i		−	0.513608i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	8.57279i		−	0.278578i	−0.990252	−	0.139289i	$-0.955518\pi$
$947$		−	8.57279i		−	0.278578i	0.990252	−	0.139289i	$-0.0444818\pi$
$948$		0			0
$949$	21.8253			0.708480
$950$		0			0
$951$	26.3376	+	44.3583i	0.854054	+	1.43842i
$952$		0			0
$953$		−	44.7977i		−	1.45114i	−0.688149	−	0.725570i	$-0.741576\pi$
$953$		−	44.7977i		−	1.45114i	0.688149	−	0.725570i	$-0.258424\pi$
$954$		0			0
$955$		−	3.21649i		−	0.104083i
$956$		0			0
$957$	24.9958	−	14.8412i	0.808000	−	0.479747i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	23.0068			0.742153
$962$		0			0
$963$	43.0275	+	23.4602i	1.38654	+	0.755995i
$964$		0			0
$965$	13.1630			0.423730
$966$		0			0
$967$	14.4085			0.463346			0.231673	−	0.972794i	$-0.425580\pi$
$967$	14.4085			0.463346			0.231673	+	0.972794i	$0.425580\pi$
$968$		0			0
$969$	−8.87065	−	14.9401i	−0.284966	−	0.479946i
$970$		0			0
$971$	−3.83048			−0.122926			−0.0614630	−	0.998109i	$-0.519577\pi$
$971$	−3.83048			−0.122926			−0.0614630	+	0.998109i	$0.519577\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	23.9563	+	40.3477i	0.767215	+	1.29216i
$976$		0			0
$977$			35.9006i			1.14856i	0.818659	+	0.574281i	$0.194718\pi$
$977$			35.9006i			1.14856i	−0.818659	+	0.574281i	$0.805282\pi$
$978$		0			0
$979$			7.95456i			0.254229i
$980$		0			0
$981$	−13.5487	+	24.8491i	−0.432576	+	0.793370i
$982$		0			0
$983$	47.4572			1.51365			0.756824	−	0.653618i	$-0.226750\pi$
$983$	47.4572			1.51365			0.756824	+	0.653618i	$0.226750\pi$
$984$		0			0
$985$		−	18.5687i		−	0.591649i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	18.6202i		−	0.592089i
$990$		0			0
$991$	56.0627			1.78089			0.890446	−	0.455089i	$-0.150393\pi$
$991$	56.0627			1.78089			0.890446	+	0.455089i	$0.150393\pi$
$992$		0			0
$993$	−18.4680	+	10.9653i	−0.586066	+	0.347974i
$994$		0			0
$995$		−	1.24071i		−	0.0393331i
$996$		0			0
$997$			52.7551i			1.67077i	0.549665	+	0.835385i	$0.314755\pi$
$997$			52.7551i			1.67077i	−0.549665	+	0.835385i	$0.685245\pi$
$998$		0			0
$999$	0.625018	+	17.0562i	0.0197747	+	0.539634i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.k.b.881.19	yes	24
3.2	odd	2	inner	1176.2.k.b.881.5	✓	24
4.3	odd	2		2352.2.k.j.881.6		24
7.2	even	3		1176.2.u.c.521.13		48
7.3	odd	6		1176.2.u.c.1097.21		48
7.4	even	3		1176.2.u.c.1097.4		48
7.5	odd	6		1176.2.u.c.521.12		48
7.6	odd	2	inner	1176.2.k.b.881.6	yes	24
12.11	even	2		2352.2.k.j.881.20		24
21.2	odd	6		1176.2.u.c.521.21		48
21.5	even	6		1176.2.u.c.521.4		48
21.11	odd	6		1176.2.u.c.1097.12		48
21.17	even	6		1176.2.u.c.1097.13		48
21.20	even	2	inner	1176.2.k.b.881.20	yes	24
28.27	even	2		2352.2.k.j.881.19		24
84.83	odd	2		2352.2.k.j.881.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.k.b.881.5	✓	24	3.2	odd	2	inner
1176.2.k.b.881.6	yes	24	7.6	odd	2	inner
1176.2.k.b.881.19	yes	24	1.1	even	1	trivial
1176.2.k.b.881.20	yes	24	21.20	even	2	inner
1176.2.u.c.521.4		48	21.5	even	6
1176.2.u.c.521.12		48	7.5	odd	6
1176.2.u.c.521.13		48	7.2	even	3
1176.2.u.c.521.21		48	21.2	odd	6
1176.2.u.c.1097.4		48	7.4	even	3
1176.2.u.c.1097.12		48	21.11	odd	6
1176.2.u.c.1097.13		48	21.17	even	6
1176.2.u.c.1097.21		48	7.3	odd	6
2352.2.k.j.881.5		24	84.83	odd	2
2352.2.k.j.881.6		24	4.3	odd	2
2352.2.k.j.881.19		24	28.27	even	2
2352.2.k.j.881.20		24	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.48931 - 0.884275i) q^{3} +0.992103 q^{5} +(1.43612 - 2.63393i) q^{9} -2.50853i q^{11} -6.74633i q^{13} +(1.47755 - 0.877291i) q^{15} -3.75663 q^{17} +2.67036i q^{19} -1.74743i q^{23} -4.01573 q^{25} +(-0.190283 - 5.19267i) q^{27} +6.69055i q^{29} -2.82723i q^{31} +(-2.21823 - 3.73599i) q^{33} -3.28467 q^{37} +(-5.96561 - 10.0474i) q^{39} +9.02583 q^{41} +10.6558 q^{43} +(1.42478 - 2.61313i) q^{45} +10.5367 q^{47} +(-5.59480 + 3.32189i) q^{51} +10.2884i q^{53} -2.48872i q^{55} +(2.36133 + 3.97701i) q^{57} +4.64320 q^{59} -8.05303i q^{61} -6.69306i q^{65} -6.69770 q^{67} +(-1.54521 - 2.60247i) q^{69} -4.56245i q^{71} +3.23514i q^{73} +(-5.98069 + 3.55101i) q^{75} -14.4669 q^{79} +(-4.87514 - 7.56525i) q^{81} +16.8613 q^{83} -3.72696 q^{85} +(5.91628 + 9.96433i) q^{87} -3.17101 q^{89} +(-2.50005 - 4.21064i) q^{93} +2.64927i q^{95} -0.989407i q^{97} +(-6.60728 - 3.60254i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 16 q^{15} + 8 q^{25} + 16 q^{37} - 64 q^{39} + 16 q^{43} + 48 q^{51} + 48 q^{57} + 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} - 56 q^{99}+O(q^{100}) \) 24 * q - 16 * q^15 + 8 * q^25 + 16 * q^37 - 64 * q^39 + 16 * q^43 + 48 * q^51 + 48 * q^57 + 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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