LMFDB - Embedded newform 9576.2.a.cn.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.a (trivial)

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:	yes
Analytic conductor:	$76.4647449756$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.401584.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 12x - 4 $ x^5 - 2x^4 - 7x^3 + 8x^2 + 12x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.06003$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73631 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+1.73631 q^{5} +1.00000 q^{7} +4.12005 q^{11} -7.04948 q^{13} -4.17833 q^{17} +1.00000 q^{19} -2.20541 q^{23} -1.98522 q^{25} -7.41435 q^{29} +3.42723 q^{31} +1.73631 q^{35} +10.1843 q^{37} -1.58918 q^{41} +3.37144 q^{43} +6.24259 q^{47} +1.00000 q^{49} -7.41435 q^{53} +7.15370 q^{55} -11.3561 q^{59} -2.00000 q^{61} -12.2401 q^{65} -0.236605 q^{67} -11.8006 q^{71} +5.47263 q^{73} +4.12005 q^{77} +2.20541 q^{79} -6.41493 q^{83} -7.25489 q^{85} -16.5098 q^{89} -7.04948 q^{91} +1.73631 q^{95} -0.149620 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} + 5 q^{7}+O(q^{10}) $ 5 * q - 2 * q^5 + 5 * q^7	$ 5 q - 2 q^{5} + 5 q^{7} - 6 q^{11} - 6 q^{17} + 5 q^{19} - 10 q^{23} + 7 q^{25} - 4 q^{29} - 4 q^{31} - 2 q^{35} + 6 q^{37} - 10 q^{41} + 4 q^{43} - 2 q^{47} + 5 q^{49} - 4 q^{53} + 8 q^{55} - 12 q^{59} - 10 q^{61} - 8 q^{65} + 2 q^{67} - 30 q^{71} + 6 q^{73} - 6 q^{77} + 10 q^{79} - 14 q^{83} - 10 q^{89} - 2 q^{95} - 8 q^{97}+O(q^{100}) $ 5 * q - 2 * q^5 + 5 * q^7 - 6 * q^11 - 6 * q^17 + 5 * q^19 - 10 * q^23 + 7 * q^25 - 4 * q^29 - 4 * q^31 - 2 * q^35 + 6 * q^37 - 10 * q^41 + 4 * q^43 - 2 * q^47 + 5 * q^49 - 4 * q^53 + 8 * q^55 - 12 * q^59 - 10 * q^61 - 8 * q^65 + 2 * q^67 - 30 * q^71 + 6 * q^73 - 6 * q^77 + 10 * q^79 - 14 * q^83 - 10 * q^89 - 2 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.73631	0.776503	0.388252	−	0.921553i	$-0.373079\pi$
$5$	1.73631	0.776503	0.388252	+	0.921553i	$0.373079\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.12005	1.24224	0.621121	−	0.783715i	$-0.286677\pi$
$11$	4.12005	1.24224	0.621121	+	0.783715i	$0.286677\pi$
$12$	0	0
$13$	−7.04948	−1.95517	−0.977587	−	0.210534i	$-0.932480\pi$
$13$	−7.04948	−1.95517	−0.977587	+	0.210534i	$0.932480\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.17833	−1.01339	−0.506697	−	0.862124i	$-0.669134\pi$
$17$	−4.17833	−1.01339	−0.506697	+	0.862124i	$0.669134\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.20541	−0.459860	−0.229930	−	0.973207i	$-0.573850\pi$
$23$	−2.20541	−0.459860	−0.229930	+	0.973207i	$0.573850\pi$
$24$	0	0
$25$	−1.98522	−0.397043
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.41435	−1.37681	−0.688405	−	0.725326i	$-0.741689\pi$
$29$	−7.41435	−1.37681	−0.688405	+	0.725326i	$0.741689\pi$
$30$	0	0
$31$	3.42723	0.615549	0.307774	−	0.951459i	$-0.400416\pi$
$31$	3.42723	0.615549	0.307774	+	0.951459i	$0.400416\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.73631	0.293491
$36$	0	0
$37$	10.1843	1.67429	0.837145	−	0.546980i	$-0.184223\pi$
$37$	10.1843	1.67429	0.837145	+	0.546980i	$0.184223\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.58918	−0.248188	−0.124094	−	0.992270i	$-0.539602\pi$
$41$	−1.58918	−0.248188	−0.124094	+	0.992270i	$0.539602\pi$
$42$	0	0
$43$	3.37144	0.514140	0.257070	−	0.966393i	$-0.417243\pi$
$43$	3.37144	0.514140	0.257070	+	0.966393i	$0.417243\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.24259	0.910575	0.455288	−	0.890344i	$-0.349536\pi$
$47$	6.24259	0.910575	0.455288	+	0.890344i	$0.349536\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.41435	−1.01844	−0.509220	−	0.860637i	$-0.670066\pi$
$53$	−7.41435	−1.01844	−0.509220	+	0.860637i	$0.670066\pi$
$54$	0	0
$55$	7.15370	0.964605
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.3561	−1.47843	−0.739217	−	0.673467i	$-0.764804\pi$
$59$	−11.3561	−1.47843	−0.739217	+	0.673467i	$0.764804\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.2401	−1.51820
$66$	0	0
$67$	−0.236605	−0.0289059	−0.0144529	−	0.999896i	$-0.504601\pi$
$67$	−0.236605	−0.0289059	−0.0144529	+	0.999896i	$0.504601\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.8006	−1.40047	−0.700235	−	0.713912i	$-0.746922\pi$
$71$	−11.8006	−1.40047	−0.700235	+	0.713912i	$0.746922\pi$
$72$	0	0
$73$	5.47263	0.640523	0.320261	−	0.947329i	$-0.396229\pi$
$73$	5.47263	0.640523	0.320261	+	0.947329i	$0.396229\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.12005	0.469523
$78$	0	0
$79$	2.20541	0.248128	0.124064	−	0.992274i	$-0.460407\pi$
$79$	2.20541	0.248128	0.124064	+	0.992274i	$0.460407\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.41493	−0.704130	−0.352065	−	0.935975i	$-0.614520\pi$
$83$	−6.41493	−0.704130	−0.352065	+	0.935975i	$0.614520\pi$
$84$	0	0
$85$	−7.25489	−0.786903
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.5098	−1.75003	−0.875016	−	0.484093i	$-0.839149\pi$
$89$	−16.5098	−1.75003	−0.875016	+	0.484093i	$0.839149\pi$
$90$	0	0
$91$	−7.04948	−0.738986
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.73631	0.178142
$96$	0	0
$97$	−0.149620	−0.0151917	−0.00759583	−	0.999971i	$-0.502418\pi$
$97$	−0.149620	−0.0151917	−0.00759583	+	0.999971i	$0.502418\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.3375	1.72514	0.862571	−	0.505936i	$-0.168853\pi$
$101$	17.3375	1.72514	0.862571	+	0.505936i	$0.168853\pi$
$102$	0	0
$103$	1.91464	0.188655	0.0943276	−	0.995541i	$-0.469930\pi$
$103$	1.91464	0.188655	0.0943276	+	0.995541i	$0.469930\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.18183	0.597620	0.298810	−	0.954313i	$-0.403410\pi$
$107$	6.18183	0.597620	0.298810	+	0.954313i	$0.403410\pi$
$108$	0	0
$109$	−18.0678	−1.73058	−0.865289	−	0.501274i	$-0.832865\pi$
$109$	−18.0678	−1.73058	−0.865289	+	0.501274i	$0.832865\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.00353	−0.658837	−0.329418	−	0.944184i	$-0.606853\pi$
$113$	−7.00353	−0.658837	−0.329418	+	0.944184i	$0.606853\pi$
$114$	0	0
$115$	−3.82928	−0.357082
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.17833	−0.383027
$120$	0	0
$121$	5.97482	0.543166
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1285	−1.08481
$126$	0	0
$127$	−12.8317	−1.13863	−0.569316	−	0.822119i	$-0.692792\pi$
$127$	−12.8317	−1.13863	−0.569316	+	0.822119i	$0.692792\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.05828	−0.179832	−0.0899162	−	0.995949i	$-0.528660\pi$
$131$	−2.05828	−0.179832	−0.0899162	+	0.995949i	$0.528660\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.3561	−1.14109	−0.570543	−	0.821268i	$-0.693267\pi$
$137$	−13.3561	−1.14109	−0.570543	+	0.821268i	$0.693267\pi$
$138$	0	0
$139$	−16.1859	−1.37287	−0.686437	−	0.727190i	$-0.740826\pi$
$139$	−16.1859	−1.37287	−0.686437	+	0.727190i	$0.740826\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−29.0442	−2.42880
$144$	0	0
$145$	−12.8736	−1.06910
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.7981	0.966537	0.483269	−	0.875472i	$-0.339449\pi$
$149$	11.7981	0.966537	0.483269	+	0.875472i	$0.339449\pi$
$150$	0	0
$151$	−16.6233	−1.35278	−0.676392	−	0.736542i	$-0.736457\pi$
$151$	−16.6233	−1.35278	−0.676392	+	0.736542i	$0.736457\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.95075	0.477975
$156$	0	0
$157$	−16.9830	−1.35539	−0.677695	−	0.735343i	$-0.737021\pi$
$157$	−16.9830	−1.35539	−0.677695	+	0.735343i	$0.737021\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.20541	−0.173811
$162$	0	0
$163$	15.6115	1.22279	0.611395	−	0.791325i	$-0.290609\pi$
$163$	15.6115	1.22279	0.611395	+	0.791325i	$0.290609\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.58860	−0.509841	−0.254921	−	0.966962i	$-0.582049\pi$
$167$	−6.58860	−0.509841	−0.254921	+	0.966962i	$0.582049\pi$
$168$	0	0
$169$	36.6951	2.82270
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.11597	0.388960	0.194480	−	0.980906i	$-0.437698\pi$
$173$	5.11597	0.388960	0.194480	+	0.980906i	$0.437698\pi$
$174$	0	0
$175$	−1.98522	−0.150068
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.9122	−0.890356	−0.445178	−	0.895442i	$-0.646860\pi$
$179$	−11.9122	−0.890356	−0.445178	+	0.895442i	$0.646860\pi$
$180$	0	0
$181$	−12.9187	−0.960241	−0.480121	−	0.877203i	$-0.659407\pi$
$181$	−12.9187	−0.960241	−0.480121	+	0.877203i	$0.659407\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.6832	1.30009
$186$	0	0
$187$	−17.2149	−1.25888
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.2496	0.813994	0.406997	−	0.913430i	$-0.366576\pi$
$191$	11.2496	0.813994	0.406997	+	0.913430i	$0.366576\pi$
$192$	0	0
$193$	18.8041	1.35355	0.676775	−	0.736190i	$-0.263377\pi$
$193$	18.8041	1.35355	0.676775	+	0.736190i	$0.263377\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.93982	0.209453	0.104727	−	0.994501i	$-0.466603\pi$
$197$	2.93982	0.209453	0.104727	+	0.994501i	$0.466603\pi$
$198$	0	0
$199$	25.1608	1.78360	0.891800	−	0.452431i	$-0.149443\pi$
$199$	25.1608	1.78360	0.891800	+	0.452431i	$0.149443\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.41435	−0.520385
$204$	0	0
$205$	−2.75932	−0.192719
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.12005	0.284990
$210$	0	0
$211$	27.1938	1.87210	0.936050	−	0.351866i	$-0.114453\pi$
$211$	27.1938	1.87210	0.936050	+	0.351866i	$0.114453\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.85388	0.399231
$216$	0	0
$217$	3.42723	0.232656
$218$	0	0
$219$	0	0
$220$	0	0
$221$	29.4550	1.98136
$222$	0	0
$223$	−18.5010	−1.23892	−0.619460	−	0.785029i	$-0.712648\pi$
$223$	−18.5010	−1.23892	−0.619460	+	0.785029i	$0.712648\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.38623	−0.158379	−0.0791897	−	0.996860i	$-0.525233\pi$
$227$	−2.38623	−0.158379	−0.0791897	+	0.996860i	$0.525233\pi$
$228$	0	0
$229$	−11.2030	−0.740312	−0.370156	−	0.928970i	$-0.620696\pi$
$229$	−11.2030	−0.740312	−0.370156	+	0.928970i	$0.620696\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.8960	0.975869	0.487935	−	0.872880i	$-0.337750\pi$
$233$	14.8960	0.975869	0.487935	+	0.872880i	$0.337750\pi$
$234$	0	0
$235$	10.8391	0.707064
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.6315	−1.72265	−0.861323	−	0.508058i	$-0.830363\pi$
$239$	−26.6315	−1.72265	−0.861323	+	0.508058i	$0.830363\pi$
$240$	0	0
$241$	6.94169	0.447154	0.223577	−	0.974686i	$-0.428227\pi$
$241$	6.94169	0.447154	0.223577	+	0.974686i	$0.428227\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.73631	0.110929
$246$	0	0
$247$	−7.04948	−0.448548
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.96690	−0.565986	−0.282993	−	0.959122i	$-0.591327\pi$
$251$	−8.96690	−0.565986	−0.282993	+	0.959122i	$0.591327\pi$
$252$	0	0
$253$	−9.08640	−0.571257
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.09954	−0.318100	−0.159050	−	0.987270i	$-0.550843\pi$
$257$	−5.09954	−0.318100	−0.159050	+	0.987270i	$0.550843\pi$
$258$	0	0
$259$	10.1843	0.632822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.9055	−1.72073	−0.860364	−	0.509680i	$-0.829764\pi$
$263$	−27.9055	−1.72073	−0.860364	+	0.509680i	$0.829764\pi$
$264$	0	0
$265$	−12.8736	−0.790821
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.1361	−1.41063	−0.705317	−	0.708893i	$-0.749195\pi$
$269$	−23.1361	−1.41063	−0.705317	+	0.708893i	$0.749195\pi$
$270$	0	0
$271$	28.4003	1.72519	0.862597	−	0.505891i	$-0.168836\pi$
$271$	28.4003	1.72519	0.862597	+	0.505891i	$0.168836\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.17919	−0.493224
$276$	0	0
$277$	21.9682	1.31994	0.659971	−	0.751291i	$-0.270569\pi$
$277$	21.9682	1.31994	0.659971	+	0.751291i	$0.270569\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	33.0778	1.97326	0.986629	−	0.162983i	$-0.0521117\pi$
$281$	33.0778	1.97326	0.986629	+	0.162983i	$0.0521117\pi$
$282$	0	0
$283$	−0.294267	−0.0174923	−0.00874616	−	0.999962i	$-0.502784\pi$
$283$	−0.294267	−0.0174923	−0.00874616	+	0.999962i	$0.502784\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.58918	−0.0938063
$288$	0	0
$289$	0.458425	0.0269662
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.12361	0.474587	0.237293	−	0.971438i	$-0.423740\pi$
$293$	8.12361	0.474587	0.237293	+	0.971438i	$0.423740\pi$
$294$	0	0
$295$	−19.7177	−1.14801
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.5470	0.899105
$300$	0	0
$301$	3.37144	0.194327
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.47263	−0.198842
$306$	0	0
$307$	−15.7023	−0.896180	−0.448090	−	0.893989i	$-0.647896\pi$
$307$	−15.7023	−0.896180	−0.448090	+	0.893989i	$0.647896\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.6786	−0.775641	−0.387821	−	0.921735i	$-0.626772\pi$
$311$	−13.6786	−0.775641	−0.387821	+	0.921735i	$0.626772\pi$
$312$	0	0
$313$	−18.3517	−1.03730	−0.518649	−	0.854987i	$-0.673565\pi$
$313$	−18.3517	−1.03730	−0.518649	+	0.854987i	$0.673565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.5013	0.758310	0.379155	−	0.925333i	$-0.376215\pi$
$317$	13.5013	0.758310	0.379155	+	0.925333i	$0.376215\pi$
$318$	0	0
$319$	−30.5475	−1.71033
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.17833	−0.232488
$324$	0	0
$325$	13.9947	0.776288
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.24259	0.344165
$330$	0	0
$331$	20.5428	1.12914	0.564568	−	0.825386i	$-0.309043\pi$
$331$	20.5428	1.12914	0.564568	+	0.825386i	$0.309043\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.410820	−0.0224455
$336$	0	0
$337$	7.17830	0.391027	0.195513	−	0.980701i	$-0.437363\pi$
$337$	7.17830	0.391027	0.195513	+	0.980701i	$0.437363\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.1204	0.764661
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.9236	−1.49902	−0.749508	−	0.661995i	$-0.769710\pi$
$347$	−27.9236	−1.49902	−0.749508	+	0.661995i	$0.769710\pi$
$348$	0	0
$349$	−30.9818	−1.65842	−0.829209	−	0.558938i	$-0.811209\pi$
$349$	−30.9818	−1.65842	−0.829209	+	0.558938i	$0.811209\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.30685	0.442129	0.221065	−	0.975259i	$-0.429047\pi$
$353$	8.30685	0.442129	0.221065	+	0.975259i	$0.429047\pi$
$354$	0	0
$355$	−20.4895	−1.08747
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.7228	0.618707	0.309354	−	0.950947i	$-0.399887\pi$
$359$	11.7228	0.618707	0.309354	+	0.950947i	$0.399887\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.50220	0.497368
$366$	0	0
$367$	−3.68107	−0.192150	−0.0960752	−	0.995374i	$-0.530629\pi$
$367$	−3.68107	−0.192150	−0.0960752	+	0.995374i	$0.530629\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.41435	−0.384934
$372$	0	0
$373$	−10.8282	−0.560665	−0.280332	−	0.959903i	$-0.590445\pi$
$373$	−10.8282	−0.560665	−0.280332	+	0.959903i	$0.590445\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	52.2673	2.69190
$378$	0	0
$379$	−34.2140	−1.75746	−0.878728	−	0.477322i	$-0.841607\pi$
$379$	−34.2140	−1.75746	−0.878728	+	0.477322i	$0.841607\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.85827	−0.452636	−0.226318	−	0.974053i	$-0.572669\pi$
$383$	−8.85827	−0.452636	−0.226318	+	0.974053i	$0.572669\pi$
$384$	0	0
$385$	7.15370	0.364586
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.09294	−0.461030	−0.230515	−	0.973069i	$-0.574041\pi$
$389$	−9.09294	−0.461030	−0.230515	+	0.973069i	$0.574041\pi$
$390$	0	0
$391$	9.21493	0.466019
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.82928	0.192672
$396$	0	0
$397$	−9.87148	−0.495435	−0.247718	−	0.968832i	$-0.579681\pi$
$397$	−9.87148	−0.495435	−0.247718	+	0.968832i	$0.579681\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.6255	0.730361	0.365180	−	0.930937i	$-0.381007\pi$
$401$	14.6255	0.730361	0.365180	+	0.930937i	$0.381007\pi$
$402$	0	0
$403$	−24.1602	−1.20350
$404$	0	0
$405$	0	0
$406$	0	0
$407$	41.9599	2.07987
$408$	0	0
$409$	−18.1966	−0.899763	−0.449881	−	0.893088i	$-0.648534\pi$
$409$	−18.1966	−0.899763	−0.449881	+	0.893088i	$0.648534\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11.3561	−0.558796
$414$	0	0
$415$	−11.1383	−0.546759
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.1654	−1.47368	−0.736838	−	0.676069i	$-0.763682\pi$
$419$	−30.1654	−1.47368	−0.736838	+	0.676069i	$0.763682\pi$
$420$	0	0
$421$	8.85283	0.431461	0.215730	−	0.976453i	$-0.430787\pi$
$421$	8.85283	0.431461	0.215730	+	0.976453i	$0.430787\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.29488	0.402361
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.8328	−1.19615	−0.598077	−	0.801438i	$-0.704068\pi$
$431$	−24.8328	−1.19615	−0.598077	+	0.801438i	$0.704068\pi$
$432$	0	0
$433$	37.5625	1.80514	0.902568	−	0.430547i	$-0.141679\pi$
$433$	37.5625	1.80514	0.902568	+	0.430547i	$0.141679\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.20541	−0.105499
$438$	0	0
$439$	−7.59795	−0.362630	−0.181315	−	0.983425i	$-0.558035\pi$
$439$	−7.59795	−0.362630	−0.181315	+	0.983425i	$0.558035\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.20587	0.389873	0.194936	−	0.980816i	$-0.437550\pi$
$443$	8.20587	0.389873	0.194936	+	0.980816i	$0.437550\pi$
$444$	0	0
$445$	−28.6661	−1.35891
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.8082	−0.887612	−0.443806	−	0.896123i	$-0.646372\pi$
$449$	−18.8082	−0.887612	−0.443806	+	0.896123i	$0.646372\pi$
$450$	0	0
$451$	−6.54750	−0.308310
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.2401	−0.573825
$456$	0	0
$457$	−8.23087	−0.385024	−0.192512	−	0.981295i	$-0.561663\pi$
$457$	−8.23087	−0.385024	−0.192512	+	0.981295i	$0.561663\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.3254	0.900075	0.450037	−	0.893010i	$-0.351411\pi$
$461$	19.3254	0.900075	0.450037	+	0.893010i	$0.351411\pi$
$462$	0	0
$463$	18.4299	0.856508	0.428254	−	0.903658i	$-0.359129\pi$
$463$	18.4299	0.856508	0.428254	+	0.903658i	$0.359129\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.3590	0.525633	0.262816	−	0.964846i	$-0.415349\pi$
$467$	11.3590	0.525633	0.262816	+	0.964846i	$0.415349\pi$
$468$	0	0
$469$	−0.236605	−0.0109254
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.8905	0.638686
$474$	0	0
$475$	−1.98522	−0.0910879
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.2508	−1.47357	−0.736787	−	0.676125i	$-0.763658\pi$
$479$	−32.2508	−1.47357	−0.736787	+	0.676125i	$0.763658\pi$
$480$	0	0
$481$	−71.7941	−3.27353
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.259788	−0.0117964
$486$	0	0
$487$	30.6434	1.38859	0.694293	−	0.719692i	$-0.255717\pi$
$487$	30.6434	1.38859	0.694293	+	0.719692i	$0.255717\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.85535	0.0837310	0.0418655	−	0.999123i	$-0.486670\pi$
$491$	1.85535	0.0837310	0.0418655	+	0.999123i	$0.486670\pi$
$492$	0	0
$493$	30.9796	1.39525
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.8006	−0.529328
$498$	0	0
$499$	18.6837	0.836399	0.418200	−	0.908355i	$-0.362661\pi$
$499$	18.6837	0.836399	0.418200	+	0.908355i	$0.362661\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.54955	−0.292030	−0.146015	−	0.989282i	$-0.546645\pi$
$503$	−6.54955	−0.292030	−0.146015	+	0.989282i	$0.546645\pi$
$504$	0	0
$505$	30.1033	1.33958
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.1166	0.980299	0.490149	−	0.871638i	$-0.336942\pi$
$509$	22.1166	0.980299	0.490149	+	0.871638i	$0.336942\pi$
$510$	0	0
$511$	5.47263	0.242095
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.32442	0.146491
$516$	0	0
$517$	25.7198	1.13116
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.08582	0.0913814	0.0456907	−	0.998956i	$-0.485451\pi$
$521$	2.08582	0.0913814	0.0456907	+	0.998956i	$0.485451\pi$
$522$	0	0
$523$	9.67277	0.422961	0.211480	−	0.977382i	$-0.432172\pi$
$523$	9.67277	0.422961	0.211480	+	0.977382i	$0.432172\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.3201	−0.623793
$528$	0	0
$529$	−18.1362	−0.788529
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.2029	0.485251
$534$	0	0
$535$	10.7336	0.464053
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.12005	0.177463
$540$	0	0
$541$	−20.3074	−0.873083	−0.436542	−	0.899684i	$-0.643797\pi$
$541$	−20.3074	−0.873083	−0.436542	+	0.899684i	$0.643797\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−31.3713	−1.34380
$546$	0	0
$547$	18.0483	0.771689	0.385845	−	0.922564i	$-0.373910\pi$
$547$	18.0483	0.771689	0.385845	+	0.922564i	$0.373910\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.41435	−0.315862
$552$	0	0
$553$	2.20541	0.0937836
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.3703	−0.778373	−0.389186	−	0.921159i	$-0.627244\pi$
$557$	−18.3703	−0.778373	−0.389186	+	0.921159i	$0.627244\pi$
$558$	0	0
$559$	−23.7669	−1.00523
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.4386	1.57785	0.788924	−	0.614490i	$-0.210638\pi$
$563$	37.4386	1.57785	0.788924	+	0.614490i	$0.210638\pi$
$564$	0	0
$565$	−12.1603	−0.511589
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.58507	0.150294	0.0751469	−	0.997172i	$-0.476057\pi$
$569$	3.58507	0.150294	0.0751469	+	0.997172i	$0.476057\pi$
$570$	0	0
$571$	14.1613	0.592634	0.296317	−	0.955090i	$-0.404242\pi$
$571$	14.1613	0.592634	0.296317	+	0.955090i	$0.404242\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.37821	0.182584
$576$	0	0
$577$	8.74288	0.363971	0.181985	−	0.983301i	$-0.441748\pi$
$577$	8.74288	0.363971	0.181985	+	0.983301i	$0.441748\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.41493	−0.266136
$582$	0	0
$583$	−30.5475	−1.26515
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.55931	−0.229457	−0.114729	−	0.993397i	$-0.536600\pi$
$587$	−5.55931	−0.229457	−0.114729	+	0.993397i	$0.536600\pi$
$588$	0	0
$589$	3.42723	0.141217
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.0190	−1.43806	−0.719029	−	0.694980i	$-0.755413\pi$
$593$	−35.0190	−1.43806	−0.719029	+	0.694980i	$0.755413\pi$
$594$	0	0
$595$	−7.25489	−0.297421
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.77101	−0.235797	−0.117898	−	0.993026i	$-0.537616\pi$
$599$	−5.77101	−0.235797	−0.117898	+	0.993026i	$0.537616\pi$
$600$	0	0
$601$	0.219006	0.00893344	0.00446672	−	0.999990i	$-0.498578\pi$
$601$	0.219006	0.00893344	0.00446672	+	0.999990i	$0.498578\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.3742	0.421770
$606$	0	0
$607$	−7.61331	−0.309015	−0.154507	−	0.987992i	$-0.549379\pi$
$607$	−7.61331	−0.309015	−0.154507	+	0.987992i	$0.549379\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−44.0070	−1.78033
$612$	0	0
$613$	44.3357	1.79070	0.895350	−	0.445363i	$-0.146925\pi$
$613$	44.3357	1.79070	0.895350	+	0.445363i	$0.146925\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.4676	−1.46813	−0.734065	−	0.679079i	$-0.762379\pi$
$617$	−36.4676	−1.46813	−0.734065	+	0.679079i	$0.762379\pi$
$618$	0	0
$619$	−17.8424	−0.717147	−0.358574	−	0.933501i	$-0.616737\pi$
$619$	−17.8424	−0.717147	−0.358574	+	0.933501i	$0.616737\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.5098	−0.661450
$624$	0	0
$625$	−11.1328	−0.445314
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−42.5534	−1.69672
$630$	0	0
$631$	−30.6192	−1.21893	−0.609465	−	0.792813i	$-0.708616\pi$
$631$	−30.6192	−1.21893	−0.609465	+	0.792813i	$0.708616\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22.2799	−0.884152
$636$	0	0
$637$	−7.04948	−0.279310
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25.9291	−1.02414	−0.512069	−	0.858944i	$-0.671121\pi$
$641$	−25.9291	−1.02414	−0.512069	+	0.858944i	$0.671121\pi$
$642$	0	0
$643$	−19.0935	−0.752973	−0.376486	−	0.926422i	$-0.622868\pi$
$643$	−19.0935	−0.752973	−0.376486	+	0.926422i	$0.622868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.7418	1.44447	0.722235	−	0.691648i	$-0.243115\pi$
$647$	36.7418	1.44447	0.722235	+	0.691648i	$0.243115\pi$
$648$	0	0
$649$	−46.7876	−1.83657
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.1843	1.18120	0.590602	−	0.806963i	$-0.298890\pi$
$653$	30.1843	1.18120	0.590602	+	0.806963i	$0.298890\pi$
$654$	0	0
$655$	−3.57381	−0.139640
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.1955	0.981479	0.490740	−	0.871306i	$-0.336727\pi$
$659$	25.1955	0.981479	0.490740	+	0.871306i	$0.336727\pi$
$660$	0	0
$661$	−21.3191	−0.829219	−0.414609	−	0.909999i	$-0.636082\pi$
$661$	−21.3191	−0.829219	−0.414609	+	0.909999i	$0.636082\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.73631	0.0673313
$666$	0	0
$667$	16.3517	0.633140
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.24010	−0.318106
$672$	0	0
$673$	−14.8111	−0.570926	−0.285463	−	0.958390i	$-0.592147\pi$
$673$	−14.8111	−0.570926	−0.285463	+	0.958390i	$0.592147\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.9168	1.03450	0.517248	−	0.855836i	$-0.326956\pi$
$677$	26.9168	1.03450	0.517248	+	0.855836i	$0.326956\pi$
$678$	0	0
$679$	−0.149620	−0.00574191
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.2676	1.15816	0.579080	−	0.815271i	$-0.303412\pi$
$683$	30.2676	1.15816	0.579080	+	0.815271i	$0.303412\pi$
$684$	0	0
$685$	−23.1903	−0.886057
$686$	0	0
$687$	0	0
$688$	0	0
$689$	52.2673	1.99123
$690$	0	0
$691$	9.68813	0.368554	0.184277	−	0.982874i	$-0.441006\pi$
$691$	9.68813	0.368554	0.184277	+	0.982874i	$0.441006\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.1039	−1.06604
$696$	0	0
$697$	6.64012	0.251512
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.61331	0.212012	0.106006	−	0.994365i	$-0.466194\pi$
$701$	5.61331	0.212012	0.106006	+	0.994365i	$0.466194\pi$
$702$	0	0
$703$	10.1843	0.384109
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.3375	0.652042
$708$	0	0
$709$	4.85106	0.182186	0.0910928	−	0.995842i	$-0.470964\pi$
$709$	4.85106	0.182186	0.0910928	+	0.995842i	$0.470964\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.55845	−0.283066
$714$	0	0
$715$	−50.4299	−1.88597
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.45254	−0.277933	−0.138966	−	0.990297i	$-0.544378\pi$
$719$	−7.45254	−0.277933	−0.138966	+	0.990297i	$0.544378\pi$
$720$	0	0
$721$	1.91464	0.0713050
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14.7191	0.546653
$726$	0	0
$727$	−36.5426	−1.35529	−0.677645	−	0.735389i	$-0.736999\pi$
$727$	−36.5426	−1.35529	−0.677645	+	0.735389i	$0.736999\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14.0870	−0.521026
$732$	0	0
$733$	−49.2974	−1.82084	−0.910421	−	0.413682i	$-0.864242\pi$
$733$	−49.2974	−1.82084	−0.910421	+	0.413682i	$0.864242\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.974823	−0.0359081
$738$	0	0
$739$	−0.520232	−0.0191370	−0.00956852	−	0.999954i	$-0.503046\pi$
$739$	−0.520232	−0.0191370	−0.00956852	+	0.999954i	$0.503046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.1818	0.960518	0.480259	−	0.877127i	$-0.340543\pi$
$743$	26.1818	0.960518	0.480259	+	0.877127i	$0.340543\pi$
$744$	0	0
$745$	20.4852	0.750519
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.18183	0.225879
$750$	0	0
$751$	−18.7398	−0.683824	−0.341912	−	0.939732i	$-0.611075\pi$
$751$	−18.7398	−0.683824	−0.341912	+	0.939732i	$0.611075\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−28.8632	−1.05044
$756$	0	0
$757$	−33.2831	−1.20970	−0.604848	−	0.796341i	$-0.706766\pi$
$757$	−33.2831	−1.20970	−0.604848	+	0.796341i	$0.706766\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−14.3681	−0.520842	−0.260421	−	0.965495i	$-0.583861\pi$
$761$	−14.3681	−0.520842	−0.260421	+	0.965495i	$0.583861\pi$
$762$	0	0
$763$	−18.0678	−0.654097
$764$	0	0
$765$	0	0
$766$	0	0
$767$	80.0544	2.89060
$768$	0	0
$769$	41.6032	1.50025	0.750124	−	0.661297i	$-0.229994\pi$
$769$	41.6032	1.50025	0.750124	+	0.661297i	$0.229994\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.2515	0.764363	0.382182	−	0.924087i	$-0.375173\pi$
$773$	21.2515	0.764363	0.382182	+	0.924087i	$0.375173\pi$
$774$	0	0
$775$	−6.80379	−0.244399
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.58918	−0.0569383
$780$	0	0
$781$	−48.6190	−1.73972
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−29.4878	−1.05246
$786$	0	0
$787$	19.7647	0.704536	0.352268	−	0.935899i	$-0.385410\pi$
$787$	19.7647	0.704536	0.352268	+	0.935899i	$0.385410\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.00353	−0.249017
$792$	0	0
$793$	14.0990	0.500669
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.5163	−0.478770	−0.239385	−	0.970925i	$-0.576946\pi$
$797$	−13.5163	−0.478770	−0.239385	+	0.970925i	$0.576946\pi$
$798$	0	0
$799$	−26.0836	−0.922771
$800$	0	0
$801$	0	0
$802$	0	0
$803$	22.5475	0.795684
$804$	0	0
$805$	−3.82928	−0.134964
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.34345	0.117549	0.0587747	−	0.998271i	$-0.481281\pi$
$809$	3.34345	0.117549	0.0587747	+	0.998271i	$0.481281\pi$
$810$	0	0
$811$	9.01580	0.316588	0.158294	−	0.987392i	$-0.449401\pi$
$811$	9.01580	0.316588	0.158294	+	0.987392i	$0.449401\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	27.1065	0.949500
$816$	0	0
$817$	3.37144	0.117952
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.8025	−0.377009	−0.188505	−	0.982072i	$-0.560364\pi$
$821$	−10.8025	−0.377009	−0.188505	+	0.982072i	$0.560364\pi$
$822$	0	0
$823$	−19.3714	−0.675246	−0.337623	−	0.941281i	$-0.609623\pi$
$823$	−19.3714	−0.675246	−0.337623	+	0.941281i	$0.609623\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.7982	−0.757998	−0.378999	−	0.925397i	$-0.623732\pi$
$827$	−21.7982	−0.757998	−0.378999	+	0.925397i	$0.623732\pi$
$828$	0	0
$829$	−36.8810	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$829$	−36.8810	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.17833	−0.144770
$834$	0	0
$835$	−11.4399	−0.395893
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.0070	0.690718	0.345359	−	0.938471i	$-0.387757\pi$
$839$	20.0070	0.690718	0.345359	+	0.938471i	$0.387757\pi$
$840$	0	0
$841$	25.9726	0.895607
$842$	0	0
$843$	0	0
$844$	0	0
$845$	63.7143	2.19184
$846$	0	0
$847$	5.97482	0.205297
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−22.4606	−0.769939
$852$	0	0
$853$	28.4014	0.972447	0.486224	−	0.873834i	$-0.338374\pi$
$853$	28.4014	0.972447	0.486224	+	0.873834i	$0.338374\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.7669	0.470268	0.235134	−	0.971963i	$-0.424447\pi$
$857$	13.7669	0.470268	0.235134	+	0.971963i	$0.424447\pi$
$858$	0	0
$859$	−23.4185	−0.799028	−0.399514	−	0.916727i	$-0.630821\pi$
$859$	−23.4185	−0.799028	−0.399514	+	0.916727i	$0.630821\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.7715	−1.25172	−0.625859	−	0.779937i	$-0.715251\pi$
$863$	−36.7715	−1.25172	−0.625859	+	0.779937i	$0.715251\pi$
$864$	0	0
$865$	8.88293	0.302029
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.08640	0.308235
$870$	0	0
$871$	1.66794	0.0565159
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.1285	−0.410019
$876$	0	0
$877$	4.47539	0.151123	0.0755617	−	0.997141i	$-0.475925\pi$
$877$	4.47539	0.151123	0.0755617	+	0.997141i	$0.475925\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.7139	−0.495723	−0.247861	−	0.968796i	$-0.579728\pi$
$881$	−14.7139	−0.495723	−0.247861	+	0.968796i	$0.579728\pi$
$882$	0	0
$883$	6.43432	0.216532	0.108266	−	0.994122i	$-0.465470\pi$
$883$	6.43432	0.216532	0.108266	+	0.994122i	$0.465470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.1929	1.34955	0.674773	−	0.738025i	$-0.264241\pi$
$887$	40.1929	1.34955	0.674773	+	0.738025i	$0.264241\pi$
$888$	0	0
$889$	−12.8317	−0.430363
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.24259	0.208900
$894$	0	0
$895$	−20.6832	−0.691364
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−25.4107	−0.847494
$900$	0	0
$901$	30.9796	1.03208
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22.4310	−0.745630
$906$	0	0
$907$	47.8825	1.58991	0.794956	−	0.606667i	$-0.207494\pi$
$907$	47.8825	1.58991	0.794956	+	0.606667i	$0.207494\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	56.9449	1.88667	0.943334	−	0.331844i	$-0.107671\pi$
$911$	56.9449	1.88667	0.943334	+	0.331844i	$0.107671\pi$
$912$	0	0
$913$	−26.4299	−0.874700
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.05828	−0.0679703
$918$	0	0
$919$	10.4299	0.344049	0.172025	−	0.985093i	$-0.444969\pi$
$919$	10.4299	0.344049	0.172025	+	0.985093i	$0.444969\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	83.1879	2.73816
$924$	0	0
$925$	−20.2181	−0.664766
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.6093	1.39797	0.698983	−	0.715139i	$-0.253636\pi$
$929$	42.6093	1.39797	0.698983	+	0.715139i	$0.253636\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29.8905	−0.977524
$936$	0	0
$937$	−32.6214	−1.06569	−0.532847	−	0.846212i	$-0.678878\pi$
$937$	−32.6214	−1.06569	−0.532847	+	0.846212i	$0.678878\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−59.3033	−1.93323	−0.966616	−	0.256230i	$-0.917520\pi$
$941$	−59.3033	−1.93323	−0.966616	+	0.256230i	$0.917520\pi$
$942$	0	0
$943$	3.50479	0.114132
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.1724	−1.01297	−0.506484	−	0.862249i	$-0.669055\pi$
$947$	−31.1724	−1.01297	−0.506484	+	0.862249i	$0.669055\pi$
$948$	0	0
$949$	−38.5792	−1.25233
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.879916	−0.0285033	−0.0142516	−	0.999898i	$-0.504537\pi$
$953$	−0.879916	−0.0285033	−0.0142516	+	0.999898i	$0.504537\pi$
$954$	0	0
$955$	19.5329	0.632069
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.3561	−0.431290
$960$	0	0
$961$	−19.2541	−0.621100
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.6498	1.05103
$966$	0	0
$967$	17.2089	0.553400	0.276700	−	0.960956i	$-0.410759\pi$
$967$	17.2089	0.553400	0.276700	+	0.960956i	$0.410759\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.0070	0.513689	0.256844	−	0.966453i	$-0.417317\pi$
$971$	16.0070	0.513689	0.256844	+	0.966453i	$0.417317\pi$
$972$	0	0
$973$	−16.1859	−0.518897
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.5485	−0.433455	−0.216727	−	0.976232i	$-0.569538\pi$
$977$	−13.5485	−0.433455	−0.216727	+	0.976232i	$0.569538\pi$
$978$	0	0
$979$	−68.0211	−2.17396
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.1493	−0.897824	−0.448912	−	0.893576i	$-0.648188\pi$
$983$	−28.1493	−0.897824	−0.448912	+	0.893576i	$0.648188\pi$
$984$	0	0
$985$	5.10445	0.162641
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.43541	−0.236432
$990$	0	0
$991$	34.3409	1.09088	0.545438	−	0.838151i	$-0.316363\pi$
$991$	34.3409	1.09088	0.545438	+	0.838151i	$0.316363\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	43.6870	1.38497
$996$	0	0
$997$	28.9830	0.917900	0.458950	−	0.888462i	$-0.348226\pi$
$997$	28.9830	0.917900	0.458950	+	0.888462i	$0.348226\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.cn.1.4		5
3.2	odd	2		3192.2.a.bb.1.2	✓	5
12.11	even	2		6384.2.a.cd.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.bb.1.2	✓	5	3.2	odd	2
6384.2.a.cd.1.2		5	12.11	even	2
9576.2.a.cn.1.4		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73631 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+1.73631 q^{5} +1.00000 q^{7} +4.12005 q^{11} -7.04948 q^{13} -4.17833 q^{17} +1.00000 q^{19} -2.20541 q^{23} -1.98522 q^{25} -7.41435 q^{29} +3.42723 q^{31} +1.73631 q^{35} +10.1843 q^{37} -1.58918 q^{41} +3.37144 q^{43} +6.24259 q^{47} +1.00000 q^{49} -7.41435 q^{53} +7.15370 q^{55} -11.3561 q^{59} -2.00000 q^{61} -12.2401 q^{65} -0.236605 q^{67} -11.8006 q^{71} +5.47263 q^{73} +4.12005 q^{77} +2.20541 q^{79} -6.41493 q^{83} -7.25489 q^{85} -16.5098 q^{89} -7.04948 q^{91} +1.73631 q^{95} -0.149620 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} + 5 q^{7}+O(q^{10}) \) 5 * q - 2 * q^5 + 5 * q^7	\( 5 q - 2 q^{5} + 5 q^{7} - 6 q^{11} - 6 q^{17} + 5 q^{19} - 10 q^{23} + 7 q^{25} - 4 q^{29} - 4 q^{31} - 2 q^{35} + 6 q^{37} - 10 q^{41} + 4 q^{43} - 2 q^{47} + 5 q^{49} - 4 q^{53} + 8 q^{55} - 12 q^{59} - 10 q^{61} - 8 q^{65} + 2 q^{67} - 30 q^{71} + 6 q^{73} - 6 q^{77} + 10 q^{79} - 14 q^{83} - 10 q^{89} - 2 q^{95} - 8 q^{97}+O(q^{100}) \) 5 * q - 2 * q^5 + 5 * q^7 - 6 * q^11 - 6 * q^17 + 5 * q^19 - 10 * q^23 + 7 * q^25 - 4 * q^29 - 4 * q^31 - 2 * q^35 + 6 * q^37 - 10 * q^41 + 4 * q^43 - 2 * q^47 + 5 * q^49 - 4 * q^53 + 8 * q^55 - 12 * q^59 - 10 * q^61 - 8 * q^65 + 2 * q^67 - 30 * q^71 + 6 * q^73 - 6 * q^77 + 10 * q^79 - 14 * q^83 - 10 * q^89 - 2 * q^95 - 8 * q^97

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