LMFDB - Embedded newform 1152.3.j.a.161.14

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			161.14
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.a.737.14

$q$-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	$0.00930769\pi$
$5$	5.14405	+	5.14405i	1.02881	+	1.02881i	0.0292368	+	0.999573i	$0.490692\pi$
$6$		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	$0.0205000\pi$
$11$	11.6851	+	11.6851i	1.06228	+	1.06228i	0.0643581	+	0.997927i	$0.479500\pi$
$12$		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	$0.0748993\pi$
$17$			7.92669i			0.466276i	−0.972444	+	0.233138i	$0.925101\pi$
$18$		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	$-0.514482\pi$
$23$	−2.09213			−0.0909621			−0.0454811	+	0.998965i	$0.514482\pi$
$24$		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	−0.170588	−	0.985343i	$-0.554567\pi$
$29$	23.6279	−	23.6279i	0.814755	−	0.814755i	0.985343	+	0.170588i	$0.0545666\pi$
$30$		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	$-0.648288\pi$
$41$	−36.8337			−0.898382			−0.449191	+	0.893436i	$0.648288\pi$
$42$		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	$-0.755087\pi$
$47$		−	65.3973i		−	1.39143i	0.718317	−	0.695716i	$-0.244913\pi$
$48$		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	0.616548	−	0.787317i	$-0.288531\pi$
$53$	−9.05079	−	9.05079i	−0.170770	−	0.170770i	−0.787317	+	0.616548i	$0.788531\pi$
$54$		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	$-0.902846\pi$
$59$	−74.0027	−	74.0027i	−1.25428	−	1.25428i	−0.300503	−	0.953781i	$-0.597154\pi$
$60$		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	$-0.590099\pi$
$71$	−39.6591			−0.558578			−0.279289	+	0.960207i	$0.590099\pi$
$72$		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	−0.650017	−	0.759920i	$-0.725238\pi$
$83$	9.12191	−	9.12191i	0.109903	−	0.109903i	0.759920	+	0.650017i	$0.225238\pi$
$84$		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	$-0.615652\pi$
$89$	−63.2593			−0.710778			−0.355389	+	0.934718i	$0.615652\pi$
$90$		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	0.566613	−	0.823984i	$-0.308254\pi$
$101$	−25.9945	−	25.9945i	−0.257372	−	0.257372i	−0.823984	+	0.566613i	$0.808254\pi$
$102$		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	0.660333	−	0.750973i	$-0.270415\pi$
$107$	−9.69852	−	9.69852i	−0.0906404	−	0.0906404i	−0.750973	+	0.660333i	$0.770415\pi$
$108$		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	$-0.978644\pi$
$113$		−	15.1517i		−	0.134086i	0.997750	−	0.0670431i	$-0.0213565\pi$
$114$		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	−0.883883	−	0.467709i	$-0.845080\pi$
$131$	−54.5188	+	54.5188i	−0.416174	+	0.416174i	0.467709	+	0.883883i	$0.345080\pi$
$132$		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	$-0.470318\pi$
$137$	25.5132			0.186228			0.0931140	+	0.995655i	$0.470318\pi$
$138$		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	0.998641	−	0.0521168i	$-0.0165968\pi$
$149$	141.032	+	141.032i	0.946524	+	0.946524i	−0.0521168	+	0.998641i	$0.516597\pi$
$150$		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	$-0.523340\pi$
$167$	−24.4685			−0.146518			−0.0732589	+	0.997313i	$0.523340\pi$
$168$		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	−0.857580	−	0.514350i	$-0.828033\pi$
$173$	−59.3789	+	59.3789i	−0.343231	+	0.343231i	0.514350	+	0.857580i	$0.328033\pi$
$174$		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	−0.388904	−	0.921278i	$-0.627146\pi$
$179$	95.2949	−	95.2949i	0.532374	−	0.532374i	0.921278	+	0.388904i	$0.127146\pi$
$180$		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	$-0.889290\pi$
$191$		−	130.199i		−	0.681669i	0.940123	−	0.340835i	$-0.110710\pi$
$192$		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	0.687603	−	0.726087i	$-0.258663\pi$
$197$	−7.58126	−	7.58126i	−0.0384836	−	0.0384836i	−0.726087	+	0.687603i	$0.758663\pi$
$198$		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	$-0.380110\pi$
$227$	294.580	−	294.580i	1.29771	−	1.29771i	0.929903	−	0.367804i	$-0.119890\pi$
$228$		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	$-0.689575\pi$
$233$	−261.416			−1.12196			−0.560979	+	0.827830i	$0.689575\pi$
$234$		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	$0.0745219\pi$
$239$			110.889i			0.463969i	−0.972719	+	0.231985i	$0.925478\pi$
$240$		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	0.439708	−	0.898141i	$-0.355082\pi$
$251$	−115.067	−	115.067i	−0.458432	−	0.458432i	−0.898141	+	0.439708i	$0.855082\pi$
$252$		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	$-0.844198\pi$
$257$		−	241.660i		−	0.940312i	0.882583	−	0.470156i	$-0.155802\pi$
$258$		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	$-0.439443\pi$
$263$	99.4659			0.378197			0.189099	+	0.981958i	$0.439443\pi$
$264$		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	−0.503115	−	0.864220i	$-0.667813\pi$
$269$	97.1372	−	97.1372i	0.361105	−	0.361105i	0.864220	+	0.503115i	$0.167813\pi$
$270$		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	$-0.779955\pi$
$281$	−432.978			−1.54085			−0.770424	+	0.637532i	$0.779955\pi$
$282$		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	0.688464	−	0.725270i	$-0.258285\pi$
$293$	−10.7843	−	10.7843i	−0.0368063	−	0.0368063i	−0.725270	+	0.688464i	$0.758285\pi$
$294$		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	$-0.143740\pi$
$311$	559.652			1.79953			0.899763	+	0.436380i	$0.143740\pi$
$312$		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	−0.726777	−	0.686874i	$-0.758982\pi$
$317$	−12.6492	+	12.6492i	−0.0399027	+	0.0399027i	0.686874	+	0.726777i	$0.258982\pi$
$318$		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	0.922996	−	0.384810i	$-0.125733\pi$
$347$	186.750	+	186.750i	0.538186	+	0.538186i	−0.384810	+	0.922996i	$0.625733\pi$
$348$		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	$-0.929478\pi$
$353$		−	155.139i		−	0.439488i	0.975558	−	0.219744i	$-0.0705222\pi$
$354$		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	$-0.372317\pi$
$359$	280.349			0.780916			0.390458	+	0.920621i	$0.372317\pi$
$360$		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	$0.0436036\pi$
$383$			104.603i			0.273114i	−0.990632	+	0.136557i	$0.956396\pi$
$384$		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	0.261273	−	0.965265i	$-0.415858\pi$
$389$	−273.853	−	273.853i	−0.703992	−	0.703992i	−0.965265	+	0.261273i	$0.915858\pi$
$390$		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	$-0.984335\pi$
$401$		−	39.4538i		−	0.0983884i	0.998789	−	0.0491942i	$-0.0156653\pi$
$402$		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	−0.985227	−	0.171255i	$-0.945218\pi$
$419$	−341.054	+	341.054i	−0.813972	+	0.813972i	0.171255	+	0.985227i	$0.445218\pi$
$420$		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	$0.0302284\pi$
$431$			81.7370i			0.189645i	−0.995494	+	0.0948225i	$0.969772\pi$
$432$		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	0.576801	−	0.816885i	$-0.304301\pi$
$443$	−106.357	−	106.357i	−0.240084	−	0.240084i	−0.816885	+	0.576801i	$0.804301\pi$
$444$		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	$-0.811489\pi$
$449$		−	501.272i		−	1.11642i	0.829700	−	0.558209i	$-0.188511\pi$
$450$		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	−0.358142	−	0.933667i	$-0.616590\pi$
$461$	265.317	−	265.317i	0.575525	−	0.575525i	0.933667	+	0.358142i	$0.116590\pi$
$462$		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	$0.530256\pi$
$467$	−509.214	+	509.214i	−1.09039	+	1.09039i	−0.995486	+	0.0949080i	$0.969744\pi$
$468$		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	$-0.890820\pi$
$479$		−	322.189i		−	0.672629i	0.941750	−	0.336314i	$-0.109180\pi$
$480$		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	0.936573	−	0.350473i	$-0.113979\pi$
$491$	287.775	+	287.775i	0.586100	+	0.586100i	−0.350473	+	0.936573i	$0.613979\pi$
$492$		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	$-0.269329\pi$
$503$	666.868			1.32578			0.662891	+	0.748716i	$0.269329\pi$
$504$		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	−0.867124	−	0.498092i	$-0.834034\pi$
$509$	−187.837	+	187.837i	−0.369032	+	0.369032i	0.498092	+	0.867124i	$0.334034\pi$
$510$		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	$-0.388528\pi$
$521$	357.495			0.686171			0.343085	+	0.939304i	$0.388528\pi$
$522$		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	−0.992243	−	0.124312i	$-0.960328\pi$
$557$	−483.437	+	483.437i	−0.867931	+	0.867931i	0.124312	+	0.992243i	$0.460328\pi$
$558$		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	−0.502067	−	0.864829i	$-0.667427\pi$
$563$	204.235	−	204.235i	0.362762	−	0.362762i	0.864829	+	0.502067i	$0.167427\pi$
$564$		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	$-0.842291\pi$
$569$	−1001.16			−1.75950			−0.879751	+	0.475434i	$0.842291\pi$
$570$		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	0.845159	−	0.534515i	$-0.179506\pi$
$587$	182.348	+	182.348i	0.310644	+	0.310644i	−0.534515	+	0.845159i	$0.679506\pi$
$588$		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	$-0.965066\pi$
$593$		−	129.900i		−	0.219056i	0.993984	−	0.109528i	$-0.0349340\pi$
$594$		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	$-0.375461\pi$
$599$	456.851			0.762689			0.381344	+	0.924433i	$0.375461\pi$
$600$		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	$-0.383925\pi$
$617$	440.085			0.713265			0.356633	+	0.934245i	$0.383925\pi$
$618$		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	$-0.859725\pi$
$641$		−	546.852i		−	0.853123i	0.904459	−	0.426561i	$-0.140275\pi$
$642$		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	$-0.796901\pi$
$647$	−1039.41			−1.60651			−0.803256	+	0.595633i	$0.796901\pi$
$648$		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	−0.977071	−	0.212913i	$-0.931705\pi$
$653$	−498.995	+	498.995i	−0.764158	+	0.764158i	0.212913	+	0.977071i	$0.431705\pi$
$654$		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	−0.996513	−	0.0834368i	$-0.973410\pi$
$659$	−601.717	+	601.717i	−0.913076	+	0.913076i	0.0834368	+	0.996513i	$0.473410\pi$
$660$		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	$-0.942682\pi$
$677$	−787.303	−	787.303i	−1.16293	−	1.16293i	−0.179097	−	0.983831i	$-0.557318\pi$
$678$		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	$-0.973217\pi$
$683$	−737.985	−	737.985i	−1.08050	−	1.08050i	−0.0840423	−	0.996462i	$-0.526783\pi$
$684$		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	−0.649743	−	0.760154i	$-0.725123\pi$
$701$	77.3987	−	77.3987i	0.110412	−	0.110412i	0.760154	+	0.649743i	$0.225123\pi$
$702$		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	$0.353074\pi$
$719$			1287.51i			1.79070i	−0.445364	+	0.895350i	$0.646926\pi$
$720$		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	$-0.719497\pi$
$743$	−945.402			−1.27241			−0.636206	+	0.771519i	$0.719497\pi$
$744$		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	$-0.362771\pi$
$761$	636.023			0.835772			0.417886	+	0.908499i	$0.362771\pi$
$762$		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	0.787235	−	0.616653i	$-0.211512\pi$
$773$	131.860	+	131.860i	0.170582	+	0.170582i	−0.616653	+	0.787235i	$0.711512\pi$
$774$		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	−0.0290618	−	0.999578i	$-0.509252\pi$
$797$	773.501	−	773.501i	0.970516	−	0.970516i	0.999578	+	0.0290618i	$0.00925197\pi$
$798$		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	$-0.719133\pi$
$809$	−1027.95			−1.27064			−0.635322	+	0.772247i	$0.719133\pi$
$810$		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	0.777660	−	0.628686i	$-0.216407\pi$
$821$	122.308	+	122.308i	0.148974	+	0.148974i	−0.628686	+	0.777660i	$0.716407\pi$
$822$		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	0.989960	−	0.141346i	$-0.0451429\pi$
$827$	701.804	+	701.804i	0.848615	+	0.848615i	−0.141346	+	0.989960i	$0.545143\pi$
$828$		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	$-0.265627\pi$
$839$	1126.87			1.34311			0.671554	+	0.740956i	$0.265627\pi$
$840$		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	$-0.441851\pi$
$857$	311.374			0.363330			0.181665	+	0.983360i	$0.441851\pi$
$858$		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	$-0.682786\pi$
$863$		−	1449.16i		−	1.67921i	0.543196	−	0.839606i	$-0.317214\pi$
$864$		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	$0.0820184\pi$
$881$			449.004i			0.509653i	−0.966987	+	0.254827i	$0.917982\pi$
$882$		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	$-0.587330\pi$
$887$	−480.623			−0.541852			−0.270926	+	0.962600i	$0.587330\pi$
$888$		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	$-0.812799\pi$
$911$		−	1010.82i		−	1.10958i	0.831992	−	0.554788i	$-0.187201\pi$
$912$		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	$0.225025\pi$
$929$			1206.78i			1.29901i	−0.760355	+	0.649507i	$0.774975\pi$
$930$		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	$-0.491475\pi$
$941$	965.861	−	965.861i	1.02642	−	1.02642i	0.999641	−	0.0267780i	$-0.00852472\pi$
$942$		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	−0.520595	−	0.853804i	$-0.674290\pi$
$947$	315.549	−	315.549i	0.333209	−	0.333209i	0.853804	+	0.520595i	$0.174290\pi$
$948$		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	$-0.568011\pi$
$953$	−404.148			−0.424080			−0.212040	+	0.977261i	$0.568011\pi$
$954$		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	0.566053	−	0.824369i	$-0.308470\pi$
$971$	−250.824	−	250.824i	−0.258315	−	0.258315i	−0.824369	+	0.566053i	$0.808470\pi$
$972$		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	$0.294279\pi$
$977$			1559.92i			1.59665i	−0.602230	+	0.798323i	$0.705721\pi$
$978$		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	$-0.373300\pi$
$983$	762.047			0.775226			0.387613	+	0.921822i	$0.373300\pi$
$984$		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.161.14		32
3.2	odd	2	inner	1152.3.j.a.161.3		32
4.3	odd	2		1152.3.j.b.161.14		32
8.3	odd	2		576.3.j.a.17.3		32
8.5	even	2		144.3.j.a.125.9	yes	32
12.11	even	2		1152.3.j.b.161.3		32
16.3	odd	4		576.3.j.a.305.14		32
16.5	even	4	inner	1152.3.j.a.737.3		32
16.11	odd	4		1152.3.j.b.737.3		32
16.13	even	4		144.3.j.a.53.8	✓	32
24.5	odd	2		144.3.j.a.125.8	yes	32
24.11	even	2		576.3.j.a.17.14		32
48.5	odd	4	inner	1152.3.j.a.737.14		32
48.11	even	4		1152.3.j.b.737.14		32
48.29	odd	4		144.3.j.a.53.9	yes	32
48.35	even	4		576.3.j.a.305.3		32

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