LMFDB - Embedded newform 720.4.f.c.289.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,4,Mod(289,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.289"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	720.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-4,0,0,0,0,0,140,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$42.4813752041$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.4.f.c.289.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 11.0000i) q^{5} +2.00000i q^{7} +70.0000 q^{11} -54.0000i q^{13} +22.0000i q^{17} +24.0000 q^{19} +100.000i q^{23} +(-117.000 + 44.0000i) q^{25} +216.000 q^{29} -208.000 q^{31} +(22.0000 - 4.00000i) q^{35} -254.000i q^{37} +206.000 q^{41} +292.000i q^{43} -320.000i q^{47} +339.000 q^{49} -402.000i q^{53} +(-140.000 - 770.000i) q^{55} +370.000 q^{59} -550.000 q^{61} +(-594.000 + 108.000i) q^{65} -728.000i q^{67} -540.000 q^{71} -604.000i q^{73} +140.000i q^{77} +792.000 q^{79} -404.000i q^{83} +(242.000 - 44.0000i) q^{85} -938.000 q^{89} +108.000 q^{91} +(-48.0000 - 264.000i) q^{95} +56.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 140 q^{11} + 48 q^{19} - 234 q^{25} + 432 q^{29} - 416 q^{31} + 44 q^{35} + 412 q^{41} + 678 q^{49} - 280 q^{55} + 740 q^{59} - 1100 q^{61} - 1188 q^{65} - 1080 q^{71} + 1584 q^{79} + 484 q^{85}+ \cdots - 96 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 + 140 * q^11 + 48 * q^19 - 234 * q^25 + 432 * q^29 - 416 * q^31 + 44 * q^35 + 412 * q^41 + 678 * q^49 - 280 * q^55 + 740 * q^59 - 1100 * q^61 - 1188 * q^65 - 1080 * q^71 + 1584 * q^79 + 484 * q^85 - 1876 * q^89 + 216 * q^91 - 96 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$5$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$6$		0			0
$7$			2.00000i			0.107990i	0.998541	+	0.0539949i	$0.0171955\pi$
$7$			2.00000i			0.107990i	−0.998541	+	0.0539949i	$0.982805\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	70.0000			1.91871			0.959354	−	0.282204i	$-0.0910657\pi$
$11$	70.0000			1.91871			0.959354	+	0.282204i	$0.0910657\pi$
$12$		0			0
$13$		−	54.0000i		−	1.15207i	−0.817425	−	0.576035i	$-0.804599\pi$
$13$		−	54.0000i		−	1.15207i	0.817425	−	0.576035i	$-0.195401\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			22.0000i			0.313870i	0.987609	+	0.156935i	$0.0501613\pi$
$17$			22.0000i			0.313870i	−0.987609	+	0.156935i	$0.949839\pi$
$18$		0			0
$19$	24.0000			0.289788			0.144894	−	0.989447i	$-0.453716\pi$
$19$	24.0000			0.289788			0.144894	+	0.989447i	$0.453716\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			100.000i			0.906584i	0.891362	+	0.453292i	$0.149751\pi$
$23$			100.000i			0.906584i	−0.891362	+	0.453292i	$0.850249\pi$
$24$		0			0
$25$	−117.000	+	44.0000i	−0.936000	+	0.352000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	216.000			1.38311			0.691555	−	0.722324i	$-0.256926\pi$
$29$	216.000			1.38311			0.691555	+	0.722324i	$0.256926\pi$
$30$		0			0
$31$	−208.000			−1.20509			−0.602547	−	0.798084i	$-0.705847\pi$
$31$	−208.000			−1.20509			−0.602547	+	0.798084i	$0.705847\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	22.0000	−	4.00000i	0.106248	−	0.0193178i
$36$		0			0
$37$		−	254.000i		−	1.12858i	−0.825578	−	0.564288i	$-0.809151\pi$
$37$		−	254.000i		−	1.12858i	0.825578	−	0.564288i	$-0.190849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	206.000			0.784678			0.392339	−	0.919821i	$-0.371666\pi$
$41$	206.000			0.784678			0.392339	+	0.919821i	$0.371666\pi$
$42$		0			0
$43$			292.000i			1.03557i	0.855510	+	0.517786i	$0.173244\pi$
$43$			292.000i			1.03557i	−0.855510	+	0.517786i	$0.826756\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	320.000i		−	0.993123i	−0.868001	−	0.496562i	$-0.834596\pi$
$47$		−	320.000i		−	0.993123i	0.868001	−	0.496562i	$-0.165404\pi$
$48$		0			0
$49$	339.000			0.988338
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	402.000i		−	1.04187i	−0.853597	−	0.520933i	$-0.825584\pi$
$53$		−	402.000i		−	1.04187i	0.853597	−	0.520933i	$-0.174416\pi$
$54$		0			0
$55$	−140.000	−	770.000i	−0.343229	−	1.88776i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	370.000			0.816439			0.408219	−	0.912884i	$-0.366150\pi$
$59$	370.000			0.816439			0.408219	+	0.912884i	$0.366150\pi$
$60$		0			0
$61$	−550.000			−1.15443			−0.577215	−	0.816592i	$-0.695861\pi$
$61$	−550.000			−1.15443			−0.577215	+	0.816592i	$0.695861\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−594.000	+	108.000i	−1.13349	+	0.206088i
$66$		0			0
$67$		−	728.000i		−	1.32745i	−0.747975	−	0.663727i	$-0.768974\pi$
$67$		−	728.000i		−	1.32745i	0.747975	−	0.663727i	$-0.231026\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−540.000			−0.902623			−0.451311	−	0.892367i	$-0.649044\pi$
$71$	−540.000			−0.902623			−0.451311	+	0.892367i	$0.649044\pi$
$72$		0			0
$73$		−	604.000i		−	0.968395i	−0.874959	−	0.484198i	$-0.839112\pi$
$73$		−	604.000i		−	0.968395i	0.874959	−	0.484198i	$-0.160888\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			140.000i			0.207201i
$78$		0			0
$79$	792.000			1.12794			0.563968	−	0.825797i	$-0.309274\pi$
$79$	792.000			1.12794			0.563968	+	0.825797i	$0.309274\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	404.000i		−	0.534274i	−0.963659	−	0.267137i	$-0.913922\pi$
$83$		−	404.000i		−	0.534274i	0.963659	−	0.267137i	$-0.0860777\pi$
$84$		0			0
$85$	242.000	−	44.0000i	0.308807	−	0.0561467i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−938.000			−1.11717			−0.558583	−	0.829449i	$-0.688655\pi$
$89$	−938.000			−1.11717			−0.558583	+	0.829449i	$0.688655\pi$
$90$		0			0
$91$	108.000			0.124412
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−48.0000	−	264.000i	−0.0518389	−	0.285114i
$96$		0			0
$97$			56.0000i			0.0586179i	0.999570	+	0.0293090i	$0.00933067\pi$
$97$			56.0000i			0.0586179i	−0.999570	+	0.0293090i	$0.990669\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	592.000			0.583230			0.291615	−	0.956536i	$-0.405807\pi$
$101$	592.000			0.583230			0.291615	+	0.956536i	$0.405807\pi$
$102$		0			0
$103$			62.0000i			0.0593111i	0.999560	+	0.0296555i	$0.00944104\pi$
$103$			62.0000i			0.0593111i	−0.999560	+	0.0296555i	$0.990559\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			84.0000i			0.0758933i	0.999280	+	0.0379467i	$0.0120817\pi$
$107$			84.0000i			0.0758933i	−0.999280	+	0.0379467i	$0.987918\pi$
$108$		0			0
$109$	−370.000			−0.325134			−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000			−0.325134			−0.162567	+	0.986698i	$0.551977\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1746.00i		−	1.45354i	−0.686882	−	0.726769i	$-0.741021\pi$
$113$		−	1746.00i		−	1.45354i	0.686882	−	0.726769i	$-0.258979\pi$
$114$		0			0
$115$	1100.00	−	200.000i	0.891961	−	0.162175i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−44.0000			−0.0338947
$120$		0			0
$121$	3569.00			2.68144
$122$		0			0
$123$		0			0
$124$		0			0
$125$	718.000	+	1199.00i	0.513759	+	0.857935i
$126$		0			0
$127$		−	1630.00i		−	1.13889i	−0.822029	−	0.569445i	$-0.807158\pi$
$127$		−	1630.00i		−	1.13889i	0.822029	−	0.569445i	$-0.192842\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−870.000			−0.580246			−0.290123	−	0.956989i	$-0.593696\pi$
$131$	−870.000			−0.580246			−0.290123	+	0.956989i	$0.593696\pi$
$132$		0			0
$133$			48.0000i			0.0312942i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	918.000i		−	0.572482i	−0.958158	−	0.286241i	$-0.907594\pi$
$137$		−	918.000i		−	0.572482i	0.958158	−	0.286241i	$-0.0924058\pi$
$138$		0			0
$139$	−596.000			−0.363684			−0.181842	−	0.983328i	$-0.558206\pi$
$139$	−596.000			−0.363684			−0.181842	+	0.983328i	$0.558206\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	3780.00i		−	2.21049i
$144$		0			0
$145$	−432.000	−	2376.00i	−0.247418	−	1.36080i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1076.00			0.591606			0.295803	−	0.955249i	$-0.404413\pi$
$149$	1076.00			0.591606			0.295803	+	0.955249i	$0.404413\pi$
$150$		0			0
$151$	32.0000			0.0172458			0.00862292	−	0.999963i	$-0.497255\pi$
$151$	32.0000			0.0172458			0.00862292	+	0.999963i	$0.497255\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	416.000	+	2288.00i	0.215574	+	1.18566i
$156$		0			0
$157$		−	2554.00i		−	1.29829i	−0.760665	−	0.649145i	$-0.775127\pi$
$157$		−	2554.00i		−	1.29829i	0.760665	−	0.649145i	$-0.224873\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−200.000			−0.0979019
$162$		0			0
$163$			752.000i			0.361357i	0.983542	+	0.180678i	$0.0578293\pi$
$163$			752.000i			0.361357i	−0.983542	+	0.180678i	$0.942171\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2700.00i			1.25109i	0.780188	+	0.625546i	$0.215124\pi$
$167$			2700.00i			1.25109i	−0.780188	+	0.625546i	$0.784876\pi$
$168$		0			0
$169$	−719.000			−0.327264
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1334.00i		−	0.586255i	−0.956073	−	0.293128i	$-0.905304\pi$
$173$		−	1334.00i		−	0.586255i	0.956073	−	0.293128i	$-0.0946961\pi$
$174$		0			0
$175$	−88.0000	−	234.000i	−0.0380124	−	0.101078i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1714.00			0.715700			0.357850	−	0.933779i	$-0.383510\pi$
$179$	1714.00			0.715700			0.357850	+	0.933779i	$0.383510\pi$
$180$		0			0
$181$	−4006.00			−1.64510			−0.822551	−	0.568691i	$-0.807450\pi$
$181$	−4006.00			−1.64510			−0.822551	+	0.568691i	$0.807450\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2794.00	+	508.000i	−1.11037	+	0.201886i
$186$		0			0
$187$			1540.00i			0.602224i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−684.000			−0.259123			−0.129562	−	0.991571i	$-0.541357\pi$
$191$	−684.000			−0.259123			−0.129562	+	0.991571i	$0.541357\pi$
$192$		0			0
$193$			4484.00i			1.67236i	0.548455	+	0.836180i	$0.315216\pi$
$193$			4484.00i			1.67236i	−0.548455	+	0.836180i	$0.684784\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1058.00i			0.382636i	0.981528	+	0.191318i	$0.0612762\pi$
$197$			1058.00i			0.382636i	−0.981528	+	0.191318i	$0.938724\pi$
$198$		0			0
$199$	−1128.00			−0.401818			−0.200909	−	0.979610i	$-0.564390\pi$
$199$	−1128.00			−0.401818			−0.200909	+	0.979610i	$0.564390\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			432.000i			0.149362i
$204$		0			0
$205$	−412.000	−	2266.00i	−0.140367	−	0.772021i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1680.00			0.556019
$210$		0			0
$211$	−780.000			−0.254490			−0.127245	−	0.991871i	$-0.540613\pi$
$211$	−780.000			−0.254490			−0.127245	+	0.991871i	$0.540613\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3212.00	−	584.000i	1.01887	−	0.185249i
$216$		0			0
$217$		−	416.000i		−	0.130138i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1188.00			0.361600
$222$		0			0
$223$		−	2570.00i		−	0.771749i	−0.922551	−	0.385874i	$-0.873900\pi$
$223$		−	2570.00i		−	0.771749i	0.922551	−	0.385874i	$-0.126100\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	2836.00i		−	0.829216i	−0.910000	−	0.414608i	$-0.863919\pi$
$227$		−	2836.00i		−	0.829216i	0.910000	−	0.414608i	$-0.136081\pi$
$228$		0			0
$229$	610.000			0.176026			0.0880130	−	0.996119i	$-0.471948\pi$
$229$	610.000			0.176026			0.0880130	+	0.996119i	$0.471948\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3514.00i			0.988025i	0.869455	+	0.494012i	$0.164470\pi$
$233$			3514.00i			0.988025i	−0.869455	+	0.494012i	$0.835530\pi$
$234$		0			0
$235$	−3520.00	+	640.000i	−0.977104	+	0.177655i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1844.00			0.499073			0.249536	−	0.968365i	$-0.419722\pi$
$239$	1844.00			0.499073			0.249536	+	0.968365i	$0.419722\pi$
$240$		0			0
$241$	982.000			0.262474			0.131237	−	0.991351i	$-0.458105\pi$
$241$	982.000			0.262474			0.131237	+	0.991351i	$0.458105\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−678.000	−	3729.00i	−0.176799	−	0.972396i
$246$		0			0
$247$		−	1296.00i		−	0.333856i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3174.00			−0.798172			−0.399086	−	0.916914i	$-0.630672\pi$
$251$	−3174.00			−0.798172			−0.399086	+	0.916914i	$0.630672\pi$
$252$		0			0
$253$			7000.00i			1.73947i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1194.00i			0.289804i	0.989446	+	0.144902i	$0.0462867\pi$
$257$			1194.00i			0.289804i	−0.989446	+	0.144902i	$0.953713\pi$
$258$		0			0
$259$	508.000			0.121875
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	140.000i		−	0.0328242i	−0.999865	−	0.0164121i	$-0.994776\pi$
$263$		−	140.000i		−	0.0328242i	0.999865	−	0.0164121i	$-0.00522437\pi$
$264$		0			0
$265$	−4422.00	+	804.000i	−1.02506	+	0.186375i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5256.00			1.19132			0.595658	−	0.803238i	$-0.296891\pi$
$269$	5256.00			1.19132			0.595658	+	0.803238i	$0.296891\pi$
$270$		0			0
$271$	−544.000			−0.121940			−0.0609698	−	0.998140i	$-0.519419\pi$
$271$	−544.000			−0.121940			−0.0609698	+	0.998140i	$0.519419\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8190.00	+	3080.00i	−1.79591	+	0.675385i
$276$		0			0
$277$			946.000i			0.205197i	0.994723	+	0.102599i	$0.0327157\pi$
$277$			946.000i			0.205197i	−0.994723	+	0.102599i	$0.967284\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1278.00			−0.271313			−0.135657	−	0.990756i	$-0.543314\pi$
$281$	−1278.00			−0.271313			−0.135657	+	0.990756i	$0.543314\pi$
$282$		0			0
$283$			7424.00i			1.55940i	0.626152	+	0.779701i	$0.284629\pi$
$283$			7424.00i			1.55940i	−0.626152	+	0.779701i	$0.715371\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			412.000i			0.0847373i
$288$		0			0
$289$	4429.00			0.901486
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1362.00i		−	0.271566i	−0.990739	−	0.135783i	$-0.956645\pi$
$293$		−	1362.00i		−	0.271566i	0.990739	−	0.135783i	$-0.0433550\pi$
$294$		0			0
$295$	−740.000	−	4070.00i	−0.146049	−	0.803270i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5400.00			1.04445
$300$		0			0
$301$	−584.000			−0.111831
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1100.00	+	6050.00i	0.206511	+	1.13581i
$306$		0			0
$307$			7740.00i			1.43891i	0.694539	+	0.719455i	$0.255608\pi$
$307$			7740.00i			1.43891i	−0.694539	+	0.719455i	$0.744392\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4980.00			0.908006			0.454003	−	0.891000i	$-0.349996\pi$
$311$	4980.00			0.908006			0.454003	+	0.891000i	$0.349996\pi$
$312$		0			0
$313$		−	604.000i		−	0.109074i	−0.998512	−	0.0545369i	$-0.982632\pi$
$313$		−	604.000i		−	0.109074i	0.998512	−	0.0545369i	$-0.0173683\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			8566.00i			1.51771i	0.651259	+	0.758856i	$0.274241\pi$
$317$			8566.00i			1.51771i	−0.651259	+	0.758856i	$0.725759\pi$
$318$		0			0
$319$	15120.0			2.65379
$320$		0			0
$321$		0			0
$322$		0			0
$323$			528.000i			0.0909557i
$324$		0			0
$325$	2376.00	+	6318.00i	0.405529	+	1.07834i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	640.000			0.107247
$330$		0			0
$331$	−3472.00			−0.576551			−0.288275	−	0.957548i	$-0.593082\pi$
$331$	−3472.00			−0.576551			−0.288275	+	0.957548i	$0.593082\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8008.00	+	1456.00i	−1.30604	+	0.237462i
$336$		0			0
$337$			5668.00i			0.916189i	0.888904	+	0.458094i	$0.151468\pi$
$337$			5668.00i			0.916189i	−0.888904	+	0.458094i	$0.848532\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14560.0			−2.31222
$342$		0			0
$343$			1364.00i			0.214720i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10836.0i			1.67639i	0.545371	+	0.838194i	$0.316389\pi$
$347$			10836.0i			1.67639i	−0.545371	+	0.838194i	$0.683611\pi$
$348$		0			0
$349$	8990.00			1.37886			0.689432	−	0.724350i	$-0.257860\pi$
$349$	8990.00			1.37886			0.689432	+	0.724350i	$0.257860\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	5078.00i		−	0.765651i	−0.923821	−	0.382825i	$-0.874951\pi$
$353$		−	5078.00i		−	0.765651i	0.923821	−	0.382825i	$-0.125049\pi$
$354$		0			0
$355$	1080.00	+	5940.00i	0.161466	+	0.888063i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3696.00			0.543363			0.271682	−	0.962387i	$-0.412420\pi$
$359$	3696.00			0.543363			0.271682	+	0.962387i	$0.412420\pi$
$360$		0			0
$361$	−6283.00			−0.916023
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6644.00	+	1208.00i	−0.952775	+	0.173232i
$366$		0			0
$367$		−	286.000i		−	0.0406787i	−0.999793	−	0.0203393i	$-0.993525\pi$
$367$		−	286.000i		−	0.0406787i	0.999793	−	0.0203393i	$-0.00647466\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	804.000			0.112511
$372$		0			0
$373$		−	8262.00i		−	1.14689i	−0.819244	−	0.573445i	$-0.805607\pi$
$373$		−	8262.00i		−	1.14689i	0.819244	−	0.573445i	$-0.194393\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	11664.0i		−	1.59344i
$378$		0			0
$379$	−2956.00			−0.400632			−0.200316	−	0.979731i	$-0.564197\pi$
$379$	−2956.00			−0.400632			−0.200316	+	0.979731i	$0.564197\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			5240.00i			0.699090i	0.936920	+	0.349545i	$0.113664\pi$
$383$			5240.00i			0.699090i	−0.936920	+	0.349545i	$0.886336\pi$
$384$		0			0
$385$	1540.00	−	280.000i	0.203859	−	0.0370653i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−884.000			−0.115220			−0.0576100	−	0.998339i	$-0.518348\pi$
$389$	−884.000			−0.115220			−0.0576100	+	0.998339i	$0.518348\pi$
$390$		0			0
$391$	−2200.00			−0.284549
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1584.00	−	8712.00i	−0.201771	−	1.10974i
$396$		0			0
$397$			3394.00i			0.429068i	0.976717	+	0.214534i	$0.0688233\pi$
$397$			3394.00i			0.429068i	−0.976717	+	0.214534i	$0.931177\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6826.00			0.850060			0.425030	−	0.905179i	$-0.360263\pi$
$401$	6826.00			0.850060			0.425030	+	0.905179i	$0.360263\pi$
$402$		0			0
$403$			11232.0i			1.38835i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	17780.0i		−	2.16541i
$408$		0			0
$409$	−7814.00			−0.944688			−0.472344	−	0.881414i	$-0.656592\pi$
$409$	−7814.00			−0.944688			−0.472344	+	0.881414i	$0.656592\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			740.000i			0.0881671i
$414$		0			0
$415$	−4444.00	+	808.000i	−0.525656	+	0.0955739i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8290.00			−0.966570			−0.483285	−	0.875463i	$-0.660557\pi$
$419$	−8290.00			−0.966570			−0.483285	+	0.875463i	$0.660557\pi$
$420$		0			0
$421$	2110.00			0.244264			0.122132	−	0.992514i	$-0.461027\pi$
$421$	2110.00			0.244264			0.122132	+	0.992514i	$0.461027\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−968.000	−	2574.00i	−0.110482	−	0.293782i
$426$		0			0
$427$		−	1100.00i		−	0.124667i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12080.0			−1.35005			−0.675027	−	0.737793i	$-0.735868\pi$
$431$	−12080.0			−1.35005			−0.675027	+	0.737793i	$0.735868\pi$
$432$		0			0
$433$		−	16492.0i		−	1.83038i	−0.403022	−	0.915190i	$-0.632040\pi$
$433$		−	16492.0i		−	1.83038i	0.403022	−	0.915190i	$-0.367960\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2400.00i			0.262718i
$438$		0			0
$439$	−15048.0			−1.63600			−0.817998	−	0.575222i	$-0.804916\pi$
$439$	−15048.0			−1.63600			−0.817998	+	0.575222i	$0.804916\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			9876.00i			1.05919i	0.848249	+	0.529597i	$0.177657\pi$
$443$			9876.00i			1.05919i	−0.848249	+	0.529597i	$0.822343\pi$
$444$		0			0
$445$	1876.00	+	10318.0i	0.199845	+	1.09915i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17166.0			1.80426			0.902131	−	0.431462i	$-0.142002\pi$
$449$	17166.0			1.80426			0.902131	+	0.431462i	$0.142002\pi$
$450$		0			0
$451$	14420.0			1.50557
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−216.000	−	1188.00i	−0.0222555	−	0.122405i
$456$		0			0
$457$			14848.0i			1.51983i	0.650025	+	0.759913i	$0.274758\pi$
$457$			14848.0i			1.51983i	−0.650025	+	0.759913i	$0.725242\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1260.00			0.127297			0.0636486	−	0.997972i	$-0.479726\pi$
$461$	1260.00			0.127297			0.0636486	+	0.997972i	$0.479726\pi$
$462$		0			0
$463$		−	11238.0i		−	1.12802i	−0.825767	−	0.564011i	$-0.809258\pi$
$463$		−	11238.0i		−	1.12802i	0.825767	−	0.564011i	$-0.190742\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			14772.0i			1.46374i	0.681444	+	0.731870i	$0.261352\pi$
$467$			14772.0i			1.46374i	−0.681444	+	0.731870i	$0.738648\pi$
$468$		0			0
$469$	1456.00			0.143351
$470$		0			0
$471$		0			0
$472$		0			0
$473$			20440.0i			1.98696i
$474$		0			0
$475$	−2808.00	+	1056.00i	−0.271242	+	0.102005i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6116.00			0.583397			0.291699	−	0.956510i	$-0.405780\pi$
$479$	6116.00			0.583397			0.291699	+	0.956510i	$0.405780\pi$
$480$		0			0
$481$	−13716.0			−1.30020
$482$		0			0
$483$		0			0
$484$		0			0
$485$	616.000	−	112.000i	0.0576724	−	0.0104859i
$486$		0			0
$487$		−	15906.0i		−	1.48002i	−0.672596	−	0.740010i	$-0.734821\pi$
$487$		−	15906.0i		−	1.48002i	0.672596	−	0.740010i	$-0.265179\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18714.0			1.72006			0.860032	−	0.510241i	$-0.170444\pi$
$491$	18714.0			1.72006			0.860032	+	0.510241i	$0.170444\pi$
$492$		0			0
$493$			4752.00i			0.434116i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1080.00i		−	0.0974741i
$498$		0			0
$499$	−4056.00			−0.363871			−0.181935	−	0.983310i	$-0.558236\pi$
$499$	−4056.00			−0.363871			−0.181935	+	0.983310i	$0.558236\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	6288.00i		−	0.557392i	−0.960379	−	0.278696i	$-0.910098\pi$
$503$		−	6288.00i		−	0.557392i	0.960379	−	0.278696i	$-0.0899021\pi$
$504$		0			0
$505$	−1184.00	−	6512.00i	−0.104331	−	0.573822i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2856.00			0.248703			0.124352	−	0.992238i	$-0.460315\pi$
$509$	2856.00			0.248703			0.124352	+	0.992238i	$0.460315\pi$
$510$		0			0
$511$	1208.00			0.104577
$512$		0			0
$513$		0			0
$514$		0			0
$515$	682.000	−	124.000i	0.0583544	−	0.0106099i
$516$		0			0
$517$		−	22400.0i		−	1.90551i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17078.0			−1.43609			−0.718043	−	0.695999i	$-0.754962\pi$
$521$	−17078.0			−1.43609			−0.718043	+	0.695999i	$0.754962\pi$
$522$		0			0
$523$		−	8560.00i		−	0.715684i	−0.933782	−	0.357842i	$-0.883513\pi$
$523$		−	8560.00i		−	0.715684i	0.933782	−	0.357842i	$-0.116487\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	4576.00i		−	0.378242i
$528$		0			0
$529$	2167.00			0.178105
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	11124.0i		−	0.904004i
$534$		0			0
$535$	924.000	−	168.000i	0.0746692	−	0.0135762i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	23730.0			1.89633
$540$		0			0
$541$	15970.0			1.26914			0.634569	−	0.772866i	$-0.281178\pi$
$541$	15970.0			1.26914			0.634569	+	0.772866i	$0.281178\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	740.000	+	4070.00i	0.0581617	+	0.319889i
$546$		0			0
$547$			15524.0i			1.21345i	0.794911	+	0.606726i	$0.207518\pi$
$547$			15524.0i			1.21345i	−0.794911	+	0.606726i	$0.792482\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5184.00			0.400809
$552$		0			0
$553$			1584.00i			0.121806i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6774.00i		−	0.515303i	−0.966238	−	0.257651i	$-0.917051\pi$
$557$		−	6774.00i		−	0.515303i	0.966238	−	0.257651i	$-0.0829486\pi$
$558$		0			0
$559$	15768.0			1.19305
$560$		0			0
$561$		0			0
$562$		0			0
$563$			10484.0i			0.784810i	0.919793	+	0.392405i	$0.128357\pi$
$563$			10484.0i			0.784810i	−0.919793	+	0.392405i	$0.871643\pi$
$564$		0			0
$565$	−19206.0	+	3492.00i	−1.43009	+	0.260017i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23302.0			1.71682			0.858410	−	0.512964i	$-0.171453\pi$
$569$	23302.0			1.71682			0.858410	+	0.512964i	$0.171453\pi$
$570$		0			0
$571$	−21520.0			−1.57720			−0.788602	−	0.614903i	$-0.789195\pi$
$571$	−21520.0			−1.57720			−0.788602	+	0.614903i	$0.789195\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4400.00	−	11700.0i	−0.319118	−	0.848563i
$576$		0			0
$577$			3856.00i			0.278210i	0.990278	+	0.139105i	$0.0444226\pi$
$577$			3856.00i			0.278210i	−0.990278	+	0.139105i	$0.955577\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	808.000			0.0576962
$582$		0			0
$583$		−	28140.0i		−	1.99904i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	26796.0i		−	1.88414i	−0.335418	−	0.942069i	$-0.608878\pi$
$587$		−	26796.0i		−	1.88414i	0.335418	−	0.942069i	$-0.391122\pi$
$588$		0			0
$589$	−4992.00			−0.349222
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9870.00i			0.683495i	0.939792	+	0.341747i	$0.111019\pi$
$593$			9870.00i			0.683495i	−0.939792	+	0.341747i	$0.888981\pi$
$594$		0			0
$595$	88.0000	+	484.000i	0.00606327	+	0.0333480i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13296.0			0.906945			0.453472	−	0.891270i	$-0.350185\pi$
$599$	13296.0			0.906945			0.453472	+	0.891270i	$0.350185\pi$
$600$		0			0
$601$	−9262.00			−0.628627			−0.314314	−	0.949319i	$-0.601774\pi$
$601$	−9262.00			−0.628627			−0.314314	+	0.949319i	$0.601774\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7138.00	−	39259.0i	−0.479671	−	2.63819i
$606$		0			0
$607$		−	5498.00i		−	0.367639i	−0.982960	−	0.183820i	$-0.941154\pi$
$607$		−	5498.00i		−	0.367639i	0.982960	−	0.183820i	$-0.0588462\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−17280.0			−1.14415
$612$		0			0
$613$		−	394.000i		−	0.0259600i	−0.999916	−	0.0129800i	$-0.995868\pi$
$613$		−	394.000i		−	0.0259600i	0.999916	−	0.0129800i	$-0.00413179\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	7370.00i		−	0.480883i	−0.970664	−	0.240442i	$-0.922708\pi$
$617$		−	7370.00i		−	0.480883i	0.970664	−	0.240442i	$-0.0772923\pi$
$618$		0			0
$619$	25316.0			1.64384			0.821919	−	0.569604i	$-0.192903\pi$
$619$	25316.0			1.64384			0.821919	+	0.569604i	$0.192903\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1876.00i		−	0.120643i
$624$		0			0
$625$	11753.0	−	10296.0i	0.752192	−	0.658944i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5588.00			0.354226
$630$		0			0
$631$	−2552.00			−0.161004			−0.0805020	−	0.996754i	$-0.525652\pi$
$631$	−2552.00			−0.161004			−0.0805020	+	0.996754i	$0.525652\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17930.0	+	3260.00i	−1.12052	+	0.203731i
$636$		0			0
$637$		−	18306.0i		−	1.13863i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8050.00			−0.496031			−0.248016	−	0.968756i	$-0.579778\pi$
$641$	−8050.00			−0.496031			−0.248016	+	0.968756i	$0.579778\pi$
$642$		0			0
$643$		−	19368.0i		−	1.18787i	−0.804514	−	0.593934i	$-0.797574\pi$
$643$		−	19368.0i		−	1.18787i	0.804514	−	0.593934i	$-0.202426\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9912.00i			0.602289i	0.953579	+	0.301144i	$0.0973686\pi$
$647$			9912.00i			0.602289i	−0.953579	+	0.301144i	$0.902631\pi$
$648$		0			0
$649$	25900.0			1.56651
$650$		0			0
$651$		0			0
$652$		0			0
$653$			27986.0i			1.67715i	0.544789	+	0.838573i	$0.316610\pi$
$653$			27986.0i			1.67715i	−0.544789	+	0.838573i	$0.683390\pi$
$654$		0			0
$655$	1740.00	+	9570.00i	0.103798	+	0.570887i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7562.00			−0.447001			−0.223501	−	0.974704i	$-0.571748\pi$
$659$	−7562.00			−0.447001			−0.223501	+	0.974704i	$0.571748\pi$
$660$		0			0
$661$	20234.0			1.19064			0.595319	−	0.803490i	$-0.297026\pi$
$661$	20234.0			1.19064			0.595319	+	0.803490i	$0.297026\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	528.000	−	96.0000i	0.0307894	−	0.00559808i
$666$		0			0
$667$			21600.0i			1.25391i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−38500.0			−2.21502
$672$		0			0
$673$		−	25332.0i		−	1.45093i	−0.688258	−	0.725466i	$-0.741624\pi$
$673$		−	25332.0i		−	1.45093i	0.688258	−	0.725466i	$-0.258376\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			18358.0i			1.04218i	0.853502	+	0.521090i	$0.174474\pi$
$677$			18358.0i			1.04218i	−0.853502	+	0.521090i	$0.825526\pi$
$678$		0			0
$679$	−112.000			−0.00633014
$680$		0			0
$681$		0			0
$682$		0			0
$683$			124.000i			0.00694689i	0.999994	+	0.00347345i	$0.00110563\pi$
$683$			124.000i			0.00694689i	−0.999994	+	0.00347345i	$0.998894\pi$
$684$		0			0
$685$	−10098.0	+	1836.00i	−0.563248	+	0.102409i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21708.0			−1.20030
$690$		0			0
$691$	17456.0			0.961009			0.480505	−	0.876992i	$-0.340453\pi$
$691$	17456.0			0.961009			0.480505	+	0.876992i	$0.340453\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1192.00	+	6556.00i	0.0650578	+	0.357818i
$696$		0			0
$697$			4532.00i			0.246287i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17816.0			0.959916			0.479958	−	0.877291i	$-0.340652\pi$
$701$	17816.0			0.959916			0.479958	+	0.877291i	$0.340652\pi$
$702$		0			0
$703$		−	6096.00i		−	0.327048i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1184.00i			0.0629829i
$708$		0			0
$709$	14298.0			0.757366			0.378683	−	0.925526i	$-0.376377\pi$
$709$	14298.0			0.757366			0.378683	+	0.925526i	$0.376377\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	20800.0i		−	1.09252i
$714$		0			0
$715$	−41580.0	+	7560.00i	−2.17483	+	0.395424i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18440.0			0.956462			0.478231	−	0.878234i	$-0.341278\pi$
$719$	18440.0			0.956462			0.478231	+	0.878234i	$0.341278\pi$
$720$		0			0
$721$	−124.000			−0.00640499
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−25272.0	+	9504.00i	−1.29459	+	0.486855i
$726$		0			0
$727$		−	9666.00i		−	0.493112i	−0.969129	−	0.246556i	$-0.920701\pi$
$727$		−	9666.00i		−	0.493112i	0.969129	−	0.246556i	$-0.0792989\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6424.00			−0.325035
$732$		0			0
$733$			6094.00i			0.307076i	0.988143	+	0.153538i	$0.0490668\pi$
$733$			6094.00i			0.307076i	−0.988143	+	0.153538i	$0.950933\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	50960.0i		−	2.54700i
$738$		0			0
$739$	9952.00			0.495386			0.247693	−	0.968839i	$-0.420328\pi$
$739$	9952.00			0.495386			0.247693	+	0.968839i	$0.420328\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2208.00i			0.109022i	0.998513	+	0.0545112i	$0.0173601\pi$
$743$			2208.00i			0.109022i	−0.998513	+	0.0545112i	$0.982640\pi$
$744$		0			0
$745$	−2152.00	−	11836.0i	−0.105830	−	0.582064i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−168.000			−0.00819571
$750$		0			0
$751$	9400.00			0.456739			0.228369	−	0.973575i	$-0.426661\pi$
$751$	9400.00			0.456739			0.228369	+	0.973575i	$0.426661\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−64.0000	−	352.000i	−0.00308503	−	0.0169677i
$756$		0			0
$757$		−	22574.0i		−	1.08384i	−0.840430	−	0.541919i	$-0.817698\pi$
$757$		−	22574.0i		−	1.08384i	0.840430	−	0.541919i	$-0.182302\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7278.00			−0.346685			−0.173343	−	0.984862i	$-0.555457\pi$
$761$	−7278.00			−0.346685			−0.173343	+	0.984862i	$0.555457\pi$
$762$		0			0
$763$		−	740.000i		−	0.0351111i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	19980.0i		−	0.940595i
$768$		0			0
$769$	16542.0			0.775708			0.387854	−	0.921721i	$-0.373216\pi$
$769$	16542.0			0.775708			0.387854	+	0.921721i	$0.373216\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	28926.0i		−	1.34592i	−0.739679	−	0.672960i	$-0.765023\pi$
$773$		−	28926.0i		−	1.34592i	0.739679	−	0.672960i	$-0.234977\pi$
$774$		0			0
$775$	24336.0	−	9152.00i	1.12797	−	0.424193i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4944.00			0.227390
$780$		0			0
$781$	−37800.0			−1.73187
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−28094.0	+	5108.00i	−1.27735	+	0.232245i
$786$		0			0
$787$		−	20608.0i		−	0.933413i	−0.884412	−	0.466706i	$-0.845440\pi$
$787$		−	20608.0i		−	0.933413i	0.884412	−	0.466706i	$-0.154560\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3492.00			0.156967
$792$		0			0
$793$			29700.0i			1.32998i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			41350.0i			1.83776i	0.394541	+	0.918878i	$0.370904\pi$
$797$			41350.0i			1.83776i	−0.394541	+	0.918878i	$0.629096\pi$
$798$		0			0
$799$	7040.00			0.311711
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	42280.0i		−	1.85807i
$804$		0			0
$805$	400.000	+	2200.00i	0.0175132	+	0.0963227i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1794.00			0.0779650			0.0389825	−	0.999240i	$-0.487588\pi$
$809$	1794.00			0.0779650			0.0389825	+	0.999240i	$0.487588\pi$
$810$		0			0
$811$	−22756.0			−0.985291			−0.492646	−	0.870230i	$-0.663970\pi$
$811$	−22756.0			−0.985291			−0.492646	+	0.870230i	$0.663970\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8272.00	−	1504.00i	0.355528	−	0.0646415i
$816$		0			0
$817$			7008.00i			0.300097i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23632.0			1.00458			0.502291	−	0.864698i	$-0.332490\pi$
$821$	23632.0			1.00458			0.502291	+	0.864698i	$0.332490\pi$
$822$		0			0
$823$			33210.0i			1.40660i	0.710896	+	0.703298i	$0.248290\pi$
$823$			33210.0i			1.40660i	−0.710896	+	0.703298i	$0.751710\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			30476.0i			1.28144i	0.767773	+	0.640722i	$0.221365\pi$
$827$			30476.0i			1.28144i	−0.767773	+	0.640722i	$0.778635\pi$
$828$		0			0
$829$	−29802.0			−1.24857			−0.624286	−	0.781196i	$-0.714610\pi$
$829$	−29802.0			−1.24857			−0.624286	+	0.781196i	$0.714610\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			7458.00i			0.310209i
$834$		0			0
$835$	29700.0	−	5400.00i	1.23091	−	0.223802i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28024.0			1.15315			0.576577	−	0.817043i	$-0.304388\pi$
$839$	28024.0			1.15315			0.576577	+	0.817043i	$0.304388\pi$
$840$		0			0
$841$	22267.0			0.912994
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1438.00	+	7909.00i	0.0585428	+	0.321986i
$846$		0			0
$847$			7138.00i			0.289569i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	25400.0			1.02315
$852$		0			0
$853$		−	3938.00i		−	0.158071i	−0.996872	−	0.0790355i	$-0.974816\pi$
$853$		−	3938.00i		−	0.158071i	0.996872	−	0.0790355i	$-0.0251840\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8094.00i			0.322621i	0.986904	+	0.161310i	$0.0515720\pi$
$857$			8094.00i			0.322621i	−0.986904	+	0.161310i	$0.948428\pi$
$858$		0			0
$859$	9044.00			0.359229			0.179614	−	0.983737i	$-0.442515\pi$
$859$	9044.00			0.359229			0.179614	+	0.983737i	$0.442515\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			6252.00i			0.246606i	0.992369	+	0.123303i	$0.0393486\pi$
$863$			6252.00i			0.246606i	−0.992369	+	0.123303i	$0.960651\pi$
$864$		0			0
$865$	−14674.0	+	2668.00i	−0.576799	+	0.104873i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	55440.0			2.16418
$870$		0			0
$871$	−39312.0			−1.52932
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2398.00	+	1436.00i	−0.0926482	+	0.0554808i
$876$		0			0
$877$			40166.0i			1.54653i	0.634081	+	0.773267i	$0.281379\pi$
$877$			40166.0i			1.54653i	−0.634081	+	0.773267i	$0.718621\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12834.0			0.490793			0.245396	−	0.969423i	$-0.421082\pi$
$881$	12834.0			0.490793			0.245396	+	0.969423i	$0.421082\pi$
$882$		0			0
$883$			27192.0i			1.03633i	0.855279	+	0.518167i	$0.173386\pi$
$883$			27192.0i			1.03633i	−0.855279	+	0.518167i	$0.826614\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			42060.0i			1.59215i	0.605198	+	0.796075i	$0.293094\pi$
$887$			42060.0i			1.59215i	−0.605198	+	0.796075i	$0.706906\pi$
$888$		0			0
$889$	3260.00			0.122989
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	7680.00i		−	0.287796i
$894$		0			0
$895$	−3428.00	−	18854.0i	−0.128028	−	0.704156i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−44928.0			−1.66678
$900$		0			0
$901$	8844.00			0.327010
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8012.00	+	44066.0i	0.294285	+	1.61857i
$906$		0			0
$907$		−	41172.0i		−	1.50727i	−0.657293	−	0.753635i	$-0.728299\pi$
$907$		−	41172.0i		−	1.50727i	0.657293	−	0.753635i	$-0.271701\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−48.0000			−0.00174568			−0.000872838	−	1.00000i	$-0.500278\pi$
$911$	−48.0000			−0.00174568			−0.000872838		1.00000i	$0.500278\pi$
$912$		0			0
$913$		−	28280.0i		−	1.02512i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1740.00i		−	0.0626607i
$918$		0			0
$919$	−34584.0			−1.24137			−0.620686	−	0.784059i	$-0.713146\pi$
$919$	−34584.0			−1.24137			−0.620686	+	0.784059i	$0.713146\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			29160.0i			1.03988i
$924$		0			0
$925$	11176.0	+	29718.0i	0.397259	+	1.05635i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3474.00			−0.122689			−0.0613446	−	0.998117i	$-0.519539\pi$
$929$	−3474.00			−0.122689			−0.0613446	+	0.998117i	$0.519539\pi$
$930$		0			0
$931$	8136.00			0.286409
$932$		0			0
$933$		0			0
$934$		0			0
$935$	16940.0	−	3080.00i	0.592510	−	0.107729i
$936$		0			0
$937$			44408.0i			1.54829i	0.633009	+	0.774144i	$0.281819\pi$
$937$			44408.0i			1.54829i	−0.633009	+	0.774144i	$0.718181\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20188.0			−0.699373			−0.349686	−	0.936867i	$-0.613712\pi$
$941$	−20188.0			−0.699373			−0.349686	+	0.936867i	$0.613712\pi$
$942$		0			0
$943$			20600.0i			0.711377i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	31212.0i		−	1.07102i	−0.844530	−	0.535509i	$-0.820120\pi$
$947$		−	31212.0i		−	1.07102i	0.844530	−	0.535509i	$-0.179880\pi$
$948$		0			0
$949$	−32616.0			−1.11566
$950$		0			0
$951$		0			0
$952$		0			0
$953$			20182.0i			0.686001i	0.939335	+	0.343001i	$0.111443\pi$
$953$			20182.0i			0.686001i	−0.939335	+	0.343001i	$0.888557\pi$
$954$		0			0
$955$	1368.00	+	7524.00i	0.0463533	+	0.254943i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1836.00			0.0618222
$960$		0			0
$961$	13473.0			0.452251
$962$		0			0
$963$		0			0
$964$		0			0
$965$	49324.0	−	8968.00i	1.64538	−	0.299161i
$966$		0			0
$967$			53722.0i			1.78654i	0.449522	+	0.893269i	$0.351594\pi$
$967$			53722.0i			1.78654i	−0.449522	+	0.893269i	$0.648406\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22554.0			0.745409			0.372705	−	0.927950i	$-0.378430\pi$
$971$	22554.0			0.745409			0.372705	+	0.927950i	$0.378430\pi$
$972$		0			0
$973$		−	1192.00i		−	0.0392742i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			18126.0i			0.593554i	0.954947	+	0.296777i	$0.0959118\pi$
$977$			18126.0i			0.593554i	−0.954947	+	0.296777i	$0.904088\pi$
$978$		0			0
$979$	−65660.0			−2.14352
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	6232.00i		−	0.202207i	−0.994876	−	0.101104i	$-0.967763\pi$
$983$		−	6232.00i		−	0.202207i	0.994876	−	0.101104i	$-0.0322374\pi$
$984$		0			0
$985$	11638.0	−	2116.00i	0.376464	−	0.0684481i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−29200.0			−0.938833
$990$		0			0
$991$	−15184.0			−0.486716			−0.243358	−	0.969937i	$-0.578249\pi$
$991$	−15184.0			−0.486716			−0.243358	+	0.969937i	$0.578249\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2256.00	+	12408.0i	0.0718794	+	0.395337i
$996$		0			0
$997$		−	29922.0i		−	0.950491i	−0.879853	−	0.475245i	$-0.842359\pi$
$997$		−	29922.0i		−	0.950491i	0.879853	−	0.475245i	$-0.157641\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.f.c.289.1		2
3.2	odd	2		240.4.f.d.49.1		2
4.3	odd	2		90.4.c.a.19.1		2
5.4	even	2	inner	720.4.f.c.289.2		2
12.11	even	2		30.4.c.a.19.2	yes	2
15.2	even	4		1200.4.a.h.1.1		1
15.8	even	4		1200.4.a.bc.1.1		1
15.14	odd	2		240.4.f.d.49.2		2
20.3	even	4		450.4.a.e.1.1		1
20.7	even	4		450.4.a.p.1.1		1
20.19	odd	2		90.4.c.a.19.2		2
24.5	odd	2		960.4.f.d.769.2		2
24.11	even	2		960.4.f.c.769.1		2
60.23	odd	4		150.4.a.f.1.1		1
60.47	odd	4		150.4.a.d.1.1		1
60.59	even	2		30.4.c.a.19.1	✓	2
120.29	odd	2		960.4.f.d.769.1		2
120.59	even	2		960.4.f.c.769.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.c.a.19.1	✓	2	60.59	even	2
30.4.c.a.19.2	yes	2	12.11	even	2
90.4.c.a.19.1		2	4.3	odd	2
90.4.c.a.19.2		2	20.19	odd	2
150.4.a.d.1.1		1	60.47	odd	4
150.4.a.f.1.1		1	60.23	odd	4
240.4.f.d.49.1		2	3.2	odd	2
240.4.f.d.49.2		2	15.14	odd	2
450.4.a.e.1.1		1	20.3	even	4
450.4.a.p.1.1		1	20.7	even	4
720.4.f.c.289.1		2	1.1	even	1	trivial
720.4.f.c.289.2		2	5.4	even	2	inner
960.4.f.c.769.1		2	24.11	even	2
960.4.f.c.769.2		2	120.59	even	2
960.4.f.d.769.1		2	120.29	odd	2
960.4.f.d.769.2		2	24.5	odd	2
1200.4.a.h.1.1		1	15.2	even	4
1200.4.a.bc.1.1		1	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 11.0000i) q^{5} +2.00000i q^{7} +70.0000 q^{11} -54.0000i q^{13} +22.0000i q^{17} +24.0000 q^{19} +100.000i q^{23} +(-117.000 + 44.0000i) q^{25} +216.000 q^{29} -208.000 q^{31} +(22.0000 - 4.00000i) q^{35} -254.000i q^{37} +206.000 q^{41} +292.000i q^{43} -320.000i q^{47} +339.000 q^{49} -402.000i q^{53} +(-140.000 - 770.000i) q^{55} +370.000 q^{59} -550.000 q^{61} +(-594.000 + 108.000i) q^{65} -728.000i q^{67} -540.000 q^{71} -604.000i q^{73} +140.000i q^{77} +792.000 q^{79} -404.000i q^{83} +(242.000 - 44.0000i) q^{85} -938.000 q^{89} +108.000 q^{91} +(-48.0000 - 264.000i) q^{95} +56.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 140 q^{11} + 48 q^{19} - 234 q^{25} + 432 q^{29} - 416 q^{31} + 44 q^{35} + 412 q^{41} + 678 q^{49} - 280 q^{55} + 740 q^{59} - 1100 q^{61} - 1188 q^{65} - 1080 q^{71} + 1584 q^{79} + 484 q^{85}+ \cdots - 96 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 + 140 * q^11 + 48 * q^19 - 234 * q^25 + 432 * q^29 - 416 * q^31 + 44 * q^35 + 412 * q^41 + 678 * q^49 - 280 * q^55 + 740 * q^59 - 1100 * q^61 - 1188 * q^65 - 1080 * q^71 + 1584 * q^79 + 484 * q^85 - 1876 * q^89 + 216 * q^91 - 96 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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