LMFDB - Embedded newform 1728.4.d.g.865.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(865,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.d (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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1731891456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.3
Root			$-1.35234 - 0.780776i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.g.865.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.3923i q^{5} +21.4243 q^{7} +O(q^{10})$	$q-10.3923i q^{5} +21.4243 q^{7} +24.7386i q^{11} -21.4243i q^{13} -123.693 q^{17} -7.00000i q^{19} +114.315 q^{23} +17.0000 q^{25} +270.200i q^{29} +214.243 q^{31} -222.648i q^{35} -235.667i q^{37} +395.818 q^{41} -92.0000i q^{43} +114.315 q^{47} +116.000 q^{49} -20.7846i q^{53} +257.091 q^{55} +173.170i q^{59} -449.910i q^{61} -222.648 q^{65} -353.000i q^{67} +789.815 q^{71} +425.000 q^{73} +530.008i q^{77} -1306.88 q^{79} +593.727i q^{83} +1285.46i q^{85} -74.2159 q^{89} -459.000i q^{91} -72.7461 q^{95} +799.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 136 q^{25} + 928 q^{49} + 3400 q^{73} + 6392 q^{97}+O(q^{100}) $ 8 * q + 136 * q^25 + 928 * q^49 + 3400 * q^73 + 6392 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 10.3923i	− 0.929516i	−0.885438	−	0.464758i	$-0.846141\pi$
$5$	− 10.3923i	− 0.929516i	0.885438	−	0.464758i	$-0.153859\pi$
$6$	0	0
$7$	21.4243	1.15680	0.578401	−	0.815752i	$-0.303677\pi$
$7$	21.4243	1.15680	0.578401	+	0.815752i	$0.303677\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	24.7386i	0.678089i	0.940770	+	0.339044i	$0.110104\pi$
$11$	24.7386i	0.678089i	−0.940770	+	0.339044i	$0.889896\pi$
$12$	0	0
$13$	− 21.4243i	− 0.457079i	−0.973535	−	0.228540i	$-0.926605\pi$
$13$	− 21.4243i	− 0.457079i	0.973535	−	0.228540i	$-0.0733950\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−123.693	−1.76471	−0.882353	−	0.470588i	$-0.844042\pi$
$17$	−123.693	−1.76471	−0.882353	+	0.470588i	$0.844042\pi$
$18$	0	0
$19$	− 7.00000i	− 0.0845216i	−0.999107	−	0.0422608i	$-0.986544\pi$
$19$	− 7.00000i	− 0.0845216i	0.999107	−	0.0422608i	$-0.0134560\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	114.315	1.03637	0.518183	−	0.855270i	$-0.326609\pi$
$23$	114.315	1.03637	0.518183	+	0.855270i	$0.326609\pi$
$24$	0	0
$25$	17.0000	0.136000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	270.200i	1.73017i	0.501627	+	0.865084i	$0.332735\pi$
$29$	270.200i	1.73017i	−0.501627	+	0.865084i	$0.667265\pi$
$30$	0	0
$31$	214.243	1.24126	0.620631	−	0.784102i	$-0.286876\pi$
$31$	214.243	1.24126	0.620631	+	0.784102i	$0.286876\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 222.648i	− 1.07527i
$36$	0	0
$37$	− 235.667i	− 1.04712i	−0.851989	−	0.523560i	$-0.824604\pi$
$37$	− 235.667i	− 1.04712i	0.851989	−	0.523560i	$-0.175396\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	395.818	1.50772	0.753859	−	0.657037i	$-0.228190\pi$
$41$	395.818	1.50772	0.753859	+	0.657037i	$0.228190\pi$
$42$	0	0
$43$	− 92.0000i	− 0.326276i	−0.986603	−	0.163138i	$-0.947838\pi$
$43$	− 92.0000i	− 0.326276i	0.986603	−	0.163138i	$-0.0521616\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	114.315	0.354779	0.177389	−	0.984141i	$-0.443235\pi$
$47$	114.315	0.354779	0.177389	+	0.984141i	$0.443235\pi$
$48$	0	0
$49$	116.000	0.338192
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 20.7846i	− 0.0538677i	−0.999637	−	0.0269338i	$-0.991426\pi$
$53$	− 20.7846i	− 0.0538677i	0.999637	−	0.0269338i	$-0.00857434\pi$
$54$	0	0
$55$	257.091	0.630295
$56$	0	0
$57$	0	0
$58$	0	0
$59$	173.170i	0.382116i	0.981579	+	0.191058i	$0.0611919\pi$
$59$	173.170i	0.382116i	−0.981579	+	0.191058i	$0.938808\pi$
$60$	0	0
$61$	− 449.910i	− 0.944345i	−0.881506	−	0.472173i	$-0.843470\pi$
$61$	− 449.910i	− 0.944345i	0.881506	−	0.472173i	$-0.156530\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−222.648	−0.424862
$66$	0	0
$67$	− 353.000i	− 0.643669i	−0.946796	−	0.321834i	$-0.895701\pi$
$67$	− 353.000i	− 0.643669i	0.946796	−	0.321834i	$-0.104299\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	789.815	1.32019	0.660097	−	0.751180i	$-0.270515\pi$
$71$	789.815	1.32019	0.660097	+	0.751180i	$0.270515\pi$
$72$	0	0
$73$	425.000	0.681404	0.340702	−	0.940171i	$-0.389335\pi$
$73$	425.000	0.681404	0.340702	+	0.940171i	$0.389335\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	530.008i	0.784415i
$78$	0	0
$79$	−1306.88	−1.86121	−0.930605	−	0.366024i	$-0.880719\pi$
$79$	−1306.88	−1.86121	−0.930605	+	0.366024i	$0.880719\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	593.727i	0.785181i	0.919713	+	0.392591i	$0.128421\pi$
$83$	593.727i	0.785181i	−0.919713	+	0.392591i	$0.871579\pi$
$84$	0	0
$85$	1285.46i	1.64032i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−74.2159	−0.0883918	−0.0441959	−	0.999023i	$-0.514073\pi$
$89$	−74.2159	−0.0883918	−0.0441959	+	0.999023i	$0.514073\pi$
$90$	0	0
$91$	− 459.000i	− 0.528750i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−72.7461	−0.0785642
$96$	0	0
$97$	799.000	0.836352	0.418176	−	0.908366i	$-0.362670\pi$
$97$	799.000	0.836352	0.418176	+	0.908366i	$0.362670\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	228.631i	0.225244i	0.993638	+	0.112622i	$0.0359249\pi$
$101$	228.631i	0.225244i	−0.993638	+	0.112622i	$0.964075\pi$
$102$	0	0
$103$	1092.64	1.04525	0.522626	−	0.852562i	$-0.324953\pi$
$103$	1092.64	1.04525	0.522626	+	0.852562i	$0.324953\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	766.898i	0.692886i	0.938071	+	0.346443i	$0.112611\pi$
$107$	766.898i	0.692886i	−0.938071	+	0.346443i	$0.887389\pi$
$108$	0	0
$109$	685.577i	0.602444i	0.953554	+	0.301222i	$0.0973945\pi$
$109$	685.577i	0.602444i	−0.953554	+	0.301222i	$0.902606\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−519.511	−0.432491	−0.216246	−	0.976339i	$-0.569381\pi$
$113$	−519.511	−0.432491	−0.216246	+	0.976339i	$0.569381\pi$
$114$	0	0
$115$	− 1188.00i	− 0.963318i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2650.04	−2.04142
$120$	0	0
$121$	719.000	0.540195
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1475.71i	− 1.05593i
$126$	0	0
$127$	−2013.88	−1.40711	−0.703556	−	0.710640i	$-0.748406\pi$
$127$	−2013.88	−1.40711	−0.703556	+	0.710640i	$0.748406\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2523.34i	− 1.68294i	−0.540303	−	0.841471i	$-0.681690\pi$
$131$	− 2523.34i	− 1.68294i	0.540303	−	0.841471i	$-0.318310\pi$
$132$	0	0
$133$	− 149.970i	− 0.0977748i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	24.7386	0.0154275	0.00771374	−	0.999970i	$-0.497545\pi$
$137$	24.7386	0.0154275	0.00771374	+	0.999970i	$0.497545\pi$
$138$	0	0
$139$	− 2185.00i	− 1.33330i	−0.745369	−	0.666652i	$-0.767727\pi$
$139$	− 2185.00i	− 1.33330i	0.745369	−	0.666652i	$-0.232273\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	530.008	0.309940
$144$	0	0
$145$	2808.00	1.60822
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3242.40i	− 1.78274i	−0.453280	−	0.891368i	$-0.649746\pi$
$149$	− 3242.40i	− 1.78274i	0.453280	−	0.891368i	$-0.350254\pi$
$150$	0	0
$151$	192.819	0.103916	0.0519581	−	0.998649i	$-0.483454\pi$
$151$	192.819	0.103916	0.0519581	+	0.998649i	$0.483454\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2226.48i	− 1.15377i
$156$	0	0
$157$	1371.15i	0.697007i	0.937308	+	0.348503i	$0.113310\pi$
$157$	1371.15i	0.697007i	−0.937308	+	0.348503i	$0.886690\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2449.12	1.19887
$162$	0	0
$163$	− 1321.00i	− 0.634777i	−0.948296	−	0.317389i	$-0.897194\pi$
$163$	− 1321.00i	− 0.634777i	0.948296	−	0.317389i	$-0.102806\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	571.577	0.264850	0.132425	−	0.991193i	$-0.457724\pi$
$167$	571.577	0.264850	0.132425	+	0.991193i	$0.457724\pi$
$168$	0	0
$169$	1738.00	0.791079
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3450.25i	1.51628i	0.652089	+	0.758142i	$0.273893\pi$
$173$	3450.25i	1.51628i	−0.652089	+	0.758142i	$0.726107\pi$
$174$	0	0
$175$	364.213	0.157325
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1484.32i	0.619794i	0.950770	+	0.309897i	$0.100295\pi$
$179$	1484.32i	0.619794i	−0.950770	+	0.309897i	$0.899705\pi$
$180$	0	0
$181$	− 1949.61i	− 0.800626i	−0.916378	−	0.400313i	$-0.868901\pi$
$181$	− 1949.61i	− 0.800626i	0.916378	−	0.400313i	$-0.131099\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2449.12	−0.973315
$186$	0	0
$187$	− 3060.00i	− 1.19663i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2047.28	−0.775583	−0.387791	−	0.921747i	$-0.626762\pi$
$191$	−2047.28	−0.775583	−0.387791	+	0.921747i	$0.626762\pi$
$192$	0	0
$193$	2459.00	0.917112	0.458556	−	0.888665i	$-0.348367\pi$
$193$	2459.00	0.917112	0.458556	+	0.888665i	$0.348367\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3959.47i	− 1.43198i	−0.698110	−	0.715991i	$-0.745975\pi$
$197$	− 3959.47i	− 1.43198i	0.698110	−	0.715991i	$-0.254025\pi$
$198$	0	0
$199$	2977.98	1.06082	0.530410	−	0.847741i	$-0.322038\pi$
$199$	2977.98	1.06082	0.530410	+	0.847741i	$0.322038\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5788.84i	2.00146i
$204$	0	0
$205$	− 4113.46i	− 1.40145i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	173.170	0.0573132
$210$	0	0
$211$	− 4763.00i	− 1.55402i	−0.629488	−	0.777011i	$-0.716735\pi$
$211$	− 4763.00i	− 1.55402i	0.629488	−	0.777011i	$-0.283265\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−956.092	−0.303279
$216$	0	0
$217$	4590.00	1.43590
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2650.04i	0.806610i
$222$	0	0
$223$	1842.49	0.553283	0.276642	−	0.960973i	$-0.410779\pi$
$223$	1842.49	0.553283	0.276642	+	0.960973i	$0.410779\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2572.82i	0.752264i	0.926566	+	0.376132i	$0.122746\pi$
$227$	2572.82i	0.752264i	−0.926566	+	0.376132i	$0.877254\pi$
$228$	0	0
$229$	− 4027.77i	− 1.16228i	−0.813803	−	0.581140i	$-0.802607\pi$
$229$	− 4027.77i	− 1.16228i	0.813803	−	0.581140i	$-0.197393\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3512.89	−0.987712	−0.493856	−	0.869544i	$-0.664413\pi$
$233$	−3512.89	−0.987712	−0.493856	+	0.869544i	$0.664413\pi$
$234$	0	0
$235$	− 1188.00i	− 0.329773i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2265.52	−0.613157	−0.306578	−	0.951845i	$-0.599184\pi$
$239$	−2265.52	−0.613157	−0.306578	+	0.951845i	$0.599184\pi$
$240$	0	0
$241$	4709.00	1.25864	0.629322	−	0.777144i	$-0.283333\pi$
$241$	4709.00	1.25864	0.629322	+	0.777144i	$0.283333\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1205.51i	− 0.314355i
$246$	0	0
$247$	−149.970	−0.0386330
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 7025.77i	− 1.76678i	−0.468635	−	0.883392i	$-0.655254\pi$
$251$	− 7025.77i	− 1.76678i	0.468635	−	0.883392i	$-0.344746\pi$
$252$	0	0
$253$	2828.01i	0.702748i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4205.57	1.02076	0.510382	−	0.859948i	$-0.329504\pi$
$257$	4205.57	1.02076	0.510382	+	0.859948i	$0.329504\pi$
$258$	0	0
$259$	− 5049.00i	− 1.21131i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5300.08	1.24265	0.621324	−	0.783553i	$-0.286595\pi$
$263$	5300.08	1.24265	0.621324	+	0.783553i	$0.286595\pi$
$264$	0	0
$265$	−216.000	−0.0500708
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4458.30i	1.01051i	0.862970	+	0.505255i	$0.168602\pi$
$269$	4458.30i	1.01051i	−0.862970	+	0.505255i	$0.831398\pi$
$270$	0	0
$271$	449.910	0.100849	0.0504245	−	0.998728i	$-0.483943\pi$
$271$	449.910	0.100849	0.0504245	+	0.998728i	$0.483943\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	420.557i	0.0922201i
$276$	0	0
$277$	7027.17i	1.52427i	0.647421	+	0.762133i	$0.275848\pi$
$277$	7027.17i	1.52427i	−0.647421	+	0.762133i	$0.724152\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6382.57	1.35499	0.677495	−	0.735528i	$-0.263066\pi$
$281$	6382.57	1.35499	0.677495	+	0.735528i	$0.263066\pi$
$282$	0	0
$283$	− 8116.00i	− 1.70476i	−0.522926	−	0.852378i	$-0.675159\pi$
$283$	− 8116.00i	− 1.70476i	0.522926	−	0.852378i	$-0.324841\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8480.12	1.74413
$288$	0	0
$289$	10387.0	2.11419
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3813.98i	0.760460i	0.924892	+	0.380230i	$0.124155\pi$
$293$	3813.98i	0.760460i	−0.924892	+	0.380230i	$0.875845\pi$
$294$	0	0
$295$	1799.64	0.355183
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2449.12i	− 0.473701i
$300$	0	0
$301$	− 1971.03i	− 0.377437i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4675.60	−0.877784
$306$	0	0
$307$	628.000i	0.116749i	0.998295	+	0.0583744i	$0.0185917\pi$
$307$	628.000i	0.116749i	−0.998295	+	0.0583744i	$0.981408\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5310.47	0.968260	0.484130	−	0.874996i	$-0.339136\pi$
$311$	5310.47	0.968260	0.484130	+	0.874996i	$0.339136\pi$
$312$	0	0
$313$	745.000	0.134536	0.0672682	−	0.997735i	$-0.478572\pi$
$313$	745.000	0.134536	0.0672682	+	0.997735i	$0.478572\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 249.415i	− 0.0441910i	−0.999756	−	0.0220955i	$-0.992966\pi$
$317$	− 249.415i	− 0.0441910i	0.999756	−	0.0220955i	$-0.00703379\pi$
$318$	0	0
$319$	−6684.38	−1.17321
$320$	0	0
$321$	0	0
$322$	0	0
$323$	865.852i	0.149156i
$324$	0	0
$325$	− 364.213i	− 0.0621628i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2449.12	0.410409
$330$	0	0
$331$	− 4057.00i	− 0.673695i	−0.941559	−	0.336847i	$-0.890639\pi$
$331$	− 4057.00i	− 0.673695i	0.941559	−	0.336847i	$-0.109361\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3668.48	−0.598301
$336$	0	0
$337$	2023.00	0.327002	0.163501	−	0.986543i	$-0.447721\pi$
$337$	2023.00	0.327002	0.163501	+	0.986543i	$0.447721\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5300.08i	0.841687i
$342$	0	0
$343$	−4863.31	−0.765581
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6283.61i	− 0.972110i	−0.873928	−	0.486055i	$-0.838436\pi$
$347$	− 6283.61i	− 0.972110i	0.873928	−	0.486055i	$-0.161564\pi$
$348$	0	0
$349$	7991.26i	1.22568i	0.790207	+	0.612840i	$0.209973\pi$
$349$	7991.26i	1.22568i	−0.790207	+	0.612840i	$0.790027\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12121.9	−1.82772	−0.913860	−	0.406029i	$-0.866913\pi$
$353$	−12121.9	−1.82772	−0.913860	+	0.406029i	$0.866913\pi$
$354$	0	0
$355$	− 8208.00i	− 1.22714i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10090.9	−1.48351	−0.741753	−	0.670673i	$-0.766005\pi$
$359$	−10090.9	−1.48351	−0.741753	+	0.670673i	$0.766005\pi$
$360$	0	0
$361$	6810.00	0.992856
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4416.73i	− 0.633376i
$366$	0	0
$367$	2892.28	0.411378	0.205689	−	0.978617i	$-0.434056\pi$
$367$	2892.28	0.411378	0.205689	+	0.978617i	$0.434056\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 445.295i	− 0.0623142i
$372$	0	0
$373$	− 4520.52i	− 0.627517i	−0.949503	−	0.313759i	$-0.898412\pi$
$373$	− 4520.52i	− 0.627517i	0.949503	−	0.313759i	$-0.101588\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5788.84	0.790823
$378$	0	0
$379$	4759.00i	0.644996i	0.946570	+	0.322498i	$0.104523\pi$
$379$	4759.00i	0.644996i	−0.946570	+	0.322498i	$0.895477\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1101.58	0.146967	0.0734835	−	0.997296i	$-0.476588\pi$
$383$	1101.58	0.146967	0.0734835	+	0.997296i	$0.476588\pi$
$384$	0	0
$385$	5508.00	0.729126
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1236.68i	− 0.161189i	−0.996747	−	0.0805943i	$-0.974318\pi$
$389$	− 1236.68i	− 0.161189i	0.996747	−	0.0805943i	$-0.0256818\pi$
$390$	0	0
$391$	−14140.0	−1.82888
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13581.5i	1.73003i
$396$	0	0
$397$	856.971i	0.108338i	0.998532	+	0.0541690i	$0.0172510\pi$
$397$	856.971i	0.108338i	−0.998532	+	0.0541690i	$0.982749\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8064.79	−1.00433	−0.502165	−	0.864772i	$-0.667463\pi$
$401$	−8064.79	−1.00433	−0.502165	+	0.864772i	$0.667463\pi$
$402$	0	0
$403$	− 4590.00i	− 0.567355i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5830.08	0.710041
$408$	0	0
$409$	−10555.0	−1.27607	−0.638033	−	0.770009i	$-0.720252\pi$
$409$	−10555.0	−1.27607	−0.638033	+	0.770009i	$0.720252\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3710.05i	0.442033i
$414$	0	0
$415$	6170.19	0.729838
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1706.97i	0.199023i	0.995036	+	0.0995116i	$0.0317280\pi$
$419$	1706.97i	0.199023i	−0.995036	+	0.0995116i	$0.968272\pi$
$420$	0	0
$421$	9662.35i	1.11856i	0.828978	+	0.559281i	$0.188923\pi$
$421$	9662.35i	1.11856i	−0.828978	+	0.559281i	$0.811077\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2102.78	−0.240000
$426$	0	0
$427$	− 9639.00i	− 1.09242i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2359.05	−0.263646	−0.131823	−	0.991273i	$-0.542083\pi$
$431$	−2359.05	−0.263646	−0.131823	+	0.991273i	$0.542083\pi$
$432$	0	0
$433$	16450.0	1.82572	0.912860	−	0.408273i	$-0.133869\pi$
$433$	16450.0	1.82572	0.912860	+	0.408273i	$0.133869\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 800.207i	− 0.0875952i
$438$	0	0
$439$	−5784.56	−0.628888	−0.314444	−	0.949276i	$-0.601818\pi$
$439$	−5784.56	−0.628888	−0.314444	+	0.949276i	$0.601818\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11627.2i	1.24700i	0.781822	+	0.623502i	$0.214291\pi$
$443$	11627.2i	1.24700i	−0.781822	+	0.623502i	$0.785709\pi$
$444$	0	0
$445$	771.274i	0.0821616i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14670.0	1.54192	0.770958	−	0.636886i	$-0.219778\pi$
$449$	14670.0	1.54192	0.770958	+	0.636886i	$0.219778\pi$
$450$	0	0
$451$	9792.00i	1.02237i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4770.07	−0.491482
$456$	0	0
$457$	−9574.00	−0.979984	−0.489992	−	0.871727i	$-0.663000\pi$
$457$	−9574.00	−0.979984	−0.489992	+	0.871727i	$0.663000\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 14310.2i	− 1.44575i	−0.690977	−	0.722877i	$-0.742819\pi$
$461$	− 14310.2i	− 1.44575i	0.690977	−	0.722877i	$-0.257181\pi$
$462$	0	0
$463$	18103.5	1.81715	0.908577	−	0.417718i	$-0.137170\pi$
$463$	18103.5	1.81715	0.908577	+	0.417718i	$0.137170\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6852.60i	− 0.679016i	−0.940603	−	0.339508i	$-0.889739\pi$
$467$	− 6852.60i	− 0.679016i	0.940603	−	0.339508i	$-0.110261\pi$
$468$	0	0
$469$	− 7562.77i	− 0.744598i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2275.95	0.221244
$474$	0	0
$475$	− 119.000i	− 0.0114949i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13156.7	1.25500	0.627498	−	0.778618i	$-0.284079\pi$
$479$	13156.7	1.25500	0.627498	+	0.778618i	$0.284079\pi$
$480$	0	0
$481$	−5049.00	−0.478617
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8303.45i	− 0.777403i
$486$	0	0
$487$	3706.40	0.344873	0.172436	−	0.985021i	$-0.444836\pi$
$487$	3706.40	0.344873	0.172436	+	0.985021i	$0.444836\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 19024.0i	− 1.74856i	−0.485425	−	0.874279i	$-0.661335\pi$
$491$	− 19024.0i	− 1.74856i	0.485425	−	0.874279i	$-0.338665\pi$
$492$	0	0
$493$	− 33421.9i	− 3.05324i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16921.2	1.52720
$498$	0	0
$499$	− 3944.00i	− 0.353823i	−0.984227	−	0.176912i	$-0.943389\pi$
$499$	− 3944.00i	− 0.353823i	0.984227	−	0.176912i	$-0.0566106\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20898.9	−1.85256	−0.926279	−	0.376838i	$-0.877011\pi$
$503$	−20898.9	−1.85256	−0.926279	+	0.376838i	$0.877011\pi$
$504$	0	0
$505$	2376.00	0.209368
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 51.9615i	− 0.00452486i	−0.999997	−	0.00226243i	$-0.999280\pi$
$509$	− 51.9615i	− 0.00452486i	0.999997	−	0.00226243i	$-0.000720155\pi$
$510$	0	0
$511$	9105.32	0.788250
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 11355.0i	− 0.971578i
$516$	0	0
$517$	2828.01i	0.240572i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9920.19	−0.834187	−0.417094	−	0.908864i	$-0.636951\pi$
$521$	−9920.19	−0.834187	−0.417094	+	0.908864i	$0.636951\pi$
$522$	0	0
$523$	20923.0i	1.74933i	0.484729	+	0.874664i	$0.338918\pi$
$523$	20923.0i	1.74933i	−0.484729	+	0.874664i	$0.661082\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26500.4	−2.19046
$528$	0	0
$529$	901.000	0.0740528
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 8480.12i	− 0.689146i
$534$	0	0
$535$	7969.83	0.644049
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2869.68i	0.229325i
$540$	0	0
$541$	19946.0i	1.58511i	0.609799	+	0.792556i	$0.291250\pi$
$541$	19946.0i	1.58511i	−0.609799	+	0.792556i	$0.708750\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7124.73	0.559981
$546$	0	0
$547$	875.000i	0.0683954i	0.999415	+	0.0341977i	$0.0108876\pi$
$547$	875.000i	0.0683954i	−0.999415	+	0.0341977i	$0.989112\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1891.40	0.146237
$552$	0	0
$553$	−27999.0	−2.15305
$554$	0	0
$555$	0	0
$556$	0	0
$557$	467.654i	0.0355747i	0.999842	+	0.0177874i	$0.00566219\pi$
$557$	467.654i	0.0355747i	−0.999842	+	0.0177874i	$0.994338\pi$
$558$	0	0
$559$	−1971.03	−0.149134
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17119.1i	1.28150i	0.767749	+	0.640751i	$0.221377\pi$
$563$	17119.1i	1.28150i	−0.767749	+	0.640751i	$0.778623\pi$
$564$	0	0
$565$	5398.92i	0.402008i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17094.4	−1.25946	−0.629731	−	0.776813i	$-0.716835\pi$
$569$	−17094.4	−1.25946	−0.629731	+	0.776813i	$0.716835\pi$
$570$	0	0
$571$	− 12823.0i	− 0.939800i	−0.882720	−	0.469900i	$-0.844290\pi$
$571$	− 12823.0i	− 0.939800i	0.882720	−	0.469900i	$-0.155710\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1943.36	0.140946
$576$	0	0
$577$	17773.0	1.28232	0.641161	−	0.767406i	$-0.278453\pi$
$577$	17773.0	1.28232	0.641161	+	0.767406i	$0.278453\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12720.2i	0.908300i
$582$	0	0
$583$	514.183	0.0365271
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4032.40i	0.283535i	0.989900	+	0.141767i	$0.0452785\pi$
$587$	4032.40i	0.283535i	−0.989900	+	0.141767i	$0.954722\pi$
$588$	0	0
$589$	− 1499.70i	− 0.104914i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18108.7	−1.25402	−0.627010	−	0.779011i	$-0.715722\pi$
$593$	−18108.7	−1.25402	−0.627010	+	0.779011i	$0.715722\pi$
$594$	0	0
$595$	27540.0i	1.89753i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22967.0	1.56662	0.783310	−	0.621631i	$-0.213530\pi$
$599$	22967.0	1.56662	0.783310	+	0.621631i	$0.213530\pi$
$600$	0	0
$601$	−21562.0	−1.46345	−0.731724	−	0.681601i	$-0.761284\pi$
$601$	−21562.0	−1.46345	−0.731724	+	0.681601i	$0.761284\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 7472.07i	− 0.502120i
$606$	0	0
$607$	2078.16	0.138962	0.0694808	−	0.997583i	$-0.477866\pi$
$607$	2078.16	0.138962	0.0694808	+	0.997583i	$0.477866\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2449.12i	− 0.162162i
$612$	0	0
$613$	− 25902.0i	− 1.70664i	−0.521388	−	0.853320i	$-0.674585\pi$
$613$	− 25902.0i	− 1.70664i	0.521388	−	0.853320i	$-0.325415\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4922.99	−0.321219	−0.160609	−	0.987018i	$-0.551346\pi$
$617$	−4922.99	−0.321219	−0.160609	+	0.987018i	$0.551346\pi$
$618$	0	0
$619$	12917.0i	0.838737i	0.907816	+	0.419368i	$0.137748\pi$
$619$	12917.0i	0.838737i	−0.907816	+	0.419368i	$0.862252\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1590.02	−0.102252
$624$	0	0
$625$	−13211.0	−0.845504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29150.4i	1.84786i
$630$	0	0
$631$	−17632.2	−1.11240	−0.556201	−	0.831047i	$-0.687742\pi$
$631$	−17632.2	−1.11240	−0.556201	+	0.831047i	$0.687742\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	20928.9i	1.30793i
$636$	0	0
$637$	− 2485.22i	− 0.154581i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	494.773	0.0304873	0.0152436	−	0.999884i	$-0.495148\pi$
$641$	494.773	0.0304873	0.0152436	+	0.999884i	$0.495148\pi$
$642$	0	0
$643$	− 1564.00i	− 0.0959225i	−0.998849	−	0.0479612i	$-0.984728\pi$
$643$	− 1564.00i	− 0.0959225i	0.998849	−	0.0479612i	$-0.0152724\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20493.6	1.24527	0.622633	−	0.782514i	$-0.286063\pi$
$647$	20493.6	1.24527	0.622633	+	0.782514i	$0.286063\pi$
$648$	0	0
$649$	−4284.00	−0.259109
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 19579.1i	− 1.17334i	−0.809827	−	0.586669i	$-0.800439\pi$
$653$	− 19579.1i	− 1.17334i	0.809827	−	0.586669i	$-0.199561\pi$
$654$	0	0
$655$	−26223.3	−1.56432
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 14991.6i	− 0.886176i	−0.896478	−	0.443088i	$-0.853883\pi$
$659$	− 14991.6i	− 0.886176i	0.896478	−	0.443088i	$-0.146117\pi$
$660$	0	0
$661$	27058.9i	1.59224i	0.605141	+	0.796118i	$0.293117\pi$
$661$	27058.9i	1.59224i	−0.605141	+	0.796118i	$0.706883\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1558.53	−0.0908832
$666$	0	0
$667$	30888.0i	1.79309i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11130.2	0.640350
$672$	0	0
$673$	−5231.00	−0.299614	−0.149807	−	0.988715i	$-0.547865\pi$
$673$	−5231.00	−0.299614	−0.149807	+	0.988715i	$0.547865\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19631.1i	1.11445i	0.830361	+	0.557225i	$0.188134\pi$
$677$	19631.1i	1.11445i	−0.830361	+	0.557225i	$0.811866\pi$
$678$	0	0
$679$	17118.0	0.967494
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 27855.7i	− 1.56057i	−0.625425	−	0.780285i	$-0.715074\pi$
$683$	− 27855.7i	− 1.56057i	0.625425	−	0.780285i	$-0.284926\pi$
$684$	0	0
$685$	− 257.091i	− 0.0143401i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−445.295	−0.0246218
$690$	0	0
$691$	4556.00i	0.250823i	0.992105	+	0.125411i	$0.0400250\pi$
$691$	4556.00i	0.250823i	−0.992105	+	0.125411i	$0.959975\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22707.2	−1.23933
$696$	0	0
$697$	−48960.0	−2.66068
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 19360.9i	− 1.04315i	−0.853205	−	0.521576i	$-0.825344\pi$
$701$	− 19360.9i	− 1.04315i	0.853205	−	0.521576i	$-0.174656\pi$
$702$	0	0
$703$	−1649.67	−0.0885042
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4898.25i	0.260562i
$708$	0	0
$709$	− 10305.1i	− 0.545861i	−0.962034	−	0.272930i	$-0.912007\pi$
$709$	− 10305.1i	− 0.545861i	0.962034	−	0.272930i	$-0.0879929\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24491.2	1.28640
$714$	0	0
$715$	− 5508.00i	− 0.288094i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9353.07	0.485133	0.242567	−	0.970135i	$-0.422011\pi$
$719$	9353.07	0.485133	0.242567	+	0.970135i	$0.422011\pi$
$720$	0	0
$721$	23409.0	1.20915
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4593.40i	0.235303i
$726$	0	0
$727$	−2956.55	−0.150829	−0.0754143	−	0.997152i	$-0.524028\pi$
$727$	−2956.55	−0.150829	−0.0754143	+	0.997152i	$0.524028\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11379.8i	0.575781i
$732$	0	0
$733$	31536.5i	1.58913i	0.607182	+	0.794563i	$0.292300\pi$
$733$	31536.5i	1.58913i	−0.607182	+	0.794563i	$0.707700\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8732.74	0.436465
$738$	0	0
$739$	33892.0i	1.68706i	0.537082	+	0.843530i	$0.319527\pi$
$739$	33892.0i	1.68706i	−0.537082	+	0.843530i	$0.680473\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17095.3	0.844101	0.422051	−	0.906572i	$-0.361310\pi$
$743$	17095.3	0.844101	0.422051	+	0.906572i	$0.361310\pi$
$744$	0	0
$745$	−33696.0	−1.65708
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16430.2i	0.801532i
$750$	0	0
$751$	−31215.2	−1.51672	−0.758361	−	0.651835i	$-0.774000\pi$
$751$	−31215.2	−1.51672	−0.758361	+	0.651835i	$0.774000\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2003.83i	− 0.0965918i
$756$	0	0
$757$	23973.8i	1.15105i	0.817786	+	0.575523i	$0.195201\pi$
$757$	23973.8i	1.15105i	−0.817786	+	0.575523i	$0.804799\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5120.90	0.243932	0.121966	−	0.992534i	$-0.461080\pi$
$761$	5120.90	0.243932	0.121966	+	0.992534i	$0.461080\pi$
$762$	0	0
$763$	14688.0i	0.696909i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3710.05	0.174657
$768$	0	0
$769$	−15649.0	−0.733833	−0.366916	−	0.930254i	$-0.619586\pi$
$769$	−15649.0	−0.733833	−0.366916	+	0.930254i	$0.619586\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8646.40i	0.402315i	0.979559	+	0.201157i	$0.0644703\pi$
$773$	8646.40i	0.402315i	−0.979559	+	0.201157i	$0.935530\pi$
$774$	0	0
$775$	3642.13	0.168812
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2770.73i	− 0.127435i
$780$	0	0
$781$	19538.9i	0.895209i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14249.5	0.647879
$786$	0	0
$787$	10217.0i	0.462766i	0.972863	+	0.231383i	$0.0743250\pi$
$787$	10217.0i	0.462766i	−0.972863	+	0.231383i	$0.925675\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11130.2	−0.500307
$792$	0	0
$793$	−9639.00	−0.431641
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32070.7i	1.42535i	0.701496	+	0.712673i	$0.252516\pi$
$797$	32070.7i	1.42535i	−0.701496	+	0.712673i	$0.747484\pi$
$798$	0	0
$799$	−14140.0	−0.626080
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10513.9i	0.462052i
$804$	0	0
$805$	− 25452.1i	− 1.11437i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26816.7	1.16542	0.582710	−	0.812680i	$-0.301992\pi$
$809$	26816.7	1.16542	0.582710	+	0.812680i	$0.301992\pi$
$810$	0	0
$811$	45556.0i	1.97249i	0.165297	+	0.986244i	$0.447142\pi$
$811$	45556.0i	1.97249i	−0.165297	+	0.986244i	$0.552858\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13728.2	−0.590036
$816$	0	0
$817$	−644.000	−0.0275774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24744.1i	1.05186i	0.850529	+	0.525928i	$0.176282\pi$
$821$	24744.1i	1.05186i	−0.850529	+	0.525928i	$0.823718\pi$
$822$	0	0
$823$	31129.5	1.31848	0.659238	−	0.751934i	$-0.270879\pi$
$823$	31129.5	1.31848	0.659238	+	0.751934i	$0.270879\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23378.0i	0.982990i	0.870880	+	0.491495i	$0.163549\pi$
$827$	23378.0i	0.982990i	−0.870880	+	0.491495i	$0.836451\pi$
$828$	0	0
$829$	− 5120.40i	− 0.214522i	−0.994231	−	0.107261i	$-0.965792\pi$
$829$	− 5120.40i	− 0.214522i	0.994231	−	0.107261i	$-0.0342081\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14348.4	−0.596810
$834$	0	0
$835$	− 5940.00i	− 0.246182i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28204.7	1.16059	0.580295	−	0.814406i	$-0.302937\pi$
$839$	28204.7	1.16059	0.580295	+	0.814406i	$0.302937\pi$
$840$	0	0
$841$	−48619.0	−1.99348
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 18061.8i	− 0.735320i
$846$	0	0
$847$	15404.1	0.624899
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 26940.4i	− 1.08520i
$852$	0	0
$853$	− 13175.9i	− 0.528881i	−0.964402	−	0.264440i	$-0.914813\pi$
$853$	− 13175.9i	− 0.528881i	0.964402	−	0.264440i	$-0.0851873\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24936.5	−0.993951	−0.496976	−	0.867765i	$-0.665556\pi$
$857$	−24936.5	−0.993951	−0.496976	+	0.867765i	$0.665556\pi$
$858$	0	0
$859$	− 25031.0i	− 0.994234i	−0.867684	−	0.497117i	$-0.834392\pi$
$859$	− 25031.0i	− 0.994234i	0.867684	−	0.497117i	$-0.165608\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46464.0	1.83274	0.916369	−	0.400335i	$-0.131106\pi$
$863$	46464.0	1.83274	0.916369	+	0.400335i	$0.131106\pi$
$864$	0	0
$865$	35856.0	1.40941
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 32330.5i	− 1.26207i
$870$	0	0
$871$	−7562.77	−0.294208
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 31616.0i	− 1.22150i
$876$	0	0
$877$	− 30315.4i	− 1.16725i	−0.812024	−	0.583624i	$-0.801634\pi$
$877$	− 30315.4i	− 1.16725i	0.812024	−	0.583624i	$-0.198366\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−28127.8	−1.07565	−0.537827	−	0.843055i	$-0.680755\pi$
$881$	−28127.8	−1.07565	−0.537827	+	0.843055i	$0.680755\pi$
$882$	0	0
$883$	34931.0i	1.33128i	0.746272	+	0.665641i	$0.231842\pi$
$883$	34931.0i	1.33128i	−0.746272	+	0.665641i	$0.768158\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29056.9	−1.09993	−0.549963	−	0.835189i	$-0.685358\pi$
$887$	−29056.9	−1.09993	−0.549963	+	0.835189i	$0.685358\pi$
$888$	0	0
$889$	−43146.0	−1.62775
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 800.207i	− 0.0299865i
$894$	0	0
$895$	15425.5	0.576109
$896$	0	0
$897$	0	0
$898$	0	0
$899$	57888.4i	2.14759i
$900$	0	0
$901$	2570.91i	0.0950606i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20260.9	−0.744195
$906$	0	0
$907$	30283.0i	1.10863i	0.832306	+	0.554317i	$0.187020\pi$
$907$	30283.0i	1.10863i	−0.832306	+	0.554317i	$0.812980\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32964.4	−1.19886	−0.599429	−	0.800428i	$-0.704605\pi$
$911$	−32964.4	−1.19886	−0.599429	+	0.800428i	$0.704605\pi$
$912$	0	0
$913$	−14688.0	−0.532423
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 54060.8i	− 1.94683i
$918$	0	0
$919$	26866.1	0.964341	0.482170	−	0.876077i	$-0.339849\pi$
$919$	26866.1	0.964341	0.482170	+	0.876077i	$0.339849\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 16921.2i	− 0.603433i
$924$	0	0
$925$	− 4006.34i	− 0.142408i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−52099.6	−1.83997	−0.919985	−	0.391955i	$-0.871799\pi$
$929$	−52099.6	−1.83997	−0.919985	+	0.391955i	$0.871799\pi$
$930$	0	0
$931$	− 812.000i	− 0.0285846i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−31800.5	−1.11228
$936$	0	0
$937$	−13367.0	−0.466041	−0.233021	−	0.972472i	$-0.574861\pi$
$937$	−13367.0	−0.466041	−0.233021	+	0.972472i	$0.574861\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4354.38i	0.150849i	0.997152	+	0.0754243i	$0.0240311\pi$
$941$	4354.38i	0.150849i	−0.997152	+	0.0754243i	$0.975969\pi$
$942$	0	0
$943$	45248.1	1.56255
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38419.1i	1.31832i	0.752001	+	0.659162i	$0.229089\pi$
$947$	38419.1i	1.31832i	−0.752001	+	0.659162i	$0.770911\pi$
$948$	0	0
$949$	− 9105.32i	− 0.311455i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35598.9	1.21003	0.605016	−	0.796213i	$-0.293167\pi$
$953$	35598.9	1.21003	0.605016	+	0.796213i	$0.293167\pi$
$954$	0	0
$955$	21276.0i	0.720916i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	530.008	0.0178465
$960$	0	0
$961$	16109.0	0.540734
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 25554.7i	− 0.852471i
$966$	0	0
$967$	−34900.2	−1.16061	−0.580307	−	0.814398i	$-0.697067\pi$
$967$	−34900.2	−1.16061	−0.580307	+	0.814398i	$0.697067\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 222.648i	− 0.00735850i	−0.999993	−	0.00367925i	$-0.998829\pi$
$971$	− 222.648i	− 0.00735850i	0.999993	−	0.00367925i	$-0.00117114\pi$
$972$	0	0
$973$	− 46812.1i	− 1.54237i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14645.3	0.479574	0.239787	−	0.970826i	$-0.422922\pi$
$977$	14645.3	0.479574	0.239787	+	0.970826i	$0.422922\pi$
$978$	0	0
$979$	− 1836.00i	− 0.0599375i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38004.7	−1.23312	−0.616562	−	0.787307i	$-0.711475\pi$
$983$	−38004.7	−1.23312	−0.616562	+	0.787307i	$0.711475\pi$
$984$	0	0
$985$	−41148.0	−1.33105
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 10517.0i	− 0.338141i
$990$	0	0
$991$	−45955.1	−1.47307	−0.736535	−	0.676400i	$-0.763539\pi$
$991$	−45955.1	−1.47307	−0.736535	+	0.676400i	$0.763539\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 30948.0i	− 0.986049i
$996$	0	0
$997$	32393.5i	1.02900i	0.857490	+	0.514500i	$0.172022\pi$
$997$	32393.5i	1.02900i	−0.857490	+	0.514500i	$0.827978\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.d.g.865.3	yes	8
3.2	odd	2	inner	1728.4.d.g.865.7	yes	8
4.3	odd	2	inner	1728.4.d.g.865.1	✓	8
8.3	odd	2	inner	1728.4.d.g.865.6	yes	8
8.5	even	2	inner	1728.4.d.g.865.8	yes	8
12.11	even	2	inner	1728.4.d.g.865.5	yes	8
24.5	odd	2	inner	1728.4.d.g.865.4	yes	8
24.11	even	2	inner	1728.4.d.g.865.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.g.865.1	✓	8	4.3	odd	2	inner
1728.4.d.g.865.2	yes	8	24.11	even	2	inner
1728.4.d.g.865.3	yes	8	1.1	even	1	trivial
1728.4.d.g.865.4	yes	8	24.5	odd	2	inner
1728.4.d.g.865.5	yes	8	12.11	even	2	inner
1728.4.d.g.865.6	yes	8	8.3	odd	2	inner
1728.4.d.g.865.7	yes	8	3.2	odd	2	inner
1728.4.d.g.865.8	yes	8	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.3923i q^{5} +21.4243 q^{7} +O(q^{10})\)	\(q-10.3923i q^{5} +21.4243 q^{7} +24.7386i q^{11} -21.4243i q^{13} -123.693 q^{17} -7.00000i q^{19} +114.315 q^{23} +17.0000 q^{25} +270.200i q^{29} +214.243 q^{31} -222.648i q^{35} -235.667i q^{37} +395.818 q^{41} -92.0000i q^{43} +114.315 q^{47} +116.000 q^{49} -20.7846i q^{53} +257.091 q^{55} +173.170i q^{59} -449.910i q^{61} -222.648 q^{65} -353.000i q^{67} +789.815 q^{71} +425.000 q^{73} +530.008i q^{77} -1306.88 q^{79} +593.727i q^{83} +1285.46i q^{85} -74.2159 q^{89} -459.000i q^{91} -72.7461 q^{95} +799.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 136 q^{25} + 928 q^{49} + 3400 q^{73} + 6392 q^{97}+O(q^{100}) \) 8 * q + 136 * q^25 + 928 * q^49 + 3400 * q^73 + 6392 * q^97

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