LMFDB - Embedded newform 3025.2.a.x.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:	$4$
Coefficient field:	4.4.105840.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 15 $ x^4 - 9*x^2 + 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48617$ 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)

$f(q)$	$=$	$q-1.48617 q^{2} +1.79129 q^{3} +0.208712 q^{4} -2.66216 q^{6} +1.48617 q^{7} +2.66216 q^{8} +0.208712 q^{9} +O(q^{10})$	\(q-1.48617 q^{2} +1.79129 q^{3} +0.208712 q^{4} -2.66216 q^{6} +1.48617 q^{7} +2.66216 q^{8} +0.208712 q^{9} +0.373864 q^{12} +5.63451 q^{13} -2.20871 q^{14} -4.37386 q^{16} -4.14834 q^{17} -0.310183 q^{18} -2.66216 q^{19} +2.66216 q^{21} -6.79129 q^{23} +4.76870 q^{24} -8.37386 q^{26} -5.00000 q^{27} +0.310183 q^{28} -10.0930 q^{29} -5.00000 q^{31} +1.17599 q^{32} +6.16515 q^{34} +0.0435608 q^{36} -8.58258 q^{37} +3.95644 q^{38} +10.0930 q^{39} +2.66216 q^{41} -3.95644 q^{42} +2.97235 q^{43} +10.0930 q^{46} -1.41742 q^{47} -7.83485 q^{48} -4.79129 q^{49} -7.43087 q^{51} +1.17599 q^{52} -6.79129 q^{53} +7.43087 q^{54} +3.95644 q^{56} -4.76870 q^{57} +15.0000 q^{58} +6.16515 q^{59} +12.7552 q^{61} +7.43087 q^{62} +0.310183 q^{63} +7.00000 q^{64} +3.58258 q^{67} -0.865809 q^{68} -12.1652 q^{69} -9.16515 q^{71} +0.555626 q^{72} -9.78285 q^{73} +12.7552 q^{74} -0.555626 q^{76} -15.0000 q^{78} +15.4174 q^{79} -9.58258 q^{81} -3.95644 q^{82} +7.12069 q^{83} +0.555626 q^{84} -4.41742 q^{86} -18.0795 q^{87} +8.20871 q^{89} +8.37386 q^{91} -1.41742 q^{92} -8.95644 q^{93} +2.10654 q^{94} +2.10654 q^{96} +0.373864 q^{97} +7.12069 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 10 q^{4} + 10 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 10 * q^4 + 10 * q^9	$ 4 q - 2 q^{3} + 10 q^{4} + 10 q^{9} - 26 q^{12} - 18 q^{14} + 10 q^{16} - 18 q^{23} - 6 q^{26} - 20 q^{27} - 20 q^{31} - 12 q^{34} + 46 q^{36} - 16 q^{37} - 30 q^{38} + 30 q^{42} - 24 q^{47} - 68 q^{48} - 10 q^{49} - 18 q^{53} - 30 q^{56} + 60 q^{58} - 12 q^{59} + 28 q^{64} - 4 q^{67} - 12 q^{69} - 60 q^{78} - 20 q^{81} + 30 q^{82} - 36 q^{86} + 42 q^{89} + 6 q^{91} - 24 q^{92} + 10 q^{93} - 26 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 + 10 * q^4 + 10 * q^9 - 26 * q^12 - 18 * q^14 + 10 * q^16 - 18 * q^23 - 6 * q^26 - 20 * q^27 - 20 * q^31 - 12 * q^34 + 46 * q^36 - 16 * q^37 - 30 * q^38 + 30 * q^42 - 24 * q^47 - 68 * q^48 - 10 * q^49 - 18 * q^53 - 30 * q^56 + 60 * q^58 - 12 * q^59 + 28 * q^64 - 4 * q^67 - 12 * q^69 - 60 * q^78 - 20 * q^81 + 30 * q^82 - 36 * q^86 + 42 * q^89 + 6 * q^91 - 24 * q^92 + 10 * q^93 - 26 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.48617	−1.05088	−0.525442	−	0.850830i	$-0.676100\pi$
$2$	−1.48617	−1.05088	−0.525442	+	0.850830i	$0.676100\pi$
$3$	1.79129	1.03420	0.517100	−	0.855925i	$-0.327011\pi$
$3$	1.79129	1.03420	0.517100	+	0.855925i	$0.327011\pi$
$4$	0.208712	0.104356
$5$	0	0
$5$	0	0
$6$	−2.66216	−1.08682
$7$	1.48617	0.561721	0.280860	−	0.959749i	$-0.409380\pi$
$7$	1.48617	0.561721	0.280860	+	0.959749i	$0.409380\pi$
$8$	2.66216	0.941217
$9$	0.208712	0.0695707
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.373864	0.107925
$13$	5.63451	1.56273	0.781366	−	0.624073i	$-0.214523\pi$
$13$	5.63451	1.56273	0.781366	+	0.624073i	$0.214523\pi$
$14$	−2.20871	−0.590303
$15$	0	0
$16$	−4.37386	−1.09347
$17$	−4.14834	−1.00612	−0.503060	−	0.864252i	$-0.667793\pi$
$17$	−4.14834	−1.00612	−0.503060	+	0.864252i	$0.667793\pi$
$18$	−0.310183	−0.0731107
$19$	−2.66216	−0.610742	−0.305371	−	0.952233i	$-0.598781\pi$
$19$	−2.66216	−0.610742	−0.305371	+	0.952233i	$0.598781\pi$
$20$	0	0
$21$	2.66216	0.580932
$22$	0	0
$23$	−6.79129	−1.41608	−0.708041	−	0.706172i	$-0.750421\pi$
$23$	−6.79129	−1.41608	−0.708041	+	0.706172i	$0.750421\pi$
$24$	4.76870	0.973408
$25$	0	0
$26$	−8.37386	−1.64225
$27$	−5.00000	−0.962250
$28$	0.310183	0.0586190
$29$	−10.0930	−1.87423	−0.937115	−	0.349022i	$-0.886514\pi$
$29$	−10.0930	−1.87423	−0.937115	+	0.349022i	$0.886514\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.17599	0.207888
$33$	0	0
$34$	6.16515	1.05731
$35$	0	0
$36$	0.0435608	0.00726013
$37$	−8.58258	−1.41097	−0.705483	−	0.708726i	$-0.749270\pi$
$37$	−8.58258	−1.41097	−0.705483	+	0.708726i	$0.749270\pi$
$38$	3.95644	0.641819
$39$	10.0930	1.61618
$40$	0	0
$41$	2.66216	0.415760	0.207880	−	0.978154i	$-0.433344\pi$
$41$	2.66216	0.415760	0.207880	+	0.978154i	$0.433344\pi$
$42$	−3.95644	−0.610492
$43$	2.97235	0.453279	0.226639	−	0.973979i	$-0.427226\pi$
$43$	2.97235	0.453279	0.226639	+	0.973979i	$0.427226\pi$
$44$	0	0
$45$	0	0
$46$	10.0930	1.48814
$47$	−1.41742	−0.206753	−0.103376	−	0.994642i	$-0.532965\pi$
$47$	−1.41742	−0.206753	−0.103376	+	0.994642i	$0.532965\pi$
$48$	−7.83485	−1.13086
$49$	−4.79129	−0.684470
$50$	0	0
$51$	−7.43087	−1.04053
$52$	1.17599	0.163081
$53$	−6.79129	−0.932855	−0.466428	−	0.884559i	$-0.654459\pi$
$53$	−6.79129	−0.932855	−0.466428	+	0.884559i	$0.654459\pi$
$54$	7.43087	1.01121
$55$	0	0
$56$	3.95644	0.528701
$57$	−4.76870	−0.631630
$58$	15.0000	1.96960
$59$	6.16515	0.802634	0.401317	−	0.915939i	$-0.368553\pi$
$59$	6.16515	0.802634	0.401317	+	0.915939i	$0.368553\pi$
$60$	0	0
$61$	12.7552	1.63314	0.816568	−	0.577249i	$-0.195874\pi$
$61$	12.7552	1.63314	0.816568	+	0.577249i	$0.195874\pi$
$62$	7.43087	0.943721
$63$	0.310183	0.0390793
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	3.58258	0.437681	0.218841	−	0.975761i	$-0.429773\pi$
$67$	3.58258	0.437681	0.218841	+	0.975761i	$0.429773\pi$
$68$	−0.865809	−0.104995
$69$	−12.1652	−1.46451
$70$	0	0
$71$	−9.16515	−1.08770	−0.543852	−	0.839181i	$-0.683035\pi$
$71$	−9.16515	−1.08770	−0.543852	+	0.839181i	$0.683035\pi$
$72$	0.555626	0.0654812
$73$	−9.78285	−1.14500	−0.572498	−	0.819906i	$-0.694026\pi$
$73$	−9.78285	−1.14500	−0.572498	+	0.819906i	$0.694026\pi$
$74$	12.7552	1.48276
$75$	0	0
$76$	−0.555626	−0.0637347
$77$	0	0
$78$	−15.0000	−1.69842
$79$	15.4174	1.73459	0.867294	−	0.497796i	$-0.165857\pi$
$79$	15.4174	1.73459	0.867294	+	0.497796i	$0.165857\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	−3.95644	−0.436916
$83$	7.12069	0.781597	0.390798	−	0.920476i	$-0.372199\pi$
$83$	7.12069	0.781597	0.390798	+	0.920476i	$0.372199\pi$
$84$	0.555626	0.0606238
$85$	0	0
$86$	−4.41742	−0.476343
$87$	−18.0795	−1.93833
$88$	0	0
$89$	8.20871	0.870122	0.435061	−	0.900401i	$-0.356727\pi$
$89$	8.20871	0.870122	0.435061	+	0.900401i	$0.356727\pi$
$90$	0	0
$91$	8.37386	0.877819
$92$	−1.41742	−0.147777
$93$	−8.95644	−0.928739
$94$	2.10654	0.217273
$95$	0	0
$96$	2.10654	0.214998
$97$	0.373864	0.0379601	0.0189800	−	0.999820i	$-0.493958\pi$
$97$	0.373864	0.0379601	0.0189800	+	0.999820i	$0.493958\pi$
$98$	7.12069	0.719298
$99$	0	0
$100$	0	0
$101$	−12.7552	−1.26919	−0.634595	−	0.772845i	$-0.718833\pi$
$101$	−12.7552	−1.26919	−0.634595	+	0.772845i	$0.718833\pi$
$102$	11.0436	1.09348
$103$	−11.7913	−1.16183	−0.580915	−	0.813964i	$-0.697305\pi$
$103$	−11.7913	−1.16183	−0.580915	+	0.813964i	$0.697305\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	10.0930	0.980322
$107$	−13.9312	−1.34678	−0.673389	−	0.739288i	$-0.735162\pi$
$107$	−13.9312	−1.34678	−0.673389	+	0.739288i	$0.735162\pi$
$108$	−1.04356	−0.100417
$109$	4.76870	0.456759	0.228379	−	0.973572i	$-0.426657\pi$
$109$	4.76870	0.456759	0.228379	+	0.973572i	$0.426657\pi$
$110$	0	0
$111$	−15.3739	−1.45922
$112$	−6.50032	−0.614223
$113$	12.1652	1.14440	0.572201	−	0.820114i	$-0.306090\pi$
$113$	12.1652	1.14440	0.572201	+	0.820114i	$0.306090\pi$
$114$	7.08712	0.663770
$115$	0	0
$116$	−2.10654	−0.195587
$117$	1.17599	0.108720
$118$	−9.16249	−0.843475
$119$	−6.16515	−0.565159
$120$	0	0
$121$	0	0
$122$	−18.9564	−1.71624
$123$	4.76870	0.429980
$124$	−1.04356	−0.0937145
$125$	0	0
$126$	−0.460985	−0.0410678
$127$	−1.17599	−0.104352	−0.0521762	−	0.998638i	$-0.516616\pi$
$127$	−1.17599	−0.104352	−0.0521762	+	0.998638i	$0.516616\pi$
$128$	−12.7552	−1.12741
$129$	5.32433	0.468781
$130$	0	0
$131$	−7.43087	−0.649238	−0.324619	−	0.945845i	$-0.605236\pi$
$131$	−7.43087	−0.649238	−0.324619	+	0.945845i	$0.605236\pi$
$132$	0	0
$133$	−3.95644	−0.343067
$134$	−5.32433	−0.459952
$135$	0	0
$136$	−11.0436	−0.946978
$137$	5.37386	0.459120	0.229560	−	0.973294i	$-0.426271\pi$
$137$	5.37386	0.459120	0.229560	+	0.973294i	$0.426271\pi$
$138$	18.0795	1.53903
$139$	−2.66216	−0.225802	−0.112901	−	0.993606i	$-0.536014\pi$
$139$	−2.66216	−0.225802	−0.112901	+	0.993606i	$0.536014\pi$
$140$	0	0
$141$	−2.53901	−0.213824
$142$	13.6210	1.14305
$143$	0	0
$144$	−0.912878	−0.0760732
$145$	0	0
$146$	14.5390	1.20326
$147$	−8.58258	−0.707879
$148$	−1.79129	−0.147243
$149$	−5.32433	−0.436186	−0.218093	−	0.975928i	$-0.569984\pi$
$149$	−5.32433	−0.436186	−0.218093	+	0.975928i	$0.569984\pi$
$150$	0	0
$151$	−17.5239	−1.42607	−0.713037	−	0.701126i	$-0.752681\pi$
$151$	−17.5239	−1.42607	−0.713037	+	0.701126i	$0.752681\pi$
$152$	−7.08712	−0.574841
$153$	−0.865809	−0.0699965
$154$	0	0
$155$	0	0
$156$	2.10654	0.168658
$157$	6.41742	0.512166	0.256083	−	0.966655i	$-0.417568\pi$
$157$	6.41742	0.512166	0.256083	+	0.966655i	$0.417568\pi$
$158$	−22.9129	−1.82285
$159$	−12.1652	−0.964759
$160$	0	0
$161$	−10.0930	−0.795442
$162$	14.2414	1.11891
$163$	−21.1216	−1.65437	−0.827185	−	0.561929i	$-0.810059\pi$
$163$	−21.1216	−1.65437	−0.827185	+	0.561929i	$0.810059\pi$
$164$	0.555626	0.0433871
$165$	0	0
$166$	−10.5826	−0.821367
$167$	−3.28253	−0.254010	−0.127005	−	0.991902i	$-0.540536\pi$
$167$	−3.28253	−0.254010	−0.127005	+	0.991902i	$0.540536\pi$
$168$	7.08712	0.546783
$169$	18.7477	1.44213
$170$	0	0
$171$	−0.555626	−0.0424898
$172$	0.620365	0.0473024
$173$	5.63451	0.428384	0.214192	−	0.976792i	$-0.431288\pi$
$173$	5.63451	0.428384	0.214192	+	0.976792i	$0.431288\pi$
$174$	26.8693	2.03696
$175$	0	0
$176$	0	0
$177$	11.0436	0.830085
$178$	−12.1996	−0.914397
$179$	−6.79129	−0.507605	−0.253802	−	0.967256i	$-0.581681\pi$
$179$	−6.79129	−0.507605	−0.253802	+	0.967256i	$0.581681\pi$
$180$	0	0
$181$	23.9564	1.78067	0.890334	−	0.455308i	$-0.150471\pi$
$181$	23.9564	1.78067	0.890334	+	0.455308i	$0.150471\pi$
$182$	−12.4450	−0.922486
$183$	22.8482	1.68899
$184$	−18.0795	−1.33284
$185$	0	0
$186$	13.3108	0.975997
$187$	0	0
$188$	−0.295834	−0.0215759
$189$	−7.43087	−0.540516
$190$	0	0
$191$	18.9564	1.37164	0.685820	−	0.727771i	$-0.259444\pi$
$191$	18.9564	1.37164	0.685820	+	0.727771i	$0.259444\pi$
$192$	12.5390	0.904925
$193$	−23.1584	−1.66698	−0.833490	−	0.552535i	$-0.813660\pi$
$193$	−23.1584	−1.66698	−0.833490	+	0.552535i	$0.813660\pi$
$194$	−0.555626	−0.0398916
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	6.81050	0.485228	0.242614	−	0.970123i	$-0.421995\pi$
$197$	6.81050	0.485228	0.242614	+	0.970123i	$0.421995\pi$
$198$	0	0
$199$	7.79129	0.552310	0.276155	−	0.961113i	$-0.410940\pi$
$199$	7.79129	0.552310	0.276155	+	0.961113i	$0.410940\pi$
$200$	0	0
$201$	6.41742	0.452650
$202$	18.9564	1.33377
$203$	−15.0000	−1.05279
$204$	−1.55091	−0.108586
$205$	0	0
$206$	17.5239	1.22095
$207$	−1.41742	−0.0985178
$208$	−24.6446	−1.70879
$209$	0	0
$210$	0	0
$211$	7.98649	0.549813	0.274906	−	0.961471i	$-0.411353\pi$
$211$	7.98649	0.549813	0.274906	+	0.961471i	$0.411353\pi$
$212$	−1.41742	−0.0973491
$213$	−16.4174	−1.12490
$214$	20.7042	1.41531
$215$	0	0
$216$	−13.3108	−0.905687
$217$	−7.43087	−0.504440
$218$	−7.08712	−0.480000
$219$	−17.5239	−1.18416
$220$	0	0
$221$	−23.3739	−1.57230
$222$	22.8482	1.53347
$223$	−2.16515	−0.144989	−0.0724946	−	0.997369i	$-0.523096\pi$
$223$	−2.16515	−0.144989	−0.0724946	+	0.997369i	$0.523096\pi$
$224$	1.74773	0.116775
$225$	0	0
$226$	−18.0795	−1.20263
$227$	−24.3344	−1.61513	−0.807566	−	0.589778i	$-0.799215\pi$
$227$	−24.3344	−1.61513	−0.807566	+	0.589778i	$0.799215\pi$
$228$	−0.995286	−0.0659145
$229$	−17.7913	−1.17568	−0.587841	−	0.808977i	$-0.700022\pi$
$229$	−17.7913	−1.17568	−0.587841	+	0.808977i	$0.700022\pi$
$230$	0	0
$231$	0	0
$232$	−26.8693	−1.76406
$233$	9.78285	0.640896	0.320448	−	0.947266i	$-0.396167\pi$
$233$	9.78285	0.640896	0.320448	+	0.947266i	$0.396167\pi$
$234$	−1.74773	−0.114252
$235$	0	0
$236$	1.28674	0.0837598
$237$	27.6169	1.79391
$238$	9.16249	0.593916
$239$	−12.7552	−0.825065	−0.412533	−	0.910943i	$-0.635356\pi$
$239$	−12.7552	−0.825065	−0.412533	+	0.910943i	$0.635356\pi$
$240$	0	0
$241$	10.0930	0.650149	0.325075	−	0.945688i	$-0.394611\pi$
$241$	10.0930	0.650149	0.325075	+	0.945688i	$0.394611\pi$
$242$	0	0
$243$	−2.16515	−0.138895
$244$	2.66216	0.170428
$245$	0	0
$246$	−7.08712	−0.451858
$247$	−15.0000	−0.954427
$248$	−13.3108	−0.845238
$249$	12.7552	0.808328
$250$	0	0
$251$	6.95644	0.439087	0.219543	−	0.975603i	$-0.429543\pi$
$251$	6.95644	0.439087	0.219543	+	0.975603i	$0.429543\pi$
$252$	0.0647389	0.00407816
$253$	0	0
$254$	1.74773	0.109662
$255$	0	0
$256$	4.95644	0.309777
$257$	−24.3303	−1.51768	−0.758841	−	0.651276i	$-0.774234\pi$
$257$	−24.3303	−1.51768	−0.758841	+	0.651276i	$0.774234\pi$
$258$	−7.91288	−0.492634
$259$	−12.7552	−0.792569
$260$	0	0
$261$	−2.10654	−0.130391
$262$	11.0436	0.682273
$263$	5.01415	0.309186	0.154593	−	0.987978i	$-0.450593\pi$
$263$	5.01415	0.309186	0.154593	+	0.987978i	$0.450593\pi$
$264$	0	0
$265$	0	0
$266$	5.87996	0.360523
$267$	14.7042	0.899880
$268$	0.747727	0.0456747
$269$	2.20871	0.134668	0.0673338	−	0.997731i	$-0.478551\pi$
$269$	2.20871	0.134668	0.0673338	+	0.997731i	$0.478551\pi$
$270$	0	0
$271$	22.8482	1.38793	0.693966	−	0.720008i	$-0.255862\pi$
$271$	22.8482	1.38793	0.693966	+	0.720008i	$0.255862\pi$
$272$	18.1443	1.10016
$273$	15.0000	0.907841
$274$	−7.98649	−0.482482
$275$	0	0
$276$	−2.53901	−0.152831
$277$	6.25488	0.375819	0.187910	−	0.982186i	$-0.439829\pi$
$277$	6.25488	0.375819	0.187910	+	0.982186i	$0.439829\pi$
$278$	3.95644	0.237291
$279$	−1.04356	−0.0624763
$280$	0	0
$281$	−2.66216	−0.158811	−0.0794057	−	0.996842i	$-0.525302\pi$
$281$	−2.66216	−0.158811	−0.0794057	+	0.996842i	$0.525302\pi$
$282$	3.77342	0.224704
$283$	27.3068	1.62322	0.811609	−	0.584201i	$-0.198592\pi$
$283$	27.3068	1.62322	0.811609	+	0.584201i	$0.198592\pi$
$284$	−1.91288	−0.113508
$285$	0	0
$286$	0	0
$287$	3.95644	0.233541
$288$	0.245444	0.0144629
$289$	0.208712	0.0122772
$290$	0	0
$291$	0.669697	0.0392583
$292$	−2.04180	−0.119487
$293$	17.2137	1.00564	0.502818	−	0.864392i	$-0.332297\pi$
$293$	17.2137	1.00564	0.502818	+	0.864392i	$0.332297\pi$
$294$	12.7552	0.743898
$295$	0	0
$296$	−22.8482	−1.32803
$297$	0	0
$298$	7.91288	0.458381
$299$	−38.2656	−2.21296
$300$	0	0
$301$	4.41742	0.254616
$302$	26.0436	1.49864
$303$	−22.8482	−1.31260
$304$	11.6439	0.667826
$305$	0	0
$306$	1.28674	0.0735581
$307$	26.6864	1.52307	0.761536	−	0.648122i	$-0.224445\pi$
$307$	26.6864	1.52307	0.761536	+	0.648122i	$0.224445\pi$
$308$	0	0
$309$	−21.1216	−1.20157
$310$	0	0
$311$	24.1652	1.37028	0.685140	−	0.728411i	$-0.259741\pi$
$311$	24.1652	1.37028	0.685140	+	0.728411i	$0.259741\pi$
$312$	26.8693	1.52118
$313$	−7.16515	−0.404998	−0.202499	−	0.979282i	$-0.564906\pi$
$313$	−7.16515	−0.404998	−0.202499	+	0.979282i	$0.564906\pi$
$314$	−9.53741	−0.538227
$315$	0	0
$316$	3.21779	0.181015
$317$	−5.37386	−0.301826	−0.150913	−	0.988547i	$-0.548221\pi$
$317$	−5.37386	−0.301826	−0.150913	+	0.988547i	$0.548221\pi$
$318$	18.0795	1.01385
$319$	0	0
$320$	0	0
$321$	−24.9548	−1.39284
$322$	15.0000	0.835917
$323$	11.0436	0.614480
$324$	−2.00000	−0.111111
$325$	0	0
$326$	31.3904	1.73855
$327$	8.54212	0.472380
$328$	7.08712	0.391321
$329$	−2.10654	−0.116137
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	1.48617	0.0815644
$333$	−1.79129	−0.0981620
$334$	4.87841	0.266935
$335$	0	0
$336$	−11.6439	−0.635229
$337$	8.91704	0.485742	0.242871	−	0.970059i	$-0.421911\pi$
$337$	8.91704	0.485742	0.242871	+	0.970059i	$0.421911\pi$
$338$	−27.8624	−1.51551
$339$	21.7913	1.18354
$340$	0	0
$341$	0	0
$342$	0.825757	0.0446518
$343$	−17.5239	−0.946202
$344$	7.91288	0.426634
$345$	0	0
$346$	−8.37386	−0.450182
$347$	3.83816	0.206043	0.103022	−	0.994679i	$-0.467149\pi$
$347$	3.83816	0.206043	0.103022	+	0.994679i	$0.467149\pi$
$348$	−3.77342	−0.202276
$349$	−17.5239	−0.938033	−0.469016	−	0.883189i	$-0.655392\pi$
$349$	−17.5239	−0.938033	−0.469016	+	0.883189i	$0.655392\pi$
$350$	0	0
$351$	−28.1726	−1.50374
$352$	0	0
$353$	12.1652	0.647486	0.323743	−	0.946145i	$-0.395059\pi$
$353$	12.1652	0.647486	0.323743	+	0.946145i	$0.395059\pi$
$354$	−16.4126	−0.872322
$355$	0	0
$356$	1.71326	0.0908025
$357$	−11.0436	−0.584487
$358$	10.0930	0.533433
$359$	30.8347	1.62739	0.813697	−	0.581289i	$-0.197451\pi$
$359$	30.8347	1.62739	0.813697	+	0.581289i	$0.197451\pi$
$360$	0	0
$361$	−11.9129	−0.626994
$362$	−35.6034	−1.87127
$363$	0	0
$364$	1.74773	0.0916058
$365$	0	0
$366$	−33.9564	−1.77493
$367$	−2.46099	−0.128462	−0.0642312	−	0.997935i	$-0.520460\pi$
$367$	−2.46099	−0.128462	−0.0642312	+	0.997935i	$0.520460\pi$
$368$	29.7042	1.54844
$369$	0.555626	0.0289247
$370$	0	0
$371$	−10.0930	−0.524004
$372$	−1.86932	−0.0969196
$373$	−9.22722	−0.477768	−0.238884	−	0.971048i	$-0.576781\pi$
$373$	−9.22722	−0.477768	−0.238884	+	0.971048i	$0.576781\pi$
$374$	0	0
$375$	0	0
$376$	−3.77342	−0.194599
$377$	−56.8693	−2.92892
$378$	11.0436	0.568019
$379$	−16.1652	−0.830348	−0.415174	−	0.909742i	$-0.636279\pi$
$379$	−16.1652	−0.830348	−0.415174	+	0.909742i	$0.636279\pi$
$380$	0	0
$381$	−2.10654	−0.107921
$382$	−28.1726	−1.44143
$383$	−27.1652	−1.38807	−0.694037	−	0.719939i	$-0.744170\pi$
$383$	−27.1652	−1.38807	−0.694037	+	0.719939i	$0.744170\pi$
$384$	−22.8482	−1.16597
$385$	0	0
$386$	34.4174	1.75180
$387$	0.620365	0.0315349
$388$	0.0780299	0.00396137
$389$	−21.1652	−1.07312	−0.536558	−	0.843864i	$-0.680276\pi$
$389$	−21.1652	−1.07312	−0.536558	+	0.843864i	$0.680276\pi$
$390$	0	0
$391$	28.1726	1.42475
$392$	−12.7552	−0.644235
$393$	−13.3108	−0.671442
$394$	−10.1216	−0.509918
$395$	0	0
$396$	0	0
$397$	15.3739	0.771592	0.385796	−	0.922584i	$-0.373927\pi$
$397$	15.3739	0.771592	0.385796	+	0.922584i	$0.373927\pi$
$398$	−11.5792	−0.580413
$399$	−7.08712	−0.354800
$400$	0	0
$401$	7.91288	0.395150	0.197575	−	0.980288i	$-0.436693\pi$
$401$	7.91288	0.395150	0.197575	+	0.980288i	$0.436693\pi$
$402$	−9.53741	−0.475683
$403$	−28.1726	−1.40338
$404$	−2.66216	−0.132448
$405$	0	0
$406$	22.2926	1.10636
$407$	0	0
$408$	−19.7822	−0.979365
$409$	−28.1726	−1.39304	−0.696522	−	0.717536i	$-0.745270\pi$
$409$	−28.1726	−1.39304	−0.696522	+	0.717536i	$0.745270\pi$
$410$	0	0
$411$	9.62614	0.474822
$412$	−2.46099	−0.121244
$413$	9.16249	0.450856
$414$	2.10654	0.103531
$415$	0	0
$416$	6.62614	0.324873
$417$	−4.76870	−0.233524
$418$	0	0
$419$	25.7477	1.25786	0.628929	−	0.777462i	$-0.283493\pi$
$419$	25.7477	1.25786	0.628929	+	0.777462i	$0.283493\pi$
$420$	0	0
$421$	−31.8693	−1.55322	−0.776608	−	0.629984i	$-0.783061\pi$
$421$	−31.8693	−1.55322	−0.776608	+	0.629984i	$0.783061\pi$
$422$	−11.8693	−0.577789
$423$	−0.295834	−0.0143839
$424$	−18.0795	−0.878019
$425$	0	0
$426$	24.3991	1.18214
$427$	18.9564	0.917366
$428$	−2.90761	−0.140545
$429$	0	0
$430$	0	0
$431$	−17.5239	−0.844097	−0.422048	−	0.906573i	$-0.638689\pi$
$431$	−17.5239	−0.844097	−0.422048	+	0.906573i	$0.638689\pi$
$432$	21.8693	1.05219
$433$	5.74773	0.276218	0.138109	−	0.990417i	$-0.455898\pi$
$433$	5.74773	0.276218	0.138109	+	0.990417i	$0.455898\pi$
$434$	11.0436	0.530108
$435$	0	0
$436$	0.995286	0.0476656
$437$	18.0795	0.864861
$438$	26.0436	1.24441
$439$	15.4174	0.735831	0.367915	−	0.929859i	$-0.380072\pi$
$439$	15.4174	0.735831	0.367915	+	0.929859i	$0.380072\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	34.7376	1.65230
$443$	−40.7477	−1.93598	−0.967991	−	0.250983i	$-0.919246\pi$
$443$	−40.7477	−1.93598	−0.967991	+	0.250983i	$0.919246\pi$
$444$	−3.20871	−0.152279
$445$	0	0
$446$	3.21779	0.152367
$447$	−9.53741	−0.451104
$448$	10.4032	0.491506
$449$	−31.1216	−1.46872	−0.734359	−	0.678761i	$-0.762517\pi$
$449$	−31.1216	−1.46872	−0.734359	+	0.678761i	$0.762517\pi$
$450$	0	0
$451$	0	0
$452$	2.53901	0.119425
$453$	−31.3904	−1.47485
$454$	36.1652	1.69731
$455$	0	0
$456$	−12.6951	−0.594501
$457$	26.9966	1.26285	0.631423	−	0.775438i	$-0.282471\pi$
$457$	26.9966	1.26285	0.631423	+	0.775438i	$0.282471\pi$
$458$	26.4409	1.23550
$459$	20.7417	0.968139
$460$	0	0
$461$	−2.66216	−0.123989	−0.0619947	−	0.998076i	$-0.519746\pi$
$461$	−2.66216	−0.123989	−0.0619947	+	0.998076i	$0.519746\pi$
$462$	0	0
$463$	5.74773	0.267120	0.133560	−	0.991041i	$-0.457359\pi$
$463$	5.74773	0.267120	0.133560	+	0.991041i	$0.457359\pi$
$464$	44.1455	2.04941
$465$	0	0
$466$	−14.5390	−0.673507
$467$	−28.5826	−1.32264	−0.661322	−	0.750102i	$-0.730004\pi$
$467$	−28.5826	−1.32264	−0.661322	+	0.750102i	$0.730004\pi$
$468$	0.245444	0.0113456
$469$	5.32433	0.245855
$470$	0	0
$471$	11.4955	0.529683
$472$	16.4126	0.755453
$473$	0	0
$474$	−41.0436	−1.88519
$475$	0	0
$476$	−1.28674	−0.0589777
$477$	−1.41742	−0.0648994
$478$	18.9564	0.867047
$479$	12.1996	0.557413	0.278706	−	0.960376i	$-0.410094\pi$
$479$	12.1996	0.557413	0.278706	+	0.960376i	$0.410094\pi$
$480$	0	0
$481$	−48.3586	−2.20496
$482$	−15.0000	−0.683231
$483$	−18.0795	−0.822647
$484$	0	0
$485$	0	0
$486$	3.21779	0.145962
$487$	−11.4174	−0.517373	−0.258686	−	0.965961i	$-0.583290\pi$
$487$	−11.4174	−0.517373	−0.258686	+	0.965961i	$0.583290\pi$
$488$	33.9564	1.53714
$489$	−37.8348	−1.71095
$490$	0	0
$491$	−38.8212	−1.75198	−0.875989	−	0.482332i	$-0.839790\pi$
$491$	−38.8212	−1.75198	−0.875989	+	0.482332i	$0.839790\pi$
$492$	0.995286	0.0448710
$493$	41.8693	1.88570
$494$	22.2926	1.00299
$495$	0	0
$496$	21.8693	0.981961
$497$	−13.6210	−0.610986
$498$	−18.9564	−0.849458
$499$	−32.7913	−1.46794	−0.733970	−	0.679182i	$-0.762335\pi$
$499$	−32.7913	−1.46794	−0.733970	+	0.679182i	$0.762335\pi$
$500$	0	0
$501$	−5.87996	−0.262697
$502$	−10.3385	−0.461429
$503$	−23.1584	−1.03258	−0.516291	−	0.856413i	$-0.672688\pi$
$503$	−23.1584	−1.03258	−0.516291	+	0.856413i	$0.672688\pi$
$504$	0.825757	0.0367821
$505$	0	0
$506$	0	0
$507$	33.5826	1.49145
$508$	−0.245444	−0.0108898
$509$	−20.7042	−0.917696	−0.458848	−	0.888515i	$-0.651738\pi$
$509$	−20.7042	−0.917696	−0.458848	+	0.888515i	$0.651738\pi$
$510$	0	0
$511$	−14.5390	−0.643168
$512$	18.1443	0.801871
$513$	13.3108	0.587687
$514$	36.1591	1.59491
$515$	0	0
$516$	1.11125	0.0489202
$517$	0	0
$518$	18.9564	0.832898
$519$	10.0930	0.443035
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	3.13068	0.137026
$523$	18.9453	0.828422	0.414211	−	0.910181i	$-0.364058\pi$
$523$	18.9453	0.828422	0.414211	+	0.910181i	$0.364058\pi$
$524$	−1.55091	−0.0677519
$525$	0	0
$526$	−7.45189	−0.324918
$527$	20.7417	0.903522
$528$	0	0
$529$	23.1216	1.00529
$530$	0	0
$531$	1.28674	0.0558398
$532$	−0.825757	−0.0358011
$533$	15.0000	0.649722
$534$	−21.8529	−0.945669
$535$	0	0
$536$	9.53741	0.411953
$537$	−12.1652	−0.524965
$538$	−3.28253	−0.141520
$539$	0	0
$540$	0	0
$541$	32.9413	1.41626	0.708128	−	0.706084i	$-0.249540\pi$
$541$	32.9413	1.41626	0.708128	+	0.706084i	$0.249540\pi$
$542$	−33.9564	−1.45855
$543$	42.9129	1.84157
$544$	−4.87841	−0.209160
$545$	0	0
$546$	−22.2926	−0.954035
$547$	−21.6722	−0.926638	−0.463319	−	0.886192i	$-0.653342\pi$
$547$	−21.6722	−0.926638	−0.463319	+	0.886192i	$0.653342\pi$
$548$	1.12159	0.0479120
$549$	2.66216	0.113618
$550$	0	0
$551$	26.8693	1.14467
$552$	−32.3856	−1.37842
$553$	22.9129	0.974355
$554$	−9.29583	−0.394942
$555$	0	0
$556$	−0.555626	−0.0235638
$557$	−8.60686	−0.364684	−0.182342	−	0.983235i	$-0.558368\pi$
$557$	−8.60686	−0.364684	−0.182342	+	0.983235i	$0.558368\pi$
$558$	1.55091	0.0656554
$559$	16.7477	0.708353
$560$	0	0
$561$	0	0
$562$	3.95644	0.166892
$563$	15.7275	0.662837	0.331419	−	0.943484i	$-0.392473\pi$
$563$	15.7275	0.662837	0.331419	+	0.943484i	$0.392473\pi$
$564$	−0.529923	−0.0223138
$565$	0	0
$566$	−40.5826	−1.70581
$567$	−14.2414	−0.598081
$568$	−24.3991	−1.02377
$569$	−12.7552	−0.534726	−0.267363	−	0.963596i	$-0.586152\pi$
$569$	−12.7552	−0.534726	−0.267363	+	0.963596i	$0.586152\pi$
$570$	0	0
$571$	32.9413	1.37855	0.689275	−	0.724500i	$-0.257929\pi$
$571$	32.9413	1.37855	0.689275	+	0.724500i	$0.257929\pi$
$572$	0	0
$573$	33.9564	1.41855
$574$	−5.87996	−0.245425
$575$	0	0
$576$	1.46099	0.0608744
$577$	−15.3739	−0.640022	−0.320011	−	0.947414i	$-0.603687\pi$
$577$	−15.3739	−0.640022	−0.320011	+	0.947414i	$0.603687\pi$
$578$	−0.310183	−0.0129019
$579$	−41.4834	−1.72399
$580$	0	0
$581$	10.5826	0.439039
$582$	−0.995286	−0.0412559
$583$	0	0
$584$	−26.0436	−1.07769
$585$	0	0
$586$	−25.5826	−1.05681
$587$	5.37386	0.221803	0.110902	−	0.993831i	$-0.464626\pi$
$587$	5.37386	0.221803	0.110902	+	0.993831i	$0.464626\pi$
$588$	−1.79129	−0.0738715
$589$	13.3108	0.548463
$590$	0	0
$591$	12.1996	0.501823
$592$	37.5390	1.54284
$593$	20.4962	0.841680	0.420840	−	0.907135i	$-0.361735\pi$
$593$	20.4962	0.841680	0.420840	+	0.907135i	$0.361735\pi$
$594$	0	0
$595$	0	0
$596$	−1.11125	−0.0455187
$597$	13.9564	0.571199
$598$	56.8693	2.32556
$599$	21.7913	0.890368	0.445184	−	0.895439i	$-0.353138\pi$
$599$	21.7913	0.890368	0.445184	+	0.895439i	$0.353138\pi$
$600$	0	0
$601$	27.6169	1.12652	0.563259	−	0.826280i	$-0.309547\pi$
$601$	27.6169	1.12652	0.563259	+	0.826280i	$0.309547\pi$
$602$	−6.56506	−0.267572
$603$	0.747727	0.0304498
$604$	−3.65745	−0.148820
$605$	0	0
$606$	33.9564	1.37939
$607$	−3.59271	−0.145824	−0.0729118	−	0.997338i	$-0.523229\pi$
$607$	−3.59271	−0.145824	−0.0729118	+	0.997338i	$0.523229\pi$
$608$	−3.13068	−0.126966
$609$	−26.8693	−1.08880
$610$	0	0
$611$	−7.98649	−0.323099
$612$	−0.180705	−0.00730456
$613$	17.7693	0.717697	0.358849	−	0.933396i	$-0.383169\pi$
$613$	17.7693	0.717697	0.358849	+	0.933396i	$0.383169\pi$
$614$	−39.6606	−1.60057
$615$	0	0
$616$	0	0
$617$	5.66970	0.228253	0.114127	−	0.993466i	$-0.463593\pi$
$617$	5.66970	0.228253	0.114127	+	0.993466i	$0.463593\pi$
$618$	31.3904	1.26271
$619$	45.7477	1.83876	0.919378	−	0.393375i	$-0.128693\pi$
$619$	45.7477	1.83876	0.919378	+	0.393375i	$0.128693\pi$
$620$	0	0
$621$	33.9564	1.36262
$622$	−35.9136	−1.44000
$623$	12.1996	0.488766
$624$	−44.1455	−1.76724
$625$	0	0
$626$	10.6487	0.425606
$627$	0	0
$628$	1.33939	0.0534477
$629$	35.6034	1.41960
$630$	0	0
$631$	33.8693	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$631$	33.8693	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$632$	41.0436	1.63263
$633$	14.3061	0.568617
$634$	7.98649	0.317184
$635$	0	0
$636$	−2.53901	−0.100678
$637$	−26.9966	−1.06964
$638$	0	0
$639$	−1.91288	−0.0756723
$640$	0	0
$641$	−22.9129	−0.905004	−0.452502	−	0.891763i	$-0.649469\pi$
$641$	−22.9129	−0.905004	−0.452502	+	0.891763i	$0.649469\pi$
$642$	37.0871	1.46371
$643$	7.16515	0.282566	0.141283	−	0.989969i	$-0.454877\pi$
$643$	7.16515	0.282566	0.141283	+	0.989969i	$0.454877\pi$
$644$	−2.10654	−0.0830093
$645$	0	0
$646$	−16.4126	−0.645747
$647$	−31.4174	−1.23515	−0.617573	−	0.786513i	$-0.711884\pi$
$647$	−31.4174	−1.23515	−0.617573	+	0.786513i	$0.711884\pi$
$648$	−25.5104	−1.00214
$649$	0	0
$650$	0	0
$651$	−13.3108	−0.521692
$652$	−4.40833	−0.172644
$653$	−21.7913	−0.852759	−0.426379	−	0.904544i	$-0.640211\pi$
$653$	−21.7913	−0.852759	−0.426379	+	0.904544i	$0.640211\pi$
$654$	−12.6951	−0.496417
$655$	0	0
$656$	−11.6439	−0.454620
$657$	−2.04180	−0.0796582
$658$	3.13068	0.122047
$659$	−20.7417	−0.807982	−0.403991	−	0.914763i	$-0.632377\pi$
$659$	−20.7417	−0.807982	−0.403991	+	0.914763i	$0.632377\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	−7.43087	−0.288809
$663$	−41.8693	−1.62607
$664$	18.9564	0.735653
$665$	0	0
$666$	2.66216	0.103157
$667$	68.5447	2.65406
$668$	−0.685104	−0.0265075
$669$	−3.87841	−0.149948
$670$	0	0
$671$	0	0
$672$	3.13068	0.120769
$673$	19.8759	0.766159	0.383080	−	0.923715i	$-0.374864\pi$
$673$	19.8759	0.766159	0.383080	+	0.923715i	$0.374864\pi$
$674$	−13.2523	−0.510458
$675$	0	0
$676$	3.91288	0.150495
$677$	−46.3168	−1.78010	−0.890050	−	0.455863i	$-0.849331\pi$
$677$	−46.3168	−1.78010	−0.890050	+	0.455863i	$0.849331\pi$
$678$	−32.3856	−1.24376
$679$	0.555626	0.0213230
$680$	0	0
$681$	−43.5899	−1.67037
$682$	0	0
$683$	−12.1652	−0.465487	−0.232743	−	0.972538i	$-0.574770\pi$
$683$	−12.1652	−0.465487	−0.232743	+	0.972538i	$0.574770\pi$
$684$	−0.115966	−0.00443407
$685$	0	0
$686$	26.0436	0.994348
$687$	−31.8693	−1.21589
$688$	−13.0006	−0.495645
$689$	−38.2656	−1.45780
$690$	0	0
$691$	22.7042	0.863707	0.431854	−	0.901944i	$-0.357860\pi$
$691$	22.7042	0.863707	0.431854	+	0.901944i	$0.357860\pi$
$692$	1.17599	0.0447045
$693$	0	0
$694$	−5.70417	−0.216527
$695$	0	0
$696$	−48.1307	−1.82439
$697$	−11.0436	−0.418305
$698$	26.0436	0.985763
$699$	17.5239	0.662815
$700$	0	0
$701$	20.1861	0.762417	0.381209	−	0.924489i	$-0.375508\pi$
$701$	20.1861	0.762417	0.381209	+	0.924489i	$0.375508\pi$
$702$	41.8693	1.58026
$703$	22.8482	0.861737
$704$	0	0
$705$	0	0
$706$	−18.0795	−0.680432
$707$	−18.9564	−0.712930
$708$	2.30493	0.0866244
$709$	−9.25227	−0.347476	−0.173738	−	0.984792i	$-0.555585\pi$
$709$	−9.25227	−0.347476	−0.173738	+	0.984792i	$0.555585\pi$
$710$	0	0
$711$	3.21779	0.120677
$712$	21.8529	0.818974
$713$	33.9564	1.27168
$714$	16.4126	0.614228
$715$	0	0
$716$	−1.41742	−0.0529716
$717$	−22.8482	−0.853283
$718$	−45.8258	−1.71020
$719$	−33.6606	−1.25533	−0.627664	−	0.778484i	$-0.715989\pi$
$719$	−33.6606	−1.25533	−0.627664	+	0.778484i	$0.715989\pi$
$720$	0	0
$721$	−17.5239	−0.652624
$722$	17.7046	0.658897
$723$	18.0795	0.672385
$724$	5.00000	0.185824
$725$	0	0
$726$	0	0
$727$	25.3739	0.941065	0.470532	−	0.882383i	$-0.344062\pi$
$727$	25.3739	0.941065	0.470532	+	0.882383i	$0.344062\pi$
$728$	22.2926	0.826219
$729$	24.8693	0.921086
$730$	0	0
$731$	−12.3303	−0.456053
$732$	4.76870	0.176256
$733$	−25.8206	−0.953705	−0.476852	−	0.878983i	$-0.658222\pi$
$733$	−25.8206	−0.953705	−0.476852	+	0.878983i	$0.658222\pi$
$734$	3.65745	0.134999
$735$	0	0
$736$	−7.98649	−0.294386
$737$	0	0
$738$	−0.825757	−0.0303965
$739$	22.2926	0.820047	0.410023	−	0.912075i	$-0.365521\pi$
$739$	22.2926	0.820047	0.410023	+	0.912075i	$0.365521\pi$
$740$	0	0
$741$	−26.8693	−0.987069
$742$	15.0000	0.550667
$743$	−0.245444	−0.00900445	−0.00450223	−	0.999990i	$-0.501433\pi$
$743$	−0.245444	−0.00900445	−0.00450223	+	0.999990i	$0.501433\pi$
$744$	−23.8435	−0.874146
$745$	0	0
$746$	13.7133	0.502078
$747$	1.48617	0.0543763
$748$	0	0
$749$	−20.7042	−0.756514
$750$	0	0
$751$	10.9564	0.399806	0.199903	−	0.979816i	$-0.435937\pi$
$751$	10.9564	0.399806	0.199903	+	0.979816i	$0.435937\pi$
$752$	6.19962	0.226077
$753$	12.4610	0.454104
$754$	84.5177	3.07795
$755$	0	0
$756$	−1.55091	−0.0564061
$757$	−16.4955	−0.599537	−0.299769	−	0.954012i	$-0.596910\pi$
$757$	−16.4955	−0.599537	−0.299769	+	0.954012i	$0.596910\pi$
$758$	24.0242	0.872599
$759$	0	0
$760$	0	0
$761$	41.4834	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$761$	41.4834	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$762$	3.13068	0.113413
$763$	7.08712	0.256571
$764$	3.95644	0.143139
$765$	0	0
$766$	40.3721	1.45870
$767$	34.7376	1.25430
$768$	8.87841	0.320372
$769$	32.3856	1.16786	0.583928	−	0.811805i	$-0.301515\pi$
$769$	32.3856	1.16786	0.583928	+	0.811805i	$0.301515\pi$
$770$	0	0
$771$	−43.5826	−1.56959
$772$	−4.83344	−0.173959
$773$	−23.2087	−0.834759	−0.417380	−	0.908732i	$-0.637051\pi$
$773$	−23.2087	−0.834759	−0.417380	+	0.908732i	$0.637051\pi$
$774$	−0.921970	−0.0331395
$775$	0	0
$776$	0.995286	0.0357287
$777$	−22.8482	−0.819676
$778$	31.4551	1.12772
$779$	−7.08712	−0.253922
$780$	0	0
$781$	0	0
$782$	−41.8693	−1.49724
$783$	50.4652	1.80348
$784$	20.9564	0.748444
$785$	0	0
$786$	19.7822	0.705608
$787$	26.4409	0.942518	0.471259	−	0.881995i	$-0.343800\pi$
$787$	26.4409	0.942518	0.471259	+	0.881995i	$0.343800\pi$
$788$	1.42143	0.0506365
$789$	8.98178	0.319760
$790$	0	0
$791$	18.0795	0.642834
$792$	0	0
$793$	71.8693	2.55215
$794$	−22.8482	−0.810853
$795$	0	0
$796$	1.62614	0.0576369
$797$	43.2867	1.53330	0.766648	−	0.642068i	$-0.221923\pi$
$797$	43.2867	1.53330	0.766648	+	0.642068i	$0.221923\pi$
$798$	10.5327	0.372853
$799$	5.87996	0.208018
$800$	0	0
$801$	1.71326	0.0605350
$802$	−11.7599	−0.415257
$803$	0	0
$804$	1.33939	0.0472368
$805$	0	0
$806$	41.8693	1.47478
$807$	3.95644	0.139273
$808$	−33.9564	−1.19458
$809$	9.53741	0.335317	0.167659	−	0.985845i	$-0.446379\pi$
$809$	9.53741	0.335317	0.167659	+	0.985845i	$0.446379\pi$
$810$	0	0
$811$	38.8212	1.36320	0.681599	−	0.731726i	$-0.261285\pi$
$811$	38.8212	1.36320	0.681599	+	0.731726i	$0.261285\pi$
$812$	−3.13068	−0.109865
$813$	40.9278	1.43540
$814$	0	0
$815$	0	0
$816$	32.5016	1.13778
$817$	−7.91288	−0.276837
$818$	41.8693	1.46393
$819$	1.74773	0.0610705
$820$	0	0
$821$	−35.0478	−1.22318	−0.611588	−	0.791176i	$-0.709469\pi$
$821$	−35.0478	−1.22318	−0.611588	+	0.791176i	$0.709469\pi$
$822$	−14.3061	−0.498983
$823$	−22.8348	−0.795973	−0.397986	−	0.917391i	$-0.630291\pi$
$823$	−22.8348	−0.795973	−0.397986	+	0.917391i	$0.630291\pi$
$824$	−31.3904	−1.09353
$825$	0	0
$826$	−13.6170	−0.473798
$827$	−0.620365	−0.0215722	−0.0107861	−	0.999942i	$-0.503433\pi$
$827$	−0.620365	−0.0215722	−0.0107861	+	0.999942i	$0.503433\pi$
$828$	−0.295834	−0.0102809
$829$	3.87841	0.134703	0.0673514	−	0.997729i	$-0.478545\pi$
$829$	3.87841	0.134703	0.0673514	+	0.997729i	$0.478545\pi$
$830$	0	0
$831$	11.2043	0.388672
$832$	39.4416	1.36739
$833$	19.8759	0.688659
$834$	7.08712	0.245407
$835$	0	0
$836$	0	0
$837$	25.0000	0.864126
$838$	−38.2656	−1.32186
$839$	12.7913	0.441604	0.220802	−	0.975319i	$-0.429132\pi$
$839$	12.7913	0.441604	0.220802	+	0.975319i	$0.429132\pi$
$840$	0	0
$841$	72.8693	2.51274
$842$	47.3633	1.63225
$843$	−4.76870	−0.164243
$844$	1.66688	0.0573763
$845$	0	0
$846$	0.439660	0.0151158
$847$	0	0
$848$	29.7042	1.02005
$849$	48.9143	1.67873
$850$	0	0
$851$	58.2867	1.99804
$852$	−3.42652	−0.117391
$853$	−27.9271	−0.956206	−0.478103	−	0.878304i	$-0.658675\pi$
$853$	−27.9271	−0.956206	−0.478103	+	0.878304i	$0.658675\pi$
$854$	−28.1726	−0.964045
$855$	0	0
$856$	−37.0871	−1.26761
$857$	−32.8765	−1.12304	−0.561520	−	0.827463i	$-0.689783\pi$
$857$	−32.8765	−1.12304	−0.561520	+	0.827463i	$0.689783\pi$
$858$	0	0
$859$	0.747727	0.0255121	0.0127561	−	0.999919i	$-0.495940\pi$
$859$	0.747727	0.0255121	0.0127561	+	0.999919i	$0.495940\pi$
$860$	0	0
$861$	7.08712	0.241528
$862$	26.0436	0.887047
$863$	−4.25227	−0.144749	−0.0723745	−	0.997378i	$-0.523058\pi$
$863$	−4.25227	−0.144749	−0.0723745	+	0.997378i	$0.523058\pi$
$864$	−5.87996	−0.200040
$865$	0	0
$866$	−8.54212	−0.290273
$867$	0.373864	0.0126971
$868$	−1.55091	−0.0526414
$869$	0	0
$870$	0	0
$871$	20.1861	0.683979
$872$	12.6951	0.429909
$873$	0.0780299	0.00264091
$874$	−26.8693	−0.908868
$875$	0	0
$876$	−3.65745	−0.123574
$877$	−42.1037	−1.42174	−0.710871	−	0.703322i	$-0.751699\pi$
$877$	−42.1037	−1.42174	−0.710871	+	0.703322i	$0.751699\pi$
$878$	−22.9129	−0.773272
$879$	30.8347	1.04003
$880$	0	0
$881$	15.9564	0.537586	0.268793	−	0.963198i	$-0.413375\pi$
$881$	15.9564	0.537586	0.268793	+	0.963198i	$0.413375\pi$
$882$	1.48617	0.0500421
$883$	15.7477	0.529953	0.264977	−	0.964255i	$-0.414636\pi$
$883$	15.7477	0.529953	0.264977	+	0.964255i	$0.414636\pi$
$884$	−4.87841	−0.164079
$885$	0	0
$886$	60.5582	2.03449
$887$	−2.04180	−0.0685569	−0.0342785	−	0.999412i	$-0.510913\pi$
$887$	−2.04180	−0.0685569	−0.0342785	+	0.999412i	$0.510913\pi$
$888$	−40.9278	−1.37345
$889$	−1.74773	−0.0586169
$890$	0	0
$891$	0	0
$892$	−0.451893	−0.0151305
$893$	3.77342	0.126273
$894$	14.1742	0.474058
$895$	0	0
$896$	−18.9564	−0.633290
$897$	−68.5447	−2.28864
$898$	46.2521	1.54345
$899$	50.4652	1.68311
$900$	0	0
$901$	28.1726	0.938564
$902$	0	0
$903$	7.91288	0.263324
$904$	32.3856	1.07713
$905$	0	0
$906$	46.6515	1.54989
$907$	−52.2432	−1.73471	−0.867353	−	0.497693i	$-0.834181\pi$
$907$	−52.2432	−1.73471	−0.867353	+	0.497693i	$0.834181\pi$
$908$	−5.07889	−0.168549
$909$	−2.66216	−0.0882984
$910$	0	0
$911$	37.9129	1.25611	0.628055	−	0.778169i	$-0.283851\pi$
$911$	37.9129	1.25611	0.628055	+	0.778169i	$0.283851\pi$
$912$	20.8577	0.690666
$913$	0	0
$914$	−40.1216	−1.32710
$915$	0	0
$916$	−3.71326	−0.122689
$917$	−11.0436	−0.364691
$918$	−30.8258	−1.01740
$919$	−9.53741	−0.314610	−0.157305	−	0.987550i	$-0.550281\pi$
$919$	−9.53741	−0.314610	−0.157305	+	0.987550i	$0.550281\pi$
$920$	0	0
$921$	47.8030	1.57516
$922$	3.95644	0.130298
$923$	−51.6412	−1.69979
$924$	0	0
$925$	0	0
$926$	−8.54212	−0.280712
$927$	−2.46099	−0.0808294
$928$	−11.8693	−0.389629
$929$	51.9909	1.70577	0.852883	−	0.522102i	$-0.174852\pi$
$929$	51.9909	1.70577	0.852883	+	0.522102i	$0.174852\pi$
$930$	0	0
$931$	12.7552	0.418035
$932$	2.04180	0.0668814
$933$	43.2867	1.41714
$934$	42.4787	1.38994
$935$	0	0
$936$	3.13068	0.102330
$937$	30.2144	0.987060	0.493530	−	0.869729i	$-0.335706\pi$
$937$	30.2144	0.987060	0.493530	+	0.869729i	$0.335706\pi$
$938$	−7.91288	−0.258365
$939$	−12.8348	−0.418849
$940$	0	0
$941$	18.6352	0.607489	0.303744	−	0.952754i	$-0.401763\pi$
$941$	18.6352	0.607489	0.303744	+	0.952754i	$0.401763\pi$
$942$	−17.0842	−0.556635
$943$	−18.0795	−0.588750
$944$	−26.9655	−0.877653
$945$	0	0
$946$	0	0
$947$	−36.4955	−1.18594	−0.592971	−	0.805223i	$-0.702045\pi$
$947$	−36.4955	−1.18594	−0.592971	+	0.805223i	$0.702045\pi$
$948$	5.76399	0.187206
$949$	−55.1216	−1.78932
$950$	0	0
$951$	−9.62614	−0.312149
$952$	−16.4126	−0.531937
$953$	−27.8624	−0.902551	−0.451275	−	0.892385i	$-0.649031\pi$
$953$	−27.8624	−0.902551	−0.451275	+	0.892385i	$0.649031\pi$
$954$	2.10654	0.0682017
$955$	0	0
$956$	−2.66216	−0.0861006
$957$	0	0
$958$	−18.1307	−0.585776
$959$	7.98649	0.257897
$960$	0	0
$961$	−6.00000	−0.193548
$962$	71.8693	2.31716
$963$	−2.90761	−0.0936964
$964$	2.10654	0.0678470
$965$	0	0
$966$	26.8693	0.864506
$967$	−46.8725	−1.50732	−0.753658	−	0.657267i	$-0.771713\pi$
$967$	−46.8725	−1.50732	−0.753658	+	0.657267i	$0.771713\pi$
$968$	0	0
$969$	19.7822	0.635496
$970$	0	0
$971$	11.8693	0.380905	0.190452	−	0.981696i	$-0.439005\pi$
$971$	11.8693	0.380905	0.190452	+	0.981696i	$0.439005\pi$
$972$	−0.451893	−0.0144945
$973$	−3.95644	−0.126838
$974$	16.9683	0.543699
$975$	0	0
$976$	−55.7895	−1.78578
$977$	20.6697	0.661282	0.330641	−	0.943757i	$-0.392735\pi$
$977$	20.6697	0.661282	0.330641	+	0.943757i	$0.392735\pi$
$978$	56.2292	1.79801
$979$	0	0
$980$	0	0
$981$	0.995286	0.0317770
$982$	57.6951	1.84112
$983$	−40.7477	−1.29965	−0.649825	−	0.760084i	$-0.725158\pi$
$983$	−40.7477	−1.29965	−0.649825	+	0.760084i	$0.725158\pi$
$984$	12.6951	0.404704
$985$	0	0
$986$	−62.2251	−1.98165
$987$	−3.77342	−0.120109
$988$	−3.13068	−0.0996003
$989$	−20.1861	−0.641880
$990$	0	0
$991$	−31.0436	−0.986131	−0.493066	−	0.869992i	$-0.664124\pi$
$991$	−31.0436	−0.986131	−0.493066	+	0.869992i	$0.664124\pi$
$992$	−5.87996	−0.186689
$993$	8.95644	0.284224
$994$	20.2432	0.642075
$995$	0	0
$996$	2.66216	0.0843539
$997$	−14.7970	−0.468626	−0.234313	−	0.972161i	$-0.575284\pi$
$997$	−14.7970	−0.468626	−0.234313	+	0.972161i	$0.575284\pi$
$998$	48.7335	1.54263
$999$	42.9129	1.35770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.x.1.2	✓	4
5.4	even	2		3025.2.a.bc.1.3	yes	4
11.10	odd	2	inner	3025.2.a.x.1.3	yes	4
55.54	odd	2		3025.2.a.bc.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3025.2.a.x.1.2	✓	4	1.1	even	1	trivial
3025.2.a.x.1.3	yes	4	11.10	odd	2	inner
3025.2.a.bc.1.2	yes	4	55.54	odd	2
3025.2.a.bc.1.3	yes	4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48617 q^{2} +1.79129 q^{3} +0.208712 q^{4} -2.66216 q^{6} +1.48617 q^{7} +2.66216 q^{8} +0.208712 q^{9} +O(q^{10})\)	\(q-1.48617 q^{2} +1.79129 q^{3} +0.208712 q^{4} -2.66216 q^{6} +1.48617 q^{7} +2.66216 q^{8} +0.208712 q^{9} +0.373864 q^{12} +5.63451 q^{13} -2.20871 q^{14} -4.37386 q^{16} -4.14834 q^{17} -0.310183 q^{18} -2.66216 q^{19} +2.66216 q^{21} -6.79129 q^{23} +4.76870 q^{24} -8.37386 q^{26} -5.00000 q^{27} +0.310183 q^{28} -10.0930 q^{29} -5.00000 q^{31} +1.17599 q^{32} +6.16515 q^{34} +0.0435608 q^{36} -8.58258 q^{37} +3.95644 q^{38} +10.0930 q^{39} +2.66216 q^{41} -3.95644 q^{42} +2.97235 q^{43} +10.0930 q^{46} -1.41742 q^{47} -7.83485 q^{48} -4.79129 q^{49} -7.43087 q^{51} +1.17599 q^{52} -6.79129 q^{53} +7.43087 q^{54} +3.95644 q^{56} -4.76870 q^{57} +15.0000 q^{58} +6.16515 q^{59} +12.7552 q^{61} +7.43087 q^{62} +0.310183 q^{63} +7.00000 q^{64} +3.58258 q^{67} -0.865809 q^{68} -12.1652 q^{69} -9.16515 q^{71} +0.555626 q^{72} -9.78285 q^{73} +12.7552 q^{74} -0.555626 q^{76} -15.0000 q^{78} +15.4174 q^{79} -9.58258 q^{81} -3.95644 q^{82} +7.12069 q^{83} +0.555626 q^{84} -4.41742 q^{86} -18.0795 q^{87} +8.20871 q^{89} +8.37386 q^{91} -1.41742 q^{92} -8.95644 q^{93} +2.10654 q^{94} +2.10654 q^{96} +0.373864 q^{97} +7.12069 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 10 q^{4} + 10 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 10 * q^4 + 10 * q^9	\( 4 q - 2 q^{3} + 10 q^{4} + 10 q^{9} - 26 q^{12} - 18 q^{14} + 10 q^{16} - 18 q^{23} - 6 q^{26} - 20 q^{27} - 20 q^{31} - 12 q^{34} + 46 q^{36} - 16 q^{37} - 30 q^{38} + 30 q^{42} - 24 q^{47} - 68 q^{48} - 10 q^{49} - 18 q^{53} - 30 q^{56} + 60 q^{58} - 12 q^{59} + 28 q^{64} - 4 q^{67} - 12 q^{69} - 60 q^{78} - 20 q^{81} + 30 q^{82} - 36 q^{86} + 42 q^{89} + 6 q^{91} - 24 q^{92} + 10 q^{93} - 26 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 + 10 * q^4 + 10 * q^9 - 26 * q^12 - 18 * q^14 + 10 * q^16 - 18 * q^23 - 6 * q^26 - 20 * q^27 - 20 * q^31 - 12 * q^34 + 46 * q^36 - 16 * q^37 - 30 * q^38 + 30 * q^42 - 24 * q^47 - 68 * q^48 - 10 * q^49 - 18 * q^53 - 30 * q^56 + 60 * q^58 - 12 * q^59 + 28 * q^64 - 4 * q^67 - 12 * q^69 - 60 * q^78 - 20 * q^81 + 30 * q^82 - 36 * q^86 + 42 * q^89 + 6 * q^91 - 24 * q^92 + 10 * q^93 - 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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