LMFDB - Embedded newform 8450.2.a.da.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, 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("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,9,7,9,0,7,1,9,8,0,4,7,0,1,0,9,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 17x^{6} + 53x^{5} - 69x^{4} - 33x^{3} + 26x^{2} + 8x - 1 $ x^9 - x^8 - 14x^7 + 17x^6 + 53x^5 - 69x^4 - 33x^3 + 26x^2 + 8*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.39737$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +3.41573 q^{3} +1.00000 q^{4} +3.41573 q^{6} -0.459770 q^{7} +1.00000 q^{8} +8.66724 q^{9} -2.03092 q^{11} +3.41573 q^{12} -0.459770 q^{14} +1.00000 q^{16} +5.39187 q^{17} +8.66724 q^{18} +3.30639 q^{19} -1.57045 q^{21} -2.03092 q^{22} -4.15544 q^{23} +3.41573 q^{24} +19.3578 q^{27} -0.459770 q^{28} +6.05566 q^{29} -6.52262 q^{31} +1.00000 q^{32} -6.93709 q^{33} +5.39187 q^{34} +8.66724 q^{36} -8.07307 q^{37} +3.30639 q^{38} +4.14511 q^{41} -1.57045 q^{42} +1.26725 q^{43} -2.03092 q^{44} -4.15544 q^{46} -1.29310 q^{47} +3.41573 q^{48} -6.78861 q^{49} +18.4172 q^{51} +5.78588 q^{53} +19.3578 q^{54} -0.459770 q^{56} +11.2937 q^{57} +6.05566 q^{58} +6.77070 q^{59} -5.98450 q^{61} -6.52262 q^{62} -3.98494 q^{63} +1.00000 q^{64} -6.93709 q^{66} +8.41427 q^{67} +5.39187 q^{68} -14.1939 q^{69} +1.47782 q^{71} +8.66724 q^{72} +8.05573 q^{73} -8.07307 q^{74} +3.30639 q^{76} +0.933756 q^{77} -1.68287 q^{79} +40.1194 q^{81} +4.14511 q^{82} +9.99259 q^{83} -1.57045 q^{84} +1.26725 q^{86} +20.6845 q^{87} -2.03092 q^{88} -3.93934 q^{89} -4.15544 q^{92} -22.2796 q^{93} -1.29310 q^{94} +3.41573 q^{96} -7.05844 q^{97} -6.78861 q^{98} -17.6025 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} + 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + 4 q^{11} + 7 q^{12} + q^{14} + 9 q^{16} + 12 q^{17} + 8 q^{18} + 6 q^{19} + 8 q^{21} + 4 q^{22} + 11 q^{23} + 7 q^{24} + 34 q^{27}+ \cdots + 16 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + 7 * q^3 + 9 * q^4 + 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 + 4 * q^11 + 7 * q^12 + q^14 + 9 * q^16 + 12 * q^17 + 8 * q^18 + 6 * q^19 + 8 * q^21 + 4 * q^22 + 11 * q^23 + 7 * q^24 + 34 * q^27 + q^28 + 15 * q^29 - 6 * q^31 + 9 * q^32 - 14 * q^33 + 12 * q^34 + 8 * q^36 - 10 * q^37 + 6 * q^38 + 27 * q^41 + 8 * q^42 + 41 * q^43 + 4 * q^44 + 11 * q^46 + 5 * q^47 + 7 * q^48 - 10 * q^49 + 12 * q^51 + 18 * q^53 + 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 + 20 * q^59 - 19 * q^61 - 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 + 12 * q^68 - 14 * q^69 - 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 + 6 * q^76 + 30 * q^77 - 14 * q^79 + 53 * q^81 + 27 * q^82 + q^83 + 8 * q^84 + 41 * q^86 + 33 * q^87 + 4 * q^88 + 43 * q^89 + 11 * q^92 - 28 * q^93 + 5 * q^94 + 7 * q^96 + 6 * q^97 - 10 * q^98 + 16 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	3.41573	1.97208	0.986038	−	0.166522i	$-0.0532538\pi$
$3$	3.41573	1.97208	0.986038	+	0.166522i	$0.0532538\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.41573	1.39447
$7$	−0.459770	−0.173777	−0.0868883	−	0.996218i	$-0.527692\pi$
$7$	−0.459770	−0.173777	−0.0868883	+	0.996218i	$0.527692\pi$
$8$	1.00000	0.353553
$9$	8.66724	2.88908
$10$	0	0
$11$	−2.03092	−0.612346	−0.306173	−	0.951976i	$-0.599049\pi$
$11$	−2.03092	−0.612346	−0.306173	+	0.951976i	$0.599049\pi$
$12$	3.41573	0.986038
$13$	0	0
$13$	0	0
$14$	−0.459770	−0.122879
$15$	0	0
$16$	1.00000	0.250000
$17$	5.39187	1.30772	0.653860	−	0.756616i	$-0.273149\pi$
$17$	5.39187	1.30772	0.653860	+	0.756616i	$0.273149\pi$
$18$	8.66724	2.04289
$19$	3.30639	0.758537	0.379269	−	0.925287i	$-0.376176\pi$
$19$	3.30639	0.758537	0.379269	+	0.925287i	$0.376176\pi$
$20$	0	0
$21$	−1.57045	−0.342701
$22$	−2.03092	−0.432994
$23$	−4.15544	−0.866468	−0.433234	−	0.901281i	$-0.642628\pi$
$23$	−4.15544	−0.866468	−0.433234	+	0.901281i	$0.642628\pi$
$24$	3.41573	0.697234
$25$	0	0
$26$	0	0
$27$	19.3578	3.72541
$28$	−0.459770	−0.0868883
$29$	6.05566	1.12451	0.562254	−	0.826965i	$-0.309934\pi$
$29$	6.05566	1.12451	0.562254	+	0.826965i	$0.309934\pi$
$30$	0	0
$31$	−6.52262	−1.17150	−0.585749	−	0.810493i	$-0.699200\pi$
$31$	−6.52262	−1.17150	−0.585749	+	0.810493i	$0.699200\pi$
$32$	1.00000	0.176777
$33$	−6.93709	−1.20759
$34$	5.39187	0.924698
$35$	0	0
$36$	8.66724	1.44454
$37$	−8.07307	−1.32720	−0.663602	−	0.748086i	$-0.730973\pi$
$37$	−8.07307	−1.32720	−0.663602	+	0.748086i	$0.730973\pi$
$38$	3.30639	0.536367
$39$	0	0
$40$	0	0
$41$	4.14511	0.647358	0.323679	−	0.946167i	$-0.395080\pi$
$41$	4.14511	0.647358	0.323679	+	0.946167i	$0.395080\pi$
$42$	−1.57045	−0.242326
$43$	1.26725	0.193253	0.0966265	−	0.995321i	$-0.469195\pi$
$43$	1.26725	0.193253	0.0966265	+	0.995321i	$0.469195\pi$
$44$	−2.03092	−0.306173
$45$	0	0
$46$	−4.15544	−0.612686
$47$	−1.29310	−0.188618	−0.0943088	−	0.995543i	$-0.530064\pi$
$47$	−1.29310	−0.188618	−0.0943088	+	0.995543i	$0.530064\pi$
$48$	3.41573	0.493019
$49$	−6.78861	−0.969802
$50$	0	0
$51$	18.4172	2.57892
$52$	0	0
$53$	5.78588	0.794751	0.397376	−	0.917656i	$-0.369921\pi$
$53$	5.78588	0.794751	0.397376	+	0.917656i	$0.369921\pi$
$54$	19.3578	2.63426
$55$	0	0
$56$	−0.459770	−0.0614393
$57$	11.2937	1.49589
$58$	6.05566	0.795147
$59$	6.77070	0.881470	0.440735	−	0.897637i	$-0.354718\pi$
$59$	6.77070	0.881470	0.440735	+	0.897637i	$0.354718\pi$
$60$	0	0
$61$	−5.98450	−0.766237	−0.383119	−	0.923699i	$-0.625150\pi$
$61$	−5.98450	−0.766237	−0.383119	+	0.923699i	$0.625150\pi$
$62$	−6.52262	−0.828374
$63$	−3.98494	−0.502055
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.93709	−0.853896
$67$	8.41427	1.02797	0.513983	−	0.857800i	$-0.328169\pi$
$67$	8.41427	1.02797	0.513983	+	0.857800i	$0.328169\pi$
$68$	5.39187	0.653860
$69$	−14.1939	−1.70874
$70$	0	0
$71$	1.47782	0.175385	0.0876927	−	0.996148i	$-0.472051\pi$
$71$	1.47782	0.175385	0.0876927	+	0.996148i	$0.472051\pi$
$72$	8.66724	1.02144
$73$	8.05573	0.942852	0.471426	−	0.881906i	$-0.343739\pi$
$73$	8.05573	0.942852	0.471426	+	0.881906i	$0.343739\pi$
$74$	−8.07307	−0.938475
$75$	0	0
$76$	3.30639	0.379269
$77$	0.933756	0.106411
$78$	0	0
$79$	−1.68287	−0.189338	−0.0946690	−	0.995509i	$-0.530179\pi$
$79$	−1.68287	−0.189338	−0.0946690	+	0.995509i	$0.530179\pi$
$80$	0	0
$81$	40.1194	4.45771
$82$	4.14511	0.457751
$83$	9.99259	1.09683	0.548415	−	0.836207i	$-0.315232\pi$
$83$	9.99259	1.09683	0.548415	+	0.836207i	$0.315232\pi$
$84$	−1.57045	−0.171350
$85$	0	0
$86$	1.26725	0.136651
$87$	20.6845	2.21761
$88$	−2.03092	−0.216497
$89$	−3.93934	−0.417569	−0.208785	−	0.977962i	$-0.566951\pi$
$89$	−3.93934	−0.417569	−0.208785	+	0.977962i	$0.566951\pi$
$90$	0	0
$91$	0	0
$92$	−4.15544	−0.433234
$93$	−22.2796	−2.31028
$94$	−1.29310	−0.133373
$95$	0	0
$96$	3.41573	0.348617
$97$	−7.05844	−0.716676	−0.358338	−	0.933592i	$-0.616657\pi$
$97$	−7.05844	−0.716676	−0.358338	+	0.933592i	$0.616657\pi$
$98$	−6.78861	−0.685753
$99$	−17.6025	−1.76912
$100$	0	0
$101$	2.37044	0.235868	0.117934	−	0.993021i	$-0.462373\pi$
$101$	2.37044	0.235868	0.117934	+	0.993021i	$0.462373\pi$
$102$	18.4172	1.82357
$103$	−6.75491	−0.665581	−0.332790	−	0.943001i	$-0.607990\pi$
$103$	−6.75491	−0.665581	−0.332790	+	0.943001i	$0.607990\pi$
$104$	0	0
$105$	0	0
$106$	5.78588	0.561974
$107$	0.0114398	0.00110593	0.000552966	−	1.00000i	$-0.499824\pi$
$107$	0.0114398	0.00110593	0.000552966		1.00000i	$0.499824\pi$
$108$	19.3578	1.86271
$109$	−3.79448	−0.363445	−0.181723	−	0.983350i	$-0.558167\pi$
$109$	−3.79448	−0.363445	−0.181723	+	0.983350i	$0.558167\pi$
$110$	0	0
$111$	−27.5755	−2.61735
$112$	−0.459770	−0.0434442
$113$	−5.57377	−0.524336	−0.262168	−	0.965022i	$-0.584438\pi$
$113$	−5.57377	−0.524336	−0.262168	+	0.965022i	$0.584438\pi$
$114$	11.2937	1.05776
$115$	0	0
$116$	6.05566	0.562254
$117$	0	0
$118$	6.77070	0.623294
$119$	−2.47902	−0.227251
$120$	0	0
$121$	−6.87536	−0.625033
$122$	−5.98450	−0.541812
$123$	14.1586	1.27664
$124$	−6.52262	−0.585749
$125$	0	0
$126$	−3.98494	−0.355006
$127$	11.4002	1.01160	0.505801	−	0.862650i	$-0.331197\pi$
$127$	11.4002	1.01160	0.505801	+	0.862650i	$0.331197\pi$
$128$	1.00000	0.0883883
$129$	4.32857	0.381110
$130$	0	0
$131$	−14.9906	−1.30974	−0.654869	−	0.755743i	$-0.727276\pi$
$131$	−14.9906	−1.30974	−0.654869	+	0.755743i	$0.727276\pi$
$132$	−6.93709	−0.603796
$133$	−1.52018	−0.131816
$134$	8.41427	0.726882
$135$	0	0
$136$	5.39187	0.462349
$137$	−13.6727	−1.16814	−0.584069	−	0.811704i	$-0.698540\pi$
$137$	−13.6727	−1.16814	−0.584069	+	0.811704i	$0.698540\pi$
$138$	−14.1939	−1.20826
$139$	11.8128	1.00195	0.500976	−	0.865461i	$-0.332974\pi$
$139$	11.8128	1.00195	0.500976	+	0.865461i	$0.332974\pi$
$140$	0	0
$141$	−4.41688	−0.371968
$142$	1.47782	0.124016
$143$	0	0
$144$	8.66724	0.722270
$145$	0	0
$146$	8.05573	0.666697
$147$	−23.1881	−1.91252
$148$	−8.07307	−0.663602
$149$	21.3478	1.74888	0.874439	−	0.485136i	$-0.161230\pi$
$149$	21.3478	1.74888	0.874439	+	0.485136i	$0.161230\pi$
$150$	0	0
$151$	−22.9456	−1.86729	−0.933644	−	0.358203i	$-0.883390\pi$
$151$	−22.9456	−1.86729	−0.933644	+	0.358203i	$0.883390\pi$
$152$	3.30639	0.268183
$153$	46.7326	3.77811
$154$	0.933756	0.0752442
$155$	0	0
$156$	0	0
$157$	−15.2264	−1.21520	−0.607599	−	0.794244i	$-0.707867\pi$
$157$	−15.2264	−1.21520	−0.607599	+	0.794244i	$0.707867\pi$
$158$	−1.68287	−0.133882
$159$	19.7630	1.56731
$160$	0	0
$161$	1.91054	0.150572
$162$	40.1194	3.15208
$163$	−5.47677	−0.428974	−0.214487	−	0.976727i	$-0.568808\pi$
$163$	−5.47677	−0.428974	−0.214487	+	0.976727i	$0.568808\pi$
$164$	4.14511	0.323679
$165$	0	0
$166$	9.99259	0.775575
$167$	2.94085	0.227569	0.113785	−	0.993505i	$-0.463703\pi$
$167$	2.94085	0.227569	0.113785	+	0.993505i	$0.463703\pi$
$168$	−1.57045	−0.121163
$169$	0	0
$170$	0	0
$171$	28.6573	2.19148
$172$	1.26725	0.0966265
$173$	−3.88650	−0.295485	−0.147742	−	0.989026i	$-0.547201\pi$
$173$	−3.88650	−0.295485	−0.147742	+	0.989026i	$0.547201\pi$
$174$	20.6845	1.56809
$175$	0	0
$176$	−2.03092	−0.153086
$177$	23.1269	1.73833
$178$	−3.93934	−0.295266
$179$	10.3238	0.771636	0.385818	−	0.922575i	$-0.373919\pi$
$179$	10.3238	0.771636	0.385818	+	0.922575i	$0.373919\pi$
$180$	0	0
$181$	3.68088	0.273598	0.136799	−	0.990599i	$-0.456319\pi$
$181$	3.68088	0.273598	0.136799	+	0.990599i	$0.456319\pi$
$182$	0	0
$183$	−20.4415	−1.51108
$184$	−4.15544	−0.306343
$185$	0	0
$186$	−22.2796	−1.63362
$187$	−10.9505	−0.800777
$188$	−1.29310	−0.0943088
$189$	−8.90013	−0.647389
$190$	0	0
$191$	25.0076	1.80948	0.904742	−	0.425959i	$-0.140063\pi$
$191$	25.0076	1.80948	0.904742	+	0.425959i	$0.140063\pi$
$192$	3.41573	0.246509
$193$	1.97401	0.142093	0.0710463	−	0.997473i	$-0.477366\pi$
$193$	1.97401	0.142093	0.0710463	+	0.997473i	$0.477366\pi$
$194$	−7.05844	−0.506767
$195$	0	0
$196$	−6.78861	−0.484901
$197$	−17.6446	−1.25712	−0.628562	−	0.777760i	$-0.716356\pi$
$197$	−17.6446	−1.25712	−0.628562	+	0.777760i	$0.716356\pi$
$198$	−17.6025	−1.25095
$199$	−16.1718	−1.14639	−0.573193	−	0.819421i	$-0.694295\pi$
$199$	−16.1718	−1.14639	−0.573193	+	0.819421i	$0.694295\pi$
$200$	0	0
$201$	28.7409	2.02723
$202$	2.37044	0.166784
$203$	−2.78421	−0.195413
$204$	18.4172	1.28946
$205$	0	0
$206$	−6.75491	−0.470637
$207$	−36.0162	−2.50330
$208$	0	0
$209$	−6.71501	−0.464487
$210$	0	0
$211$	−13.3757	−0.920819	−0.460410	−	0.887707i	$-0.652297\pi$
$211$	−13.3757	−0.920819	−0.460410	+	0.887707i	$0.652297\pi$
$212$	5.78588	0.397376
$213$	5.04785	0.345873
$214$	0.0114398	0.000782011 0
$215$	0	0
$216$	19.3578	1.31713
$217$	2.99890	0.203579
$218$	−3.79448	−0.256995
$219$	27.5162	1.85937
$220$	0	0
$221$	0	0
$222$	−27.5755	−1.85074
$223$	−3.02345	−0.202465	−0.101233	−	0.994863i	$-0.532279\pi$
$223$	−3.02345	−0.202465	−0.101233	+	0.994863i	$0.532279\pi$
$224$	−0.459770	−0.0307197
$225$	0	0
$226$	−5.57377	−0.370762
$227$	−14.6074	−0.969525	−0.484762	−	0.874646i	$-0.661094\pi$
$227$	−14.6074	−0.969525	−0.484762	+	0.874646i	$0.661094\pi$
$228$	11.2937	0.747946
$229$	14.0176	0.926308	0.463154	−	0.886278i	$-0.346718\pi$
$229$	14.0176	0.926308	0.463154	+	0.886278i	$0.346718\pi$
$230$	0	0
$231$	3.18946	0.209851
$232$	6.05566	0.397573
$233$	−26.5904	−1.74200	−0.870998	−	0.491286i	$-0.836527\pi$
$233$	−26.5904	−1.74200	−0.870998	+	0.491286i	$0.836527\pi$
$234$	0	0
$235$	0	0
$236$	6.77070	0.440735
$237$	−5.74825	−0.373389
$238$	−2.47902	−0.160691
$239$	−12.3423	−0.798356	−0.399178	−	0.916874i	$-0.630704\pi$
$239$	−12.3423	−0.798356	−0.399178	+	0.916874i	$0.630704\pi$
$240$	0	0
$241$	3.79308	0.244334	0.122167	−	0.992510i	$-0.461016\pi$
$241$	3.79308	0.244334	0.122167	+	0.992510i	$0.461016\pi$
$242$	−6.87536	−0.441965
$243$	78.9638	5.06553
$244$	−5.98450	−0.383119
$245$	0	0
$246$	14.1586	0.902719
$247$	0	0
$248$	−6.52262	−0.414187
$249$	34.1320	2.16303
$250$	0	0
$251$	−0.547631	−0.0345662	−0.0172831	−	0.999851i	$-0.505502\pi$
$251$	−0.547631	−0.0345662	−0.0172831	+	0.999851i	$0.505502\pi$
$252$	−3.98494	−0.251027
$253$	8.43936	0.530578
$254$	11.4002	0.715311
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.4033	1.08559	0.542793	−	0.839866i	$-0.317367\pi$
$257$	17.4033	1.08559	0.542793	+	0.839866i	$0.317367\pi$
$258$	4.32857	0.269485
$259$	3.71175	0.230637
$260$	0	0
$261$	52.4859	3.24879
$262$	−14.9906	−0.926124
$263$	11.6459	0.718114	0.359057	−	0.933316i	$-0.383098\pi$
$263$	11.6459	0.718114	0.359057	+	0.933316i	$0.383098\pi$
$264$	−6.93709	−0.426948
$265$	0	0
$266$	−1.52018	−0.0932080
$267$	−13.4557	−0.823478
$268$	8.41427	0.513983
$269$	−16.9075	−1.03087	−0.515434	−	0.856929i	$-0.672369\pi$
$269$	−16.9075	−1.03087	−0.515434	+	0.856929i	$0.672369\pi$
$270$	0	0
$271$	−9.94128	−0.603889	−0.301945	−	0.953325i	$-0.597636\pi$
$271$	−9.94128	−0.603889	−0.301945	+	0.953325i	$0.597636\pi$
$272$	5.39187	0.326930
$273$	0	0
$274$	−13.6727	−0.825999
$275$	0	0
$276$	−14.1939	−0.854370
$277$	−4.24476	−0.255043	−0.127521	−	0.991836i	$-0.540702\pi$
$277$	−4.24476	−0.255043	−0.127521	+	0.991836i	$0.540702\pi$
$278$	11.8128	0.708487
$279$	−56.5332	−3.38455
$280$	0	0
$281$	24.6567	1.47090	0.735449	−	0.677580i	$-0.236971\pi$
$281$	24.6567	1.47090	0.735449	+	0.677580i	$0.236971\pi$
$282$	−4.41688	−0.263021
$283$	9.64034	0.573059	0.286529	−	0.958071i	$-0.407498\pi$
$283$	9.64034	0.573059	0.286529	+	0.958071i	$0.407498\pi$
$284$	1.47782	0.0876927
$285$	0	0
$286$	0	0
$287$	−1.90580	−0.112496
$288$	8.66724	0.510722
$289$	12.0722	0.710131
$290$	0	0
$291$	−24.1098	−1.41334
$292$	8.05573	0.471426
$293$	−6.02494	−0.351981	−0.175990	−	0.984392i	$-0.556313\pi$
$293$	−6.02494	−0.351981	−0.175990	+	0.984392i	$0.556313\pi$
$294$	−23.1881	−1.35236
$295$	0	0
$296$	−8.07307	−0.469238
$297$	−39.3142	−2.28124
$298$	21.3478	1.23664
$299$	0	0
$300$	0	0
$301$	−0.582641	−0.0335829
$302$	−22.9456	−1.32037
$303$	8.09679	0.465149
$304$	3.30639	0.189634
$305$	0	0
$306$	46.7326	2.67153
$307$	−24.5247	−1.39970	−0.699850	−	0.714290i	$-0.746750\pi$
$307$	−24.5247	−1.39970	−0.699850	+	0.714290i	$0.746750\pi$
$308$	0.933756	0.0532057
$309$	−23.0730	−1.31258
$310$	0	0
$311$	−20.6188	−1.16918	−0.584591	−	0.811328i	$-0.698745\pi$
$311$	−20.6188	−1.16918	−0.584591	+	0.811328i	$0.698745\pi$
$312$	0	0
$313$	7.35529	0.415746	0.207873	−	0.978156i	$-0.433346\pi$
$313$	7.35529	0.415746	0.207873	+	0.978156i	$0.433346\pi$
$314$	−15.2264	−0.859275
$315$	0	0
$316$	−1.68287	−0.0946690
$317$	−17.4476	−0.979954	−0.489977	−	0.871735i	$-0.662995\pi$
$317$	−17.4476	−0.979954	−0.489977	+	0.871735i	$0.662995\pi$
$318$	19.7630	1.10826
$319$	−12.2986	−0.688587
$320$	0	0
$321$	0.0390755	0.00218098
$322$	1.91054	0.106470
$323$	17.8276	0.991954
$324$	40.1194	2.22885
$325$	0	0
$326$	−5.47677	−0.303330
$327$	−12.9609	−0.716742
$328$	4.14511	0.228875
$329$	0.594527	0.0327773
$330$	0	0
$331$	−18.5906	−1.02183	−0.510917	−	0.859630i	$-0.670694\pi$
$331$	−18.5906	−1.02183	−0.510917	+	0.859630i	$0.670694\pi$
$332$	9.99259	0.548415
$333$	−69.9713	−3.83440
$334$	2.94085	0.160916
$335$	0	0
$336$	−1.57045	−0.0856752
$337$	11.2984	0.615463	0.307732	−	0.951473i	$-0.400430\pi$
$337$	11.2984	0.615463	0.307732	+	0.951473i	$0.400430\pi$
$338$	0	0
$339$	−19.0385	−1.03403
$340$	0	0
$341$	13.2469	0.717362
$342$	28.6573	1.54961
$343$	6.33959	0.342305
$344$	1.26725	0.0683253
$345$	0	0
$346$	−3.88650	−0.208939
$347$	−22.2403	−1.19392	−0.596961	−	0.802270i	$-0.703625\pi$
$347$	−22.2403	−1.19392	−0.596961	+	0.802270i	$0.703625\pi$
$348$	20.6845	1.10881
$349$	22.6265	1.21117	0.605585	−	0.795781i	$-0.292939\pi$
$349$	22.6265	1.21117	0.605585	+	0.795781i	$0.292939\pi$
$350$	0	0
$351$	0	0
$352$	−2.03092	−0.108248
$353$	7.66644	0.408044	0.204022	−	0.978966i	$-0.434599\pi$
$353$	7.66644	0.408044	0.204022	+	0.978966i	$0.434599\pi$
$354$	23.1269	1.22918
$355$	0	0
$356$	−3.93934	−0.208785
$357$	−8.46766	−0.448156
$358$	10.3238	0.545629
$359$	31.5137	1.66323	0.831615	−	0.555353i	$-0.187417\pi$
$359$	31.5137	1.66323	0.831615	+	0.555353i	$0.187417\pi$
$360$	0	0
$361$	−8.06780	−0.424621
$362$	3.68088	0.193463
$363$	−23.4844	−1.23261
$364$	0	0
$365$	0	0
$366$	−20.4415	−1.06849
$367$	25.5000	1.33109	0.665544	−	0.746359i	$-0.268200\pi$
$367$	25.5000	1.33109	0.665544	+	0.746359i	$0.268200\pi$
$368$	−4.15544	−0.216617
$369$	35.9267	1.87027
$370$	0	0
$371$	−2.66017	−0.138109
$372$	−22.2796	−1.15514
$373$	15.2112	0.787604	0.393802	−	0.919195i	$-0.371159\pi$
$373$	15.2112	0.787604	0.393802	+	0.919195i	$0.371159\pi$
$374$	−10.9505	−0.566235
$375$	0	0
$376$	−1.29310	−0.0666864
$377$	0	0
$378$	−8.90013	−0.457773
$379$	−17.2766	−0.887439	−0.443719	−	0.896166i	$-0.646341\pi$
$379$	−17.2766	−0.887439	−0.443719	+	0.896166i	$0.646341\pi$
$380$	0	0
$381$	38.9400	1.99496
$382$	25.0076	1.27950
$383$	−15.1262	−0.772915	−0.386458	−	0.922307i	$-0.626301\pi$
$383$	−15.1262	−0.772915	−0.386458	+	0.922307i	$0.626301\pi$
$384$	3.41573	0.174308
$385$	0	0
$386$	1.97401	0.100475
$387$	10.9835	0.558324
$388$	−7.05844	−0.358338
$389$	−32.0553	−1.62527	−0.812635	−	0.582774i	$-0.801967\pi$
$389$	−32.0553	−1.62527	−0.812635	+	0.582774i	$0.801967\pi$
$390$	0	0
$391$	−22.4056	−1.13310
$392$	−6.78861	−0.342877
$393$	−51.2040	−2.58290
$394$	−17.6446	−0.888920
$395$	0	0
$396$	−17.6025	−0.884558
$397$	−7.52503	−0.377670	−0.188835	−	0.982009i	$-0.560471\pi$
$397$	−7.52503	−0.377670	−0.188835	+	0.982009i	$0.560471\pi$
$398$	−16.1718	−0.810617
$399$	−5.19252	−0.259951
$400$	0	0
$401$	37.3359	1.86447	0.932233	−	0.361860i	$-0.117858\pi$
$401$	37.3359	1.86447	0.932233	+	0.361860i	$0.117858\pi$
$402$	28.7409	1.43347
$403$	0	0
$404$	2.37044	0.117934
$405$	0	0
$406$	−2.78421	−0.138178
$407$	16.3958	0.812708
$408$	18.4172	0.911787
$409$	19.7733	0.977729	0.488864	−	0.872360i	$-0.337411\pi$
$409$	19.7733	0.977729	0.488864	+	0.872360i	$0.337411\pi$
$410$	0	0
$411$	−46.7024	−2.30366
$412$	−6.75491	−0.332790
$413$	−3.11296	−0.153179
$414$	−36.0162	−1.77010
$415$	0	0
$416$	0	0
$417$	40.3495	1.97592
$418$	−6.71501	−0.328442
$419$	−12.4677	−0.609086	−0.304543	−	0.952499i	$-0.598504\pi$
$419$	−12.4677	−0.609086	−0.304543	+	0.952499i	$0.598504\pi$
$420$	0	0
$421$	1.36944	0.0667423	0.0333712	−	0.999443i	$-0.489376\pi$
$421$	1.36944	0.0667423	0.0333712	+	0.999443i	$0.489376\pi$
$422$	−13.3757	−0.651118
$423$	−11.2076	−0.544932
$424$	5.78588	0.280987
$425$	0	0
$426$	5.04785	0.244569
$427$	2.75149	0.133154
$428$	0.0114398	0.000552966 0
$429$	0	0
$430$	0	0
$431$	−11.1142	−0.535354	−0.267677	−	0.963509i	$-0.586256\pi$
$431$	−11.1142	−0.535354	−0.267677	+	0.963509i	$0.586256\pi$
$432$	19.3578	0.931353
$433$	30.2038	1.45150	0.725752	−	0.687957i	$-0.241492\pi$
$433$	30.2038	1.45150	0.725752	+	0.687957i	$0.241492\pi$
$434$	2.99890	0.143952
$435$	0	0
$436$	−3.79448	−0.181723
$437$	−13.7395	−0.657249
$438$	27.5162	1.31478
$439$	30.0084	1.43223	0.716113	−	0.697985i	$-0.245920\pi$
$439$	30.0084	1.43223	0.716113	+	0.697985i	$0.245920\pi$
$440$	0	0
$441$	−58.8386	−2.80184
$442$	0	0
$443$	−10.0957	−0.479661	−0.239830	−	0.970815i	$-0.577092\pi$
$443$	−10.0957	−0.479661	−0.239830	+	0.970815i	$0.577092\pi$
$444$	−27.5755	−1.30867
$445$	0	0
$446$	−3.02345	−0.143165
$447$	72.9183	3.44892
$448$	−0.459770	−0.0217221
$449$	−28.9563	−1.36653	−0.683266	−	0.730169i	$-0.739441\pi$
$449$	−28.9563	−1.36653	−0.683266	+	0.730169i	$0.739441\pi$
$450$	0	0
$451$	−8.41839	−0.396407
$452$	−5.57377	−0.262168
$453$	−78.3761	−3.68243
$454$	−14.6074	−0.685558
$455$	0	0
$456$	11.2937	0.528878
$457$	40.4620	1.89273	0.946367	−	0.323094i	$-0.104723\pi$
$457$	40.4620	1.89273	0.946367	+	0.323094i	$0.104723\pi$
$458$	14.0176	0.654998
$459$	104.375	4.87179
$460$	0	0
$461$	−6.39717	−0.297946	−0.148973	−	0.988841i	$-0.547597\pi$
$461$	−6.39717	−0.297946	−0.148973	+	0.988841i	$0.547597\pi$
$462$	3.18946	0.148387
$463$	15.3751	0.714541	0.357270	−	0.934001i	$-0.383707\pi$
$463$	15.3751	0.714541	0.357270	+	0.934001i	$0.383707\pi$
$464$	6.05566	0.281127
$465$	0	0
$466$	−26.5904	−1.23178
$467$	9.41379	0.435618	0.217809	−	0.975991i	$-0.430109\pi$
$467$	9.41379	0.435618	0.217809	+	0.975991i	$0.430109\pi$
$468$	0	0
$469$	−3.86863	−0.178637
$470$	0	0
$471$	−52.0093	−2.39646
$472$	6.77070	0.311647
$473$	−2.57367	−0.118338
$474$	−5.74825	−0.264026
$475$	0	0
$476$	−2.47902	−0.113626
$477$	50.1476	2.29610
$478$	−12.3423	−0.564523
$479$	−24.7651	−1.13155	−0.565774	−	0.824560i	$-0.691423\pi$
$479$	−24.7651	−1.13155	−0.565774	+	0.824560i	$0.691423\pi$
$480$	0	0
$481$	0	0
$482$	3.79308	0.172770
$483$	6.52591	0.296939
$484$	−6.87536	−0.312516
$485$	0	0
$486$	78.9638	3.58187
$487$	−27.2037	−1.23272	−0.616358	−	0.787466i	$-0.711393\pi$
$487$	−27.2037	−1.23272	−0.616358	+	0.787466i	$0.711393\pi$
$488$	−5.98450	−0.270906
$489$	−18.7072	−0.845969
$490$	0	0
$491$	−27.4578	−1.23915	−0.619576	−	0.784937i	$-0.712695\pi$
$491$	−27.4578	−1.23915	−0.619576	+	0.784937i	$0.712695\pi$
$492$	14.1586	0.638319
$493$	32.6513	1.47054
$494$	0	0
$495$	0	0
$496$	−6.52262	−0.292874
$497$	−0.679459	−0.0304779
$498$	34.1320	1.52949
$499$	−2.92681	−0.131022	−0.0655109	−	0.997852i	$-0.520868\pi$
$499$	−2.92681	−0.131022	−0.0655109	+	0.997852i	$0.520868\pi$
$500$	0	0
$501$	10.0451	0.448784
$502$	−0.547631	−0.0244420
$503$	−0.113123	−0.00504392	−0.00252196	−	0.999997i	$-0.500803\pi$
$503$	−0.113123	−0.00504392	−0.00252196	+	0.999997i	$0.500803\pi$
$504$	−3.98494	−0.177503
$505$	0	0
$506$	8.43936	0.375175
$507$	0	0
$508$	11.4002	0.505801
$509$	−19.0806	−0.845731	−0.422865	−	0.906193i	$-0.638976\pi$
$509$	−19.0806	−0.845731	−0.422865	+	0.906193i	$0.638976\pi$
$510$	0	0
$511$	−3.70378	−0.163846
$512$	1.00000	0.0441942
$513$	64.0044	2.82586
$514$	17.4033	0.767626
$515$	0	0
$516$	4.32857	0.190555
$517$	2.62618	0.115499
$518$	3.71175	0.163085
$519$	−13.2752	−0.582718
$520$	0	0
$521$	−6.88125	−0.301473	−0.150736	−	0.988574i	$-0.548164\pi$
$521$	−6.88125	−0.301473	−0.150736	+	0.988574i	$0.548164\pi$
$522$	52.4859	2.29724
$523$	19.7227	0.862414	0.431207	−	0.902253i	$-0.358088\pi$
$523$	19.7227	0.862414	0.431207	+	0.902253i	$0.358088\pi$
$524$	−14.9906	−0.654869
$525$	0	0
$526$	11.6459	0.507783
$527$	−35.1691	−1.53199
$528$	−6.93709	−0.301898
$529$	−5.73235	−0.249233
$530$	0	0
$531$	58.6833	2.54664
$532$	−1.52018	−0.0659080
$533$	0	0
$534$	−13.4557	−0.582287
$535$	0	0
$536$	8.41427	0.363441
$537$	35.2633	1.52173
$538$	−16.9075	−0.728934
$539$	13.7871	0.593854
$540$	0	0
$541$	41.8747	1.80034	0.900168	−	0.435543i	$-0.143444\pi$
$541$	41.8747	1.80034	0.900168	+	0.435543i	$0.143444\pi$
$542$	−9.94128	−0.427014
$543$	12.5729	0.539555
$544$	5.39187	0.231174
$545$	0	0
$546$	0	0
$547$	−28.0881	−1.20096	−0.600481	−	0.799639i	$-0.705024\pi$
$547$	−28.0881	−1.20096	−0.600481	+	0.799639i	$0.705024\pi$
$548$	−13.6727	−0.584069
$549$	−51.8692	−2.21372
$550$	0	0
$551$	20.0224	0.852981
$552$	−14.1939	−0.604131
$553$	0.773734	0.0329025
$554$	−4.24476	−0.180343
$555$	0	0
$556$	11.8128	0.500976
$557$	−29.3170	−1.24220	−0.621101	−	0.783730i	$-0.713315\pi$
$557$	−29.3170	−1.24220	−0.621101	+	0.783730i	$0.713315\pi$
$558$	−56.5332	−2.39324
$559$	0	0
$560$	0	0
$561$	−37.4038	−1.57919
$562$	24.6567	1.04008
$563$	2.10937	0.0888995	0.0444497	−	0.999012i	$-0.485847\pi$
$563$	2.10937	0.0888995	0.0444497	+	0.999012i	$0.485847\pi$
$564$	−4.41688	−0.185984
$565$	0	0
$566$	9.64034	0.405214
$567$	−18.4457	−0.774646
$568$	1.47782	0.0620081
$569$	14.0188	0.587697	0.293849	−	0.955852i	$-0.405064\pi$
$569$	14.0188	0.587697	0.293849	+	0.955852i	$0.405064\pi$
$570$	0	0
$571$	−27.5675	−1.15366	−0.576831	−	0.816864i	$-0.695711\pi$
$571$	−27.5675	−1.15366	−0.576831	+	0.816864i	$0.695711\pi$
$572$	0	0
$573$	85.4192	3.56844
$574$	−1.90580	−0.0795464
$575$	0	0
$576$	8.66724	0.361135
$577$	19.0434	0.792788	0.396394	−	0.918080i	$-0.370261\pi$
$577$	19.0434	0.792788	0.396394	+	0.918080i	$0.370261\pi$
$578$	12.0722	0.502138
$579$	6.74271	0.280217
$580$	0	0
$581$	−4.59429	−0.190603
$582$	−24.1098	−0.999382
$583$	−11.7507	−0.486663
$584$	8.05573	0.333348
$585$	0	0
$586$	−6.02494	−0.248888
$587$	−16.0453	−0.662260	−0.331130	−	0.943585i	$-0.607430\pi$
$587$	−16.0453	−0.662260	−0.331130	+	0.943585i	$0.607430\pi$
$588$	−23.1881	−0.956261
$589$	−21.5663	−0.888625
$590$	0	0
$591$	−60.2691	−2.47914
$592$	−8.07307	−0.331801
$593$	43.3422	1.77985	0.889925	−	0.456107i	$-0.150756\pi$
$593$	43.3422	1.77985	0.889925	+	0.456107i	$0.150756\pi$
$594$	−39.3142	−1.61308
$595$	0	0
$596$	21.3478	0.874439
$597$	−55.2384	−2.26076
$598$	0	0
$599$	−8.53779	−0.348845	−0.174422	−	0.984671i	$-0.555806\pi$
$599$	−8.53779	−0.348845	−0.174422	+	0.984671i	$0.555806\pi$
$600$	0	0
$601$	−31.5044	−1.28509	−0.642545	−	0.766248i	$-0.722121\pi$
$601$	−31.5044	−1.28509	−0.642545	+	0.766248i	$0.722121\pi$
$602$	−0.582641	−0.0237467
$603$	72.9285	2.96988
$604$	−22.9456	−0.933644
$605$	0	0
$606$	8.09679	0.328910
$607$	−14.8586	−0.603093	−0.301546	−	0.953451i	$-0.597503\pi$
$607$	−14.8586	−0.603093	−0.301546	+	0.953451i	$0.597503\pi$
$608$	3.30639	0.134092
$609$	−9.51012	−0.385369
$610$	0	0
$611$	0	0
$612$	46.7326	1.88905
$613$	−24.9342	−1.00708	−0.503541	−	0.863971i	$-0.667970\pi$
$613$	−24.9342	−1.00708	−0.503541	+	0.863971i	$0.667970\pi$
$614$	−24.5247	−0.989737
$615$	0	0
$616$	0.933756	0.0376221
$617$	20.7921	0.837057	0.418528	−	0.908204i	$-0.362546\pi$
$617$	20.7921	0.837057	0.418528	+	0.908204i	$0.362546\pi$
$618$	−23.0730	−0.928131
$619$	−22.8769	−0.919501	−0.459751	−	0.888048i	$-0.652061\pi$
$619$	−22.8769	−0.919501	−0.459751	+	0.888048i	$0.652061\pi$
$620$	0	0
$621$	−80.4401	−3.22795
$622$	−20.6188	−0.826737
$623$	1.81119	0.0725638
$624$	0	0
$625$	0	0
$626$	7.35529	0.293977
$627$	−22.9367	−0.916004
$628$	−15.2264	−0.607599
$629$	−43.5289	−1.73561
$630$	0	0
$631$	22.6688	0.902429	0.451214	−	0.892416i	$-0.350991\pi$
$631$	22.6688	0.902429	0.451214	+	0.892416i	$0.350991\pi$
$632$	−1.68287	−0.0669411
$633$	−45.6877	−1.81592
$634$	−17.4476	−0.692932
$635$	0	0
$636$	19.7630	0.783655
$637$	0	0
$638$	−12.2986	−0.486905
$639$	12.8087	0.506703
$640$	0	0
$641$	−11.5650	−0.456789	−0.228394	−	0.973569i	$-0.573348\pi$
$641$	−11.5650	−0.456789	−0.228394	+	0.973569i	$0.573348\pi$
$642$	0.0390755	0.00154219
$643$	−25.4887	−1.00518	−0.502588	−	0.864526i	$-0.667619\pi$
$643$	−25.4887	−1.00518	−0.502588	+	0.864526i	$0.667619\pi$
$644$	1.91054	0.0752860
$645$	0	0
$646$	17.8276	0.701418
$647$	−0.686256	−0.0269795	−0.0134897	−	0.999909i	$-0.504294\pi$
$647$	−0.686256	−0.0269795	−0.0134897	+	0.999909i	$0.504294\pi$
$648$	40.1194	1.57604
$649$	−13.7508	−0.539764
$650$	0	0
$651$	10.2435	0.401473
$652$	−5.47677	−0.214487
$653$	−20.7584	−0.812338	−0.406169	−	0.913798i	$-0.633136\pi$
$653$	−20.7584	−0.812338	−0.406169	+	0.913798i	$0.633136\pi$
$654$	−12.9609	−0.506813
$655$	0	0
$656$	4.14511	0.161839
$657$	69.8210	2.72398
$658$	0.594527	0.0231771
$659$	−23.2146	−0.904312	−0.452156	−	0.891939i	$-0.649345\pi$
$659$	−23.2146	−0.904312	−0.452156	+	0.891939i	$0.649345\pi$
$660$	0	0
$661$	−14.9740	−0.582422	−0.291211	−	0.956659i	$-0.594058\pi$
$661$	−14.9740	−0.582422	−0.291211	+	0.956659i	$0.594058\pi$
$662$	−18.5906	−0.722546
$663$	0	0
$664$	9.99259	0.387788
$665$	0	0
$666$	−69.9713	−2.71133
$667$	−25.1639	−0.974350
$668$	2.94085	0.113785
$669$	−10.3273	−0.399277
$670$	0	0
$671$	12.1541	0.469202
$672$	−1.57045	−0.0605815
$673$	17.2347	0.664348	0.332174	−	0.943218i	$-0.392218\pi$
$673$	17.2347	0.664348	0.332174	+	0.943218i	$0.392218\pi$
$674$	11.2984	0.435198
$675$	0	0
$676$	0	0
$677$	12.1285	0.466136	0.233068	−	0.972460i	$-0.425124\pi$
$677$	12.1285	0.466136	0.233068	+	0.972460i	$0.425124\pi$
$678$	−19.0385	−0.731170
$679$	3.24526	0.124542
$680$	0	0
$681$	−49.8949	−1.91198
$682$	13.2469	0.507251
$683$	20.0828	0.768449	0.384224	−	0.923240i	$-0.374469\pi$
$683$	20.0828	0.768449	0.384224	+	0.923240i	$0.374469\pi$
$684$	28.6573	1.09574
$685$	0	0
$686$	6.33959	0.242047
$687$	47.8803	1.82675
$688$	1.26725	0.0483133
$689$	0	0
$690$	0	0
$691$	−24.8495	−0.945320	−0.472660	−	0.881245i	$-0.656706\pi$
$691$	−24.8495	−0.945320	−0.472660	+	0.881245i	$0.656706\pi$
$692$	−3.88650	−0.147742
$693$	8.09309	0.307431
$694$	−22.2403	−0.844231
$695$	0	0
$696$	20.6845	0.784045
$697$	22.3499	0.846562
$698$	22.6265	0.856427
$699$	−90.8258	−3.43535
$700$	0	0
$701$	−26.3896	−0.996720	−0.498360	−	0.866970i	$-0.666064\pi$
$701$	−26.3896	−0.996720	−0.498360	+	0.866970i	$0.666064\pi$
$702$	0	0
$703$	−26.6927	−1.00673
$704$	−2.03092	−0.0765432
$705$	0	0
$706$	7.66644	0.288530
$707$	−1.08986	−0.0409883
$708$	23.1269	0.869163
$709$	28.0118	1.05201	0.526003	−	0.850483i	$-0.323690\pi$
$709$	28.0118	1.05201	0.526003	+	0.850483i	$0.323690\pi$
$710$	0	0
$711$	−14.5859	−0.547013
$712$	−3.93934	−0.147633
$713$	27.1043	1.01507
$714$	−8.46766	−0.316894
$715$	0	0
$716$	10.3238	0.385818
$717$	−42.1580	−1.57442
$718$	31.5137	1.17608
$719$	−40.0185	−1.49244	−0.746219	−	0.665700i	$-0.768133\pi$
$719$	−40.0185	−1.49244	−0.746219	+	0.665700i	$0.768133\pi$
$720$	0	0
$721$	3.10570	0.115662
$722$	−8.06780	−0.300252
$723$	12.9562	0.481845
$724$	3.68088	0.136799
$725$	0	0
$726$	−23.4844	−0.871588
$727$	37.8003	1.40193	0.700967	−	0.713194i	$-0.252752\pi$
$727$	37.8003	1.40193	0.700967	+	0.713194i	$0.252752\pi$
$728$	0	0
$729$	149.361	5.53189
$730$	0	0
$731$	6.83282	0.252721
$732$	−20.4415	−0.755539
$733$	−6.65212	−0.245701	−0.122851	−	0.992425i	$-0.539204\pi$
$733$	−6.65212	−0.245701	−0.122851	+	0.992425i	$0.539204\pi$
$734$	25.5000	0.941221
$735$	0	0
$736$	−4.15544	−0.153171
$737$	−17.0887	−0.629471
$738$	35.9267	1.32248
$739$	28.4479	1.04647	0.523236	−	0.852188i	$-0.324725\pi$
$739$	28.4479	1.04647	0.523236	+	0.852188i	$0.324725\pi$
$740$	0	0
$741$	0	0
$742$	−2.66017	−0.0976580
$743$	16.3224	0.598809	0.299405	−	0.954126i	$-0.403212\pi$
$743$	16.3224	0.598809	0.299405	+	0.954126i	$0.403212\pi$
$744$	−22.2796	−0.816808
$745$	0	0
$746$	15.2112	0.556920
$747$	86.6082	3.16883
$748$	−10.9505	−0.400388
$749$	−0.00525969	−0.000192185 0
$750$	0	0
$751$	−0.398628	−0.0145461	−0.00727307	−	0.999974i	$-0.502315\pi$
$751$	−0.398628	−0.0145461	−0.00727307	+	0.999974i	$0.502315\pi$
$752$	−1.29310	−0.0471544
$753$	−1.87056	−0.0681671
$754$	0	0
$755$	0	0
$756$	−8.90013	−0.323695
$757$	40.1310	1.45859	0.729293	−	0.684201i	$-0.239849\pi$
$757$	40.1310	1.45859	0.729293	+	0.684201i	$0.239849\pi$
$758$	−17.2766	−0.627514
$759$	28.8266	1.04634
$760$	0	0
$761$	3.64226	0.132032	0.0660158	−	0.997819i	$-0.478971\pi$
$761$	3.64226	0.132032	0.0660158	+	0.997819i	$0.478971\pi$
$762$	38.9400	1.41065
$763$	1.74459	0.0631583
$764$	25.0076	0.904742
$765$	0	0
$766$	−15.1262	−0.546533
$767$	0	0
$768$	3.41573	0.123255
$769$	−16.4037	−0.591533	−0.295767	−	0.955260i	$-0.595575\pi$
$769$	−16.4037	−0.591533	−0.295767	+	0.955260i	$0.595575\pi$
$770$	0	0
$771$	59.4450	2.14086
$772$	1.97401	0.0710463
$773$	11.3166	0.407028	0.203514	−	0.979072i	$-0.434764\pi$
$773$	11.3166	0.407028	0.203514	+	0.979072i	$0.434764\pi$
$774$	10.9835	0.394795
$775$	0	0
$776$	−7.05844	−0.253383
$777$	12.6784	0.454834
$778$	−32.0553	−1.14924
$779$	13.7053	0.491045
$780$	0	0
$781$	−3.00134	−0.107396
$782$	−22.4056	−0.801221
$783$	117.224	4.18925
$784$	−6.78861	−0.242450
$785$	0	0
$786$	−51.2040	−1.82639
$787$	−23.6730	−0.843850	−0.421925	−	0.906631i	$-0.638646\pi$
$787$	−23.6730	−0.843850	−0.421925	+	0.906631i	$0.638646\pi$
$788$	−17.6446	−0.628562
$789$	39.7791	1.41618
$790$	0	0
$791$	2.56265	0.0911174
$792$	−17.6025	−0.625477
$793$	0	0
$794$	−7.52503	−0.267053
$795$	0	0
$796$	−16.1718	−0.573193
$797$	−4.35054	−0.154104	−0.0770520	−	0.997027i	$-0.524551\pi$
$797$	−4.35054	−0.154104	−0.0770520	+	0.997027i	$0.524551\pi$
$798$	−5.19252	−0.183813
$799$	−6.97221	−0.246659
$800$	0	0
$801$	−34.1432	−1.20639
$802$	37.3359	1.31838
$803$	−16.3605	−0.577351
$804$	28.7409	1.01361
$805$	0	0
$806$	0	0
$807$	−57.7516	−2.03295
$808$	2.37044	0.0833918
$809$	−24.9033	−0.875552	−0.437776	−	0.899084i	$-0.644234\pi$
$809$	−24.9033	−0.875552	−0.437776	+	0.899084i	$0.644234\pi$
$810$	0	0
$811$	−30.1105	−1.05732	−0.528661	−	0.848833i	$-0.677306\pi$
$811$	−30.1105	−1.05732	−0.528661	+	0.848833i	$0.677306\pi$
$812$	−2.78421	−0.0977065
$813$	−33.9568	−1.19092
$814$	16.3958	0.574671
$815$	0	0
$816$	18.4172	0.644730
$817$	4.19000	0.146590
$818$	19.7733	0.691359
$819$	0	0
$820$	0	0
$821$	−33.9761	−1.18577	−0.592887	−	0.805286i	$-0.702012\pi$
$821$	−33.9761	−1.18577	−0.592887	+	0.805286i	$0.702012\pi$
$822$	−46.7024	−1.62893
$823$	18.7804	0.654644	0.327322	−	0.944913i	$-0.393854\pi$
$823$	18.7804	0.654644	0.327322	+	0.944913i	$0.393854\pi$
$824$	−6.75491	−0.235318
$825$	0	0
$826$	−3.11296	−0.108314
$827$	56.4739	1.96379	0.981895	−	0.189426i	$-0.0606627\pi$
$827$	56.4739	1.96379	0.981895	+	0.189426i	$0.0606627\pi$
$828$	−36.0162	−1.25165
$829$	37.9617	1.31846	0.659231	−	0.751940i	$-0.270882\pi$
$829$	37.9617	1.31846	0.659231	+	0.751940i	$0.270882\pi$
$830$	0	0
$831$	−14.4990	−0.502964
$832$	0	0
$833$	−36.6033	−1.26823
$834$	40.3495	1.39719
$835$	0	0
$836$	−6.71501	−0.232244
$837$	−126.264	−4.36431
$838$	−12.4677	−0.430689
$839$	−17.7895	−0.614162	−0.307081	−	0.951683i	$-0.599352\pi$
$839$	−17.7895	−0.614162	−0.307081	+	0.951683i	$0.599352\pi$
$840$	0	0
$841$	7.67098	0.264517
$842$	1.36944	0.0471940
$843$	84.2209	2.90072
$844$	−13.3757	−0.460410
$845$	0	0
$846$	−11.2076	−0.385325
$847$	3.16108	0.108616
$848$	5.78588	0.198688
$849$	32.9289	1.13012
$850$	0	0
$851$	33.5471	1.14998
$852$	5.04785	0.172937
$853$	−12.0381	−0.412178	−0.206089	−	0.978533i	$-0.566074\pi$
$853$	−12.0381	−0.412178	−0.206089	+	0.978533i	$0.566074\pi$
$854$	2.75149	0.0941542
$855$	0	0
$856$	0.0114398	0.000391006 0
$857$	−21.6745	−0.740388	−0.370194	−	0.928954i	$-0.620709\pi$
$857$	−21.6745	−0.740388	−0.370194	+	0.928954i	$0.620709\pi$
$858$	0	0
$859$	−54.1515	−1.84762	−0.923812	−	0.382846i	$-0.874944\pi$
$859$	−54.1515	−1.84762	−0.923812	+	0.382846i	$0.874944\pi$
$860$	0	0
$861$	−6.50969	−0.221850
$862$	−11.1142	−0.378553
$863$	41.6531	1.41789	0.708943	−	0.705265i	$-0.249172\pi$
$863$	41.6531	1.41789	0.708943	+	0.705265i	$0.249172\pi$
$864$	19.3578	0.658566
$865$	0	0
$866$	30.2038	1.02637
$867$	41.2355	1.40043
$868$	2.99890	0.101789
$869$	3.41778	0.115940
$870$	0	0
$871$	0	0
$872$	−3.79448	−0.128497
$873$	−61.1773	−2.07054
$874$	−13.7395	−0.464745
$875$	0	0
$876$	27.5162	0.929687
$877$	51.8500	1.75085	0.875425	−	0.483355i	$-0.160582\pi$
$877$	51.8500	1.75085	0.875425	+	0.483355i	$0.160582\pi$
$878$	30.0084	1.01274
$879$	−20.5796	−0.694133
$880$	0	0
$881$	11.0112	0.370977	0.185488	−	0.982646i	$-0.440613\pi$
$881$	11.0112	0.370977	0.185488	+	0.982646i	$0.440613\pi$
$882$	−58.8386	−1.98120
$883$	−42.0418	−1.41482	−0.707410	−	0.706803i	$-0.750137\pi$
$883$	−42.0418	−1.41482	−0.707410	+	0.706803i	$0.750137\pi$
$884$	0	0
$885$	0	0
$886$	−10.0957	−0.339171
$887$	−50.3024	−1.68899	−0.844495	−	0.535563i	$-0.820100\pi$
$887$	−50.3024	−1.68899	−0.844495	+	0.535563i	$0.820100\pi$
$888$	−27.5755	−0.925372
$889$	−5.24146	−0.175793
$890$	0	0
$891$	−81.4793	−2.72966
$892$	−3.02345	−0.101233
$893$	−4.27548	−0.143073
$894$	72.9183	2.43875
$895$	0	0
$896$	−0.459770	−0.0153598
$897$	0	0
$898$	−28.9563	−0.966284
$899$	−39.4988	−1.31736
$900$	0	0
$901$	31.1967	1.03931
$902$	−8.41839	−0.280302
$903$	−1.99015	−0.0662279
$904$	−5.57377	−0.185381
$905$	0	0
$906$	−78.3761	−2.60387
$907$	−1.54390	−0.0512642	−0.0256321	−	0.999671i	$-0.508160\pi$
$907$	−1.54390	−0.0512642	−0.0256321	+	0.999671i	$0.508160\pi$
$908$	−14.6074	−0.484762
$909$	20.5452	0.681440
$910$	0	0
$911$	25.5772	0.847411	0.423705	−	0.905800i	$-0.360729\pi$
$911$	25.5772	0.847411	0.423705	+	0.905800i	$0.360729\pi$
$912$	11.2937	0.373973
$913$	−20.2942	−0.671639
$914$	40.4620	1.33836
$915$	0	0
$916$	14.0176	0.463154
$917$	6.89224	0.227602
$918$	104.375	3.44488
$919$	49.1848	1.62246	0.811228	−	0.584730i	$-0.198799\pi$
$919$	49.1848	1.62246	0.811228	+	0.584730i	$0.198799\pi$
$920$	0	0
$921$	−83.7700	−2.76031
$922$	−6.39717	−0.210680
$923$	0	0
$924$	3.18946	0.104926
$925$	0	0
$926$	15.3751	0.505257
$927$	−58.5464	−1.92292
$928$	6.05566	0.198787
$929$	−9.91513	−0.325305	−0.162652	−	0.986683i	$-0.552005\pi$
$929$	−9.91513	−0.325305	−0.162652	+	0.986683i	$0.552005\pi$
$930$	0	0
$931$	−22.4458	−0.735631
$932$	−26.5904	−0.870998
$933$	−70.4282	−2.30572
$934$	9.41379	0.308029
$935$	0	0
$936$	0	0
$937$	−24.7485	−0.808497	−0.404248	−	0.914649i	$-0.632467\pi$
$937$	−24.7485	−0.808497	−0.404248	+	0.914649i	$0.632467\pi$
$938$	−3.86863	−0.126315
$939$	25.1237	0.819882
$940$	0	0
$941$	40.4541	1.31877	0.659383	−	0.751807i	$-0.270817\pi$
$941$	40.4541	1.31877	0.659383	+	0.751807i	$0.270817\pi$
$942$	−52.0093	−1.69455
$943$	−17.2247	−0.560915
$944$	6.77070	0.220368
$945$	0	0
$946$	−2.57367	−0.0836774
$947$	18.3367	0.595864	0.297932	−	0.954587i	$-0.403703\pi$
$947$	18.3367	0.595864	0.297932	+	0.954587i	$0.403703\pi$
$948$	−5.74825	−0.186694
$949$	0	0
$950$	0	0
$951$	−59.5963	−1.93254
$952$	−2.47902	−0.0803454
$953$	−4.21731	−0.136612	−0.0683060	−	0.997664i	$-0.521759\pi$
$953$	−4.21731	−0.136612	−0.0683060	+	0.997664i	$0.521759\pi$
$954$	50.1476	1.62359
$955$	0	0
$956$	−12.3423	−0.399178
$957$	−42.0086	−1.35795
$958$	−24.7651	−0.800125
$959$	6.28630	0.202995
$960$	0	0
$961$	11.5446	0.372407
$962$	0	0
$963$	0.0991519	0.00319512
$964$	3.79308	0.122167
$965$	0	0
$966$	6.52591	0.209968
$967$	−46.9135	−1.50864	−0.754318	−	0.656509i	$-0.772033\pi$
$967$	−46.9135	−1.50864	−0.754318	+	0.656509i	$0.772033\pi$
$968$	−6.87536	−0.220982
$969$	60.8944	1.95621
$970$	0	0
$971$	−40.5157	−1.30021	−0.650105	−	0.759844i	$-0.725275\pi$
$971$	−40.5157	−1.30021	−0.650105	+	0.759844i	$0.725275\pi$
$972$	78.9638	2.53276
$973$	−5.43119	−0.174116
$974$	−27.2037	−0.871662
$975$	0	0
$976$	−5.98450	−0.191559
$977$	−51.0911	−1.63455	−0.817275	−	0.576248i	$-0.804516\pi$
$977$	−51.0911	−1.63455	−0.817275	+	0.576248i	$0.804516\pi$
$978$	−18.7072	−0.598190
$979$	8.00049	0.255697
$980$	0	0
$981$	−32.8877	−1.05002
$982$	−27.4578	−0.876213
$983$	−47.7672	−1.52354	−0.761768	−	0.647850i	$-0.775668\pi$
$983$	−47.7672	−1.52354	−0.761768	+	0.647850i	$0.775668\pi$
$984$	14.1586	0.451360
$985$	0	0
$986$	32.6513	1.03983
$987$	2.03075	0.0646394
$988$	0	0
$989$	−5.26596	−0.167448
$990$	0	0
$991$	36.5994	1.16262	0.581309	−	0.813683i	$-0.302541\pi$
$991$	36.5994	1.16262	0.581309	+	0.813683i	$0.302541\pi$
$992$	−6.52262	−0.207094
$993$	−63.5007	−2.01513
$994$	−0.679459	−0.0215511
$995$	0	0
$996$	34.1320	1.08151
$997$	−52.9031	−1.67546	−0.837729	−	0.546087i	$-0.816117\pi$
$997$	−52.9031	−1.67546	−0.837729	+	0.546087i	$0.816117\pi$
$998$	−2.92681	−0.0926464
$999$	−156.277	−4.94438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.da.1.9		9
5.2	odd	4		1690.2.b.g.339.10	yes	18
5.3	odd	4		1690.2.b.g.339.9	yes	18
5.4	even	2		8450.2.a.ct.1.1		9
13.12	even	2		8450.2.a.cw.1.9		9
65.8	even	4		1690.2.c.h.1689.18		18
65.12	odd	4		1690.2.b.f.339.1	✓	18
65.18	even	4		1690.2.c.g.1689.18		18
65.38	odd	4		1690.2.b.f.339.18	yes	18
65.47	even	4		1690.2.c.g.1689.1		18
65.57	even	4		1690.2.c.h.1689.1		18
65.64	even	2		8450.2.a.cx.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.b.f.339.1	✓	18	65.12	odd	4
1690.2.b.f.339.18	yes	18	65.38	odd	4
1690.2.b.g.339.9	yes	18	5.3	odd	4
1690.2.b.g.339.10	yes	18	5.2	odd	4
1690.2.c.g.1689.1		18	65.47	even	4
1690.2.c.g.1689.18		18	65.18	even	4
1690.2.c.h.1689.1		18	65.57	even	4
1690.2.c.h.1689.18		18	65.8	even	4
8450.2.a.ct.1.1		9	5.4	even	2
8450.2.a.cw.1.9		9	13.12	even	2
8450.2.a.cx.1.1		9	65.64	even	2
8450.2.a.da.1.9		9	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.41573 q^{3} +1.00000 q^{4} +3.41573 q^{6} -0.459770 q^{7} +1.00000 q^{8} +8.66724 q^{9} -2.03092 q^{11} +3.41573 q^{12} -0.459770 q^{14} +1.00000 q^{16} +5.39187 q^{17} +8.66724 q^{18} +3.30639 q^{19} -1.57045 q^{21} -2.03092 q^{22} -4.15544 q^{23} +3.41573 q^{24} +19.3578 q^{27} -0.459770 q^{28} +6.05566 q^{29} -6.52262 q^{31} +1.00000 q^{32} -6.93709 q^{33} +5.39187 q^{34} +8.66724 q^{36} -8.07307 q^{37} +3.30639 q^{38} +4.14511 q^{41} -1.57045 q^{42} +1.26725 q^{43} -2.03092 q^{44} -4.15544 q^{46} -1.29310 q^{47} +3.41573 q^{48} -6.78861 q^{49} +18.4172 q^{51} +5.78588 q^{53} +19.3578 q^{54} -0.459770 q^{56} +11.2937 q^{57} +6.05566 q^{58} +6.77070 q^{59} -5.98450 q^{61} -6.52262 q^{62} -3.98494 q^{63} +1.00000 q^{64} -6.93709 q^{66} +8.41427 q^{67} +5.39187 q^{68} -14.1939 q^{69} +1.47782 q^{71} +8.66724 q^{72} +8.05573 q^{73} -8.07307 q^{74} +3.30639 q^{76} +0.933756 q^{77} -1.68287 q^{79} +40.1194 q^{81} +4.14511 q^{82} +9.99259 q^{83} -1.57045 q^{84} +1.26725 q^{86} +20.6845 q^{87} -2.03092 q^{88} -3.93934 q^{89} -4.15544 q^{92} -22.2796 q^{93} -1.29310 q^{94} +3.41573 q^{96} -7.05844 q^{97} -6.78861 q^{98} -17.6025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} + 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + 4 q^{11} + 7 q^{12} + q^{14} + 9 q^{16} + 12 q^{17} + 8 q^{18} + 6 q^{19} + 8 q^{21} + 4 q^{22} + 11 q^{23} + 7 q^{24} + 34 q^{27}+ \cdots + 16 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + 7 * q^3 + 9 * q^4 + 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 + 4 * q^11 + 7 * q^12 + q^14 + 9 * q^16 + 12 * q^17 + 8 * q^18 + 6 * q^19 + 8 * q^21 + 4 * q^22 + 11 * q^23 + 7 * q^24 + 34 * q^27 + q^28 + 15 * q^29 - 6 * q^31 + 9 * q^32 - 14 * q^33 + 12 * q^34 + 8 * q^36 - 10 * q^37 + 6 * q^38 + 27 * q^41 + 8 * q^42 + 41 * q^43 + 4 * q^44 + 11 * q^46 + 5 * q^47 + 7 * q^48 - 10 * q^49 + 12 * q^51 + 18 * q^53 + 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 + 20 * q^59 - 19 * q^61 - 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 + 12 * q^68 - 14 * q^69 - 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 + 6 * q^76 + 30 * q^77 - 14 * q^79 + 53 * q^81 + 27 * q^82 + q^83 + 8 * q^84 + 41 * q^86 + 33 * q^87 + 4 * q^88 + 43 * q^89 + 11 * q^92 - 28 * q^93 + 5 * q^94 + 7 * q^96 + 6 * q^97 - 10 * q^98 + 16 * q^99

Introduction

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