LMFDB - Embedded newform 3025.2.a.bk.1.1

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			1.1
Root			$-2.02368$ 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)

$f(q)$	$=$	$q-2.02368 q^{2} +2.62059 q^{3} +2.09529 q^{4} -5.30325 q^{6} +0.965823 q^{7} -0.192845 q^{8} +3.86752 q^{9} +O(q^{10})$	\(q-2.02368 q^{2} +2.62059 q^{3} +2.09529 q^{4} -5.30325 q^{6} +0.965823 q^{7} -0.192845 q^{8} +3.86752 q^{9} +5.49092 q^{12} -4.52509 q^{13} -1.95452 q^{14} -3.80033 q^{16} -3.33669 q^{17} -7.82663 q^{18} -3.27759 q^{19} +2.53103 q^{21} +3.36643 q^{23} -0.505368 q^{24} +9.15736 q^{26} +2.27341 q^{27} +2.02368 q^{28} -4.91300 q^{29} -0.418365 q^{31} +8.07636 q^{32} +6.75241 q^{34} +8.10358 q^{36} -6.33755 q^{37} +6.63281 q^{38} -11.8584 q^{39} -5.78564 q^{41} -5.12200 q^{42} -2.26205 q^{43} -6.81258 q^{46} -4.32424 q^{47} -9.95913 q^{48} -6.06719 q^{49} -8.74411 q^{51} -9.48140 q^{52} +2.66070 q^{53} -4.60066 q^{54} -0.186254 q^{56} -8.58924 q^{57} +9.94235 q^{58} -10.1253 q^{59} +2.47086 q^{61} +0.846638 q^{62} +3.73534 q^{63} -8.74332 q^{64} +9.60059 q^{67} -6.99135 q^{68} +8.82204 q^{69} -5.45311 q^{71} -0.745831 q^{72} +1.43554 q^{73} +12.8252 q^{74} -6.86752 q^{76} +23.9977 q^{78} -1.00396 q^{79} -5.64486 q^{81} +11.7083 q^{82} +7.39542 q^{83} +5.30325 q^{84} +4.57768 q^{86} -12.8750 q^{87} -12.1964 q^{89} -4.37044 q^{91} +7.05365 q^{92} -1.09637 q^{93} +8.75089 q^{94} +21.1649 q^{96} +3.01924 q^{97} +12.2781 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.02368	−1.43096	−0.715480	−	0.698633i	$-0.753792\pi$
$2$	−2.02368	−1.43096	−0.715480	+	0.698633i	$0.753792\pi$
$3$	2.62059	1.51300	0.756501	−	0.653993i	$-0.226907\pi$
$3$	2.62059	1.51300	0.756501	+	0.653993i	$0.226907\pi$
$4$	2.09529	1.04765
$5$	0	0
$5$	0	0
$6$	−5.30325	−2.16504
$7$	0.965823	0.365047	0.182523	−	0.983202i	$-0.441573\pi$
$7$	0.965823	0.365047	0.182523	+	0.983202i	$0.441573\pi$
$8$	−0.192845	−0.0681809
$9$	3.86752	1.28917
$10$	0	0
$11$	0	0
$11$	0	0
$12$	5.49092	1.58509
$13$	−4.52509	−1.25504	−0.627518	−	0.778602i	$-0.715929\pi$
$13$	−4.52509	−1.25504	−0.627518	+	0.778602i	$0.715929\pi$
$14$	−1.95452	−0.522367
$15$	0	0
$16$	−3.80033	−0.950083
$17$	−3.33669	−0.809266	−0.404633	−	0.914479i	$-0.632601\pi$
$17$	−3.33669	−0.809266	−0.404633	+	0.914479i	$0.632601\pi$
$18$	−7.82663	−1.84475
$19$	−3.27759	−0.751931	−0.375965	−	0.926634i	$-0.622689\pi$
$19$	−3.27759	−0.751931	−0.375965	+	0.926634i	$0.622689\pi$
$20$	0	0
$21$	2.53103	0.552316
$22$	0	0
$23$	3.36643	0.701948	0.350974	−	0.936385i	$-0.385851\pi$
$23$	3.36643	0.701948	0.350974	+	0.936385i	$0.385851\pi$
$24$	−0.505368	−0.103158
$25$	0	0
$26$	9.15736	1.79591
$27$	2.27341	0.437518
$28$	2.02368	0.382440
$29$	−4.91300	−0.912321	−0.456160	−	0.889898i	$-0.650776\pi$
$29$	−4.91300	−0.912321	−0.456160	+	0.889898i	$0.650776\pi$
$30$	0	0
$31$	−0.418365	−0.0751406	−0.0375703	−	0.999294i	$-0.511962\pi$
$31$	−0.418365	−0.0751406	−0.0375703	+	0.999294i	$0.511962\pi$
$32$	8.07636	1.42771
$33$	0	0
$34$	6.75241	1.15803
$35$	0	0
$36$	8.10358	1.35060
$37$	−6.33755	−1.04189	−0.520944	−	0.853591i	$-0.674420\pi$
$37$	−6.33755	−1.04189	−0.520944	+	0.853591i	$0.674420\pi$
$38$	6.63281	1.07598
$39$	−11.8584	−1.89887
$40$	0	0
$41$	−5.78564	−0.903565	−0.451782	−	0.892128i	$-0.649212\pi$
$41$	−5.78564	−0.903565	−0.451782	+	0.892128i	$0.649212\pi$
$42$	−5.12200	−0.790343
$43$	−2.26205	−0.344960	−0.172480	−	0.985013i	$-0.555178\pi$
$43$	−2.26205	−0.344960	−0.172480	+	0.985013i	$0.555178\pi$
$44$	0	0
$45$	0	0
$46$	−6.81258	−1.00446
$47$	−4.32424	−0.630755	−0.315378	−	0.948966i	$-0.602131\pi$
$47$	−4.32424	−0.630755	−0.315378	+	0.948966i	$0.602131\pi$
$48$	−9.95913	−1.43748
$49$	−6.06719	−0.866741
$50$	0	0
$51$	−8.74411	−1.22442
$52$	−9.48140	−1.31483
$53$	2.66070	0.365475	0.182738	−	0.983162i	$-0.441504\pi$
$53$	2.66070	0.365475	0.182738	+	0.983162i	$0.441504\pi$
$54$	−4.60066	−0.626071
$55$	0	0
$56$	−0.186254	−0.0248892
$57$	−8.58924	−1.13767
$58$	9.94235	1.30549
$59$	−10.1253	−1.31820	−0.659100	−	0.752055i	$-0.729063\pi$
$59$	−10.1253	−1.31820	−0.659100	+	0.752055i	$0.729063\pi$
$60$	0	0
$61$	2.47086	0.316361	0.158180	−	0.987410i	$-0.449437\pi$
$61$	2.47086	0.316361	0.158180	+	0.987410i	$0.449437\pi$
$62$	0.846638	0.107523
$63$	3.73534	0.470608
$64$	−8.74332	−1.09292
$65$	0	0
$66$	0	0
$67$	9.60059	1.17290	0.586449	−	0.809986i	$-0.300525\pi$
$67$	9.60059	1.17290	0.586449	+	0.809986i	$0.300525\pi$
$68$	−6.99135	−0.847826
$69$	8.82204	1.06205
$70$	0	0
$71$	−5.45311	−0.647165	−0.323582	−	0.946200i	$-0.604887\pi$
$71$	−5.45311	−0.647165	−0.323582	+	0.946200i	$0.604887\pi$
$72$	−0.745831	−0.0878970
$73$	1.43554	0.168018	0.0840088	−	0.996465i	$-0.473228\pi$
$73$	1.43554	0.168018	0.0840088	+	0.996465i	$0.473228\pi$
$74$	12.8252	1.49090
$75$	0	0
$76$	−6.86752	−0.787758
$77$	0	0
$78$	23.9977	2.71721
$79$	−1.00396	−0.112954	−0.0564770	−	0.998404i	$-0.517987\pi$
$79$	−1.00396	−0.112954	−0.0564770	+	0.998404i	$0.517987\pi$
$80$	0	0
$81$	−5.64486	−0.627207
$82$	11.7083	1.29297
$83$	7.39542	0.811752	0.405876	−	0.913928i	$-0.366966\pi$
$83$	7.39542	0.811752	0.405876	+	0.913928i	$0.366966\pi$
$84$	5.30325	0.578632
$85$	0	0
$86$	4.57768	0.493624
$87$	−12.8750	−1.38034
$88$	0	0
$89$	−12.1964	−1.29282	−0.646410	−	0.762991i	$-0.723730\pi$
$89$	−12.1964	−1.29282	−0.646410	+	0.762991i	$0.723730\pi$
$90$	0	0
$91$	−4.37044	−0.458147
$92$	7.05365	0.735394
$93$	−1.09637	−0.113688
$94$	8.75089	0.902586
$95$	0	0
$96$	21.1649	2.16013
$97$	3.01924	0.306557	0.153279	−	0.988183i	$-0.451017\pi$
$97$	3.01924	0.306557	0.153279	+	0.988183i	$0.451017\pi$
$98$	12.2781	1.24027
$99$	0	0
$100$	0	0
$101$	−15.1962	−1.51208	−0.756039	−	0.654526i	$-0.772868\pi$
$101$	−15.1962	−1.51208	−0.756039	+	0.654526i	$0.772868\pi$
$102$	17.6953	1.75210
$103$	9.74692	0.960392	0.480196	−	0.877161i	$-0.340565\pi$
$103$	9.74692	0.960392	0.480196	+	0.877161i	$0.340565\pi$
$104$	0.872641	0.0855695
$105$	0	0
$106$	−5.38441	−0.522980
$107$	7.64179	0.738760	0.369380	−	0.929278i	$-0.379570\pi$
$107$	7.64179	0.738760	0.369380	+	0.929278i	$0.379570\pi$
$108$	4.76346	0.458364
$109$	9.85576	0.944010	0.472005	−	0.881596i	$-0.343530\pi$
$109$	9.85576	0.944010	0.472005	+	0.881596i	$0.343530\pi$
$110$	0	0
$111$	−16.6082	−1.57638
$112$	−3.67045	−0.346825
$113$	−5.01861	−0.472111	−0.236056	−	0.971740i	$-0.575855\pi$
$113$	−5.01861	−0.472111	−0.236056	+	0.971740i	$0.575855\pi$
$114$	17.3819	1.62796
$115$	0	0
$116$	−10.2942	−0.955790
$117$	−17.5009	−1.61796
$118$	20.4904	1.88629
$119$	−3.22265	−0.295420
$120$	0	0
$121$	0	0
$122$	−5.00023	−0.452700
$123$	−15.1618	−1.36709
$124$	−0.876598	−0.0787208
$125$	0	0
$126$	−7.55914	−0.673422
$127$	7.09040	0.629172	0.314586	−	0.949229i	$-0.398134\pi$
$127$	7.09040	0.629172	0.314586	+	0.949229i	$0.398134\pi$
$128$	1.54101	0.136207
$129$	−5.92792	−0.521925
$130$	0	0
$131$	−7.21704	−0.630556	−0.315278	−	0.948999i	$-0.602098\pi$
$131$	−7.21704	−0.630556	−0.315278	+	0.948999i	$0.602098\pi$
$132$	0	0
$133$	−3.16557	−0.274490
$134$	−19.4285	−1.67837
$135$	0	0
$136$	0.643464	0.0551766
$137$	−8.49079	−0.725417	−0.362709	−	0.931903i	$-0.618148\pi$
$137$	−8.49079	−0.725417	−0.362709	+	0.931903i	$0.618148\pi$
$138$	−17.8530	−1.51975
$139$	−7.74016	−0.656512	−0.328256	−	0.944589i	$-0.606461\pi$
$139$	−7.74016	−0.656512	−0.328256	+	0.944589i	$0.606461\pi$
$140$	0	0
$141$	−11.3321	−0.954334
$142$	11.0354	0.926067
$143$	0	0
$144$	−14.6978	−1.22482
$145$	0	0
$146$	−2.90508	−0.240426
$147$	−15.8996	−1.31138
$148$	−13.2790	−1.09153
$149$	16.8833	1.38313	0.691567	−	0.722312i	$-0.256921\pi$
$149$	16.8833	1.38313	0.691567	+	0.722312i	$0.256921\pi$
$150$	0	0
$151$	12.2826	0.999548	0.499774	−	0.866156i	$-0.333416\pi$
$151$	12.2826	0.999548	0.499774	+	0.866156i	$0.333416\pi$
$152$	0.632067	0.0512674
$153$	−12.9047	−1.04328
$154$	0	0
$155$	0	0
$156$	−24.8469	−1.98934
$157$	−4.20971	−0.335972	−0.167986	−	0.985789i	$-0.553726\pi$
$157$	−4.20971	−0.335972	−0.167986	+	0.985789i	$0.553726\pi$
$158$	2.03169	0.161633
$159$	6.97261	0.552964
$160$	0	0
$161$	3.25137	0.256244
$162$	11.4234	0.897508
$163$	16.5736	1.29814	0.649071	−	0.760728i	$-0.275158\pi$
$163$	16.5736	1.29814	0.649071	+	0.760728i	$0.275158\pi$
$164$	−12.1226	−0.946617
$165$	0	0
$166$	−14.9660	−1.16159
$167$	9.16678	0.709347	0.354674	−	0.934990i	$-0.384592\pi$
$167$	9.16678	0.709347	0.354674	+	0.934990i	$0.384592\pi$
$168$	−0.488096	−0.0376574
$169$	7.47647	0.575113
$170$	0	0
$171$	−12.6761	−0.969369
$172$	−4.73967	−0.361396
$173$	−4.10967	−0.312452	−0.156226	−	0.987721i	$-0.549933\pi$
$173$	−4.10967	−0.312452	−0.156226	+	0.987721i	$0.549933\pi$
$174$	26.0549	1.97521
$175$	0	0
$176$	0	0
$177$	−26.5343	−1.99444
$178$	24.6817	1.84997
$179$	−16.2961	−1.21802	−0.609012	−	0.793161i	$-0.708434\pi$
$179$	−16.2961	−1.21802	−0.609012	+	0.793161i	$0.708434\pi$
$180$	0	0
$181$	−5.54689	−0.412297	−0.206149	−	0.978521i	$-0.566093\pi$
$181$	−5.54689	−0.412297	−0.206149	+	0.978521i	$0.566093\pi$
$182$	8.84439	0.655589
$183$	6.47512	0.478654
$184$	−0.649198	−0.0478595
$185$	0	0
$186$	2.21870	0.162683
$187$	0	0
$188$	−9.06056	−0.660809
$189$	2.19571	0.159715
$190$	0	0
$191$	21.8161	1.57856	0.789280	−	0.614034i	$-0.210454\pi$
$191$	21.8161	1.57856	0.789280	+	0.614034i	$0.210454\pi$
$192$	−22.9127	−1.65358
$193$	22.4789	1.61807	0.809034	−	0.587762i	$-0.199991\pi$
$193$	22.4789	1.61807	0.809034	+	0.587762i	$0.199991\pi$
$194$	−6.10999	−0.438672
$195$	0	0
$196$	−12.7125	−0.908038
$197$	−25.7479	−1.83446	−0.917230	−	0.398358i	$-0.869580\pi$
$197$	−25.7479	−1.83446	−0.917230	+	0.398358i	$0.869580\pi$
$198$	0	0
$199$	16.9671	1.20277	0.601385	−	0.798960i	$-0.294616\pi$
$199$	16.9671	1.20277	0.601385	+	0.798960i	$0.294616\pi$
$200$	0	0
$201$	25.1592	1.77460
$202$	30.7523	2.16372
$203$	−4.74509	−0.333040
$204$	−18.3215	−1.28276
$205$	0	0
$206$	−19.7247	−1.37428
$207$	13.0197	0.904932
$208$	17.1969	1.19239
$209$	0	0
$210$	0	0
$211$	5.82637	0.401104	0.200552	−	0.979683i	$-0.435726\pi$
$211$	5.82637	0.401104	0.200552	+	0.979683i	$0.435726\pi$
$212$	5.57495	0.382889
$213$	−14.2904	−0.979161
$214$	−15.4646	−1.05714
$215$	0	0
$216$	−0.438415	−0.0298304
$217$	−0.404067	−0.0274298
$218$	−19.9449	−1.35084
$219$	3.76197	0.254211
$220$	0	0
$221$	15.0988	1.01566
$222$	33.6097	2.25573
$223$	−19.0223	−1.27383	−0.636913	−	0.770936i	$-0.719789\pi$
$223$	−19.0223	−1.27383	−0.636913	+	0.770936i	$0.719789\pi$
$224$	7.80033	0.521182
$225$	0	0
$226$	10.1561	0.675572
$227$	28.4659	1.88935	0.944675	−	0.328008i	$-0.106377\pi$
$227$	28.4659	1.88935	0.944675	+	0.328008i	$0.106377\pi$
$228$	−17.9970	−1.19188
$229$	−25.3914	−1.67791	−0.838954	−	0.544202i	$-0.816832\pi$
$229$	−25.3914	−1.67791	−0.838954	+	0.544202i	$0.816832\pi$
$230$	0	0
$231$	0	0
$232$	0.947446	0.0622029
$233$	12.5174	0.820043	0.410022	−	0.912076i	$-0.365521\pi$
$233$	12.5174	0.820043	0.410022	+	0.912076i	$0.365521\pi$
$234$	35.4162	2.31523
$235$	0	0
$236$	−21.2155	−1.38101
$237$	−2.63096	−0.170899
$238$	6.52163	0.422734
$239$	−20.3876	−1.31876	−0.659381	−	0.751809i	$-0.729182\pi$
$239$	−20.3876	−1.31876	−0.659381	+	0.751809i	$0.729182\pi$
$240$	0	0
$241$	22.7935	1.46826	0.734129	−	0.679010i	$-0.237591\pi$
$241$	22.7935	1.46826	0.734129	+	0.679010i	$0.237591\pi$
$242$	0	0
$243$	−21.6131	−1.38648
$244$	5.17717	0.331435
$245$	0	0
$246$	30.6827	1.95626
$247$	14.8314	0.943700
$248$	0.0806795	0.00512316
$249$	19.3804	1.22818
$250$	0	0
$251$	16.9528	1.07005	0.535025	−	0.844836i	$-0.320302\pi$
$251$	16.9528	1.07005	0.535025	+	0.844836i	$0.320302\pi$
$252$	7.82663	0.493031
$253$	0	0
$254$	−14.3487	−0.900320
$255$	0	0
$256$	14.3681	0.898009
$257$	−4.73584	−0.295414	−0.147707	−	0.989031i	$-0.547189\pi$
$257$	−4.73584	−0.295414	−0.147707	+	0.989031i	$0.547189\pi$
$258$	11.9962	0.746853
$259$	−6.12096	−0.380338
$260$	0	0
$261$	−19.0011	−1.17614
$262$	14.6050	0.902300
$263$	−18.1037	−1.11632	−0.558162	−	0.829732i	$-0.688493\pi$
$263$	−18.1037	−1.11632	−0.558162	+	0.829732i	$0.688493\pi$
$264$	0	0
$265$	0	0
$266$	6.40612	0.392784
$267$	−31.9619	−1.95604
$268$	20.1160	1.22878
$269$	−2.29802	−0.140113	−0.0700563	−	0.997543i	$-0.522318\pi$
$269$	−2.29802	−0.140113	−0.0700563	+	0.997543i	$0.522318\pi$
$270$	0	0
$271$	−0.805688	−0.0489421	−0.0244710	−	0.999701i	$-0.507790\pi$
$271$	−0.805688	−0.0489421	−0.0244710	+	0.999701i	$0.507790\pi$
$272$	12.6805	0.768870
$273$	−11.4532	−0.693176
$274$	17.1827	1.03804
$275$	0	0
$276$	18.4848	1.11265
$277$	−11.8329	−0.710970	−0.355485	−	0.934682i	$-0.615684\pi$
$277$	−11.8329	−0.710970	−0.355485	+	0.934682i	$0.615684\pi$
$278$	15.6636	0.939442
$279$	−1.61803	−0.0968692
$280$	0	0
$281$	−6.22929	−0.371608	−0.185804	−	0.982587i	$-0.559489\pi$
$281$	−6.22929	−0.371608	−0.185804	+	0.982587i	$0.559489\pi$
$282$	22.9325	1.36561
$283$	−8.64565	−0.513930	−0.256965	−	0.966421i	$-0.582723\pi$
$283$	−8.64565	−0.513930	−0.256965	+	0.966421i	$0.582723\pi$
$284$	−11.4259	−0.678000
$285$	0	0
$286$	0	0
$287$	−5.58790	−0.329843
$288$	31.2354	1.84057
$289$	−5.86649	−0.345088
$290$	0	0
$291$	7.91221	0.463822
$292$	3.00788	0.176023
$293$	5.81048	0.339452	0.169726	−	0.985491i	$-0.445712\pi$
$293$	5.81048	0.339452	0.169726	+	0.985491i	$0.445712\pi$
$294$	32.1758	1.87653
$295$	0	0
$296$	1.22216	0.0710369
$297$	0	0
$298$	−34.1665	−1.97921
$299$	−15.2334	−0.880970
$300$	0	0
$301$	−2.18474	−0.125926
$302$	−24.8562	−1.43031
$303$	−39.8231	−2.28778
$304$	12.4559	0.714397
$305$	0	0
$306$	26.1150	1.49290
$307$	18.4721	1.05426	0.527130	−	0.849785i	$-0.323268\pi$
$307$	18.4721	1.05426	0.527130	+	0.849785i	$0.323268\pi$
$308$	0	0
$309$	25.5427	1.45307
$310$	0	0
$311$	−11.3314	−0.642543	−0.321271	−	0.946987i	$-0.604110\pi$
$311$	−11.3314	−0.642543	−0.321271	+	0.946987i	$0.604110\pi$
$312$	2.28684	0.129467
$313$	5.11499	0.289116	0.144558	−	0.989496i	$-0.453824\pi$
$313$	5.11499	0.289116	0.144558	+	0.989496i	$0.453824\pi$
$314$	8.51913	0.480762
$315$	0	0
$316$	−2.10358	−0.118336
$317$	−11.8133	−0.663501	−0.331750	−	0.943367i	$-0.607639\pi$
$317$	−11.8133	−0.663501	−0.331750	+	0.943367i	$0.607639\pi$
$318$	−14.1104	−0.791270
$319$	0	0
$320$	0	0
$321$	20.0260	1.11774
$322$	−6.57975	−0.366675
$323$	10.9363	0.608512
$324$	−11.8276	−0.657092
$325$	0	0
$326$	−33.5396	−1.85759
$327$	25.8279	1.42829
$328$	1.11573	0.0616059
$329$	−4.17645	−0.230255
$330$	0	0
$331$	29.2692	1.60878	0.804391	−	0.594100i	$-0.202492\pi$
$331$	29.2692	1.60878	0.804391	+	0.594100i	$0.202492\pi$
$332$	15.4956	0.850430
$333$	−24.5106	−1.34317
$334$	−18.5507	−1.01505
$335$	0	0
$336$	−9.61876	−0.524746
$337$	−26.6441	−1.45140	−0.725698	−	0.688013i	$-0.758483\pi$
$337$	−26.6441	−1.45140	−0.725698	+	0.688013i	$0.758483\pi$
$338$	−15.1300	−0.822964
$339$	−13.1517	−0.714305
$340$	0	0
$341$	0	0
$342$	25.6525	1.38713
$343$	−12.6206	−0.681448
$344$	0.436225	0.0235197
$345$	0	0
$346$	8.31667	0.447107
$347$	4.31128	0.231442	0.115721	−	0.993282i	$-0.463082\pi$
$347$	4.31128	0.231442	0.115721	+	0.993282i	$0.463082\pi$
$348$	−26.9769	−1.44611
$349$	13.1695	0.704947	0.352473	−	0.935822i	$-0.385341\pi$
$349$	13.1695	0.704947	0.352473	+	0.935822i	$0.385341\pi$
$350$	0	0
$351$	−10.2874	−0.549100
$352$	0	0
$353$	−25.4904	−1.35672	−0.678358	−	0.734732i	$-0.737308\pi$
$353$	−25.4904	−1.35672	−0.678358	+	0.734732i	$0.737308\pi$
$354$	53.6970	2.85396
$355$	0	0
$356$	−25.5551	−1.35442
$357$	−8.44527	−0.446971
$358$	32.9781	1.74294
$359$	−26.2741	−1.38670	−0.693348	−	0.720603i	$-0.743865\pi$
$359$	−26.2741	−1.38670	−0.693348	+	0.720603i	$0.743865\pi$
$360$	0	0
$361$	−8.25740	−0.434600
$362$	11.2252	0.589981
$363$	0	0
$364$	−9.15736	−0.479976
$365$	0	0
$366$	−13.1036	−0.684935
$367$	−2.01873	−0.105377	−0.0526885	−	0.998611i	$-0.516779\pi$
$367$	−2.01873	−0.105377	−0.0526885	+	0.998611i	$0.516779\pi$
$368$	−12.7935	−0.666909
$369$	−22.3761	−1.16485
$370$	0	0
$371$	2.56976	0.133416
$372$	−2.29721	−0.119105
$373$	−8.87153	−0.459351	−0.229675	−	0.973267i	$-0.573766\pi$
$373$	−8.87153	−0.459351	−0.229675	+	0.973267i	$0.573766\pi$
$374$	0	0
$375$	0	0
$376$	0.833908	0.0430055
$377$	22.2318	1.14499
$378$	−4.44343	−0.228545
$379$	21.0641	1.08199	0.540995	−	0.841026i	$-0.318048\pi$
$379$	21.0641	1.08199	0.540995	+	0.841026i	$0.318048\pi$
$380$	0	0
$381$	18.5811	0.951937
$382$	−44.1489	−2.25886
$383$	−25.9795	−1.32749	−0.663746	−	0.747958i	$-0.731035\pi$
$383$	−25.9795	−1.32749	−0.663746	+	0.747958i	$0.731035\pi$
$384$	4.03836	0.206082
$385$	0	0
$386$	−45.4902	−2.31539
$387$	−8.74853	−0.444713
$388$	6.32620	0.321164
$389$	−1.64381	−0.0833444	−0.0416722	−	0.999131i	$-0.513269\pi$
$389$	−1.64381	−0.0833444	−0.0416722	+	0.999131i	$0.513269\pi$
$390$	0	0
$391$	−11.2327	−0.568063
$392$	1.17003	0.0590952
$393$	−18.9129	−0.954032
$394$	52.1055	2.62504
$395$	0	0
$396$	0	0
$397$	−16.7088	−0.838588	−0.419294	−	0.907850i	$-0.637722\pi$
$397$	−16.7088	−0.838588	−0.419294	+	0.907850i	$0.637722\pi$
$398$	−34.3361	−1.72111
$399$	−8.29568	−0.415304
$400$	0	0
$401$	27.7920	1.38786	0.693932	−	0.720040i	$-0.255877\pi$
$401$	27.7920	1.38786	0.693932	+	0.720040i	$0.255877\pi$
$402$	−50.9143	−2.53938
$403$	1.89314	0.0943041
$404$	−31.8405	−1.58412
$405$	0	0
$406$	9.60255	0.476567
$407$	0	0
$408$	1.68626	0.0834822
$409$	−39.4330	−1.94983	−0.974917	−	0.222569i	$-0.928556\pi$
$409$	−39.4330	−1.94983	−0.974917	+	0.222569i	$0.928556\pi$
$410$	0	0
$411$	−22.2509	−1.09756
$412$	20.4227	1.00615
$413$	−9.77924	−0.481205
$414$	−26.3478	−1.29492
$415$	0	0
$416$	−36.5463	−1.79183
$417$	−20.2838	−0.993303
$418$	0	0
$419$	14.7812	0.722111	0.361055	−	0.932544i	$-0.382417\pi$
$419$	14.7812	0.722111	0.361055	+	0.932544i	$0.382417\pi$
$420$	0	0
$421$	−10.7773	−0.525252	−0.262626	−	0.964898i	$-0.584589\pi$
$421$	−10.7773	−0.525252	−0.262626	+	0.964898i	$0.584589\pi$
$422$	−11.7907	−0.573964
$423$	−16.7241	−0.813152
$424$	−0.513102	−0.0249184
$425$	0	0
$426$	28.9192	1.40114
$427$	2.38641	0.115487
$428$	16.0118	0.773960
$429$	0	0
$430$	0	0
$431$	−32.1916	−1.55061	−0.775307	−	0.631585i	$-0.782405\pi$
$431$	−32.1916	−1.55061	−0.775307	+	0.631585i	$0.782405\pi$
$432$	−8.63971	−0.415678
$433$	−0.382631	−0.0183881	−0.00919404	−	0.999958i	$-0.502927\pi$
$433$	−0.382631	−0.0183881	−0.00919404	+	0.999958i	$0.502927\pi$
$434$	0.817703	0.0392510
$435$	0	0
$436$	20.6507	0.988990
$437$	−11.0338	−0.527817
$438$	−7.61305	−0.363765
$439$	−26.5331	−1.26635	−0.633177	−	0.774007i	$-0.718249\pi$
$439$	−26.5331	−1.26635	−0.633177	+	0.774007i	$0.718249\pi$
$440$	0	0
$441$	−23.4649	−1.11738
$442$	−30.5553	−1.45337
$443$	14.5010	0.688963	0.344482	−	0.938793i	$-0.388055\pi$
$443$	14.5010	0.688963	0.344482	+	0.938793i	$0.388055\pi$
$444$	−34.7990	−1.65149
$445$	0	0
$446$	38.4951	1.82279
$447$	44.2443	2.09268
$448$	−8.44450	−0.398965
$449$	10.1140	0.477309	0.238655	−	0.971105i	$-0.423294\pi$
$449$	10.1140	0.477309	0.238655	+	0.971105i	$0.423294\pi$
$450$	0	0
$451$	0	0
$452$	−10.5155	−0.494606
$453$	32.1878	1.51232
$454$	−57.6060	−2.70358
$455$	0	0
$456$	1.65639	0.0775676
$457$	39.0887	1.82849	0.914247	−	0.405157i	$-0.132783\pi$
$457$	39.0887	1.82849	0.914247	+	0.405157i	$0.132783\pi$
$458$	51.3841	2.40102
$459$	−7.58567	−0.354069
$460$	0	0
$461$	−39.1322	−1.82257	−0.911285	−	0.411776i	$-0.864908\pi$
$461$	−39.1322	−1.82257	−0.911285	+	0.411776i	$0.864908\pi$
$462$	0	0
$463$	12.9189	0.600392	0.300196	−	0.953878i	$-0.402948\pi$
$463$	12.9189	0.600392	0.300196	+	0.953878i	$0.402948\pi$
$464$	18.6710	0.866780
$465$	0	0
$466$	−25.3313	−1.17345
$467$	11.2491	0.520546	0.260273	−	0.965535i	$-0.416187\pi$
$467$	11.2491	0.520546	0.260273	+	0.965535i	$0.416187\pi$
$468$	−36.6695	−1.69505
$469$	9.27247	0.428163
$470$	0	0
$471$	−11.0320	−0.508326
$472$	1.95261	0.0898762
$473$	0	0
$474$	5.32424	0.244550
$475$	0	0
$476$	−6.75241	−0.309496
$477$	10.2903	0.471160
$478$	41.2580	1.88710
$479$	−20.9663	−0.957974	−0.478987	−	0.877822i	$-0.658996\pi$
$479$	−20.9663	−0.957974	−0.478987	+	0.877822i	$0.658996\pi$
$480$	0	0
$481$	28.6780	1.30761
$482$	−46.1268	−2.10102
$483$	8.52053	0.387697
$484$	0	0
$485$	0	0
$486$	43.7381	1.98400
$487$	−22.5691	−1.02270	−0.511352	−	0.859371i	$-0.670855\pi$
$487$	−22.5691	−1.02270	−0.511352	+	0.859371i	$0.670855\pi$
$488$	−0.476492	−0.0215698
$489$	43.4326	1.96409
$490$	0	0
$491$	20.0762	0.906026	0.453013	−	0.891504i	$-0.350349\pi$
$491$	20.0762	0.906026	0.453013	+	0.891504i	$0.350349\pi$
$492$	−31.7685	−1.43223
$493$	16.3932	0.738310
$494$	−30.0141	−1.35040
$495$	0	0
$496$	1.58993	0.0713898
$497$	−5.26674	−0.236245
$498$	−39.2198	−1.75748
$499$	4.65184	0.208245	0.104123	−	0.994564i	$-0.466797\pi$
$499$	4.65184	0.208245	0.104123	+	0.994564i	$0.466797\pi$
$500$	0	0
$501$	24.0224	1.07324
$502$	−34.3071	−1.53120
$503$	12.9972	0.579517	0.289759	−	0.957100i	$-0.406425\pi$
$503$	12.9972	0.579517	0.289759	+	0.957100i	$0.406425\pi$
$504$	−0.720340	−0.0320865
$505$	0	0
$506$	0	0
$507$	19.5928	0.870147
$508$	14.8565	0.659150
$509$	29.5445	1.30954	0.654769	−	0.755829i	$-0.272766\pi$
$509$	29.5445	1.30954	0.654769	+	0.755829i	$0.272766\pi$
$510$	0	0
$511$	1.38648	0.0613343
$512$	−32.1586	−1.42122
$513$	−7.45131	−0.328983
$514$	9.58385	0.422725
$515$	0	0
$516$	−12.4207	−0.546793
$517$	0	0
$518$	12.3869	0.544248
$519$	−10.7698	−0.472741
$520$	0	0
$521$	−31.4512	−1.37790	−0.688952	−	0.724807i	$-0.741929\pi$
$521$	−31.4512	−1.37790	−0.688952	+	0.724807i	$0.741929\pi$
$522$	38.4522	1.68301
$523$	9.02021	0.394426	0.197213	−	0.980361i	$-0.436811\pi$
$523$	9.02021	0.394426	0.197213	+	0.980361i	$0.436811\pi$
$524$	−15.1218	−0.660600
$525$	0	0
$526$	36.6362	1.59741
$527$	1.39595	0.0608088
$528$	0	0
$529$	−11.6672	−0.507269
$530$	0	0
$531$	−39.1597	−1.69939
$532$	−6.63281	−0.287569
$533$	26.1806	1.13401
$534$	64.6808	2.79901
$535$	0	0
$536$	−1.85142	−0.0799693
$537$	−42.7054	−1.84287
$538$	4.65046	0.200496
$539$	0	0
$540$	0	0
$541$	8.55013	0.367599	0.183799	−	0.982964i	$-0.441160\pi$
$541$	8.55013	0.367599	0.183799	+	0.982964i	$0.441160\pi$
$542$	1.63046	0.0700342
$543$	−14.5362	−0.623806
$544$	−26.9483	−1.15540
$545$	0	0
$546$	23.1776	0.991908
$547$	−33.9300	−1.45074	−0.725371	−	0.688358i	$-0.758332\pi$
$547$	−33.9300	−1.45074	−0.725371	+	0.688358i	$0.758332\pi$
$548$	−17.7907	−0.759981
$549$	9.55608	0.407844
$550$	0	0
$551$	16.1028	0.686002
$552$	−1.70128	−0.0724115
$553$	−0.969645	−0.0412335
$554$	23.9460	1.01737
$555$	0	0
$556$	−16.2179	−0.687792
$557$	−22.9406	−0.972025	−0.486013	−	0.873952i	$-0.661549\pi$
$557$	−22.9406	−0.972025	−0.486013	+	0.873952i	$0.661549\pi$
$558$	3.27439	0.138616
$559$	10.2360	0.432937
$560$	0	0
$561$	0	0
$562$	12.6061	0.531757
$563$	9.28262	0.391216	0.195608	−	0.980682i	$-0.437332\pi$
$563$	9.28262	0.391216	0.195608	+	0.980682i	$0.437332\pi$
$564$	−23.7440	−0.999805
$565$	0	0
$566$	17.4961	0.735414
$567$	−5.45194	−0.228960
$568$	1.05160	0.0441243
$569$	−30.9271	−1.29653	−0.648266	−	0.761414i	$-0.724506\pi$
$569$	−30.9271	−1.29653	−0.648266	+	0.761414i	$0.724506\pi$
$570$	0	0
$571$	2.63736	0.110370	0.0551851	−	0.998476i	$-0.482425\pi$
$571$	2.63736	0.110370	0.0551851	+	0.998476i	$0.482425\pi$
$572$	0	0
$573$	57.1712	2.38836
$574$	11.3081	0.471993
$575$	0	0
$576$	−33.8150	−1.40896
$577$	36.6305	1.52495	0.762474	−	0.647019i	$-0.223985\pi$
$577$	36.6305	1.52495	0.762474	+	0.647019i	$0.223985\pi$
$578$	11.8719	0.493807
$579$	58.9081	2.44814
$580$	0	0
$581$	7.14266	0.296328
$582$	−16.0118	−0.663710
$583$	0	0
$584$	−0.276837	−0.0114556
$585$	0	0
$586$	−11.7586	−0.485742
$587$	−13.7524	−0.567623	−0.283812	−	0.958880i	$-0.591599\pi$
$587$	−13.7524	−0.567623	−0.283812	+	0.958880i	$0.591599\pi$
$588$	−33.3144	−1.37386
$589$	1.37123	0.0565005
$590$	0	0
$591$	−67.4748	−2.77554
$592$	24.0848	0.989879
$593$	20.1550	0.827668	0.413834	−	0.910352i	$-0.364189\pi$
$593$	20.1550	0.827668	0.413834	+	0.910352i	$0.364189\pi$
$594$	0	0
$595$	0	0
$596$	35.3755	1.44904
$597$	44.4640	1.81979
$598$	30.8276	1.26063
$599$	4.32318	0.176640	0.0883202	−	0.996092i	$-0.471850\pi$
$599$	4.32318	0.176640	0.0883202	+	0.996092i	$0.471850\pi$
$600$	0	0
$601$	−35.9895	−1.46804	−0.734021	−	0.679127i	$-0.762359\pi$
$601$	−35.9895	−1.46804	−0.734021	+	0.679127i	$0.762359\pi$
$602$	4.42123	0.180196
$603$	37.1304	1.51207
$604$	25.7358	1.04717
$605$	0	0
$606$	80.5893	3.27372
$607$	39.6498	1.60934	0.804669	−	0.593724i	$-0.202343\pi$
$607$	39.6498	1.60934	0.804669	+	0.593724i	$0.202343\pi$
$608$	−26.4710	−1.07354
$609$	−12.4349	−0.503889
$610$	0	0
$611$	19.5676	0.791620
$612$	−27.0392	−1.09299
$613$	−3.81489	−0.154082	−0.0770410	−	0.997028i	$-0.524547\pi$
$613$	−3.81489	−0.154082	−0.0770410	+	0.997028i	$0.524547\pi$
$614$	−37.3817	−1.50860
$615$	0	0
$616$	0	0
$617$	−28.4055	−1.14356	−0.571781	−	0.820407i	$-0.693747\pi$
$617$	−28.4055	−1.14356	−0.571781	+	0.820407i	$0.693747\pi$
$618$	−51.6904	−2.07929
$619$	24.0709	0.967489	0.483745	−	0.875209i	$-0.339276\pi$
$619$	24.0709	0.967489	0.483745	+	0.875209i	$0.339276\pi$
$620$	0	0
$621$	7.65327	0.307115
$622$	22.9311	0.919453
$623$	−11.7796	−0.471940
$624$	45.0660	1.80408
$625$	0	0
$626$	−10.3511	−0.413714
$627$	0	0
$628$	−8.82059	−0.351980
$629$	21.1465	0.843165
$630$	0	0
$631$	19.1216	0.761218	0.380609	−	0.924736i	$-0.375714\pi$
$631$	19.1216	0.761218	0.380609	+	0.924736i	$0.375714\pi$
$632$	0.193608	0.00770131
$633$	15.2686	0.606871
$634$	23.9064	0.949443
$635$	0	0
$636$	14.6097	0.579311
$637$	27.4546	1.08779
$638$	0	0
$639$	−21.0900	−0.834307
$640$	0	0
$641$	11.9790	0.473143	0.236571	−	0.971614i	$-0.423976\pi$
$641$	11.9790	0.473143	0.236571	+	0.971614i	$0.423976\pi$
$642$	−40.5264	−1.59945
$643$	−25.7408	−1.01512	−0.507559	−	0.861617i	$-0.669452\pi$
$643$	−25.7408	−1.01512	−0.507559	+	0.861617i	$0.669452\pi$
$644$	6.81258	0.268453
$645$	0	0
$646$	−22.1316	−0.870757
$647$	9.30228	0.365710	0.182855	−	0.983140i	$-0.441466\pi$
$647$	9.30228	0.365710	0.182855	+	0.983140i	$0.441466\pi$
$648$	1.08858	0.0427636
$649$	0	0
$650$	0	0
$651$	−1.05889	−0.0415014
$652$	34.7265	1.35999
$653$	−37.4506	−1.46556	−0.732778	−	0.680468i	$-0.761777\pi$
$653$	−37.4506	−1.46556	−0.732778	+	0.680468i	$0.761777\pi$
$654$	−52.2676	−2.04382
$655$	0	0
$656$	21.9873	0.858461
$657$	5.55198	0.216604
$658$	8.45182	0.329486
$659$	4.93753	0.192339	0.0961693	−	0.995365i	$-0.469341\pi$
$659$	4.93753	0.192339	0.0961693	+	0.995365i	$0.469341\pi$
$660$	0	0
$661$	−4.82155	−0.187537	−0.0937683	−	0.995594i	$-0.529891\pi$
$661$	−4.82155	−0.187537	−0.0937683	+	0.995594i	$0.529891\pi$
$662$	−59.2317	−2.30210
$663$	39.5679	1.53669
$664$	−1.42617	−0.0553460
$665$	0	0
$666$	49.6017	1.92203
$667$	−16.5392	−0.640402
$668$	19.2071	0.743145
$669$	−49.8497	−1.92730
$670$	0	0
$671$	0	0
$672$	20.4415	0.788548
$673$	−32.9141	−1.26874	−0.634372	−	0.773028i	$-0.718741\pi$
$673$	−32.9141	−1.26874	−0.634372	+	0.773028i	$0.718741\pi$
$674$	53.9192	2.07689
$675$	0	0
$676$	15.6654	0.602515
$677$	−17.1426	−0.658842	−0.329421	−	0.944183i	$-0.606854\pi$
$677$	−17.1426	−0.658842	−0.329421	+	0.944183i	$0.606854\pi$
$678$	26.6150	1.02214
$679$	2.91605	0.111908
$680$	0	0
$681$	74.5977	2.85859
$682$	0	0
$683$	19.3586	0.740737	0.370368	−	0.928885i	$-0.379231\pi$
$683$	19.3586	0.740737	0.370368	+	0.928885i	$0.379231\pi$
$684$	−26.5602	−1.01556
$685$	0	0
$686$	25.5401	0.975125
$687$	−66.5405	−2.53868
$688$	8.59655	0.327740
$689$	−12.0399	−0.458684
$690$	0	0
$691$	−9.14145	−0.347757	−0.173879	−	0.984767i	$-0.555630\pi$
$691$	−9.14145	−0.347757	−0.173879	+	0.984767i	$0.555630\pi$
$692$	−8.61097	−0.327340
$693$	0	0
$694$	−8.72467	−0.331184
$695$	0	0
$696$	2.48287	0.0941130
$697$	19.3049	0.731225
$698$	−26.6509	−1.00875
$699$	32.8031	1.24073
$700$	0	0
$701$	15.2935	0.577626	0.288813	−	0.957385i	$-0.406739\pi$
$701$	15.2935	0.577626	0.288813	+	0.957385i	$0.406739\pi$
$702$	20.8184	0.785741
$703$	20.7719	0.783428
$704$	0	0
$705$	0	0
$706$	51.5844	1.94141
$707$	−14.6768	−0.551979
$708$	−55.5971	−2.08947
$709$	42.9129	1.61163	0.805813	−	0.592170i	$-0.201729\pi$
$709$	42.9129	1.61163	0.805813	+	0.592170i	$0.201729\pi$
$710$	0	0
$711$	−3.88282	−0.145617
$712$	2.35202	0.0881456
$713$	−1.40839	−0.0527448
$714$	17.0905	0.639598
$715$	0	0
$716$	−34.1450	−1.27606
$717$	−53.4276	−1.99529
$718$	53.1705	1.98431
$719$	4.81326	0.179504	0.0897521	−	0.995964i	$-0.471393\pi$
$719$	4.81326	0.179504	0.0897521	+	0.995964i	$0.471393\pi$
$720$	0	0
$721$	9.41380	0.350588
$722$	16.7104	0.621895
$723$	59.7325	2.22147
$724$	−11.6224	−0.431942
$725$	0	0
$726$	0	0
$727$	21.8922	0.811937	0.405969	−	0.913887i	$-0.366934\pi$
$727$	21.8922	0.811937	0.405969	+	0.913887i	$0.366934\pi$
$728$	0.842817	0.0312369
$729$	−39.7047	−1.47054
$730$	0	0
$731$	7.54777	0.279164
$732$	13.5673	0.501461
$733$	−48.0295	−1.77401	−0.887005	−	0.461759i	$-0.847218\pi$
$733$	−48.0295	−1.77401	−0.887005	+	0.461759i	$0.847218\pi$
$734$	4.08528	0.150790
$735$	0	0
$736$	27.1885	1.00218
$737$	0	0
$738$	45.2820	1.66685
$739$	34.1074	1.25466	0.627330	−	0.778754i	$-0.284148\pi$
$739$	34.1074	1.25466	0.627330	+	0.778754i	$0.284148\pi$
$740$	0	0
$741$	38.8671	1.42782
$742$	−5.20039	−0.190912
$743$	−26.4402	−0.969996	−0.484998	−	0.874515i	$-0.661180\pi$
$743$	−26.4402	−0.969996	−0.484998	+	0.874515i	$0.661180\pi$
$744$	0.211428	0.00775134
$745$	0	0
$746$	17.9532	0.657312
$747$	28.6019	1.04649
$748$	0	0
$749$	7.38062	0.269682
$750$	0	0
$751$	45.9403	1.67639	0.838193	−	0.545374i	$-0.183612\pi$
$751$	45.9403	1.67639	0.838193	+	0.545374i	$0.183612\pi$
$752$	16.4335	0.599270
$753$	44.4264	1.61899
$754$	−44.9901	−1.63844
$755$	0	0
$756$	4.60066	0.167324
$757$	30.8557	1.12147	0.560735	−	0.827996i	$-0.310519\pi$
$757$	30.8557	1.12147	0.560735	+	0.827996i	$0.310519\pi$
$758$	−42.6271	−1.54828
$759$	0	0
$760$	0	0
$761$	14.1975	0.514658	0.257329	−	0.966324i	$-0.417158\pi$
$761$	14.1975	0.514658	0.257329	+	0.966324i	$0.417158\pi$
$762$	−37.6022	−1.36218
$763$	9.51892	0.344608
$764$	45.7112	1.65377
$765$	0	0
$766$	52.5744	1.89959
$767$	45.8179	1.65439
$768$	37.6531	1.35869
$769$	30.0208	1.08258	0.541290	−	0.840836i	$-0.317936\pi$
$769$	30.0208	1.08258	0.541290	+	0.840836i	$0.317936\pi$
$770$	0	0
$771$	−12.4107	−0.446961
$772$	47.0999	1.69516
$773$	30.4294	1.09447	0.547234	−	0.836979i	$-0.315681\pi$
$773$	30.4294	1.09447	0.547234	+	0.836979i	$0.315681\pi$
$774$	17.7042	0.636366
$775$	0	0
$776$	−0.582245	−0.0209014
$777$	−16.0405	−0.575451
$778$	3.32655	0.119262
$779$	18.9630	0.679418
$780$	0	0
$781$	0	0
$782$	22.7315	0.812876
$783$	−11.1693	−0.399157
$784$	23.0573	0.823476
$785$	0	0
$786$	38.2738	1.36518
$787$	30.4648	1.08595	0.542976	−	0.839748i	$-0.317298\pi$
$787$	30.4648	1.08595	0.542976	+	0.839748i	$0.317298\pi$
$788$	−53.9494	−1.92187
$789$	−47.4425	−1.68900
$790$	0	0
$791$	−4.84709	−0.172343
$792$	0	0
$793$	−11.1809	−0.397044
$794$	33.8132	1.19999
$795$	0	0
$796$	35.5512	1.26008
$797$	33.0452	1.17052	0.585260	−	0.810846i	$-0.300992\pi$
$797$	33.0452	1.17052	0.585260	+	0.810846i	$0.300992\pi$
$798$	16.7878	0.594283
$799$	14.4287	0.510449
$800$	0	0
$801$	−47.1699	−1.66667
$802$	−56.2421	−1.98598
$803$	0	0
$804$	52.7160	1.85915
$805$	0	0
$806$	−3.83112	−0.134945
$807$	−6.02218	−0.211991
$808$	2.93051	0.103095
$809$	−10.9895	−0.386371	−0.193186	−	0.981162i	$-0.561882\pi$
$809$	−10.9895	−0.386371	−0.193186	+	0.981162i	$0.561882\pi$
$810$	0	0
$811$	−40.7587	−1.43123	−0.715615	−	0.698495i	$-0.753854\pi$
$811$	−40.7587	−1.43123	−0.715615	+	0.698495i	$0.753854\pi$
$812$	−9.94235	−0.348908
$813$	−2.11138	−0.0740494
$814$	0	0
$815$	0	0
$816$	33.2305	1.16330
$817$	7.41408	0.259386
$818$	79.7998	2.79013
$819$	−16.9027	−0.590630
$820$	0	0
$821$	−15.1076	−0.527259	−0.263630	−	0.964624i	$-0.584920\pi$
$821$	−15.1076	−0.527259	−0.263630	+	0.964624i	$0.584920\pi$
$822$	45.0288	1.57056
$823$	−36.4313	−1.26991	−0.634957	−	0.772548i	$-0.718982\pi$
$823$	−36.4313	−1.26991	−0.634957	+	0.772548i	$0.718982\pi$
$824$	−1.87964	−0.0654805
$825$	0	0
$826$	19.7901	0.688585
$827$	3.80342	0.132258	0.0661290	−	0.997811i	$-0.478935\pi$
$827$	3.80342	0.132258	0.0661290	+	0.997811i	$0.478935\pi$
$828$	27.2801	0.948050
$829$	−27.3063	−0.948386	−0.474193	−	0.880421i	$-0.657260\pi$
$829$	−27.3063	−0.948386	−0.474193	+	0.880421i	$0.657260\pi$
$830$	0	0
$831$	−31.0092	−1.07570
$832$	39.5644	1.37165
$833$	20.2443	0.701424
$834$	41.0480	1.42138
$835$	0	0
$836$	0	0
$837$	−0.951115	−0.0328754
$838$	−29.9125	−1.03331
$839$	28.3250	0.977888	0.488944	−	0.872315i	$-0.337382\pi$
$839$	28.3250	0.977888	0.488944	+	0.872315i	$0.337382\pi$
$840$	0	0
$841$	−4.86246	−0.167671
$842$	21.8098	0.751615
$843$	−16.3244	−0.562244
$844$	12.2080	0.420215
$845$	0	0
$846$	33.8442	1.16359
$847$	0	0
$848$	−10.1115	−0.347232
$849$	−22.6567	−0.777577
$850$	0	0
$851$	−21.3349	−0.731351
$852$	−29.9426	−1.02582
$853$	8.34318	0.285665	0.142833	−	0.989747i	$-0.454379\pi$
$853$	8.34318	0.285665	0.142833	+	0.989747i	$0.454379\pi$
$854$	−4.82934	−0.165257
$855$	0	0
$856$	−1.47368	−0.0503694
$857$	−8.59547	−0.293616	−0.146808	−	0.989165i	$-0.546900\pi$
$857$	−8.59547	−0.293616	−0.146808	+	0.989165i	$0.546900\pi$
$858$	0	0
$859$	9.40807	0.320999	0.160500	−	0.987036i	$-0.448689\pi$
$859$	9.40807	0.320999	0.160500	+	0.987036i	$0.448689\pi$
$860$	0	0
$861$	−14.6436	−0.499054
$862$	65.1456	2.21887
$863$	10.3261	0.351503	0.175752	−	0.984435i	$-0.443764\pi$
$863$	10.3261	0.351503	0.175752	+	0.984435i	$0.443764\pi$
$864$	18.3609	0.624649
$865$	0	0
$866$	0.774324	0.0263126
$867$	−15.3737	−0.522118
$868$	−0.846638	−0.0287368
$869$	0	0
$870$	0	0
$871$	−43.4435	−1.47203
$872$	−1.90063	−0.0643635
$873$	11.6770	0.395205
$874$	22.3288	0.755285
$875$	0	0
$876$	7.88244	0.266323
$877$	29.8032	1.00638	0.503191	−	0.864175i	$-0.332159\pi$
$877$	29.8032	1.00638	0.503191	+	0.864175i	$0.332159\pi$
$878$	53.6945	1.81210
$879$	15.2269	0.513591
$880$	0	0
$881$	30.1175	1.01469	0.507343	−	0.861744i	$-0.330628\pi$
$881$	30.1175	1.01469	0.507343	+	0.861744i	$0.330628\pi$
$882$	47.4856	1.59892
$883$	−50.3619	−1.69481	−0.847406	−	0.530945i	$-0.821837\pi$
$883$	−50.3619	−1.69481	−0.847406	+	0.530945i	$0.821837\pi$
$884$	31.6365	1.06405
$885$	0	0
$886$	−29.3454	−0.985879
$887$	41.4054	1.39026	0.695129	−	0.718885i	$-0.255347\pi$
$887$	41.4054	1.39026	0.695129	+	0.718885i	$0.255347\pi$
$888$	3.20280	0.107479
$889$	6.84808	0.229677
$890$	0	0
$891$	0	0
$892$	−39.8572	−1.33452
$893$	14.1731	0.474284
$894$	−89.5365	−2.99455
$895$	0	0
$896$	1.48834	0.0497220
$897$	−39.9205	−1.33291
$898$	−20.4675	−0.683010
$899$	2.05543	0.0685523
$900$	0	0
$901$	−8.87793	−0.295767
$902$	0	0
$903$	−5.72532	−0.190527
$904$	0.967813	0.0321890
$905$	0	0
$906$	−65.1380	−2.16406
$907$	4.63216	0.153808	0.0769041	−	0.997038i	$-0.475496\pi$
$907$	4.63216	0.153808	0.0769041	+	0.997038i	$0.475496\pi$
$908$	59.6445	1.97937
$909$	−58.7716	−1.94933
$910$	0	0
$911$	6.97855	0.231210	0.115605	−	0.993295i	$-0.463119\pi$
$911$	6.97855	0.231210	0.115605	+	0.993295i	$0.463119\pi$
$912$	32.6419	1.08088
$913$	0	0
$914$	−79.1032	−2.61650
$915$	0	0
$916$	−53.2024	−1.75786
$917$	−6.97038	−0.230182
$918$	15.3510	0.506658
$919$	23.4207	0.772576	0.386288	−	0.922378i	$-0.373757\pi$
$919$	23.4207	0.772576	0.386288	+	0.922378i	$0.373757\pi$
$920$	0	0
$921$	48.4079	1.59510
$922$	79.1912	2.60802
$923$	24.6758	0.812215
$924$	0	0
$925$	0	0
$926$	−26.1438	−0.859137
$927$	37.6964	1.23811
$928$	−39.6791	−1.30253
$929$	−18.0905	−0.593529	−0.296765	−	0.954951i	$-0.595908\pi$
$929$	−18.0905	−0.593529	−0.296765	+	0.954951i	$0.595908\pi$
$930$	0	0
$931$	19.8858	0.651729
$932$	26.2277	0.859116
$933$	−29.6949	−0.972168
$934$	−22.7646	−0.744881
$935$	0	0
$936$	3.37495	0.110314
$937$	11.0096	0.359668	0.179834	−	0.983697i	$-0.442444\pi$
$937$	11.0096	0.359668	0.179834	+	0.983697i	$0.442444\pi$
$938$	−18.7645	−0.612684
$939$	13.4043	0.437433
$940$	0	0
$941$	−33.7808	−1.10122	−0.550612	−	0.834762i	$-0.685605\pi$
$941$	−33.7808	−1.10122	−0.550612	+	0.834762i	$0.685605\pi$
$942$	22.3252	0.727394
$943$	−19.4769	−0.634256
$944$	38.4795	1.25240
$945$	0	0
$946$	0	0
$947$	46.9853	1.52682	0.763409	−	0.645915i	$-0.223524\pi$
$947$	46.9853	1.52682	0.763409	+	0.645915i	$0.223524\pi$
$948$	−5.51264	−0.179042
$949$	−6.49596	−0.210868
$950$	0	0
$951$	−30.9579	−1.00388
$952$	0.621472	0.0201420
$953$	42.4907	1.37641	0.688204	−	0.725517i	$-0.258400\pi$
$953$	42.4907	1.37641	0.688204	+	0.725517i	$0.258400\pi$
$954$	−20.8243	−0.674212
$955$	0	0
$956$	−42.7180	−1.38160
$957$	0	0
$958$	42.4291	1.37082
$959$	−8.20060	−0.264811
$960$	0	0
$961$	−30.8250	−0.994354
$962$	−58.0352	−1.87113
$963$	29.5548	0.952389
$964$	47.7590	1.53822
$965$	0	0
$966$	−17.2428	−0.554780
$967$	54.1642	1.74180	0.870901	−	0.491458i	$-0.163536\pi$
$967$	54.1642	1.74180	0.870901	+	0.491458i	$0.163536\pi$
$968$	0	0
$969$	28.6596	0.920680
$970$	0	0
$971$	5.23848	0.168111	0.0840554	−	0.996461i	$-0.473213\pi$
$971$	5.23848	0.168111	0.0840554	+	0.996461i	$0.473213\pi$
$972$	−45.2859	−1.45254
$973$	−7.47562	−0.239657
$974$	45.6727	1.46345
$975$	0	0
$976$	−9.39008	−0.300569
$977$	7.78142	0.248950	0.124475	−	0.992223i	$-0.460275\pi$
$977$	7.78142	0.248950	0.124475	+	0.992223i	$0.460275\pi$
$978$	−87.8938	−2.81053
$979$	0	0
$980$	0	0
$981$	38.1173	1.21699
$982$	−40.6279	−1.29649
$983$	−51.5136	−1.64303	−0.821515	−	0.570188i	$-0.806871\pi$
$983$	−51.5136	−1.64303	−0.821515	+	0.570188i	$0.806871\pi$
$984$	2.92388	0.0932098
$985$	0	0
$986$	−33.1745	−1.05649
$987$	−10.9448	−0.348376
$988$	31.0762	0.988664
$989$	−7.61503	−0.242144
$990$	0	0
$991$	−22.3382	−0.709596	−0.354798	−	0.934943i	$-0.615450\pi$
$991$	−22.3382	−0.709596	−0.354798	+	0.934943i	$0.615450\pi$
$992$	−3.37887	−0.107279
$993$	76.7028	2.43409
$994$	10.6582	0.338058
$995$	0	0
$996$	40.6076	1.28670
$997$	0.184546	0.00584463	0.00292231	−	0.999996i	$-0.499070\pi$
$997$	0.184546	0.00584463	0.00292231	+	0.999996i	$0.499070\pi$
$998$	−9.41386	−0.297990
$999$	−14.4079	−0.455845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bk.1.1		8
5.2	odd	4		605.2.b.f.364.1		8
5.3	odd	4		605.2.b.f.364.8		8
5.4	even	2	inner	3025.2.a.bk.1.8		8
11.2	odd	10		275.2.h.d.26.1		16
11.6	odd	10		275.2.h.d.201.1		16
11.10	odd	2		3025.2.a.bl.1.8		8
55.2	even	20		55.2.j.a.4.4	yes	16
55.3	odd	20		605.2.j.g.9.4		16
55.7	even	20		605.2.j.h.269.1		16
55.8	even	20		605.2.j.h.9.1		16
55.13	even	20		55.2.j.a.4.1	✓	16
55.17	even	20		55.2.j.a.14.1	yes	16
55.18	even	20		605.2.j.h.269.4		16
55.24	odd	10		275.2.h.d.26.4		16
55.27	odd	20		605.2.j.d.124.4		16
55.28	even	20		55.2.j.a.14.4	yes	16
55.32	even	4		605.2.b.g.364.8		8
55.37	odd	20		605.2.j.g.269.4		16
55.38	odd	20		605.2.j.d.124.1		16
55.39	odd	10		275.2.h.d.201.4		16
55.42	odd	20		605.2.j.d.444.1		16
55.43	even	4		605.2.b.g.364.1		8
55.47	odd	20		605.2.j.g.9.1		16
55.48	odd	20		605.2.j.g.269.1		16
55.52	even	20		605.2.j.h.9.4		16
55.53	odd	20		605.2.j.d.444.4		16
55.54	odd	2		3025.2.a.bl.1.1		8
165.2	odd	20		495.2.ba.a.334.1		16
165.17	odd	20		495.2.ba.a.289.4		16
165.68	odd	20		495.2.ba.a.334.4		16
165.83	odd	20		495.2.ba.a.289.1		16
220.83	odd	20		880.2.cd.c.289.4		16
220.123	odd	20		880.2.cd.c.609.1		16
220.127	odd	20		880.2.cd.c.289.1		16
220.167	odd	20		880.2.cd.c.609.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.j.a.4.1	✓	16	55.13	even	20
55.2.j.a.4.4	yes	16	55.2	even	20
55.2.j.a.14.1	yes	16	55.17	even	20
55.2.j.a.14.4	yes	16	55.28	even	20
275.2.h.d.26.1		16	11.2	odd	10
275.2.h.d.26.4		16	55.24	odd	10
275.2.h.d.201.1		16	11.6	odd	10
275.2.h.d.201.4		16	55.39	odd	10
495.2.ba.a.289.1		16	165.83	odd	20
495.2.ba.a.289.4		16	165.17	odd	20
495.2.ba.a.334.1		16	165.2	odd	20
495.2.ba.a.334.4		16	165.68	odd	20
605.2.b.f.364.1		8	5.2	odd	4
605.2.b.f.364.8		8	5.3	odd	4
605.2.b.g.364.1		8	55.43	even	4
605.2.b.g.364.8		8	55.32	even	4
605.2.j.d.124.1		16	55.38	odd	20
605.2.j.d.124.4		16	55.27	odd	20
605.2.j.d.444.1		16	55.42	odd	20
605.2.j.d.444.4		16	55.53	odd	20
605.2.j.g.9.1		16	55.47	odd	20
605.2.j.g.9.4		16	55.3	odd	20
605.2.j.g.269.1		16	55.48	odd	20
605.2.j.g.269.4		16	55.37	odd	20
605.2.j.h.9.1		16	55.8	even	20
605.2.j.h.9.4		16	55.52	even	20
605.2.j.h.269.1		16	55.7	even	20
605.2.j.h.269.4		16	55.18	even	20
880.2.cd.c.289.1		16	220.127	odd	20
880.2.cd.c.289.4		16	220.83	odd	20
880.2.cd.c.609.1		16	220.123	odd	20
880.2.cd.c.609.4		16	220.167	odd	20
3025.2.a.bk.1.1		8	1.1	even	1	trivial
3025.2.a.bk.1.8		8	5.4	even	2	inner
3025.2.a.bl.1.1		8	55.54	odd	2
3025.2.a.bl.1.8		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)

\(f(q)\)	\(=\)	\(q-2.02368 q^{2} +2.62059 q^{3} +2.09529 q^{4} -5.30325 q^{6} +0.965823 q^{7} -0.192845 q^{8} +3.86752 q^{9} +O(q^{10})\)	\(q-2.02368 q^{2} +2.62059 q^{3} +2.09529 q^{4} -5.30325 q^{6} +0.965823 q^{7} -0.192845 q^{8} +3.86752 q^{9} +5.49092 q^{12} -4.52509 q^{13} -1.95452 q^{14} -3.80033 q^{16} -3.33669 q^{17} -7.82663 q^{18} -3.27759 q^{19} +2.53103 q^{21} +3.36643 q^{23} -0.505368 q^{24} +9.15736 q^{26} +2.27341 q^{27} +2.02368 q^{28} -4.91300 q^{29} -0.418365 q^{31} +8.07636 q^{32} +6.75241 q^{34} +8.10358 q^{36} -6.33755 q^{37} +6.63281 q^{38} -11.8584 q^{39} -5.78564 q^{41} -5.12200 q^{42} -2.26205 q^{43} -6.81258 q^{46} -4.32424 q^{47} -9.95913 q^{48} -6.06719 q^{49} -8.74411 q^{51} -9.48140 q^{52} +2.66070 q^{53} -4.60066 q^{54} -0.186254 q^{56} -8.58924 q^{57} +9.94235 q^{58} -10.1253 q^{59} +2.47086 q^{61} +0.846638 q^{62} +3.73534 q^{63} -8.74332 q^{64} +9.60059 q^{67} -6.99135 q^{68} +8.82204 q^{69} -5.45311 q^{71} -0.745831 q^{72} +1.43554 q^{73} +12.8252 q^{74} -6.86752 q^{76} +23.9977 q^{78} -1.00396 q^{79} -5.64486 q^{81} +11.7083 q^{82} +7.39542 q^{83} +5.30325 q^{84} +4.57768 q^{86} -12.8750 q^{87} -12.1964 q^{89} -4.37044 q^{91} +7.05365 q^{92} -1.09637 q^{93} +8.75089 q^{94} +21.1649 q^{96} +3.01924 q^{97} +12.2781 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10}) \) 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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