LMFDB - Embedded newform 588.4.f.d.293.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,4,Mod(293,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("588.293");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	588.f (of order $2$, degree $1$, 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:	$34.6931230834$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			293.5
Character	$\chi$	$=$	588.293
Dual form			588.4.f.d.293.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.20506 - 4.08994i) q^{3} -7.66547 q^{5} +(-6.45517 + 26.2170i) q^{9} +O(q^{10})$	$q+(-3.20506 - 4.08994i) q^{3} -7.66547 q^{5} +(-6.45517 + 26.2170i) q^{9} -1.31471i q^{11} -23.9754i q^{13} +(24.5683 + 31.3513i) q^{15} -59.0980 q^{17} +28.2485i q^{19} +75.8909i q^{23} -66.2406 q^{25} +(127.915 - 57.6258i) q^{27} +302.001i q^{29} -93.4786i q^{31} +(-5.37707 + 4.21372i) q^{33} +266.866 q^{37} +(-98.0577 + 76.8425i) q^{39} +142.471 q^{41} +284.654 q^{43} +(49.4819 - 200.965i) q^{45} -209.752 q^{47} +(189.413 + 241.707i) q^{51} -629.531i q^{53} +10.0779i q^{55} +(115.535 - 90.5383i) q^{57} +730.372 q^{59} -544.977i q^{61} +183.782i q^{65} -481.112 q^{67} +(310.389 - 243.235i) q^{69} -46.5477i q^{71} -963.130i q^{73} +(212.305 + 270.920i) q^{75} +1261.64 q^{79} +(-645.662 - 338.470i) q^{81} -841.130 q^{83} +453.014 q^{85} +(1235.17 - 967.933i) q^{87} +1282.34 q^{89} +(-382.322 + 299.605i) q^{93} -216.538i q^{95} +60.2806i q^{97} +(34.4677 + 8.48666i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 64 q^{9}+O(q^{10}) $ 24 * q - 64 * q^9	$ 24 q - 64 q^{9} - 96 q^{15} + 456 q^{25} - 432 q^{37} + 688 q^{39} + 624 q^{43} - 1536 q^{51} - 1360 q^{57} - 528 q^{67} + 3744 q^{79} + 3408 q^{81} + 6912 q^{85} - 5088 q^{93} - 7736 q^{99}+O(q^{100}) $ 24 * q - 64 * q^9 - 96 * q^15 + 456 * q^25 - 432 * q^37 + 688 * q^39 + 624 * q^43 - 1536 * q^51 - 1360 * q^57 - 528 * q^67 + 3744 * q^79 + 3408 * q^81 + 6912 * q^85 - 5088 * q^93 - 7736 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−3.20506	−	4.08994i	−0.616814	−	0.787109i
$3$	−3.20506	−	4.08994i	−0.616814	−	0.787109i
$4$		0			0
$5$	−7.66547			−0.685620			−0.342810	−	0.939405i	$-0.611379\pi$
$5$	−7.66547			−0.685620			−0.342810	+	0.939405i	$0.611379\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−6.45517	+	26.2170i	−0.239080	+	0.971000i
$10$		0			0
$11$		−	1.31471i		−	0.0360363i	−0.999838	−	0.0180182i	$-0.994264\pi$
$11$		−	1.31471i		−	0.0360363i	0.999838	−	0.0180182i	$-0.00573567\pi$
$12$		0			0
$13$		−	23.9754i		−	0.511505i	−0.966742	−	0.255753i	$-0.917677\pi$
$13$		−	23.9754i		−	0.511505i	0.966742	−	0.255753i	$-0.0823233\pi$
$14$		0			0
$15$	24.5683	+	31.3513i	0.422900	+	0.539658i
$16$		0			0
$17$	−59.0980			−0.843140			−0.421570	−	0.906796i	$-0.638521\pi$
$17$	−59.0980			−0.843140			−0.421570	+	0.906796i	$0.638521\pi$
$18$		0			0
$19$			28.2485i			0.341087i	0.985350	+	0.170544i	$0.0545524\pi$
$19$			28.2485i			0.341087i	−0.985350	+	0.170544i	$0.945448\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			75.8909i			0.688015i	0.938967	+	0.344008i	$0.111785\pi$
$23$			75.8909i			0.688015i	−0.938967	+	0.344008i	$0.888215\pi$
$24$		0			0
$25$	−66.2406			−0.529925
$26$		0			0
$27$	127.915	−	57.6258i	0.911751	−	0.410744i
$28$		0			0
$29$			302.001i			1.93380i	0.255151	+	0.966901i	$0.417875\pi$
$29$			302.001i			1.93380i	−0.255151	+	0.966901i	$0.582125\pi$
$30$		0			0
$31$		−	93.4786i		−	0.541589i	−0.962637	−	0.270795i	$-0.912714\pi$
$31$		−	93.4786i		−	0.541589i	0.962637	−	0.270795i	$-0.0872864\pi$
$32$		0			0
$33$	−5.37707	+	4.21372i	−0.0283645	+	0.0222277i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	266.866			1.18574			0.592871	−	0.805297i	$-0.297994\pi$
$37$	266.866			1.18574			0.592871	+	0.805297i	$0.297994\pi$
$38$		0			0
$39$	−98.0577	+	76.8425i	−0.402610	+	0.315504i
$40$		0			0
$41$	142.471			0.542688			0.271344	−	0.962482i	$-0.412532\pi$
$41$	142.471			0.542688			0.271344	+	0.962482i	$0.412532\pi$
$42$		0			0
$43$	284.654			1.00952			0.504760	−	0.863260i	$-0.331581\pi$
$43$	284.654			1.00952			0.504760	+	0.863260i	$0.331581\pi$
$44$		0			0
$45$	49.4819	−	200.965i	0.163918	−	0.665737i
$46$		0			0
$47$	−209.752			−0.650969			−0.325484	−	0.945547i	$-0.605527\pi$
$47$	−209.752			−0.650969			−0.325484	+	0.945547i	$0.605527\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	189.413	+	241.707i	0.520061	+	0.663643i
$52$		0			0
$53$		−	629.531i		−	1.63156i	−0.578362	−	0.815780i	$-0.696308\pi$
$53$		−	629.531i		−	1.63156i	0.578362	−	0.815780i	$-0.303692\pi$
$54$		0			0
$55$			10.0779i			0.0247072i
$56$		0			0
$57$	115.535	−	90.5383i	0.268473	−	0.210387i
$58$		0			0
$59$	730.372			1.61163			0.805817	−	0.592165i	$-0.201727\pi$
$59$	730.372			1.61163			0.805817	+	0.592165i	$0.201727\pi$
$60$		0			0
$61$		−	544.977i		−	1.14389i	−0.820293	−	0.571943i	$-0.806190\pi$
$61$		−	544.977i		−	1.14389i	0.820293	−	0.571943i	$-0.193810\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			183.782i			0.350698i
$66$		0			0
$67$	−481.112			−0.877271			−0.438635	−	0.898665i	$-0.644538\pi$
$67$	−481.112			−0.877271			−0.438635	+	0.898665i	$0.644538\pi$
$68$		0			0
$69$	310.389	−	243.235i	0.541543	−	0.424377i
$70$		0			0
$71$		−	46.5477i		−	0.0778056i	−0.999243	−	0.0389028i	$-0.987614\pi$
$71$		−	46.5477i		−	0.0778056i	0.999243	−	0.0389028i	$-0.0123863\pi$
$72$		0			0
$73$		−	963.130i		−	1.54419i	−0.635507	−	0.772095i	$-0.719209\pi$
$73$		−	963.130i		−	1.54419i	0.635507	−	0.772095i	$-0.280791\pi$
$74$		0			0
$75$	212.305	+	270.920i	0.326865	+	0.417109i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1261.64			1.79677			0.898386	−	0.439206i	$-0.144740\pi$
$79$	1261.64			1.79677			0.898386	+	0.439206i	$0.144740\pi$
$80$		0			0
$81$	−645.662	−	338.470i	−0.885681	−	0.464294i
$82$		0			0
$83$	−841.130			−1.11236			−0.556181	−	0.831061i	$-0.687734\pi$
$83$	−841.130			−1.11236			−0.556181	+	0.831061i	$0.687734\pi$
$84$		0			0
$85$	453.014			0.578074
$86$		0			0
$87$	1235.17	−	967.933i	1.52211	−	1.19280i
$88$		0			0
$89$	1282.34			1.52728			0.763638	−	0.645645i	$-0.223411\pi$
$89$	1282.34			1.52728			0.763638	+	0.645645i	$0.223411\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−382.322	+	299.605i	−0.426289	+	0.334060i
$94$		0			0
$95$		−	216.538i		−	0.233856i
$96$		0			0
$97$			60.2806i			0.0630986i	0.999502	+	0.0315493i	$0.0100441\pi$
$97$			60.2806i			0.0630986i	−0.999502	+	0.0315493i	$0.989956\pi$
$98$		0			0
$99$	34.4677	+	8.48666i	0.0349913	+	0.00861557i
$100$		0			0
$101$	1168.72			1.15140			0.575701	−	0.817660i	$-0.304729\pi$
$101$	1168.72			1.15140			0.575701	+	0.817660i	$0.304729\pi$
$102$		0			0
$103$			18.3235i			0.0175288i	0.999962	+	0.00876439i	$0.00278983\pi$
$103$			18.3235i			0.0175288i	−0.999962	+	0.00876439i	$0.997210\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			952.591i			0.860658i	0.902672	+	0.430329i	$0.141602\pi$
$107$			952.591i			0.860658i	−0.902672	+	0.430329i	$0.858398\pi$
$108$		0			0
$109$	−707.880			−0.622042			−0.311021	−	0.950403i	$-0.600671\pi$
$109$	−707.880			−0.622042			−0.311021	+	0.950403i	$0.600671\pi$
$110$		0			0
$111$	−855.322	−	1091.46i	−0.731383	−	0.933309i
$112$		0			0
$113$			9.01287i			0.00750318i	0.999993	+	0.00375159i	$0.00119417\pi$
$113$			9.01287i			0.00750318i	−0.999993	+	0.00375159i	$0.998806\pi$
$114$		0			0
$115$		−	581.739i		−	0.471717i
$116$		0			0
$117$	628.562	+	154.765i	0.496672	+	0.122291i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1329.27			0.998701
$122$		0			0
$123$	−456.628	−	582.697i	−0.334738	−	0.427154i
$124$		0			0
$125$	1465.95			1.04895
$126$		0			0
$127$	387.174			0.270521			0.135260	−	0.990810i	$-0.456813\pi$
$127$	387.174			0.270521			0.135260	+	0.990810i	$0.456813\pi$
$128$		0			0
$129$	−912.334	−	1164.22i	−0.622686	−	0.794601i
$130$		0			0
$131$	−1411.56			−0.941437			−0.470718	−	0.882283i	$-0.656005\pi$
$131$	−1411.56			−0.941437			−0.470718	+	0.882283i	$0.656005\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−980.528	+	441.729i	−0.625114	+	0.281615i
$136$		0			0
$137$		−	728.843i		−	0.454520i	−0.973834	−	0.227260i	$-0.927023\pi$
$137$		−	728.843i		−	0.454520i	0.973834	−	0.227260i	$-0.0729768\pi$
$138$		0			0
$139$			1262.40i			0.770324i	0.922849	+	0.385162i	$0.125854\pi$
$139$			1262.40i			0.770324i	−0.922849	+	0.385162i	$0.874146\pi$
$140$		0			0
$141$	672.269	+	857.874i	0.401527	+	0.512383i
$142$		0			0
$143$	−31.5206			−0.0184328
$144$		0			0
$145$		−	2314.98i		−	1.32585i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			943.979i			0.519018i	0.965741	+	0.259509i	$0.0835608\pi$
$149$			943.979i			0.519018i	−0.965741	+	0.259509i	$0.916439\pi$
$150$		0			0
$151$	307.542			0.165744			0.0828722	−	0.996560i	$-0.473591\pi$
$151$	307.542			0.165744			0.0828722	+	0.996560i	$0.473591\pi$
$152$		0			0
$153$	381.488	−	1549.37i	0.201578	−	0.818689i
$154$		0			0
$155$			716.557i			0.371324i
$156$		0			0
$157$			1864.22i			0.947651i	0.880619	+	0.473825i	$0.157127\pi$
$157$			1864.22i			0.947651i	−0.880619	+	0.473825i	$0.842873\pi$
$158$		0			0
$159$	−2574.74	+	2017.68i	−1.28422	+	1.00637i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3241.22			1.55750			0.778749	−	0.627335i	$-0.215854\pi$
$163$	3241.22			1.55750			0.778749	+	0.627335i	$0.215854\pi$
$164$		0			0
$165$	41.2178	−	32.3001i	0.0194473	−	0.0152398i
$166$		0			0
$167$	2021.13			0.936523			0.468262	−	0.883590i	$-0.344881\pi$
$167$	2021.13			0.936523			0.468262	+	0.883590i	$0.344881\pi$
$168$		0			0
$169$	1622.18			0.738362
$170$		0			0
$171$	−740.591	−	182.349i	−0.331196	−	0.0815472i
$172$		0			0
$173$	700.588			0.307889			0.153944	−	0.988080i	$-0.450802\pi$
$173$	700.588			0.307889			0.153944	+	0.988080i	$0.450802\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2340.89	−	2987.18i	−0.994079	−	1.26853i
$178$		0			0
$179$			2869.24i			1.19809i	0.800717	+	0.599043i	$0.204452\pi$
$179$			2869.24i			1.19809i	−0.800717	+	0.599043i	$0.795548\pi$
$180$		0			0
$181$			1974.33i			0.810778i	0.914144	+	0.405389i	$0.132864\pi$
$181$			1974.33i			0.810778i	−0.914144	+	0.405389i	$0.867136\pi$
$182$		0			0
$183$	−2228.92	+	1746.68i	−0.900363	+	0.705566i
$184$		0			0
$185$	−2045.65			−0.812969
$186$		0			0
$187$			77.6967i			0.0303837i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1087.70i		−	0.412060i	−0.978546	−	0.206030i	$-0.933946\pi$
$191$		−	1087.70i		−	0.412060i	0.978546	−	0.206030i	$-0.0660544\pi$
$192$		0			0
$193$	3835.04			1.43032			0.715161	−	0.698959i	$-0.246353\pi$
$193$	3835.04			1.43032			0.715161	+	0.698959i	$0.246353\pi$
$194$		0			0
$195$	751.658	−	589.034i	0.276038	−	0.216316i
$196$		0			0
$197$			1635.24i			0.591400i	0.955281	+	0.295700i	$0.0955528\pi$
$197$			1635.24i			0.591400i	−0.955281	+	0.295700i	$0.904447\pi$
$198$		0			0
$199$			4312.67i			1.53627i	0.640289	+	0.768134i	$0.278815\pi$
$199$			4312.67i			1.53627i	−0.640289	+	0.768134i	$0.721185\pi$
$200$		0			0
$201$	1541.99	+	1967.72i	0.541113	+	0.690507i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1092.11			−0.372078
$206$		0			0
$207$	−1989.63	−	489.888i	−0.668062	−	0.164491i
$208$		0			0
$209$	37.1386			0.0122915
$210$		0			0
$211$	−2882.49			−0.940468			−0.470234	−	0.882542i	$-0.655831\pi$
$211$	−2882.49			−0.940468			−0.470234	+	0.882542i	$0.655831\pi$
$212$		0			0
$213$	−190.377	+	149.188i	−0.0612415	+	0.0479916i
$214$		0			0
$215$	−2182.01			−0.692147
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−3939.14	+	3086.89i	−1.21545	+	0.952478i
$220$		0			0
$221$			1416.90i			0.431271i
$222$		0			0
$223$			3378.49i			1.01453i	0.861790	+	0.507265i	$0.169344\pi$
$223$			3378.49i			1.01453i	−0.861790	+	0.507265i	$0.830656\pi$
$224$		0			0
$225$	427.594	−	1736.63i	0.126695	−	0.514557i
$226$		0			0
$227$	−5077.67			−1.48466			−0.742328	−	0.670037i	$-0.766278\pi$
$227$	−5077.67			−1.48466			−0.742328	+	0.670037i	$0.766278\pi$
$228$		0			0
$229$		−	4370.47i		−	1.26117i	−0.776118	−	0.630587i	$-0.782814\pi$
$229$		−	4370.47i		−	1.26117i	0.776118	−	0.630587i	$-0.217186\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3851.58i		−	1.08294i	−0.840720	−	0.541470i	$-0.817868\pi$
$233$		−	3851.58i		−	1.08294i	0.840720	−	0.541470i	$-0.182132\pi$
$234$		0			0
$235$	1607.85			0.446317
$236$		0			0
$237$	−4043.62	−	5160.01i	−1.10827	−	1.41426i
$238$		0			0
$239$		−	7067.79i		−	1.91288i	−0.291937	−	0.956438i	$-0.594300\pi$
$239$		−	7067.79i		−	1.91288i	0.291937	−	0.956438i	$-0.405700\pi$
$240$		0			0
$241$			3044.53i			0.813756i	0.913483	+	0.406878i	$0.133383\pi$
$241$			3044.53i			0.813756i	−0.913483	+	0.406878i	$0.866617\pi$
$242$		0			0
$243$	685.063	+	3725.53i	0.180851	+	0.983510i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	677.269			0.174468
$248$		0			0
$249$	2695.87	+	3440.17i	0.686120	+	0.875549i
$250$		0			0
$251$	−5439.29			−1.36783			−0.683914	−	0.729562i	$-0.739724\pi$
$251$	−5439.29			−1.36783			−0.683914	+	0.729562i	$0.739724\pi$
$252$		0			0
$253$	99.7744			0.0247935
$254$		0			0
$255$	−1451.94	−	1852.80i	−0.356564	−	0.455007i
$256$		0			0
$257$	2278.75			0.553091			0.276545	−	0.961001i	$-0.410810\pi$
$257$	2278.75			0.553091			0.276545	+	0.961001i	$0.410810\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−7917.57	−	1949.47i	−1.87772	−	0.462334i
$262$		0			0
$263$		−	1159.29i		−	0.271805i	−0.990722	−	0.135902i	$-0.956607\pi$
$263$		−	1159.29i		−	0.271805i	0.990722	−	0.135902i	$-0.0433933\pi$
$264$		0			0
$265$			4825.65i			1.11863i
$266$		0			0
$267$	−4109.97	−	5244.68i	−0.942045	−	1.20213i
$268$		0			0
$269$	3747.89			0.849490			0.424745	−	0.905313i	$-0.360364\pi$
$269$	3747.89			0.849490			0.424745	+	0.905313i	$0.360364\pi$
$270$		0			0
$271$			2045.83i			0.458580i	0.973358	+	0.229290i	$0.0736405\pi$
$271$			2045.83i			0.458580i	−0.973358	+	0.229290i	$0.926360\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			87.0871i			0.0190965i
$276$		0			0
$277$	4843.15			1.05053			0.525265	−	0.850939i	$-0.323966\pi$
$277$	4843.15			1.05053			0.525265	+	0.850939i	$0.323966\pi$
$278$		0			0
$279$	2450.73	+	603.420i	0.525883	+	0.129483i
$280$		0			0
$281$		−	827.961i		−	0.175772i	−0.996131	−	0.0878862i	$-0.971989\pi$
$281$		−	827.961i		−	0.175772i	0.996131	−	0.0878862i	$-0.0280112\pi$
$282$		0			0
$283$			8547.60i			1.79541i	0.440594	+	0.897707i	$0.354768\pi$
$283$			8547.60i			1.79541i	−0.440594	+	0.897707i	$0.645232\pi$
$284$		0			0
$285$	−885.627	+	694.018i	−0.184070	+	0.144246i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1420.42			−0.289115
$290$		0			0
$291$	246.544	−	193.203i	0.0496655	−	0.0389201i
$292$		0			0
$293$	−7266.94			−1.44894			−0.724469	−	0.689307i	$-0.757915\pi$
$293$	−7266.94			−1.44894			−0.724469	+	0.689307i	$0.757915\pi$
$294$		0			0
$295$	−5598.64			−1.10497
$296$		0			0
$297$	−75.7612	−	168.171i	−0.0148017	−	0.0328561i
$298$		0			0
$299$	1819.51			0.351923
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3745.81	−	4779.98i	−0.710202	−	0.906279i
$304$		0			0
$305$			4177.50i			0.784272i
$306$		0			0
$307$			4124.73i			0.766811i	0.923580	+	0.383406i	$0.125249\pi$
$307$			4124.73i			0.766811i	−0.923580	+	0.383406i	$0.874751\pi$
$308$		0			0
$309$	74.9418	−	58.7278i	0.0137971	−	0.0108120i
$310$		0			0
$311$	−3298.30			−0.601381			−0.300691	−	0.953722i	$-0.597217\pi$
$311$	−3298.30			−0.601381			−0.300691	+	0.953722i	$0.597217\pi$
$312$		0			0
$313$		−	5990.94i		−	1.08188i	−0.841062	−	0.540939i	$-0.818069\pi$
$313$		−	5990.94i		−	1.08188i	0.841062	−	0.540939i	$-0.181931\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7094.89i		−	1.25706i	−0.777784	−	0.628531i	$-0.783656\pi$
$317$		−	7094.89i		−	1.25706i	0.777784	−	0.628531i	$-0.216344\pi$
$318$		0			0
$319$	397.044			0.0696871
$320$		0			0
$321$	3896.04	−	3053.11i	0.677431	−	0.530866i
$322$		0			0
$323$		−	1669.43i		−	0.287584i
$324$		0			0
$325$			1588.14i			0.271060i
$326$		0			0
$327$	2268.80	+	2895.18i	0.383685	+	0.489615i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	9163.82			1.52172			0.760860	−	0.648916i	$-0.224778\pi$
$331$	9163.82			1.52172			0.760860	+	0.648916i	$0.224778\pi$
$332$		0			0
$333$	−1722.66	+	6996.42i	−0.283488	+	1.15136i
$334$		0			0
$335$	3687.94			0.601474
$336$		0			0
$337$	1546.78			0.250025			0.125012	−	0.992155i	$-0.460103\pi$
$337$	1546.78			0.250025			0.125012	+	0.992155i	$0.460103\pi$
$338$		0			0
$339$	36.8621	−	28.8868i	0.00590582	−	0.00462807i
$340$		0			0
$341$	−122.897			−0.0195169
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2379.28	+	1864.51i	−0.371292	+	0.290962i
$346$		0			0
$347$		−	9198.55i		−	1.42307i	−0.702653	−	0.711533i	$-0.748001\pi$
$347$		−	9198.55i		−	1.42307i	0.702653	−	0.711533i	$-0.251999\pi$
$348$		0			0
$349$			6916.38i			1.06082i	0.847742	+	0.530409i	$0.177962\pi$
$349$			6916.38i			1.06082i	−0.847742	+	0.530409i	$0.822038\pi$
$350$		0			0
$351$	−1381.60	−	3066.81i	−0.210098	−	0.466365i
$352$		0			0
$353$	2753.41			0.415154			0.207577	−	0.978219i	$-0.433442\pi$
$353$	2753.41			0.415154			0.207577	+	0.978219i	$0.433442\pi$
$354$		0			0
$355$			356.810i			0.0533451i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	3556.30i		−	0.522825i	−0.965227	−	0.261413i	$-0.915812\pi$
$359$		−	3556.30i		−	0.522825i	0.965227	−	0.261413i	$-0.0841883\pi$
$360$		0			0
$361$	6061.02			0.883660
$362$		0			0
$363$	−4260.40	−	5436.64i	−0.616013	−	0.786087i
$364$		0			0
$365$			7382.84i			1.05873i
$366$		0			0
$367$			5529.67i			0.786502i	0.919431	+	0.393251i	$0.128650\pi$
$367$			5529.67i			0.786502i	−0.919431	+	0.393251i	$0.871350\pi$
$368$		0			0
$369$	−919.673	+	3735.16i	−0.129746	+	0.526950i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1133.94			−0.157408			−0.0787039	−	0.996898i	$-0.525078\pi$
$373$	−1133.94			−0.157408			−0.0787039	+	0.996898i	$0.525078\pi$
$374$		0			0
$375$	−4698.45	−	5995.64i	−0.647006	−	0.825636i
$376$		0			0
$377$	7240.60			0.989150
$378$		0			0
$379$	4071.36			0.551799			0.275900	−	0.961186i	$-0.411024\pi$
$379$	4071.36			0.551799			0.275900	+	0.961186i	$0.411024\pi$
$380$		0			0
$381$	−1240.92	−	1583.52i	−0.166861	−	0.212929i
$382$		0			0
$383$	−14138.5			−1.88628			−0.943140	−	0.332395i	$-0.892143\pi$
$383$	−14138.5			−1.88628			−0.943140	+	0.332395i	$0.892143\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1837.49	+	7462.77i	−0.241356	+	0.980243i
$388$		0			0
$389$			10689.3i			1.39323i	0.717445	+	0.696615i	$0.245311\pi$
$389$			10689.3i			1.39323i	−0.717445	+	0.696615i	$0.754689\pi$
$390$		0			0
$391$		−	4485.00i		−	0.580093i
$392$		0			0
$393$	4524.12	+	5773.18i	0.580692	+	0.741013i
$394$		0			0
$395$	−9671.02			−1.23190
$396$		0			0
$397$		−	10137.0i		−	1.28152i	−0.767742	−	0.640760i	$-0.778619\pi$
$397$		−	10137.0i		−	1.28152i	0.767742	−	0.640760i	$-0.221381\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			8420.32i			1.04861i	0.851532	+	0.524303i	$0.175674\pi$
$401$			8420.32i			1.04861i	−0.851532	+	0.524303i	$0.824326\pi$
$402$		0			0
$403$	−2241.18			−0.277026
$404$		0			0
$405$	4949.30	+	2594.53i	0.607241	+	0.318329i
$406$		0			0
$407$		−	350.851i		−	0.0427298i
$408$		0			0
$409$		−	9262.60i		−	1.11982i	−0.828554	−	0.559909i	$-0.810836\pi$
$409$		−	9262.60i		−	1.11982i	0.828554	−	0.559909i	$-0.189164\pi$
$410$		0			0
$411$	−2980.92	+	2335.99i	−0.357757	+	0.280355i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6447.65			0.762657
$416$		0			0
$417$	5163.12	−	4046.06i	0.606329	−	0.475147i
$418$		0			0
$419$	−883.038			−0.102958			−0.0514788	−	0.998674i	$-0.516393\pi$
$419$	−883.038			−0.102958			−0.0514788	+	0.998674i	$0.516393\pi$
$420$		0			0
$421$	6313.53			0.730886			0.365443	−	0.930834i	$-0.380918\pi$
$421$	6313.53			0.730886			0.365443	+	0.930834i	$0.380918\pi$
$422$		0			0
$423$	1353.99	−	5499.08i	0.155634	−	0.632090i
$424$		0			0
$425$	3914.69			0.446801
$426$		0			0
$427$		0			0
$428$		0			0
$429$	101.025	+	128.917i	0.0113696	+	0.0145086i
$430$		0			0
$431$		−	7264.48i		−	0.811874i	−0.913901	−	0.405937i	$-0.866945\pi$
$431$		−	7264.48i		−	0.811874i	0.913901	−	0.405937i	$-0.133055\pi$
$432$		0			0
$433$		−	4242.72i		−	0.470883i	−0.971889	−	0.235441i	$-0.924346\pi$
$433$		−	4242.72i		−	0.470883i	0.971889	−	0.235441i	$-0.0756536\pi$
$434$		0			0
$435$	−9468.13	+	7419.66i	−1.04359	+	0.817806i
$436$		0			0
$437$	−2143.81			−0.234673
$438$		0			0
$439$			10889.5i			1.18389i	0.805980	+	0.591943i	$0.201639\pi$
$439$			10889.5i			1.18389i	−0.805980	+	0.591943i	$0.798361\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			16079.8i			1.72455i	0.506438	+	0.862276i	$0.330962\pi$
$443$			16079.8i			1.72455i	−0.506438	+	0.862276i	$0.669038\pi$
$444$		0			0
$445$	−9829.71			−1.04713
$446$		0			0
$447$	3860.81	−	3025.51i	0.408524	−	0.320138i
$448$		0			0
$449$			13638.3i			1.43348i	0.697341	+	0.716740i	$0.254366\pi$
$449$			13638.3i			1.43348i	−0.697341	+	0.716740i	$0.745634\pi$
$450$		0			0
$451$		−	187.308i		−	0.0195565i
$452$		0			0
$453$	−985.691	−	1257.83i	−0.102234	−	0.130459i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1878.80			0.192312			0.0961559	−	0.995366i	$-0.469345\pi$
$457$	1878.80			0.192312			0.0961559	+	0.995366i	$0.469345\pi$
$458$		0			0
$459$	−7559.53	+	3405.57i	−0.768733	+	0.346315i
$460$		0			0
$461$	2579.48			0.260604			0.130302	−	0.991474i	$-0.458405\pi$
$461$	2579.48			0.260604			0.130302	+	0.991474i	$0.458405\pi$
$462$		0			0
$463$	6099.88			0.612280			0.306140	−	0.951986i	$-0.400962\pi$
$463$	6099.88			0.612280			0.306140	+	0.951986i	$0.400962\pi$
$464$		0			0
$465$	2930.67	−	2296.61i	0.292273	−	0.229038i
$466$		0			0
$467$	19053.5			1.88799			0.943993	−	0.329965i	$-0.107037\pi$
$467$	19053.5			1.88799			0.943993	+	0.329965i	$0.107037\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	7624.55	−	5974.95i	0.745904	−	0.584525i
$472$		0			0
$473$		−	374.237i		−	0.0363794i
$474$		0			0
$475$		−	1871.20i		−	0.180751i
$476$		0			0
$477$	16504.4	+	4063.73i	1.58425	+	0.390074i
$478$		0			0
$479$	17980.1			1.71510			0.857551	−	0.514400i	$-0.171985\pi$
$479$	17980.1			1.71510			0.857551	+	0.514400i	$0.171985\pi$
$480$		0			0
$481$		−	6398.21i		−	0.606514i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	462.079i		−	0.0432617i
$486$		0			0
$487$	−12314.6			−1.14585			−0.572924	−	0.819609i	$-0.694191\pi$
$487$	−12314.6			−1.14585			−0.572924	+	0.819609i	$0.694191\pi$
$488$		0			0
$489$	−10388.3	−	13256.4i	−0.960688	−	1.22592i
$490$		0			0
$491$		−	6650.90i		−	0.611305i	−0.952143	−	0.305653i	$-0.901125\pi$
$491$		−	6650.90i		−	0.611305i	0.952143	−	0.305653i	$-0.0988747\pi$
$492$		0			0
$493$		−	17847.7i		−	1.63047i
$494$		0			0
$495$	−264.211	−	65.0542i	−0.0239907	−	0.00590701i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12753.0			1.14409			0.572046	−	0.820221i	$-0.306150\pi$
$499$	12753.0			1.14409			0.572046	+	0.820221i	$0.306150\pi$
$500$		0			0
$501$	−6477.83	−	8266.27i	−0.577661	−	0.737145i
$502$		0			0
$503$	7044.47			0.624448			0.312224	−	0.950009i	$-0.398926\pi$
$503$	7044.47			0.624448			0.312224	+	0.950009i	$0.398926\pi$
$504$		0			0
$505$	−8958.76			−0.789425
$506$		0			0
$507$	−5199.19	−	6634.62i	−0.455432	−	0.581171i
$508$		0			0
$509$	−4772.94			−0.415632			−0.207816	−	0.978168i	$-0.566636\pi$
$509$	−4772.94			−0.415632			−0.207816	+	0.978168i	$0.566636\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1627.85	+	3613.41i	0.140100	+	0.310986i
$514$		0			0
$515$		−	140.458i		−	0.0120181i
$516$		0			0
$517$			275.763i			0.0234585i
$518$		0			0
$519$	−2245.43	−	2865.36i	−0.189910	−	0.242342i
$520$		0			0
$521$	14983.1			1.25992			0.629961	−	0.776626i	$-0.283071\pi$
$521$	14983.1			1.25992			0.629961	+	0.776626i	$0.283071\pi$
$522$		0			0
$523$			5356.14i			0.447816i	0.974610	+	0.223908i	$0.0718815\pi$
$523$			5356.14i			0.447816i	−0.974610	+	0.223908i	$0.928118\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5524.40i			0.456635i
$528$		0			0
$529$	6407.57			0.526635
$530$		0			0
$531$	−4714.68	+	19148.2i	−0.385310	+	1.56490i
$532$		0			0
$533$		−	3415.79i		−	0.277588i
$534$		0			0
$535$		−	7302.05i		−	0.590084i
$536$		0			0
$537$	11735.0	−	9196.10i	0.943024	−	0.738996i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.37252			−0.000665366			−0.000332683	−	1.00000i	$-0.500106\pi$
$541$	−8.37252			−0.000665366			−0.000332683		1.00000i	$0.500106\pi$
$542$		0			0
$543$	8074.88	−	6327.85i	0.638170	−	0.500099i
$544$		0			0
$545$	5426.23			0.426485
$546$		0			0
$547$	10274.7			0.803137			0.401568	−	0.915829i	$-0.368465\pi$
$547$	10274.7			0.803137			0.401568	+	0.915829i	$0.368465\pi$
$548$		0			0
$549$	14287.6	+	3517.91i	1.11071	+	0.273481i
$550$		0			0
$551$	−8531.10			−0.659595
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6556.44	+	8366.59i	0.501451	+	0.639895i
$556$		0			0
$557$		−	15376.1i		−	1.16967i	−0.811152	−	0.584835i	$-0.801159\pi$
$557$		−	15376.1i		−	1.16967i	0.811152	−	0.584835i	$-0.198841\pi$
$558$		0			0
$559$		−	6824.68i		−	0.516375i
$560$		0			0
$561$	317.775	−	249.023i	0.0239152	−	0.0187411i
$562$		0			0
$563$	18459.9			1.38187			0.690936	−	0.722916i	$-0.257199\pi$
$563$	18459.9			1.38187			0.690936	+	0.722916i	$0.257199\pi$
$564$		0			0
$565$		−	69.0878i		−	0.00514433i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	3502.34i		−	0.258042i	−0.991642	−	0.129021i	$-0.958817\pi$
$569$		−	3502.34i		−	0.258042i	0.991642	−	0.129021i	$-0.0411834\pi$
$570$		0			0
$571$	−11187.5			−0.819932			−0.409966	−	0.912101i	$-0.634459\pi$
$571$	−11187.5			−0.819932			−0.409966	+	0.912101i	$0.634459\pi$
$572$		0			0
$573$	−4448.64	+	3486.15i	−0.324336	+	0.254164i
$574$		0			0
$575$		−	5027.06i		−	0.364596i
$576$		0			0
$577$		−	9637.84i		−	0.695370i	−0.937611	−	0.347685i	$-0.886968\pi$
$577$		−	9637.84i		−	0.695370i	0.937611	−	0.347685i	$-0.113032\pi$
$578$		0			0
$579$	−12291.5	−	15685.1i	−0.882244	−	1.12582i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−827.649			−0.0587954
$584$		0			0
$585$	−4818.22	−	1186.35i	−0.340528	−	0.0838451i
$586$		0			0
$587$	−3998.01			−0.281117			−0.140559	−	0.990072i	$-0.544890\pi$
$587$	−3998.01			−0.281117			−0.140559	+	0.990072i	$0.544890\pi$
$588$		0			0
$589$	2640.63			0.184729
$590$		0			0
$591$	6688.01	−	5241.03i	0.465496	−	0.364784i
$592$		0			0
$593$	11432.0			0.791663			0.395832	−	0.918323i	$-0.370456\pi$
$593$	11432.0			0.791663			0.395832	+	0.918323i	$0.370456\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	17638.6	−	13822.4i	1.20921	−	0.947592i
$598$		0			0
$599$		−	808.857i		−	0.0551736i	−0.999619	−	0.0275868i	$-0.991218\pi$
$599$		−	808.857i		−	0.0551736i	0.999619	−	0.0275868i	$-0.00878227\pi$
$600$		0			0
$601$		−	6813.76i		−	0.462461i	−0.972899	−	0.231231i	$-0.925725\pi$
$601$		−	6813.76i		−	0.462461i	0.972899	−	0.231231i	$-0.0742752\pi$
$602$		0			0
$603$	3105.66	−	12613.3i	0.209738	−	0.851830i
$604$		0			0
$605$	−10189.5			−0.684730
$606$		0			0
$607$		−	3232.04i		−	0.216119i	−0.994144	−	0.108060i	$-0.965536\pi$
$607$		−	3232.04i		−	0.216119i	0.994144	−	0.108060i	$-0.0344638\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			5028.89i			0.332974i
$612$		0			0
$613$	−13215.8			−0.870771			−0.435385	−	0.900244i	$-0.643388\pi$
$613$	−13215.8			−0.870771			−0.435385	+	0.900244i	$0.643388\pi$
$614$		0			0
$615$	3500.26	+	4466.64i	0.229503	+	0.292866i
$616$		0			0
$617$		−	23412.7i		−	1.52765i	−0.645422	−	0.763826i	$-0.723319\pi$
$617$		−	23412.7i		−	1.52765i	0.645422	−	0.763826i	$-0.276681\pi$
$618$		0			0
$619$		−	10440.9i		−	0.677955i	−0.940794	−	0.338977i	$-0.889919\pi$
$619$		−	10440.9i		−	0.677955i	0.940794	−	0.338977i	$-0.110081\pi$
$620$		0			0
$621$	4373.28	+	9707.59i	0.282598	+	0.627298i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2957.10			−0.189254
$626$		0			0
$627$	−119.031	−	151.894i	−0.00758159	−	0.00967477i
$628$		0			0
$629$	−15771.3			−0.999747
$630$		0			0
$631$	−1116.76			−0.0704556			−0.0352278	−	0.999379i	$-0.511216\pi$
$631$	−1116.76			−0.0704556			−0.0352278	+	0.999379i	$0.511216\pi$
$632$		0			0
$633$	9238.56	+	11789.2i	0.580094	+	0.740251i
$634$		0			0
$635$	−2967.87			−0.185474
$636$		0			0
$637$		0			0
$638$		0			0
$639$	1220.34	+	300.473i	0.0755492	+	0.0186018i
$640$		0			0
$641$		−	1383.26i		−	0.0852348i	−0.999091	−	0.0426174i	$-0.986430\pi$
$641$		−	1383.26i		−	0.0852348i	0.999091	−	0.0426174i	$-0.0135697\pi$
$642$		0			0
$643$		−	13841.5i		−	0.848918i	−0.905447	−	0.424459i	$-0.860464\pi$
$643$		−	13841.5i		−	0.848918i	0.905447	−	0.424459i	$-0.139536\pi$
$644$		0			0
$645$	6993.46	+	8924.27i	0.426926	+	0.544795i
$646$		0			0
$647$	2377.08			0.144440			0.0722201	−	0.997389i	$-0.476992\pi$
$647$	2377.08			0.144440			0.0722201	+	0.997389i	$0.476992\pi$
$648$		0			0
$649$		−	960.227i		−	0.0580773i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	5557.13i		−	0.333028i	−0.986039	−	0.166514i	$-0.946749\pi$
$653$		−	5557.13i		−	0.333028i	0.986039	−	0.166514i	$-0.0532511\pi$
$654$		0			0
$655$	10820.2			0.645468
$656$		0			0
$657$	25250.4	+	6217.17i	1.49941	+	0.369185i
$658$		0			0
$659$			32145.5i			1.90017i	0.311991	+	0.950085i	$0.399004\pi$
$659$			32145.5i			1.90017i	−0.311991	+	0.950085i	$0.600996\pi$
$660$		0			0
$661$			696.068i			0.0409590i	0.999790	+	0.0204795i	$0.00651929\pi$
$661$			696.068i			0.0409590i	−0.999790	+	0.0204795i	$0.993481\pi$
$662$		0			0
$663$	5795.02	−	4541.24i	0.339457	−	0.266014i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−22919.2			−1.33049
$668$		0			0
$669$	13817.8	−	10828.3i	0.798546	−	0.625777i
$670$		0			0
$671$	−716.485			−0.0412215
$672$		0			0
$673$	−12403.6			−0.710436			−0.355218	−	0.934783i	$-0.615593\pi$
$673$	−12403.6			−0.710436			−0.355218	+	0.934783i	$0.615593\pi$
$674$		0			0
$675$	−8473.18	+	3817.17i	−0.483159	+	0.217664i
$676$		0			0
$677$	19505.7			1.10733			0.553666	−	0.832739i	$-0.313229\pi$
$677$	19505.7			1.10733			0.553666	+	0.832739i	$0.313229\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	16274.2	+	20767.4i	0.915757	+	1.16859i
$682$		0			0
$683$			29389.4i			1.64649i	0.567683	+	0.823247i	$0.307840\pi$
$683$			29389.4i			1.64649i	−0.567683	+	0.823247i	$0.692160\pi$
$684$		0			0
$685$			5586.92i			0.311628i
$686$		0			0
$687$	−17875.0	+	14007.6i	−0.992682	+	0.777911i
$688$		0			0
$689$	−15093.2			−0.834552
$690$		0			0
$691$			31657.9i			1.74287i	0.490509	+	0.871436i	$0.336811\pi$
$691$			31657.9i			1.74287i	−0.490509	+	0.871436i	$0.663189\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	9676.86i		−	0.528150i
$696$		0			0
$697$	−8419.75			−0.457562
$698$		0			0
$699$	−15752.7	+	12344.5i	−0.852392	+	0.667973i
$700$		0			0
$701$			5372.77i			0.289482i	0.989470	+	0.144741i	$0.0462349\pi$
$701$			5372.77i			0.289482i	−0.989470	+	0.144741i	$0.953765\pi$
$702$		0			0
$703$			7538.57i			0.404442i
$704$		0			0
$705$	−5153.26	−	6576.00i	−0.275295	−	0.351300i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−33427.5			−1.77066			−0.885328	−	0.464967i	$-0.846066\pi$
$709$	−33427.5			−1.77066			−0.885328	+	0.464967i	$0.846066\pi$
$710$		0			0
$711$	−8144.06	+	33076.3i	−0.429573	+	1.74467i
$712$		0			0
$713$	7094.18			0.372621
$714$		0			0
$715$	241.620			0.0126379
$716$		0			0
$717$	−28906.8	+	22652.7i	−1.50564	+	1.17989i
$718$		0			0
$719$	−27591.4			−1.43113			−0.715567	−	0.698544i	$-0.753832\pi$
$719$	−27591.4			−1.43113			−0.715567	+	0.698544i	$0.753832\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	12451.9	−	9757.89i	0.640514	−	0.501936i
$724$		0			0
$725$		−	20004.8i		−	1.02477i
$726$		0			0
$727$		−	16801.8i		−	0.857143i	−0.903508	−	0.428572i	$-0.859017\pi$
$727$		−	16801.8i		−	0.857143i	0.903508	−	0.428572i	$-0.140983\pi$
$728$		0			0
$729$	13041.5	−	14742.4i	0.662578	−	0.748993i
$730$		0			0
$731$	−16822.5			−0.851166
$732$		0			0
$733$			17113.1i			0.862331i	0.902273	+	0.431165i	$0.141897\pi$
$733$			17113.1i			0.862331i	−0.902273	+	0.431165i	$0.858103\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			632.521i			0.0316136i
$738$		0			0
$739$	19711.7			0.981198			0.490599	−	0.871385i	$-0.336778\pi$
$739$	19711.7			0.981198			0.490599	+	0.871385i	$0.336778\pi$
$740$		0			0
$741$	−2170.69	−	2769.99i	−0.107614	−	0.137325i
$742$		0			0
$743$			17443.3i			0.861283i	0.902523	+	0.430642i	$0.141713\pi$
$743$			17443.3i			0.861283i	−0.902523	+	0.430642i	$0.858287\pi$
$744$		0			0
$745$		−	7236.04i		−	0.355849i
$746$		0			0
$747$	5429.63	−	22051.9i	0.265944	−	1.08010i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−19587.6			−0.951744			−0.475872	−	0.879514i	$-0.657868\pi$
$751$	−19587.6			−0.951744			−0.475872	+	0.879514i	$0.657868\pi$
$752$		0			0
$753$	17433.3	+	22246.4i	0.843696	+	1.07663i
$754$		0			0
$755$	−2357.45			−0.113638
$756$		0			0
$757$	−4635.88			−0.222581			−0.111291	−	0.993788i	$-0.535498\pi$
$757$	−4635.88			−0.222581			−0.111291	+	0.993788i	$0.535498\pi$
$758$		0			0
$759$	−319.783	−	408.071i	−0.0152930	−	0.0195152i
$760$		0			0
$761$	−19170.9			−0.913199			−0.456600	−	0.889672i	$-0.650933\pi$
$761$	−19170.9			−0.913199			−0.456600	+	0.889672i	$0.650933\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2924.28	+	11876.7i	−0.138206	+	0.561309i
$766$		0			0
$767$		−	17510.9i		−	0.824359i
$768$		0			0
$769$		−	22223.5i		−	1.04213i	−0.853517	−	0.521065i	$-0.825535\pi$
$769$		−	22223.5i		−	1.04213i	0.853517	−	0.521065i	$-0.174465\pi$
$770$		0			0
$771$	−7303.52	−	9319.93i	−0.341154	−	0.435342i
$772$		0			0
$773$	−20995.6			−0.976920			−0.488460	−	0.872586i	$-0.662441\pi$
$773$	−20995.6			−0.976920			−0.488460	+	0.872586i	$0.662441\pi$
$774$		0			0
$775$			6192.08i			0.287002i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			4024.59i			0.185104i
$780$		0			0
$781$	−61.1967			−0.00280383
$782$		0			0
$783$	17403.1	+	38630.5i	0.794299	+	1.76315i
$784$		0			0
$785$		−	14290.1i		−	0.649728i
$786$		0			0
$787$			23377.5i			1.05885i	0.848356	+	0.529427i	$0.177593\pi$
$787$			23377.5i			1.05885i	−0.848356	+	0.529427i	$0.822407\pi$
$788$		0			0
$789$	−4741.41	+	3715.58i	−0.213940	+	0.167653i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−13066.0			−0.585104
$794$		0			0
$795$	19736.6	−	15466.5i	0.880484	−	0.689987i
$796$		0			0
$797$	5445.93			0.242039			0.121019	−	0.992650i	$-0.461384\pi$
$797$	5445.93			0.242039			0.121019	+	0.992650i	$0.461384\pi$
$798$		0			0
$799$	12396.0			0.548858
$800$		0			0
$801$	−8277.70	+	33619.0i	−0.365141	+	1.48298i
$802$		0			0
$803$	−1266.24			−0.0556469
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−12012.2	−	15328.6i	−0.523978	−	0.668641i
$808$		0			0
$809$			32518.0i			1.41319i	0.707617	+	0.706597i	$0.249770\pi$
$809$			32518.0i			1.41319i	−0.707617	+	0.706597i	$0.750230\pi$
$810$		0			0
$811$		−	16408.0i		−	0.710437i	−0.934783	−	0.355218i	$-0.884406\pi$
$811$		−	16408.0i		−	0.710437i	0.934783	−	0.355218i	$-0.115594\pi$
$812$		0			0
$813$	8367.31	−	6557.01i	0.360953	−	0.282859i
$814$		0			0
$815$	−24845.5			−1.06785
$816$		0			0
$817$			8041.06i			0.344334i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	12182.3i		−	0.517861i	−0.965896	−	0.258931i	$-0.916630\pi$
$821$		−	12182.3i		−	0.517861i	0.965896	−	0.258931i	$-0.0833701\pi$
$822$		0			0
$823$	4462.63			0.189013			0.0945064	−	0.995524i	$-0.469873\pi$
$823$	4462.63			0.189013			0.0945064	+	0.995524i	$0.469873\pi$
$824$		0			0
$825$	356.181	−	279.120i	0.0150311	−	0.0117790i
$826$		0			0
$827$			2442.53i			0.102703i	0.998681	+	0.0513513i	$0.0163528\pi$
$827$			2442.53i			0.102703i	−0.998681	+	0.0513513i	$0.983647\pi$
$828$		0			0
$829$			9615.24i			0.402836i	0.979505	+	0.201418i	$0.0645550\pi$
$829$			9615.24i			0.402836i	−0.979505	+	0.201418i	$0.935445\pi$
$830$		0			0
$831$	−15522.6	−	19808.2i	−0.647982	−	0.826881i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−15492.9			−0.642099
$836$		0			0
$837$	−5386.79	−	11957.3i	−0.222455	−	0.493794i
$838$		0			0
$839$	−19611.2			−0.806979			−0.403489	−	0.914984i	$-0.632203\pi$
$839$	−19611.2			−0.806979			−0.403489	+	0.914984i	$0.632203\pi$
$840$		0			0
$841$	−66815.9			−2.73959
$842$		0			0
$843$	−3386.31	+	2653.67i	−0.138352	+	0.108419i
$844$		0			0
$845$	−12434.8			−0.506236
$846$		0			0
$847$		0			0
$848$		0			0
$849$	34959.1	−	27395.6i	1.41319	−	1.10744i
$850$		0			0
$851$			20252.7i			0.815809i
$852$		0			0
$853$			49160.6i			1.97330i	0.162848	+	0.986651i	$0.447932\pi$
$853$			49160.6i			1.97330i	−0.162848	+	0.986651i	$0.552068\pi$
$854$		0			0
$855$	5676.98	+	1397.79i	0.227074	+	0.0559104i
$856$		0			0
$857$	−11368.6			−0.453145			−0.226572	−	0.973994i	$-0.572752\pi$
$857$	−11368.6			−0.453145			−0.226572	+	0.973994i	$0.572752\pi$
$858$		0			0
$859$		−	7067.35i		−	0.280716i	−0.990101	−	0.140358i	$-0.955175\pi$
$859$		−	7067.35i		−	0.280716i	0.990101	−	0.140358i	$-0.0448253\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	26962.6i		−	1.06352i	−0.846895	−	0.531760i	$-0.821531\pi$
$863$		−	26962.6i		−	1.06352i	0.846895	−	0.531760i	$-0.178469\pi$
$864$		0			0
$865$	−5370.33			−0.211095
$866$		0			0
$867$	4552.54	+	5809.44i	0.178330	+	0.227565i
$868$		0			0
$869$		−	1658.68i		−	0.0647491i
$870$		0			0
$871$			11534.8i			0.448729i
$872$		0			0
$873$	−1580.38	−	389.121i	−0.0612687	−	0.0150856i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−38607.2			−1.48652			−0.743258	−	0.669005i	$-0.766720\pi$
$877$	−38607.2			−1.48652			−0.743258	+	0.669005i	$0.766720\pi$
$878$		0			0
$879$	23291.0	+	29721.3i	0.893726	+	1.14047i
$880$		0			0
$881$	3460.09			0.132320			0.0661598	−	0.997809i	$-0.478925\pi$
$881$	3460.09			0.132320			0.0661598	+	0.997809i	$0.478925\pi$
$882$		0			0
$883$	10380.8			0.395631			0.197815	−	0.980239i	$-0.436615\pi$
$883$	10380.8			0.395631			0.197815	+	0.980239i	$0.436615\pi$
$884$		0			0
$885$	17944.0	+	22898.1i	0.681560	+	0.869730i
$886$		0			0
$887$	45131.9			1.70843			0.854217	−	0.519916i	$-0.174037\pi$
$887$	45131.9			1.70843			0.854217	+	0.519916i	$0.174037\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−444.990	+	848.857i	−0.0167314	+	0.0319167i
$892$		0			0
$893$		−	5925.19i		−	0.222037i
$894$		0			0
$895$		−	21994.1i		−	0.821431i
$896$		0			0
$897$	−5831.65	−	7441.69i	−0.217071	−	0.277002i
$898$		0			0
$899$	28230.7			1.04733
$900$		0			0
$901$			37204.0i			1.37563i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	15134.2i		−	0.555885i
$906$		0			0
$907$	42983.8			1.57360			0.786800	−	0.617208i	$-0.211736\pi$
$907$	42983.8			1.57360			0.786800	+	0.617208i	$0.211736\pi$
$908$		0			0
$909$	−7544.26	+	30640.2i	−0.275278	+	1.11801i
$910$		0			0
$911$		−	45726.0i		−	1.66297i	−0.555545	−	0.831487i	$-0.687490\pi$
$911$		−	45726.0i		−	1.66297i	0.555545	−	0.831487i	$-0.312510\pi$
$912$		0			0
$913$			1105.84i			0.0400854i
$914$		0			0
$915$	17085.7	−	13389.1i	0.617307	−	0.483750i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−25163.2			−0.903216			−0.451608	−	0.892216i	$-0.649150\pi$
$919$	−25163.2			−0.903216			−0.451608	+	0.892216i	$0.649150\pi$
$920$		0			0
$921$	16869.9	−	13220.0i	0.603564	−	0.472980i
$922$		0			0
$923$	−1116.00			−0.0397980
$924$		0			0
$925$	−17677.4			−0.628355
$926$		0			0
$927$	−480.386	−	118.281i	−0.0170204	−	0.00419078i
$928$		0			0
$929$	−20291.3			−0.716617			−0.358309	−	0.933603i	$-0.616646\pi$
$929$	−20291.3			−0.716617			−0.358309	+	0.933603i	$0.616646\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	10571.3	+	13489.9i	0.370941	+	0.473353i
$934$		0			0
$935$		−	595.581i		−	0.0208316i
$936$		0			0
$937$		−	25708.8i		−	0.896339i	−0.893949	−	0.448170i	$-0.852076\pi$
$937$		−	25708.8i		−	0.896339i	0.893949	−	0.448170i	$-0.147924\pi$
$938$		0			0
$939$	−24502.5	+	19201.3i	−0.851555	+	0.667318i
$940$		0			0
$941$	9143.86			0.316771			0.158385	−	0.987377i	$-0.449371\pi$
$941$	9143.86			0.316771			0.158385	+	0.987377i	$0.449371\pi$
$942$		0			0
$943$			10812.2i			0.373378i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	43884.1i		−	1.50585i	−0.658106	−	0.752925i	$-0.728642\pi$
$947$		−	43884.1i		−	1.50585i	0.658106	−	0.752925i	$-0.271358\pi$
$948$		0			0
$949$	−23091.4			−0.789861
$950$		0			0
$951$	−29017.7	+	22739.6i	−0.989445	+	0.775374i
$952$		0			0
$953$		−	29621.5i		−	1.00686i	−0.864037	−	0.503429i	$-0.832072\pi$
$953$		−	29621.5i		−	1.00686i	0.864037	−	0.503429i	$-0.167928\pi$
$954$		0			0
$955$			8337.75i			0.282516i
$956$		0			0
$957$	−1272.55	−	1623.88i	−0.0429840	−	0.0548513i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	21052.7			0.706681
$962$		0			0
$963$	−24974.1	−	6149.13i	−0.835699	−	0.205766i
$964$		0			0
$965$	−29397.4			−0.980658
$966$		0			0
$967$	39437.0			1.31149			0.655744	−	0.754983i	$-0.272355\pi$
$967$	39437.0			1.31149			0.655744	+	0.754983i	$0.272355\pi$
$968$		0			0
$969$	−6827.87	+	5350.63i	−0.226360	+	0.177386i
$970$		0			0
$971$	22462.7			0.742393			0.371197	−	0.928554i	$-0.378948\pi$
$971$	22462.7			0.742393			0.371197	+	0.928554i	$0.378948\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	6495.41	−	5090.10i	0.213353	−	0.167193i
$976$		0			0
$977$			27239.2i			0.891976i	0.895039	+	0.445988i	$0.147148\pi$
$977$			27239.2i			0.891976i	−0.895039	+	0.445988i	$0.852852\pi$
$978$		0			0
$979$		−	1685.90i		−	0.0550374i
$980$		0			0
$981$	4569.48	−	18558.5i	0.148718	−	0.604003i
$982$		0			0
$983$	8716.81			0.282831			0.141416	−	0.989950i	$-0.454835\pi$
$983$	8716.81			0.282831			0.141416	+	0.989950i	$0.454835\pi$
$984$		0			0
$985$		−	12534.8i		−	0.405475i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			21602.6i			0.694564i
$990$		0			0
$991$	−29777.5			−0.954505			−0.477252	−	0.878766i	$-0.658367\pi$
$991$	−29777.5			−0.954505			−0.477252	+	0.878766i	$0.658367\pi$
$992$		0			0
$993$	−29370.6	−	37479.4i	−0.938618	−	1.19776i
$994$		0			0
$995$		−	33058.7i		−	1.05330i
$996$		0			0
$997$		−	53698.9i		−	1.70578i	−0.522091	−	0.852890i	$-0.674848\pi$
$997$		−	53698.9i		−	1.70578i	0.522091	−	0.852890i	$-0.325152\pi$
$998$		0			0
$999$	34136.2	−	15378.4i	1.08110	−	0.487037i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.4.f.d.293.5	✓	24
3.2	odd	2	inner	588.4.f.d.293.19	yes	24
7.2	even	3		588.4.k.e.521.22		48
7.3	odd	6		588.4.k.e.509.15		48
7.4	even	3		588.4.k.e.509.10		48
7.5	odd	6		588.4.k.e.521.3		48
7.6	odd	2	inner	588.4.f.d.293.20	yes	24
21.2	odd	6		588.4.k.e.521.15		48
21.5	even	6		588.4.k.e.521.10		48
21.11	odd	6		588.4.k.e.509.3		48
21.17	even	6		588.4.k.e.509.22		48
21.20	even	2	inner	588.4.f.d.293.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.f.d.293.5	✓	24	1.1	even	1	trivial
588.4.f.d.293.6	yes	24	21.20	even	2	inner
588.4.f.d.293.19	yes	24	3.2	odd	2	inner
588.4.f.d.293.20	yes	24	7.6	odd	2	inner
588.4.k.e.509.3		48	21.11	odd	6
588.4.k.e.509.10		48	7.4	even	3
588.4.k.e.509.15		48	7.3	odd	6
588.4.k.e.509.22		48	21.17	even	6
588.4.k.e.521.3		48	7.5	odd	6
588.4.k.e.521.10		48	21.5	even	6
588.4.k.e.521.15		48	21.2	odd	6
588.4.k.e.521.22		48	7.2	even	3

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.20506 - 4.08994i) q^{3} -7.66547 q^{5} +(-6.45517 + 26.2170i) q^{9} +O(q^{10})\)	\(q+(-3.20506 - 4.08994i) q^{3} -7.66547 q^{5} +(-6.45517 + 26.2170i) q^{9} -1.31471i q^{11} -23.9754i q^{13} +(24.5683 + 31.3513i) q^{15} -59.0980 q^{17} +28.2485i q^{19} +75.8909i q^{23} -66.2406 q^{25} +(127.915 - 57.6258i) q^{27} +302.001i q^{29} -93.4786i q^{31} +(-5.37707 + 4.21372i) q^{33} +266.866 q^{37} +(-98.0577 + 76.8425i) q^{39} +142.471 q^{41} +284.654 q^{43} +(49.4819 - 200.965i) q^{45} -209.752 q^{47} +(189.413 + 241.707i) q^{51} -629.531i q^{53} +10.0779i q^{55} +(115.535 - 90.5383i) q^{57} +730.372 q^{59} -544.977i q^{61} +183.782i q^{65} -481.112 q^{67} +(310.389 - 243.235i) q^{69} -46.5477i q^{71} -963.130i q^{73} +(212.305 + 270.920i) q^{75} +1261.64 q^{79} +(-645.662 - 338.470i) q^{81} -841.130 q^{83} +453.014 q^{85} +(1235.17 - 967.933i) q^{87} +1282.34 q^{89} +(-382.322 + 299.605i) q^{93} -216.538i q^{95} +60.2806i q^{97} +(34.4677 + 8.48666i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 64 q^{9}+O(q^{10}) \) 24 * q - 64 * q^9	\( 24 q - 64 q^{9} - 96 q^{15} + 456 q^{25} - 432 q^{37} + 688 q^{39} + 624 q^{43} - 1536 q^{51} - 1360 q^{57} - 528 q^{67} + 3744 q^{79} + 3408 q^{81} + 6912 q^{85} - 5088 q^{93} - 7736 q^{99}+O(q^{100}) \) 24 * q - 64 * q^9 - 96 * q^15 + 456 * q^25 - 432 * q^37 + 688 * q^39 + 624 * q^43 - 1536 * q^51 - 1360 * q^57 - 528 * q^67 + 3744 * q^79 + 3408 * q^81 + 6912 * q^85 - 5088 * q^93 - 7736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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