LMFDB - Embedded newform 324.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(1,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("324.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.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:	$19.1166188419$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.47735$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.8439 q^{5} +30.7080 q^{7} +O(q^{10})$	$q-13.8439 q^{5} +30.7080 q^{7} -43.9046 q^{11} -12.2371 q^{13} +76.0286 q^{17} -44.1789 q^{19} -78.6271 q^{23} +66.6530 q^{25} -92.7806 q^{29} -143.066 q^{31} -425.118 q^{35} -32.4741 q^{37} -335.555 q^{41} -498.331 q^{43} -281.765 q^{47} +599.981 q^{49} -628.565 q^{53} +607.810 q^{55} +504.654 q^{59} +371.901 q^{61} +169.408 q^{65} -162.661 q^{67} +433.512 q^{71} -629.645 q^{73} -1348.22 q^{77} +172.730 q^{79} -174.858 q^{83} -1052.53 q^{85} +336.716 q^{89} -375.775 q^{91} +611.608 q^{95} +84.3187 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{5} + 6 q^{7}+O(q^{10}) $ 3 * q - 6 * q^5 + 6 * q^7	$ 3 q - 6 q^{5} + 6 q^{7} - 51 q^{11} - 12 q^{13} - 111 q^{17} + 15 q^{19} - 210 q^{23} + 3 q^{25} - 456 q^{29} - 48 q^{31} - 552 q^{35} - 48 q^{37} - 897 q^{41} - 129 q^{43} - 522 q^{47} + 225 q^{49} - 1104 q^{53} - 108 q^{55} - 453 q^{59} + 402 q^{61} - 1110 q^{65} + 213 q^{67} + 60 q^{71} + 375 q^{73} - 1128 q^{77} - 552 q^{79} + 612 q^{83} - 1188 q^{85} - 462 q^{89} - 132 q^{91} + 2184 q^{95} - 93 q^{97}+O(q^{100}) $ 3 * q - 6 * q^5 + 6 * q^7 - 51 * q^11 - 12 * q^13 - 111 * q^17 + 15 * q^19 - 210 * q^23 + 3 * q^25 - 456 * q^29 - 48 * q^31 - 552 * q^35 - 48 * q^37 - 897 * q^41 - 129 * q^43 - 522 * q^47 + 225 * q^49 - 1104 * q^53 - 108 * q^55 - 453 * q^59 + 402 * q^61 - 1110 * q^65 + 213 * q^67 + 60 * q^71 + 375 * q^73 - 1128 * q^77 - 552 * q^79 + 612 * q^83 - 1188 * q^85 - 462 * q^89 - 132 * q^91 + 2184 * q^95 - 93 * 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
$5$	−13.8439	−1.23823	−0.619117	−	0.785299i	$-0.712509\pi$
$5$	−13.8439	−1.23823	−0.619117	+	0.785299i	$0.712509\pi$
$6$	0	0
$7$	30.7080	1.65808	0.829038	−	0.559192i	$-0.188889\pi$
$7$	30.7080	1.65808	0.829038	+	0.559192i	$0.188889\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−43.9046	−1.20343	−0.601715	−	0.798711i	$-0.705516\pi$
$11$	−43.9046	−1.20343	−0.601715	+	0.798711i	$0.705516\pi$
$12$	0	0
$13$	−12.2371	−0.261073	−0.130536	−	0.991444i	$-0.541670\pi$
$13$	−12.2371	−0.261073	−0.130536	+	0.991444i	$0.541670\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	76.0286	1.08468	0.542342	−	0.840158i	$-0.317538\pi$
$17$	76.0286	1.08468	0.542342	+	0.840158i	$0.317538\pi$
$18$	0	0
$19$	−44.1789	−0.533439	−0.266720	−	0.963774i	$-0.585940\pi$
$19$	−44.1789	−0.533439	−0.266720	+	0.963774i	$0.585940\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−78.6271	−0.712821	−0.356410	−	0.934330i	$-0.615999\pi$
$23$	−78.6271	−0.712821	−0.356410	+	0.934330i	$0.615999\pi$
$24$	0	0
$25$	66.6530	0.533224
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−92.7806	−0.594101	−0.297050	−	0.954862i	$-0.596003\pi$
$29$	−92.7806	−0.594101	−0.297050	+	0.954862i	$0.596003\pi$
$30$	0	0
$31$	−143.066	−0.828884	−0.414442	−	0.910076i	$-0.636023\pi$
$31$	−143.066	−0.828884	−0.414442	+	0.910076i	$0.636023\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−425.118	−2.05309
$36$	0	0
$37$	−32.4741	−0.144289	−0.0721447	−	0.997394i	$-0.522984\pi$
$37$	−32.4741	−0.144289	−0.0721447	+	0.997394i	$0.522984\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−335.555	−1.27817	−0.639084	−	0.769137i	$-0.720686\pi$
$41$	−335.555	−1.27817	−0.639084	+	0.769137i	$0.720686\pi$
$42$	0	0
$43$	−498.331	−1.76732	−0.883660	−	0.468129i	$-0.844928\pi$
$43$	−498.331	−1.76732	−0.883660	+	0.468129i	$0.844928\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−281.765	−0.874461	−0.437230	−	0.899350i	$-0.644041\pi$
$47$	−281.765	−0.874461	−0.437230	+	0.899350i	$0.644041\pi$
$48$	0	0
$49$	599.981	1.74922
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−628.565	−1.62906	−0.814529	−	0.580123i	$-0.803005\pi$
$53$	−628.565	−1.62906	−0.814529	+	0.580123i	$0.803005\pi$
$54$	0	0
$55$	607.810	1.49013
$56$	0	0
$57$	0	0
$58$	0	0
$59$	504.654	1.11356	0.556782	−	0.830658i	$-0.312036\pi$
$59$	504.654	1.11356	0.556782	+	0.830658i	$0.312036\pi$
$60$	0	0
$61$	371.901	0.780607	0.390304	−	0.920686i	$-0.372370\pi$
$61$	371.901	0.780607	0.390304	+	0.920686i	$0.372370\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	169.408	0.323269
$66$	0	0
$67$	−162.661	−0.296600	−0.148300	−	0.988942i	$-0.547380\pi$
$67$	−162.661	−0.296600	−0.148300	+	0.988942i	$0.547380\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	433.512	0.724626	0.362313	−	0.932056i	$-0.381987\pi$
$71$	433.512	0.724626	0.362313	+	0.932056i	$0.381987\pi$
$72$	0	0
$73$	−629.645	−1.00951	−0.504756	−	0.863262i	$-0.668418\pi$
$73$	−629.645	−1.00951	−0.504756	+	0.863262i	$0.668418\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1348.22	−1.99538
$78$	0	0
$79$	172.730	0.245995	0.122998	−	0.992407i	$-0.460749\pi$
$79$	172.730	0.245995	0.122998	+	0.992407i	$0.460749\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−174.858	−0.231242	−0.115621	−	0.993293i	$-0.536886\pi$
$83$	−174.858	−0.231242	−0.115621	+	0.993293i	$0.536886\pi$
$84$	0	0
$85$	−1052.53	−1.34309
$86$	0	0
$87$	0	0
$88$	0	0
$89$	336.716	0.401032	0.200516	−	0.979690i	$-0.435738\pi$
$89$	336.716	0.401032	0.200516	+	0.979690i	$0.435738\pi$
$90$	0	0
$91$	−375.775	−0.432879
$92$	0	0
$93$	0	0
$94$	0	0
$95$	611.608	0.660523
$96$	0	0
$97$	84.3187	0.0882605	0.0441303	−	0.999026i	$-0.485948\pi$
$97$	84.3187	0.0882605	0.0441303	+	0.999026i	$0.485948\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1749.80	−1.72388	−0.861941	−	0.507008i	$-0.830751\pi$
$101$	−1749.80	−1.72388	−0.861941	+	0.507008i	$0.830751\pi$
$102$	0	0
$103$	111.396	0.106564	0.0532822	−	0.998579i	$-0.483032\pi$
$103$	111.396	0.106564	0.0532822	+	0.998579i	$0.483032\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	895.520	0.809095	0.404548	−	0.914517i	$-0.367429\pi$
$107$	895.520	0.809095	0.404548	+	0.914517i	$0.367429\pi$
$108$	0	0
$109$	−716.957	−0.630019	−0.315009	−	0.949089i	$-0.602008\pi$
$109$	−716.957	−0.630019	−0.315009	+	0.949089i	$0.602008\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	231.051	0.192349	0.0961746	−	0.995364i	$-0.469339\pi$
$113$	231.051	0.192349	0.0961746	+	0.995364i	$0.469339\pi$
$114$	0	0
$115$	1088.50	0.882639
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2334.68	1.79849
$120$	0	0
$121$	596.612	0.448244
$122$	0	0
$123$	0	0
$124$	0	0
$125$	807.748	0.577978
$126$	0	0
$127$	1715.22	1.19843	0.599217	−	0.800586i	$-0.295478\pi$
$127$	1715.22	1.19843	0.599217	+	0.800586i	$0.295478\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1332.87	0.888957	0.444479	−	0.895789i	$-0.353389\pi$
$131$	1332.87	0.888957	0.444479	+	0.895789i	$0.353389\pi$
$132$	0	0
$133$	−1356.65	−0.884483
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2518.98	1.57089	0.785443	−	0.618935i	$-0.212435\pi$
$137$	2518.98	1.57089	0.785443	+	0.618935i	$0.212435\pi$
$138$	0	0
$139$	622.883	0.380088	0.190044	−	0.981776i	$-0.439137\pi$
$139$	622.883	0.380088	0.190044	+	0.981776i	$0.439137\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	537.263	0.314183
$144$	0	0
$145$	1284.44	0.735636
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2408.95	1.32449	0.662243	−	0.749289i	$-0.269604\pi$
$149$	2408.95	1.32449	0.662243	+	0.749289i	$0.269604\pi$
$150$	0	0
$151$	67.6436	0.0364553	0.0182277	−	0.999834i	$-0.494198\pi$
$151$	67.6436	0.0364553	0.0182277	+	0.999834i	$0.494198\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1980.59	1.02635
$156$	0	0
$157$	−4.39352	−0.00223338	−0.00111669	−	0.999999i	$-0.500355\pi$
$157$	−4.39352	−0.00223338	−0.00111669	+	0.999999i	$0.500355\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2414.48	−1.18191
$162$	0	0
$163$	−2863.88	−1.37617	−0.688087	−	0.725628i	$-0.741549\pi$
$163$	−2863.88	−1.37617	−0.688087	+	0.725628i	$0.741549\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	429.514	0.199023	0.0995114	−	0.995036i	$-0.468272\pi$
$167$	429.514	0.199023	0.0995114	+	0.995036i	$0.468272\pi$
$168$	0	0
$169$	−2047.25	−0.931841
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2574.29	−1.13133	−0.565663	−	0.824637i	$-0.691380\pi$
$173$	−2574.29	−1.13133	−0.565663	+	0.824637i	$0.691380\pi$
$174$	0	0
$175$	2046.78	0.884126
$176$	0	0
$177$	0	0
$178$	0	0
$179$	807.448	0.337159	0.168580	−	0.985688i	$-0.446082\pi$
$179$	807.448	0.337159	0.168580	+	0.985688i	$0.446082\pi$
$180$	0	0
$181$	4296.57	1.76443	0.882215	−	0.470847i	$-0.156052\pi$
$181$	4296.57	1.76443	0.882215	+	0.470847i	$0.156052\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	449.568	0.178664
$186$	0	0
$187$	−3338.00	−1.30534
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−306.295	−0.116035	−0.0580176	−	0.998316i	$-0.518478\pi$
$191$	−306.295	−0.116035	−0.0580176	+	0.998316i	$0.518478\pi$
$192$	0	0
$193$	−1712.35	−0.638642	−0.319321	−	0.947647i	$-0.603455\pi$
$193$	−1712.35	−0.638642	−0.319321	+	0.947647i	$0.603455\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−263.636	−0.0953466	−0.0476733	−	0.998863i	$-0.515181\pi$
$197$	−263.636	−0.0953466	−0.0476733	+	0.998863i	$0.515181\pi$
$198$	0	0
$199$	−3835.87	−1.36642	−0.683211	−	0.730221i	$-0.739417\pi$
$199$	−3835.87	−1.36642	−0.683211	+	0.730221i	$0.739417\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2849.11	−0.985064
$204$	0	0
$205$	4645.38	1.58267
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1939.66	0.641957
$210$	0	0
$211$	−4069.32	−1.32770	−0.663848	−	0.747867i	$-0.731078\pi$
$211$	−4069.32	−1.32770	−0.663848	+	0.747867i	$0.731078\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6898.84	2.18836
$216$	0	0
$217$	−4393.27	−1.37435
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−930.365	−0.283182
$222$	0	0
$223$	6156.14	1.84864	0.924318	−	0.381622i	$-0.124635\pi$
$223$	6156.14	1.84864	0.924318	+	0.381622i	$0.124635\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.2614	0.00709378	0.00354689	−	0.999994i	$-0.498871\pi$
$227$	24.2614	0.00709378	0.00354689	+	0.999994i	$0.498871\pi$
$228$	0	0
$229$	−632.419	−0.182495	−0.0912476	−	0.995828i	$-0.529085\pi$
$229$	−632.419	−0.182495	−0.0912476	+	0.995828i	$0.529085\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3163.59	−0.889502	−0.444751	−	0.895654i	$-0.646708\pi$
$233$	−3163.59	−0.889502	−0.444751	+	0.895654i	$0.646708\pi$
$234$	0	0
$235$	3900.72	1.08279
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2221.15	0.601147	0.300573	−	0.953759i	$-0.402822\pi$
$239$	2221.15	0.601147	0.300573	+	0.953759i	$0.402822\pi$
$240$	0	0
$241$	−601.459	−0.160761	−0.0803805	−	0.996764i	$-0.525614\pi$
$241$	−601.459	−0.160761	−0.0803805	+	0.996764i	$0.525614\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8306.07	−2.16594
$246$	0	0
$247$	540.620	0.139266
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1350.71	0.339665	0.169833	−	0.985473i	$-0.445677\pi$
$251$	1350.71	0.339665	0.169833	+	0.985473i	$0.445677\pi$
$252$	0	0
$253$	3452.09	0.857830
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7043.91	−1.70968	−0.854838	−	0.518894i	$-0.826344\pi$
$257$	−7043.91	−1.70968	−0.854838	+	0.518894i	$0.826344\pi$
$258$	0	0
$259$	−997.215	−0.239243
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2324.26	0.544944	0.272472	−	0.962164i	$-0.412159\pi$
$263$	2324.26	0.544944	0.272472	+	0.962164i	$0.412159\pi$
$264$	0	0
$265$	8701.78	2.01716
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3675.87	−0.833167	−0.416584	−	0.909097i	$-0.636773\pi$
$269$	−3675.87	−0.833167	−0.416584	+	0.909097i	$0.636773\pi$
$270$	0	0
$271$	−1881.48	−0.421741	−0.210871	−	0.977514i	$-0.567630\pi$
$271$	−1881.48	−0.421741	−0.210871	+	0.977514i	$0.567630\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2926.37	−0.641698
$276$	0	0
$277$	1464.02	0.317560	0.158780	−	0.987314i	$-0.449244\pi$
$277$	1464.02	0.317560	0.158780	+	0.987314i	$0.449244\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8232.56	−1.74773	−0.873867	−	0.486165i	$-0.838395\pi$
$281$	−8232.56	−1.74773	−0.873867	+	0.486165i	$0.838395\pi$
$282$	0	0
$283$	1688.73	0.354715	0.177358	−	0.984146i	$-0.443245\pi$
$283$	1688.73	0.354715	0.177358	+	0.984146i	$0.443245\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10304.2	−2.11930
$288$	0	0
$289$	867.343	0.176540
$290$	0	0
$291$	0	0
$292$	0	0
$293$	511.732	0.102033	0.0510166	−	0.998698i	$-0.483754\pi$
$293$	511.732	0.102033	0.0510166	+	0.998698i	$0.483754\pi$
$294$	0	0
$295$	−6986.37	−1.37885
$296$	0	0
$297$	0	0
$298$	0	0
$299$	962.163	0.186098
$300$	0	0
$301$	−15302.7	−2.93035
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5148.55	−0.966575
$306$	0	0
$307$	−3171.98	−0.589690	−0.294845	−	0.955545i	$-0.595268\pi$
$307$	−3171.98	−0.589690	−0.294845	+	0.955545i	$0.595268\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5060.78	0.922735	0.461367	−	0.887209i	$-0.347359\pi$
$311$	5060.78	0.922735	0.461367	+	0.887209i	$0.347359\pi$
$312$	0	0
$313$	−9465.84	−1.70940	−0.854698	−	0.519125i	$-0.826258\pi$
$313$	−9465.84	−1.70940	−0.854698	+	0.519125i	$0.826258\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6417.13	1.13698	0.568489	−	0.822691i	$-0.307528\pi$
$317$	6417.13	1.13698	0.568489	+	0.822691i	$0.307528\pi$
$318$	0	0
$319$	4073.49	0.714959
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3358.86	−0.578613
$324$	0	0
$325$	−815.637	−0.139210
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8652.44	−1.44992
$330$	0	0
$331$	6664.74	1.10673	0.553364	−	0.832940i	$-0.313344\pi$
$331$	6664.74	1.10673	0.553364	+	0.832940i	$0.313344\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2251.86	0.367260
$336$	0	0
$337$	10488.1	1.69532	0.847660	−	0.530540i	$-0.178011\pi$
$337$	10488.1	1.69532	0.847660	+	0.530540i	$0.178011\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6281.25	0.997503
$342$	0	0
$343$	7891.37	1.24226
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2148.96	0.332455	0.166228	−	0.986087i	$-0.446841\pi$
$347$	2148.96	0.332455	0.166228	+	0.986087i	$0.446841\pi$
$348$	0	0
$349$	−2777.06	−0.425939	−0.212969	−	0.977059i	$-0.568313\pi$
$349$	−2777.06	−0.425939	−0.212969	+	0.977059i	$0.568313\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7797.44	1.17568	0.587841	−	0.808976i	$-0.299978\pi$
$353$	7797.44	1.17568	0.587841	+	0.808976i	$0.299978\pi$
$354$	0	0
$355$	−6001.49	−0.897257
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2309.31	0.339500	0.169750	−	0.985487i	$-0.445704\pi$
$359$	2309.31	0.339500	0.169750	+	0.985487i	$0.445704\pi$
$360$	0	0
$361$	−4907.22	−0.715443
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8716.73	1.25001
$366$	0	0
$367$	−6211.32	−0.883455	−0.441728	−	0.897149i	$-0.645634\pi$
$367$	−6211.32	−0.883455	−0.441728	+	0.897149i	$0.645634\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−19302.0	−2.70110
$372$	0	0
$373$	8699.22	1.20758	0.603792	−	0.797142i	$-0.293656\pi$
$373$	8699.22	1.20758	0.603792	+	0.797142i	$0.293656\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1135.36	0.155104
$378$	0	0
$379$	5954.77	0.807061	0.403531	−	0.914966i	$-0.367783\pi$
$379$	5954.77	0.807061	0.403531	+	0.914966i	$0.367783\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4256.70	0.567904	0.283952	−	0.958839i	$-0.408354\pi$
$383$	4256.70	0.567904	0.283952	+	0.958839i	$0.408354\pi$
$384$	0	0
$385$	18664.6	2.47075
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1602.04	0.208809	0.104404	−	0.994535i	$-0.466706\pi$
$389$	1602.04	0.208809	0.104404	+	0.994535i	$0.466706\pi$
$390$	0	0
$391$	−5977.90	−0.773186
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2391.25	−0.304600
$396$	0	0
$397$	−7987.87	−1.00982	−0.504912	−	0.863171i	$-0.668475\pi$
$397$	−7987.87	−1.00982	−0.504912	+	0.863171i	$0.668475\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7069.49	−0.880383	−0.440191	−	0.897904i	$-0.645089\pi$
$401$	−7069.49	−0.880383	−0.440191	+	0.897904i	$0.645089\pi$
$402$	0	0
$403$	1750.70	0.216399
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1425.76	0.173642
$408$	0	0
$409$	−11864.8	−1.43441	−0.717207	−	0.696861i	$-0.754580\pi$
$409$	−11864.8	−1.43441	−0.717207	+	0.696861i	$0.754580\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15496.9	1.84637
$414$	0	0
$415$	2420.71	0.286332
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12342.6	−1.43908	−0.719541	−	0.694450i	$-0.755648\pi$
$419$	−12342.6	−1.43908	−0.719541	+	0.694450i	$0.755648\pi$
$420$	0	0
$421$	5465.86	0.632755	0.316377	−	0.948633i	$-0.397533\pi$
$421$	5465.86	0.632755	0.316377	+	0.948633i	$0.397533\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5067.54	0.578380
$426$	0	0
$427$	11420.3	1.29431
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8997.67	1.00557	0.502787	−	0.864410i	$-0.332308\pi$
$431$	8997.67	1.00557	0.502787	+	0.864410i	$0.332308\pi$
$432$	0	0
$433$	−10967.4	−1.21723	−0.608615	−	0.793466i	$-0.708275\pi$
$433$	−10967.4	−1.21723	−0.608615	+	0.793466i	$0.708275\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3473.66	0.380246
$438$	0	0
$439$	7127.02	0.774838	0.387419	−	0.921904i	$-0.373367\pi$
$439$	7127.02	0.774838	0.387419	+	0.921904i	$0.373367\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8868.55	0.951146	0.475573	−	0.879676i	$-0.342241\pi$
$443$	8868.55	0.951146	0.475573	+	0.879676i	$0.342241\pi$
$444$	0	0
$445$	−4661.46	−0.496571
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9835.05	1.03373	0.516865	−	0.856067i	$-0.327099\pi$
$449$	9835.05	1.03373	0.516865	+	0.856067i	$0.327099\pi$
$450$	0	0
$451$	14732.4	1.53819
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5202.19	0.536005
$456$	0	0
$457$	−8886.49	−0.909612	−0.454806	−	0.890591i	$-0.650291\pi$
$457$	−8886.49	−0.909612	−0.454806	+	0.890591i	$0.650291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1865.79	0.188500	0.0942499	−	0.995549i	$-0.469955\pi$
$461$	1865.79	0.188500	0.0942499	+	0.995549i	$0.469955\pi$
$462$	0	0
$463$	−13623.4	−1.36746	−0.683729	−	0.729736i	$-0.739643\pi$
$463$	−13623.4	−1.36746	−0.683729	+	0.729736i	$0.739643\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9328.97	−0.924397	−0.462199	−	0.886776i	$-0.652939\pi$
$467$	−9328.97	−0.924397	−0.462199	+	0.886776i	$0.652939\pi$
$468$	0	0
$469$	−4994.99	−0.491785
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21879.0	2.12685
$474$	0	0
$475$	−2944.66	−0.284443
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2660.30	0.253762	0.126881	−	0.991918i	$-0.459503\pi$
$479$	2660.30	0.253762	0.126881	+	0.991918i	$0.459503\pi$
$480$	0	0
$481$	397.387	0.0376701
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1167.30	−0.109287
$486$	0	0
$487$	−20450.7	−1.90289	−0.951447	−	0.307813i	$-0.900403\pi$
$487$	−20450.7	−1.90289	−0.951447	+	0.307813i	$0.900403\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20654.9	1.89846	0.949228	−	0.314588i	$-0.101866\pi$
$491$	20654.9	1.89846	0.949228	+	0.314588i	$0.101866\pi$
$492$	0	0
$493$	−7053.98	−0.644412
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13312.3	1.20149
$498$	0	0
$499$	17466.6	1.56696	0.783480	−	0.621417i	$-0.213443\pi$
$499$	17466.6	1.56696	0.783480	+	0.621417i	$0.213443\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19951.2	1.76855	0.884275	−	0.466966i	$-0.154653\pi$
$503$	19951.2	1.76855	0.884275	+	0.466966i	$0.154653\pi$
$504$	0	0
$505$	24224.1	2.13457
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7277.77	−0.633755	−0.316878	−	0.948466i	$-0.602634\pi$
$509$	−7277.77	−0.633755	−0.316878	+	0.948466i	$0.602634\pi$
$510$	0	0
$511$	−19335.1	−1.67385
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1542.15	−0.131952
$516$	0	0
$517$	12370.8	1.05235
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11664.5	−0.980866	−0.490433	−	0.871479i	$-0.663161\pi$
$521$	−11664.5	−0.980866	−0.490433	+	0.871479i	$0.663161\pi$
$522$	0	0
$523$	−8925.51	−0.746243	−0.373122	−	0.927782i	$-0.621713\pi$
$523$	−8925.51	−0.746243	−0.373122	+	0.927782i	$0.621713\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10877.1	−0.899077
$528$	0	0
$529$	−5984.78	−0.491887
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4106.20	0.333695
$534$	0	0
$535$	−12397.5	−1.00185
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−26341.9	−2.10506
$540$	0	0
$541$	12334.1	0.980193	0.490097	−	0.871668i	$-0.336962\pi$
$541$	12334.1	0.980193	0.490097	+	0.871668i	$0.336962\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9925.47	0.780111
$546$	0	0
$547$	4569.63	0.357190	0.178595	−	0.983923i	$-0.442845\pi$
$547$	4569.63	0.357190	0.178595	+	0.983923i	$0.442845\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4098.95	0.316917
$552$	0	0
$553$	5304.19	0.407879
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8154.40	−0.620310	−0.310155	−	0.950686i	$-0.600381\pi$
$557$	−8154.40	−0.620310	−0.310155	+	0.950686i	$0.600381\pi$
$558$	0	0
$559$	6098.10	0.461399
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16173.6	−1.21072	−0.605362	−	0.795950i	$-0.706972\pi$
$563$	−16173.6	−1.21072	−0.605362	+	0.795950i	$0.706972\pi$
$564$	0	0
$565$	−3198.65	−0.238174
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9097.53	−0.670278	−0.335139	−	0.942169i	$-0.608783\pi$
$569$	−9097.53	−0.670278	−0.335139	+	0.942169i	$0.608783\pi$
$570$	0	0
$571$	−3295.09	−0.241498	−0.120749	−	0.992683i	$-0.538530\pi$
$571$	−3295.09	−0.241498	−0.120749	+	0.992683i	$0.538530\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5240.73	−0.380093
$576$	0	0
$577$	15544.7	1.12155	0.560775	−	0.827968i	$-0.310503\pi$
$577$	15544.7	1.12155	0.560775	+	0.827968i	$0.310503\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5369.52	−0.383417
$582$	0	0
$583$	27596.9	1.96046
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4688.66	−0.329680	−0.164840	−	0.986320i	$-0.552711\pi$
$587$	−4688.66	−0.329680	−0.164840	+	0.986320i	$0.552711\pi$
$588$	0	0
$589$	6320.50	0.442159
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16637.6	1.15215	0.576075	−	0.817397i	$-0.304584\pi$
$593$	16637.6	1.15215	0.576075	+	0.817397i	$0.304584\pi$
$594$	0	0
$595$	−32321.1	−2.22695
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5867.46	−0.400230	−0.200115	−	0.979772i	$-0.564132\pi$
$599$	−5867.46	−0.400230	−0.200115	+	0.979772i	$0.564132\pi$
$600$	0	0
$601$	20675.8	1.40330	0.701649	−	0.712522i	$-0.252447\pi$
$601$	20675.8	1.40330	0.701649	+	0.712522i	$0.252447\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8259.43	−0.555031
$606$	0	0
$607$	12607.7	0.843051	0.421525	−	0.906817i	$-0.361495\pi$
$607$	12607.7	0.843051	0.421525	+	0.906817i	$0.361495\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3447.97	0.228298
$612$	0	0
$613$	12469.7	0.821608	0.410804	−	0.911724i	$-0.365248\pi$
$613$	12469.7	0.821608	0.410804	+	0.911724i	$0.365248\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18587.0	1.21278	0.606388	−	0.795169i	$-0.292618\pi$
$617$	18587.0	1.21278	0.606388	+	0.795169i	$0.292618\pi$
$618$	0	0
$619$	−11945.9	−0.775682	−0.387841	−	0.921726i	$-0.626779\pi$
$619$	−11945.9	−0.775682	−0.387841	+	0.921726i	$0.626779\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10339.9	0.664941
$624$	0	0
$625$	−19514.0	−1.24890
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2468.96	−0.156509
$630$	0	0
$631$	4285.35	0.270360	0.135180	−	0.990821i	$-0.456839\pi$
$631$	4285.35	0.270360	0.135180	+	0.990821i	$0.456839\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−23745.3	−1.48394
$636$	0	0
$637$	−7342.00	−0.456673
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21758.2	−1.34071	−0.670357	−	0.742039i	$-0.733859\pi$
$641$	−21758.2	−1.34071	−0.670357	+	0.742039i	$0.733859\pi$
$642$	0	0
$643$	−7343.09	−0.450363	−0.225181	−	0.974317i	$-0.572297\pi$
$643$	−7343.09	−0.450363	−0.225181	+	0.974317i	$0.572297\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9483.24	0.576236	0.288118	−	0.957595i	$-0.406970\pi$
$647$	9483.24	0.576236	0.288118	+	0.957595i	$0.406970\pi$
$648$	0	0
$649$	−22156.6	−1.34010
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4784.18	0.286707	0.143353	−	0.989672i	$-0.454211\pi$
$653$	4784.18	0.286707	0.143353	+	0.989672i	$0.454211\pi$
$654$	0	0
$655$	−18452.1	−1.10074
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−32067.9	−1.89558	−0.947791	−	0.318892i	$-0.896689\pi$
$659$	−32067.9	−1.89558	−0.947791	+	0.318892i	$0.896689\pi$
$660$	0	0
$661$	−7525.28	−0.442813	−0.221407	−	0.975182i	$-0.571065\pi$
$661$	−7525.28	−0.442813	−0.221407	+	0.975182i	$0.571065\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	18781.3	1.09520
$666$	0	0
$667$	7295.07	0.423487
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16328.2	−0.939406
$672$	0	0
$673$	−7341.95	−0.420522	−0.210261	−	0.977645i	$-0.567431\pi$
$673$	−7341.95	−0.420522	−0.210261	+	0.977645i	$0.567431\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9521.03	0.540507	0.270253	−	0.962789i	$-0.412893\pi$
$677$	9521.03	0.540507	0.270253	+	0.962789i	$0.412893\pi$
$678$	0	0
$679$	2589.26	0.146343
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1475.06	−0.0826377	−0.0413188	−	0.999146i	$-0.513156\pi$
$683$	−1475.06	−0.0826377	−0.0413188	+	0.999146i	$0.513156\pi$
$684$	0	0
$685$	−34872.5	−1.94512
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7691.79	0.425303
$690$	0	0
$691$	34160.7	1.88066	0.940329	−	0.340267i	$-0.110518\pi$
$691$	34160.7	1.88066	0.940329	+	0.340267i	$0.110518\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8623.12	−0.470638
$696$	0	0
$697$	−25511.8	−1.38641
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29427.2	−1.58552	−0.792759	−	0.609535i	$-0.791356\pi$
$701$	−29427.2	−1.58552	−0.792759	+	0.609535i	$0.791356\pi$
$702$	0	0
$703$	1434.67	0.0769696
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−53733.0	−2.85833
$708$	0	0
$709$	19692.9	1.04313	0.521566	−	0.853211i	$-0.325348\pi$
$709$	19692.9	1.04313	0.521566	+	0.853211i	$0.325348\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11248.8	0.590845
$714$	0	0
$715$	−7437.80	−0.389032
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33746.9	−1.75041	−0.875206	−	0.483751i	$-0.839274\pi$
$719$	−33746.9	−1.75041	−0.875206	+	0.483751i	$0.839274\pi$
$720$	0	0
$721$	3420.74	0.176692
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6184.11	−0.316789
$726$	0	0
$727$	20760.8	1.05912	0.529558	−	0.848274i	$-0.322358\pi$
$727$	20760.8	1.05912	0.529558	+	0.848274i	$0.322358\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−37887.4	−1.91698
$732$	0	0
$733$	14244.0	0.717757	0.358878	−	0.933384i	$-0.383159\pi$
$733$	14244.0	0.717757	0.358878	+	0.933384i	$0.383159\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7141.55	0.356937
$738$	0	0
$739$	27490.2	1.36839	0.684196	−	0.729298i	$-0.260153\pi$
$739$	27490.2	1.36839	0.684196	+	0.729298i	$0.260153\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36468.1	1.80065	0.900326	−	0.435217i	$-0.143328\pi$
$743$	36468.1	1.80065	0.900326	+	0.435217i	$0.143328\pi$
$744$	0	0
$745$	−33349.2	−1.64003
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27499.6	1.34154
$750$	0	0
$751$	38683.1	1.87958	0.939791	−	0.341751i	$-0.111020\pi$
$751$	38683.1	1.87958	0.939791	+	0.341751i	$0.111020\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−936.449	−0.0451402
$756$	0	0
$757$	37768.4	1.81337	0.906683	−	0.421813i	$-0.138606\pi$
$757$	37768.4	1.81337	0.906683	+	0.421813i	$0.138606\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20813.2	−0.991429	−0.495714	−	0.868486i	$-0.665094\pi$
$761$	−20813.2	−0.991429	−0.495714	+	0.868486i	$0.665094\pi$
$762$	0	0
$763$	−22016.3	−1.04462
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6175.47	−0.290722
$768$	0	0
$769$	6121.70	0.287067	0.143533	−	0.989645i	$-0.454154\pi$
$769$	6121.70	0.287067	0.143533	+	0.989645i	$0.454154\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34251.2	−1.59370	−0.796850	−	0.604177i	$-0.793502\pi$
$773$	−34251.2	−1.59370	−0.796850	+	0.604177i	$0.793502\pi$
$774$	0	0
$775$	−9535.78	−0.441981
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14824.5	0.681825
$780$	0	0
$781$	−19033.2	−0.872037
$782$	0	0
$783$	0	0
$784$	0	0
$785$	60.8234	0.00276545
$786$	0	0
$787$	−9122.68	−0.413200	−0.206600	−	0.978425i	$-0.566240\pi$
$787$	−9122.68	−0.413200	−0.206600	+	0.978425i	$0.566240\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7095.12	0.318930
$792$	0	0
$793$	−4550.97	−0.203795
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10012.2	−0.444983	−0.222492	−	0.974935i	$-0.571419\pi$
$797$	−10012.2	−0.444983	−0.222492	+	0.974935i	$0.571419\pi$
$798$	0	0
$799$	−21422.2	−0.948514
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27644.3	1.21488
$804$	0	0
$805$	33425.8	1.46348
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22375.5	0.972413	0.486207	−	0.873844i	$-0.338380\pi$
$809$	22375.5	0.972413	0.486207	+	0.873844i	$0.338380\pi$
$810$	0	0
$811$	970.867	0.0420367	0.0210183	−	0.999779i	$-0.493309\pi$
$811$	970.867	0.0420367	0.0210183	+	0.999779i	$0.493309\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	39647.2	1.70403
$816$	0	0
$817$	22015.7	0.942758
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3334.40	0.141743	0.0708717	−	0.997485i	$-0.477422\pi$
$821$	3334.40	0.141743	0.0708717	+	0.997485i	$0.477422\pi$
$822$	0	0
$823$	3714.29	0.157317	0.0786585	−	0.996902i	$-0.474936\pi$
$823$	3714.29	0.157317	0.0786585	+	0.996902i	$0.474936\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30744.8	−1.29275	−0.646374	−	0.763020i	$-0.723716\pi$
$827$	−30744.8	−1.29275	−0.646374	+	0.763020i	$0.723716\pi$
$828$	0	0
$829$	−29789.6	−1.24805	−0.624026	−	0.781404i	$-0.714504\pi$
$829$	−29789.6	−1.24805	−0.624026	+	0.781404i	$0.714504\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	45615.7	1.89735
$834$	0	0
$835$	−5946.14	−0.246437
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32304.0	1.32927	0.664635	−	0.747168i	$-0.268587\pi$
$839$	32304.0	1.32927	0.664635	+	0.747168i	$0.268587\pi$
$840$	0	0
$841$	−15780.8	−0.647044
$842$	0	0
$843$	0	0
$844$	0	0
$845$	28341.9	1.15384
$846$	0	0
$847$	18320.8	0.743222
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2553.34	0.102853
$852$	0	0
$853$	−17420.6	−0.699260	−0.349630	−	0.936888i	$-0.613693\pi$
$853$	−17420.6	−0.699260	−0.349630	+	0.936888i	$0.613693\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34483.8	−1.37450	−0.687248	−	0.726423i	$-0.741182\pi$
$857$	−34483.8	−1.37450	−0.687248	+	0.726423i	$0.741182\pi$
$858$	0	0
$859$	−15174.4	−0.602730	−0.301365	−	0.953509i	$-0.597442\pi$
$859$	−15174.4	−0.602730	−0.301365	+	0.953509i	$0.597442\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19314.7	0.761852	0.380926	−	0.924605i	$-0.375605\pi$
$863$	19314.7	0.761852	0.380926	+	0.924605i	$0.375605\pi$
$864$	0	0
$865$	35638.1	1.40085
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7583.63	−0.296038
$870$	0	0
$871$	1990.49	0.0774341
$872$	0	0
$873$	0	0
$874$	0	0
$875$	24804.3	0.958331
$876$	0	0
$877$	−13744.8	−0.529222	−0.264611	−	0.964355i	$-0.585244\pi$
$877$	−13744.8	−0.529222	−0.264611	+	0.964355i	$0.585244\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13269.0	−0.507429	−0.253714	−	0.967279i	$-0.581652\pi$
$881$	−13269.0	−0.507429	−0.253714	+	0.967279i	$0.581652\pi$
$882$	0	0
$883$	−5785.26	−0.220487	−0.110243	−	0.993905i	$-0.535163\pi$
$883$	−5785.26	−0.220487	−0.110243	+	0.993905i	$0.535163\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1338.82	0.0506802	0.0253401	−	0.999679i	$-0.491933\pi$
$887$	1338.82	0.0506802	0.0253401	+	0.999679i	$0.491933\pi$
$888$	0	0
$889$	52671.0	1.98710
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12448.1	0.466471
$894$	0	0
$895$	−11178.2	−0.417482
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13273.7	0.492440
$900$	0	0
$901$	−47788.9	−1.76701
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−59481.3	−2.18478
$906$	0	0
$907$	−13647.5	−0.499622	−0.249811	−	0.968295i	$-0.580368\pi$
$907$	−13647.5	−0.499622	−0.249811	+	0.968295i	$0.580368\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39201.3	1.42568	0.712842	−	0.701325i	$-0.247408\pi$
$911$	39201.3	1.42568	0.712842	+	0.701325i	$0.247408\pi$
$912$	0	0
$913$	7677.05	0.278284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40929.8	1.47396
$918$	0	0
$919$	−4733.10	−0.169892	−0.0849459	−	0.996386i	$-0.527072\pi$
$919$	−4733.10	−0.169892	−0.0849459	+	0.996386i	$0.527072\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5304.91	−0.189180
$924$	0	0
$925$	−2164.50	−0.0769386
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29332.8	−1.03593	−0.517964	−	0.855402i	$-0.673310\pi$
$929$	−29332.8	−1.03593	−0.517964	+	0.855402i	$0.673310\pi$
$930$	0	0
$931$	−26506.5	−0.933100
$932$	0	0
$933$	0	0
$934$	0	0
$935$	46210.9	1.61632
$936$	0	0
$937$	−39028.8	−1.36074	−0.680371	−	0.732867i	$-0.738182\pi$
$937$	−39028.8	−1.36074	−0.680371	+	0.732867i	$0.738182\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26632.5	−0.922629	−0.461314	−	0.887237i	$-0.652622\pi$
$941$	−26632.5	−0.922629	−0.461314	+	0.887237i	$0.652622\pi$
$942$	0	0
$943$	26383.7	0.911105
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−51212.3	−1.75731	−0.878657	−	0.477454i	$-0.841560\pi$
$947$	−51212.3	−1.75731	−0.878657	+	0.477454i	$0.841560\pi$
$948$	0	0
$949$	7705.00	0.263556
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6350.46	−0.215857	−0.107928	−	0.994159i	$-0.534422\pi$
$953$	−6350.46	−0.215857	−0.107928	+	0.994159i	$0.534422\pi$
$954$	0	0
$955$	4240.31	0.143679
$956$	0	0
$957$	0	0
$958$	0	0
$959$	77352.9	2.60465
$960$	0	0
$961$	−9323.16	−0.312952
$962$	0	0
$963$	0	0
$964$	0	0
$965$	23705.6	0.790788
$966$	0	0
$967$	−39546.1	−1.31512	−0.657558	−	0.753404i	$-0.728411\pi$
$967$	−39546.1	−1.31512	−0.657558	+	0.753404i	$0.728411\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21043.4	−0.695483	−0.347742	−	0.937590i	$-0.613051\pi$
$971$	−21043.4	−0.695483	−0.347742	+	0.937590i	$0.613051\pi$
$972$	0	0
$973$	19127.5	0.630215
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28818.1	0.943677	0.471838	−	0.881685i	$-0.343591\pi$
$977$	28818.1	0.943677	0.471838	+	0.881685i	$0.343591\pi$
$978$	0	0
$979$	−14783.4	−0.482614
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−29223.7	−0.948210	−0.474105	−	0.880468i	$-0.657228\pi$
$983$	−29223.7	−0.948210	−0.474105	+	0.880468i	$0.657228\pi$
$984$	0	0
$985$	3649.75	0.118061
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39182.3	1.25978
$990$	0	0
$991$	1667.46	0.0534496	0.0267248	−	0.999643i	$-0.491492\pi$
$991$	1667.46	0.0534496	0.0267248	+	0.999643i	$0.491492\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	53103.4	1.69195
$996$	0	0
$997$	33972.0	1.07914	0.539571	−	0.841940i	$-0.318587\pi$
$997$	33972.0	1.07914	0.539571	+	0.841940i	$0.318587\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.a.c.1.1		3
3.2	odd	2		324.4.a.d.1.3		3
4.3	odd	2		1296.4.a.v.1.1		3
9.2	odd	6		108.4.e.a.37.1		6
9.4	even	3		36.4.e.a.25.1	yes	6
9.5	odd	6		108.4.e.a.73.1		6
9.7	even	3		36.4.e.a.13.1	✓	6
12.11	even	2		1296.4.a.w.1.3		3
36.7	odd	6		144.4.i.d.49.3		6
36.11	even	6		432.4.i.d.145.1		6
36.23	even	6		432.4.i.d.289.1		6
36.31	odd	6		144.4.i.d.97.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.4.e.a.13.1	✓	6	9.7	even	3
36.4.e.a.25.1	yes	6	9.4	even	3
108.4.e.a.37.1		6	9.2	odd	6
108.4.e.a.73.1		6	9.5	odd	6
144.4.i.d.49.3		6	36.7	odd	6
144.4.i.d.97.3		6	36.31	odd	6
324.4.a.c.1.1		3	1.1	even	1	trivial
324.4.a.d.1.3		3	3.2	odd	2
432.4.i.d.145.1		6	36.11	even	6
432.4.i.d.289.1		6	36.23	even	6
1296.4.a.v.1.1		3	4.3	odd	2
1296.4.a.w.1.3		3	12.11	even	2

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q-13.8439 q^{5} +30.7080 q^{7} +O(q^{10})\)	\(q-13.8439 q^{5} +30.7080 q^{7} -43.9046 q^{11} -12.2371 q^{13} +76.0286 q^{17} -44.1789 q^{19} -78.6271 q^{23} +66.6530 q^{25} -92.7806 q^{29} -143.066 q^{31} -425.118 q^{35} -32.4741 q^{37} -335.555 q^{41} -498.331 q^{43} -281.765 q^{47} +599.981 q^{49} -628.565 q^{53} +607.810 q^{55} +504.654 q^{59} +371.901 q^{61} +169.408 q^{65} -162.661 q^{67} +433.512 q^{71} -629.645 q^{73} -1348.22 q^{77} +172.730 q^{79} -174.858 q^{83} -1052.53 q^{85} +336.716 q^{89} -375.775 q^{91} +611.608 q^{95} +84.3187 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{5} + 6 q^{7}+O(q^{10}) \) 3 * q - 6 * q^5 + 6 * q^7	\( 3 q - 6 q^{5} + 6 q^{7} - 51 q^{11} - 12 q^{13} - 111 q^{17} + 15 q^{19} - 210 q^{23} + 3 q^{25} - 456 q^{29} - 48 q^{31} - 552 q^{35} - 48 q^{37} - 897 q^{41} - 129 q^{43} - 522 q^{47} + 225 q^{49} - 1104 q^{53} - 108 q^{55} - 453 q^{59} + 402 q^{61} - 1110 q^{65} + 213 q^{67} + 60 q^{71} + 375 q^{73} - 1128 q^{77} - 552 q^{79} + 612 q^{83} - 1188 q^{85} - 462 q^{89} - 132 q^{91} + 2184 q^{95} - 93 q^{97}+O(q^{100}) \) 3 * q - 6 * q^5 + 6 * q^7 - 51 * q^11 - 12 * q^13 - 111 * q^17 + 15 * q^19 - 210 * q^23 + 3 * q^25 - 456 * q^29 - 48 * q^31 - 552 * q^35 - 48 * q^37 - 897 * q^41 - 129 * q^43 - 522 * q^47 + 225 * q^49 - 1104 * q^53 - 108 * q^55 - 453 * q^59 + 402 * q^61 - 1110 * q^65 + 213 * q^67 + 60 * q^71 + 375 * q^73 - 1128 * q^77 - 552 * q^79 + 612 * q^83 - 1188 * q^85 - 462 * q^89 - 132 * q^91 + 2184 * q^95 - 93 * q^97

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