LMFDB - Embedded newform 1296.4.a.ba.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:	$4$
Coefficient field:	4.4.72153.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 3x + 12 $ x^4 - x^3 - 8x^2 + 3x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.90825$ 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)

$f(q)$	$=$	$q+1.69184 q^{5} +17.1413 q^{7} +O(q^{10})$	$q+1.69184 q^{5} +17.1413 q^{7} +4.00463 q^{11} +41.0449 q^{13} +3.32758 q^{17} -108.713 q^{19} +142.585 q^{23} -122.138 q^{25} +295.230 q^{29} +239.871 q^{31} +29.0005 q^{35} -121.735 q^{37} -344.941 q^{41} +420.750 q^{43} +92.7799 q^{47} -49.1746 q^{49} +191.286 q^{53} +6.77521 q^{55} -661.952 q^{59} -359.535 q^{61} +69.4417 q^{65} +273.668 q^{67} +344.143 q^{71} -824.894 q^{73} +68.6447 q^{77} -289.206 q^{79} +1320.12 q^{83} +5.62975 q^{85} +328.626 q^{89} +703.565 q^{91} -183.926 q^{95} +767.992 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{5} + 3 q^{7}+O(q^{10}) $ 4 * q + 5 * q^5 + 3 * q^7	$ 4 q + 5 q^{5} + 3 q^{7} + 16 q^{11} - 29 q^{13} - 17 q^{17} + 109 q^{19} + 37 q^{23} - 97 q^{25} + 3 q^{29} + 331 q^{31} + 171 q^{35} - 366 q^{37} + 378 q^{41} + 506 q^{43} + 171 q^{47} - 829 q^{49} + 410 q^{53} + 1163 q^{55} + 616 q^{59} - 1331 q^{61} + 815 q^{65} + 1162 q^{67} + 344 q^{71} - 1307 q^{73} + 741 q^{77} + 1853 q^{79} + 1421 q^{83} - 2074 q^{85} + 816 q^{89} + 1995 q^{91} + 1292 q^{95} - 2506 q^{97}+O(q^{100}) $ 4 * q + 5 * q^5 + 3 * q^7 + 16 * q^11 - 29 * q^13 - 17 * q^17 + 109 * q^19 + 37 * q^23 - 97 * q^25 + 3 * q^29 + 331 * q^31 + 171 * q^35 - 366 * q^37 + 378 * q^41 + 506 * q^43 + 171 * q^47 - 829 * q^49 + 410 * q^53 + 1163 * q^55 + 616 * q^59 - 1331 * q^61 + 815 * q^65 + 1162 * q^67 + 344 * q^71 - 1307 * q^73 + 741 * q^77 + 1853 * q^79 + 1421 * q^83 - 2074 * q^85 + 816 * q^89 + 1995 * q^91 + 1292 * q^95 - 2506 * 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
$5$	1.69184	0.151323	0.0756616	−	0.997134i	$-0.475893\pi$
$5$	1.69184	0.151323	0.0756616	+	0.997134i	$0.475893\pi$
$6$	0	0
$7$	17.1413	0.925545	0.462773	−	0.886477i	$-0.346855\pi$
$7$	17.1413	0.925545	0.462773	+	0.886477i	$0.346855\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00463	0.109767	0.0548837	−	0.998493i	$-0.482521\pi$
$11$	4.00463	0.109767	0.0548837	+	0.998493i	$0.482521\pi$
$12$	0	0
$13$	41.0449	0.875678	0.437839	−	0.899053i	$-0.355744\pi$
$13$	41.0449	0.875678	0.437839	+	0.899053i	$0.355744\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.32758	0.0474739	0.0237370	−	0.999718i	$-0.492444\pi$
$17$	3.32758	0.0474739	0.0237370	+	0.999718i	$0.492444\pi$
$18$	0	0
$19$	−108.713	−1.31266	−0.656330	−	0.754474i	$-0.727892\pi$
$19$	−108.713	−1.31266	−0.656330	+	0.754474i	$0.727892\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	142.585	1.29266	0.646328	−	0.763060i	$-0.276304\pi$
$23$	142.585	1.29266	0.646328	+	0.763060i	$0.276304\pi$
$24$	0	0
$25$	−122.138	−0.977101
$26$	0	0
$27$	0	0
$28$	0	0
$29$	295.230	1.89045	0.945223	−	0.326426i	$-0.105844\pi$
$29$	295.230	1.89045	0.945223	+	0.326426i	$0.105844\pi$
$30$	0	0
$31$	239.871	1.38974	0.694872	−	0.719133i	$-0.255461\pi$
$31$	239.871	1.38974	0.694872	+	0.719133i	$0.255461\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	29.0005	0.140056
$36$	0	0
$37$	−121.735	−0.540894	−0.270447	−	0.962735i	$-0.587172\pi$
$37$	−121.735	−0.540894	−0.270447	+	0.962735i	$0.587172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−344.941	−1.31392	−0.656960	−	0.753926i	$-0.728158\pi$
$41$	−344.941	−1.31392	−0.656960	+	0.753926i	$0.728158\pi$
$42$	0	0
$43$	420.750	1.49218	0.746090	−	0.665845i	$-0.231929\pi$
$43$	420.750	1.49218	0.746090	+	0.665845i	$0.231929\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	92.7799	0.287943	0.143972	−	0.989582i	$-0.454013\pi$
$47$	92.7799	0.287943	0.143972	+	0.989582i	$0.454013\pi$
$48$	0	0
$49$	−49.1746	−0.143366
$50$	0	0
$51$	0	0
$52$	0	0
$53$	191.286	0.495758	0.247879	−	0.968791i	$-0.420266\pi$
$53$	191.286	0.495758	0.247879	+	0.968791i	$0.420266\pi$
$54$	0	0
$55$	6.77521	0.0166103
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−661.952	−1.46066	−0.730329	−	0.683096i	$-0.760633\pi$
$59$	−661.952	−1.46066	−0.730329	+	0.683096i	$0.760633\pi$
$60$	0	0
$61$	−359.535	−0.754650	−0.377325	−	0.926081i	$-0.623156\pi$
$61$	−359.535	−0.754650	−0.377325	+	0.926081i	$0.623156\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	69.4417	0.132510
$66$	0	0
$67$	273.668	0.499013	0.249507	−	0.968373i	$-0.419732\pi$
$67$	273.668	0.499013	0.249507	+	0.968373i	$0.419732\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	344.143	0.575244	0.287622	−	0.957744i	$-0.407135\pi$
$71$	344.143	0.575244	0.287622	+	0.957744i	$0.407135\pi$
$72$	0	0
$73$	−824.894	−1.32256	−0.661278	−	0.750141i	$-0.729986\pi$
$73$	−824.894	−1.32256	−0.661278	+	0.750141i	$0.729986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	68.6447	0.101595
$78$	0	0
$79$	−289.206	−0.411876	−0.205938	−	0.978565i	$-0.566024\pi$
$79$	−289.206	−0.411876	−0.205938	+	0.978565i	$0.566024\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1320.12	1.74581	0.872905	−	0.487891i	$-0.162234\pi$
$83$	1320.12	1.74581	0.872905	+	0.487891i	$0.162234\pi$
$84$	0	0
$85$	5.62975	0.00718390
$86$	0	0
$87$	0	0
$88$	0	0
$89$	328.626	0.391396	0.195698	−	0.980664i	$-0.437303\pi$
$89$	328.626	0.391396	0.195698	+	0.980664i	$0.437303\pi$
$90$	0	0
$91$	703.565	0.810480
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−183.926	−0.198636
$96$	0	0
$97$	767.992	0.803894	0.401947	−	0.915663i	$-0.368334\pi$
$97$	767.992	0.803894	0.401947	+	0.915663i	$0.368334\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	725.221	0.714477	0.357238	−	0.934013i	$-0.383718\pi$
$101$	725.221	0.714477	0.357238	+	0.934013i	$0.383718\pi$
$102$	0	0
$103$	1589.51	1.52058	0.760288	−	0.649586i	$-0.225058\pi$
$103$	1589.51	1.52058	0.760288	+	0.649586i	$0.225058\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	209.744	0.189502	0.0947512	−	0.995501i	$-0.469794\pi$
$107$	209.744	0.189502	0.0947512	+	0.995501i	$0.469794\pi$
$108$	0	0
$109$	−1756.44	−1.54345	−0.771727	−	0.635953i	$-0.780607\pi$
$109$	−1756.44	−1.54345	−0.771727	+	0.635953i	$0.780607\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1154.80	0.961366	0.480683	−	0.876894i	$-0.340389\pi$
$113$	1154.80	0.961366	0.480683	+	0.876894i	$0.340389\pi$
$114$	0	0
$115$	241.232	0.195609
$116$	0	0
$117$	0	0
$118$	0	0
$119$	57.0391	0.0439392
$120$	0	0
$121$	−1314.96	−0.987951
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−418.119	−0.299181
$126$	0	0
$127$	834.642	0.583170	0.291585	−	0.956545i	$-0.405817\pi$
$127$	834.642	0.583170	0.291585	+	0.956545i	$0.405817\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1467.85	0.978979	0.489490	−	0.872009i	$-0.337183\pi$
$131$	1467.85	0.978979	0.489490	+	0.872009i	$0.337183\pi$
$132$	0	0
$133$	−1863.49	−1.21493
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1604.49	−1.00059	−0.500296	−	0.865855i	$-0.666775\pi$
$137$	−1604.49	−1.00059	−0.500296	+	0.865855i	$0.666775\pi$
$138$	0	0
$139$	−460.387	−0.280932	−0.140466	−	0.990086i	$-0.544860\pi$
$139$	−460.387	−0.280932	−0.140466	+	0.990086i	$0.544860\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	164.370	0.0961209
$144$	0	0
$145$	499.484	0.286068
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2150.18	1.18221	0.591107	−	0.806593i	$-0.298691\pi$
$149$	2150.18	1.18221	0.591107	+	0.806593i	$0.298691\pi$
$150$	0	0
$151$	1490.90	0.803493	0.401746	−	0.915751i	$-0.368403\pi$
$151$	1490.90	0.803493	0.401746	+	0.915751i	$0.368403\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	405.824	0.210301
$156$	0	0
$157$	1091.07	0.554628	0.277314	−	0.960779i	$-0.410556\pi$
$157$	1091.07	0.554628	0.277314	+	0.960779i	$0.410556\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2444.10	1.19641
$162$	0	0
$163$	−429.947	−0.206601	−0.103301	−	0.994650i	$-0.532940\pi$
$163$	−429.947	−0.206601	−0.103301	+	0.994650i	$0.532940\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2673.90	1.23900	0.619498	−	0.784998i	$-0.287336\pi$
$167$	2673.90	1.23900	0.619498	+	0.784998i	$0.287336\pi$
$168$	0	0
$169$	−512.312	−0.233187
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4457.98	1.95916	0.979578	−	0.201065i	$-0.0644402\pi$
$173$	4457.98	1.95916	0.979578	+	0.201065i	$0.0644402\pi$
$174$	0	0
$175$	−2093.60	−0.904351
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2607.97	1.08899	0.544493	−	0.838765i	$-0.316722\pi$
$179$	2607.97	1.08899	0.544493	+	0.838765i	$0.316722\pi$
$180$	0	0
$181$	4090.68	1.67988	0.839939	−	0.542680i	$-0.182590\pi$
$181$	4090.68	1.67988	0.839939	+	0.542680i	$0.182590\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−205.956	−0.0818498
$186$	0	0
$187$	13.3257	0.00521108
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1386.05	−0.525083	−0.262541	−	0.964921i	$-0.584561\pi$
$191$	−1386.05	−0.525083	−0.262541	+	0.964921i	$0.584561\pi$
$192$	0	0
$193$	−4306.46	−1.60615	−0.803073	−	0.595881i	$-0.796803\pi$
$193$	−4306.46	−1.60615	−0.803073	+	0.595881i	$0.796803\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	636.919	0.230348	0.115174	−	0.993345i	$-0.463257\pi$
$197$	636.919	0.230348	0.115174	+	0.993345i	$0.463257\pi$
$198$	0	0
$199$	663.859	0.236481	0.118240	−	0.992985i	$-0.462275\pi$
$199$	663.859	0.236481	0.118240	+	0.992985i	$0.462275\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5060.64	1.74969
$204$	0	0
$205$	−583.586	−0.198826
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−435.356	−0.144087
$210$	0	0
$211$	−1478.64	−0.482435	−0.241217	−	0.970471i	$-0.577547\pi$
$211$	−1478.64	−0.482435	−0.241217	+	0.970471i	$0.577547\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	711.843	0.225801
$216$	0	0
$217$	4111.71	1.28627
$218$	0	0
$219$	0	0
$220$	0	0
$221$	136.580	0.0415719
$222$	0	0
$223$	−908.345	−0.272768	−0.136384	−	0.990656i	$-0.543548\pi$
$223$	−908.345	−0.272768	−0.136384	+	0.990656i	$0.543548\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2334.06	−0.682454	−0.341227	−	0.939981i	$-0.610842\pi$
$227$	−2334.06	−0.682454	−0.341227	+	0.939981i	$0.610842\pi$
$228$	0	0
$229$	2677.11	0.772526	0.386263	−	0.922389i	$-0.373766\pi$
$229$	2677.11	0.772526	0.386263	+	0.922389i	$0.373766\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4541.81	1.27701	0.638506	−	0.769617i	$-0.279553\pi$
$233$	4541.81	1.27701	0.638506	+	0.769617i	$0.279553\pi$
$234$	0	0
$235$	156.969	0.0435725
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1190.05	−0.322083	−0.161041	−	0.986948i	$-0.551485\pi$
$239$	−1190.05	−0.322083	−0.161041	+	0.986948i	$0.551485\pi$
$240$	0	0
$241$	5158.22	1.37871	0.689357	−	0.724422i	$-0.257893\pi$
$241$	5158.22	1.37871	0.689357	+	0.724422i	$0.257893\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−83.1958	−0.0216946
$246$	0	0
$247$	−4462.13	−1.14947
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2609.95	0.656328	0.328164	−	0.944621i	$-0.393570\pi$
$251$	2609.95	0.656328	0.328164	+	0.944621i	$0.393570\pi$
$252$	0	0
$253$	571.001	0.141891
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4470.66	1.08510	0.542552	−	0.840022i	$-0.317458\pi$
$257$	4470.66	1.08510	0.542552	+	0.840022i	$0.317458\pi$
$258$	0	0
$259$	−2086.70	−0.500622
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3828.71	0.897675	0.448838	−	0.893613i	$-0.351838\pi$
$263$	3828.71	0.897675	0.448838	+	0.893613i	$0.351838\pi$
$264$	0	0
$265$	323.627	0.0750197
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6082.35	−1.37862	−0.689308	−	0.724468i	$-0.742085\pi$
$269$	−6082.35	−1.37862	−0.689308	+	0.724468i	$0.742085\pi$
$270$	0	0
$271$	−1339.55	−0.300264	−0.150132	−	0.988666i	$-0.547970\pi$
$271$	−1339.55	−0.300264	−0.150132	+	0.988666i	$0.547970\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−489.116	−0.107254
$276$	0	0
$277$	−3280.41	−0.711555	−0.355778	−	0.934571i	$-0.615784\pi$
$277$	−3280.41	−0.711555	−0.355778	+	0.934571i	$0.615784\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5176.68	−1.09899	−0.549493	−	0.835498i	$-0.685179\pi$
$281$	−5176.68	−1.09899	−0.549493	+	0.835498i	$0.685179\pi$
$282$	0	0
$283$	3064.51	0.643697	0.321849	−	0.946791i	$-0.395696\pi$
$283$	3064.51	0.643697	0.321849	+	0.946791i	$0.395696\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5912.75	−1.21609
$288$	0	0
$289$	−4901.93	−0.997746
$290$	0	0
$291$	0	0
$292$	0	0
$293$	489.906	0.0976813	0.0488407	−	0.998807i	$-0.484447\pi$
$293$	489.906	0.0976813	0.0488407	+	0.998807i	$0.484447\pi$
$294$	0	0
$295$	−1119.92	−0.221031
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5852.41	1.13195
$300$	0	0
$301$	7212.21	1.38108
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−608.277	−0.114196
$306$	0	0
$307$	−1287.40	−0.239334	−0.119667	−	0.992814i	$-0.538183\pi$
$307$	−1287.40	−0.239334	−0.119667	+	0.992814i	$0.538183\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8570.79	−1.56272	−0.781359	−	0.624082i	$-0.785473\pi$
$311$	−8570.79	−1.56272	−0.781359	+	0.624082i	$0.785473\pi$
$312$	0	0
$313$	8852.32	1.59860	0.799302	−	0.600930i	$-0.205203\pi$
$313$	8852.32	1.59860	0.799302	+	0.600930i	$0.205203\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2530.00	0.448262	0.224131	−	0.974559i	$-0.428046\pi$
$317$	2530.00	0.448262	0.224131	+	0.974559i	$0.428046\pi$
$318$	0	0
$319$	1182.29	0.207509
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−361.752	−0.0623170
$324$	0	0
$325$	−5013.13	−0.855627
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1590.37	0.266505
$330$	0	0
$331$	−11329.8	−1.88140	−0.940699	−	0.339244i	$-0.889829\pi$
$331$	−11329.8	−1.88140	−0.940699	+	0.339244i	$0.889829\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	463.004	0.0755123
$336$	0	0
$337$	3406.13	0.550574	0.275287	−	0.961362i	$-0.411227\pi$
$337$	3406.13	0.550574	0.275287	+	0.961362i	$0.411227\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	960.593	0.152549
$342$	0	0
$343$	−6722.40	−1.05824
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6612.80	1.02304	0.511519	−	0.859272i	$-0.329083\pi$
$347$	6612.80	1.02304	0.511519	+	0.859272i	$0.329083\pi$
$348$	0	0
$349$	4891.86	0.750302	0.375151	−	0.926964i	$-0.377591\pi$
$349$	4891.86	0.750302	0.375151	+	0.926964i	$0.377591\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5088.41	0.767219	0.383610	−	0.923495i	$-0.374681\pi$
$353$	5088.41	0.767219	0.383610	+	0.923495i	$0.374681\pi$
$354$	0	0
$355$	582.237	0.0870477
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9904.80	1.45614	0.728071	−	0.685502i	$-0.240417\pi$
$359$	9904.80	1.45614	0.728071	+	0.685502i	$0.240417\pi$
$360$	0	0
$361$	4959.56	0.723074
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1395.59	−0.200133
$366$	0	0
$367$	−436.789	−0.0621259	−0.0310629	−	0.999517i	$-0.509889\pi$
$367$	−436.789	−0.0621259	−0.0310629	+	0.999517i	$0.509889\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3278.90	0.458847
$372$	0	0
$373$	1992.21	0.276549	0.138275	−	0.990394i	$-0.455844\pi$
$373$	1992.21	0.276549	0.138275	+	0.990394i	$0.455844\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12117.7	1.65542
$378$	0	0
$379$	−4262.74	−0.577737	−0.288868	−	0.957369i	$-0.593279\pi$
$379$	−4262.74	−0.577737	−0.288868	+	0.957369i	$0.593279\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1528.41	−0.203911	−0.101956	−	0.994789i	$-0.532510\pi$
$383$	−1528.41	−0.203911	−0.101956	+	0.994789i	$0.532510\pi$
$384$	0	0
$385$	116.136	0.0153736
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6630.11	0.864164	0.432082	−	0.901834i	$-0.357779\pi$
$389$	6630.11	0.864164	0.432082	+	0.901834i	$0.357779\pi$
$390$	0	0
$391$	474.464	0.0613674
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−489.291	−0.0623263
$396$	0	0
$397$	−227.783	−0.0287962	−0.0143981	−	0.999896i	$-0.504583\pi$
$397$	−227.783	−0.0287962	−0.0143981	+	0.999896i	$0.504583\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8698.94	−1.08330	−0.541651	−	0.840603i	$-0.682201\pi$
$401$	−8698.94	−1.08330	−0.541651	+	0.840603i	$0.682201\pi$
$402$	0	0
$403$	9845.49	1.21697
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−487.503	−0.0593725
$408$	0	0
$409$	−2901.99	−0.350842	−0.175421	−	0.984494i	$-0.556129\pi$
$409$	−2901.99	−0.350842	−0.175421	+	0.984494i	$0.556129\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11346.7	−1.35190
$414$	0	0
$415$	2233.44	0.264181
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14380.9	−1.67674	−0.838368	−	0.545105i	$-0.816490\pi$
$419$	−14380.9	−1.67674	−0.838368	+	0.545105i	$0.816490\pi$
$420$	0	0
$421$	−9689.26	−1.12168	−0.560838	−	0.827926i	$-0.689521\pi$
$421$	−9689.26	−1.12168	−0.560838	+	0.827926i	$0.689521\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−406.423	−0.0463868
$426$	0	0
$427$	−6162.90	−0.698463
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11100.2	−1.24055	−0.620274	−	0.784385i	$-0.712979\pi$
$431$	−11100.2	−1.24055	−0.620274	+	0.784385i	$0.712979\pi$
$432$	0	0
$433$	−3933.85	−0.436602	−0.218301	−	0.975881i	$-0.570051\pi$
$433$	−3933.85	−0.436602	−0.218301	+	0.975881i	$0.570051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15500.9	−1.69682
$438$	0	0
$439$	6074.10	0.660367	0.330183	−	0.943917i	$-0.392889\pi$
$439$	6074.10	0.660367	0.330183	+	0.943917i	$0.392889\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1625.58	0.174342	0.0871710	−	0.996193i	$-0.472217\pi$
$443$	1625.58	0.174342	0.0871710	+	0.996193i	$0.472217\pi$
$444$	0	0
$445$	555.984	0.0592273
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1062.76	0.111703	0.0558515	−	0.998439i	$-0.482213\pi$
$449$	1062.76	0.111703	0.0558515	+	0.998439i	$0.482213\pi$
$450$	0	0
$451$	−1381.36	−0.144225
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1190.32	0.122644
$456$	0	0
$457$	−16024.6	−1.64026	−0.820132	−	0.572174i	$-0.806100\pi$
$457$	−16024.6	−1.64026	−0.820132	+	0.572174i	$0.806100\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3632.19	−0.366959	−0.183479	−	0.983024i	$-0.558736\pi$
$461$	−3632.19	−0.366959	−0.183479	+	0.983024i	$0.558736\pi$
$462$	0	0
$463$	11454.6	1.14977	0.574884	−	0.818235i	$-0.305047\pi$
$463$	11454.6	1.14977	0.574884	+	0.818235i	$0.305047\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7874.65	0.780290	0.390145	−	0.920754i	$-0.372425\pi$
$467$	7874.65	0.780290	0.390145	+	0.920754i	$0.372425\pi$
$468$	0	0
$469$	4691.04	0.461859
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1684.95	0.163793
$474$	0	0
$475$	13278.0	1.28260
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18364.6	−1.75178	−0.875888	−	0.482514i	$-0.839724\pi$
$479$	−18364.6	−1.75178	−0.875888	+	0.482514i	$0.839724\pi$
$480$	0	0
$481$	−4996.60	−0.473649
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1299.32	0.121648
$486$	0	0
$487$	−3708.90	−0.345105	−0.172552	−	0.985000i	$-0.555201\pi$
$487$	−3708.90	−0.345105	−0.172552	+	0.985000i	$0.555201\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10339.7	0.950358	0.475179	−	0.879889i	$-0.342383\pi$
$491$	10339.7	0.950358	0.475179	+	0.879889i	$0.342383\pi$
$492$	0	0
$493$	982.403	0.0897468
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5899.08	0.532414
$498$	0	0
$499$	6150.13	0.551739	0.275869	−	0.961195i	$-0.411034\pi$
$499$	6150.13	0.551739	0.275869	+	0.961195i	$0.411034\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13657.7	−1.21067	−0.605336	−	0.795970i	$-0.706961\pi$
$503$	−13657.7	−1.21067	−0.605336	+	0.795970i	$0.706961\pi$
$504$	0	0
$505$	1226.96	0.108117
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3901.70	−0.339764	−0.169882	−	0.985464i	$-0.554339\pi$
$509$	−3901.70	−0.339764	−0.169882	+	0.985464i	$0.554339\pi$
$510$	0	0
$511$	−14139.8	−1.22408
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2689.21	0.230098
$516$	0	0
$517$	371.549	0.0316068
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22839.2	−1.92054	−0.960271	−	0.279068i	$-0.909975\pi$
$521$	−22839.2	−1.92054	−0.960271	+	0.279068i	$0.909975\pi$
$522$	0	0
$523$	8467.66	0.707964	0.353982	−	0.935252i	$-0.384827\pi$
$523$	8467.66	0.707964	0.353982	+	0.935252i	$0.384827\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	798.189	0.0659766
$528$	0	0
$529$	8163.56	0.670959
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14158.1	−1.15057
$534$	0	0
$535$	354.855	0.0286761
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−196.926	−0.0157369
$540$	0	0
$541$	−6327.34	−0.502835	−0.251417	−	0.967879i	$-0.580897\pi$
$541$	−6327.34	−0.502835	−0.251417	+	0.967879i	$0.580897\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2971.63	−0.233560
$546$	0	0
$547$	12262.7	0.958525	0.479263	−	0.877671i	$-0.340904\pi$
$547$	12262.7	0.958525	0.479263	+	0.877671i	$0.340904\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−32095.5	−2.48151
$552$	0	0
$553$	−4957.37	−0.381209
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12501.2	0.950972	0.475486	−	0.879723i	$-0.342272\pi$
$557$	12501.2	0.950972	0.475486	+	0.879723i	$0.342272\pi$
$558$	0	0
$559$	17269.6	1.30667
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2634.68	0.197226	0.0986131	−	0.995126i	$-0.468559\pi$
$563$	2634.68	0.197226	0.0986131	+	0.995126i	$0.468559\pi$
$564$	0	0
$565$	1953.74	0.145477
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24259.9	−1.78739	−0.893696	−	0.448673i	$-0.851897\pi$
$569$	−24259.9	−1.78739	−0.893696	+	0.448673i	$0.851897\pi$
$570$	0	0
$571$	2196.39	0.160974	0.0804871	−	0.996756i	$-0.474352\pi$
$571$	2196.39	0.160974	0.0804871	+	0.996756i	$0.474352\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−17415.0	−1.26306
$576$	0	0
$577$	−11167.6	−0.805745	−0.402872	−	0.915256i	$-0.631988\pi$
$577$	−11167.6	−0.805745	−0.402872	+	0.915256i	$0.631988\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22628.6	1.61583
$582$	0	0
$583$	766.030	0.0544180
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17340.4	1.21928	0.609638	−	0.792680i	$-0.291315\pi$
$587$	17340.4	1.21928	0.609638	+	0.792680i	$0.291315\pi$
$588$	0	0
$589$	−26077.1	−1.82426
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21672.7	−1.50083	−0.750414	−	0.660968i	$-0.770146\pi$
$593$	−21672.7	−1.50083	−0.750414	+	0.660968i	$0.770146\pi$
$594$	0	0
$595$	96.5014	0.00664903
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24124.7	−1.64559	−0.822794	−	0.568340i	$-0.807586\pi$
$599$	−24124.7	−1.64559	−0.822794	+	0.568340i	$0.807586\pi$
$600$	0	0
$601$	−7945.41	−0.539268	−0.269634	−	0.962963i	$-0.586903\pi$
$601$	−7945.41	−0.539268	−0.269634	+	0.962963i	$0.586903\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2224.71	−0.149500
$606$	0	0
$607$	−6295.91	−0.420994	−0.210497	−	0.977595i	$-0.567508\pi$
$607$	−6295.91	−0.420994	−0.210497	+	0.977595i	$0.567508\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3808.15	0.252146
$612$	0	0
$613$	1806.97	0.119058	0.0595291	−	0.998227i	$-0.481040\pi$
$613$	1806.97	0.119058	0.0595291	+	0.998227i	$0.481040\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9142.53	−0.596539	−0.298269	−	0.954482i	$-0.596409\pi$
$617$	−9142.53	−0.596539	−0.298269	+	0.954482i	$0.596409\pi$
$618$	0	0
$619$	−13843.0	−0.898862	−0.449431	−	0.893315i	$-0.648373\pi$
$619$	−13843.0	−0.898862	−0.449431	+	0.893315i	$0.648373\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5633.09	0.362255
$624$	0	0
$625$	14559.8	0.931828
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−405.082	−0.0256784
$630$	0	0
$631$	8019.33	0.505934	0.252967	−	0.967475i	$-0.418594\pi$
$631$	8019.33	0.505934	0.252967	+	0.967475i	$0.418594\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1412.09	0.0882471
$636$	0	0
$637$	−2018.37	−0.125543
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13637.1	−0.840299	−0.420149	−	0.907455i	$-0.638022\pi$
$641$	−13637.1	−0.840299	−0.420149	+	0.907455i	$0.638022\pi$
$642$	0	0
$643$	−10948.2	−0.671471	−0.335736	−	0.941956i	$-0.608985\pi$
$643$	−10948.2	−0.671471	−0.335736	+	0.941956i	$0.608985\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7720.91	−0.469150	−0.234575	−	0.972098i	$-0.575370\pi$
$647$	−7720.91	−0.469150	−0.234575	+	0.972098i	$0.575370\pi$
$648$	0	0
$649$	−2650.87	−0.160332
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26065.9	1.56208	0.781040	−	0.624481i	$-0.214689\pi$
$653$	26065.9	1.56208	0.781040	+	0.624481i	$0.214689\pi$
$654$	0	0
$655$	2483.37	0.148142
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9441.58	−0.558106	−0.279053	−	0.960276i	$-0.590021\pi$
$659$	−9441.58	−0.558106	−0.279053	+	0.960276i	$0.590021\pi$
$660$	0	0
$661$	−18098.2	−1.06496	−0.532479	−	0.846443i	$-0.678739\pi$
$661$	−18098.2	−1.06496	−0.532479	+	0.846443i	$0.678739\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3152.74	−0.183846
$666$	0	0
$667$	42095.5	2.44370
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1439.80	−0.0828360
$672$	0	0
$673$	−15168.8	−0.868818	−0.434409	−	0.900716i	$-0.643043\pi$
$673$	−15168.8	−0.868818	−0.434409	+	0.900716i	$0.643043\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28029.1	−1.59120	−0.795602	−	0.605820i	$-0.792845\pi$
$677$	−28029.1	−1.59120	−0.795602	+	0.605820i	$0.792845\pi$
$678$	0	0
$679$	13164.4	0.744041
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13655.7	0.765037	0.382518	−	0.923948i	$-0.375057\pi$
$683$	13655.7	0.765037	0.382518	+	0.923948i	$0.375057\pi$
$684$	0	0
$685$	−2714.55	−0.151413
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7851.33	0.434125
$690$	0	0
$691$	1193.35	0.0656975	0.0328488	−	0.999460i	$-0.489542\pi$
$691$	1193.35	0.0656975	0.0328488	+	0.999460i	$0.489542\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−778.903	−0.0425115
$696$	0	0
$697$	−1147.82	−0.0623769
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25648.9	1.38195	0.690975	−	0.722879i	$-0.257182\pi$
$701$	25648.9	1.38195	0.690975	+	0.722879i	$0.257182\pi$
$702$	0	0
$703$	13234.2	0.710010
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12431.3	0.661281
$708$	0	0
$709$	28955.6	1.53378	0.766890	−	0.641779i	$-0.221803\pi$
$709$	28955.6	1.53378	0.766890	+	0.641779i	$0.221803\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34202.1	1.79646
$714$	0	0
$715$	278.088	0.0145453
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1167.91	−0.0605781	−0.0302890	−	0.999541i	$-0.509643\pi$
$719$	−1167.91	−0.0605781	−0.0302890	+	0.999541i	$0.509643\pi$
$720$	0	0
$721$	27246.4	1.40736
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−36058.8	−1.84716
$726$	0	0
$727$	24274.0	1.23834	0.619170	−	0.785257i	$-0.287469\pi$
$727$	24274.0	1.23834	0.619170	+	0.785257i	$0.287469\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1400.08	0.0708396
$732$	0	0
$733$	−11429.8	−0.575946	−0.287973	−	0.957638i	$-0.592981\pi$
$733$	−11429.8	−0.575946	−0.287973	+	0.957638i	$0.592981\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1095.94	0.0547754
$738$	0	0
$739$	−34238.4	−1.70430	−0.852152	−	0.523294i	$-0.824703\pi$
$739$	−34238.4	−1.70430	−0.852152	+	0.523294i	$0.824703\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29843.6	−1.47356	−0.736781	−	0.676131i	$-0.763655\pi$
$743$	−29843.6	−1.47356	−0.736781	+	0.676131i	$0.763655\pi$
$744$	0	0
$745$	3637.77	0.178896
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3595.30	0.175393
$750$	0	0
$751$	−24246.8	−1.17813	−0.589066	−	0.808085i	$-0.700504\pi$
$751$	−24246.8	−1.17813	−0.589066	+	0.808085i	$0.700504\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2522.36	0.121587
$756$	0	0
$757$	−2988.55	−0.143489	−0.0717443	−	0.997423i	$-0.522857\pi$
$757$	−2988.55	−0.143489	−0.0717443	+	0.997423i	$0.522857\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15332.4	−0.730355	−0.365178	−	0.930938i	$-0.618992\pi$
$761$	−15332.4	−0.730355	−0.365178	+	0.930938i	$0.618992\pi$
$762$	0	0
$763$	−30107.8	−1.42854
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27169.8	−1.27907
$768$	0	0
$769$	−2714.31	−0.127283	−0.0636415	−	0.997973i	$-0.520271\pi$
$769$	−2714.31	−0.127283	−0.0636415	+	0.997973i	$0.520271\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18179.4	−0.845882	−0.422941	−	0.906157i	$-0.639002\pi$
$773$	−18179.4	−0.845882	−0.422941	+	0.906157i	$0.639002\pi$
$774$	0	0
$775$	−29297.3	−1.35792
$776$	0	0
$777$	0	0
$778$	0	0
$779$	37499.6	1.72473
$780$	0	0
$781$	1378.17	0.0631430
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1845.92	0.0839281
$786$	0	0
$787$	7231.05	0.327521	0.163760	−	0.986500i	$-0.447638\pi$
$787$	7231.05	0.327521	0.163760	+	0.986500i	$0.447638\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19794.8	0.889788
$792$	0	0
$793$	−14757.1	−0.660831
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10616.6	−0.471844	−0.235922	−	0.971772i	$-0.575811\pi$
$797$	−10616.6	−0.471844	−0.235922	+	0.971772i	$0.575811\pi$
$798$	0	0
$799$	308.733	0.0136698
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3303.39	−0.145173
$804$	0	0
$805$	4135.04	0.181045
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6223.80	−0.270478	−0.135239	−	0.990813i	$-0.543180\pi$
$809$	−6223.80	−0.270478	−0.135239	+	0.990813i	$0.543180\pi$
$810$	0	0
$811$	−15669.2	−0.678448	−0.339224	−	0.940706i	$-0.610164\pi$
$811$	−15669.2	−0.678448	−0.339224	+	0.940706i	$0.610164\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−727.403	−0.0312636
$816$	0	0
$817$	−45741.1	−1.95872
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3545.01	0.150696	0.0753482	−	0.997157i	$-0.475993\pi$
$821$	3545.01	0.150696	0.0753482	+	0.997157i	$0.475993\pi$
$822$	0	0
$823$	−6035.45	−0.255629	−0.127814	−	0.991798i	$-0.540796\pi$
$823$	−6035.45	−0.255629	−0.127814	+	0.991798i	$0.540796\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35595.0	−1.49668	−0.748342	−	0.663313i	$-0.769150\pi$
$827$	−35595.0	−1.49668	−0.748342	+	0.663313i	$0.769150\pi$
$828$	0	0
$829$	21599.6	0.904929	0.452465	−	0.891782i	$-0.350545\pi$
$829$	21599.6	0.904929	0.452465	+	0.891782i	$0.350545\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−163.632	−0.00680615
$834$	0	0
$835$	4523.82	0.187489
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31775.5	1.30752	0.653762	−	0.756701i	$-0.273190\pi$
$839$	31775.5	1.30752	0.653762	+	0.756701i	$0.273190\pi$
$840$	0	0
$841$	62772.0	2.57378
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−866.753	−0.0352866
$846$	0	0
$847$	−22540.2	−0.914393
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−17357.6	−0.699190
$852$	0	0
$853$	14668.4	0.588788	0.294394	−	0.955684i	$-0.404882\pi$
$853$	14668.4	0.588788	0.294394	+	0.955684i	$0.404882\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31006.9	1.23591	0.617955	−	0.786213i	$-0.287961\pi$
$857$	31006.9	1.23591	0.617955	+	0.786213i	$0.287961\pi$
$858$	0	0
$859$	19411.3	0.771020	0.385510	−	0.922704i	$-0.374025\pi$
$859$	19411.3	0.771020	0.385510	+	0.922704i	$0.374025\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46474.7	−1.83316	−0.916579	−	0.399853i	$-0.869061\pi$
$863$	−46474.7	−1.83316	−0.916579	+	0.399853i	$0.869061\pi$
$864$	0	0
$865$	7542.21	0.296466
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1158.16	−0.0452105
$870$	0	0
$871$	11232.7	0.436975
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7167.11	−0.276906
$876$	0	0
$877$	50151.1	1.93099	0.965497	−	0.260415i	$-0.0838594\pi$
$877$	50151.1	1.93099	0.965497	+	0.260415i	$0.0838594\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	133.690	0.00511253	0.00255627	−	0.999997i	$-0.499186\pi$
$881$	133.690	0.00511253	0.00255627	+	0.999997i	$0.499186\pi$
$882$	0	0
$883$	−21513.9	−0.819931	−0.409965	−	0.912101i	$-0.634459\pi$
$883$	−21513.9	−0.819931	−0.409965	+	0.912101i	$0.634459\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8261.34	0.312727	0.156363	−	0.987700i	$-0.450023\pi$
$887$	8261.34	0.312727	0.156363	+	0.987700i	$0.450023\pi$
$888$	0	0
$889$	14306.9	0.539750
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10086.4	−0.377972
$894$	0	0
$895$	4412.27	0.164789
$896$	0	0
$897$	0	0
$898$	0	0
$899$	70817.2	2.62724
$900$	0	0
$901$	636.520	0.0235356
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6920.80	0.254205
$906$	0	0
$907$	13659.6	0.500066	0.250033	−	0.968237i	$-0.419559\pi$
$907$	13659.6	0.500066	0.250033	+	0.968237i	$0.419559\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10765.5	0.391524	0.195762	−	0.980651i	$-0.437282\pi$
$911$	10765.5	0.391524	0.195762	+	0.980651i	$0.437282\pi$
$912$	0	0
$913$	5286.59	0.191633
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25160.8	0.906089
$918$	0	0
$919$	−33843.9	−1.21481	−0.607404	−	0.794393i	$-0.707789\pi$
$919$	−33843.9	−1.21481	−0.607404	+	0.794393i	$0.707789\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14125.3	0.503729
$924$	0	0
$925$	14868.4	0.528508
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42751.1	1.50981	0.754907	−	0.655831i	$-0.227682\pi$
$929$	42751.1	1.50981	0.754907	+	0.655831i	$0.227682\pi$
$930$	0	0
$931$	5345.93	0.188191
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.5450	0.000788558 0
$936$	0	0
$937$	−15886.0	−0.553865	−0.276933	−	0.960889i	$-0.589318\pi$
$937$	−15886.0	−0.553865	−0.276933	+	0.960889i	$0.589318\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10233.2	−0.354510	−0.177255	−	0.984165i	$-0.556722\pi$
$941$	−10233.2	−0.354510	−0.177255	+	0.984165i	$0.556722\pi$
$942$	0	0
$943$	−49183.5	−1.69845
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37156.0	1.27498	0.637491	−	0.770457i	$-0.279972\pi$
$947$	37156.0	1.27498	0.637491	+	0.770457i	$0.279972\pi$
$948$	0	0
$949$	−33857.7	−1.15813
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48165.0	−1.63716	−0.818582	−	0.574389i	$-0.805240\pi$
$953$	−48165.0	−1.63716	−0.818582	+	0.574389i	$0.805240\pi$
$954$	0	0
$955$	−2344.98	−0.0794572
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−27503.1	−0.926093
$960$	0	0
$961$	27747.0	0.931390
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7285.87	−0.243047
$966$	0	0
$967$	−20583.5	−0.684508	−0.342254	−	0.939607i	$-0.611190\pi$
$967$	−20583.5	−0.684508	−0.342254	+	0.939607i	$0.611190\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2053.65	−0.0678730	−0.0339365	−	0.999424i	$-0.510804\pi$
$971$	−2053.65	−0.0678730	−0.0339365	+	0.999424i	$0.510804\pi$
$972$	0	0
$973$	−7891.64	−0.260015
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10377.0	−0.339805	−0.169902	−	0.985461i	$-0.554345\pi$
$977$	−10377.0	−0.339805	−0.169902	+	0.985461i	$0.554345\pi$
$978$	0	0
$979$	1316.02	0.0429625
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−15507.3	−0.503159	−0.251580	−	0.967837i	$-0.580950\pi$
$983$	−15507.3	−0.503159	−0.251580	+	0.967837i	$0.580950\pi$
$984$	0	0
$985$	1077.57	0.0348571
$986$	0	0
$987$	0	0
$988$	0	0
$989$	59992.7	1.92888
$990$	0	0
$991$	43480.9	1.39376	0.696880	−	0.717188i	$-0.254571\pi$
$991$	43480.9	1.39376	0.696880	+	0.717188i	$0.254571\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1123.15	0.0357850
$996$	0	0
$997$	−4692.04	−0.149046	−0.0745228	−	0.997219i	$-0.523743\pi$
$997$	−4692.04	−0.149046	−0.0745228	+	0.997219i	$0.523743\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.ba.1.3		4
3.2	odd	2		1296.4.a.y.1.2		4
4.3	odd	2		648.4.a.i.1.3		4
9.2	odd	6		432.4.i.e.145.3		8
9.4	even	3		144.4.i.e.97.2		8
9.5	odd	6		432.4.i.e.289.3		8
9.7	even	3		144.4.i.e.49.2		8
12.11	even	2		648.4.a.h.1.2		4
36.7	odd	6		72.4.i.a.49.3	yes	8
36.11	even	6		216.4.i.a.145.3		8
36.23	even	6		216.4.i.a.73.3		8
36.31	odd	6		72.4.i.a.25.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.a.25.3	✓	8	36.31	odd	6
72.4.i.a.49.3	yes	8	36.7	odd	6
144.4.i.e.49.2		8	9.7	even	3
144.4.i.e.97.2		8	9.4	even	3
216.4.i.a.73.3		8	36.23	even	6
216.4.i.a.145.3		8	36.11	even	6
432.4.i.e.145.3		8	9.2	odd	6
432.4.i.e.289.3		8	9.5	odd	6
648.4.a.h.1.2		4	12.11	even	2
648.4.a.i.1.3		4	4.3	odd	2
1296.4.a.y.1.2		4	3.2	odd	2
1296.4.a.ba.1.3		4	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.69184 q^{5} +17.1413 q^{7} +O(q^{10})\)	\(q+1.69184 q^{5} +17.1413 q^{7} +4.00463 q^{11} +41.0449 q^{13} +3.32758 q^{17} -108.713 q^{19} +142.585 q^{23} -122.138 q^{25} +295.230 q^{29} +239.871 q^{31} +29.0005 q^{35} -121.735 q^{37} -344.941 q^{41} +420.750 q^{43} +92.7799 q^{47} -49.1746 q^{49} +191.286 q^{53} +6.77521 q^{55} -661.952 q^{59} -359.535 q^{61} +69.4417 q^{65} +273.668 q^{67} +344.143 q^{71} -824.894 q^{73} +68.6447 q^{77} -289.206 q^{79} +1320.12 q^{83} +5.62975 q^{85} +328.626 q^{89} +703.565 q^{91} -183.926 q^{95} +767.992 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{5} + 3 q^{7}+O(q^{10}) \) 4 * q + 5 * q^5 + 3 * q^7	\( 4 q + 5 q^{5} + 3 q^{7} + 16 q^{11} - 29 q^{13} - 17 q^{17} + 109 q^{19} + 37 q^{23} - 97 q^{25} + 3 q^{29} + 331 q^{31} + 171 q^{35} - 366 q^{37} + 378 q^{41} + 506 q^{43} + 171 q^{47} - 829 q^{49} + 410 q^{53} + 1163 q^{55} + 616 q^{59} - 1331 q^{61} + 815 q^{65} + 1162 q^{67} + 344 q^{71} - 1307 q^{73} + 741 q^{77} + 1853 q^{79} + 1421 q^{83} - 2074 q^{85} + 816 q^{89} + 1995 q^{91} + 1292 q^{95} - 2506 q^{97}+O(q^{100}) \) 4 * q + 5 * q^5 + 3 * q^7 + 16 * q^11 - 29 * q^13 - 17 * q^17 + 109 * q^19 + 37 * q^23 - 97 * q^25 + 3 * q^29 + 331 * q^31 + 171 * q^35 - 366 * q^37 + 378 * q^41 + 506 * q^43 + 171 * q^47 - 829 * q^49 + 410 * q^53 + 1163 * q^55 + 616 * q^59 - 1331 * q^61 + 815 * q^65 + 1162 * q^67 + 344 * q^71 - 1307 * q^73 + 741 * q^77 + 1853 * q^79 + 1421 * q^83 - 2074 * q^85 + 816 * q^89 + 1995 * q^91 + 1292 * q^95 - 2506 * q^97

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