LMFDB - Embedded newform 9350.2.a.dc.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9350,2,Mod(1,9350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("9350.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9350 = 2 \cdot 5^{2} \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,7,-3,7,0,-3,-4,7,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.6601258899$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 20x^{4} + 3x^{3} - 25x^{2} + 8 $ x^7 - 3x^6 - 6x^5 + 20x^4 + 3x^3 - 25*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1870)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.724668$ of defining polynomial
Character	$\chi$	$=$	9350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.724668 q^{3} +1.00000 q^{4} -0.724668 q^{6} +3.02718 q^{7} +1.00000 q^{8} -2.47486 q^{9} -1.00000 q^{11} -0.724668 q^{12} +1.39587 q^{13} +3.02718 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.47486 q^{18} +0.0534645 q^{19} -2.19370 q^{21} -1.00000 q^{22} -8.40277 q^{23} -0.724668 q^{24} +1.39587 q^{26} +3.96745 q^{27} +3.02718 q^{28} +2.16970 q^{29} -2.74831 q^{31} +1.00000 q^{32} +0.724668 q^{33} -1.00000 q^{34} -2.47486 q^{36} -2.67359 q^{37} +0.0534645 q^{38} -1.01154 q^{39} +5.95641 q^{41} -2.19370 q^{42} -10.8302 q^{43} -1.00000 q^{44} -8.40277 q^{46} -3.10280 q^{47} -0.724668 q^{48} +2.16380 q^{49} +0.724668 q^{51} +1.39587 q^{52} +7.41679 q^{53} +3.96745 q^{54} +3.02718 q^{56} -0.0387440 q^{57} +2.16970 q^{58} -8.18848 q^{59} -5.86969 q^{61} -2.74831 q^{62} -7.49183 q^{63} +1.00000 q^{64} +0.724668 q^{66} -6.60195 q^{67} -1.00000 q^{68} +6.08922 q^{69} -3.28974 q^{71} -2.47486 q^{72} -6.31206 q^{73} -2.67359 q^{74} +0.0534645 q^{76} -3.02718 q^{77} -1.01154 q^{78} -0.551928 q^{79} +4.54949 q^{81} +5.95641 q^{82} +6.86419 q^{83} -2.19370 q^{84} -10.8302 q^{86} -1.57231 q^{87} -1.00000 q^{88} +6.03841 q^{89} +4.22555 q^{91} -8.40277 q^{92} +1.99161 q^{93} -3.10280 q^{94} -0.724668 q^{96} -1.74048 q^{97} +2.16380 q^{98} +2.47486 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 3 q^{6} - 4 q^{7} + 7 q^{8} - 7 q^{11} - 3 q^{12} + 9 q^{13} - 4 q^{14} + 7 q^{16} - 7 q^{17} - 3 q^{19} - 4 q^{21} - 7 q^{22} - 6 q^{23} - 3 q^{24} + 9 q^{26} - 3 q^{27}+ \cdots + 3 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 - 3 * q^3 + 7 * q^4 - 3 * q^6 - 4 * q^7 + 7 * q^8 - 7 * q^11 - 3 * q^12 + 9 * q^13 - 4 * q^14 + 7 * q^16 - 7 * q^17 - 3 * q^19 - 4 * q^21 - 7 * q^22 - 6 * q^23 - 3 * q^24 + 9 * q^26 - 3 * q^27 - 4 * q^28 - 9 * q^29 - 9 * q^31 + 7 * q^32 + 3 * q^33 - 7 * q^34 - 6 * q^37 - 3 * q^38 - 13 * q^39 - 4 * q^42 - 7 * q^44 - 6 * q^46 - 15 * q^47 - 3 * q^48 + 3 * q^49 + 3 * q^51 + 9 * q^52 + 17 * q^53 - 3 * q^54 - 4 * q^56 - 29 * q^57 - 9 * q^58 - 27 * q^59 - 33 * q^61 - 9 * q^62 - 4 * q^63 + 7 * q^64 + 3 * q^66 - 16 * q^67 - 7 * q^68 + 2 * q^69 - q^71 + q^73 - 6 * q^74 - 3 * q^76 + 4 * q^77 - 13 * q^78 - 38 * q^79 - 9 * q^81 - 8 * q^83 - 4 * q^84 + 21 * q^87 - 7 * q^88 + 31 * q^89 - 12 * q^91 - 6 * q^92 + 29 * q^93 - 15 * q^94 - 3 * q^96 + 3 * q^97 + 3 * q^98

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
$3$	−0.724668	−0.418387	−0.209194	−	0.977874i	$-0.567084\pi$
$3$	−0.724668	−0.418387	−0.209194	+	0.977874i	$0.567084\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.724668	−0.295844
$7$	3.02718	1.14417	0.572083	−	0.820196i	$-0.306136\pi$
$7$	3.02718	1.14417	0.572083	+	0.820196i	$0.306136\pi$
$8$	1.00000	0.353553
$9$	−2.47486	−0.824952
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.724668	−0.209194
$13$	1.39587	0.387145	0.193572	−	0.981086i	$-0.437993\pi$
$13$	1.39587	0.387145	0.193572	+	0.981086i	$0.437993\pi$
$14$	3.02718	0.809047
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.47486	−0.583329
$19$	0.0534645	0.0122656	0.00613280	−	0.999981i	$-0.498048\pi$
$19$	0.0534645	0.0122656	0.00613280	+	0.999981i	$0.498048\pi$
$20$	0	0
$21$	−2.19370	−0.478704
$22$	−1.00000	−0.213201
$23$	−8.40277	−1.75210	−0.876049	−	0.482221i	$-0.839830\pi$
$23$	−8.40277	−1.75210	−0.876049	+	0.482221i	$0.839830\pi$
$24$	−0.724668	−0.147922
$25$	0	0
$26$	1.39587	0.273753
$27$	3.96745	0.763536
$28$	3.02718	0.572083
$29$	2.16970	0.402903	0.201451	−	0.979499i	$-0.435434\pi$
$29$	2.16970	0.402903	0.201451	+	0.979499i	$0.435434\pi$
$30$	0	0
$31$	−2.74831	−0.493610	−0.246805	−	0.969065i	$-0.579381\pi$
$31$	−2.74831	−0.493610	−0.246805	+	0.969065i	$0.579381\pi$
$32$	1.00000	0.176777
$33$	0.724668	0.126148
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.47486	−0.412476
$37$	−2.67359	−0.439535	−0.219768	−	0.975552i	$-0.570530\pi$
$37$	−2.67359	−0.439535	−0.219768	+	0.975552i	$0.570530\pi$
$38$	0.0534645	0.00867309
$39$	−1.01154	−0.161976
$40$	0	0
$41$	5.95641	0.930234	0.465117	−	0.885249i	$-0.346012\pi$
$41$	5.95641	0.930234	0.465117	+	0.885249i	$0.346012\pi$
$42$	−2.19370	−0.338495
$43$	−10.8302	−1.65159	−0.825795	−	0.563970i	$-0.809273\pi$
$43$	−10.8302	−1.65159	−0.825795	+	0.563970i	$0.809273\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−8.40277	−1.23892
$47$	−3.10280	−0.452590	−0.226295	−	0.974059i	$-0.572661\pi$
$47$	−3.10280	−0.452590	−0.226295	+	0.974059i	$0.572661\pi$
$48$	−0.724668	−0.104597
$49$	2.16380	0.309114
$50$	0	0
$51$	0.724668	0.101474
$52$	1.39587	0.193572
$53$	7.41679	1.01877	0.509387	−	0.860538i	$-0.329872\pi$
$53$	7.41679	1.01877	0.509387	+	0.860538i	$0.329872\pi$
$54$	3.96745	0.539902
$55$	0	0
$56$	3.02718	0.404523
$57$	−0.0387440	−0.00513177
$58$	2.16970	0.284895
$59$	−8.18848	−1.06605	−0.533025	−	0.846100i	$-0.678945\pi$
$59$	−8.18848	−1.06605	−0.533025	+	0.846100i	$0.678945\pi$
$60$	0	0
$61$	−5.86969	−0.751536	−0.375768	−	0.926714i	$-0.622621\pi$
$61$	−5.86969	−0.751536	−0.375768	+	0.926714i	$0.622621\pi$
$62$	−2.74831	−0.349035
$63$	−7.49183	−0.943882
$64$	1.00000	0.125000
$65$	0	0
$66$	0.724668	0.0892004
$67$	−6.60195	−0.806557	−0.403278	−	0.915077i	$-0.632129\pi$
$67$	−6.60195	−0.806557	−0.403278	+	0.915077i	$0.632129\pi$
$68$	−1.00000	−0.121268
$69$	6.08922	0.733056
$70$	0	0
$71$	−3.28974	−0.390420	−0.195210	−	0.980761i	$-0.562539\pi$
$71$	−3.28974	−0.390420	−0.195210	+	0.980761i	$0.562539\pi$
$72$	−2.47486	−0.291665
$73$	−6.31206	−0.738771	−0.369385	−	0.929276i	$-0.620432\pi$
$73$	−6.31206	−0.738771	−0.369385	+	0.929276i	$0.620432\pi$
$74$	−2.67359	−0.310798
$75$	0	0
$76$	0.0534645	0.00613280
$77$	−3.02718	−0.344979
$78$	−1.01154	−0.114535
$79$	−0.551928	−0.0620968	−0.0310484	−	0.999518i	$-0.509885\pi$
$79$	−0.551928	−0.0620968	−0.0310484	+	0.999518i	$0.509885\pi$
$80$	0	0
$81$	4.54949	0.505498
$82$	5.95641	0.657775
$83$	6.86419	0.753443	0.376722	−	0.926327i	$-0.377051\pi$
$83$	6.86419	0.753443	0.376722	+	0.926327i	$0.377051\pi$
$84$	−2.19370	−0.239352
$85$	0	0
$86$	−10.8302	−1.16785
$87$	−1.57231	−0.168569
$88$	−1.00000	−0.106600
$89$	6.03841	0.640070	0.320035	−	0.947406i	$-0.396305\pi$
$89$	6.03841	0.640070	0.320035	+	0.947406i	$0.396305\pi$
$90$	0	0
$91$	4.22555	0.442958
$92$	−8.40277	−0.876049
$93$	1.99161	0.206520
$94$	−3.10280	−0.320029
$95$	0	0
$96$	−0.724668	−0.0739611
$97$	−1.74048	−0.176719	−0.0883594	−	0.996089i	$-0.528162\pi$
$97$	−1.74048	−0.176719	−0.0883594	+	0.996089i	$0.528162\pi$
$98$	2.16380	0.218577
$99$	2.47486	0.248732
$100$	0	0
$101$	−14.4049	−1.43334	−0.716672	−	0.697410i	$-0.754336\pi$
$101$	−14.4049	−1.43334	−0.716672	+	0.697410i	$0.754336\pi$
$102$	0.724668	0.0717528
$103$	−0.264784	−0.0260899	−0.0130450	−	0.999915i	$-0.504152\pi$
$103$	−0.264784	−0.0260899	−0.0130450	+	0.999915i	$0.504152\pi$
$104$	1.39587	0.136876
$105$	0	0
$106$	7.41679	0.720382
$107$	12.0412	1.16406	0.582031	−	0.813166i	$-0.302258\pi$
$107$	12.0412	1.16406	0.582031	+	0.813166i	$0.302258\pi$
$108$	3.96745	0.381768
$109$	3.73438	0.357688	0.178844	−	0.983877i	$-0.442764\pi$
$109$	3.73438	0.357688	0.178844	+	0.983877i	$0.442764\pi$
$110$	0	0
$111$	1.93746	0.183896
$112$	3.02718	0.286041
$113$	−12.0191	−1.13066	−0.565331	−	0.824864i	$-0.691252\pi$
$113$	−12.0191	−1.13066	−0.565331	+	0.824864i	$0.691252\pi$
$114$	−0.0387440	−0.00362871
$115$	0	0
$116$	2.16970	0.201451
$117$	−3.45458	−0.319376
$118$	−8.18848	−0.753811
$119$	−3.02718	−0.277501
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.86969	−0.531416
$123$	−4.31641	−0.389198
$124$	−2.74831	−0.246805
$125$	0	0
$126$	−7.49183	−0.667425
$127$	−4.16210	−0.369326	−0.184663	−	0.982802i	$-0.559119\pi$
$127$	−4.16210	−0.369326	−0.184663	+	0.982802i	$0.559119\pi$
$128$	1.00000	0.0883883
$129$	7.84830	0.691004
$130$	0	0
$131$	−20.2513	−1.76936	−0.884680	−	0.466199i	$-0.845623\pi$
$131$	−20.2513	−1.76936	−0.884680	+	0.466199i	$0.845623\pi$
$132$	0.724668	0.0630742
$133$	0.161846	0.0140339
$134$	−6.60195	−0.570322
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	1.24462	0.106335	0.0531673	−	0.998586i	$-0.483068\pi$
$137$	1.24462	0.106335	0.0531673	+	0.998586i	$0.483068\pi$
$138$	6.08922	0.518349
$139$	1.56712	0.132921	0.0664605	−	0.997789i	$-0.478829\pi$
$139$	1.56712	0.132921	0.0664605	+	0.997789i	$0.478829\pi$
$140$	0	0
$141$	2.24850	0.189358
$142$	−3.28974	−0.276069
$143$	−1.39587	−0.116729
$144$	−2.47486	−0.206238
$145$	0	0
$146$	−6.31206	−0.522390
$147$	−1.56803	−0.129329
$148$	−2.67359	−0.219768
$149$	3.42956	0.280961	0.140480	−	0.990083i	$-0.455135\pi$
$149$	3.42956	0.280961	0.140480	+	0.990083i	$0.455135\pi$
$150$	0	0
$151$	15.3510	1.24925	0.624623	−	0.780926i	$-0.285253\pi$
$151$	15.3510	1.24925	0.624623	+	0.780926i	$0.285253\pi$
$152$	0.0534645	0.00433654
$153$	2.47486	0.200080
$154$	−3.02718	−0.243937
$155$	0	0
$156$	−1.01154	−0.0809882
$157$	−12.0923	−0.965067	−0.482534	−	0.875877i	$-0.660283\pi$
$157$	−12.0923	−0.965067	−0.482534	+	0.875877i	$0.660283\pi$
$158$	−0.551928	−0.0439090
$159$	−5.37471	−0.426242
$160$	0	0
$161$	−25.4367	−2.00469
$162$	4.54949	0.357441
$163$	−5.33332	−0.417738	−0.208869	−	0.977944i	$-0.566978\pi$
$163$	−5.33332	−0.417738	−0.208869	+	0.977944i	$0.566978\pi$
$164$	5.95641	0.465117
$165$	0	0
$166$	6.86419	0.532765
$167$	−6.51295	−0.503987	−0.251994	−	0.967729i	$-0.581086\pi$
$167$	−6.51295	−0.503987	−0.251994	+	0.967729i	$0.581086\pi$
$168$	−2.19370	−0.169247
$169$	−11.0515	−0.850119
$170$	0	0
$171$	−0.132317	−0.0101185
$172$	−10.8302	−0.825795
$173$	12.1605	0.924547	0.462273	−	0.886737i	$-0.347034\pi$
$173$	12.1605	0.924547	0.462273	+	0.886737i	$0.347034\pi$
$174$	−1.57231	−0.119197
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	5.93393	0.446021
$178$	6.03841	0.452598
$179$	−10.4443	−0.780642	−0.390321	−	0.920679i	$-0.627636\pi$
$179$	−10.4443	−0.780642	−0.390321	+	0.920679i	$0.627636\pi$
$180$	0	0
$181$	3.20921	0.238539	0.119270	−	0.992862i	$-0.461945\pi$
$181$	3.20921	0.238539	0.119270	+	0.992862i	$0.461945\pi$
$182$	4.22555	0.313218
$183$	4.25357	0.314433
$184$	−8.40277	−0.619461
$185$	0	0
$186$	1.99161	0.146032
$187$	1.00000	0.0731272
$188$	−3.10280	−0.226295
$189$	12.0102	0.873612
$190$	0	0
$191$	1.15482	0.0835596	0.0417798	−	0.999127i	$-0.486697\pi$
$191$	1.15482	0.0835596	0.0417798	+	0.999127i	$0.486697\pi$
$192$	−0.724668	−0.0522984
$193$	23.6016	1.69888	0.849442	−	0.527682i	$-0.176939\pi$
$193$	23.6016	1.69888	0.849442	+	0.527682i	$0.176939\pi$
$194$	−1.74048	−0.124959
$195$	0	0
$196$	2.16380	0.154557
$197$	−0.923402	−0.0657897	−0.0328948	−	0.999459i	$-0.510473\pi$
$197$	−0.923402	−0.0657897	−0.0328948	+	0.999459i	$0.510473\pi$
$198$	2.47486	0.175880
$199$	−11.4023	−0.808287	−0.404143	−	0.914696i	$-0.632430\pi$
$199$	−11.4023	−0.808287	−0.404143	+	0.914696i	$0.632430\pi$
$200$	0	0
$201$	4.78422	0.337453
$202$	−14.4049	−1.01353
$203$	6.56806	0.460987
$204$	0.724668	0.0507369
$205$	0	0
$206$	−0.264784	−0.0184483
$207$	20.7957	1.44540
$208$	1.39587	0.0967862
$209$	−0.0534645	−0.00369822
$210$	0	0
$211$	−18.3851	−1.26568	−0.632841	−	0.774282i	$-0.718111\pi$
$211$	−18.3851	−1.26568	−0.632841	+	0.774282i	$0.718111\pi$
$212$	7.41679	0.509387
$213$	2.38397	0.163347
$214$	12.0412	0.823117
$215$	0	0
$216$	3.96745	0.269951
$217$	−8.31961	−0.564772
$218$	3.73438	0.252924
$219$	4.57414	0.309092
$220$	0	0
$221$	−1.39587	−0.0938964
$222$	1.93746	0.130034
$223$	6.79286	0.454884	0.227442	−	0.973792i	$-0.426964\pi$
$223$	6.79286	0.454884	0.227442	+	0.973792i	$0.426964\pi$
$224$	3.02718	0.202262
$225$	0	0
$226$	−12.0191	−0.799499
$227$	−1.83089	−0.121520	−0.0607602	−	0.998152i	$-0.519352\pi$
$227$	−1.83089	−0.121520	−0.0607602	+	0.998152i	$0.519352\pi$
$228$	−0.0387440	−0.00256588
$229$	−11.5899	−0.765884	−0.382942	−	0.923772i	$-0.625089\pi$
$229$	−11.5899	−0.765884	−0.382942	+	0.923772i	$0.625089\pi$
$230$	0	0
$231$	2.19370	0.144335
$232$	2.16970	0.142448
$233$	−3.95494	−0.259096	−0.129548	−	0.991573i	$-0.541353\pi$
$233$	−3.95494	−0.259096	−0.129548	+	0.991573i	$0.541353\pi$
$234$	−3.45458	−0.225833
$235$	0	0
$236$	−8.18848	−0.533025
$237$	0.399964	0.0259805
$238$	−3.02718	−0.196223
$239$	−24.6785	−1.59632	−0.798159	−	0.602447i	$-0.794192\pi$
$239$	−24.6785	−1.59632	−0.798159	+	0.602447i	$0.794192\pi$
$240$	0	0
$241$	−19.9073	−1.28234	−0.641170	−	0.767399i	$-0.721551\pi$
$241$	−19.9073	−1.28234	−0.641170	+	0.767399i	$0.721551\pi$
$242$	1.00000	0.0642824
$243$	−15.1992	−0.975030
$244$	−5.86969	−0.375768
$245$	0	0
$246$	−4.31641	−0.275205
$247$	0.0746295	0.00474856
$248$	−2.74831	−0.174518
$249$	−4.97426	−0.315231
$250$	0	0
$251$	20.6043	1.30053	0.650267	−	0.759705i	$-0.274657\pi$
$251$	20.6043	1.30053	0.650267	+	0.759705i	$0.274657\pi$
$252$	−7.49183	−0.471941
$253$	8.40277	0.528278
$254$	−4.16210	−0.261153
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.395876	−0.0246940	−0.0123470	−	0.999924i	$-0.503930\pi$
$257$	−0.395876	−0.0246940	−0.0123470	+	0.999924i	$0.503930\pi$
$258$	7.84830	0.488614
$259$	−8.09342	−0.502901
$260$	0	0
$261$	−5.36969	−0.332376
$262$	−20.2513	−1.25113
$263$	20.9577	1.29230	0.646152	−	0.763208i	$-0.276377\pi$
$263$	20.9577	1.29230	0.646152	+	0.763208i	$0.276377\pi$
$264$	0.724668	0.0446002
$265$	0	0
$266$	0.161846	0.00992344
$267$	−4.37584	−0.267797
$268$	−6.60195	−0.403278
$269$	−28.9880	−1.76743	−0.883716	−	0.468023i	$-0.844966\pi$
$269$	−28.9880	−1.76743	−0.883716	+	0.468023i	$0.844966\pi$
$270$	0	0
$271$	−1.00582	−0.0610992	−0.0305496	−	0.999533i	$-0.509726\pi$
$271$	−1.00582	−0.0610992	−0.0305496	+	0.999533i	$0.509726\pi$
$272$	−1.00000	−0.0606339
$273$	−3.06212	−0.185328
$274$	1.24462	0.0751900
$275$	0	0
$276$	6.08922	0.366528
$277$	17.5108	1.05212	0.526062	−	0.850446i	$-0.323668\pi$
$277$	17.5108	1.05212	0.526062	+	0.850446i	$0.323668\pi$
$278$	1.56712	0.0939894
$279$	6.80166	0.407205
$280$	0	0
$281$	33.1304	1.97639	0.988197	−	0.153188i	$-0.0489539\pi$
$281$	33.1304	1.97639	0.988197	+	0.153188i	$0.0489539\pi$
$282$	2.24850	0.133896
$283$	29.0481	1.72673	0.863367	−	0.504577i	$-0.168352\pi$
$283$	29.0481	1.72673	0.863367	+	0.504577i	$0.168352\pi$
$284$	−3.28974	−0.195210
$285$	0	0
$286$	−1.39587	−0.0825396
$287$	18.0311	1.06434
$288$	−2.47486	−0.145832
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.26127	0.0739368
$292$	−6.31206	−0.369385
$293$	10.7790	0.629717	0.314859	−	0.949139i	$-0.398043\pi$
$293$	10.7790	0.629717	0.314859	+	0.949139i	$0.398043\pi$
$294$	−1.56803	−0.0914496
$295$	0	0
$296$	−2.67359	−0.155399
$297$	−3.96745	−0.230215
$298$	3.42956	0.198669
$299$	−11.7292	−0.678316
$300$	0	0
$301$	−32.7849	−1.88969
$302$	15.3510	0.883351
$303$	10.4388	0.599693
$304$	0.0534645	0.00306640
$305$	0	0
$306$	2.47486	0.141478
$307$	4.78632	0.273170	0.136585	−	0.990628i	$-0.456387\pi$
$307$	4.78632	0.273170	0.136585	+	0.990628i	$0.456387\pi$
$308$	−3.02718	−0.172489
$309$	0.191880	0.0109157
$310$	0	0
$311$	27.4662	1.55746	0.778731	−	0.627358i	$-0.215864\pi$
$311$	27.4662	1.55746	0.778731	+	0.627358i	$0.215864\pi$
$312$	−1.01154	−0.0572673
$313$	−22.3627	−1.26401	−0.632006	−	0.774964i	$-0.717768\pi$
$313$	−22.3627	−1.26401	−0.632006	+	0.774964i	$0.717768\pi$
$314$	−12.0923	−0.682406
$315$	0	0
$316$	−0.551928	−0.0310484
$317$	−3.26388	−0.183318	−0.0916589	−	0.995790i	$-0.529217\pi$
$317$	−3.26388	−0.183318	−0.0916589	+	0.995790i	$0.529217\pi$
$318$	−5.37471	−0.301399
$319$	−2.16970	−0.121480
$320$	0	0
$321$	−8.72584	−0.487029
$322$	−25.4367	−1.41753
$323$	−0.0534645	−0.00297484
$324$	4.54949	0.252749
$325$	0	0
$326$	−5.33332	−0.295385
$327$	−2.70618	−0.149652
$328$	5.95641	0.328888
$329$	−9.39272	−0.517838
$330$	0	0
$331$	−34.9394	−1.92044	−0.960221	−	0.279240i	$-0.909917\pi$
$331$	−34.9394	−1.92044	−0.960221	+	0.279240i	$0.909917\pi$
$332$	6.86419	0.376722
$333$	6.61675	0.362595
$334$	−6.51295	−0.356373
$335$	0	0
$336$	−2.19370	−0.119676
$337$	13.2745	0.723108	0.361554	−	0.932351i	$-0.382246\pi$
$337$	13.2745	0.723108	0.361554	+	0.932351i	$0.382246\pi$
$338$	−11.0515	−0.601125
$339$	8.70986	0.473055
$340$	0	0
$341$	2.74831	0.148829
$342$	−0.132317	−0.00715488
$343$	−14.6400	−0.790488
$344$	−10.8302	−0.583925
$345$	0	0
$346$	12.1605	0.653753
$347$	21.6176	1.16049	0.580247	−	0.814441i	$-0.302956\pi$
$347$	21.6176	1.16049	0.580247	+	0.814441i	$0.302956\pi$
$348$	−1.57231	−0.0842847
$349$	−13.2983	−0.711844	−0.355922	−	0.934516i	$-0.615833\pi$
$349$	−13.2983	−0.711844	−0.355922	+	0.934516i	$0.615833\pi$
$350$	0	0
$351$	5.53805	0.295599
$352$	−1.00000	−0.0533002
$353$	−19.5775	−1.04201	−0.521003	−	0.853555i	$-0.674442\pi$
$353$	−19.5775	−1.04201	−0.521003	+	0.853555i	$0.674442\pi$
$354$	5.93393	0.315385
$355$	0	0
$356$	6.03841	0.320035
$357$	2.19370	0.116103
$358$	−10.4443	−0.551998
$359$	3.63893	0.192055	0.0960276	−	0.995379i	$-0.469386\pi$
$359$	3.63893	0.192055	0.0960276	+	0.995379i	$0.469386\pi$
$360$	0	0
$361$	−18.9971	−0.999850
$362$	3.20921	0.168673
$363$	−0.724668	−0.0380352
$364$	4.22555	0.221479
$365$	0	0
$366$	4.25357	0.222338
$367$	−20.6075	−1.07570	−0.537852	−	0.843039i	$-0.680764\pi$
$367$	−20.6075	−1.07570	−0.537852	+	0.843039i	$0.680764\pi$
$368$	−8.40277	−0.438025
$369$	−14.7413	−0.767399
$370$	0	0
$371$	22.4519	1.16565
$372$	1.99161	0.103260
$373$	27.3516	1.41621	0.708107	−	0.706105i	$-0.249549\pi$
$373$	27.3516	1.41621	0.708107	+	0.706105i	$0.249549\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	−3.10280	−0.160015
$377$	3.02862	0.155982
$378$	12.0102	0.617737
$379$	−24.7166	−1.26961	−0.634803	−	0.772674i	$-0.718919\pi$
$379$	−24.7166	−1.26961	−0.634803	+	0.772674i	$0.718919\pi$
$380$	0	0
$381$	3.01614	0.154521
$382$	1.15482	0.0590855
$383$	15.2868	0.781121	0.390561	−	0.920577i	$-0.372281\pi$
$383$	15.2868	0.781121	0.390561	+	0.920577i	$0.372281\pi$
$384$	−0.724668	−0.0369805
$385$	0	0
$386$	23.6016	1.20129
$387$	26.8032	1.36248
$388$	−1.74048	−0.0883594
$389$	−31.3293	−1.58846	−0.794230	−	0.607617i	$-0.792126\pi$
$389$	−31.3293	−1.58846	−0.794230	+	0.607617i	$0.792126\pi$
$390$	0	0
$391$	8.40277	0.424946
$392$	2.16380	0.109288
$393$	14.6754	0.740277
$394$	−0.923402	−0.0465203
$395$	0	0
$396$	2.47486	0.124366
$397$	−0.421597	−0.0211593	−0.0105797	−	0.999944i	$-0.503368\pi$
$397$	−0.421597	−0.0211593	−0.0105797	+	0.999944i	$0.503368\pi$
$398$	−11.4023	−0.571545
$399$	−0.117285	−0.00587159
$400$	0	0
$401$	4.99225	0.249301	0.124651	−	0.992201i	$-0.460219\pi$
$401$	4.99225	0.249301	0.124651	+	0.992201i	$0.460219\pi$
$402$	4.78422	0.238615
$403$	−3.83628	−0.191099
$404$	−14.4049	−0.716672
$405$	0	0
$406$	6.56806	0.325967
$407$	2.67359	0.132525
$408$	0.724668	0.0358764
$409$	−18.2776	−0.903769	−0.451885	−	0.892076i	$-0.649248\pi$
$409$	−18.2776	−0.903769	−0.451885	+	0.892076i	$0.649248\pi$
$410$	0	0
$411$	−0.901932	−0.0444890
$412$	−0.264784	−0.0130450
$413$	−24.7880	−1.21974
$414$	20.7957	1.02205
$415$	0	0
$416$	1.39587	0.0684382
$417$	−1.13564	−0.0556125
$418$	−0.0534645	−0.00261503
$419$	18.6601	0.911605	0.455803	−	0.890081i	$-0.349352\pi$
$419$	18.6601	0.911605	0.455803	+	0.890081i	$0.349352\pi$
$420$	0	0
$421$	9.32531	0.454488	0.227244	−	0.973838i	$-0.427029\pi$
$421$	9.32531	0.454488	0.227244	+	0.973838i	$0.427029\pi$
$422$	−18.3851	−0.894973
$423$	7.67899	0.373365
$424$	7.41679	0.360191
$425$	0	0
$426$	2.38397	0.115504
$427$	−17.7686	−0.859882
$428$	12.0412	0.582031
$429$	1.01154	0.0488377
$430$	0	0
$431$	12.0838	0.582057	0.291029	−	0.956714i	$-0.406003\pi$
$431$	12.0838	0.582057	0.291029	+	0.956714i	$0.406003\pi$
$432$	3.96745	0.190884
$433$	−28.6421	−1.37645	−0.688225	−	0.725497i	$-0.741610\pi$
$433$	−28.6421	−1.37645	−0.688225	+	0.725497i	$0.741610\pi$
$434$	−8.31961	−0.399354
$435$	0	0
$436$	3.73438	0.178844
$437$	−0.449250	−0.0214905
$438$	4.57414	0.218561
$439$	−31.8119	−1.51830	−0.759151	−	0.650915i	$-0.774385\pi$
$439$	−31.8119	−1.51830	−0.759151	+	0.650915i	$0.774385\pi$
$440$	0	0
$441$	−5.35509	−0.255004
$442$	−1.39587	−0.0663948
$443$	−28.2282	−1.34116	−0.670581	−	0.741836i	$-0.733955\pi$
$443$	−28.2282	−1.34116	−0.670581	+	0.741836i	$0.733955\pi$
$444$	1.93746	0.0919479
$445$	0	0
$446$	6.79286	0.321651
$447$	−2.48529	−0.117550
$448$	3.02718	0.143021
$449$	−29.6729	−1.40035	−0.700175	−	0.713972i	$-0.746895\pi$
$449$	−29.6729	−1.40035	−0.700175	+	0.713972i	$0.746895\pi$
$450$	0	0
$451$	−5.95641	−0.280476
$452$	−12.0191	−0.565331
$453$	−11.1244	−0.522669
$454$	−1.83089	−0.0859279
$455$	0	0
$456$	−0.0387440	−0.00181435
$457$	8.96831	0.419520	0.209760	−	0.977753i	$-0.432732\pi$
$457$	8.96831	0.419520	0.209760	+	0.977753i	$0.432732\pi$
$458$	−11.5899	−0.541562
$459$	−3.96745	−0.185185
$460$	0	0
$461$	0.685658	0.0319343	0.0159671	−	0.999873i	$-0.494917\pi$
$461$	0.685658	0.0319343	0.0159671	+	0.999873i	$0.494917\pi$
$462$	2.19370	0.102060
$463$	−1.17487	−0.0546008	−0.0273004	−	0.999627i	$-0.508691\pi$
$463$	−1.17487	−0.0546008	−0.0273004	+	0.999627i	$0.508691\pi$
$464$	2.16970	0.100726
$465$	0	0
$466$	−3.95494	−0.183209
$467$	−15.8853	−0.735086	−0.367543	−	0.930007i	$-0.619801\pi$
$467$	−15.8853	−0.735086	−0.367543	+	0.930007i	$0.619801\pi$
$468$	−3.45458	−0.159688
$469$	−19.9853	−0.922834
$470$	0	0
$471$	8.76287	0.403772
$472$	−8.18848	−0.376905
$473$	10.8302	0.497973
$474$	0.399964	0.0183710
$475$	0	0
$476$	−3.02718	−0.138750
$477$	−18.3555	−0.840440
$478$	−24.6785	−1.12877
$479$	16.9229	0.773227	0.386613	−	0.922242i	$-0.373645\pi$
$479$	16.9229	0.773227	0.386613	+	0.922242i	$0.373645\pi$
$480$	0	0
$481$	−3.73198	−0.170164
$482$	−19.9073	−0.906752
$483$	18.4331	0.838737
$484$	1.00000	0.0454545
$485$	0	0
$486$	−15.1992	−0.689451
$487$	−28.1832	−1.27710	−0.638551	−	0.769580i	$-0.720466\pi$
$487$	−28.1832	−1.27710	−0.638551	+	0.769580i	$0.720466\pi$
$488$	−5.86969	−0.265708
$489$	3.86488	0.174776
$490$	0	0
$491$	14.6487	0.661087	0.330544	−	0.943791i	$-0.392768\pi$
$491$	14.6487	0.661087	0.330544	+	0.943791i	$0.392768\pi$
$492$	−4.31641	−0.194599
$493$	−2.16970	−0.0977183
$494$	0.0746295	0.00335774
$495$	0	0
$496$	−2.74831	−0.123403
$497$	−9.95862	−0.446705
$498$	−4.97426	−0.222902
$499$	13.8988	0.622196	0.311098	−	0.950378i	$-0.399303\pi$
$499$	13.8988	0.622196	0.311098	+	0.950378i	$0.399303\pi$
$500$	0	0
$501$	4.71972	0.210862
$502$	20.6043	0.919617
$503$	−4.38882	−0.195688	−0.0978439	−	0.995202i	$-0.531195\pi$
$503$	−4.38882	−0.195688	−0.0978439	+	0.995202i	$0.531195\pi$
$504$	−7.49183	−0.333713
$505$	0	0
$506$	8.40277	0.373549
$507$	8.00870	0.355679
$508$	−4.16210	−0.184663
$509$	0.992746	0.0440027	0.0220013	−	0.999758i	$-0.492996\pi$
$509$	0.992746	0.0440027	0.0220013	+	0.999758i	$0.492996\pi$
$510$	0	0
$511$	−19.1077	−0.845276
$512$	1.00000	0.0441942
$513$	0.212118	0.00936523
$514$	−0.395876	−0.0174613
$515$	0	0
$516$	7.84830	0.345502
$517$	3.10280	0.136461
$518$	−8.09342	−0.355605
$519$	−8.81233	−0.386818
$520$	0	0
$521$	25.4548	1.11520	0.557598	−	0.830111i	$-0.311723\pi$
$521$	25.4548	1.11520	0.557598	+	0.830111i	$0.311723\pi$
$522$	−5.36969	−0.235025
$523$	−39.1522	−1.71201	−0.856003	−	0.516970i	$-0.827060\pi$
$523$	−39.1522	−1.71201	−0.856003	+	0.516970i	$0.827060\pi$
$524$	−20.2513	−0.884680
$525$	0	0
$526$	20.9577	0.913798
$527$	2.74831	0.119718
$528$	0.724668	0.0315371
$529$	47.6066	2.06985
$530$	0	0
$531$	20.2653	0.879440
$532$	0.161846	0.00701693
$533$	8.31437	0.360136
$534$	−4.37584	−0.189361
$535$	0	0
$536$	−6.60195	−0.285161
$537$	7.56864	0.326611
$538$	−28.9880	−1.24976
$539$	−2.16380	−0.0932014
$540$	0	0
$541$	−32.6588	−1.40411	−0.702055	−	0.712123i	$-0.747734\pi$
$541$	−32.6588	−1.40411	−0.702055	+	0.712123i	$0.747734\pi$
$542$	−1.00582	−0.0432037
$543$	−2.32561	−0.0998017
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−3.06212	−0.131047
$547$	2.59218	0.110834	0.0554168	−	0.998463i	$-0.482351\pi$
$547$	2.59218	0.110834	0.0554168	+	0.998463i	$0.482351\pi$
$548$	1.24462	0.0531673
$549$	14.5266	0.619981
$550$	0	0
$551$	0.116002	0.00494184
$552$	6.08922	0.259174
$553$	−1.67078	−0.0710489
$554$	17.5108	0.743965
$555$	0	0
$556$	1.56712	0.0664605
$557$	−10.1094	−0.428350	−0.214175	−	0.976795i	$-0.568706\pi$
$557$	−10.1094	−0.428350	−0.214175	+	0.976795i	$0.568706\pi$
$558$	6.80166	0.287937
$559$	−15.1176	−0.639405
$560$	0	0
$561$	−0.724668	−0.0305955
$562$	33.1304	1.39752
$563$	−10.5762	−0.445732	−0.222866	−	0.974849i	$-0.571541\pi$
$563$	−10.5762	−0.445732	−0.222866	+	0.974849i	$0.571541\pi$
$564$	2.24850	0.0946789
$565$	0	0
$566$	29.0481	1.22098
$567$	13.7721	0.578374
$568$	−3.28974	−0.138034
$569$	3.76468	0.157824	0.0789118	−	0.996882i	$-0.474855\pi$
$569$	3.76468	0.157824	0.0789118	+	0.996882i	$0.474855\pi$
$570$	0	0
$571$	−14.9685	−0.626411	−0.313206	−	0.949685i	$-0.601403\pi$
$571$	−14.9685	−0.626411	−0.313206	+	0.949685i	$0.601403\pi$
$572$	−1.39587	−0.0583643
$573$	−0.836858	−0.0349602
$574$	18.0311	0.752603
$575$	0	0
$576$	−2.47486	−0.103119
$577$	−36.9847	−1.53969	−0.769847	−	0.638228i	$-0.779668\pi$
$577$	−36.9847	−1.53969	−0.769847	+	0.638228i	$0.779668\pi$
$578$	1.00000	0.0415945
$579$	−17.1034	−0.710791
$580$	0	0
$581$	20.7791	0.862063
$582$	1.26127	0.0522812
$583$	−7.41679	−0.307172
$584$	−6.31206	−0.261195
$585$	0	0
$586$	10.7790	0.445277
$587$	16.7220	0.690190	0.345095	−	0.938568i	$-0.387847\pi$
$587$	16.7220	0.690190	0.345095	+	0.938568i	$0.387847\pi$
$588$	−1.56803	−0.0646647
$589$	−0.146937	−0.00605443
$590$	0	0
$591$	0.669160	0.0275255
$592$	−2.67359	−0.109884
$593$	24.0675	0.988334	0.494167	−	0.869367i	$-0.335473\pi$
$593$	24.0675	0.988334	0.494167	+	0.869367i	$0.335473\pi$
$594$	−3.96745	−0.162787
$595$	0	0
$596$	3.42956	0.140480
$597$	8.26287	0.338177
$598$	−11.7292	−0.479642
$599$	−26.6896	−1.09051	−0.545254	−	0.838271i	$-0.683567\pi$
$599$	−26.6896	−1.09051	−0.545254	+	0.838271i	$0.683567\pi$
$600$	0	0
$601$	−30.0183	−1.22447	−0.612236	−	0.790675i	$-0.709730\pi$
$601$	−30.0183	−1.22447	−0.612236	+	0.790675i	$0.709730\pi$
$602$	−32.7849	−1.33621
$603$	16.3389	0.665371
$604$	15.3510	0.624623
$605$	0	0
$606$	10.4388	0.424047
$607$	2.28769	0.0928545	0.0464272	−	0.998922i	$-0.485216\pi$
$607$	2.28769	0.0928545	0.0464272	+	0.998922i	$0.485216\pi$
$608$	0.0534645	0.00216827
$609$	−4.75966	−0.192871
$610$	0	0
$611$	−4.33111	−0.175218
$612$	2.47486	0.100040
$613$	31.3655	1.26684	0.633420	−	0.773808i	$-0.281650\pi$
$613$	31.3655	1.26684	0.633420	+	0.773808i	$0.281650\pi$
$614$	4.78632	0.193160
$615$	0	0
$616$	−3.02718	−0.121968
$617$	14.6167	0.588448	0.294224	−	0.955737i	$-0.404939\pi$
$617$	14.6167	0.588448	0.294224	+	0.955737i	$0.404939\pi$
$618$	0.191880	0.00771855
$619$	−1.57326	−0.0632346	−0.0316173	−	0.999500i	$-0.510066\pi$
$619$	−1.57326	−0.0632346	−0.0316173	+	0.999500i	$0.510066\pi$
$620$	0	0
$621$	−33.3376	−1.33779
$622$	27.4662	1.10129
$623$	18.2793	0.732346
$624$	−1.01154	−0.0404941
$625$	0	0
$626$	−22.3627	−0.893791
$627$	0.0387440	0.00154729
$628$	−12.0923	−0.482534
$629$	2.67359	0.106603
$630$	0	0
$631$	−4.65673	−0.185381	−0.0926907	−	0.995695i	$-0.529547\pi$
$631$	−4.65673	−0.185381	−0.0926907	+	0.995695i	$0.529547\pi$
$632$	−0.551928	−0.0219545
$633$	13.3231	0.529545
$634$	−3.26388	−0.129625
$635$	0	0
$636$	−5.37471	−0.213121
$637$	3.02038	0.119672
$638$	−2.16970	−0.0858992
$639$	8.14164	0.322078
$640$	0	0
$641$	20.8853	0.824920	0.412460	−	0.910976i	$-0.364670\pi$
$641$	20.8853	0.824920	0.412460	+	0.910976i	$0.364670\pi$
$642$	−8.72584	−0.344381
$643$	36.7808	1.45049	0.725247	−	0.688489i	$-0.241726\pi$
$643$	36.7808	1.45049	0.725247	+	0.688489i	$0.241726\pi$
$644$	−25.4367	−1.00235
$645$	0	0
$646$	−0.0534645	−0.00210353
$647$	7.36088	0.289386	0.144693	−	0.989477i	$-0.453781\pi$
$647$	7.36088	0.289386	0.144693	+	0.989477i	$0.453781\pi$
$648$	4.54949	0.178721
$649$	8.18848	0.321426
$650$	0	0
$651$	6.02895	0.236293
$652$	−5.33332	−0.208869
$653$	12.6554	0.495242	0.247621	−	0.968857i	$-0.420351\pi$
$653$	12.6554	0.495242	0.247621	+	0.968857i	$0.420351\pi$
$654$	−2.70618	−0.105820
$655$	0	0
$656$	5.95641	0.232559
$657$	15.6214	0.609450
$658$	−9.39272	−0.366167
$659$	−1.31820	−0.0513496	−0.0256748	−	0.999670i	$-0.508173\pi$
$659$	−1.31820	−0.0513496	−0.0256748	+	0.999670i	$0.508173\pi$
$660$	0	0
$661$	30.2120	1.17511	0.587555	−	0.809184i	$-0.300091\pi$
$661$	30.2120	1.17511	0.587555	+	0.809184i	$0.300091\pi$
$662$	−34.9394	−1.35796
$663$	1.01154	0.0392851
$664$	6.86419	0.266382
$665$	0	0
$666$	6.61675	0.256394
$667$	−18.2315	−0.705926
$668$	−6.51295	−0.251994
$669$	−4.92257	−0.190317
$670$	0	0
$671$	5.86969	0.226597
$672$	−2.19370	−0.0846237
$673$	39.6417	1.52808	0.764038	−	0.645171i	$-0.223214\pi$
$673$	39.6417	1.52808	0.764038	+	0.645171i	$0.223214\pi$
$674$	13.2745	0.511315
$675$	0	0
$676$	−11.0515	−0.425059
$677$	−3.27436	−0.125844	−0.0629219	−	0.998018i	$-0.520042\pi$
$677$	−3.27436	−0.125844	−0.0629219	+	0.998018i	$0.520042\pi$
$678$	8.70986	0.334500
$679$	−5.26873	−0.202195
$680$	0	0
$681$	1.32679	0.0508426
$682$	2.74831	0.105238
$683$	17.6334	0.674725	0.337362	−	0.941375i	$-0.390465\pi$
$683$	17.6334	0.674725	0.337362	+	0.941375i	$0.390465\pi$
$684$	−0.132317	−0.00505927
$685$	0	0
$686$	−14.6400	−0.558959
$687$	8.39885	0.320436
$688$	−10.8302	−0.412897
$689$	10.3529	0.394413
$690$	0	0
$691$	21.2663	0.809007	0.404504	−	0.914536i	$-0.367444\pi$
$691$	21.2663	0.809007	0.404504	+	0.914536i	$0.367444\pi$
$692$	12.1605	0.462273
$693$	7.49183	0.284591
$694$	21.6176	0.820593
$695$	0	0
$696$	−1.57231	−0.0595983
$697$	−5.95641	−0.225615
$698$	−13.2983	−0.503350
$699$	2.86601	0.108403
$700$	0	0
$701$	28.9152	1.09211	0.546055	−	0.837749i	$-0.316129\pi$
$701$	28.9152	1.09211	0.546055	+	0.837749i	$0.316129\pi$
$702$	5.53805	0.209020
$703$	−0.142942	−0.00539116
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−19.5775	−0.736810
$707$	−43.6063	−1.63998
$708$	5.93393	0.223011
$709$	−44.2276	−1.66100	−0.830501	−	0.557018i	$-0.811946\pi$
$709$	−44.2276	−1.66100	−0.830501	+	0.557018i	$0.811946\pi$
$710$	0	0
$711$	1.36594	0.0512269
$712$	6.03841	0.226299
$713$	23.0934	0.864854
$714$	2.19370	0.0820970
$715$	0	0
$716$	−10.4443	−0.390321
$717$	17.8837	0.667879
$718$	3.63893	0.135804
$719$	20.7101	0.772356	0.386178	−	0.922424i	$-0.373795\pi$
$719$	20.7101	0.772356	0.386178	+	0.922424i	$0.373795\pi$
$720$	0	0
$721$	−0.801547	−0.0298512
$722$	−18.9971	−0.707000
$723$	14.4262	0.536515
$724$	3.20921	0.119270
$725$	0	0
$726$	−0.724668	−0.0268949
$727$	16.8122	0.623531	0.311766	−	0.950159i	$-0.399080\pi$
$727$	16.8122	0.623531	0.311766	+	0.950159i	$0.399080\pi$
$728$	4.22555	0.156609
$729$	−2.63407	−0.0975583
$730$	0	0
$731$	10.8302	0.400569
$732$	4.25357	0.157217
$733$	38.6736	1.42844	0.714221	−	0.699920i	$-0.246781\pi$
$733$	38.6736	1.42844	0.714221	+	0.699920i	$0.246781\pi$
$734$	−20.6075	−0.760638
$735$	0	0
$736$	−8.40277	−0.309730
$737$	6.60195	0.243186
$738$	−14.7413	−0.542633
$739$	−21.5690	−0.793427	−0.396713	−	0.917943i	$-0.629849\pi$
$739$	−21.5690	−0.793427	−0.396713	+	0.917943i	$0.629849\pi$
$740$	0	0
$741$	−0.0540816	−0.00198674
$742$	22.4519	0.824236
$743$	−25.3310	−0.929304	−0.464652	−	0.885493i	$-0.653821\pi$
$743$	−25.3310	−0.929304	−0.464652	+	0.885493i	$0.653821\pi$
$744$	1.99161	0.0730159
$745$	0	0
$746$	27.3516	1.00141
$747$	−16.9879	−0.621555
$748$	1.00000	0.0365636
$749$	36.4507	1.33188
$750$	0	0
$751$	−53.1871	−1.94082	−0.970412	−	0.241457i	$-0.922375\pi$
$751$	−53.1871	−1.94082	−0.970412	+	0.241457i	$0.922375\pi$
$752$	−3.10280	−0.113147
$753$	−14.9313	−0.544127
$754$	3.02862	0.110296
$755$	0	0
$756$	12.0102	0.436806
$757$	−4.36305	−0.158578	−0.0792889	−	0.996852i	$-0.525265\pi$
$757$	−4.36305	−0.158578	−0.0792889	+	0.996852i	$0.525265\pi$
$758$	−24.7166	−0.897747
$759$	−6.08922	−0.221025
$760$	0	0
$761$	−7.13617	−0.258686	−0.129343	−	0.991600i	$-0.541287\pi$
$761$	−7.13617	−0.258686	−0.129343	+	0.991600i	$0.541287\pi$
$762$	3.01614	0.109263
$763$	11.3046	0.409255
$764$	1.15482	0.0417798
$765$	0	0
$766$	15.2868	0.552336
$767$	−11.4301	−0.412716
$768$	−0.724668	−0.0261492
$769$	46.7698	1.68656	0.843281	−	0.537473i	$-0.180621\pi$
$769$	46.7698	1.68656	0.843281	+	0.537473i	$0.180621\pi$
$770$	0	0
$771$	0.286878	0.0103317
$772$	23.6016	0.849442
$773$	−26.7223	−0.961134	−0.480567	−	0.876958i	$-0.659569\pi$
$773$	−26.7223	−0.961134	−0.480567	+	0.876958i	$0.659569\pi$
$774$	26.8032	0.963421
$775$	0	0
$776$	−1.74048	−0.0624795
$777$	5.86504	0.210407
$778$	−31.3293	−1.12321
$779$	0.318456	0.0114099
$780$	0	0
$781$	3.28974	0.117716
$782$	8.40277	0.300482
$783$	8.60817	0.307631
$784$	2.16380	0.0772785
$785$	0	0
$786$	14.6754	0.523455
$787$	14.8532	0.529460	0.264730	−	0.964323i	$-0.414717\pi$
$787$	14.8532	0.529460	0.264730	+	0.964323i	$0.414717\pi$
$788$	−0.923402	−0.0328948
$789$	−15.1873	−0.540684
$790$	0	0
$791$	−36.3840	−1.29366
$792$	2.47486	0.0879402
$793$	−8.19332	−0.290953
$794$	−0.421597	−0.0149619
$795$	0	0
$796$	−11.4023	−0.404143
$797$	−1.85452	−0.0656904	−0.0328452	−	0.999460i	$-0.510457\pi$
$797$	−1.85452	−0.0656904	−0.0328452	+	0.999460i	$0.510457\pi$
$798$	−0.117285	−0.00415184
$799$	3.10280	0.109769
$800$	0	0
$801$	−14.9442	−0.528027
$802$	4.99225	0.176283
$803$	6.31206	0.222748
$804$	4.78422	0.168726
$805$	0	0
$806$	−3.83628	−0.135127
$807$	21.0067	0.739471
$808$	−14.4049	−0.506764
$809$	−23.4811	−0.825550	−0.412775	−	0.910833i	$-0.635440\pi$
$809$	−23.4811	−0.825550	−0.412775	+	0.910833i	$0.635440\pi$
$810$	0	0
$811$	18.8763	0.662837	0.331419	−	0.943484i	$-0.392473\pi$
$811$	18.8763	0.662837	0.331419	+	0.943484i	$0.392473\pi$
$812$	6.56806	0.230494
$813$	0.728885	0.0255631
$814$	2.67359	0.0937092
$815$	0	0
$816$	0.724668	0.0253684
$817$	−0.579031	−0.0202577
$818$	−18.2776	−0.639061
$819$	−10.4576	−0.365419
$820$	0	0
$821$	41.5288	1.44937	0.724683	−	0.689083i	$-0.241986\pi$
$821$	41.5288	1.44937	0.724683	+	0.689083i	$0.241986\pi$
$822$	−0.901932	−0.0314585
$823$	−22.7171	−0.791868	−0.395934	−	0.918279i	$-0.629579\pi$
$823$	−22.7171	−0.791868	−0.395934	+	0.918279i	$0.629579\pi$
$824$	−0.264784	−0.00922417
$825$	0	0
$826$	−24.7880	−0.862484
$827$	−27.0014	−0.938930	−0.469465	−	0.882951i	$-0.655553\pi$
$827$	−27.0014	−0.938930	−0.469465	+	0.882951i	$0.655553\pi$
$828$	20.7957	0.722699
$829$	12.8741	0.447136	0.223568	−	0.974688i	$-0.428229\pi$
$829$	12.8741	0.447136	0.223568	+	0.974688i	$0.428229\pi$
$830$	0	0
$831$	−12.6895	−0.440195
$832$	1.39587	0.0483931
$833$	−2.16380	−0.0749712
$834$	−1.13564	−0.0393240
$835$	0	0
$836$	−0.0534645	−0.00184911
$837$	−10.9038	−0.376889
$838$	18.6601	0.644602
$839$	−10.3177	−0.356206	−0.178103	−	0.984012i	$-0.556996\pi$
$839$	−10.3177	−0.356206	−0.178103	+	0.984012i	$0.556996\pi$
$840$	0	0
$841$	−24.2924	−0.837669
$842$	9.32531	0.321371
$843$	−24.0085	−0.826898
$844$	−18.3851	−0.632841
$845$	0	0
$846$	7.67899	0.264009
$847$	3.02718	0.104015
$848$	7.41679	0.254694
$849$	−21.0503	−0.722443
$850$	0	0
$851$	22.4655	0.770109
$852$	2.38397	0.0816734
$853$	−21.1747	−0.725008	−0.362504	−	0.931982i	$-0.618078\pi$
$853$	−21.1747	−0.725008	−0.362504	+	0.931982i	$0.618078\pi$
$854$	−17.7686	−0.608028
$855$	0	0
$856$	12.0412	0.411558
$857$	−16.9109	−0.577665	−0.288833	−	0.957380i	$-0.593267\pi$
$857$	−16.9109	−0.577665	−0.288833	+	0.957380i	$0.593267\pi$
$858$	1.01154	0.0345335
$859$	−18.4243	−0.628630	−0.314315	−	0.949319i	$-0.601775\pi$
$859$	−18.4243	−0.628630	−0.314315	+	0.949319i	$0.601775\pi$
$860$	0	0
$861$	−13.0666	−0.445307
$862$	12.0838	0.411577
$863$	−6.89704	−0.234778	−0.117389	−	0.993086i	$-0.537452\pi$
$863$	−6.89704	−0.234778	−0.117389	+	0.993086i	$0.537452\pi$
$864$	3.96745	0.134975
$865$	0	0
$866$	−28.6421	−0.973297
$867$	−0.724668	−0.0246110
$868$	−8.31961	−0.282386
$869$	0.551928	0.0187229
$870$	0	0
$871$	−9.21547	−0.312254
$872$	3.73438	0.126462
$873$	4.30743	0.145785
$874$	−0.449250	−0.0151961
$875$	0	0
$876$	4.57414	0.154546
$877$	−29.8559	−1.00816	−0.504082	−	0.863656i	$-0.668169\pi$
$877$	−29.8559	−1.00816	−0.504082	+	0.863656i	$0.668169\pi$
$878$	−31.8119	−1.07360
$879$	−7.81121	−0.263466
$880$	0	0
$881$	−31.1015	−1.04784	−0.523918	−	0.851769i	$-0.675530\pi$
$881$	−31.1015	−1.04784	−0.523918	+	0.851769i	$0.675530\pi$
$882$	−5.35509	−0.180315
$883$	8.75992	0.294795	0.147397	−	0.989077i	$-0.452910\pi$
$883$	8.75992	0.294795	0.147397	+	0.989077i	$0.452910\pi$
$884$	−1.39587	−0.0469482
$885$	0	0
$886$	−28.2282	−0.948345
$887$	−43.6395	−1.46527	−0.732635	−	0.680621i	$-0.761710\pi$
$887$	−43.6395	−1.46527	−0.732635	+	0.680621i	$0.761710\pi$
$888$	1.93746	0.0650170
$889$	−12.5994	−0.422570
$890$	0	0
$891$	−4.54949	−0.152414
$892$	6.79286	0.227442
$893$	−0.165890	−0.00555129
$894$	−2.48529	−0.0831206
$895$	0	0
$896$	3.02718	0.101131
$897$	8.49976	0.283799
$898$	−29.6729	−0.990196
$899$	−5.96300	−0.198877
$900$	0	0
$901$	−7.41679	−0.247089
$902$	−5.95641	−0.198327
$903$	23.7582	0.790623
$904$	−12.0191	−0.399750
$905$	0	0
$906$	−11.1244	−0.369582
$907$	−32.3656	−1.07468	−0.537342	−	0.843365i	$-0.680571\pi$
$907$	−32.3656	−1.07468	−0.537342	+	0.843365i	$0.680571\pi$
$908$	−1.83089	−0.0607602
$909$	35.6501	1.18244
$910$	0	0
$911$	27.5847	0.913921	0.456961	−	0.889487i	$-0.348938\pi$
$911$	27.5847	0.913921	0.456961	+	0.889487i	$0.348938\pi$
$912$	−0.0387440	−0.00128294
$913$	−6.86419	−0.227172
$914$	8.96831	0.296645
$915$	0	0
$916$	−11.5899	−0.382942
$917$	−61.3041	−2.02444
$918$	−3.96745	−0.130945
$919$	41.8655	1.38102	0.690508	−	0.723325i	$-0.257387\pi$
$919$	41.8655	1.38102	0.690508	+	0.723325i	$0.257387\pi$
$920$	0	0
$921$	−3.46849	−0.114291
$922$	0.685658	0.0225809
$923$	−4.59205	−0.151149
$924$	2.19370	0.0721673
$925$	0	0
$926$	−1.17487	−0.0386086
$927$	0.655302	0.0215229
$928$	2.16970	0.0712238
$929$	−8.76334	−0.287516	−0.143758	−	0.989613i	$-0.545919\pi$
$929$	−8.76334	−0.287516	−0.143758	+	0.989613i	$0.545919\pi$
$930$	0	0
$931$	0.115686	0.00379147
$932$	−3.95494	−0.129548
$933$	−19.9038	−0.651622
$934$	−15.8853	−0.519784
$935$	0	0
$936$	−3.45458	−0.112916
$937$	38.8017	1.26760	0.633798	−	0.773499i	$-0.281495\pi$
$937$	38.8017	1.26760	0.633798	+	0.773499i	$0.281495\pi$
$938$	−19.9853	−0.652542
$939$	16.2055	0.528846
$940$	0	0
$941$	−34.3084	−1.11842	−0.559211	−	0.829025i	$-0.688896\pi$
$941$	−34.3084	−1.11842	−0.559211	+	0.829025i	$0.688896\pi$
$942$	8.76287	0.285510
$943$	−50.0503	−1.62986
$944$	−8.18848	−0.266512
$945$	0	0
$946$	10.8302	0.352120
$947$	13.5470	0.440219	0.220110	−	0.975475i	$-0.429358\pi$
$947$	13.5470	0.440219	0.220110	+	0.975475i	$0.429358\pi$
$948$	0.399964	0.0129902
$949$	−8.81082	−0.286011
$950$	0	0
$951$	2.36523	0.0766978
$952$	−3.02718	−0.0981114
$953$	51.1084	1.65556	0.827782	−	0.561050i	$-0.189603\pi$
$953$	51.1084	1.65556	0.827782	+	0.561050i	$0.189603\pi$
$954$	−18.3555	−0.594281
$955$	0	0
$956$	−24.6785	−0.798159
$957$	1.57231	0.0508256
$958$	16.9229	0.546754
$959$	3.76767	0.121664
$960$	0	0
$961$	−23.4468	−0.756349
$962$	−3.73198	−0.120324
$963$	−29.8001	−0.960296
$964$	−19.9073	−0.641170
$965$	0	0
$966$	18.4331	0.593076
$967$	51.3932	1.65269	0.826347	−	0.563161i	$-0.190415\pi$
$967$	51.3932	1.65269	0.826347	+	0.563161i	$0.190415\pi$
$968$	1.00000	0.0321412
$969$	0.0387440	0.00124464
$970$	0	0
$971$	8.91971	0.286247	0.143124	−	0.989705i	$-0.454285\pi$
$971$	8.91971	0.286247	0.143124	+	0.989705i	$0.454285\pi$
$972$	−15.1992	−0.487515
$973$	4.74394	0.152084
$974$	−28.1832	−0.903047
$975$	0	0
$976$	−5.86969	−0.187884
$977$	35.7429	1.14352	0.571758	−	0.820423i	$-0.306262\pi$
$977$	35.7429	1.14352	0.571758	+	0.820423i	$0.306262\pi$
$978$	3.86488	0.123585
$979$	−6.03841	−0.192988
$980$	0	0
$981$	−9.24205	−0.295076
$982$	14.6487	0.467459
$983$	−55.8385	−1.78097	−0.890485	−	0.455012i	$-0.849635\pi$
$983$	−55.8385	−1.78097	−0.890485	+	0.455012i	$0.849635\pi$
$984$	−4.31641	−0.137602
$985$	0	0
$986$	−2.16970	−0.0690973
$987$	6.80660	0.216657
$988$	0.0746295	0.00237428
$989$	91.0037	2.89375
$990$	0	0
$991$	−17.6297	−0.560025	−0.280012	−	0.959996i	$-0.590339\pi$
$991$	−17.6297	−0.560025	−0.280012	+	0.959996i	$0.590339\pi$
$992$	−2.74831	−0.0872588
$993$	25.3194	0.803488
$994$	−9.95862	−0.315868
$995$	0	0
$996$	−4.97426	−0.157615
$997$	18.2497	0.577974	0.288987	−	0.957333i	$-0.406682\pi$
$997$	18.2497	0.577974	0.288987	+	0.957333i	$0.406682\pi$
$998$	13.8988	0.439959
$999$	−10.6073	−0.335601

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9350.2.a.dc.1.4		7
5.2	odd	4		1870.2.b.d.749.11	yes	14
5.3	odd	4		1870.2.b.d.749.4	✓	14
5.4	even	2		9350.2.a.db.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1870.2.b.d.749.4	✓	14	5.3	odd	4
1870.2.b.d.749.11	yes	14	5.2	odd	4
9350.2.a.db.1.4		7	5.4	even	2
9350.2.a.dc.1.4		7	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:	\( N \)	\(=\)	\( 9350 = 2 \cdot 5^{2} \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9350.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.724668 q^{3} +1.00000 q^{4} -0.724668 q^{6} +3.02718 q^{7} +1.00000 q^{8} -2.47486 q^{9} -1.00000 q^{11} -0.724668 q^{12} +1.39587 q^{13} +3.02718 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.47486 q^{18} +0.0534645 q^{19} -2.19370 q^{21} -1.00000 q^{22} -8.40277 q^{23} -0.724668 q^{24} +1.39587 q^{26} +3.96745 q^{27} +3.02718 q^{28} +2.16970 q^{29} -2.74831 q^{31} +1.00000 q^{32} +0.724668 q^{33} -1.00000 q^{34} -2.47486 q^{36} -2.67359 q^{37} +0.0534645 q^{38} -1.01154 q^{39} +5.95641 q^{41} -2.19370 q^{42} -10.8302 q^{43} -1.00000 q^{44} -8.40277 q^{46} -3.10280 q^{47} -0.724668 q^{48} +2.16380 q^{49} +0.724668 q^{51} +1.39587 q^{52} +7.41679 q^{53} +3.96745 q^{54} +3.02718 q^{56} -0.0387440 q^{57} +2.16970 q^{58} -8.18848 q^{59} -5.86969 q^{61} -2.74831 q^{62} -7.49183 q^{63} +1.00000 q^{64} +0.724668 q^{66} -6.60195 q^{67} -1.00000 q^{68} +6.08922 q^{69} -3.28974 q^{71} -2.47486 q^{72} -6.31206 q^{73} -2.67359 q^{74} +0.0534645 q^{76} -3.02718 q^{77} -1.01154 q^{78} -0.551928 q^{79} +4.54949 q^{81} +5.95641 q^{82} +6.86419 q^{83} -2.19370 q^{84} -10.8302 q^{86} -1.57231 q^{87} -1.00000 q^{88} +6.03841 q^{89} +4.22555 q^{91} -8.40277 q^{92} +1.99161 q^{93} -3.10280 q^{94} -0.724668 q^{96} -1.74048 q^{97} +2.16380 q^{98} +2.47486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 3 q^{6} - 4 q^{7} + 7 q^{8} - 7 q^{11} - 3 q^{12} + 9 q^{13} - 4 q^{14} + 7 q^{16} - 7 q^{17} - 3 q^{19} - 4 q^{21} - 7 q^{22} - 6 q^{23} - 3 q^{24} + 9 q^{26} - 3 q^{27}+ \cdots + 3 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 - 3 * q^3 + 7 * q^4 - 3 * q^6 - 4 * q^7 + 7 * q^8 - 7 * q^11 - 3 * q^12 + 9 * q^13 - 4 * q^14 + 7 * q^16 - 7 * q^17 - 3 * q^19 - 4 * q^21 - 7 * q^22 - 6 * q^23 - 3 * q^24 + 9 * q^26 - 3 * q^27 - 4 * q^28 - 9 * q^29 - 9 * q^31 + 7 * q^32 + 3 * q^33 - 7 * q^34 - 6 * q^37 - 3 * q^38 - 13 * q^39 - 4 * q^42 - 7 * q^44 - 6 * q^46 - 15 * q^47 - 3 * q^48 + 3 * q^49 + 3 * q^51 + 9 * q^52 + 17 * q^53 - 3 * q^54 - 4 * q^56 - 29 * q^57 - 9 * q^58 - 27 * q^59 - 33 * q^61 - 9 * q^62 - 4 * q^63 + 7 * q^64 + 3 * q^66 - 16 * q^67 - 7 * q^68 + 2 * q^69 - q^71 + q^73 - 6 * q^74 - 3 * q^76 + 4 * q^77 - 13 * q^78 - 38 * q^79 - 9 * q^81 - 8 * q^83 - 4 * q^84 + 21 * q^87 - 7 * q^88 + 31 * q^89 - 12 * q^91 - 6 * q^92 + 29 * q^93 - 15 * q^94 - 3 * q^96 + 3 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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