LMFDB - Embedded newform 1152.3.j.b.737.1

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			737.1
Character	$\chi$	$=$	1152.737
Dual form			1152.3.j.b.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.08688 + 6.08688i) q^{5} +9.40026i q^{7} +O(q^{10})$	$q+(-6.08688 + 6.08688i) q^{5} +9.40026i q^{7} +(-11.9531 + 11.9531i) q^{11} +(-3.47284 + 3.47284i) q^{13} +28.5398i q^{17} +(-3.08620 + 3.08620i) q^{19} +2.57437 q^{23} -49.1003i q^{25} +(3.49628 + 3.49628i) q^{29} +21.0909 q^{31} +(-57.2183 - 57.2183i) q^{35} +(13.2847 + 13.2847i) q^{37} +11.2365 q^{41} +(8.19150 + 8.19150i) q^{43} -17.2381i q^{47} -39.3649 q^{49} +(-30.5182 + 30.5182i) q^{53} -145.514i q^{55} +(-14.7367 + 14.7367i) q^{59} +(-39.3037 + 39.3037i) q^{61} -42.2775i q^{65} +(79.6470 - 79.6470i) q^{67} +73.1921 q^{71} +1.23261i q^{73} +(-112.362 - 112.362i) q^{77} +110.191 q^{79} +(13.7800 + 13.7800i) q^{83} +(-173.718 - 173.718i) q^{85} +39.2555 q^{89} +(-32.6456 - 32.6456i) q^{91} -37.5706i q^{95} -109.383 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

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

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
$5$	−6.08688	+	6.08688i	−1.21738	+	1.21738i	−0.248829	+	0.968547i	$0.580046\pi$
$5$	−6.08688	+	6.08688i	−1.21738	+	1.21738i	−0.968547	+	0.248829i	$0.919954\pi$
$6$		0			0
$7$			9.40026i			1.34289i	0.741053	+	0.671447i	$0.234327\pi$
$7$			9.40026i			1.34289i	−0.741053	+	0.671447i	$0.765673\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−11.9531	+	11.9531i	−1.08665	+	1.08665i	−0.0907749	+	0.995871i	$0.528934\pi$
$11$	−11.9531	+	11.9531i	−1.08665	+	1.08665i	−0.995871	+	0.0907749i	$0.971066\pi$
$12$		0			0
$13$	−3.47284	+	3.47284i	−0.267142	+	0.267142i	−0.827947	−	0.560806i	$-0.810491\pi$
$13$	−3.47284	+	3.47284i	−0.267142	+	0.267142i	0.560806	+	0.827947i	$0.310491\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			28.5398i			1.67881i	0.543506	+	0.839405i	$0.317097\pi$
$17$			28.5398i			1.67881i	−0.543506	+	0.839405i	$0.682903\pi$
$18$		0			0
$19$	−3.08620	+	3.08620i	−0.162431	+	0.162431i	−0.783643	−	0.621212i	$-0.786641\pi$
$19$	−3.08620	+	3.08620i	−0.162431	+	0.162431i	0.621212	+	0.783643i	$0.286641\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.57437			0.111929			0.0559645	−	0.998433i	$-0.482177\pi$
$23$	2.57437			0.111929			0.0559645	+	0.998433i	$0.482177\pi$
$24$		0			0
$25$		−	49.1003i		−	1.96401i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.49628	+	3.49628i	0.120561	+	0.120561i	0.764813	−	0.644252i	$-0.222831\pi$
$29$	3.49628	+	3.49628i	0.120561	+	0.120561i	−0.644252	+	0.764813i	$0.722831\pi$
$30$		0			0
$31$	21.0909			0.680351			0.340176	−	0.940362i	$-0.389513\pi$
$31$	21.0909			0.680351			0.340176	+	0.940362i	$0.389513\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−57.2183	−	57.2183i	−1.63481	−	1.63481i
$36$		0			0
$37$	13.2847	+	13.2847i	0.359047	+	0.359047i	0.863462	−	0.504414i	$-0.168292\pi$
$37$	13.2847	+	13.2847i	0.359047	+	0.359047i	−0.504414	+	0.863462i	$0.668292\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.2365			0.274060			0.137030	−	0.990567i	$-0.456244\pi$
$41$	11.2365			0.274060			0.137030	+	0.990567i	$0.456244\pi$
$42$		0			0
$43$	8.19150	+	8.19150i	0.190500	+	0.190500i	0.795912	−	0.605412i	$-0.206992\pi$
$43$	8.19150	+	8.19150i	0.190500	+	0.190500i	−0.605412	+	0.795912i	$0.706992\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	17.2381i		−	0.366769i	−0.983041	−	0.183384i	$-0.941295\pi$
$47$		−	17.2381i		−	0.366769i	0.983041	−	0.183384i	$-0.0587053\pi$
$48$		0			0
$49$	−39.3649			−0.803364
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−30.5182	+	30.5182i	−0.575816	+	0.575816i	−0.933748	−	0.357932i	$-0.883482\pi$
$53$	−30.5182	+	30.5182i	−0.575816	+	0.575816i	0.357932	+	0.933748i	$0.383482\pi$
$54$		0			0
$55$		−	145.514i		−	2.64572i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−14.7367	+	14.7367i	−0.249774	+	0.249774i	−0.820878	−	0.571104i	$-0.806515\pi$
$59$	−14.7367	+	14.7367i	−0.249774	+	0.249774i	0.571104	+	0.820878i	$0.306515\pi$
$60$		0			0
$61$	−39.3037	+	39.3037i	−0.644323	+	0.644323i	−0.951615	−	0.307292i	$-0.900577\pi$
$61$	−39.3037	+	39.3037i	−0.644323	+	0.644323i	0.307292	+	0.951615i	$0.400577\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	42.2775i		−	0.650424i
$66$		0			0
$67$	79.6470	−	79.6470i	1.18876	−	1.18876i	0.211351	−	0.977410i	$-0.432214\pi$
$67$	79.6470	−	79.6470i	1.18876	−	1.18876i	0.977410	−	0.211351i	$-0.0677863\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	73.1921			1.03087			0.515437	−	0.856927i	$-0.327630\pi$
$71$	73.1921			1.03087			0.515437	+	0.856927i	$0.327630\pi$
$72$		0			0
$73$			1.23261i			0.0168851i	0.999964	+	0.00844256i	$0.00268738\pi$
$73$			1.23261i			0.0168851i	−0.999964	+	0.00844256i	$0.997313\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−112.362	−	112.362i	−1.45925	−	1.45925i
$78$		0			0
$79$	110.191			1.39482			0.697411	−	0.716672i	$-0.254335\pi$
$79$	110.191			1.39482			0.697411	+	0.716672i	$0.254335\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.7800	+	13.7800i	0.166025	+	0.166025i	0.785229	−	0.619205i	$-0.212545\pi$
$83$	13.7800	+	13.7800i	0.166025	+	0.166025i	−0.619205	+	0.785229i	$0.712545\pi$
$84$		0			0
$85$	−173.718	−	173.718i	−2.04374	−	2.04374i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	39.2555			0.441073			0.220537	−	0.975379i	$-0.429219\pi$
$89$	39.2555			0.441073			0.220537	+	0.975379i	$0.429219\pi$
$90$		0			0
$91$	−32.6456	−	32.6456i	−0.358743	−	0.358743i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	37.5706i		−	0.395480i
$96$		0			0
$97$	−109.383			−1.12766			−0.563830	−	0.825891i	$-0.690673\pi$
$97$	−109.383			−1.12766			−0.563830	+	0.825891i	$0.690673\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−102.843	+	102.843i	−1.01824	+	1.01824i	−0.0184123	+	0.999830i	$0.505861\pi$
$101$	−102.843	+	102.843i	−1.01824	+	1.01824i	−0.999830	+	0.0184123i	$0.994139\pi$
$102$		0			0
$103$		−	66.0152i		−	0.640924i	−0.947261	−	0.320462i	$-0.896162\pi$
$103$		−	66.0152i		−	0.640924i	0.947261	−	0.320462i	$-0.103838\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	87.3982	−	87.3982i	0.816805	−	0.816805i	−0.168838	−	0.985644i	$-0.554002\pi$
$107$	87.3982	−	87.3982i	0.816805	−	0.816805i	0.985644	+	0.168838i	$0.0540016\pi$
$108$		0			0
$109$	−0.493298	+	0.493298i	−0.00452567	+	0.00452567i	−0.709366	−	0.704840i	$-0.751019\pi$
$109$	−0.493298	+	0.493298i	−0.00452567	+	0.00452567i	0.704840	+	0.709366i	$0.251019\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			181.667i			1.60768i	0.594848	+	0.803838i	$0.297212\pi$
$113$			181.667i			1.60768i	−0.594848	+	0.803838i	$0.702788\pi$
$114$		0			0
$115$	−15.6699	+	15.6699i	−0.136260	+	0.136260i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−268.281			−2.25447
$120$		0			0
$121$		−	164.754i		−	1.36160i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	146.696	+	146.696i	1.17356	+	1.17356i
$126$		0			0
$127$	−188.223			−1.48207			−0.741035	−	0.671466i	$-0.765665\pi$
$127$	−188.223			−1.48207			−0.741035	+	0.671466i	$0.765665\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−79.5518	−	79.5518i	−0.607266	−	0.607266i	0.334965	−	0.942231i	$-0.391275\pi$
$131$	−79.5518	−	79.5518i	−0.607266	−	0.607266i	−0.942231	+	0.334965i	$0.891275\pi$
$132$		0			0
$133$	−29.0110	−	29.0110i	−0.218128	−	0.218128i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−204.894			−1.49557			−0.747787	−	0.663939i	$-0.768883\pi$
$137$	−204.894			−1.49557			−0.747787	+	0.663939i	$0.768883\pi$
$138$		0			0
$139$	153.586	+	153.586i	1.10494	+	1.10494i	0.993806	+	0.111133i	$0.0354478\pi$
$139$	153.586	+	153.586i	1.10494	+	1.10494i	0.111133	+	0.993806i	$0.464552\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	83.0225i		−	0.580577i
$144$		0			0
$145$	−42.5629			−0.293537
$146$		0			0
$147$		0			0
$148$		0			0
$149$	24.6527	−	24.6527i	0.165455	−	0.165455i	−0.619523	−	0.784978i	$-0.712674\pi$
$149$	24.6527	−	24.6527i	0.165455	−	0.165455i	0.784978	+	0.619523i	$0.212674\pi$
$150$		0			0
$151$			56.2047i			0.372216i	0.982529	+	0.186108i	$0.0595875\pi$
$151$			56.2047i			0.372216i	−0.982529	+	0.186108i	$0.940412\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−128.378	+	128.378i	−0.828244	+	0.828244i
$156$		0			0
$157$	152.225	−	152.225i	0.969589	−	0.969589i	−0.0299624	−	0.999551i	$-0.509539\pi$
$157$	152.225	−	152.225i	0.969589	−	0.969589i	0.999551	+	0.0299624i	$0.00953875\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			24.1997i			0.150309i
$162$		0			0
$163$	147.071	−	147.071i	0.902276	−	0.902276i	−0.0933563	−	0.995633i	$-0.529760\pi$
$163$	147.071	−	147.071i	0.902276	−	0.902276i	0.995633	+	0.0933563i	$0.0297596\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−100.239			−0.600232			−0.300116	−	0.953903i	$-0.597025\pi$
$167$	−100.239			−0.600232			−0.300116	+	0.953903i	$0.597025\pi$
$168$		0			0
$169$			144.879i			0.857271i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	179.056	+	179.056i	1.03501	+	1.03501i	0.999365	+	0.0356433i	$0.0113480\pi$
$173$	179.056	+	179.056i	1.03501	+	1.03501i	0.0356433	+	0.999365i	$0.488652\pi$
$174$		0			0
$175$	461.555			2.63746
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.84637	+	8.84637i	0.0494211	+	0.0494211i	0.731385	−	0.681964i	$-0.238874\pi$
$179$	8.84637	+	8.84637i	0.0494211	+	0.0494211i	−0.681964	+	0.731385i	$0.738874\pi$
$180$		0			0
$181$	135.353	+	135.353i	0.747807	+	0.747807i	0.974067	−	0.226260i	$-0.0726499\pi$
$181$	135.353	+	135.353i	0.747807	+	0.747807i	−0.226260	+	0.974067i	$0.572650\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−161.725			−0.874191
$186$		0			0
$187$	−341.139	−	341.139i	−1.82427	−	1.82427i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			117.914i			0.617351i	0.951167	+	0.308675i	$0.0998856\pi$
$191$			117.914i			0.617351i	−0.951167	+	0.308675i	$0.900114\pi$
$192$		0			0
$193$	65.1940			0.337793			0.168896	−	0.985634i	$-0.445980\pi$
$193$	65.1940			0.337793			0.168896	+	0.985634i	$0.445980\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	26.9481	−	26.9481i	0.136793	−	0.136793i	−0.635395	−	0.772187i	$-0.719163\pi$
$197$	26.9481	−	26.9481i	0.136793	−	0.136793i	0.772187	+	0.635395i	$0.219163\pi$
$198$		0			0
$199$			290.806i			1.46133i	0.682734	+	0.730667i	$0.260791\pi$
$199$			290.806i			1.46133i	−0.682734	+	0.730667i	$0.739209\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−32.8660	+	32.8660i	−0.161901	+	0.161901i
$204$		0			0
$205$	−68.3951	+	68.3951i	−0.333634	+	0.333634i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	73.7793i		−	0.353011i
$210$		0			0
$211$	143.972	−	143.972i	0.682334	−	0.682334i	−0.278192	−	0.960526i	$-0.589735\pi$
$211$	143.972	−	143.972i	0.682334	−	0.682334i	0.960526	+	0.278192i	$0.0897351\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−99.7214			−0.463821
$216$		0			0
$217$			198.260i			0.913640i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−99.1141	−	99.1141i	−0.448480	−	0.448480i
$222$		0			0
$223$	−200.870			−0.900761			−0.450381	−	0.892837i	$-0.648712\pi$
$223$	−200.870			−0.900761			−0.450381	+	0.892837i	$0.648712\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−37.3792	−	37.3792i	−0.164666	−	0.164666i	0.619964	−	0.784630i	$-0.287147\pi$
$227$	−37.3792	−	37.3792i	−0.164666	−	0.164666i	−0.784630	+	0.619964i	$0.787147\pi$
$228$		0			0
$229$	−101.806	−	101.806i	−0.444569	−	0.444569i	0.448975	−	0.893544i	$-0.351789\pi$
$229$	−101.806	−	101.806i	−0.444569	−	0.444569i	−0.893544	+	0.448975i	$0.851789\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−80.6982			−0.346344			−0.173172	−	0.984892i	$-0.555402\pi$
$233$	−80.6982			−0.346344			−0.173172	+	0.984892i	$0.555402\pi$
$234$		0			0
$235$	104.927	+	104.927i	0.446496	+	0.446496i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	301.011i		−	1.25946i	−0.776814	−	0.629730i	$-0.783166\pi$
$239$		−	301.011i		−	1.25946i	0.776814	−	0.629730i	$-0.216834\pi$
$240$		0			0
$241$	159.477			0.661732			0.330866	−	0.943678i	$-0.392659\pi$
$241$	159.477			0.661732			0.330866	+	0.943678i	$0.392659\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	239.609	−	239.609i	0.977997	−	0.977997i
$246$		0			0
$247$		−	21.4357i		−	0.0867844i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	291.951	−	291.951i	1.16315	−	1.16315i	0.179368	−	0.983782i	$-0.442595\pi$
$251$	291.951	−	291.951i	1.16315	−	1.16315i	0.983782	−	0.179368i	$-0.0574053\pi$
$252$		0			0
$253$	−30.7717	+	30.7717i	−0.121627	+	0.121627i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	136.807i		−	0.532323i	−0.963929	−	0.266161i	$-0.914245\pi$
$257$		−	136.807i		−	0.532323i	0.963929	−	0.266161i	$-0.0857554\pi$
$258$		0			0
$259$	−124.880	+	124.880i	−0.482162	+	0.482162i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	449.794			1.71024			0.855121	−	0.518429i	$-0.173483\pi$
$263$	449.794			1.71024			0.855121	+	0.518429i	$0.173483\pi$
$264$		0			0
$265$		−	371.522i		−	1.40197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−77.2032	−	77.2032i	−0.287001	−	0.287001i	0.548892	−	0.835893i	$-0.315050\pi$
$269$	−77.2032	−	77.2032i	−0.287001	−	0.287001i	−0.835893	+	0.548892i	$0.815050\pi$
$270$		0			0
$271$	−496.378			−1.83165			−0.915827	−	0.401574i	$-0.868463\pi$
$271$	−496.378			−1.83165			−0.915827	+	0.401574i	$0.868463\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	586.901	+	586.901i	2.13419	+	2.13419i
$276$		0			0
$277$	25.7238	+	25.7238i	0.0928658	+	0.0928658i	0.752014	−	0.659148i	$-0.229083\pi$
$277$	25.7238	+	25.7238i	0.0928658	+	0.0928658i	−0.659148	+	0.752014i	$0.729083\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	380.796			1.35515			0.677573	−	0.735456i	$-0.263032\pi$
$281$	380.796			1.35515			0.677573	+	0.735456i	$0.263032\pi$
$282$		0			0
$283$	−36.5558	−	36.5558i	−0.129172	−	0.129172i	0.639565	−	0.768737i	$-0.279115\pi$
$283$	−36.5558	−	36.5558i	−0.129172	−	0.129172i	−0.768737	+	0.639565i	$0.779115\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			105.626i			0.368034i
$288$		0			0
$289$	−525.519			−1.81841
$290$		0			0
$291$		0			0
$292$		0			0
$293$	119.828	−	119.828i	0.408970	−	0.408970i	−0.472409	−	0.881379i	$-0.656615\pi$
$293$	119.828	−	119.828i	0.408970	−	0.408970i	0.881379	+	0.472409i	$0.156615\pi$
$294$		0			0
$295$		−	179.401i		−	0.608138i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.94036	+	8.94036i	−0.0299009	+	0.0299009i
$300$		0			0
$301$	−77.0023	+	77.0023i	−0.255821	+	0.255821i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	478.474i		−	1.56877i
$306$		0			0
$307$	−368.177	+	368.177i	−1.19927	+	1.19927i	−0.224889	+	0.974384i	$0.572202\pi$
$307$	−368.177	+	368.177i	−1.19927	+	1.19927i	−0.974384	+	0.224889i	$0.927798\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	82.3959			0.264939			0.132469	−	0.991187i	$-0.457709\pi$
$311$	82.3959			0.264939			0.132469	+	0.991187i	$0.457709\pi$
$312$		0			0
$313$			368.385i			1.17695i	0.808516	+	0.588474i	$0.200271\pi$
$313$			368.385i			1.17695i	−0.808516	+	0.588474i	$0.799729\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−185.382	−	185.382i	−0.584800	−	0.584800i	0.351419	−	0.936218i	$-0.385699\pi$
$317$	−185.382	−	185.382i	−0.584800	−	0.584800i	−0.936218	+	0.351419i	$0.885699\pi$
$318$		0			0
$319$	−83.5829			−0.262015
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−88.0794	−	88.0794i	−0.272692	−	0.272692i
$324$		0			0
$325$	170.517	+	170.517i	0.524669	+	0.524669i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	162.043			0.492532
$330$		0			0
$331$	209.678	+	209.678i	0.633467	+	0.633467i	0.948936	−	0.315469i	$-0.102162\pi$
$331$	209.678	+	209.678i	0.633467	+	0.633467i	−0.315469	+	0.948936i	$0.602162\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			969.604i			2.89434i
$336$		0			0
$337$	417.353			1.23844			0.619218	−	0.785219i	$-0.287450\pi$
$337$	417.353			1.23844			0.619218	+	0.785219i	$0.287450\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−252.102	+	252.102i	−0.739301	+	0.739301i
$342$		0			0
$343$			90.5729i			0.264061i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	99.3706	−	99.3706i	0.286371	−	0.286371i	−0.549273	−	0.835643i	$-0.685095\pi$
$347$	99.3706	−	99.3706i	0.286371	−	0.286371i	0.835643	+	0.549273i	$0.185095\pi$
$348$		0			0
$349$	9.95933	−	9.95933i	0.0285368	−	0.0285368i	−0.692694	−	0.721231i	$-0.743577\pi$
$349$	9.95933	−	9.95933i	0.0285368	−	0.0285368i	0.721231	+	0.692694i	$0.243577\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			509.355i			1.44293i	0.692451	+	0.721465i	$0.256531\pi$
$353$			509.355i			1.44293i	−0.692451	+	0.721465i	$0.743469\pi$
$354$		0			0
$355$	−445.511	+	445.511i	−1.25496	+	1.25496i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	401.516			1.11843			0.559214	−	0.829023i	$-0.311103\pi$
$359$	401.516			1.11843			0.559214	+	0.829023i	$0.311103\pi$
$360$		0			0
$361$			341.951i			0.947232i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.50278	−	7.50278i	−0.0205556	−	0.0205556i
$366$		0			0
$367$	338.335			0.921894			0.460947	−	0.887428i	$-0.347510\pi$
$367$	338.335			0.921894			0.460947	+	0.887428i	$0.347510\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−286.879	−	286.879i	−0.773259	−	0.773259i
$372$		0			0
$373$	342.158	+	342.158i	0.917314	+	0.917314i	0.996833	−	0.0795197i	$-0.0253387\pi$
$373$	342.158	+	342.158i	0.917314	+	0.917314i	−0.0795197	+	0.996833i	$0.525339\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.2841			−0.0644140
$378$		0			0
$379$	400.188	+	400.188i	1.05591	+	1.05591i	0.998342	+	0.0575645i	$0.0183335\pi$
$379$	400.188	+	400.188i	1.05591	+	1.05591i	0.0575645	+	0.998342i	$0.481667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	595.773i		−	1.55554i	−0.628548	−	0.777771i	$-0.716350\pi$
$383$		−	595.773i		−	1.55554i	0.628548	−	0.777771i	$-0.283650\pi$
$384$		0			0
$385$	1367.87			3.55292
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−83.6879	+	83.6879i	−0.215136	+	0.215136i	−0.806445	−	0.591309i	$-0.798611\pi$
$389$	−83.6879	+	83.6879i	−0.215136	+	0.215136i	0.591309	+	0.806445i	$0.298611\pi$
$390$		0			0
$391$			73.4718i			0.187908i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−670.719	+	670.719i	−1.69802	+	1.69802i
$396$		0			0
$397$	311.065	−	311.065i	0.783539	−	0.783539i	−0.196887	−	0.980426i	$-0.563083\pi$
$397$	311.065	−	311.065i	0.783539	−	0.783539i	0.980426	+	0.196887i	$0.0630832\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	279.299i		−	0.696505i	−0.937401	−	0.348253i	$-0.886775\pi$
$401$		−	279.299i		−	0.696505i	0.937401	−	0.348253i	$-0.113225\pi$
$402$		0			0
$403$	−73.2453	+	73.2453i	−0.181750	+	0.181750i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−317.588			−0.780315
$408$		0			0
$409$			446.244i			1.09106i	0.838091	+	0.545530i	$0.183672\pi$
$409$			446.244i			1.09106i	−0.838091	+	0.545530i	$0.816328\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−138.529	−	138.529i	−0.335420	−	0.335420i
$414$		0			0
$415$	−167.755			−0.404229
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−331.372	−	331.372i	−0.790864	−	0.790864i	0.190770	−	0.981635i	$-0.438901\pi$
$419$	−331.372	−	331.372i	−0.790864	−	0.790864i	−0.981635	+	0.190770i	$0.938901\pi$
$420$		0			0
$421$	74.8932	+	74.8932i	0.177893	+	0.177893i	0.790437	−	0.612543i	$-0.209854\pi$
$421$	74.8932	+	74.8932i	0.177893	+	0.177893i	−0.612543	+	0.790437i	$0.709854\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1401.31			3.29720
$426$		0			0
$427$	−369.465	−	369.465i	−0.865258	−	0.865258i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			433.486i			1.00577i	0.864354	+	0.502884i	$0.167728\pi$
$431$			433.486i			1.00577i	−0.864354	+	0.502884i	$0.832272\pi$
$432$		0			0
$433$	−279.207			−0.644820			−0.322410	−	0.946600i	$-0.604493\pi$
$433$	−279.207			−0.644820			−0.322410	+	0.946600i	$0.604493\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.94500	+	7.94500i	−0.0181808	+	0.0181808i
$438$		0			0
$439$			564.253i			1.28531i	0.766154	+	0.642657i	$0.222168\pi$
$439$			564.253i			1.28531i	−0.766154	+	0.642657i	$0.777832\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−39.2113	+	39.2113i	−0.0885131	+	0.0885131i	−0.749977	−	0.661464i	$-0.769935\pi$
$443$	−39.2113	+	39.2113i	−0.0885131	+	0.0885131i	0.661464	+	0.749977i	$0.269935\pi$
$444$		0			0
$445$	−238.944	+	238.944i	−0.536952	+	0.536952i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	277.135i		−	0.617228i	−0.951187	−	0.308614i	$-0.900135\pi$
$449$		−	277.135i		−	0.617228i	0.951187	−	0.308614i	$-0.0998651\pi$
$450$		0			0
$451$	−134.311	+	134.311i	−0.297806	+	0.297806i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	397.420			0.873450
$456$		0			0
$457$		−	631.173i		−	1.38112i	−0.723274	−	0.690561i	$-0.757364\pi$
$457$		−	631.173i		−	1.38112i	0.723274	−	0.690561i	$-0.242636\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	494.475	+	494.475i	1.07261	+	1.07261i	0.997149	+	0.0754643i	$0.0240439\pi$
$461$	494.475	+	494.475i	1.07261	+	1.07261i	0.0754643	+	0.997149i	$0.475956\pi$
$462$		0			0
$463$	−394.376			−0.851784			−0.425892	−	0.904774i	$-0.640040\pi$
$463$	−394.376			−0.851784			−0.425892	+	0.904774i	$0.640040\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	214.554	+	214.554i	0.459430	+	0.459430i	0.898468	−	0.439039i	$-0.144681\pi$
$467$	214.554	+	214.554i	0.459430	+	0.459430i	−0.439039	+	0.898468i	$0.644681\pi$
$468$		0			0
$469$	748.702	+	748.702i	1.59638	+	1.59638i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−195.828			−0.414012
$474$		0			0
$475$	151.533	+	151.533i	0.319017	+	0.319017i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	636.488i		−	1.32879i	−0.747383	−	0.664393i	$-0.768690\pi$
$479$		−	636.488i		−	1.32879i	0.747383	−	0.664393i	$-0.231310\pi$
$480$		0			0
$481$	−92.2716			−0.191833
$482$		0			0
$483$		0			0
$484$		0			0
$485$	665.802	−	665.802i	1.37279	−	1.37279i
$486$		0			0
$487$		−	550.556i		−	1.13051i	−0.824918	−	0.565253i	$-0.808779\pi$
$487$		−	550.556i		−	1.13051i	0.824918	−	0.565253i	$-0.191221\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	192.006	−	192.006i	0.391051	−	0.391051i	−0.484011	−	0.875062i	$-0.660821\pi$
$491$	192.006	−	192.006i	0.391051	−	0.391051i	0.875062	+	0.484011i	$0.160821\pi$
$492$		0			0
$493$	−99.7832	+	99.7832i	−0.202400	+	0.202400i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			688.024i			1.38435i
$498$		0			0
$499$	156.794	−	156.794i	0.314215	−	0.314215i	−0.532325	−	0.846540i	$-0.678681\pi$
$499$	156.794	−	156.794i	0.314215	−	0.314215i	0.846540	+	0.532325i	$0.178681\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	88.5520			0.176048			0.0880239	−	0.996118i	$-0.471945\pi$
$503$	88.5520			0.176048			0.0880239	+	0.996118i	$0.471945\pi$
$504$		0			0
$505$		−	1251.98i		−	2.47917i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−430.228	−	430.228i	−0.845242	−	0.845242i	0.144293	−	0.989535i	$-0.453909\pi$
$509$	−430.228	−	430.228i	−0.845242	−	0.845242i	−0.989535	+	0.144293i	$0.953909\pi$
$510$		0			0
$511$	−11.5869			−0.0226749
$512$		0			0
$513$		0			0
$514$		0			0
$515$	401.827	+	401.827i	0.780246	+	0.780246i
$516$		0			0
$517$	206.049	+	206.049i	0.398548	+	0.398548i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−889.748			−1.70777			−0.853885	−	0.520462i	$-0.825760\pi$
$521$	−889.748			−1.70777			−0.853885	+	0.520462i	$0.825760\pi$
$522$		0			0
$523$	134.417	+	134.417i	0.257012	+	0.257012i	0.823838	−	0.566826i	$-0.191829\pi$
$523$	134.417	+	134.417i	0.257012	+	0.257012i	−0.566826	+	0.823838i	$0.691829\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			601.929i			1.14218i
$528$		0			0
$529$	−522.373			−0.987472
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−39.0225	+	39.0225i	−0.0732129	+	0.0732129i
$534$		0			0
$535$			1063.96i			1.98872i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	470.532	−	470.532i	0.872973	−	0.872973i
$540$		0			0
$541$	−309.758	+	309.758i	−0.572565	+	0.572565i	−0.932844	−	0.360279i	$-0.882681\pi$
$541$	−309.758	+	309.758i	−0.572565	+	0.572565i	0.360279	+	0.932844i	$0.382681\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	6.00529i		−	0.0110189i
$546$		0			0
$547$	−609.425	+	609.425i	−1.11412	+	1.11412i	−0.121535	+	0.992587i	$0.538782\pi$
$547$	−609.425	+	609.425i	−1.11412	+	1.11412i	−0.992587	+	0.121535i	$0.961218\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21.5804			−0.0391659
$552$		0			0
$553$			1035.82i			1.87310i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−605.408	−	605.408i	−1.08691	−	1.08691i	−0.995845	−	0.0910624i	$-0.970974\pi$
$557$	−605.408	−	605.408i	−1.08691	−	1.08691i	−0.0910624	−	0.995845i	$-0.529026\pi$
$558$		0			0
$559$	−56.8956			−0.101781
$560$		0			0
$561$		0			0
$562$		0			0
$563$	68.2515	+	68.2515i	0.121228	+	0.121228i	0.765118	−	0.643890i	$-0.222681\pi$
$563$	68.2515	+	68.2515i	0.121228	+	0.121228i	−0.643890	+	0.765118i	$0.722681\pi$
$564$		0			0
$565$	−1105.79	−	1105.79i	−1.95715	−	1.95715i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−758.298			−1.33269			−0.666343	−	0.745646i	$-0.732141\pi$
$569$	−758.298			−1.33269			−0.666343	+	0.745646i	$0.732141\pi$
$570$		0			0
$571$	−663.635	−	663.635i	−1.16223	−	1.16223i	−0.983986	−	0.178248i	$-0.942957\pi$
$571$	−663.635	−	663.635i	−1.16223	−	1.16223i	−0.178248	−	0.983986i	$-0.557043\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	126.402i		−	0.219830i
$576$		0			0
$577$	−68.8744			−0.119366			−0.0596832	−	0.998217i	$-0.519009\pi$
$577$	−68.8744			−0.119366			−0.0596832	+	0.998217i	$0.519009\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−129.536	+	129.536i	−0.222953	+	0.222953i
$582$		0			0
$583$		−	729.575i		−	1.25142i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−720.353	+	720.353i	−1.22718	+	1.22718i	−0.262151	+	0.965027i	$0.584432\pi$
$587$	−720.353	+	720.353i	−1.22718	+	1.22718i	−0.965027	+	0.262151i	$0.915568\pi$
$588$		0			0
$589$	−65.0906	+	65.0906i	−0.110510	+	0.110510i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	585.630i		−	0.987572i	−0.869583	−	0.493786i	$-0.835613\pi$
$593$		−	585.630i		−	0.987572i	0.869583	−	0.493786i	$-0.164387\pi$
$594$		0			0
$595$	1633.00	−	1633.00i	2.74453	−	2.74453i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1006.31			1.67999			0.839995	−	0.542594i	$-0.182558\pi$
$599$	1006.31			1.67999			0.839995	+	0.542594i	$0.182558\pi$
$600$		0			0
$601$			534.875i			0.889976i	0.895537	+	0.444988i	$0.146792\pi$
$601$			534.875i			0.889976i	−0.895537	+	0.444988i	$0.853208\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1002.84	+	1002.84i	1.65758	+	1.65758i
$606$		0			0
$607$	219.896			0.362267			0.181133	−	0.983459i	$-0.442023\pi$
$607$	219.896			0.362267			0.181133	+	0.983459i	$0.442023\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	59.8653	+	59.8653i	0.0979792	+	0.0979792i
$612$		0			0
$613$	−389.389	−	389.389i	−0.635219	−	0.635219i	0.314153	−	0.949372i	$-0.398279\pi$
$613$	−389.389	−	389.389i	−0.635219	−	0.635219i	−0.949372	+	0.314153i	$0.898279\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−261.466			−0.423771			−0.211885	−	0.977295i	$-0.567960\pi$
$617$	−261.466			−0.423771			−0.211885	+	0.977295i	$0.567960\pi$
$618$		0			0
$619$	−275.915	−	275.915i	−0.445743	−	0.445743i	0.448193	−	0.893937i	$-0.352068\pi$
$619$	−275.915	−	275.915i	−0.445743	−	0.445743i	−0.893937	+	0.448193i	$0.852068\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			369.012i			0.592315i
$624$		0			0
$625$	−558.331			−0.893329
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−379.144	+	379.144i	−0.602772	+	0.602772i
$630$		0			0
$631$			237.889i			0.377003i	0.982073	+	0.188501i	$0.0603630\pi$
$631$			237.889i			0.377003i	−0.982073	+	0.188501i	$0.939637\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1145.69	−	1145.69i	1.80424	−	1.80424i
$636$		0			0
$637$	136.708	−	136.708i	0.214612	−	0.214612i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	858.416i		−	1.33918i	−0.742730	−	0.669591i	$-0.766469\pi$
$641$		−	858.416i		−	1.33918i	0.742730	−	0.669591i	$-0.233531\pi$
$642$		0			0
$643$	−643.410	+	643.410i	−1.00064	+	1.00064i	−0.000638349		1.00000i	$0.500203\pi$
$643$	−643.410	+	643.410i	−1.00064	+	1.00064i	−1.00000		0.000638349i	$0.999797\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−597.840			−0.924019			−0.462009	−	0.886875i	$-0.652871\pi$
$647$	−597.840			−0.924019			−0.462009	+	0.886875i	$0.652871\pi$
$648$		0			0
$649$		−	352.298i		−	0.542832i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−91.3671	−	91.3671i	−0.139919	−	0.139919i	0.633678	−	0.773597i	$-0.281544\pi$
$653$	−91.3671	−	91.3671i	−0.139919	−	0.139919i	−0.773597	+	0.633678i	$0.781544\pi$
$654$		0			0
$655$	968.445			1.47854
$656$		0			0
$657$		0			0
$658$		0			0
$659$	445.487	+	445.487i	0.676004	+	0.676004i	0.959094	−	0.283089i	$-0.0913593\pi$
$659$	445.487	+	445.487i	0.676004	+	0.676004i	−0.283089	+	0.959094i	$0.591359\pi$
$660$		0			0
$661$	−387.304	−	387.304i	−0.585937	−	0.585937i	0.350592	−	0.936528i	$-0.385981\pi$
$661$	−387.304	−	387.304i	−0.585937	−	0.585937i	−0.936528	+	0.350592i	$0.885981\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	353.174			0.531088
$666$		0			0
$667$	9.00071	+	9.00071i	0.0134943	+	0.0134943i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	939.603i		−	1.40030i
$672$		0			0
$673$	−845.703			−1.25662			−0.628309	−	0.777964i	$-0.716252\pi$
$673$	−845.703			−1.25662			−0.628309	+	0.777964i	$0.716252\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	311.755	−	311.755i	0.460495	−	0.460495i	−0.438323	−	0.898818i	$-0.644427\pi$
$677$	311.755	−	311.755i	0.460495	−	0.460495i	0.898818	+	0.438323i	$0.144427\pi$
$678$		0			0
$679$		−	1028.23i		−	1.51433i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−214.466	+	214.466i	−0.314005	+	0.314005i	−0.846459	−	0.532454i	$-0.821270\pi$
$683$	−214.466	+	214.466i	−0.314005	+	0.314005i	0.532454	+	0.846459i	$0.321270\pi$
$684$		0			0
$685$	1247.16	−	1247.16i	1.82068	−	1.82068i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	211.970i		−	0.307649i
$690$		0			0
$691$	812.184	−	812.184i	1.17537	−	1.17537i	0.194465	−	0.980909i	$-0.437703\pi$
$691$	812.184	−	812.184i	1.17537	−	1.17537i	0.980909	−	0.194465i	$-0.0622972\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1869.72			−2.69025
$696$		0			0
$697$			320.686i			0.460095i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	105.563	+	105.563i	0.150589	+	0.150589i	0.778381	−	0.627792i	$-0.216041\pi$
$701$	105.563	+	105.563i	0.150589	+	0.150589i	−0.627792	+	0.778381i	$0.716041\pi$
$702$		0			0
$703$	−81.9987			−0.116641
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−966.746	−	966.746i	−1.36739	−	1.36739i
$708$		0			0
$709$	106.375	+	106.375i	0.150035	+	0.150035i	0.778134	−	0.628099i	$-0.216167\pi$
$709$	106.375	+	106.375i	0.150035	+	0.150035i	−0.628099	+	0.778134i	$0.716167\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	54.2957			0.0761510
$714$		0			0
$715$	505.348	+	505.348i	0.706781	+	0.706781i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1119.16i		−	1.55656i	−0.627919	−	0.778278i	$-0.716093\pi$
$719$		−	1119.16i		−	1.55656i	0.627919	−	0.778278i	$-0.283907\pi$
$720$		0			0
$721$	620.560			0.860693
$722$		0			0
$723$		0			0
$724$		0			0
$725$	171.668	−	171.668i	0.236784	−	0.236784i
$726$		0			0
$727$			334.759i			0.460466i	0.973136	+	0.230233i	$0.0739489\pi$
$727$			334.759i			0.460466i	−0.973136	+	0.230233i	$0.926051\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−233.784	+	233.784i	−0.319814	+	0.319814i
$732$		0			0
$733$	−97.8371	+	97.8371i	−0.133475	+	0.133475i	−0.770688	−	0.637213i	$-0.780087\pi$
$733$	−97.8371	+	97.8371i	−0.133475	+	0.133475i	0.637213	+	0.770688i	$0.280087\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1904.06i			2.58353i
$738$		0			0
$739$	249.678	−	249.678i	0.337860	−	0.337860i	−0.517702	−	0.855561i	$-0.673212\pi$
$739$	249.678	−	249.678i	0.337860	−	0.337860i	0.855561	+	0.517702i	$0.173212\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1176.88			−1.58396			−0.791981	−	0.610545i	$-0.790950\pi$
$743$	−1176.88			−1.58396			−0.791981	+	0.610545i	$0.790950\pi$
$744$		0			0
$745$			300.117i			0.402841i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	821.565	+	821.565i	1.09688	+	1.09688i
$750$		0			0
$751$	−909.942			−1.21164			−0.605821	−	0.795601i	$-0.707155\pi$
$751$	−909.942			−1.21164			−0.605821	+	0.795601i	$0.707155\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−342.111	−	342.111i	−0.453128	−	0.453128i
$756$		0			0
$757$	−418.277	−	418.277i	−0.552545	−	0.552545i	0.374629	−	0.927175i	$-0.377770\pi$
$757$	−418.277	−	418.277i	−0.552545	−	0.552545i	−0.927175	+	0.374629i	$0.877770\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−670.931			−0.881644			−0.440822	−	0.897594i	$-0.645313\pi$
$761$	−670.931			−0.881644			−0.440822	+	0.897594i	$0.645313\pi$
$762$		0			0
$763$	−4.63713	−	4.63713i	−0.00607750	−	0.00607750i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	102.356i		−	0.133450i
$768$		0			0
$769$	1054.59			1.37138			0.685689	−	0.727895i	$-0.259501\pi$
$769$	1054.59			1.37138			0.685689	+	0.727895i	$0.259501\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−877.952	+	877.952i	−1.13577	+	1.13577i	−0.146573	+	0.989200i	$0.546824\pi$
$773$	−877.952	+	877.952i	−1.13577	+	1.13577i	−0.989200	+	0.146573i	$0.953176\pi$
$774$		0			0
$775$		−	1035.57i		−	1.33622i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−34.6780	+	34.6780i	−0.0445160	+	0.0445160i
$780$		0			0
$781$	−874.873	+	874.873i	−1.12020	+	1.12020i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1853.16i			2.36071i
$786$		0			0
$787$	18.3967	−	18.3967i	0.0233758	−	0.0233758i	−0.695322	−	0.718698i	$-0.744738\pi$
$787$	18.3967	−	18.3967i	0.0233758	−	0.0233758i	0.718698	+	0.695322i	$0.244738\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1707.72			−2.15894
$792$		0			0
$793$		−	272.991i		−	0.344251i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−27.8374	−	27.8374i	−0.0349278	−	0.0349278i	0.689427	−	0.724355i	$-0.257862\pi$
$797$	−27.8374	−	27.8374i	−0.0349278	−	0.0349278i	−0.724355	+	0.689427i	$0.757862\pi$
$798$		0			0
$799$	491.973			0.615736
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.7336	−	14.7336i	−0.0183482	−	0.0183482i
$804$		0			0
$805$	−147.301	−	147.301i	−0.182982	−	0.182982i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−416.938			−0.515374			−0.257687	−	0.966228i	$-0.582960\pi$
$809$	−416.938			−0.515374			−0.257687	+	0.966228i	$0.582960\pi$
$810$		0			0
$811$	699.525	+	699.525i	0.862546	+	0.862546i	0.991633	−	0.129087i	$-0.0412047\pi$
$811$	699.525	+	699.525i	0.862546	+	0.862546i	−0.129087	+	0.991633i	$0.541205\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			1790.41i			2.19682i
$816$		0			0
$817$	−50.5612			−0.0618864
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−200.751	+	200.751i	−0.244520	+	0.244520i	−0.818717	−	0.574197i	$-0.805314\pi$
$821$	−200.751	+	200.751i	−0.244520	+	0.244520i	0.574197	+	0.818717i	$0.305314\pi$
$822$		0			0
$823$			159.123i			0.193345i	0.995316	+	0.0966724i	$0.0308199\pi$
$823$			159.123i			0.193345i	−0.995316	+	0.0966724i	$0.969180\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	790.803	−	790.803i	0.956231	−	0.956231i	−0.0428505	−	0.999081i	$-0.513644\pi$
$827$	790.803	−	790.803i	0.956231	−	0.956231i	0.999081	+	0.0428505i	$0.0136439\pi$
$828$		0			0
$829$	827.815	−	827.815i	0.998570	−	0.998570i	−0.00142868	−	0.999999i	$-0.500455\pi$
$829$	827.815	−	827.815i	0.998570	−	0.998570i	0.999999	+	0.00142868i	$0.000454762\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1123.46i		−	1.34870i
$834$		0			0
$835$	610.141	−	610.141i	0.730708	−	0.730708i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	65.9441			0.0785985			0.0392992	−	0.999227i	$-0.487487\pi$
$839$	65.9441			0.0785985			0.0392992	+	0.999227i	$0.487487\pi$
$840$		0			0
$841$		−	816.552i		−	0.970930i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−881.860	−	881.860i	−1.04362	−	1.04362i
$846$		0			0
$847$	1548.73			1.82849
$848$		0			0
$849$		0			0
$850$		0			0
$851$	34.1998	+	34.1998i	0.0401878	+	0.0401878i
$852$		0			0
$853$	22.8417	+	22.8417i	0.0267780	+	0.0267780i	0.720369	−	0.693591i	$-0.243972\pi$
$853$	22.8417	+	22.8417i	0.0267780	+	0.0267780i	−0.693591	+	0.720369i	$0.743972\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	630.654			0.735886			0.367943	−	0.929848i	$-0.380062\pi$
$857$	630.654			0.735886			0.367943	+	0.929848i	$0.380062\pi$
$858$		0			0
$859$	161.861	+	161.861i	0.188430	+	0.188430i	0.795017	−	0.606587i	$-0.207462\pi$
$859$	161.861	+	161.861i	0.188430	+	0.188430i	−0.606587	+	0.795017i	$0.707462\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1070.23i		−	1.24013i	−0.784549	−	0.620066i	$-0.787106\pi$
$863$		−	1070.23i		−	1.24013i	0.784549	−	0.620066i	$-0.212894\pi$
$864$		0			0
$865$	−2179.79			−2.51999
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1317.12	+	1317.12i	−1.51568	+	1.51568i
$870$		0			0
$871$			553.203i			0.635135i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1378.98	+	1378.98i	−1.57597	+	1.57597i
$876$		0			0
$877$	−773.573	+	773.573i	−0.882067	+	0.882067i	−0.993745	−	0.111677i	$-0.964378\pi$
$877$	−773.573	+	773.573i	−0.882067	+	0.882067i	0.111677	+	0.993745i	$0.464378\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			495.482i			0.562408i	0.959648	+	0.281204i	$0.0907338\pi$
$881$			495.482i			0.562408i	−0.959648	+	0.281204i	$0.909266\pi$
$882$		0			0
$883$	−119.779	+	119.779i	−0.135650	+	0.135650i	−0.771671	−	0.636021i	$-0.780579\pi$
$883$	−119.779	+	119.779i	−0.135650	+	0.135650i	0.636021	+	0.771671i	$0.280579\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	895.932			1.01007			0.505035	−	0.863099i	$-0.331480\pi$
$887$	895.932			1.01007			0.505035	+	0.863099i	$0.331480\pi$
$888$		0			0
$889$		−	1769.34i		−	1.99026i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	53.2003	+	53.2003i	0.0595748	+	0.0595748i
$894$		0			0
$895$	−107.694			−0.120328
$896$		0			0
$897$		0			0
$898$		0			0
$899$	73.7397	+	73.7397i	0.0820242	+	0.0820242i
$900$		0			0
$901$	−870.984	−	870.984i	−0.966686	−	0.966686i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1647.76			−1.82073
$906$		0			0
$907$	−852.347	−	852.347i	−0.939743	−	0.939743i	0.0585418	−	0.998285i	$-0.481355\pi$
$907$	−852.347	−	852.347i	−0.939743	−	0.939743i	−0.998285	+	0.0585418i	$0.981355\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			748.289i			0.821393i	0.911772	+	0.410697i	$0.134714\pi$
$911$			748.289i			0.821393i	−0.911772	+	0.410697i	$0.865286\pi$
$912$		0			0
$913$	−329.429			−0.360820
$914$		0			0
$915$		0			0
$916$		0			0
$917$	747.807	−	747.807i	0.815493	−	0.815493i
$918$		0			0
$919$		−	85.7074i		−	0.0932616i	−0.998912	−	0.0466308i	$-0.985152\pi$
$919$		−	85.7074i		−	0.0932616i	0.998912	−	0.0466308i	$-0.0148484\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−254.184	+	254.184i	−0.275389	+	0.275389i
$924$		0			0
$925$	652.285	−	652.285i	0.705173	−	0.705173i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	119.970i		−	0.129139i	−0.997913	−	0.0645696i	$-0.979433\pi$
$929$		−	119.970i		−	0.129139i	0.997913	−	0.0645696i	$-0.0205675\pi$
$930$		0			0
$931$	121.488	−	121.488i	0.130492	−	0.130492i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4152.95			4.44166
$936$		0			0
$937$			631.867i			0.674351i	0.941442	+	0.337176i	$0.109472\pi$
$937$			631.867i			0.674351i	−0.941442	+	0.337176i	$0.890528\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	370.590	+	370.590i	0.393825	+	0.393825i	0.876048	−	0.482223i	$-0.160171\pi$
$941$	370.590	+	370.590i	0.393825	+	0.393825i	−0.482223	+	0.876048i	$0.660171\pi$
$942$		0			0
$943$	28.9268			0.0306753
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.2368	+	16.2368i	0.0171455	+	0.0171455i	0.715628	−	0.698482i	$-0.246141\pi$
$947$	16.2368	+	16.2368i	0.0171455	+	0.0171455i	−0.698482	+	0.715628i	$0.746141\pi$
$948$		0			0
$949$	−4.28067	−	4.28067i	−0.00451072	−	0.00451072i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1216.35			1.27634			0.638168	−	0.769897i	$-0.279693\pi$
$953$	1216.35			1.27634			0.638168	+	0.769897i	$0.279693\pi$
$954$		0			0
$955$	−717.728	−	717.728i	−0.751548	−	0.751548i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1926.05i		−	2.00840i
$960$		0			0
$961$	−516.174			−0.537122
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−396.828	+	396.828i	−0.411221	+	0.411221i
$966$		0			0
$967$		−	781.540i		−	0.808211i	−0.914712	−	0.404105i	$-0.867583\pi$
$967$		−	781.540i		−	0.808211i	0.914712	−	0.404105i	$-0.132417\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.3700	+	15.3700i	−0.0158290	+	0.0158290i	−0.714977	−	0.699148i	$-0.753563\pi$
$971$	−15.3700	+	15.3700i	−0.0158290	+	0.0158290i	0.699148	+	0.714977i	$0.253563\pi$
$972$		0			0
$973$	−1443.75	+	1443.75i	−1.48381	+	1.48381i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	806.751i		−	0.825743i	−0.910789	−	0.412871i	$-0.864526\pi$
$977$		−	806.751i		−	0.825743i	0.910789	−	0.412871i	$-0.135474\pi$
$978$		0			0
$979$	−469.226	+	469.226i	−0.479291	+	0.479291i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1060.81			−1.07916			−0.539579	−	0.841935i	$-0.681417\pi$
$983$	−1060.81			−1.07916			−0.539579	+	0.841935i	$0.681417\pi$
$984$		0			0
$985$			328.060i			0.333056i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21.0879	+	21.0879i	0.0213225	+	0.0213225i
$990$		0			0
$991$	−1036.12			−1.04553			−0.522766	−	0.852476i	$-0.675100\pi$
$991$	−1036.12			−1.04553			−0.522766	+	0.852476i	$0.675100\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1770.10	−	1770.10i	−1.77899	−	1.77899i
$996$		0			0
$997$	−44.6898	−	44.6898i	−0.0448242	−	0.0448242i	0.684339	−	0.729164i	$-0.260091\pi$
$997$	−44.6898	−	44.6898i	−0.0448242	−	0.0448242i	−0.729164	+	0.684339i	$0.760091\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.j.b.737.1		32
3.2	odd	2	inner	1152.3.j.b.737.16		32
4.3	odd	2		1152.3.j.a.737.1		32
8.3	odd	2		144.3.j.a.53.14	yes	32
8.5	even	2		576.3.j.a.305.16		32
12.11	even	2		1152.3.j.a.737.16		32
16.3	odd	4		1152.3.j.a.161.16		32
16.5	even	4		576.3.j.a.17.1		32
16.11	odd	4		144.3.j.a.125.3	yes	32
16.13	even	4	inner	1152.3.j.b.161.16		32
24.5	odd	2		576.3.j.a.305.1		32
24.11	even	2		144.3.j.a.53.3	✓	32
48.5	odd	4		576.3.j.a.17.16		32
48.11	even	4		144.3.j.a.125.14	yes	32
48.29	odd	4	inner	1152.3.j.b.161.1		32
48.35	even	4		1152.3.j.a.161.1		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.3	✓	32	24.11	even	2
144.3.j.a.53.14	yes	32	8.3	odd	2
144.3.j.a.125.3	yes	32	16.11	odd	4
144.3.j.a.125.14	yes	32	48.11	even	4
576.3.j.a.17.1		32	16.5	even	4
576.3.j.a.17.16		32	48.5	odd	4
576.3.j.a.305.1		32	24.5	odd	2
576.3.j.a.305.16		32	8.5	even	2
1152.3.j.a.161.1		32	48.35	even	4
1152.3.j.a.161.16		32	16.3	odd	4
1152.3.j.a.737.1		32	4.3	odd	2
1152.3.j.a.737.16		32	12.11	even	2
1152.3.j.b.161.1		32	48.29	odd	4	inner
1152.3.j.b.161.16		32	16.13	even	4	inner
1152.3.j.b.737.1		32	1.1	even	1	trivial
1152.3.j.b.737.16		32	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+(-6.08688 + 6.08688i) q^{5} +9.40026i q^{7} +O(q^{10})\)	\(q+(-6.08688 + 6.08688i) q^{5} +9.40026i q^{7} +(-11.9531 + 11.9531i) q^{11} +(-3.47284 + 3.47284i) q^{13} +28.5398i q^{17} +(-3.08620 + 3.08620i) q^{19} +2.57437 q^{23} -49.1003i q^{25} +(3.49628 + 3.49628i) q^{29} +21.0909 q^{31} +(-57.2183 - 57.2183i) q^{35} +(13.2847 + 13.2847i) q^{37} +11.2365 q^{41} +(8.19150 + 8.19150i) q^{43} -17.2381i q^{47} -39.3649 q^{49} +(-30.5182 + 30.5182i) q^{53} -145.514i q^{55} +(-14.7367 + 14.7367i) q^{59} +(-39.3037 + 39.3037i) q^{61} -42.2775i q^{65} +(79.6470 - 79.6470i) q^{67} +73.1921 q^{71} +1.23261i q^{73} +(-112.362 - 112.362i) q^{77} +110.191 q^{79} +(13.7800 + 13.7800i) q^{83} +(-173.718 - 173.718i) q^{85} +39.2555 q^{89} +(-32.6456 - 32.6456i) q^{91} -37.5706i q^{95} -109.383 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more