LMFDB - Embedded newform 1296.4.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.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:	$76.4664753674$
Analytic rank:	$0$
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			1.3
Root			$-2.47735$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-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	$-0.287491\pi$
$5$	13.8439	1.23823	0.619117	+	0.785299i	$0.287491\pi$
$6$	0	0
$7$	−30.7080	−1.65808	−0.829038	−	0.559192i	$-0.811111\pi$
$7$	−30.7080	−1.65808	−0.829038	+	0.559192i	$0.811111\pi$
$8$	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	$-0.682462\pi$
$17$	−76.0286	−1.08468	−0.542342	+	0.840158i	$0.682462\pi$
$18$	0	0
$19$	44.1789	0.533439	0.266720	−	0.963774i	$-0.414060\pi$
$19$	44.1789	0.533439	0.266720	+	0.963774i	$0.414060\pi$
$20$	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	$-0.403997\pi$
$29$	92.7806	0.594101	0.297050	+	0.954862i	$0.403997\pi$
$30$	0	0
$31$	143.066	0.828884	0.414442	−	0.910076i	$-0.363977\pi$
$31$	143.066	0.828884	0.414442	+	0.910076i	$0.363977\pi$
$32$	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	$-0.279314\pi$
$41$	335.555	1.27817	0.639084	+	0.769137i	$0.279314\pi$
$42$	0	0
$43$	498.331	1.76732	0.883660	−	0.468129i	$-0.155072\pi$
$43$	498.331	1.76732	0.883660	+	0.468129i	$0.155072\pi$
$44$	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	$-0.196995\pi$
$53$	628.565	1.62906	0.814529	+	0.580123i	$0.196995\pi$
$54$	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	$-0.452620\pi$
$67$	162.661	0.296600	0.148300	+	0.988942i	$0.452620\pi$
$68$	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	$-0.539251\pi$
$79$	−172.730	−0.245995	−0.122998	+	0.992407i	$0.539251\pi$
$80$	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	$-0.564262\pi$
$89$	−336.716	−0.401032	−0.200516	+	0.979690i	$0.564262\pi$
$90$	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	$-0.169249\pi$
$101$	1749.80	1.72388	0.861941	+	0.507008i	$0.169249\pi$
$102$	0	0
$103$	−111.396	−0.106564	−0.0532822	−	0.998579i	$-0.516968\pi$
$103$	−111.396	−0.106564	−0.0532822	+	0.998579i	$0.516968\pi$
$104$	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	$-0.530661\pi$
$113$	−231.051	−0.192349	−0.0961746	+	0.995364i	$0.530661\pi$
$114$	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	$-0.704522\pi$
$127$	−1715.22	−1.19843	−0.599217	+	0.800586i	$0.704522\pi$
$128$	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	$-0.787565\pi$
$137$	−2518.98	−1.57089	−0.785443	+	0.618935i	$0.787565\pi$
$138$	0	0
$139$	−622.883	−0.380088	−0.190044	−	0.981776i	$-0.560863\pi$
$139$	−622.883	−0.380088	−0.190044	+	0.981776i	$0.560863\pi$
$140$	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	$-0.730396\pi$
$149$	−2408.95	−1.32449	−0.662243	+	0.749289i	$0.730396\pi$
$150$	0	0
$151$	−67.6436	−0.0364553	−0.0182277	−	0.999834i	$-0.505802\pi$
$151$	−67.6436	−0.0364553	−0.0182277	+	0.999834i	$0.505802\pi$
$152$	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	$-0.258451\pi$
$163$	2863.88	1.37617	0.688087	+	0.725628i	$0.258451\pi$
$164$	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	$-0.308620\pi$
$173$	2574.29	1.13133	0.565663	+	0.824637i	$0.308620\pi$
$174$	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	$-0.484819\pi$
$197$	263.636	0.0953466	0.0476733	+	0.998863i	$0.484819\pi$
$198$	0	0
$199$	3835.87	1.36642	0.683211	−	0.730221i	$-0.260583\pi$
$199$	3835.87	1.36642	0.683211	+	0.730221i	$0.260583\pi$
$200$	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	$-0.268922\pi$
$211$	4069.32	1.32770	0.663848	+	0.747867i	$0.268922\pi$
$212$	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	$-0.875365\pi$
$223$	−6156.14	−1.84864	−0.924318	+	0.381622i	$0.875365\pi$
$224$	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	$-0.353292\pi$
$233$	3163.59	0.889502	0.444751	+	0.895654i	$0.353292\pi$
$234$	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	$-0.173656\pi$
$257$	7043.91	1.70968	0.854838	+	0.518894i	$0.173656\pi$
$258$	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	$-0.363227\pi$
$269$	3675.87	0.833167	0.416584	+	0.909097i	$0.363227\pi$
$270$	0	0
$271$	1881.48	0.421741	0.210871	−	0.977514i	$-0.432370\pi$
$271$	1881.48	0.421741	0.210871	+	0.977514i	$0.432370\pi$
$272$	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	$-0.161605\pi$
$281$	8232.56	1.74773	0.873867	+	0.486165i	$0.161605\pi$
$282$	0	0
$283$	−1688.73	−0.354715	−0.177358	−	0.984146i	$-0.556755\pi$
$283$	−1688.73	−0.354715	−0.177358	+	0.984146i	$0.556755\pi$
$284$	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	$-0.516246\pi$
$293$	−511.732	−0.102033	−0.0510166	+	0.998698i	$0.516246\pi$
$294$	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	$-0.404732\pi$
$307$	3171.98	0.589690	0.294845	+	0.955545i	$0.404732\pi$
$308$	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	$-0.692472\pi$
$317$	−6417.13	−1.13698	−0.568489	+	0.822691i	$0.692472\pi$
$318$	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	$-0.686656\pi$
$331$	−6664.74	−1.10673	−0.553364	+	0.832940i	$0.686656\pi$
$332$	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	$-0.700022\pi$
$353$	−7797.44	−1.17568	−0.587841	+	0.808976i	$0.700022\pi$
$354$	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	$-0.354366\pi$
$367$	6211.32	0.883455	0.441728	+	0.897149i	$0.354366\pi$
$368$	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	$-0.632217\pi$
$379$	−5954.77	−0.807061	−0.403531	+	0.914966i	$0.632217\pi$
$380$	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	$-0.533294\pi$
$389$	−1602.04	−0.208809	−0.104404	+	0.994535i	$0.533294\pi$
$390$	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	$-0.354911\pi$
$401$	7069.49	0.880383	0.440191	+	0.897904i	$0.354911\pi$
$402$	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	$-0.626633\pi$
$439$	−7127.02	−0.774838	−0.387419	+	0.921904i	$0.626633\pi$
$440$	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	$-0.672901\pi$
$449$	−9835.05	−1.03373	−0.516865	+	0.856067i	$0.672901\pi$
$450$	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	$-0.530045\pi$
$461$	−1865.79	−0.188500	−0.0942499	+	0.995549i	$0.530045\pi$
$462$	0	0
$463$	13623.4	1.36746	0.683729	−	0.729736i	$-0.260357\pi$
$463$	13623.4	1.36746	0.683729	+	0.729736i	$0.260357\pi$
$464$	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	$-0.0995971\pi$
$487$	20450.7	1.90289	0.951447	+	0.307813i	$0.0995971\pi$
$488$	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	$-0.786557\pi$
$499$	−17466.6	−1.56696	−0.783480	+	0.621417i	$0.786557\pi$
$500$	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	$-0.397366\pi$
$509$	7277.77	0.633755	0.316878	+	0.948466i	$0.397366\pi$
$510$	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	$-0.336839\pi$
$521$	11664.5	0.980866	0.490433	+	0.871479i	$0.336839\pi$
$522$	0	0
$523$	8925.51	0.746243	0.373122	−	0.927782i	$-0.378287\pi$
$523$	8925.51	0.746243	0.373122	+	0.927782i	$0.378287\pi$
$524$	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	$-0.557155\pi$
$547$	−4569.63	−0.357190	−0.178595	+	0.983923i	$0.557155\pi$
$548$	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	$-0.399619\pi$
$557$	8154.40	0.620310	0.310155	+	0.950686i	$0.399619\pi$
$558$	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	$-0.391217\pi$
$569$	9097.53	0.670278	0.335139	+	0.942169i	$0.391217\pi$
$570$	0	0
$571$	3295.09	0.241498	0.120749	−	0.992683i	$-0.461470\pi$
$571$	3295.09	0.241498	0.120749	+	0.992683i	$0.461470\pi$
$572$	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	$-0.695416\pi$
$593$	−16637.6	−1.15215	−0.576075	+	0.817397i	$0.695416\pi$
$594$	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	$-0.638505\pi$
$607$	−12607.7	−0.843051	−0.421525	+	0.906817i	$0.638505\pi$
$608$	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	$-0.707382\pi$
$617$	−18587.0	−1.21278	−0.606388	+	0.795169i	$0.707382\pi$
$618$	0	0
$619$	11945.9	0.775682	0.387841	−	0.921726i	$-0.373221\pi$
$619$	11945.9	0.775682	0.387841	+	0.921726i	$0.373221\pi$
$620$	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	$-0.543161\pi$
$631$	−4285.35	−0.270360	−0.135180	+	0.990821i	$0.543161\pi$
$632$	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	$-0.266141\pi$
$641$	21758.2	1.34071	0.670357	+	0.742039i	$0.266141\pi$
$642$	0	0
$643$	7343.09	0.450363	0.225181	−	0.974317i	$-0.427703\pi$
$643$	7343.09	0.450363	0.225181	+	0.974317i	$0.427703\pi$
$644$	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	$-0.545789\pi$
$653$	−4784.18	−0.286707	−0.143353	+	0.989672i	$0.545789\pi$
$654$	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	$-0.587107\pi$
$677$	−9521.03	−0.540507	−0.270253	+	0.962789i	$0.587107\pi$
$678$	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	$-0.889482\pi$
$691$	−34160.7	−1.88066	−0.940329	+	0.340267i	$0.889482\pi$
$692$	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	$-0.208644\pi$
$701$	29427.2	1.58552	0.792759	+	0.609535i	$0.208644\pi$
$702$	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	$-0.677642\pi$
$727$	−20760.8	−1.05912	−0.529558	+	0.848274i	$0.677642\pi$
$728$	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	$-0.739847\pi$
$739$	−27490.2	−1.36839	−0.684196	+	0.729298i	$0.739847\pi$
$740$	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	$-0.888980\pi$
$751$	−38683.1	−1.87958	−0.939791	+	0.341751i	$0.888980\pi$
$752$	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	$-0.334906\pi$
$761$	20813.2	0.991429	0.495714	+	0.868486i	$0.334906\pi$
$762$	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	$-0.206498\pi$
$773$	34251.2	1.59370	0.796850	+	0.604177i	$0.206498\pi$
$774$	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	$-0.433760\pi$
$787$	9122.68	0.413200	0.206600	+	0.978425i	$0.433760\pi$
$788$	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	$-0.428581\pi$
$797$	10012.2	0.444983	0.222492	+	0.974935i	$0.428581\pi$
$798$	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	$-0.661620\pi$
$809$	−22375.5	−0.972413	−0.486207	+	0.873844i	$0.661620\pi$
$810$	0	0
$811$	−970.867	−0.0420367	−0.0210183	−	0.999779i	$-0.506691\pi$
$811$	−970.867	−0.0420367	−0.0210183	+	0.999779i	$0.506691\pi$
$812$	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	$-0.522578\pi$
$821$	−3334.40	−0.141743	−0.0708717	+	0.997485i	$0.522578\pi$
$822$	0	0
$823$	−3714.29	−0.157317	−0.0786585	−	0.996902i	$-0.525064\pi$
$823$	−3714.29	−0.157317	−0.0786585	+	0.996902i	$0.525064\pi$
$824$	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	$-0.258818\pi$
$857$	34483.8	1.37450	0.687248	+	0.726423i	$0.258818\pi$
$858$	0	0
$859$	15174.4	0.602730	0.301365	−	0.953509i	$-0.402558\pi$
$859$	15174.4	0.602730	0.301365	+	0.953509i	$0.402558\pi$
$860$	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	$-0.418348\pi$
$881$	13269.0	0.507429	0.253714	+	0.967279i	$0.418348\pi$
$882$	0	0
$883$	5785.26	0.220487	0.110243	−	0.993905i	$-0.464837\pi$
$883$	5785.26	0.220487	0.110243	+	0.993905i	$0.464837\pi$
$884$	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	$-0.419632\pi$
$907$	13647.5	0.499622	0.249811	+	0.968295i	$0.419632\pi$
$908$	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	$-0.472928\pi$
$919$	4733.10	0.169892	0.0849459	+	0.996386i	$0.472928\pi$
$920$	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	$-0.326690\pi$
$929$	29332.8	1.03593	0.517964	+	0.855402i	$0.326690\pi$
$930$	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	$-0.347378\pi$
$941$	26632.5	0.922629	0.461314	+	0.887237i	$0.347378\pi$
$942$	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	$-0.465578\pi$
$953$	6350.46	0.215857	0.107928	+	0.994159i	$0.465578\pi$
$954$	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	$-0.271589\pi$
$967$	39546.1	1.31512	0.657558	+	0.753404i	$0.271589\pi$
$968$	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	$-0.656409\pi$
$977$	−28818.1	−0.943677	−0.471838	+	0.881685i	$0.656409\pi$
$978$	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	$-0.508508\pi$
$991$	−1667.46	−0.0534496	−0.0267248	+	0.999643i	$0.508508\pi$
$992$	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	1296.4.a.w.1.3		3
3.2	odd	2		1296.4.a.v.1.1		3
4.3	odd	2		324.4.a.d.1.3		3
9.2	odd	6		144.4.i.d.49.3		6
9.4	even	3		432.4.i.d.289.1		6
9.5	odd	6		144.4.i.d.97.3		6
9.7	even	3		432.4.i.d.145.1		6
12.11	even	2		324.4.a.c.1.1		3
36.7	odd	6		108.4.e.a.37.1		6
36.11	even	6		36.4.e.a.13.1	✓	6
36.23	even	6		36.4.e.a.25.1	yes	6
36.31	odd	6		108.4.e.a.73.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.4.e.a.13.1	✓	6	36.11	even	6
36.4.e.a.25.1	yes	6	36.23	even	6
108.4.e.a.37.1		6	36.7	odd	6
108.4.e.a.73.1		6	36.31	odd	6
144.4.i.d.49.3		6	9.2	odd	6
144.4.i.d.97.3		6	9.5	odd	6
324.4.a.c.1.1		3	12.11	even	2
324.4.a.d.1.3		3	4.3	odd	2
432.4.i.d.145.1		6	9.7	even	3
432.4.i.d.289.1		6	9.4	even	3
1296.4.a.v.1.1		3	3.2	odd	2
1296.4.a.w.1.3		3	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.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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