LMFDB - Embedded newform 8649.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.1413480448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - x^{4} - 16x^{3} + 2x^{2} + 4x - 1 $ x^8 - 4x^7 - 2x^6 + 16x^5 - x^4 - 16x^3 + 2x^2 + 4x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.733914$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.680299 q^{2} -1.53719 q^{4} -0.966483 q^{5} -4.84476 q^{7} +2.40635 q^{8} +O(q^{10})$	\(q-0.680299 q^{2} -1.53719 q^{4} -0.966483 q^{5} -4.84476 q^{7} +2.40635 q^{8} +0.657497 q^{10} -4.81756 q^{11} +0.875151 q^{13} +3.29588 q^{14} +1.43735 q^{16} -2.95917 q^{17} -3.40195 q^{19} +1.48567 q^{20} +3.27738 q^{22} -0.246832 q^{23} -4.06591 q^{25} -0.595364 q^{26} +7.44733 q^{28} +8.57976 q^{29} -5.79053 q^{32} +2.01312 q^{34} +4.68237 q^{35} +5.56649 q^{37} +2.31435 q^{38} -2.32569 q^{40} +8.25354 q^{41} -4.19197 q^{43} +7.40552 q^{44} +0.167919 q^{46} +11.6222 q^{47} +16.4717 q^{49} +2.76604 q^{50} -1.34528 q^{52} +12.1858 q^{53} +4.65609 q^{55} -11.6582 q^{56} -5.83681 q^{58} -2.28620 q^{59} +0.815070 q^{61} +1.06459 q^{64} -0.845818 q^{65} +2.96756 q^{67} +4.54881 q^{68} -3.18541 q^{70} +4.11205 q^{71} -15.3094 q^{73} -3.78688 q^{74} +5.22946 q^{76} +23.3399 q^{77} -3.22375 q^{79} -1.38917 q^{80} -5.61487 q^{82} -11.3210 q^{83} +2.85998 q^{85} +2.85180 q^{86} -11.5927 q^{88} +1.60611 q^{89} -4.23989 q^{91} +0.379428 q^{92} -7.90657 q^{94} +3.28793 q^{95} +0.978949 q^{97} -11.2057 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 4 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q - 4 * q^2 + 4 * q^4 - 8 * q^7	$ 8 q - 4 q^{2} + 4 q^{4} - 8 q^{7} - 20 q^{10} + 8 q^{11} - 16 q^{13} + 12 q^{14} + 4 q^{16} - 16 q^{19} + 12 q^{20} + 16 q^{23} + 8 q^{25} + 8 q^{26} - 4 q^{28} + 16 q^{29} - 4 q^{32} - 16 q^{34} + 24 q^{35} - 16 q^{37} + 12 q^{38} - 16 q^{40} - 32 q^{43} + 24 q^{44} - 16 q^{46} + 8 q^{47} + 40 q^{49} - 8 q^{50} - 32 q^{52} + 16 q^{53} + 16 q^{55} + 24 q^{56} + 16 q^{58} - 24 q^{59} - 48 q^{61} - 16 q^{64} + 24 q^{65} - 40 q^{67} - 24 q^{68} + 4 q^{70} - 16 q^{71} - 48 q^{73} + 48 q^{74} - 20 q^{76} + 48 q^{77} + 36 q^{80} + 52 q^{82} + 24 q^{83} - 16 q^{88} + 24 q^{89} + 16 q^{91} + 80 q^{92} - 24 q^{94} - 40 q^{95} - 64 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 4 * q^4 - 8 * q^7 - 20 * q^10 + 8 * q^11 - 16 * q^13 + 12 * q^14 + 4 * q^16 - 16 * q^19 + 12 * q^20 + 16 * q^23 + 8 * q^25 + 8 * q^26 - 4 * q^28 + 16 * q^29 - 4 * q^32 - 16 * q^34 + 24 * q^35 - 16 * q^37 + 12 * q^38 - 16 * q^40 - 32 * q^43 + 24 * q^44 - 16 * q^46 + 8 * q^47 + 40 * q^49 - 8 * q^50 - 32 * q^52 + 16 * q^53 + 16 * q^55 + 24 * q^56 + 16 * q^58 - 24 * q^59 - 48 * q^61 - 16 * q^64 + 24 * q^65 - 40 * q^67 - 24 * q^68 + 4 * q^70 - 16 * q^71 - 48 * q^73 + 48 * q^74 - 20 * q^76 + 48 * q^77 + 36 * q^80 + 52 * q^82 + 24 * q^83 - 16 * q^88 + 24 * q^89 + 16 * q^91 + 80 * q^92 - 24 * q^94 - 40 * q^95 - 64 * q^98

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 $

$2$	−0.680299	−0.481044	−0.240522	−	0.970644i	$-0.577319\pi$
$2$	−0.680299	−0.481044	−0.240522	+	0.970644i	$0.577319\pi$
$3$	0	0
$3$	0	0
$4$	−1.53719	−0.768597
$5$	−0.966483	−0.432224	−0.216112	−	0.976369i	$-0.569338\pi$
$5$	−0.966483	−0.432224	−0.216112	+	0.976369i	$0.569338\pi$
$6$	0	0
$7$	−4.84476	−1.83115	−0.915573	−	0.402152i	$-0.868262\pi$
$7$	−4.84476	−1.83115	−0.915573	+	0.402152i	$0.868262\pi$
$8$	2.40635	0.850773
$9$	0	0
$10$	0.657497	0.207919
$11$	−4.81756	−1.45255	−0.726275	−	0.687405i	$-0.758750\pi$
$11$	−4.81756	−1.45255	−0.726275	+	0.687405i	$0.758750\pi$
$12$	0	0
$13$	0.875151	0.242723	0.121362	−	0.992608i	$-0.461274\pi$
$13$	0.875151	0.242723	0.121362	+	0.992608i	$0.461274\pi$
$14$	3.29588	0.880862
$15$	0	0
$16$	1.43735	0.359337
$17$	−2.95917	−0.717704	−0.358852	−	0.933395i	$-0.616832\pi$
$17$	−2.95917	−0.717704	−0.358852	+	0.933395i	$0.616832\pi$
$18$	0	0
$19$	−3.40195	−0.780462	−0.390231	−	0.920717i	$-0.627605\pi$
$19$	−3.40195	−0.780462	−0.390231	+	0.920717i	$0.627605\pi$
$20$	1.48567	0.332206
$21$	0	0
$22$	3.27738	0.698740
$23$	−0.246832	−0.0514680	−0.0257340	−	0.999669i	$-0.508192\pi$
$23$	−0.246832	−0.0514680	−0.0257340	+	0.999669i	$0.508192\pi$
$24$	0	0
$25$	−4.06591	−0.813182
$26$	−0.595364	−0.116761
$27$	0	0
$28$	7.44733	1.40741
$29$	8.57976	1.59322	0.796611	−	0.604492i	$-0.206624\pi$
$29$	8.57976	1.59322	0.796611	+	0.604492i	$0.206624\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−5.79053	−1.02363
$33$	0	0
$34$	2.01312	0.345247
$35$	4.68237	0.791465
$36$	0	0
$37$	5.56649	0.915125	0.457563	−	0.889177i	$-0.348723\pi$
$37$	5.56649	0.915125	0.457563	+	0.889177i	$0.348723\pi$
$38$	2.31435	0.375437
$39$	0	0
$40$	−2.32569	−0.367725
$41$	8.25354	1.28899	0.644493	−	0.764610i	$-0.277069\pi$
$41$	8.25354	1.28899	0.644493	+	0.764610i	$0.277069\pi$
$42$	0	0
$43$	−4.19197	−0.639270	−0.319635	−	0.947541i	$-0.603560\pi$
$43$	−4.19197	−0.639270	−0.319635	+	0.947541i	$0.603560\pi$
$44$	7.40552	1.11642
$45$	0	0
$46$	0.167919	0.0247584
$47$	11.6222	1.69527	0.847636	−	0.530578i	$-0.178025\pi$
$47$	11.6222	1.69527	0.847636	+	0.530578i	$0.178025\pi$
$48$	0	0
$49$	16.4717	2.35310
$50$	2.76604	0.391177
$51$	0	0
$52$	−1.34528	−0.186556
$53$	12.1858	1.67385	0.836923	−	0.547321i	$-0.184352\pi$
$53$	12.1858	1.67385	0.836923	+	0.547321i	$0.184352\pi$
$54$	0	0
$55$	4.65609	0.627827
$56$	−11.6582	−1.55789
$57$	0	0
$58$	−5.83681	−0.766410
$59$	−2.28620	−0.297638	−0.148819	−	0.988864i	$-0.547547\pi$
$59$	−2.28620	−0.297638	−0.148819	+	0.988864i	$0.547547\pi$
$60$	0	0
$61$	0.815070	0.104359	0.0521795	−	0.998638i	$-0.483383\pi$
$61$	0.815070	0.104359	0.0521795	+	0.998638i	$0.483383\pi$
$62$	0	0
$63$	0	0
$64$	1.06459	0.133074
$65$	−0.845818	−0.104911
$66$	0	0
$67$	2.96756	0.362545	0.181273	−	0.983433i	$-0.441978\pi$
$67$	2.96756	0.362545	0.181273	+	0.983433i	$0.441978\pi$
$68$	4.54881	0.551625
$69$	0	0
$70$	−3.18541	−0.380730
$71$	4.11205	0.488010	0.244005	−	0.969774i	$-0.421539\pi$
$71$	4.11205	0.488010	0.244005	+	0.969774i	$0.421539\pi$
$72$	0	0
$73$	−15.3094	−1.79183	−0.895915	−	0.444225i	$-0.853479\pi$
$73$	−15.3094	−1.79183	−0.895915	+	0.444225i	$0.853479\pi$
$74$	−3.78688	−0.440216
$75$	0	0
$76$	5.22946	0.599860
$77$	23.3399	2.65983
$78$	0	0
$79$	−3.22375	−0.362700	−0.181350	−	0.983419i	$-0.558047\pi$
$79$	−3.22375	−0.362700	−0.181350	+	0.983419i	$0.558047\pi$
$80$	−1.38917	−0.155314
$81$	0	0
$82$	−5.61487	−0.620059
$83$	−11.3210	−1.24264	−0.621319	−	0.783558i	$-0.713403\pi$
$83$	−11.3210	−1.24264	−0.621319	+	0.783558i	$0.713403\pi$
$84$	0	0
$85$	2.85998	0.310209
$86$	2.85180	0.307517
$87$	0	0
$88$	−11.5927	−1.23579
$89$	1.60611	0.170247	0.0851236	−	0.996370i	$-0.472871\pi$
$89$	1.60611	0.170247	0.0851236	+	0.996370i	$0.472871\pi$
$90$	0	0
$91$	−4.23989	−0.444462
$92$	0.379428	0.0395581
$93$	0	0
$94$	−7.90657	−0.815501
$95$	3.28793	0.337334
$96$	0	0
$97$	0.978949	0.0993972	0.0496986	−	0.998764i	$-0.484174\pi$
$97$	0.978949	0.0993972	0.0496986	+	0.998764i	$0.484174\pi$
$98$	−11.2057	−1.13194
$99$	0	0
$100$	6.25009	0.625009
$101$	4.68349	0.466024	0.233012	−	0.972474i	$-0.425142\pi$
$101$	4.68349	0.466024	0.233012	+	0.972474i	$0.425142\pi$
$102$	0	0
$103$	2.32141	0.228735	0.114368	−	0.993438i	$-0.463516\pi$
$103$	2.32141	0.228735	0.114368	+	0.993438i	$0.463516\pi$
$104$	2.10592	0.206502
$105$	0	0
$106$	−8.28998	−0.805194
$107$	−4.40981	−0.426313	−0.213156	−	0.977018i	$-0.568374\pi$
$107$	−4.40981	−0.426313	−0.213156	+	0.977018i	$0.568374\pi$
$108$	0	0
$109$	−19.2658	−1.84533	−0.922663	−	0.385608i	$-0.873992\pi$
$109$	−19.2658	−1.84533	−0.922663	+	0.385608i	$0.873992\pi$
$110$	−3.16753	−0.302012
$111$	0	0
$112$	−6.96361	−0.657999
$113$	−4.12706	−0.388241	−0.194121	−	0.980978i	$-0.562185\pi$
$113$	−4.12706	−0.388241	−0.194121	+	0.980978i	$0.562185\pi$
$114$	0	0
$115$	0.238559	0.0222457
$116$	−13.1888	−1.22455
$117$	0	0
$118$	1.55530	0.143177
$119$	14.3364	1.31422
$120$	0	0
$121$	12.2089	1.10990
$122$	−0.554491	−0.0502013
$123$	0	0
$124$	0	0
$125$	8.76205	0.783701
$126$	0	0
$127$	−6.81786	−0.604987	−0.302493	−	0.953151i	$-0.597819\pi$
$127$	−6.81786	−0.604987	−0.302493	+	0.953151i	$0.597819\pi$
$128$	10.8568	0.959616
$129$	0	0
$130$	0.575409	0.0504667
$131$	16.3689	1.43016	0.715080	−	0.699043i	$-0.246390\pi$
$131$	16.3689	1.43016	0.715080	+	0.699043i	$0.246390\pi$
$132$	0	0
$133$	16.4816	1.42914
$134$	−2.01883	−0.174400
$135$	0	0
$136$	−7.12079	−0.610603
$137$	4.40002	0.375919	0.187959	−	0.982177i	$-0.439813\pi$
$137$	4.40002	0.375919	0.187959	+	0.982177i	$0.439813\pi$
$138$	0	0
$139$	−9.08569	−0.770638	−0.385319	−	0.922783i	$-0.625909\pi$
$139$	−9.08569	−0.770638	−0.385319	+	0.922783i	$0.625909\pi$
$140$	−7.19771	−0.608318
$141$	0	0
$142$	−2.79742	−0.234755
$143$	−4.21609	−0.352567
$144$	0	0
$145$	−8.29219	−0.688629
$146$	10.4150	0.861950
$147$	0	0
$148$	−8.55677	−0.703362
$149$	3.02411	0.247745	0.123872	−	0.992298i	$-0.460469\pi$
$149$	3.02411	0.247745	0.123872	+	0.992298i	$0.460469\pi$
$150$	0	0
$151$	−8.63438	−0.702656	−0.351328	−	0.936253i	$-0.614270\pi$
$151$	−8.63438	−0.702656	−0.351328	+	0.936253i	$0.614270\pi$
$152$	−8.18629	−0.663996
$153$	0	0
$154$	−15.8781	−1.27950
$155$	0	0
$156$	0	0
$157$	15.4682	1.23450	0.617250	−	0.786767i	$-0.288247\pi$
$157$	15.4682	1.23450	0.617250	+	0.786767i	$0.288247\pi$
$158$	2.19311	0.174475
$159$	0	0
$160$	5.59644	0.442438
$161$	1.19584	0.0942454
$162$	0	0
$163$	−0.959116	−0.0751237	−0.0375619	−	0.999294i	$-0.511959\pi$
$163$	−0.959116	−0.0751237	−0.0375619	+	0.999294i	$0.511959\pi$
$164$	−12.6873	−0.990710
$165$	0	0
$166$	7.70164	0.597764
$167$	17.3131	1.33973	0.669864	−	0.742483i	$-0.266352\pi$
$167$	17.3131	1.33973	0.669864	+	0.742483i	$0.266352\pi$
$168$	0	0
$169$	−12.2341	−0.941085
$170$	−1.94564	−0.149224
$171$	0	0
$172$	6.44387	0.491341
$173$	17.5627	1.33527	0.667634	−	0.744489i	$-0.267307\pi$
$173$	17.5627	1.33527	0.667634	+	0.744489i	$0.267307\pi$
$174$	0	0
$175$	19.6984	1.48906
$176$	−6.92452	−0.521955
$177$	0	0
$178$	−1.09264	−0.0818965
$179$	−10.0214	−0.749036	−0.374518	−	0.927220i	$-0.622192\pi$
$179$	−10.0214	−0.749036	−0.374518	+	0.927220i	$0.622192\pi$
$180$	0	0
$181$	12.7166	0.945218	0.472609	−	0.881272i	$-0.343312\pi$
$181$	12.7166	0.945218	0.472609	+	0.881272i	$0.343312\pi$
$182$	2.88440	0.213806
$183$	0	0
$184$	−0.593964	−0.0437876
$185$	−5.37992	−0.395539
$186$	0	0
$187$	14.2560	1.04250
$188$	−17.8656	−1.30298
$189$	0	0
$190$	−2.23678	−0.162273
$191$	−25.9353	−1.87661	−0.938307	−	0.345803i	$-0.887606\pi$
$191$	−25.9353	−1.87661	−0.938307	+	0.345803i	$0.887606\pi$
$192$	0	0
$193$	22.6819	1.63268	0.816339	−	0.577574i	$-0.196000\pi$
$193$	22.6819	1.63268	0.816339	+	0.577574i	$0.196000\pi$
$194$	−0.665978	−0.0478144
$195$	0	0
$196$	−25.3201	−1.80858
$197$	3.51683	0.250564	0.125282	−	0.992121i	$-0.460016\pi$
$197$	3.51683	0.250564	0.125282	+	0.992121i	$0.460016\pi$
$198$	0	0
$199$	5.44788	0.386190	0.193095	−	0.981180i	$-0.438147\pi$
$199$	5.44788	0.386190	0.193095	+	0.981180i	$0.438147\pi$
$200$	−9.78400	−0.691833
$201$	0	0
$202$	−3.18617	−0.224178
$203$	−41.5669	−2.91742
$204$	0	0
$205$	−7.97690	−0.557131
$206$	−1.57925	−0.110032
$207$	0	0
$208$	1.25790	0.0872195
$209$	16.3891	1.13366
$210$	0	0
$211$	13.1941	0.908318	0.454159	−	0.890921i	$-0.349940\pi$
$211$	13.1941	0.908318	0.454159	+	0.890921i	$0.349940\pi$
$212$	−18.7319	−1.28651
$213$	0	0
$214$	2.99999	0.205075
$215$	4.05147	0.276308
$216$	0	0
$217$	0	0
$218$	13.1065	0.887683
$219$	0	0
$220$	−7.15731	−0.482546
$221$	−2.58972	−0.174203
$222$	0	0
$223$	26.9707	1.80609	0.903045	−	0.429546i	$-0.141326\pi$
$223$	26.9707	1.80609	0.903045	+	0.429546i	$0.141326\pi$
$224$	28.0537	1.87442
$225$	0	0
$226$	2.80764	0.186761
$227$	−16.1362	−1.07100	−0.535499	−	0.844536i	$-0.679877\pi$
$227$	−16.1362	−1.07100	−0.535499	+	0.844536i	$0.679877\pi$
$228$	0	0
$229$	0.687404	0.0454249	0.0227125	−	0.999742i	$-0.492770\pi$
$229$	0.687404	0.0454249	0.0227125	+	0.999742i	$0.492770\pi$
$230$	−0.162291	−0.0107012
$231$	0	0
$232$	20.6459	1.35547
$233$	−14.9971	−0.982491	−0.491246	−	0.871021i	$-0.663458\pi$
$233$	−14.9971	−0.982491	−0.491246	+	0.871021i	$0.663458\pi$
$234$	0	0
$235$	−11.2327	−0.732738
$236$	3.51433	0.228763
$237$	0	0
$238$	−9.75307	−0.632198
$239$	−10.2223	−0.661228	−0.330614	−	0.943766i	$-0.607256\pi$
$239$	−10.2223	−0.661228	−0.330614	+	0.943766i	$0.607256\pi$
$240$	0	0
$241$	14.3035	0.921371	0.460686	−	0.887563i	$-0.347604\pi$
$241$	14.3035	0.921371	0.460686	+	0.887563i	$0.347604\pi$
$242$	−8.30570	−0.533910
$243$	0	0
$244$	−1.25292	−0.0802100
$245$	−15.9196	−1.01706
$246$	0	0
$247$	−2.97722	−0.189436
$248$	0	0
$249$	0	0
$250$	−5.96081	−0.376995
$251$	−14.4580	−0.912581	−0.456291	−	0.889831i	$-0.650822\pi$
$251$	−14.4580	−0.912581	−0.456291	+	0.889831i	$0.650822\pi$
$252$	0	0
$253$	1.18913	0.0747598
$254$	4.63818	0.291025
$255$	0	0
$256$	−9.51506	−0.594691
$257$	9.98082	0.622586	0.311293	−	0.950314i	$-0.399238\pi$
$257$	9.98082	0.622586	0.311293	+	0.950314i	$0.399238\pi$
$258$	0	0
$259$	−26.9683	−1.67573
$260$	1.30019	0.0806341
$261$	0	0
$262$	−11.1358	−0.687970
$263$	12.9852	0.800702	0.400351	−	0.916362i	$-0.368888\pi$
$263$	12.9852	0.800702	0.400351	+	0.916362i	$0.368888\pi$
$264$	0	0
$265$	−11.7773	−0.723477
$266$	−11.2124	−0.687479
$267$	0	0
$268$	−4.56172	−0.278651
$269$	11.6848	0.712433	0.356217	−	0.934403i	$-0.384066\pi$
$269$	11.6848	0.712433	0.356217	+	0.934403i	$0.384066\pi$
$270$	0	0
$271$	20.3075	1.23359	0.616796	−	0.787123i	$-0.288430\pi$
$271$	20.3075	1.23359	0.616796	+	0.787123i	$0.288430\pi$
$272$	−4.25336	−0.257898
$273$	0	0
$274$	−2.99333	−0.180833
$275$	19.5878	1.18119
$276$	0	0
$277$	−23.5956	−1.41772	−0.708862	−	0.705347i	$-0.750791\pi$
$277$	−23.5956	−1.41772	−0.708862	+	0.705347i	$0.750791\pi$
$278$	6.18099	0.370711
$279$	0	0
$280$	11.2674	0.673357
$281$	−17.1323	−1.02203	−0.511015	−	0.859572i	$-0.670730\pi$
$281$	−17.1323	−1.02203	−0.511015	+	0.859572i	$0.670730\pi$
$282$	0	0
$283$	−7.97956	−0.474335	−0.237168	−	0.971469i	$-0.576219\pi$
$283$	−7.97956	−0.474335	−0.237168	+	0.971469i	$0.576219\pi$
$284$	−6.32101	−0.375083
$285$	0	0
$286$	2.86820	0.169600
$287$	−39.9864	−2.36032
$288$	0	0
$289$	−8.24333	−0.484902
$290$	5.64117	0.331261
$291$	0	0
$292$	23.5335	1.37720
$293$	−14.4006	−0.841292	−0.420646	−	0.907225i	$-0.638197\pi$
$293$	−14.4006	−0.841292	−0.420646	+	0.907225i	$0.638197\pi$
$294$	0	0
$295$	2.20957	0.128646
$296$	13.3949	0.778564
$297$	0	0
$298$	−2.05730	−0.119176
$299$	−0.216015	−0.0124925
$300$	0	0
$301$	20.3091	1.17060
$302$	5.87396	0.338008
$303$	0	0
$304$	−4.88980	−0.280449
$305$	−0.787751	−0.0451065
$306$	0	0
$307$	−19.6649	−1.12234	−0.561168	−	0.827702i	$-0.689648\pi$
$307$	−19.6649	−1.12234	−0.561168	+	0.827702i	$0.689648\pi$
$308$	−35.8780	−2.04434
$309$	0	0
$310$	0	0
$311$	4.93106	0.279615	0.139807	−	0.990179i	$-0.455352\pi$
$311$	4.93106	0.279615	0.139807	+	0.990179i	$0.455352\pi$
$312$	0	0
$313$	−14.3904	−0.813394	−0.406697	−	0.913563i	$-0.633320\pi$
$313$	−14.3904	−0.813394	−0.406697	+	0.913563i	$0.633320\pi$
$314$	−10.5230	−0.593849
$315$	0	0
$316$	4.95552	0.278770
$317$	9.57788	0.537947	0.268974	−	0.963148i	$-0.413316\pi$
$317$	9.57788	0.537947	0.268974	+	0.963148i	$0.413316\pi$
$318$	0	0
$319$	−41.3335	−2.31423
$320$	−1.02891	−0.0575177
$321$	0	0
$322$	−0.813529	−0.0453362
$323$	10.0670	0.560140
$324$	0	0
$325$	−3.55829	−0.197378
$326$	0.652486	0.0361378
$327$	0	0
$328$	19.8609	1.09663
$329$	−56.3067	−3.10429
$330$	0	0
$331$	25.0527	1.37702	0.688510	−	0.725227i	$-0.258265\pi$
$331$	25.0527	1.37702	0.688510	+	0.725227i	$0.258265\pi$
$332$	17.4025	0.955087
$333$	0	0
$334$	−11.7781	−0.644469
$335$	−2.86810	−0.156701
$336$	0	0
$337$	15.1770	0.826743	0.413372	−	0.910562i	$-0.364351\pi$
$337$	15.1770	0.826743	0.413372	+	0.910562i	$0.364351\pi$
$338$	8.32285	0.452704
$339$	0	0
$340$	−4.39635	−0.238425
$341$	0	0
$342$	0	0
$343$	−45.8879	−2.47772
$344$	−10.0874	−0.543874
$345$	0	0
$346$	−11.9479	−0.642323
$347$	3.45991	0.185738	0.0928689	−	0.995678i	$-0.470396\pi$
$347$	3.45991	0.185738	0.0928689	+	0.995678i	$0.470396\pi$
$348$	0	0
$349$	15.6981	0.840300	0.420150	−	0.907455i	$-0.361978\pi$
$349$	15.6981	0.840300	0.420150	+	0.907455i	$0.361978\pi$
$350$	−13.4008	−0.716301
$351$	0	0
$352$	27.8962	1.48687
$353$	−2.69094	−0.143224	−0.0716122	−	0.997433i	$-0.522814\pi$
$353$	−2.69094	−0.143224	−0.0716122	+	0.997433i	$0.522814\pi$
$354$	0	0
$355$	−3.97422	−0.210930
$356$	−2.46890	−0.130851
$357$	0	0
$358$	6.81756	0.360319
$359$	15.1167	0.797830	0.398915	−	0.916988i	$-0.369387\pi$
$359$	15.1167	0.797830	0.398915	+	0.916988i	$0.369387\pi$
$360$	0	0
$361$	−7.42670	−0.390879
$362$	−8.65109	−0.454691
$363$	0	0
$364$	6.51753	0.341612
$365$	14.7963	0.774473
$366$	0	0
$367$	−11.3881	−0.594456	−0.297228	−	0.954806i	$-0.596062\pi$
$367$	−11.3881	−0.594456	−0.297228	+	0.954806i	$0.596062\pi$
$368$	−0.354784	−0.0184944
$369$	0	0
$370$	3.65995	0.190272
$371$	−59.0371	−3.06506
$372$	0	0
$373$	−28.3331	−1.46703	−0.733516	−	0.679672i	$-0.762122\pi$
$373$	−28.3331	−1.46703	−0.733516	+	0.679672i	$0.762122\pi$
$374$	−9.69832	−0.501488
$375$	0	0
$376$	27.9671	1.44229
$377$	7.50859	0.386712
$378$	0	0
$379$	−5.10171	−0.262057	−0.131029	−	0.991379i	$-0.541828\pi$
$379$	−5.10171	−0.262057	−0.131029	+	0.991379i	$0.541828\pi$
$380$	−5.05418	−0.259274
$381$	0	0
$382$	17.6438	0.902734
$383$	0.122621	0.00626564	0.00313282	−	0.999995i	$-0.499003\pi$
$383$	0.122621	0.00626564	0.00313282	+	0.999995i	$0.499003\pi$
$384$	0	0
$385$	−22.5576	−1.14964
$386$	−15.4305	−0.785390
$387$	0	0
$388$	−1.50483	−0.0763963
$389$	7.67839	0.389310	0.194655	−	0.980872i	$-0.437641\pi$
$389$	7.67839	0.389310	0.194655	+	0.980872i	$0.437641\pi$
$390$	0	0
$391$	0.730417	0.0369388
$392$	39.6366	2.00195
$393$	0	0
$394$	−2.39250	−0.120532
$395$	3.11569	0.156768
$396$	0	0
$397$	−2.85031	−0.143053	−0.0715265	−	0.997439i	$-0.522787\pi$
$397$	−2.85031	−0.143053	−0.0715265	+	0.997439i	$0.522787\pi$
$398$	−3.70619	−0.185774
$399$	0	0
$400$	−5.84413	−0.292207
$401$	24.6673	1.23183	0.615913	−	0.787814i	$-0.288787\pi$
$401$	24.6673	1.23183	0.615913	+	0.787814i	$0.288787\pi$
$402$	0	0
$403$	0	0
$404$	−7.19942	−0.358185
$405$	0	0
$406$	28.2779	1.40341
$407$	−26.8169	−1.32926
$408$	0	0
$409$	−31.5914	−1.56210	−0.781048	−	0.624471i	$-0.785314\pi$
$409$	−31.5914	−1.56210	−0.781048	+	0.624471i	$0.785314\pi$
$410$	5.42668	0.268004
$411$	0	0
$412$	−3.56846	−0.175805
$413$	11.0761	0.545018
$414$	0	0
$415$	10.9415	0.537098
$416$	−5.06758	−0.248459
$417$	0	0
$418$	−11.1495	−0.545340
$419$	−26.5104	−1.29512	−0.647559	−	0.762015i	$-0.724210\pi$
$419$	−26.5104	−1.29512	−0.647559	+	0.762015i	$0.724210\pi$
$420$	0	0
$421$	5.13278	0.250157	0.125078	−	0.992147i	$-0.460082\pi$
$421$	5.13278	0.250157	0.125078	+	0.992147i	$0.460082\pi$
$422$	−8.97592	−0.436941
$423$	0	0
$424$	29.3232	1.42406
$425$	12.0317	0.583624
$426$	0	0
$427$	−3.94882	−0.191097
$428$	6.77873	0.327662
$429$	0	0
$430$	−2.75621	−0.132916
$431$	10.3856	0.500257	0.250128	−	0.968213i	$-0.419527\pi$
$431$	10.3856	0.500257	0.250128	+	0.968213i	$0.419527\pi$
$432$	0	0
$433$	11.2659	0.541406	0.270703	−	0.962663i	$-0.412744\pi$
$433$	11.2659	0.541406	0.270703	+	0.962663i	$0.412744\pi$
$434$	0	0
$435$	0	0
$436$	29.6152	1.41831
$437$	0.839711	0.0401688
$438$	0	0
$439$	−21.1928	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$439$	−21.1928	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$440$	11.2042	0.534138
$441$	0	0
$442$	1.76178	0.0837995
$443$	7.87489	0.374148	0.187074	−	0.982346i	$-0.440100\pi$
$443$	7.87489	0.374148	0.187074	+	0.982346i	$0.440100\pi$
$444$	0	0
$445$	−1.55228	−0.0735850
$446$	−18.3481	−0.868809
$447$	0	0
$448$	−5.15769	−0.243678
$449$	29.2154	1.37876	0.689380	−	0.724400i	$-0.257883\pi$
$449$	29.2154	1.37876	0.689380	+	0.724400i	$0.257883\pi$
$450$	0	0
$451$	−39.7619	−1.87232
$452$	6.34409	0.298401
$453$	0	0
$454$	10.9775	0.515198
$455$	4.09778	0.192107
$456$	0	0
$457$	−8.23720	−0.385320	−0.192660	−	0.981266i	$-0.561711\pi$
$457$	−8.23720	−0.385320	−0.192660	+	0.981266i	$0.561711\pi$
$458$	−0.467640	−0.0218514
$459$	0	0
$460$	−0.366711	−0.0170980
$461$	−29.9085	−1.39298	−0.696489	−	0.717568i	$-0.745255\pi$
$461$	−29.9085	−1.39298	−0.696489	+	0.717568i	$0.745255\pi$
$462$	0	0
$463$	38.4868	1.78863	0.894316	−	0.447435i	$-0.147662\pi$
$463$	38.4868	1.78863	0.894316	+	0.447435i	$0.147662\pi$
$464$	12.3321	0.572504
$465$	0	0
$466$	10.2025	0.472621
$467$	−10.7229	−0.496199	−0.248099	−	0.968735i	$-0.579806\pi$
$467$	−10.7229	−0.496199	−0.248099	+	0.968735i	$0.579806\pi$
$468$	0	0
$469$	−14.3771	−0.663874
$470$	7.64157	0.352479
$471$	0	0
$472$	−5.50139	−0.253222
$473$	20.1951	0.928571
$474$	0	0
$475$	13.8320	0.634658
$476$	−22.0379	−1.01011
$477$	0	0
$478$	6.95425	0.318080
$479$	−18.8650	−0.861963	−0.430982	−	0.902361i	$-0.641833\pi$
$479$	−18.8650	−0.861963	−0.430982	+	0.902361i	$0.641833\pi$
$480$	0	0
$481$	4.87152	0.222122
$482$	−9.73068	−0.443220
$483$	0	0
$484$	−18.7674	−0.853065
$485$	−0.946137	−0.0429619
$486$	0	0
$487$	−1.37611	−0.0623577	−0.0311789	−	0.999514i	$-0.509926\pi$
$487$	−1.37611	−0.0623577	−0.0311789	+	0.999514i	$0.509926\pi$
$488$	1.96134	0.0887858
$489$	0	0
$490$	10.8301	0.489253
$491$	−2.95058	−0.133158	−0.0665788	−	0.997781i	$-0.521208\pi$
$491$	−2.95058	−0.133158	−0.0665788	+	0.997781i	$0.521208\pi$
$492$	0	0
$493$	−25.3890	−1.14346
$494$	2.02540	0.0911272
$495$	0	0
$496$	0	0
$497$	−19.9219	−0.893618
$498$	0	0
$499$	−12.0616	−0.539950	−0.269975	−	0.962867i	$-0.587015\pi$
$499$	−12.0616	−0.539950	−0.269975	+	0.962867i	$0.587015\pi$
$500$	−13.4690	−0.602350
$501$	0	0
$502$	9.83577	0.438992
$503$	−30.0097	−1.33807	−0.669033	−	0.743233i	$-0.733291\pi$
$503$	−30.0097	−1.33807	−0.669033	+	0.743233i	$0.733291\pi$
$504$	0	0
$505$	−4.52651	−0.201427
$506$	−0.808962	−0.0359628
$507$	0	0
$508$	10.4804	0.464991
$509$	−21.2766	−0.943071	−0.471535	−	0.881847i	$-0.656300\pi$
$509$	−21.2766	−0.943071	−0.471535	+	0.881847i	$0.656300\pi$
$510$	0	0
$511$	74.1704	3.28110
$512$	−15.2405	−0.673543
$513$	0	0
$514$	−6.78994	−0.299491
$515$	−2.24360	−0.0988649
$516$	0	0
$517$	−55.9907	−2.46247
$518$	18.3465	0.806099
$519$	0	0
$520$	−2.03533	−0.0892553
$521$	−18.5837	−0.814168	−0.407084	−	0.913391i	$-0.633454\pi$
$521$	−18.5837	−0.814168	−0.407084	+	0.913391i	$0.633454\pi$
$522$	0	0
$523$	−21.7668	−0.951795	−0.475898	−	0.879501i	$-0.657877\pi$
$523$	−21.7668	−0.951795	−0.475898	+	0.879501i	$0.657877\pi$
$524$	−25.1622	−1.09922
$525$	0	0
$526$	−8.83382	−0.385173
$527$	0	0
$528$	0	0
$529$	−22.9391	−0.997351
$530$	8.01212	0.348024
$531$	0	0
$532$	−25.3355	−1.09843
$533$	7.22309	0.312867
$534$	0	0
$535$	4.26201	0.184263
$536$	7.14099	0.308444
$537$	0	0
$538$	−7.94914	−0.342712
$539$	−79.3533	−3.41799
$540$	0	0
$541$	−4.72886	−0.203309	−0.101655	−	0.994820i	$-0.532414\pi$
$541$	−4.72886	−0.203309	−0.101655	+	0.994820i	$0.532414\pi$
$542$	−13.8152	−0.593413
$543$	0	0
$544$	17.1351	0.734663
$545$	18.6200	0.797594
$546$	0	0
$547$	0.343385	0.0146821	0.00734105	−	0.999973i	$-0.497663\pi$
$547$	0.343385	0.0146821	0.00734105	+	0.999973i	$0.497663\pi$
$548$	−6.76367	−0.288930
$549$	0	0
$550$	−13.3255	−0.568203
$551$	−29.1880	−1.24345
$552$	0	0
$553$	15.6183	0.664156
$554$	16.0521	0.681988
$555$	0	0
$556$	13.9665	0.592310
$557$	7.36693	0.312147	0.156073	−	0.987745i	$-0.450116\pi$
$557$	7.36693	0.312147	0.156073	+	0.987745i	$0.450116\pi$
$558$	0	0
$559$	−3.66861	−0.155166
$560$	6.73020	0.284403
$561$	0	0
$562$	11.6551	0.491641
$563$	−1.20688	−0.0508638	−0.0254319	−	0.999677i	$-0.508096\pi$
$563$	−1.20688	−0.0508638	−0.0254319	+	0.999677i	$0.508096\pi$
$564$	0	0
$565$	3.98873	0.167807
$566$	5.42849	0.228176
$567$	0	0
$568$	9.89503	0.415186
$569$	−36.4315	−1.52729	−0.763643	−	0.645638i	$-0.776591\pi$
$569$	−36.4315	−1.52729	−0.763643	+	0.645638i	$0.776591\pi$
$570$	0	0
$571$	30.0033	1.25560	0.627799	−	0.778375i	$-0.283956\pi$
$571$	30.0033	1.25560	0.627799	+	0.778375i	$0.283956\pi$
$572$	6.48095	0.270982
$573$	0	0
$574$	27.2027	1.13542
$575$	1.00360	0.0418529
$576$	0	0
$577$	−25.9415	−1.07996	−0.539980	−	0.841678i	$-0.681568\pi$
$577$	−25.9415	−1.07996	−0.539980	+	0.841678i	$0.681568\pi$
$578$	5.60793	0.233259
$579$	0	0
$580$	12.7467	0.529278
$581$	54.8473	2.27545
$582$	0	0
$583$	−58.7057	−2.43134
$584$	−36.8398	−1.52444
$585$	0	0
$586$	9.79672	0.404699
$587$	23.9802	0.989767	0.494883	−	0.868959i	$-0.335211\pi$
$587$	23.9802	0.989767	0.494883	+	0.868959i	$0.335211\pi$
$588$	0	0
$589$	0	0
$590$	−1.50317	−0.0618845
$591$	0	0
$592$	8.00099	0.328839
$593$	39.5584	1.62447	0.812235	−	0.583330i	$-0.198251\pi$
$593$	39.5584	1.62447	0.812235	+	0.583330i	$0.198251\pi$
$594$	0	0
$595$	−13.8559	−0.568038
$596$	−4.64864	−0.190416
$597$	0	0
$598$	0.146955	0.00600943
$599$	9.96386	0.407112	0.203556	−	0.979063i	$-0.434750\pi$
$599$	9.96386	0.407112	0.203556	+	0.979063i	$0.434750\pi$
$600$	0	0
$601$	−18.4274	−0.751671	−0.375835	−	0.926686i	$-0.622644\pi$
$601$	−18.4274	−0.751671	−0.375835	+	0.926686i	$0.622644\pi$
$602$	−13.8163	−0.563109
$603$	0	0
$604$	13.2727	0.540059
$605$	−11.7997	−0.479725
$606$	0	0
$607$	8.75532	0.355368	0.177684	−	0.984088i	$-0.443140\pi$
$607$	8.75532	0.355368	0.177684	+	0.984088i	$0.443140\pi$
$608$	19.6991	0.798904
$609$	0	0
$610$	0.535906	0.0216982
$611$	10.1712	0.411482
$612$	0	0
$613$	18.1505	0.733092	0.366546	−	0.930400i	$-0.380540\pi$
$613$	18.1505	0.733092	0.366546	+	0.930400i	$0.380540\pi$
$614$	13.3780	0.539893
$615$	0	0
$616$	56.1640	2.26291
$617$	21.7866	0.877094	0.438547	−	0.898708i	$-0.355493\pi$
$617$	21.7866	0.877094	0.438547	+	0.898708i	$0.355493\pi$
$618$	0	0
$619$	31.1895	1.25361	0.626806	−	0.779175i	$-0.284362\pi$
$619$	31.1895	1.25361	0.626806	+	0.779175i	$0.284362\pi$
$620$	0	0
$621$	0	0
$622$	−3.35460	−0.134507
$623$	−7.78121	−0.311748
$624$	0	0
$625$	11.8612	0.474448
$626$	9.78979	0.391279
$627$	0	0
$628$	−23.7777	−0.948833
$629$	−16.4722	−0.656789
$630$	0	0
$631$	25.4446	1.01293	0.506467	−	0.862259i	$-0.330951\pi$
$631$	25.4446	1.01293	0.506467	+	0.862259i	$0.330951\pi$
$632$	−7.75746	−0.308575
$633$	0	0
$634$	−6.51582	−0.258776
$635$	6.58934	0.261490
$636$	0	0
$637$	14.4152	0.571151
$638$	28.1192	1.11325
$639$	0	0
$640$	−10.4929	−0.414769
$641$	−23.3667	−0.922931	−0.461465	−	0.887158i	$-0.652676\pi$
$641$	−23.3667	−0.922931	−0.461465	+	0.887158i	$0.652676\pi$
$642$	0	0
$643$	−35.4360	−1.39746	−0.698729	−	0.715386i	$-0.746251\pi$
$643$	−35.4360	−1.39746	−0.698729	+	0.715386i	$0.746251\pi$
$644$	−1.83824	−0.0724367
$645$	0	0
$646$	−6.84854	−0.269452
$647$	22.2654	0.875342	0.437671	−	0.899135i	$-0.355804\pi$
$647$	22.2654	0.875342	0.437671	+	0.899135i	$0.355804\pi$
$648$	0	0
$649$	11.0139	0.432334
$650$	2.42070	0.0949476
$651$	0	0
$652$	1.47435	0.0577399
$653$	44.5179	1.74212	0.871061	−	0.491176i	$-0.163433\pi$
$653$	44.5179	1.74212	0.871061	+	0.491176i	$0.163433\pi$
$654$	0	0
$655$	−15.8203	−0.618149
$656$	11.8632	0.463181
$657$	0	0
$658$	38.3054	1.49330
$659$	−35.8028	−1.39468	−0.697339	−	0.716742i	$-0.745633\pi$
$659$	−35.8028	−1.39468	−0.697339	+	0.716742i	$0.745633\pi$
$660$	0	0
$661$	32.5606	1.26646	0.633231	−	0.773963i	$-0.281728\pi$
$661$	32.5606	1.26646	0.633231	+	0.773963i	$0.281728\pi$
$662$	−17.0433	−0.662407
$663$	0	0
$664$	−27.2422	−1.05720
$665$	−15.9292	−0.617709
$666$	0	0
$667$	−2.11776	−0.0820000
$668$	−26.6136	−1.02971
$669$	0	0
$670$	1.95116	0.0753800
$671$	−3.92665	−0.151587
$672$	0	0
$673$	−22.1828	−0.855083	−0.427541	−	0.903996i	$-0.640620\pi$
$673$	−22.1828	−0.855083	−0.427541	+	0.903996i	$0.640620\pi$
$674$	−10.3249	−0.397700
$675$	0	0
$676$	18.8062	0.723315
$677$	−11.9156	−0.457955	−0.228978	−	0.973432i	$-0.573538\pi$
$677$	−11.9156	−0.457955	−0.228978	+	0.973432i	$0.573538\pi$
$678$	0	0
$679$	−4.74277	−0.182011
$680$	6.88212	0.263917
$681$	0	0
$682$	0	0
$683$	−40.1932	−1.53795	−0.768976	−	0.639278i	$-0.779233\pi$
$683$	−40.1932	−1.53795	−0.768976	+	0.639278i	$0.779233\pi$
$684$	0	0
$685$	−4.25254	−0.162481
$686$	31.2175	1.19189
$687$	0	0
$688$	−6.02533	−0.229714
$689$	10.6644	0.406281
$690$	0	0
$691$	38.9529	1.48184	0.740918	−	0.671595i	$-0.234391\pi$
$691$	38.9529	1.48184	0.740918	+	0.671595i	$0.234391\pi$
$692$	−26.9973	−1.02628
$693$	0	0
$694$	−2.35377	−0.0893480
$695$	8.78116	0.333088
$696$	0	0
$697$	−24.4236	−0.925110
$698$	−10.6794	−0.404221
$699$	0	0
$700$	−30.2802	−1.14448
$701$	−30.0377	−1.13451	−0.567255	−	0.823543i	$-0.691994\pi$
$701$	−30.0377	−1.13451	−0.567255	+	0.823543i	$0.691994\pi$
$702$	0	0
$703$	−18.9370	−0.714221
$704$	−5.12873	−0.193296
$705$	0	0
$706$	1.83065	0.0688973
$707$	−22.6904	−0.853359
$708$	0	0
$709$	37.5066	1.40859	0.704295	−	0.709907i	$-0.251263\pi$
$709$	37.5066	1.40859	0.704295	+	0.709907i	$0.251263\pi$
$710$	2.70366	0.101467
$711$	0	0
$712$	3.86486	0.144842
$713$	0	0
$714$	0	0
$715$	4.07478	0.152388
$716$	15.4049	0.575706
$717$	0	0
$718$	−10.2839	−0.383791
$719$	−9.26528	−0.345537	−0.172768	−	0.984962i	$-0.555271\pi$
$719$	−9.26528	−0.345537	−0.172768	+	0.984962i	$0.555271\pi$
$720$	0	0
$721$	−11.2467	−0.418848
$722$	5.05238	0.188030
$723$	0	0
$724$	−19.5479	−0.726491
$725$	−34.8846	−1.29558
$726$	0	0
$727$	2.94255	0.109133	0.0545665	−	0.998510i	$-0.482622\pi$
$727$	2.94255	0.109133	0.0545665	+	0.998510i	$0.482622\pi$
$728$	−10.2027	−0.378136
$729$	0	0
$730$	−10.0659	−0.372555
$731$	12.4048	0.458806
$732$	0	0
$733$	34.5697	1.27686	0.638430	−	0.769680i	$-0.279584\pi$
$733$	34.5697	1.27686	0.638430	+	0.769680i	$0.279584\pi$
$734$	7.74735	0.285960
$735$	0	0
$736$	1.42929	0.0526842
$737$	−14.2964	−0.526615
$738$	0	0
$739$	23.2016	0.853484	0.426742	−	0.904373i	$-0.359661\pi$
$739$	23.2016	0.853484	0.426742	+	0.904373i	$0.359661\pi$
$740$	8.26997	0.304010
$741$	0	0
$742$	40.1629	1.47443
$743$	29.1855	1.07071	0.535356	−	0.844627i	$-0.320178\pi$
$743$	29.1855	1.07071	0.535356	+	0.844627i	$0.320178\pi$
$744$	0	0
$745$	−2.92275	−0.107081
$746$	19.2750	0.705707
$747$	0	0
$748$	−21.9142	−0.801262
$749$	21.3645	0.780641
$750$	0	0
$751$	−0.860503	−0.0314002	−0.0157001	−	0.999877i	$-0.504998\pi$
$751$	−0.860503	−0.0314002	−0.0157001	+	0.999877i	$0.504998\pi$
$752$	16.7052	0.609175
$753$	0	0
$754$	−5.10808	−0.186025
$755$	8.34497	0.303705
$756$	0	0
$757$	−14.7910	−0.537588	−0.268794	−	0.963198i	$-0.586625\pi$
$757$	−14.7910	−0.537588	−0.268794	+	0.963198i	$0.586625\pi$
$758$	3.47069	0.126061
$759$	0	0
$760$	7.91191	0.286995
$761$	39.1117	1.41780	0.708898	−	0.705311i	$-0.249192\pi$
$761$	39.1117	1.41780	0.708898	+	0.705311i	$0.249192\pi$
$762$	0	0
$763$	93.3379	3.37906
$764$	39.8676	1.44236
$765$	0	0
$766$	−0.0834189	−0.00301405
$767$	−2.00077	−0.0722436
$768$	0	0
$769$	−32.8678	−1.18524	−0.592622	−	0.805480i	$-0.701907\pi$
$769$	−32.8678	−1.18524	−0.592622	+	0.805480i	$0.701907\pi$
$770$	15.3459	0.553029
$771$	0	0
$772$	−34.8664	−1.25487
$773$	−4.26241	−0.153308	−0.0766541	−	0.997058i	$-0.524424\pi$
$773$	−4.26241	−0.153308	−0.0766541	+	0.997058i	$0.524424\pi$
$774$	0	0
$775$	0	0
$776$	2.35569	0.0845644
$777$	0	0
$778$	−5.22360	−0.187275
$779$	−28.0782	−1.00600
$780$	0	0
$781$	−19.8100	−0.708859
$782$	−0.496902	−0.0177692
$783$	0	0
$784$	23.6755	0.845555
$785$	−14.9498	−0.533581
$786$	0	0
$787$	−29.5817	−1.05447	−0.527236	−	0.849719i	$-0.676772\pi$
$787$	−29.5817	−1.05447	−0.527236	+	0.849719i	$0.676772\pi$
$788$	−5.40605	−0.192583
$789$	0	0
$790$	−2.11960	−0.0754121
$791$	19.9946	0.710926
$792$	0	0
$793$	0.713309	0.0253304
$794$	1.93906	0.0688148
$795$	0	0
$796$	−8.37444	−0.296824
$797$	17.1036	0.605840	0.302920	−	0.953016i	$-0.402038\pi$
$797$	17.1036	0.605840	0.302920	+	0.953016i	$0.402038\pi$
$798$	0	0
$799$	−34.3920	−1.21670
$800$	23.5438	0.832398
$801$	0	0
$802$	−16.7811	−0.592562
$803$	73.7540	2.60272
$804$	0	0
$805$	−1.15576	−0.0407351
$806$	0	0
$807$	0	0
$808$	11.2701	0.396481
$809$	−19.6488	−0.690815	−0.345407	−	0.938453i	$-0.612259\pi$
$809$	−19.6488	−0.690815	−0.345407	+	0.938453i	$0.612259\pi$
$810$	0	0
$811$	−15.7253	−0.552191	−0.276095	−	0.961130i	$-0.589041\pi$
$811$	−15.7253	−0.552191	−0.276095	+	0.961130i	$0.589041\pi$
$812$	63.8963	2.24232
$813$	0	0
$814$	18.2435	0.639435
$815$	0.926969	0.0324703
$816$	0	0
$817$	14.2609	0.498926
$818$	21.4916	0.751437
$819$	0	0
$820$	12.2620	0.428209
$821$	14.3502	0.500824	0.250412	−	0.968139i	$-0.419434\pi$
$821$	14.3502	0.500824	0.250412	+	0.968139i	$0.419434\pi$
$822$	0	0
$823$	−37.8189	−1.31829	−0.659143	−	0.752018i	$-0.729081\pi$
$823$	−37.8189	−1.31829	−0.659143	+	0.752018i	$0.729081\pi$
$824$	5.58612	0.194602
$825$	0	0
$826$	−7.53505	−0.262178
$827$	−0.274555	−0.00954721	−0.00477361	−	0.999989i	$-0.501519\pi$
$827$	−0.274555	−0.00954721	−0.00477361	+	0.999989i	$0.501519\pi$
$828$	0	0
$829$	−32.0155	−1.11194	−0.555972	−	0.831201i	$-0.687654\pi$
$829$	−32.0155	−1.11194	−0.555972	+	0.831201i	$0.687654\pi$
$830$	−7.44350	−0.258368
$831$	0	0
$832$	0.931678	0.0323001
$833$	−48.7424	−1.68883
$834$	0	0
$835$	−16.7328	−0.579063
$836$	−25.1932	−0.871327
$837$	0	0
$838$	18.0350	0.623009
$839$	−2.54412	−0.0878327	−0.0439164	−	0.999035i	$-0.513984\pi$
$839$	−2.54412	−0.0878327	−0.0439164	+	0.999035i	$0.513984\pi$
$840$	0	0
$841$	44.6123	1.53836
$842$	−3.49183	−0.120336
$843$	0	0
$844$	−20.2818	−0.698130
$845$	11.8241	0.406760
$846$	0	0
$847$	−59.1491	−2.03239
$848$	17.5152	0.601475
$849$	0	0
$850$	−8.18516	−0.280749
$851$	−1.37399	−0.0470997
$852$	0	0
$853$	−10.0609	−0.344480	−0.172240	−	0.985055i	$-0.555100\pi$
$853$	−10.0609	−0.344480	−0.172240	+	0.985055i	$0.555100\pi$
$854$	2.68638	0.0919259
$855$	0	0
$856$	−10.6115	−0.362695
$857$	−41.1782	−1.40662	−0.703311	−	0.710882i	$-0.748296\pi$
$857$	−41.1782	−1.40662	−0.703311	+	0.710882i	$0.748296\pi$
$858$	0	0
$859$	22.9225	0.782106	0.391053	−	0.920368i	$-0.372111\pi$
$859$	22.9225	0.782106	0.391053	+	0.920368i	$0.372111\pi$
$860$	−6.22789	−0.212369
$861$	0	0
$862$	−7.06531	−0.240646
$863$	−0.116956	−0.00398123	−0.00199061	−	0.999998i	$-0.500634\pi$
$863$	−0.116956	−0.00398123	−0.00199061	+	0.999998i	$0.500634\pi$
$864$	0	0
$865$	−16.9741	−0.577135
$866$	−7.66420	−0.260440
$867$	0	0
$868$	0	0
$869$	15.5306	0.526839
$870$	0	0
$871$	2.59706	0.0879982
$872$	−46.3602	−1.56995
$873$	0	0
$874$	−0.571255	−0.0193230
$875$	−42.4500	−1.43507
$876$	0	0
$877$	−5.62462	−0.189930	−0.0949650	−	0.995481i	$-0.530274\pi$
$877$	−5.62462	−0.189930	−0.0949650	+	0.995481i	$0.530274\pi$
$878$	14.4175	0.486566
$879$	0	0
$880$	6.69242	0.225602
$881$	17.5385	0.590889	0.295444	−	0.955360i	$-0.404532\pi$
$881$	17.5385	0.590889	0.295444	+	0.955360i	$0.404532\pi$
$882$	0	0
$883$	−34.2506	−1.15263	−0.576313	−	0.817229i	$-0.695509\pi$
$883$	−34.2506	−1.15263	−0.576313	+	0.817229i	$0.695509\pi$
$884$	3.98090	0.133892
$885$	0	0
$886$	−5.35728	−0.179981
$887$	−6.57610	−0.220804	−0.110402	−	0.993887i	$-0.535214\pi$
$887$	−6.57610	−0.220804	−0.110402	+	0.993887i	$0.535214\pi$
$888$	0	0
$889$	33.0309	1.10782
$890$	1.05601	0.0353976
$891$	0	0
$892$	−41.4591	−1.38815
$893$	−39.5382	−1.32310
$894$	0	0
$895$	9.68552	0.323751
$896$	−52.5986	−1.75720
$897$	0	0
$898$	−19.8752	−0.663245
$899$	0	0
$900$	0	0
$901$	−36.0598	−1.20133
$902$	27.0500	0.900666
$903$	0	0
$904$	−9.93115	−0.330305
$905$	−12.2904	−0.408546
$906$	0	0
$907$	−9.28071	−0.308161	−0.154080	−	0.988058i	$-0.549241\pi$
$907$	−9.28071	−0.308161	−0.154080	+	0.988058i	$0.549241\pi$
$908$	24.8045	0.823166
$909$	0	0
$910$	−2.78772	−0.0924119
$911$	40.1965	1.33177	0.665885	−	0.746054i	$-0.268054\pi$
$911$	40.1965	1.33177	0.665885	+	0.746054i	$0.268054\pi$
$912$	0	0
$913$	54.5395	1.80499
$914$	5.60376	0.185356
$915$	0	0
$916$	−1.05667	−0.0349134
$917$	−79.3034	−2.61883
$918$	0	0
$919$	−38.9886	−1.28612	−0.643058	−	0.765817i	$-0.722335\pi$
$919$	−38.9886	−1.28612	−0.643058	+	0.765817i	$0.722335\pi$
$920$	0.574056	0.0189261
$921$	0	0
$922$	20.3467	0.670084
$923$	3.59866	0.118451
$924$	0	0
$925$	−22.6329	−0.744164
$926$	−26.1825	−0.860411
$927$	0	0
$928$	−49.6813	−1.63087
$929$	52.3078	1.71616	0.858082	−	0.513512i	$-0.171656\pi$
$929$	52.3078	1.71616	0.858082	+	0.513512i	$0.171656\pi$
$930$	0	0
$931$	−56.0359	−1.83650
$932$	23.0534	0.755139
$933$	0	0
$934$	7.29481	0.238694
$935$	−13.7781	−0.450594
$936$	0	0
$937$	−1.97424	−0.0644957	−0.0322479	−	0.999480i	$-0.510267\pi$
$937$	−1.97424	−0.0644957	−0.0322479	+	0.999480i	$0.510267\pi$
$938$	9.78074	0.319352
$939$	0	0
$940$	17.2668	0.563180
$941$	23.4133	0.763250	0.381625	−	0.924317i	$-0.375365\pi$
$941$	23.4133	0.763250	0.381625	+	0.924317i	$0.375365\pi$
$942$	0	0
$943$	−2.03724	−0.0663415
$944$	−3.28607	−0.106952
$945$	0	0
$946$	−13.7387	−0.446684
$947$	−31.7480	−1.03167	−0.515836	−	0.856688i	$-0.672518\pi$
$947$	−31.7480	−1.03167	−0.515836	+	0.856688i	$0.672518\pi$
$948$	0	0
$949$	−13.3980	−0.434919
$950$	−9.40993	−0.305298
$951$	0	0
$952$	34.4985	1.11810
$953$	2.70185	0.0875214	0.0437607	−	0.999042i	$-0.486066\pi$
$953$	2.70185	0.0875214	0.0437607	+	0.999042i	$0.486066\pi$
$954$	0	0
$955$	25.0660	0.811118
$956$	15.7137	0.508218
$957$	0	0
$958$	12.8338	0.414642
$959$	−21.3170	−0.688362
$960$	0	0
$961$	0	0
$962$	−3.31409	−0.106851
$963$	0	0
$964$	−21.9873	−0.708163
$965$	−21.9216	−0.705682
$966$	0	0
$967$	5.72961	0.184252	0.0921260	−	0.995747i	$-0.470634\pi$
$967$	5.72961	0.184252	0.0921260	+	0.995747i	$0.470634\pi$
$968$	29.3789	0.944272
$969$	0	0
$970$	0.643656	0.0206666
$971$	−19.5288	−0.626708	−0.313354	−	0.949636i	$-0.601453\pi$
$971$	−19.5288	−0.626708	−0.313354	+	0.949636i	$0.601453\pi$
$972$	0	0
$973$	44.0179	1.41115
$974$	0.936170	0.0299968
$975$	0	0
$976$	1.17154	0.0375001
$977$	−10.1317	−0.324143	−0.162071	−	0.986779i	$-0.551817\pi$
$977$	−10.1317	−0.324143	−0.162071	+	0.986779i	$0.551817\pi$
$978$	0	0
$979$	−7.73753	−0.247293
$980$	24.4715	0.781712
$981$	0	0
$982$	2.00727	0.0640547
$983$	27.0850	0.863876	0.431938	−	0.901903i	$-0.357830\pi$
$983$	27.0850	0.863876	0.431938	+	0.901903i	$0.357830\pi$
$984$	0	0
$985$	−3.39896	−0.108300
$986$	17.2721	0.550055
$987$	0	0
$988$	4.57657	0.145600
$989$	1.03471	0.0329019
$990$	0	0
$991$	−20.7450	−0.658986	−0.329493	−	0.944158i	$-0.606878\pi$
$991$	−20.7450	−0.658986	−0.329493	+	0.944158i	$0.606878\pi$
$992$	0	0
$993$	0	0
$994$	13.5528	0.429870
$995$	−5.26528	−0.166921
$996$	0	0
$997$	15.5965	0.493946	0.246973	−	0.969022i	$-0.420564\pi$
$997$	15.5965	0.493946	0.246973	+	0.969022i	$0.420564\pi$
$998$	8.20548	0.259740
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bd.1.4		8
3.2	odd	2		2883.2.a.r.1.5	yes	8
31.30	odd	2		8649.2.a.bc.1.4		8
93.92	even	2		2883.2.a.q.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.q.1.5	✓	8	93.92	even	2
2883.2.a.r.1.5	yes	8	3.2	odd	2
8649.2.a.bc.1.4		8	31.30	odd	2
8649.2.a.bd.1.4		8	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-0.680299 q^{2} -1.53719 q^{4} -0.966483 q^{5} -4.84476 q^{7} +2.40635 q^{8} +O(q^{10})\)	\(q-0.680299 q^{2} -1.53719 q^{4} -0.966483 q^{5} -4.84476 q^{7} +2.40635 q^{8} +0.657497 q^{10} -4.81756 q^{11} +0.875151 q^{13} +3.29588 q^{14} +1.43735 q^{16} -2.95917 q^{17} -3.40195 q^{19} +1.48567 q^{20} +3.27738 q^{22} -0.246832 q^{23} -4.06591 q^{25} -0.595364 q^{26} +7.44733 q^{28} +8.57976 q^{29} -5.79053 q^{32} +2.01312 q^{34} +4.68237 q^{35} +5.56649 q^{37} +2.31435 q^{38} -2.32569 q^{40} +8.25354 q^{41} -4.19197 q^{43} +7.40552 q^{44} +0.167919 q^{46} +11.6222 q^{47} +16.4717 q^{49} +2.76604 q^{50} -1.34528 q^{52} +12.1858 q^{53} +4.65609 q^{55} -11.6582 q^{56} -5.83681 q^{58} -2.28620 q^{59} +0.815070 q^{61} +1.06459 q^{64} -0.845818 q^{65} +2.96756 q^{67} +4.54881 q^{68} -3.18541 q^{70} +4.11205 q^{71} -15.3094 q^{73} -3.78688 q^{74} +5.22946 q^{76} +23.3399 q^{77} -3.22375 q^{79} -1.38917 q^{80} -5.61487 q^{82} -11.3210 q^{83} +2.85998 q^{85} +2.85180 q^{86} -11.5927 q^{88} +1.60611 q^{89} -4.23989 q^{91} +0.379428 q^{92} -7.90657 q^{94} +3.28793 q^{95} +0.978949 q^{97} -11.2057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 4 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q - 4 * q^2 + 4 * q^4 - 8 * q^7	\( 8 q - 4 q^{2} + 4 q^{4} - 8 q^{7} - 20 q^{10} + 8 q^{11} - 16 q^{13} + 12 q^{14} + 4 q^{16} - 16 q^{19} + 12 q^{20} + 16 q^{23} + 8 q^{25} + 8 q^{26} - 4 q^{28} + 16 q^{29} - 4 q^{32} - 16 q^{34} + 24 q^{35} - 16 q^{37} + 12 q^{38} - 16 q^{40} - 32 q^{43} + 24 q^{44} - 16 q^{46} + 8 q^{47} + 40 q^{49} - 8 q^{50} - 32 q^{52} + 16 q^{53} + 16 q^{55} + 24 q^{56} + 16 q^{58} - 24 q^{59} - 48 q^{61} - 16 q^{64} + 24 q^{65} - 40 q^{67} - 24 q^{68} + 4 q^{70} - 16 q^{71} - 48 q^{73} + 48 q^{74} - 20 q^{76} + 48 q^{77} + 36 q^{80} + 52 q^{82} + 24 q^{83} - 16 q^{88} + 24 q^{89} + 16 q^{91} + 80 q^{92} - 24 q^{94} - 40 q^{95} - 64 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 4 * q^4 - 8 * q^7 - 20 * q^10 + 8 * q^11 - 16 * q^13 + 12 * q^14 + 4 * q^16 - 16 * q^19 + 12 * q^20 + 16 * q^23 + 8 * q^25 + 8 * q^26 - 4 * q^28 + 16 * q^29 - 4 * q^32 - 16 * q^34 + 24 * q^35 - 16 * q^37 + 12 * q^38 - 16 * q^40 - 32 * q^43 + 24 * q^44 - 16 * q^46 + 8 * q^47 + 40 * q^49 - 8 * q^50 - 32 * q^52 + 16 * q^53 + 16 * q^55 + 24 * q^56 + 16 * q^58 - 24 * q^59 - 48 * q^61 - 16 * q^64 + 24 * q^65 - 40 * q^67 - 24 * q^68 + 4 * q^70 - 16 * q^71 - 48 * q^73 + 48 * q^74 - 20 * q^76 + 48 * q^77 + 36 * q^80 + 52 * q^82 + 24 * q^83 - 16 * q^88 + 24 * q^89 + 16 * q^91 + 80 * q^92 - 24 * q^94 - 40 * q^95 - 64 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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