LMFDB - Embedded newform 1296.4.a.z.1.4

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:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 324)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.456850$ 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)

$f(q)$	$=$	$q+21.0707 q^{5} +31.4955 q^{7} +O(q^{10})$	$q+21.0707 q^{5} +31.4955 q^{7} -36.6591 q^{11} +56.4955 q^{13} -35.8010 q^{17} -83.4955 q^{19} +69.5523 q^{23} +318.973 q^{25} +81.7082 q^{29} +72.9727 q^{31} +663.630 q^{35} -25.4682 q^{37} +399.484 q^{41} +83.4591 q^{43} -311.769 q^{47} +648.964 q^{49} -4.09919 q^{53} -772.432 q^{55} -352.194 q^{59} +3.54091 q^{61} +1190.40 q^{65} -492.432 q^{67} +154.502 q^{71} +305.000 q^{73} -1154.60 q^{77} -671.386 q^{79} +1293.70 q^{83} -754.350 q^{85} -1183.91 q^{89} +1779.35 q^{91} -1759.30 q^{95} +615.982 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7}+O(q^{10}) $ 4 * q + 16 * q^7	$ 4 q + 16 q^{7} + 116 q^{13} - 224 q^{19} + 616 q^{25} - 368 q^{31} + 668 q^{37} - 656 q^{43} + 1716 q^{49} - 1440 q^{55} + 1004 q^{61} - 320 q^{67} + 1220 q^{73} + 64 q^{79} + 612 q^{85} + 3488 q^{91} + 2024 q^{97}+O(q^{100}) $ 4 * q + 16 * q^7 + 116 * q^13 - 224 * q^19 + 616 * q^25 - 368 * q^31 + 668 * q^37 - 656 * q^43 + 1716 * q^49 - 1440 * q^55 + 1004 * q^61 - 320 * q^67 + 1220 * q^73 + 64 * q^79 + 612 * q^85 + 3488 * q^91 + 2024 * 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
$5$	21.0707	1.88462	0.942309	−	0.334746i	$-0.108650\pi$
$5$	21.0707	1.88462	0.942309	+	0.334746i	$0.108650\pi$
$6$	0	0
$7$	31.4955	1.70059	0.850297	−	0.526303i	$-0.176422\pi$
$7$	31.4955	1.70059	0.850297	+	0.526303i	$0.176422\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−36.6591	−1.00483	−0.502415	−	0.864626i	$-0.667555\pi$
$11$	−36.6591	−1.00483	−0.502415	+	0.864626i	$0.667555\pi$
$12$	0	0
$13$	56.4955	1.20531	0.602655	−	0.798002i	$-0.294110\pi$
$13$	56.4955	1.20531	0.602655	+	0.798002i	$0.294110\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−35.8010	−0.510765	−0.255383	−	0.966840i	$-0.582201\pi$
$17$	−35.8010	−0.510765	−0.255383	+	0.966840i	$0.582201\pi$
$18$	0	0
$19$	−83.4955	−1.00817	−0.504083	−	0.863655i	$-0.668170\pi$
$19$	−83.4955	−1.00817	−0.504083	+	0.863655i	$0.668170\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	69.5523	0.630551	0.315275	−	0.949000i	$-0.397903\pi$
$23$	69.5523	0.630551	0.315275	+	0.949000i	$0.397903\pi$
$24$	0	0
$25$	318.973	2.55178
$26$	0	0
$27$	0	0
$28$	0	0
$29$	81.7082	0.523201	0.261601	−	0.965176i	$-0.415750\pi$
$29$	81.7082	0.523201	0.261601	+	0.965176i	$0.415750\pi$
$30$	0	0
$31$	72.9727	0.422783	0.211392	−	0.977401i	$-0.432200\pi$
$31$	72.9727	0.422783	0.211392	+	0.977401i	$0.432200\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	663.630	3.20497
$36$	0	0
$37$	−25.4682	−0.113161	−0.0565803	−	0.998398i	$-0.518020\pi$
$37$	−25.4682	−0.113161	−0.0565803	+	0.998398i	$0.518020\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	399.484	1.52168	0.760841	−	0.648938i	$-0.224787\pi$
$41$	399.484	1.52168	0.760841	+	0.648938i	$0.224787\pi$
$42$	0	0
$43$	83.4591	0.295986	0.147993	−	0.988988i	$-0.452719\pi$
$43$	83.4591	0.295986	0.147993	+	0.988988i	$0.452719\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−311.769	−0.967579	−0.483789	−	0.875184i	$-0.660740\pi$
$47$	−311.769	−0.967579	−0.483789	+	0.875184i	$0.660740\pi$
$48$	0	0
$49$	648.964	1.89202
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.09919	−0.0106239	−0.00531196	−	0.999986i	$-0.501691\pi$
$53$	−4.09919	−0.0106239	−0.00531196	+	0.999986i	$0.501691\pi$
$54$	0	0
$55$	−772.432	−1.89372
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−352.194	−0.777149	−0.388574	−	0.921417i	$-0.627032\pi$
$59$	−352.194	−0.777149	−0.388574	+	0.921417i	$0.627032\pi$
$60$	0	0
$61$	3.54091	0.00743225	0.00371613	−	0.999993i	$-0.498817\pi$
$61$	3.54091	0.00743225	0.00371613	+	0.999993i	$0.498817\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1190.40	2.27155
$66$	0	0
$67$	−492.432	−0.897912	−0.448956	−	0.893554i	$-0.648204\pi$
$67$	−492.432	−0.897912	−0.448956	+	0.893554i	$0.648204\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	154.502	0.258253	0.129126	−	0.991628i	$-0.458783\pi$
$71$	154.502	0.258253	0.129126	+	0.991628i	$0.458783\pi$
$72$	0	0
$73$	305.000	0.489008	0.244504	−	0.969648i	$-0.421375\pi$
$73$	305.000	0.489008	0.244504	+	0.969648i	$0.421375\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1154.60	−1.70881
$78$	0	0
$79$	−671.386	−0.956163	−0.478081	−	0.878316i	$-0.658668\pi$
$79$	−671.386	−0.956163	−0.478081	+	0.878316i	$0.658668\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1293.70	1.71087	0.855434	−	0.517912i	$-0.173290\pi$
$83$	1293.70	1.71087	0.855434	+	0.517912i	$0.173290\pi$
$84$	0	0
$85$	−754.350	−0.962597
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1183.91	−1.41005	−0.705026	−	0.709181i	$-0.749065\pi$
$89$	−1183.91	−1.41005	−0.705026	+	0.709181i	$0.749065\pi$
$90$	0	0
$91$	1779.35	2.04974
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1759.30	−1.90001
$96$	0	0
$97$	615.982	0.644778	0.322389	−	0.946607i	$-0.395514\pi$
$97$	615.982	0.644778	0.322389	+	0.946607i	$0.395514\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1187.49	−1.16990	−0.584948	−	0.811071i	$-0.698885\pi$
$101$	−1187.49	−1.16990	−0.584948	+	0.811071i	$0.698885\pi$
$102$	0	0
$103$	1386.92	1.32677	0.663384	−	0.748279i	$-0.269120\pi$
$103$	1386.92	1.32677	0.663384	+	0.748279i	$0.269120\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−678.360	−0.612893	−0.306447	−	0.951888i	$-0.599140\pi$
$107$	−678.360	−0.612893	−0.306447	+	0.951888i	$0.599140\pi$
$108$	0	0
$109$	945.432	0.830788	0.415394	−	0.909641i	$-0.363644\pi$
$109$	945.432	0.830788	0.415394	+	0.909641i	$0.363644\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1986.60	−1.65384	−0.826918	−	0.562322i	$-0.809908\pi$
$113$	−1986.60	−1.65384	−0.826918	+	0.562322i	$0.809908\pi$
$114$	0	0
$115$	1465.51	1.18835
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1127.57	−0.868605
$120$	0	0
$121$	12.8909	0.00968512
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4087.13	2.92451
$126$	0	0
$127$	1496.54	1.04564	0.522821	−	0.852442i	$-0.324880\pi$
$127$	1496.54	1.04564	0.522821	+	0.852442i	$0.324880\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−85.2825	−0.0568791	−0.0284396	−	0.999596i	$-0.509054\pi$
$131$	−85.2825	−0.0568791	−0.0284396	+	0.999596i	$0.509054\pi$
$132$	0	0
$133$	−2629.73	−1.71448
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−899.506	−0.560949	−0.280475	−	0.959861i	$-0.590492\pi$
$137$	−899.506	−0.560949	−0.280475	+	0.959861i	$0.590492\pi$
$138$	0	0
$139$	−2000.61	−1.22079	−0.610394	−	0.792098i	$-0.708989\pi$
$139$	−2000.61	−1.22079	−0.610394	+	0.792098i	$0.708989\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2071.07	−1.21113
$144$	0	0
$145$	1721.65	0.986034
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1474.37	−0.810639	−0.405320	−	0.914175i	$-0.632840\pi$
$149$	−1474.37	−0.810639	−0.405320	+	0.914175i	$0.632840\pi$
$150$	0	0
$151$	1092.97	0.589039	0.294519	−	0.955646i	$-0.404840\pi$
$151$	1092.97	0.589039	0.294519	+	0.955646i	$0.404840\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1537.58	0.796785
$156$	0	0
$157$	540.514	0.274762	0.137381	−	0.990518i	$-0.456131\pi$
$157$	540.514	0.274762	0.137381	+	0.990518i	$0.456131\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2190.58	1.07231
$162$	0	0
$163$	3251.60	1.56248	0.781242	−	0.624228i	$-0.214586\pi$
$163$	3251.60	1.56248	0.781242	+	0.624228i	$0.214586\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1708.58	−0.791701	−0.395850	−	0.918315i	$-0.629550\pi$
$167$	−1708.58	−0.791701	−0.395850	+	0.918315i	$0.629550\pi$
$168$	0	0
$169$	994.736	0.452770
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2629.73	−1.15569	−0.577847	−	0.816145i	$-0.696107\pi$
$173$	−2629.73	−1.15569	−0.577847	+	0.816145i	$0.696107\pi$
$174$	0	0
$175$	10046.2	4.33955
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4532.77	1.89271	0.946354	−	0.323131i	$-0.104735\pi$
$179$	4532.77	1.89271	0.946354	+	0.323131i	$0.104735\pi$
$180$	0	0
$181$	−2327.65	−0.955874	−0.477937	−	0.878394i	$-0.658615\pi$
$181$	−2327.65	−0.955874	−0.477937	+	0.878394i	$0.658615\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−536.631	−0.213264
$186$	0	0
$187$	1312.43	0.513233
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1231.68	0.466603	0.233302	−	0.972404i	$-0.425047\pi$
$191$	1231.68	0.466603	0.233302	+	0.972404i	$0.425047\pi$
$192$	0	0
$193$	2143.97	0.799619	0.399810	−	0.916598i	$-0.369076\pi$
$193$	2143.97	0.799619	0.399810	+	0.916598i	$0.369076\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1984.55	0.717733	0.358866	−	0.933389i	$-0.383163\pi$
$197$	1984.55	0.717733	0.358866	+	0.933389i	$0.383163\pi$
$198$	0	0
$199$	224.400	0.0799362	0.0399681	−	0.999201i	$-0.487274\pi$
$199$	224.400	0.0799362	0.0399681	+	0.999201i	$0.487274\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2573.44	0.889753
$204$	0	0
$205$	8417.40	2.86779
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3060.87	1.01304
$210$	0	0
$211$	−3785.22	−1.23500	−0.617501	−	0.786570i	$-0.711855\pi$
$211$	−3785.22	−1.23500	−0.617501	+	0.786570i	$0.711855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1758.54	0.557820
$216$	0	0
$217$	2298.31	0.718983
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2022.59	−0.615630
$222$	0	0
$223$	5408.32	1.62407	0.812036	−	0.583607i	$-0.198359\pi$
$223$	5408.32	1.62407	0.812036	+	0.583607i	$0.198359\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4489.91	1.31280	0.656400	−	0.754413i	$-0.272078\pi$
$227$	4489.91	1.31280	0.656400	+	0.754413i	$0.272078\pi$
$228$	0	0
$229$	3117.00	0.899462	0.449731	−	0.893164i	$-0.351520\pi$
$229$	3117.00	0.899462	0.449731	+	0.893164i	$0.351520\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1675.60	0.471124	0.235562	−	0.971859i	$-0.424307\pi$
$233$	1675.60	0.471124	0.235562	+	0.971859i	$0.424307\pi$
$234$	0	0
$235$	−6569.18	−1.82352
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5751.72	−1.55668	−0.778342	−	0.627840i	$-0.783939\pi$
$239$	−5751.72	−1.55668	−0.778342	+	0.627840i	$0.783939\pi$
$240$	0	0
$241$	−1805.41	−0.482559	−0.241279	−	0.970456i	$-0.577567\pi$
$241$	−1805.41	−0.482559	−0.241279	+	0.970456i	$0.577567\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	13674.1	3.56574
$246$	0	0
$247$	−4717.11	−1.21515
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−32.6596	−0.00821297	−0.00410649	−	0.999992i	$-0.501307\pi$
$251$	−32.6596	−0.00821297	−0.00410649	+	0.999992i	$0.501307\pi$
$252$	0	0
$253$	−2549.73	−0.633597
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1148.26	−0.278701	−0.139351	−	0.990243i	$-0.544501\pi$
$257$	−1148.26	−0.278701	−0.139351	+	0.990243i	$0.544501\pi$
$258$	0	0
$259$	−802.132	−0.192440
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4634.01	−1.08648	−0.543242	−	0.839576i	$-0.682804\pi$
$263$	−4634.01	−1.08648	−0.543242	+	0.839576i	$0.682804\pi$
$264$	0	0
$265$	−86.3727	−0.0200220
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8134.42	−1.84373	−0.921866	−	0.387508i	$-0.873336\pi$
$269$	−8134.42	−1.84373	−0.921866	+	0.387508i	$0.873336\pi$
$270$	0	0
$271$	−6891.35	−1.54472	−0.772361	−	0.635184i	$-0.780924\pi$
$271$	−6891.35	−1.54472	−0.772361	+	0.635184i	$0.780924\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11693.3	−2.56411
$276$	0	0
$277$	−1821.78	−0.395163	−0.197582	−	0.980286i	$-0.563309\pi$
$277$	−1821.78	−0.395163	−0.197582	+	0.980286i	$0.563309\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4541.21	−0.964077	−0.482039	−	0.876150i	$-0.660103\pi$
$281$	−4541.21	−0.964077	−0.482039	+	0.876150i	$0.660103\pi$
$282$	0	0
$283$	2757.95	0.579303	0.289652	−	0.957132i	$-0.406461\pi$
$283$	2757.95	0.579303	0.289652	+	0.957132i	$0.406461\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12581.9	2.58777
$288$	0	0
$289$	−3631.29	−0.739119
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1669.06	0.332791	0.166396	−	0.986059i	$-0.446787\pi$
$293$	1669.06	0.332791	0.166396	+	0.986059i	$0.446787\pi$
$294$	0	0
$295$	−7420.96	−1.46463
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3929.39	0.760008
$300$	0	0
$301$	2628.58	0.503352
$302$	0	0
$303$	0	0
$304$	0	0
$305$	74.6094	0.0140069
$306$	0	0
$307$	7500.21	1.39433	0.697165	−	0.716910i	$-0.254444\pi$
$307$	7500.21	1.39433	0.697165	+	0.716910i	$0.254444\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8486.52	1.54735	0.773676	−	0.633582i	$-0.218416\pi$
$311$	8486.52	1.54735	0.773676	+	0.633582i	$0.218416\pi$
$312$	0	0
$313$	9365.66	1.69131	0.845653	−	0.533734i	$-0.179211\pi$
$313$	9365.66	1.69131	0.845653	+	0.533734i	$0.179211\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−505.887	−0.0896324	−0.0448162	−	0.998995i	$-0.514270\pi$
$317$	−505.887	−0.0896324	−0.0448162	+	0.998995i	$0.514270\pi$
$318$	0	0
$319$	−2995.35	−0.525729
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2989.22	0.514937
$324$	0	0
$325$	18020.5	3.07569
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9819.31	−1.64546
$330$	0	0
$331$	−6598.51	−1.09573	−0.547866	−	0.836566i	$-0.684560\pi$
$331$	−6598.51	−1.09573	−0.547866	+	0.836566i	$0.684560\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10375.9	−1.69222
$336$	0	0
$337$	−3612.42	−0.583920	−0.291960	−	0.956431i	$-0.594307\pi$
$337$	−3612.42	−0.583920	−0.291960	+	0.956431i	$0.594307\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2675.12	−0.424826
$342$	0	0
$343$	9636.46	1.51697
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2609.56	−0.403714	−0.201857	−	0.979415i	$-0.564698\pi$
$347$	−2609.56	−0.403714	−0.201857	+	0.979415i	$0.564698\pi$
$348$	0	0
$349$	677.527	0.103917	0.0519587	−	0.998649i	$-0.483454\pi$
$349$	677.527	0.103917	0.0519587	+	0.998649i	$0.483454\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1549.88	0.233688	0.116844	−	0.993150i	$-0.462722\pi$
$353$	1549.88	0.233688	0.116844	+	0.993150i	$0.462722\pi$
$354$	0	0
$355$	3255.45	0.486708
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8381.31	−1.23217	−0.616084	−	0.787680i	$-0.711282\pi$
$359$	−8381.31	−1.23217	−0.616084	+	0.787680i	$0.711282\pi$
$360$	0	0
$361$	112.491	0.0164005
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6426.55	0.921592
$366$	0	0
$367$	4205.83	0.598208	0.299104	−	0.954220i	$-0.403312\pi$
$367$	4205.83	0.598208	0.299104	+	0.954220i	$0.403312\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−129.106	−0.0180670
$372$	0	0
$373$	8012.84	1.11230	0.556151	−	0.831081i	$-0.312277\pi$
$373$	8012.84	1.11230	0.556151	+	0.831081i	$0.312277\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4616.14	0.630619
$378$	0	0
$379$	−8049.26	−1.09093	−0.545466	−	0.838133i	$-0.683647\pi$
$379$	−8049.26	−1.09093	−0.545466	+	0.838133i	$0.683647\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3227.47	0.430590	0.215295	−	0.976549i	$-0.430929\pi$
$383$	3227.47	0.430590	0.215295	+	0.976549i	$0.430929\pi$
$384$	0	0
$385$	−24328.1	−3.22045
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8911.93	−1.16158	−0.580788	−	0.814055i	$-0.697255\pi$
$389$	−8911.93	−1.16158	−0.580788	+	0.814055i	$0.697255\pi$
$390$	0	0
$391$	−2490.04	−0.322063
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14146.6	−1.80200
$396$	0	0
$397$	2972.50	0.375783	0.187891	−	0.982190i	$-0.439835\pi$
$397$	2972.50	0.375783	0.187891	+	0.982190i	$0.439835\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12273.2	1.52841	0.764206	−	0.644972i	$-0.223131\pi$
$401$	12273.2	1.52841	0.764206	+	0.644972i	$0.223131\pi$
$402$	0	0
$403$	4122.63	0.509585
$404$	0	0
$405$	0	0
$406$	0	0
$407$	933.641	0.113707
$408$	0	0
$409$	−2249.30	−0.271933	−0.135967	−	0.990713i	$-0.543414\pi$
$409$	−2249.30	−0.271933	−0.135967	+	0.990713i	$0.543414\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11092.5	−1.32161
$414$	0	0
$415$	27259.1	3.22433
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9460.92	−1.10309	−0.551547	−	0.834144i	$-0.685962\pi$
$419$	−9460.92	−1.10309	−0.551547	+	0.834144i	$0.685962\pi$
$420$	0	0
$421$	1534.91	0.177689	0.0888446	−	0.996046i	$-0.471683\pi$
$421$	1534.91	0.177689	0.0888446	+	0.996046i	$0.471683\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11419.5	−1.30336
$426$	0	0
$427$	111.523	0.0126392
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3991.88	−0.446130	−0.223065	−	0.974804i	$-0.571606\pi$
$431$	−3991.88	−0.446130	−0.223065	+	0.974804i	$0.571606\pi$
$432$	0	0
$433$	−3058.08	−0.339404	−0.169702	−	0.985495i	$-0.554281\pi$
$433$	−3058.08	−0.339404	−0.169702	+	0.985495i	$0.554281\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5807.30	−0.635700
$438$	0	0
$439$	14763.8	1.60510	0.802548	−	0.596588i	$-0.203477\pi$
$439$	14763.8	1.60510	0.802548	+	0.596588i	$0.203477\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11564.7	−1.24030	−0.620150	−	0.784483i	$-0.712928\pi$
$443$	−11564.7	−1.24030	−0.620150	+	0.784483i	$0.712928\pi$
$444$	0	0
$445$	−24945.9	−2.65741
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17266.9	1.81486	0.907432	−	0.420198i	$-0.138039\pi$
$449$	17266.9	1.81486	0.907432	+	0.420198i	$0.138039\pi$
$450$	0	0
$451$	−14644.7	−1.52903
$452$	0	0
$453$	0	0
$454$	0	0
$455$	37492.1	3.86298
$456$	0	0
$457$	12701.5	1.30011	0.650057	−	0.759886i	$-0.274745\pi$
$457$	12701.5	1.30011	0.650057	+	0.759886i	$0.274745\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16499.8	−1.66697	−0.833486	−	0.552541i	$-0.813658\pi$
$461$	−16499.8	−1.66697	−0.833486	+	0.552541i	$0.813658\pi$
$462$	0	0
$463$	1364.84	0.136996	0.0684982	−	0.997651i	$-0.478179\pi$
$463$	1364.84	0.136996	0.0684982	+	0.997651i	$0.478179\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10333.5	−1.02393	−0.511966	−	0.859006i	$-0.671083\pi$
$467$	−10333.5	−1.02393	−0.511966	+	0.859006i	$0.671083\pi$
$468$	0	0
$469$	−15509.4	−1.52698
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3059.54	−0.297416
$474$	0	0
$475$	−26632.8	−2.57262
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12188.6	1.16266	0.581328	−	0.813669i	$-0.302533\pi$
$479$	12188.6	1.16266	0.581328	+	0.813669i	$0.302533\pi$
$480$	0	0
$481$	−1438.84	−0.136394
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12979.1	1.21516
$486$	0	0
$487$	−9816.46	−0.913401	−0.456701	−	0.889620i	$-0.650969\pi$
$487$	−9816.46	−0.913401	−0.456701	+	0.889620i	$0.650969\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3464.02	−0.318389	−0.159195	−	0.987247i	$-0.550890\pi$
$491$	−3464.02	−0.318389	−0.159195	+	0.987247i	$0.550890\pi$
$492$	0	0
$493$	−2925.23	−0.267233
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4866.10	0.439184
$498$	0	0
$499$	12366.9	1.10946	0.554728	−	0.832032i	$-0.312822\pi$
$499$	12366.9	1.10946	0.554728	+	0.832032i	$0.312822\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6249.98	0.554021	0.277011	−	0.960867i	$-0.410656\pi$
$503$	6249.98	0.554021	0.277011	+	0.960867i	$0.410656\pi$
$504$	0	0
$505$	−25021.2	−2.20481
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7481.43	−0.651490	−0.325745	−	0.945458i	$-0.605615\pi$
$509$	−7481.43	−0.651490	−0.325745	+	0.945458i	$0.605615\pi$
$510$	0	0
$511$	9606.11	0.831604
$512$	0	0
$513$	0	0
$514$	0	0
$515$	29223.3	2.50045
$516$	0	0
$517$	11429.2	0.972253
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17700.1	−1.48840	−0.744200	−	0.667957i	$-0.767169\pi$
$521$	−17700.1	−1.48840	−0.744200	+	0.667957i	$0.767169\pi$
$522$	0	0
$523$	−21707.9	−1.81495	−0.907475	−	0.420107i	$-0.861993\pi$
$523$	−21707.9	−1.81495	−0.907475	+	0.420107i	$0.861993\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2612.49	−0.215943
$528$	0	0
$529$	−7329.47	−0.602406
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22569.1	1.83410
$534$	0	0
$535$	−14293.5	−1.15507
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23790.4	−1.90116
$540$	0	0
$541$	−19282.0	−1.53235	−0.766173	−	0.642634i	$-0.777841\pi$
$541$	−19282.0	−1.53235	−0.766173	+	0.642634i	$0.777841\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19920.9	1.56572
$546$	0	0
$547$	7784.53	0.608487	0.304243	−	0.952594i	$-0.401596\pi$
$547$	7784.53	0.608487	0.304243	+	0.952594i	$0.401596\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6822.26	−0.527474
$552$	0	0
$553$	−21145.6	−1.62605
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8556.03	0.650863	0.325431	−	0.945566i	$-0.394490\pi$
$557$	8556.03	0.650863	0.325431	+	0.945566i	$0.394490\pi$
$558$	0	0
$559$	4715.06	0.356754
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2963.22	0.221821	0.110910	−	0.993830i	$-0.464623\pi$
$563$	2963.22	0.221821	0.110910	+	0.993830i	$0.464623\pi$
$564$	0	0
$565$	−41859.0	−3.11685
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14484.1	1.06714	0.533571	−	0.845755i	$-0.320850\pi$
$569$	14484.1	1.06714	0.533571	+	0.845755i	$0.320850\pi$
$570$	0	0
$571$	−16649.8	−1.22027	−0.610133	−	0.792299i	$-0.708884\pi$
$571$	−16649.8	−1.22027	−0.610133	+	0.792299i	$0.708884\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	22185.3	1.60903
$576$	0	0
$577$	14355.4	1.03574	0.517872	−	0.855458i	$-0.326724\pi$
$577$	14355.4	1.03574	0.517872	+	0.855458i	$0.326724\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	40745.7	2.90949
$582$	0	0
$583$	150.273	0.0106752
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8006.22	0.562951	0.281475	−	0.959568i	$-0.409176\pi$
$587$	8006.22	0.562951	0.281475	+	0.959568i	$0.409176\pi$
$588$	0	0
$589$	−6092.89	−0.426236
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17866.7	−1.23726	−0.618632	−	0.785681i	$-0.712313\pi$
$593$	−17866.7	−1.23726	−0.618632	+	0.785681i	$0.712313\pi$
$594$	0	0
$595$	−23758.6	−1.63699
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15812.0	1.07857	0.539284	−	0.842124i	$-0.318695\pi$
$599$	15812.0	1.07857	0.539284	+	0.842124i	$0.318695\pi$
$600$	0	0
$601$	22910.3	1.55496	0.777481	−	0.628907i	$-0.216497\pi$
$601$	22910.3	1.55496	0.777481	+	0.628907i	$0.216497\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	271.620	0.0182528
$606$	0	0
$607$	−16.7227	−0.00111821	−0.000559107	−	1.00000i	$-0.500178\pi$
$607$	−16.7227	−0.00111821	−0.000559107		1.00000i	$0.500178\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17613.5	−1.16623
$612$	0	0
$613$	162.745	0.0107230	0.00536152	−	0.999986i	$-0.498293\pi$
$613$	162.745	0.0107230	0.00536152	+	0.999986i	$0.498293\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8893.18	0.580269	0.290134	−	0.956986i	$-0.406300\pi$
$617$	8893.18	0.580269	0.290134	+	0.956986i	$0.406300\pi$
$618$	0	0
$619$	−21557.6	−1.39980	−0.699898	−	0.714242i	$-0.746771\pi$
$619$	−21557.6	−1.39980	−0.699898	+	0.714242i	$0.746771\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−37287.9	−2.39793
$624$	0	0
$625$	46247.0	2.95981
$626$	0	0
$627$	0	0
$628$	0	0
$629$	911.785	0.0577985
$630$	0	0
$631$	−3085.90	−0.194687	−0.0973437	−	0.995251i	$-0.531035\pi$
$631$	−3085.90	−0.194687	−0.0973437	+	0.995251i	$0.531035\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	31533.1	1.97063
$636$	0	0
$637$	36663.5	2.28047
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12566.5	−0.774334	−0.387167	−	0.922010i	$-0.626546\pi$
$641$	−12566.5	−0.774334	−0.387167	+	0.922010i	$0.626546\pi$
$642$	0	0
$643$	−876.227	−0.0537403	−0.0268702	−	0.999639i	$-0.508554\pi$
$643$	−876.227	−0.0537403	−0.0268702	+	0.999639i	$0.508554\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11125.4	0.676022	0.338011	−	0.941142i	$-0.390246\pi$
$647$	11125.4	0.676022	0.338011	+	0.941142i	$0.390246\pi$
$648$	0	0
$649$	12911.1	0.780903
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20660.8	−1.23816	−0.619080	−	0.785328i	$-0.712494\pi$
$653$	−20660.8	−1.23816	−0.619080	+	0.785328i	$0.712494\pi$
$654$	0	0
$655$	−1796.96	−0.107195
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3653.75	0.215978	0.107989	−	0.994152i	$-0.465559\pi$
$659$	3653.75	0.215978	0.107989	+	0.994152i	$0.465559\pi$
$660$	0	0
$661$	−13340.8	−0.785016	−0.392508	−	0.919749i	$-0.628392\pi$
$661$	−13340.8	−0.785016	−0.392508	+	0.919749i	$0.628392\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−55410.1	−3.23114
$666$	0	0
$667$	5683.00	0.329905
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−129.807	−0.00746816
$672$	0	0
$673$	−4221.95	−0.241819	−0.120910	−	0.992664i	$-0.538581\pi$
$673$	−4221.95	−0.241819	−0.120910	+	0.992664i	$0.538581\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18682.1	−1.06058	−0.530288	−	0.847818i	$-0.677916\pi$
$677$	−18682.1	−1.06058	−0.530288	+	0.847818i	$0.677916\pi$
$678$	0	0
$679$	19400.6	1.09651
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18674.1	1.04619	0.523093	−	0.852275i	$-0.324778\pi$
$683$	18674.1	1.04619	0.523093	+	0.852275i	$0.324778\pi$
$684$	0	0
$685$	−18953.2	−1.05717
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−231.586	−0.0128051
$690$	0	0
$691$	−35528.3	−1.95595	−0.977974	−	0.208724i	$-0.933069\pi$
$691$	−35528.3	−1.95595	−0.977974	+	0.208724i	$0.933069\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−42154.2	−2.30072
$696$	0	0
$697$	−14301.9	−0.777223
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13722.2	−0.739346	−0.369673	−	0.929162i	$-0.620530\pi$
$701$	−13722.2	−0.739346	−0.369673	+	0.929162i	$0.620530\pi$
$702$	0	0
$703$	2126.48	0.114085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−37400.5	−1.98952
$708$	0	0
$709$	2952.83	0.156412	0.0782059	−	0.996937i	$-0.475081\pi$
$709$	2952.83	0.156412	0.0782059	+	0.996937i	$0.475081\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5075.42	0.266586
$714$	0	0
$715$	−43638.9	−2.28252
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8101.57	0.420219	0.210109	−	0.977678i	$-0.432618\pi$
$719$	8101.57	0.420219	0.210109	+	0.977678i	$0.432618\pi$
$720$	0	0
$721$	43681.6	2.25629
$722$	0	0
$723$	0	0
$724$	0	0
$725$	26062.7	1.33509
$726$	0	0
$727$	−3400.44	−0.173474	−0.0867368	−	0.996231i	$-0.527644\pi$
$727$	−3400.44	−0.173474	−0.0867368	+	0.996231i	$0.527644\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2987.92	−0.151179
$732$	0	0
$733$	−20414.0	−1.02866	−0.514330	−	0.857592i	$-0.671959\pi$
$733$	−20414.0	−1.02866	−0.514330	+	0.857592i	$0.671959\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18052.1	0.902250
$738$	0	0
$739$	9505.82	0.473176	0.236588	−	0.971610i	$-0.423971\pi$
$739$	9505.82	0.473176	0.236588	+	0.971610i	$0.423971\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	37.1606	0.00183484	0.000917422	−	1.00000i	$-0.499708\pi$
$743$	37.1606	0.00183484	0.000917422		1.00000i	$0.499708\pi$
$744$	0	0
$745$	−31066.0	−1.52774
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−21365.3	−1.04228
$750$	0	0
$751$	−10177.8	−0.494532	−0.247266	−	0.968948i	$-0.579532\pi$
$751$	−10177.8	−0.494532	−0.247266	+	0.968948i	$0.579532\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23029.7	1.11011
$756$	0	0
$757$	17129.3	0.822425	0.411213	−	0.911539i	$-0.365105\pi$
$757$	17129.3	0.822425	0.411213	+	0.911539i	$0.365105\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22937.8	1.09263	0.546317	−	0.837579i	$-0.316030\pi$
$761$	22937.8	1.09263	0.546317	+	0.837579i	$0.316030\pi$
$762$	0	0
$763$	29776.8	1.41283
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19897.4	−0.936705
$768$	0	0
$769$	5270.15	0.247135	0.123567	−	0.992336i	$-0.460566\pi$
$769$	5270.15	0.247135	0.123567	+	0.992336i	$0.460566\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.1847	0.00145102	0.000725509	−	1.00000i	$-0.499769\pi$
$773$	31.1847	0.00145102	0.000725509		1.00000i	$0.499769\pi$
$774$	0	0
$775$	23276.3	1.07885
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−33355.1	−1.53411
$780$	0	0
$781$	−5663.89	−0.259501
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11389.0	0.517822
$786$	0	0
$787$	−23063.1	−1.04461	−0.522307	−	0.852757i	$-0.674929\pi$
$787$	−23063.1	−1.04461	−0.522307	+	0.852757i	$0.674929\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−62568.8	−2.81251
$792$	0	0
$793$	200.045	0.00895816
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34044.1	−1.51305	−0.756527	−	0.653963i	$-0.773105\pi$
$797$	−34044.1	−1.51305	−0.756527	+	0.653963i	$0.773105\pi$
$798$	0	0
$799$	11161.6	0.494206
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11181.0	−0.491370
$804$	0	0
$805$	46157.0	2.02090
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16017.6	−0.696103	−0.348052	−	0.937475i	$-0.613157\pi$
$809$	−16017.6	−0.696103	−0.348052	+	0.937475i	$0.613157\pi$
$810$	0	0
$811$	−15085.2	−0.653160	−0.326580	−	0.945170i	$-0.605896\pi$
$811$	−15085.2	−0.653160	−0.326580	+	0.945170i	$0.605896\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	68513.4	2.94468
$816$	0	0
$817$	−6968.45	−0.298403
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10695.3	−0.454651	−0.227325	−	0.973819i	$-0.572998\pi$
$821$	−10695.3	−0.454651	−0.227325	+	0.973819i	$0.572998\pi$
$822$	0	0
$823$	29383.7	1.24454	0.622268	−	0.782804i	$-0.286211\pi$
$823$	29383.7	1.24454	0.622268	+	0.782804i	$0.286211\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5402.42	−0.227159	−0.113580	−	0.993529i	$-0.536232\pi$
$827$	−5402.42	−0.227159	−0.113580	+	0.993529i	$0.536232\pi$
$828$	0	0
$829$	25212.9	1.05631	0.528155	−	0.849148i	$-0.322884\pi$
$829$	25212.9	1.05631	0.528155	+	0.849148i	$0.322884\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−23233.5	−0.966379
$834$	0	0
$835$	−36000.9	−1.49205
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30316.7	−1.24750	−0.623748	−	0.781625i	$-0.714391\pi$
$839$	−30316.7	−1.24750	−0.623748	+	0.781625i	$0.714391\pi$
$840$	0	0
$841$	−17712.8	−0.726261
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20959.8	0.853299
$846$	0	0
$847$	406.005	0.0164705
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1771.37	−0.0713535
$852$	0	0
$853$	−25328.9	−1.01670	−0.508350	−	0.861150i	$-0.669744\pi$
$853$	−25328.9	−1.01670	−0.508350	+	0.861150i	$0.669744\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2010.93	0.0801540	0.0400770	−	0.999197i	$-0.487240\pi$
$857$	2010.93	0.0801540	0.0400770	+	0.999197i	$0.487240\pi$
$858$	0	0
$859$	−6964.49	−0.276630	−0.138315	−	0.990388i	$-0.544169\pi$
$859$	−6964.49	−0.276630	−0.138315	+	0.990388i	$0.544169\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14772.4	−0.582686	−0.291343	−	0.956619i	$-0.594102\pi$
$863$	−14772.4	−0.582686	−0.291343	+	0.956619i	$0.594102\pi$
$864$	0	0
$865$	−55410.2	−2.17804
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24612.4	0.960782
$870$	0	0
$871$	−27820.2	−1.08226
$872$	0	0
$873$	0	0
$874$	0	0
$875$	128726.	4.97341
$876$	0	0
$877$	−25122.9	−0.967322	−0.483661	−	0.875255i	$-0.660693\pi$
$877$	−25122.9	−0.967322	−0.483661	+	0.875255i	$0.660693\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20881.3	0.798533	0.399267	−	0.916835i	$-0.369265\pi$
$881$	20881.3	0.798533	0.399267	+	0.916835i	$0.369265\pi$
$882$	0	0
$883$	38366.4	1.46221	0.731106	−	0.682264i	$-0.239005\pi$
$883$	38366.4	1.46221	0.731106	+	0.682264i	$0.239005\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40986.6	−1.55152	−0.775758	−	0.631030i	$-0.782632\pi$
$887$	−40986.6	−1.55152	−0.775758	+	0.631030i	$0.782632\pi$
$888$	0	0
$889$	47134.2	1.77821
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26031.3	0.975481
$894$	0	0
$895$	95508.4	3.56703
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5962.47	0.221201
$900$	0	0
$901$	146.755	0.00542632
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−49045.2	−1.80146
$906$	0	0
$907$	33795.7	1.23723	0.618616	−	0.785694i	$-0.287694\pi$
$907$	33795.7	1.23723	0.618616	+	0.785694i	$0.287694\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52137.7	1.89616	0.948078	−	0.318037i	$-0.103023\pi$
$911$	52137.7	1.89616	0.948078	+	0.318037i	$0.103023\pi$
$912$	0	0
$913$	−47425.9	−1.71913
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2686.01	−0.0967284
$918$	0	0
$919$	−28749.5	−1.03194	−0.515972	−	0.856605i	$-0.672569\pi$
$919$	−28749.5	−1.03194	−0.515972	+	0.856605i	$0.672569\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8728.64	0.311275
$924$	0	0
$925$	−8123.65	−0.288761
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8061.26	0.284695	0.142347	−	0.989817i	$-0.454535\pi$
$929$	8061.26	0.284695	0.142347	+	0.989817i	$0.454535\pi$
$930$	0	0
$931$	−54185.5	−1.90747
$932$	0	0
$933$	0	0
$934$	0	0
$935$	27653.8	0.967247
$936$	0	0
$937$	−38362.0	−1.33749	−0.668746	−	0.743491i	$-0.733169\pi$
$937$	−38362.0	−1.33749	−0.668746	+	0.743491i	$0.733169\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25782.6	−0.893188	−0.446594	−	0.894737i	$-0.647363\pi$
$941$	−25782.6	−0.893188	−0.446594	+	0.894737i	$0.647363\pi$
$942$	0	0
$943$	27785.1	0.959498
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48058.9	1.64911	0.824554	−	0.565783i	$-0.191426\pi$
$947$	48058.9	1.64911	0.824554	+	0.565783i	$0.191426\pi$
$948$	0	0
$949$	17231.1	0.589405
$950$	0	0
$951$	0	0
$952$	0	0
$953$	49570.9	1.68495	0.842476	−	0.538733i	$-0.181097\pi$
$953$	49570.9	1.68495	0.842476	+	0.538733i	$0.181097\pi$
$954$	0	0
$955$	25952.3	0.879368
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28330.4	−0.953947
$960$	0	0
$961$	−24466.0	−0.821254
$962$	0	0
$963$	0	0
$964$	0	0
$965$	45174.9	1.50698
$966$	0	0
$967$	43179.2	1.43594	0.717968	−	0.696076i	$-0.245072\pi$
$967$	43179.2	1.43594	0.717968	+	0.696076i	$0.245072\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23494.7	−0.776498	−0.388249	−	0.921554i	$-0.626920\pi$
$971$	−23494.7	−0.776498	−0.388249	+	0.921554i	$0.626920\pi$
$972$	0	0
$973$	−63010.1	−2.07606
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20192.7	0.661231	0.330616	−	0.943765i	$-0.392744\pi$
$977$	20192.7	0.661231	0.330616	+	0.943765i	$0.392744\pi$
$978$	0	0
$979$	43401.3	1.41686
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−49221.9	−1.59708	−0.798542	−	0.601939i	$-0.794395\pi$
$983$	−49221.9	−1.59708	−0.798542	+	0.601939i	$0.794395\pi$
$984$	0	0
$985$	41815.8	1.35265
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5804.77	0.186634
$990$	0	0
$991$	−38703.5	−1.24062	−0.620312	−	0.784355i	$-0.712994\pi$
$991$	−38703.5	−1.24062	−0.620312	+	0.784355i	$0.712994\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4728.26	0.150649
$996$	0	0
$997$	8342.78	0.265013	0.132507	−	0.991182i	$-0.457697\pi$
$997$	8342.78	0.265013	0.132507	+	0.991182i	$0.457697\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.z.1.4		4
3.2	odd	2	inner	1296.4.a.z.1.1		4
4.3	odd	2		324.4.a.e.1.4	yes	4
12.11	even	2		324.4.a.e.1.1	✓	4
36.7	odd	6		324.4.e.i.109.1		8
36.11	even	6		324.4.e.i.109.4		8
36.23	even	6		324.4.e.i.217.4		8
36.31	odd	6		324.4.e.i.217.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.4.a.e.1.1	✓	4	12.11	even	2
324.4.a.e.1.4	yes	4	4.3	odd	2
324.4.e.i.109.1		8	36.7	odd	6
324.4.e.i.109.4		8	36.11	even	6
324.4.e.i.217.1		8	36.31	odd	6
324.4.e.i.217.4		8	36.23	even	6
1296.4.a.z.1.1		4	3.2	odd	2	inner
1296.4.a.z.1.4		4	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)

\(f(q)\)	\(=\)	\(q+21.0707 q^{5} +31.4955 q^{7} +O(q^{10})\)	\(q+21.0707 q^{5} +31.4955 q^{7} -36.6591 q^{11} +56.4955 q^{13} -35.8010 q^{17} -83.4955 q^{19} +69.5523 q^{23} +318.973 q^{25} +81.7082 q^{29} +72.9727 q^{31} +663.630 q^{35} -25.4682 q^{37} +399.484 q^{41} +83.4591 q^{43} -311.769 q^{47} +648.964 q^{49} -4.09919 q^{53} -772.432 q^{55} -352.194 q^{59} +3.54091 q^{61} +1190.40 q^{65} -492.432 q^{67} +154.502 q^{71} +305.000 q^{73} -1154.60 q^{77} -671.386 q^{79} +1293.70 q^{83} -754.350 q^{85} -1183.91 q^{89} +1779.35 q^{91} -1759.30 q^{95} +615.982 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 4 * q + 16 * q^7	\( 4 q + 16 q^{7} + 116 q^{13} - 224 q^{19} + 616 q^{25} - 368 q^{31} + 668 q^{37} - 656 q^{43} + 1716 q^{49} - 1440 q^{55} + 1004 q^{61} - 320 q^{67} + 1220 q^{73} + 64 q^{79} + 612 q^{85} + 3488 q^{91} + 2024 q^{97}+O(q^{100}) \) 4 * q + 16 * q^7 + 116 * q^13 - 224 * q^19 + 616 * q^25 - 368 * q^31 + 668 * q^37 - 656 * q^43 + 1716 * q^49 - 1440 * q^55 + 1004 * q^61 - 320 * q^67 + 1220 * q^73 + 64 * q^79 + 612 * q^85 + 3488 * q^91 + 2024 * q^97

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