LMFDB - Embedded newform 576.4.k.a.145.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,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, 4, names="a")

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

chi := DirichletCharacter("576.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 $ x^10 - 2x^9 - x^8 + 6x^7 + 14x^6 - 80x^5 + 56x^4 + 96x^3 - 64x^2 - 512x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			145.3
Root			$0.932438 - 1.76934i$ of defining polynomial
Character	$\chi$	$=$	576.145
Dual form			576.4.k.a.433.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.596848 - 0.596848i) q^{5} -29.0828i q^{7} +O(q^{10})$	$q+(0.596848 - 0.596848i) q^{5} -29.0828i q^{7} +(12.1291 - 12.1291i) q^{11} +(-48.5658 - 48.5658i) q^{13} -86.7193 q^{17} +(54.8442 + 54.8442i) q^{19} +70.2145i q^{23} +124.288i q^{25} +(-63.4021 - 63.4021i) q^{29} +8.86868 q^{31} +(-17.3580 - 17.3580i) q^{35} +(-21.7145 + 21.7145i) q^{37} +153.274i q^{41} +(120.951 - 120.951i) q^{43} -99.9792 q^{47} -502.812 q^{49} +(-389.132 + 389.132i) q^{53} -14.4785i q^{55} +(-324.819 + 324.819i) q^{59} +(-0.339194 - 0.339194i) q^{61} -57.9728 q^{65} +(-565.288 - 565.288i) q^{67} -419.500i q^{71} +374.833i q^{73} +(-352.750 - 352.750i) q^{77} +705.750 q^{79} +(-947.092 - 947.092i) q^{83} +(-51.7582 + 51.7582i) q^{85} +4.72918i q^{89} +(-1412.43 + 1412.43i) q^{91} +65.4673 q^{95} +379.542 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5}+O(q^{10}) $ 10 * q + 2 * q^5	$ 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	0.596848	−	0.596848i	0.0533837	−	0.0533837i	−0.679911	−	0.733295i	$-0.737982\pi$
$5$	0.596848	−	0.596848i	0.0533837	−	0.0533837i	0.733295	+	0.679911i	$0.237982\pi$
$6$		0			0
$7$		−	29.0828i		−	1.57033i	−0.619289	−	0.785163i	$-0.712579\pi$
$7$		−	29.0828i		−	1.57033i	0.619289	−	0.785163i	$-0.287421\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.1291	−	12.1291i	0.332461	−	0.332461i	−0.521059	−	0.853520i	$-0.674463\pi$
$11$	12.1291	−	12.1291i	0.332461	−	0.332461i	0.853520	+	0.521059i	$0.174463\pi$
$12$		0			0
$13$	−48.5658	−	48.5658i	−1.03613	−	1.03613i	−0.999322	−	0.0368113i	$-0.988280\pi$
$13$	−48.5658	−	48.5658i	−1.03613	−	1.03613i	−0.0368113	−	0.999322i	$-0.511720\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−86.7193			−1.23721			−0.618604	−	0.785703i	$-0.712301\pi$
$17$	−86.7193			−1.23721			−0.618604	+	0.785703i	$0.712301\pi$
$18$		0			0
$19$	54.8442	+	54.8442i	0.662217	+	0.662217i	0.955902	−	0.293685i	$-0.0948817\pi$
$19$	54.8442	+	54.8442i	0.662217	+	0.662217i	−0.293685	+	0.955902i	$0.594882\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			70.2145i			0.636554i	0.947998	+	0.318277i	$0.103104\pi$
$23$			70.2145i			0.636554i	−0.947998	+	0.318277i	$0.896896\pi$
$24$		0			0
$25$			124.288i			0.994300i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−63.4021	−	63.4021i	−0.405982	−	0.405982i	0.474353	−	0.880335i	$-0.342682\pi$
$29$	−63.4021	−	63.4021i	−0.405982	−	0.405982i	−0.880335	+	0.474353i	$0.842682\pi$
$30$		0			0
$31$	8.86868			0.0513826			0.0256913	−	0.999670i	$-0.491821\pi$
$31$	8.86868			0.0513826			0.0256913	+	0.999670i	$0.491821\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−17.3580	−	17.3580i	−0.0838298	−	0.0838298i
$36$		0			0
$37$	−21.7145	+	21.7145i	−0.0964820	+	0.0964820i	−0.753700	−	0.657218i	$-0.771733\pi$
$37$	−21.7145	+	21.7145i	−0.0964820	+	0.0964820i	0.657218	+	0.753700i	$0.271733\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			153.274i			0.583840i	0.956443	+	0.291920i	$0.0942941\pi$
$41$			153.274i			0.583840i	−0.956443	+	0.291920i	$0.905706\pi$
$42$		0			0
$43$	120.951	−	120.951i	0.428949	−	0.428949i	−0.459322	−	0.888270i	$-0.651907\pi$
$43$	120.951	−	120.951i	0.428949	−	0.428949i	0.888270	+	0.459322i	$0.151907\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−99.9792			−0.310286			−0.155143	−	0.987892i	$-0.549584\pi$
$47$	−99.9792			−0.310286			−0.155143	+	0.987892i	$0.549584\pi$
$48$		0			0
$49$	−502.812			−1.46592
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−389.132	+	389.132i	−1.00852	+	1.00852i	−0.00855213	+	0.999963i	$0.502722\pi$
$53$	−389.132	+	389.132i	−1.00852	+	1.00852i	−0.999963	+	0.00855213i	$0.997278\pi$
$54$		0			0
$55$		−	14.4785i		−	0.0354960i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−324.819	+	324.819i	−0.716744	+	0.716744i	−0.967937	−	0.251193i	$-0.919177\pi$
$59$	−324.819	+	324.819i	−0.716744	+	0.716744i	0.251193	+	0.967937i	$0.419177\pi$
$60$		0			0
$61$	−0.339194	−	0.339194i	−0.000711957	−	0.000711957i	0.706751	−	0.707463i	$-0.250160\pi$
$61$	−0.339194	−	0.339194i	−0.000711957	−	0.000711957i	−0.707463	+	0.706751i	$0.750160\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−57.9728			−0.110625
$66$		0			0
$67$	−565.288	−	565.288i	−1.03076	−	1.03076i	−0.999512	−	0.0312478i	$-0.990052\pi$
$67$	−565.288	−	565.288i	−1.03076	−	1.03076i	−0.0312478	−	0.999512i	$-0.509948\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	419.500i		−	0.701205i	−0.936524	−	0.350602i	$-0.885977\pi$
$71$		−	419.500i		−	0.701205i	0.936524	−	0.350602i	$-0.114023\pi$
$72$		0			0
$73$			374.833i			0.600971i	0.953786	+	0.300485i	$0.0971487\pi$
$73$			374.833i			0.600971i	−0.953786	+	0.300485i	$0.902851\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−352.750	−	352.750i	−0.522072	−	0.522072i
$78$		0			0
$79$	705.750			1.00510			0.502551	−	0.864547i	$-0.332395\pi$
$79$	705.750			1.00510			0.502551	+	0.864547i	$0.332395\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−947.092	−	947.092i	−1.25249	−	1.25249i	−0.954599	−	0.297893i	$-0.903716\pi$
$83$	−947.092	−	947.092i	−1.25249	−	1.25249i	−0.297893	−	0.954599i	$-0.596284\pi$
$84$		0			0
$85$	−51.7582	+	51.7582i	−0.0660467	+	0.0660467i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			4.72918i			0.00563249i	0.999996	+	0.00281625i	$0.000896440\pi$
$89$			4.72918i			0.00563249i	−0.999996	+	0.00281625i	$0.999104\pi$
$90$		0			0
$91$	−1412.43	+	1412.43i	−1.62707	+	1.62707i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	65.4673			0.0707032
$96$		0			0
$97$	379.542			0.397285			0.198643	−	0.980072i	$-0.436347\pi$
$97$	379.542			0.397285			0.198643	+	0.980072i	$0.436347\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	391.005	−	391.005i	0.385212	−	0.385212i	−0.487764	−	0.872976i	$-0.662187\pi$
$101$	391.005	−	391.005i	0.385212	−	0.385212i	0.872976	+	0.487764i	$0.162187\pi$
$102$		0			0
$103$			307.935i			0.294580i	0.989093	+	0.147290i	$0.0470551\pi$
$103$			307.935i			0.294580i	−0.989093	+	0.147290i	$0.952945\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−601.607	+	601.607i	−0.543548	+	0.543548i	−0.924567	−	0.381019i	$-0.875573\pi$
$107$	−601.607	+	601.607i	−0.543548	+	0.543548i	0.381019	+	0.924567i	$0.375573\pi$
$108$		0			0
$109$	−948.890	−	948.890i	−0.833827	−	0.833827i	0.154211	−	0.988038i	$-0.450717\pi$
$109$	−948.890	−	948.890i	−0.833827	−	0.833827i	−0.988038	+	0.154211i	$0.950717\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1824.02			−1.51849			−0.759244	−	0.650807i	$-0.774431\pi$
$113$	−1824.02			−1.51849			−0.759244	+	0.650807i	$0.774431\pi$
$114$		0			0
$115$	41.9074	+	41.9074i	0.0339816	+	0.0339816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2522.04i			1.94282i
$120$		0			0
$121$			1036.77i			0.778939i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	148.787	+	148.787i	0.106463	+	0.106463i
$126$		0			0
$127$	−988.748			−0.690844			−0.345422	−	0.938447i	$-0.612264\pi$
$127$	−988.748			−0.690844			−0.345422	+	0.938447i	$0.612264\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	793.572	+	793.572i	0.529273	+	0.529273i	0.920356	−	0.391083i	$-0.127899\pi$
$131$	793.572	+	793.572i	0.529273	+	0.529273i	−0.391083	+	0.920356i	$0.627899\pi$
$132$		0			0
$133$	1595.03	−	1595.03i	1.03990	−	1.03990i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1595.30i			0.994856i	0.867505	+	0.497428i	$0.165722\pi$
$137$			1595.30i			0.994856i	−0.867505	+	0.497428i	$0.834278\pi$
$138$		0			0
$139$	277.696	−	277.696i	0.169452	−	0.169452i	−0.617286	−	0.786738i	$-0.711768\pi$
$139$	277.696	−	277.696i	0.169452	−	0.169452i	0.786738	+	0.617286i	$0.211768\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1178.12			−0.688948
$144$		0			0
$145$	−75.6828			−0.0433456
$146$		0			0
$147$		0			0
$148$		0			0
$149$	593.272	−	593.272i	0.326193	−	0.326193i	−0.524944	−	0.851137i	$-0.675914\pi$
$149$	593.272	−	593.272i	0.326193	−	0.326193i	0.851137	+	0.524944i	$0.175914\pi$
$150$		0			0
$151$			160.655i			0.0865821i	0.999063	+	0.0432911i	$0.0137843\pi$
$151$			160.655i			0.0865821i	−0.999063	+	0.0432911i	$0.986216\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.29325	−	5.29325i	0.00274299	−	0.00274299i
$156$		0			0
$157$	−705.762	−	705.762i	−0.358764	−	0.358764i	0.504593	−	0.863357i	$-0.331642\pi$
$157$	−705.762	−	705.762i	−0.358764	−	0.358764i	−0.863357	+	0.504593i	$0.831642\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2042.04			0.999598
$162$		0			0
$163$	−1872.64	−	1872.64i	−0.899855	−	0.899855i	0.0955676	−	0.995423i	$-0.469533\pi$
$163$	−1872.64	−	1872.64i	−0.899855	−	0.899855i	−0.995423	+	0.0955676i	$0.969533\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3852.19i		−	1.78498i	−0.451066	−	0.892490i	$-0.648956\pi$
$167$		−	3852.19i		−	1.78498i	0.451066	−	0.892490i	$-0.351044\pi$
$168$		0			0
$169$			2520.28i			1.14715i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2625.61	−	2625.61i	−1.15388	−	1.15388i	−0.985768	−	0.168112i	$-0.946233\pi$
$173$	−2625.61	−	2625.61i	−1.15388	−	1.15388i	−0.168112	−	0.985768i	$-0.553767\pi$
$174$		0			0
$175$	3614.64			1.56138
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1236.73	−	1236.73i	−0.516413	−	0.516413i	0.400071	−	0.916484i	$-0.368985\pi$
$179$	−1236.73	−	1236.73i	−0.516413	−	0.516413i	−0.916484	+	0.400071i	$0.868985\pi$
$180$		0			0
$181$	1574.90	−	1574.90i	0.646748	−	0.646748i	−0.305458	−	0.952206i	$-0.598809\pi$
$181$	1574.90	−	1574.90i	0.646748	−	0.646748i	0.952206	+	0.305458i	$0.0988094\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			25.9204i			0.0103011i
$186$		0			0
$187$	−1051.83	+	1051.83i	−0.411323	+	0.411323i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3585.92			1.35847			0.679236	−	0.733920i	$-0.262311\pi$
$191$	3585.92			1.35847			0.679236	+	0.733920i	$0.262311\pi$
$192$		0			0
$193$	523.601			0.195283			0.0976415	−	0.995222i	$-0.468870\pi$
$193$	523.601			0.195283			0.0976415	+	0.995222i	$0.468870\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1125.64	−	1125.64i	0.407098	−	0.407098i	−0.473627	−	0.880725i	$-0.657056\pi$
$197$	1125.64	−	1125.64i	0.407098	−	0.407098i	0.880725	+	0.473627i	$0.157056\pi$
$198$		0			0
$199$			2312.48i			0.823757i	0.911239	+	0.411878i	$0.135127\pi$
$199$			2312.48i			0.823757i	−0.911239	+	0.411878i	$0.864873\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1843.91	+	1843.91i	−0.637524	+	0.637524i
$204$		0			0
$205$	91.4815	+	91.4815i	0.0311675	+	0.0311675i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1330.43			0.440323
$210$		0			0
$211$	−1418.59	−	1418.59i	−0.462842	−	0.462842i	0.436744	−	0.899586i	$-0.356132\pi$
$211$	−1418.59	−	1418.59i	−0.462842	−	0.462842i	−0.899586	+	0.436744i	$0.856132\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	144.378i		−	0.0457977i
$216$		0			0
$217$		−	257.926i		−	0.0806875i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4211.60	+	4211.60i	1.28191	+	1.28191i
$222$		0			0
$223$	4315.08			1.29578			0.647890	−	0.761734i	$-0.275651\pi$
$223$	4315.08			1.29578			0.647890	+	0.761734i	$0.275651\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	701.203	+	701.203i	0.205024	+	0.205024i	0.802149	−	0.597124i	$-0.203690\pi$
$227$	701.203	+	701.203i	0.205024	+	0.205024i	−0.597124	+	0.802149i	$0.703690\pi$
$228$		0			0
$229$	−663.351	+	663.351i	−0.191421	+	0.191421i	−0.796310	−	0.604889i	$-0.793218\pi$
$229$	−663.351	+	663.351i	−0.191421	+	0.191421i	0.604889	+	0.796310i	$0.293218\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3490.15i			0.981318i	0.871352	+	0.490659i	$0.163244\pi$
$233$			3490.15i			0.981318i	−0.871352	+	0.490659i	$0.836756\pi$
$234$		0			0
$235$	−59.6724	+	59.6724i	−0.0165642	+	0.0165642i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2950.43			0.798525			0.399263	−	0.916837i	$-0.369266\pi$
$239$	2950.43			0.798525			0.399263	+	0.916837i	$0.369266\pi$
$240$		0			0
$241$	−1128.96			−0.301755			−0.150877	−	0.988552i	$-0.548210\pi$
$241$	−1128.96			−0.301755			−0.150877	+	0.988552i	$0.548210\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−300.102	+	300.102i	−0.0782565	+	0.0782565i
$246$		0			0
$247$		−	5327.11i		−	1.37229i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4621.86	−	4621.86i	1.16227	−	1.16227i	0.178291	−	0.983978i	$-0.442943\pi$
$251$	4621.86	−	4621.86i	1.16227	−	1.16227i	0.983978	−	0.178291i	$-0.0570569\pi$
$252$		0			0
$253$	851.642	+	851.642i	0.211630	+	0.211630i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−610.977			−0.148295			−0.0741473	−	0.997247i	$-0.523623\pi$
$257$	−610.977			−0.148295			−0.0741473	+	0.997247i	$0.523623\pi$
$258$		0			0
$259$	631.518	+	631.518i	0.151508	+	0.151508i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4973.57i			1.16610i	0.812438	+	0.583048i	$0.198140\pi$
$263$			4973.57i			1.16610i	−0.812438	+	0.583048i	$0.801860\pi$
$264$		0			0
$265$			464.505i			0.107677i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	938.415	+	938.415i	0.212700	+	0.212700i	0.805413	−	0.592714i	$-0.201943\pi$
$269$	938.415	+	938.415i	0.212700	+	0.212700i	−0.592714	+	0.805413i	$0.701943\pi$
$270$		0			0
$271$	4010.64			0.898999			0.449500	−	0.893280i	$-0.351602\pi$
$271$	4010.64			0.898999			0.449500	+	0.893280i	$0.351602\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1507.50	+	1507.50i	0.330566	+	0.330566i
$276$		0			0
$277$	−3534.99	+	3534.99i	−0.766776	+	0.766776i	−0.977538	−	0.210762i	$-0.932406\pi$
$277$	−3534.99	+	3534.99i	−0.766776	+	0.766776i	0.210762	+	0.977538i	$0.432406\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7468.35i		−	1.58550i	−0.609550	−	0.792748i	$-0.708650\pi$
$281$		−	7468.35i		−	1.58550i	0.609550	−	0.792748i	$-0.291350\pi$
$282$		0			0
$283$	2249.22	−	2249.22i	0.472447	−	0.472447i	−0.430259	−	0.902705i	$-0.641578\pi$
$283$	2249.22	−	2249.22i	0.472447	−	0.472447i	0.902705	+	0.430259i	$0.141578\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4457.66			0.916819
$288$		0			0
$289$	2607.24			0.530682
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3952.79	−	3952.79i	0.788139	−	0.788139i	−0.193050	−	0.981189i	$-0.561838\pi$
$293$	3952.79	−	3952.79i	0.788139	−	0.788139i	0.981189	+	0.193050i	$0.0618381\pi$
$294$		0			0
$295$			387.736i			0.0765249i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3410.03	−	3410.03i	0.659555	−	0.659555i
$300$		0			0
$301$	−3517.59	−	3517.59i	−0.673589	−	0.673589i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.404895			−7.60138e−5
$306$		0			0
$307$	3855.24	+	3855.24i	0.716711	+	0.716711i	0.967930	−	0.251219i	$-0.0808316\pi$
$307$	3855.24	+	3855.24i	0.716711	+	0.716711i	−0.251219	+	0.967930i	$0.580832\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	5194.39i		−	0.947096i	−0.880768	−	0.473548i	$-0.842973\pi$
$311$		−	5194.39i		−	0.947096i	0.880768	−	0.473548i	$-0.157027\pi$
$312$		0			0
$313$			4710.01i			0.850561i	0.905062	+	0.425281i	$0.139825\pi$
$313$			4710.01i			0.850561i	−0.905062	+	0.425281i	$0.860175\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5680.21	−	5680.21i	−1.00641	−	1.00641i	−0.999979	−	0.00643263i	$-0.997952\pi$
$317$	−5680.21	−	5680.21i	−1.00641	−	1.00641i	−0.00643263	−	0.999979i	$-0.502048\pi$
$318$		0			0
$319$	−1538.03			−0.269946
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4756.05	−	4756.05i	−0.819300	−	0.819300i
$324$		0			0
$325$	6036.13	−	6036.13i	1.03023	−	1.03023i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2907.68i			0.487251i
$330$		0			0
$331$	−1815.80	+	1815.80i	−0.301526	+	0.301526i	−0.841611	−	0.540084i	$-0.818392\pi$
$331$	−1815.80	+	1815.80i	−0.301526	+	0.301526i	0.540084	+	0.841611i	$0.318392\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−674.782			−0.110052
$336$		0			0
$337$	−2683.29			−0.433733			−0.216867	−	0.976201i	$-0.569584\pi$
$337$	−2683.29			−0.433733			−0.216867	+	0.976201i	$0.569584\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	107.569	−	107.569i	0.0170827	−	0.0170827i
$342$		0			0
$343$			4647.79i			0.731653i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5291.81	+	5291.81i	−0.818671	+	0.818671i	−0.985916	−	0.167244i	$-0.946513\pi$
$347$	−5291.81	+	5291.81i	−0.818671	+	0.818671i	0.167244	+	0.985916i	$0.446513\pi$
$348$		0			0
$349$	73.7084	+	73.7084i	0.0113052	+	0.0113052i	0.712737	−	0.701432i	$-0.247455\pi$
$349$	73.7084	+	73.7084i	0.0113052	+	0.0113052i	−0.701432	+	0.712737i	$0.747455\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5067.25			0.764030			0.382015	−	0.924156i	$-0.375230\pi$
$353$	5067.25			0.764030			0.382015	+	0.924156i	$0.375230\pi$
$354$		0			0
$355$	−250.378	−	250.378i	−0.0374329	−	0.0374329i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			970.230i			0.142637i	0.997454	+	0.0713186i	$0.0227207\pi$
$359$			970.230i			0.142637i	−0.997454	+	0.0713186i	$0.977279\pi$
$360$		0			0
$361$		−	843.224i		−	0.122937i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	223.718	+	223.718i	0.0320821	+	0.0320821i
$366$		0			0
$367$	−13451.4			−1.91323			−0.956617	−	0.291347i	$-0.905896\pi$
$367$	−13451.4			−1.91323			−0.956617	+	0.291347i	$0.905896\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	11317.1	+	11317.1i	1.58370	+	1.58370i
$372$		0			0
$373$	5898.22	−	5898.22i	0.818762	−	0.818762i	−0.167167	−	0.985929i	$-0.553462\pi$
$373$	5898.22	−	5898.22i	0.818762	−	0.818762i	0.985929	+	0.167167i	$0.0534619\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6158.35i			0.841302i
$378$		0			0
$379$	4446.72	−	4446.72i	0.602673	−	0.602673i	−0.338348	−	0.941021i	$-0.609868\pi$
$379$	4446.72	−	4446.72i	0.602673	−	0.602673i	0.941021	+	0.338348i	$0.109868\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6417.68			−0.856209			−0.428105	−	0.903729i	$-0.640819\pi$
$383$	−6417.68			−0.856209			−0.428105	+	0.903729i	$0.640819\pi$
$384$		0			0
$385$	−421.076			−0.0557403
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6555.61	+	6555.61i	−0.854455	+	0.854455i	−0.990678	−	0.136223i	$-0.956503\pi$
$389$	−6555.61	+	6555.61i	−0.854455	+	0.854455i	0.136223	+	0.990678i	$0.456503\pi$
$390$		0			0
$391$		−	6088.96i		−	0.787549i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	421.226	−	421.226i	0.0536561	−	0.0536561i
$396$		0			0
$397$	8902.51	+	8902.51i	1.12545	+	1.12545i	0.990908	+	0.134543i	$0.0429566\pi$
$397$	8902.51	+	8902.51i	1.12545	+	1.12545i	0.134543	+	0.990908i	$0.457043\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6425.77			0.800218			0.400109	−	0.916468i	$-0.368972\pi$
$401$	6425.77			0.800218			0.400109	+	0.916468i	$0.368972\pi$
$402$		0			0
$403$	−430.715	−	430.715i	−0.0532393	−	0.0532393i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			526.755i			0.0641530i
$408$		0			0
$409$		−	12796.0i		−	1.54699i	−0.633801	−	0.773496i	$-0.718506\pi$
$409$		−	12796.0i		−	1.54699i	0.633801	−	0.773496i	$-0.281494\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9446.67	+	9446.67i	1.12552	+	1.12552i
$414$		0			0
$415$	−1130.54			−0.133725
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6545.21	−	6545.21i	−0.763137	−	0.763137i	0.213751	−	0.976888i	$-0.431432\pi$
$419$	−6545.21	−	6545.21i	−0.763137	−	0.763137i	−0.976888	+	0.213751i	$0.931432\pi$
$420$		0			0
$421$	−6390.00	+	6390.00i	−0.739738	+	0.739738i	−0.972527	−	0.232789i	$-0.925215\pi$
$421$	−6390.00	+	6390.00i	−0.739738	+	0.739738i	0.232789	+	0.972527i	$0.425215\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	10778.1i		−	1.23016i
$426$		0			0
$427$	−9.86474	+	9.86474i	−0.00111801	+	0.00111801i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10639.3			−1.18904			−0.594519	−	0.804081i	$-0.702658\pi$
$431$	−10639.3			−1.18904			−0.594519	+	0.804081i	$0.702658\pi$
$432$		0			0
$433$	3806.14			0.422428			0.211214	−	0.977440i	$-0.432258\pi$
$433$	3806.14			0.422428			0.211214	+	0.977440i	$0.432258\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3850.86	+	3850.86i	−0.421537	+	0.421537i
$438$		0			0
$439$			14102.8i			1.53323i	0.642106	+	0.766616i	$0.278061\pi$
$439$			14102.8i			1.53323i	−0.642106	+	0.766616i	$0.721939\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7662.45	+	7662.45i	−0.821792	+	0.821792i	−0.986365	−	0.164573i	$-0.947375\pi$
$443$	−7662.45	+	7662.45i	−0.821792	+	0.821792i	0.164573	+	0.986365i	$0.447375\pi$
$444$		0			0
$445$	2.82260	+	2.82260i	0.000300683	+	0.000300683i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13679.4			−1.43779			−0.718897	−	0.695117i	$-0.755353\pi$
$449$	−13679.4			−1.43779			−0.718897	+	0.695117i	$0.755353\pi$
$450$		0			0
$451$	1859.09	+	1859.09i	0.194104	+	0.194104i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			1686.01i			0.173718i
$456$		0			0
$457$		−	7913.48i		−	0.810016i	−0.914313	−	0.405008i	$-0.867269\pi$
$457$		−	7913.48i		−	0.810016i	0.914313	−	0.405008i	$-0.132731\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	580.215	+	580.215i	0.0586189	+	0.0586189i	0.735809	−	0.677190i	$-0.236802\pi$
$461$	580.215	+	580.215i	0.0586189	+	0.0586189i	−0.677190	+	0.735809i	$0.736802\pi$
$462$		0			0
$463$	−14236.5			−1.42899			−0.714497	−	0.699638i	$-0.753344\pi$
$463$	−14236.5			−1.42899			−0.714497	+	0.699638i	$0.753344\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8344.57	−	8344.57i	−0.826853	−	0.826853i	0.160227	−	0.987080i	$-0.448777\pi$
$467$	−8344.57	−	8344.57i	−0.826853	−	0.826853i	−0.987080	+	0.160227i	$0.948777\pi$
$468$		0			0
$469$	−16440.2	+	16440.2i	−1.61863	+	1.61863i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2934.05i		−	0.285217i
$474$		0			0
$475$	−6816.45	+	6816.45i	−0.658443	+	0.658443i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5563.77			−0.530720			−0.265360	−	0.964149i	$-0.585491\pi$
$479$	−5563.77			−0.530720			−0.265360	+	0.964149i	$0.585491\pi$
$480$		0			0
$481$	2109.16			0.199936
$482$		0			0
$483$		0			0
$484$		0			0
$485$	226.529	−	226.529i	0.0212085	−	0.0212085i
$486$		0			0
$487$		−	18150.5i		−	1.68886i	−0.535662	−	0.844432i	$-0.679938\pi$
$487$		−	18150.5i		−	1.68886i	0.535662	−	0.844432i	$-0.320062\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11593.0	−	11593.0i	1.06555	−	1.06555i	0.0678570	−	0.997695i	$-0.478384\pi$
$491$	11593.0	−	11593.0i	1.06555	−	1.06555i	0.997695	−	0.0678570i	$-0.0216162\pi$
$492$		0			0
$493$	5498.18	+	5498.18i	0.502284	+	0.502284i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12200.3			−1.10112
$498$		0			0
$499$	−3109.58	−	3109.58i	−0.278966	−	0.278966i	0.553730	−	0.832696i	$-0.313204\pi$
$499$	−3109.58	−	3109.58i	−0.278966	−	0.278966i	−0.832696	+	0.553730i	$0.813204\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6221.21i			0.551471i	0.961233	+	0.275736i	$0.0889215\pi$
$503$			6221.21i			0.551471i	−0.961233	+	0.275736i	$0.911079\pi$
$504$		0			0
$505$		−	466.740i		−	0.0411281i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13800.4	−	13800.4i	−1.20176	−	1.20176i	−0.973632	−	0.228124i	$-0.926741\pi$
$509$	−13800.4	−	13800.4i	−1.20176	−	1.20176i	−0.228124	−	0.973632i	$-0.573259\pi$
$510$		0			0
$511$	10901.2			0.943720
$512$		0			0
$513$		0			0
$514$		0			0
$515$	183.791	+	183.791i	0.0157258	+	0.0157258i
$516$		0			0
$517$	−1212.66	+	1212.66i	−0.103158	+	0.103158i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	6874.63i		−	0.578086i	−0.957316	−	0.289043i	$-0.906663\pi$
$521$		−	6874.63i		−	0.578086i	0.957316	−	0.289043i	$-0.0933371\pi$
$522$		0			0
$523$	−2306.52	+	2306.52i	−0.192843	+	0.192843i	−0.796924	−	0.604080i	$-0.793541\pi$
$523$	−2306.52	+	2306.52i	−0.192843	+	0.192843i	0.604080	+	0.796924i	$0.293541\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−769.086			−0.0635710
$528$		0			0
$529$	7236.92			0.594799
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7443.90	−	7443.90i	0.604936	−	0.604936i
$534$		0			0
$535$			718.136i			0.0580332i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6098.68	+	6098.68i	−0.487363	+	0.487363i
$540$		0			0
$541$	−13240.0	−	13240.0i	−1.05218	−	1.05218i	−0.998561	−	0.0536210i	$-0.982924\pi$
$541$	−13240.0	−	13240.0i	−1.05218	−	1.05218i	−0.0536210	−	0.998561i	$-0.517076\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1132.69			−0.0890256
$546$		0			0
$547$	13271.3	+	13271.3i	1.03737	+	1.03737i	0.999274	+	0.0380940i	$0.0121286\pi$
$547$	13271.3	+	13271.3i	1.03737	+	1.03737i	0.0380940	+	0.999274i	$0.487871\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	6954.47i		−	0.537696i
$552$		0			0
$553$		−	20525.2i		−	1.57834i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8500.61	+	8500.61i	0.646647	+	0.646647i	0.952181	−	0.305534i	$-0.0988349\pi$
$557$	8500.61	+	8500.61i	0.646647	+	0.646647i	−0.305534	+	0.952181i	$0.598835\pi$
$558$		0			0
$559$	−11748.1			−0.888896
$560$		0			0
$561$		0			0
$562$		0			0
$563$	17327.2	+	17327.2i	1.29708	+	1.29708i	0.930314	+	0.366763i	$0.119534\pi$
$563$	17327.2	+	17327.2i	1.29708	+	1.29708i	0.366763	+	0.930314i	$0.380466\pi$
$564$		0			0
$565$	−1088.66	+	1088.66i	−0.0810625	+	0.0810625i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			8998.54i			0.662985i	0.943458	+	0.331492i	$0.107552\pi$
$569$			8998.54i			0.662985i	−0.943458	+	0.331492i	$0.892448\pi$
$570$		0			0
$571$	9849.25	−	9849.25i	0.721853	−	0.721853i	−0.247129	−	0.968983i	$-0.579487\pi$
$571$	9849.25	−	9849.25i	0.721853	−	0.721853i	0.968983	+	0.247129i	$0.0794872\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8726.79			−0.632926
$576$		0			0
$577$	−20584.4			−1.48516			−0.742580	−	0.669757i	$-0.766398\pi$
$577$	−20584.4			−1.48516			−0.742580	+	0.669757i	$0.766398\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−27544.1	+	27544.1i	−1.96682	+	1.96682i
$582$		0			0
$583$			9439.66i			0.670585i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2586.77	+	2586.77i	−0.181887	+	0.181887i	−0.792177	−	0.610291i	$-0.791053\pi$
$587$	−2586.77	+	2586.77i	−0.181887	+	0.181887i	0.610291	+	0.792177i	$0.291053\pi$
$588$		0			0
$589$	486.396	+	486.396i	0.0340265	+	0.0340265i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6035.89			0.417984			0.208992	−	0.977917i	$-0.432982\pi$
$593$	6035.89			0.417984			0.208992	+	0.977917i	$0.432982\pi$
$594$		0			0
$595$	1505.28	+	1505.28i	0.103715	+	0.103715i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			5427.20i			0.370199i	0.982720	+	0.185100i	$0.0592608\pi$
$599$			5427.20i			0.370199i	−0.982720	+	0.185100i	$0.940739\pi$
$600$		0			0
$601$			17725.7i			1.20307i	0.798847	+	0.601535i	$0.205444\pi$
$601$			17725.7i			1.20307i	−0.798847	+	0.601535i	$0.794556\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	618.793	+	618.793i	0.0415827	+	0.0415827i
$606$		0			0
$607$	−13487.6			−0.901884			−0.450942	−	0.892553i	$-0.648912\pi$
$607$	−13487.6			−0.901884			−0.450942	+	0.892553i	$0.648912\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4855.57	+	4855.57i	0.321498	+	0.321498i
$612$		0			0
$613$	−16850.4	+	16850.4i	−1.11025	+	1.11025i	−0.117129	+	0.993117i	$0.537369\pi$
$613$	−16850.4	+	16850.4i	−1.11025	+	1.11025i	−0.993117	+	0.117129i	$0.962631\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			535.243i			0.0349239i	0.999848	+	0.0174620i	$0.00555860\pi$
$617$			535.243i			0.0349239i	−0.999848	+	0.0174620i	$0.994441\pi$
$618$		0			0
$619$	19691.0	−	19691.0i	1.27859	−	1.27859i	0.337130	−	0.941458i	$-0.390544\pi$
$619$	19691.0	−	19691.0i	1.27859	−	1.27859i	0.941458	−	0.337130i	$-0.109456\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	137.538			0.00884485
$624$		0			0
$625$	−15358.3			−0.982934
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1883.06	−	1883.06i	0.119368	−	0.119368i
$630$		0			0
$631$			11880.2i			0.749511i	0.927124	+	0.374755i	$0.122273\pi$
$631$			11880.2i			0.749511i	−0.927124	+	0.374755i	$0.877727\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−590.132	+	590.132i	−0.0368798	+	0.0368798i
$636$		0			0
$637$	24419.5	+	24419.5i	1.51889	+	1.51889i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18341.0			−1.13015			−0.565074	−	0.825040i	$-0.691152\pi$
$641$	−18341.0			−1.13015			−0.565074	+	0.825040i	$0.691152\pi$
$642$		0			0
$643$	−7026.21	−	7026.21i	−0.430928	−	0.430928i	0.458016	−	0.888944i	$-0.348560\pi$
$643$	−7026.21	−	7026.21i	−0.430928	−	0.430928i	−0.888944	+	0.458016i	$0.848560\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	21429.7i		−	1.30215i	−0.759015	−	0.651073i	$-0.774319\pi$
$647$		−	21429.7i		−	1.30215i	0.759015	−	0.651073i	$-0.225681\pi$
$648$		0			0
$649$			7879.56i			0.476579i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8681.22	−	8681.22i	−0.520248	−	0.520248i	0.397398	−	0.917646i	$-0.369913\pi$
$653$	−8681.22	−	8681.22i	−0.520248	−	0.520248i	−0.917646	+	0.397398i	$0.869913\pi$
$654$		0			0
$655$	947.284			0.0565091
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4151.60	−	4151.60i	−0.245407	−	0.245407i	0.573675	−	0.819083i	$-0.305517\pi$
$659$	−4151.60	−	4151.60i	−0.245407	−	0.245407i	−0.819083	+	0.573675i	$0.805517\pi$
$660$		0			0
$661$	12239.0	−	12239.0i	0.720182	−	0.720182i	−0.248460	−	0.968642i	$-0.579924\pi$
$661$	12239.0	−	12239.0i	0.720182	−	0.720182i	0.968642	+	0.248460i	$0.0799244\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1903.98i		−	0.111027i
$666$		0			0
$667$	4451.75	−	4451.75i	0.258429	−	0.258429i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.22827			−0.000473396
$672$		0			0
$673$	6528.62			0.373937			0.186969	−	0.982366i	$-0.440134\pi$
$673$	6528.62			0.373937			0.186969	+	0.982366i	$0.440134\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14220.6	−	14220.6i	0.807299	−	0.807299i	−0.176925	−	0.984224i	$-0.556615\pi$
$677$	14220.6	−	14220.6i	0.807299	−	0.807299i	0.984224	+	0.176925i	$0.0566151\pi$
$678$		0			0
$679$		−	11038.2i		−	0.623867i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21419.5	−	21419.5i	1.19999	−	1.19999i	0.225827	−	0.974167i	$-0.427492\pi$
$683$	21419.5	−	21419.5i	1.19999	−	1.19999i	0.974167	−	0.225827i	$-0.0725084\pi$
$684$		0			0
$685$	952.149	+	952.149i	0.0531091	+	0.0531091i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	37797.0			2.08991
$690$		0			0
$691$	16537.1	+	16537.1i	0.910423	+	0.910423i	0.996305	−	0.0858818i	$-0.0273708\pi$
$691$	16537.1	+	16537.1i	0.910423	+	0.910423i	−0.0858818	+	0.996305i	$0.527371\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	331.484i		−	0.0180920i
$696$		0			0
$697$		−	13291.8i		−	0.722331i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3026.52	+	3026.52i	0.163067	+	0.163067i	0.783924	−	0.620857i	$-0.213215\pi$
$701$	3026.52	+	3026.52i	0.163067	+	0.163067i	−0.620857	+	0.783924i	$0.713215\pi$
$702$		0			0
$703$	−2381.82			−0.127784
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11371.5	−	11371.5i	−0.604908	−	0.604908i
$708$		0			0
$709$	3981.14	−	3981.14i	0.210881	−	0.210881i	−0.593761	−	0.804642i	$-0.702357\pi$
$709$	3981.14	−	3981.14i	0.210881	−	0.210881i	0.804642	+	0.593761i	$0.202357\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			622.710i			0.0327078i
$714$		0			0
$715$	−703.160	+	703.160i	−0.0367786	+	0.0367786i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5682.25			−0.294732			−0.147366	−	0.989082i	$-0.547079\pi$
$719$	−5682.25			−0.294732			−0.147366	+	0.989082i	$0.547079\pi$
$720$		0			0
$721$	8955.64			0.462587
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7880.09	−	7880.09i	0.403668	−	0.403668i
$726$		0			0
$727$		−	18883.0i		−	0.963317i	−0.876359	−	0.481658i	$-0.840035\pi$
$727$		−	18883.0i		−	0.963317i	0.876359	−	0.481658i	$-0.159965\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10488.8	+	10488.8i	−0.530698	+	0.530698i
$732$		0			0
$733$	−24962.5	−	24962.5i	−1.25786	−	1.25786i	−0.952113	−	0.305748i	$-0.901094\pi$
$733$	−24962.5	−	24962.5i	−1.25786	−	1.25786i	−0.305748	−	0.952113i	$-0.598906\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13712.9			−0.685375
$738$		0			0
$739$	−6202.38	−	6202.38i	−0.308739	−	0.308739i	0.535681	−	0.844420i	$-0.320055\pi$
$739$	−6202.38	−	6202.38i	−0.308739	−	0.308739i	−0.844420	+	0.535681i	$0.820055\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	30.9140i		−	0.00152641i	−1.00000		0.000763205i	$-0.999757\pi$
$743$		−	30.9140i		−	0.00152641i	1.00000		0.000763205i	$-0.000242936\pi$
$744$		0			0
$745$		−	708.187i		−	0.0348268i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17496.5	+	17496.5i	0.853547	+	0.853547i
$750$		0			0
$751$	16318.5			0.792905			0.396453	−	0.918055i	$-0.370241\pi$
$751$	16318.5			0.792905			0.396453	+	0.918055i	$0.370241\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	95.8865	+	95.8865i	0.00462208	+	0.00462208i
$756$		0			0
$757$	9854.59	−	9854.59i	0.473146	−	0.473146i	−0.429786	−	0.902931i	$-0.641411\pi$
$757$	9854.59	−	9854.59i	0.473146	−	0.473146i	0.902931	+	0.429786i	$0.141411\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	3823.42i		−	0.182127i	−0.995845	−	0.0910637i	$-0.970973\pi$
$761$		−	3823.42i		−	0.182127i	0.995845	−	0.0910637i	$-0.0290267\pi$
$762$		0			0
$763$	−27596.4	+	27596.4i	−1.30938	+	1.30938i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	31550.2			1.48528
$768$		0			0
$769$	31689.1			1.48601			0.743003	−	0.669288i	$-0.233401\pi$
$769$	31689.1			1.48601			0.743003	+	0.669288i	$0.233401\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1305.84	−	1305.84i	0.0607604	−	0.0607604i	−0.676074	−	0.736834i	$-0.736320\pi$
$773$	1305.84	−	1305.84i	0.0607604	−	0.0607604i	0.736834	+	0.676074i	$0.236320\pi$
$774$		0			0
$775$			1102.27i			0.0510898i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8406.21	+	8406.21i	−0.386629	+	0.386629i
$780$		0			0
$781$	−5088.18	−	5088.18i	−0.233123	−	0.233123i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−842.465			−0.0383043
$786$		0			0
$787$	−14399.5	−	14399.5i	−0.652209	−	0.652209i	0.301316	−	0.953524i	$-0.402574\pi$
$787$	−14399.5	−	14399.5i	−0.652209	−	0.652209i	−0.953524	+	0.301316i	$0.902574\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			53047.6i			2.38452i
$792$		0			0
$793$			32.9465i			0.00147537i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5572.19	+	5572.19i	0.247650	+	0.247650i	0.820006	−	0.572356i	$-0.193970\pi$
$797$	5572.19	+	5572.19i	0.247650	+	0.247650i	−0.572356	+	0.820006i	$0.693970\pi$
$798$		0			0
$799$	8670.13			0.383889
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4546.40	+	4546.40i	0.199800	+	0.199800i
$804$		0			0
$805$	1218.79	−	1218.79i	0.0533622	−	0.0533622i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	5081.28i		−	0.220826i	−0.993886	−	0.110413i	$-0.964783\pi$
$809$		−	5081.28i		−	0.220826i	0.993886	−	0.110413i	$-0.0352174\pi$
$810$		0			0
$811$	−4493.19	+	4493.19i	−0.194546	+	0.194546i	−0.797657	−	0.603111i	$-0.793928\pi$
$811$	−4493.19	+	4493.19i	−0.194546	+	0.194546i	0.603111	+	0.797657i	$0.293928\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2235.36			−0.0960752
$816$		0			0
$817$	13266.9			0.568114
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2115.54	+	2115.54i	−0.0899304	+	0.0899304i	−0.750641	−	0.660710i	$-0.770255\pi$
$821$	−2115.54	+	2115.54i	−0.0899304	+	0.0899304i	0.660710	+	0.750641i	$0.270255\pi$
$822$		0			0
$823$		−	24432.6i		−	1.03483i	−0.855734	−	0.517416i	$-0.826894\pi$
$823$		−	24432.6i		−	1.03483i	0.855734	−	0.517416i	$-0.173106\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	21051.4	−	21051.4i	0.885162	−	0.885162i	−0.108892	−	0.994054i	$-0.534730\pi$
$827$	21051.4	−	21051.4i	0.885162	−	0.885162i	0.994054	+	0.108892i	$0.0347302\pi$
$828$		0			0
$829$	25442.0	+	25442.0i	1.06591	+	1.06591i	0.997669	+	0.0682392i	$0.0217381\pi$
$829$	25442.0	+	25442.0i	1.06591	+	1.06591i	0.0682392	+	0.997669i	$0.478262\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	43603.5			1.81365
$834$		0			0
$835$	−2299.17	−	2299.17i	−0.0952889	−	0.0952889i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	18757.2i		−	0.771837i	−0.922533	−	0.385919i	$-0.873885\pi$
$839$		−	18757.2i		−	0.771837i	0.922533	−	0.385919i	$-0.126115\pi$
$840$		0			0
$841$		−	16349.4i		−	0.670358i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1504.22	+	1504.22i	0.0612389	+	0.0612389i
$846$		0			0
$847$	30152.2			1.22319
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1524.67	−	1524.67i	−0.0614160	−	0.0614160i
$852$		0			0
$853$	−6100.17	+	6100.17i	−0.244860	+	0.244860i	−0.818857	−	0.573997i	$-0.805392\pi$
$853$	−6100.17	+	6100.17i	−0.244860	+	0.244860i	0.573997	+	0.818857i	$0.305392\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	2079.04i		−	0.0828687i	−0.999141	−	0.0414344i	$-0.986807\pi$
$857$		−	2079.04i		−	0.0828687i	0.999141	−	0.0414344i	$-0.0131927\pi$
$858$		0			0
$859$	−8301.95	+	8301.95i	−0.329754	+	0.329754i	−0.852493	−	0.522739i	$-0.824910\pi$
$859$	−8301.95	+	8301.95i	−0.329754	+	0.329754i	0.522739	+	0.852493i	$0.324910\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	43830.8			1.72887			0.864436	−	0.502743i	$-0.167676\pi$
$863$	43830.8			1.72887			0.864436	+	0.502743i	$0.167676\pi$
$864$		0			0
$865$	−3134.18			−0.123197
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8560.14	−	8560.14i	0.334158	−	0.334158i
$870$		0			0
$871$			54907.3i			2.13601i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4327.14	−	4327.14i	0.167182	−	0.167182i
$876$		0			0
$877$	−2565.89	−	2565.89i	−0.0987958	−	0.0987958i	0.655981	−	0.754777i	$-0.272255\pi$
$877$	−2565.89	−	2565.89i	−0.0987958	−	0.0987958i	−0.754777	+	0.655981i	$0.772255\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−26429.8			−1.01072			−0.505360	−	0.862909i	$-0.668640\pi$
$881$	−26429.8			−1.01072			−0.505360	+	0.862909i	$0.668640\pi$
$882$		0			0
$883$	20708.4	+	20708.4i	0.789232	+	0.789232i	0.981368	−	0.192136i	$-0.0615415\pi$
$883$	20708.4	+	20708.4i	0.789232	+	0.789232i	−0.192136	+	0.981368i	$0.561542\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	22350.0i		−	0.846042i	−0.906120	−	0.423021i	$-0.860970\pi$
$887$		−	22350.0i		−	0.846042i	0.906120	−	0.423021i	$-0.139030\pi$
$888$		0			0
$889$			28755.6i			1.08485i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5483.28	−	5483.28i	−0.205477	−	0.205477i
$894$		0			0
$895$	−1476.28			−0.0551360
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−562.292	−	562.292i	−0.0208604	−	0.0208604i
$900$		0			0
$901$	33745.2	−	33745.2i	1.24774	−	1.24774i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1879.95i		−	0.0690516i
$906$		0			0
$907$	4086.93	−	4086.93i	0.149619	−	0.149619i	−0.628329	−	0.777948i	$-0.716261\pi$
$907$	4086.93	−	4086.93i	0.149619	−	0.149619i	0.777948	+	0.628329i	$0.216261\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−20024.6			−0.728259			−0.364130	−	0.931348i	$-0.618633\pi$
$911$	−20024.6			−0.728259			−0.364130	+	0.931348i	$0.618633\pi$
$912$		0			0
$913$	−22974.8			−0.832810
$914$		0			0
$915$		0			0
$916$		0			0
$917$	23079.3	−	23079.3i	0.831131	−	0.831131i
$918$		0			0
$919$			26977.7i			0.968349i	0.874971	+	0.484175i	$0.160880\pi$
$919$			26977.7i			0.968349i	−0.874971	+	0.484175i	$0.839120\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−20373.4	+	20373.4i	−0.726542	+	0.726542i
$924$		0			0
$925$	−2698.84	−	2698.84i	−0.0959321	−	0.0959321i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	34371.4			1.21387			0.606937	−	0.794750i	$-0.292398\pi$
$929$	34371.4			1.21387			0.606937	+	0.794750i	$0.292398\pi$
$930$		0			0
$931$	−27576.3	−	27576.3i	−0.970760	−	0.970760i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			1255.57i			0.0439159i
$936$		0			0
$937$			32901.8i			1.14712i	0.819163	+	0.573561i	$0.194439\pi$
$937$			32901.8i			1.14712i	−0.819163	+	0.573561i	$0.805561\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−33381.6	−	33381.6i	−1.15644	−	1.15644i	−0.985236	−	0.171204i	$-0.945234\pi$
$941$	−33381.6	−	33381.6i	−1.15644	−	1.15644i	−0.171204	−	0.985236i	$-0.554766\pi$
$942$		0			0
$943$	−10762.1			−0.371646
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13611.9	+	13611.9i	0.467084	+	0.467084i	0.900969	−	0.433884i	$-0.142857\pi$
$947$	13611.9	+	13611.9i	0.467084	+	0.467084i	−0.433884	+	0.900969i	$0.642857\pi$
$948$		0			0
$949$	18204.1	−	18204.1i	0.622686	−	0.622686i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	4572.49i		−	0.155422i	−0.996976	−	0.0777112i	$-0.975239\pi$
$953$		−	4572.49i		−	0.155422i	0.996976	−	0.0777112i	$-0.0247612\pi$
$954$		0			0
$955$	2140.25	−	2140.25i	0.0725202	−	0.0725202i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	46395.7			1.56225
$960$		0			0
$961$	−29712.3			−0.997360
$962$		0			0
$963$		0			0
$964$		0			0
$965$	312.510	−	312.510i	0.0104249	−	0.0104249i
$966$		0			0
$967$		−	33060.9i		−	1.09945i	−0.835346	−	0.549725i	$-0.814732\pi$
$967$		−	33060.9i		−	1.09945i	0.835346	−	0.549725i	$-0.185268\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15553.4	+	15553.4i	−0.514041	+	0.514041i	−0.915762	−	0.401721i	$-0.868412\pi$
$971$	−15553.4	+	15553.4i	−0.514041	+	0.514041i	0.401721	+	0.915762i	$0.368412\pi$
$972$		0			0
$973$	−8076.18	−	8076.18i	−0.266095	−	0.266095i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1241.54			−0.0406555			−0.0203277	−	0.999793i	$-0.506471\pi$
$977$	−1241.54			−0.0406555			−0.0203277	+	0.999793i	$0.506471\pi$
$978$		0			0
$979$	57.3608	+	57.3608i	0.00187258	+	0.00187258i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	10797.7i		−	0.350349i	−0.984537	−	0.175175i	$-0.943951\pi$
$983$		−	10797.7i		−	0.350349i	0.984537	−	0.175175i	$-0.0560490\pi$
$984$		0			0
$985$		−	1343.67i		−	0.0434648i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8492.49	+	8492.49i	0.273049	+	0.273049i
$990$		0			0
$991$	10204.8			0.327112			0.163556	−	0.986534i	$-0.447704\pi$
$991$	10204.8			0.327112			0.163556	+	0.986534i	$0.447704\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1380.20	+	1380.20i	0.0439752	+	0.0439752i
$996$		0			0
$997$	15046.7	−	15046.7i	0.477967	−	0.477967i	−0.426514	−	0.904481i	$-0.640259\pi$
$997$	15046.7	−	15046.7i	0.477967	−	0.477967i	0.904481	+	0.426514i	$0.140259\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.a.145.3		10
3.2	odd	2		64.4.e.a.17.2		10
4.3	odd	2		144.4.k.a.109.2		10
12.11	even	2		16.4.e.a.13.4	yes	10
16.5	even	4	inner	576.4.k.a.433.3		10
16.11	odd	4		144.4.k.a.37.2		10
24.5	odd	2		128.4.e.a.33.4		10
24.11	even	2		128.4.e.b.33.2		10
48.5	odd	4		64.4.e.a.49.2		10
48.11	even	4		16.4.e.a.5.4	✓	10
48.29	odd	4		128.4.e.a.97.4		10
48.35	even	4		128.4.e.b.97.2		10
96.5	odd	8		1024.4.a.m.1.7		10
96.11	even	8		1024.4.a.n.1.7		10
96.29	odd	8		1024.4.b.k.513.4		10
96.35	even	8		1024.4.b.j.513.7		10
96.53	odd	8		1024.4.a.m.1.4		10
96.59	even	8		1024.4.a.n.1.4		10
96.77	odd	8		1024.4.b.k.513.7		10
96.83	even	8		1024.4.b.j.513.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.4	✓	10	48.11	even	4
16.4.e.a.13.4	yes	10	12.11	even	2
64.4.e.a.17.2		10	3.2	odd	2
64.4.e.a.49.2		10	48.5	odd	4
128.4.e.a.33.4		10	24.5	odd	2
128.4.e.a.97.4		10	48.29	odd	4
128.4.e.b.33.2		10	24.11	even	2
128.4.e.b.97.2		10	48.35	even	4
144.4.k.a.37.2		10	16.11	odd	4
144.4.k.a.109.2		10	4.3	odd	2
576.4.k.a.145.3		10	1.1	even	1	trivial
576.4.k.a.433.3		10	16.5	even	4	inner
1024.4.a.m.1.4		10	96.53	odd	8
1024.4.a.m.1.7		10	96.5	odd	8
1024.4.a.n.1.4		10	96.59	even	8
1024.4.a.n.1.7		10	96.11	even	8
1024.4.b.j.513.4		10	96.83	even	8
1024.4.b.j.513.7		10	96.35	even	8
1024.4.b.k.513.4		10	96.29	odd	8
1024.4.b.k.513.7		10	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.596848 - 0.596848i) q^{5} -29.0828i q^{7} +O(q^{10})\)	\(q+(0.596848 - 0.596848i) q^{5} -29.0828i q^{7} +(12.1291 - 12.1291i) q^{11} +(-48.5658 - 48.5658i) q^{13} -86.7193 q^{17} +(54.8442 + 54.8442i) q^{19} +70.2145i q^{23} +124.288i q^{25} +(-63.4021 - 63.4021i) q^{29} +8.86868 q^{31} +(-17.3580 - 17.3580i) q^{35} +(-21.7145 + 21.7145i) q^{37} +153.274i q^{41} +(120.951 - 120.951i) q^{43} -99.9792 q^{47} -502.812 q^{49} +(-389.132 + 389.132i) q^{53} -14.4785i q^{55} +(-324.819 + 324.819i) q^{59} +(-0.339194 - 0.339194i) q^{61} -57.9728 q^{65} +(-565.288 - 565.288i) q^{67} -419.500i q^{71} +374.833i q^{73} +(-352.750 - 352.750i) q^{77} +705.750 q^{79} +(-947.092 - 947.092i) q^{83} +(-51.7582 + 51.7582i) q^{85} +4.72918i q^{89} +(-1412.43 + 1412.43i) q^{91} +65.4673 q^{95} +379.542 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5}+O(q^{10}) \) 10 * q + 2 * q^5	\( 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more