LMFDB - Embedded newform 7203.2.a.k.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7203,2,Mod(1,7203)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7203, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7203.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7203 = 3 \cdot 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7203.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,-6,24,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.5162445759$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	7203.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.839793 q^{2} +1.00000 q^{3} -1.29475 q^{4} +1.55513 q^{5} -0.839793 q^{6} +2.76691 q^{8} +1.00000 q^{9} -1.30599 q^{10} -1.97493 q^{11} -1.29475 q^{12} -3.72295 q^{13} +1.55513 q^{15} +0.265864 q^{16} +4.12576 q^{17} -0.839793 q^{18} -3.76986 q^{19} -2.01350 q^{20} +1.65853 q^{22} -2.95485 q^{23} +2.76691 q^{24} -2.58158 q^{25} +3.12651 q^{26} +1.00000 q^{27} -2.16007 q^{29} -1.30599 q^{30} +0.843379 q^{31} -5.75708 q^{32} -1.97493 q^{33} -3.46479 q^{34} -1.29475 q^{36} +10.3415 q^{37} +3.16591 q^{38} -3.72295 q^{39} +4.30289 q^{40} +10.4271 q^{41} +7.92726 q^{43} +2.55703 q^{44} +1.55513 q^{45} +2.48147 q^{46} -1.66788 q^{47} +0.265864 q^{48} +2.16799 q^{50} +4.12576 q^{51} +4.82028 q^{52} -4.92947 q^{53} -0.839793 q^{54} -3.07126 q^{55} -3.76986 q^{57} +1.81402 q^{58} -8.20223 q^{59} -2.01350 q^{60} +2.64053 q^{61} -0.708264 q^{62} +4.30303 q^{64} -5.78966 q^{65} +1.65853 q^{66} +4.26159 q^{67} -5.34181 q^{68} -2.95485 q^{69} -14.6555 q^{71} +2.76691 q^{72} -4.71061 q^{73} -8.68474 q^{74} -2.58158 q^{75} +4.88102 q^{76} +3.12651 q^{78} -14.2452 q^{79} +0.413452 q^{80} +1.00000 q^{81} -8.75661 q^{82} -2.86098 q^{83} +6.41608 q^{85} -6.65726 q^{86} -2.16007 q^{87} -5.46444 q^{88} -17.0826 q^{89} -1.30599 q^{90} +3.82579 q^{92} +0.843379 q^{93} +1.40067 q^{94} -5.86262 q^{95} -5.75708 q^{96} -5.74864 q^{97} -1.97493 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{2} + 24 q^{3} + 18 q^{4} - 6 q^{6} - 18 q^{8} + 24 q^{9} - 12 q^{10} - 12 q^{11} + 18 q^{12} - 14 q^{13} + 6 q^{16} + 2 q^{17} - 6 q^{18} - 26 q^{19} + 6 q^{20} - 24 q^{22} - 24 q^{23} - 18 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) $ 24 * q - 6 * q^2 + 24 * q^3 + 18 * q^4 - 6 * q^6 - 18 * q^8 + 24 * q^9 - 12 * q^10 - 12 * q^11 + 18 * q^12 - 14 * q^13 + 6 * q^16 + 2 * q^17 - 6 * q^18 - 26 * q^19 + 6 * q^20 - 24 * q^22 - 24 * q^23 - 18 * q^24 - 13 * q^26 + 24 * q^27 - 36 * q^29 - 12 * q^30 - 36 * q^31 - 42 * q^32 - 12 * q^33 - 4 * q^34 + 18 * q^36 - 50 * q^37 + 38 * q^38 - 14 * q^39 - 28 * q^40 + 25 * q^41 - 36 * q^43 - 30 * q^44 - 30 * q^46 + 6 * q^47 + 6 * q^48 - 6 * q^50 + 2 * q^51 - 41 * q^52 - 63 * q^53 - 6 * q^54 - 44 * q^55 - 26 * q^57 - 3 * q^58 - 41 * q^59 + 6 * q^60 - 38 * q^61 + 22 * q^62 - 6 * q^64 - 14 * q^65 - 24 * q^66 - 34 * q^67 + 22 * q^68 - 24 * q^69 - 32 * q^71 - 18 * q^72 - 13 * q^73 - 43 * q^74 - 105 * q^76 - 13 * q^78 - 34 * q^79 + 55 * q^80 + 24 * q^81 + 28 * q^82 + 42 * q^83 - 92 * q^85 + 40 * q^86 - 36 * q^87 - 10 * q^88 + 32 * q^89 - 12 * q^90 - 14 * q^92 - 36 * q^93 - 64 * q^94 - 38 * q^95 - 42 * q^96 - 62 * q^97 - 12 * 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 $

$2$	−0.839793	−0.593824	−0.296912	−	0.954905i	$-0.595957\pi$
$2$	−0.839793	−0.593824	−0.296912	+	0.954905i	$0.595957\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.29475	−0.647373
$5$	1.55513	0.695474	0.347737	−	0.937592i	$-0.386950\pi$
$5$	1.55513	0.695474	0.347737	+	0.937592i	$0.386950\pi$
$6$	−0.839793	−0.342844
$7$	0	0
$7$	0	0
$8$	2.76691	0.978249
$9$	1.00000	0.333333
$10$	−1.30599	−0.412989
$11$	−1.97493	−0.595463	−0.297731	−	0.954650i	$-0.596230\pi$
$11$	−1.97493	−0.595463	−0.297731	+	0.954650i	$0.596230\pi$
$12$	−1.29475	−0.373761
$13$	−3.72295	−1.03256	−0.516280	−	0.856420i	$-0.672684\pi$
$13$	−3.72295	−1.03256	−0.516280	+	0.856420i	$0.672684\pi$
$14$	0	0
$15$	1.55513	0.401532
$16$	0.265864	0.0664659
$17$	4.12576	1.00064	0.500322	−	0.865840i	$-0.333215\pi$
$17$	4.12576	1.00064	0.500322	+	0.865840i	$0.333215\pi$
$18$	−0.839793	−0.197941
$19$	−3.76986	−0.864866	−0.432433	−	0.901666i	$-0.642345\pi$
$19$	−3.76986	−0.864866	−0.432433	+	0.901666i	$0.642345\pi$
$20$	−2.01350	−0.450232
$21$	0	0
$22$	1.65853	0.353600
$23$	−2.95485	−0.616129	−0.308065	−	0.951365i	$-0.599681\pi$
$23$	−2.95485	−0.616129	−0.308065	+	0.951365i	$0.599681\pi$
$24$	2.76691	0.564793
$25$	−2.58158	−0.516315
$26$	3.12651	0.613159
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.16007	−0.401115	−0.200558	−	0.979682i	$-0.564275\pi$
$29$	−2.16007	−0.401115	−0.200558	+	0.979682i	$0.564275\pi$
$30$	−1.30599	−0.238439
$31$	0.843379	0.151475	0.0757377	−	0.997128i	$-0.475869\pi$
$31$	0.843379	0.151475	0.0757377	+	0.997128i	$0.475869\pi$
$32$	−5.75708	−1.01772
$33$	−1.97493	−0.343791
$34$	−3.46479	−0.594206
$35$	0	0
$36$	−1.29475	−0.215791
$37$	10.3415	1.70014	0.850068	−	0.526674i	$-0.176561\pi$
$37$	10.3415	1.70014	0.850068	+	0.526674i	$0.176561\pi$
$38$	3.16591	0.513578
$39$	−3.72295	−0.596149
$40$	4.30289	0.680347
$41$	10.4271	1.62844	0.814220	−	0.580557i	$-0.197165\pi$
$41$	10.4271	1.62844	0.814220	+	0.580557i	$0.197165\pi$
$42$	0	0
$43$	7.92726	1.20890	0.604448	−	0.796645i	$-0.293394\pi$
$43$	7.92726	1.20890	0.604448	+	0.796645i	$0.293394\pi$
$44$	2.55703	0.385487
$45$	1.55513	0.231825
$46$	2.48147	0.365872
$47$	−1.66788	−0.243285	−0.121642	−	0.992574i	$-0.538816\pi$
$47$	−1.66788	−0.243285	−0.121642	+	0.992574i	$0.538816\pi$
$48$	0.265864	0.0383741
$49$	0	0
$50$	2.16799	0.306600
$51$	4.12576	0.577722
$52$	4.82028	0.668452
$53$	−4.92947	−0.677114	−0.338557	−	0.940946i	$-0.609939\pi$
$53$	−4.92947	−0.677114	−0.338557	+	0.940946i	$0.609939\pi$
$54$	−0.839793	−0.114281
$55$	−3.07126	−0.414129
$56$	0	0
$57$	−3.76986	−0.499331
$58$	1.81402	0.238192
$59$	−8.20223	−1.06784	−0.533920	−	0.845535i	$-0.679282\pi$
$59$	−8.20223	−1.06784	−0.533920	+	0.845535i	$0.679282\pi$
$60$	−2.01350	−0.259941
$61$	2.64053	0.338085	0.169042	−	0.985609i	$-0.445933\pi$
$61$	2.64053	0.338085	0.169042	+	0.985609i	$0.445933\pi$
$62$	−0.708264	−0.0899496
$63$	0	0
$64$	4.30303	0.537879
$65$	−5.78966	−0.718119
$66$	1.65853	0.204151
$67$	4.26159	0.520636	0.260318	−	0.965523i	$-0.416173\pi$
$67$	4.26159	0.520636	0.260318	+	0.965523i	$0.416173\pi$
$68$	−5.34181	−0.647790
$69$	−2.95485	−0.355723
$70$	0	0
$71$	−14.6555	−1.73928	−0.869642	−	0.493683i	$-0.835650\pi$
$71$	−14.6555	−1.73928	−0.869642	+	0.493683i	$0.835650\pi$
$72$	2.76691	0.326083
$73$	−4.71061	−0.551335	−0.275667	−	0.961253i	$-0.588899\pi$
$73$	−4.71061	−0.551335	−0.275667	+	0.961253i	$0.588899\pi$
$74$	−8.68474	−1.00958
$75$	−2.58158	−0.298095
$76$	4.88102	0.559891
$77$	0	0
$78$	3.12651	0.354007
$79$	−14.2452	−1.60271	−0.801357	−	0.598186i	$-0.795888\pi$
$79$	−14.2452	−1.60271	−0.801357	+	0.598186i	$0.795888\pi$
$80$	0.413452	0.0462253
$81$	1.00000	0.111111
$82$	−8.75661	−0.967006
$83$	−2.86098	−0.314034	−0.157017	−	0.987596i	$-0.550188\pi$
$83$	−2.86098	−0.314034	−0.157017	+	0.987596i	$0.550188\pi$
$84$	0	0
$85$	6.41608	0.695922
$86$	−6.65726	−0.717871
$87$	−2.16007	−0.231584
$88$	−5.46444	−0.582511
$89$	−17.0826	−1.81075	−0.905377	−	0.424608i	$-0.860412\pi$
$89$	−17.0826	−1.81075	−0.905377	+	0.424608i	$0.860412\pi$
$90$	−1.30599	−0.137663
$91$	0	0
$92$	3.82579	0.398866
$93$	0.843379	0.0874543
$94$	1.40067	0.144468
$95$	−5.86262	−0.601492
$96$	−5.75708	−0.587580
$97$	−5.74864	−0.583686	−0.291843	−	0.956466i	$-0.594268\pi$
$97$	−5.74864	−0.583686	−0.291843	+	0.956466i	$0.594268\pi$
$98$	0	0
$99$	−1.97493	−0.198488
$100$	3.34249	0.334249
$101$	−15.4205	−1.53440	−0.767200	−	0.641408i	$-0.778351\pi$
$101$	−15.4205	−1.53440	−0.767200	+	0.641408i	$0.778351\pi$
$102$	−3.46479	−0.343065
$103$	10.3174	1.01660	0.508301	−	0.861179i	$-0.330274\pi$
$103$	10.3174	1.01660	0.508301	+	0.861179i	$0.330274\pi$
$104$	−10.3011	−1.01010
$105$	0	0
$106$	4.13973	0.402086
$107$	14.8735	1.43788	0.718939	−	0.695074i	$-0.244628\pi$
$107$	14.8735	1.43788	0.718939	+	0.695074i	$0.244628\pi$
$108$	−1.29475	−0.124587
$109$	−12.0261	−1.15189	−0.575944	−	0.817489i	$-0.695365\pi$
$109$	−12.0261	−1.15189	−0.575944	+	0.817489i	$0.695365\pi$
$110$	2.57923	0.245920
$111$	10.3415	0.981574
$112$	0	0
$113$	9.52106	0.895666	0.447833	−	0.894117i	$-0.352196\pi$
$113$	9.52106	0.895666	0.447833	+	0.894117i	$0.352196\pi$
$114$	3.16591	0.296514
$115$	−4.59517	−0.428502
$116$	2.79675	0.259672
$117$	−3.72295	−0.344187
$118$	6.88818	0.634108
$119$	0	0
$120$	4.30289	0.392799
$121$	−7.09967	−0.645424
$122$	−2.21750	−0.200763
$123$	10.4271	0.940180
$124$	−1.09196	−0.0980611
$125$	−11.7903	−1.05456
$126$	0	0
$127$	2.68618	0.238360	0.119180	−	0.992873i	$-0.461973\pi$
$127$	2.68618	0.238360	0.119180	+	0.992873i	$0.461973\pi$
$128$	7.90051	0.698313
$129$	7.92726	0.697956
$130$	4.86212	0.426436
$131$	18.1812	1.58850	0.794250	−	0.607591i	$-0.207864\pi$
$131$	18.1812	1.58850	0.794250	+	0.607591i	$0.207864\pi$
$132$	2.55703	0.222561
$133$	0	0
$134$	−3.57886	−0.309166
$135$	1.55513	0.133844
$136$	11.4156	0.978879
$137$	−10.1300	−0.865463	−0.432732	−	0.901523i	$-0.642450\pi$
$137$	−10.1300	−0.865463	−0.432732	+	0.901523i	$0.642450\pi$
$138$	2.48147	0.211236
$139$	−11.7778	−0.998978	−0.499489	−	0.866320i	$-0.666479\pi$
$139$	−11.7778	−0.998978	−0.499489	+	0.866320i	$0.666479\pi$
$140$	0	0
$141$	−1.66788	−0.140460
$142$	12.3076	1.03283
$143$	7.35255	0.614851
$144$	0.265864	0.0221553
$145$	−3.35919	−0.278966
$146$	3.95594	0.327396
$147$	0	0
$148$	−13.3897	−1.10062
$149$	15.5676	1.27535	0.637673	−	0.770307i	$-0.279897\pi$
$149$	15.5676	1.27535	0.637673	+	0.770307i	$0.279897\pi$
$150$	2.16799	0.177016
$151$	9.07107	0.738194	0.369097	−	0.929391i	$-0.379667\pi$
$151$	9.07107	0.738194	0.369097	+	0.929391i	$0.379667\pi$
$152$	−10.4309	−0.846055
$153$	4.12576	0.333548
$154$	0	0
$155$	1.31156	0.105347
$156$	4.82028	0.385931
$157$	−21.7234	−1.73372	−0.866858	−	0.498554i	$-0.833864\pi$
$157$	−21.7234	−1.73372	−0.866858	+	0.498554i	$0.833864\pi$
$158$	11.9631	0.951729
$159$	−4.92947	−0.390932
$160$	−8.95300	−0.707797
$161$	0	0
$162$	−0.839793	−0.0659804
$163$	22.3661	1.75185	0.875923	−	0.482452i	$-0.160254\pi$
$163$	22.3661	1.75185	0.875923	+	0.482452i	$0.160254\pi$
$164$	−13.5005	−1.05421
$165$	−3.07126	−0.239098
$166$	2.40263	0.186481
$167$	−8.97216	−0.694287	−0.347143	−	0.937812i	$-0.612848\pi$
$167$	−8.97216	−0.694287	−0.347143	+	0.937812i	$0.612848\pi$
$168$	0	0
$169$	0.860357	0.0661813
$170$	−5.38819	−0.413255
$171$	−3.76986	−0.288289
$172$	−10.2638	−0.782607
$173$	6.73401	0.511977	0.255989	−	0.966680i	$-0.417599\pi$
$173$	6.73401	0.511977	0.255989	+	0.966680i	$0.417599\pi$
$174$	1.81402	0.137520
$175$	0	0
$176$	−0.525061	−0.0395779
$177$	−8.20223	−0.616517
$178$	14.3459	1.07527
$179$	−1.59942	−0.119546	−0.0597731	−	0.998212i	$-0.519038\pi$
$179$	−1.59942	−0.119546	−0.0597731	+	0.998212i	$0.519038\pi$
$180$	−2.01350	−0.150077
$181$	−18.4398	−1.37062	−0.685308	−	0.728253i	$-0.740333\pi$
$181$	−18.4398	−1.37062	−0.685308	+	0.728253i	$0.740333\pi$
$182$	0	0
$183$	2.64053	0.195193
$184$	−8.17580	−0.602728
$185$	16.0824	1.18240
$186$	−0.708264	−0.0519324
$187$	−8.14807	−0.595846
$188$	2.15948	0.157496
$189$	0	0
$190$	4.92339	0.357180
$191$	4.09123	0.296031	0.148016	−	0.988985i	$-0.452711\pi$
$191$	4.09123	0.296031	0.148016	+	0.988985i	$0.452711\pi$
$192$	4.30303	0.310545
$193$	−16.5235	−1.18939	−0.594693	−	0.803953i	$-0.702726\pi$
$193$	−16.5235	−1.18939	−0.594693	+	0.803953i	$0.702726\pi$
$194$	4.82767	0.346606
$195$	−5.78966	−0.414606
$196$	0	0
$197$	−11.1608	−0.795175	−0.397587	−	0.917564i	$-0.630152\pi$
$197$	−11.1608	−0.795175	−0.397587	+	0.917564i	$0.630152\pi$
$198$	1.65853	0.117867
$199$	7.20432	0.510701	0.255350	−	0.966849i	$-0.417809\pi$
$199$	7.20432	0.510701	0.255350	+	0.966849i	$0.417809\pi$
$200$	−7.14298	−0.505085
$201$	4.26159	0.300589
$202$	12.9501	0.911163
$203$	0	0
$204$	−5.34181	−0.374002
$205$	16.2155	1.13254
$206$	−8.66447	−0.603682
$207$	−2.95485	−0.205376
$208$	−0.989797	−0.0686300
$209$	7.44520	0.514995
$210$	0	0
$211$	−2.48747	−0.171244	−0.0856222	−	0.996328i	$-0.527288\pi$
$211$	−2.48747	−0.171244	−0.0856222	+	0.996328i	$0.527288\pi$
$212$	6.38241	0.438346
$213$	−14.6555	−1.00418
$214$	−12.4907	−0.853845
$215$	12.3279	0.840756
$216$	2.76691	0.188264
$217$	0	0
$218$	10.0994	0.684019
$219$	−4.71061	−0.318313
$220$	3.97651	0.268096
$221$	−15.3600	−1.03323
$222$	−8.68474	−0.582882
$223$	−1.38300	−0.0926127	−0.0463063	−	0.998927i	$-0.514745\pi$
$223$	−1.38300	−0.0926127	−0.0463063	+	0.998927i	$0.514745\pi$
$224$	0	0
$225$	−2.58158	−0.172105
$226$	−7.99572	−0.531867
$227$	13.4520	0.892839	0.446419	−	0.894824i	$-0.352699\pi$
$227$	13.4520	0.892839	0.446419	+	0.894824i	$0.352699\pi$
$228$	4.88102	0.323253
$229$	23.2176	1.53426	0.767130	−	0.641491i	$-0.221684\pi$
$229$	23.2176	1.53426	0.767130	+	0.641491i	$0.221684\pi$
$230$	3.85900	0.254455
$231$	0	0
$232$	−5.97672	−0.392391
$233$	−24.6966	−1.61793	−0.808963	−	0.587859i	$-0.799971\pi$
$233$	−24.6966	−1.61793	−0.808963	+	0.587859i	$0.799971\pi$
$234$	3.12651	0.204386
$235$	−2.59376	−0.169198
$236$	10.6198	0.691291
$237$	−14.2452	−0.925327
$238$	0	0
$239$	−3.80550	−0.246157	−0.123079	−	0.992397i	$-0.539277\pi$
$239$	−3.80550	−0.246157	−0.123079	+	0.992397i	$0.539277\pi$
$240$	0.413452	0.0266882
$241$	−12.4874	−0.804386	−0.402193	−	0.915555i	$-0.631752\pi$
$241$	−12.4874	−0.804386	−0.402193	+	0.915555i	$0.631752\pi$
$242$	5.96225	0.383268
$243$	1.00000	0.0641500
$244$	−3.41881	−0.218867
$245$	0	0
$246$	−8.75661	−0.558301
$247$	14.0350	0.893027
$248$	2.33355	0.148181
$249$	−2.86098	−0.181307
$250$	9.90144	0.626222
$251$	−4.71580	−0.297659	−0.148829	−	0.988863i	$-0.547551\pi$
$251$	−4.71580	−0.297659	−0.148829	+	0.988863i	$0.547551\pi$
$252$	0	0
$253$	5.83562	0.366882
$254$	−2.25584	−0.141544
$255$	6.41608	0.401791
$256$	−15.2409	−0.952554
$257$	5.11230	0.318896	0.159448	−	0.987206i	$-0.449029\pi$
$257$	5.11230	0.318896	0.159448	+	0.987206i	$0.449029\pi$
$258$	−6.65726	−0.414463
$259$	0	0
$260$	7.49615	0.464891
$261$	−2.16007	−0.133705
$262$	−15.2685	−0.943289
$263$	−21.9571	−1.35393	−0.676966	−	0.736014i	$-0.736706\pi$
$263$	−21.9571	−1.35393	−0.676966	+	0.736014i	$0.736706\pi$
$264$	−5.46444	−0.336313
$265$	−7.66595	−0.470916
$266$	0	0
$267$	−17.0826	−1.04544
$268$	−5.51768	−0.337046
$269$	−27.2105	−1.65906	−0.829528	−	0.558465i	$-0.811390\pi$
$269$	−27.2105	−1.65906	−0.829528	+	0.558465i	$0.811390\pi$
$270$	−1.30599	−0.0794798
$271$	−2.48901	−0.151196	−0.0755982	−	0.997138i	$-0.524087\pi$
$271$	−2.48901	−0.151196	−0.0755982	+	0.997138i	$0.524087\pi$
$272$	1.09689	0.0665087
$273$	0	0
$274$	8.50710	0.513933
$275$	5.09842	0.307446
$276$	3.82579	0.230285
$277$	−3.54233	−0.212838	−0.106419	−	0.994321i	$-0.533938\pi$
$277$	−3.54233	−0.212838	−0.106419	+	0.994321i	$0.533938\pi$
$278$	9.89090	0.593217
$279$	0.843379	0.0504918
$280$	0	0
$281$	11.4519	0.683163	0.341581	−	0.939852i	$-0.389037\pi$
$281$	11.4519	0.683163	0.341581	+	0.939852i	$0.389037\pi$
$282$	1.40067	0.0834087
$283$	27.6149	1.64153	0.820767	−	0.571263i	$-0.193547\pi$
$283$	27.6149	1.64153	0.820767	+	0.571263i	$0.193547\pi$
$284$	18.9751	1.12597
$285$	−5.86262	−0.347272
$286$	−6.17462	−0.365113
$287$	0	0
$288$	−5.75708	−0.339239
$289$	0.0218917	0.00128775
$290$	2.82103	0.165656
$291$	−5.74864	−0.336991
$292$	6.09905	0.356920
$293$	9.10545	0.531946	0.265973	−	0.963981i	$-0.414307\pi$
$293$	9.10545	0.531946	0.265973	+	0.963981i	$0.414307\pi$
$294$	0	0
$295$	−12.7555	−0.742655
$296$	28.6140	1.66316
$297$	−1.97493	−0.114597
$298$	−13.0736	−0.757331
$299$	11.0008	0.636191
$300$	3.34249	0.192979
$301$	0	0
$302$	−7.61783	−0.438357
$303$	−15.4205	−0.885886
$304$	−1.00227	−0.0574841
$305$	4.10636	0.235129
$306$	−3.46479	−0.198069
$307$	−8.39817	−0.479309	−0.239654	−	0.970858i	$-0.577034\pi$
$307$	−8.39817	−0.479309	−0.239654	+	0.970858i	$0.577034\pi$
$308$	0	0
$309$	10.3174	0.586935
$310$	−1.10144	−0.0625577
$311$	−12.2380	−0.693954	−0.346977	−	0.937874i	$-0.612792\pi$
$311$	−12.2380	−0.693954	−0.346977	+	0.937874i	$0.612792\pi$
$312$	−10.3011	−0.583182
$313$	−17.6525	−0.997781	−0.498890	−	0.866665i	$-0.666259\pi$
$313$	−17.6525	−0.997781	−0.498890	+	0.866665i	$0.666259\pi$
$314$	18.2432	1.02952
$315$	0	0
$316$	18.4440	1.03755
$317$	−8.56447	−0.481028	−0.240514	−	0.970646i	$-0.577316\pi$
$317$	−8.56447	−0.481028	−0.240514	+	0.970646i	$0.577316\pi$
$318$	4.13973	0.232145
$319$	4.26598	0.238849
$320$	6.69177	0.374081
$321$	14.8735	0.830159
$322$	0	0
$323$	−15.5536	−0.865423
$324$	−1.29475	−0.0719304
$325$	9.61108	0.533127
$326$	−18.7829	−1.04029
$327$	−12.0261	−0.665043
$328$	28.8508	1.59302
$329$	0	0
$330$	2.57923	0.141982
$331$	−14.3796	−0.790376	−0.395188	−	0.918600i	$-0.629321\pi$
$331$	−14.3796	−0.790376	−0.395188	+	0.918600i	$0.629321\pi$
$332$	3.70425	0.203297
$333$	10.3415	0.566712
$334$	7.53476	0.412284
$335$	6.62732	0.362089
$336$	0	0
$337$	−16.9818	−0.925059	−0.462529	−	0.886604i	$-0.653058\pi$
$337$	−16.9818	−0.925059	−0.462529	+	0.886604i	$0.653058\pi$
$338$	−0.722522	−0.0393000
$339$	9.52106	0.517113
$340$	−8.30721	−0.450521
$341$	−1.66561	−0.0901979
$342$	3.16591	0.171193
$343$	0	0
$344$	21.9340	1.18260
$345$	−4.59517	−0.247396
$346$	−5.65518	−0.304024
$347$	10.0770	0.540962	0.270481	−	0.962725i	$-0.412817\pi$
$347$	10.0770	0.540962	0.270481	+	0.962725i	$0.412817\pi$
$348$	2.79675	0.149921
$349$	18.5426	0.992561	0.496281	−	0.868162i	$-0.334699\pi$
$349$	18.5426	0.992561	0.496281	+	0.868162i	$0.334699\pi$
$350$	0	0
$351$	−3.72295	−0.198716
$352$	11.3698	0.606013
$353$	−10.0935	−0.537224	−0.268612	−	0.963248i	$-0.586565\pi$
$353$	−10.0935	−0.537224	−0.268612	+	0.963248i	$0.586565\pi$
$354$	6.88818	0.366103
$355$	−22.7911	−1.20963
$356$	22.1177	1.17223
$357$	0	0
$358$	1.34318	0.0709894
$359$	−33.3546	−1.76039	−0.880193	−	0.474615i	$-0.842587\pi$
$359$	−33.3546	−1.76039	−0.880193	+	0.474615i	$0.842587\pi$
$360$	4.30289	0.226782
$361$	−4.78812	−0.252007
$362$	15.4856	0.813905
$363$	−7.09967	−0.372636
$364$	0	0
$365$	−7.32560	−0.383439
$366$	−2.21750	−0.115910
$367$	17.5852	0.917941	0.458970	−	0.888452i	$-0.348218\pi$
$367$	17.5852	0.917941	0.458970	+	0.888452i	$0.348218\pi$
$368$	−0.785588	−0.0409516
$369$	10.4271	0.542813
$370$	−13.5059	−0.702137
$371$	0	0
$372$	−1.09196	−0.0566156
$373$	−15.3968	−0.797215	−0.398607	−	0.917122i	$-0.630506\pi$
$373$	−15.3968	−0.797215	−0.398607	+	0.917122i	$0.630506\pi$
$374$	6.84270	0.353827
$375$	−11.7903	−0.608850
$376$	−4.61486	−0.237993
$377$	8.04184	0.414176
$378$	0	0
$379$	7.92553	0.407107	0.203554	−	0.979064i	$-0.434751\pi$
$379$	7.92553	0.407107	0.203554	+	0.979064i	$0.434751\pi$
$380$	7.59061	0.389390
$381$	2.68618	0.137617
$382$	−3.43579	−0.175790
$383$	21.4950	1.09834	0.549171	−	0.835710i	$-0.314944\pi$
$383$	21.4950	1.09834	0.549171	+	0.835710i	$0.314944\pi$
$384$	7.90051	0.403171
$385$	0	0
$386$	13.8763	0.706285
$387$	7.92726	0.402965
$388$	7.44303	0.377863
$389$	−9.55401	−0.484408	−0.242204	−	0.970225i	$-0.577870\pi$
$389$	−9.55401	−0.484408	−0.242204	+	0.970225i	$0.577870\pi$
$390$	4.86212	0.246203
$391$	−12.1910	−0.616526
$392$	0	0
$393$	18.1812	0.917121
$394$	9.37277	0.472194
$395$	−22.1532	−1.11465
$396$	2.55703	0.128496
$397$	−19.0564	−0.956413	−0.478207	−	0.878247i	$-0.658713\pi$
$397$	−19.0564	−0.956413	−0.478207	+	0.878247i	$0.658713\pi$
$398$	−6.05014	−0.303266
$399$	0	0
$400$	−0.686347	−0.0343174
$401$	32.3394	1.61495	0.807477	−	0.589899i	$-0.200833\pi$
$401$	32.3394	1.61495	0.807477	+	0.589899i	$0.200833\pi$
$402$	−3.57886	−0.178497
$403$	−3.13986	−0.156407
$404$	19.9657	0.993330
$405$	1.55513	0.0772749
$406$	0	0
$407$	−20.4237	−1.01237
$408$	11.4156	0.565156
$409$	14.0715	0.695793	0.347897	−	0.937533i	$-0.386896\pi$
$409$	14.0715	0.695793	0.347897	+	0.937533i	$0.386896\pi$
$410$	−13.6176	−0.672528
$411$	−10.1300	−0.499675
$412$	−13.3584	−0.658121
$413$	0	0
$414$	2.48147	0.121957
$415$	−4.44919	−0.218402
$416$	21.4333	1.05086
$417$	−11.7778	−0.576760
$418$	−6.25243	−0.305816
$419$	0.974002	0.0475831	0.0237916	−	0.999717i	$-0.492426\pi$
$419$	0.974002	0.0475831	0.0237916	+	0.999717i	$0.492426\pi$
$420$	0	0
$421$	−16.5696	−0.807552	−0.403776	−	0.914858i	$-0.632302\pi$
$421$	−16.5696	−0.807552	−0.403776	+	0.914858i	$0.632302\pi$
$422$	2.08896	0.101689
$423$	−1.66788	−0.0810949
$424$	−13.6394	−0.662386
$425$	−10.6510	−0.516648
$426$	12.3076	0.596303
$427$	0	0
$428$	−19.2574	−0.930844
$429$	7.35255	0.354985
$430$	−10.3529	−0.499261
$431$	−31.0706	−1.49662	−0.748310	−	0.663349i	$-0.769134\pi$
$431$	−31.0706	−1.49662	−0.748310	+	0.663349i	$0.769134\pi$
$432$	0.265864	0.0127914
$433$	−9.62165	−0.462387	−0.231194	−	0.972908i	$-0.574263\pi$
$433$	−9.62165	−0.462387	−0.231194	+	0.972908i	$0.574263\pi$
$434$	0	0
$435$	−3.35919	−0.161061
$436$	15.5707	0.745702
$437$	11.1394	0.532869
$438$	3.95594	0.189022
$439$	−7.98407	−0.381059	−0.190530	−	0.981681i	$-0.561020\pi$
$439$	−7.98407	−0.381059	−0.190530	+	0.981681i	$0.561020\pi$
$440$	−8.49790	−0.405121
$441$	0	0
$442$	12.8992	0.613554
$443$	26.7679	1.27178	0.635891	−	0.771779i	$-0.280633\pi$
$443$	26.7679	1.27178	0.635891	+	0.771779i	$0.280633\pi$
$444$	−13.3897	−0.635445
$445$	−26.5657	−1.25933
$446$	1.16144	0.0549956
$447$	15.5676	0.736321
$448$	0	0
$449$	−21.4758	−1.01351	−0.506754	−	0.862091i	$-0.669155\pi$
$449$	−21.4758	−1.01351	−0.506754	+	0.862091i	$0.669155\pi$
$450$	2.16799	0.102200
$451$	−20.5927	−0.969675
$452$	−12.3274	−0.579830
$453$	9.07107	0.426196
$454$	−11.2969	−0.530189
$455$	0	0
$456$	−10.4309	−0.488470
$457$	−23.2566	−1.08790	−0.543948	−	0.839119i	$-0.683071\pi$
$457$	−23.2566	−1.08790	−0.543948	+	0.839119i	$0.683071\pi$
$458$	−19.4980	−0.911080
$459$	4.12576	0.192574
$460$	5.94959	0.277401
$461$	37.4300	1.74329	0.871646	−	0.490137i	$-0.163053\pi$
$461$	37.4300	1.74329	0.871646	+	0.490137i	$0.163053\pi$
$462$	0	0
$463$	−1.75490	−0.0815573	−0.0407786	−	0.999168i	$-0.512984\pi$
$463$	−1.75490	−0.0815573	−0.0407786	+	0.999168i	$0.512984\pi$
$464$	−0.574285	−0.0266605
$465$	1.31156	0.0608222
$466$	20.7400	0.960763
$467$	16.5659	0.766580	0.383290	−	0.923628i	$-0.374791\pi$
$467$	16.5659	0.766580	0.383290	+	0.923628i	$0.374791\pi$
$468$	4.82028	0.222817
$469$	0	0
$470$	2.17822	0.100474
$471$	−21.7234	−1.00096
$472$	−22.6948	−1.04461
$473$	−15.6558	−0.719852
$474$	11.9631	0.549481
$475$	9.73219	0.446544
$476$	0	0
$477$	−4.92947	−0.225705
$478$	3.19583	0.146174
$479$	−25.4105	−1.16104	−0.580518	−	0.814248i	$-0.697150\pi$
$479$	−25.4105	−1.16104	−0.580518	+	0.814248i	$0.697150\pi$
$480$	−8.95300	−0.408647
$481$	−38.5010	−1.75549
$482$	10.4869	0.477663
$483$	0	0
$484$	9.19227	0.417831
$485$	−8.93987	−0.405939
$486$	−0.839793	−0.0380938
$487$	−11.9215	−0.540216	−0.270108	−	0.962830i	$-0.587059\pi$
$487$	−11.9215	−0.540216	−0.270108	+	0.962830i	$0.587059\pi$
$488$	7.30609	0.330731
$489$	22.3661	1.01143
$490$	0	0
$491$	2.54465	0.114838	0.0574191	−	0.998350i	$-0.481713\pi$
$491$	2.54465	0.114838	0.0574191	+	0.998350i	$0.481713\pi$
$492$	−13.5005	−0.608647
$493$	−8.91194	−0.401374
$494$	−11.7865	−0.530300
$495$	−3.07126	−0.138043
$496$	0.224224	0.0100679
$497$	0	0
$498$	2.40263	0.107665
$499$	−32.6805	−1.46298	−0.731490	−	0.681852i	$-0.761175\pi$
$499$	−32.6805	−1.46298	−0.731490	+	0.681852i	$0.761175\pi$
$500$	15.2655	0.682693
$501$	−8.97216	−0.400847
$502$	3.96030	0.176757
$503$	−7.86982	−0.350898	−0.175449	−	0.984489i	$-0.556138\pi$
$503$	−7.86982	−0.350898	−0.175449	+	0.984489i	$0.556138\pi$
$504$	0	0
$505$	−23.9809	−1.06714
$506$	−4.90071	−0.217863
$507$	0.860357	0.0382098
$508$	−3.47793	−0.154308
$509$	13.7079	0.607591	0.303795	−	0.952737i	$-0.401746\pi$
$509$	13.7079	0.607591	0.303795	+	0.952737i	$0.401746\pi$
$510$	−5.38819	−0.238593
$511$	0	0
$512$	−3.00184	−0.132664
$513$	−3.76986	−0.166444
$514$	−4.29327	−0.189368
$515$	16.0449	0.707021
$516$	−10.2638	−0.451839
$517$	3.29393	0.144867
$518$	0	0
$519$	6.73401	0.295590
$520$	−16.0195	−0.702500
$521$	−20.9426	−0.917512	−0.458756	−	0.888562i	$-0.651705\pi$
$521$	−20.9426	−0.917512	−0.458756	+	0.888562i	$0.651705\pi$
$522$	1.81402	0.0793973
$523$	−31.6495	−1.38393	−0.691967	−	0.721929i	$-0.743256\pi$
$523$	−31.6495	−1.38393	−0.691967	+	0.721929i	$0.743256\pi$
$524$	−23.5401	−1.02835
$525$	0	0
$526$	18.4394	0.803997
$527$	3.47958	0.151573
$528$	−0.525061	−0.0228503
$529$	−14.2688	−0.620384
$530$	6.43781	0.279641
$531$	−8.20223	−0.355947
$532$	0	0
$533$	−38.8196	−1.68146
$534$	14.3459	0.620807
$535$	23.1302	1.00001
$536$	11.7914	0.509312
$537$	−1.59942	−0.0690200
$538$	22.8512	0.985186
$539$	0	0
$540$	−2.01350	−0.0866471
$541$	−21.1516	−0.909377	−0.454688	−	0.890651i	$-0.650249\pi$
$541$	−21.1516	−0.909377	−0.454688	+	0.890651i	$0.650249\pi$
$542$	2.09025	0.0897840
$543$	−18.4398	−0.791326
$544$	−23.7523	−1.01837
$545$	−18.7021	−0.801109
$546$	0	0
$547$	43.0476	1.84058	0.920292	−	0.391233i	$-0.127951\pi$
$547$	43.0476	1.84058	0.920292	+	0.391233i	$0.127951\pi$
$548$	13.1158	0.560278
$549$	2.64053	0.112695
$550$	−4.28162	−0.182569
$551$	8.14318	0.346911
$552$	−8.17580	−0.347985
$553$	0	0
$554$	2.97482	0.126388
$555$	16.0824	0.682659
$556$	15.2492	0.646712
$557$	28.7820	1.21953	0.609767	−	0.792581i	$-0.291263\pi$
$557$	28.7820	1.21953	0.609767	+	0.792581i	$0.291263\pi$
$558$	−0.708264	−0.0299832
$559$	−29.5128	−1.24826
$560$	0	0
$561$	−8.14807	−0.344012
$562$	−9.61723	−0.405678
$563$	14.9378	0.629553	0.314777	−	0.949166i	$-0.398070\pi$
$563$	14.9378	0.629553	0.314777	+	0.949166i	$0.398070\pi$
$564$	2.15948	0.0909304
$565$	14.8065	0.622913
$566$	−23.1908	−0.974781
$567$	0	0
$568$	−40.5503	−1.70145
$569$	−31.7429	−1.33073	−0.665365	−	0.746518i	$-0.731724\pi$
$569$	−31.7429	−1.33073	−0.665365	+	0.746518i	$0.731724\pi$
$570$	4.92339	0.206218
$571$	−23.4475	−0.981246	−0.490623	−	0.871372i	$-0.663231\pi$
$571$	−23.4475	−0.981246	−0.490623	+	0.871372i	$0.663231\pi$
$572$	−9.51969	−0.398038
$573$	4.09123	0.170914
$574$	0	0
$575$	7.62818	0.318117
$576$	4.30303	0.179293
$577$	31.2782	1.30213	0.651064	−	0.759023i	$-0.274323\pi$
$577$	31.2782	1.30213	0.651064	+	0.759023i	$0.274323\pi$
$578$	−0.0183845	−0.000764694 0
$579$	−16.5235	−0.686692
$580$	4.34930	0.180595
$581$	0	0
$582$	4.82767	0.200113
$583$	9.73533	0.403196
$584$	−13.0338	−0.539343
$585$	−5.78966	−0.239373
$586$	−7.64669	−0.315882
$587$	11.4659	0.473249	0.236624	−	0.971601i	$-0.423959\pi$
$587$	11.4659	0.473249	0.236624	+	0.971601i	$0.423959\pi$
$588$	0	0
$589$	−3.17942	−0.131006
$590$	10.7120	0.441006
$591$	−11.1608	−0.459094
$592$	2.74943	0.113001
$593$	40.2254	1.65186	0.825929	−	0.563774i	$-0.190651\pi$
$593$	40.2254	1.65186	0.825929	+	0.563774i	$0.190651\pi$
$594$	1.65853	0.0680503
$595$	0	0
$596$	−20.1561	−0.825625
$597$	7.20432	0.294853
$598$	−9.23837	−0.377785
$599$	−39.2772	−1.60482	−0.802411	−	0.596772i	$-0.796450\pi$
$599$	−39.2772	−1.60482	−0.802411	+	0.596772i	$0.796450\pi$
$600$	−7.14298	−0.291611
$601$	13.6601	0.557206	0.278603	−	0.960406i	$-0.410129\pi$
$601$	13.6601	0.557206	0.278603	+	0.960406i	$0.410129\pi$
$602$	0	0
$603$	4.26159	0.173545
$604$	−11.7447	−0.477887
$605$	−11.0409	−0.448876
$606$	12.9501	0.526060
$607$	−1.96729	−0.0798500	−0.0399250	−	0.999203i	$-0.512712\pi$
$607$	−1.96729	−0.0798500	−0.0399250	+	0.999203i	$0.512712\pi$
$608$	21.7034	0.880190
$609$	0	0
$610$	−3.44849	−0.139625
$611$	6.20942	0.251206
$612$	−5.34181	−0.215930
$613$	−31.9180	−1.28916	−0.644578	−	0.764539i	$-0.722967\pi$
$613$	−31.9180	−1.28916	−0.644578	+	0.764539i	$0.722967\pi$
$614$	7.05273	0.284625
$615$	16.2155	0.653871
$616$	0	0
$617$	30.3534	1.22198	0.610992	−	0.791637i	$-0.290771\pi$
$617$	30.3534	1.22198	0.610992	+	0.791637i	$0.290771\pi$
$618$	−8.66447	−0.348536
$619$	12.7902	0.514080	0.257040	−	0.966401i	$-0.417253\pi$
$619$	12.7902	0.514080	0.257040	+	0.966401i	$0.417253\pi$
$620$	−1.69814	−0.0681990
$621$	−2.95485	−0.118574
$622$	10.2774	0.412087
$623$	0	0
$624$	−0.989797	−0.0396236
$625$	−5.42758	−0.217103
$626$	14.8245	0.592506
$627$	7.44520	0.297333
$628$	28.1263	1.12236
$629$	42.6666	1.70123
$630$	0	0
$631$	11.8293	0.470916	0.235458	−	0.971885i	$-0.424341\pi$
$631$	11.8293	0.470916	0.235458	+	0.971885i	$0.424341\pi$
$632$	−39.4152	−1.56785
$633$	−2.48747	−0.0988680
$634$	7.19238	0.285646
$635$	4.17736	0.165774
$636$	6.38241	0.253079
$637$	0	0
$638$	−3.58255	−0.141834
$639$	−14.6555	−0.579761
$640$	12.2863	0.485659
$641$	18.6471	0.736517	0.368259	−	0.929723i	$-0.379954\pi$
$641$	18.6471	0.736517	0.368259	+	0.929723i	$0.379954\pi$
$642$	−12.4907	−0.492968
$643$	−45.7833	−1.80552	−0.902758	−	0.430148i	$-0.858461\pi$
$643$	−45.7833	−1.80552	−0.902758	+	0.430148i	$0.858461\pi$
$644$	0	0
$645$	12.3279	0.485411
$646$	13.0618	0.513909
$647$	−13.1863	−0.518407	−0.259204	−	0.965823i	$-0.583460\pi$
$647$	−13.1863	−0.518407	−0.259204	+	0.965823i	$0.583460\pi$
$648$	2.76691	0.108694
$649$	16.1988	0.635859
$650$	−8.07132	−0.316583
$651$	0	0
$652$	−28.9584	−1.13410
$653$	−18.7848	−0.735108	−0.367554	−	0.930002i	$-0.619805\pi$
$653$	−18.7848	−0.735108	−0.367554	+	0.930002i	$0.619805\pi$
$654$	10.0994	0.394918
$655$	28.2741	1.10476
$656$	2.77218	0.108236
$657$	−4.71061	−0.183778
$658$	0	0
$659$	23.1125	0.900336	0.450168	−	0.892944i	$-0.351364\pi$
$659$	23.1125	0.900336	0.450168	+	0.892944i	$0.351364\pi$
$660$	3.97651	0.154785
$661$	3.19997	0.124465	0.0622323	−	0.998062i	$-0.480178\pi$
$661$	3.19997	0.124465	0.0622323	+	0.998062i	$0.480178\pi$
$662$	12.0759	0.469344
$663$	−15.3600	−0.596533
$664$	−7.91607	−0.307203
$665$	0	0
$666$	−8.68474	−0.336527
$667$	6.38270	0.247139
$668$	11.6167	0.449463
$669$	−1.38300	−0.0534700
$670$	−5.56558	−0.215017
$671$	−5.21484	−0.201317
$672$	0	0
$673$	−38.5773	−1.48705	−0.743524	−	0.668710i	$-0.766847\pi$
$673$	−38.5773	−1.48705	−0.743524	+	0.668710i	$0.766847\pi$
$674$	14.2612	0.549322
$675$	−2.58158	−0.0993649
$676$	−1.11394	−0.0428440
$677$	31.0852	1.19470	0.597350	−	0.801981i	$-0.296220\pi$
$677$	31.0852	1.19470	0.597350	+	0.801981i	$0.296220\pi$
$678$	−7.99572	−0.307074
$679$	0	0
$680$	17.7527	0.680785
$681$	13.4520	0.515481
$682$	1.39877	0.0535616
$683$	−24.8684	−0.951563	−0.475782	−	0.879563i	$-0.657835\pi$
$683$	−24.8684	−0.951563	−0.475782	+	0.879563i	$0.657835\pi$
$684$	4.88102	0.186630
$685$	−15.7534	−0.601908
$686$	0	0
$687$	23.2176	0.885806
$688$	2.10757	0.0803503
$689$	18.3522	0.699161
$690$	3.85900	0.146910
$691$	−17.2740	−0.657134	−0.328567	−	0.944481i	$-0.606566\pi$
$691$	−17.2740	−0.657134	−0.328567	+	0.944481i	$0.606566\pi$
$692$	−8.71884	−0.331441
$693$	0	0
$694$	−8.46260	−0.321236
$695$	−18.3160	−0.694764
$696$	−5.97672	−0.226547
$697$	43.0197	1.62949
$698$	−15.5719	−0.589406
$699$	−24.6966	−0.934111
$700$	0	0
$701$	3.79801	0.143449	0.0717244	−	0.997424i	$-0.477150\pi$
$701$	3.79801	0.143449	0.0717244	+	0.997424i	$0.477150\pi$
$702$	3.12651	0.118002
$703$	−38.9861	−1.47039
$704$	−8.49818	−0.320287
$705$	−2.59376	−0.0976866
$706$	8.47648	0.319017
$707$	0	0
$708$	10.6198	0.399117
$709$	−12.8807	−0.483744	−0.241872	−	0.970308i	$-0.577761\pi$
$709$	−12.8807	−0.483744	−0.241872	+	0.970308i	$0.577761\pi$
$710$	19.1398	0.718305
$711$	−14.2452	−0.534238
$712$	−47.2660	−1.77137
$713$	−2.49206	−0.0933284
$714$	0	0
$715$	11.4342	0.427613
$716$	2.07084	0.0773911
$717$	−3.80550	−0.142119
$718$	28.0109	1.04536
$719$	2.00632	0.0748231	0.0374116	−	0.999300i	$-0.488089\pi$
$719$	2.00632	0.0748231	0.0374116	+	0.999300i	$0.488089\pi$
$720$	0.413452	0.0154084
$721$	0	0
$722$	4.02104	0.149647
$723$	−12.4874	−0.464412
$724$	23.8748	0.887301
$725$	5.57639	0.207102
$726$	5.96225	0.221280
$727$	−1.15772	−0.0429373	−0.0214687	−	0.999770i	$-0.506834\pi$
$727$	−1.15772	−0.0429373	−0.0214687	+	0.999770i	$0.506834\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.15199	0.227695
$731$	32.7060	1.20967
$732$	−3.41881	−0.126363
$733$	−17.7744	−0.656511	−0.328255	−	0.944589i	$-0.606461\pi$
$733$	−17.7744	−0.656511	−0.328255	+	0.944589i	$0.606461\pi$
$734$	−14.7680	−0.545095
$735$	0	0
$736$	17.0113	0.627046
$737$	−8.41633	−0.310019
$738$	−8.75661	−0.322335
$739$	12.4249	0.457056	0.228528	−	0.973537i	$-0.426609\pi$
$739$	12.4249	0.457056	0.228528	+	0.973537i	$0.426609\pi$
$740$	−20.8226	−0.765455
$741$	14.0350	0.515589
$742$	0	0
$743$	28.7754	1.05567	0.527834	−	0.849348i	$-0.323004\pi$
$743$	28.7754	1.05567	0.527834	+	0.849348i	$0.323004\pi$
$744$	2.33355	0.0855521
$745$	24.2096	0.886971
$746$	12.9301	0.473405
$747$	−2.86098	−0.104678
$748$	10.5497	0.385735
$749$	0	0
$750$	9.90144	0.361549
$751$	38.0584	1.38877	0.694385	−	0.719604i	$-0.255676\pi$
$751$	38.0584	1.38877	0.694385	+	0.719604i	$0.255676\pi$
$752$	−0.443427	−0.0161701
$753$	−4.71580	−0.171853
$754$	−6.75349	−0.245948
$755$	14.1067	0.513395
$756$	0	0
$757$	−4.53831	−0.164948	−0.0824738	−	0.996593i	$-0.526282\pi$
$757$	−4.53831	−0.164948	−0.0824738	+	0.996593i	$0.526282\pi$
$758$	−6.65581	−0.241750
$759$	5.83562	0.211819
$760$	−16.2213	−0.588409
$761$	11.5503	0.418698	0.209349	−	0.977841i	$-0.432866\pi$
$761$	11.5503	0.418698	0.209349	+	0.977841i	$0.432866\pi$
$762$	−2.25584	−0.0817205
$763$	0	0
$764$	−5.29711	−0.191643
$765$	6.41608	0.231974
$766$	−18.0513	−0.652221
$767$	30.5365	1.10261
$768$	−15.2409	−0.549957
$769$	−7.50232	−0.270541	−0.135270	−	0.990809i	$-0.543190\pi$
$769$	−7.50232	−0.270541	−0.135270	+	0.990809i	$0.543190\pi$
$770$	0	0
$771$	5.11230	0.184115
$772$	21.3937	0.769976
$773$	−13.9081	−0.500240	−0.250120	−	0.968215i	$-0.580470\pi$
$773$	−13.9081	−0.500240	−0.250120	+	0.968215i	$0.580470\pi$
$774$	−6.65726	−0.239290
$775$	−2.17725	−0.0782090
$776$	−15.9059	−0.570990
$777$	0	0
$778$	8.02340	0.287653
$779$	−39.3087	−1.40838
$780$	7.49615	0.268405
$781$	28.9435	1.03568
$782$	10.2379	0.366108
$783$	−2.16007	−0.0771947
$784$	0	0
$785$	−33.7827	−1.20576
$786$	−15.2685	−0.544608
$787$	10.1926	0.363326	0.181663	−	0.983361i	$-0.441852\pi$
$787$	10.1926	0.363326	0.181663	+	0.983361i	$0.441852\pi$
$788$	14.4504	0.514775
$789$	−21.9571	−0.781693
$790$	18.6041	0.661903
$791$	0	0
$792$	−5.46444	−0.194170
$793$	−9.83055	−0.349093
$794$	16.0034	0.567941
$795$	−7.66595	−0.271883
$796$	−9.32777	−0.330614
$797$	44.6992	1.58333	0.791664	−	0.610957i	$-0.209215\pi$
$797$	44.6992	1.58333	0.791664	+	0.610957i	$0.209215\pi$
$798$	0	0
$799$	−6.88125	−0.243441
$800$	14.8624	0.525464
$801$	−17.0826	−0.603585
$802$	−27.1584	−0.958998
$803$	9.30310	0.328299
$804$	−5.51768	−0.194594
$805$	0	0
$806$	2.63683	0.0928784
$807$	−27.2105	−0.957856
$808$	−42.6672	−1.50103
$809$	−15.1947	−0.534216	−0.267108	−	0.963667i	$-0.586068\pi$
$809$	−15.1947	−0.534216	−0.267108	+	0.963667i	$0.586068\pi$
$810$	−1.30599	−0.0458877
$811$	37.0114	1.29965	0.649824	−	0.760085i	$-0.274843\pi$
$811$	37.0114	1.29965	0.649824	+	0.760085i	$0.274843\pi$
$812$	0	0
$813$	−2.48901	−0.0872933
$814$	17.1517	0.601168
$815$	34.7821	1.21836
$816$	1.09689	0.0383988
$817$	−29.8847	−1.04553
$818$	−11.8172	−0.413178
$819$	0	0
$820$	−20.9949	−0.733175
$821$	−37.5491	−1.31047	−0.655236	−	0.755424i	$-0.727431\pi$
$821$	−37.5491	−1.31047	−0.655236	+	0.755424i	$0.727431\pi$
$822$	8.50710	0.296719
$823$	51.2765	1.78738	0.893692	−	0.448680i	$-0.148106\pi$
$823$	51.2765	1.78738	0.893692	+	0.448680i	$0.148106\pi$
$824$	28.5472	0.994490
$825$	5.09842	0.177504
$826$	0	0
$827$	−43.3868	−1.50871	−0.754354	−	0.656468i	$-0.772050\pi$
$827$	−43.3868	−1.50871	−0.754354	+	0.656468i	$0.772050\pi$
$828$	3.82579	0.132955
$829$	−44.0733	−1.53073	−0.765364	−	0.643598i	$-0.777441\pi$
$829$	−44.0733	−1.53073	−0.765364	+	0.643598i	$0.777441\pi$
$830$	3.73640	0.129692
$831$	−3.54233	−0.122882
$832$	−16.0200	−0.555393
$833$	0	0
$834$	9.89090	0.342494
$835$	−13.9529	−0.482859
$836$	−9.63965	−0.333394
$837$	0.843379	0.0291514
$838$	−0.817960	−0.0282560
$839$	−25.0824	−0.865940	−0.432970	−	0.901408i	$-0.642534\pi$
$839$	−25.0824	−0.865940	−0.432970	+	0.901408i	$0.642534\pi$
$840$	0	0
$841$	−24.3341	−0.839106
$842$	13.9150	0.479543
$843$	11.4519	0.394424
$844$	3.22064	0.110859
$845$	1.33797	0.0460274
$846$	1.40067	0.0481561
$847$	0	0
$848$	−1.31056	−0.0450050
$849$	27.6149	0.947740
$850$	8.94461	0.306798
$851$	−30.5577	−1.04750
$852$	18.9751	0.650077
$853$	−21.3025	−0.729383	−0.364691	−	0.931128i	$-0.618826\pi$
$853$	−21.3025	−0.729383	−0.364691	+	0.931128i	$0.618826\pi$
$854$	0	0
$855$	−5.86262	−0.200497
$856$	41.1536	1.40660
$857$	−15.6908	−0.535987	−0.267993	−	0.963421i	$-0.586361\pi$
$857$	−15.6908	−0.535987	−0.267993	+	0.963421i	$0.586361\pi$
$858$	−6.17462	−0.210798
$859$	−18.0571	−0.616102	−0.308051	−	0.951370i	$-0.599677\pi$
$859$	−18.0571	−0.616102	−0.308051	+	0.951370i	$0.599677\pi$
$860$	−15.9615	−0.544283
$861$	0	0
$862$	26.0929	0.888728
$863$	39.1902	1.33405	0.667026	−	0.745035i	$-0.267567\pi$
$863$	39.1902	1.33405	0.667026	+	0.745035i	$0.267567\pi$
$864$	−5.75708	−0.195860
$865$	10.4723	0.356067
$866$	8.08020	0.274576
$867$	0.0218917	0.000743481 0
$868$	0	0
$869$	28.1333	0.954356
$870$	2.82103	0.0956417
$871$	−15.8657	−0.537588
$872$	−33.2750	−1.12683
$873$	−5.74864	−0.194562
$874$	−9.35479	−0.316431
$875$	0	0
$876$	6.09905	0.206068
$877$	−33.5667	−1.13347	−0.566733	−	0.823901i	$-0.691793\pi$
$877$	−33.5667	−1.13347	−0.566733	+	0.823901i	$0.691793\pi$
$878$	6.70497	0.226282
$879$	9.10545	0.307119
$880$	−0.816537	−0.0275254
$881$	−4.10302	−0.138234	−0.0691171	−	0.997609i	$-0.522018\pi$
$881$	−4.10302	−0.138234	−0.0691171	+	0.997609i	$0.522018\pi$
$882$	0	0
$883$	8.81326	0.296590	0.148295	−	0.988943i	$-0.452622\pi$
$883$	8.81326	0.296590	0.148295	+	0.988943i	$0.452622\pi$
$884$	19.8873	0.668883
$885$	−12.7555	−0.428772
$886$	−22.4795	−0.755215
$887$	42.7798	1.43641	0.718203	−	0.695834i	$-0.244965\pi$
$887$	42.7798	1.43641	0.718203	+	0.695834i	$0.244965\pi$
$888$	28.6140	0.960224
$889$	0	0
$890$	22.3097	0.747822
$891$	−1.97493	−0.0661625
$892$	1.79064	0.0599550
$893$	6.28766	0.210409
$894$	−13.0736	−0.437245
$895$	−2.48730	−0.0831413
$896$	0	0
$897$	11.0008	0.367305
$898$	18.0353	0.601845
$899$	−1.82176	−0.0607591
$900$	3.34249	0.111416
$901$	−20.3378	−0.677550
$902$	17.2937	0.575816
$903$	0	0
$904$	26.3439	0.876184
$905$	−28.6762	−0.953229
$906$	−7.61783	−0.253085
$907$	28.1370	0.934274	0.467137	−	0.884185i	$-0.345286\pi$
$907$	28.1370	0.934274	0.467137	+	0.884185i	$0.345286\pi$
$908$	−17.4169	−0.578000
$909$	−15.4205	−0.511467
$910$	0	0
$911$	49.7583	1.64857	0.824284	−	0.566177i	$-0.191578\pi$
$911$	49.7583	1.64857	0.824284	+	0.566177i	$0.191578\pi$
$912$	−1.00227	−0.0331885
$913$	5.65023	0.186995
$914$	19.5307	0.646018
$915$	4.10636	0.135752
$916$	−30.0609	−0.993240
$917$	0	0
$918$	−3.46479	−0.114355
$919$	−26.7020	−0.880818	−0.440409	−	0.897797i	$-0.645166\pi$
$919$	−26.7020	−0.880818	−0.440409	+	0.897797i	$0.645166\pi$
$920$	−12.7144	−0.419182
$921$	−8.39817	−0.276729
$922$	−31.4335	−1.03521
$923$	54.5616	1.79592
$924$	0	0
$925$	−26.6974	−0.877806
$926$	1.47376	0.0484307
$927$	10.3174	0.338867
$928$	12.4357	0.408223
$929$	16.1865	0.531062	0.265531	−	0.964102i	$-0.414453\pi$
$929$	16.1865	0.531062	0.265531	+	0.964102i	$0.414453\pi$
$930$	−1.10144	−0.0361177
$931$	0	0
$932$	31.9758	1.04740
$933$	−12.2380	−0.400655
$934$	−13.9120	−0.455213
$935$	−12.6713	−0.414396
$936$	−10.3011	−0.336701
$937$	20.7859	0.679047	0.339524	−	0.940598i	$-0.389734\pi$
$937$	20.7859	0.679047	0.339524	+	0.940598i	$0.389734\pi$
$938$	0	0
$939$	−17.6525	−0.576069
$940$	3.35826	0.109534
$941$	−3.91637	−0.127670	−0.0638351	−	0.997960i	$-0.520333\pi$
$941$	−3.91637	−0.127670	−0.0638351	+	0.997960i	$0.520333\pi$
$942$	18.2432	0.594395
$943$	−30.8105	−1.00333
$944$	−2.18067	−0.0709749
$945$	0	0
$946$	13.1476	0.427465
$947$	−51.2509	−1.66543	−0.832714	−	0.553703i	$-0.813214\pi$
$947$	−51.2509	−1.66543	−0.832714	+	0.553703i	$0.813214\pi$
$948$	18.4440	0.599032
$949$	17.5374	0.569287
$950$	−8.17303	−0.265168
$951$	−8.56447	−0.277722
$952$	0	0
$953$	18.4457	0.597516	0.298758	−	0.954329i	$-0.403428\pi$
$953$	18.4457	0.597516	0.298758	+	0.954329i	$0.403428\pi$
$954$	4.13973	0.134029
$955$	6.36239	0.205882
$956$	4.92715	0.159356
$957$	4.26598	0.137900
$958$	21.3396	0.689450
$959$	0	0
$960$	6.69177	0.215976
$961$	−30.2887	−0.977055
$962$	32.3329	1.04245
$963$	14.8735	0.479292
$964$	16.1681	0.520738
$965$	−25.6961	−0.827187
$966$	0	0
$967$	21.9504	0.705879	0.352939	−	0.935646i	$-0.385182\pi$
$967$	21.9504	0.705879	0.352939	+	0.935646i	$0.385182\pi$
$968$	−19.6441	−0.631386
$969$	−15.5536	−0.499652
$970$	7.50764	0.241056
$971$	−39.9353	−1.28159	−0.640793	−	0.767714i	$-0.721394\pi$
$971$	−39.9353	−1.28159	−0.640793	+	0.767714i	$0.721394\pi$
$972$	−1.29475	−0.0415290
$973$	0	0
$974$	10.0116	0.320793
$975$	9.61108	0.307801
$976$	0.702019	0.0224711
$977$	24.2096	0.774533	0.387266	−	0.921968i	$-0.373419\pi$
$977$	24.2096	0.774533	0.387266	+	0.921968i	$0.373419\pi$
$978$	−18.7829	−0.600610
$979$	33.7369	1.07824
$980$	0	0
$981$	−12.0261	−0.383963
$982$	−2.13698	−0.0681937
$983$	52.4490	1.67286	0.836432	−	0.548070i	$-0.184637\pi$
$983$	52.4490	1.67286	0.836432	+	0.548070i	$0.184637\pi$
$984$	28.8508	0.919730
$985$	−17.3565	−0.553024
$986$	7.48419	0.238345
$987$	0	0
$988$	−18.1718	−0.578122
$989$	−23.4239	−0.744836
$990$	2.57923	0.0819732
$991$	−0.271740	−0.00863210	−0.00431605	−	0.999991i	$-0.501374\pi$
$991$	−0.271740	−0.00863210	−0.00431605	+	0.999991i	$0.501374\pi$
$992$	−4.85540	−0.154159
$993$	−14.3796	−0.456324
$994$	0	0
$995$	11.2036	0.355179
$996$	3.70425	0.117374
$997$	1.00370	0.0317873	0.0158937	−	0.999874i	$-0.494941\pi$
$997$	1.00370	0.0317873	0.0158937	+	0.999874i	$0.494941\pi$
$998$	27.4449	0.868752
$999$	10.3415	0.327191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7203.2.a.k.1.10		24
7.6	odd	2		7203.2.a.i.1.10		24
49.3	odd	42		147.2.m.a.58.3	✓	48
49.33	odd	42		147.2.m.a.109.3	yes	48
147.101	even	42		441.2.bb.c.352.2		48
147.131	even	42		441.2.bb.c.109.2		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.m.a.58.3	✓	48	49.3	odd	42
147.2.m.a.109.3	yes	48	49.33	odd	42
441.2.bb.c.109.2		48	147.131	even	42
441.2.bb.c.352.2		48	147.101	even	42
7203.2.a.i.1.10		24	7.6	odd	2
7203.2.a.k.1.10		24	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:	\( N \)	\(=\)	\( 7203 = 3 \cdot 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7203.a (trivial)

\(f(q)\)	\(=\)	\(q-0.839793 q^{2} +1.00000 q^{3} -1.29475 q^{4} +1.55513 q^{5} -0.839793 q^{6} +2.76691 q^{8} +1.00000 q^{9} -1.30599 q^{10} -1.97493 q^{11} -1.29475 q^{12} -3.72295 q^{13} +1.55513 q^{15} +0.265864 q^{16} +4.12576 q^{17} -0.839793 q^{18} -3.76986 q^{19} -2.01350 q^{20} +1.65853 q^{22} -2.95485 q^{23} +2.76691 q^{24} -2.58158 q^{25} +3.12651 q^{26} +1.00000 q^{27} -2.16007 q^{29} -1.30599 q^{30} +0.843379 q^{31} -5.75708 q^{32} -1.97493 q^{33} -3.46479 q^{34} -1.29475 q^{36} +10.3415 q^{37} +3.16591 q^{38} -3.72295 q^{39} +4.30289 q^{40} +10.4271 q^{41} +7.92726 q^{43} +2.55703 q^{44} +1.55513 q^{45} +2.48147 q^{46} -1.66788 q^{47} +0.265864 q^{48} +2.16799 q^{50} +4.12576 q^{51} +4.82028 q^{52} -4.92947 q^{53} -0.839793 q^{54} -3.07126 q^{55} -3.76986 q^{57} +1.81402 q^{58} -8.20223 q^{59} -2.01350 q^{60} +2.64053 q^{61} -0.708264 q^{62} +4.30303 q^{64} -5.78966 q^{65} +1.65853 q^{66} +4.26159 q^{67} -5.34181 q^{68} -2.95485 q^{69} -14.6555 q^{71} +2.76691 q^{72} -4.71061 q^{73} -8.68474 q^{74} -2.58158 q^{75} +4.88102 q^{76} +3.12651 q^{78} -14.2452 q^{79} +0.413452 q^{80} +1.00000 q^{81} -8.75661 q^{82} -2.86098 q^{83} +6.41608 q^{85} -6.65726 q^{86} -2.16007 q^{87} -5.46444 q^{88} -17.0826 q^{89} -1.30599 q^{90} +3.82579 q^{92} +0.843379 q^{93} +1.40067 q^{94} -5.86262 q^{95} -5.75708 q^{96} -5.74864 q^{97} -1.97493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} + 24 q^{3} + 18 q^{4} - 6 q^{6} - 18 q^{8} + 24 q^{9} - 12 q^{10} - 12 q^{11} + 18 q^{12} - 14 q^{13} + 6 q^{16} + 2 q^{17} - 6 q^{18} - 26 q^{19} + 6 q^{20} - 24 q^{22} - 24 q^{23} - 18 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) 24 * q - 6 * q^2 + 24 * q^3 + 18 * q^4 - 6 * q^6 - 18 * q^8 + 24 * q^9 - 12 * q^10 - 12 * q^11 + 18 * q^12 - 14 * q^13 + 6 * q^16 + 2 * q^17 - 6 * q^18 - 26 * q^19 + 6 * q^20 - 24 * q^22 - 24 * q^23 - 18 * q^24 - 13 * q^26 + 24 * q^27 - 36 * q^29 - 12 * q^30 - 36 * q^31 - 42 * q^32 - 12 * q^33 - 4 * q^34 + 18 * q^36 - 50 * q^37 + 38 * q^38 - 14 * q^39 - 28 * q^40 + 25 * q^41 - 36 * q^43 - 30 * q^44 - 30 * q^46 + 6 * q^47 + 6 * q^48 - 6 * q^50 + 2 * q^51 - 41 * q^52 - 63 * q^53 - 6 * q^54 - 44 * q^55 - 26 * q^57 - 3 * q^58 - 41 * q^59 + 6 * q^60 - 38 * q^61 + 22 * q^62 - 6 * q^64 - 14 * q^65 - 24 * q^66 - 34 * q^67 + 22 * q^68 - 24 * q^69 - 32 * q^71 - 18 * q^72 - 13 * q^73 - 43 * q^74 - 105 * q^76 - 13 * q^78 - 34 * q^79 + 55 * q^80 + 24 * q^81 + 28 * q^82 + 42 * q^83 - 92 * q^85 + 40 * q^86 - 36 * q^87 - 10 * q^88 + 32 * q^89 - 12 * q^90 - 14 * q^92 - 36 * q^93 - 64 * q^94 - 38 * q^95 - 42 * q^96 - 62 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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