LMFDB - Embedded newform 1152.3.j.a.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(161,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.j (of order $4$, degree $2$, 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.3
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.a.737.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.14405 - 5.14405i) q^{5} +7.48880i q^{7} +O(q^{10})$	$q+(-5.14405 - 5.14405i) q^{5} +7.48880i q^{7} +(-11.6851 - 11.6851i) q^{11} +(14.0697 + 14.0697i) q^{13} -7.92669i q^{17} +(-10.7920 - 10.7920i) q^{19} +2.09213 q^{23} +27.9224i q^{25} +(-23.6279 + 23.6279i) q^{29} -17.8620 q^{31} +(38.5228 - 38.5228i) q^{35} +(30.8593 - 30.8593i) q^{37} +36.8337 q^{41} +(-28.4050 + 28.4050i) q^{43} +65.3973i q^{47} -7.08218 q^{49} +(9.05079 + 9.05079i) q^{53} +120.218i q^{55} +(74.0027 + 74.0027i) q^{59} +(53.4704 + 53.4704i) q^{61} -144.750i q^{65} +(20.6746 + 20.6746i) q^{67} +39.6591 q^{71} -91.3758i q^{73} +(87.5077 - 87.5077i) q^{77} +92.1359 q^{79} +(-9.12191 + 9.12191i) q^{83} +(-40.7752 + 40.7752i) q^{85} +63.2593 q^{89} +(-105.365 + 105.365i) q^{91} +111.029i q^{95} +152.715 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 64 q^{19} - 128 q^{43} - 224 q^{49} - 64 q^{61} + 64 q^{67} - 512 q^{79} - 320 q^{85} + 192 q^{91}+O(q^{100}) $ 32 * q - 64 * q^19 - 128 * q^43 - 224 * q^49 - 64 * q^61 + 64 * q^67 - 512 * q^79 - 320 * q^85 + 192 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5.14405	−	5.14405i	−1.02881	−	1.02881i	−0.999573	−	0.0292368i	$-0.990692\pi$
$5$	−5.14405	−	5.14405i	−1.02881	−	1.02881i	−0.0292368	−	0.999573i	$-0.509308\pi$
$6$		0			0
$7$			7.48880i			1.06983i	0.844906	+	0.534915i	$0.179656\pi$
$7$			7.48880i			1.06983i	−0.844906	+	0.534915i	$0.820344\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−11.6851	−	11.6851i	−1.06228	−	1.06228i	−0.997927	−	0.0643581i	$-0.979500\pi$
$11$	−11.6851	−	11.6851i	−1.06228	−	1.06228i	−0.0643581	−	0.997927i	$-0.520500\pi$
$12$		0			0
$13$	14.0697	+	14.0697i	1.08228	+	1.08228i	0.996296	+	0.0859869i	$0.0274043\pi$
$13$	14.0697	+	14.0697i	1.08228	+	1.08228i	0.0859869	+	0.996296i	$0.472596\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.92669i		−	0.466276i	−0.972444	−	0.233138i	$-0.925101\pi$
$17$		−	7.92669i		−	0.466276i	0.972444	−	0.233138i	$-0.0748993\pi$
$18$		0			0
$19$	−10.7920	−	10.7920i	−0.567998	−	0.567998i	0.363569	−	0.931567i	$-0.381558\pi$
$19$	−10.7920	−	10.7920i	−0.567998	−	0.567998i	−0.931567	+	0.363569i	$0.881558\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.09213			0.0909621			0.0454811	−	0.998965i	$-0.485518\pi$
$23$	2.09213			0.0909621			0.0454811	+	0.998965i	$0.485518\pi$
$24$		0			0
$25$			27.9224i			1.11690i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−23.6279	+	23.6279i	−0.814755	+	0.814755i	−0.985343	−	0.170588i	$-0.945433\pi$
$29$	−23.6279	+	23.6279i	−0.814755	+	0.814755i	0.170588	+	0.985343i	$0.445433\pi$
$30$		0			0
$31$	−17.8620			−0.576193			−0.288096	−	0.957601i	$-0.593022\pi$
$31$	−17.8620			−0.576193			−0.288096	+	0.957601i	$0.593022\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	38.5228	−	38.5228i	1.10065	−	1.10065i
$36$		0			0
$37$	30.8593	−	30.8593i	0.834036	−	0.834036i	−0.154030	−	0.988066i	$-0.549225\pi$
$37$	30.8593	−	30.8593i	0.834036	−	0.834036i	0.988066	+	0.154030i	$0.0492252\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	36.8337			0.898382			0.449191	−	0.893436i	$-0.351712\pi$
$41$	36.8337			0.898382			0.449191	+	0.893436i	$0.351712\pi$
$42$		0			0
$43$	−28.4050	+	28.4050i	−0.660582	+	0.660582i	−0.955517	−	0.294935i	$-0.904702\pi$
$43$	−28.4050	+	28.4050i	−0.660582	+	0.660582i	0.294935	+	0.955517i	$0.404702\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			65.3973i			1.39143i	0.718317	+	0.695716i	$0.244913\pi$
$47$			65.3973i			1.39143i	−0.718317	+	0.695716i	$0.755087\pi$
$48$		0			0
$49$	−7.08218			−0.144534
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.05079	+	9.05079i	0.170770	+	0.170770i	0.787317	−	0.616548i	$-0.211469\pi$
$53$	9.05079	+	9.05079i	0.170770	+	0.170770i	−0.616548	+	0.787317i	$0.711469\pi$
$54$		0			0
$55$			120.218i			2.18578i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	74.0027	+	74.0027i	1.25428	+	1.25428i	0.953781	+	0.300503i	$0.0971545\pi$
$59$	74.0027	+	74.0027i	1.25428	+	1.25428i	0.300503	+	0.953781i	$0.402846\pi$
$60$		0			0
$61$	53.4704	+	53.4704i	0.876564	+	0.876564i	0.993177	−	0.116614i	$-0.0372039\pi$
$61$	53.4704	+	53.4704i	0.876564	+	0.876564i	−0.116614	+	0.993177i	$0.537204\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	144.750i		−	2.22693i
$66$		0			0
$67$	20.6746	+	20.6746i	0.308576	+	0.308576i	0.844357	−	0.535781i	$-0.179983\pi$
$67$	20.6746	+	20.6746i	0.308576	+	0.308576i	−0.535781	+	0.844357i	$0.679983\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	39.6591			0.558578			0.279289	−	0.960207i	$-0.409901\pi$
$71$	39.6591			0.558578			0.279289	+	0.960207i	$0.409901\pi$
$72$		0			0
$73$		−	91.3758i		−	1.25172i	−0.779934	−	0.625861i	$-0.784748\pi$
$73$		−	91.3758i		−	1.25172i	0.779934	−	0.625861i	$-0.215252\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	87.5077	−	87.5077i	1.13646	−	1.13646i
$78$		0			0
$79$	92.1359			1.16628			0.583138	−	0.812373i	$-0.301825\pi$
$79$	92.1359			1.16628			0.583138	+	0.812373i	$0.301825\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.12191	+	9.12191i	−0.109903	+	0.109903i	−0.759920	−	0.650017i	$-0.774762\pi$
$83$	−9.12191	+	9.12191i	−0.109903	+	0.109903i	0.650017	+	0.759920i	$0.274762\pi$
$84$		0			0
$85$	−40.7752	+	40.7752i	−0.479709	+	0.479709i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	63.2593			0.710778			0.355389	−	0.934718i	$-0.384348\pi$
$89$	63.2593			0.710778			0.355389	+	0.934718i	$0.384348\pi$
$90$		0			0
$91$	−105.365	+	105.365i	−1.15786	+	1.15786i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			111.029i			1.16872i
$96$		0			0
$97$	152.715			1.57438			0.787192	−	0.616708i	$-0.211534\pi$
$97$	152.715			1.57438			0.787192	+	0.616708i	$0.211534\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	25.9945	+	25.9945i	0.257372	+	0.257372i	0.823984	−	0.566613i	$-0.191746\pi$
$101$	25.9945	+	25.9945i	0.257372	+	0.257372i	−0.566613	+	0.823984i	$0.691746\pi$
$102$		0			0
$103$		−	111.000i		−	1.07767i	−0.842411	−	0.538836i	$-0.818864\pi$
$103$		−	111.000i		−	1.07767i	0.842411	−	0.538836i	$-0.181136\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.69852	+	9.69852i	0.0906404	+	0.0906404i	0.750973	−	0.660333i	$-0.229585\pi$
$107$	9.69852	+	9.69852i	0.0906404	+	0.0906404i	−0.660333	+	0.750973i	$0.729585\pi$
$108$		0			0
$109$	100.334	+	100.334i	0.920494	+	0.920494i	0.997064	−	0.0765702i	$-0.0243969\pi$
$109$	100.334	+	100.334i	0.920494	+	0.920494i	−0.0765702	+	0.997064i	$0.524397\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			15.1517i			0.134086i	0.997750	+	0.0670431i	$0.0213565\pi$
$113$			15.1517i			0.134086i	−0.997750	+	0.0670431i	$0.978644\pi$
$114$		0			0
$115$	−10.7620	−	10.7620i	−0.0935827	−	0.0935827i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	59.3614			0.498835
$120$		0			0
$121$			152.085i			1.25690i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	15.0331	−	15.0331i	0.120265	−	0.120265i
$126$		0			0
$127$	−187.811			−1.47883			−0.739414	−	0.673252i	$-0.764897\pi$
$127$	−187.811			−1.47883			−0.739414	+	0.673252i	$0.764897\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	54.5188	−	54.5188i	0.416174	−	0.416174i	−0.467709	−	0.883883i	$-0.654920\pi$
$131$	54.5188	−	54.5188i	0.416174	−	0.416174i	0.883883	+	0.467709i	$0.154920\pi$
$132$		0			0
$133$	80.8189	−	80.8189i	0.607661	−	0.607661i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−25.5132			−0.186228			−0.0931140	−	0.995655i	$-0.529682\pi$
$137$	−25.5132			−0.186228			−0.0931140	+	0.995655i	$0.529682\pi$
$138$		0			0
$139$	−17.6891	+	17.6891i	−0.127260	+	0.127260i	−0.767868	−	0.640608i	$-0.778682\pi$
$139$	−17.6891	+	17.6891i	−0.127260	+	0.127260i	0.640608	+	0.767868i	$0.278682\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	328.812i		−	2.29939i
$144$		0			0
$145$	243.086			1.67645
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−141.032	−	141.032i	−0.946524	−	0.946524i	0.0521168	−	0.998641i	$-0.483403\pi$
$149$	−141.032	−	141.032i	−0.946524	−	0.946524i	−0.998641	+	0.0521168i	$0.983403\pi$
$150$		0			0
$151$			241.064i			1.59645i	0.602358	+	0.798226i	$0.294228\pi$
$151$			241.064i			1.59645i	−0.602358	+	0.798226i	$0.705772\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	91.8828	+	91.8828i	0.592792	+	0.592792i
$156$		0			0
$157$	13.5949	+	13.5949i	0.0865915	+	0.0865915i	0.749076	−	0.662484i	$-0.230498\pi$
$157$	13.5949	+	13.5949i	0.0865915	+	0.0865915i	−0.662484	+	0.749076i	$0.730498\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			15.6675i			0.0973140i
$162$		0			0
$163$	194.538	+	194.538i	1.19348	+	1.19348i	0.976082	+	0.217401i	$0.0697579\pi$
$163$	194.538	+	194.538i	1.19348	+	1.19348i	0.217401	+	0.976082i	$0.430242\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	24.4685			0.146518			0.0732589	−	0.997313i	$-0.476660\pi$
$167$	24.4685			0.146518			0.0732589	+	0.997313i	$0.476660\pi$
$168$		0			0
$169$			226.912i			1.34267i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	59.3789	−	59.3789i	0.343231	−	0.343231i	−0.514350	−	0.857580i	$-0.671967\pi$
$173$	59.3789	−	59.3789i	0.343231	−	0.343231i	0.857580	+	0.514350i	$0.171967\pi$
$174$		0			0
$175$	−209.106			−1.19489
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−95.2949	+	95.2949i	−0.532374	+	0.532374i	−0.921278	−	0.388904i	$-0.872854\pi$
$179$	−95.2949	+	95.2949i	−0.532374	+	0.532374i	0.388904	+	0.921278i	$0.372854\pi$
$180$		0			0
$181$	−18.2965	+	18.2965i	−0.101086	+	0.101086i	−0.755841	−	0.654755i	$-0.772772\pi$
$181$	−18.2965	+	18.2965i	−0.101086	+	0.101086i	0.654755	+	0.755841i	$0.272772\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−317.484			−1.71613
$186$		0			0
$187$	−92.6244	+	92.6244i	−0.495318	+	0.495318i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			130.199i			0.681669i	0.940123	+	0.340835i	$0.110710\pi$
$191$			130.199i			0.681669i	−0.940123	+	0.340835i	$0.889290\pi$
$192$		0			0
$193$	−280.445			−1.45308			−0.726540	−	0.687124i	$-0.758873\pi$
$193$	−280.445			−1.45308			−0.726540	+	0.687124i	$0.758873\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.58126	+	7.58126i	0.0384836	+	0.0384836i	0.726087	−	0.687603i	$-0.241337\pi$
$197$	7.58126	+	7.58126i	0.0384836	+	0.0384836i	−0.687603	+	0.726087i	$0.741337\pi$
$198$		0			0
$199$			157.616i			0.792040i	0.918242	+	0.396020i	$0.129609\pi$
$199$			157.616i			0.792040i	−0.918242	+	0.396020i	$0.870391\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−176.945	−	176.945i	−0.871649	−	0.871649i
$204$		0			0
$205$	−189.474	−	189.474i	−0.924264	−	0.924264i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			252.211i			1.20675i
$210$		0			0
$211$	−211.249	−	211.249i	−1.00118	−	1.00118i	−0.999999	−	0.00118241i	$-0.999624\pi$
$211$	−211.249	−	211.249i	−1.00118	−	1.00118i	−0.00118241	−	0.999999i	$-0.500376\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	292.233			1.35923
$216$		0			0
$217$		−	133.765i		−	0.616428i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	111.526	−	111.526i	0.504642	−	0.504642i
$222$		0			0
$223$	−7.05679			−0.0316448			−0.0158224	−	0.999875i	$-0.505037\pi$
$223$	−7.05679			−0.0316448			−0.0158224	+	0.999875i	$0.505037\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−294.580	+	294.580i	−1.29771	+	1.29771i	−0.367804	+	0.929903i	$0.619890\pi$
$227$	−294.580	+	294.580i	−1.29771	+	1.29771i	−0.929903	+	0.367804i	$0.880110\pi$
$228$		0			0
$229$	−66.7027	+	66.7027i	−0.291278	+	0.291278i	−0.837585	−	0.546307i	$-0.816033\pi$
$229$	−66.7027	+	66.7027i	−0.291278	+	0.291278i	0.546307	+	0.837585i	$0.316033\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	261.416			1.12196			0.560979	−	0.827830i	$-0.310425\pi$
$233$	261.416			1.12196			0.560979	+	0.827830i	$0.310425\pi$
$234$		0			0
$235$	336.407	−	336.407i	1.43152	−	1.43152i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	110.889i		−	0.463969i	−0.972719	−	0.231985i	$-0.925478\pi$
$239$		−	110.889i		−	0.463969i	0.972719	−	0.231985i	$-0.0745219\pi$
$240$		0			0
$241$	1.69179			0.00701988			0.00350994	−	0.999994i	$-0.498883\pi$
$241$	1.69179			0.00701988			0.00350994	+	0.999994i	$0.498883\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	36.4311	+	36.4311i	0.148698	+	0.148698i
$246$		0			0
$247$		−	303.679i		−	1.22947i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	115.067	+	115.067i	0.458432	+	0.458432i	0.898141	−	0.439708i	$-0.144918\pi$
$251$	115.067	+	115.067i	0.458432	+	0.458432i	−0.439708	+	0.898141i	$0.644918\pi$
$252$		0			0
$253$	−24.4468	−	24.4468i	−0.0966277	−	0.0966277i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			241.660i			0.940312i	0.882583	+	0.470156i	$0.155802\pi$
$257$			241.660i			0.940312i	−0.882583	+	0.470156i	$0.844198\pi$
$258$		0			0
$259$	231.100	+	231.100i	0.892276	+	0.892276i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−99.4659			−0.378197			−0.189099	−	0.981958i	$-0.560557\pi$
$263$	−99.4659			−0.378197			−0.189099	+	0.981958i	$0.560557\pi$
$264$		0			0
$265$		−	93.1154i		−	0.351379i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−97.1372	+	97.1372i	−0.361105	+	0.361105i	−0.864220	−	0.503115i	$-0.832187\pi$
$269$	−97.1372	+	97.1372i	−0.361105	+	0.361105i	0.503115	+	0.864220i	$0.332187\pi$
$270$		0			0
$271$	273.817			1.01040			0.505198	−	0.863004i	$-0.331420\pi$
$271$	273.817			1.01040			0.505198	+	0.863004i	$0.331420\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	326.277	−	326.277i	1.18646	−	1.18646i
$276$		0			0
$277$	222.564	−	222.564i	0.803481	−	0.803481i	−0.180157	−	0.983638i	$-0.557661\pi$
$277$	222.564	−	222.564i	0.803481	−	0.803481i	0.983638	+	0.180157i	$0.0576605\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	432.978			1.54085			0.770424	−	0.637532i	$-0.220045\pi$
$281$	432.978			1.54085			0.770424	+	0.637532i	$0.220045\pi$
$282$		0			0
$283$	117.222	−	117.222i	0.414213	−	0.414213i	−0.468990	−	0.883203i	$-0.655382\pi$
$283$	117.222	−	117.222i	0.414213	−	0.414213i	0.883203	+	0.468990i	$0.155382\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			275.840i			0.961116i
$288$		0			0
$289$	226.168			0.782587
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.7843	+	10.7843i	0.0368063	+	0.0368063i	0.725270	−	0.688464i	$-0.241715\pi$
$293$	10.7843	+	10.7843i	0.0368063	+	0.0368063i	−0.688464	+	0.725270i	$0.741715\pi$
$294$		0			0
$295$		−	761.347i		−	2.58084i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	29.4356	+	29.4356i	0.0984468	+	0.0984468i
$300$		0			0
$301$	−212.720	−	212.720i	−0.706710	−	0.706710i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	550.108i		−	1.80363i
$306$		0			0
$307$	−135.224	−	135.224i	−0.440470	−	0.440470i	0.451700	−	0.892170i	$-0.350818\pi$
$307$	−135.224	−	135.224i	−0.440470	−	0.440470i	−0.892170	+	0.451700i	$0.850818\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−559.652			−1.79953			−0.899763	−	0.436380i	$-0.856260\pi$
$311$	−559.652			−1.79953			−0.899763	+	0.436380i	$0.856260\pi$
$312$		0			0
$313$		−	355.506i		−	1.13580i	−0.823097	−	0.567901i	$-0.807756\pi$
$313$		−	355.506i		−	1.13580i	0.823097	−	0.567901i	$-0.192244\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.6492	−	12.6492i	0.0399027	−	0.0399027i	−0.686874	−	0.726777i	$-0.741018\pi$
$317$	12.6492	−	12.6492i	0.0399027	−	0.0399027i	0.726777	+	0.686874i	$0.241018\pi$
$318$		0			0
$319$	552.190			1.73100
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−85.5446	+	85.5446i	−0.264844	+	0.264844i
$324$		0			0
$325$	−392.860	+	392.860i	−1.20880	+	1.20880i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−489.748			−1.48860
$330$		0			0
$331$	−239.087	+	239.087i	−0.722317	+	0.722317i	−0.969077	−	0.246759i	$-0.920634\pi$
$331$	−239.087	+	239.087i	−0.722317	+	0.722317i	0.246759	+	0.969077i	$0.420634\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	212.702i		−	0.634931i
$336$		0			0
$337$	−485.512			−1.44069			−0.720344	−	0.693617i	$-0.756016\pi$
$337$	−485.512			−1.44069			−0.720344	+	0.693617i	$0.756016\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	208.720	+	208.720i	0.612081	+	0.612081i
$342$		0			0
$343$			313.914i			0.915202i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−186.750	−	186.750i	−0.538186	−	0.538186i	0.384810	−	0.922996i	$-0.374267\pi$
$347$	−186.750	−	186.750i	−0.538186	−	0.538186i	−0.922996	+	0.384810i	$0.874267\pi$
$348$		0			0
$349$	27.8498	+	27.8498i	0.0797988	+	0.0797988i	0.745880	−	0.666081i	$-0.232029\pi$
$349$	27.8498	+	27.8498i	0.0797988	+	0.0797988i	−0.666081	+	0.745880i	$0.732029\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			155.139i			0.439488i	0.975558	+	0.219744i	$0.0705222\pi$
$353$			155.139i			0.439488i	−0.975558	+	0.219744i	$0.929478\pi$
$354$		0			0
$355$	−204.008	−	204.008i	−0.574670	−	0.574670i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−280.349			−0.780916			−0.390458	−	0.920621i	$-0.627683\pi$
$359$	−280.349			−0.780916			−0.390458	+	0.920621i	$0.627683\pi$
$360$		0			0
$361$		−	128.067i		−	0.354756i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−470.041	+	470.041i	−1.28778	+	1.28778i
$366$		0			0
$367$	−153.450			−0.418120			−0.209060	−	0.977903i	$-0.567040\pi$
$367$	−153.450			−0.418120			−0.209060	+	0.977903i	$0.567040\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−67.7796	+	67.7796i	−0.182694	+	0.182694i
$372$		0			0
$373$	−70.9904	+	70.9904i	−0.190323	+	0.190323i	−0.795836	−	0.605513i	$-0.792968\pi$
$373$	−70.9904	+	70.9904i	−0.190323	+	0.190323i	0.605513	+	0.795836i	$0.292968\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−664.874			−1.76359
$378$		0			0
$379$	508.260	−	508.260i	1.34106	−	1.34106i	0.446047	−	0.895009i	$-0.352831\pi$
$379$	508.260	−	508.260i	1.34106	−	1.34106i	0.895009	−	0.446047i	$-0.147169\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	104.603i		−	0.273114i	−0.990632	−	0.136557i	$-0.956396\pi$
$383$		−	104.603i		−	0.273114i	0.990632	−	0.136557i	$-0.0436036\pi$
$384$		0			0
$385$	−900.287			−2.33841
$386$		0			0
$387$		0			0
$388$		0			0
$389$	273.853	+	273.853i	0.703992	+	0.703992i	0.965265	−	0.261273i	$-0.0841422\pi$
$389$	273.853	+	273.853i	0.703992	+	0.703992i	−0.261273	+	0.965265i	$0.584142\pi$
$390$		0			0
$391$		−	16.5837i		−	0.0424134i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−473.951	−	473.951i	−1.19988	−	1.19988i
$396$		0			0
$397$	153.531	+	153.531i	0.386728	+	0.386728i	0.873518	−	0.486791i	$-0.161833\pi$
$397$	153.531	+	153.531i	0.386728	+	0.386728i	−0.486791	+	0.873518i	$0.661833\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			39.4538i			0.0983884i	0.998789	+	0.0491942i	$0.0156653\pi$
$401$			39.4538i			0.0983884i	−0.998789	+	0.0491942i	$0.984335\pi$
$402$		0			0
$403$	−251.312	−	251.312i	−0.623604	−	0.623604i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−721.191			−1.77197
$408$		0			0
$409$		−	47.0882i		−	0.115130i	−0.998342	−	0.0575650i	$-0.981666\pi$
$409$		−	47.0882i		−	0.115130i	0.998342	−	0.0575650i	$-0.0183336\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−554.192	+	554.192i	−1.34187	+	1.34187i
$414$		0			0
$415$	93.8471			0.226138
$416$		0			0
$417$		0			0
$418$		0			0
$419$	341.054	−	341.054i	0.813972	−	0.813972i	−0.171255	−	0.985227i	$-0.554782\pi$
$419$	341.054	−	341.054i	0.813972	−	0.813972i	0.985227	+	0.171255i	$0.0547822\pi$
$420$		0			0
$421$	−34.9805	+	34.9805i	−0.0830891	+	0.0830891i	−0.747430	−	0.664341i	$-0.768712\pi$
$421$	−34.9805	+	34.9805i	−0.0830891	+	0.0830891i	0.664341	+	0.747430i	$0.268712\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	221.332			0.520782
$426$		0			0
$427$	−400.429	+	400.429i	−0.937774	+	0.937774i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	81.7370i		−	0.189645i	−0.995494	−	0.0948225i	$-0.969772\pi$
$431$		−	81.7370i		−	0.189645i	0.995494	−	0.0948225i	$-0.0302284\pi$
$432$		0			0
$433$	−258.726			−0.597520			−0.298760	−	0.954328i	$-0.596573\pi$
$433$	−258.726			−0.597520			−0.298760	+	0.954328i	$0.596573\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.5782	−	22.5782i	−0.0516664	−	0.0516664i
$438$		0			0
$439$		−	107.775i		−	0.245502i	−0.992438	−	0.122751i	$-0.960828\pi$
$439$		−	107.775i		−	0.245502i	0.992438	−	0.122751i	$-0.0391716\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	106.357	+	106.357i	0.240084	+	0.240084i	0.816885	−	0.576801i	$-0.195699\pi$
$443$	106.357	+	106.357i	0.240084	+	0.240084i	−0.576801	+	0.816885i	$0.695699\pi$
$444$		0			0
$445$	−325.409	−	325.409i	−0.731255	−	0.731255i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			501.272i			1.11642i	0.829700	+	0.558209i	$0.188511\pi$
$449$			501.272i			1.11642i	−0.829700	+	0.558209i	$0.811489\pi$
$450$		0			0
$451$	−430.406	−	430.406i	−0.954338	−	0.954338i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1084.01			2.38243
$456$		0			0
$457$		−	10.5786i		−	0.0231478i	−0.999933	−	0.0115739i	$-0.996316\pi$
$457$		−	10.5786i		−	0.0231478i	0.999933	−	0.0115739i	$-0.00368418\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−265.317	+	265.317i	−0.575525	+	0.575525i	−0.933667	−	0.358142i	$-0.883410\pi$
$461$	−265.317	+	265.317i	−0.575525	+	0.575525i	0.358142	+	0.933667i	$0.383410\pi$
$462$		0			0
$463$	−203.374			−0.439253			−0.219627	−	0.975584i	$-0.570484\pi$
$463$	−203.374			−0.439253			−0.219627	+	0.975584i	$0.570484\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	509.214	−	509.214i	1.09039	−	1.09039i	0.0949080	−	0.995486i	$-0.469744\pi$
$467$	509.214	−	509.214i	1.09039	−	1.09039i	0.995486	−	0.0949080i	$-0.0302557\pi$
$468$		0			0
$469$	−154.828	+	154.828i	−0.330123	+	0.330123i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	663.833			1.40345
$474$		0			0
$475$	301.338	−	301.338i	0.634396	−	0.634396i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			322.189i			0.672629i	0.941750	+	0.336314i	$0.109180\pi$
$479$			322.189i			0.672629i	−0.941750	+	0.336314i	$0.890820\pi$
$480$		0			0
$481$	868.362			1.80533
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−785.574	−	785.574i	−1.61974	−	1.61974i
$486$		0			0
$487$		−	52.6960i		−	0.108205i	−0.998535	−	0.0541026i	$-0.982770\pi$
$487$		−	52.6960i		−	0.108205i	0.998535	−	0.0541026i	$-0.0172298\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−287.775	−	287.775i	−0.586100	−	0.586100i	0.350473	−	0.936573i	$-0.386021\pi$
$491$	−287.775	−	287.775i	−0.586100	−	0.586100i	−0.936573	+	0.350473i	$0.886021\pi$
$492$		0			0
$493$	187.291	+	187.291i	0.379900	+	0.379900i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			296.999i			0.597583i
$498$		0			0
$499$	574.350	+	574.350i	1.15100	+	1.15100i	0.986352	+	0.164649i	$0.0526492\pi$
$499$	574.350	+	574.350i	1.15100	+	1.15100i	0.164649	+	0.986352i	$0.447351\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−666.868			−1.32578			−0.662891	−	0.748716i	$-0.730671\pi$
$503$	−666.868			−1.32578			−0.662891	+	0.748716i	$0.730671\pi$
$504$		0			0
$505$		−	267.434i		−	0.529573i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	187.837	−	187.837i	0.369032	−	0.369032i	−0.498092	−	0.867124i	$-0.665966\pi$
$509$	187.837	−	187.837i	0.369032	−	0.369032i	0.867124	+	0.498092i	$0.165966\pi$
$510$		0			0
$511$	684.295			1.33913
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−570.990	+	570.990i	−1.10872	+	1.10872i
$516$		0			0
$517$	764.177	−	764.177i	1.47810	−	1.47810i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−357.495			−0.686171			−0.343085	−	0.939304i	$-0.611472\pi$
$521$	−357.495			−0.686171			−0.343085	+	0.939304i	$0.611472\pi$
$522$		0			0
$523$	40.5076	−	40.5076i	0.0774525	−	0.0774525i	−0.667319	−	0.744772i	$-0.732558\pi$
$523$	40.5076	−	40.5076i	0.0774525	−	0.0774525i	0.744772	+	0.667319i	$0.232558\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			141.586i			0.268665i
$528$		0			0
$529$	−524.623			−0.991726
$530$		0			0
$531$		0			0
$532$		0			0
$533$	518.238	+	518.238i	0.972304	+	0.972304i
$534$		0			0
$535$		−	99.7793i		−	0.186503i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	82.7563	+	82.7563i	0.153537	+	0.153537i
$540$		0			0
$541$	−433.787	−	433.787i	−0.801825	−	0.801825i	0.181556	−	0.983381i	$-0.441887\pi$
$541$	−433.787	−	433.787i	−0.801825	−	0.801825i	−0.983381	+	0.181556i	$0.941887\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1032.24i		−	1.89403i
$546$		0			0
$547$	187.846	+	187.846i	0.343412	+	0.343412i	0.857649	−	0.514236i	$-0.171925\pi$
$547$	187.846	+	187.846i	0.343412	+	0.343412i	−0.514236	+	0.857649i	$0.671925\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	509.983			0.925559
$552$		0			0
$553$			689.987i			1.24772i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	483.437	−	483.437i	0.867931	−	0.867931i	−0.124312	−	0.992243i	$-0.539672\pi$
$557$	483.437	−	483.437i	0.867931	−	0.867931i	0.992243	+	0.124312i	$0.0396725\pi$
$558$		0			0
$559$	−799.299			−1.42987
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−204.235	+	204.235i	−0.362762	+	0.362762i	−0.864829	−	0.502067i	$-0.832573\pi$
$563$	−204.235	+	204.235i	−0.362762	+	0.362762i	0.502067	+	0.864829i	$0.332573\pi$
$564$		0			0
$565$	77.9412	−	77.9412i	0.137949	−	0.137949i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1001.16			1.75950			0.879751	−	0.475434i	$-0.157709\pi$
$569$	1001.16			1.75950			0.879751	+	0.475434i	$0.157709\pi$
$570$		0			0
$571$	709.935	−	709.935i	1.24332	−	1.24332i	0.284703	−	0.958616i	$-0.408105\pi$
$571$	709.935	−	709.935i	1.24332	−	1.24332i	0.958616	−	0.284703i	$-0.0918951\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			58.4173i			0.101595i
$576$		0			0
$577$	−348.321			−0.603676			−0.301838	−	0.953359i	$-0.597600\pi$
$577$	−348.321			−0.603676			−0.301838	+	0.953359i	$0.597600\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−68.3122	−	68.3122i	−0.117577	−	0.117577i
$582$		0			0
$583$		−	211.519i		−	0.362812i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−182.348	−	182.348i	−0.310644	−	0.310644i	0.534515	−	0.845159i	$-0.320494\pi$
$587$	−182.348	−	182.348i	−0.310644	−	0.310644i	−0.845159	+	0.534515i	$0.820494\pi$
$588$		0			0
$589$	192.766	+	192.766i	0.327276	+	0.327276i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			129.900i			0.219056i	0.993984	+	0.109528i	$0.0349340\pi$
$593$			129.900i			0.219056i	−0.993984	+	0.109528i	$0.965066\pi$
$594$		0			0
$595$	−305.358	−	305.358i	−0.513206	−	0.513206i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−456.851			−0.762689			−0.381344	−	0.924433i	$-0.624539\pi$
$599$	−456.851			−0.762689			−0.381344	+	0.924433i	$0.624539\pi$
$600$		0			0
$601$			965.221i			1.60603i	0.595962	+	0.803013i	$0.296771\pi$
$601$			965.221i			1.60603i	−0.595962	+	0.803013i	$0.703229\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	782.331	−	782.331i	1.29311	−	1.29311i
$606$		0			0
$607$	563.934			0.929050			0.464525	−	0.885560i	$-0.346225\pi$
$607$	563.934			0.929050			0.464525	+	0.885560i	$0.346225\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−920.120	+	920.120i	−1.50592	+	1.50592i
$612$		0			0
$613$	−537.334	+	537.334i	−0.876564	+	0.876564i	−0.993177	−	0.116613i	$-0.962796\pi$
$613$	−537.334	+	537.334i	−0.876564	+	0.876564i	0.116613	+	0.993177i	$0.462796\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−440.085			−0.713265			−0.356633	−	0.934245i	$-0.616075\pi$
$617$	−440.085			−0.713265			−0.356633	+	0.934245i	$0.616075\pi$
$618$		0			0
$619$	−173.277	+	173.277i	−0.279931	+	0.279931i	−0.833081	−	0.553150i	$-0.813425\pi$
$619$	−173.277	+	173.277i	−0.279931	+	0.279931i	0.553150	+	0.833081i	$0.313425\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			473.736i			0.760411i
$624$		0			0
$625$	543.399			0.869438
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−244.612	−	244.612i	−0.388891	−	0.388891i
$630$		0			0
$631$			385.341i			0.610683i	0.952243	+	0.305341i	$0.0987706\pi$
$631$			385.341i			0.610683i	−0.952243	+	0.305341i	$0.901229\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	966.109	+	966.109i	1.52143	+	1.52143i
$636$		0			0
$637$	−99.6441	−	99.6441i	−0.156427	−	0.156427i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			546.852i			0.853123i	0.904459	+	0.426561i	$0.140275\pi$
$641$			546.852i			0.853123i	−0.904459	+	0.426561i	$0.859725\pi$
$642$		0			0
$643$	350.269	+	350.269i	0.544742	+	0.544742i	0.924915	−	0.380173i	$-0.124136\pi$
$643$	350.269	+	350.269i	0.544742	+	0.544742i	−0.380173	+	0.924915i	$0.624136\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1039.41			1.60651			0.803256	−	0.595633i	$-0.203099\pi$
$647$	1039.41			1.60651			0.803256	+	0.595633i	$0.203099\pi$
$648$		0			0
$649$		−	1729.46i		−	2.66481i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	498.995	−	498.995i	0.764158	−	0.764158i	−0.212913	−	0.977071i	$-0.568295\pi$
$653$	498.995	−	498.995i	0.764158	−	0.764158i	0.977071	+	0.212913i	$0.0682951\pi$
$654$		0			0
$655$	−560.895			−0.856328
$656$		0			0
$657$		0			0
$658$		0			0
$659$	601.717	−	601.717i	0.913076	−	0.913076i	−0.0834368	−	0.996513i	$-0.526590\pi$
$659$	601.717	−	601.717i	0.913076	−	0.913076i	0.996513	+	0.0834368i	$0.0265897\pi$
$660$		0			0
$661$	−32.4484	+	32.4484i	−0.0490898	+	0.0490898i	−0.731226	−	0.682136i	$-0.761051\pi$
$661$	−32.4484	+	32.4484i	−0.0490898	+	0.0490898i	0.682136	+	0.731226i	$0.261051\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−831.473			−1.25034
$666$		0			0
$667$	−49.4326	+	49.4326i	−0.0741119	+	0.0741119i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1249.62i		−	1.86232i
$672$		0			0
$673$	1246.60			1.85230			0.926151	−	0.377153i	$-0.123097\pi$
$673$	1246.60			1.85230			0.926151	+	0.377153i	$0.123097\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	787.303	+	787.303i	1.16293	+	1.16293i	0.983831	+	0.179097i	$0.0573177\pi$
$677$	787.303	+	787.303i	1.16293	+	1.16293i	0.179097	+	0.983831i	$0.442682\pi$
$678$		0			0
$679$			1143.65i			1.68432i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	737.985	+	737.985i	1.08050	+	1.08050i	0.996462	+	0.0840423i	$0.0267831\pi$
$683$	737.985	+	737.985i	1.08050	+	1.08050i	0.0840423	+	0.996462i	$0.473217\pi$
$684$		0			0
$685$	131.241	+	131.241i	0.191593	+	0.191593i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			254.684i			0.369642i
$690$		0			0
$691$	185.700	+	185.700i	0.268741	+	0.268741i	0.828593	−	0.559852i	$-0.189142\pi$
$691$	185.700	+	185.700i	0.268741	+	0.268741i	−0.559852	+	0.828593i	$0.689142\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	181.987			0.261852
$696$		0			0
$697$		−	291.969i		−	0.418894i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−77.3987	+	77.3987i	−0.110412	+	0.110412i	−0.760154	−	0.649743i	$-0.774877\pi$
$701$	−77.3987	+	77.3987i	−0.110412	+	0.110412i	0.649743	+	0.760154i	$0.274877\pi$
$702$		0			0
$703$	−666.066			−0.947463
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−194.668	+	194.668i	−0.275344	+	0.275344i
$708$		0			0
$709$	281.705	−	281.705i	0.397327	−	0.397327i	−0.479962	−	0.877289i	$-0.659350\pi$
$709$	281.705	−	281.705i	0.397327	−	0.397327i	0.877289	+	0.479962i	$0.159350\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−37.3696			−0.0524117
$714$		0			0
$715$	−1691.43	+	1691.43i	−2.36563	+	2.36563i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1287.51i		−	1.79070i	−0.445364	−	0.895350i	$-0.646926\pi$
$719$		−	1287.51i		−	1.79070i	0.445364	−	0.895350i	$-0.353074\pi$
$720$		0			0
$721$	831.259			1.15292
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−659.748	−	659.748i	−0.909997	−	0.909997i
$726$		0			0
$727$		−	549.191i		−	0.755421i	−0.925924	−	0.377711i	$-0.876711\pi$
$727$		−	549.191i		−	0.755421i	0.925924	−	0.377711i	$-0.123289\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	225.158	+	225.158i	0.308013	+	0.308013i
$732$		0			0
$733$	889.166	+	889.166i	1.21305	+	1.21305i	0.970018	+	0.243031i	$0.0781418\pi$
$733$	889.166	+	889.166i	1.21305	+	1.21305i	0.243031	+	0.970018i	$0.421858\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	483.171i		−	0.655591i
$738$		0			0
$739$	561.713	+	561.713i	0.760099	+	0.760099i	0.976340	−	0.216241i	$-0.0693798\pi$
$739$	561.713	+	561.713i	0.760099	+	0.760099i	−0.216241	+	0.976340i	$0.569380\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	945.402			1.27241			0.636206	−	0.771519i	$-0.280503\pi$
$743$	945.402			1.27241			0.636206	+	0.771519i	$0.280503\pi$
$744$		0			0
$745$			1450.95i			1.94759i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−72.6303	+	72.6303i	−0.0969697	+	0.0969697i
$750$		0			0
$751$	−751.358			−1.00048			−0.500238	−	0.865888i	$-0.666754\pi$
$751$	−751.358			−1.00048			−0.500238	+	0.865888i	$0.666754\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1240.05	−	1240.05i	1.64245	−	1.64245i
$756$		0			0
$757$	−326.074	+	326.074i	−0.430745	+	0.430745i	−0.888882	−	0.458137i	$-0.848517\pi$
$757$	−326.074	+	326.074i	−0.430745	+	0.430745i	0.458137	+	0.888882i	$0.348517\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−636.023			−0.835772			−0.417886	−	0.908499i	$-0.637229\pi$
$761$	−636.023			−0.835772			−0.417886	+	0.908499i	$0.637229\pi$
$762$		0			0
$763$	−751.381	+	751.381i	−0.984771	+	0.984771i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			2082.39i			2.71498i
$768$		0			0
$769$	463.385			0.602582			0.301291	−	0.953532i	$-0.402583\pi$
$769$	463.385			0.602582			0.301291	+	0.953532i	$0.402583\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−131.860	−	131.860i	−0.170582	−	0.170582i	0.616653	−	0.787235i	$-0.288488\pi$
$773$	−131.860	−	131.860i	−0.170582	−	0.170582i	−0.787235	+	0.616653i	$0.788488\pi$
$774$		0			0
$775$		−	498.750i		−	0.643548i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−397.508	−	397.508i	−0.510280	−	0.510280i
$780$		0			0
$781$	−463.421	−	463.421i	−0.593369	−	0.593369i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	139.865i		−	0.178172i
$786$		0			0
$787$	478.975	+	478.975i	0.608609	+	0.608609i	0.942583	−	0.333973i	$-0.108390\pi$
$787$	478.975	+	478.975i	0.608609	+	0.608609i	−0.333973	+	0.942583i	$0.608390\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−113.468			−0.143449
$792$		0			0
$793$			1504.62i			1.89738i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−773.501	+	773.501i	−0.970516	+	0.970516i	−0.999578	−	0.0290618i	$-0.990748\pi$
$797$	−773.501	+	773.501i	−0.970516	+	0.970516i	0.0290618	+	0.999578i	$0.490748\pi$
$798$		0			0
$799$	518.384			0.648791
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1067.74	+	1067.74i	−1.32969	+	1.32969i
$804$		0			0
$805$	80.5946	−	80.5946i	0.100118	−	0.100118i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1027.95			1.27064			0.635322	−	0.772247i	$-0.280867\pi$
$809$	1027.95			1.27064			0.635322	+	0.772247i	$0.280867\pi$
$810$		0			0
$811$	−398.337	+	398.337i	−0.491168	+	0.491168i	−0.908674	−	0.417506i	$-0.862904\pi$
$811$	−398.337	+	398.337i	−0.491168	+	0.491168i	0.417506	+	0.908674i	$0.362904\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	2001.42i		−	2.45573i
$816$		0			0
$817$	613.092			0.750419
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−122.308	−	122.308i	−0.148974	−	0.148974i	0.628686	−	0.777660i	$-0.283593\pi$
$821$	−122.308	−	122.308i	−0.148974	−	0.148974i	−0.777660	+	0.628686i	$0.783593\pi$
$822$		0			0
$823$			767.738i			0.932853i	0.884560	+	0.466426i	$0.154459\pi$
$823$			767.738i			0.932853i	−0.884560	+	0.466426i	$0.845541\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−701.804	−	701.804i	−0.848615	−	0.848615i	0.141346	−	0.989960i	$-0.454857\pi$
$827$	−701.804	−	701.804i	−0.848615	−	0.848615i	−0.989960	+	0.141346i	$0.954857\pi$
$828$		0			0
$829$	433.827	+	433.827i	0.523314	+	0.523314i	0.918571	−	0.395257i	$-0.129344\pi$
$829$	433.827	+	433.827i	0.523314	+	0.523314i	−0.395257	+	0.918571i	$0.629344\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			56.1382i			0.0673929i
$834$		0			0
$835$	−125.867	−	125.867i	−0.150739	−	0.150739i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1126.87			−1.34311			−0.671554	−	0.740956i	$-0.734373\pi$
$839$	−1126.87			−1.34311			−0.671554	+	0.740956i	$0.734373\pi$
$840$		0			0
$841$		−	275.555i		−	0.327651i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1167.25	−	1167.25i	1.38136	−	1.38136i
$846$		0			0
$847$	−1138.93			−1.34467
$848$		0			0
$849$		0			0
$850$		0			0
$851$	64.5617	−	64.5617i	0.0758657	−	0.0758657i
$852$		0			0
$853$	−409.568	+	409.568i	−0.480150	+	0.480150i	−0.905179	−	0.425030i	$-0.860264\pi$
$853$	−409.568	+	409.568i	−0.480150	+	0.480150i	0.425030	+	0.905179i	$0.360264\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−311.374			−0.363330			−0.181665	−	0.983360i	$-0.558149\pi$
$857$	−311.374			−0.363330			−0.181665	+	0.983360i	$0.558149\pi$
$858$		0			0
$859$	223.590	−	223.590i	0.260291	−	0.260291i	−0.564881	−	0.825172i	$-0.691078\pi$
$859$	223.590	−	223.590i	0.260291	−	0.260291i	0.825172	+	0.564881i	$0.191078\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1449.16i			1.67921i	0.543196	+	0.839606i	$0.317214\pi$
$863$			1449.16i			1.67921i	−0.543196	+	0.839606i	$0.682786\pi$
$864$		0			0
$865$	−610.896			−0.706238
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1076.62	−	1076.62i	−1.23892	−	1.23892i
$870$		0			0
$871$			581.770i			0.667933i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	112.580	+	112.580i	0.128663	+	0.128663i
$876$		0			0
$877$	752.642	+	752.642i	0.858201	+	0.858201i	0.991126	−	0.132925i	$-0.0424370\pi$
$877$	752.642	+	752.642i	0.858201	+	0.858201i	−0.132925	+	0.991126i	$0.542437\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	449.004i		−	0.509653i	−0.966987	−	0.254827i	$-0.917982\pi$
$881$		−	449.004i		−	0.509653i	0.966987	−	0.254827i	$-0.0820184\pi$
$882$		0			0
$883$	−1009.33	−	1009.33i	−1.14307	−	1.14307i	−0.987886	−	0.155179i	$-0.950404\pi$
$883$	−1009.33	−	1009.33i	−1.14307	−	1.14307i	−0.155179	−	0.987886i	$-0.549596\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	480.623			0.541852			0.270926	−	0.962600i	$-0.412670\pi$
$887$	480.623			0.541852			0.270926	+	0.962600i	$0.412670\pi$
$888$		0			0
$889$		−	1406.48i		−	1.58209i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	705.766	−	705.766i	0.790332	−	0.790332i
$894$		0			0
$895$	980.403			1.09542
$896$		0			0
$897$		0			0
$898$		0			0
$899$	422.041	−	422.041i	0.469456	−	0.469456i
$900$		0			0
$901$	71.7428	−	71.7428i	0.0796257	−	0.0796257i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	188.236			0.207996
$906$		0			0
$907$	529.858	−	529.858i	0.584188	−	0.584188i	−0.351863	−	0.936051i	$-0.614452\pi$
$907$	529.858	−	529.858i	0.584188	−	0.584188i	0.936051	+	0.351863i	$0.114452\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1010.82i			1.10958i	0.831992	+	0.554788i	$0.187201\pi$
$911$			1010.82i			1.10958i	−0.831992	+	0.554788i	$0.812799\pi$
$912$		0			0
$913$	213.182			0.233496
$914$		0			0
$915$		0			0
$916$		0			0
$917$	408.281	+	408.281i	0.445235	+	0.445235i
$918$		0			0
$919$			1314.55i			1.43042i	0.698911	+	0.715208i	$0.253668\pi$
$919$			1314.55i			1.43042i	−0.698911	+	0.715208i	$0.746332\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	557.990	+	557.990i	0.604540	+	0.604540i
$924$		0			0
$925$	861.668	+	861.668i	0.931533	+	0.931533i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1206.78i		−	1.29901i	−0.760355	−	0.649507i	$-0.774975\pi$
$929$		−	1206.78i		−	1.29901i	0.760355	−	0.649507i	$-0.225025\pi$
$930$		0			0
$931$	76.4307	+	76.4307i	0.0820953	+	0.0820953i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	952.928			1.01917
$936$		0			0
$937$			926.373i			0.988659i	0.869275	+	0.494329i	$0.164586\pi$
$937$			926.373i			0.988659i	−0.869275	+	0.494329i	$0.835414\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−965.861	+	965.861i	−1.02642	+	1.02642i	−0.0267780	+	0.999641i	$0.508525\pi$
$941$	−965.861	+	965.861i	−1.02642	+	1.02642i	−0.999641	+	0.0267780i	$0.991475\pi$
$942$		0			0
$943$	77.0608			0.0817188
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−315.549	+	315.549i	−0.333209	+	0.333209i	−0.853804	−	0.520595i	$-0.825710\pi$
$947$	−315.549	+	315.549i	−0.333209	+	0.333209i	0.520595	+	0.853804i	$0.325710\pi$
$948$		0			0
$949$	1285.63	−	1285.63i	1.35472	−	1.35472i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	404.148			0.424080			0.212040	−	0.977261i	$-0.431989\pi$
$953$	404.148			0.424080			0.212040	+	0.977261i	$0.431989\pi$
$954$		0			0
$955$	669.749	−	669.749i	0.701308	−	0.701308i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	191.064i		−	0.199232i
$960$		0			0
$961$	−641.950			−0.668002
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1442.62	+	1442.62i	1.49494	+	1.49494i
$966$		0			0
$967$		−	571.435i		−	0.590936i	−0.955353	−	0.295468i	$-0.904524\pi$
$967$		−	571.435i		−	0.590936i	0.955353	−	0.295468i	$-0.0954756\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	250.824	+	250.824i	0.258315	+	0.258315i	0.824369	−	0.566053i	$-0.191530\pi$
$971$	250.824	+	250.824i	0.258315	+	0.258315i	−0.566053	+	0.824369i	$0.691530\pi$
$972$		0			0
$973$	−132.470	−	132.470i	−0.136146	−	0.136146i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1559.92i		−	1.59665i	−0.602230	−	0.798323i	$-0.705721\pi$
$977$		−	1559.92i		−	1.59665i	0.602230	−	0.798323i	$-0.294279\pi$
$978$		0			0
$979$	−739.193	−	739.193i	−0.755049	−	0.755049i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−762.047			−0.775226			−0.387613	−	0.921822i	$-0.626700\pi$
$983$	−762.047			−0.775226			−0.387613	+	0.921822i	$0.626700\pi$
$984$		0			0
$985$		−	77.9967i		−	0.0791845i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−59.4270	+	59.4270i	−0.0600879	+	0.0600879i
$990$		0			0
$991$	895.296			0.903427			0.451713	−	0.892163i	$-0.350813\pi$
$991$	895.296			0.903427			0.451713	+	0.892163i	$0.350813\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	810.784	−	810.784i	0.814858	−	0.814858i
$996$		0			0
$997$	539.874	−	539.874i	0.541498	−	0.541498i	−0.382470	−	0.923968i	$-0.624926\pi$
$997$	539.874	−	539.874i	0.541498	−	0.541498i	0.923968	+	0.382470i	$0.124926\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.j.a.161.3		32
3.2	odd	2	inner	1152.3.j.a.161.14		32
4.3	odd	2		1152.3.j.b.161.3		32
8.3	odd	2		576.3.j.a.17.14		32
8.5	even	2		144.3.j.a.125.8	yes	32
12.11	even	2		1152.3.j.b.161.14		32
16.3	odd	4		576.3.j.a.305.3		32
16.5	even	4	inner	1152.3.j.a.737.14		32
16.11	odd	4		1152.3.j.b.737.14		32
16.13	even	4		144.3.j.a.53.9	yes	32
24.5	odd	2		144.3.j.a.125.9	yes	32
24.11	even	2		576.3.j.a.17.3		32
48.5	odd	4	inner	1152.3.j.a.737.3		32
48.11	even	4		1152.3.j.b.737.3		32
48.29	odd	4		144.3.j.a.53.8	✓	32
48.35	even	4		576.3.j.a.305.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.8	✓	32	48.29	odd	4
144.3.j.a.53.9	yes	32	16.13	even	4
144.3.j.a.125.8	yes	32	8.5	even	2
144.3.j.a.125.9	yes	32	24.5	odd	2
576.3.j.a.17.3		32	24.11	even	2
576.3.j.a.17.14		32	8.3	odd	2
576.3.j.a.305.3		32	16.3	odd	4
576.3.j.a.305.14		32	48.35	even	4
1152.3.j.a.161.3		32	1.1	even	1	trivial
1152.3.j.a.161.14		32	3.2	odd	2	inner
1152.3.j.a.737.3		32	48.5	odd	4	inner
1152.3.j.a.737.14		32	16.5	even	4	inner
1152.3.j.b.161.3		32	4.3	odd	2
1152.3.j.b.161.14		32	12.11	even	2
1152.3.j.b.737.3		32	48.11	even	4
1152.3.j.b.737.14		32	16.11	odd	4

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.14405 - 5.14405i) q^{5} +7.48880i q^{7} +O(q^{10})\)	\(q+(-5.14405 - 5.14405i) q^{5} +7.48880i q^{7} +(-11.6851 - 11.6851i) q^{11} +(14.0697 + 14.0697i) q^{13} -7.92669i q^{17} +(-10.7920 - 10.7920i) q^{19} +2.09213 q^{23} +27.9224i q^{25} +(-23.6279 + 23.6279i) q^{29} -17.8620 q^{31} +(38.5228 - 38.5228i) q^{35} +(30.8593 - 30.8593i) q^{37} +36.8337 q^{41} +(-28.4050 + 28.4050i) q^{43} +65.3973i q^{47} -7.08218 q^{49} +(9.05079 + 9.05079i) q^{53} +120.218i q^{55} +(74.0027 + 74.0027i) q^{59} +(53.4704 + 53.4704i) q^{61} -144.750i q^{65} +(20.6746 + 20.6746i) q^{67} +39.6591 q^{71} -91.3758i q^{73} +(87.5077 - 87.5077i) q^{77} +92.1359 q^{79} +(-9.12191 + 9.12191i) q^{83} +(-40.7752 + 40.7752i) q^{85} +63.2593 q^{89} +(-105.365 + 105.365i) q^{91} +111.029i q^{95} +152.715 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 64 q^{19} - 128 q^{43} - 224 q^{49} - 64 q^{61} + 64 q^{67} - 512 q^{79} - 320 q^{85} + 192 q^{91}+O(q^{100}) \) 32 * q - 64 * q^19 - 128 * q^43 - 224 * q^49 - 64 * q^61 + 64 * q^67 - 512 * q^79 - 320 * q^85 + 192 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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