LMFDB - Embedded newform 3025.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.1480160000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11 $ x^8 - 9x^6 + 27x^4 - 31*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.65458$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65458 q^{2} -1.97479 q^{3} +0.737640 q^{4} +3.26745 q^{6} +2.24307 q^{7} +2.08868 q^{8} +0.899788 q^{9} +O(q^{10})$	\(q-1.65458 q^{2} -1.97479 q^{3} +0.737640 q^{4} +3.26745 q^{6} +2.24307 q^{7} +2.08868 q^{8} +0.899788 q^{9} -1.45668 q^{12} +3.69976 q^{13} -3.71135 q^{14} -4.93117 q^{16} +2.22461 q^{17} -1.48877 q^{18} -5.28684 q^{19} -4.42960 q^{21} -3.85415 q^{23} -4.12469 q^{24} -6.12155 q^{26} +4.14747 q^{27} +1.65458 q^{28} -0.188439 q^{29} +0.686867 q^{31} +3.98166 q^{32} -3.68079 q^{34} +0.663720 q^{36} +2.59316 q^{37} +8.74751 q^{38} -7.30624 q^{39} -7.91604 q^{41} +7.32913 q^{42} -8.41368 q^{43} +6.37701 q^{46} -12.0132 q^{47} +9.73801 q^{48} -1.96862 q^{49} -4.39313 q^{51} +2.72909 q^{52} +12.6566 q^{53} -6.86233 q^{54} +4.68506 q^{56} +10.4404 q^{57} +0.311788 q^{58} -0.343688 q^{59} -1.73338 q^{61} -1.13648 q^{62} +2.01829 q^{63} +3.27435 q^{64} +0.650461 q^{67} +1.64096 q^{68} +7.61114 q^{69} +4.64760 q^{71} +1.87937 q^{72} +8.85841 q^{73} -4.29059 q^{74} -3.89979 q^{76} +12.0888 q^{78} +7.23426 q^{79} -10.8897 q^{81} +13.0977 q^{82} -3.18165 q^{83} -3.26745 q^{84} +13.9211 q^{86} +0.372127 q^{87} +9.92195 q^{89} +8.29883 q^{91} -2.84298 q^{92} -1.35642 q^{93} +19.8768 q^{94} -7.86294 q^{96} +2.26811 q^{97} +3.25724 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10}) $ 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9	$ 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} - 12 q^{19} - 4 q^{21} - 2 q^{24} + 10 q^{26} - 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} - 34 q^{41} - 24 q^{46} - 30 q^{49} - 54 q^{51} - 20 q^{54} - 10 q^{56} - 6 q^{59} - 20 q^{61} + 14 q^{64} + 32 q^{69} - 42 q^{71} + 4 q^{74} - 28 q^{76} - 16 q^{79} - 36 q^{81} + 6 q^{84} + 46 q^{86} - 12 q^{89} + 20 q^{91} + 42 q^{94} + 8 q^{96}+O(q^{100}) $ 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9 - 4 * q^14 - 22 * q^16 - 12 * q^19 - 4 * q^21 - 2 * q^24 + 10 * q^26 - 24 * q^29 + 14 * q^31 + 8 * q^34 + 20 * q^36 - 30 * q^39 - 34 * q^41 - 24 * q^46 - 30 * q^49 - 54 * q^51 - 20 * q^54 - 10 * q^56 - 6 * q^59 - 20 * q^61 + 14 * q^64 + 32 * q^69 - 42 * q^71 + 4 * q^74 - 28 * q^76 - 16 * q^79 - 36 * q^81 + 6 * q^84 + 46 * q^86 - 12 * q^89 + 20 * q^91 + 42 * q^94 + 8 * q^96

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 $

$2$	−1.65458	−1.16997	−0.584983	−	0.811046i	$-0.698899\pi$
$2$	−1.65458	−1.16997	−0.584983	+	0.811046i	$0.698899\pi$
$3$	−1.97479	−1.14014	−0.570072	−	0.821595i	$-0.693085\pi$
$3$	−1.97479	−1.14014	−0.570072	+	0.821595i	$0.693085\pi$
$4$	0.737640	0.368820
$5$	0	0
$5$	0	0
$6$	3.26745	1.33393
$7$	2.24307	0.847802	0.423901	−	0.905708i	$-0.360660\pi$
$7$	2.24307	0.847802	0.423901	+	0.905708i	$0.360660\pi$
$8$	2.08868	0.738459
$9$	0.899788	0.299929
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.45668	−0.420508
$13$	3.69976	1.02613	0.513064	−	0.858350i	$-0.328510\pi$
$13$	3.69976	1.02613	0.513064	+	0.858350i	$0.328510\pi$
$14$	−3.71135	−0.991900
$15$	0	0
$16$	−4.93117	−1.23279
$17$	2.22461	0.539546	0.269773	−	0.962924i	$-0.413051\pi$
$17$	2.22461	0.539546	0.269773	+	0.962924i	$0.413051\pi$
$18$	−1.48877	−0.350907
$19$	−5.28684	−1.21288	−0.606442	−	0.795127i	$-0.707404\pi$
$19$	−5.28684	−1.21288	−0.606442	+	0.795127i	$0.707404\pi$
$20$	0	0
$21$	−4.42960	−0.966617
$22$	0	0
$23$	−3.85415	−0.803647	−0.401823	−	0.915717i	$-0.631623\pi$
$23$	−3.85415	−0.803647	−0.401823	+	0.915717i	$0.631623\pi$
$24$	−4.12469	−0.841950
$25$	0	0
$26$	−6.12155	−1.20053
$27$	4.14747	0.798182
$28$	1.65458	0.312687
$29$	−0.188439	−0.0349922	−0.0174961	−	0.999847i	$-0.505569\pi$
$29$	−0.188439	−0.0349922	−0.0174961	+	0.999847i	$0.505569\pi$
$30$	0	0
$31$	0.686867	0.123365	0.0616824	−	0.998096i	$-0.480353\pi$
$31$	0.686867	0.123365	0.0616824	+	0.998096i	$0.480353\pi$
$32$	3.98166	0.703866
$33$	0	0
$34$	−3.68079	−0.631251
$35$	0	0
$36$	0.663720	0.110620
$37$	2.59316	0.426313	0.213156	−	0.977018i	$-0.431626\pi$
$37$	2.59316	0.426313	0.213156	+	0.977018i	$0.431626\pi$
$38$	8.74751	1.41903
$39$	−7.30624	−1.16993
$40$	0	0
$41$	−7.91604	−1.23628	−0.618139	−	0.786069i	$-0.712113\pi$
$41$	−7.91604	−1.23628	−0.618139	+	0.786069i	$0.712113\pi$
$42$	7.32913	1.13091
$43$	−8.41368	−1.28307	−0.641537	−	0.767092i	$-0.721703\pi$
$43$	−8.41368	−1.28307	−0.641537	+	0.767092i	$0.721703\pi$
$44$	0	0
$45$	0	0
$46$	6.37701	0.940239
$47$	−12.0132	−1.75230	−0.876151	−	0.482037i	$-0.839897\pi$
$47$	−12.0132	−1.75230	−0.876151	+	0.482037i	$0.839897\pi$
$48$	9.73801	1.40556
$49$	−1.96862	−0.281231
$50$	0	0
$51$	−4.39313	−0.615161
$52$	2.72909	0.378457
$53$	12.6566	1.73851	0.869257	−	0.494360i	$-0.164597\pi$
$53$	12.6566	1.73851	0.869257	+	0.494360i	$0.164597\pi$
$54$	−6.86233	−0.933845
$55$	0	0
$56$	4.68506	0.626067
$57$	10.4404	1.38286
$58$	0.311788	0.0409397
$59$	−0.343688	−0.0447444	−0.0223722	−	0.999750i	$-0.507122\pi$
$59$	−0.343688	−0.0447444	−0.0223722	+	0.999750i	$0.507122\pi$
$60$	0	0
$61$	−1.73338	−0.221936	−0.110968	−	0.993824i	$-0.535395\pi$
$61$	−1.73338	−0.221936	−0.110968	+	0.993824i	$0.535395\pi$
$62$	−1.13648	−0.144333
$63$	2.01829	0.254281
$64$	3.27435	0.409293
$65$	0	0
$66$	0	0
$67$	0.650461	0.0794664	0.0397332	−	0.999210i	$-0.487349\pi$
$67$	0.650461	0.0794664	0.0397332	+	0.999210i	$0.487349\pi$
$68$	1.64096	0.198996
$69$	7.61114	0.916273
$70$	0	0
$71$	4.64760	0.551569	0.275785	−	0.961219i	$-0.411062\pi$
$71$	4.64760	0.551569	0.275785	+	0.961219i	$0.411062\pi$
$72$	1.87937	0.221485
$73$	8.85841	1.03680	0.518399	−	0.855139i	$-0.326528\pi$
$73$	8.85841	1.03680	0.518399	+	0.855139i	$0.326528\pi$
$74$	−4.29059	−0.498772
$75$	0	0
$76$	−3.89979	−0.447336
$77$	0	0
$78$	12.0888	1.36878
$79$	7.23426	0.813918	0.406959	−	0.913447i	$-0.366589\pi$
$79$	7.23426	0.813918	0.406959	+	0.913447i	$0.366589\pi$
$80$	0	0
$81$	−10.8897	−1.20997
$82$	13.0977	1.44640
$83$	−3.18165	−0.349232	−0.174616	−	0.984637i	$-0.555868\pi$
$83$	−3.18165	−0.349232	−0.174616	+	0.984637i	$0.555868\pi$
$84$	−3.26745	−0.356508
$85$	0	0
$86$	13.9211	1.50115
$87$	0.372127	0.0398962
$88$	0	0
$89$	9.92195	1.05172	0.525862	−	0.850570i	$-0.323743\pi$
$89$	9.92195	1.05172	0.525862	+	0.850570i	$0.323743\pi$
$90$	0	0
$91$	8.29883	0.869954
$92$	−2.84298	−0.296401
$93$	−1.35642	−0.140654
$94$	19.8768	2.05013
$95$	0	0
$96$	−7.86294	−0.802508
$97$	2.26811	0.230292	0.115146	−	0.993349i	$-0.463266\pi$
$97$	2.26811	0.230292	0.115146	+	0.993349i	$0.463266\pi$
$98$	3.25724	0.329031
$99$	0	0
$100$	0	0
$101$	−9.89686	−0.984774	−0.492387	−	0.870376i	$-0.663876\pi$
$101$	−9.89686	−0.984774	−0.492387	+	0.870376i	$0.663876\pi$
$102$	7.26879	0.719717
$103$	10.2411	1.00909	0.504544	−	0.863386i	$-0.331661\pi$
$103$	10.2411	1.00909	0.504544	+	0.863386i	$0.331661\pi$
$104$	7.72760	0.757753
$105$	0	0
$106$	−20.9413	−2.03400
$107$	−10.0468	−0.971261	−0.485631	−	0.874164i	$-0.661410\pi$
$107$	−10.0468	−0.971261	−0.485631	+	0.874164i	$0.661410\pi$
$108$	3.05934	0.294386
$109$	8.80173	0.843053	0.421527	−	0.906816i	$-0.361494\pi$
$109$	8.80173	0.843053	0.421527	+	0.906816i	$0.361494\pi$
$110$	0	0
$111$	−5.12094	−0.486058
$112$	−11.0610	−1.04516
$113$	0.231352	0.0217638	0.0108819	−	0.999941i	$-0.496536\pi$
$113$	0.231352	0.0217638	0.0108819	+	0.999941i	$0.496536\pi$
$114$	−17.2745	−1.61790
$115$	0	0
$116$	−0.139000	−0.0129058
$117$	3.32900	0.307766
$118$	0.568660	0.0523494
$119$	4.98996	0.457429
$120$	0	0
$121$	0	0
$122$	2.86801	0.259658
$123$	15.6325	1.40953
$124$	0.506660	0.0454995
$125$	0	0
$126$	−3.33943	−0.297500
$127$	2.43034	0.215658	0.107829	−	0.994169i	$-0.465610\pi$
$127$	2.43034	0.215658	0.107829	+	0.994169i	$0.465610\pi$
$128$	−13.3810	−1.18272
$129$	16.6152	1.46289
$130$	0	0
$131$	−1.58846	−0.138785	−0.0693924	−	0.997589i	$-0.522106\pi$
$131$	−1.58846	−0.138785	−0.0693924	+	0.997589i	$0.522106\pi$
$132$	0	0
$133$	−11.8588	−1.02829
$134$	−1.07624	−0.0929730
$135$	0	0
$136$	4.64649	0.398433
$137$	−18.7019	−1.59781	−0.798905	−	0.601457i	$-0.794587\pi$
$137$	−18.7019	−1.59781	−0.798905	+	0.601457i	$0.794587\pi$
$138$	−12.5932	−1.07201
$139$	−11.6274	−0.986222	−0.493111	−	0.869966i	$-0.664140\pi$
$139$	−11.6274	−0.986222	−0.493111	+	0.869966i	$0.664140\pi$
$140$	0	0
$141$	23.7235	1.99788
$142$	−7.68984	−0.645317
$143$	0	0
$144$	−4.43700	−0.369750
$145$	0	0
$146$	−14.6570	−1.21302
$147$	3.88761	0.320644
$148$	1.91282	0.157233
$149$	−5.91553	−0.484619	−0.242309	−	0.970199i	$-0.577905\pi$
$149$	−5.91553	−0.484619	−0.242309	+	0.970199i	$0.577905\pi$
$150$	0	0
$151$	−12.7779	−1.03985	−0.519924	−	0.854213i	$-0.674040\pi$
$151$	−12.7779	−1.03985	−0.519924	+	0.854213i	$0.674040\pi$
$152$	−11.0425	−0.895665
$153$	2.00167	0.161826
$154$	0	0
$155$	0	0
$156$	−5.38937	−0.431495
$157$	14.3487	1.14515	0.572574	−	0.819853i	$-0.305945\pi$
$157$	14.3487	1.14515	0.572574	+	0.819853i	$0.305945\pi$
$158$	−11.9697	−0.952256
$159$	−24.9941	−1.98216
$160$	0	0
$161$	−8.64515	−0.681333
$162$	18.0180	1.41563
$163$	−3.62716	−0.284101	−0.142051	−	0.989859i	$-0.545370\pi$
$163$	−3.62716	−0.284101	−0.142051	+	0.989859i	$0.545370\pi$
$164$	−5.83919	−0.455964
$165$	0	0
$166$	5.26430	0.408589
$167$	3.82070	0.295655	0.147827	−	0.989013i	$-0.452772\pi$
$167$	3.82070	0.295655	0.147827	+	0.989013i	$0.452772\pi$
$168$	−9.25199	−0.713807
$169$	0.688202	0.0529386
$170$	0	0
$171$	−4.75703	−0.363779
$172$	−6.20627	−0.473224
$173$	−2.10714	−0.160203	−0.0801016	−	0.996787i	$-0.525524\pi$
$173$	−2.10714	−0.160203	−0.0801016	+	0.996787i	$0.525524\pi$
$174$	−0.615714	−0.0466772
$175$	0	0
$176$	0	0
$177$	0.678711	0.0510151
$178$	−16.4167	−1.23048
$179$	5.02397	0.375509	0.187755	−	0.982216i	$-0.439879\pi$
$179$	5.02397	0.375509	0.187755	+	0.982216i	$0.439879\pi$
$180$	0	0
$181$	−15.6476	−1.16308	−0.581539	−	0.813519i	$-0.697549\pi$
$181$	−15.6476	−1.16308	−0.581539	+	0.813519i	$0.697549\pi$
$182$	−13.7311	−1.01782
$183$	3.42305	0.253039
$184$	−8.05008	−0.593460
$185$	0	0
$186$	2.24430	0.164560
$187$	0	0
$188$	−8.86140	−0.646284
$189$	9.30309	0.676700
$190$	0	0
$191$	3.11585	0.225455	0.112728	−	0.993626i	$-0.464041\pi$
$191$	3.11585	0.225455	0.112728	+	0.993626i	$0.464041\pi$
$192$	−6.46614	−0.466653
$193$	9.63638	0.693642	0.346821	−	0.937931i	$-0.387261\pi$
$193$	9.63638	0.693642	0.346821	+	0.937931i	$0.387261\pi$
$194$	−3.75277	−0.269433
$195$	0	0
$196$	−1.45213	−0.103724
$197$	−14.3974	−1.02577	−0.512885	−	0.858457i	$-0.671423\pi$
$197$	−14.3974	−1.02577	−0.512885	+	0.858457i	$0.671423\pi$
$198$	0	0
$199$	14.7978	1.04899	0.524493	−	0.851415i	$-0.324255\pi$
$199$	14.7978	1.04899	0.524493	+	0.851415i	$0.324255\pi$
$200$	0	0
$201$	−1.28452	−0.0906032
$202$	16.3752	1.15215
$203$	−0.422682	−0.0296665
$204$	−3.24055	−0.226884
$205$	0	0
$206$	−16.9448	−1.18060
$207$	−3.46792	−0.241037
$208$	−18.2441	−1.26500
$209$	0	0
$210$	0	0
$211$	6.77147	0.466167	0.233084	−	0.972457i	$-0.425118\pi$
$211$	6.77147	0.466167	0.233084	+	0.972457i	$0.425118\pi$
$212$	9.33600	0.641199
$213$	−9.17803	−0.628868
$214$	16.6233	1.13634
$215$	0	0
$216$	8.66273	0.589424
$217$	1.54069	0.104589
$218$	−14.5632	−0.986344
$219$	−17.4935	−1.18210
$220$	0	0
$221$	8.23051	0.553644
$222$	8.47302	0.568672
$223$	8.71727	0.583752	0.291876	−	0.956456i	$-0.405721\pi$
$223$	8.71727	0.583752	0.291876	+	0.956456i	$0.405721\pi$
$224$	8.93117	0.596739
$225$	0	0
$226$	−0.382791	−0.0254629
$227$	3.80744	0.252708	0.126354	−	0.991985i	$-0.459672\pi$
$227$	3.80744	0.252708	0.126354	+	0.991985i	$0.459672\pi$
$228$	7.70125	0.510028
$229$	−2.71367	−0.179324	−0.0896621	−	0.995972i	$-0.528579\pi$
$229$	−2.71367	−0.179324	−0.0896621	+	0.995972i	$0.528579\pi$
$230$	0	0
$231$	0	0
$232$	−0.393588	−0.0258403
$233$	−10.5108	−0.688584	−0.344292	−	0.938863i	$-0.611881\pi$
$233$	−10.5108	−0.688584	−0.344292	+	0.938863i	$0.611881\pi$
$234$	−5.50809	−0.360075
$235$	0	0
$236$	−0.253518	−0.0165026
$237$	−14.2861	−0.927984
$238$	−8.25629	−0.535176
$239$	−20.0396	−1.29625	−0.648127	−	0.761532i	$-0.724448\pi$
$239$	−20.0396	−1.29625	−0.648127	+	0.761532i	$0.724448\pi$
$240$	0	0
$241$	−28.4450	−1.83230	−0.916152	−	0.400832i	$-0.868721\pi$
$241$	−28.4450	−1.83230	−0.916152	+	0.400832i	$0.868721\pi$
$242$	0	0
$243$	9.06251	0.581361
$244$	−1.27861	−0.0818545
$245$	0	0
$246$	−25.8652	−1.64911
$247$	−19.5600	−1.24457
$248$	1.43464	0.0910999
$249$	6.28309	0.398175
$250$	0	0
$251$	−23.8370	−1.50458	−0.752289	−	0.658833i	$-0.771050\pi$
$251$	−23.8370	−1.50458	−0.752289	+	0.658833i	$0.771050\pi$
$252$	1.48877	0.0937838
$253$	0	0
$254$	−4.02120	−0.252313
$255$	0	0
$256$	15.5913	0.974454
$257$	−24.6763	−1.53927	−0.769633	−	0.638486i	$-0.779561\pi$
$257$	−24.6763	−1.53927	−0.769633	+	0.638486i	$0.779561\pi$
$258$	−27.4913	−1.71153
$259$	5.81665	0.361429
$260$	0	0
$261$	−0.169555	−0.0104952
$262$	2.62824	0.162373
$263$	5.44098	0.335505	0.167753	−	0.985829i	$-0.446349\pi$
$263$	5.44098	0.335505	0.167753	+	0.985829i	$0.446349\pi$
$264$	0	0
$265$	0	0
$266$	19.6213	1.20306
$267$	−19.5937	−1.19912
$268$	0.479806	0.0293088
$269$	−6.70557	−0.408846	−0.204423	−	0.978883i	$-0.565532\pi$
$269$	−6.70557	−0.408846	−0.204423	+	0.978883i	$0.565532\pi$
$270$	0	0
$271$	5.05983	0.307363	0.153681	−	0.988120i	$-0.450887\pi$
$271$	5.05983	0.307363	0.153681	+	0.988120i	$0.450887\pi$
$272$	−10.9699	−0.665148
$273$	−16.3884	−0.991873
$274$	30.9438	1.86938
$275$	0	0
$276$	5.61428	0.337940
$277$	−10.6937	−0.642522	−0.321261	−	0.946991i	$-0.604107\pi$
$277$	−10.6937	−0.642522	−0.321261	+	0.946991i	$0.604107\pi$
$278$	19.2385	1.15385
$279$	0.618034	0.0370007
$280$	0	0
$281$	13.7197	0.818450	0.409225	−	0.912434i	$-0.365799\pi$
$281$	13.7197	0.818450	0.409225	+	0.912434i	$0.365799\pi$
$282$	−39.2524	−2.33745
$283$	−21.9693	−1.30594	−0.652969	−	0.757385i	$-0.726477\pi$
$283$	−21.9693	−1.30594	−0.652969	+	0.757385i	$0.726477\pi$
$284$	3.42826	0.203430
$285$	0	0
$286$	0	0
$287$	−17.7563	−1.04812
$288$	3.58265	0.211110
$289$	−12.0511	−0.708890
$290$	0	0
$291$	−4.47903	−0.262566
$292$	6.53432	0.382392
$293$	14.0380	0.820110	0.410055	−	0.912061i	$-0.365509\pi$
$293$	14.0380	0.820110	0.410055	+	0.912061i	$0.365509\pi$
$294$	−6.43236	−0.375143
$295$	0	0
$296$	5.41627	0.314815
$297$	0	0
$298$	9.78772	0.566988
$299$	−14.2594	−0.824644
$300$	0	0
$301$	−18.8725	−1.08779
$302$	21.1420	1.21659
$303$	19.5442	1.12278
$304$	26.0703	1.49523
$305$	0	0
$306$	−3.31193	−0.189331
$307$	−6.86951	−0.392064	−0.196032	−	0.980598i	$-0.562806\pi$
$307$	−6.86951	−0.392064	−0.196032	+	0.980598i	$0.562806\pi$
$308$	0	0
$309$	−20.2241	−1.15051
$310$	0	0
$311$	−5.50157	−0.311966	−0.155983	−	0.987760i	$-0.549854\pi$
$311$	−5.50157	−0.311966	−0.155983	+	0.987760i	$0.549854\pi$
$312$	−15.2604	−0.863948
$313$	−14.2320	−0.804440	−0.402220	−	0.915543i	$-0.631761\pi$
$313$	−14.2320	−0.804440	−0.402220	+	0.915543i	$0.631761\pi$
$314$	−23.7411	−1.33979
$315$	0	0
$316$	5.33628	0.300189
$317$	−18.6864	−1.04953	−0.524767	−	0.851246i	$-0.675847\pi$
$317$	−18.6864	−1.04953	−0.524767	+	0.851246i	$0.675847\pi$
$318$	41.3547	2.31906
$319$	0	0
$320$	0	0
$321$	19.8403	1.10738
$322$	14.3041	0.797137
$323$	−11.7611	−0.654408
$324$	−8.03271	−0.446262
$325$	0	0
$326$	6.00143	0.332389
$327$	−17.3816	−0.961202
$328$	−16.5340	−0.912940
$329$	−26.9464	−1.48560
$330$	0	0
$331$	0.468249	0.0257373	0.0128686	−	0.999917i	$-0.495904\pi$
$331$	0.468249	0.0257373	0.0128686	+	0.999917i	$0.495904\pi$
$332$	−2.34691	−0.128804
$333$	2.33329	0.127864
$334$	−6.32166	−0.345906
$335$	0	0
$336$	21.8431	1.19164
$337$	34.0872	1.85685	0.928424	−	0.371522i	$-0.121164\pi$
$337$	34.0872	1.85685	0.928424	+	0.371522i	$0.121164\pi$
$338$	−1.13869	−0.0619363
$339$	−0.456871	−0.0248139
$340$	0	0
$341$	0	0
$342$	7.87090	0.425610
$343$	−20.1173	−1.08623
$344$	−17.5735	−0.947498
$345$	0	0
$346$	3.48644	0.187432
$347$	3.59292	0.192878	0.0964391	−	0.995339i	$-0.469255\pi$
$347$	3.59292	0.192878	0.0964391	+	0.995339i	$0.469255\pi$
$348$	0.274496	0.0147145
$349$	6.37110	0.341037	0.170519	−	0.985354i	$-0.445456\pi$
$349$	6.37110	0.341037	0.170519	+	0.985354i	$0.445456\pi$
$350$	0	0
$351$	15.3446	0.819037
$352$	0	0
$353$	12.1971	0.649186	0.324593	−	0.945854i	$-0.394773\pi$
$353$	12.1971	0.649186	0.324593	+	0.945854i	$0.394773\pi$
$354$	−1.12298	−0.0596859
$355$	0	0
$356$	7.31883	0.387897
$357$	−9.85411	−0.521535
$358$	−8.31257	−0.439333
$359$	−24.1149	−1.27273	−0.636367	−	0.771386i	$-0.719564\pi$
$359$	−24.1149	−1.27273	−0.636367	+	0.771386i	$0.719564\pi$
$360$	0	0
$361$	8.95069	0.471089
$362$	25.8902	1.36076
$363$	0	0
$364$	6.12155	0.320856
$365$	0	0
$366$	−5.66372	−0.296047
$367$	20.3899	1.06435	0.532173	−	0.846636i	$-0.321376\pi$
$367$	20.3899	1.06435	0.532173	+	0.846636i	$0.321376\pi$
$368$	19.0055	0.990729
$369$	−7.12275	−0.370796
$370$	0	0
$371$	28.3896	1.47392
$372$	−1.00055	−0.0518759
$373$	7.51997	0.389369	0.194685	−	0.980866i	$-0.437632\pi$
$373$	7.51997	0.389369	0.194685	+	0.980866i	$0.437632\pi$
$374$	0	0
$375$	0	0
$376$	−25.0916	−1.29400
$377$	−0.697178	−0.0359065
$378$	−15.3927	−0.791716
$379$	23.1912	1.19125	0.595627	−	0.803261i	$-0.296904\pi$
$379$	23.1912	1.19125	0.595627	+	0.803261i	$0.296904\pi$
$380$	0	0
$381$	−4.79942	−0.245881
$382$	−5.15543	−0.263775
$383$	−2.44039	−0.124698	−0.0623491	−	0.998054i	$-0.519859\pi$
$383$	−2.44039	−0.124698	−0.0623491	+	0.998054i	$0.519859\pi$
$384$	26.4246	1.34848
$385$	0	0
$386$	−15.9442	−0.811537
$387$	−7.57053	−0.384832
$388$	1.67305	0.0849362
$389$	33.9732	1.72251	0.861254	−	0.508175i	$-0.169680\pi$
$389$	33.9732	1.72251	0.861254	+	0.508175i	$0.169680\pi$
$390$	0	0
$391$	−8.57398	−0.433605
$392$	−4.11181	−0.207678
$393$	3.13688	0.158235
$394$	23.8216	1.20012
$395$	0	0
$396$	0	0
$397$	27.4961	1.37999	0.689995	−	0.723814i	$-0.257613\pi$
$397$	27.4961	1.37999	0.689995	+	0.723814i	$0.257613\pi$
$398$	−24.4841	−1.22728
$399$	23.4186	1.17239
$400$	0	0
$401$	−1.88743	−0.0942535	−0.0471268	−	0.998889i	$-0.515006\pi$
$401$	−1.88743	−0.0942535	−0.0471268	+	0.998889i	$0.515006\pi$
$402$	2.12535	0.106003
$403$	2.54124	0.126588
$404$	−7.30032	−0.363205
$405$	0	0
$406$	0.699363	0.0347088
$407$	0	0
$408$	−9.17582	−0.454271
$409$	−13.5575	−0.670374	−0.335187	−	0.942152i	$-0.608799\pi$
$409$	−13.5575	−0.670374	−0.335187	+	0.942152i	$0.608799\pi$
$410$	0	0
$411$	36.9323	1.82173
$412$	7.55427	0.372172
$413$	−0.770918	−0.0379344
$414$	5.73796	0.282005
$415$	0	0
$416$	14.7312	0.722256
$417$	22.9616	1.12444
$418$	0	0
$419$	−22.1368	−1.08145	−0.540727	−	0.841198i	$-0.681851\pi$
$419$	−22.1368	−1.08145	−0.540727	+	0.841198i	$0.681851\pi$
$420$	0	0
$421$	17.9026	0.872517	0.436259	−	0.899821i	$-0.356303\pi$
$421$	17.9026	0.872517	0.436259	+	0.899821i	$0.356303\pi$
$422$	−11.2040	−0.545400
$423$	−10.8093	−0.525566
$424$	26.4355	1.28382
$425$	0	0
$426$	15.1858	0.735755
$427$	−3.88809	−0.188158
$428$	−7.41093	−0.358221
$429$	0	0
$430$	0	0
$431$	−33.4457	−1.61102	−0.805510	−	0.592582i	$-0.798109\pi$
$431$	−33.4457	−1.61102	−0.805510	+	0.592582i	$0.798109\pi$
$432$	−20.4519	−0.983992
$433$	−31.4914	−1.51338	−0.756690	−	0.653774i	$-0.773185\pi$
$433$	−31.4914	−1.51338	−0.756690	+	0.653774i	$0.773185\pi$
$434$	−2.54920	−0.122366
$435$	0	0
$436$	6.49251	0.310935
$437$	20.3763	0.974731
$438$	28.9444	1.38302
$439$	−35.6208	−1.70009	−0.850045	−	0.526710i	$-0.823425\pi$
$439$	−35.6208	−1.70009	−0.850045	+	0.526710i	$0.823425\pi$
$440$	0	0
$441$	−1.77134	−0.0843495
$442$	−13.6180	−0.647744
$443$	−23.4876	−1.11593	−0.557964	−	0.829865i	$-0.688417\pi$
$443$	−23.4876	−1.11593	−0.557964	+	0.829865i	$0.688417\pi$
$444$	−3.77741	−0.179268
$445$	0	0
$446$	−14.4234	−0.682970
$447$	11.6819	0.552535
$448$	7.34460	0.347000
$449$	−31.3920	−1.48148	−0.740740	−	0.671792i	$-0.765525\pi$
$449$	−31.3920	−1.48148	−0.740740	+	0.671792i	$0.765525\pi$
$450$	0	0
$451$	0	0
$452$	0.170655	0.00802692
$453$	25.2336	1.18558
$454$	−6.29971	−0.295660
$455$	0	0
$456$	21.8066	1.02119
$457$	39.1106	1.82952	0.914759	−	0.404000i	$-0.132380\pi$
$457$	39.1106	1.82952	0.914759	+	0.404000i	$0.132380\pi$
$458$	4.48999	0.209803
$459$	9.22650	0.430656
$460$	0	0
$461$	8.88399	0.413769	0.206884	−	0.978365i	$-0.433668\pi$
$461$	8.88399	0.413769	0.206884	+	0.978365i	$0.433668\pi$
$462$	0	0
$463$	−4.21081	−0.195693	−0.0978464	−	0.995202i	$-0.531195\pi$
$463$	−4.21081	−0.195693	−0.0978464	+	0.995202i	$0.531195\pi$
$464$	0.929224	0.0431381
$465$	0	0
$466$	17.3909	0.805620
$467$	6.72844	0.311355	0.155677	−	0.987808i	$-0.450244\pi$
$467$	6.72844	0.311355	0.155677	+	0.987808i	$0.450244\pi$
$468$	2.45560	0.113510
$469$	1.45903	0.0673718
$470$	0	0
$471$	−28.3356	−1.30564
$472$	−0.717854	−0.0330419
$473$	0	0
$474$	23.6376	1.08571
$475$	0	0
$476$	3.68079	0.168709
$477$	11.3882	0.521431
$478$	33.1572	1.51657
$479$	−20.8094	−0.950806	−0.475403	−	0.879768i	$-0.657698\pi$
$479$	−20.8094	−0.950806	−0.475403	+	0.879768i	$0.657698\pi$
$480$	0	0
$481$	9.59406	0.437452
$482$	47.0646	2.14373
$483$	17.0723	0.776818
$484$	0	0
$485$	0	0
$486$	−14.9947	−0.680172
$487$	15.7794	0.715032	0.357516	−	0.933907i	$-0.383624\pi$
$487$	15.7794	0.715032	0.357516	+	0.933907i	$0.383624\pi$
$488$	−3.62047	−0.163891
$489$	7.16287	0.323916
$490$	0	0
$491$	−19.3303	−0.872366	−0.436183	−	0.899858i	$-0.643670\pi$
$491$	−19.3303	−0.872366	−0.436183	+	0.899858i	$0.643670\pi$
$492$	11.5312	0.519865
$493$	−0.419203	−0.0188799
$494$	32.3637	1.45611
$495$	0	0
$496$	−3.38705	−0.152083
$497$	10.4249	0.467622
$498$	−10.3959	−0.465851
$499$	−41.4596	−1.85599	−0.927994	−	0.372594i	$-0.878468\pi$
$499$	−41.4596	−1.85599	−0.927994	+	0.372594i	$0.878468\pi$
$500$	0	0
$501$	−7.54507	−0.337089
$502$	39.4403	1.76031
$503$	−32.6613	−1.45630	−0.728148	−	0.685420i	$-0.759619\pi$
$503$	−32.6613	−1.45630	−0.728148	+	0.685420i	$0.759619\pi$
$504$	4.21556	0.187776
$505$	0	0
$506$	0	0
$507$	−1.35905	−0.0603576
$508$	1.79272	0.0795391
$509$	16.6452	0.737785	0.368892	−	0.929472i	$-0.379737\pi$
$509$	16.6452	0.737785	0.368892	+	0.929472i	$0.379737\pi$
$510$	0	0
$511$	19.8701	0.879000
$512$	0.964978	0.0426464
$513$	−21.9270	−0.968102
$514$	40.8290	1.80089
$515$	0	0
$516$	12.2561	0.539543
$517$	0	0
$518$	−9.62412	−0.422860
$519$	4.16116	0.182655
$520$	0	0
$521$	14.0563	0.615816	0.307908	−	0.951416i	$-0.400371\pi$
$521$	14.0563	0.615816	0.307908	+	0.951416i	$0.400371\pi$
$522$	0.280543	0.0122790
$523$	−15.6677	−0.685101	−0.342550	−	0.939499i	$-0.611291\pi$
$523$	−15.6677	−0.685101	−0.342550	+	0.939499i	$0.611291\pi$
$524$	−1.17171	−0.0511866
$525$	0	0
$526$	−9.00255	−0.392530
$527$	1.52801	0.0665611
$528$	0	0
$529$	−8.14550	−0.354152
$530$	0	0
$531$	−0.309246	−0.0134202
$532$	−8.74751	−0.379253
$533$	−29.2874	−1.26858
$534$	32.4195	1.40293
$535$	0	0
$536$	1.35860	0.0586827
$537$	−9.92128	−0.428135
$538$	11.0949	0.478336
$539$	0	0
$540$	0	0
$541$	39.6384	1.70419	0.852094	−	0.523389i	$-0.175333\pi$
$541$	39.6384	1.70419	0.852094	+	0.523389i	$0.175333\pi$
$542$	−8.37190	−0.359604
$543$	30.9007	1.32608
$544$	8.85764	0.379768
$545$	0	0
$546$	27.1160	1.16046
$547$	−41.1664	−1.76015	−0.880075	−	0.474835i	$-0.842508\pi$
$547$	−41.1664	−1.76015	−0.880075	+	0.474835i	$0.842508\pi$
$548$	−13.7953	−0.589305
$549$	−1.55967	−0.0665651
$550$	0	0
$551$	0.996247	0.0424415
$552$	15.8972	0.676630
$553$	16.2270	0.690041
$554$	17.6936	0.751728
$555$	0	0
$556$	−8.57683	−0.363739
$557$	−29.8760	−1.26589	−0.632943	−	0.774199i	$-0.718153\pi$
$557$	−29.8760	−1.26589	−0.632943	+	0.774199i	$0.718153\pi$
$558$	−1.02259	−0.0432896
$559$	−31.1286	−1.31660
$560$	0	0
$561$	0	0
$562$	−22.7004	−0.957558
$563$	−2.15779	−0.0909401	−0.0454701	−	0.998966i	$-0.514479\pi$
$563$	−2.15779	−0.0909401	−0.0454701	+	0.998966i	$0.514479\pi$
$564$	17.4994	0.736857
$565$	0	0
$566$	36.3500	1.52790
$567$	−24.4265	−1.02582
$568$	9.70734	0.407311
$569$	−0.717288	−0.0300703	−0.0150351	−	0.999887i	$-0.504786\pi$
$569$	−0.717288	−0.0300703	−0.0150351	+	0.999887i	$0.504786\pi$
$570$	0	0
$571$	21.6311	0.905235	0.452617	−	0.891705i	$-0.350490\pi$
$571$	21.6311	0.905235	0.452617	+	0.891705i	$0.350490\pi$
$572$	0	0
$573$	−6.15315	−0.257051
$574$	29.3792	1.22626
$575$	0	0
$576$	2.94622	0.122759
$577$	23.4276	0.975303	0.487652	−	0.873038i	$-0.337854\pi$
$577$	23.4276	0.975303	0.487652	+	0.873038i	$0.337854\pi$
$578$	19.9396	0.829377
$579$	−19.0298	−0.790852
$580$	0	0
$581$	−7.13668	−0.296079
$582$	7.41093	0.307193
$583$	0	0
$584$	18.5023	0.765633
$585$	0	0
$586$	−23.2271	−0.959501
$587$	−2.24734	−0.0927576	−0.0463788	−	0.998924i	$-0.514768\pi$
$587$	−2.24734	−0.0927576	−0.0463788	+	0.998924i	$0.514768\pi$
$588$	2.86766	0.118260
$589$	−3.63135	−0.149627
$590$	0	0
$591$	28.4318	1.16953
$592$	−12.7873	−0.525555
$593$	−25.4034	−1.04319	−0.521596	−	0.853193i	$-0.674663\pi$
$593$	−25.4034	−1.04319	−0.521596	+	0.853193i	$0.674663\pi$
$594$	0	0
$595$	0	0
$596$	−4.36353	−0.178737
$597$	−29.2224	−1.19600
$598$	23.5934	0.964806
$599$	−18.2253	−0.744667	−0.372333	−	0.928099i	$-0.621442\pi$
$599$	−18.2253	−0.744667	−0.372333	+	0.928099i	$0.621442\pi$
$600$	0	0
$601$	34.6398	1.41299	0.706494	−	0.707719i	$-0.250276\pi$
$601$	34.6398	1.41299	0.706494	+	0.707719i	$0.250276\pi$
$602$	31.2261	1.27268
$603$	0.585276	0.0238343
$604$	−9.42547	−0.383517
$605$	0	0
$606$	−32.3375	−1.31362
$607$	−25.4482	−1.03291	−0.516456	−	0.856314i	$-0.672749\pi$
$607$	−25.4482	−1.03291	−0.516456	+	0.856314i	$0.672749\pi$
$608$	−21.0504	−0.853708
$609$	0.834708	0.0338241
$610$	0	0
$611$	−44.4458	−1.79809
$612$	1.47652	0.0596846
$613$	37.0616	1.49690	0.748452	−	0.663189i	$-0.230797\pi$
$613$	37.0616	1.49690	0.748452	+	0.663189i	$0.230797\pi$
$614$	11.3662	0.458701
$615$	0	0
$616$	0	0
$617$	−27.5937	−1.11088	−0.555439	−	0.831557i	$-0.687450\pi$
$617$	−27.5937	−1.11088	−0.555439	+	0.831557i	$0.687450\pi$
$618$	33.4624	1.34605
$619$	20.4435	0.821694	0.410847	−	0.911704i	$-0.365233\pi$
$619$	20.4435	0.821694	0.410847	+	0.911704i	$0.365233\pi$
$620$	0	0
$621$	−15.9850	−0.641456
$622$	9.10280	0.364989
$623$	22.2557	0.891654
$624$	36.0283	1.44229
$625$	0	0
$626$	23.5480	0.941167
$627$	0	0
$628$	10.5842	0.422354
$629$	5.76876	0.230016
$630$	0	0
$631$	0.759137	0.0302208	0.0151104	−	0.999886i	$-0.495190\pi$
$631$	0.759137	0.0302208	0.0151104	+	0.999886i	$0.495190\pi$
$632$	15.1100	0.601045
$633$	−13.3722	−0.531498
$634$	30.9182	1.22792
$635$	0	0
$636$	−18.4366	−0.731060
$637$	−7.28342	−0.288579
$638$	0	0
$639$	4.18186	0.165432
$640$	0	0
$641$	−14.9050	−0.588712	−0.294356	−	0.955696i	$-0.595105\pi$
$641$	−14.9050	−0.588712	−0.294356	+	0.955696i	$0.595105\pi$
$642$	−32.8274	−1.29559
$643$	27.6346	1.08980	0.544900	−	0.838501i	$-0.316568\pi$
$643$	27.6346	1.08980	0.544900	+	0.838501i	$0.316568\pi$
$644$	−6.37701	−0.251289
$645$	0	0
$646$	19.4598	0.765635
$647$	24.9785	0.982008	0.491004	−	0.871157i	$-0.336630\pi$
$647$	24.9785	0.982008	0.491004	+	0.871157i	$0.336630\pi$
$648$	−22.7452	−0.893514
$649$	0	0
$650$	0	0
$651$	−3.04254	−0.119247
$652$	−2.67554	−0.104782
$653$	27.7630	1.08645	0.543225	−	0.839587i	$-0.317203\pi$
$653$	27.7630	1.08645	0.543225	+	0.839587i	$0.317203\pi$
$654$	28.7592	1.12457
$655$	0	0
$656$	39.0353	1.52407
$657$	7.97068	0.310966
$658$	44.5851	1.73811
$659$	21.5863	0.840883	0.420442	−	0.907320i	$-0.361875\pi$
$659$	21.5863	0.840883	0.420442	+	0.907320i	$0.361875\pi$
$660$	0	0
$661$	−16.0174	−0.623003	−0.311502	−	0.950246i	$-0.600832\pi$
$661$	−16.0174	−0.623003	−0.311502	+	0.950246i	$0.600832\pi$
$662$	−0.774756	−0.0301118
$663$	−16.2535	−0.631234
$664$	−6.64544	−0.257893
$665$	0	0
$666$	−3.86062	−0.149596
$667$	0.726273	0.0281214
$668$	2.81830	0.109043
$669$	−17.2148	−0.665561
$670$	0	0
$671$	0	0
$672$	−17.6372	−0.680368
$673$	−31.3469	−1.20834	−0.604168	−	0.796857i	$-0.706494\pi$
$673$	−31.3469	−1.20834	−0.604168	+	0.796857i	$0.706494\pi$
$674$	−56.4001	−2.17245
$675$	0	0
$676$	0.507645	0.0195248
$677$	30.6664	1.17860	0.589302	−	0.807913i	$-0.299403\pi$
$677$	30.6664	1.17860	0.589302	+	0.807913i	$0.299403\pi$
$678$	0.755931	0.0290314
$679$	5.08754	0.195242
$680$	0	0
$681$	−7.51888	−0.288124
$682$	0	0
$683$	−3.27236	−0.125213	−0.0626066	−	0.998038i	$-0.519941\pi$
$683$	−3.27236	−0.125213	−0.0626066	+	0.998038i	$0.519941\pi$
$684$	−3.50898	−0.134169
$685$	0	0
$686$	33.2857	1.27085
$687$	5.35892	0.204456
$688$	41.4893	1.58176
$689$	46.8263	1.78394
$690$	0	0
$691$	−36.4946	−1.38832	−0.694160	−	0.719821i	$-0.744224\pi$
$691$	−36.4946	−1.38832	−0.694160	+	0.719821i	$0.744224\pi$
$692$	−1.55431	−0.0590862
$693$	0	0
$694$	−5.94478	−0.225661
$695$	0	0
$696$	0.777253	0.0294617
$697$	−17.6101	−0.667029
$698$	−10.5415	−0.399002
$699$	20.7566	0.785085
$700$	0	0
$701$	−46.5607	−1.75857	−0.879286	−	0.476293i	$-0.841980\pi$
$701$	−46.5607	−1.75857	−0.879286	+	0.476293i	$0.841980\pi$
$702$	−25.3890	−0.958245
$703$	−13.7096	−0.517068
$704$	0	0
$705$	0	0
$706$	−20.1811	−0.759525
$707$	−22.1994	−0.834894
$708$	0.500645	0.0188154
$709$	−35.5966	−1.33686	−0.668429	−	0.743776i	$-0.733033\pi$
$709$	−35.5966	−1.33686	−0.668429	+	0.743776i	$0.733033\pi$
$710$	0	0
$711$	6.50930	0.244118
$712$	20.7237	0.776655
$713$	−2.64729	−0.0991418
$714$	16.3044	0.610178
$715$	0	0
$716$	3.70588	0.138495
$717$	39.5740	1.47792
$718$	39.9000	1.48906
$719$	22.0913	0.823866	0.411933	−	0.911214i	$-0.364854\pi$
$719$	22.0913	0.823866	0.411933	+	0.911214i	$0.364854\pi$
$720$	0	0
$721$	22.9716	0.855507
$722$	−14.8097	−0.551158
$723$	56.1728	2.08909
$724$	−11.5423	−0.428966
$725$	0	0
$726$	0	0
$727$	45.5415	1.68904	0.844521	−	0.535522i	$-0.179885\pi$
$727$	45.5415	1.68904	0.844521	+	0.535522i	$0.179885\pi$
$728$	17.3336	0.642425
$729$	14.7727	0.547137
$730$	0	0
$731$	−18.7171	−0.692278
$732$	2.52498	0.0933260
$733$	−11.3789	−0.420289	−0.210145	−	0.977670i	$-0.567393\pi$
$733$	−11.3789	−0.420289	−0.210145	+	0.977670i	$0.567393\pi$
$734$	−33.7368	−1.24525
$735$	0	0
$736$	−15.3459	−0.565659
$737$	0	0
$738$	11.7852	0.433818
$739$	4.33778	0.159568	0.0797838	−	0.996812i	$-0.474577\pi$
$739$	4.33778	0.159568	0.0797838	+	0.996812i	$0.474577\pi$
$740$	0	0
$741$	38.6269	1.41900
$742$	−46.9730	−1.72443
$743$	−17.2945	−0.634473	−0.317237	−	0.948346i	$-0.602755\pi$
$743$	−17.2945	−0.634473	−0.317237	+	0.948346i	$0.602755\pi$
$744$	−2.83311	−0.103867
$745$	0	0
$746$	−12.4424	−0.455549
$747$	−2.86281	−0.104745
$748$	0	0
$749$	−22.5357	−0.823437
$750$	0	0
$751$	−31.5130	−1.14993	−0.574963	−	0.818179i	$-0.694984\pi$
$751$	−31.5130	−1.14993	−0.574963	+	0.818179i	$0.694984\pi$
$752$	59.2390	2.16022
$753$	47.0730	1.71544
$754$	1.15354	0.0420094
$755$	0	0
$756$	6.86233	0.249581
$757$	9.27739	0.337192	0.168596	−	0.985685i	$-0.446077\pi$
$757$	9.27739	0.337192	0.168596	+	0.985685i	$0.446077\pi$
$758$	−38.3718	−1.39373
$759$	0	0
$760$	0	0
$761$	4.15810	0.150731	0.0753655	−	0.997156i	$-0.475988\pi$
$761$	4.15810	0.150731	0.0753655	+	0.997156i	$0.475988\pi$
$762$	7.94102	0.287673
$763$	19.7429	0.714742
$764$	2.29838	0.0831524
$765$	0	0
$766$	4.03783	0.145893
$767$	−1.27156	−0.0459135
$768$	−30.7894	−1.11102
$769$	−16.8800	−0.608709	−0.304355	−	0.952559i	$-0.598441\pi$
$769$	−16.8800	−0.608709	−0.304355	+	0.952559i	$0.598441\pi$
$770$	0	0
$771$	48.7305	1.75499
$772$	7.10818	0.255829
$773$	−8.47760	−0.304918	−0.152459	−	0.988310i	$-0.548719\pi$
$773$	−8.47760	−0.304918	−0.152459	+	0.988310i	$0.548719\pi$
$774$	12.5261	0.450240
$775$	0	0
$776$	4.73735	0.170061
$777$	−11.4866	−0.412081
$778$	−56.2114	−2.01527
$779$	41.8508	1.49946
$780$	0	0
$781$	0	0
$782$	14.1863	0.507303
$783$	−0.781546	−0.0279302
$784$	9.70760	0.346700
$785$	0	0
$786$	−5.19022	−0.185129
$787$	−53.6166	−1.91122	−0.955612	−	0.294628i	$-0.904804\pi$
$787$	−53.6166	−1.91122	−0.955612	+	0.294628i	$0.904804\pi$
$788$	−10.6201	−0.378325
$789$	−10.7448	−0.382525
$790$	0	0
$791$	0.518940	0.0184514
$792$	0	0
$793$	−6.41307	−0.227735
$794$	−45.4946	−1.61454
$795$	0	0
$796$	10.9154	0.386887
$797$	28.5448	1.01111	0.505554	−	0.862795i	$-0.331288\pi$
$797$	28.5448	1.01111	0.505554	+	0.862795i	$0.331288\pi$
$798$	−38.7479	−1.37166
$799$	−26.7246	−0.945448
$800$	0	0
$801$	8.92765	0.315443
$802$	3.12290	0.110273
$803$	0	0
$804$	−0.947515	−0.0334163
$805$	0	0
$806$	−4.20469	−0.148104
$807$	13.2421	0.466143
$808$	−20.6713	−0.727215
$809$	−36.9460	−1.29895	−0.649477	−	0.760382i	$-0.725012\pi$
$809$	−36.9460	−1.29895	−0.649477	+	0.760382i	$0.725012\pi$
$810$	0	0
$811$	38.3768	1.34759	0.673795	−	0.738918i	$-0.264663\pi$
$811$	38.3768	1.34759	0.673795	+	0.738918i	$0.264663\pi$
$812$	−0.311788	−0.0109416
$813$	−9.99209	−0.350438
$814$	0	0
$815$	0	0
$816$	21.6632	0.758365
$817$	44.4818	1.55622
$818$	22.4319	0.784314
$819$	7.46718	0.260924
$820$	0	0
$821$	10.2496	0.357715	0.178858	−	0.983875i	$-0.442760\pi$
$821$	10.2496	0.357715	0.178858	+	0.983875i	$0.442760\pi$
$822$	−61.1074	−2.13137
$823$	25.2296	0.879448	0.439724	−	0.898133i	$-0.355076\pi$
$823$	25.2296	0.879448	0.439724	+	0.898133i	$0.355076\pi$
$824$	21.3904	0.745170
$825$	0	0
$826$	1.27555	0.0443820
$827$	18.3485	0.638041	0.319020	−	0.947748i	$-0.396646\pi$
$827$	18.3485	0.638041	0.319020	+	0.947748i	$0.396646\pi$
$828$	−2.55808	−0.0888993
$829$	24.3826	0.846842	0.423421	−	0.905933i	$-0.360829\pi$
$829$	24.3826	0.846842	0.423421	+	0.905933i	$0.360829\pi$
$830$	0	0
$831$	21.1178	0.732567
$832$	12.1143	0.419987
$833$	−4.37941	−0.151737
$834$	−37.9919	−1.31555
$835$	0	0
$836$	0	0
$837$	2.84876	0.0984676
$838$	36.6272	1.26526
$839$	−42.2808	−1.45970	−0.729848	−	0.683609i	$-0.760409\pi$
$839$	−42.2808	−1.45970	−0.729848	+	0.683609i	$0.760409\pi$
$840$	0	0
$841$	−28.9645	−0.998776
$842$	−29.6212	−1.02082
$843$	−27.0935	−0.933151
$844$	4.99491	0.171932
$845$	0	0
$846$	17.8849	0.614895
$847$	0	0
$848$	−62.4117	−2.14323
$849$	43.3847	1.48896
$850$	0	0
$851$	−9.99444	−0.342605
$852$	−6.77009	−0.231939
$853$	−15.3885	−0.526891	−0.263445	−	0.964674i	$-0.584859\pi$
$853$	−15.3885	−0.526891	−0.263445	+	0.964674i	$0.584859\pi$
$854$	6.43317	0.220138
$855$	0	0
$856$	−20.9845	−0.717236
$857$	−36.1038	−1.23328	−0.616641	−	0.787245i	$-0.711507\pi$
$857$	−36.1038	−1.23328	−0.616641	+	0.787245i	$0.711507\pi$
$858$	0	0
$859$	48.3509	1.64971	0.824855	−	0.565344i	$-0.191257\pi$
$859$	48.3509	1.64971	0.824855	+	0.565344i	$0.191257\pi$
$860$	0	0
$861$	35.0648	1.19501
$862$	55.3386	1.88484
$863$	37.1887	1.26592	0.632959	−	0.774186i	$-0.281840\pi$
$863$	37.1887	1.26592	0.632959	+	0.774186i	$0.281840\pi$
$864$	16.5139	0.561813
$865$	0	0
$866$	52.1051	1.77060
$867$	23.7984	0.808237
$868$	1.13648	0.0385745
$869$	0	0
$870$	0	0
$871$	2.40655	0.0815427
$872$	18.3840	0.622560
$873$	2.04082	0.0690712
$874$	−33.7143	−1.14040
$875$	0	0
$876$	−12.9039	−0.435982
$877$	25.7932	0.870976	0.435488	−	0.900195i	$-0.356576\pi$
$877$	25.7932	0.870976	0.435488	+	0.900195i	$0.356576\pi$
$878$	58.9376	1.98905
$879$	−27.7221	−0.935044
$880$	0	0
$881$	−45.6820	−1.53906	−0.769532	−	0.638608i	$-0.779511\pi$
$881$	−45.6820	−1.53906	−0.769532	+	0.638608i	$0.779511\pi$
$882$	2.93083	0.0986861
$883$	−4.96631	−0.167130	−0.0835648	−	0.996502i	$-0.526631\pi$
$883$	−4.96631	−0.167130	−0.0835648	+	0.996502i	$0.526631\pi$
$884$	6.07115	0.204195
$885$	0	0
$886$	38.8621	1.30560
$887$	−28.9232	−0.971147	−0.485573	−	0.874196i	$-0.661389\pi$
$887$	−28.9232	−0.971147	−0.485573	+	0.874196i	$0.661389\pi$
$888$	−10.6960	−0.358934
$889$	5.45144	0.182836
$890$	0	0
$891$	0	0
$892$	6.43021	0.215299
$893$	63.5117	2.12534
$894$	−19.3287	−0.646448
$895$	0	0
$896$	−30.0146	−1.00272
$897$	28.1594	0.940213
$898$	51.9406	1.73328
$899$	−0.129432	−0.00431681
$900$	0	0
$901$	28.1559	0.938010
$902$	0	0
$903$	37.2692	1.24024
$904$	0.483220	0.0160717
$905$	0	0
$906$	−41.7510	−1.38708
$907$	−13.2527	−0.440049	−0.220024	−	0.975494i	$-0.570614\pi$
$907$	−13.2527	−0.440049	−0.220024	+	0.975494i	$0.570614\pi$
$908$	2.80852	0.0932040
$909$	−8.90507	−0.295363
$910$	0	0
$911$	13.7326	0.454982	0.227491	−	0.973780i	$-0.426948\pi$
$911$	13.7326	0.454982	0.227491	+	0.973780i	$0.426948\pi$
$912$	−51.4833	−1.70478
$913$	0	0
$914$	−64.7117	−2.14047
$915$	0	0
$916$	−2.00171	−0.0661384
$917$	−3.56304	−0.117662
$918$	−15.2660	−0.503853
$919$	−59.3800	−1.95876	−0.979382	−	0.202016i	$-0.935251\pi$
$919$	−59.3800	−1.95876	−0.979382	+	0.202016i	$0.935251\pi$
$920$	0	0
$921$	13.5658	0.447009
$922$	−14.6993	−0.484095
$923$	17.1950	0.565981
$924$	0	0
$925$	0	0
$926$	6.96713	0.228954
$927$	9.21484	0.302655
$928$	−0.750301	−0.0246298
$929$	19.5881	0.642664	0.321332	−	0.946967i	$-0.395869\pi$
$929$	19.5881	0.642664	0.321332	+	0.946967i	$0.395869\pi$
$930$	0	0
$931$	10.4078	0.341101
$932$	−7.75317	−0.253964
$933$	10.8644	0.355686
$934$	−11.1327	−0.364275
$935$	0	0
$936$	6.95320	0.227272
$937$	−40.5452	−1.32455	−0.662277	−	0.749259i	$-0.730410\pi$
$937$	−40.5452	−1.32455	−0.662277	+	0.749259i	$0.730410\pi$
$938$	−2.41409	−0.0788227
$939$	28.1052	0.917178
$940$	0	0
$941$	0.409691	0.0133556	0.00667778	−	0.999978i	$-0.497874\pi$
$941$	0.409691	0.0133556	0.00667778	+	0.999978i	$0.497874\pi$
$942$	46.8835	1.52755
$943$	30.5096	0.993530
$944$	1.69478	0.0551605
$945$	0	0
$946$	0	0
$947$	2.45729	0.0798511	0.0399256	−	0.999203i	$-0.487288\pi$
$947$	2.45729	0.0798511	0.0399256	+	0.999203i	$0.487288\pi$
$948$	−10.5380	−0.342259
$949$	32.7739	1.06389
$950$	0	0
$951$	36.9017	1.19662
$952$	10.4224	0.337792
$953$	−61.0264	−1.97684	−0.988420	−	0.151744i	$-0.951511\pi$
$953$	−61.0264	−1.97684	−0.988420	+	0.151744i	$0.951511\pi$
$954$	−18.8428	−0.610057
$955$	0	0
$956$	−14.7820	−0.478085
$957$	0	0
$958$	34.4309	1.11241
$959$	−41.9497	−1.35463
$960$	0	0
$961$	−30.5282	−0.984781
$962$	−15.8742	−0.511803
$963$	−9.03999	−0.291310
$964$	−20.9822	−0.675790
$965$	0	0
$966$	−28.2476	−0.908851
$967$	−17.1997	−0.553106	−0.276553	−	0.960999i	$-0.589192\pi$
$967$	−17.1997	−0.553106	−0.276553	+	0.960999i	$0.589192\pi$
$968$	0	0
$969$	23.2258	0.746119
$970$	0	0
$971$	27.2090	0.873177	0.436589	−	0.899661i	$-0.356187\pi$
$971$	27.2090	0.873177	0.436589	+	0.899661i	$0.356187\pi$
$972$	6.68488	0.214417
$973$	−26.0811	−0.836121
$974$	−26.1083	−0.836563
$975$	0	0
$976$	8.54757	0.273601
$977$	−19.1722	−0.613374	−0.306687	−	0.951810i	$-0.599220\pi$
$977$	−19.1722	−0.613374	−0.306687	+	0.951810i	$0.599220\pi$
$978$	−11.8516	−0.378971
$979$	0	0
$980$	0	0
$981$	7.91969	0.252856
$982$	31.9836	1.02064
$983$	−24.1305	−0.769642	−0.384821	−	0.922991i	$-0.625737\pi$
$983$	−24.1305	−0.769642	−0.384821	+	0.922991i	$0.625737\pi$
$984$	32.6512	1.04088
$985$	0	0
$986$	0.693605	0.0220889
$987$	53.2135	1.69380
$988$	−14.4283	−0.459024
$989$	32.4276	1.03114
$990$	0	0
$991$	27.7081	0.880177	0.440089	−	0.897954i	$-0.354947\pi$
$991$	27.7081	0.880177	0.440089	+	0.897954i	$0.354947\pi$
$992$	2.73487	0.0868323
$993$	−0.924692	−0.0293442
$994$	−17.2489	−0.547101
$995$	0	0
$996$	4.63466	0.146855
$997$	−33.4912	−1.06068	−0.530339	−	0.847786i	$-0.677935\pi$
$997$	−33.4912	−1.06068	−0.530339	+	0.847786i	$0.677935\pi$
$998$	68.5984	2.17144
$999$	10.7551	0.340275

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bk.1.2		8
5.2	odd	4		605.2.b.f.364.2		8
5.3	odd	4		605.2.b.f.364.7		8
5.4	even	2	inner	3025.2.a.bk.1.7		8
11.7	odd	10		275.2.h.d.126.4		16
11.8	odd	10		275.2.h.d.251.4		16
11.10	odd	2		3025.2.a.bl.1.7		8
55.2	even	20		605.2.j.h.444.4		16
55.3	odd	20		605.2.j.d.9.4		16
55.7	even	20		55.2.j.a.49.1	yes	16
55.8	even	20		55.2.j.a.9.1	✓	16
55.13	even	20		605.2.j.h.444.1		16
55.17	even	20		605.2.j.h.124.1		16
55.18	even	20		55.2.j.a.49.4	yes	16
55.19	odd	10		275.2.h.d.251.1		16
55.27	odd	20		605.2.j.g.124.4		16
55.28	even	20		605.2.j.h.124.4		16
55.29	odd	10		275.2.h.d.126.1		16
55.32	even	4		605.2.b.g.364.7		8
55.37	odd	20		605.2.j.d.269.4		16
55.38	odd	20		605.2.j.g.124.1		16
55.42	odd	20		605.2.j.g.444.1		16
55.43	even	4		605.2.b.g.364.2		8
55.47	odd	20		605.2.j.d.9.1		16
55.48	odd	20		605.2.j.d.269.1		16
55.52	even	20		55.2.j.a.9.4	yes	16
55.53	odd	20		605.2.j.g.444.4		16
55.54	odd	2		3025.2.a.bl.1.2		8
165.8	odd	20		495.2.ba.a.64.4		16
165.62	odd	20		495.2.ba.a.379.4		16
165.107	odd	20		495.2.ba.a.64.1		16
165.128	odd	20		495.2.ba.a.379.1		16
220.7	odd	20		880.2.cd.c.49.1		16
220.63	odd	20		880.2.cd.c.449.1		16
220.107	odd	20		880.2.cd.c.449.4		16
220.183	odd	20		880.2.cd.c.49.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.j.a.9.1	✓	16	55.8	even	20
55.2.j.a.9.4	yes	16	55.52	even	20
55.2.j.a.49.1	yes	16	55.7	even	20
55.2.j.a.49.4	yes	16	55.18	even	20
275.2.h.d.126.1		16	55.29	odd	10
275.2.h.d.126.4		16	11.7	odd	10
275.2.h.d.251.1		16	55.19	odd	10
275.2.h.d.251.4		16	11.8	odd	10
495.2.ba.a.64.1		16	165.107	odd	20
495.2.ba.a.64.4		16	165.8	odd	20
495.2.ba.a.379.1		16	165.128	odd	20
495.2.ba.a.379.4		16	165.62	odd	20
605.2.b.f.364.2		8	5.2	odd	4
605.2.b.f.364.7		8	5.3	odd	4
605.2.b.g.364.2		8	55.43	even	4
605.2.b.g.364.7		8	55.32	even	4
605.2.j.d.9.1		16	55.47	odd	20
605.2.j.d.9.4		16	55.3	odd	20
605.2.j.d.269.1		16	55.48	odd	20
605.2.j.d.269.4		16	55.37	odd	20
605.2.j.g.124.1		16	55.38	odd	20
605.2.j.g.124.4		16	55.27	odd	20
605.2.j.g.444.1		16	55.42	odd	20
605.2.j.g.444.4		16	55.53	odd	20
605.2.j.h.124.1		16	55.17	even	20
605.2.j.h.124.4		16	55.28	even	20
605.2.j.h.444.1		16	55.13	even	20
605.2.j.h.444.4		16	55.2	even	20
880.2.cd.c.49.1		16	220.7	odd	20
880.2.cd.c.49.4		16	220.183	odd	20
880.2.cd.c.449.1		16	220.63	odd	20
880.2.cd.c.449.4		16	220.107	odd	20
3025.2.a.bk.1.2		8	1.1	even	1	trivial
3025.2.a.bk.1.7		8	5.4	even	2	inner
3025.2.a.bl.1.2		8	55.54	odd	2
3025.2.a.bl.1.7		8	11.10	odd	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65458 q^{2} -1.97479 q^{3} +0.737640 q^{4} +3.26745 q^{6} +2.24307 q^{7} +2.08868 q^{8} +0.899788 q^{9} +O(q^{10})\)	\(q-1.65458 q^{2} -1.97479 q^{3} +0.737640 q^{4} +3.26745 q^{6} +2.24307 q^{7} +2.08868 q^{8} +0.899788 q^{9} -1.45668 q^{12} +3.69976 q^{13} -3.71135 q^{14} -4.93117 q^{16} +2.22461 q^{17} -1.48877 q^{18} -5.28684 q^{19} -4.42960 q^{21} -3.85415 q^{23} -4.12469 q^{24} -6.12155 q^{26} +4.14747 q^{27} +1.65458 q^{28} -0.188439 q^{29} +0.686867 q^{31} +3.98166 q^{32} -3.68079 q^{34} +0.663720 q^{36} +2.59316 q^{37} +8.74751 q^{38} -7.30624 q^{39} -7.91604 q^{41} +7.32913 q^{42} -8.41368 q^{43} +6.37701 q^{46} -12.0132 q^{47} +9.73801 q^{48} -1.96862 q^{49} -4.39313 q^{51} +2.72909 q^{52} +12.6566 q^{53} -6.86233 q^{54} +4.68506 q^{56} +10.4404 q^{57} +0.311788 q^{58} -0.343688 q^{59} -1.73338 q^{61} -1.13648 q^{62} +2.01829 q^{63} +3.27435 q^{64} +0.650461 q^{67} +1.64096 q^{68} +7.61114 q^{69} +4.64760 q^{71} +1.87937 q^{72} +8.85841 q^{73} -4.29059 q^{74} -3.89979 q^{76} +12.0888 q^{78} +7.23426 q^{79} -10.8897 q^{81} +13.0977 q^{82} -3.18165 q^{83} -3.26745 q^{84} +13.9211 q^{86} +0.372127 q^{87} +9.92195 q^{89} +8.29883 q^{91} -2.84298 q^{92} -1.35642 q^{93} +19.8768 q^{94} -7.86294 q^{96} +2.26811 q^{97} +3.25724 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10}) \) 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9	\( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} - 12 q^{19} - 4 q^{21} - 2 q^{24} + 10 q^{26} - 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} - 34 q^{41} - 24 q^{46} - 30 q^{49} - 54 q^{51} - 20 q^{54} - 10 q^{56} - 6 q^{59} - 20 q^{61} + 14 q^{64} + 32 q^{69} - 42 q^{71} + 4 q^{74} - 28 q^{76} - 16 q^{79} - 36 q^{81} + 6 q^{84} + 46 q^{86} - 12 q^{89} + 20 q^{91} + 42 q^{94} + 8 q^{96}+O(q^{100}) \) 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9 - 4 * q^14 - 22 * q^16 - 12 * q^19 - 4 * q^21 - 2 * q^24 + 10 * q^26 - 24 * q^29 + 14 * q^31 + 8 * q^34 + 20 * q^36 - 30 * q^39 - 34 * q^41 - 24 * q^46 - 30 * q^49 - 54 * q^51 - 20 * q^54 - 10 * q^56 - 6 * q^59 - 20 * q^61 + 14 * q^64 + 32 * q^69 - 42 * q^71 + 4 * q^74 - 28 * q^76 - 16 * q^79 - 36 * q^81 + 6 * q^84 + 46 * q^86 - 12 * q^89 + 20 * q^91 + 42 * q^94 + 8 * q^96

Introduction

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