LMFDB - Embedded newform 1296.4.a.u.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ 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)

$f(q)$	$=$	$q+4.62772 q^{5} -12.1168 q^{7} +O(q^{10})$	$q+4.62772 q^{5} -12.1168 q^{7} -10.0217 q^{11} -48.5842 q^{13} +75.3505 q^{17} +116.052 q^{19} +38.0733 q^{23} -103.584 q^{25} -22.6277 q^{29} -30.1168 q^{31} -56.0733 q^{35} +130.103 q^{37} +347.484 q^{41} -26.7663 q^{43} -460.878 q^{47} -196.182 q^{49} -438.310 q^{53} -46.3778 q^{55} -8.36974 q^{59} -82.0895 q^{61} -224.834 q^{65} -683.571 q^{67} -1097.61 q^{71} +470.464 q^{73} +121.432 q^{77} -486.035 q^{79} -99.1659 q^{83} +348.701 q^{85} +8.80426 q^{89} +588.687 q^{91} +537.054 q^{95} +660.973 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 15 q^{5} - 7 q^{7}+O(q^{10}) $ 2 * q + 15 * q^5 - 7 * q^7	$ 2 q + 15 q^{5} - 7 q^{7} - 66 q^{11} - 11 q^{13} + 99 q^{17} + 77 q^{19} - 33 q^{23} - 121 q^{25} - 51 q^{29} - 43 q^{31} - 3 q^{35} - 50 q^{37} + 132 q^{41} - 88 q^{43} - 399 q^{47} - 513 q^{49} + 54 q^{53} - 627 q^{55} - 798 q^{59} + 439 q^{61} + 165 q^{65} - 988 q^{67} - 1368 q^{71} - 455 q^{73} - 165 q^{77} + 803 q^{79} - 813 q^{83} + 594 q^{85} - 396 q^{89} + 781 q^{91} + 132 q^{95} + 736 q^{97}+O(q^{100}) $ 2 * q + 15 * q^5 - 7 * q^7 - 66 * q^11 - 11 * q^13 + 99 * q^17 + 77 * q^19 - 33 * q^23 - 121 * q^25 - 51 * q^29 - 43 * q^31 - 3 * q^35 - 50 * q^37 + 132 * q^41 - 88 * q^43 - 399 * q^47 - 513 * q^49 + 54 * q^53 - 627 * q^55 - 798 * q^59 + 439 * q^61 + 165 * q^65 - 988 * q^67 - 1368 * q^71 - 455 * q^73 - 165 * q^77 + 803 * q^79 - 813 * q^83 + 594 * q^85 - 396 * q^89 + 781 * q^91 + 132 * q^95 + 736 * 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
$5$	4.62772	0.413916	0.206958	−	0.978350i	$-0.433644\pi$
$5$	4.62772	0.413916	0.206958	+	0.978350i	$0.433644\pi$
$6$	0	0
$7$	−12.1168	−0.654248	−0.327124	−	0.944981i	$-0.606079\pi$
$7$	−12.1168	−0.654248	−0.327124	+	0.944981i	$0.606079\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−10.0217	−0.274697	−0.137349	−	0.990523i	$-0.543858\pi$
$11$	−10.0217	−0.274697	−0.137349	+	0.990523i	$0.543858\pi$
$12$	0	0
$13$	−48.5842	−1.03653	−0.518263	−	0.855221i	$-0.673421\pi$
$13$	−48.5842	−1.03653	−0.518263	+	0.855221i	$0.673421\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	75.3505	1.07501	0.537506	−	0.843260i	$-0.319367\pi$
$17$	75.3505	1.07501	0.537506	+	0.843260i	$0.319367\pi$
$18$	0	0
$19$	116.052	1.40127	0.700633	−	0.713522i	$-0.252901\pi$
$19$	116.052	1.40127	0.700633	+	0.713522i	$0.252901\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	38.0733	0.345167	0.172584	−	0.984995i	$-0.444789\pi$
$23$	38.0733	0.345167	0.172584	+	0.984995i	$0.444789\pi$
$24$	0	0
$25$	−103.584	−0.828674
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−22.6277	−0.144892	−0.0724459	−	0.997372i	$-0.523080\pi$
$29$	−22.6277	−0.144892	−0.0724459	+	0.997372i	$0.523080\pi$
$30$	0	0
$31$	−30.1168	−0.174489	−0.0872443	−	0.996187i	$-0.527806\pi$
$31$	−30.1168	−0.174489	−0.0872443	+	0.996187i	$0.527806\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−56.0733	−0.270804
$36$	0	0
$37$	130.103	0.578077	0.289038	−	0.957318i	$-0.406665\pi$
$37$	130.103	0.578077	0.289038	+	0.957318i	$0.406665\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	347.484	1.32361	0.661803	−	0.749678i	$-0.269792\pi$
$41$	347.484	1.32361	0.661803	+	0.749678i	$0.269792\pi$
$42$	0	0
$43$	−26.7663	−0.0949261	−0.0474631	−	0.998873i	$-0.515114\pi$
$43$	−26.7663	−0.0949261	−0.0474631	+	0.998873i	$0.515114\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−460.878	−1.43034	−0.715169	−	0.698951i	$-0.753650\pi$
$47$	−460.878	−1.43034	−0.715169	+	0.698951i	$0.753650\pi$
$48$	0	0
$49$	−196.182	−0.571959
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−438.310	−1.13597	−0.567985	−	0.823039i	$-0.692277\pi$
$53$	−438.310	−1.13597	−0.567985	+	0.823039i	$0.692277\pi$
$54$	0	0
$55$	−46.3778	−0.113702
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.36974	−0.0184686	−0.00923430	−	0.999957i	$-0.502939\pi$
$59$	−8.36974	−0.0184686	−0.00923430	+	0.999957i	$0.502939\pi$
$60$	0	0
$61$	−82.0895	−0.172303	−0.0861515	−	0.996282i	$-0.527457\pi$
$61$	−82.0895	−0.172303	−0.0861515	+	0.996282i	$0.527457\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−224.834	−0.429034
$66$	0	0
$67$	−683.571	−1.24644	−0.623220	−	0.782047i	$-0.714176\pi$
$67$	−683.571	−1.24644	−0.623220	+	0.782047i	$0.714176\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1097.61	−1.83468	−0.917339	−	0.398107i	$-0.869667\pi$
$71$	−1097.61	−1.83468	−0.917339	+	0.398107i	$0.869667\pi$
$72$	0	0
$73$	470.464	0.754297	0.377149	−	0.926153i	$-0.376905\pi$
$73$	470.464	0.754297	0.377149	+	0.926153i	$0.376905\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	121.432	0.179720
$78$	0	0
$79$	−486.035	−0.692192	−0.346096	−	0.938199i	$-0.612493\pi$
$79$	−486.035	−0.692192	−0.346096	+	0.938199i	$0.612493\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−99.1659	−0.131143	−0.0655715	−	0.997848i	$-0.520887\pi$
$83$	−99.1659	−0.131143	−0.0655715	+	0.997848i	$0.520887\pi$
$84$	0	0
$85$	348.701	0.444964
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.80426	0.0104859	0.00524297	−	0.999986i	$-0.498331\pi$
$89$	8.80426	0.0104859	0.00524297	+	0.999986i	$0.498331\pi$
$90$	0	0
$91$	588.687	0.678145
$92$	0	0
$93$	0	0
$94$	0	0
$95$	537.054	0.580006
$96$	0	0
$97$	660.973	0.691872	0.345936	−	0.938258i	$-0.387561\pi$
$97$	660.973	0.691872	0.345936	+	0.938258i	$0.387561\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−565.122	−0.556750	−0.278375	−	0.960472i	$-0.589796\pi$
$101$	−565.122	−0.556750	−0.278375	+	0.960472i	$0.589796\pi$
$102$	0	0
$103$	971.182	0.929062	0.464531	−	0.885557i	$-0.346223\pi$
$103$	971.182	0.929062	0.464531	+	0.885557i	$0.346223\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	563.845	0.509430	0.254715	−	0.967016i	$-0.418018\pi$
$107$	563.845	0.509430	0.254715	+	0.967016i	$0.418018\pi$
$108$	0	0
$109$	225.484	0.198142	0.0990709	−	0.995080i	$-0.468413\pi$
$109$	225.484	0.198142	0.0990709	+	0.995080i	$0.468413\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−691.063	−0.575307	−0.287654	−	0.957735i	$-0.592875\pi$
$113$	−691.063	−0.575307	−0.287654	+	0.957735i	$0.592875\pi$
$114$	0	0
$115$	176.193	0.142870
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−913.011	−0.703324
$120$	0	0
$121$	−1230.56	−0.924541
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1057.82	−0.756917
$126$	0	0
$127$	895.897	0.625968	0.312984	−	0.949758i	$-0.398671\pi$
$127$	895.897	0.625968	0.312984	+	0.949758i	$0.398671\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1654.86	−1.10371	−0.551853	−	0.833942i	$-0.686079\pi$
$131$	−1654.86	−1.10371	−0.551853	+	0.833942i	$0.686079\pi$
$132$	0	0
$133$	−1406.18	−0.916776
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2651.09	−1.65327	−0.826635	−	0.562738i	$-0.809748\pi$
$137$	−2651.09	−1.65327	−0.826635	+	0.562738i	$0.809748\pi$
$138$	0	0
$139$	−634.168	−0.386975	−0.193487	−	0.981103i	$-0.561980\pi$
$139$	−634.168	−0.386975	−0.193487	+	0.981103i	$0.561980\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	486.899	0.284731
$144$	0	0
$145$	−104.715	−0.0599730
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3406.32	−1.87286	−0.936432	−	0.350849i	$-0.885893\pi$
$149$	−3406.32	−1.87286	−0.936432	+	0.350849i	$0.885893\pi$
$150$	0	0
$151$	1750.32	0.943303	0.471652	−	0.881785i	$-0.343658\pi$
$151$	1750.32	0.943303	0.471652	+	0.881785i	$0.343658\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−139.372	−0.0722236
$156$	0	0
$157$	2178.57	1.10745	0.553723	−	0.832701i	$-0.313207\pi$
$157$	2178.57	1.10745	0.553723	+	0.832701i	$0.313207\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−461.329	−0.225825
$162$	0	0
$163$	−2188.41	−1.05159	−0.525797	−	0.850610i	$-0.676233\pi$
$163$	−2188.41	−1.05159	−0.525797	+	0.850610i	$0.676233\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1921.04	−0.890147	−0.445074	−	0.895494i	$-0.646822\pi$
$167$	−1921.04	−0.890147	−0.445074	+	0.895494i	$0.646822\pi$
$168$	0	0
$169$	163.426	0.0743862
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3169.83	1.39305	0.696525	−	0.717533i	$-0.254729\pi$
$173$	3169.83	1.39305	0.696525	+	0.717533i	$0.254729\pi$
$174$	0	0
$175$	1255.11	0.542158
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1368.78	−0.571551	−0.285776	−	0.958297i	$-0.592251\pi$
$179$	−1368.78	−0.571551	−0.285776	+	0.958297i	$0.592251\pi$
$180$	0	0
$181$	−3951.44	−1.62270	−0.811350	−	0.584561i	$-0.801267\pi$
$181$	−3951.44	−1.62270	−0.811350	+	0.584561i	$0.801267\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	602.081	0.239275
$186$	0	0
$187$	−755.144	−0.295303
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2402.59	0.910184	0.455092	−	0.890444i	$-0.349606\pi$
$191$	2402.59	0.910184	0.455092	+	0.890444i	$0.349606\pi$
$192$	0	0
$193$	−1335.07	−0.497930	−0.248965	−	0.968513i	$-0.580090\pi$
$193$	−1335.07	−0.497930	−0.248965	+	0.968513i	$0.580090\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2630.89	−0.951487	−0.475743	−	0.879584i	$-0.657821\pi$
$197$	−2630.89	−0.951487	−0.475743	+	0.879584i	$0.657821\pi$
$198$	0	0
$199$	−2477.34	−0.882483	−0.441241	−	0.897388i	$-0.645462\pi$
$199$	−2477.34	−0.882483	−0.441241	+	0.897388i	$0.645462\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	274.177	0.0947952
$204$	0	0
$205$	1608.06	0.547861
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1163.04	−0.384924
$210$	0	0
$211$	2784.73	0.908572	0.454286	−	0.890856i	$-0.349894\pi$
$211$	2784.73	0.908572	0.454286	+	0.890856i	$0.349894\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−123.867	−0.0392914
$216$	0	0
$217$	364.921	0.114159
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3660.85	−1.11428
$222$	0	0
$223$	43.0576	0.0129298	0.00646491	−	0.999979i	$-0.497942\pi$
$223$	43.0576	0.0129298	0.00646491	+	0.999979i	$0.497942\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−683.260	−0.199778	−0.0998889	−	0.994999i	$-0.531849\pi$
$227$	−683.260	−0.199778	−0.0998889	+	0.994999i	$0.531849\pi$
$228$	0	0
$229$	4294.31	1.23920	0.619598	−	0.784920i	$-0.287296\pi$
$229$	4294.31	1.23920	0.619598	+	0.784920i	$0.287296\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3466.34	0.974625	0.487313	−	0.873228i	$-0.337977\pi$
$233$	3466.34	0.974625	0.487313	+	0.873228i	$0.337977\pi$
$234$	0	0
$235$	−2132.81	−0.592040
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5643.37	−1.52736	−0.763681	−	0.645594i	$-0.776610\pi$
$239$	−5643.37	−1.52736	−0.763681	+	0.645594i	$0.776610\pi$
$240$	0	0
$241$	6589.43	1.76125	0.880627	−	0.473810i	$-0.157121\pi$
$241$	6589.43	1.76125	0.880627	+	0.473810i	$0.157121\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−907.876	−0.236743
$246$	0	0
$247$	−5638.28	−1.45245
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4135.47	−1.03996	−0.519978	−	0.854180i	$-0.674060\pi$
$251$	−4135.47	−1.03996	−0.519978	+	0.854180i	$0.674060\pi$
$252$	0	0
$253$	−381.562	−0.0948165
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3344.62	−0.811796	−0.405898	−	0.913918i	$-0.633041\pi$
$257$	−3344.62	−0.811796	−0.405898	+	0.913918i	$0.633041\pi$
$258$	0	0
$259$	−1576.44	−0.378205
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6521.50	−1.52902	−0.764511	−	0.644611i	$-0.777019\pi$
$263$	−6521.50	−1.52902	−0.764511	+	0.644611i	$0.777019\pi$
$264$	0	0
$265$	−2028.37	−0.470196
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2904.99	−0.658441	−0.329220	−	0.944253i	$-0.606786\pi$
$269$	−2904.99	−0.658441	−0.329220	+	0.944253i	$0.606786\pi$
$270$	0	0
$271$	1335.38	0.299331	0.149665	−	0.988737i	$-0.452180\pi$
$271$	1335.38	0.299331	0.149665	+	0.988737i	$0.452180\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1038.10	0.227635
$276$	0	0
$277$	−8375.64	−1.81676	−0.908381	−	0.418143i	$-0.862681\pi$
$277$	−8375.64	−1.81676	−0.908381	+	0.418143i	$0.862681\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5179.33	−1.09955	−0.549774	−	0.835313i	$-0.685286\pi$
$281$	−5179.33	−1.09955	−0.549774	+	0.835313i	$0.685286\pi$
$282$	0	0
$283$	3080.07	0.646965	0.323482	−	0.946234i	$-0.395146\pi$
$283$	3080.07	0.646965	0.323482	+	0.946234i	$0.395146\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4210.40	−0.865966
$288$	0	0
$289$	764.703	0.155649
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3648.89	0.727545	0.363773	−	0.931488i	$-0.381489\pi$
$293$	3648.89	0.727545	0.363773	+	0.931488i	$0.381489\pi$
$294$	0	0
$295$	−38.7328	−0.00764444
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1849.76	−0.357775
$300$	0	0
$301$	324.323	0.0621052
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−379.887	−0.0713190
$306$	0	0
$307$	3439.25	0.639376	0.319688	−	0.947523i	$-0.396422\pi$
$307$	3439.25	0.639376	0.319688	+	0.947523i	$0.396422\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7175.33	−1.30828	−0.654141	−	0.756373i	$-0.726970\pi$
$311$	−7175.33	−1.30828	−0.654141	+	0.756373i	$0.726970\pi$
$312$	0	0
$313$	−4857.30	−0.877159	−0.438579	−	0.898692i	$-0.644518\pi$
$313$	−4857.30	−0.877159	−0.438579	+	0.898692i	$0.644518\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7547.93	1.33733	0.668666	−	0.743563i	$-0.266866\pi$
$317$	7547.93	1.33733	0.668666	+	0.743563i	$0.266866\pi$
$318$	0	0
$319$	226.769	0.0398014
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8744.55	1.50638
$324$	0	0
$325$	5032.56	0.858942
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5584.38	0.935796
$330$	0	0
$331$	3129.72	0.519712	0.259856	−	0.965647i	$-0.416325\pi$
$331$	3129.72	0.519712	0.259856	+	0.965647i	$0.416325\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3163.37	−0.515921
$336$	0	0
$337$	−9229.98	−1.49196	−0.745978	−	0.665971i	$-0.768017\pi$
$337$	−9229.98	−1.49196	−0.745978	+	0.665971i	$0.768017\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	301.823	0.0479315
$342$	0	0
$343$	6533.19	1.02845
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8104.40	1.25379	0.626897	−	0.779102i	$-0.284325\pi$
$347$	8104.40	1.25379	0.626897	+	0.779102i	$0.284325\pi$
$348$	0	0
$349$	3076.42	0.471854	0.235927	−	0.971771i	$-0.424187\pi$
$349$	3076.42	0.471854	0.235927	+	0.971771i	$0.424187\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6150.08	−0.927297	−0.463649	−	0.886019i	$-0.653460\pi$
$353$	−6150.08	−0.927297	−0.463649	+	0.886019i	$0.653460\pi$
$354$	0	0
$355$	−5079.42	−0.759402
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3307.94	−0.486313	−0.243156	−	0.969987i	$-0.578183\pi$
$359$	−3307.94	−0.486313	−0.243156	+	0.969987i	$0.578183\pi$
$360$	0	0
$361$	6608.97	0.963548
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2177.18	0.312215
$366$	0	0
$367$	−2949.29	−0.419488	−0.209744	−	0.977756i	$-0.567263\pi$
$367$	−2949.29	−0.419488	−0.209744	+	0.977756i	$0.567263\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5310.93	0.743207
$372$	0	0
$373$	1163.58	0.161522	0.0807612	−	0.996733i	$-0.474265\pi$
$373$	1163.58	0.161522	0.0807612	+	0.996733i	$0.474265\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1099.35	0.150184
$378$	0	0
$379$	4016.67	0.544387	0.272193	−	0.962243i	$-0.412251\pi$
$379$	4016.67	0.544387	0.272193	+	0.962243i	$0.412251\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10803.3	1.44131	0.720656	−	0.693293i	$-0.243841\pi$
$383$	10803.3	1.44131	0.720656	+	0.693293i	$0.243841\pi$
$384$	0	0
$385$	561.953	0.0743890
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2064.59	0.269097	0.134549	−	0.990907i	$-0.457042\pi$
$389$	2064.59	0.269097	0.134549	+	0.990907i	$0.457042\pi$
$390$	0	0
$391$	2868.85	0.371058
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2249.23	−0.286509
$396$	0	0
$397$	−7937.61	−1.00347	−0.501735	−	0.865022i	$-0.667305\pi$
$397$	−7937.61	−1.00347	−0.501735	+	0.865022i	$0.667305\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2578.21	0.321071	0.160536	−	0.987030i	$-0.448678\pi$
$401$	2578.21	0.321071	0.160536	+	0.987030i	$0.448678\pi$
$402$	0	0
$403$	1463.20	0.180862
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1303.86	−0.158796
$408$	0	0
$409$	5845.76	0.706734	0.353367	−	0.935485i	$-0.385037\pi$
$409$	5845.76	0.706734	0.353367	+	0.935485i	$0.385037\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	101.415	0.0120830
$414$	0	0
$415$	−458.912	−0.0542822
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8741.31	1.01919	0.509596	−	0.860414i	$-0.329795\pi$
$419$	8741.31	1.01919	0.509596	+	0.860414i	$0.329795\pi$
$420$	0	0
$421$	1056.51	0.122307	0.0611533	−	0.998128i	$-0.480522\pi$
$421$	1056.51	0.122307	0.0611533	+	0.998128i	$0.480522\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7805.13	−0.890833
$426$	0	0
$427$	994.666	0.112729
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9868.64	−1.10291	−0.551457	−	0.834203i	$-0.685928\pi$
$431$	−9868.64	−1.10291	−0.551457	+	0.834203i	$0.685928\pi$
$432$	0	0
$433$	477.948	0.0530456	0.0265228	−	0.999648i	$-0.491557\pi$
$433$	477.948	0.0530456	0.0265228	+	0.999648i	$0.491557\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4418.47	0.483671
$438$	0	0
$439$	1052.48	0.114424	0.0572119	−	0.998362i	$-0.481779\pi$
$439$	1052.48	0.114424	0.0572119	+	0.998362i	$0.481779\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10498.7	1.12598	0.562989	−	0.826464i	$-0.309651\pi$
$443$	10498.7	1.12598	0.562989	+	0.826464i	$0.309651\pi$
$444$	0	0
$445$	40.7436	0.00434030
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7329.40	−0.770369	−0.385184	−	0.922840i	$-0.625862\pi$
$449$	−7329.40	−0.770369	−0.385184	+	0.922840i	$0.625862\pi$
$450$	0	0
$451$	−3482.39	−0.363591
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2724.28	0.280695
$456$	0	0
$457$	7144.40	0.731293	0.365646	−	0.930754i	$-0.380848\pi$
$457$	7144.40	0.731293	0.365646	+	0.930754i	$0.380848\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5718.69	0.577757	0.288878	−	0.957366i	$-0.406718\pi$
$461$	5718.69	0.577757	0.288878	+	0.957366i	$0.406718\pi$
$462$	0	0
$463$	−788.466	−0.0791428	−0.0395714	−	0.999217i	$-0.512599\pi$
$463$	−788.466	−0.0791428	−0.0395714	+	0.999217i	$0.512599\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17068.0	1.69125	0.845626	−	0.533776i	$-0.179228\pi$
$467$	17068.0	1.69125	0.845626	+	0.533776i	$0.179228\pi$
$468$	0	0
$469$	8282.72	0.815481
$470$	0	0
$471$	0	0
$472$	0	0
$473$	268.245	0.0260760
$474$	0	0
$475$	−12021.1	−1.16119
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1517.99	−0.144799	−0.0723994	−	0.997376i	$-0.523066\pi$
$479$	−1517.99	−0.144799	−0.0723994	+	0.997376i	$0.523066\pi$
$480$	0	0
$481$	−6320.96	−0.599191
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3058.80	0.286377
$486$	0	0
$487$	−12737.3	−1.18518	−0.592591	−	0.805503i	$-0.701895\pi$
$487$	−12737.3	−1.18518	−0.592591	+	0.805503i	$0.701895\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5647.70	−0.519098	−0.259549	−	0.965730i	$-0.583574\pi$
$491$	−5647.70	−0.519098	−0.259549	+	0.965730i	$0.583574\pi$
$492$	0	0
$493$	−1705.01	−0.155760
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13299.6	1.20033
$498$	0	0
$499$	−10846.3	−0.973038	−0.486519	−	0.873670i	$-0.661734\pi$
$499$	−10846.3	−0.973038	−0.486519	+	0.873670i	$0.661734\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12345.7	1.09437	0.547186	−	0.837011i	$-0.315699\pi$
$503$	12345.7	1.09437	0.547186	+	0.837011i	$0.315699\pi$
$504$	0	0
$505$	−2615.23	−0.230448
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5895.75	−0.513408	−0.256704	−	0.966490i	$-0.582636\pi$
$509$	−5895.75	−0.513408	−0.256704	+	0.966490i	$0.582636\pi$
$510$	0	0
$511$	−5700.54	−0.493497
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4494.36	0.384554
$516$	0	0
$517$	4618.80	0.392910
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5211.51	0.438235	0.219118	−	0.975698i	$-0.429682\pi$
$521$	5211.51	0.438235	0.219118	+	0.975698i	$0.429682\pi$
$522$	0	0
$523$	9809.86	0.820182	0.410091	−	0.912045i	$-0.365497\pi$
$523$	9809.86	0.820182	0.410091	+	0.912045i	$0.365497\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2269.32	−0.187577
$528$	0	0
$529$	−10717.4	−0.880860
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16882.2	−1.37195
$534$	0	0
$535$	2609.32	0.210861
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1966.09	0.157116
$540$	0	0
$541$	8084.25	0.642456	0.321228	−	0.947002i	$-0.395904\pi$
$541$	8084.25	0.642456	0.321228	+	0.947002i	$0.395904\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1043.48	0.0820140
$546$	0	0
$547$	24067.1	1.88123	0.940617	−	0.339470i	$-0.110248\pi$
$547$	24067.1	1.88123	0.940617	+	0.339470i	$0.110248\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2625.98	−0.203032
$552$	0	0
$553$	5889.21	0.452866
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4582.37	0.348584	0.174292	−	0.984694i	$-0.444236\pi$
$557$	4582.37	0.348584	0.174292	+	0.984694i	$0.444236\pi$
$558$	0	0
$559$	1300.42	0.0983934
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3095.31	0.231709	0.115854	−	0.993266i	$-0.463039\pi$
$563$	3095.31	0.231709	0.115854	+	0.993266i	$0.463039\pi$
$564$	0	0
$565$	−3198.04	−0.238129
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20565.1	1.51517	0.757586	−	0.652735i	$-0.226379\pi$
$569$	20565.1	1.51517	0.757586	+	0.652735i	$0.226379\pi$
$570$	0	0
$571$	1169.98	0.0857483	0.0428742	−	0.999080i	$-0.486349\pi$
$571$	1169.98	0.0857483	0.0428742	+	0.999080i	$0.486349\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3943.80	−0.286031
$576$	0	0
$577$	−13073.0	−0.943214	−0.471607	−	0.881809i	$-0.656326\pi$
$577$	−13073.0	−0.943214	−0.471607	+	0.881809i	$0.656326\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1201.58	0.0858001
$582$	0	0
$583$	4392.63	0.312048
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14547.8	1.02292	0.511459	−	0.859308i	$-0.329105\pi$
$587$	14547.8	1.02292	0.511459	+	0.859308i	$0.329105\pi$
$588$	0	0
$589$	−3495.11	−0.244505
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19018.0	1.31699	0.658494	−	0.752586i	$-0.271194\pi$
$593$	19018.0	1.31699	0.658494	+	0.752586i	$0.271194\pi$
$594$	0	0
$595$	−4225.16	−0.291117
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11174.4	−0.762227	−0.381113	−	0.924528i	$-0.624459\pi$
$599$	−11174.4	−0.762227	−0.381113	+	0.924528i	$0.624459\pi$
$600$	0	0
$601$	4588.54	0.311432	0.155716	−	0.987802i	$-0.450232\pi$
$601$	4588.54	0.311432	0.155716	+	0.987802i	$0.450232\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5694.71	−0.382682
$606$	0	0
$607$	11490.0	0.768309	0.384155	−	0.923269i	$-0.374493\pi$
$607$	11490.0	0.768309	0.384155	+	0.923269i	$0.374493\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	22391.4	1.48258
$612$	0	0
$613$	−22966.9	−1.51326	−0.756628	−	0.653846i	$-0.773154\pi$
$613$	−22966.9	−1.51326	−0.756628	+	0.653846i	$0.773154\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8496.63	−0.554394	−0.277197	−	0.960813i	$-0.589406\pi$
$617$	−8496.63	−0.554394	−0.277197	+	0.960813i	$0.589406\pi$
$618$	0	0
$619$	−14339.9	−0.931129	−0.465564	−	0.885014i	$-0.654149\pi$
$619$	−14339.9	−0.931129	−0.465564	+	0.885014i	$0.654149\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−106.680	−0.00686041
$624$	0	0
$625$	8052.72	0.515374
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9803.34	0.621439
$630$	0	0
$631$	−17834.3	−1.12516	−0.562578	−	0.826744i	$-0.690190\pi$
$631$	−17834.3	−1.12516	−0.562578	+	0.826744i	$0.690190\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4145.96	0.259098
$636$	0	0
$637$	9531.35	0.592851
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26346.0	1.62341	0.811705	−	0.584068i	$-0.198540\pi$
$641$	26346.0	1.62341	0.811705	+	0.584068i	$0.198540\pi$
$642$	0	0
$643$	21565.1	1.32262	0.661309	−	0.750114i	$-0.270001\pi$
$643$	21565.1	1.32262	0.661309	+	0.750114i	$0.270001\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4186.32	−0.254376	−0.127188	−	0.991879i	$-0.540595\pi$
$647$	−4186.32	−0.254376	−0.127188	+	0.991879i	$0.540595\pi$
$648$	0	0
$649$	83.8794	0.00507328
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2805.14	0.168107	0.0840534	−	0.996461i	$-0.473213\pi$
$653$	2805.14	0.168107	0.0840534	+	0.996461i	$0.473213\pi$
$654$	0	0
$655$	−7658.21	−0.456841
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19072.3	1.12739	0.563696	−	0.825982i	$-0.309379\pi$
$659$	19072.3	1.12739	0.563696	+	0.825982i	$0.309379\pi$
$660$	0	0
$661$	−24513.1	−1.44243	−0.721216	−	0.692710i	$-0.756417\pi$
$661$	−24513.1	−1.44243	−0.721216	+	0.692710i	$0.756417\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6507.40	−0.379468
$666$	0	0
$667$	−861.513	−0.0500119
$668$	0	0
$669$	0	0
$670$	0	0
$671$	822.681	0.0473312
$672$	0	0
$673$	−3546.65	−0.203140	−0.101570	−	0.994828i	$-0.532387\pi$
$673$	−3546.65	−0.203140	−0.101570	+	0.994828i	$0.532387\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17875.0	1.01476	0.507380	−	0.861723i	$-0.330614\pi$
$677$	17875.0	1.01476	0.507380	+	0.861723i	$0.330614\pi$
$678$	0	0
$679$	−8008.90	−0.452656
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2857.23	−0.160072	−0.0800358	−	0.996792i	$-0.525503\pi$
$683$	−2857.23	−0.160072	−0.0800358	+	0.996792i	$0.525503\pi$
$684$	0	0
$685$	−12268.5	−0.684315
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21294.9	1.17746
$690$	0	0
$691$	15627.6	0.860349	0.430175	−	0.902746i	$-0.358452\pi$
$691$	15627.6	0.860349	0.430175	+	0.902746i	$0.358452\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2934.75	−0.160175
$696$	0	0
$697$	26183.1	1.42289
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17562.6	0.946264	0.473132	−	0.880992i	$-0.343123\pi$
$701$	17562.6	0.946264	0.473132	+	0.880992i	$0.343123\pi$
$702$	0	0
$703$	15098.7	0.810039
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6847.50	0.364253
$708$	0	0
$709$	20003.6	1.05959	0.529795	−	0.848126i	$-0.322269\pi$
$709$	20003.6	1.05959	0.529795	+	0.848126i	$0.322269\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1146.65	−0.0602277
$714$	0	0
$715$	2253.23	0.117855
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25504.1	−1.32287	−0.661435	−	0.750002i	$-0.730052\pi$
$719$	−25504.1	−1.32287	−0.661435	+	0.750002i	$0.730052\pi$
$720$	0	0
$721$	−11767.7	−0.607837
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2343.87	0.120068
$726$	0	0
$727$	23909.8	1.21976	0.609879	−	0.792494i	$-0.291218\pi$
$727$	23909.8	1.21976	0.609879	+	0.792494i	$0.291218\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2016.86	−0.102047
$732$	0	0
$733$	−8505.54	−0.428594	−0.214297	−	0.976769i	$-0.568746\pi$
$733$	−8505.54	−0.428594	−0.214297	+	0.976769i	$0.568746\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6850.57	0.342394
$738$	0	0
$739$	−25802.5	−1.28438	−0.642192	−	0.766544i	$-0.721975\pi$
$739$	−25802.5	−1.28438	−0.642192	+	0.766544i	$0.721975\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27337.6	−1.34982	−0.674911	−	0.737899i	$-0.735818\pi$
$743$	−27337.6	−1.34982	−0.674911	+	0.737899i	$0.735818\pi$
$744$	0	0
$745$	−15763.5	−0.775208
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6832.02	−0.333293
$750$	0	0
$751$	−3336.64	−0.162125	−0.0810623	−	0.996709i	$-0.525831\pi$
$751$	−3336.64	−0.162125	−0.0810623	+	0.996709i	$0.525831\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8099.98	0.390448
$756$	0	0
$757$	33149.5	1.59160	0.795798	−	0.605562i	$-0.207052\pi$
$757$	33149.5	1.59160	0.795798	+	0.605562i	$0.207052\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10996.8	−0.523831	−0.261915	−	0.965091i	$-0.584354\pi$
$761$	−10996.8	−0.523831	−0.261915	+	0.965091i	$0.584354\pi$
$762$	0	0
$763$	−2732.15	−0.129634
$764$	0	0
$765$	0	0
$766$	0	0
$767$	406.637	0.0191432
$768$	0	0
$769$	33285.0	1.56084	0.780420	−	0.625255i	$-0.215005\pi$
$769$	33285.0	1.56084	0.780420	+	0.625255i	$0.215005\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7242.46	−0.336990	−0.168495	−	0.985703i	$-0.553891\pi$
$773$	−7242.46	−0.336990	−0.168495	+	0.985703i	$0.553891\pi$
$774$	0	0
$775$	3119.63	0.144594
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40326.0	1.85472
$780$	0	0
$781$	11000.0	0.503981
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10081.8	0.458389
$786$	0	0
$787$	−17414.3	−0.788758	−0.394379	−	0.918948i	$-0.629040\pi$
$787$	−17414.3	−0.788758	−0.394379	+	0.918948i	$0.629040\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8373.50	0.376394
$792$	0	0
$793$	3988.26	0.178597
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29132.9	1.29478	0.647391	−	0.762158i	$-0.275860\pi$
$797$	29132.9	1.29478	0.647391	+	0.762158i	$0.275860\pi$
$798$	0	0
$799$	−34727.4	−1.53763
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4714.88	−0.207203
$804$	0	0
$805$	−2134.90	−0.0934725
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36440.1	−1.58364	−0.791820	−	0.610754i	$-0.790866\pi$
$809$	−36440.1	−1.58364	−0.791820	+	0.610754i	$0.790866\pi$
$810$	0	0
$811$	18922.0	0.819286	0.409643	−	0.912246i	$-0.365653\pi$
$811$	18922.0	0.819286	0.409643	+	0.912246i	$0.365653\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10127.4	−0.435271
$816$	0	0
$817$	−3106.27	−0.133017
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17910.7	−0.761374	−0.380687	−	0.924704i	$-0.624312\pi$
$821$	−17910.7	−0.761374	−0.380687	+	0.924704i	$0.624312\pi$
$822$	0	0
$823$	−17524.4	−0.742240	−0.371120	−	0.928585i	$-0.621026\pi$
$823$	−17524.4	−0.742240	−0.371120	+	0.928585i	$0.621026\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17643.9	0.741885	0.370943	−	0.928656i	$-0.379035\pi$
$827$	17643.9	0.741885	0.370943	+	0.928656i	$0.379035\pi$
$828$	0	0
$829$	45178.6	1.89278	0.946391	−	0.323023i	$-0.104699\pi$
$829$	45178.6	1.89278	0.946391	+	0.323023i	$0.104699\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14782.4	−0.614863
$834$	0	0
$835$	−8890.03	−0.368446
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26776.0	−1.10180	−0.550901	−	0.834571i	$-0.685716\pi$
$839$	−26776.0	−1.10180	−0.550901	+	0.834571i	$0.685716\pi$
$840$	0	0
$841$	−23877.0	−0.979006
$842$	0	0
$843$	0	0
$844$	0	0
$845$	756.291	0.0307896
$846$	0	0
$847$	14910.6	0.604879
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4953.46	0.199533
$852$	0	0
$853$	−7911.75	−0.317577	−0.158789	−	0.987313i	$-0.550759\pi$
$853$	−7911.75	−0.317577	−0.158789	+	0.987313i	$0.550759\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1848.30	0.0736717	0.0368359	−	0.999321i	$-0.488272\pi$
$857$	1848.30	0.0736717	0.0368359	+	0.999321i	$0.488272\pi$
$858$	0	0
$859$	18854.2	0.748890	0.374445	−	0.927249i	$-0.377833\pi$
$859$	18854.2	0.748890	0.374445	+	0.927249i	$0.377833\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2086.03	−0.0822821	−0.0411410	−	0.999153i	$-0.513099\pi$
$863$	−2086.03	−0.0822821	−0.0411410	+	0.999153i	$0.513099\pi$
$864$	0	0
$865$	14669.1	0.576605
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4870.92	0.190143
$870$	0	0
$871$	33210.7	1.29197
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12817.5	0.495211
$876$	0	0
$877$	−24777.2	−0.954010	−0.477005	−	0.878901i	$-0.658278\pi$
$877$	−24777.2	−0.954010	−0.477005	+	0.878901i	$0.658278\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3741.26	0.143072	0.0715359	−	0.997438i	$-0.477210\pi$
$881$	3741.26	0.143072	0.0715359	+	0.997438i	$0.477210\pi$
$882$	0	0
$883$	−14131.6	−0.538580	−0.269290	−	0.963059i	$-0.586789\pi$
$883$	−14131.6	−0.538580	−0.269290	+	0.963059i	$0.586789\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26622.5	−1.00778	−0.503888	−	0.863769i	$-0.668097\pi$
$887$	−26622.5	−1.00778	−0.503888	+	0.863769i	$0.668097\pi$
$888$	0	0
$889$	−10855.4	−0.409539
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−53485.6	−2.00429
$894$	0	0
$895$	−6334.34	−0.236574
$896$	0	0
$897$	0	0
$898$	0	0
$899$	681.475	0.0252820
$900$	0	0
$901$	−33026.9	−1.22118
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18286.2	−0.671661
$906$	0	0
$907$	−52619.4	−1.92635	−0.963174	−	0.268878i	$-0.913347\pi$
$907$	−52619.4	−1.92635	−0.963174	+	0.268878i	$0.913347\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10180.4	−0.370245	−0.185122	−	0.982715i	$-0.559268\pi$
$911$	−10180.4	−0.370245	−0.185122	+	0.982715i	$0.559268\pi$
$912$	0	0
$913$	993.816	0.0360246
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20051.6	0.722097
$918$	0	0
$919$	45618.2	1.63744	0.818718	−	0.574195i	$-0.194685\pi$
$919$	45618.2	1.63744	0.818718	+	0.574195i	$0.194685\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	53326.5	1.90169
$924$	0	0
$925$	−13476.6	−0.479037
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13200.4	−0.466190	−0.233095	−	0.972454i	$-0.574885\pi$
$929$	−13200.4	−0.466190	−0.233095	+	0.972454i	$0.574885\pi$
$930$	0	0
$931$	−22767.2	−0.801468
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3494.59	−0.122230
$936$	0	0
$937$	−13468.1	−0.469565	−0.234783	−	0.972048i	$-0.575438\pi$
$937$	−13468.1	−0.469565	−0.234783	+	0.972048i	$0.575438\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13884.1	0.480987	0.240493	−	0.970651i	$-0.422691\pi$
$941$	13884.1	0.480987	0.240493	+	0.970651i	$0.422691\pi$
$942$	0	0
$943$	13229.9	0.456865
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5276.64	−0.181064	−0.0905320	−	0.995894i	$-0.528857\pi$
$947$	−5276.64	−0.181064	−0.0905320	+	0.995894i	$0.528857\pi$
$948$	0	0
$949$	−22857.1	−0.781849
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26131.4	−0.888225	−0.444112	−	0.895971i	$-0.646481\pi$
$953$	−26131.4	−0.888225	−0.444112	+	0.895971i	$0.646481\pi$
$954$	0	0
$955$	11118.5	0.376740
$956$	0	0
$957$	0	0
$958$	0	0
$959$	32122.9	1.08165
$960$	0	0
$961$	−28884.0	−0.969554
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6178.33	−0.206101
$966$	0	0
$967$	3997.53	0.132939	0.0664695	−	0.997788i	$-0.478827\pi$
$967$	3997.53	0.132939	0.0664695	+	0.997788i	$0.478827\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41785.1	−1.38100	−0.690499	−	0.723334i	$-0.742609\pi$
$971$	−41785.1	−1.38100	−0.690499	+	0.723334i	$0.742609\pi$
$972$	0	0
$973$	7684.12	0.253177
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12245.1	0.400978	0.200489	−	0.979696i	$-0.435747\pi$
$977$	12245.1	0.400978	0.200489	+	0.979696i	$0.435747\pi$
$978$	0	0
$979$	−88.2340	−0.00288046
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43783.1	−1.42061	−0.710307	−	0.703892i	$-0.751444\pi$
$983$	−43783.1	−1.42061	−0.710307	+	0.703892i	$0.751444\pi$
$984$	0	0
$985$	−12175.0	−0.393835
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1019.08	−0.0327654
$990$	0	0
$991$	−5178.38	−0.165991	−0.0829953	−	0.996550i	$-0.526449\pi$
$991$	−5178.38	−0.165991	−0.0829953	+	0.996550i	$0.526449\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11464.4	−0.365274
$996$	0	0
$997$	25720.9	0.817041	0.408520	−	0.912749i	$-0.366045\pi$
$997$	25720.9	0.817041	0.408520	+	0.912749i	$0.366045\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.u.1.1		2
3.2	odd	2		1296.4.a.i.1.2		2
4.3	odd	2		81.4.a.d.1.2		2
9.2	odd	6		432.4.i.c.145.1		4
9.4	even	3		144.4.i.c.97.1		4
9.5	odd	6		432.4.i.c.289.1		4
9.7	even	3		144.4.i.c.49.1		4
12.11	even	2		81.4.a.a.1.1		2
20.19	odd	2		2025.4.a.g.1.1		2
36.7	odd	6		9.4.c.a.4.1	✓	4
36.11	even	6		27.4.c.a.10.2		4
36.23	even	6		27.4.c.a.19.2		4
36.31	odd	6		9.4.c.a.7.1	yes	4
60.59	even	2		2025.4.a.n.1.2		2
180.7	even	12		225.4.k.b.49.4		8
180.43	even	12		225.4.k.b.49.1		8
180.67	even	12		225.4.k.b.124.1		8
180.79	odd	6		225.4.e.b.76.2		4
180.103	even	12		225.4.k.b.124.4		8
180.139	odd	6		225.4.e.b.151.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.4.c.a.4.1	✓	4	36.7	odd	6
9.4.c.a.7.1	yes	4	36.31	odd	6
27.4.c.a.10.2		4	36.11	even	6
27.4.c.a.19.2		4	36.23	even	6
81.4.a.a.1.1		2	12.11	even	2
81.4.a.d.1.2		2	4.3	odd	2
144.4.i.c.49.1		4	9.7	even	3
144.4.i.c.97.1		4	9.4	even	3
225.4.e.b.76.2		4	180.79	odd	6
225.4.e.b.151.2		4	180.139	odd	6
225.4.k.b.49.1		8	180.43	even	12
225.4.k.b.49.4		8	180.7	even	12
225.4.k.b.124.1		8	180.67	even	12
225.4.k.b.124.4		8	180.103	even	12
432.4.i.c.145.1		4	9.2	odd	6
432.4.i.c.289.1		4	9.5	odd	6
1296.4.a.i.1.2		2	3.2	odd	2
1296.4.a.u.1.1		2	1.1	even	1	trivial
2025.4.a.g.1.1		2	20.19	odd	2
2025.4.a.n.1.2		2	60.59	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q+4.62772 q^{5} -12.1168 q^{7} +O(q^{10})\)	\(q+4.62772 q^{5} -12.1168 q^{7} -10.0217 q^{11} -48.5842 q^{13} +75.3505 q^{17} +116.052 q^{19} +38.0733 q^{23} -103.584 q^{25} -22.6277 q^{29} -30.1168 q^{31} -56.0733 q^{35} +130.103 q^{37} +347.484 q^{41} -26.7663 q^{43} -460.878 q^{47} -196.182 q^{49} -438.310 q^{53} -46.3778 q^{55} -8.36974 q^{59} -82.0895 q^{61} -224.834 q^{65} -683.571 q^{67} -1097.61 q^{71} +470.464 q^{73} +121.432 q^{77} -486.035 q^{79} -99.1659 q^{83} +348.701 q^{85} +8.80426 q^{89} +588.687 q^{91} +537.054 q^{95} +660.973 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 15 q^{5} - 7 q^{7}+O(q^{10}) \) 2 * q + 15 * q^5 - 7 * q^7	\( 2 q + 15 q^{5} - 7 q^{7} - 66 q^{11} - 11 q^{13} + 99 q^{17} + 77 q^{19} - 33 q^{23} - 121 q^{25} - 51 q^{29} - 43 q^{31} - 3 q^{35} - 50 q^{37} + 132 q^{41} - 88 q^{43} - 399 q^{47} - 513 q^{49} + 54 q^{53} - 627 q^{55} - 798 q^{59} + 439 q^{61} + 165 q^{65} - 988 q^{67} - 1368 q^{71} - 455 q^{73} - 165 q^{77} + 803 q^{79} - 813 q^{83} + 594 q^{85} - 396 q^{89} + 781 q^{91} + 132 q^{95} + 736 q^{97}+O(q^{100}) \) 2 * q + 15 * q^5 - 7 * q^7 - 66 * q^11 - 11 * q^13 + 99 * q^17 + 77 * q^19 - 33 * q^23 - 121 * q^25 - 51 * q^29 - 43 * q^31 - 3 * q^35 - 50 * q^37 + 132 * q^41 - 88 * q^43 - 399 * q^47 - 513 * q^49 + 54 * q^53 - 627 * q^55 - 798 * q^59 + 439 * q^61 + 165 * q^65 - 988 * q^67 - 1368 * q^71 - 455 * q^73 - 165 * q^77 + 803 * q^79 - 813 * q^83 + 594 * q^85 - 396 * q^89 + 781 * q^91 + 132 * q^95 + 736 * q^97

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