LMFDB - Embedded newform 576.2.k.c.433.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,2,Mod(145,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("576.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	576.k (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:	$4.59938315643$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.629407744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 2x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.2
Root			$-1.38255 - 0.297594i$ of defining polynomial
Character	$\chi$	$=$	576.433
Dual form			576.2.k.c.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.595188 - 0.595188i) q^{5} -1.64575i q^{7} +O(q^{10})$	$q+(-0.595188 - 0.595188i) q^{5} -1.64575i q^{7} +(3.36028 + 3.36028i) q^{11} +(2.64575 - 2.64575i) q^{13} -5.53019 q^{17} +(3.64575 - 3.64575i) q^{19} -4.33981i q^{23} -4.29150i q^{25} +(6.12538 - 6.12538i) q^{29} +5.64575 q^{31} +(-0.979531 + 0.979531i) q^{35} +(-0.645751 - 0.645751i) q^{37} +7.91094i q^{41} +(0.354249 + 0.354249i) q^{43} -9.10132 q^{47} +4.29150 q^{49} +(4.93500 + 4.93500i) q^{53} -4.00000i q^{55} +(4.33981 + 4.33981i) q^{59} +(-0.645751 + 0.645751i) q^{61} -3.14944 q^{65} +(-4.00000 + 4.00000i) q^{67} -13.4411i q^{71} +3.29150i q^{73} +(5.53019 - 5.53019i) q^{77} -9.64575 q^{79} +(-3.36028 + 3.36028i) q^{83} +(3.29150 + 3.29150i) q^{85} +2.38075i q^{89} +(-4.35425 - 4.35425i) q^{91} -4.33981 q^{95} -10.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{19} + 24 q^{31} + 16 q^{37} + 24 q^{43} - 8 q^{49} + 16 q^{61} - 32 q^{67} - 56 q^{79} - 16 q^{85} - 56 q^{91}+O(q^{100}) $ 8 * q + 8 * q^19 + 24 * q^31 + 16 * q^37 + 24 * q^43 - 8 * q^49 + 16 * q^61 - 32 * q^67 - 56 * q^79 - 16 * q^85 - 56 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$	−0.595188	−	0.595188i	−0.266176	−	0.266176i	0.561381	−	0.827557i	$-0.310270\pi$
$5$	−0.595188	−	0.595188i	−0.266176	−	0.266176i	−0.827557	+	0.561381i	$0.810270\pi$
$6$		0			0
$7$		−	1.64575i		−	0.622036i	−0.950404	−	0.311018i	$-0.899330\pi$
$7$		−	1.64575i		−	0.622036i	0.950404	−	0.311018i	$-0.100670\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.36028	+	3.36028i	1.01316	+	1.01316i	0.999912	+	0.0132513i	$0.00421814\pi$
$11$	3.36028	+	3.36028i	1.01316	+	1.01316i	0.0132513	+	0.999912i	$0.495782\pi$
$12$		0			0
$13$	2.64575	−	2.64575i	0.733799	−	0.733799i	−0.237571	−	0.971370i	$-0.576351\pi$
$13$	2.64575	−	2.64575i	0.733799	−	0.733799i	0.971370	+	0.237571i	$0.0763512\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.53019			−1.34127			−0.670634	−	0.741788i	$-0.733978\pi$
$17$	−5.53019			−1.34127			−0.670634	+	0.741788i	$0.733978\pi$
$18$		0			0
$19$	3.64575	−	3.64575i	0.836393	−	0.836393i	−0.151989	−	0.988382i	$-0.548568\pi$
$19$	3.64575	−	3.64575i	0.836393	−	0.836393i	0.988382	+	0.151989i	$0.0485680\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.33981i		−	0.904914i	−0.891786	−	0.452457i	$-0.850548\pi$
$23$		−	4.33981i		−	0.904914i	0.891786	−	0.452457i	$-0.149452\pi$
$24$		0			0
$25$		−	4.29150i		−	0.858301i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.12538	−	6.12538i	1.13745	−	1.13745i	0.148549	−	0.988905i	$-0.452540\pi$
$29$	6.12538	−	6.12538i	1.13745	−	1.13745i	0.988905	−	0.148549i	$-0.0474603\pi$
$30$		0			0
$31$	5.64575			1.01401			0.507003	−	0.861944i	$-0.330753\pi$
$31$	5.64575			1.01401			0.507003	+	0.861944i	$0.330753\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.979531	+	0.979531i	−0.165571	+	0.165571i
$36$		0			0
$37$	−0.645751	−	0.645751i	−0.106161	−	0.106161i	0.652031	−	0.758192i	$-0.273917\pi$
$37$	−0.645751	−	0.645751i	−0.106161	−	0.106161i	−0.758192	+	0.652031i	$0.773917\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.91094i			1.23548i	0.786382	+	0.617741i	$0.211952\pi$
$41$			7.91094i			1.23548i	−0.786382	+	0.617741i	$0.788048\pi$
$42$		0			0
$43$	0.354249	+	0.354249i	0.0540224	+	0.0540224i	0.733602	−	0.679579i	$-0.237838\pi$
$43$	0.354249	+	0.354249i	0.0540224	+	0.0540224i	−0.679579	+	0.733602i	$0.737838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.10132			−1.32756			−0.663782	−	0.747926i	$-0.731050\pi$
$47$	−9.10132			−1.32756			−0.663782	+	0.747926i	$0.731050\pi$
$48$		0			0
$49$	4.29150			0.613072
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.93500	+	4.93500i	0.677875	+	0.677875i	0.959519	−	0.281644i	$-0.0908798\pi$
$53$	4.93500	+	4.93500i	0.677875	+	0.677875i	−0.281644	+	0.959519i	$0.590880\pi$
$54$		0			0
$55$		−	4.00000i		−	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.33981	+	4.33981i	0.564996	+	0.564996i	0.930722	−	0.365727i	$-0.119179\pi$
$59$	4.33981	+	4.33981i	0.564996	+	0.564996i	−0.365727	+	0.930722i	$0.619179\pi$
$60$		0			0
$61$	−0.645751	+	0.645751i	−0.0826800	+	0.0826800i	−0.747237	−	0.664557i	$-0.768620\pi$
$61$	−0.645751	+	0.645751i	−0.0826800	+	0.0826800i	0.664557	+	0.747237i	$0.268620\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.14944			−0.390640
$66$		0			0
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	−0.907889	−	0.419211i	$-0.862307\pi$
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	0.419211	+	0.907889i	$0.362307\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.4411i		−	1.59517i	−0.603207	−	0.797584i	$-0.706111\pi$
$71$		−	13.4411i		−	1.59517i	0.603207	−	0.797584i	$-0.293889\pi$
$72$		0			0
$73$			3.29150i			0.385241i	0.981273	+	0.192621i	$0.0616987\pi$
$73$			3.29150i			0.385241i	−0.981273	+	0.192621i	$0.938301\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.53019	−	5.53019i	0.630224	−	0.630224i
$78$		0			0
$79$	−9.64575			−1.08523			−0.542616	−	0.839981i	$-0.682566\pi$
$79$	−9.64575			−1.08523			−0.542616	+	0.839981i	$0.682566\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.36028	+	3.36028i	−0.368839	+	0.368839i	−0.867054	−	0.498215i	$-0.833989\pi$
$83$	−3.36028	+	3.36028i	−0.368839	+	0.368839i	0.498215	+	0.867054i	$0.333989\pi$
$84$		0			0
$85$	3.29150	+	3.29150i	0.357014	+	0.357014i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.38075i			0.252359i	0.992007	+	0.126180i	$0.0402716\pi$
$89$			2.38075i			0.252359i	−0.992007	+	0.126180i	$0.959728\pi$
$90$		0			0
$91$	−4.35425	−	4.35425i	−0.456449	−	0.456449i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.33981			−0.445256
$96$		0			0
$97$	−10.5830			−1.07454			−0.537271	−	0.843410i	$-0.680545\pi$
$97$	−10.5830			−1.07454			−0.537271	+	0.843410i	$0.680545\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.595188	−	0.595188i	−0.0592234	−	0.0592234i	0.676875	−	0.736098i	$-0.263334\pi$
$101$	−0.595188	−	0.595188i	−0.0592234	−	0.0592234i	−0.736098	+	0.676875i	$0.763334\pi$
$102$		0			0
$103$			16.9373i			1.66888i	0.551101	+	0.834439i	$0.314208\pi$
$103$			16.9373i			1.66888i	−0.551101	+	0.834439i	$0.685792\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.38075	+	2.38075i	0.230156	+	0.230156i	0.812758	−	0.582602i	$-0.197965\pi$
$107$	2.38075	+	2.38075i	0.230156	+	0.230156i	−0.582602	+	0.812758i	$0.697965\pi$
$108$		0			0
$109$	−6.64575	+	6.64575i	−0.636548	+	0.636548i	−0.949702	−	0.313155i	$-0.898614\pi$
$109$	−6.64575	+	6.64575i	−0.636548	+	0.636548i	0.313155	+	0.949702i	$0.398614\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−2.58301	+	2.58301i	−0.240866	+	0.240866i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			9.10132i			0.834316i
$120$		0			0
$121$			11.5830i			1.05300i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.53019	+	5.53019i	−0.494635	+	0.494635i
$126$		0			0
$127$	−0.937254			−0.0831678			−0.0415839	−	0.999135i	$-0.513240\pi$
$127$	−0.937254			−0.0831678			−0.0415839	+	0.999135i	$0.513240\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.72057	−	6.72057i	0.587179	−	0.587179i	−0.349688	−	0.936866i	$-0.613712\pi$
$131$	6.72057	−	6.72057i	0.587179	−	0.587179i	0.936866	+	0.349688i	$0.113712\pi$
$132$		0			0
$133$	−6.00000	−	6.00000i	−0.520266	−	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.2917i		−	0.879279i	−0.898174	−	0.439639i	$-0.855106\pi$
$137$		−	10.2917i		−	0.879279i	0.898174	−	0.439639i	$-0.144894\pi$
$138$		0			0
$139$	14.5830	+	14.5830i	1.23691	+	1.23691i	0.961256	+	0.275659i	$0.0888960\pi$
$139$	14.5830	+	14.5830i	1.23691	+	1.23691i	0.275659	+	0.961256i	$0.411104\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.7809			1.48692
$144$		0			0
$145$	−7.29150			−0.605526
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.6556	−	11.6556i	−0.954861	−	0.954861i	0.0441630	−	0.999024i	$-0.485938\pi$
$149$	−11.6556	−	11.6556i	−0.954861	−	0.954861i	−0.999024	+	0.0441630i	$0.985938\pi$
$150$		0			0
$151$		−	10.3542i		−	0.842617i	−0.906917	−	0.421308i	$-0.861571\pi$
$151$		−	10.3542i		−	0.842617i	0.906917	−	0.421308i	$-0.138429\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.36028	−	3.36028i	−0.269904	−	0.269904i
$156$		0			0
$157$	−6.64575	+	6.64575i	−0.530389	+	0.530389i	−0.920688	−	0.390299i	$-0.872372\pi$
$157$	−6.64575	+	6.64575i	−0.530389	+	0.530389i	0.390299	+	0.920688i	$0.372372\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.14226			−0.562889
$162$		0			0
$163$	−11.6458	+	11.6458i	−0.912166	+	0.912166i	−0.996442	−	0.0842767i	$-0.973142\pi$
$163$	−11.6458	+	11.6458i	−0.912166	+	0.912166i	0.0842767	+	0.996442i	$0.473142\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.7809i			1.37593i	0.725743	+	0.687965i	$0.241496\pi$
$167$			17.7809i			1.37593i	−0.725743	+	0.687965i	$0.758504\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.93500	+	4.93500i	−0.375201	+	0.375201i	−0.869367	−	0.494166i	$-0.835473\pi$
$173$	−4.93500	+	4.93500i	−0.375201	+	0.375201i	0.494166	+	0.869367i	$0.335473\pi$
$174$		0			0
$175$	−7.06275			−0.533893
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.38075	+	2.38075i	−0.177946	+	0.177946i	−0.790460	−	0.612514i	$-0.790158\pi$
$179$	−2.38075	+	2.38075i	−0.177946	+	0.177946i	0.612514	+	0.790460i	$0.290158\pi$
$180$		0			0
$181$	−0.645751	−	0.645751i	−0.0479983	−	0.0479983i	0.682700	−	0.730699i	$-0.260805\pi$
$181$	−0.645751	−	0.645751i	−0.0479983	−	0.0479983i	−0.730699	+	0.682700i	$0.760805\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.768687i			0.0565150i
$186$		0			0
$187$	−18.5830	−	18.5830i	−1.35892	−	1.35892i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.10132			0.658548			0.329274	−	0.944234i	$-0.393196\pi$
$191$	9.10132			0.658548			0.329274	+	0.944234i	$0.393196\pi$
$192$		0			0
$193$	11.8745			0.854746			0.427373	−	0.904075i	$-0.359439\pi$
$193$	11.8745			0.854746			0.427373	+	0.904075i	$0.359439\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.50613	+	8.50613i	0.606037	+	0.606037i	0.941908	−	0.335871i	$-0.109031\pi$
$197$	8.50613	+	8.50613i	0.606037	+	0.606037i	−0.335871	+	0.941908i	$0.609031\pi$
$198$		0			0
$199$			13.6458i			0.967322i	0.875256	+	0.483661i	$0.160693\pi$
$199$			13.6458i			0.967322i	−0.875256	+	0.483661i	$0.839307\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.0808	−	10.0808i	−0.707537	−	0.707537i
$204$		0			0
$205$	4.70850	−	4.70850i	0.328856	−	0.328856i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.5015			1.69481
$210$		0			0
$211$	14.5830	−	14.5830i	1.00394	−	1.00394i	0.00394326	−	0.999992i	$-0.498745\pi$
$211$	14.5830	−	14.5830i	1.00394	−	1.00394i	0.999992	−	0.00394326i	$-0.00125518\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.421689i		−	0.0287590i
$216$		0			0
$217$		−	9.29150i		−	0.630748i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−14.6315	+	14.6315i	−0.984222	+	0.984222i
$222$		0			0
$223$	14.3542			0.961232			0.480616	−	0.876931i	$-0.340413\pi$
$223$	14.3542			0.961232			0.480616	+	0.876931i	$0.340413\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.40122	+	1.40122i	−0.0930023	+	0.0930023i	−0.752077	−	0.659075i	$-0.770948\pi$
$227$	−1.40122	+	1.40122i	−0.0930023	+	0.0930023i	0.659075	+	0.752077i	$0.270948\pi$
$228$		0			0
$229$	17.9373	+	17.9373i	1.18533	+	1.18533i	0.978345	+	0.206982i	$0.0663643\pi$
$229$	17.9373	+	17.9373i	1.18533	+	1.18533i	0.206982	+	0.978345i	$0.433636\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	19.7400i		−	1.29321i	−0.762825	−	0.646606i	$-0.776188\pi$
$233$		−	19.7400i		−	1.29321i	0.762825	−	0.646606i	$-0.223812\pi$
$234$		0			0
$235$	5.41699	+	5.41699i	0.353366	+	0.353366i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.0194			−0.842158			−0.421079	−	0.907024i	$-0.638348\pi$
$239$	−13.0194			−0.842158			−0.421079	+	0.907024i	$0.638348\pi$
$240$		0			0
$241$	−1.29150			−0.0831930			−0.0415965	−	0.999134i	$-0.513244\pi$
$241$	−1.29150			−0.0831930			−0.0415965	+	0.999134i	$0.513244\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.55425	−	2.55425i	−0.163185	−	0.163185i
$246$		0			0
$247$		−	19.2915i		−	1.22749i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.7605	−	18.7605i	−1.18415	−	1.18415i	−0.978659	−	0.205492i	$-0.934121\pi$
$251$	−18.7605	−	18.7605i	−1.18415	−	1.18415i	−0.205492	−	0.978659i	$-0.565879\pi$
$252$		0			0
$253$	14.5830	−	14.5830i	0.916826	−	0.916826i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	29.2630			1.82538			0.912688	−	0.408656i	$-0.134002\pi$
$257$	29.2630			1.82538			0.912688	+	0.408656i	$0.134002\pi$
$258$		0			0
$259$	−1.06275	+	1.06275i	−0.0660358	+	0.0660358i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			8.67963i			0.535209i	0.963529	+	0.267604i	$0.0862320\pi$
$263$			8.67963i			0.535209i	−0.963529	+	0.267604i	$0.913768\pi$
$264$		0			0
$265$		−	5.87451i		−	0.360868i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.4652	+	10.4652i	−0.638074	+	0.638074i	−0.950080	−	0.312006i	$-0.898999\pi$
$269$	−10.4652	+	10.4652i	−0.638074	+	0.638074i	0.312006	+	0.950080i	$0.398999\pi$
$270$		0			0
$271$	−6.35425			−0.385993			−0.192997	−	0.981199i	$-0.561821\pi$
$271$	−6.35425			−0.385993			−0.192997	+	0.981199i	$0.561821\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	14.4207	−	14.4207i	0.869599	−	0.869599i
$276$		0			0
$277$	−16.5203	−	16.5203i	−0.992606	−	0.992606i	0.00736669	−	0.999973i	$-0.497655\pi$
$277$	−16.5203	−	16.5203i	−0.992606	−	0.992606i	−0.999973	+	0.00736669i	$0.997655\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.38075i			0.142024i	0.997475	+	0.0710119i	$0.0226228\pi$
$281$			2.38075i			0.142024i	−0.997475	+	0.0710119i	$0.977377\pi$
$282$		0			0
$283$	2.58301	+	2.58301i	0.153544	+	0.153544i	0.779699	−	0.626155i	$-0.215372\pi$
$283$	2.58301	+	2.58301i	0.153544	+	0.153544i	−0.626155	+	0.779699i	$0.715372\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.0194			0.768513
$288$		0			0
$289$	13.5830			0.799000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.55425	−	2.55425i	−0.149221	−	0.149221i	0.628549	−	0.777770i	$-0.283649\pi$
$293$	−2.55425	−	2.55425i	−0.149221	−	0.149221i	−0.777770	+	0.628549i	$0.783649\pi$
$294$		0			0
$295$		−	5.16601i		−	0.300777i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11.4821	−	11.4821i	−0.664025	−	0.664025i
$300$		0			0
$301$	0.583005	−	0.583005i	0.0336039	−	0.0336039i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.768687			0.0440149
$306$		0			0
$307$	20.0000	−	20.0000i	1.14146	−	1.14146i	0.153277	−	0.988183i	$-0.451017\pi$
$307$	20.0000	−	20.0000i	1.14146	−	1.14146i	0.988183	−	0.153277i	$-0.0489827\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			8.67963i			0.492177i	0.969247	+	0.246088i	$0.0791453\pi$
$311$			8.67963i			0.492177i	−0.969247	+	0.246088i	$0.920855\pi$
$312$		0			0
$313$		−	9.29150i		−	0.525187i	−0.964907	−	0.262593i	$-0.915422\pi$
$313$		−	9.29150i		−	0.525187i	0.964907	−	0.262593i	$-0.0845778\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.595188	−	0.595188i	0.0334291	−	0.0334291i	−0.690195	−	0.723624i	$-0.742475\pi$
$317$	0.595188	−	0.595188i	0.0334291	−	0.0334291i	0.723624	+	0.690195i	$0.242475\pi$
$318$		0			0
$319$	41.1660			2.30485
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20.1617	+	20.1617i	−1.12183	+	1.12183i
$324$		0			0
$325$	−11.3542	−	11.3542i	−0.629820	−	0.629820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			14.9785i			0.825792i
$330$		0			0
$331$	8.00000	+	8.00000i	0.439720	+	0.439720i	0.891918	−	0.452198i	$-0.149360\pi$
$331$	8.00000	+	8.00000i	0.439720	+	0.439720i	−0.452198	+	0.891918i	$0.649360\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.76150			0.260149
$336$		0			0
$337$	4.70850			0.256488			0.128244	−	0.991743i	$-0.459066\pi$
$337$	4.70850			0.256488			0.128244	+	0.991743i	$0.459066\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.9713	+	18.9713i	1.02735	+	1.02735i
$342$		0			0
$343$		−	18.5830i		−	1.00339i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.36028	+	3.36028i	0.180389	+	0.180389i	0.791526	−	0.611136i	$-0.209287\pi$
$347$	3.36028	+	3.36028i	0.180389	+	0.180389i	−0.611136	+	0.791526i	$0.709287\pi$
$348$		0			0
$349$	3.22876	−	3.22876i	0.172831	−	0.172831i	−0.615391	−	0.788222i	$-0.711002\pi$
$349$	3.22876	−	3.22876i	0.172831	−	0.172831i	0.788222	+	0.615391i	$0.211002\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.14226			−0.380144			−0.190072	−	0.981770i	$-0.560872\pi$
$353$	−7.14226			−0.380144			−0.190072	+	0.981770i	$0.560872\pi$
$354$		0			0
$355$	−8.00000	+	8.00000i	−0.424596	+	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.76150i			0.251303i	0.992074	+	0.125651i	$0.0401020\pi$
$359$			4.76150i			0.251303i	−0.992074	+	0.125651i	$0.959898\pi$
$360$		0			0
$361$		−	7.58301i		−	0.399106i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.95906	−	1.95906i	0.102542	−	0.102542i
$366$		0			0
$367$	−34.8118			−1.81716			−0.908580	−	0.417712i	$-0.862832\pi$
$367$	−34.8118			−1.81716			−0.908580	+	0.417712i	$0.862832\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.12179	−	8.12179i	0.421662	−	0.421662i
$372$		0			0
$373$	11.9373	+	11.9373i	0.618088	+	0.618088i	0.945041	−	0.326953i	$-0.106022\pi$
$373$	11.9373	+	11.9373i	0.618088	+	0.618088i	−0.326953	+	0.945041i	$0.606022\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	32.4125i		−	1.66933i
$378$		0			0
$379$	−8.35425	−	8.35425i	−0.429129	−	0.429129i	0.459203	−	0.888332i	$-0.348135\pi$
$379$	−8.35425	−	8.35425i	−0.429129	−	0.429129i	−0.888332	+	0.459203i	$0.848135\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.10132			−0.465056			−0.232528	−	0.972590i	$-0.574700\pi$
$383$	−9.10132			−0.465056			−0.232528	+	0.972590i	$0.574700\pi$
$384$		0			0
$385$	−6.58301			−0.335501
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.7569	−	20.7569i	−1.05242	−	1.05242i	−0.998548	−	0.0538679i	$-0.982845\pi$
$389$	−20.7569	−	20.7569i	−1.05242	−	1.05242i	−0.0538679	−	0.998548i	$-0.517155\pi$
$390$		0			0
$391$			24.0000i			1.21373i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.74103	+	5.74103i	0.288863	+	0.288863i
$396$		0			0
$397$	8.06275	−	8.06275i	0.404658	−	0.404658i	−0.475213	−	0.879871i	$-0.657629\pi$
$397$	8.06275	−	8.06275i	0.404658	−	0.404658i	0.879871	+	0.475213i	$0.157629\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.5906			−0.828494			−0.414247	−	0.910165i	$-0.635955\pi$
$401$	−16.5906			−0.828494			−0.414247	+	0.910165i	$0.635955\pi$
$402$		0			0
$403$	14.9373	−	14.9373i	0.744078	−	0.744078i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.33981i		−	0.215117i
$408$		0			0
$409$			25.1660i			1.24438i	0.782867	+	0.622190i	$0.213757\pi$
$409$			25.1660i			1.24438i	−0.782867	+	0.622190i	$0.786243\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.14226	−	7.14226i	0.351447	−	0.351447i
$414$		0			0
$415$	4.00000			0.196352
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.7605	−	18.7605i	0.916509	−	0.916509i	−0.0802643	−	0.996774i	$-0.525576\pi$
$419$	18.7605	−	18.7605i	0.916509	−	0.916509i	0.996774	+	0.0802643i	$0.0255764\pi$
$420$		0			0
$421$	11.3542	+	11.3542i	0.553372	+	0.553372i	0.927412	−	0.374040i	$-0.122028\pi$
$421$	11.3542	+	11.3542i	0.553372	+	0.553372i	−0.374040	+	0.927412i	$0.622028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			23.7328i			1.15121i
$426$		0			0
$427$	1.06275	+	1.06275i	0.0514299	+	0.0514299i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	31.2221			1.50391			0.751957	−	0.659212i	$-0.229110\pi$
$431$	31.2221			1.50391			0.751957	+	0.659212i	$0.229110\pi$
$432$		0			0
$433$	−4.00000			−0.192228			−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000			−0.192228			−0.0961139	+	0.995370i	$0.530641\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.8219	−	15.8219i	−0.756863	−	0.756863i
$438$		0			0
$439$		−	0.479741i		−	0.0228968i	−0.999934	−	0.0114484i	$-0.996356\pi$
$439$		−	0.479741i		−	0.0228968i	0.999934	−	0.0114484i	$-0.00364422\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.5629	+	21.5629i	1.02449	+	1.02449i	0.999693	+	0.0247926i	$0.00789253\pi$
$443$	21.5629	+	21.5629i	1.02449	+	1.02449i	0.0247926	+	0.999693i	$0.492107\pi$
$444$		0			0
$445$	1.41699	−	1.41699i	0.0671720	−	0.0671720i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.44832			−0.445894			−0.222947	−	0.974831i	$-0.571568\pi$
$449$	−9.44832			−0.445894			−0.222947	+	0.974831i	$0.571568\pi$
$450$		0			0
$451$	−26.5830	+	26.5830i	−1.25174	+	1.25174i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			5.18319i			0.242992i
$456$		0			0
$457$		−	5.41699i		−	0.253396i	−0.991941	−	0.126698i	$-0.959562\pi$
$457$		−	5.41699i		−	0.253396i	0.991941	−	0.126698i	$-0.0404380\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.50613	+	8.50613i	−0.396170	+	0.396170i	−0.876880	−	0.480710i	$-0.840379\pi$
$461$	−8.50613	+	8.50613i	−0.396170	+	0.396170i	0.480710	+	0.876880i	$0.340379\pi$
$462$		0			0
$463$	−15.0627			−0.700025			−0.350013	−	0.936745i	$-0.613823\pi$
$463$	−15.0627			−0.700025			−0.350013	+	0.936745i	$0.613823\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.4810	+	25.4810i	−1.17912	+	1.17912i	−0.199154	+	0.979968i	$0.563819\pi$
$467$	−25.4810	+	25.4810i	−1.17912	+	1.17912i	−0.979968	+	0.199154i	$0.936181\pi$
$468$		0			0
$469$	6.58301	+	6.58301i	0.303975	+	0.303975i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.38075i			0.109467i
$474$		0			0
$475$	−15.6458	−	15.6458i	−0.717876	−	0.717876i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.1208			−1.01072			−0.505362	−	0.862908i	$-0.668641\pi$
$479$	−22.1208			−1.01072			−0.505362	+	0.862908i	$0.668641\pi$
$480$		0			0
$481$	−3.41699			−0.155802
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.29888	+	6.29888i	0.286017	+	0.286017i
$486$		0			0
$487$			8.22876i			0.372881i	0.982466	+	0.186440i	$0.0596951\pi$
$487$			8.22876i			0.372881i	−0.982466	+	0.186440i	$0.940305\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.4002	+	15.4002i	0.695001	+	0.695001i	0.963328	−	0.268327i	$-0.0864707\pi$
$491$	15.4002	+	15.4002i	0.695001	+	0.695001i	−0.268327	+	0.963328i	$0.586471\pi$
$492$		0			0
$493$	−33.8745	+	33.8745i	−1.52563	+	1.52563i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−22.1208			−0.992252
$498$		0			0
$499$	−10.5830	+	10.5830i	−0.473760	+	0.473760i	−0.903129	−	0.429369i	$-0.858736\pi$
$499$	−10.5830	+	10.5830i	−0.473760	+	0.473760i	0.429369	+	0.903129i	$0.358736\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	31.6438i		−	1.41093i	−0.708747	−	0.705463i	$-0.750739\pi$
$503$		−	31.6438i		−	1.41093i	0.708747	−	0.705463i	$-0.249261\pi$
$504$		0			0
$505$			0.708497i			0.0315277i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	24.6750	−	24.6750i	1.09370	−	1.09370i	0.0985706	−	0.995130i	$-0.468573\pi$
$509$	24.6750	−	24.6750i	1.09370	−	1.09370i	0.995130	−	0.0985706i	$-0.0314270\pi$
$510$		0			0
$511$	5.41699			0.239634
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.0808	−	10.0808i	0.444215	−	0.444215i
$516$		0			0
$517$	−30.5830	−	30.5830i	−1.34504	−	1.34504i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	14.2098i		−	0.622543i	−0.950321	−	0.311272i	$-0.899245\pi$
$521$		−	14.2098i		−	0.622543i	0.950321	−	0.311272i	$-0.100755\pi$
$522$		0			0
$523$	18.9373	+	18.9373i	0.828068	+	0.828068i	0.987249	−	0.159181i	$-0.0508854\pi$
$523$	18.9373	+	18.9373i	0.828068	+	0.828068i	−0.159181	+	0.987249i	$0.550885\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−31.2221			−1.36006
$528$		0			0
$529$	4.16601			0.181131
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20.9304	+	20.9304i	0.906596	+	0.906596i
$534$		0			0
$535$		−	2.83399i		−	0.122524i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	14.4207	+	14.4207i	0.621142	+	0.621142i
$540$		0			0
$541$	−28.5203	+	28.5203i	−1.22618	+	1.22618i	−0.260785	+	0.965397i	$0.583981\pi$
$541$	−28.5203	+	28.5203i	−1.22618	+	1.22618i	−0.965397	+	0.260785i	$0.916019\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.91094			0.338868
$546$		0			0
$547$	−14.9373	+	14.9373i	−0.638671	+	0.638671i	−0.950228	−	0.311557i	$-0.899150\pi$
$547$	−14.9373	+	14.9373i	−0.638671	+	0.638671i	0.311557	+	0.950228i	$0.399150\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	44.6632i		−	1.90272i
$552$		0			0
$553$			15.8745i			0.675053i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.7978	−	18.7978i	0.796489	−	0.796489i	−0.186051	−	0.982540i	$-0.559569\pi$
$557$	18.7978	−	18.7978i	0.796489	−	0.796489i	0.982540	+	0.186051i	$0.0595689\pi$
$558$		0			0
$559$	1.87451			0.0792832
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.31935	+	5.31935i	−0.224184	+	0.224184i	−0.810258	−	0.586074i	$-0.800673\pi$
$563$	−5.31935	+	5.31935i	−0.224184	+	0.224184i	0.586074	+	0.810258i	$0.300673\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			41.0921i			1.72267i	0.508038	+	0.861335i	$0.330371\pi$
$569$			41.0921i			1.72267i	−0.508038	+	0.861335i	$0.669629\pi$
$570$		0			0
$571$	−4.00000	−	4.00000i	−0.167395	−	0.167395i	0.618438	−	0.785833i	$-0.287766\pi$
$571$	−4.00000	−	4.00000i	−0.167395	−	0.167395i	−0.785833	+	0.618438i	$0.787766\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−18.6243			−0.776688
$576$		0			0
$577$	−39.0405			−1.62528			−0.812639	−	0.582767i	$-0.801970\pi$
$577$	−39.0405			−1.62528			−0.812639	+	0.582767i	$0.801970\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.53019	+	5.53019i	0.229431	+	0.229431i
$582$		0			0
$583$			33.1660i			1.37360i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.76150	−	4.76150i	−0.196528	−	0.196528i	0.601982	−	0.798510i	$-0.294378\pi$
$587$	−4.76150	−	4.76150i	−0.196528	−	0.196528i	−0.798510	+	0.601982i	$0.794378\pi$
$588$		0			0
$589$	20.5830	−	20.5830i	0.848108	−	0.848108i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.1208			−0.908391			−0.454195	−	0.890902i	$-0.650073\pi$
$593$	−22.1208			−0.908391			−0.454195	+	0.890902i	$0.650073\pi$
$594$		0			0
$595$	5.41699	−	5.41699i	0.222075	−	0.222075i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			13.8628i			0.566420i	0.959058	+	0.283210i	$0.0913993\pi$
$599$			13.8628i			0.566420i	−0.959058	+	0.283210i	$0.908601\pi$
$600$		0			0
$601$			3.29150i			0.134263i	0.997744	+	0.0671316i	$0.0213847\pi$
$601$			3.29150i			0.134263i	−0.997744	+	0.0671316i	$0.978615\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.89407	−	6.89407i	0.280284	−	0.280284i
$606$		0			0
$607$	−14.1033			−0.572434			−0.286217	−	0.958165i	$-0.592398\pi$
$607$	−14.1033			−0.572434			−0.286217	+	0.958165i	$0.592398\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0798	+	24.0798i	−0.974165	+	0.974165i
$612$		0			0
$613$	26.6458	+	26.6458i	1.07621	+	1.07621i	0.996846	+	0.0793662i	$0.0252897\pi$
$613$	26.6458	+	26.6458i	1.07621	+	1.07621i	0.0793662	+	0.996846i	$0.474710\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			31.6438i			1.27393i	0.770893	+	0.636965i	$0.219810\pi$
$617$			31.6438i			1.27393i	−0.770893	+	0.636965i	$0.780190\pi$
$618$		0			0
$619$	21.1660	+	21.1660i	0.850734	+	0.850734i	0.990224	−	0.139490i	$-0.0445462\pi$
$619$	21.1660	+	21.1660i	0.850734	+	0.850734i	−0.139490	+	0.990224i	$0.544546\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.91813			0.156976
$624$		0			0
$625$	−14.8745			−0.594980
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.57113	+	3.57113i	0.142390	+	0.142390i
$630$		0			0
$631$		−	8.22876i		−	0.327582i	−0.986495	−	0.163791i	$-0.947628\pi$
$631$		−	8.22876i		−	0.327582i	0.986495	−	0.163791i	$-0.0523722\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.557842	+	0.557842i	0.0221373	+	0.0221373i
$636$		0			0
$637$	11.3542	−	11.3542i	0.449872	−	0.449872i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−16.5906			−0.655288			−0.327644	−	0.944801i	$-0.606255\pi$
$641$	−16.5906			−0.655288			−0.327644	+	0.944801i	$0.606255\pi$
$642$		0			0
$643$	3.64575	−	3.64575i	0.143774	−	0.143774i	−0.631556	−	0.775330i	$-0.717583\pi$
$643$	3.64575	−	3.64575i	0.143774	−	0.143774i	0.775330	+	0.631556i	$0.217583\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	4.33981i		−	0.170616i	−0.996355	−	0.0853079i	$-0.972813\pi$
$647$		−	4.33981i		−	0.170616i	0.996355	−	0.0853079i	$-0.0271874\pi$
$648$		0			0
$649$			29.1660i			1.14487i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.7569	−	20.7569i	0.812280	−	0.812280i	−0.172696	−	0.984975i	$-0.555248\pi$
$653$	20.7569	−	20.7569i	0.812280	−	0.812280i	0.984975	+	0.172696i	$0.0552477\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.9232	−	24.9232i	0.970870	−	0.970870i	−0.0287175	−	0.999588i	$-0.509142\pi$
$659$	24.9232	−	24.9232i	0.970870	−	0.970870i	0.999588	+	0.0287175i	$0.00914231\pi$
$660$		0			0
$661$	−2.77124	−	2.77124i	−0.107789	−	0.107789i	0.651155	−	0.758944i	$-0.274285\pi$
$661$	−2.77124	−	2.77124i	−0.107789	−	0.107789i	−0.758944	+	0.651155i	$0.774285\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			7.14226i			0.276965i
$666$		0			0
$667$	−26.5830	−	26.5830i	−1.02930	−	1.02930i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.33981			−0.167537
$672$		0			0
$673$	20.0000			0.770943			0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000			0.770943			0.385472	+	0.922720i	$0.374039\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.0363	+	14.0363i	0.539460	+	0.539460i	0.923370	−	0.383911i	$-0.125423\pi$
$677$	14.0363	+	14.0363i	0.539460	+	0.539460i	−0.383911	+	0.923370i	$0.625423\pi$
$678$		0			0
$679$			17.4170i			0.668403i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.557842	−	0.557842i	−0.0213452	−	0.0213452i	0.696354	−	0.717699i	$-0.254805\pi$
$683$	−0.557842	−	0.557842i	−0.0213452	−	0.0213452i	−0.717699	+	0.696354i	$0.754805\pi$
$684$		0			0
$685$	−6.12549	+	6.12549i	−0.234043	+	0.234043i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	26.1136			0.994848
$690$		0			0
$691$	3.64575	−	3.64575i	0.138691	−	0.138691i	−0.634353	−	0.773044i	$-0.718733\pi$
$691$	3.64575	−	3.64575i	0.138691	−	0.138691i	0.773044	+	0.634353i	$0.218733\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	17.3593i		−	0.658474i
$696$		0			0
$697$		−	43.7490i		−	1.65711i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.16632	−	4.16632i	0.157360	−	0.157360i	−0.624036	−	0.781396i	$-0.714508\pi$
$701$	4.16632	−	4.16632i	0.157360	−	0.157360i	0.781396	+	0.624036i	$0.214508\pi$
$702$		0			0
$703$	−4.70850			−0.177584
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.979531	+	0.979531i	−0.0368391	+	0.0368391i
$708$		0			0
$709$	−8.77124	−	8.77124i	−0.329411	−	0.329411i	0.522951	−	0.852362i	$-0.324831\pi$
$709$	−8.77124	−	8.77124i	−0.329411	−	0.329411i	−0.852362	+	0.522951i	$0.824831\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	24.5015i		−	0.917589i
$714$		0			0
$715$	−10.5830	−	10.5830i	−0.395782	−	0.395782i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.3234			−1.50381			−0.751904	−	0.659272i	$-0.770865\pi$
$719$	−40.3234			−1.50381			−0.751904	+	0.659272i	$0.770865\pi$
$720$		0			0
$721$	27.8745			1.03810
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−26.2871	−	26.2871i	−0.976278	−	0.976278i
$726$		0			0
$727$		−	33.3948i		−	1.23854i	−0.785177	−	0.619272i	$-0.787428\pi$
$727$		−	33.3948i		−	1.23854i	0.785177	−	0.619272i	$-0.212572\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.95906	−	1.95906i	−0.0724586	−	0.0724586i
$732$		0			0
$733$	−12.6458	+	12.6458i	−0.467081	+	0.467081i	−0.900968	−	0.433886i	$-0.857142\pi$
$733$	−12.6458	+	12.6458i	−0.467081	+	0.467081i	0.433886	+	0.900968i	$0.357142\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−26.8823			−0.990221
$738$		0			0
$739$	9.16601	−	9.16601i	0.337177	−	0.337177i	−0.518127	−	0.855304i	$-0.673370\pi$
$739$	9.16601	−	9.16601i	0.337177	−	0.337177i	0.855304	+	0.518127i	$0.173370\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	22.5425i		−	0.827002i	−0.910504	−	0.413501i	$-0.864306\pi$
$743$		−	22.5425i		−	0.827002i	0.910504	−	0.413501i	$-0.135694\pi$
$744$		0			0
$745$			13.8745i			0.508323i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.91813	−	3.91813i	0.143165	−	0.143165i
$750$		0			0
$751$	−12.9373			−0.472087			−0.236044	−	0.971742i	$-0.575851\pi$
$751$	−12.9373			−0.472087			−0.236044	+	0.971742i	$0.575851\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.16272	+	6.16272i	−0.224284	+	0.224284i
$756$		0			0
$757$	−7.22876	−	7.22876i	−0.262734	−	0.262734i	0.563430	−	0.826164i	$-0.309481\pi$
$757$	−7.22876	−	7.22876i	−0.262734	−	0.262734i	−0.826164	+	0.563430i	$0.809481\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			7.91094i			0.286771i	0.989667	+	0.143386i	$0.0457989\pi$
$761$			7.91094i			0.286771i	−0.989667	+	0.143386i	$0.954201\pi$
$762$		0			0
$763$	10.9373	+	10.9373i	0.395955	+	0.395955i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.9641			0.829187
$768$		0			0
$769$	17.2915			0.623548			0.311774	−	0.950156i	$-0.399077\pi$
$769$	17.2915			0.623548			0.311774	+	0.950156i	$0.399077\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.5256	+	21.5256i	0.774221	+	0.774221i	0.978841	−	0.204620i	$-0.0655960\pi$
$773$	21.5256	+	21.5256i	0.774221	+	0.774221i	−0.204620	+	0.978841i	$0.565596\pi$
$774$		0			0
$775$		−	24.2288i		−	0.870323i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.8413	+	28.8413i	1.03335	+	1.03335i
$780$		0			0
$781$	45.1660	−	45.1660i	1.61617	−	1.61617i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.91094			0.282354
$786$		0			0
$787$	−6.22876	+	6.22876i	−0.222031	+	0.222031i	−0.809353	−	0.587322i	$-0.800182\pi$
$787$	−6.22876	+	6.22876i	−0.222031	+	0.222031i	0.587322	+	0.809353i	$0.300182\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			3.41699i			0.121341i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.5256	+	21.5256i	−0.762475	+	0.762475i	−0.976769	−	0.214294i	$-0.931255\pi$
$797$	−21.5256	+	21.5256i	−0.762475	+	0.762475i	0.214294	+	0.976769i	$0.431255\pi$
$798$		0			0
$799$	50.3320			1.78062
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.0604	+	11.0604i	−0.390312	+	0.390312i
$804$		0			0
$805$	4.25098	+	4.25098i	0.149828	+	0.149828i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	17.4339i		−	0.612945i	−0.951880	−	0.306473i	$-0.900851\pi$
$809$		−	17.4339i		−	0.612945i	0.951880	−	0.306473i	$-0.0991488\pi$
$810$		0			0
$811$	15.6458	+	15.6458i	0.549397	+	0.549397i	0.926266	−	0.376870i	$-0.123000\pi$
$811$	15.6458	+	15.6458i	0.549397	+	0.549397i	−0.376870	+	0.926266i	$0.623000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.8628			0.485593
$816$		0			0
$817$	2.58301			0.0903679
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.73744	−	7.73744i	−0.270039	−	0.270039i	0.559077	−	0.829116i	$-0.311156\pi$
$821$	−7.73744	−	7.73744i	−0.270039	−	0.270039i	−0.829116	+	0.559077i	$0.811156\pi$
$822$		0			0
$823$		−	32.2288i		−	1.12342i	−0.827333	−	0.561712i	$-0.810143\pi$
$823$		−	32.2288i		−	1.12342i	0.827333	−	0.561712i	$-0.189857\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.33981	+	4.33981i	0.150910	+	0.150910i	0.778524	−	0.627614i	$-0.215968\pi$
$827$	4.33981	+	4.33981i	0.150910	+	0.150910i	−0.627614	+	0.778524i	$0.715968\pi$
$828$		0			0
$829$	14.6458	−	14.6458i	0.508668	−	0.508668i	−0.405449	−	0.914117i	$-0.632885\pi$
$829$	14.6458	−	14.6458i	0.508668	−	0.508668i	0.914117	+	0.405449i	$0.132885\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23.7328			−0.822294
$834$		0			0
$835$	10.5830	−	10.5830i	0.366240	−	0.366240i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			52.9212i			1.82704i	0.406793	+	0.913521i	$0.366647\pi$
$839$			52.9212i			1.82704i	−0.406793	+	0.913521i	$0.633353\pi$
$840$		0			0
$841$		−	46.0405i		−	1.58760i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.595188	+	0.595188i	−0.0204751	+	0.0204751i
$846$		0			0
$847$	19.0627			0.655004
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.80244	+	2.80244i	−0.0960664	+	0.0960664i
$852$		0			0
$853$	11.9373	+	11.9373i	0.408724	+	0.408724i	0.881293	−	0.472570i	$-0.156673\pi$
$853$	11.9373	+	11.9373i	0.408724	+	0.408724i	−0.472570	+	0.881293i	$0.656673\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	10.2917i		−	0.351558i	−0.984430	−	0.175779i	$-0.943756\pi$
$857$		−	10.2917i		−	0.351558i	0.984430	−	0.175779i	$-0.0562443\pi$
$858$		0			0
$859$	5.77124	+	5.77124i	0.196912	+	0.196912i	0.798675	−	0.601763i	$-0.205535\pi$
$859$	5.77124	+	5.77124i	0.196912	+	0.196912i	−0.601763	+	0.798675i	$0.705535\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	53.3428			1.81581			0.907906	−	0.419174i	$-0.137680\pi$
$863$	53.3428			1.81581			0.907906	+	0.419174i	$0.137680\pi$
$864$		0			0
$865$	5.87451			0.199739
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−32.4125	−	32.4125i	−1.09952	−	1.09952i
$870$		0			0
$871$			21.1660i			0.717183i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.10132	+	9.10132i	0.307681	+	0.307681i
$876$		0			0
$877$	−21.9373	+	21.9373i	−0.740768	+	0.740768i	−0.972726	−	0.231957i	$-0.925487\pi$
$877$	−21.9373	+	21.9373i	−0.740768	+	0.740768i	0.231957	+	0.972726i	$0.425487\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14.9785			0.504639			0.252319	−	0.967644i	$-0.418807\pi$
$881$	14.9785			0.504639			0.252319	+	0.967644i	$0.418807\pi$
$882$		0			0
$883$	35.3948	−	35.3948i	1.19113	−	1.19113i	0.214378	−	0.976751i	$-0.431228\pi$
$883$	35.3948	−	35.3948i	1.19113	−	1.19113i	0.976751	−	0.214378i	$-0.0687724\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.76150i			0.159876i	0.996800	+	0.0799378i	$0.0254722\pi$
$887$			4.76150i			0.159876i	−0.996800	+	0.0799378i	$0.974528\pi$
$888$		0			0
$889$			1.54249i			0.0517333i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.1811	+	33.1811i	−1.11036	+	1.11036i
$894$		0			0
$895$	2.83399			0.0947298
$896$		0			0
$897$		0			0
$898$		0			0
$899$	34.5824	−	34.5824i	1.15339	−	1.15339i
$900$		0			0
$901$	−27.2915	−	27.2915i	−0.909212	−	0.909212i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			0.768687i			0.0255520i
$906$		0			0
$907$	−16.1033	−	16.1033i	−0.534700	−	0.534700i	0.387267	−	0.921967i	$-0.373419\pi$
$907$	−16.1033	−	16.1033i	−0.534700	−	0.534700i	−0.921967	+	0.387267i	$0.873419\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.18319			−0.171727			−0.0858634	−	0.996307i	$-0.527365\pi$
$911$	−5.18319			−0.171727			−0.0858634	+	0.996307i	$0.527365\pi$
$912$		0			0
$913$	−22.5830			−0.747388
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11.0604	−	11.0604i	−0.365246	−	0.365246i
$918$		0			0
$919$			42.1033i			1.38886i	0.719561	+	0.694429i	$0.244343\pi$
$919$			42.1033i			1.38886i	−0.719561	+	0.694429i	$0.755657\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−35.5619	−	35.5619i	−1.17053	−	1.17053i
$924$		0			0
$925$	−2.77124	+	2.77124i	−0.0911179	+	0.0911179i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.7328			0.778649			0.389324	−	0.921101i	$-0.372709\pi$
$929$	23.7328			0.778649			0.389324	+	0.921101i	$0.372709\pi$
$930$		0			0
$931$	15.6458	−	15.6458i	0.512769	−	0.512769i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			22.1208i			0.723426i
$936$		0			0
$937$		−	31.1660i		−	1.01815i	−0.860722	−	0.509075i	$-0.829988\pi$
$937$		−	31.1660i		−	1.01815i	0.860722	−	0.509075i	$-0.170012\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.4243	+	12.4243i	−0.405019	+	0.405019i	−0.879997	−	0.474978i	$-0.842456\pi$
$941$	−12.4243	+	12.4243i	−0.405019	+	0.405019i	0.474978	+	0.879997i	$0.342456\pi$
$942$		0			0
$943$	34.3320			1.11800
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.38075	+	2.38075i	−0.0773640	+	0.0773640i	−0.744730	−	0.667366i	$-0.767422\pi$
$947$	−2.38075	+	2.38075i	−0.0773640	+	0.0773640i	0.667366	+	0.744730i	$0.267422\pi$
$948$		0			0
$949$	8.70850	+	8.70850i	0.282690	+	0.282690i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	25.2702i		−	0.818582i	−0.912404	−	0.409291i	$-0.865776\pi$
$953$		−	25.2702i		−	0.818582i	0.912404	−	0.409291i	$-0.134224\pi$
$954$		0			0
$955$	−5.41699	−	5.41699i	−0.175290	−	0.175290i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16.9376			−0.546943
$960$		0			0
$961$	0.874508			0.0282099
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.06756	−	7.06756i	−0.227513	−	0.227513i
$966$		0			0
$967$		−	45.3948i		−	1.45980i	−0.683555	−	0.729899i	$-0.739567\pi$
$967$		−	45.3948i		−	1.45980i	0.683555	−	0.729899i	$-0.260433\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.7195	−	20.7195i	−0.664922	−	0.664922i	0.291614	−	0.956536i	$-0.405808\pi$
$971$	−20.7195	−	20.7195i	−0.664922	−	0.664922i	−0.956536	+	0.291614i	$0.905808\pi$
$972$		0			0
$973$	24.0000	−	24.0000i	0.769405	−	0.769405i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.5087			0.656131			0.328066	−	0.944655i	$-0.393603\pi$
$977$	20.5087			0.656131			0.328066	+	0.944655i	$0.393603\pi$
$978$		0			0
$979$	−8.00000	+	8.00000i	−0.255681	+	0.255681i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			26.8823i			0.857411i	0.903444	+	0.428706i	$0.141030\pi$
$983$			26.8823i			0.857411i	−0.903444	+	0.428706i	$0.858970\pi$
$984$		0			0
$985$		−	10.1255i		−	0.322625i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.53737	−	1.53737i	0.0488856	−	0.0488856i
$990$		0			0
$991$	−3.06275			−0.0972913			−0.0486457	−	0.998816i	$-0.515491\pi$
$991$	−3.06275			−0.0972913			−0.0486457	+	0.998816i	$0.515491\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.12179	−	8.12179i	0.257478	−	0.257478i
$996$		0			0
$997$	36.5203	+	36.5203i	1.15661	+	1.15661i	0.985200	+	0.171408i	$0.0548317\pi$
$997$	36.5203	+	36.5203i	1.15661	+	1.15661i	0.171408	+	0.985200i	$0.445168\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.k.c.433.2		8
3.2	odd	2	inner	576.2.k.c.433.3		8
4.3	odd	2		144.2.k.c.37.4	yes	8
8.3	odd	2		1152.2.k.e.865.3		8
8.5	even	2		1152.2.k.d.865.3		8
12.11	even	2		144.2.k.c.37.1	✓	8
16.3	odd	4		144.2.k.c.109.4	yes	8
16.5	even	4		1152.2.k.d.289.3		8
16.11	odd	4		1152.2.k.e.289.3		8
16.13	even	4	inner	576.2.k.c.145.2		8
24.5	odd	2		1152.2.k.d.865.2		8
24.11	even	2		1152.2.k.e.865.2		8
32.3	odd	8		9216.2.a.bq.1.5		8
32.13	even	8		9216.2.a.bt.1.4		8
32.19	odd	8		9216.2.a.bq.1.4		8
32.29	even	8		9216.2.a.bt.1.5		8
48.5	odd	4		1152.2.k.d.289.2		8
48.11	even	4		1152.2.k.e.289.2		8
48.29	odd	4	inner	576.2.k.c.145.3		8
48.35	even	4		144.2.k.c.109.1	yes	8
96.29	odd	8		9216.2.a.bt.1.3		8
96.35	even	8		9216.2.a.bq.1.3		8
96.77	odd	8		9216.2.a.bt.1.6		8
96.83	even	8		9216.2.a.bq.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.k.c.37.1	✓	8	12.11	even	2
144.2.k.c.37.4	yes	8	4.3	odd	2
144.2.k.c.109.1	yes	8	48.35	even	4
144.2.k.c.109.4	yes	8	16.3	odd	4
576.2.k.c.145.2		8	16.13	even	4	inner
576.2.k.c.145.3		8	48.29	odd	4	inner
576.2.k.c.433.2		8	1.1	even	1	trivial
576.2.k.c.433.3		8	3.2	odd	2	inner
1152.2.k.d.289.2		8	48.5	odd	4
1152.2.k.d.289.3		8	16.5	even	4
1152.2.k.d.865.2		8	24.5	odd	2
1152.2.k.d.865.3		8	8.5	even	2
1152.2.k.e.289.2		8	48.11	even	4
1152.2.k.e.289.3		8	16.11	odd	4
1152.2.k.e.865.2		8	24.11	even	2
1152.2.k.e.865.3		8	8.3	odd	2
9216.2.a.bq.1.3		8	96.35	even	8
9216.2.a.bq.1.4		8	32.19	odd	8
9216.2.a.bq.1.5		8	32.3	odd	8
9216.2.a.bq.1.6		8	96.83	even	8
9216.2.a.bt.1.3		8	96.29	odd	8
9216.2.a.bt.1.4		8	32.13	even	8
9216.2.a.bt.1.5		8	32.29	even	8
9216.2.a.bt.1.6		8	96.77	odd	8

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.595188 - 0.595188i) q^{5} -1.64575i q^{7} +O(q^{10})\)	\(q+(-0.595188 - 0.595188i) q^{5} -1.64575i q^{7} +(3.36028 + 3.36028i) q^{11} +(2.64575 - 2.64575i) q^{13} -5.53019 q^{17} +(3.64575 - 3.64575i) q^{19} -4.33981i q^{23} -4.29150i q^{25} +(6.12538 - 6.12538i) q^{29} +5.64575 q^{31} +(-0.979531 + 0.979531i) q^{35} +(-0.645751 - 0.645751i) q^{37} +7.91094i q^{41} +(0.354249 + 0.354249i) q^{43} -9.10132 q^{47} +4.29150 q^{49} +(4.93500 + 4.93500i) q^{53} -4.00000i q^{55} +(4.33981 + 4.33981i) q^{59} +(-0.645751 + 0.645751i) q^{61} -3.14944 q^{65} +(-4.00000 + 4.00000i) q^{67} -13.4411i q^{71} +3.29150i q^{73} +(5.53019 - 5.53019i) q^{77} -9.64575 q^{79} +(-3.36028 + 3.36028i) q^{83} +(3.29150 + 3.29150i) q^{85} +2.38075i q^{89} +(-4.35425 - 4.35425i) q^{91} -4.33981 q^{95} -10.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{19} + 24 q^{31} + 16 q^{37} + 24 q^{43} - 8 q^{49} + 16 q^{61} - 32 q^{67} - 56 q^{79} - 16 q^{85} - 56 q^{91}+O(q^{100}) \) 8 * q + 8 * q^19 + 24 * q^31 + 16 * q^37 + 24 * q^43 - 8 * q^49 + 16 * q^61 - 32 * q^67 - 56 * q^79 - 16 * q^85 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

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