LMFDB - Embedded newform 7350.2.a.ds.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.16053$ of defining polynomial
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.46967 q^{11} +1.00000 q^{12} -5.88388 q^{13} +1.00000 q^{16} +7.73528 q^{17} -1.00000 q^{18} -6.61827 q^{19} -4.46967 q^{22} +2.61827 q^{23} -1.00000 q^{24} +5.88388 q^{26} +1.00000 q^{27} -8.17246 q^{29} -8.46967 q^{31} -1.00000 q^{32} +4.46967 q^{33} -7.73528 q^{34} +1.00000 q^{36} +3.18719 q^{37} +6.61827 q^{38} -5.88388 q^{39} -3.56282 q^{41} +1.43108 q^{43} +4.46967 q^{44} -2.61827 q^{46} -6.79073 q^{47} +1.00000 q^{48} +7.73528 q^{51} -5.88388 q^{52} -1.38173 q^{53} -1.00000 q^{54} -6.61827 q^{57} +8.17246 q^{58} -4.66421 q^{59} -9.79683 q^{61} +8.46967 q^{62} +1.00000 q^{64} -4.46967 q^{66} +1.85140 q^{67} +7.73528 q^{68} +2.61827 q^{69} +2.02386 q^{71} -1.00000 q^{72} -4.01687 q^{73} -3.18719 q^{74} -6.61827 q^{76} +5.88388 q^{78} -6.98527 q^{79} +1.00000 q^{81} +3.56282 q^{82} -5.35965 q^{83} -1.43108 q^{86} -8.17246 q^{87} -4.46967 q^{88} -14.3912 q^{89} +2.61827 q^{92} -8.46967 q^{93} +6.79073 q^{94} -1.00000 q^{96} +7.71966 q^{97} +4.46967 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{12} + 4 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} - 4 q^{23} - 4 q^{24} + 4 q^{27} - 8 q^{29} - 16 q^{31} - 4 q^{32} - 4 q^{34} + 4 q^{36} + 8 q^{37} + 12 q^{38} - 12 q^{41} - 4 q^{43} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 4 q^{51} - 20 q^{53} - 4 q^{54} - 12 q^{57} + 8 q^{58} - 20 q^{59} - 12 q^{61} + 16 q^{62} + 4 q^{64} + 4 q^{67} + 4 q^{68} - 4 q^{69} - 20 q^{71} - 4 q^{72} - 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} + 12 q^{82} + 8 q^{83} + 4 q^{86} - 8 q^{87} - 44 q^{89} - 4 q^{92} - 16 q^{93} - 12 q^{94} - 4 q^{96} + 20 q^{97}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^12 + 4 * q^16 + 4 * q^17 - 4 * q^18 - 12 * q^19 - 4 * q^23 - 4 * q^24 + 4 * q^27 - 8 * q^29 - 16 * q^31 - 4 * q^32 - 4 * q^34 + 4 * q^36 + 8 * q^37 + 12 * q^38 - 12 * q^41 - 4 * q^43 + 4 * q^46 + 12 * q^47 + 4 * q^48 + 4 * q^51 - 20 * q^53 - 4 * q^54 - 12 * q^57 + 8 * q^58 - 20 * q^59 - 12 * q^61 + 16 * q^62 + 4 * q^64 + 4 * q^67 + 4 * q^68 - 4 * q^69 - 20 * q^71 - 4 * q^72 - 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 + 12 * q^82 + 8 * q^83 + 4 * q^86 - 8 * q^87 - 44 * q^89 - 4 * q^92 - 16 * q^93 - 12 * q^94 - 4 * q^96 + 20 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	4.46967	1.34766	0.673828	−	0.738889i	$-0.264649\pi$
$11$	4.46967	1.34766	0.673828	+	0.738889i	$0.264649\pi$
$12$	1.00000	0.288675
$13$	−5.88388	−1.63189	−0.815947	−	0.578126i	$-0.803784\pi$
$13$	−5.88388	−1.63189	−0.815947	+	0.578126i	$0.803784\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.73528	1.87608	0.938040	−	0.346527i	$-0.112639\pi$
$17$	7.73528	1.87608	0.938040	+	0.346527i	$0.112639\pi$
$18$	−1.00000	−0.235702
$19$	−6.61827	−1.51834	−0.759168	−	0.650895i	$-0.774394\pi$
$19$	−6.61827	−1.51834	−0.759168	+	0.650895i	$0.774394\pi$
$20$	0	0
$21$	0	0
$22$	−4.46967	−0.952936
$23$	2.61827	0.545947	0.272974	−	0.962022i	$-0.411993\pi$
$23$	2.61827	0.545947	0.272974	+	0.962022i	$0.411993\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	5.88388	1.15392
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.17246	−1.51759	−0.758794	−	0.651331i	$-0.774211\pi$
$29$	−8.17246	−1.51759	−0.758794	+	0.651331i	$0.774211\pi$
$30$	0	0
$31$	−8.46967	−1.52120	−0.760598	−	0.649223i	$-0.775094\pi$
$31$	−8.46967	−1.52120	−0.760598	+	0.649223i	$0.775094\pi$
$32$	−1.00000	−0.176777
$33$	4.46967	0.778069
$34$	−7.73528	−1.32659
$35$	0	0
$36$	1.00000	0.166667
$37$	3.18719	0.523970	0.261985	−	0.965072i	$-0.415623\pi$
$37$	3.18719	0.523970	0.261985	+	0.965072i	$0.415623\pi$
$38$	6.61827	1.07363
$39$	−5.88388	−0.942175
$40$	0	0
$41$	−3.56282	−0.556419	−0.278209	−	0.960520i	$-0.589741\pi$
$41$	−3.56282	−0.556419	−0.278209	+	0.960520i	$0.589741\pi$
$42$	0	0
$43$	1.43108	0.218238	0.109119	−	0.994029i	$-0.465197\pi$
$43$	1.43108	0.218238	0.109119	+	0.994029i	$0.465197\pi$
$44$	4.46967	0.673828
$45$	0	0
$46$	−2.61827	−0.386043
$47$	−6.79073	−0.990530	−0.495265	−	0.868742i	$-0.664929\pi$
$47$	−6.79073	−0.990530	−0.495265	+	0.868742i	$0.664929\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	7.73528	1.08316
$52$	−5.88388	−0.815947
$53$	−1.38173	−0.189795	−0.0948976	−	0.995487i	$-0.530252\pi$
$53$	−1.38173	−0.189795	−0.0948976	+	0.995487i	$0.530252\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−6.61827	−0.876611
$58$	8.17246	1.07310
$59$	−4.66421	−0.607228	−0.303614	−	0.952795i	$-0.598193\pi$
$59$	−4.66421	−0.607228	−0.303614	+	0.952795i	$0.598193\pi$
$60$	0	0
$61$	−9.79683	−1.25436	−0.627178	−	0.778876i	$-0.715790\pi$
$61$	−9.79683	−1.25436	−0.627178	+	0.778876i	$0.715790\pi$
$62$	8.46967	1.07565
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.46967	−0.550178
$67$	1.85140	0.226184	0.113092	−	0.993585i	$-0.463925\pi$
$67$	1.85140	0.226184	0.113092	+	0.993585i	$0.463925\pi$
$68$	7.73528	0.938040
$69$	2.61827	0.315203
$70$	0	0
$71$	2.02386	0.240187	0.120094	−	0.992763i	$-0.461680\pi$
$71$	2.02386	0.240187	0.120094	+	0.992763i	$0.461680\pi$
$72$	−1.00000	−0.117851
$73$	−4.01687	−0.470139	−0.235069	−	0.971979i	$-0.575532\pi$
$73$	−4.01687	−0.470139	−0.235069	+	0.971979i	$0.575532\pi$
$74$	−3.18719	−0.370503
$75$	0	0
$76$	−6.61827	−0.759168
$77$	0	0
$78$	5.88388	0.666218
$79$	−6.98527	−0.785904	−0.392952	−	0.919559i	$-0.628546\pi$
$79$	−6.98527	−0.785904	−0.392952	+	0.919559i	$0.628546\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.56282	0.393447
$83$	−5.35965	−0.588298	−0.294149	−	0.955760i	$-0.595036\pi$
$83$	−5.35965	−0.588298	−0.294149	+	0.955760i	$0.595036\pi$
$84$	0	0
$85$	0	0
$86$	−1.43108	−0.154318
$87$	−8.17246	−0.876180
$88$	−4.46967	−0.476468
$89$	−14.3912	−1.52547	−0.762734	−	0.646712i	$-0.776144\pi$
$89$	−14.3912	−1.52547	−0.762734	+	0.646712i	$0.776144\pi$
$90$	0	0
$91$	0	0
$92$	2.61827	0.272974
$93$	−8.46967	−0.878263
$94$	6.79073	0.700410
$95$	0	0
$96$	−1.00000	−0.102062
$97$	7.71966	0.783813	0.391906	−	0.920005i	$-0.371816\pi$
$97$	7.71966	0.783813	0.391906	+	0.920005i	$0.371816\pi$
$98$	0	0
$99$	4.46967	0.449218
$100$	0	0
$101$	13.3544	1.32882	0.664408	−	0.747370i	$-0.268684\pi$
$101$	13.3544	1.32882	0.664408	+	0.747370i	$0.268684\pi$
$102$	−7.73528	−0.765906
$103$	9.35965	0.922233	0.461117	−	0.887339i	$-0.347449\pi$
$103$	9.35965	0.922233	0.461117	+	0.887339i	$0.347449\pi$
$104$	5.88388	0.576962
$105$	0	0
$106$	1.38173	0.134205
$107$	11.0165	1.06501	0.532503	−	0.846428i	$-0.321252\pi$
$107$	11.0165	1.06501	0.532503	+	0.846428i	$0.321252\pi$
$108$	1.00000	0.0962250
$109$	−4.69544	−0.449741	−0.224871	−	0.974389i	$-0.572196\pi$
$109$	−4.69544	−0.449741	−0.224871	+	0.974389i	$0.572196\pi$
$110$	0	0
$111$	3.18719	0.302514
$112$	0	0
$113$	−12.5723	−1.18271	−0.591353	−	0.806413i	$-0.701406\pi$
$113$	−12.5723	−1.18271	−0.591353	+	0.806413i	$0.701406\pi$
$114$	6.61827	0.619858
$115$	0	0
$116$	−8.17246	−0.758794
$117$	−5.88388	−0.543965
$118$	4.66421	0.429375
$119$	0	0
$120$	0	0
$121$	8.97792	0.816174
$122$	9.79683	0.886963
$123$	−3.56282	−0.321248
$124$	−8.46967	−0.760598
$125$	0	0
$126$	0	0
$127$	−6.59619	−0.585317	−0.292658	−	0.956217i	$-0.594540\pi$
$127$	−6.59619	−0.585317	−0.292658	+	0.956217i	$0.594540\pi$
$128$	−1.00000	−0.0883883
$129$	1.43108	0.126000
$130$	0	0
$131$	2.32106	0.202792	0.101396	−	0.994846i	$-0.467669\pi$
$131$	2.32106	0.202792	0.101396	+	0.994846i	$0.467669\pi$
$132$	4.46967	0.389035
$133$	0	0
$134$	−1.85140	−0.159936
$135$	0	0
$136$	−7.73528	−0.663294
$137$	−14.2751	−1.21961	−0.609803	−	0.792553i	$-0.708751\pi$
$137$	−14.2751	−1.21961	−0.609803	+	0.792553i	$0.708751\pi$
$138$	−2.61827	−0.222882
$139$	14.9632	1.26916	0.634581	−	0.772857i	$-0.281173\pi$
$139$	14.9632	1.26916	0.634581	+	0.772857i	$0.281173\pi$
$140$	0	0
$141$	−6.79073	−0.571883
$142$	−2.02386	−0.169838
$143$	−26.2990	−2.19923
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.01687	0.332438
$147$	0	0
$148$	3.18719	0.261985
$149$	11.1651	0.914681	0.457341	−	0.889292i	$-0.348802\pi$
$149$	11.1651	0.914681	0.457341	+	0.889292i	$0.348802\pi$
$150$	0	0
$151$	0.664208	0.0540525	0.0270263	−	0.999635i	$-0.491396\pi$
$151$	0.664208	0.0540525	0.0270263	+	0.999635i	$0.491396\pi$
$152$	6.61827	0.536813
$153$	7.73528	0.625360
$154$	0	0
$155$	0	0
$156$	−5.88388	−0.471087
$157$	8.08184	0.645001	0.322500	−	0.946569i	$-0.395477\pi$
$157$	8.08184	0.645001	0.322500	+	0.946569i	$0.395477\pi$
$158$	6.98527	0.555718
$159$	−1.38173	−0.109578
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−0.836668	−0.0655329	−0.0327664	−	0.999463i	$-0.510432\pi$
$163$	−0.836668	−0.0655329	−0.0327664	+	0.999463i	$0.510432\pi$
$164$	−3.56282	−0.278209
$165$	0	0
$166$	5.35965	0.415989
$167$	−2.66762	−0.206427	−0.103213	−	0.994659i	$-0.532912\pi$
$167$	−2.66762	−0.206427	−0.103213	+	0.994659i	$0.532912\pi$
$168$	0	0
$169$	21.6200	1.66308
$170$	0	0
$171$	−6.61827	−0.506112
$172$	1.43108	0.109119
$173$	−0.647340	−0.0492164	−0.0246082	−	0.999697i	$-0.507834\pi$
$173$	−0.647340	−0.0492164	−0.0246082	+	0.999697i	$0.507834\pi$
$174$	8.17246	0.619553
$175$	0	0
$176$	4.46967	0.336914
$177$	−4.66421	−0.350583
$178$	14.3912	1.07867
$179$	−25.7374	−1.92371	−0.961853	−	0.273566i	$-0.911797\pi$
$179$	−25.7374	−1.92371	−0.961853	+	0.273566i	$0.911797\pi$
$180$	0	0
$181$	−7.42245	−0.551707	−0.275853	−	0.961200i	$-0.588960\pi$
$181$	−7.42245	−0.551707	−0.275853	+	0.961200i	$0.588960\pi$
$182$	0	0
$183$	−9.79683	−0.724203
$184$	−2.61827	−0.193021
$185$	0	0
$186$	8.46967	0.621026
$187$	34.5741	2.52831
$188$	−6.79073	−0.495265
$189$	0	0
$190$	0	0
$191$	−1.08452	−0.0784733	−0.0392367	−	0.999230i	$-0.512493\pi$
$191$	−1.08452	−0.0784733	−0.0392367	+	0.999230i	$0.512493\pi$
$192$	1.00000	0.0721688
$193$	−8.93933	−0.643467	−0.321734	−	0.946830i	$-0.604266\pi$
$193$	−8.93933	−0.643467	−0.321734	+	0.946830i	$0.604266\pi$
$194$	−7.71966	−0.554239
$195$	0	0
$196$	0	0
$197$	−7.45890	−0.531425	−0.265712	−	0.964052i	$-0.585607\pi$
$197$	−7.45890	−0.531425	−0.265712	+	0.964052i	$0.585607\pi$
$198$	−4.46967	−0.317645
$199$	−25.8293	−1.83099	−0.915496	−	0.402328i	$-0.868201\pi$
$199$	−25.8293	−1.83099	−0.915496	+	0.402328i	$0.868201\pi$
$200$	0	0
$201$	1.85140	0.130587
$202$	−13.3544	−0.939615
$203$	0	0
$204$	7.73528	0.541578
$205$	0	0
$206$	−9.35965	−0.652118
$207$	2.61827	0.181982
$208$	−5.88388	−0.407974
$209$	−29.5815	−2.04619
$210$	0	0
$211$	5.53375	0.380959	0.190479	−	0.981691i	$-0.438996\pi$
$211$	5.53375	0.380959	0.190479	+	0.981691i	$0.438996\pi$
$212$	−1.38173	−0.0948976
$213$	2.02386	0.138672
$214$	−11.0165	−0.753073
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	4.69544	0.318015
$219$	−4.01687	−0.271435
$220$	0	0
$221$	−45.5134	−3.06157
$222$	−3.18719	−0.213910
$223$	3.28248	0.219811	0.109906	−	0.993942i	$-0.464945\pi$
$223$	3.28248	0.219811	0.109906	+	0.993942i	$0.464945\pi$
$224$	0	0
$225$	0	0
$226$	12.5723	0.836299
$227$	4.42031	0.293387	0.146693	−	0.989182i	$-0.453137\pi$
$227$	4.42031	0.293387	0.146693	+	0.989182i	$0.453137\pi$
$228$	−6.61827	−0.438306
$229$	7.97270	0.526851	0.263426	−	0.964680i	$-0.415148\pi$
$229$	7.97270	0.526851	0.263426	+	0.964680i	$0.415148\pi$
$230$	0	0
$231$	0	0
$232$	8.17246	0.536548
$233$	17.9558	1.17633	0.588163	−	0.808742i	$-0.299851\pi$
$233$	17.9558	1.17633	0.588163	+	0.808742i	$0.299851\pi$
$234$	5.88388	0.384641
$235$	0	0
$236$	−4.66421	−0.303614
$237$	−6.98527	−0.453742
$238$	0	0
$239$	5.41296	0.350135	0.175068	−	0.984556i	$-0.443986\pi$
$239$	5.41296	0.350135	0.175068	+	0.984556i	$0.443986\pi$
$240$	0	0
$241$	4.69669	0.302541	0.151270	−	0.988492i	$-0.451664\pi$
$241$	4.69669	0.302541	0.151270	+	0.988492i	$0.451664\pi$
$242$	−8.97792	−0.577122
$243$	1.00000	0.0641500
$244$	−9.79683	−0.627178
$245$	0	0
$246$	3.56282	0.227157
$247$	38.9411	2.47776
$248$	8.46967	0.537824
$249$	−5.35965	−0.339654
$250$	0	0
$251$	−7.73402	−0.488167	−0.244084	−	0.969754i	$-0.578487\pi$
$251$	−7.73402	−0.488167	−0.244084	+	0.969754i	$0.578487\pi$
$252$	0	0
$253$	11.7028	0.735748
$254$	6.59619	0.413882
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.34442	−0.0838626	−0.0419313	−	0.999120i	$-0.513351\pi$
$257$	−1.34442	−0.0838626	−0.0419313	+	0.999120i	$0.513351\pi$
$258$	−1.43108	−0.0890953
$259$	0	0
$260$	0	0
$261$	−8.17246	−0.505863
$262$	−2.32106	−0.143396
$263$	−4.36700	−0.269281	−0.134640	−	0.990895i	$-0.542988\pi$
$263$	−4.36700	−0.269281	−0.134640	+	0.990895i	$0.542988\pi$
$264$	−4.46967	−0.275089
$265$	0	0
$266$	0	0
$267$	−14.3912	−0.880730
$268$	1.85140	0.113092
$269$	−23.0671	−1.40643	−0.703213	−	0.710979i	$-0.748252\pi$
$269$	−23.0671	−1.40643	−0.703213	+	0.710979i	$0.748252\pi$
$270$	0	0
$271$	−14.7209	−0.894233	−0.447117	−	0.894476i	$-0.647549\pi$
$271$	−14.7209	−0.894233	−0.447117	+	0.894476i	$0.647549\pi$
$272$	7.73528	0.469020
$273$	0	0
$274$	14.2751	0.862392
$275$	0	0
$276$	2.61827	0.157601
$277$	−3.21104	−0.192933	−0.0964665	−	0.995336i	$-0.530754\pi$
$277$	−3.21104	−0.192933	−0.0964665	+	0.995336i	$0.530754\pi$
$278$	−14.9632	−0.897432
$279$	−8.46967	−0.507066
$280$	0	0
$281$	−7.72488	−0.460827	−0.230414	−	0.973093i	$-0.574008\pi$
$281$	−7.72488	−0.460827	−0.230414	+	0.973093i	$0.574008\pi$
$282$	6.79073	0.404382
$283$	29.5594	1.75712	0.878561	−	0.477630i	$-0.158504\pi$
$283$	29.5594	1.75712	0.878561	+	0.477630i	$0.158504\pi$
$284$	2.02386	0.120094
$285$	0	0
$286$	26.2990	1.55509
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	42.8345	2.51968
$290$	0	0
$291$	7.71966	0.452535
$292$	−4.01687	−0.235069
$293$	−1.17679	−0.0687487	−0.0343743	−	0.999409i	$-0.510944\pi$
$293$	−1.17679	−0.0687487	−0.0343743	+	0.999409i	$0.510944\pi$
$294$	0	0
$295$	0	0
$296$	−3.18719	−0.185252
$297$	4.46967	0.259356
$298$	−11.1651	−0.646777
$299$	−15.4056	−0.890928
$300$	0	0
$301$	0	0
$302$	−0.664208	−0.0382209
$303$	13.3544	0.767192
$304$	−6.61827	−0.379584
$305$	0	0
$306$	−7.73528	−0.442196
$307$	−20.3761	−1.16293	−0.581464	−	0.813572i	$-0.697520\pi$
$307$	−20.3761	−1.16293	−0.581464	+	0.813572i	$0.697520\pi$
$308$	0	0
$309$	9.35965	0.532452
$310$	0	0
$311$	18.8934	1.07135	0.535673	−	0.844425i	$-0.320058\pi$
$311$	18.8934	1.07135	0.535673	+	0.844425i	$0.320058\pi$
$312$	5.88388	0.333109
$313$	−16.4849	−0.931781	−0.465891	−	0.884842i	$-0.654266\pi$
$313$	−16.4849	−0.931781	−0.465891	+	0.884842i	$0.654266\pi$
$314$	−8.08184	−0.456084
$315$	0	0
$316$	−6.98527	−0.392952
$317$	−10.3211	−0.579689	−0.289844	−	0.957074i	$-0.593604\pi$
$317$	−10.3211	−0.579689	−0.289844	+	0.957074i	$0.593604\pi$
$318$	1.38173	0.0774836
$319$	−36.5282	−2.04518
$320$	0	0
$321$	11.0165	0.614881
$322$	0	0
$323$	−51.1941	−2.84852
$324$	1.00000	0.0555556
$325$	0	0
$326$	0.836668	0.0463387
$327$	−4.69544	−0.259658
$328$	3.56282	0.196724
$329$	0	0
$330$	0	0
$331$	−19.1906	−1.05481	−0.527405	−	0.849614i	$-0.676835\pi$
$331$	−19.1906	−1.05481	−0.527405	+	0.849614i	$0.676835\pi$
$332$	−5.35965	−0.294149
$333$	3.18719	0.174657
$334$	2.66762	0.145966
$335$	0	0
$336$	0	0
$337$	27.6035	1.50366	0.751830	−	0.659357i	$-0.229171\pi$
$337$	27.6035	1.50366	0.751830	+	0.659357i	$0.229171\pi$
$338$	−21.6200	−1.17598
$339$	−12.5723	−0.682835
$340$	0	0
$341$	−37.8566	−2.05005
$342$	6.61827	0.357875
$343$	0	0
$344$	−1.43108	−0.0771588
$345$	0	0
$346$	0.647340	0.0348012
$347$	20.3761	1.09385	0.546924	−	0.837182i	$-0.315798\pi$
$347$	20.3761	1.09385	0.546924	+	0.837182i	$0.315798\pi$
$348$	−8.17246	−0.438090
$349$	17.8281	0.954314	0.477157	−	0.878818i	$-0.341667\pi$
$349$	17.8281	0.954314	0.477157	+	0.878818i	$0.341667\pi$
$350$	0	0
$351$	−5.88388	−0.314058
$352$	−4.46967	−0.238234
$353$	−3.51347	−0.187003	−0.0935014	−	0.995619i	$-0.529806\pi$
$353$	−3.51347	−0.187003	−0.0935014	+	0.995619i	$0.529806\pi$
$354$	4.66421	0.247900
$355$	0	0
$356$	−14.3912	−0.762734
$357$	0	0
$358$	25.7374	1.36027
$359$	−2.19796	−0.116004	−0.0580018	−	0.998316i	$-0.518473\pi$
$359$	−2.19796	−0.116004	−0.0580018	+	0.998316i	$0.518473\pi$
$360$	0	0
$361$	24.8015	1.30534
$362$	7.42245	0.390116
$363$	8.97792	0.471218
$364$	0	0
$365$	0	0
$366$	9.79683	0.512089
$367$	−28.7721	−1.50189	−0.750945	−	0.660365i	$-0.770402\pi$
$367$	−28.7721	−1.50189	−0.750945	+	0.660365i	$0.770402\pi$
$368$	2.61827	0.136487
$369$	−3.56282	−0.185473
$370$	0	0
$371$	0	0
$372$	−8.46967	−0.439132
$373$	14.5689	0.754350	0.377175	−	0.926142i	$-0.376896\pi$
$373$	14.5689	0.754350	0.377175	+	0.926142i	$0.376896\pi$
$374$	−34.5741	−1.78778
$375$	0	0
$376$	6.79073	0.350205
$377$	48.0858	2.47654
$378$	0	0
$379$	−30.7868	−1.58141	−0.790705	−	0.612197i	$-0.790286\pi$
$379$	−30.7868	−1.58141	−0.790705	+	0.612197i	$0.790286\pi$
$380$	0	0
$381$	−6.59619	−0.337933
$382$	1.08452	0.0554890
$383$	−23.3410	−1.19267	−0.596334	−	0.802736i	$-0.703377\pi$
$383$	−23.3410	−1.19267	−0.596334	+	0.802736i	$0.703377\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	8.93933	0.455000
$387$	1.43108	0.0727460
$388$	7.71966	0.391906
$389$	−1.21842	−0.0617762	−0.0308881	−	0.999523i	$-0.509834\pi$
$389$	−1.21842	−0.0617762	−0.0308881	+	0.999523i	$0.509834\pi$
$390$	0	0
$391$	20.2530	1.02424
$392$	0	0
$393$	2.32106	0.117082
$394$	7.45890	0.375774
$395$	0	0
$396$	4.46967	0.224609
$397$	−8.40595	−0.421883	−0.210941	−	0.977499i	$-0.567653\pi$
$397$	−8.40595	−0.421883	−0.210941	+	0.977499i	$0.567653\pi$
$398$	25.8293	1.29471
$399$	0	0
$400$	0	0
$401$	−0.664208	−0.0331690	−0.0165845	−	0.999862i	$-0.505279\pi$
$401$	−0.664208	−0.0331690	−0.0165845	+	0.999862i	$0.505279\pi$
$402$	−1.85140	−0.0923393
$403$	49.8345	2.48243
$404$	13.3544	0.664408
$405$	0	0
$406$	0	0
$407$	14.2457	0.706131
$408$	−7.73528	−0.382953
$409$	7.38476	0.365153	0.182576	−	0.983192i	$-0.441556\pi$
$409$	7.38476	0.365153	0.182576	+	0.983192i	$0.441556\pi$
$410$	0	0
$411$	−14.2751	−0.704140
$412$	9.35965	0.461117
$413$	0	0
$414$	−2.61827	−0.128681
$415$	0	0
$416$	5.88388	0.288481
$417$	14.9632	0.732750
$418$	29.5815	1.44688
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	−13.5043	−0.658159	−0.329079	−	0.944302i	$-0.606738\pi$
$421$	−13.5043	−0.658159	−0.329079	+	0.944302i	$0.606738\pi$
$422$	−5.53375	−0.269379
$423$	−6.79073	−0.330177
$424$	1.38173	0.0671027
$425$	0	0
$426$	−2.02386	−0.0980561
$427$	0	0
$428$	11.0165	0.532503
$429$	−26.2990	−1.26973
$430$	0	0
$431$	−31.6807	−1.52601	−0.763003	−	0.646395i	$-0.776276\pi$
$431$	−31.6807	−1.52601	−0.763003	+	0.646395i	$0.776276\pi$
$432$	1.00000	0.0481125
$433$	−25.3184	−1.21672	−0.608362	−	0.793660i	$-0.708173\pi$
$433$	−25.3184	−1.21672	−0.608362	+	0.793660i	$0.708173\pi$
$434$	0	0
$435$	0	0
$436$	−4.69544	−0.224871
$437$	−17.3284	−0.828931
$438$	4.01687	0.191933
$439$	−32.8146	−1.56615	−0.783077	−	0.621924i	$-0.786351\pi$
$439$	−32.8146	−1.56615	−0.783077	+	0.621924i	$0.786351\pi$
$440$	0	0
$441$	0	0
$442$	45.5134	2.16485
$443$	−16.6421	−0.790691	−0.395346	−	0.918532i	$-0.629375\pi$
$443$	−16.6421	−0.790691	−0.395346	+	0.918532i	$0.629375\pi$
$444$	3.18719	0.151257
$445$	0	0
$446$	−3.28248	−0.155430
$447$	11.1651	0.528091
$448$	0	0
$449$	20.1759	0.952158	0.476079	−	0.879402i	$-0.342058\pi$
$449$	20.1759	0.952158	0.476079	+	0.879402i	$0.342058\pi$
$450$	0	0
$451$	−15.9246	−0.749860
$452$	−12.5723	−0.591353
$453$	0.664208	0.0312072
$454$	−4.42031	−0.207456
$455$	0	0
$456$	6.61827	0.309929
$457$	35.4124	1.65652	0.828261	−	0.560342i	$-0.189330\pi$
$457$	35.4124	1.65652	0.828261	+	0.560342i	$0.189330\pi$
$458$	−7.97270	−0.372540
$459$	7.73528	0.361052
$460$	0	0
$461$	12.5940	0.586562	0.293281	−	0.956026i	$-0.405253\pi$
$461$	12.5940	0.586562	0.293281	+	0.956026i	$0.405253\pi$
$462$	0	0
$463$	−32.5061	−1.51069	−0.755343	−	0.655330i	$-0.772530\pi$
$463$	−32.5061	−1.51069	−0.755343	+	0.655330i	$0.772530\pi$
$464$	−8.17246	−0.379397
$465$	0	0
$466$	−17.9558	−0.831788
$467$	9.20531	0.425971	0.212985	−	0.977055i	$-0.431681\pi$
$467$	9.20531	0.425971	0.212985	+	0.977055i	$0.431681\pi$
$468$	−5.88388	−0.271982
$469$	0	0
$470$	0	0
$471$	8.08184	0.372391
$472$	4.66421	0.214688
$473$	6.39646	0.294109
$474$	6.98527	0.320844
$475$	0	0
$476$	0	0
$477$	−1.38173	−0.0632651
$478$	−5.41296	−0.247583
$479$	39.4914	1.80441	0.902203	−	0.431312i	$-0.141949\pi$
$479$	39.4914	1.80441	0.902203	+	0.431312i	$0.141949\pi$
$480$	0	0
$481$	−18.7530	−0.855065
$482$	−4.69669	−0.213928
$483$	0	0
$484$	8.97792	0.408087
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	29.2678	1.32625	0.663125	−	0.748509i	$-0.269230\pi$
$487$	29.2678	1.32625	0.663125	+	0.748509i	$0.269230\pi$
$488$	9.79683	0.443482
$489$	−0.836668	−0.0378354
$490$	0	0
$491$	15.3631	0.693325	0.346663	−	0.937990i	$-0.387315\pi$
$491$	15.3631	0.693325	0.346663	+	0.937990i	$0.387315\pi$
$492$	−3.56282	−0.160624
$493$	−63.2162	−2.84712
$494$	−38.9411	−1.75204
$495$	0	0
$496$	−8.46967	−0.380299
$497$	0	0
$498$	5.35965	0.240172
$499$	20.5502	0.919955	0.459978	−	0.887930i	$-0.347857\pi$
$499$	20.5502	0.919955	0.459978	+	0.887930i	$0.347857\pi$
$500$	0	0
$501$	−2.66762	−0.119181
$502$	7.73402	0.345186
$503$	20.4788	0.913105	0.456553	−	0.889696i	$-0.349084\pi$
$503$	20.4788	0.913105	0.456553	+	0.889696i	$0.349084\pi$
$504$	0	0
$505$	0	0
$506$	−11.7028	−0.520253
$507$	21.6200	0.960180
$508$	−6.59619	−0.292658
$509$	−12.0702	−0.535001	−0.267501	−	0.963558i	$-0.586198\pi$
$509$	−12.0702	−0.535001	−0.267501	+	0.963558i	$0.586198\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−6.61827	−0.292204
$514$	1.34442	0.0592998
$515$	0	0
$516$	1.43108	0.0629999
$517$	−30.3523	−1.33489
$518$	0	0
$519$	−0.647340	−0.0284151
$520$	0	0
$521$	−14.7049	−0.644235	−0.322117	−	0.946700i	$-0.604395\pi$
$521$	−14.7049	−0.644235	−0.322117	+	0.946700i	$0.604395\pi$
$522$	8.17246	0.357699
$523$	6.44417	0.281784	0.140892	−	0.990025i	$-0.455003\pi$
$523$	6.44417	0.281784	0.140892	+	0.990025i	$0.455003\pi$
$524$	2.32106	0.101396
$525$	0	0
$526$	4.36700	0.190410
$527$	−65.5152	−2.85389
$528$	4.46967	0.194517
$529$	−16.1447	−0.701942
$530$	0	0
$531$	−4.66421	−0.202409
$532$	0	0
$533$	20.9632	0.908016
$534$	14.3912	0.622770
$535$	0	0
$536$	−1.85140	−0.0799681
$537$	−25.7374	−1.11065
$538$	23.0671	0.994494
$539$	0	0
$540$	0	0
$541$	−11.5815	−0.497926	−0.248963	−	0.968513i	$-0.580090\pi$
$541$	−11.5815	−0.497926	−0.248963	+	0.968513i	$0.580090\pi$
$542$	14.7209	0.632318
$543$	−7.42245	−0.318528
$544$	−7.73528	−0.331647
$545$	0	0
$546$	0	0
$547$	−19.2656	−0.823737	−0.411868	−	0.911243i	$-0.635124\pi$
$547$	−19.2656	−0.823737	−0.411868	+	0.911243i	$0.635124\pi$
$548$	−14.2751	−0.609803
$549$	−9.79683	−0.418119
$550$	0	0
$551$	54.0875	2.30421
$552$	−2.61827	−0.111441
$553$	0	0
$554$	3.21104	0.136424
$555$	0	0
$556$	14.9632	0.634581
$557$	−21.0404	−0.891509	−0.445754	−	0.895155i	$-0.647064\pi$
$557$	−21.0404	−0.891509	−0.445754	+	0.895155i	$0.647064\pi$
$558$	8.46967	0.358550
$559$	−8.42031	−0.356141
$560$	0	0
$561$	34.5741	1.45972
$562$	7.72488	0.325854
$563$	3.61092	0.152182	0.0760910	−	0.997101i	$-0.475756\pi$
$563$	3.61092	0.152182	0.0760910	+	0.997101i	$0.475756\pi$
$564$	−6.79073	−0.285941
$565$	0	0
$566$	−29.5594	−1.24247
$567$	0	0
$568$	−2.02386	−0.0849191
$569$	33.2145	1.39242	0.696211	−	0.717837i	$-0.254868\pi$
$569$	33.2145	1.39242	0.696211	+	0.717837i	$0.254868\pi$
$570$	0	0
$571$	−32.1776	−1.34659	−0.673296	−	0.739373i	$-0.735122\pi$
$571$	−32.1776	−1.34659	−0.673296	+	0.739373i	$0.735122\pi$
$572$	−26.2990	−1.09962
$573$	−1.08452	−0.0453066
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−34.1902	−1.42336	−0.711679	−	0.702505i	$-0.752065\pi$
$577$	−34.1902	−1.42336	−0.711679	+	0.702505i	$0.752065\pi$
$578$	−42.8345	−1.78168
$579$	−8.93933	−0.371506
$580$	0	0
$581$	0	0
$582$	−7.71966	−0.319990
$583$	−6.17587	−0.255779
$584$	4.01687	0.166219
$585$	0	0
$586$	1.17679	0.0486126
$587$	−34.5962	−1.42794	−0.713969	−	0.700178i	$-0.753104\pi$
$587$	−34.5962	−1.42794	−0.713969	+	0.700178i	$0.753104\pi$
$588$	0	0
$589$	56.0545	2.30969
$590$	0	0
$591$	−7.45890	−0.306818
$592$	3.18719	0.130993
$593$	33.0845	1.35862	0.679309	−	0.733853i	$-0.262280\pi$
$593$	33.0845	1.35862	0.679309	+	0.733853i	$0.262280\pi$
$594$	−4.46967	−0.183393
$595$	0	0
$596$	11.1651	0.457341
$597$	−25.8293	−1.05712
$598$	15.4056	0.629981
$599$	6.07485	0.248212	0.124106	−	0.992269i	$-0.460394\pi$
$599$	6.07485	0.248212	0.124106	+	0.992269i	$0.460394\pi$
$600$	0	0
$601$	11.7469	0.479167	0.239584	−	0.970876i	$-0.422989\pi$
$601$	11.7469	0.479167	0.239584	+	0.970876i	$0.422989\pi$
$602$	0	0
$603$	1.85140	0.0753947
$604$	0.664208	0.0270263
$605$	0	0
$606$	−13.3544	−0.542487
$607$	45.8180	1.85970	0.929848	−	0.367944i	$-0.119938\pi$
$607$	45.8180	1.85970	0.929848	+	0.367944i	$0.119938\pi$
$608$	6.61827	0.268406
$609$	0	0
$610$	0	0
$611$	39.9558	1.61644
$612$	7.73528	0.312680
$613$	34.2514	1.38340	0.691701	−	0.722184i	$-0.256862\pi$
$613$	34.2514	1.38340	0.691701	+	0.722184i	$0.256862\pi$
$614$	20.3761	0.822314
$615$	0	0
$616$	0	0
$617$	−38.8106	−1.56246	−0.781228	−	0.624245i	$-0.785407\pi$
$617$	−38.8106	−1.56246	−0.781228	+	0.624245i	$0.785407\pi$
$618$	−9.35965	−0.376500
$619$	4.27512	0.171832	0.0859159	−	0.996302i	$-0.472618\pi$
$619$	4.27512	0.171832	0.0859159	+	0.996302i	$0.472618\pi$
$620$	0	0
$621$	2.61827	0.105068
$622$	−18.8934	−0.757556
$623$	0	0
$624$	−5.88388	−0.235544
$625$	0	0
$626$	16.4849	0.658869
$627$	−29.5815	−1.18137
$628$	8.08184	0.322500
$629$	24.6538	0.983011
$630$	0	0
$631$	36.8419	1.46665	0.733326	−	0.679878i	$-0.237967\pi$
$631$	36.8419	1.46665	0.733326	+	0.679878i	$0.237967\pi$
$632$	6.98527	0.277859
$633$	5.53375	0.219947
$634$	10.3211	0.409902
$635$	0	0
$636$	−1.38173	−0.0547892
$637$	0	0
$638$	36.5282	1.44616
$639$	2.02386	0.0800625
$640$	0	0
$641$	49.9043	1.97110	0.985551	−	0.169382i	$-0.0541770\pi$
$641$	49.9043	1.97110	0.985551	+	0.169382i	$0.0541770\pi$
$642$	−11.0165	−0.434787
$643$	−29.0091	−1.14401	−0.572004	−	0.820251i	$-0.693834\pi$
$643$	−29.0091	−1.14401	−0.572004	+	0.820251i	$0.693834\pi$
$644$	0	0
$645$	0	0
$646$	51.1941	2.01421
$647$	−4.94329	−0.194341	−0.0971705	−	0.995268i	$-0.530979\pi$
$647$	−4.94329	−0.194341	−0.0971705	+	0.995268i	$0.530979\pi$
$648$	−1.00000	−0.0392837
$649$	−20.8475	−0.818334
$650$	0	0
$651$	0	0
$652$	−0.836668	−0.0327664
$653$	9.40559	0.368069	0.184035	−	0.982920i	$-0.441084\pi$
$653$	9.40559	0.368069	0.184035	+	0.982920i	$0.441084\pi$
$654$	4.69544	0.183606
$655$	0	0
$656$	−3.56282	−0.139105
$657$	−4.01687	−0.156713
$658$	0	0
$659$	13.4090	0.522340	0.261170	−	0.965293i	$-0.415892\pi$
$659$	13.4090	0.522340	0.261170	+	0.965293i	$0.415892\pi$
$660$	0	0
$661$	−13.2509	−0.515400	−0.257700	−	0.966225i	$-0.582965\pi$
$661$	−13.2509	−0.515400	−0.257700	+	0.966225i	$0.582965\pi$
$662$	19.1906	0.745864
$663$	−45.5134	−1.76760
$664$	5.35965	0.207995
$665$	0	0
$666$	−3.18719	−0.123501
$667$	−21.3977	−0.828522
$668$	−2.66762	−0.103213
$669$	3.28248	0.126908
$670$	0	0
$671$	−43.7886	−1.69044
$672$	0	0
$673$	21.8236	0.841237	0.420619	−	0.907238i	$-0.361813\pi$
$673$	21.8236	0.841237	0.420619	+	0.907238i	$0.361813\pi$
$674$	−27.6035	−1.06325
$675$	0	0
$676$	21.6200	0.831540
$677$	−41.0572	−1.57796	−0.788979	−	0.614421i	$-0.789390\pi$
$677$	−41.0572	−1.57796	−0.788979	+	0.614421i	$0.789390\pi$
$678$	12.5723	0.482837
$679$	0	0
$680$	0	0
$681$	4.42031	0.169387
$682$	37.8566	1.44960
$683$	31.0642	1.18864	0.594320	−	0.804229i	$-0.297421\pi$
$683$	31.0642	1.18864	0.594320	+	0.804229i	$0.297421\pi$
$684$	−6.61827	−0.253056
$685$	0	0
$686$	0	0
$687$	7.97270	0.304178
$688$	1.43108	0.0545595
$689$	8.12993	0.309726
$690$	0	0
$691$	32.0012	1.21738	0.608692	−	0.793407i	$-0.291695\pi$
$691$	32.0012	1.21738	0.608692	+	0.793407i	$0.291695\pi$
$692$	−0.647340	−0.0246082
$693$	0	0
$694$	−20.3761	−0.773468
$695$	0	0
$696$	8.17246	0.309776
$697$	−27.5594	−1.04389
$698$	−17.8281	−0.674802
$699$	17.9558	0.679152
$700$	0	0
$701$	−45.4635	−1.71713	−0.858567	−	0.512701i	$-0.828645\pi$
$701$	−45.4635	−1.71713	−0.858567	+	0.512701i	$0.828645\pi$
$702$	5.88388	0.222073
$703$	−21.0937	−0.795563
$704$	4.46967	0.168457
$705$	0	0
$706$	3.51347	0.132231
$707$	0	0
$708$	−4.66421	−0.175292
$709$	−47.7597	−1.79365	−0.896826	−	0.442384i	$-0.854133\pi$
$709$	−47.7597	−1.79365	−0.896826	+	0.442384i	$0.854133\pi$
$710$	0	0
$711$	−6.98527	−0.261968
$712$	14.3912	0.539335
$713$	−22.1759	−0.830493
$714$	0	0
$715$	0	0
$716$	−25.7374	−0.961853
$717$	5.41296	0.202151
$718$	2.19796	0.0820270
$719$	−13.1611	−0.490828	−0.245414	−	0.969418i	$-0.578924\pi$
$719$	−13.1611	−0.490828	−0.245414	+	0.969418i	$0.578924\pi$
$720$	0	0
$721$	0	0
$722$	−24.8015	−0.923016
$723$	4.69669	0.174672
$724$	−7.42245	−0.275853
$725$	0	0
$726$	−8.97792	−0.333202
$727$	8.17082	0.303039	0.151519	−	0.988454i	$-0.451583\pi$
$727$	8.17082	0.303039	0.151519	+	0.988454i	$0.451583\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.0698	0.409432
$732$	−9.79683	−0.362101
$733$	−2.01510	−0.0744293	−0.0372146	−	0.999307i	$-0.511849\pi$
$733$	−2.01510	−0.0744293	−0.0372146	+	0.999307i	$0.511849\pi$
$734$	28.7721	1.06200
$735$	0	0
$736$	−2.61827	−0.0965107
$737$	8.27512	0.304818
$738$	3.56282	0.131149
$739$	−3.82590	−0.140738	−0.0703690	−	0.997521i	$-0.522418\pi$
$739$	−3.82590	−0.140738	−0.0703690	+	0.997521i	$0.522418\pi$
$740$	0	0
$741$	38.9411	1.43054
$742$	0	0
$743$	36.6654	1.34512	0.672562	−	0.740041i	$-0.265194\pi$
$743$	36.6654	1.34512	0.672562	+	0.740041i	$0.265194\pi$
$744$	8.46967	0.310513
$745$	0	0
$746$	−14.5689	−0.533406
$747$	−5.35965	−0.196099
$748$	34.5741	1.26415
$749$	0	0
$750$	0	0
$751$	−20.4657	−0.746804	−0.373402	−	0.927670i	$-0.621809\pi$
$751$	−20.4657	−0.746804	−0.373402	+	0.927670i	$0.621809\pi$
$752$	−6.79073	−0.247632
$753$	−7.73402	−0.281843
$754$	−48.0858	−1.75118
$755$	0	0
$756$	0	0
$757$	37.9575	1.37959	0.689794	−	0.724006i	$-0.257701\pi$
$757$	37.9575	1.37959	0.689794	+	0.724006i	$0.257701\pi$
$758$	30.7868	1.11823
$759$	11.7028	0.424784
$760$	0	0
$761$	34.6131	1.25472	0.627361	−	0.778728i	$-0.284135\pi$
$761$	34.6131	1.25472	0.627361	+	0.778728i	$0.284135\pi$
$762$	6.59619	0.238955
$763$	0	0
$764$	−1.08452	−0.0392367
$765$	0	0
$766$	23.3410	0.843344
$767$	27.4436	0.990932
$768$	1.00000	0.0360844
$769$	1.48961	0.0537167	0.0268584	−	0.999639i	$-0.491450\pi$
$769$	1.48961	0.0537167	0.0268584	+	0.999639i	$0.491450\pi$
$770$	0	0
$771$	−1.34442	−0.0484181
$772$	−8.93933	−0.321734
$773$	0.992258	0.0356891	0.0178445	−	0.999841i	$-0.494320\pi$
$773$	0.992258	0.0356891	0.0178445	+	0.999841i	$0.494320\pi$
$774$	−1.43108	−0.0514392
$775$	0	0
$776$	−7.71966	−0.277120
$777$	0	0
$778$	1.21842	0.0436824
$779$	23.5797	0.844830
$780$	0	0
$781$	9.04596	0.323690
$782$	−20.2530	−0.724247
$783$	−8.17246	−0.292060
$784$	0	0
$785$	0	0
$786$	−2.32106	−0.0827896
$787$	−10.2513	−0.365418	−0.182709	−	0.983167i	$-0.558487\pi$
$787$	−10.2513	−0.365418	−0.182709	+	0.983167i	$0.558487\pi$
$788$	−7.45890	−0.265712
$789$	−4.36700	−0.155469
$790$	0	0
$791$	0	0
$792$	−4.46967	−0.158823
$793$	57.6434	2.04698
$794$	8.40595	0.298316
$795$	0	0
$796$	−25.8293	−0.915496
$797$	−32.6811	−1.15762	−0.578812	−	0.815461i	$-0.696483\pi$
$797$	−32.6811	−1.15762	−0.578812	+	0.815461i	$0.696483\pi$
$798$	0	0
$799$	−52.5282	−1.85831
$800$	0	0
$801$	−14.3912	−0.508490
$802$	0.664208	0.0234540
$803$	−17.9541	−0.633585
$804$	1.85140	0.0652937
$805$	0	0
$806$	−49.8345	−1.75535
$807$	−23.0671	−0.812001
$808$	−13.3544	−0.469807
$809$	22.4515	0.789354	0.394677	−	0.918820i	$-0.370856\pi$
$809$	22.4515	0.789354	0.394677	+	0.918820i	$0.370856\pi$
$810$	0	0
$811$	−37.6581	−1.32235	−0.661177	−	0.750230i	$-0.729943\pi$
$811$	−37.6581	−1.32235	−0.661177	+	0.750230i	$0.729943\pi$
$812$	0	0
$813$	−14.7209	−0.516286
$814$	−14.2457	−0.499310
$815$	0	0
$816$	7.73528	0.270789
$817$	−9.47129	−0.331358
$818$	−7.38476	−0.258202
$819$	0	0
$820$	0	0
$821$	−21.4838	−0.749791	−0.374896	−	0.927067i	$-0.622321\pi$
$821$	−21.4838	−0.749791	−0.374896	+	0.927067i	$0.622321\pi$
$822$	14.2751	0.497902
$823$	−18.6127	−0.648797	−0.324398	−	0.945921i	$-0.605162\pi$
$823$	−18.6127	−0.648797	−0.324398	+	0.945921i	$0.605162\pi$
$824$	−9.35965	−0.326059
$825$	0	0
$826$	0	0
$827$	−18.4254	−0.640713	−0.320356	−	0.947297i	$-0.603803\pi$
$827$	−18.4254	−0.640713	−0.320356	+	0.947297i	$0.603803\pi$
$828$	2.61827	0.0909912
$829$	−32.0377	−1.11271	−0.556357	−	0.830943i	$-0.687801\pi$
$829$	−32.0377	−1.11271	−0.556357	+	0.830943i	$0.687801\pi$
$830$	0	0
$831$	−3.21104	−0.111390
$832$	−5.88388	−0.203987
$833$	0	0
$834$	−14.9632	−0.518133
$835$	0	0
$836$	−29.5815	−1.02310
$837$	−8.46967	−0.292754
$838$	18.6274	0.643473
$839$	−8.45154	−0.291780	−0.145890	−	0.989301i	$-0.546605\pi$
$839$	−8.45154	−0.291780	−0.145890	+	0.989301i	$0.546605\pi$
$840$	0	0
$841$	37.7891	1.30307
$842$	13.5043	0.465389
$843$	−7.72488	−0.266059
$844$	5.53375	0.190479
$845$	0	0
$846$	6.79073	0.233470
$847$	0	0
$848$	−1.38173	−0.0474488
$849$	29.5594	1.01448
$850$	0	0
$851$	8.34492	0.286060
$852$	2.02386	0.0693361
$853$	−50.6270	−1.73344	−0.866718	−	0.498798i	$-0.833775\pi$
$853$	−50.6270	−1.73344	−0.866718	+	0.498798i	$0.833775\pi$
$854$	0	0
$855$	0	0
$856$	−11.0165	−0.376536
$857$	−12.0013	−0.409955	−0.204978	−	0.978767i	$-0.565712\pi$
$857$	−12.0013	−0.409955	−0.204978	+	0.978767i	$0.565712\pi$
$858$	26.2990	0.897832
$859$	35.7119	1.21848	0.609238	−	0.792988i	$-0.291475\pi$
$859$	35.7119	1.21848	0.609238	+	0.792988i	$0.291475\pi$
$860$	0	0
$861$	0	0
$862$	31.6807	1.07905
$863$	1.28425	0.0437164	0.0218582	−	0.999761i	$-0.493042\pi$
$863$	1.28425	0.0437164	0.0218582	+	0.999761i	$0.493042\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	25.3184	0.860353
$867$	42.8345	1.45474
$868$	0	0
$869$	−31.2218	−1.05913
$870$	0	0
$871$	−10.8934	−0.369109
$872$	4.69544	0.159008
$873$	7.71966	0.261271
$874$	17.3284	0.586142
$875$	0	0
$876$	−4.01687	−0.135717
$877$	−38.3501	−1.29499	−0.647496	−	0.762069i	$-0.724184\pi$
$877$	−38.3501	−1.29499	−0.647496	+	0.762069i	$0.724184\pi$
$878$	32.8146	1.10744
$879$	−1.17679	−0.0396921
$880$	0	0
$881$	−0.389472	−0.0131216	−0.00656082	−	0.999978i	$-0.502088\pi$
$881$	−0.389472	−0.0131216	−0.00656082	+	0.999978i	$0.502088\pi$
$882$	0	0
$883$	18.5412	0.623962	0.311981	−	0.950088i	$-0.399007\pi$
$883$	18.5412	0.623962	0.311981	+	0.950088i	$0.399007\pi$
$884$	−45.5134	−1.53078
$885$	0	0
$886$	16.6421	0.559103
$887$	−42.4806	−1.42636	−0.713179	−	0.700982i	$-0.752745\pi$
$887$	−42.4806	−1.42636	−0.713179	+	0.700982i	$0.752745\pi$
$888$	−3.18719	−0.106955
$889$	0	0
$890$	0	0
$891$	4.46967	0.149739
$892$	3.28248	0.109906
$893$	44.9429	1.50396
$894$	−11.1651	−0.373417
$895$	0	0
$896$	0	0
$897$	−15.4056	−0.514378
$898$	−20.1759	−0.673278
$899$	69.2180	2.30855
$900$	0	0
$901$	−10.6881	−0.356071
$902$	15.9246	0.530231
$903$	0	0
$904$	12.5723	0.418150
$905$	0	0
$906$	−0.664208	−0.0220668
$907$	41.1686	1.36698	0.683491	−	0.729959i	$-0.260461\pi$
$907$	41.1686	1.36698	0.683491	+	0.729959i	$0.260461\pi$
$908$	4.42031	0.146693
$909$	13.3544	0.442939
$910$	0	0
$911$	−25.1703	−0.833929	−0.416964	−	0.908923i	$-0.636906\pi$
$911$	−25.1703	−0.833929	−0.416964	+	0.908923i	$0.636906\pi$
$912$	−6.61827	−0.219153
$913$	−23.9558	−0.792822
$914$	−35.4124	−1.17134
$915$	0	0
$916$	7.97270	0.263426
$917$	0	0
$918$	−7.73528	−0.255302
$919$	−5.03681	−0.166149	−0.0830745	−	0.996543i	$-0.526474\pi$
$919$	−5.03681	−0.166149	−0.0830745	+	0.996543i	$0.526474\pi$
$920$	0	0
$921$	−20.3761	−0.671417
$922$	−12.5940	−0.414762
$923$	−11.9081	−0.391961
$924$	0	0
$925$	0	0
$926$	32.5061	1.06822
$927$	9.35965	0.307411
$928$	8.17246	0.268274
$929$	14.2846	0.468663	0.234332	−	0.972157i	$-0.424710\pi$
$929$	14.2846	0.468663	0.234332	+	0.972157i	$0.424710\pi$
$930$	0	0
$931$	0	0
$932$	17.9558	0.588163
$933$	18.8934	0.618542
$934$	−9.20531	−0.301207
$935$	0	0
$936$	5.88388	0.192321
$937$	−19.8403	−0.648153	−0.324077	−	0.946031i	$-0.605054\pi$
$937$	−19.8403	−0.648153	−0.324077	+	0.946031i	$0.605054\pi$
$938$	0	0
$939$	−16.4849	−0.537964
$940$	0	0
$941$	−15.9054	−0.518502	−0.259251	−	0.965810i	$-0.583476\pi$
$941$	−15.9054	−0.518502	−0.259251	+	0.965810i	$0.583476\pi$
$942$	−8.08184	−0.263320
$943$	−9.32842	−0.303775
$944$	−4.66421	−0.151807
$945$	0	0
$946$	−6.39646	−0.207967
$947$	−9.55815	−0.310598	−0.155299	−	0.987867i	$-0.549634\pi$
$947$	−9.55815	−0.310598	−0.155299	+	0.987867i	$0.549634\pi$
$948$	−6.98527	−0.226871
$949$	23.6348	0.767217
$950$	0	0
$951$	−10.3211	−0.334683
$952$	0	0
$953$	−23.7505	−0.769354	−0.384677	−	0.923051i	$-0.625687\pi$
$953$	−23.7505	−0.769354	−0.384677	+	0.923051i	$0.625687\pi$
$954$	1.38173	0.0447352
$955$	0	0
$956$	5.41296	0.175068
$957$	−36.5282	−1.18079
$958$	−39.4914	−1.27591
$959$	0	0
$960$	0	0
$961$	40.7352	1.31404
$962$	18.7530	0.604622
$963$	11.0165	0.355002
$964$	4.69669	0.151270
$965$	0	0
$966$	0	0
$967$	56.9706	1.83205	0.916025	−	0.401121i	$-0.131379\pi$
$967$	56.9706	1.83205	0.916025	+	0.401121i	$0.131379\pi$
$968$	−8.97792	−0.288561
$969$	−51.1941	−1.64459
$970$	0	0
$971$	−33.0857	−1.06177	−0.530886	−	0.847443i	$-0.678141\pi$
$971$	−33.0857	−1.06177	−0.530886	+	0.847443i	$0.678141\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−29.2678	−0.937800
$975$	0	0
$976$	−9.79683	−0.313589
$977$	24.6127	0.787429	0.393715	−	0.919233i	$-0.371190\pi$
$977$	24.6127	0.787429	0.393715	+	0.919233i	$0.371190\pi$
$978$	0.836668	0.0267537
$979$	−64.3241	−2.05581
$980$	0	0
$981$	−4.69544	−0.149914
$982$	−15.3631	−0.490255
$983$	42.4806	1.35492	0.677460	−	0.735559i	$-0.263081\pi$
$983$	42.4806	1.35492	0.677460	+	0.735559i	$0.263081\pi$
$984$	3.56282	0.113578
$985$	0	0
$986$	63.2162	2.01321
$987$	0	0
$988$	38.9411	1.23888
$989$	3.74696	0.119146
$990$	0	0
$991$	−0.838308	−0.0266297	−0.0133149	−	0.999911i	$-0.504238\pi$
$991$	−0.838308	−0.0266297	−0.0133149	+	0.999911i	$0.504238\pi$
$992$	8.46967	0.268912
$993$	−19.1906	−0.608995
$994$	0	0
$995$	0	0
$996$	−5.35965	−0.169827
$997$	−31.5940	−1.00059	−0.500297	−	0.865854i	$-0.666776\pi$
$997$	−31.5940	−1.00059	−0.500297	+	0.865854i	$0.666776\pi$
$998$	−20.5502	−0.650507
$999$	3.18719	0.100838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.ds.1.4		4
5.2	odd	4		1470.2.g.j.589.2	✓	8
5.3	odd	4		1470.2.g.j.589.6	yes	8
5.4	even	2		7350.2.a.dt.1.4		4
7.6	odd	2		7350.2.a.dr.1.4		4
35.2	odd	12		1470.2.n.l.949.2		16
35.3	even	12		1470.2.n.k.79.3		16
35.12	even	12		1470.2.n.k.949.3		16
35.13	even	4		1470.2.g.k.589.7	yes	8
35.17	even	12		1470.2.n.k.79.5		16
35.18	odd	12		1470.2.n.l.79.2		16
35.23	odd	12		1470.2.n.l.949.8		16
35.27	even	4		1470.2.g.k.589.3	yes	8
35.32	odd	12		1470.2.n.l.79.8		16
35.33	even	12		1470.2.n.k.949.5		16
35.34	odd	2		7350.2.a.du.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.g.j.589.2	✓	8	5.2	odd	4
1470.2.g.j.589.6	yes	8	5.3	odd	4
1470.2.g.k.589.3	yes	8	35.27	even	4
1470.2.g.k.589.7	yes	8	35.13	even	4
1470.2.n.k.79.3		16	35.3	even	12
1470.2.n.k.79.5		16	35.17	even	12
1470.2.n.k.949.3		16	35.12	even	12
1470.2.n.k.949.5		16	35.33	even	12
1470.2.n.l.79.2		16	35.18	odd	12
1470.2.n.l.79.8		16	35.32	odd	12
1470.2.n.l.949.2		16	35.2	odd	12
1470.2.n.l.949.8		16	35.23	odd	12
7350.2.a.dr.1.4		4	7.6	odd	2
7350.2.a.ds.1.4		4	1.1	even	1	trivial
7350.2.a.dt.1.4		4	5.4	even	2
7350.2.a.du.1.4		4	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.46967 q^{11} +1.00000 q^{12} -5.88388 q^{13} +1.00000 q^{16} +7.73528 q^{17} -1.00000 q^{18} -6.61827 q^{19} -4.46967 q^{22} +2.61827 q^{23} -1.00000 q^{24} +5.88388 q^{26} +1.00000 q^{27} -8.17246 q^{29} -8.46967 q^{31} -1.00000 q^{32} +4.46967 q^{33} -7.73528 q^{34} +1.00000 q^{36} +3.18719 q^{37} +6.61827 q^{38} -5.88388 q^{39} -3.56282 q^{41} +1.43108 q^{43} +4.46967 q^{44} -2.61827 q^{46} -6.79073 q^{47} +1.00000 q^{48} +7.73528 q^{51} -5.88388 q^{52} -1.38173 q^{53} -1.00000 q^{54} -6.61827 q^{57} +8.17246 q^{58} -4.66421 q^{59} -9.79683 q^{61} +8.46967 q^{62} +1.00000 q^{64} -4.46967 q^{66} +1.85140 q^{67} +7.73528 q^{68} +2.61827 q^{69} +2.02386 q^{71} -1.00000 q^{72} -4.01687 q^{73} -3.18719 q^{74} -6.61827 q^{76} +5.88388 q^{78} -6.98527 q^{79} +1.00000 q^{81} +3.56282 q^{82} -5.35965 q^{83} -1.43108 q^{86} -8.17246 q^{87} -4.46967 q^{88} -14.3912 q^{89} +2.61827 q^{92} -8.46967 q^{93} +6.79073 q^{94} -1.00000 q^{96} +7.71966 q^{97} +4.46967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{12} + 4 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} - 4 q^{23} - 4 q^{24} + 4 q^{27} - 8 q^{29} - 16 q^{31} - 4 q^{32} - 4 q^{34} + 4 q^{36} + 8 q^{37} + 12 q^{38} - 12 q^{41} - 4 q^{43} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 4 q^{51} - 20 q^{53} - 4 q^{54} - 12 q^{57} + 8 q^{58} - 20 q^{59} - 12 q^{61} + 16 q^{62} + 4 q^{64} + 4 q^{67} + 4 q^{68} - 4 q^{69} - 20 q^{71} - 4 q^{72} - 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} + 12 q^{82} + 8 q^{83} + 4 q^{86} - 8 q^{87} - 44 q^{89} - 4 q^{92} - 16 q^{93} - 12 q^{94} - 4 q^{96} + 20 q^{97}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^12 + 4 * q^16 + 4 * q^17 - 4 * q^18 - 12 * q^19 - 4 * q^23 - 4 * q^24 + 4 * q^27 - 8 * q^29 - 16 * q^31 - 4 * q^32 - 4 * q^34 + 4 * q^36 + 8 * q^37 + 12 * q^38 - 12 * q^41 - 4 * q^43 + 4 * q^46 + 12 * q^47 + 4 * q^48 + 4 * q^51 - 20 * q^53 - 4 * q^54 - 12 * q^57 + 8 * q^58 - 20 * q^59 - 12 * q^61 + 16 * q^62 + 4 * q^64 + 4 * q^67 + 4 * q^68 - 4 * q^69 - 20 * q^71 - 4 * q^72 - 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 + 12 * q^82 + 8 * q^83 + 4 * q^86 - 8 * q^87 - 44 * q^89 - 4 * q^92 - 16 * q^93 - 12 * q^94 - 4 * q^96 + 20 * q^97

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