LMFDB - Embedded newform 912.2.cc.a.641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(257,912)]

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([0, 0, 9, 17]))

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

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

chi := DirichletCharacter("912.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.cc (of order $18$, degree $6$, not 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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			641.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	912.641
Dual form			912.2.cc.a.737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70574 - 0.300767i) q^{3} +(2.05303 + 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})$	$q+(-1.70574 - 0.300767i) q^{3} +(2.05303 + 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-3.96064 + 0.698367i) q^{13} +(-0.500000 - 4.33013i) q^{19} +(-2.43242 - 6.68302i) q^{21} +(0.868241 + 4.92404i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-9.64203 + 5.56683i) q^{31} +3.09018i q^{37} +6.96585 q^{39} +(8.48158 + 7.11689i) q^{43} +(-4.92989 + 8.53882i) q^{49} +(-0.449493 + 7.53644i) q^{57} +(-11.6270 + 9.75622i) q^{61} +(2.13903 + 12.1311i) q^{63} +(-5.18345 + 14.2414i) q^{67} +(0.216415 - 1.22735i) q^{73} -8.66025i q^{75} +(-0.917404 - 0.161763i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-10.6147 - 12.6501i) q^{91} +(18.1211 - 6.59553i) q^{93} +(-1.77719 - 4.88279i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{7}{18}\right)$	$1$	$-1$	$1$

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
$7$	2.05303	+	3.55596i	0.775974	+	1.34403i	0.934246	+	0.356630i	$0.116074\pi$
$7$	2.05303	+	3.55596i	0.775974	+	1.34403i	−0.158272	+	0.987396i	$0.550592\pi$
$8$		0			0
$9$	2.81908	+	1.02606i	0.939693	+	0.342020i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−3.96064	+	0.698367i	−1.09848	+	0.193692i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−3.96064	+	0.698367i	−1.09848	+	0.193692i	−0.405108	+	0.914269i	$0.632766\pi$
$14$		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
$21$	−2.43242	−	6.68302i	−0.530797	−	1.45835i
$22$		0			0
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$		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		0.342020	−	0.939693i	$-0.388889\pi$
$29$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$30$		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
$37$			3.09018i			0.508022i	0.967201	+	0.254011i	$0.0817500\pi$
$37$			3.09018i			0.508022i	−0.967201	+	0.254011i	$0.918250\pi$
$38$		0			0
$39$	6.96585			1.11543
$40$		0			0
$41$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$41$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$42$		0			0
$43$	8.48158	+	7.11689i	1.29343	+	1.08532i	0.991241	+	0.132068i	$0.0421616\pi$
$43$	8.48158	+	7.11689i	1.29343	+	1.08532i	0.302188	+	0.953248i	$0.402283\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$47$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$48$		0			0
$49$	−4.92989	+	8.53882i	−0.704270	+	1.21983i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$53$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$54$		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	$0.701831\pi$
$61$	−11.6270	+	9.75622i	−1.48869	+	1.24916i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	2.13903	+	12.1311i	0.269493	+	1.52837i
$64$		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	−0.969520	−	0.245011i	$-0.921208\pi$
$73$	0.216415	−	1.22735i	0.0253294	−	0.143650i	0.994850	+	0.101361i	$0.0323196\pi$
$74$		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	0.121802	−	0.992554i	$-0.461133\pi$
$79$	−0.917404	−	0.161763i	−0.103216	−	0.0181998i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	6.89440	+	5.78509i	0.766044	+	0.642788i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$89$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$90$		0			0
$91$	−10.6147	−	12.6501i	−1.11272	−	1.32609i
$92$		0			0
$93$	18.1211	−	6.59553i	1.87907	−	0.683925i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.77719	−	4.88279i	−0.180446	−	0.495772i	0.816185	−	0.577791i	$-0.196085\pi$
$97$	−1.77719	−	4.88279i	−0.180446	−	0.495772i	−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	0.867536	−	0.497374i	$-0.165702\pi$
$103$	8.83527	+	5.10105i	0.870565	+	0.502621i	0.00302937	+	0.999995i	$0.499036\pi$
$104$		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
$109$	5.56670	−	6.63414i	0.533194	−	0.635435i	−0.430454	−	0.902613i	$-0.641646\pi$
$109$	5.56670	−	6.63414i	0.533194	−	0.635435i	0.963647	+	0.267177i	$0.0860909\pi$
$110$		0			0
$111$	0.929426	−	5.27103i	0.0882172	−	0.500304i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.8819	−	2.09510i	−1.09848	−	0.193692i
$118$		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
$129$	−12.3268	−	14.6905i	−1.08532	−	1.29343i
$130$		0			0
$131$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$131$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$132$		0			0
$133$	14.3712	−	10.6679i	1.24614	−	0.925022i
$134$		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
$139$	−4.06165	−	23.0348i	−0.344505	−	1.95378i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−4.06165	−	23.0348i	−0.344505	−	1.95378i	−0.0476387	−	0.998865i	$-0.515170\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	10.9773	−	13.0822i	0.905393	−	1.07900i
$148$		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	$0.218706\pi$
$151$			15.5885i			1.26857i	−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	13.8177	+	11.5945i	1.10278	+	0.925338i	0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	13.8177	+	11.5945i	1.10278	+	0.925338i	0.105167	+	0.994455i	$0.466462\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.72921	−	8.19123i	0.370420	−	0.641587i	−0.619210	−	0.785225i	$-0.712547\pi$
$163$	4.72921	−	8.19123i	0.370420	−	0.641587i	0.989630	+	0.143639i	$0.0458804\pi$
$164$		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
$169$	2.98293	−	1.08570i	0.229456	−	0.0835151i
$170$		0			0
$171$	3.03343	−	12.7200i	0.231972	−	0.972722i
$172$		0			0
$173$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$173$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$174$		0			0
$175$	−15.7271	+	13.1966i	−1.18886	+	0.997573i
$176$		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
$181$	−6.51636	+	17.9035i	−0.484357	+	1.33076i	0.421366	+	0.906891i	$0.361551\pi$
$181$	−6.51636	+	17.9035i	−0.484357	+	1.33076i	−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
$189$		−	21.3357i		−	1.55195i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	24.9722	+	4.40328i	1.79754	+	0.316955i	0.969754	−	0.244086i	$-0.0784878\pi$
$193$	24.9722	+	4.40328i	1.79754	+	0.316955i	0.827788	+	0.561041i	$0.189599\pi$
$194$		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
$199$	25.8380	+	9.40425i	1.83160	+	0.666650i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	25.8380	+	9.40425i	1.83160	+	0.666650i	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
$217$	−39.5908	−	22.8578i	−2.68760	−	1.55169i
$218$		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	$-0.335348\pi$
$223$	18.8846	−	22.5058i	1.26461	−	1.50710i	0.770097	−	0.637927i	$-0.220208\pi$
$224$		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		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	8.83868	−	1.55850i	0.569349	−	0.100392i	0.118440	−	0.992961i	$-0.462211\pi$
$241$	8.83868	−	1.55850i	0.569349	−	0.100392i	0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	−10.0201	−	11.9415i	−0.642788	−	0.766044i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.00434	+	16.8009i	0.318418	+	1.06901i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$251$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$257$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$258$		0			0
$259$	−10.9886	+	6.34424i	−0.682795	+	0.394212i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$269$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$270$		0			0
$271$	0.766044	+	0.642788i	0.0465339	+	0.0390466i	0.665758	−	0.746168i	$-0.268108\pi$
$271$	0.766044	+	0.642788i	0.0465339	+	0.0390466i	−0.619224	+	0.785214i	$0.712553\pi$
$272$		0			0
$273$	14.3011	+	24.7703i	0.865544	+	1.49917i
$274$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$281$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$282$		0			0
$283$	−6.57785	+	2.39414i	−0.391012	+	0.142317i	−0.530042	−	0.847971i	$-0.677824\pi$
$283$	−6.57785	+	2.39414i	−0.391012	+	0.142317i	0.139030	+	0.990288i	$0.455602\pi$
$284$		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
$291$	1.56283	+	8.86327i	0.0916149	+	0.519574i
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−7.89440	+	44.7714i	−0.455026	+	2.58058i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	28.9975	+	5.11305i	1.65498	+	0.291817i	0.921639	−	0.388048i	$-0.126851\pi$
$307$	28.9975	+	5.11305i	1.65498	+	0.291817i	0.733337	+	0.679865i	$0.237962\pi$
$308$		0			0
$309$	−13.5364	−	11.3584i	−0.770060	−	0.646157i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−12.2160	−	4.44626i	−0.690489	−	0.251318i	−0.0271446	−	0.999632i	$-0.508641\pi$
$313$	−12.2160	−	4.44626i	−0.690489	−	0.251318i	−0.663345	+	0.748314i	$0.730864\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$317$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−6.87757	−	18.8960i	−0.381499	−	1.04816i
$326$		0			0
$327$	−11.4907	+	9.64181i	−0.635435	+	0.533194i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−23.5737	−	13.6103i	−1.29573	−	0.748090i	−0.316066	−	0.948737i	$-0.602362\pi$
$331$	−23.5737	−	13.6103i	−1.29573	−	0.748090i	−0.979663	+	0.200648i	$0.935695\pi$
$332$		0			0
$333$	−3.17071	+	8.71146i	−0.173754	+	0.477385i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.8709	−	24.8730i	1.13691	−	1.35492i	0.210863	−	0.977516i	$-0.432373\pi$
$337$	20.8709	−	24.8730i	1.13691	−	1.35492i	0.926049	−	0.377403i	$-0.123183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−11.7425			−0.634034
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$347$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$348$		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
$351$	19.6373	+	7.14738i	1.04816	+	0.381499i
$352$		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		0.939693	−	0.342020i	$-0.111111\pi$
$359$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$360$		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
$367$	−2.91581	−	16.5364i	−0.152204	−	0.863192i	−0.961298	−	0.275512i	$-0.911152\pi$
$367$	−2.91581	−	16.5364i	−0.152204	−	0.863192i	0.809093	−	0.587680i	$-0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	0.336557	+	0.941663i	$0.390737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			34.3325i			1.76354i	0.471677	+	0.881771i	$0.343649\pi$
$379$			34.3325i			1.76354i	−0.471677	+	0.881771i	$0.656351\pi$
$380$		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
$387$	16.6079	+	28.7657i	0.844226	+	1.46224i
$388$		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
$399$	−27.7221	+	13.8742i	−1.38784	+	0.694578i
$400$		0			0
$401$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$401$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$402$		0			0
$403$	34.3009	−	28.7819i	1.70865	−	1.43373i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.73917	−	13.0208i	0.234337	−	0.643835i	−0.765663	−	0.643242i	$-0.777589\pi$
$409$	4.73917	−	13.0208i	0.234337	−	0.643835i	1.00000		0.000593299i	$-0.000188853\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			40.5129i			1.98392i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	35.8205	+	6.31612i	1.74578	+	0.307829i	0.953291	−	0.302053i	$-0.0976721\pi$
$421$	35.8205	+	6.31612i	1.74578	+	0.307829i	0.792492	+	0.609882i	$0.208783\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−58.5634	−	21.3153i	−2.83408	−	1.03152i
$428$		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
$433$	3.37387	+	4.02082i	0.162138	+	0.193228i	0.840996	−	0.541041i	$-0.181970\pi$
$433$	3.37387	+	4.02082i	0.162138	+	0.193228i	−0.678859	+	0.734269i	$0.737525\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−2.01274	−	5.52995i	−0.0960627	−	0.263930i	0.882349	−	0.470596i	$-0.155961\pi$
$439$	−2.01274	−	5.52995i	−0.0960627	−	0.263930i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$	−22.6591	+	19.0132i	−1.07900	+	0.905393i
$442$		0			0
$443$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$443$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$449$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$450$		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	$-0.980410\pi$
$457$	−42.6742			−1.99621			−0.998107	+	0.0615051i	$0.980410\pi$
$458$		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
$463$	19.9072	+	34.4803i	0.925166	+	1.60243i	0.791294	+	0.611435i	$0.209408\pi$
$463$	19.9072	+	34.4803i	0.925166	+	1.60243i	0.133871	+	0.990999i	$0.457259\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	−61.2836	+	10.8060i	−2.82982	+	0.498973i
$470$		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		0.766044	−	0.642788i	$-0.222222\pi$
$479$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$480$		0			0
$481$	−2.15808	−	12.2391i	−0.0984000	−	0.558054i
$482$		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
$489$	−10.5304	+	12.5497i	−0.476203	+	0.567517i
$490$		0			0
$491$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$491$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3.42674	+	2.87537i	0.153402	+	0.128719i	0.716258	−	0.697835i	$-0.245853\pi$
$499$	3.42674	+	2.87537i	0.153402	+	0.128719i	−0.562857	+	0.826555i	$0.690298\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$503$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.41463	+	0.954745i	−0.240472	+	0.0424017i
$508$		0			0
$509$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$509$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$510$		0			0
$511$	4.80871	−	1.75023i	0.212725	−	0.0774254i
$512$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		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
$525$	30.7955	−	17.7798i	1.34403	−	0.775974i
$526$		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	0.623404	−	0.781900i	$-0.285749\pi$
$541$	13.9176	+	5.06558i	0.598363	+	0.217786i	−0.0250408	+	0.999686i	$0.507972\pi$
$542$		0			0
$543$	16.5000	−	28.5788i	0.708083	−	1.22644i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	28.0275	+	33.4019i	1.19837	+	1.42816i	0.876517	+	0.481371i	$0.159861\pi$
$547$	28.0275	+	33.4019i	1.19837	+	1.42816i	0.321853	+	0.946790i	$0.395694\pi$
$548$		0			0
$549$	−42.7879	+	15.5735i	−1.82614	+	0.664662i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.30824	−	3.59435i	−0.0556319	−	0.152848i
$554$		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
$559$	−38.5627	−	22.2642i	−1.63103	−	0.941674i
$560$		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
$567$	−6.41710	+	36.3932i	−0.269493	+	1.52837i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	0.367221			0.0153677			0.00768386	−	0.999970i	$-0.497554\pi$
$571$	0.367221			0.0153677			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	$0.0931316\pi$
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	−41.2717	−	15.0217i	−1.71519	−	0.624280i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$587$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$588$		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
$597$	−41.2443	−	23.8124i	−1.68802	−	0.974576i
$598$		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
$601$	42.0510	−	24.2782i	1.71530	−	0.990327i	0.788273	−	0.615325i	$-0.210975\pi$
$601$	42.0510	−	24.2782i	1.71530	−	0.990327i	0.927024	−	0.375002i	$-0.122358\pi$
$602$		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	0.999585	−	0.0288097i	$-0.00917168\pi$
$613$	28.3436	+	23.7831i	1.14479	+	0.960592i	0.145204	+	0.989402i	$0.453616\pi$
$614$		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
$619$	15.9847	−	27.6864i	0.642481	−	1.11281i	−0.342396	−	0.939556i	$-0.611239\pi$
$619$	15.9847	−	27.6864i	0.642481	−	1.11281i	0.984877	−	0.173254i	$-0.0554281\pi$
$620$		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
$631$	−38.0265	+	31.9080i	−1.51381	+	1.27024i	−0.657908	+	0.753098i	$0.728558\pi$
$631$	−38.0265	+	31.9080i	−1.51381	+	1.27024i	−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
$637$	13.5623	−	37.2621i	0.537357	−	1.47638i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$641$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$642$		0			0
$643$	−4.12542	+	23.3964i	−0.162691	+	0.922664i	0.788723	+	0.614749i	$0.210743\pi$
$643$	−4.12542	+	23.3964i	−0.162691	+	0.922664i	−0.951414	+	0.307916i	$0.900369\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	60.6566	+	50.8970i	2.37732	+	1.99481i
$652$		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
$661$	−22.2668	−	26.5366i	−0.866079	−	1.03215i	−0.999157	−	0.0410470i	$-0.986931\pi$
$661$	−22.2668	−	26.5366i	−0.866079	−	1.03215i	0.133078	−	0.991106i	$-0.457514\pi$
$662$		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
$673$	44.7272	+	25.8233i	1.72411	+	0.995414i	0.909886	+	0.414859i	$0.136169\pi$
$673$	44.7272	+	25.8233i	1.72411	+	0.995414i	0.814221	+	0.580554i	$0.197164\pi$
$674$		0			0
$675$	8.88594	−	24.4139i	0.342020	−	0.939693i
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	13.7144	−	16.3441i	0.526309	−	0.627230i
$680$		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
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	0.932024	+	0.362397i	$0.118041\pi$
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	−0.152167	+	0.988355i	$0.548625\pi$
$692$		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
$703$	13.3809	−	1.54509i	0.504669	−	0.0582742i
$704$		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	$-0.697775\pi$
$709$	−8.33972	−	47.2969i	−0.313205	−	1.77627i	0.268910	−	0.963165i	$-0.413337\pi$
$710$		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		0.173648	−	0.984808i	$-0.444444\pi$
$719$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$720$		0			0
$721$			41.8905i			1.56008i
$722$		0			0
$723$	−15.5452			−0.578132
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−41.1719	−	34.5473i	−1.52698	−	1.28129i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−41.1719	−	34.5473i	−1.52698	−	1.28129i	−0.711046	−	0.703145i	$-0.751778\pi$
$728$		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	$-0.541265\pi$
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−45.4752	+	16.5516i	−1.67283	+	0.608862i	−0.992300	−	0.123855i	$-0.960474\pi$
$739$	−45.4752	+	16.5516i	−1.67283	+	0.608862i	−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$	−3.48293	−	30.1630i	−0.127948	−	1.10807i
$742$		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	$-0.564321\pi$
$751$	15.9700	−	43.8773i	0.582755	−	1.60110i	0.783452	−	0.621452i	$-0.213457\pi$
$752$		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
$763$	35.0194	+	6.17486i	1.26779	+	0.223545i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−32.0386	−	11.6611i	−1.15534	−	0.420511i	−0.307912	−	0.951415i	$-0.599630\pi$
$769$	−32.0386	−	11.6611i	−1.15534	−	0.420511i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$773$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$774$		0			0
$775$	−35.7829	−	42.6444i	−1.28536	−	1.53183i
$776$		0			0
$777$	20.6517	−	7.51661i	0.740876	−	0.269657i
$778$		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
$793$	39.2369	−	46.7608i	1.39334	−	1.66052i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		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	0.870469	−	0.492223i	$-0.163816\pi$
$811$	52.8778	−	9.32379i	1.85679	−	0.327403i	0.986324	+	0.164821i	$0.0527046\pi$
$812$		0			0
$813$	−1.11334	−	1.32683i	−0.0390466	−	0.0465339i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	26.5763	−	40.2848i	0.929785	−	1.40939i
$818$		0			0
$819$	−16.9439	−	46.5529i	−0.592067	−	1.62669i
$820$		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
$823$	8.16146	+	46.2860i	0.284491	+	1.61343i	0.707099	+	0.707115i	$0.250004\pi$
$823$	8.16146	+	46.2860i	0.284491	+	1.61343i	−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
$829$	−34.0225	+	19.6429i	−1.18165	+	0.682226i	−0.956396	−	0.292074i	$-0.905655\pi$
$829$	−34.0225	+	19.6429i	−1.18165	+	0.682226i	−0.225255	+	0.974300i	$0.572321\pi$
$830$		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
$847$	22.5834	−	39.1155i	0.775974	−	1.34403i
$848$		0			0
$849$	11.9402	−	2.10537i	0.409785	−	0.0722562i
$850$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$857$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$858$		0			0
$859$	−5.61746	+	4.71361i	−0.191665	+	0.160826i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−5.61746	+	4.71361i	−0.191665	+	0.160826i	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
$871$	10.5840	−	60.0250i	0.358626	−	2.03387i
$872$		0			0
$873$		−	15.5885i		−	0.527589i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−24.6812	−	4.35197i	−0.833426	−	0.146955i	−0.259377	−	0.965776i	$-0.583517\pi$
$877$	−24.6812	−	4.35197i	−0.833426	−	0.146955i	−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
$883$	53.1921	+	19.3603i	1.79006	+	0.651528i	0.999219	+	0.0395021i	$0.0125772\pi$
$883$	53.1921	+	19.3603i	1.79006	+	0.651528i	0.790838	+	0.612026i	$0.209645\pi$
$884$		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
$889$	−32.0002	−	38.1363i	−1.07325	−	1.27905i
$890$		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
$903$	26.9315	−	73.9938i	0.896226	−	2.46236i
$904$		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
$919$	−19.1004	−	33.0829i	−0.630065	−	1.09131i	−0.987538	−	0.157382i	$-0.949695\pi$
$919$	−19.1004	−	33.0829i	−0.630065	−	1.09131i	0.357472	−	0.933924i	$-0.383639\pi$
$920$		0			0
$921$	−47.9243	−	17.4430i	−1.57916	−	0.574767i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−15.2162	+	2.68302i	−0.500304	+	0.0882172i
$926$		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
$931$	39.4391	+	17.0776i	1.29257	+	0.559697i
$932$		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		0.342020	−	0.939693i	$-0.388889\pi$
$941$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$947$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$948$		0			0
$949$			5.01222i			0.162703i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$953$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$954$		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
$967$	36.5872	−	13.3167i	1.17657	−	0.428235i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	36.5872	−	13.3167i	1.17657	−	0.428235i	0.854988	+	0.518648i	$0.173564\pi$
$968$		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
$973$	73.5720	−	61.7342i	2.35861	−	1.97911i
$974$		0			0
$975$	6.04804	+	34.3001i	0.193692	+	1.09848i
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	22.5000	−	12.9904i	0.718370	−	0.414751i
$982$		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	0.755982	−	0.654593i	$-0.227160\pi$
$997$	28.8704	+	10.5080i	0.914333	+	0.332790i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	8.02852	−	13.9058i	0.254011	−	0.439960i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.cc.a.641.1		6
3.2	odd	2	CM	912.2.cc.a.641.1		6
4.3	odd	2		57.2.j.a.14.1	✓	6
12.11	even	2		57.2.j.a.14.1	✓	6
19.15	odd	18	inner	912.2.cc.a.737.1		6
57.53	even	18	inner	912.2.cc.a.737.1		6
76.15	even	18		57.2.j.a.53.1	yes	6
76.55	odd	18		1083.2.d.a.1082.6		6
76.59	even	18		1083.2.d.a.1082.3		6
228.59	odd	18		1083.2.d.a.1082.3		6
228.131	even	18		1083.2.d.a.1082.6		6
228.167	odd	18		57.2.j.a.53.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.j.a.14.1	✓	6	4.3	odd	2
57.2.j.a.14.1	✓	6	12.11	even	2
57.2.j.a.53.1	yes	6	76.15	even	18
57.2.j.a.53.1	yes	6	228.167	odd	18
912.2.cc.a.641.1		6	1.1	even	1	trivial
912.2.cc.a.641.1		6	3.2	odd	2	CM
912.2.cc.a.737.1		6	19.15	odd	18	inner
912.2.cc.a.737.1		6	57.53	even	18	inner
1083.2.d.a.1082.3		6	76.59	even	18
1083.2.d.a.1082.3		6	228.59	odd	18
1083.2.d.a.1082.6		6	76.55	odd	18
1083.2.d.a.1082.6		6	228.131	even	18

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.cc (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.70574 - 0.300767i) q^{3} +(2.05303 + 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)	\(q+(-1.70574 - 0.300767i) q^{3} +(2.05303 + 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-3.96064 + 0.698367i) q^{13} +(-0.500000 - 4.33013i) q^{19} +(-2.43242 - 6.68302i) q^{21} +(0.868241 + 4.92404i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-9.64203 + 5.56683i) q^{31} +3.09018i q^{37} +6.96585 q^{39} +(8.48158 + 7.11689i) q^{43} +(-4.92989 + 8.53882i) q^{49} +(-0.449493 + 7.53644i) q^{57} +(-11.6270 + 9.75622i) q^{61} +(2.13903 + 12.1311i) q^{63} +(-5.18345 + 14.2414i) q^{67} +(0.216415 - 1.22735i) q^{73} -8.66025i q^{75} +(-0.917404 - 0.161763i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-10.6147 - 12.6501i) q^{91} +(18.1211 - 6.59553i) q^{93} +(-1.77719 - 4.88279i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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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