LMFDB - Embedded newform 912.2.ch.a.575.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(47,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 0, 9, 8]))

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

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

chi := DirichletCharacter("912.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.ch (of order $18$, degree $6$, 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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			575.1
Root			$0.939693 - 0.342020i$ of defining polynomial
Character	$\chi$	$=$	912.575
Dual form			912.2.ch.a.479.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.70574 + 0.300767i) q^{3} +(2.89053 - 1.66885i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})$	$q+(1.70574 + 0.300767i) q^{3} +(2.89053 - 1.66885i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-1.03936 - 5.89452i) q^{13} +(-0.500000 - 4.33013i) q^{19} +(5.43242 - 1.97724i) q^{21} +(0.868241 + 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(-9.64203 + 5.56683i) q^{31} +11.7665 q^{37} -10.3671i q^{39} +(-4.51842 + 5.38484i) q^{43} +(2.07011 - 3.58553i) q^{49} +(0.449493 - 7.53644i) q^{57} +(11.6270 - 9.75622i) q^{61} +(9.86097 - 1.73875i) q^{63} +(-5.18345 + 14.2414i) q^{67} +(-0.216415 + 1.22735i) q^{73} +8.66025i q^{75} +(0.917404 + 0.161763i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-12.8414 - 15.3037i) q^{91} +(-18.1211 + 6.59553i) q^{93} +(-17.8542 + 6.49838i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 15 q^{13} - 3 q^{19} + 9 q^{21} + 27 q^{27} - 39 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 33 q^{67} - 51 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93}+O(q^{100}) $ 6 * q - 15 * q^13 - 3 * q^19 + 9 * q^21 + 27 * q^27 - 39 * q^43 + 21 * q^49 + 42 * q^61 + 36 * q^63 - 33 * q^67 - 51 * q^73 + 12 * q^79 - 48 * q^91 - 54 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{8}{9}\right)$	$1$	$-1$	$-1$

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
$3$	1.70574	+	0.300767i	0.984808	+	0.173648i
$3$	1.70574	+	0.300767i	0.984808	+	0.173648i
$4$		0			0
$5$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$5$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$6$		0			0
$7$	2.89053	−	1.66885i	1.09252	−	0.630765i	0.158272	−	0.987396i	$-0.449408\pi$
$7$	2.89053	−	1.66885i	1.09252	−	0.630765i	0.934246	+	0.356630i	$0.116074\pi$
$8$		0			0
$9$	2.81908	+	1.02606i	0.939693	+	0.342020i
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−1.03936	−	5.89452i	−0.288267	−	1.63485i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−1.03936	−	5.89452i	−0.288267	−	1.63485i	0.405108	−	0.914269i	$-0.367234\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$17$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$18$		0			0
$19$	−0.500000	−	4.33013i	−0.114708	−	0.993399i
$19$	−0.500000	−	4.33013i	−0.114708	−	0.993399i
$20$		0			0
$21$	5.43242	−	1.97724i	1.18545	−	0.431469i
$22$		0			0
$23$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$23$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$24$		0			0
$25$	0.868241	+	4.92404i	0.173648	+	0.984808i
$26$		0			0
$27$	4.50000	+	2.59808i	0.866025	+	0.500000i
$28$		0			0
$29$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$29$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$30$		0			0
$31$	−9.64203	+	5.56683i	−1.73176	+	0.999832i	−0.856702	+	0.515812i	$0.827490\pi$
$31$	−9.64203	+	5.56683i	−1.73176	+	0.999832i	−0.875057	+	0.484020i	$0.839176\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	11.7665			1.93440			0.967201	−	0.254011i	$-0.0817500\pi$
$37$	11.7665			1.93440			0.967201	+	0.254011i	$0.0817500\pi$
$38$		0			0
$39$		−	10.3671i		−	1.66007i
$40$		0			0
$41$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$41$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$42$		0			0
$43$	−4.51842	+	5.38484i	−0.689052	+	0.821181i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.51842	+	5.38484i	−0.689052	+	0.821181i	0.302188	+	0.953248i	$0.402283\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$47$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$48$		0			0
$49$	2.07011	−	3.58553i	0.295730	−	0.512219i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$53$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.449493	−	7.53644i	0.0595368	−	0.998226i
$58$		0			0
$59$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$59$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$60$		0			0
$61$	11.6270	−	9.75622i	1.48869	−	1.24916i	0.592428	−	0.805623i	$-0.298169\pi$
$61$	11.6270	−	9.75622i	1.48869	−	1.24916i	0.896258	−	0.443533i	$-0.146275\pi$
$62$		0			0
$63$	9.86097	−	1.73875i	1.24237	−	0.219062i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.18345	+	14.2414i	−0.633259	+	1.73986i	0.0386729	+	0.999252i	$0.487687\pi$
$67$	−5.18345	+	14.2414i	−0.633259	+	1.73986i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$71$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$72$		0			0
$73$	−0.216415	+	1.22735i	−0.0253294	+	0.143650i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−0.216415	+	1.22735i	−0.0253294	+	0.143650i	0.969520	+	0.245011i	$0.0787915\pi$
$74$		0			0
$75$			8.66025i			1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.917404	+	0.161763i	0.103216	+	0.0181998i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	0.917404	+	0.161763i	0.103216	+	0.0181998i	−0.121802	+	0.992554i	$0.538867\pi$
$80$		0			0
$81$	6.89440	+	5.78509i	0.766044	+	0.642788i
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$89$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$90$		0			0
$91$	−12.8414	−	15.3037i	−1.34614	−	1.60427i
$92$		0			0
$93$	−18.1211	+	6.59553i	−1.87907	+	0.683925i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−17.8542	+	6.49838i	−1.81282	+	0.659811i	−0.816185	+	0.577791i	$0.803915\pi$
$97$	−17.8542	+	6.49838i	−1.81282	+	0.659811i	−0.996631	+	0.0820195i	$0.973863\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$101$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$102$		0			0
$103$	−8.83527	−	5.10105i	−0.870565	−	0.502621i	−0.00302937	−	0.999995i	$-0.500964\pi$
$103$	−8.83527	−	5.10105i	−0.870565	−	0.502621i	−0.867536	+	0.497374i	$0.834298\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	14.5548	+	12.2130i	1.39410	+	1.16979i	0.963647	+	0.267177i	$0.0860909\pi$
$109$	14.5548	+	12.2130i	1.39410	+	1.16979i	0.430454	+	0.902613i	$0.358354\pi$
$110$		0			0
$111$	20.0706	+	3.53898i	1.90501	+	0.335905i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.11809	−	17.6836i	0.288267	−	1.63485i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−11.9402	+	2.10537i	−1.05952	+	0.186822i	−0.676142	−	0.736771i	$-0.736350\pi$
$127$	−11.9402	+	2.10537i	−1.05952	+	0.186822i	−0.383375	+	0.923593i	$0.625238\pi$
$128$		0			0
$129$	−9.32682	+	7.82613i	−0.821181	+	0.689052i
$130$		0			0
$131$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$131$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$132$		0			0
$133$	−8.67159	−	11.6819i	−0.751922	−	1.01295i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$137$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$138$		0			0
$139$	−2.93835	+	0.518110i	−0.249227	+	0.0439455i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−2.93835	+	0.518110i	−0.249227	+	0.0439455i	0.0476387	+	0.998865i	$0.484830\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.60947	−	5.49335i	0.380183	−	0.453084i
$148$		0			0
$149$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$149$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$150$		0			0
$151$		−	15.5885i		−	1.26857i	−0.773099	−	0.634285i	$-0.781294\pi$
$151$		−	15.5885i		−	1.26857i	0.773099	−	0.634285i	$-0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−13.8177	−	11.5945i	−1.10278	−	0.925338i	−0.105167	−	0.994455i	$-0.533538\pi$
$157$	−13.8177	−	11.5945i	−1.10278	−	0.925338i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	20.5403	+	11.8589i	1.60884	+	0.928864i	0.989630	+	0.143639i	$0.0458804\pi$
$163$	20.5403	+	11.8589i	1.60884	+	0.928864i	0.619210	+	0.785225i	$0.287453\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$167$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$168$		0			0
$169$	−21.4491	+	7.80683i	−1.64993	+	0.600525i
$170$		0			0
$171$	3.03343	−	12.7200i	0.231972	−	0.972722i
$172$		0			0
$173$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$173$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$174$		0			0
$175$	10.7271	+	12.7841i	0.810896	+	0.966388i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−17.8542	−	6.49838i	−1.32709	−	0.483021i	−0.421366	−	0.906891i	$-0.638449\pi$
$181$	−17.8542	−	6.49838i	−1.32709	−	0.483021i	−0.905723	+	0.423870i	$0.860671\pi$
$182$		0			0
$183$	22.7670	−	13.1445i	1.68298	−	0.971671i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	17.3432			1.26153
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−1.97225	+	11.1852i	−0.141966	+	0.805127i	0.827788	+	0.561041i	$0.189599\pi$
$193$	−1.97225	+	11.1852i	−0.141966	+	0.805127i	−0.969754	+	0.244086i	$0.921512\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−2.16204	+	5.94015i	−0.153263	+	0.421086i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−2.16204	+	5.94015i	−0.153263	+	0.421086i	0.839171	+	0.543868i	$0.183041\pi$
$200$		0			0
$201$	−13.1250	+	22.7331i	−0.925763	+	1.60347i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−5.92174	−	16.2699i	−0.407669	−	1.12006i	−0.958412	−	0.285388i	$-0.907878\pi$
$211$	−5.92174	−	16.2699i	−0.407669	−	1.12006i	0.550743	−	0.834675i	$-0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−18.5804	+	32.1822i	−1.26132	+	2.18467i
$218$		0			0
$219$	−0.738293	+	2.02844i	−0.0498892	+	0.137069i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−18.8846	+	22.5058i	−1.26461	+	1.50710i	−0.494509	+	0.869172i	$0.664652\pi$
$223$	−18.8846	+	22.5058i	−1.26461	+	1.50710i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$	−2.60472	+	14.7721i	−0.173648	+	0.984808i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−13.5645			−0.896366			−0.448183	−	0.893942i	$-0.647929\pi$
$229$	−13.5645			−0.896366			−0.448183	+	0.893942i	$0.647929\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$233$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.51620	+	0.551851i	0.0984876	+	0.0358465i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−5.16132	−	29.2713i	−0.332470	−	1.88553i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−5.16132	−	29.2713i	−0.332470	−	1.88553i	0.118440	−	0.992961i	$-0.462211\pi$
$242$		0			0
$243$	10.0201	+	11.9415i	0.642788	+	0.766044i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−25.0043	+	7.44783i	−1.59099	+	0.473894i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$251$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$257$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$258$		0			0
$259$	34.0114	−	19.6365i	2.11337	−	1.22015i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$263$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$269$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$270$		0			0
$271$	−21.1535	+	25.2097i	−1.28498	+	1.53138i	−0.619224	+	0.785214i	$0.712553\pi$
$271$	−21.1535	+	25.2097i	−1.28498	+	1.53138i	−0.665758	+	0.746168i	$0.731892\pi$
$272$		0			0
$273$	−17.3011	−	29.9664i	−1.04711	−	1.81365i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	0.931305	+	0.364241i	$0.118672\pi$
$278$		0			0
$279$	−32.8935	+	5.80002i	−1.96928	+	0.347238i
$280$		0			0
$281$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$281$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$282$		0			0
$283$	−11.2555	−	30.9243i	−0.669072	−	1.83826i	−0.530042	−	0.847971i	$-0.677824\pi$
$283$	−11.2555	−	30.9243i	−0.669072	−	1.83826i	−0.139030	−	0.990288i	$-0.544398\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0228	−	10.9274i	0.766044	−	0.642788i
$290$		0			0
$291$	−32.4090	+	5.71458i	−1.89985	+	0.334995i
$292$		0			0
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−4.07414	+	23.1056i	−0.234830	+	1.33178i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−28.9975	−	5.11305i	−1.65498	−	0.291817i	−0.733337	−	0.679865i	$-0.762038\pi$
$307$	−28.9975	−	5.11305i	−1.65498	−	0.291817i	−0.921639	+	0.388048i	$0.873149\pi$
$308$		0			0
$309$	−13.5364	−	11.3584i	−0.770060	−	0.646157i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	12.2160	+	4.44626i	0.690489	+	0.251318i	0.663345	−	0.748314i	$-0.269136\pi$
$313$	12.2160	+	4.44626i	0.690489	+	0.251318i	0.0271446	+	0.999632i	$0.491359\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$317$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	28.1224	−	10.2357i	1.55995	−	0.567776i
$326$		0			0
$327$	21.1535	+	25.2097i	1.16979	+	1.39410i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	23.5737	+	13.6103i	1.29573	+	0.748090i	0.979663	−	0.200648i	$-0.0643046\pi$
$331$	23.5737	+	13.6103i	1.29573	+	0.748090i	0.316066	+	0.948737i	$0.397638\pi$
$332$		0			0
$333$	33.1707	+	12.0732i	1.81774	+	0.661605i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.1291	+	11.0166i	0.715186	+	0.600112i	0.926049	−	0.377403i	$-0.123183\pi$
$337$	13.1291	+	11.0166i	0.715186	+	0.600112i	−0.210863	+	0.977516i	$0.567627\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			9.54509i			0.515387i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$347$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$348$		0			0
$349$	12.5018	+	21.6538i	0.669207	+	1.15910i	0.978126	+	0.208012i	$0.0666992\pi$
$349$	12.5018	+	21.6538i	0.669207	+	1.15910i	−0.308920	+	0.951088i	$0.599967\pi$
$350$		0			0
$351$	10.6373	−	29.2257i	0.567776	−	1.55995i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$359$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$360$		0			0
$361$	−18.5000	+	4.33013i	−0.973684	+	0.227901i
$362$		0			0
$363$	6.51636	+	17.9035i	0.342020	+	0.939693i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	33.9158	−	5.98027i	1.77039	−	0.312168i	0.809093	−	0.587680i	$-0.199959\pi$
$367$	33.9158	−	5.98027i	1.77039	−	0.312168i	0.961298	+	0.275512i	$0.0888475\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−19.0000	−	32.9090i	−0.983783	−	1.70396i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−19.0000	−	32.9090i	−0.983783	−	1.70396i	−0.336557	−	0.941663i	$-0.609263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	34.3325i		−	1.76354i	−0.471677	−	0.881771i	$-0.656351\pi$
$379$		−	34.3325i		−	1.76354i	0.471677	−	0.881771i	$-0.343649\pi$
$380$		0			0
$381$	−21.0000			−1.07586
$382$		0			0
$383$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$383$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−18.2629	+	10.5441i	−0.928358	+	0.535988i
$388$		0			0
$389$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$389$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.9204	−	4.33867i	0.598268	−	0.217752i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	11.9204	−	4.33867i	0.598268	−	0.217752i	0.623362	+	0.781933i	$0.285766\pi$
$398$		0			0
$399$	−11.2779	−	22.5344i	−0.564601	−	1.12813i
$400$		0			0
$401$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$401$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$402$		0			0
$403$	42.8353	+	51.0492i	2.13378	+	2.54294i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	35.7083	+	12.9968i	1.76566	+	0.642649i	1.00000		0.000593299i	$-0.000188853\pi$
$409$	35.7083	+	12.9968i	1.76566	+	0.642649i	0.765663	+	0.643242i	$0.222411\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.16788			−0.253072
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	3.29932	−	18.7113i	0.160799	−	0.911935i	−0.792492	−	0.609882i	$-0.791217\pi$
$421$	3.29932	−	18.7113i	0.160799	−	0.911935i	0.953291	−	0.302053i	$-0.0976721\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	17.3266	−	47.6044i	0.838492	−	2.30374i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$431$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$432$		0			0
$433$	31.6261	−	26.5375i	1.51986	−	1.27531i	0.678859	−	0.734269i	$-0.262475\pi$
$433$	31.6261	−	26.5375i	1.51986	−	1.27531i	0.840996	−	0.541041i	$-0.181970\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	2.01274	+	5.52995i	0.0960627	+	0.263930i	0.978412	−	0.206666i	$-0.0662612\pi$
$439$	2.01274	+	5.52995i	0.0960627	+	0.263930i	−0.882349	+	0.470596i	$0.844039\pi$
$440$		0			0
$441$	9.51477	−	7.98384i	0.453084	−	0.380183i
$442$		0			0
$443$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$443$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	4.68850	−	26.5898i	0.220285	−	1.24930i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	42.6742			1.99621			0.998107	−	0.0615051i	$-0.0195901\pi$
$457$	42.6742			1.99621			0.998107	+	0.0615051i	$0.0195901\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$461$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$462$		0			0
$463$	14.1461	−	8.16723i	0.657423	−	0.379563i	−0.133871	−	0.990999i	$-0.542741\pi$
$463$	14.1461	−	8.16723i	0.657423	−	0.379563i	0.791294	+	0.611435i	$0.209408\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	8.78383	+	49.8156i	0.405600	+	2.30027i
$470$		0			0
$471$	−20.0822	−	23.9330i	−0.925338	−	1.10278i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	20.8876	−	6.22161i	0.958388	−	0.285467i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$479$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$480$		0			0
$481$	−12.2297	−	69.3579i	−0.557625	−	3.16245i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	34.5000	−	19.9186i	1.56334	−	0.902597i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	34.5000	−	19.9186i	1.56334	−	0.902597i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$		0			0
$489$	31.4696	+	26.4061i	1.42310	+	1.19412i
$490$		0			0
$491$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$491$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−28.5733	+	34.0523i	−1.27911	+	1.52439i	−0.562857	+	0.826555i	$0.690298\pi$
$499$	−28.5733	+	34.0523i	−1.27911	+	1.52439i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$503$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−38.9345	+	6.86521i	−1.72914	+	0.304895i
$508$		0			0
$509$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$509$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$510$		0			0
$511$	1.42270	+	3.90885i	0.0629368	+	0.172917i
$512$		0			0
$513$	9.00000	−	20.7846i	0.397360	−	0.917663i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−13.5376	+	37.1943i	−0.591958	+	1.62639i	0.174908	+	0.984585i	$0.444037\pi$
$523$	−13.5376	+	37.1943i	−0.591958	+	1.62639i	−0.766866	+	0.641807i	$0.778185\pi$
$524$		0			0
$525$	14.4526	+	25.0327i	0.630765	+	1.09252i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.99391	+	22.6506i	−0.173648	+	0.984808i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.9176	−	5.06558i	−0.598363	−	0.217786i	0.0250408	−	0.999686i	$-0.492028\pi$
$541$	−13.9176	−	5.06558i	−0.598363	−	0.217786i	−0.623404	+	0.781900i	$0.714251\pi$
$542$		0			0
$543$	−28.5000	−	16.4545i	−1.22305	−	0.706129i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−28.0275	−	33.4019i	−1.19837	−	1.42816i	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−28.0275	−	33.4019i	−1.19837	−	1.42816i	−0.321853	−	0.946790i	$-0.604306\pi$
$548$		0			0
$549$	42.7879	−	15.5735i	1.82614	−	0.664662i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.92174	−	1.06343i	0.124245	−	0.0452215i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$557$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$558$		0			0
$559$	36.4373	+	21.0371i	1.54113	+	0.889775i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	29.5829	+	5.21626i	1.24237	+	0.219062i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		−	47.7898i		−	1.99994i	−0.00768386	−	0.999970i	$-0.502446\pi$
$571$		−	47.7898i		−	1.99994i	0.00768386	−	0.999970i	$-0.497554\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−17.5000	−	30.3109i	−0.728535	−	1.26186i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−17.5000	−	30.3109i	−0.728535	−	1.26186i	0.228968	−	0.973434i	$-0.426465\pi$
$578$		0			0
$579$	−6.72827	+	18.4858i	−0.279618	+	0.768243i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$587$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$588$		0			0
$589$	28.9261	+	38.9678i	1.19188	+	1.60564i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$593$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.47447	+	9.48206i	−0.224055	+	0.388075i
$598$		0			0
$599$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$599$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$600$		0			0
$601$	3.40151	+	5.89159i	0.138751	+	0.240323i	0.927024	−	0.375002i	$-0.122358\pi$
$601$	3.40151	+	5.89159i	0.138751	+	0.240323i	−0.788273	+	0.615325i	$0.789025\pi$
$602$		0			0
$603$	−29.2251	+	34.8291i	−1.19014	+	1.41835i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	27.5161i		−	1.11684i	−0.829557	−	0.558422i	$-0.811407\pi$
$607$		−	27.5161i		−	1.11684i	0.829557	−	0.558422i	$-0.188593\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−28.3436	−	23.7831i	−1.14479	−	0.960592i	−0.145204	−	0.989402i	$-0.546384\pi$
$613$	−28.3436	−	23.7831i	−1.14479	−	0.960592i	−0.999585	+	0.0288097i	$0.990828\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$617$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$618$		0			0
$619$	33.0222	+	19.0654i	1.32727	+	0.766302i	0.984877	−	0.173254i	$-0.0554281\pi$
$619$	33.0222	+	19.0654i	1.32727	+	0.766302i	0.342396	+	0.939556i	$0.388761\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.4923	+	8.55050i	−0.939693	+	0.342020i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−4.97354	−	5.92723i	−0.197993	−	0.235959i	0.657908	−	0.753098i	$-0.271442\pi$
$631$	−4.97354	−	5.92723i	−0.197993	−	0.235959i	−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$	−5.20749	−	29.5332i	−0.206979	−	1.17384i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−23.2866	−	8.47562i	−0.922648	−	0.335816i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$641$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$642$		0			0
$643$	44.1254	+	7.78050i	1.74014	+	0.306833i	0.951414	−	0.307916i	$-0.0996315\pi$
$643$	44.1254	+	7.78050i	1.74014	+	0.306833i	0.788723	+	0.614749i	$0.210743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−41.3726	+	49.3059i	−1.62152	+	1.93245i
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.86942	+	3.23794i	−0.0729331	+	0.126324i
$658$		0			0
$659$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$659$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$660$		0			0
$661$	29.1097	−	24.4259i	1.13224	−	0.950059i	0.133078	−	0.991106i	$-0.457514\pi$
$661$	29.1097	−	24.4259i	1.13224	−	0.950059i	0.999157	+	0.0410470i	$0.0130693\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−38.9812	+	32.7091i	−1.50710	+	1.26461i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	2.48174	−	4.29851i	0.0956642	−	0.165695i	−0.814221	−	0.580554i	$-0.802836\pi$
$673$	2.48174	−	4.29851i	0.0956642	−	0.165695i	0.909886	+	0.414859i	$0.136169\pi$
$674$		0			0
$675$	−8.88594	+	24.4139i	−0.342020	+	0.939693i
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−40.7632	+	48.5796i	−1.56435	+	1.86432i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−23.1374	−	4.07976i	−0.882748	−	0.155652i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	28.5000	−	16.4545i	1.08419	−	0.625958i	0.152167	−	0.988355i	$-0.451375\pi$
$691$	28.5000	−	16.4545i	1.08419	−	0.625958i	0.932024	+	0.362397i	$0.118041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$701$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$702$		0			0
$703$	−5.88326	−	50.9505i	−0.221891	−	1.92163i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	8.33972	+	47.2969i	0.313205	+	1.77627i	0.582115	+	0.813107i	$0.302225\pi$
$709$	8.33972	+	47.2969i	0.313205	+	1.77627i	−0.268910	+	0.963165i	$0.586663\pi$
$710$		0			0
$711$	2.42025	+	1.39733i	0.0907666	+	0.0524041i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$719$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$720$		0			0
$721$	−34.0515			−1.26814
$722$		0			0
$723$		−	51.4815i		−	1.91462i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−2.82811	+	3.37041i	−0.104889	+	0.125001i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−2.82811	+	3.37041i	−0.104889	+	0.125001i	0.711046	+	0.703145i	$0.248222\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−21.5000	+	37.2391i	−0.794121	+	1.37546i	0.129275	+	0.991609i	$0.458735\pi$
$733$	−21.5000	+	37.2391i	−0.794121	+	1.37546i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	8.47524	+	23.2855i	0.311767	+	0.856572i	0.992300	+	0.123855i	$0.0395259\pi$
$739$	8.47524	+	23.2855i	0.311767	+	0.856572i	−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$	−44.8909	+	5.18355i	−1.64911	+	0.190423i
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.9700	+	43.8773i	−0.582755	+	1.60110i	0.200698	+	0.979653i	$0.435679\pi$
$751$	−15.9700	+	43.8773i	−0.582755	+	1.60110i	−0.783452	+	0.621452i	$0.786543\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.36225	−	7.72573i	0.0495120	−	0.280796i	−0.949993	−	0.312273i	$-0.898910\pi$
$757$	1.36225	−	7.72573i	0.0495120	−	0.280796i	0.999505	+	0.0314762i	$0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	62.4528	+	11.0121i	2.26094	+	0.398665i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	32.0386	+	11.6611i	1.15534	+	0.420511i	0.847432	−	0.530904i	$-0.178148\pi$
$769$	32.0386	+	11.6611i	1.15534	+	0.420511i	0.307912	+	0.951415i	$0.400370\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$773$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$774$		0			0
$775$	−35.7829	−	42.6444i	−1.28536	−	1.53183i
$776$		0			0
$777$	63.9206	−	23.2652i	2.29314	−	0.834634i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−33.4717	−	19.3249i	−1.19314	−	0.688858i	−0.234120	−	0.972208i	$-0.575221\pi$
$787$	−33.4717	−	19.3249i	−1.19314	−	0.688858i	−0.959017	+	0.283350i	$0.908554\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−69.5929	−	58.3954i	−2.47132	−	2.07368i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	−52.8778	+	9.32379i	−1.85679	+	0.327403i	−0.986324	−	0.164821i	$-0.947295\pi$
$811$	−52.8778	+	9.32379i	−1.85679	+	0.327403i	−0.870469	+	0.492223i	$0.836184\pi$
$812$		0			0
$813$	−43.6645	+	36.6389i	−1.53138	+	1.28498i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	25.5763	+	16.8729i	0.894800	+	0.590308i
$818$		0			0
$819$	−20.4982	−	56.3185i	−0.716267	−	1.96793i
$820$		0			0
$821$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$821$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$822$		0			0
$823$	−32.4090	+	5.71458i	−1.12971	+	0.199198i	−0.707099	−	0.707115i	$-0.749996\pi$
$823$	−32.4090	+	5.71458i	−1.12971	+	0.199198i	−0.422608	+	0.906313i	$0.638885\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$827$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$828$		0			0
$829$	−21.0513	−	36.4619i	−0.731141	−	1.26637i	−0.956396	−	0.292074i	$-0.905655\pi$
$829$	−21.0513	−	36.4619i	−0.731141	−	1.26637i	0.225255	−	0.974300i	$-0.427679\pi$
$830$		0			0
$831$	5.56670	−	6.63414i	0.193107	−	0.230136i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−57.8522			−1.99966
$838$		0			0
$839$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$839$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$840$		0			0
$841$	22.2153	+	18.6408i	0.766044	+	0.642788i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	31.7958	+	18.3573i	1.09252	+	0.630765i
$848$		0			0
$849$	−9.89795	−	56.1340i	−0.339697	−	1.92652i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−48.9886	+	17.8304i	−1.67734	+	0.610501i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−48.9886	+	17.8304i	−1.67734	+	0.610501i	−0.684397	+	0.729110i	$0.739934\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$857$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$858$		0			0
$859$	37.3825	+	44.5508i	1.27548	+	1.52005i	0.733571	+	0.679613i	$0.237852\pi$
$859$	37.3825	+	44.5508i	1.27548	+	1.52005i	0.541905	+	0.840440i	$0.317703\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25.5000	−	14.7224i	0.866025	−	0.500000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	89.3337	+	15.7519i	3.02696	+	0.533734i
$872$		0			0
$873$	−57.0000			−1.92916
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.31877	+	52.8494i	−0.314673	+	1.78460i	0.259377	+	0.965776i	$0.416483\pi$
$877$	−9.31877	+	52.8494i	−0.314673	+	1.78460i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	6.19212	−	17.0127i	0.208382	−	0.572524i	−0.790838	−	0.612026i	$-0.790355\pi$
$883$	6.19212	−	17.0127i	0.208382	−	0.572524i	0.999219	+	0.0395021i	$0.0125772\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$887$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$888$		0			0
$889$	−30.9998	+	26.0120i	−1.03970	+	0.872413i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−13.8988	+	38.1867i	−0.462524	+	1.27077i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	36.7402	−	43.7853i	1.21994	−	1.45387i	0.368327	−	0.929696i	$-0.379931\pi$
$907$	36.7402	−	43.7853i	1.21994	−	1.45387i	0.851613	−	0.524171i	$-0.175625\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	40.7740	−	23.5409i	1.34501	−	0.776542i	0.357472	−	0.933924i	$-0.383639\pi$
$919$	40.7740	−	23.5409i	1.34501	−	0.776542i	0.987538	+	0.157382i	$0.0503053\pi$
$920$		0			0
$921$	−47.9243	−	17.4430i	−1.57916	−	0.574767i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	10.2162	+	57.9388i	0.335905	+	1.90501i
$926$		0			0
$927$	−19.6733	−	23.4458i	−0.646157	−	0.770060i
$928$		0			0
$929$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$929$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$930$		0			0
$931$	−16.5609	−	7.17106i	−0.542760	−	0.235022i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	10.2623	+	58.2006i	0.335256	+	1.90133i	0.424691	+	0.905338i	$0.360383\pi$
$937$	10.2623	+	58.2006i	0.335256	+	1.90133i	−0.0894356	+	0.995993i	$0.528506\pi$
$938$		0			0
$939$	19.5000	+	11.2583i	0.636358	+	0.367402i
$940$		0			0
$941$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$941$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$947$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$948$		0			0
$949$	7.45956			0.242148
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$953$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	46.4791	−	80.5042i	1.49933	−	2.59691i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−16.5872	−	45.5731i	−0.533410	−	1.46553i	−0.854988	−	0.518648i	$-0.826436\pi$
$967$	−16.5872	−	45.5731i	−0.533410	−	1.46553i	0.321578	−	0.946883i	$-0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$971$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$972$		0			0
$973$	−7.62874	+	6.40127i	−0.244566	+	0.205215i
$974$		0			0
$975$	51.0480	−	9.00115i	1.63485	−	0.288267i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	28.5000	+	49.3634i	0.909935	+	1.57605i
$982$		0			0
$983$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$983$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−61.8264	−	10.9017i	−1.96398	−	0.346303i	−0.995116	−	0.0987109i	$-0.968528\pi$
$991$	−61.8264	−	10.9017i	−1.96398	−	0.346303i	−0.968864	−	0.247592i	$-0.920361\pi$
$992$		0			0
$993$	36.1170	+	30.3058i	1.14614	+	0.961726i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−28.8704	−	10.5080i	−0.914333	−	0.332790i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−28.8704	−	10.5080i	−0.914333	−	0.332790i	−0.755982	+	0.654593i	$0.772840\pi$
$998$		0			0
$999$	52.9493	+	30.5703i	1.67524	+	0.967201i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.ch.a.575.1	yes	6
3.2	odd	2	CM	912.2.ch.a.575.1	yes	6
4.3	odd	2		912.2.ch.b.575.1	yes	6
12.11	even	2		912.2.ch.b.575.1	yes	6
19.4	even	9		912.2.ch.b.479.1	yes	6
57.23	odd	18		912.2.ch.b.479.1	yes	6
76.23	odd	18	inner	912.2.ch.a.479.1	✓	6
228.23	even	18	inner	912.2.ch.a.479.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.ch.a.479.1	✓	6	76.23	odd	18	inner
912.2.ch.a.479.1	✓	6	228.23	even	18	inner
912.2.ch.a.575.1	yes	6	1.1	even	1	trivial
912.2.ch.a.575.1	yes	6	3.2	odd	2	CM
912.2.ch.b.479.1	yes	6	19.4	even	9
912.2.ch.b.479.1	yes	6	57.23	odd	18
912.2.ch.b.575.1	yes	6	4.3	odd	2
912.2.ch.b.575.1	yes	6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.ch (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(1.70574 + 0.300767i) q^{3} +(2.89053 - 1.66885i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)	\(q+(1.70574 + 0.300767i) q^{3} +(2.89053 - 1.66885i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-1.03936 - 5.89452i) q^{13} +(-0.500000 - 4.33013i) q^{19} +(5.43242 - 1.97724i) q^{21} +(0.868241 + 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(-9.64203 + 5.56683i) q^{31} +11.7665 q^{37} -10.3671i q^{39} +(-4.51842 + 5.38484i) q^{43} +(2.07011 - 3.58553i) q^{49} +(0.449493 - 7.53644i) q^{57} +(11.6270 - 9.75622i) q^{61} +(9.86097 - 1.73875i) q^{63} +(-5.18345 + 14.2414i) q^{67} +(-0.216415 + 1.22735i) q^{73} +8.66025i q^{75} +(0.917404 + 0.161763i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-12.8414 - 15.3037i) q^{91} +(-18.1211 + 6.59553i) q^{93} +(-17.8542 + 6.49838i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 15 q^{13} - 3 q^{19} + 9 q^{21} + 27 q^{27} - 39 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 33 q^{67} - 51 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93}+O(q^{100}) \) 6 * q - 15 * q^13 - 3 * q^19 + 9 * q^21 + 27 * q^27 - 39 * q^43 + 21 * q^49 + 42 * q^61 + 36 * q^63 - 33 * q^67 - 51 * q^73 + 12 * q^79 - 48 * q^91 - 54 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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