LMFDB - Embedded newform 1176.2.k.a.881.7

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 = [16] 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 $ x^16 - 6x^15 + 19x^14 - 42x^13 + 65x^12 - 48x^11 - 94x^10 + 444x^9 - 962x^8 + 1332x^7 - 846x^6 - 1296x^5 + 5265x^4 - 10206x^3 + 13851x^2 - 13122*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$1.22961 - 1.21986i$ of defining polynomial
Character	$\chi$	$=$	1176.881
Dual form			1176.2.k.a.881.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.454941 - 1.67124i) q^{3} +2.80795 q^{5} +(-2.58606 + 1.52063i) q^{9} -5.48072i q^{11} +1.35669i q^{13} +(-1.27745 - 4.69274i) q^{15} -5.77506 q^{17} -1.98578i q^{19} -2.42404i q^{23} +2.88457 q^{25} +(3.71783 + 3.63012i) q^{27} -7.05668i q^{29} -3.55181i q^{31} +(-9.15958 + 2.49340i) q^{33} +4.28755 q^{37} +(2.26735 - 0.617215i) q^{39} -1.81976 q^{41} +11.2288 q^{43} +(-7.26151 + 4.26984i) q^{45} +0.402427 q^{47} +(2.62731 + 9.65149i) q^{51} -6.09794i q^{53} -15.3896i q^{55} +(-3.31870 + 0.903412i) q^{57} -2.56469 q^{59} +5.49426i q^{61} +3.80952i q^{65} -6.90476 q^{67} +(-4.05114 + 1.10279i) q^{69} -2.08251i q^{71} -0.341440i q^{73} +(-1.31231 - 4.82079i) q^{75} -2.38278 q^{79} +(4.37539 - 7.86486i) q^{81} -11.8717 q^{83} -16.2161 q^{85} +(-11.7934 + 3.21037i) q^{87} +1.15314 q^{89} +(-5.93592 + 1.61587i) q^{93} -5.57596i q^{95} +16.0187i q^{97} +(8.33413 + 14.1735i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{9} + 8 q^{15} + 36 q^{25} + 4 q^{37} + 44 q^{39} + 20 q^{43} - 12 q^{51} - 8 q^{57} - 28 q^{67} - 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} + 20 q^{99}+O(q^{100}) $ 16 * q - 4 * q^9 + 8 * q^15 + 36 * q^25 + 4 * q^37 + 44 * q^39 + 20 * q^43 - 12 * q^51 - 8 * q^57 - 28 * q^67 - 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 + 20 * 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
$3$	−0.454941	−	1.67124i	−0.262660	−	0.964888i
$3$	−0.454941	−	1.67124i	−0.262660	−	0.964888i
$4$		0			0
$5$	2.80795			1.25575			0.627876	−	0.778313i	$-0.283924\pi$
$5$	2.80795			1.25575			0.627876	+	0.778313i	$0.283924\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.58606	+	1.52063i	−0.862019	+	0.506876i
$10$		0			0
$11$		−	5.48072i		−	1.65250i	−0.563304	−	0.826250i	$-0.690470\pi$
$11$		−	5.48072i		−	1.65250i	0.563304	−	0.826250i	$-0.309530\pi$
$12$		0			0
$13$			1.35669i			0.376279i	0.982142	+	0.188139i	$0.0602457\pi$
$13$			1.35669i			0.376279i	−0.982142	+	0.188139i	$0.939754\pi$
$14$		0			0
$15$	−1.27745	−	4.69274i	−0.329836	−	1.21166i
$16$		0			0
$17$	−5.77506			−1.40066			−0.700329	−	0.713820i	$-0.746963\pi$
$17$	−5.77506			−1.40066			−0.700329	+	0.713820i	$0.746963\pi$
$18$		0			0
$19$		−	1.98578i		−	0.455569i	−0.973712	−	0.227784i	$-0.926852\pi$
$19$		−	1.98578i		−	0.455569i	0.973712	−	0.227784i	$-0.0731481\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.42404i		−	0.505447i	−0.967539	−	0.252723i	$-0.918674\pi$
$23$		−	2.42404i		−	0.505447i	0.967539	−	0.252723i	$-0.0813263\pi$
$24$		0			0
$25$	2.88457			0.576914
$26$		0			0
$27$	3.71783	+	3.63012i	0.715497	+	0.698616i
$28$		0			0
$29$		−	7.05668i		−	1.31039i	−0.755458	−	0.655197i	$-0.772586\pi$
$29$		−	7.05668i		−	1.31039i	0.755458	−	0.655197i	$-0.227414\pi$
$30$		0			0
$31$		−	3.55181i		−	0.637925i	−0.947767	−	0.318962i	$-0.896666\pi$
$31$		−	3.55181i		−	0.637925i	0.947767	−	0.318962i	$-0.103334\pi$
$32$		0			0
$33$	−9.15958	+	2.49340i	−1.59448	+	0.434046i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.28755			0.704868			0.352434	−	0.935837i	$-0.385354\pi$
$37$	4.28755			0.704868			0.352434	+	0.935837i	$0.385354\pi$
$38$		0			0
$39$	2.26735	−	0.617215i	0.363067	−	0.0988335i
$40$		0			0
$41$	−1.81976			−0.284199			−0.142100	−	0.989852i	$-0.545385\pi$
$41$	−1.81976			−0.284199			−0.142100	+	0.989852i	$0.545385\pi$
$42$		0			0
$43$	11.2288			1.71238			0.856188	−	0.516665i	$-0.172827\pi$
$43$	11.2288			1.71238			0.856188	+	0.516665i	$0.172827\pi$
$44$		0			0
$45$	−7.26151	+	4.26984i	−1.08248	+	0.636510i
$46$		0			0
$47$	0.402427			0.0586999			0.0293500	−	0.999569i	$-0.490656\pi$
$47$	0.402427			0.0586999			0.0293500	+	0.999569i	$0.490656\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	2.62731	+	9.65149i	0.367897	+	1.35148i
$52$		0			0
$53$		−	6.09794i		−	0.837616i	−0.908075	−	0.418808i	$-0.862448\pi$
$53$		−	6.09794i		−	0.837616i	0.908075	−	0.418808i	$-0.137552\pi$
$54$		0			0
$55$		−	15.3896i		−	2.07513i
$56$		0			0
$57$	−3.31870	+	0.903412i	−0.439573	+	0.119660i
$58$		0			0
$59$	−2.56469			−0.333894			−0.166947	−	0.985966i	$-0.553391\pi$
$59$	−2.56469			−0.333894			−0.166947	+	0.985966i	$0.553391\pi$
$60$		0			0
$61$			5.49426i			0.703469i	0.936100	+	0.351734i	$0.114408\pi$
$61$			5.49426i			0.703469i	−0.936100	+	0.351734i	$0.885592\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			3.80952i			0.472513i
$66$		0			0
$67$	−6.90476			−0.843551			−0.421775	−	0.906700i	$-0.638593\pi$
$67$	−6.90476			−0.843551			−0.421775	+	0.906700i	$0.638593\pi$
$68$		0			0
$69$	−4.05114	+	1.10279i	−0.487700	+	0.132761i
$70$		0			0
$71$		−	2.08251i		−	0.247148i	−0.992335	−	0.123574i	$-0.960564\pi$
$71$		−	2.08251i		−	0.247148i	0.992335	−	0.123574i	$-0.0394357\pi$
$72$		0			0
$73$		−	0.341440i		−	0.0399626i	−0.999800	−	0.0199813i	$-0.993639\pi$
$73$		−	0.341440i		−	0.0399626i	0.999800	−	0.0199813i	$-0.00636066\pi$
$74$		0			0
$75$	−1.31231	−	4.82079i	−0.151532	−	0.556657i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.38278			−0.268084			−0.134042	−	0.990976i	$-0.542796\pi$
$79$	−2.38278			−0.268084			−0.134042	+	0.990976i	$0.542796\pi$
$80$		0			0
$81$	4.37539	−	7.86486i	0.486154	−	0.873873i
$82$		0			0
$83$	−11.8717			−1.30309			−0.651543	−	0.758611i	$-0.725878\pi$
$83$	−11.8717			−1.30309			−0.651543	+	0.758611i	$0.725878\pi$
$84$		0			0
$85$	−16.2161			−1.75888
$86$		0			0
$87$	−11.7934	+	3.21037i	−1.26438	+	0.344188i
$88$		0			0
$89$	1.15314			0.122233			0.0611164	−	0.998131i	$-0.480534\pi$
$89$	1.15314			0.122233			0.0611164	+	0.998131i	$0.480534\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−5.93592	+	1.61587i	−0.615526	+	0.167557i
$94$		0			0
$95$		−	5.57596i		−	0.572082i
$96$		0			0
$97$			16.0187i			1.62645i	0.581950	+	0.813225i	$0.302290\pi$
$97$			16.0187i			1.62645i	−0.581950	+	0.813225i	$0.697710\pi$
$98$		0			0
$99$	8.33413	+	14.1735i	0.837612	+	1.42449i
$100$		0			0
$101$	14.6796			1.46068			0.730339	−	0.683085i	$-0.239362\pi$
$101$	14.6796			1.46068			0.730339	+	0.683085i	$0.239362\pi$
$102$		0			0
$103$		−	4.69916i		−	0.463022i	−0.972832	−	0.231511i	$-0.925633\pi$
$103$		−	4.69916i		−	0.463022i	0.972832	−	0.231511i	$-0.0743670\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.24629i		−	0.797199i	−0.917125	−	0.398600i	$-0.869496\pi$
$107$		−	8.24629i		−	0.797199i	0.917125	−	0.398600i	$-0.130504\pi$
$108$		0			0
$109$	8.82225			0.845019			0.422509	−	0.906359i	$-0.361149\pi$
$109$	8.82225			0.845019			0.422509	+	0.906359i	$0.361149\pi$
$110$		0			0
$111$	−1.95058	−	7.16550i	−0.185141	−	0.680119i
$112$		0			0
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$		−	6.80657i		−	0.634716i
$116$		0			0
$117$	−2.06302	−	3.50849i	−0.190727	−	0.324360i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−19.0383			−1.73076
$122$		0			0
$123$	0.827885	+	3.04125i	0.0746479	+	0.274221i
$124$		0			0
$125$	−5.94002			−0.531291
$126$		0			0
$127$	−6.93769			−0.615620			−0.307810	−	0.951448i	$-0.599596\pi$
$127$	−6.93769			−0.615620			−0.307810	+	0.951448i	$0.599596\pi$
$128$		0			0
$129$	−5.10844	−	18.7660i	−0.449773	−	1.65225i
$130$		0			0
$131$	−0.237468			−0.0207477			−0.0103739	−	0.999946i	$-0.503302\pi$
$131$	−0.237468			−0.0207477			−0.0103739	+	0.999946i	$0.503302\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	10.4395	+	10.1932i	0.898486	+	0.877289i
$136$		0			0
$137$			11.0721i			0.945955i	0.881075	+	0.472977i	$0.156821\pi$
$137$			11.0721i			0.945955i	−0.881075	+	0.472977i	$0.843179\pi$
$138$		0			0
$139$		−	1.02466i		−	0.0869108i	−0.999055	−	0.0434554i	$-0.986163\pi$
$139$		−	1.02466i		−	0.0869108i	0.999055	−	0.0434554i	$-0.0138366\pi$
$140$		0			0
$141$	−0.183080	−	0.672550i	−0.0154181	−	0.0566389i
$142$		0			0
$143$	7.43566			0.621801
$144$		0			0
$145$		−	19.8148i		−	1.64553i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	22.0027i		−	1.80253i	−0.433267	−	0.901266i	$-0.642639\pi$
$149$		−	22.0027i		−	1.80253i	0.433267	−	0.901266i	$-0.357361\pi$
$150$		0			0
$151$	−7.26735			−0.591409			−0.295704	−	0.955279i	$-0.595554\pi$
$151$	−7.26735			−0.591409			−0.295704	+	0.955279i	$0.595554\pi$
$152$		0			0
$153$	14.9346	−	8.78171i	1.20739	−	0.709959i
$154$		0			0
$155$		−	9.97331i		−	0.801075i
$156$		0			0
$157$			22.7469i			1.81540i	0.419616	+	0.907702i	$0.362165\pi$
$157$			22.7469i			1.81540i	−0.419616	+	0.907702i	$0.637835\pi$
$158$		0			0
$159$	−10.1911	+	2.77420i	−0.808206	+	0.220008i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	18.1336			1.42033			0.710165	−	0.704035i	$-0.248620\pi$
$163$	18.1336			1.42033			0.710165	+	0.704035i	$0.248620\pi$
$164$		0			0
$165$	−25.7196	+	7.00135i	−2.00227	+	0.545054i
$166$		0			0
$167$	24.0942			1.86447			0.932233	−	0.361858i	$-0.117857\pi$
$167$	24.0942			1.86447			0.932233	+	0.361858i	$0.117857\pi$
$168$		0			0
$169$	11.1594			0.858414
$170$		0			0
$171$	3.01963	+	5.13534i	0.230917	+	0.392709i
$172$		0			0
$173$	10.3760			0.788876			0.394438	−	0.918923i	$-0.370939\pi$
$173$	10.3760			0.788876			0.394438	+	0.918923i	$0.370939\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.16678	+	4.28620i	0.0877006	+	0.322170i
$178$		0			0
$179$			13.3855i			1.00048i	0.865887	+	0.500239i	$0.166755\pi$
$179$			13.3855i			1.00048i	−0.865887	+	0.500239i	$0.833245\pi$
$180$		0			0
$181$			18.4339i			1.37018i	0.728457	+	0.685092i	$0.240238\pi$
$181$			18.4339i			1.37018i	−0.728457	+	0.685092i	$0.759762\pi$
$182$		0			0
$183$	9.18221	−	2.49957i	0.678769	−	0.184773i
$184$		0			0
$185$	12.0392			0.885140
$186$		0			0
$187$			31.6515i			2.31459i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			4.15105i			0.300360i	0.988659	+	0.150180i	$0.0479853\pi$
$191$			4.15105i			0.300360i	−0.988659	+	0.150180i	$0.952015\pi$
$192$		0			0
$193$	19.5092			1.40431			0.702153	−	0.712026i	$-0.252222\pi$
$193$	19.5092			1.40431			0.702153	+	0.712026i	$0.252222\pi$
$194$		0			0
$195$	6.36661	−	1.73311i	0.455922	−	0.124110i
$196$		0			0
$197$			3.80952i			0.271417i	0.990749	+	0.135709i	$0.0433311\pi$
$197$			3.80952i			0.271417i	−0.990749	+	0.135709i	$0.956669\pi$
$198$		0			0
$199$		−	6.12369i		−	0.434097i	−0.976161	−	0.217049i	$-0.930357\pi$
$199$		−	6.12369i		−	0.434097i	0.976161	−	0.217049i	$-0.0696430\pi$
$200$		0			0
$201$	3.14126	+	11.5395i	0.221567	+	0.813932i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−5.10980			−0.356884
$206$		0			0
$207$	3.68606	+	6.26870i	0.256199	+	0.435705i
$208$		0			0
$209$	−10.8835			−0.752827
$210$		0			0
$211$	2.93058			0.201750			0.100875	−	0.994899i	$-0.467836\pi$
$211$	2.93058			0.201750			0.100875	+	0.994899i	$0.467836\pi$
$212$		0			0
$213$	−3.48036	+	0.947418i	−0.238470	+	0.0649160i
$214$		0			0
$215$	31.5299			2.15032
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.570627	+	0.155335i	−0.0385594	+	0.0104966i
$220$		0			0
$221$		−	7.83499i		−	0.527038i
$222$		0			0
$223$			4.61145i			0.308806i	0.988008	+	0.154403i	$0.0493454\pi$
$223$			4.61145i			0.308806i	−0.988008	+	0.154403i	$0.950655\pi$
$224$		0			0
$225$	−7.45966	+	4.38635i	−0.497311	+	0.292424i
$226$		0			0
$227$	17.2469			1.14472			0.572358	−	0.820004i	$-0.306029\pi$
$227$	17.2469			1.14472			0.572358	+	0.820004i	$0.306029\pi$
$228$		0			0
$229$			13.3605i			0.882885i	0.897290	+	0.441443i	$0.145533\pi$
$229$			13.3605i			0.882885i	−0.897290	+	0.441443i	$0.854467\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.0027i		−	1.17940i	−0.807624	−	0.589698i	$-0.799247\pi$
$233$		−	18.0027i		−	1.17940i	0.807624	−	0.589698i	$-0.200753\pi$
$234$		0			0
$235$	1.12999			0.0737126
$236$		0			0
$237$	1.08403	+	3.98220i	0.0704151	+	0.258671i
$238$		0			0
$239$			23.6499i			1.52979i	0.644158	+	0.764893i	$0.277208\pi$
$239$			23.6499i			1.52979i	−0.644158	+	0.764893i	$0.722792\pi$
$240$		0			0
$241$			4.08272i			0.262991i	0.991317	+	0.131496i	$0.0419779\pi$
$241$			4.08272i			0.262991i	−0.991317	+	0.131496i	$0.958022\pi$
$242$		0			0
$243$	−15.1346	−	3.73426i	−0.970883	−	0.239553i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.69409			0.171421
$248$		0			0
$249$	5.40091	+	19.8404i	0.342269	+	1.25733i
$250$		0			0
$251$	−5.78085			−0.364884			−0.182442	−	0.983217i	$-0.558400\pi$
$251$	−5.78085			−0.364884			−0.182442	+	0.983217i	$0.558400\pi$
$252$		0			0
$253$	−13.2855			−0.835251
$254$		0			0
$255$	7.37735	+	27.1009i	0.461988	+	1.69712i
$256$		0			0
$257$	20.9647			1.30774			0.653871	−	0.756606i	$-0.273144\pi$
$257$	20.9647			1.30774			0.653871	+	0.756606i	$0.273144\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	10.7306	+	18.2490i	0.664206	+	1.12958i
$262$		0			0
$263$		−	4.99913i		−	0.308260i	−0.988051	−	0.154130i	$-0.950743\pi$
$263$		−	4.99913i		−	0.308260i	0.988051	−	0.154130i	$-0.0492574\pi$
$264$		0			0
$265$		−	17.1227i		−	1.05184i
$266$		0			0
$267$	−0.524611	−	1.92717i	−0.0321057	−	0.117941i
$268$		0			0
$269$	−15.3520			−0.936031			−0.468015	−	0.883720i	$-0.655031\pi$
$269$	−15.3520			−0.936031			−0.468015	+	0.883720i	$0.655031\pi$
$270$		0			0
$271$		−	16.7156i		−	1.01540i	−0.861534	−	0.507700i	$-0.830496\pi$
$271$		−	16.7156i		−	1.01540i	0.861534	−	0.507700i	$-0.169504\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	15.8095i		−	0.953350i
$276$		0			0
$277$	−22.5143			−1.35275			−0.676376	−	0.736556i	$-0.736451\pi$
$277$	−22.5143			−1.35275			−0.676376	+	0.736556i	$0.736451\pi$
$278$		0			0
$279$	5.40098	+	9.18520i	0.323348	+	0.549903i
$280$		0			0
$281$			18.1134i			1.08055i	0.841488	+	0.540276i	$0.181680\pi$
$281$			18.1134i			1.08055i	−0.841488	+	0.540276i	$0.818320\pi$
$282$		0			0
$283$		−	5.78191i		−	0.343699i	−0.985123	−	0.171849i	$-0.945026\pi$
$283$		−	5.78191i		−	0.343699i	0.985123	−	0.171849i	$-0.0549743\pi$
$284$		0			0
$285$	−9.31875	+	2.53673i	−0.551995	+	0.150263i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.3513			0.961843
$290$		0			0
$291$	26.7710	−	7.28755i	1.56934	−	0.427204i
$292$		0			0
$293$	9.38786			0.548445			0.274222	−	0.961666i	$-0.411580\pi$
$293$	9.38786			0.548445			0.274222	+	0.961666i	$0.411580\pi$
$294$		0			0
$295$	−7.20151			−0.419288
$296$		0			0
$297$	19.8957	−	20.3764i	1.15446	−	1.18236i
$298$		0			0
$299$	3.28868			0.190189
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−6.67836	−	24.5331i	−0.383662	−	1.40939i
$304$		0			0
$305$			15.4276i			0.883382i
$306$		0			0
$307$		−	19.7599i		−	1.12776i	−0.825857	−	0.563880i	$-0.809308\pi$
$307$		−	19.7599i		−	1.12776i	0.825857	−	0.563880i	$-0.190692\pi$
$308$		0			0
$309$	−7.85341	+	2.13784i	−0.446765	+	0.121618i
$310$		0			0
$311$	20.3822			1.15577			0.577884	−	0.816119i	$-0.303879\pi$
$311$	20.3822			1.15577			0.577884	+	0.816119i	$0.303879\pi$
$312$		0			0
$313$		−	7.15882i		−	0.404641i	−0.979319	−	0.202320i	$-0.935152\pi$
$313$		−	7.15882i		−	0.404641i	0.979319	−	0.202320i	$-0.0648482\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			11.3324i			0.636489i	0.948009	+	0.318245i	$0.103093\pi$
$317$			11.3324i			0.636489i	−0.948009	+	0.318245i	$0.896907\pi$
$318$		0			0
$319$	−38.6757			−2.16543
$320$		0			0
$321$	−13.7815	+	3.75158i	−0.769208	+	0.209393i
$322$		0			0
$323$			11.4680i			0.638096i
$324$		0			0
$325$			3.91347i			0.217081i
$326$		0			0
$327$	−4.01360	−	14.7441i	−0.221953	−	0.815349i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−18.8277			−1.03486			−0.517431	−	0.855725i	$-0.673111\pi$
$331$	−18.8277			−1.03486			−0.517431	+	0.855725i	$0.673111\pi$
$332$		0			0
$333$	−11.0878	+	6.51976i	−0.607610	+	0.357281i
$334$		0			0
$335$	−19.3882			−1.05929
$336$		0			0
$337$	28.9739			1.57831			0.789156	−	0.614193i	$-0.210518\pi$
$337$	28.9739			1.57831			0.789156	+	0.614193i	$0.210518\pi$
$338$		0			0
$339$	−6.68494	+	1.81976i	−0.363076	+	0.0988360i
$340$		0			0
$341$	−19.4665			−1.05417
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−11.3754	+	3.09659i	−0.612430	+	0.166715i
$346$		0			0
$347$			18.0739i			0.970260i	0.874442	+	0.485130i	$0.161228\pi$
$347$			18.0739i			0.970260i	−0.874442	+	0.485130i	$0.838772\pi$
$348$		0			0
$349$		−	12.8624i		−	0.688510i	−0.938876	−	0.344255i	$-0.888132\pi$
$349$		−	12.8624i		−	0.688510i	0.938876	−	0.344255i	$-0.111868\pi$
$350$		0			0
$351$	−4.92495	+	5.04395i	−0.262875	+	0.269226i
$352$		0			0
$353$	27.2772			1.45182			0.725909	−	0.687791i	$-0.241419\pi$
$353$	27.2772			1.45182			0.725909	+	0.687791i	$0.241419\pi$
$354$		0			0
$355$		−	5.84757i		−	0.310357i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	0.892899i		−	0.0471254i	−0.999722	−	0.0235627i	$-0.992499\pi$
$359$		−	0.892899i		−	0.0471254i	0.999722	−	0.0235627i	$-0.00750094\pi$
$360$		0			0
$361$	15.0567			0.792457
$362$		0			0
$363$	8.66131	+	31.8175i	0.454601	+	1.66999i
$364$		0			0
$365$		−	0.958746i		−	0.0501831i
$366$		0			0
$367$		−	11.0553i		−	0.577082i	−0.957467	−	0.288541i	$-0.906830\pi$
$367$		−	11.0553i		−	0.577082i	0.957467	−	0.288541i	$-0.0931702\pi$
$368$		0			0
$369$	4.70601	−	2.76718i	0.244985	−	0.144054i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	23.1006			1.19611			0.598053	−	0.801457i	$-0.295941\pi$
$373$	23.1006			1.19611			0.598053	+	0.801457i	$0.295941\pi$
$374$		0			0
$375$	2.70236	+	9.92717i	0.139549	+	0.512637i
$376$		0			0
$377$	9.57375			0.493073
$378$		0			0
$379$	−23.3938			−1.20166			−0.600830	−	0.799377i	$-0.705163\pi$
$379$	−23.3938			−1.20166			−0.600830	+	0.799377i	$0.705163\pi$
$380$		0			0
$381$	3.15624	+	11.5945i	0.161699	+	0.594005i
$382$		0			0
$383$	−23.0277			−1.17666			−0.588331	−	0.808620i	$-0.700215\pi$
$383$	−23.0277			−1.17666			−0.588331	+	0.808620i	$0.700215\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−29.0383	+	17.0748i	−1.47610	+	0.867962i
$388$		0			0
$389$			6.29941i			0.319393i	0.987166	+	0.159696i	$0.0510515\pi$
$389$			6.29941i			0.319393i	−0.987166	+	0.159696i	$0.948948\pi$
$390$		0			0
$391$			13.9990i			0.707958i
$392$		0			0
$393$	0.108034	+	0.396866i	0.00544960	+	0.0200192i
$394$		0			0
$395$	−6.69073			−0.336647
$396$		0			0
$397$		−	7.25082i		−	0.363908i	−0.983307	−	0.181954i	$-0.941758\pi$
$397$		−	7.25082i		−	0.363908i	0.983307	−	0.181954i	$-0.0582423\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	13.6472i		−	0.681509i	−0.940152	−	0.340755i	$-0.889318\pi$
$401$		−	13.6472i		−	0.681509i	0.940152	−	0.340755i	$-0.110682\pi$
$402$		0			0
$403$	4.81872			0.240038
$404$		0			0
$405$	12.2859	−	22.0841i	0.610489	−	1.09737i
$406$		0			0
$407$		−	23.4989i		−	1.16479i
$408$		0			0
$409$		−	13.7759i		−	0.681176i	−0.940213	−	0.340588i	$-0.889374\pi$
$409$		−	13.7759i		−	0.681176i	0.940213	−	0.340588i	$-0.110626\pi$
$410$		0			0
$411$	18.5041	−	5.03716i	0.912741	−	0.248465i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−33.3351			−1.63635
$416$		0			0
$417$	−1.71245	+	0.466161i	−0.0838592	+	0.0228280i
$418$		0			0
$419$	−6.94914			−0.339488			−0.169744	−	0.985488i	$-0.554294\pi$
$419$	−6.94914			−0.339488			−0.169744	+	0.985488i	$0.554294\pi$
$420$		0			0
$421$	−0.349861			−0.0170512			−0.00852560	−	0.999964i	$-0.502714\pi$
$421$	−0.349861			−0.0170512			−0.00852560	+	0.999964i	$0.502714\pi$
$422$		0			0
$423$	−1.04070	+	0.611941i	−0.0506005	+	0.0297536i
$424$		0			0
$425$	−16.6586			−0.808059
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−3.38278	−	12.4267i	−0.163322	−	0.599969i
$430$		0			0
$431$			20.1511i			0.970642i	0.874336	+	0.485321i	$0.161297\pi$
$431$			20.1511i			0.970642i	−0.874336	+	0.485321i	$0.838703\pi$
$432$		0			0
$433$			1.42453i			0.0684585i	0.999414	+	0.0342292i	$0.0108976\pi$
$433$			1.42453i			0.0684585i	−0.999414	+	0.0342292i	$0.989102\pi$
$434$		0			0
$435$	−33.1152	+	9.01456i	−1.58775	+	0.432215i
$436$		0			0
$437$	−4.81360			−0.230266
$438$		0			0
$439$		−	2.03852i		−	0.0972931i	−0.998816	−	0.0486465i	$-0.984509\pi$
$439$		−	2.03852i		−	0.0972931i	0.998816	−	0.0486465i	$-0.0154908\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.9039i			0.613082i	0.951857	+	0.306541i	$0.0991717\pi$
$443$			12.9039i			0.613082i	−0.951857	+	0.306541i	$0.900828\pi$
$444$		0			0
$445$	3.23796			0.153494
$446$		0			0
$447$	−36.7717	+	10.0099i	−1.73924	+	0.473453i
$448$		0			0
$449$			2.49432i			0.117714i	0.998266	+	0.0588572i	$0.0187457\pi$
$449$			2.49432i			0.117714i	−0.998266	+	0.0588572i	$0.981254\pi$
$450$		0			0
$451$			9.97362i			0.469639i
$452$		0			0
$453$	3.30622	+	12.1455i	0.155340	+	0.570644i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.6190			0.683850			0.341925	−	0.939727i	$-0.388921\pi$
$457$	14.6190			0.683850			0.341925	+	0.939727i	$0.388921\pi$
$458$		0			0
$459$	−21.4707	−	20.9641i	−1.00217	−	0.978522i
$460$		0			0
$461$	2.83467			0.132024			0.0660120	−	0.997819i	$-0.478972\pi$
$461$	2.83467			0.132024			0.0660120	+	0.997819i	$0.478972\pi$
$462$		0			0
$463$	14.1594			0.658042			0.329021	−	0.944323i	$-0.393281\pi$
$463$	14.1594			0.658042			0.329021	+	0.944323i	$0.393281\pi$
$464$		0			0
$465$	−16.6677	+	4.53727i	−0.772948	+	0.210411i
$466$		0			0
$467$	9.97618			0.461643			0.230821	−	0.972996i	$-0.425859\pi$
$467$	9.97618			0.461643			0.230821	+	0.972996i	$0.425859\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	38.0155	−	10.3485i	1.75166	−	0.476834i
$472$		0			0
$473$		−	61.5419i		−	2.82970i
$474$		0			0
$475$		−	5.72811i		−	0.262824i
$476$		0			0
$477$	9.27269	+	15.7696i	0.424567	+	0.722041i
$478$		0			0
$479$	−43.5149			−1.98825			−0.994124	−	0.108244i	$-0.965477\pi$
$479$	−43.5149			−1.98825			−0.994124	+	0.108244i	$0.965477\pi$
$480$		0			0
$481$			5.81688i			0.265227i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			44.9796i			2.04242i
$486$		0			0
$487$	37.1592			1.68384			0.841921	−	0.539601i	$-0.181425\pi$
$487$	37.1592			1.68384			0.841921	+	0.539601i	$0.181425\pi$
$488$		0			0
$489$	−8.24970	−	30.3055i	−0.373064	−	1.37046i
$490$		0			0
$491$		−	22.1831i		−	1.00111i	−0.865704	−	0.500556i	$-0.833129\pi$
$491$		−	22.1831i		−	1.00111i	0.865704	−	0.500556i	$-0.166871\pi$
$492$		0			0
$493$			40.7528i			1.83541i
$494$		0			0
$495$	23.4018	+	39.7983i	1.05183	+	1.78880i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	16.6739			0.746426			0.373213	−	0.927746i	$-0.378256\pi$
$499$	16.6739			0.746426			0.373213	+	0.927746i	$0.378256\pi$
$500$		0			0
$501$	−10.9614	−	40.2671i	−0.489721	−	1.79900i
$502$		0			0
$503$	−8.55884			−0.381620			−0.190810	−	0.981627i	$-0.561111\pi$
$503$	−8.55884			−0.381620			−0.190810	+	0.981627i	$0.561111\pi$
$504$		0			0
$505$	41.2196			1.83425
$506$		0			0
$507$	−5.07686	−	18.6500i	−0.225471	−	0.828274i
$508$		0			0
$509$	−28.2145			−1.25058			−0.625292	−	0.780391i	$-0.715020\pi$
$509$	−28.2145			−1.25058			−0.625292	+	0.780391i	$0.715020\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	7.20861	−	7.38278i	0.318268	−	0.325958i
$514$		0			0
$515$		−	13.1950i		−	0.581441i
$516$		0			0
$517$		−	2.20559i		−	0.0970017i
$518$		0			0
$519$	−4.72049	−	17.3408i	−0.207206	−	0.761177i
$520$		0			0
$521$	−18.0008			−0.788631			−0.394315	−	0.918975i	$-0.629018\pi$
$521$	−18.0008			−0.788631			−0.394315	+	0.918975i	$0.629018\pi$
$522$		0			0
$523$			13.7466i			0.601099i	0.953766	+	0.300549i	$0.0971700\pi$
$523$			13.7466i			0.601099i	−0.953766	+	0.300549i	$0.902830\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			20.5119i			0.893514i
$528$		0			0
$529$	17.1240			0.744523
$530$		0			0
$531$	6.63243	−	3.89993i	0.287823	−	0.169243i
$532$		0			0
$533$		−	2.46886i		−	0.106938i
$534$		0			0
$535$		−	23.1552i		−	1.00108i
$536$		0			0
$537$	22.3703	−	6.08960i	0.965350	−	0.262786i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	39.2544			1.68768			0.843839	−	0.536596i	$-0.180290\pi$
$541$	39.2544			1.68768			0.843839	+	0.536596i	$0.180290\pi$
$542$		0			0
$543$	30.8075	−	8.38635i	1.32207	−	0.359893i
$544$		0			0
$545$	24.7724			1.06113
$546$		0			0
$547$	−12.4980			−0.534375			−0.267188	−	0.963645i	$-0.586094\pi$
$547$	−12.4980			−0.534375			−0.267188	+	0.963645i	$0.586094\pi$
$548$		0			0
$549$	−8.35473	−	14.2085i	−0.356571	−	0.606403i
$550$		0			0
$551$	−14.0130			−0.596974
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−5.47713	−	20.1203i	−0.232491	−	0.854061i
$556$		0			0
$557$			17.8766i			0.757456i	0.925508	+	0.378728i	$0.123638\pi$
$557$			17.8766i			0.757456i	−0.925508	+	0.378728i	$0.876362\pi$
$558$		0			0
$559$			15.2340i			0.644331i
$560$		0			0
$561$	52.8971	−	14.3996i	2.23332	−	0.607950i
$562$		0			0
$563$	2.73288			0.115177			0.0575885	−	0.998340i	$-0.481659\pi$
$563$	2.73288			0.115177			0.0575885	+	0.998340i	$0.481659\pi$
$564$		0			0
$565$		−	11.2318i		−	0.472525i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1.99730i			0.0837312i	0.999123	+	0.0418656i	$0.0133301\pi$
$569$			1.99730i			0.0837312i	−0.999123	+	0.0418656i	$0.986670\pi$
$570$		0			0
$571$	−2.01456			−0.0843068			−0.0421534	−	0.999111i	$-0.513422\pi$
$571$	−2.01456			−0.0843068			−0.0421534	+	0.999111i	$0.513422\pi$
$572$		0			0
$573$	6.93739	−	1.88848i	0.289814	−	0.0788926i
$574$		0			0
$575$		−	6.99231i		−	0.291599i
$576$		0			0
$577$		−	25.4264i		−	1.05852i	−0.848461	−	0.529258i	$-0.822471\pi$
$577$		−	25.4264i		−	1.05852i	0.848461	−	0.529258i	$-0.177529\pi$
$578$		0			0
$579$	−8.87555	−	32.6045i	−0.368855	−	1.35500i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−33.4211			−1.38416
$584$		0			0
$585$	−5.79286	−	9.85165i	−0.239505	−	0.407315i
$586$		0			0
$587$	34.4645			1.42250			0.711251	−	0.702939i	$-0.248129\pi$
$587$	34.4645			1.42250			0.711251	+	0.702939i	$0.248129\pi$
$588$		0			0
$589$	−7.05312			−0.290619
$590$		0			0
$591$	6.36661	−	1.73311i	0.261887	−	0.0712905i
$592$		0			0
$593$	7.24397			0.297474			0.148737	−	0.988877i	$-0.452479\pi$
$593$	7.24397			0.297474			0.148737	+	0.988877i	$0.452479\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−10.2341	+	2.78592i	−0.418855	+	0.114020i
$598$		0			0
$599$		−	37.5814i		−	1.53553i	−0.640730	−	0.767766i	$-0.721368\pi$
$599$		−	37.5814i		−	1.53553i	0.640730	−	0.767766i	$-0.278632\pi$
$600$		0			0
$601$		−	3.78103i		−	0.154232i	−0.997022	−	0.0771158i	$-0.975429\pi$
$601$		−	3.78103i		−	0.154232i	0.997022	−	0.0771158i	$-0.0245711\pi$
$602$		0			0
$603$	17.8561	−	10.4996i	0.727157	−	0.427575i
$604$		0			0
$605$	−53.4586			−2.17340
$606$		0			0
$607$			27.7536i			1.12648i	0.826292	+	0.563242i	$0.190446\pi$
$607$			27.7536i			1.12648i	−0.826292	+	0.563242i	$0.809554\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			0.545969i			0.0220876i
$612$		0			0
$613$	−30.7141			−1.24053			−0.620264	−	0.784393i	$-0.712975\pi$
$613$	−30.7141			−1.24053			−0.620264	+	0.784393i	$0.712975\pi$
$614$		0			0
$615$	2.32466	+	8.53968i	0.0937392	+	0.344353i
$616$		0			0
$617$		−	44.3075i		−	1.78375i	−0.452279	−	0.891877i	$-0.649389\pi$
$617$		−	44.3075i		−	1.78375i	0.452279	−	0.891877i	$-0.350611\pi$
$618$		0			0
$619$			31.6418i			1.27179i	0.771776	+	0.635895i	$0.219369\pi$
$619$			31.6418i			1.27179i	−0.771776	+	0.635895i	$0.780631\pi$
$620$		0			0
$621$	8.79954	−	9.01216i	0.353113	−	0.361646i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−31.1021			−1.24408
$626$		0			0
$627$	4.95135	+	18.1889i	0.197738	+	0.726395i
$628$		0			0
$629$	−24.7608			−0.987279
$630$		0			0
$631$	−20.7528			−0.826157			−0.413079	−	0.910695i	$-0.635547\pi$
$631$	−20.7528			−0.826157			−0.413079	+	0.910695i	$0.635547\pi$
$632$		0			0
$633$	−1.33324	−	4.89770i	−0.0529916	−	0.194666i
$634$		0			0
$635$	−19.4807			−0.773066
$636$		0			0
$637$		0			0
$638$		0			0
$639$	3.16672	+	5.38548i	0.125273	+	0.213046i
$640$		0			0
$641$			38.1089i			1.50521i	0.658471	+	0.752606i	$0.271204\pi$
$641$			38.1089i			1.50521i	−0.658471	+	0.752606i	$0.728796\pi$
$642$		0			0
$643$			29.5791i			1.16648i	0.812298	+	0.583242i	$0.198216\pi$
$643$			29.5791i			1.16648i	−0.812298	+	0.583242i	$0.801784\pi$
$644$		0			0
$645$	−14.3442	−	52.6939i	−0.564804	−	2.07482i
$646$		0			0
$647$	−21.1870			−0.832948			−0.416474	−	0.909148i	$-0.636734\pi$
$647$	−21.1870			−0.832948			−0.416474	+	0.909148i	$0.636734\pi$
$648$		0			0
$649$			14.0563i			0.551760i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			26.6214i			1.04178i	0.853625	+	0.520888i	$0.174399\pi$
$653$			26.6214i			1.04178i	−0.853625	+	0.520888i	$0.825601\pi$
$654$		0			0
$655$	−0.666799			−0.0260540
$656$		0			0
$657$	0.519203	+	0.882984i	0.0202560	+	0.0344485i
$658$		0			0
$659$		−	16.3864i		−	0.638322i	−0.947701	−	0.319161i	$-0.896599\pi$
$659$		−	16.3864i		−	0.638322i	0.947701	−	0.319161i	$-0.103401\pi$
$660$		0			0
$661$		−	18.5014i		−	0.719622i	−0.933025	−	0.359811i	$-0.882841\pi$
$661$		−	18.5014i		−	0.719622i	0.933025	−	0.359811i	$-0.117159\pi$
$662$		0			0
$663$	−13.0941	+	3.56445i	−0.508533	+	0.138432i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−17.1057			−0.662334
$668$		0			0
$669$	7.70682	−	2.09794i	0.297963	−	0.0811110i
$670$		0			0
$671$	30.1125			1.16248
$672$		0			0
$673$	−45.4357			−1.75142			−0.875708	−	0.482841i	$-0.839605\pi$
$673$	−45.4357			−1.75142			−0.875708	+	0.482841i	$0.839605\pi$
$674$		0			0
$675$	10.7243	+	10.4713i	0.412780	+	0.403041i
$676$		0			0
$677$	31.7132			1.21884			0.609419	−	0.792848i	$-0.291403\pi$
$677$	31.7132			1.21884			0.609419	+	0.792848i	$0.291403\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−7.84631	−	28.8236i	−0.300671	−	1.10452i
$682$		0			0
$683$		−	35.9016i		−	1.37374i	−0.726782	−	0.686868i	$-0.758985\pi$
$683$		−	35.9016i		−	1.37374i	0.726782	−	0.686868i	$-0.241015\pi$
$684$		0			0
$685$			31.0899i			1.18788i
$686$		0			0
$687$	22.3285	−	6.07823i	0.851886	−	0.231899i
$688$		0			0
$689$	8.27303			0.315177
$690$		0			0
$691$		−	25.6822i		−	0.976999i	−0.872564	−	0.488499i	$-0.837544\pi$
$691$		−	25.6822i		−	0.976999i	0.872564	−	0.488499i	$-0.162456\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	2.87720i		−	0.109138i
$696$		0			0
$697$	10.5092			0.398066
$698$		0			0
$699$	−30.0868	+	8.19016i	−1.13799	+	0.309780i
$700$		0			0
$701$			1.29881i			0.0490553i	0.999699	+	0.0245276i	$0.00780818\pi$
$701$			1.29881i			0.0490553i	−0.999699	+	0.0245276i	$0.992192\pi$
$702$		0			0
$703$		−	8.51412i		−	0.321116i
$704$		0			0
$705$	−0.514080	−	1.88848i	−0.0193614	−	0.0711244i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−27.7217			−1.04111			−0.520556	−	0.853828i	$-0.674275\pi$
$709$	−27.7217			−1.04111			−0.520556	+	0.853828i	$0.674275\pi$
$710$		0			0
$711$	6.16202	−	3.62333i	0.231094	−	0.135885i
$712$		0			0
$713$	−8.60973			−0.322437
$714$		0			0
$715$	20.8789			0.780828
$716$		0			0
$717$	39.5246	−	10.7593i	1.47607	−	0.401814i
$718$		0			0
$719$	−41.8244			−1.55979			−0.779893	−	0.625912i	$-0.784727\pi$
$719$	−41.8244			−1.55979			−0.779893	+	0.625912i	$0.784727\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	6.82319	−	1.85740i	0.253757	−	0.0690773i
$724$		0			0
$725$		−	20.3555i		−	0.755984i
$726$		0			0
$727$		−	2.19295i		−	0.0813319i	−0.999173	−	0.0406660i	$-0.987052\pi$
$727$		−	2.19295i		−	0.0813319i	0.999173	−	0.0406660i	$-0.0129479\pi$
$728$		0			0
$729$	0.644508	+	26.9923i	0.0238707	+	0.999715i
$730$		0			0
$731$	−64.8470			−2.39845
$732$		0			0
$733$		−	20.8828i		−	0.771323i	−0.922640	−	0.385662i	$-0.873973\pi$
$733$		−	20.8828i		−	0.771323i	0.922640	−	0.385662i	$-0.126027\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			37.8431i			1.39397i
$738$		0			0
$739$	−13.3006			−0.489272			−0.244636	−	0.969615i	$-0.578668\pi$
$739$	−13.3006			−0.489272			−0.244636	+	0.969615i	$0.578668\pi$
$740$		0			0
$741$	−1.22565	−	4.50246i	−0.0450255	−	0.165402i
$742$		0			0
$743$			24.8226i			0.910653i	0.890324	+	0.455327i	$0.150478\pi$
$743$			24.8226i			0.910653i	−0.890324	+	0.455327i	$0.849522\pi$
$744$		0			0
$745$		−	61.7824i		−	2.26353i
$746$		0			0
$747$	30.7009	−	18.0524i	1.12329	−	0.660503i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−11.9727			−0.436890			−0.218445	−	0.975849i	$-0.570098\pi$
$751$	−11.9727			−0.436890			−0.218445	+	0.975849i	$0.570098\pi$
$752$		0			0
$753$	2.62995	+	9.66117i	0.0958406	+	0.352073i
$754$		0			0
$755$	−20.4063			−0.742663
$756$		0			0
$757$	29.8095			1.08345			0.541723	−	0.840557i	$-0.317772\pi$
$757$	29.8095			1.08345			0.541723	+	0.840557i	$0.317772\pi$
$758$		0			0
$759$	6.04411	+	22.2032i	0.219387	+	0.805924i
$760$		0			0
$761$	−33.4879			−1.21393			−0.606967	−	0.794727i	$-0.707614\pi$
$761$	−33.4879			−1.21393			−0.606967	+	0.794727i	$0.707614\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	41.9357	−	24.6586i	1.51619	−	0.891533i
$766$		0			0
$767$		−	3.47949i		−	0.125637i
$768$		0			0
$769$		−	19.6491i		−	0.708566i	−0.935138	−	0.354283i	$-0.884725\pi$
$769$		−	19.6491i		−	0.708566i	0.935138	−	0.354283i	$-0.115275\pi$
$770$		0			0
$771$	−9.53770	−	35.0370i	−0.343492	−	1.26183i
$772$		0			0
$773$	−13.0332			−0.468771			−0.234385	−	0.972144i	$-0.575308\pi$
$773$	−13.0332			−0.468771			−0.234385	+	0.972144i	$0.575308\pi$
$774$		0			0
$775$		−	10.2455i		−	0.368028i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			3.61365i			0.129472i
$780$		0			0
$781$	−11.4136			−0.408412
$782$		0			0
$783$	25.6166	−	26.2355i	0.915462	−	0.937582i
$784$		0			0
$785$			63.8722i			2.27970i
$786$		0			0
$787$			24.3703i			0.868709i	0.900742	+	0.434354i	$0.143023\pi$
$787$			24.3703i			0.868709i	−0.900742	+	0.434354i	$0.856977\pi$
$788$		0			0
$789$	−8.35473	+	2.27431i	−0.297436	+	0.0809675i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.45403			−0.264700
$794$		0			0
$795$	−28.6160	+	7.78981i	−1.01491	+	0.276276i
$796$		0			0
$797$	−3.14465			−0.111389			−0.0556947	−	0.998448i	$-0.517737\pi$
$797$	−3.14465			−0.111389			−0.0556947	+	0.998448i	$0.517737\pi$
$798$		0			0
$799$	−2.32404			−0.0822186
$800$		0			0
$801$	−2.98209	+	1.75350i	−0.105367	+	0.0619568i
$802$		0			0
$803$	−1.87134			−0.0660381
$804$		0			0
$805$		0			0
$806$		0			0
$807$	6.98427	+	25.6569i	0.245858	+	0.903165i
$808$		0			0
$809$		−	6.66030i		−	0.234164i	−0.993122	−	0.117082i	$-0.962646\pi$
$809$		−	6.66030i		−	0.234164i	0.993122	−	0.117082i	$-0.0373540\pi$
$810$		0			0
$811$		−	48.8504i		−	1.71537i	−0.514176	−	0.857685i	$-0.671902\pi$
$811$		−	48.8504i		−	1.71537i	0.514176	−	0.857685i	$-0.328098\pi$
$812$		0			0
$813$	−27.9357	+	7.60460i	−0.979747	+	0.266705i
$814$		0			0
$815$	50.9181			1.78358
$816$		0			0
$817$		−	22.2979i		−	0.780105i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	40.4880i		−	1.41304i	−0.707693	−	0.706520i	$-0.750264\pi$
$821$		−	40.4880i		−	1.41304i	0.707693	−	0.706520i	$-0.249736\pi$
$822$		0			0
$823$	34.5912			1.20577			0.602886	−	0.797827i	$-0.294017\pi$
$823$	34.5912			1.20577			0.602886	+	0.797827i	$0.294017\pi$
$824$		0			0
$825$	−26.4214	+	7.19240i	−0.919876	+	0.250407i
$826$		0			0
$827$			29.3071i			1.01911i	0.860438	+	0.509555i	$0.170190\pi$
$827$			29.3071i			1.01911i	−0.860438	+	0.509555i	$0.829810\pi$
$828$		0			0
$829$			14.7753i			0.513166i	0.966522	+	0.256583i	$0.0825966\pi$
$829$			14.7753i			0.513166i	−0.966522	+	0.256583i	$0.917403\pi$
$830$		0			0
$831$	10.2427	+	37.6267i	0.355314	+	1.30525i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	67.6553			2.34131
$836$		0			0
$837$	12.8935	−	13.2050i	0.445665	−	0.456433i
$838$		0			0
$839$	−32.0373			−1.10605			−0.553026	−	0.833164i	$-0.686527\pi$
$839$	−32.0373			−1.10605			−0.553026	+	0.833164i	$0.686527\pi$
$840$		0			0
$841$	−20.7968			−0.717131
$842$		0			0
$843$	30.2717	−	8.24051i	1.04261	−	0.283818i
$844$		0			0
$845$	31.3350			1.07796
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−9.66293	+	2.63043i	−0.331631	+	0.0902760i
$850$		0			0
$851$		−	10.3932i		−	0.356274i
$852$		0			0
$853$		−	33.1110i		−	1.13370i	−0.823821	−	0.566850i	$-0.808162\pi$
$853$		−	33.1110i		−	1.13370i	0.823821	−	0.566850i	$-0.191838\pi$
$854$		0			0
$855$	8.47896	+	14.4198i	0.289974	+	0.493145i
$856$		0			0
$857$	−18.0008			−0.614897			−0.307448	−	0.951565i	$-0.599475\pi$
$857$	−18.0008			−0.614897			−0.307448	+	0.951565i	$0.599475\pi$
$858$		0			0
$859$			26.6860i			0.910514i	0.890360	+	0.455257i	$0.150453\pi$
$859$			26.6860i			0.910514i	−0.890360	+	0.455257i	$0.849547\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			10.0377i			0.341687i	0.985298	+	0.170843i	$0.0546492\pi$
$863$			10.0377i			0.341687i	−0.985298	+	0.170843i	$0.945351\pi$
$864$		0			0
$865$	29.1354			0.990633
$866$		0			0
$867$	−7.43889	−	27.3269i	−0.252638	−	0.928071i
$868$		0			0
$869$			13.0594i			0.443009i
$870$		0			0
$871$		−	9.36764i		−	0.317410i
$872$		0			0
$873$	−24.3584	−	41.4252i	−0.824407	−	1.40203i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.7620			−0.498479			−0.249239	−	0.968442i	$-0.580181\pi$
$877$	−14.7620			−0.498479			−0.249239	+	0.968442i	$0.580181\pi$
$878$		0			0
$879$	−4.27092	−	15.6893i	−0.144055	−	0.529188i
$880$		0			0
$881$	−4.42345			−0.149030			−0.0745148	−	0.997220i	$-0.523741\pi$
$881$	−4.42345			−0.149030			−0.0745148	+	0.997220i	$0.523741\pi$
$882$		0			0
$883$	−10.5403			−0.354711			−0.177355	−	0.984147i	$-0.556754\pi$
$883$	−10.5403			−0.354711			−0.177355	+	0.984147i	$0.556754\pi$
$884$		0			0
$885$	3.27626	+	12.0354i	0.110130	+	0.404566i
$886$		0			0
$887$	9.84051			0.330412			0.165206	−	0.986259i	$-0.447171\pi$
$887$	9.84051			0.330412			0.165206	+	0.986259i	$0.447171\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−43.1051	−	23.9803i	−1.44408	−	0.803370i
$892$		0			0
$893$		−	0.799130i		−	0.0267419i
$894$		0			0
$895$			37.5857i			1.25635i
$896$		0			0
$897$	−1.49615	−	5.49615i	−0.0499551	−	0.183511i
$898$		0			0
$899$	−25.0640			−0.835932
$900$		0			0
$901$			35.2160i			1.17321i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			51.7615i			1.72061i
$906$		0			0
$907$	−8.21218			−0.272681			−0.136340	−	0.990662i	$-0.543534\pi$
$907$	−8.21218			−0.272681			−0.136340	+	0.990662i	$0.543534\pi$
$908$		0			0
$909$	−37.9624	+	22.3222i	−1.25913	+	0.740382i
$910$		0			0
$911$			44.6131i			1.47810i	0.673652	+	0.739049i	$0.264725\pi$
$911$			44.6131i			1.47810i	−0.673652	+	0.739049i	$0.735275\pi$
$912$		0			0
$913$			65.0654i			2.15335i
$914$		0			0
$915$	25.7832	−	7.01865i	0.852365	−	0.232029i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	30.6443			1.01086			0.505431	−	0.862867i	$-0.331333\pi$
$919$	30.6443			1.01086			0.505431	+	0.862867i	$0.331333\pi$
$920$		0			0
$921$	−33.0235	+	8.98961i	−1.08816	+	0.296218i
$922$		0			0
$923$	2.82532			0.0929966
$924$		0			0
$925$	12.3677			0.406648
$926$		0			0
$927$	7.14568	+	12.1523i	0.234695	+	0.399134i
$928$		0			0
$929$	−21.4396			−0.703412			−0.351706	−	0.936111i	$-0.614398\pi$
$929$	−21.4396			−0.703412			−0.351706	+	0.936111i	$0.614398\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−9.27269	−	34.0634i	−0.303574	−	1.11519i
$934$		0			0
$935$			88.8758i			2.90655i
$936$		0			0
$937$			25.1409i			0.821319i	0.911789	+	0.410659i	$0.134701\pi$
$937$			25.1409i			0.821319i	−0.911789	+	0.410659i	$0.865299\pi$
$938$		0			0
$939$	−11.9641	+	3.25684i	−0.390433	+	0.106283i
$940$		0			0
$941$	52.1196			1.69905			0.849525	−	0.527549i	$-0.176889\pi$
$941$	52.1196			1.69905			0.849525	+	0.527549i	$0.176889\pi$
$942$		0			0
$943$			4.41118i			0.143648i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			41.4472i			1.34685i	0.739254	+	0.673427i	$0.235178\pi$
$947$			41.4472i			1.34685i	−0.739254	+	0.673427i	$0.764822\pi$
$948$		0			0
$949$	0.463230			0.0150371
$950$		0			0
$951$	18.9391	−	5.15556i	0.614141	−	0.167180i
$952$		0			0
$953$		−	29.7579i		−	0.963952i	−0.876184	−	0.481976i	$-0.839919\pi$
$953$		−	29.7579i		−	0.963952i	0.876184	−	0.481976i	$-0.160081\pi$
$954$		0			0
$955$			11.6559i			0.377177i
$956$		0			0
$957$	17.5952	+	64.6363i	0.568771	+	2.08939i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	18.3846			0.593052
$962$		0			0
$963$	12.5395	+	21.3254i	0.404081	+	0.687201i
$964$		0			0
$965$	54.7809			1.76346
$966$		0			0
$967$	−16.6814			−0.536436			−0.268218	−	0.963358i	$-0.586435\pi$
$967$	−16.6814			−0.536436			−0.268218	+	0.963358i	$0.586435\pi$
$968$		0			0
$969$	19.1657	−	5.21726i	0.615692	−	0.167602i
$970$		0			0
$971$	−51.2931			−1.64607			−0.823037	−	0.567988i	$-0.807722\pi$
$971$	−51.2931			−1.64607			−0.823037	+	0.567988i	$0.807722\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	6.54034	−	1.78040i	0.209458	−	0.0570184i
$976$		0			0
$977$			19.2734i			0.616610i	0.951288	+	0.308305i	$0.0997618\pi$
$977$			19.2734i			0.616610i	−0.951288	+	0.308305i	$0.900238\pi$
$978$		0			0
$979$		−	6.32005i		−	0.201990i
$980$		0			0
$981$	−22.8149	+	13.4154i	−0.728422	+	0.428319i
$982$		0			0
$983$	−3.71850			−0.118602			−0.0593008	−	0.998240i	$-0.518887\pi$
$983$	−3.71850			−0.118602			−0.0593008	+	0.998240i	$0.518887\pi$
$984$		0			0
$985$			10.6969i			0.340833i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	27.2190i		−	0.865515i
$990$		0			0
$991$	−50.9829			−1.61952			−0.809762	−	0.586758i	$-0.800404\pi$
$991$	−50.9829			−1.61952			−0.809762	+	0.586758i	$0.800404\pi$
$992$		0			0
$993$	8.56547	+	31.4654i	0.271817	+	0.998526i
$994$		0			0
$995$		−	17.1950i		−	0.545118i
$996$		0			0
$997$			52.7468i			1.67051i	0.549864	+	0.835254i	$0.314680\pi$
$997$			52.7468i			1.67051i	−0.549864	+	0.835254i	$0.685320\pi$
$998$		0			0
$999$	15.9404	+	15.5643i	0.504331	+	0.492432i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.k.a.881.7		16
3.2	odd	2	inner	1176.2.k.a.881.9		16
4.3	odd	2		2352.2.k.i.881.10		16
7.2	even	3		1176.2.u.b.521.8		16
7.3	odd	6		1176.2.u.b.1097.6		16
7.4	even	3		168.2.u.a.89.3	yes	16
7.5	odd	6		168.2.u.a.17.1	✓	16
7.6	odd	2	inner	1176.2.k.a.881.10		16
12.11	even	2		2352.2.k.i.881.8		16
21.2	odd	6		1176.2.u.b.521.6		16
21.5	even	6		168.2.u.a.17.3	yes	16
21.11	odd	6		168.2.u.a.89.1	yes	16
21.17	even	6		1176.2.u.b.1097.8		16
21.20	even	2	inner	1176.2.k.a.881.8		16
28.11	odd	6		336.2.bc.f.257.6		16
28.19	even	6		336.2.bc.f.17.8		16
28.27	even	2		2352.2.k.i.881.7		16
84.11	even	6		336.2.bc.f.257.8		16
84.47	odd	6		336.2.bc.f.17.6		16
84.83	odd	2		2352.2.k.i.881.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.u.a.17.1	✓	16	7.5	odd	6
168.2.u.a.17.3	yes	16	21.5	even	6
168.2.u.a.89.1	yes	16	21.11	odd	6
168.2.u.a.89.3	yes	16	7.4	even	3
336.2.bc.f.17.6		16	84.47	odd	6
336.2.bc.f.17.8		16	28.19	even	6
336.2.bc.f.257.6		16	28.11	odd	6
336.2.bc.f.257.8		16	84.11	even	6
1176.2.k.a.881.7		16	1.1	even	1	trivial
1176.2.k.a.881.8		16	21.20	even	2	inner
1176.2.k.a.881.9		16	3.2	odd	2	inner
1176.2.k.a.881.10		16	7.6	odd	2	inner
1176.2.u.b.521.6		16	21.2	odd	6
1176.2.u.b.521.8		16	7.2	even	3
1176.2.u.b.1097.6		16	7.3	odd	6
1176.2.u.b.1097.8		16	21.17	even	6
2352.2.k.i.881.7		16	28.27	even	2
2352.2.k.i.881.8		16	12.11	even	2
2352.2.k.i.881.9		16	84.83	odd	2
2352.2.k.i.881.10		16	4.3	odd	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)

\(f(q)\)	\(=\)	\(q+(-0.454941 - 1.67124i) q^{3} +2.80795 q^{5} +(-2.58606 + 1.52063i) q^{9} -5.48072i q^{11} +1.35669i q^{13} +(-1.27745 - 4.69274i) q^{15} -5.77506 q^{17} -1.98578i q^{19} -2.42404i q^{23} +2.88457 q^{25} +(3.71783 + 3.63012i) q^{27} -7.05668i q^{29} -3.55181i q^{31} +(-9.15958 + 2.49340i) q^{33} +4.28755 q^{37} +(2.26735 - 0.617215i) q^{39} -1.81976 q^{41} +11.2288 q^{43} +(-7.26151 + 4.26984i) q^{45} +0.402427 q^{47} +(2.62731 + 9.65149i) q^{51} -6.09794i q^{53} -15.3896i q^{55} +(-3.31870 + 0.903412i) q^{57} -2.56469 q^{59} +5.49426i q^{61} +3.80952i q^{65} -6.90476 q^{67} +(-4.05114 + 1.10279i) q^{69} -2.08251i q^{71} -0.341440i q^{73} +(-1.31231 - 4.82079i) q^{75} -2.38278 q^{79} +(4.37539 - 7.86486i) q^{81} -11.8717 q^{83} -16.2161 q^{85} +(-11.7934 + 3.21037i) q^{87} +1.15314 q^{89} +(-5.93592 + 1.61587i) q^{93} -5.57596i q^{95} +16.0187i q^{97} +(8.33413 + 14.1735i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{9} + 8 q^{15} + 36 q^{25} + 4 q^{37} + 44 q^{39} + 20 q^{43} - 12 q^{51} - 8 q^{57} - 28 q^{67} - 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} + 20 q^{99}+O(q^{100}) \) 16 * q - 4 * q^9 + 8 * q^15 + 36 * q^25 + 4 * q^37 + 44 * q^39 + 20 * q^43 - 12 * q^51 - 8 * q^57 - 28 * q^67 - 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more