LMFDB - Embedded newform 5950.2.a.cc.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5950,2,Mod(1,5950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5950, 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("5950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,7,-2,7,0,-2,-7,7,7,0,-7,-2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.5109892027$
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} - 2x^{6} - 12x^{5} + 17x^{4} + 40x^{3} - 32x^{2} - 40x + 4 $ x^7 - 2x^6 - 12x^5 + 17x^4 + 40x^3 - 32x^2 - 40x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.944189$ of defining polynomial
Character	$\chi$	$=$	5950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.944189 q^{3} +1.00000 q^{4} +0.944189 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.10851 q^{9} -6.47760 q^{11} +0.944189 q^{12} +6.60115 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.10851 q^{18} +3.14658 q^{19} -0.944189 q^{21} -6.47760 q^{22} +0.715465 q^{23} +0.944189 q^{24} +6.60115 q^{26} -4.82340 q^{27} -1.00000 q^{28} -5.52962 q^{29} -10.3233 q^{31} +1.00000 q^{32} -6.11608 q^{33} -1.00000 q^{34} -2.10851 q^{36} +8.27542 q^{37} +3.14658 q^{38} +6.23274 q^{39} -7.08117 q^{41} -0.944189 q^{42} -0.272148 q^{43} -6.47760 q^{44} +0.715465 q^{46} -7.39269 q^{47} +0.944189 q^{48} +1.00000 q^{49} -0.944189 q^{51} +6.60115 q^{52} +1.24260 q^{53} -4.82340 q^{54} -1.00000 q^{56} +2.97097 q^{57} -5.52962 q^{58} -6.52216 q^{59} -5.44476 q^{61} -10.3233 q^{62} +2.10851 q^{63} +1.00000 q^{64} -6.11608 q^{66} -8.75002 q^{67} -1.00000 q^{68} +0.675535 q^{69} +12.4820 q^{71} -2.10851 q^{72} +10.5962 q^{73} +8.27542 q^{74} +3.14658 q^{76} +6.47760 q^{77} +6.23274 q^{78} -15.8721 q^{79} +1.77132 q^{81} -7.08117 q^{82} -11.0408 q^{83} -0.944189 q^{84} -0.272148 q^{86} -5.22101 q^{87} -6.47760 q^{88} -6.41380 q^{89} -6.60115 q^{91} +0.715465 q^{92} -9.74715 q^{93} -7.39269 q^{94} +0.944189 q^{96} -7.55828 q^{97} +1.00000 q^{98} +13.6581 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} - 2 q^{3} + 7 q^{4} - 2 q^{6} - 7 q^{7} + 7 q^{8} + 7 q^{9} - 7 q^{11} - 2 q^{12} + q^{13} - 7 q^{14} + 7 q^{16} - 7 q^{17} + 7 q^{18} - 2 q^{19} + 2 q^{21} - 7 q^{22} - 27 q^{23} - 2 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) $ 7 * q + 7 * q^2 - 2 * q^3 + 7 * q^4 - 2 * q^6 - 7 * q^7 + 7 * q^8 + 7 * q^9 - 7 * q^11 - 2 * q^12 + q^13 - 7 * q^14 + 7 * q^16 - 7 * q^17 + 7 * q^18 - 2 * q^19 + 2 * q^21 - 7 * q^22 - 27 * q^23 - 2 * q^24 + q^26 - 17 * q^27 - 7 * q^28 - 10 * q^29 - 18 * q^31 + 7 * q^32 - 8 * q^33 - 7 * q^34 + 7 * q^36 - 12 * q^37 - 2 * q^38 - 25 * q^39 + 17 * q^41 + 2 * q^42 - 7 * q^43 - 7 * q^44 - 27 * q^46 - 8 * q^47 - 2 * q^48 + 7 * q^49 + 2 * q^51 + q^52 - 20 * q^53 - 17 * q^54 - 7 * q^56 - 25 * q^57 - 10 * q^58 + q^61 - 18 * q^62 - 7 * q^63 + 7 * q^64 - 8 * q^66 - 7 * q^67 - 7 * q^68 + 3 * q^69 - 11 * q^71 + 7 * q^72 + 7 * q^73 - 12 * q^74 - 2 * q^76 + 7 * q^77 - 25 * q^78 - 26 * q^79 + 11 * q^81 + 17 * q^82 - 22 * q^83 + 2 * q^84 - 7 * q^86 - 20 * q^87 - 7 * q^88 + 21 * q^89 - q^91 - 27 * q^92 - 25 * q^93 - 8 * q^94 - 2 * q^96 + 19 * q^97 + 7 * q^98 - 44 * 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
$3$	0.944189	0.545128	0.272564	−	0.962138i	$-0.412128\pi$
$3$	0.944189	0.545128	0.272564	+	0.962138i	$0.412128\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.944189	0.385464
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.10851	−0.702836
$10$	0	0
$11$	−6.47760	−1.95307	−0.976535	−	0.215359i	$-0.930908\pi$
$11$	−6.47760	−1.95307	−0.976535	+	0.215359i	$0.930908\pi$
$12$	0.944189	0.272564
$13$	6.60115	1.83083	0.915415	−	0.402510i	$-0.131862\pi$
$13$	6.60115	1.83083	0.915415	+	0.402510i	$0.131862\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.10851	−0.496980
$19$	3.14658	0.721875	0.360938	−	0.932590i	$-0.382457\pi$
$19$	3.14658	0.721875	0.360938	+	0.932590i	$0.382457\pi$
$20$	0	0
$21$	−0.944189	−0.206039
$22$	−6.47760	−1.38103
$23$	0.715465	0.149185	0.0745924	−	0.997214i	$-0.476234\pi$
$23$	0.715465	0.149185	0.0745924	+	0.997214i	$0.476234\pi$
$24$	0.944189	0.192732
$25$	0	0
$26$	6.60115	1.29459
$27$	−4.82340	−0.928263
$28$	−1.00000	−0.188982
$29$	−5.52962	−1.02682	−0.513412	−	0.858142i	$-0.671619\pi$
$29$	−5.52962	−1.02682	−0.513412	+	0.858142i	$0.671619\pi$
$30$	0	0
$31$	−10.3233	−1.85412	−0.927060	−	0.374913i	$-0.877672\pi$
$31$	−10.3233	−1.85412	−0.927060	+	0.374913i	$0.877672\pi$
$32$	1.00000	0.176777
$33$	−6.11608	−1.06467
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.10851	−0.351418
$37$	8.27542	1.36047	0.680235	−	0.732994i	$-0.261878\pi$
$37$	8.27542	1.36047	0.680235	+	0.732994i	$0.261878\pi$
$38$	3.14658	0.510443
$39$	6.23274	0.998037
$40$	0	0
$41$	−7.08117	−1.10589	−0.552947	−	0.833217i	$-0.686497\pi$
$41$	−7.08117	−1.10589	−0.552947	+	0.833217i	$0.686497\pi$
$42$	−0.944189	−0.145692
$43$	−0.272148	−0.0415021	−0.0207511	−	0.999785i	$-0.506606\pi$
$43$	−0.272148	−0.0415021	−0.0207511	+	0.999785i	$0.506606\pi$
$44$	−6.47760	−0.976535
$45$	0	0
$46$	0.715465	0.105490
$47$	−7.39269	−1.07833	−0.539167	−	0.842199i	$-0.681261\pi$
$47$	−7.39269	−1.07833	−0.539167	+	0.842199i	$0.681261\pi$
$48$	0.944189	0.136282
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.944189	−0.132213
$52$	6.60115	0.915415
$53$	1.24260	0.170684	0.0853418	−	0.996352i	$-0.472802\pi$
$53$	1.24260	0.170684	0.0853418	+	0.996352i	$0.472802\pi$
$54$	−4.82340	−0.656381
$55$	0	0
$56$	−1.00000	−0.133631
$57$	2.97097	0.393514
$58$	−5.52962	−0.726075
$59$	−6.52216	−0.849112	−0.424556	−	0.905402i	$-0.639570\pi$
$59$	−6.52216	−0.849112	−0.424556	+	0.905402i	$0.639570\pi$
$60$	0	0
$61$	−5.44476	−0.697130	−0.348565	−	0.937285i	$-0.613331\pi$
$61$	−5.44476	−0.697130	−0.348565	+	0.937285i	$0.613331\pi$
$62$	−10.3233	−1.31106
$63$	2.10851	0.265647
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.11608	−0.752837
$67$	−8.75002	−1.06898	−0.534492	−	0.845173i	$-0.679497\pi$
$67$	−8.75002	−1.06898	−0.534492	+	0.845173i	$0.679497\pi$
$68$	−1.00000	−0.121268
$69$	0.675535	0.0813248
$70$	0	0
$71$	12.4820	1.48134	0.740672	−	0.671867i	$-0.234507\pi$
$71$	12.4820	1.48134	0.740672	+	0.671867i	$0.234507\pi$
$72$	−2.10851	−0.248490
$73$	10.5962	1.24019	0.620094	−	0.784527i	$-0.287094\pi$
$73$	10.5962	1.24019	0.620094	+	0.784527i	$0.287094\pi$
$74$	8.27542	0.961997
$75$	0	0
$76$	3.14658	0.360938
$77$	6.47760	0.738191
$78$	6.23274	0.705719
$79$	−15.8721	−1.78575	−0.892873	−	0.450308i	$-0.851314\pi$
$79$	−15.8721	−1.78575	−0.892873	+	0.450308i	$0.851314\pi$
$80$	0	0
$81$	1.77132	0.196814
$82$	−7.08117	−0.781985
$83$	−11.0408	−1.21189	−0.605944	−	0.795507i	$-0.707205\pi$
$83$	−11.0408	−1.21189	−0.605944	+	0.795507i	$0.707205\pi$
$84$	−0.944189	−0.103019
$85$	0	0
$86$	−0.272148	−0.0293464
$87$	−5.22101	−0.559751
$88$	−6.47760	−0.690515
$89$	−6.41380	−0.679862	−0.339931	−	0.940450i	$-0.610404\pi$
$89$	−6.41380	−0.679862	−0.339931	+	0.940450i	$0.610404\pi$
$90$	0	0
$91$	−6.60115	−0.691989
$92$	0.715465	0.0745924
$93$	−9.74715	−1.01073
$94$	−7.39269	−0.762498
$95$	0	0
$96$	0.944189	0.0963659
$97$	−7.55828	−0.767427	−0.383713	−	0.923452i	$-0.625355\pi$
$97$	−7.55828	−0.767427	−0.383713	+	0.923452i	$0.625355\pi$
$98$	1.00000	0.101015
$99$	13.6581	1.37269
$100$	0	0
$101$	0.605246	0.0602242	0.0301121	−	0.999547i	$-0.490414\pi$
$101$	0.605246	0.0602242	0.0301121	+	0.999547i	$0.490414\pi$
$102$	−0.944189	−0.0934887
$103$	10.6708	1.05142	0.525711	−	0.850663i	$-0.323799\pi$
$103$	10.6708	1.05142	0.525711	+	0.850663i	$0.323799\pi$
$104$	6.60115	0.647296
$105$	0	0
$106$	1.24260	0.120692
$107$	−2.47479	−0.239247	−0.119624	−	0.992819i	$-0.538169\pi$
$107$	−2.47479	−0.239247	−0.119624	+	0.992819i	$0.538169\pi$
$108$	−4.82340	−0.464132
$109$	−2.40704	−0.230553	−0.115276	−	0.993333i	$-0.536775\pi$
$109$	−2.40704	−0.230553	−0.115276	+	0.993333i	$0.536775\pi$
$110$	0	0
$111$	7.81356	0.741630
$112$	−1.00000	−0.0944911
$113$	−5.04159	−0.474272	−0.237136	−	0.971476i	$-0.576209\pi$
$113$	−5.04159	−0.474272	−0.237136	+	0.971476i	$0.576209\pi$
$114$	2.97097	0.278257
$115$	0	0
$116$	−5.52962	−0.513412
$117$	−13.9186	−1.28677
$118$	−6.52216	−0.600413
$119$	1.00000	0.0916698
$120$	0	0
$121$	30.9593	2.81448
$122$	−5.44476	−0.492946
$123$	−6.68597	−0.602853
$124$	−10.3233	−0.927060
$125$	0	0
$126$	2.10851	0.187841
$127$	10.7877	0.957252	0.478626	−	0.878019i	$-0.341135\pi$
$127$	10.7877	0.957252	0.478626	+	0.878019i	$0.341135\pi$
$128$	1.00000	0.0883883
$129$	−0.256959	−0.0226240
$130$	0	0
$131$	−1.16568	−0.101846	−0.0509229	−	0.998703i	$-0.516216\pi$
$131$	−1.16568	−0.101846	−0.0509229	+	0.998703i	$0.516216\pi$
$132$	−6.11608	−0.532336
$133$	−3.14658	−0.272843
$134$	−8.75002	−0.755886
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−20.9397	−1.78900	−0.894498	−	0.447073i	$-0.852467\pi$
$137$	−20.9397	−1.78900	−0.894498	+	0.447073i	$0.852467\pi$
$138$	0.675535	0.0575053
$139$	−3.72458	−0.315915	−0.157957	−	0.987446i	$-0.550491\pi$
$139$	−3.72458	−0.315915	−0.157957	+	0.987446i	$0.550491\pi$
$140$	0	0
$141$	−6.98010	−0.587830
$142$	12.4820	1.04747
$143$	−42.7596	−3.57574
$144$	−2.10851	−0.175709
$145$	0	0
$146$	10.5962	0.876945
$147$	0.944189	0.0778754
$148$	8.27542	0.680235
$149$	2.93671	0.240585	0.120293	−	0.992738i	$-0.461617\pi$
$149$	2.93671	0.240585	0.120293	+	0.992738i	$0.461617\pi$
$150$	0	0
$151$	−8.47874	−0.689990	−0.344995	−	0.938605i	$-0.612119\pi$
$151$	−8.47874	−0.689990	−0.344995	+	0.938605i	$0.612119\pi$
$152$	3.14658	0.255221
$153$	2.10851	0.170463
$154$	6.47760	0.521980
$155$	0	0
$156$	6.23274	0.499018
$157$	4.58849	0.366201	0.183101	−	0.983094i	$-0.441387\pi$
$157$	4.58849	0.366201	0.183101	+	0.983094i	$0.441387\pi$
$158$	−15.8721	−1.26271
$159$	1.17325	0.0930444
$160$	0	0
$161$	−0.715465	−0.0563866
$162$	1.77132	0.139168
$163$	−12.9615	−1.01523	−0.507613	−	0.861585i	$-0.669472\pi$
$163$	−12.9615	−1.01523	−0.507613	+	0.861585i	$0.669472\pi$
$164$	−7.08117	−0.552947
$165$	0	0
$166$	−11.0408	−0.856935
$167$	9.29757	0.719467	0.359734	−	0.933055i	$-0.382868\pi$
$167$	9.29757	0.719467	0.359734	+	0.933055i	$0.382868\pi$
$168$	−0.944189	−0.0728458
$169$	30.5752	2.35194
$170$	0	0
$171$	−6.63459	−0.507360
$172$	−0.272148	−0.0207511
$173$	−9.43425	−0.717272	−0.358636	−	0.933477i	$-0.616758\pi$
$173$	−9.43425	−0.717272	−0.358636	+	0.933477i	$0.616758\pi$
$174$	−5.22101	−0.395803
$175$	0	0
$176$	−6.47760	−0.488267
$177$	−6.15815	−0.462875
$178$	−6.41380	−0.480735
$179$	−2.84058	−0.212315	−0.106157	−	0.994349i	$-0.533855\pi$
$179$	−2.84058	−0.212315	−0.106157	+	0.994349i	$0.533855\pi$
$180$	0	0
$181$	−3.01811	−0.224334	−0.112167	−	0.993689i	$-0.535779\pi$
$181$	−3.01811	−0.224334	−0.112167	+	0.993689i	$0.535779\pi$
$182$	−6.60115	−0.489310
$183$	−5.14088	−0.380025
$184$	0.715465	0.0527448
$185$	0	0
$186$	−9.74715	−0.714696
$187$	6.47760	0.473689
$188$	−7.39269	−0.539167
$189$	4.82340	0.350850
$190$	0	0
$191$	7.85353	0.568261	0.284131	−	0.958786i	$-0.408295\pi$
$191$	7.85353	0.568261	0.284131	+	0.958786i	$0.408295\pi$
$192$	0.944189	0.0681410
$193$	−6.63482	−0.477584	−0.238792	−	0.971071i	$-0.576751\pi$
$193$	−6.63482	−0.477584	−0.238792	+	0.971071i	$0.576751\pi$
$194$	−7.55828	−0.542653
$195$	0	0
$196$	1.00000	0.0714286
$197$	11.5437	0.822454	0.411227	−	0.911533i	$-0.365100\pi$
$197$	11.5437	0.822454	0.411227	+	0.911533i	$0.365100\pi$
$198$	13.6581	0.970636
$199$	−26.0150	−1.84416	−0.922078	−	0.387004i	$-0.873510\pi$
$199$	−26.0150	−1.84416	−0.922078	+	0.387004i	$0.873510\pi$
$200$	0	0
$201$	−8.26167	−0.582733
$202$	0.605246	0.0425849
$203$	5.52962	0.388103
$204$	−0.944189	−0.0661065
$205$	0	0
$206$	10.6708	0.743467
$207$	−1.50856	−0.104852
$208$	6.60115	0.457708
$209$	−20.3823	−1.40987
$210$	0	0
$211$	−19.0340	−1.31036	−0.655179	−	0.755474i	$-0.727407\pi$
$211$	−19.0340	−1.31036	−0.655179	+	0.755474i	$0.727407\pi$
$212$	1.24260	0.0853418
$213$	11.7854	0.807522
$214$	−2.47479	−0.169173
$215$	0	0
$216$	−4.82340	−0.328191
$217$	10.3233	0.700791
$218$	−2.40704	−0.163025
$219$	10.0048	0.676061
$220$	0	0
$221$	−6.60115	−0.444042
$222$	7.81356	0.524412
$223$	−2.99408	−0.200498	−0.100249	−	0.994962i	$-0.531964\pi$
$223$	−2.99408	−0.200498	−0.100249	+	0.994962i	$0.531964\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−5.04159	−0.335361
$227$	−7.17860	−0.476460	−0.238230	−	0.971209i	$-0.576567\pi$
$227$	−7.17860	−0.476460	−0.238230	+	0.971209i	$0.576567\pi$
$228$	2.97097	0.196757
$229$	20.4722	1.35284	0.676422	−	0.736515i	$-0.263530\pi$
$229$	20.4722	1.35284	0.676422	+	0.736515i	$0.263530\pi$
$230$	0	0
$231$	6.11608	0.402408
$232$	−5.52962	−0.363037
$233$	−14.5803	−0.955185	−0.477592	−	0.878582i	$-0.658490\pi$
$233$	−14.5803	−0.955185	−0.477592	+	0.878582i	$0.658490\pi$
$234$	−13.9186	−0.909886
$235$	0	0
$236$	−6.52216	−0.424556
$237$	−14.9862	−0.973460
$238$	1.00000	0.0648204
$239$	−27.8402	−1.80083	−0.900415	−	0.435031i	$-0.856737\pi$
$239$	−27.8402	−1.80083	−0.900415	+	0.435031i	$0.856737\pi$
$240$	0	0
$241$	2.46386	0.158711	0.0793555	−	0.996846i	$-0.474714\pi$
$241$	2.46386	0.158711	0.0793555	+	0.996846i	$0.474714\pi$
$242$	30.9593	1.99014
$243$	16.1427	1.03555
$244$	−5.44476	−0.348565
$245$	0	0
$246$	−6.68597	−0.426282
$247$	20.7711	1.32163
$248$	−10.3233	−0.655530
$249$	−10.4246	−0.660634
$250$	0	0
$251$	7.79037	0.491724	0.245862	−	0.969305i	$-0.420929\pi$
$251$	7.79037	0.491724	0.245862	+	0.969305i	$0.420929\pi$
$252$	2.10851	0.132823
$253$	−4.63450	−0.291368
$254$	10.7877	0.676879
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.2883	−1.88933	−0.944667	−	0.328030i	$-0.893615\pi$
$257$	−30.2883	−1.88933	−0.944667	+	0.328030i	$0.893615\pi$
$258$	−0.256959	−0.0159976
$259$	−8.27542	−0.514209
$260$	0	0
$261$	11.6592	0.721689
$262$	−1.16568	−0.0720159
$263$	13.8039	0.851182	0.425591	−	0.904916i	$-0.360066\pi$
$263$	13.8039	0.851182	0.425591	+	0.904916i	$0.360066\pi$
$264$	−6.11608	−0.376419
$265$	0	0
$266$	−3.14658	−0.192929
$267$	−6.05584	−0.370612
$268$	−8.75002	−0.534492
$269$	−12.4186	−0.757178	−0.378589	−	0.925565i	$-0.623591\pi$
$269$	−12.4186	−0.757178	−0.378589	+	0.925565i	$0.623591\pi$
$270$	0	0
$271$	12.1674	0.739114	0.369557	−	0.929208i	$-0.379509\pi$
$271$	12.1674	0.739114	0.369557	+	0.929208i	$0.379509\pi$
$272$	−1.00000	−0.0606339
$273$	−6.23274	−0.377222
$274$	−20.9397	−1.26501
$275$	0	0
$276$	0.675535	0.0406624
$277$	26.5516	1.59533	0.797665	−	0.603101i	$-0.206069\pi$
$277$	26.5516	1.59533	0.797665	+	0.603101i	$0.206069\pi$
$278$	−3.72458	−0.223385
$279$	21.7668	1.30314
$280$	0	0
$281$	−9.09912	−0.542808	−0.271404	−	0.962465i	$-0.587488\pi$
$281$	−9.09912	−0.542808	−0.271404	+	0.962465i	$0.587488\pi$
$282$	−6.98010	−0.415659
$283$	−6.80753	−0.404666	−0.202333	−	0.979317i	$-0.564852\pi$
$283$	−6.80753	−0.404666	−0.202333	+	0.979317i	$0.564852\pi$
$284$	12.4820	0.740672
$285$	0	0
$286$	−42.7596	−2.52843
$287$	7.08117	0.417989
$288$	−2.10851	−0.124245
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.13644	−0.418346
$292$	10.5962	0.620094
$293$	27.1352	1.58526	0.792628	−	0.609706i	$-0.208713\pi$
$293$	27.1352	1.58526	0.792628	+	0.609706i	$0.208713\pi$
$294$	0.944189	0.0550662
$295$	0	0
$296$	8.27542	0.480999
$297$	31.2440	1.81296
$298$	2.93671	0.170119
$299$	4.72290	0.273132
$300$	0	0
$301$	0.272148	0.0156863
$302$	−8.47874	−0.487897
$303$	0.571466	0.0328299
$304$	3.14658	0.180469
$305$	0	0
$306$	2.10851	0.120535
$307$	24.9786	1.42560	0.712802	−	0.701365i	$-0.247426\pi$
$307$	24.9786	1.42560	0.712802	+	0.701365i	$0.247426\pi$
$308$	6.47760	0.369096
$309$	10.0752	0.573159
$310$	0	0
$311$	20.4702	1.16076	0.580378	−	0.814347i	$-0.302905\pi$
$311$	20.4702	1.16076	0.580378	+	0.814347i	$0.302905\pi$
$312$	6.23274	0.352859
$313$	−7.84620	−0.443493	−0.221747	−	0.975104i	$-0.571176\pi$
$313$	−7.84620	−0.443493	−0.221747	+	0.975104i	$0.571176\pi$
$314$	4.58849	0.258943
$315$	0	0
$316$	−15.8721	−0.892873
$317$	31.3528	1.76095	0.880476	−	0.474091i	$-0.157223\pi$
$317$	31.3528	1.76095	0.880476	+	0.474091i	$0.157223\pi$
$318$	1.17325	0.0657923
$319$	35.8187	2.00546
$320$	0	0
$321$	−2.33667	−0.130420
$322$	−0.715465	−0.0398713
$323$	−3.14658	−0.175080
$324$	1.77132	0.0984068
$325$	0	0
$326$	−12.9615	−0.717873
$327$	−2.27270	−0.125681
$328$	−7.08117	−0.390992
$329$	7.39269	0.407572
$330$	0	0
$331$	8.54057	0.469432	0.234716	−	0.972064i	$-0.424584\pi$
$331$	8.54057	0.469432	0.234716	+	0.972064i	$0.424584\pi$
$332$	−11.0408	−0.605944
$333$	−17.4488	−0.956187
$334$	9.29757	0.508740
$335$	0	0
$336$	−0.944189	−0.0515097
$337$	8.28173	0.451134	0.225567	−	0.974228i	$-0.427576\pi$
$337$	8.28173	0.451134	0.225567	+	0.974228i	$0.427576\pi$
$338$	30.5752	1.66307
$339$	−4.76021	−0.258539
$340$	0	0
$341$	66.8702	3.62123
$342$	−6.63459	−0.358757
$343$	−1.00000	−0.0539949
$344$	−0.272148	−0.0146732
$345$	0	0
$346$	−9.43425	−0.507188
$347$	−11.7797	−0.632365	−0.316183	−	0.948698i	$-0.602401\pi$
$347$	−11.7797	−0.632365	−0.316183	+	0.948698i	$0.602401\pi$
$348$	−5.22101	−0.279875
$349$	28.7276	1.53776	0.768878	−	0.639396i	$-0.220815\pi$
$349$	28.7276	1.53776	0.768878	+	0.639396i	$0.220815\pi$
$350$	0	0
$351$	−31.8400	−1.69949
$352$	−6.47760	−0.345257
$353$	−9.10674	−0.484703	−0.242352	−	0.970188i	$-0.577919\pi$
$353$	−9.10674	−0.484703	−0.242352	+	0.970188i	$0.577919\pi$
$354$	−6.15815	−0.327302
$355$	0	0
$356$	−6.41380	−0.339931
$357$	0.944189	0.0499718
$358$	−2.84058	−0.150129
$359$	10.5395	0.556252	0.278126	−	0.960545i	$-0.410287\pi$
$359$	10.5395	0.556252	0.278126	+	0.960545i	$0.410287\pi$
$360$	0	0
$361$	−9.09903	−0.478896
$362$	−3.01811	−0.158628
$363$	29.2314	1.53425
$364$	−6.60115	−0.345994
$365$	0	0
$366$	−5.14088	−0.268718
$367$	−6.19187	−0.323213	−0.161607	−	0.986855i	$-0.551668\pi$
$367$	−6.19187	−0.323213	−0.161607	+	0.986855i	$0.551668\pi$
$368$	0.715465	0.0372962
$369$	14.9307	0.777261
$370$	0	0
$371$	−1.24260	−0.0645123
$372$	−9.74715	−0.505366
$373$	6.40599	0.331690	0.165845	−	0.986152i	$-0.446965\pi$
$373$	6.40599	0.331690	0.165845	+	0.986152i	$0.446965\pi$
$374$	6.47760	0.334949
$375$	0	0
$376$	−7.39269	−0.381249
$377$	−36.5019	−1.87994
$378$	4.82340	0.248089
$379$	−4.47057	−0.229638	−0.114819	−	0.993386i	$-0.536629\pi$
$379$	−4.47057	−0.229638	−0.114819	+	0.993386i	$0.536629\pi$
$380$	0	0
$381$	10.1856	0.521824
$382$	7.85353	0.401821
$383$	−24.9589	−1.27534	−0.637670	−	0.770310i	$-0.720102\pi$
$383$	−24.9589	−1.27534	−0.637670	+	0.770310i	$0.720102\pi$
$384$	0.944189	0.0481829
$385$	0	0
$386$	−6.63482	−0.337703
$387$	0.573825	0.0291692
$388$	−7.55828	−0.383713
$389$	12.9696	0.657585	0.328793	−	0.944402i	$-0.393358\pi$
$389$	12.9696	0.657585	0.328793	+	0.944402i	$0.393358\pi$
$390$	0	0
$391$	−0.715465	−0.0361826
$392$	1.00000	0.0505076
$393$	−1.10062	−0.0555190
$394$	11.5437	0.581563
$395$	0	0
$396$	13.6581	0.686344
$397$	29.5224	1.48169	0.740844	−	0.671678i	$-0.234426\pi$
$397$	29.5224	1.48169	0.740844	+	0.671678i	$0.234426\pi$
$398$	−26.0150	−1.30402
$399$	−2.97097	−0.148734
$400$	0	0
$401$	33.3973	1.66778	0.833890	−	0.551931i	$-0.186109\pi$
$401$	33.3973	1.66778	0.833890	+	0.551931i	$0.186109\pi$
$402$	−8.26167	−0.412055
$403$	−68.1457	−3.39458
$404$	0.605246	0.0301121
$405$	0	0
$406$	5.52962	0.274430
$407$	−53.6048	−2.65709
$408$	−0.944189	−0.0467443
$409$	7.96757	0.393971	0.196986	−	0.980406i	$-0.436885\pi$
$409$	7.96757	0.393971	0.196986	+	0.980406i	$0.436885\pi$
$410$	0	0
$411$	−19.7710	−0.975231
$412$	10.6708	0.525711
$413$	6.52216	0.320934
$414$	−1.50856	−0.0741419
$415$	0	0
$416$	6.60115	0.323648
$417$	−3.51671	−0.172214
$418$	−20.3823	−0.996931
$419$	−4.82584	−0.235757	−0.117879	−	0.993028i	$-0.537609\pi$
$419$	−4.82584	−0.235757	−0.117879	+	0.993028i	$0.537609\pi$
$420$	0	0
$421$	0.992428	0.0483680	0.0241840	−	0.999708i	$-0.492301\pi$
$421$	0.992428	0.0483680	0.0241840	+	0.999708i	$0.492301\pi$
$422$	−19.0340	−0.926563
$423$	15.5875	0.757892
$424$	1.24260	0.0603458
$425$	0	0
$426$	11.7854	0.571004
$427$	5.44476	0.263490
$428$	−2.47479	−0.119624
$429$	−40.3732	−1.94924
$430$	0	0
$431$	−4.23083	−0.203792	−0.101896	−	0.994795i	$-0.532491\pi$
$431$	−4.23083	−0.203792	−0.101896	+	0.994795i	$0.532491\pi$
$432$	−4.82340	−0.232066
$433$	−4.54830	−0.218577	−0.109289	−	0.994010i	$-0.534857\pi$
$433$	−4.54830	−0.218577	−0.109289	+	0.994010i	$0.534857\pi$
$434$	10.3233	0.495534
$435$	0	0
$436$	−2.40704	−0.115276
$437$	2.25127	0.107693
$438$	10.0048	0.478047
$439$	20.1943	0.963823	0.481911	−	0.876220i	$-0.339943\pi$
$439$	20.1943	0.963823	0.481911	+	0.876220i	$0.339943\pi$
$440$	0	0
$441$	−2.10851	−0.100405
$442$	−6.60115	−0.313985
$443$	7.95689	0.378043	0.189022	−	0.981973i	$-0.439468\pi$
$443$	7.95689	0.378043	0.189022	+	0.981973i	$0.439468\pi$
$444$	7.81356	0.370815
$445$	0	0
$446$	−2.99408	−0.141774
$447$	2.77281	0.131150
$448$	−1.00000	−0.0472456
$449$	−5.34314	−0.252159	−0.126079	−	0.992020i	$-0.540239\pi$
$449$	−5.34314	−0.252159	−0.126079	+	0.992020i	$0.540239\pi$
$450$	0	0
$451$	45.8690	2.15989
$452$	−5.04159	−0.237136
$453$	−8.00553	−0.376133
$454$	−7.17860	−0.336908
$455$	0	0
$456$	2.97097	0.139128
$457$	−8.78868	−0.411117	−0.205559	−	0.978645i	$-0.565901\pi$
$457$	−8.78868	−0.411117	−0.205559	+	0.978645i	$0.565901\pi$
$458$	20.4722	0.956605
$459$	4.82340	0.225137
$460$	0	0
$461$	21.7515	1.01307	0.506534	−	0.862220i	$-0.330927\pi$
$461$	21.7515	1.01307	0.506534	+	0.862220i	$0.330927\pi$
$462$	6.11608	0.284546
$463$	18.4736	0.858541	0.429270	−	0.903176i	$-0.358771\pi$
$463$	18.4736	0.858541	0.429270	+	0.903176i	$0.358771\pi$
$464$	−5.52962	−0.256706
$465$	0	0
$466$	−14.5803	−0.675418
$467$	0.163658	0.00757320	0.00378660	−	0.999993i	$-0.498795\pi$
$467$	0.163658	0.00757320	0.00378660	+	0.999993i	$0.498795\pi$
$468$	−13.9186	−0.643387
$469$	8.75002	0.404038
$470$	0	0
$471$	4.33240	0.199627
$472$	−6.52216	−0.300207
$473$	1.76286	0.0810565
$474$	−14.9862	−0.688340
$475$	0	0
$476$	1.00000	0.0458349
$477$	−2.62002	−0.119963
$478$	−27.8402	−1.27338
$479$	25.1540	1.14931	0.574657	−	0.818394i	$-0.305135\pi$
$479$	25.1540	1.14931	0.574657	+	0.818394i	$0.305135\pi$
$480$	0	0
$481$	54.6273	2.49079
$482$	2.46386	0.112226
$483$	−0.675535	−0.0307379
$484$	30.9593	1.40724
$485$	0	0
$486$	16.1427	0.732246
$487$	−28.8860	−1.30895	−0.654474	−	0.756084i	$-0.727110\pi$
$487$	−28.8860	−1.30895	−0.654474	+	0.756084i	$0.727110\pi$
$488$	−5.44476	−0.246473
$489$	−12.2381	−0.553428
$490$	0	0
$491$	−4.17428	−0.188383	−0.0941913	−	0.995554i	$-0.530027\pi$
$491$	−4.17428	−0.188383	−0.0941913	+	0.995554i	$0.530027\pi$
$492$	−6.68597	−0.301427
$493$	5.52962	0.249042
$494$	20.7711	0.934535
$495$	0	0
$496$	−10.3233	−0.463530
$497$	−12.4820	−0.559896
$498$	−10.4246	−0.467139
$499$	20.3782	0.912253	0.456126	−	0.889915i	$-0.349237\pi$
$499$	20.3782	0.912253	0.456126	+	0.889915i	$0.349237\pi$
$500$	0	0
$501$	8.77866	0.392202
$502$	7.79037	0.347701
$503$	13.0966	0.583947	0.291973	−	0.956426i	$-0.405688\pi$
$503$	13.0966	0.583947	0.291973	+	0.956426i	$0.405688\pi$
$504$	2.10851	0.0939204
$505$	0	0
$506$	−4.63450	−0.206029
$507$	28.8688	1.28211
$508$	10.7877	0.478626
$509$	−22.4005	−0.992883	−0.496442	−	0.868070i	$-0.665360\pi$
$509$	−22.4005	−0.992883	−0.496442	+	0.868070i	$0.665360\pi$
$510$	0	0
$511$	−10.5962	−0.468747
$512$	1.00000	0.0441942
$513$	−15.1772	−0.670090
$514$	−30.2883	−1.33596
$515$	0	0
$516$	−0.256959	−0.0113120
$517$	47.8869	2.10606
$518$	−8.27542	−0.363601
$519$	−8.90771	−0.391005
$520$	0	0
$521$	12.8800	0.564284	0.282142	−	0.959373i	$-0.408955\pi$
$521$	12.8800	0.564284	0.282142	+	0.959373i	$0.408955\pi$
$522$	11.6592	0.510311
$523$	−5.02369	−0.219670	−0.109835	−	0.993950i	$-0.535032\pi$
$523$	−5.02369	−0.219670	−0.109835	+	0.993950i	$0.535032\pi$
$524$	−1.16568	−0.0509229
$525$	0	0
$526$	13.8039	0.601877
$527$	10.3233	0.449690
$528$	−6.11608	−0.266168
$529$	−22.4881	−0.977744
$530$	0	0
$531$	13.7520	0.596786
$532$	−3.14658	−0.136422
$533$	−46.7439	−2.02470
$534$	−6.05584	−0.262062
$535$	0	0
$536$	−8.75002	−0.377943
$537$	−2.68205	−0.115739
$538$	−12.4186	−0.535406
$539$	−6.47760	−0.279010
$540$	0	0
$541$	−21.0348	−0.904355	−0.452178	−	0.891928i	$-0.649353\pi$
$541$	−21.0348	−0.904355	−0.452178	+	0.891928i	$0.649353\pi$
$542$	12.1674	0.522633
$543$	−2.84966	−0.122291
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−6.23274	−0.266737
$547$	10.3391	0.442068	0.221034	−	0.975266i	$-0.429057\pi$
$547$	10.3391	0.442068	0.221034	+	0.975266i	$0.429057\pi$
$548$	−20.9397	−0.894498
$549$	11.4803	0.489968
$550$	0	0
$551$	−17.3994	−0.741239
$552$	0.675535	0.0287527
$553$	15.8721	0.674949
$554$	26.5516	1.12807
$555$	0	0
$556$	−3.72458	−0.157957
$557$	−39.0942	−1.65647	−0.828237	−	0.560377i	$-0.810656\pi$
$557$	−39.0942	−1.65647	−0.828237	+	0.560377i	$0.810656\pi$
$558$	21.7668	0.921460
$559$	−1.79649	−0.0759833
$560$	0	0
$561$	6.11608	0.258221
$562$	−9.09912	−0.383823
$563$	24.8465	1.04716	0.523579	−	0.851977i	$-0.324597\pi$
$563$	24.8465	1.04716	0.523579	+	0.851977i	$0.324597\pi$
$564$	−6.98010	−0.293915
$565$	0	0
$566$	−6.80753	−0.286142
$567$	−1.77132	−0.0743885
$568$	12.4820	0.523734
$569$	−7.84639	−0.328938	−0.164469	−	0.986382i	$-0.552591\pi$
$569$	−7.84639	−0.328938	−0.164469	+	0.986382i	$0.552591\pi$
$570$	0	0
$571$	33.9814	1.42208	0.711040	−	0.703152i	$-0.248225\pi$
$571$	33.9814	1.42208	0.711040	+	0.703152i	$0.248225\pi$
$572$	−42.7596	−1.78787
$573$	7.41522	0.309775
$574$	7.08117	0.295563
$575$	0	0
$576$	−2.10851	−0.0878545
$577$	10.2218	0.425538	0.212769	−	0.977103i	$-0.431752\pi$
$577$	10.2218	0.425538	0.212769	+	0.977103i	$0.431752\pi$
$578$	1.00000	0.0415945
$579$	−6.26452	−0.260345
$580$	0	0
$581$	11.0408	0.458051
$582$	−7.13644	−0.295815
$583$	−8.04904	−0.333357
$584$	10.5962	0.438473
$585$	0	0
$586$	27.1352	1.12094
$587$	1.62318	0.0669960	0.0334980	−	0.999439i	$-0.489335\pi$
$587$	1.62318	0.0669960	0.0334980	+	0.999439i	$0.489335\pi$
$588$	0.944189	0.0389377
$589$	−32.4831	−1.33844
$590$	0	0
$591$	10.8994	0.448343
$592$	8.27542	0.340117
$593$	−38.4450	−1.57875	−0.789373	−	0.613914i	$-0.789594\pi$
$593$	−38.4450	−1.57875	−0.789373	+	0.613914i	$0.789594\pi$
$594$	31.2440	1.28196
$595$	0	0
$596$	2.93671	0.120293
$597$	−24.5631	−1.00530
$598$	4.72290	0.193134
$599$	−35.5424	−1.45222	−0.726112	−	0.687576i	$-0.758675\pi$
$599$	−35.5424	−1.45222	−0.726112	+	0.687576i	$0.758675\pi$
$600$	0	0
$601$	−2.27757	−0.0929039	−0.0464520	−	0.998921i	$-0.514791\pi$
$601$	−2.27757	−0.0929039	−0.0464520	+	0.998921i	$0.514791\pi$
$602$	0.272148	0.0110919
$603$	18.4495	0.751321
$604$	−8.47874	−0.344995
$605$	0	0
$606$	0.571466	0.0232142
$607$	−26.2815	−1.06673	−0.533367	−	0.845884i	$-0.679074\pi$
$607$	−26.2815	−1.06673	−0.533367	+	0.845884i	$0.679074\pi$
$608$	3.14658	0.127611
$609$	5.22101	0.211566
$610$	0	0
$611$	−48.8003	−1.97425
$612$	2.10851	0.0852313
$613$	5.46121	0.220576	0.110288	−	0.993900i	$-0.464823\pi$
$613$	5.46121	0.220576	0.110288	+	0.993900i	$0.464823\pi$
$614$	24.9786	1.00805
$615$	0	0
$616$	6.47760	0.260990
$617$	−37.4904	−1.50930	−0.754652	−	0.656125i	$-0.772194\pi$
$617$	−37.4904	−1.50930	−0.754652	+	0.656125i	$0.772194\pi$
$618$	10.0752	0.405285
$619$	−25.9784	−1.04416	−0.522080	−	0.852896i	$-0.674844\pi$
$619$	−25.9784	−1.04416	−0.522080	+	0.852896i	$0.674844\pi$
$620$	0	0
$621$	−3.45097	−0.138483
$622$	20.4702	0.820779
$623$	6.41380	0.256964
$624$	6.23274	0.249509
$625$	0	0
$626$	−7.84620	−0.313597
$627$	−19.2447	−0.768561
$628$	4.58849	0.183101
$629$	−8.27542	−0.329962
$630$	0	0
$631$	−19.8832	−0.791539	−0.395769	−	0.918350i	$-0.629522\pi$
$631$	−19.8832	−0.791539	−0.395769	+	0.918350i	$0.629522\pi$
$632$	−15.8721	−0.631357
$633$	−17.9717	−0.714312
$634$	31.3528	1.24518
$635$	0	0
$636$	1.17325	0.0465222
$637$	6.60115	0.261547
$638$	35.8187	1.41807
$639$	−26.3184	−1.04114
$640$	0	0
$641$	0.590754	0.0233334	0.0116667	−	0.999932i	$-0.496286\pi$
$641$	0.590754	0.0233334	0.0116667	+	0.999932i	$0.496286\pi$
$642$	−2.33667	−0.0922212
$643$	28.1565	1.11038	0.555192	−	0.831722i	$-0.312645\pi$
$643$	28.1565	1.11038	0.555192	+	0.831722i	$0.312645\pi$
$644$	−0.715465	−0.0281933
$645$	0	0
$646$	−3.14658	−0.123801
$647$	−11.0568	−0.434688	−0.217344	−	0.976095i	$-0.569739\pi$
$647$	−11.0568	−0.434688	−0.217344	+	0.976095i	$0.569739\pi$
$648$	1.77132	0.0695841
$649$	42.2479	1.65838
$650$	0	0
$651$	9.74715	0.382021
$652$	−12.9615	−0.507613
$653$	27.8917	1.09149	0.545744	−	0.837952i	$-0.316247\pi$
$653$	27.8917	1.09149	0.545744	+	0.837952i	$0.316247\pi$
$654$	−2.27270	−0.0888696
$655$	0	0
$656$	−7.08117	−0.276473
$657$	−22.3421	−0.871648
$658$	7.39269	0.288197
$659$	−12.7077	−0.495022	−0.247511	−	0.968885i	$-0.579613\pi$
$659$	−12.7077	−0.495022	−0.247511	+	0.968885i	$0.579613\pi$
$660$	0	0
$661$	−17.5119	−0.681135	−0.340568	−	0.940220i	$-0.610619\pi$
$661$	−17.5119	−0.681135	−0.340568	+	0.940220i	$0.610619\pi$
$662$	8.54057	0.331939
$663$	−6.23274	−0.242059
$664$	−11.0408	−0.428467
$665$	0	0
$666$	−17.4488	−0.676126
$667$	−3.95625	−0.153187
$668$	9.29757	0.359734
$669$	−2.82698	−0.109297
$670$	0	0
$671$	35.2690	1.36154
$672$	−0.944189	−0.0364229
$673$	−40.2926	−1.55317	−0.776584	−	0.630014i	$-0.783049\pi$
$673$	−40.2926	−1.55317	−0.776584	+	0.630014i	$0.783049\pi$
$674$	8.28173	0.319000
$675$	0	0
$676$	30.5752	1.17597
$677$	−38.6926	−1.48708	−0.743539	−	0.668692i	$-0.766854\pi$
$677$	−38.6926	−1.48708	−0.743539	+	0.668692i	$0.766854\pi$
$678$	−4.76021	−0.182815
$679$	7.55828	0.290060
$680$	0	0
$681$	−6.77795	−0.259732
$682$	66.8702	2.56059
$683$	21.9118	0.838431	0.419215	−	0.907887i	$-0.362305\pi$
$683$	21.9118	0.838431	0.419215	+	0.907887i	$0.362305\pi$
$684$	−6.63459	−0.253680
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	19.3297	0.737472
$688$	−0.272148	−0.0103755
$689$	8.20256	0.312493
$690$	0	0
$691$	27.5037	1.04629	0.523144	−	0.852244i	$-0.324759\pi$
$691$	27.5037	1.04629	0.523144	+	0.852244i	$0.324759\pi$
$692$	−9.43425	−0.358636
$693$	−13.6581	−0.518827
$694$	−11.7797	−0.447150
$695$	0	0
$696$	−5.22101	−0.197902
$697$	7.08117	0.268219
$698$	28.7276	1.08736
$699$	−13.7665	−0.520698
$700$	0	0
$701$	−19.8851	−0.751050	−0.375525	−	0.926812i	$-0.622537\pi$
$701$	−19.8851	−0.751050	−0.375525	+	0.926812i	$0.622537\pi$
$702$	−31.8400	−1.20172
$703$	26.0393	0.982090
$704$	−6.47760	−0.244134
$705$	0	0
$706$	−9.10674	−0.342737
$707$	−0.605246	−0.0227626
$708$	−6.15815	−0.231437
$709$	23.0108	0.864188	0.432094	−	0.901828i	$-0.357775\pi$
$709$	23.0108	0.864188	0.432094	+	0.901828i	$0.357775\pi$
$710$	0	0
$711$	33.4664	1.25509
$712$	−6.41380	−0.240367
$713$	−7.38596	−0.276607
$714$	0.944189	0.0353354
$715$	0	0
$716$	−2.84058	−0.106157
$717$	−26.2864	−0.981683
$718$	10.5395	0.393330
$719$	30.8148	1.14920	0.574599	−	0.818435i	$-0.305158\pi$
$719$	30.8148	1.14920	0.574599	+	0.818435i	$0.305158\pi$
$720$	0	0
$721$	−10.6708	−0.397400
$722$	−9.09903	−0.338631
$723$	2.32635	0.0865177
$724$	−3.01811	−0.112167
$725$	0	0
$726$	29.2314	1.08488
$727$	0.215845	0.00800526	0.00400263	−	0.999992i	$-0.498726\pi$
$727$	0.215845	0.00800526	0.00400263	+	0.999992i	$0.498726\pi$
$728$	−6.60115	−0.244655
$729$	9.92775	0.367694
$730$	0	0
$731$	0.272148	0.0100657
$732$	−5.14088	−0.190013
$733$	4.16143	0.153706	0.0768530	−	0.997042i	$-0.475513\pi$
$733$	4.16143	0.153706	0.0768530	+	0.997042i	$0.475513\pi$
$734$	−6.19187	−0.228546
$735$	0	0
$736$	0.715465	0.0263724
$737$	56.6791	2.08780
$738$	14.9307	0.549607
$739$	22.2623	0.818931	0.409465	−	0.912326i	$-0.365715\pi$
$739$	22.2623	0.818931	0.409465	+	0.912326i	$0.365715\pi$
$740$	0	0
$741$	19.6118	0.720458
$742$	−1.24260	−0.0456171
$743$	35.4588	1.30086	0.650429	−	0.759567i	$-0.274589\pi$
$743$	35.4588	1.30086	0.650429	+	0.759567i	$0.274589\pi$
$744$	−9.74715	−0.357348
$745$	0	0
$746$	6.40599	0.234540
$747$	23.2797	0.851759
$748$	6.47760	0.236845
$749$	2.47479	0.0904270
$750$	0	0
$751$	−21.8355	−0.796790	−0.398395	−	0.917214i	$-0.630433\pi$
$751$	−21.8355	−0.796790	−0.398395	+	0.917214i	$0.630433\pi$
$752$	−7.39269	−0.269584
$753$	7.35559	0.268052
$754$	−36.5019	−1.32932
$755$	0	0
$756$	4.82340	0.175425
$757$	−32.4582	−1.17971	−0.589857	−	0.807508i	$-0.700816\pi$
$757$	−32.4582	−1.17971	−0.589857	+	0.807508i	$0.700816\pi$
$758$	−4.47057	−0.162378
$759$	−4.37584	−0.158833
$760$	0	0
$761$	36.5248	1.32402	0.662012	−	0.749493i	$-0.269703\pi$
$761$	36.5248	1.32402	0.662012	+	0.749493i	$0.269703\pi$
$762$	10.1856	0.368986
$763$	2.40704	0.0871407
$764$	7.85353	0.284131
$765$	0	0
$766$	−24.9589	−0.901801
$767$	−43.0538	−1.55458
$768$	0.944189	0.0340705
$769$	−22.8778	−0.824994	−0.412497	−	0.910959i	$-0.635343\pi$
$769$	−22.8778	−0.824994	−0.412497	+	0.910959i	$0.635343\pi$
$770$	0	0
$771$	−28.5979	−1.02993
$772$	−6.63482	−0.238792
$773$	34.1082	1.22679	0.613394	−	0.789777i	$-0.289804\pi$
$773$	34.1082	1.22679	0.613394	+	0.789777i	$0.289804\pi$
$774$	0.573825	0.0206257
$775$	0	0
$776$	−7.55828	−0.271326
$777$	−7.81356	−0.280310
$778$	12.9696	0.464983
$779$	−22.2815	−0.798317
$780$	0	0
$781$	−80.8536	−2.89317
$782$	−0.715465	−0.0255850
$783$	26.6715	0.953163
$784$	1.00000	0.0357143
$785$	0	0
$786$	−1.10062	−0.0392579
$787$	14.8271	0.528530	0.264265	−	0.964450i	$-0.414871\pi$
$787$	14.8271	0.528530	0.264265	+	0.964450i	$0.414871\pi$
$788$	11.5437	0.411227
$789$	13.0335	0.464003
$790$	0	0
$791$	5.04159	0.179258
$792$	13.6581	0.485318
$793$	−35.9417	−1.27633
$794$	29.5224	1.04771
$795$	0	0
$796$	−26.0150	−0.922078
$797$	47.0924	1.66810	0.834049	−	0.551691i	$-0.186017\pi$
$797$	47.0924	1.66810	0.834049	+	0.551691i	$0.186017\pi$
$798$	−2.97097	−0.105171
$799$	7.39269	0.261535
$800$	0	0
$801$	13.5235	0.477831
$802$	33.3973	1.17930
$803$	−68.6378	−2.42217
$804$	−8.26167	−0.291367
$805$	0	0
$806$	−68.1457	−2.40033
$807$	−11.7255	−0.412759
$808$	0.605246	0.0212925
$809$	−18.0195	−0.633530	−0.316765	−	0.948504i	$-0.602597\pi$
$809$	−18.0195	−0.633530	−0.316765	+	0.948504i	$0.602597\pi$
$810$	0	0
$811$	−37.6389	−1.32168	−0.660841	−	0.750526i	$-0.729800\pi$
$811$	−37.6389	−1.32168	−0.660841	+	0.750526i	$0.729800\pi$
$812$	5.52962	0.194052
$813$	11.4883	0.402912
$814$	−53.6048	−1.87885
$815$	0	0
$816$	−0.944189	−0.0330532
$817$	−0.856334	−0.0299593
$818$	7.96757	0.278580
$819$	13.9186	0.486355
$820$	0	0
$821$	−7.43512	−0.259487	−0.129744	−	0.991548i	$-0.541415\pi$
$821$	−7.43512	−0.259487	−0.129744	+	0.991548i	$0.541415\pi$
$822$	−19.7710	−0.689593
$823$	6.52273	0.227368	0.113684	−	0.993517i	$-0.463735\pi$
$823$	6.52273	0.227368	0.113684	+	0.993517i	$0.463735\pi$
$824$	10.6708	0.371734
$825$	0	0
$826$	6.52216	0.226935
$827$	27.0754	0.941504	0.470752	−	0.882266i	$-0.343983\pi$
$827$	27.0754	0.941504	0.470752	+	0.882266i	$0.343983\pi$
$828$	−1.50856	−0.0524262
$829$	−34.5150	−1.19876	−0.599379	−	0.800466i	$-0.704586\pi$
$829$	−34.5150	−1.19876	−0.599379	+	0.800466i	$0.704586\pi$
$830$	0	0
$831$	25.0697	0.869658
$832$	6.60115	0.228854
$833$	−1.00000	−0.0346479
$834$	−3.51671	−0.121774
$835$	0	0
$836$	−20.3823	−0.704936
$837$	49.7934	1.72111
$838$	−4.82584	−0.166706
$839$	37.8184	1.30563	0.652817	−	0.757515i	$-0.273587\pi$
$839$	37.8184	1.30563	0.652817	+	0.757515i	$0.273587\pi$
$840$	0	0
$841$	1.57669	0.0543685
$842$	0.992428	0.0342013
$843$	−8.59129	−0.295900
$844$	−19.0340	−0.655179
$845$	0	0
$846$	15.5875	0.535910
$847$	−30.9593	−1.06377
$848$	1.24260	0.0426709
$849$	−6.42759	−0.220594
$850$	0	0
$851$	5.92077	0.202961
$852$	11.7854	0.403761
$853$	−2.31438	−0.0792427	−0.0396214	−	0.999215i	$-0.512615\pi$
$853$	−2.31438	−0.0792427	−0.0396214	+	0.999215i	$0.512615\pi$
$854$	5.44476	0.186316
$855$	0	0
$856$	−2.47479	−0.0845867
$857$	−17.5385	−0.599105	−0.299553	−	0.954080i	$-0.596837\pi$
$857$	−17.5385	−0.599105	−0.299553	+	0.954080i	$0.596837\pi$
$858$	−40.3732	−1.37832
$859$	−17.2802	−0.589593	−0.294796	−	0.955560i	$-0.595252\pi$
$859$	−17.2802	−0.589593	−0.294796	+	0.955560i	$0.595252\pi$
$860$	0	0
$861$	6.68597	0.227857
$862$	−4.23083	−0.144103
$863$	15.4600	0.526264	0.263132	−	0.964760i	$-0.415245\pi$
$863$	15.4600	0.526264	0.263132	+	0.964760i	$0.415245\pi$
$864$	−4.82340	−0.164095
$865$	0	0
$866$	−4.54830	−0.154557
$867$	0.944189	0.0320663
$868$	10.3233	0.350396
$869$	102.813	3.48769
$870$	0	0
$871$	−57.7602	−1.95713
$872$	−2.40704	−0.0815127
$873$	15.9367	0.539375
$874$	2.25127	0.0761503
$875$	0	0
$876$	10.0048	0.338031
$877$	−48.6209	−1.64181	−0.820906	−	0.571063i	$-0.806531\pi$
$877$	−48.6209	−1.64181	−0.820906	+	0.571063i	$0.806531\pi$
$878$	20.1943	0.681526
$879$	25.6208	0.864167
$880$	0	0
$881$	−18.8100	−0.633724	−0.316862	−	0.948472i	$-0.602629\pi$
$881$	−18.8100	−0.633724	−0.316862	+	0.948472i	$0.602629\pi$
$882$	−2.10851	−0.0709971
$883$	−2.98271	−0.100376	−0.0501881	−	0.998740i	$-0.515982\pi$
$883$	−2.98271	−0.100376	−0.0501881	+	0.998740i	$0.515982\pi$
$884$	−6.60115	−0.222021
$885$	0	0
$886$	7.95689	0.267317
$887$	8.81549	0.295995	0.147998	−	0.988988i	$-0.452717\pi$
$887$	8.81549	0.295995	0.147998	+	0.988988i	$0.452717\pi$
$888$	7.81356	0.262206
$889$	−10.7877	−0.361807
$890$	0	0
$891$	−11.4739	−0.384391
$892$	−2.99408	−0.100249
$893$	−23.2617	−0.778423
$894$	2.77281	0.0927368
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	4.45931	0.148892
$898$	−5.34314	−0.178303
$899$	57.0839	1.90386
$900$	0	0
$901$	−1.24260	−0.0413969
$902$	45.8690	1.52727
$903$	0.256959	0.00855105
$904$	−5.04159	−0.167681
$905$	0	0
$906$	−8.00553	−0.265966
$907$	−23.3580	−0.775589	−0.387794	−	0.921746i	$-0.626763\pi$
$907$	−23.3580	−0.775589	−0.387794	+	0.921746i	$0.626763\pi$
$908$	−7.17860	−0.238230
$909$	−1.27616	−0.0423277
$910$	0	0
$911$	40.5921	1.34488	0.672438	−	0.740153i	$-0.265247\pi$
$911$	40.5921	1.34488	0.672438	+	0.740153i	$0.265247\pi$
$912$	2.97097	0.0983786
$913$	71.5181	2.36690
$914$	−8.78868	−0.290704
$915$	0	0
$916$	20.4722	0.676422
$917$	1.16568	0.0384941
$918$	4.82340	0.159196
$919$	−18.4602	−0.608947	−0.304473	−	0.952521i	$-0.598480\pi$
$919$	−18.4602	−0.608947	−0.304473	+	0.952521i	$0.598480\pi$
$920$	0	0
$921$	23.5845	0.777137
$922$	21.7515	0.716347
$923$	82.3958	2.71209
$924$	6.11608	0.201204
$925$	0	0
$926$	18.4736	0.607080
$927$	−22.4994	−0.738977
$928$	−5.52962	−0.181519
$929$	34.7395	1.13977	0.569884	−	0.821725i	$-0.306988\pi$
$929$	34.7395	1.13977	0.569884	+	0.821725i	$0.306988\pi$
$930$	0	0
$931$	3.14658	0.103125
$932$	−14.5803	−0.477592
$933$	19.3277	0.632761
$934$	0.163658	0.00535506
$935$	0	0
$936$	−13.9186	−0.454943
$937$	−28.8524	−0.942566	−0.471283	−	0.881982i	$-0.656209\pi$
$937$	−28.8524	−0.942566	−0.471283	+	0.881982i	$0.656209\pi$
$938$	8.75002	0.285698
$939$	−7.40830	−0.241761
$940$	0	0
$941$	33.3643	1.08764	0.543822	−	0.839200i	$-0.316977\pi$
$941$	33.3643	1.08764	0.543822	+	0.839200i	$0.316977\pi$
$942$	4.33240	0.141157
$943$	−5.06633	−0.164983
$944$	−6.52216	−0.212278
$945$	0	0
$946$	1.76286	0.0573156
$947$	59.0329	1.91831	0.959156	−	0.282876i	$-0.0912886\pi$
$947$	59.0329	1.91831	0.959156	+	0.282876i	$0.0912886\pi$
$948$	−14.9862	−0.486730
$949$	69.9470	2.27057
$950$	0	0
$951$	29.6030	0.959944
$952$	1.00000	0.0324102
$953$	−13.5500	−0.438928	−0.219464	−	0.975621i	$-0.570431\pi$
$953$	−13.5500	−0.438928	−0.219464	+	0.975621i	$0.570431\pi$
$954$	−2.62002	−0.0848263
$955$	0	0
$956$	−27.8402	−0.900415
$957$	33.8196	1.09323
$958$	25.1540	0.812688
$959$	20.9397	0.676177
$960$	0	0
$961$	75.5706	2.43776
$962$	54.6273	1.76125
$963$	5.21812	0.168152
$964$	2.46386	0.0793555
$965$	0	0
$966$	−0.675535	−0.0217350
$967$	54.7241	1.75981	0.879904	−	0.475151i	$-0.157607\pi$
$967$	54.7241	1.75981	0.879904	+	0.475151i	$0.157607\pi$
$968$	30.9593	0.995070
$969$	−2.97097	−0.0954412
$970$	0	0
$971$	−11.2224	−0.360145	−0.180072	−	0.983653i	$-0.557633\pi$
$971$	−11.2224	−0.360145	−0.180072	+	0.983653i	$0.557633\pi$
$972$	16.1427	0.517776
$973$	3.72458	0.119405
$974$	−28.8860	−0.925567
$975$	0	0
$976$	−5.44476	−0.174283
$977$	44.3382	1.41850	0.709252	−	0.704955i	$-0.249033\pi$
$977$	44.3382	1.41850	0.709252	+	0.704955i	$0.249033\pi$
$978$	−12.2381	−0.391333
$979$	41.5460	1.32782
$980$	0	0
$981$	5.07526	0.162041
$982$	−4.17428	−0.133207
$983$	5.47897	0.174752	0.0873759	−	0.996175i	$-0.472152\pi$
$983$	5.47897	0.174752	0.0873759	+	0.996175i	$0.472152\pi$
$984$	−6.68597	−0.213141
$985$	0	0
$986$	5.52962	0.176099
$987$	6.98010	0.222179
$988$	20.7711	0.660816
$989$	−0.194712	−0.00619149
$990$	0	0
$991$	−3.48863	−0.110820	−0.0554100	−	0.998464i	$-0.517647\pi$
$991$	−3.48863	−0.110820	−0.0554100	+	0.998464i	$0.517647\pi$
$992$	−10.3233	−0.327765
$993$	8.06391	0.255901
$994$	−12.4820	−0.395906
$995$	0	0
$996$	−10.4246	−0.330317
$997$	−35.4114	−1.12149	−0.560746	−	0.827988i	$-0.689485\pi$
$997$	−35.4114	−1.12149	−0.560746	+	0.827988i	$0.689485\pi$
$998$	20.3782	0.645060
$999$	−39.9156	−1.26287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5950.2.a.cc.1.5		7
5.2	odd	4		1190.2.e.g.239.10	yes	14
5.3	odd	4		1190.2.e.g.239.5	✓	14
5.4	even	2		5950.2.a.cb.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.e.g.239.5	✓	14	5.3	odd	4
1190.2.e.g.239.10	yes	14	5.2	odd	4
5950.2.a.cb.1.3		7	5.4	even	2
5950.2.a.cc.1.5		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:	\( N \)	\(=\)	\( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5950.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.944189 q^{3} +1.00000 q^{4} +0.944189 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.10851 q^{9} -6.47760 q^{11} +0.944189 q^{12} +6.60115 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.10851 q^{18} +3.14658 q^{19} -0.944189 q^{21} -6.47760 q^{22} +0.715465 q^{23} +0.944189 q^{24} +6.60115 q^{26} -4.82340 q^{27} -1.00000 q^{28} -5.52962 q^{29} -10.3233 q^{31} +1.00000 q^{32} -6.11608 q^{33} -1.00000 q^{34} -2.10851 q^{36} +8.27542 q^{37} +3.14658 q^{38} +6.23274 q^{39} -7.08117 q^{41} -0.944189 q^{42} -0.272148 q^{43} -6.47760 q^{44} +0.715465 q^{46} -7.39269 q^{47} +0.944189 q^{48} +1.00000 q^{49} -0.944189 q^{51} +6.60115 q^{52} +1.24260 q^{53} -4.82340 q^{54} -1.00000 q^{56} +2.97097 q^{57} -5.52962 q^{58} -6.52216 q^{59} -5.44476 q^{61} -10.3233 q^{62} +2.10851 q^{63} +1.00000 q^{64} -6.11608 q^{66} -8.75002 q^{67} -1.00000 q^{68} +0.675535 q^{69} +12.4820 q^{71} -2.10851 q^{72} +10.5962 q^{73} +8.27542 q^{74} +3.14658 q^{76} +6.47760 q^{77} +6.23274 q^{78} -15.8721 q^{79} +1.77132 q^{81} -7.08117 q^{82} -11.0408 q^{83} -0.944189 q^{84} -0.272148 q^{86} -5.22101 q^{87} -6.47760 q^{88} -6.41380 q^{89} -6.60115 q^{91} +0.715465 q^{92} -9.74715 q^{93} -7.39269 q^{94} +0.944189 q^{96} -7.55828 q^{97} +1.00000 q^{98} +13.6581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} - 2 q^{3} + 7 q^{4} - 2 q^{6} - 7 q^{7} + 7 q^{8} + 7 q^{9} - 7 q^{11} - 2 q^{12} + q^{13} - 7 q^{14} + 7 q^{16} - 7 q^{17} + 7 q^{18} - 2 q^{19} + 2 q^{21} - 7 q^{22} - 27 q^{23} - 2 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 - 2 * q^3 + 7 * q^4 - 2 * q^6 - 7 * q^7 + 7 * q^8 + 7 * q^9 - 7 * q^11 - 2 * q^12 + q^13 - 7 * q^14 + 7 * q^16 - 7 * q^17 + 7 * q^18 - 2 * q^19 + 2 * q^21 - 7 * q^22 - 27 * q^23 - 2 * q^24 + q^26 - 17 * q^27 - 7 * q^28 - 10 * q^29 - 18 * q^31 + 7 * q^32 - 8 * q^33 - 7 * q^34 + 7 * q^36 - 12 * q^37 - 2 * q^38 - 25 * q^39 + 17 * q^41 + 2 * q^42 - 7 * q^43 - 7 * q^44 - 27 * q^46 - 8 * q^47 - 2 * q^48 + 7 * q^49 + 2 * q^51 + q^52 - 20 * q^53 - 17 * q^54 - 7 * q^56 - 25 * q^57 - 10 * q^58 + q^61 - 18 * q^62 - 7 * q^63 + 7 * q^64 - 8 * q^66 - 7 * q^67 - 7 * q^68 + 3 * q^69 - 11 * q^71 + 7 * q^72 + 7 * q^73 - 12 * q^74 - 2 * q^76 + 7 * q^77 - 25 * q^78 - 26 * q^79 + 11 * q^81 + 17 * q^82 - 22 * q^83 + 2 * q^84 - 7 * q^86 - 20 * q^87 - 7 * q^88 + 21 * q^89 - q^91 - 27 * q^92 - 25 * q^93 - 8 * q^94 - 2 * q^96 + 19 * q^97 + 7 * q^98 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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