LMFDB - Embedded newform 5225.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.51908$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.82669 q^{2} +0.563416 q^{3} +1.33679 q^{4} -1.02918 q^{6} -1.69239 q^{7} +1.21147 q^{8} -2.68256 q^{9} +1.00000 q^{11} +0.753170 q^{12} -4.11168 q^{13} +3.09147 q^{14} -4.88657 q^{16} +6.16499 q^{17} +4.90021 q^{18} -1.00000 q^{19} -0.953520 q^{21} -1.82669 q^{22} -3.52199 q^{23} +0.682563 q^{24} +7.51076 q^{26} -3.20164 q^{27} -2.26238 q^{28} -8.10336 q^{29} -2.30144 q^{31} +6.50330 q^{32} +0.563416 q^{33} -11.2615 q^{34} -3.58603 q^{36} +6.56016 q^{37} +1.82669 q^{38} -2.31659 q^{39} +7.75013 q^{41} +1.74178 q^{42} -7.75102 q^{43} +1.33679 q^{44} +6.43359 q^{46} +10.8969 q^{47} -2.75317 q^{48} -4.13581 q^{49} +3.47345 q^{51} -5.49647 q^{52} -7.93511 q^{53} +5.84841 q^{54} -2.05029 q^{56} -0.563416 q^{57} +14.8023 q^{58} +10.9247 q^{59} -4.51162 q^{61} +4.20401 q^{62} +4.53995 q^{63} -2.10636 q^{64} -1.02918 q^{66} -14.7201 q^{67} +8.24132 q^{68} -1.98435 q^{69} +3.12026 q^{71} -3.24985 q^{72} -11.5827 q^{73} -11.9834 q^{74} -1.33679 q^{76} -1.69239 q^{77} +4.23168 q^{78} +4.96184 q^{79} +6.24383 q^{81} -14.1571 q^{82} +1.82905 q^{83} -1.27466 q^{84} +14.1587 q^{86} -4.56556 q^{87} +1.21147 q^{88} -9.37496 q^{89} +6.95858 q^{91} -4.70818 q^{92} -1.29666 q^{93} -19.9053 q^{94} +3.66406 q^{96} +10.9937 q^{97} +7.55484 q^{98} -2.68256 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 + 5 * q^11 - 6 * q^12 - 4 * q^13 - 14 * q^14 + 8 * q^16 + 4 * q^17 + 20 * q^18 - 5 * q^19 + 10 * q^21 - 2 * q^22 - 3 * q^23 - 14 * q^24 - 6 * q^26 + 11 * q^27 + 10 * q^28 + 10 * q^29 + 11 * q^31 - 14 * q^32 - q^33 - 4 * q^34 - 26 * q^36 - q^37 + 2 * q^38 + 2 * q^39 + 2 * q^41 + 16 * q^42 - 20 * q^43 + 6 * q^44 - 4 * q^46 + 20 * q^47 - 4 * q^48 + 3 * q^49 + 24 * q^51 - 6 * q^52 + 14 * q^53 + 16 * q^54 - 38 * q^56 + q^57 + 6 * q^58 + 3 * q^59 - 10 * q^61 + 6 * q^62 - 24 * q^63 - 2 * q^66 - 9 * q^67 - 24 * q^68 - 5 * q^69 + 23 * q^71 + 12 * q^72 + 8 * q^74 - 6 * q^76 - 6 * q^77 + 22 * q^78 + 44 * q^79 + q^81 + 30 * q^82 + 14 * q^83 + 14 * q^84 + 52 * q^86 - 28 * q^87 - 6 * q^88 - 27 * q^89 + 24 * q^91 - 58 * q^92 + 27 * q^93 - 8 * q^94 + 50 * q^96 - 15 * q^97 + 10 * q^98 + 4 * q^99

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 $

$2$	−1.82669	−1.29166	−0.645832	−	0.763479i	$-0.723489\pi$
$2$	−1.82669	−1.29166	−0.645832	+	0.763479i	$0.723489\pi$
$3$	0.563416	0.325288	0.162644	−	0.986685i	$-0.447998\pi$
$3$	0.563416	0.325288	0.162644	+	0.986685i	$0.447998\pi$
$4$	1.33679	0.668396
$5$	0	0
$5$	0	0
$6$	−1.02918	−0.420163
$7$	−1.69239	−0.639664	−0.319832	−	0.947474i	$-0.603626\pi$
$7$	−1.69239	−0.639664	−0.319832	+	0.947474i	$0.603626\pi$
$8$	1.21147	0.428321
$9$	−2.68256	−0.894188
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.753170	0.217421
$13$	−4.11168	−1.14038	−0.570188	−	0.821514i	$-0.693130\pi$
$13$	−4.11168	−1.14038	−0.570188	+	0.821514i	$0.693130\pi$
$14$	3.09147	0.826231
$15$	0	0
$16$	−4.88657	−1.22164
$17$	6.16499	1.49523	0.747615	−	0.664132i	$-0.231199\pi$
$17$	6.16499	1.49523	0.747615	+	0.664132i	$0.231199\pi$
$18$	4.90021	1.15499
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.953520	−0.208075
$22$	−1.82669	−0.389451
$23$	−3.52199	−0.734386	−0.367193	−	0.930145i	$-0.619681\pi$
$23$	−3.52199	−0.734386	−0.367193	+	0.930145i	$0.619681\pi$
$24$	0.682563	0.139328
$25$	0	0
$26$	7.51076	1.47298
$27$	−3.20164	−0.616157
$28$	−2.26238	−0.427549
$29$	−8.10336	−1.50476	−0.752378	−	0.658731i	$-0.771094\pi$
$29$	−8.10336	−1.50476	−0.752378	+	0.658731i	$0.771094\pi$
$30$	0	0
$31$	−2.30144	−0.413350	−0.206675	−	0.978410i	$-0.566264\pi$
$31$	−2.30144	−0.413350	−0.206675	+	0.978410i	$0.566264\pi$
$32$	6.50330	1.14963
$33$	0.563416	0.0980781
$34$	−11.2615	−1.93134
$35$	0	0
$36$	−3.58603	−0.597672
$37$	6.56016	1.07848	0.539242	−	0.842151i	$-0.318711\pi$
$37$	6.56016	1.07848	0.539242	+	0.842151i	$0.318711\pi$
$38$	1.82669	0.296328
$39$	−2.31659	−0.370951
$40$	0	0
$41$	7.75013	1.21037	0.605183	−	0.796086i	$-0.293100\pi$
$41$	7.75013	1.21037	0.605183	+	0.796086i	$0.293100\pi$
$42$	1.74178	0.268763
$43$	−7.75102	−1.18202	−0.591010	−	0.806664i	$-0.701271\pi$
$43$	−7.75102	−1.18202	−0.591010	+	0.806664i	$0.701271\pi$
$44$	1.33679	0.201529
$45$	0	0
$46$	6.43359	0.948581
$47$	10.8969	1.58948	0.794742	−	0.606948i	$-0.207606\pi$
$47$	10.8969	1.58948	0.794742	+	0.606948i	$0.207606\pi$
$48$	−2.75317	−0.397386
$49$	−4.13581	−0.590830
$50$	0	0
$51$	3.47345	0.486381
$52$	−5.49647	−0.762223
$53$	−7.93511	−1.08997	−0.544986	−	0.838445i	$-0.683465\pi$
$53$	−7.93511	−1.08997	−0.544986	+	0.838445i	$0.683465\pi$
$54$	5.84841	0.795868
$55$	0	0
$56$	−2.05029	−0.273981
$57$	−0.563416	−0.0746262
$58$	14.8023	1.94364
$59$	10.9247	1.42228	0.711140	−	0.703051i	$-0.248179\pi$
$59$	10.9247	1.42228	0.711140	+	0.703051i	$0.248179\pi$
$60$	0	0
$61$	−4.51162	−0.577653	−0.288827	−	0.957381i	$-0.593265\pi$
$61$	−4.51162	−0.577653	−0.288827	+	0.957381i	$0.593265\pi$
$62$	4.20401	0.533910
$63$	4.53995	0.571980
$64$	−2.10636	−0.263295
$65$	0	0
$66$	−1.02918	−0.126684
$67$	−14.7201	−1.79835	−0.899173	−	0.437594i	$-0.855831\pi$
$67$	−14.7201	−1.79835	−0.899173	+	0.437594i	$0.855831\pi$
$68$	8.24132	0.999407
$69$	−1.98435	−0.238887
$70$	0	0
$71$	3.12026	0.370307	0.185153	−	0.982710i	$-0.440722\pi$
$71$	3.12026	0.370307	0.185153	+	0.982710i	$0.440722\pi$
$72$	−3.24985	−0.382999
$73$	−11.5827	−1.35565	−0.677824	−	0.735224i	$-0.737077\pi$
$73$	−11.5827	−1.35565	−0.677824	+	0.735224i	$0.737077\pi$
$74$	−11.9834	−1.39304
$75$	0	0
$76$	−1.33679	−0.153341
$77$	−1.69239	−0.192866
$78$	4.23168	0.479144
$79$	4.96184	0.558250	0.279125	−	0.960255i	$-0.409956\pi$
$79$	4.96184	0.558250	0.279125	+	0.960255i	$0.409956\pi$
$80$	0	0
$81$	6.24383	0.693759
$82$	−14.1571	−1.56339
$83$	1.82905	0.200765	0.100382	−	0.994949i	$-0.467993\pi$
$83$	1.82905	0.200765	0.100382	+	0.994949i	$0.467993\pi$
$84$	−1.27466	−0.139077
$85$	0	0
$86$	14.1587	1.52677
$87$	−4.56556	−0.489480
$88$	1.21147	0.129144
$89$	−9.37496	−0.993743	−0.496872	−	0.867824i	$-0.665518\pi$
$89$	−9.37496	−0.993743	−0.496872	+	0.867824i	$0.665518\pi$
$90$	0	0
$91$	6.95858	0.729457
$92$	−4.70818	−0.490861
$93$	−1.29666	−0.134458
$94$	−19.9053	−2.05308
$95$	0	0
$96$	3.66406	0.373961
$97$	10.9937	1.11625	0.558123	−	0.829758i	$-0.311522\pi$
$97$	10.9937	1.11625	0.558123	+	0.829758i	$0.311522\pi$
$98$	7.55484	0.763154
$99$	−2.68256	−0.269608
$100$	0	0
$101$	2.30621	0.229476	0.114738	−	0.993396i	$-0.463397\pi$
$101$	2.30621	0.229476	0.114738	+	0.993396i	$0.463397\pi$
$102$	−6.34492	−0.628241
$103$	−6.12000	−0.603021	−0.301511	−	0.953463i	$-0.597491\pi$
$103$	−6.12000	−0.603021	−0.301511	+	0.953463i	$0.597491\pi$
$104$	−4.98119	−0.488446
$105$	0	0
$106$	14.4950	1.40788
$107$	−0.422947	−0.0408878	−0.0204439	−	0.999791i	$-0.506508\pi$
$107$	−0.422947	−0.0408878	−0.0204439	+	0.999791i	$0.506508\pi$
$108$	−4.27993	−0.411837
$109$	14.0180	1.34268	0.671338	−	0.741151i	$-0.265720\pi$
$109$	14.0180	1.34268	0.671338	+	0.741151i	$0.265720\pi$
$110$	0	0
$111$	3.69609	0.350818
$112$	8.26999	0.781441
$113$	2.53638	0.238602	0.119301	−	0.992858i	$-0.461935\pi$
$113$	2.53638	0.238602	0.119301	+	0.992858i	$0.461935\pi$
$114$	1.02918	0.0963920
$115$	0	0
$116$	−10.8325	−1.00577
$117$	11.0298	1.01971
$118$	−19.9561	−1.83711
$119$	−10.4336	−0.956445
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.24132	0.746134
$123$	4.36654	0.393718
$124$	−3.07654	−0.276282
$125$	0	0
$126$	−8.29307	−0.738806
$127$	−1.07275	−0.0951914	−0.0475957	−	0.998867i	$-0.515156\pi$
$127$	−1.07275	−0.0951914	−0.0475957	+	0.998867i	$0.515156\pi$
$128$	−9.15893	−0.809543
$129$	−4.36705	−0.384497
$130$	0	0
$131$	−7.82709	−0.683856	−0.341928	−	0.939726i	$-0.611080\pi$
$131$	−7.82709	−0.683856	−0.341928	+	0.939726i	$0.611080\pi$
$132$	0.753170	0.0655550
$133$	1.69239	0.146749
$134$	26.8890	2.32286
$135$	0	0
$136$	7.46873	0.640438
$137$	11.2473	0.960919	0.480460	−	0.877017i	$-0.340470\pi$
$137$	11.2473	0.960919	0.480460	+	0.877017i	$0.340470\pi$
$138$	3.62478	0.308562
$139$	−0.905926	−0.0768397	−0.0384198	−	0.999262i	$-0.512232\pi$
$139$	−0.905926	−0.0768397	−0.0384198	+	0.999262i	$0.512232\pi$
$140$	0	0
$141$	6.13951	0.517040
$142$	−5.69975	−0.478312
$143$	−4.11168	−0.343836
$144$	13.1085	1.09238
$145$	0	0
$146$	21.1579	1.75104
$147$	−2.33018	−0.192190
$148$	8.76957	0.720854
$149$	10.5174	0.861622	0.430811	−	0.902442i	$-0.358228\pi$
$149$	10.5174	0.861622	0.430811	+	0.902442i	$0.358228\pi$
$150$	0	0
$151$	13.2436	1.07775	0.538873	−	0.842387i	$-0.318850\pi$
$151$	13.2436	1.07775	0.538873	+	0.842387i	$0.318850\pi$
$152$	−1.21147	−0.0982635
$153$	−16.5380	−1.33702
$154$	3.09147	0.249118
$155$	0	0
$156$	−3.09679	−0.247942
$157$	−1.99915	−0.159549	−0.0797747	−	0.996813i	$-0.525420\pi$
$157$	−1.99915	−0.159549	−0.0797747	+	0.996813i	$0.525420\pi$
$158$	−9.06373	−0.721072
$159$	−4.47076	−0.354555
$160$	0	0
$161$	5.96059	0.469761
$162$	−11.4055	−0.896104
$163$	18.7557	1.46906	0.734531	−	0.678575i	$-0.237402\pi$
$163$	18.7557	1.46906	0.734531	+	0.678575i	$0.237402\pi$
$164$	10.3603	0.809005
$165$	0	0
$166$	−3.34111	−0.259320
$167$	6.31203	0.488440	0.244220	−	0.969720i	$-0.421468\pi$
$167$	6.31203	0.488440	0.244220	+	0.969720i	$0.421468\pi$
$168$	−1.15516	−0.0891229
$169$	3.90593	0.300456
$170$	0	0
$171$	2.68256	0.205141
$172$	−10.3615	−0.790058
$173$	21.5269	1.63666	0.818328	−	0.574751i	$-0.194901\pi$
$173$	21.5269	1.63666	0.818328	+	0.574751i	$0.194901\pi$
$174$	8.33986	0.632243
$175$	0	0
$176$	−4.88657	−0.368339
$177$	6.15516	0.462650
$178$	17.1251	1.28358
$179$	−7.97018	−0.595719	−0.297860	−	0.954610i	$-0.596273\pi$
$179$	−7.97018	−0.595719	−0.297860	+	0.954610i	$0.596273\pi$
$180$	0	0
$181$	−1.16955	−0.0869317	−0.0434659	−	0.999055i	$-0.513840\pi$
$181$	−1.16955	−0.0869317	−0.0434659	+	0.999055i	$0.513840\pi$
$182$	−12.7112	−0.942214
$183$	−2.54191	−0.187904
$184$	−4.26680	−0.314553
$185$	0	0
$186$	2.36860	0.173674
$187$	6.16499	0.450829
$188$	14.5670	1.06240
$189$	5.41844	0.394133
$190$	0	0
$191$	−15.6673	−1.13365	−0.566824	−	0.823839i	$-0.691828\pi$
$191$	−15.6673	−1.13365	−0.566824	+	0.823839i	$0.691828\pi$
$192$	−1.18676	−0.0856468
$193$	−21.6769	−1.56034	−0.780169	−	0.625568i	$-0.784867\pi$
$193$	−21.6769	−1.56034	−0.780169	+	0.625568i	$0.784867\pi$
$194$	−20.0821	−1.44181
$195$	0	0
$196$	−5.52872	−0.394908
$197$	−4.58794	−0.326877	−0.163439	−	0.986554i	$-0.552259\pi$
$197$	−4.58794	−0.326877	−0.163439	+	0.986554i	$0.552259\pi$
$198$	4.90021	0.348243
$199$	13.0619	0.925937	0.462968	−	0.886375i	$-0.346784\pi$
$199$	13.0619	0.925937	0.462968	+	0.886375i	$0.346784\pi$
$200$	0	0
$201$	−8.29353	−0.584980
$202$	−4.21272	−0.296406
$203$	13.7141	0.962539
$204$	4.64329	0.325095
$205$	0	0
$206$	11.1793	0.778901
$207$	9.44797	0.656679
$208$	20.0920	1.39313
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	2.91188	0.200462	0.100231	−	0.994964i	$-0.468042\pi$
$211$	2.91188	0.200462	0.100231	+	0.994964i	$0.468042\pi$
$212$	−10.6076	−0.728533
$213$	1.75800	0.120456
$214$	0.772592	0.0528133
$215$	0	0
$216$	−3.87871	−0.263913
$217$	3.89493	0.264405
$218$	−25.6064	−1.73429
$219$	−6.52585	−0.440976
$220$	0	0
$221$	−25.3485	−1.70512
$222$	−6.75161	−0.453139
$223$	−6.34161	−0.424665	−0.212333	−	0.977197i	$-0.568106\pi$
$223$	−6.34161	−0.424665	−0.212333	+	0.977197i	$0.568106\pi$
$224$	−11.0061	−0.735378
$225$	0	0
$226$	−4.63317	−0.308194
$227$	2.00429	0.133029	0.0665147	−	0.997785i	$-0.478812\pi$
$227$	2.00429	0.133029	0.0665147	+	0.997785i	$0.478812\pi$
$228$	−0.753170	−0.0498799
$229$	−20.9893	−1.38701	−0.693507	−	0.720450i	$-0.743935\pi$
$229$	−20.9893	−1.38701	−0.693507	+	0.720450i	$0.743935\pi$
$230$	0	0
$231$	−0.953520	−0.0627370
$232$	−9.81701	−0.644518
$233$	17.6006	1.15305	0.576527	−	0.817078i	$-0.304407\pi$
$233$	17.6006	1.15305	0.576527	+	0.817078i	$0.304407\pi$
$234$	−20.1481	−1.31712
$235$	0	0
$236$	14.6041	0.950646
$237$	2.79558	0.181592
$238$	19.0589	1.23541
$239$	26.6207	1.72195	0.860975	−	0.508647i	$-0.169854\pi$
$239$	26.6207	1.72195	0.860975	+	0.508647i	$0.169854\pi$
$240$	0	0
$241$	13.1342	0.846049	0.423024	−	0.906118i	$-0.360968\pi$
$241$	13.1342	0.846049	0.423024	+	0.906118i	$0.360968\pi$
$242$	−1.82669	−0.117424
$243$	13.1228	0.841828
$244$	−6.03109	−0.386101
$245$	0	0
$246$	−7.97632	−0.508551
$247$	4.11168	0.261620
$248$	−2.78813	−0.177046
$249$	1.03052	0.0653063
$250$	0	0
$251$	−26.1636	−1.65143	−0.825715	−	0.564087i	$-0.809228\pi$
$251$	−26.1636	−1.65143	−0.825715	+	0.564087i	$0.809228\pi$
$252$	6.06897	0.382309
$253$	−3.52199	−0.221426
$254$	1.95959	0.122955
$255$	0	0
$256$	20.9432	1.30895
$257$	−12.6117	−0.786697	−0.393349	−	0.919389i	$-0.628683\pi$
$257$	−12.6117	−0.786697	−0.393349	+	0.919389i	$0.628683\pi$
$258$	7.97724	0.496641
$259$	−11.1024	−0.689867
$260$	0	0
$261$	21.7378	1.34554
$262$	14.2977	0.883312
$263$	20.0550	1.23664	0.618322	−	0.785925i	$-0.287813\pi$
$263$	20.0550	1.23664	0.618322	+	0.785925i	$0.287813\pi$
$264$	0.682563	0.0420088
$265$	0	0
$266$	−3.09147	−0.189550
$267$	−5.28200	−0.323253
$268$	−19.6777	−1.20201
$269$	−20.1150	−1.22644	−0.613218	−	0.789914i	$-0.710125\pi$
$269$	−20.1150	−1.22644	−0.613218	+	0.789914i	$0.710125\pi$
$270$	0	0
$271$	15.5586	0.945117	0.472558	−	0.881299i	$-0.343331\pi$
$271$	15.5586	0.945117	0.472558	+	0.881299i	$0.343331\pi$
$272$	−30.1257	−1.82664
$273$	3.92057	0.237284
$274$	−20.5453	−1.24119
$275$	0	0
$276$	−2.65266	−0.159671
$277$	−1.89211	−0.113686	−0.0568429	−	0.998383i	$-0.518103\pi$
$277$	−1.89211	−0.113686	−0.0568429	+	0.998383i	$0.518103\pi$
$278$	1.65485	0.0992510
$279$	6.17375	0.369613
$280$	0	0
$281$	4.59299	0.273995	0.136997	−	0.990571i	$-0.456255\pi$
$281$	4.59299	0.273995	0.136997	+	0.990571i	$0.456255\pi$
$282$	−11.2150	−0.667842
$283$	1.17798	0.0700234	0.0350117	−	0.999387i	$-0.488853\pi$
$283$	1.17798	0.0700234	0.0350117	+	0.999387i	$0.488853\pi$
$284$	4.17114	0.247512
$285$	0	0
$286$	7.51076	0.444121
$287$	−13.1163	−0.774228
$288$	−17.4455	−1.02799
$289$	21.0071	1.23571
$290$	0	0
$291$	6.19405	0.363101
$292$	−15.4836	−0.906110
$293$	11.2606	0.657851	0.328925	−	0.944356i	$-0.393314\pi$
$293$	11.2606	0.657851	0.328925	+	0.944356i	$0.393314\pi$
$294$	4.25651	0.248245
$295$	0	0
$296$	7.94745	0.461936
$297$	−3.20164	−0.185778
$298$	−19.2121	−1.11293
$299$	14.4813	0.837476
$300$	0	0
$301$	13.1178	0.756096
$302$	−24.1919	−1.39209
$303$	1.29935	0.0746459
$304$	4.88657	0.280264
$305$	0	0
$306$	30.2098	1.72698
$307$	−4.35245	−0.248407	−0.124204	−	0.992257i	$-0.539638\pi$
$307$	−4.35245	−0.248407	−0.124204	+	0.992257i	$0.539638\pi$
$308$	−2.26238	−0.128911
$309$	−3.44810	−0.196156
$310$	0	0
$311$	7.61725	0.431935	0.215967	−	0.976401i	$-0.430709\pi$
$311$	7.61725	0.431935	0.215967	+	0.976401i	$0.430709\pi$
$312$	−2.80648	−0.158886
$313$	17.5654	0.992855	0.496427	−	0.868078i	$-0.334645\pi$
$313$	17.5654	0.992855	0.496427	+	0.868078i	$0.334645\pi$
$314$	3.65182	0.206084
$315$	0	0
$316$	6.63295	0.373132
$317$	1.94192	0.109069	0.0545345	−	0.998512i	$-0.482633\pi$
$317$	1.94192	0.109069	0.0545345	+	0.998512i	$0.482633\pi$
$318$	8.16670	0.457966
$319$	−8.10336	−0.453701
$320$	0	0
$321$	−0.238295	−0.0133003
$322$	−10.8882	−0.606773
$323$	−6.16499	−0.343029
$324$	8.34671	0.463706
$325$	0	0
$326$	−34.2609	−1.89754
$327$	7.89793	0.436757
$328$	9.38907	0.518425
$329$	−18.4419	−1.01674
$330$	0	0
$331$	13.3988	0.736462	0.368231	−	0.929734i	$-0.379964\pi$
$331$	13.3988	0.736462	0.368231	+	0.929734i	$0.379964\pi$
$332$	2.44506	0.134190
$333$	−17.5980	−0.964366
$334$	−11.5301	−0.630900
$335$	0	0
$336$	4.65944	0.254193
$337$	−33.1631	−1.80651	−0.903254	−	0.429107i	$-0.858828\pi$
$337$	−33.1631	−1.80651	−0.903254	+	0.429107i	$0.858828\pi$
$338$	−7.13491	−0.388088
$339$	1.42903	0.0776145
$340$	0	0
$341$	−2.30144	−0.124630
$342$	−4.90021	−0.264973
$343$	18.8462	1.01760
$344$	−9.39016	−0.506283
$345$	0	0
$346$	−39.3229	−2.11401
$347$	−18.0268	−0.967730	−0.483865	−	0.875143i	$-0.660767\pi$
$347$	−18.0268	−0.967730	−0.483865	+	0.875143i	$0.660767\pi$
$348$	−6.10321	−0.327166
$349$	8.81411	0.471808	0.235904	−	0.971776i	$-0.424195\pi$
$349$	8.81411	0.471808	0.235904	+	0.971776i	$0.424195\pi$
$350$	0	0
$351$	13.1641	0.702650
$352$	6.50330	0.346627
$353$	1.59249	0.0847599	0.0423799	−	0.999102i	$-0.486506\pi$
$353$	1.59249	0.0847599	0.0423799	+	0.999102i	$0.486506\pi$
$354$	−11.2436	−0.597589
$355$	0	0
$356$	−12.5324	−0.664214
$357$	−5.87844	−0.311120
$358$	14.5590	0.769469
$359$	36.3774	1.91993	0.959963	−	0.280126i	$-0.0903763\pi$
$359$	36.3774	1.91993	0.959963	+	0.280126i	$0.0903763\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.13640	0.112287
$363$	0.563416	0.0295716
$364$	9.30218	0.487567
$365$	0	0
$366$	4.64329	0.242708
$367$	5.00276	0.261142	0.130571	−	0.991439i	$-0.458319\pi$
$367$	5.00276	0.261142	0.130571	+	0.991439i	$0.458319\pi$
$368$	17.2105	0.897158
$369$	−20.7902	−1.08229
$370$	0	0
$371$	13.4293	0.697216
$372$	−1.73337	−0.0898712
$373$	34.1313	1.76725	0.883625	−	0.468195i	$-0.155096\pi$
$373$	34.1313	1.76725	0.883625	+	0.468195i	$0.155096\pi$
$374$	−11.2615	−0.582320
$375$	0	0
$376$	13.2014	0.680808
$377$	33.3185	1.71599
$378$	−9.89780	−0.509088
$379$	15.7749	0.810302	0.405151	−	0.914250i	$-0.367219\pi$
$379$	15.7749	0.810302	0.405151	+	0.914250i	$0.367219\pi$
$380$	0	0
$381$	−0.604405	−0.0309646
$382$	28.6193	1.46429
$383$	22.4260	1.14592	0.572958	−	0.819585i	$-0.305796\pi$
$383$	22.4260	1.14592	0.572958	+	0.819585i	$0.305796\pi$
$384$	−5.16028	−0.263335
$385$	0	0
$386$	39.5970	2.01543
$387$	20.7926	1.05695
$388$	14.6964	0.746094
$389$	−18.3488	−0.930321	−0.465160	−	0.885226i	$-0.654003\pi$
$389$	−18.3488	−0.930321	−0.465160	+	0.885226i	$0.654003\pi$
$390$	0	0
$391$	−21.7131	−1.09808
$392$	−5.01042	−0.253065
$393$	−4.40990	−0.222450
$394$	8.38074	0.422216
$395$	0	0
$396$	−3.58603	−0.180205
$397$	0.511483	0.0256706	0.0128353	−	0.999918i	$-0.495914\pi$
$397$	0.511483	0.0256706	0.0128353	+	0.999918i	$0.495914\pi$
$398$	−23.8601	−1.19600
$399$	0.953520	0.0477357
$400$	0	0
$401$	18.9824	0.947935	0.473967	−	0.880542i	$-0.342821\pi$
$401$	18.9824	0.947935	0.473967	+	0.880542i	$0.342821\pi$
$402$	15.1497	0.755598
$403$	9.46277	0.471374
$404$	3.08292	0.153381
$405$	0	0
$406$	−25.0513	−1.24328
$407$	6.56016	0.325175
$408$	4.20800	0.208327
$409$	1.10089	0.0544355	0.0272177	−	0.999630i	$-0.491335\pi$
$409$	1.10089	0.0544355	0.0272177	+	0.999630i	$0.491335\pi$
$410$	0	0
$411$	6.33689	0.312576
$412$	−8.18117	−0.403057
$413$	−18.4889	−0.909781
$414$	−17.2585	−0.848209
$415$	0	0
$416$	−26.7395	−1.31101
$417$	−0.510413	−0.0249950
$418$	1.82669	0.0893463
$419$	−18.7929	−0.918094	−0.459047	−	0.888412i	$-0.651809\pi$
$419$	−18.7929	−0.918094	−0.459047	+	0.888412i	$0.651809\pi$
$420$	0	0
$421$	−24.7618	−1.20682	−0.603408	−	0.797433i	$-0.706191\pi$
$421$	−24.7618	−1.20682	−0.603408	+	0.797433i	$0.706191\pi$
$422$	−5.31910	−0.258930
$423$	−29.2318	−1.42130
$424$	−9.61318	−0.466857
$425$	0	0
$426$	−3.21133	−0.155589
$427$	7.63542	0.369504
$428$	−0.565392	−0.0273293
$429$	−2.31659	−0.111846
$430$	0	0
$431$	−32.1985	−1.55095	−0.775474	−	0.631380i	$-0.782489\pi$
$431$	−32.1985	−1.55095	−0.775474	+	0.631380i	$0.782489\pi$
$432$	15.6451	0.752723
$433$	15.3042	0.735475	0.367737	−	0.929930i	$-0.380133\pi$
$433$	15.3042	0.735475	0.367737	+	0.929930i	$0.380133\pi$
$434$	−7.11483	−0.341523
$435$	0	0
$436$	18.7391	0.897440
$437$	3.52199	0.168480
$438$	11.9207	0.569593
$439$	26.1812	1.24956	0.624781	−	0.780800i	$-0.285188\pi$
$439$	26.1812	1.24956	0.624781	+	0.780800i	$0.285188\pi$
$440$	0	0
$441$	11.0946	0.528313
$442$	46.3038	2.20245
$443$	−21.5579	−1.02425	−0.512123	−	0.858912i	$-0.671141\pi$
$443$	−21.5579	−1.02425	−0.512123	+	0.858912i	$0.671141\pi$
$444$	4.94091	0.234485
$445$	0	0
$446$	11.5841	0.548525
$447$	5.92569	0.280275
$448$	3.56479	0.168420
$449$	11.7990	0.556829	0.278415	−	0.960461i	$-0.410191\pi$
$449$	11.7990	0.556829	0.278415	+	0.960461i	$0.410191\pi$
$450$	0	0
$451$	7.75013	0.364939
$452$	3.39061	0.159481
$453$	7.46163	0.350578
$454$	−3.66122	−0.171829
$455$	0	0
$456$	−0.682563	−0.0319639
$457$	15.6863	0.733773	0.366886	−	0.930266i	$-0.380424\pi$
$457$	15.6863	0.733773	0.366886	+	0.930266i	$0.380424\pi$
$458$	38.3410	1.79156
$459$	−19.7381	−0.921296
$460$	0	0
$461$	27.2451	1.26893	0.634466	−	0.772951i	$-0.281220\pi$
$461$	27.2451	1.26893	0.634466	+	0.772951i	$0.281220\pi$
$462$	1.74178	0.0810352
$463$	17.9364	0.833573	0.416787	−	0.909004i	$-0.363156\pi$
$463$	17.9364	0.833573	0.416787	+	0.909004i	$0.363156\pi$
$464$	39.5977	1.83828
$465$	0	0
$466$	−32.1509	−1.48936
$467$	−29.0100	−1.34242	−0.671211	−	0.741266i	$-0.734226\pi$
$467$	−29.0100	−1.34242	−0.671211	+	0.741266i	$0.734226\pi$
$468$	14.7446	0.681570
$469$	24.9122	1.15034
$470$	0	0
$471$	−1.12635	−0.0518995
$472$	13.2350	0.609191
$473$	−7.75102	−0.356392
$474$	−5.10665	−0.234556
$475$	0	0
$476$	−13.9475	−0.639285
$477$	21.2864	0.974639
$478$	−48.6277	−2.22418
$479$	34.0896	1.55759	0.778797	−	0.627276i	$-0.215830\pi$
$479$	34.0896	1.55759	0.778797	+	0.627276i	$0.215830\pi$
$480$	0	0
$481$	−26.9733	−1.22988
$482$	−23.9921	−1.09281
$483$	3.35829	0.152808
$484$	1.33679	0.0607633
$485$	0	0
$486$	−23.9713	−1.08736
$487$	−6.38349	−0.289264	−0.144632	−	0.989486i	$-0.546200\pi$
$487$	−6.38349	−0.289264	−0.144632	+	0.989486i	$0.546200\pi$
$488$	−5.46570	−0.247421
$489$	10.5673	0.477869
$490$	0	0
$491$	−31.4552	−1.41956	−0.709778	−	0.704426i	$-0.751205\pi$
$491$	−31.4552	−1.41956	−0.709778	+	0.704426i	$0.751205\pi$
$492$	5.83716	0.263160
$493$	−49.9572	−2.24996
$494$	−7.51076	−0.337925
$495$	0	0
$496$	11.2461	0.504966
$497$	−5.28071	−0.236872
$498$	−1.88243	−0.0843539
$499$	−31.9667	−1.43103	−0.715514	−	0.698599i	$-0.753807\pi$
$499$	−31.9667	−1.43103	−0.715514	+	0.698599i	$0.753807\pi$
$500$	0	0
$501$	3.55630	0.158884
$502$	47.7927	2.13309
$503$	34.0522	1.51831	0.759157	−	0.650907i	$-0.225611\pi$
$503$	34.0522	1.51831	0.759157	+	0.650907i	$0.225611\pi$
$504$	5.50003	0.244991
$505$	0	0
$506$	6.43359	0.286008
$507$	2.20066	0.0977347
$508$	−1.43405	−0.0636256
$509$	25.7904	1.14314	0.571569	−	0.820554i	$-0.306335\pi$
$509$	25.7904	1.14314	0.571569	+	0.820554i	$0.306335\pi$
$510$	0	0
$511$	19.6024	0.867160
$512$	−19.9389	−0.881184
$513$	3.20164	0.141356
$514$	23.0377	1.01615
$515$	0	0
$516$	−5.83784	−0.256996
$517$	10.8969	0.479247
$518$	20.2806	0.891076
$519$	12.1286	0.532385
$520$	0	0
$521$	31.3774	1.37467	0.687334	−	0.726342i	$-0.258781\pi$
$521$	31.3774	1.37467	0.687334	+	0.726342i	$0.258781\pi$
$522$	−39.7082	−1.73798
$523$	23.2388	1.01616	0.508082	−	0.861309i	$-0.330355\pi$
$523$	23.2388	1.01616	0.508082	+	0.861309i	$0.330355\pi$
$524$	−10.4632	−0.457087
$525$	0	0
$526$	−36.6343	−1.59733
$527$	−14.1883	−0.618054
$528$	−2.75317	−0.119816
$529$	−10.5956	−0.460677
$530$	0	0
$531$	−29.3063	−1.27178
$532$	2.26238	0.0980865
$533$	−31.8661	−1.38027
$534$	9.64856	0.417534
$535$	0	0
$536$	−17.8330	−0.770268
$537$	−4.49052	−0.193780
$538$	36.7439	1.58414
$539$	−4.13581	−0.178142
$540$	0	0
$541$	36.0085	1.54812	0.774062	−	0.633109i	$-0.218222\pi$
$541$	36.0085	1.54812	0.774062	+	0.633109i	$0.218222\pi$
$542$	−28.4207	−1.22077
$543$	−0.658941	−0.0282779
$544$	40.0928	1.71896
$545$	0	0
$546$	−7.16166	−0.306491
$547$	28.1855	1.20513	0.602563	−	0.798071i	$-0.294146\pi$
$547$	28.1855	1.20513	0.602563	+	0.798071i	$0.294146\pi$
$548$	15.0353	0.642275
$549$	12.1027	0.516530
$550$	0	0
$551$	8.10336	0.345215
$552$	−2.40398	−0.102320
$553$	−8.39738	−0.357093
$554$	3.45629	0.146844
$555$	0	0
$556$	−1.21104	−0.0513594
$557$	−21.6248	−0.916274	−0.458137	−	0.888882i	$-0.651483\pi$
$557$	−21.6248	−0.916274	−0.458137	+	0.888882i	$0.651483\pi$
$558$	−11.2775	−0.477415
$559$	31.8697	1.34795
$560$	0	0
$561$	3.47345	0.146649
$562$	−8.38997	−0.353909
$563$	−13.1791	−0.555434	−0.277717	−	0.960663i	$-0.589578\pi$
$563$	−13.1791	−0.555434	−0.277717	+	0.960663i	$0.589578\pi$
$564$	8.20725	0.345588
$565$	0	0
$566$	−2.15179	−0.0904467
$567$	−10.5670	−0.443773
$568$	3.78011	0.158610
$569$	8.61143	0.361010	0.180505	−	0.983574i	$-0.442227\pi$
$569$	8.61143	0.361010	0.180505	+	0.983574i	$0.442227\pi$
$570$	0	0
$571$	−17.6546	−0.738821	−0.369410	−	0.929266i	$-0.620440\pi$
$571$	−17.6546	−0.738821	−0.369410	+	0.929266i	$0.620440\pi$
$572$	−5.49647	−0.229819
$573$	−8.82722	−0.368762
$574$	23.9593	1.00004
$575$	0	0
$576$	5.65045	0.235435
$577$	31.0907	1.29432	0.647162	−	0.762352i	$-0.275956\pi$
$577$	31.0907	1.29432	0.647162	+	0.762352i	$0.275956\pi$
$578$	−38.3735	−1.59613
$579$	−12.2131	−0.507560
$580$	0	0
$581$	−3.09547	−0.128422
$582$	−11.3146	−0.469005
$583$	−7.93511	−0.328639
$584$	−14.0321	−0.580652
$585$	0	0
$586$	−20.5696	−0.849722
$587$	26.7211	1.10290	0.551450	−	0.834208i	$-0.314075\pi$
$587$	26.7211	1.10290	0.551450	+	0.834208i	$0.314075\pi$
$588$	−3.11497	−0.128459
$589$	2.30144	0.0948290
$590$	0	0
$591$	−2.58492	−0.106329
$592$	−32.0567	−1.31752
$593$	24.2460	0.995666	0.497833	−	0.867273i	$-0.334129\pi$
$593$	24.2460	0.995666	0.497833	+	0.867273i	$0.334129\pi$
$594$	5.84841	0.239963
$595$	0	0
$596$	14.0596	0.575905
$597$	7.35930	0.301196
$598$	−26.4529	−1.08174
$599$	−18.2095	−0.744019	−0.372009	−	0.928229i	$-0.621331\pi$
$599$	−18.2095	−0.744019	−0.372009	+	0.928229i	$0.621331\pi$
$600$	0	0
$601$	37.9824	1.54934	0.774668	−	0.632368i	$-0.217917\pi$
$601$	37.9824	1.54934	0.774668	+	0.632368i	$0.217917\pi$
$602$	−23.9621	−0.976622
$603$	39.4876	1.60806
$604$	17.7039	0.720362
$605$	0	0
$606$	−2.37351	−0.0964174
$607$	20.0130	0.812301	0.406151	−	0.913806i	$-0.366871\pi$
$607$	20.0130	0.812301	0.406151	+	0.913806i	$0.366871\pi$
$608$	−6.50330	−0.263744
$609$	7.72672	0.313103
$610$	0	0
$611$	−44.8048	−1.81261
$612$	−22.1079	−0.893657
$613$	42.1414	1.70208	0.851038	−	0.525105i	$-0.175974\pi$
$613$	42.1414	1.70208	0.851038	+	0.525105i	$0.175974\pi$
$614$	7.95057	0.320859
$615$	0	0
$616$	−2.05029	−0.0826085
$617$	−16.9044	−0.680545	−0.340273	−	0.940327i	$-0.610519\pi$
$617$	−16.9044	−0.680545	−0.340273	+	0.940327i	$0.610519\pi$
$618$	6.29861	0.253367
$619$	5.31177	0.213498	0.106749	−	0.994286i	$-0.465956\pi$
$619$	5.31177	0.213498	0.106749	+	0.994286i	$0.465956\pi$
$620$	0	0
$621$	11.2762	0.452497
$622$	−13.9144	−0.557915
$623$	15.8661	0.635662
$624$	11.3202	0.453169
$625$	0	0
$626$	−32.0865	−1.28243
$627$	−0.563416	−0.0225006
$628$	−2.67245	−0.106642
$629$	40.4433	1.61258
$630$	0	0
$631$	34.0425	1.35521	0.677604	−	0.735427i	$-0.263018\pi$
$631$	34.0425	1.35521	0.677604	+	0.735427i	$0.263018\pi$
$632$	6.01113	0.239110
$633$	1.64060	0.0652079
$634$	−3.54728	−0.140880
$635$	0	0
$636$	−5.97649	−0.236983
$637$	17.0051	0.673768
$638$	14.8023	0.586030
$639$	−8.37030	−0.331124
$640$	0	0
$641$	−15.0405	−0.594063	−0.297031	−	0.954868i	$-0.595997\pi$
$641$	−15.0405	−0.594063	−0.297031	+	0.954868i	$0.595997\pi$
$642$	0.435291	0.0171795
$643$	−32.3618	−1.27622	−0.638112	−	0.769943i	$-0.720285\pi$
$643$	−32.3618	−1.27622	−0.638112	+	0.769943i	$0.720285\pi$
$644$	7.96808	0.313986
$645$	0	0
$646$	11.2615	0.443079
$647$	49.0042	1.92655	0.963276	−	0.268512i	$-0.0865319\pi$
$647$	49.0042	1.92655	0.963276	+	0.268512i	$0.0865319\pi$
$648$	7.56424	0.297151
$649$	10.9247	0.428833
$650$	0	0
$651$	2.19447	0.0860079
$652$	25.0725	0.981916
$653$	−31.1992	−1.22092	−0.610460	−	0.792047i	$-0.709016\pi$
$653$	−31.1992	−1.22092	−0.610460	+	0.792047i	$0.709016\pi$
$654$	−14.4271	−0.564143
$655$	0	0
$656$	−37.8715	−1.47864
$657$	31.0712	1.21220
$658$	33.6876	1.31328
$659$	−35.7237	−1.39160	−0.695799	−	0.718237i	$-0.744950\pi$
$659$	−35.7237	−1.39160	−0.695799	+	0.718237i	$0.744950\pi$
$660$	0	0
$661$	33.8677	1.31730	0.658651	−	0.752449i	$-0.271128\pi$
$661$	33.8677	1.31730	0.658651	+	0.752449i	$0.271128\pi$
$662$	−24.4754	−0.951262
$663$	−14.2817	−0.554657
$664$	2.21585	0.0859916
$665$	0	0
$666$	32.1461	1.24564
$667$	28.5400	1.10507
$668$	8.43788	0.326471
$669$	−3.57296	−0.138139
$670$	0	0
$671$	−4.51162	−0.174169
$672$	−6.20103	−0.239210
$673$	−21.1648	−0.815842	−0.407921	−	0.913017i	$-0.633746\pi$
$673$	−21.1648	−0.815842	−0.407921	+	0.913017i	$0.633746\pi$
$674$	60.5786	2.33340
$675$	0	0
$676$	5.22141	0.200824
$677$	34.5239	1.32686	0.663431	−	0.748238i	$-0.269100\pi$
$677$	34.5239	1.32686	0.663431	+	0.748238i	$0.269100\pi$
$678$	−2.61040	−0.100252
$679$	−18.6057	−0.714022
$680$	0	0
$681$	1.12925	0.0432729
$682$	4.20401	0.160980
$683$	−12.8022	−0.489862	−0.244931	−	0.969540i	$-0.578765\pi$
$683$	−12.8022	−0.489862	−0.244931	+	0.969540i	$0.578765\pi$
$684$	3.58603	0.137115
$685$	0	0
$686$	−34.4261	−1.31439
$687$	−11.8257	−0.451179
$688$	37.8759	1.44401
$689$	32.6267	1.24298
$690$	0	0
$691$	39.5406	1.50419	0.752097	−	0.659052i	$-0.229042\pi$
$691$	39.5406	1.50419	0.752097	+	0.659052i	$0.229042\pi$
$692$	28.7769	1.09394
$693$	4.53995	0.172458
$694$	32.9294	1.24998
$695$	0	0
$696$	−5.53106	−0.209654
$697$	47.7795	1.80978
$698$	−16.1006	−0.609418
$699$	9.91646	0.375075
$700$	0	0
$701$	22.4546	0.848097	0.424049	−	0.905639i	$-0.360609\pi$
$701$	22.4546	0.848097	0.424049	+	0.905639i	$0.360609\pi$
$702$	−24.0468	−0.907588
$703$	−6.56016	−0.247421
$704$	−2.10636	−0.0793865
$705$	0	0
$706$	−2.90899	−0.109481
$707$	−3.90301	−0.146788
$708$	8.22818	0.309234
$709$	−1.41244	−0.0530451	−0.0265226	−	0.999648i	$-0.508443\pi$
$709$	−1.41244	−0.0530451	−0.0265226	+	0.999648i	$0.508443\pi$
$710$	0	0
$711$	−13.3104	−0.499181
$712$	−11.3575	−0.425641
$713$	8.10564	0.303559
$714$	10.7381	0.401863
$715$	0	0
$716$	−10.6545	−0.398177
$717$	14.9985	0.560130
$718$	−66.4502	−2.47990
$719$	−39.0879	−1.45773	−0.728867	−	0.684655i	$-0.759953\pi$
$719$	−39.0879	−1.45773	−0.728867	+	0.684655i	$0.759953\pi$
$720$	0	0
$721$	10.3574	0.385731
$722$	−1.82669	−0.0679823
$723$	7.40002	0.275210
$724$	−1.56344	−0.0581049
$725$	0	0
$726$	−1.02918	−0.0381966
$727$	−9.42098	−0.349405	−0.174702	−	0.984621i	$-0.555896\pi$
$727$	−9.42098	−0.349405	−0.174702	+	0.984621i	$0.555896\pi$
$728$	8.43013	0.312442
$729$	−11.3379	−0.419922
$730$	0	0
$731$	−47.7850	−1.76739
$732$	−3.39801	−0.125594
$733$	−4.63361	−0.171146	−0.0855731	−	0.996332i	$-0.527272\pi$
$733$	−4.63361	−0.171146	−0.0855731	+	0.996332i	$0.527272\pi$
$734$	−9.13849	−0.337308
$735$	0	0
$736$	−22.9046	−0.844274
$737$	−14.7201	−0.542222
$738$	37.9772	1.39796
$739$	−13.1654	−0.484298	−0.242149	−	0.970239i	$-0.577852\pi$
$739$	−13.1654	−0.484298	−0.242149	+	0.970239i	$0.577852\pi$
$740$	0	0
$741$	2.31659	0.0851019
$742$	−24.5312	−0.900568
$743$	14.7689	0.541817	0.270909	−	0.962605i	$-0.412676\pi$
$743$	14.7689	0.541817	0.270909	+	0.962605i	$0.412676\pi$
$744$	−1.57087	−0.0575911
$745$	0	0
$746$	−62.3472	−2.28269
$747$	−4.90655	−0.179521
$748$	8.24132	0.301332
$749$	0.715792	0.0261545
$750$	0	0
$751$	35.0415	1.27868	0.639341	−	0.768924i	$-0.279207\pi$
$751$	35.0415	1.27868	0.639341	+	0.768924i	$0.279207\pi$
$752$	−53.2487	−1.94178
$753$	−14.7410	−0.537191
$754$	−60.8625	−2.21648
$755$	0	0
$756$	7.24333	0.263437
$757$	23.8005	0.865044	0.432522	−	0.901623i	$-0.357624\pi$
$757$	23.8005	0.865044	0.432522	+	0.901623i	$0.357624\pi$
$758$	−28.8158	−1.04664
$759$	−1.98435	−0.0720272
$760$	0	0
$761$	−39.0282	−1.41477	−0.707386	−	0.706827i	$-0.750126\pi$
$761$	−39.0282	−1.41477	−0.707386	+	0.706827i	$0.750126\pi$
$762$	1.10406	0.0399959
$763$	−23.7239	−0.858862
$764$	−20.9440	−0.757726
$765$	0	0
$766$	−40.9654	−1.48014
$767$	−44.9190	−1.62193
$768$	11.7997	0.425787
$769$	9.84757	0.355113	0.177556	−	0.984111i	$-0.443181\pi$
$769$	9.84757	0.355113	0.177556	+	0.984111i	$0.443181\pi$
$770$	0	0
$771$	−7.10564	−0.255903
$772$	−28.9776	−1.04292
$773$	42.9837	1.54602	0.773008	−	0.634396i	$-0.218751\pi$
$773$	42.9837	1.54602	0.773008	+	0.634396i	$0.218751\pi$
$774$	−37.9816	−1.36522
$775$	0	0
$776$	13.3186	0.478111
$777$	−6.25524	−0.224405
$778$	33.5175	1.20166
$779$	−7.75013	−0.277677
$780$	0	0
$781$	3.12026	0.111652
$782$	39.6630	1.41835
$783$	25.9441	0.927166
$784$	20.2099	0.721783
$785$	0	0
$786$	8.05552	0.287331
$787$	25.8709	0.922199	0.461099	−	0.887349i	$-0.347455\pi$
$787$	25.8709	0.922199	0.461099	+	0.887349i	$0.347455\pi$
$788$	−6.13313	−0.218484
$789$	11.2993	0.402266
$790$	0	0
$791$	−4.29254	−0.152625
$792$	−3.24985	−0.115479
$793$	18.5503	0.658741
$794$	−0.934320	−0.0331578
$795$	0	0
$796$	17.4611	0.618893
$797$	−3.62989	−0.128577	−0.0642886	−	0.997931i	$-0.520478\pi$
$797$	−3.62989	−0.128577	−0.0642886	+	0.997931i	$0.520478\pi$
$798$	−1.74178	−0.0616585
$799$	67.1796	2.37664
$800$	0	0
$801$	25.1489	0.888593
$802$	−34.6749	−1.22441
$803$	−11.5827	−0.408743
$804$	−11.0867	−0.390999
$805$	0	0
$806$	−17.2855	−0.608857
$807$	−11.3331	−0.398945
$808$	2.79391	0.0982894
$809$	−13.1327	−0.461720	−0.230860	−	0.972987i	$-0.574154\pi$
$809$	−13.1327	−0.461720	−0.230860	+	0.972987i	$0.574154\pi$
$810$	0	0
$811$	9.40230	0.330160	0.165080	−	0.986280i	$-0.447212\pi$
$811$	9.40230	0.330160	0.165080	+	0.986280i	$0.447212\pi$
$812$	18.3329	0.643358
$813$	8.76595	0.307435
$814$	−11.9834	−0.420017
$815$	0	0
$816$	−16.9733	−0.594183
$817$	7.75102	0.271174
$818$	−2.01098	−0.0703124
$819$	−18.6668	−0.652272
$820$	0	0
$821$	−21.1701	−0.738843	−0.369422	−	0.929262i	$-0.620444\pi$
$821$	−21.1701	−0.738843	−0.369422	+	0.929262i	$0.620444\pi$
$822$	−11.5755	−0.403743
$823$	−25.2277	−0.879384	−0.439692	−	0.898149i	$-0.644912\pi$
$823$	−25.2277	−0.879384	−0.439692	+	0.898149i	$0.644912\pi$
$824$	−7.41422	−0.258286
$825$	0	0
$826$	33.7735	1.17513
$827$	39.4460	1.37167	0.685836	−	0.727756i	$-0.259437\pi$
$827$	39.4460	1.37167	0.685836	+	0.727756i	$0.259437\pi$
$828$	12.6300	0.438922
$829$	−36.5068	−1.26793	−0.633967	−	0.773360i	$-0.718574\pi$
$829$	−36.5068	−1.26793	−0.633967	+	0.773360i	$0.718574\pi$
$830$	0	0
$831$	−1.06604	−0.0369806
$832$	8.66069	0.300255
$833$	−25.4972	−0.883427
$834$	0.932366	0.0322852
$835$	0	0
$836$	−1.33679	−0.0462339
$837$	7.36838	0.254688
$838$	34.3288	1.18587
$839$	42.4670	1.46612	0.733061	−	0.680163i	$-0.238091\pi$
$839$	42.4670	1.46612	0.733061	+	0.680163i	$0.238091\pi$
$840$	0	0
$841$	36.6645	1.26429
$842$	45.2321	1.55880
$843$	2.58776	0.0891273
$844$	3.89258	0.133988
$845$	0	0
$846$	53.3973	1.83584
$847$	−1.69239	−0.0581513
$848$	38.7755	1.33156
$849$	0.663690	0.0227778
$850$	0	0
$851$	−23.1048	−0.792023
$852$	2.35009	0.0805126
$853$	−21.5088	−0.736446	−0.368223	−	0.929738i	$-0.620034\pi$
$853$	−21.5088	−0.736446	−0.368223	+	0.929738i	$0.620034\pi$
$854$	−13.9475	−0.477275
$855$	0	0
$856$	−0.512389	−0.0175131
$857$	12.6370	0.431670	0.215835	−	0.976430i	$-0.430753\pi$
$857$	12.6370	0.431670	0.215835	+	0.976430i	$0.430753\pi$
$858$	4.23168	0.144467
$859$	−3.39720	−0.115911	−0.0579555	−	0.998319i	$-0.518458\pi$
$859$	−3.39720	−0.115911	−0.0579555	+	0.998319i	$0.518458\pi$
$860$	0	0
$861$	−7.38990	−0.251847
$862$	58.8167	2.00330
$863$	10.9308	0.372088	0.186044	−	0.982541i	$-0.440433\pi$
$863$	10.9308	0.372088	0.186044	+	0.982541i	$0.440433\pi$
$864$	−20.8212	−0.708353
$865$	0	0
$866$	−27.9561	−0.949987
$867$	11.8358	0.401963
$868$	5.20672	0.176727
$869$	4.96184	0.168319
$870$	0	0
$871$	60.5243	2.05079
$872$	16.9824	0.575096
$873$	−29.4914	−0.998133
$874$	−6.43359	−0.217619
$875$	0	0
$876$	−8.72371	−0.294747
$877$	−31.7230	−1.07121	−0.535605	−	0.844468i	$-0.679917\pi$
$877$	−31.7230	−1.07121	−0.535605	+	0.844468i	$0.679917\pi$
$878$	−47.8250	−1.61402
$879$	6.34439	0.213991
$880$	0	0
$881$	−6.45839	−0.217589	−0.108794	−	0.994064i	$-0.534699\pi$
$881$	−6.45839	−0.217589	−0.108794	+	0.994064i	$0.534699\pi$
$882$	−20.2663	−0.682403
$883$	48.5242	1.63297	0.816485	−	0.577367i	$-0.195920\pi$
$883$	48.5242	1.63297	0.816485	+	0.577367i	$0.195920\pi$
$884$	−33.8857	−1.13970
$885$	0	0
$886$	39.3796	1.32298
$887$	44.0097	1.47770	0.738851	−	0.673869i	$-0.235369\pi$
$887$	44.0097	1.47770	0.738851	+	0.673869i	$0.235369\pi$
$888$	4.47772	0.150262
$889$	1.81552	0.0608905
$890$	0	0
$891$	6.24383	0.209176
$892$	−8.47742	−0.283845
$893$	−10.8969	−0.364652
$894$	−10.8244	−0.362022
$895$	0	0
$896$	15.5005	0.517835
$897$	8.15900	0.272421
$898$	−21.5531	−0.719236
$899$	18.6494	0.621991
$900$	0	0
$901$	−48.9199	−1.62976
$902$	−14.1571	−0.471379
$903$	7.39076	0.245949
$904$	3.07275	0.102198
$905$	0	0
$906$	−13.6301	−0.452829
$907$	2.05084	0.0680971	0.0340485	−	0.999420i	$-0.489160\pi$
$907$	2.05084	0.0680971	0.0340485	+	0.999420i	$0.489160\pi$
$908$	2.67932	0.0889164
$909$	−6.18655	−0.205195
$910$	0	0
$911$	−16.8417	−0.557990	−0.278995	−	0.960293i	$-0.590001\pi$
$911$	−16.8417	−0.557990	−0.278995	+	0.960293i	$0.590001\pi$
$912$	2.75317	0.0911666
$913$	1.82905	0.0605328
$914$	−28.6539	−0.947788
$915$	0	0
$916$	−28.0584	−0.927075
$917$	13.2465	0.437438
$918$	36.0554	1.19001
$919$	8.01221	0.264299	0.132149	−	0.991230i	$-0.457812\pi$
$919$	8.01221	0.264299	0.132149	+	0.991230i	$0.457812\pi$
$920$	0	0
$921$	−2.45224	−0.0808039
$922$	−49.7683	−1.63903
$923$	−12.8295	−0.422289
$924$	−1.27466	−0.0419332
$925$	0	0
$926$	−32.7641	−1.07670
$927$	16.4173	0.539214
$928$	−52.6986	−1.72992
$929$	16.7897	0.550853	0.275426	−	0.961322i	$-0.411181\pi$
$929$	16.7897	0.550853	0.275426	+	0.961322i	$0.411181\pi$
$930$	0	0
$931$	4.13581	0.135546
$932$	23.5284	0.770698
$933$	4.29168	0.140503
$934$	52.9922	1.73396
$935$	0	0
$936$	13.3624	0.436763
$937$	−21.8471	−0.713713	−0.356856	−	0.934159i	$-0.616151\pi$
$937$	−21.8471	−0.713713	−0.356856	+	0.934159i	$0.616151\pi$
$938$	−45.5068	−1.48585
$939$	9.89661	0.322964
$940$	0	0
$941$	17.8421	0.581635	0.290817	−	0.956779i	$-0.406073\pi$
$941$	17.8421	0.581635	0.290817	+	0.956779i	$0.406073\pi$
$942$	2.05749	0.0670368
$943$	−27.2959	−0.888877
$944$	−53.3845	−1.73752
$945$	0	0
$946$	14.1587	0.460339
$947$	−41.9736	−1.36396	−0.681980	−	0.731371i	$-0.738881\pi$
$947$	−41.9736	−1.36396	−0.681980	+	0.731371i	$0.738881\pi$
$948$	3.73711	0.121376
$949$	47.6242	1.54595
$950$	0	0
$951$	1.09411	0.0354788
$952$	−12.6400	−0.409665
$953$	8.06945	0.261395	0.130697	−	0.991422i	$-0.458278\pi$
$953$	8.06945	0.261395	0.130697	+	0.991422i	$0.458278\pi$
$954$	−38.8837	−1.25891
$955$	0	0
$956$	35.5864	1.15094
$957$	−4.56556	−0.147584
$958$	−62.2711	−2.01189
$959$	−19.0348	−0.614666
$960$	0	0
$961$	−25.7034	−0.829142
$962$	49.2718	1.58859
$963$	1.13458	0.0365614
$964$	17.5577	0.565496
$965$	0	0
$966$	−6.13455	−0.197376
$967$	3.86360	0.124245	0.0621225	−	0.998069i	$-0.480213\pi$
$967$	3.86360	0.124245	0.0621225	+	0.998069i	$0.480213\pi$
$968$	1.21147	0.0389382
$969$	−3.47345	−0.111583
$970$	0	0
$971$	42.7787	1.37283	0.686417	−	0.727208i	$-0.259182\pi$
$971$	42.7787	1.37283	0.686417	+	0.727208i	$0.259182\pi$
$972$	17.5425	0.562675
$973$	1.53318	0.0491516
$974$	11.6607	0.373631
$975$	0	0
$976$	22.0463	0.705686
$977$	−54.5146	−1.74408	−0.872039	−	0.489437i	$-0.837202\pi$
$977$	−54.5146	−1.74408	−0.872039	+	0.489437i	$0.837202\pi$
$978$	−19.3031	−0.617246
$979$	−9.37496	−0.299625
$980$	0	0
$981$	−37.6040	−1.20060
$982$	57.4590	1.83359
$983$	−33.5422	−1.06983	−0.534916	−	0.844905i	$-0.679657\pi$
$983$	−33.5422	−1.06983	−0.534916	+	0.844905i	$0.679657\pi$
$984$	5.28995	0.168637
$985$	0	0
$986$	91.2562	2.90619
$987$	−10.3905	−0.330732
$988$	5.49647	0.174866
$989$	27.2991	0.868060
$990$	0	0
$991$	31.7767	1.00942	0.504710	−	0.863289i	$-0.331600\pi$
$991$	31.7767	1.00942	0.504710	+	0.863289i	$0.331600\pi$
$992$	−14.9669	−0.475200
$993$	7.54907	0.239562
$994$	9.64621	0.305959
$995$	0	0
$996$	1.37759	0.0436505
$997$	47.5668	1.50646	0.753228	−	0.657759i	$-0.228496\pi$
$997$	47.5668	1.50646	0.753228	+	0.657759i	$0.228496\pi$
$998$	58.3933	1.84841
$999$	−21.0033	−0.664515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.h.1.2		5
5.4	even	2		209.2.a.c.1.4	✓	5
15.14	odd	2		1881.2.a.k.1.2		5
20.19	odd	2		3344.2.a.t.1.4		5
55.54	odd	2		2299.2.a.n.1.2		5
95.94	odd	2		3971.2.a.h.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.4	✓	5	5.4	even	2
1881.2.a.k.1.2		5	15.14	odd	2
2299.2.a.n.1.2		5	55.54	odd	2
3344.2.a.t.1.4		5	20.19	odd	2
3971.2.a.h.1.2		5	95.94	odd	2
5225.2.a.h.1.2		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82669 q^{2} +0.563416 q^{3} +1.33679 q^{4} -1.02918 q^{6} -1.69239 q^{7} +1.21147 q^{8} -2.68256 q^{9} +1.00000 q^{11} +0.753170 q^{12} -4.11168 q^{13} +3.09147 q^{14} -4.88657 q^{16} +6.16499 q^{17} +4.90021 q^{18} -1.00000 q^{19} -0.953520 q^{21} -1.82669 q^{22} -3.52199 q^{23} +0.682563 q^{24} +7.51076 q^{26} -3.20164 q^{27} -2.26238 q^{28} -8.10336 q^{29} -2.30144 q^{31} +6.50330 q^{32} +0.563416 q^{33} -11.2615 q^{34} -3.58603 q^{36} +6.56016 q^{37} +1.82669 q^{38} -2.31659 q^{39} +7.75013 q^{41} +1.74178 q^{42} -7.75102 q^{43} +1.33679 q^{44} +6.43359 q^{46} +10.8969 q^{47} -2.75317 q^{48} -4.13581 q^{49} +3.47345 q^{51} -5.49647 q^{52} -7.93511 q^{53} +5.84841 q^{54} -2.05029 q^{56} -0.563416 q^{57} +14.8023 q^{58} +10.9247 q^{59} -4.51162 q^{61} +4.20401 q^{62} +4.53995 q^{63} -2.10636 q^{64} -1.02918 q^{66} -14.7201 q^{67} +8.24132 q^{68} -1.98435 q^{69} +3.12026 q^{71} -3.24985 q^{72} -11.5827 q^{73} -11.9834 q^{74} -1.33679 q^{76} -1.69239 q^{77} +4.23168 q^{78} +4.96184 q^{79} +6.24383 q^{81} -14.1571 q^{82} +1.82905 q^{83} -1.27466 q^{84} +14.1587 q^{86} -4.56556 q^{87} +1.21147 q^{88} -9.37496 q^{89} +6.95858 q^{91} -4.70818 q^{92} -1.29666 q^{93} -19.9053 q^{94} +3.66406 q^{96} +10.9937 q^{97} +7.55484 q^{98} -2.68256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 + 5 * q^11 - 6 * q^12 - 4 * q^13 - 14 * q^14 + 8 * q^16 + 4 * q^17 + 20 * q^18 - 5 * q^19 + 10 * q^21 - 2 * q^22 - 3 * q^23 - 14 * q^24 - 6 * q^26 + 11 * q^27 + 10 * q^28 + 10 * q^29 + 11 * q^31 - 14 * q^32 - q^33 - 4 * q^34 - 26 * q^36 - q^37 + 2 * q^38 + 2 * q^39 + 2 * q^41 + 16 * q^42 - 20 * q^43 + 6 * q^44 - 4 * q^46 + 20 * q^47 - 4 * q^48 + 3 * q^49 + 24 * q^51 - 6 * q^52 + 14 * q^53 + 16 * q^54 - 38 * q^56 + q^57 + 6 * q^58 + 3 * q^59 - 10 * q^61 + 6 * q^62 - 24 * q^63 - 2 * q^66 - 9 * q^67 - 24 * q^68 - 5 * q^69 + 23 * q^71 + 12 * q^72 + 8 * q^74 - 6 * q^76 - 6 * q^77 + 22 * q^78 + 44 * q^79 + q^81 + 30 * q^82 + 14 * q^83 + 14 * q^84 + 52 * q^86 - 28 * q^87 - 6 * q^88 - 27 * q^89 + 24 * q^91 - 58 * q^92 + 27 * q^93 - 8 * q^94 + 50 * q^96 - 15 * q^97 + 10 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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