LMFDB - Embedded newform 4650.2.a.cp.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(1,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4650.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,3,3,0,3,6,3,3,0,8,3,10] 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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1708.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x - 2 $ x^3 - x^2 - 8*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.21018$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.21018 q^{11} +1.00000 q^{12} +0.789816 q^{13} +2.00000 q^{14} +1.00000 q^{16} -0.115086 q^{17} +1.00000 q^{18} +0.115086 q^{19} +2.00000 q^{21} +5.21018 q^{22} -4.42037 q^{23} +1.00000 q^{24} +0.789816 q^{26} +1.00000 q^{27} +2.00000 q^{28} +4.42037 q^{29} -1.00000 q^{31} +1.00000 q^{32} +5.21018 q^{33} -0.115086 q^{34} +1.00000 q^{36} +6.61056 q^{37} +0.115086 q^{38} +0.789816 q^{39} +2.00000 q^{41} +2.00000 q^{42} -8.61056 q^{43} +5.21018 q^{44} -4.42037 q^{46} +0.115086 q^{47} +1.00000 q^{48} -3.00000 q^{49} -0.115086 q^{51} +0.789816 q^{52} -0.190196 q^{53} +1.00000 q^{54} +2.00000 q^{56} +0.115086 q^{57} +4.42037 q^{58} -4.19020 q^{59} +12.4955 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +5.21018 q^{66} +5.82075 q^{67} -0.115086 q^{68} -4.42037 q^{69} -13.8207 q^{71} +1.00000 q^{72} -10.8407 q^{73} +6.61056 q^{74} +0.115086 q^{76} +10.4204 q^{77} +0.789816 q^{78} +2.30528 q^{79} +1.00000 q^{81} +2.00000 q^{82} +15.1460 q^{83} +2.00000 q^{84} -8.61056 q^{86} +4.42037 q^{87} +5.21018 q^{88} -2.23017 q^{89} +1.57963 q^{91} -4.42037 q^{92} -1.00000 q^{93} +0.115086 q^{94} +1.00000 q^{96} -9.21018 q^{97} -3.00000 q^{98} +5.21018 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 8 q^{11} + 3 q^{12} + 10 q^{13} + 6 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{18} - 2 q^{19} + 6 q^{21} + 8 q^{22} + 2 q^{23} + 3 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 8 * q^11 + 3 * q^12 + 10 * q^13 + 6 * q^14 + 3 * q^16 + 2 * q^17 + 3 * q^18 - 2 * q^19 + 6 * q^21 + 8 * q^22 + 2 * q^23 + 3 * q^24 + 10 * q^26 + 3 * q^27 + 6 * q^28 - 2 * q^29 - 3 * q^31 + 3 * q^32 + 8 * q^33 + 2 * q^34 + 3 * q^36 - 6 * q^37 - 2 * q^38 + 10 * q^39 + 6 * q^41 + 6 * q^42 + 8 * q^44 + 2 * q^46 - 2 * q^47 + 3 * q^48 - 9 * q^49 + 2 * q^51 + 10 * q^52 + 10 * q^53 + 3 * q^54 + 6 * q^56 - 2 * q^57 - 2 * q^58 - 2 * q^59 + 14 * q^61 - 3 * q^62 + 6 * q^63 + 3 * q^64 + 8 * q^66 - 16 * q^67 + 2 * q^68 + 2 * q^69 - 8 * q^71 + 3 * q^72 - 2 * q^73 - 6 * q^74 - 2 * q^76 + 16 * q^77 + 10 * q^78 - 6 * q^79 + 3 * q^81 + 6 * q^82 + 2 * q^83 + 6 * q^84 - 2 * q^87 + 8 * q^88 - 2 * q^89 + 20 * q^91 + 2 * q^92 - 3 * q^93 - 2 * q^94 + 3 * q^96 - 20 * q^97 - 9 * q^98 + 8 * 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
$3$	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
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	5.21018	1.57093	0.785465	−	0.618906i	$-0.212424\pi$
$11$	5.21018	1.57093	0.785465	+	0.618906i	$0.212424\pi$
$12$	1.00000	0.288675
$13$	0.789816	0.219056	0.109528	−	0.993984i	$-0.465066\pi$
$13$	0.789816	0.219056	0.109528	+	0.993984i	$0.465066\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.115086	−0.0279125	−0.0139562	−	0.999903i	$-0.504443\pi$
$17$	−0.115086	−0.0279125	−0.0139562	+	0.999903i	$0.504443\pi$
$18$	1.00000	0.235702
$19$	0.115086	0.0264026	0.0132013	−	0.999913i	$-0.495798\pi$
$19$	0.115086	0.0264026	0.0132013	+	0.999913i	$0.495798\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	5.21018	1.11081
$23$	−4.42037	−0.921710	−0.460855	−	0.887475i	$-0.652457\pi$
$23$	−4.42037	−0.921710	−0.460855	+	0.887475i	$0.652457\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	0.789816	0.154896
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	4.42037	0.820842	0.410421	−	0.911896i	$-0.365382\pi$
$29$	4.42037	0.820842	0.410421	+	0.911896i	$0.365382\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	5.21018	0.906977
$34$	−0.115086	−0.0197371
$35$	0	0
$36$	1.00000	0.166667
$37$	6.61056	1.08677	0.543385	−	0.839484i	$-0.317142\pi$
$37$	6.61056	1.08677	0.543385	+	0.839484i	$0.317142\pi$
$38$	0.115086	0.0186694
$39$	0.789816	0.126472
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	2.00000	0.308607
$43$	−8.61056	−1.31310	−0.656549	−	0.754283i	$-0.727985\pi$
$43$	−8.61056	−1.31310	−0.656549	+	0.754283i	$0.727985\pi$
$44$	5.21018	0.785465
$45$	0	0
$46$	−4.42037	−0.651748
$47$	0.115086	0.0167870	0.00839352	−	0.999965i	$-0.497328\pi$
$47$	0.115086	0.0167870	0.00839352	+	0.999965i	$0.497328\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−0.115086	−0.0161153
$52$	0.789816	0.109528
$53$	−0.190196	−0.0261254	−0.0130627	−	0.999915i	$-0.504158\pi$
$53$	−0.190196	−0.0261254	−0.0130627	+	0.999915i	$0.504158\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	0.115086	0.0152435
$58$	4.42037	0.580423
$59$	−4.19020	−0.545517	−0.272759	−	0.962083i	$-0.587936\pi$
$59$	−4.19020	−0.545517	−0.272759	+	0.962083i	$0.587936\pi$
$60$	0	0
$61$	12.4955	1.59988	0.799941	−	0.600079i	$-0.204864\pi$
$61$	12.4955	1.59988	0.799941	+	0.600079i	$0.204864\pi$
$62$	−1.00000	−0.127000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	5.21018	0.641329
$67$	5.82075	0.711118	0.355559	−	0.934654i	$-0.384291\pi$
$67$	5.82075	0.711118	0.355559	+	0.934654i	$0.384291\pi$
$68$	−0.115086	−0.0139562
$69$	−4.42037	−0.532150
$70$	0	0
$71$	−13.8207	−1.64022	−0.820111	−	0.572205i	$-0.806088\pi$
$71$	−13.8207	−1.64022	−0.820111	+	0.572205i	$0.806088\pi$
$72$	1.00000	0.117851
$73$	−10.8407	−1.26881	−0.634406	−	0.773000i	$-0.718755\pi$
$73$	−10.8407	−1.26881	−0.634406	+	0.773000i	$0.718755\pi$
$74$	6.61056	0.768462
$75$	0	0
$76$	0.115086	0.0132013
$77$	10.4204	1.18751
$78$	0.789816	0.0894290
$79$	2.30528	0.259365	0.129682	−	0.991556i	$-0.458604\pi$
$79$	2.30528	0.259365	0.129682	+	0.991556i	$0.458604\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	15.1460	1.66249	0.831246	−	0.555905i	$-0.187628\pi$
$83$	15.1460	1.66249	0.831246	+	0.555905i	$0.187628\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−8.61056	−0.928501
$87$	4.42037	0.473913
$88$	5.21018	0.555407
$89$	−2.23017	−0.236398	−0.118199	−	0.992990i	$-0.537712\pi$
$89$	−2.23017	−0.236398	−0.118199	+	0.992990i	$0.537712\pi$
$90$	0	0
$91$	1.57963	0.165590
$92$	−4.42037	−0.460855
$93$	−1.00000	−0.103695
$94$	0.115086	0.0118702
$95$	0	0
$96$	1.00000	0.102062
$97$	−9.21018	−0.935153	−0.467576	−	0.883953i	$-0.654873\pi$
$97$	−9.21018	−0.935153	−0.467576	+	0.883953i	$0.654873\pi$
$98$	−3.00000	−0.303046
$99$	5.21018	0.523643
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−0.115086	−0.0113952
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0.789816	0.0774478
$105$	0	0
$106$	−0.190196	−0.0184735
$107$	−3.03093	−0.293011	−0.146506	−	0.989210i	$-0.546803\pi$
$107$	−3.03093	−0.293011	−0.146506	+	0.989210i	$0.546803\pi$
$108$	1.00000	0.0962250
$109$	−8.19020	−0.784479	−0.392239	−	0.919863i	$-0.628299\pi$
$109$	−8.19020	−0.784479	−0.392239	+	0.919863i	$0.628299\pi$
$110$	0	0
$111$	6.61056	0.627447
$112$	2.00000	0.188982
$113$	10.3804	0.976505	0.488253	−	0.872702i	$-0.337634\pi$
$113$	10.3804	0.976505	0.488253	+	0.872702i	$0.337634\pi$
$114$	0.115086	0.0107788
$115$	0	0
$116$	4.42037	0.410421
$117$	0.789816	0.0730185
$118$	−4.19020	−0.385739
$119$	−0.230172	−0.0210999
$120$	0	0
$121$	16.1460	1.46782
$122$	12.4955	1.13129
$123$	2.00000	0.180334
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	2.00000	0.178174
$127$	−10.4204	−0.924658	−0.462329	−	0.886708i	$-0.652986\pi$
$127$	−10.4204	−0.924658	−0.462329	+	0.886708i	$0.652986\pi$
$128$	1.00000	0.0883883
$129$	−8.61056	−0.758118
$130$	0	0
$131$	−0.190196	−0.0166175	−0.00830875	−	0.999965i	$-0.502645\pi$
$131$	−0.190196	−0.0166175	−0.00830875	+	0.999965i	$0.502645\pi$
$132$	5.21018	0.453488
$133$	0.230172	0.0199585
$134$	5.82075	0.502836
$135$	0	0
$136$	−0.115086	−0.00986855
$137$	−21.4513	−1.83271	−0.916354	−	0.400369i	$-0.868882\pi$
$137$	−21.4513	−1.83271	−0.916354	+	0.400369i	$0.868882\pi$
$138$	−4.42037	−0.376287
$139$	16.4204	1.39276	0.696379	−	0.717674i	$-0.254793\pi$
$139$	16.4204	1.39276	0.696379	+	0.717674i	$0.254793\pi$
$140$	0	0
$141$	0.115086	0.00969200
$142$	−13.8207	−1.15981
$143$	4.11509	0.344121
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.8407	−0.897186
$147$	−3.00000	−0.247436
$148$	6.61056	0.543385
$149$	−3.63055	−0.297426	−0.148713	−	0.988880i	$-0.547513\pi$
$149$	−3.63055	−0.297426	−0.148713	+	0.988880i	$0.547513\pi$
$150$	0	0
$151$	−15.7566	−1.28225	−0.641126	−	0.767435i	$-0.721533\pi$
$151$	−15.7566	−1.28225	−0.641126	+	0.767435i	$0.721533\pi$
$152$	0.115086	0.00933472
$153$	−0.115086	−0.00930416
$154$	10.4204	0.839697
$155$	0	0
$156$	0.789816	0.0632359
$157$	−3.96002	−0.316044	−0.158022	−	0.987436i	$-0.550512\pi$
$157$	−3.96002	−0.316044	−0.158022	+	0.987436i	$0.550512\pi$
$158$	2.30528	0.183398
$159$	−0.190196	−0.0150835
$160$	0	0
$161$	−8.84074	−0.696748
$162$	1.00000	0.0785674
$163$	9.82075	0.769220	0.384610	−	0.923079i	$-0.374336\pi$
$163$	9.82075	0.769220	0.384610	+	0.923079i	$0.374336\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	15.1460	1.17556
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	2.00000	0.154303
$169$	−12.3762	−0.952015
$170$	0	0
$171$	0.115086	0.00880085
$172$	−8.61056	−0.656549
$173$	−0.535454	−0.0407098	−0.0203549	−	0.999793i	$-0.506480\pi$
$173$	−0.535454	−0.0407098	−0.0203549	+	0.999793i	$0.506480\pi$
$174$	4.42037	0.335107
$175$	0	0
$176$	5.21018	0.392732
$177$	−4.19020	−0.314954
$178$	−2.23017	−0.167158
$179$	10.4313	0.779673	0.389836	−	0.920884i	$-0.372532\pi$
$179$	10.4313	0.779673	0.389836	+	0.920884i	$0.372532\pi$
$180$	0	0
$181$	17.4513	1.29714	0.648572	−	0.761153i	$-0.275366\pi$
$181$	17.4513	1.29714	0.648572	+	0.761153i	$0.275366\pi$
$182$	1.57963	0.117090
$183$	12.4955	0.923692
$184$	−4.42037	−0.325874
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	−0.599620	−0.0438485
$188$	0.115086	0.00839352
$189$	2.00000	0.145479
$190$	0	0
$191$	19.2611	1.39368	0.696842	−	0.717224i	$-0.254588\pi$
$191$	19.2611	1.39368	0.696842	+	0.717224i	$0.254588\pi$
$192$	1.00000	0.0721688
$193$	14.4313	1.03879	0.519394	−	0.854535i	$-0.326158\pi$
$193$	14.4313	1.03879	0.519394	+	0.854535i	$0.326158\pi$
$194$	−9.21018	−0.661253
$195$	0	0
$196$	−3.00000	−0.214286
$197$	10.3804	0.739572	0.369786	−	0.929117i	$-0.379431\pi$
$197$	10.3804	0.739572	0.369786	+	0.929117i	$0.379431\pi$
$198$	5.21018	0.370272
$199$	14.5355	1.03039	0.515196	−	0.857073i	$-0.327719\pi$
$199$	14.5355	1.03039	0.515196	+	0.857073i	$0.327719\pi$
$200$	0	0
$201$	5.82075	0.410564
$202$	0	0
$203$	8.84074	0.620498
$204$	−0.115086	−0.00805764
$205$	0	0
$206$	−6.00000	−0.418040
$207$	−4.42037	−0.307237
$208$	0.789816	0.0547639
$209$	0.599620	0.0414766
$210$	0	0
$211$	24.8407	1.71011	0.855053	−	0.518540i	$-0.173524\pi$
$211$	24.8407	1.71011	0.855053	+	0.518540i	$0.173524\pi$
$212$	−0.190196	−0.0130627
$213$	−13.8207	−0.946982
$214$	−3.03093	−0.207190
$215$	0	0
$216$	1.00000	0.0680414
$217$	−2.00000	−0.135769
$218$	−8.19020	−0.554710
$219$	−10.8407	−0.732549
$220$	0	0
$221$	−0.0908968	−0.00611438
$222$	6.61056	0.443672
$223$	−19.6306	−1.31456	−0.657280	−	0.753647i	$-0.728293\pi$
$223$	−19.6306	−1.31456	−0.657280	+	0.753647i	$0.728293\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	10.3804	0.690493
$227$	−6.42037	−0.426135	−0.213067	−	0.977038i	$-0.568345\pi$
$227$	−6.42037	−0.426135	−0.213067	+	0.977038i	$0.568345\pi$
$228$	0.115086	0.00762176
$229$	−29.7166	−1.96373	−0.981864	−	0.189585i	$-0.939286\pi$
$229$	−29.7166	−1.96373	−0.981864	+	0.189585i	$0.939286\pi$
$230$	0	0
$231$	10.4204	0.685610
$232$	4.42037	0.290211
$233$	5.38944	0.353074	0.176537	−	0.984294i	$-0.443511\pi$
$233$	5.38944	0.353074	0.176537	+	0.984294i	$0.443511\pi$
$234$	0.789816	0.0516319
$235$	0	0
$236$	−4.19020	−0.272759
$237$	2.30528	0.149744
$238$	−0.230172	−0.0149198
$239$	7.03093	0.454793	0.227397	−	0.973802i	$-0.426979\pi$
$239$	7.03093	0.454793	0.227397	+	0.973802i	$0.426979\pi$
$240$	0	0
$241$	−0.190196	−0.0122516	−0.00612580	−	0.999981i	$-0.501950\pi$
$241$	−0.190196	−0.0122516	−0.00612580	+	0.999981i	$0.501950\pi$
$242$	16.1460	1.03791
$243$	1.00000	0.0641500
$244$	12.4955	0.799941
$245$	0	0
$246$	2.00000	0.127515
$247$	0.0908968	0.00578363
$248$	−1.00000	−0.0635001
$249$	15.1460	0.959840
$250$	0	0
$251$	14.4204	0.910206	0.455103	−	0.890439i	$-0.349603\pi$
$251$	14.4204	0.910206	0.455103	+	0.890439i	$0.349603\pi$
$252$	2.00000	0.125988
$253$	−23.0309	−1.44794
$254$	−10.4204	−0.653832
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.76983	−0.110399	−0.0551994	−	0.998475i	$-0.517579\pi$
$257$	−1.76983	−0.110399	−0.0551994	+	0.998475i	$0.517579\pi$
$258$	−8.61056	−0.536070
$259$	13.2211	0.821521
$260$	0	0
$261$	4.42037	0.273614
$262$	−0.190196	−0.0117504
$263$	14.2302	0.877470	0.438735	−	0.898616i	$-0.355427\pi$
$263$	14.2302	0.877470	0.438735	+	0.898616i	$0.355427\pi$
$264$	5.21018	0.320665
$265$	0	0
$266$	0.230172	0.0141128
$267$	−2.23017	−0.136484
$268$	5.82075	0.355559
$269$	−1.15926	−0.0706815	−0.0353408	−	0.999375i	$-0.511252\pi$
$269$	−1.15926	−0.0706815	−0.0353408	+	0.999375i	$0.511252\pi$
$270$	0	0
$271$	22.6857	1.37806	0.689028	−	0.724734i	$-0.258038\pi$
$271$	22.6857	1.37806	0.689028	+	0.724734i	$0.258038\pi$
$272$	−0.115086	−0.00697812
$273$	1.57963	0.0956037
$274$	−21.4513	−1.29592
$275$	0	0
$276$	−4.42037	−0.266075
$277$	−9.63055	−0.578644	−0.289322	−	0.957232i	$-0.593430\pi$
$277$	−9.63055	−0.578644	−0.289322	+	0.957232i	$0.593430\pi$
$278$	16.4204	0.984828
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	10.8407	0.646704	0.323352	−	0.946279i	$-0.395190\pi$
$281$	10.8407	0.646704	0.323352	+	0.946279i	$0.395190\pi$
$282$	0.115086	0.00685328
$283$	−19.7808	−1.17584	−0.587922	−	0.808917i	$-0.700054\pi$
$283$	−19.7808	−1.17584	−0.587922	+	0.808917i	$0.700054\pi$
$284$	−13.8207	−0.820111
$285$	0	0
$286$	4.11509	0.243330
$287$	4.00000	0.236113
$288$	1.00000	0.0589256
$289$	−16.9868	−0.999221
$290$	0	0
$291$	−9.21018	−0.539911
$292$	−10.8407	−0.634406
$293$	31.2211	1.82396	0.911979	−	0.410237i	$-0.134554\pi$
$293$	31.2211	1.82396	0.911979	+	0.410237i	$0.134554\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	6.61056	0.384231
$297$	5.21018	0.302326
$298$	−3.63055	−0.210312
$299$	−3.49128	−0.201906
$300$	0	0
$301$	−17.2211	−0.992609
$302$	−15.7566	−0.906689
$303$	0	0
$304$	0.115086	0.00660064
$305$	0	0
$306$	−0.115086	−0.00657904
$307$	−13.5796	−0.775031	−0.387515	−	0.921863i	$-0.626667\pi$
$307$	−13.5796	−0.775031	−0.387515	+	0.921863i	$0.626667\pi$
$308$	10.4204	0.593756
$309$	−6.00000	−0.341328
$310$	0	0
$311$	18.2811	1.03663	0.518313	−	0.855191i	$-0.326560\pi$
$311$	18.2811	1.03663	0.518313	+	0.855191i	$0.326560\pi$
$312$	0.789816	0.0447145
$313$	−14.4603	−0.817347	−0.408673	−	0.912681i	$-0.634009\pi$
$313$	−14.4603	−0.817347	−0.408673	+	0.912681i	$0.634009\pi$
$314$	−3.96002	−0.223477
$315$	0	0
$316$	2.30528	0.129682
$317$	−3.46455	−0.194588	−0.0972941	−	0.995256i	$-0.531019\pi$
$317$	−3.46455	−0.194588	−0.0972941	+	0.995256i	$0.531019\pi$
$318$	−0.190196	−0.0106657
$319$	23.0309	1.28948
$320$	0	0
$321$	−3.03093	−0.169170
$322$	−8.84074	−0.492675
$323$	−0.0132448	−0.000736961 0
$324$	1.00000	0.0555556
$325$	0	0
$326$	9.82075	0.543921
$327$	−8.19020	−0.452919
$328$	2.00000	0.110432
$329$	0.230172	0.0126898
$330$	0	0
$331$	−20.8008	−1.14331	−0.571657	−	0.820493i	$-0.693699\pi$
$331$	−20.8008	−1.14331	−0.571657	+	0.820493i	$0.693699\pi$
$332$	15.1460	0.831246
$333$	6.61056	0.362257
$334$	2.00000	0.109435
$335$	0	0
$336$	2.00000	0.109109
$337$	−23.0709	−1.25675	−0.628376	−	0.777910i	$-0.716280\pi$
$337$	−23.0709	−1.25675	−0.628376	+	0.777910i	$0.716280\pi$
$338$	−12.3762	−0.673176
$339$	10.3804	0.563786
$340$	0	0
$341$	−5.21018	−0.282147
$342$	0.115086	0.00622314
$343$	−20.0000	−1.07990
$344$	−8.61056	−0.464251
$345$	0	0
$346$	−0.535454	−0.0287862
$347$	−31.9868	−1.71714	−0.858569	−	0.512697i	$-0.828646\pi$
$347$	−31.9868	−1.71714	−0.858569	+	0.512697i	$0.828646\pi$
$348$	4.42037	0.236957
$349$	23.4513	1.25532	0.627660	−	0.778488i	$-0.284013\pi$
$349$	23.4513	1.25532	0.627660	+	0.778488i	$0.284013\pi$
$350$	0	0
$351$	0.789816	0.0421573
$352$	5.21018	0.277704
$353$	3.88491	0.206773	0.103387	−	0.994641i	$-0.467032\pi$
$353$	3.88491	0.206773	0.103387	+	0.994641i	$0.467032\pi$
$354$	−4.19020	−0.222706
$355$	0	0
$356$	−2.23017	−0.118199
$357$	−0.230172	−0.0121820
$358$	10.4313	0.551312
$359$	−21.3604	−1.12736	−0.563680	−	0.825994i	$-0.690615\pi$
$359$	−21.3604	−1.12736	−0.563680	+	0.825994i	$0.690615\pi$
$360$	0	0
$361$	−18.9868	−0.999303
$362$	17.4513	0.917220
$363$	16.1460	0.847446
$364$	1.57963	0.0827952
$365$	0	0
$366$	12.4955	0.653149
$367$	−20.4713	−1.06859	−0.534296	−	0.845297i	$-0.679423\pi$
$367$	−20.4713	−1.06859	−0.534296	+	0.845297i	$0.679423\pi$
$368$	−4.42037	−0.230428
$369$	2.00000	0.104116
$370$	0	0
$371$	−0.380392	−0.0197490
$372$	−1.00000	−0.0518476
$373$	−9.76983	−0.505863	−0.252931	−	0.967484i	$-0.581395\pi$
$373$	−9.76983	−0.505863	−0.252931	+	0.967484i	$0.581395\pi$
$374$	−0.599620	−0.0310056
$375$	0	0
$376$	0.115086	0.00593511
$377$	3.49128	0.179810
$378$	2.00000	0.102869
$379$	3.65474	0.187731	0.0938657	−	0.995585i	$-0.470078\pi$
$379$	3.65474	0.187731	0.0938657	+	0.995585i	$0.470078\pi$
$380$	0	0
$381$	−10.4204	−0.533852
$382$	19.2611	0.985484
$383$	−25.4913	−1.30254	−0.651272	−	0.758845i	$-0.725764\pi$
$383$	−25.4913	−1.30254	−0.651272	+	0.758845i	$0.725764\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.4313	0.734534
$387$	−8.61056	−0.437700
$388$	−9.21018	−0.467576
$389$	4.65054	0.235792	0.117896	−	0.993026i	$-0.462385\pi$
$389$	4.65054	0.235792	0.117896	+	0.993026i	$0.462385\pi$
$390$	0	0
$391$	0.508723	0.0257272
$392$	−3.00000	−0.151523
$393$	−0.190196	−0.00959412
$394$	10.3804	0.522957
$395$	0	0
$396$	5.21018	0.261822
$397$	26.7124	1.34066	0.670329	−	0.742064i	$-0.266153\pi$
$397$	26.7124	1.34066	0.670329	+	0.742064i	$0.266153\pi$
$398$	14.5355	0.728596
$399$	0.230172	0.0115230
$400$	0	0
$401$	16.0509	0.801545	0.400772	−	0.916178i	$-0.368742\pi$
$401$	16.0509	0.801545	0.400772	+	0.916178i	$0.368742\pi$
$402$	5.82075	0.290313
$403$	−0.789816	−0.0393435
$404$	0	0
$405$	0	0
$406$	8.84074	0.438758
$407$	34.4423	1.70724
$408$	−0.115086	−0.00569761
$409$	9.87167	0.488123	0.244061	−	0.969760i	$-0.421520\pi$
$409$	9.87167	0.488123	0.244061	+	0.969760i	$0.421520\pi$
$410$	0	0
$411$	−21.4513	−1.05811
$412$	−6.00000	−0.295599
$413$	−8.38039	−0.412372
$414$	−4.42037	−0.217249
$415$	0	0
$416$	0.789816	0.0387239
$417$	16.4204	0.804109
$418$	0.599620	0.0293284
$419$	−0.420368	−0.0205363	−0.0102682	−	0.999947i	$-0.503269\pi$
$419$	−0.420368	−0.0205363	−0.0102682	+	0.999947i	$0.503269\pi$
$420$	0	0
$421$	−39.2211	−1.91152	−0.955760	−	0.294146i	$-0.904965\pi$
$421$	−39.2211	−1.91152	−0.955760	+	0.294146i	$0.904965\pi$
$422$	24.8407	1.20923
$423$	0.115086	0.00559568
$424$	−0.190196	−0.00923674
$425$	0	0
$426$	−13.8207	−0.669617
$427$	24.9910	1.20940
$428$	−3.03093	−0.146506
$429$	4.11509	0.198678
$430$	0	0
$431$	−6.88071	−0.331432	−0.165716	−	0.986173i	$-0.552994\pi$
$431$	−6.88071	−0.331432	−0.165716	+	0.986173i	$0.552994\pi$
$432$	1.00000	0.0481125
$433$	−1.38944	−0.0667720	−0.0333860	−	0.999443i	$-0.510629\pi$
$433$	−1.38944	−0.0667720	−0.0333860	+	0.999443i	$0.510629\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−8.19020	−0.392239
$437$	−0.508723	−0.0243355
$438$	−10.8407	−0.517990
$439$	19.8717	0.948423	0.474212	−	0.880411i	$-0.342733\pi$
$439$	19.8717	0.948423	0.474212	+	0.880411i	$0.342733\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−0.0908968	−0.00432352
$443$	8.61056	0.409100	0.204550	−	0.978856i	$-0.434427\pi$
$443$	8.61056	0.409100	0.204550	+	0.978856i	$0.434427\pi$
$444$	6.61056	0.313723
$445$	0	0
$446$	−19.6306	−0.929534
$447$	−3.63055	−0.171719
$448$	2.00000	0.0944911
$449$	−10.3694	−0.489364	−0.244682	−	0.969603i	$-0.578684\pi$
$449$	−10.3694	−0.489364	−0.244682	+	0.969603i	$0.578684\pi$
$450$	0	0
$451$	10.4204	0.490676
$452$	10.3804	0.488253
$453$	−15.7566	−0.740309
$454$	−6.42037	−0.301323
$455$	0	0
$456$	0.115086	0.00538940
$457$	−21.4913	−1.00532	−0.502660	−	0.864484i	$-0.667645\pi$
$457$	−21.4913	−1.00532	−0.502660	+	0.864484i	$0.667645\pi$
$458$	−29.7166	−1.38857
$459$	−0.115086	−0.00537176
$460$	0	0
$461$	3.34946	0.156000	0.0779999	−	0.996953i	$-0.475147\pi$
$461$	3.34946	0.156000	0.0779999	+	0.996953i	$0.475147\pi$
$462$	10.4204	0.484799
$463$	−28.0109	−1.30178	−0.650889	−	0.759172i	$-0.725604\pi$
$463$	−28.0109	−1.30178	−0.650889	+	0.759172i	$0.725604\pi$
$464$	4.42037	0.205210
$465$	0	0
$466$	5.38944	0.249661
$467$	−37.6815	−1.74369	−0.871845	−	0.489781i	$-0.837077\pi$
$467$	−37.6815	−1.74369	−0.871845	+	0.489781i	$0.837077\pi$
$468$	0.789816	0.0365093
$469$	11.6415	0.537554
$470$	0	0
$471$	−3.96002	−0.182468
$472$	−4.19020	−0.192869
$473$	−44.8626	−2.06279
$474$	2.30528	0.105885
$475$	0	0
$476$	−0.230172	−0.0105499
$477$	−0.190196	−0.00870848
$478$	7.03093	0.321587
$479$	−10.2811	−0.469755	−0.234878	−	0.972025i	$-0.575469\pi$
$479$	−10.2811	−0.469755	−0.234878	+	0.972025i	$0.575469\pi$
$480$	0	0
$481$	5.22113	0.238063
$482$	−0.190196	−0.00866319
$483$	−8.84074	−0.402267
$484$	16.1460	0.733910
$485$	0	0
$486$	1.00000	0.0453609
$487$	−32.9316	−1.49227	−0.746137	−	0.665792i	$-0.768094\pi$
$487$	−32.9316	−1.49227	−0.746137	+	0.665792i	$0.768094\pi$
$488$	12.4955	0.565644
$489$	9.82075	0.444110
$490$	0	0
$491$	−41.7034	−1.88205	−0.941023	−	0.338342i	$-0.890134\pi$
$491$	−41.7034	−1.88205	−0.941023	+	0.338342i	$0.890134\pi$
$492$	2.00000	0.0901670
$493$	−0.508723	−0.0229117
$494$	0.0908968	0.00408964
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−27.6415	−1.23989
$498$	15.1460	0.678709
$499$	22.2302	0.995159	0.497580	−	0.867418i	$-0.334222\pi$
$499$	22.2302	0.995159	0.497580	+	0.867418i	$0.334222\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	14.4204	0.643613
$503$	−12.3453	−0.550448	−0.275224	−	0.961380i	$-0.588752\pi$
$503$	−12.3453	−0.550448	−0.275224	+	0.961380i	$0.588752\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	−23.0309	−1.02385
$507$	−12.3762	−0.549646
$508$	−10.4204	−0.462329
$509$	30.9910	1.37365	0.686825	−	0.726823i	$-0.259004\pi$
$509$	30.9910	1.37365	0.686825	+	0.726823i	$0.259004\pi$
$510$	0	0
$511$	−21.6815	−0.959132
$512$	1.00000	0.0441942
$513$	0.115086	0.00508118
$514$	−1.76983	−0.0780638
$515$	0	0
$516$	−8.61056	−0.379059
$517$	0.599620	0.0263713
$518$	13.2211	0.580903
$519$	−0.535454	−0.0235038
$520$	0	0
$521$	28.8008	1.26178	0.630892	−	0.775871i	$-0.282689\pi$
$521$	28.8008	1.26178	0.630892	+	0.775871i	$0.282689\pi$
$522$	4.42037	0.193474
$523$	−27.7698	−1.21429	−0.607145	−	0.794591i	$-0.707685\pi$
$523$	−27.7698	−1.21429	−0.607145	+	0.794591i	$0.707685\pi$
$524$	−0.190196	−0.00830875
$525$	0	0
$526$	14.2302	0.620465
$527$	0.115086	0.00501323
$528$	5.21018	0.226744
$529$	−3.46034	−0.150450
$530$	0	0
$531$	−4.19020	−0.181839
$532$	0.230172	0.00997923
$533$	1.57963	0.0684214
$534$	−2.23017	−0.0965090
$535$	0	0
$536$	5.82075	0.251418
$537$	10.4313	0.450144
$538$	−1.15926	−0.0499794
$539$	−15.6306	−0.673256
$540$	0	0
$541$	34.9426	1.50230	0.751149	−	0.660132i	$-0.229500\pi$
$541$	34.9426	1.50230	0.751149	+	0.660132i	$0.229500\pi$
$542$	22.6857	0.974433
$543$	17.4513	0.748907
$544$	−0.115086	−0.00493428
$545$	0	0
$546$	1.57963	0.0676020
$547$	−23.6415	−1.01084	−0.505419	−	0.862874i	$-0.668662\pi$
$547$	−23.6415	−1.01084	−0.505419	+	0.862874i	$0.668662\pi$
$548$	−21.4513	−0.916354
$549$	12.4955	0.533294
$550$	0	0
$551$	0.508723	0.0216723
$552$	−4.42037	−0.188143
$553$	4.61056	0.196061
$554$	−9.63055	−0.409163
$555$	0	0
$556$	16.4204	0.696379
$557$	22.9910	0.974158	0.487079	−	0.873358i	$-0.338062\pi$
$557$	22.9910	0.974158	0.487079	+	0.873358i	$0.338062\pi$
$558$	−1.00000	−0.0423334
$559$	−6.80076	−0.287642
$560$	0	0
$561$	−0.599620	−0.0253160
$562$	10.8407	0.457289
$563$	7.87167	0.331751	0.165876	−	0.986147i	$-0.446955\pi$
$563$	7.87167	0.331751	0.165876	+	0.986147i	$0.446955\pi$
$564$	0.115086	0.00484600
$565$	0	0
$566$	−19.7808	−0.831448
$567$	2.00000	0.0839921
$568$	−13.8207	−0.579906
$569$	−40.6724	−1.70508	−0.852538	−	0.522664i	$-0.824938\pi$
$569$	−40.6724	−1.70508	−0.852538	+	0.522664i	$0.824938\pi$
$570$	0	0
$571$	14.2302	0.595514	0.297757	−	0.954642i	$-0.403761\pi$
$571$	14.2302	0.595514	0.297757	+	0.954642i	$0.403761\pi$
$572$	4.11509	0.172060
$573$	19.2611	0.804644
$574$	4.00000	0.166957
$575$	0	0
$576$	1.00000	0.0416667
$577$	7.63055	0.317664	0.158832	−	0.987306i	$-0.449227\pi$
$577$	7.63055	0.317664	0.158832	+	0.987306i	$0.449227\pi$
$578$	−16.9868	−0.706556
$579$	14.4313	0.599745
$580$	0	0
$581$	30.2920	1.25673
$582$	−9.21018	−0.381774
$583$	−0.990956	−0.0410412
$584$	−10.8407	−0.448593
$585$	0	0
$586$	31.2211	1.28973
$587$	−17.8449	−0.736539	−0.368270	−	0.929719i	$-0.620050\pi$
$587$	−17.8449	−0.736539	−0.368270	+	0.929719i	$0.620050\pi$
$588$	−3.00000	−0.123718
$589$	−0.115086	−0.00474204
$590$	0	0
$591$	10.3804	0.426992
$592$	6.61056	0.271693
$593$	−47.4732	−1.94949	−0.974745	−	0.223320i	$-0.928310\pi$
$593$	−47.4732	−1.94949	−0.974745	+	0.223320i	$0.928310\pi$
$594$	5.21018	0.213776
$595$	0	0
$596$	−3.63055	−0.148713
$597$	14.5355	0.594897
$598$	−3.49128	−0.142769
$599$	41.4622	1.69410	0.847051	−	0.531512i	$-0.178376\pi$
$599$	41.4622	1.69410	0.847051	+	0.531512i	$0.178376\pi$
$600$	0	0
$601$	−8.77887	−0.358098	−0.179049	−	0.983840i	$-0.557302\pi$
$601$	−8.77887	−0.358098	−0.179049	+	0.983840i	$0.557302\pi$
$602$	−17.2211	−0.701881
$603$	5.82075	0.237039
$604$	−15.7566	−0.641126
$605$	0	0
$606$	0	0
$607$	32.0619	1.30135	0.650675	−	0.759356i	$-0.274486\pi$
$607$	32.0619	1.30135	0.650675	+	0.759356i	$0.274486\pi$
$608$	0.115086	0.00466736
$609$	8.84074	0.358245
$610$	0	0
$611$	0.0908968	0.00367729
$612$	−0.115086	−0.00465208
$613$	−23.2321	−0.938335	−0.469167	−	0.883109i	$-0.655446\pi$
$613$	−23.2321	−0.938335	−0.469167	+	0.883109i	$0.655446\pi$
$614$	−13.5796	−0.548029
$615$	0	0
$616$	10.4204	0.419849
$617$	−13.6196	−0.548305	−0.274152	−	0.961686i	$-0.588397\pi$
$617$	−13.6196	−0.548305	−0.274152	+	0.961686i	$0.588397\pi$
$618$	−6.00000	−0.241355
$619$	25.3894	1.02049	0.510244	−	0.860030i	$-0.329555\pi$
$619$	25.3894	1.02049	0.510244	+	0.860030i	$0.329555\pi$
$620$	0	0
$621$	−4.42037	−0.177383
$622$	18.2811	0.733005
$623$	−4.46034	−0.178700
$624$	0.789816	0.0316179
$625$	0	0
$626$	−14.4603	−0.577952
$627$	0.599620	0.0239465
$628$	−3.96002	−0.158022
$629$	−0.760784	−0.0303345
$630$	0	0
$631$	−28.3804	−1.12981	−0.564903	−	0.825157i	$-0.691086\pi$
$631$	−28.3804	−1.12981	−0.564903	+	0.825157i	$0.691086\pi$
$632$	2.30528	0.0916992
$633$	24.8407	0.987331
$634$	−3.46455	−0.137595
$635$	0	0
$636$	−0.190196	−0.00754176
$637$	−2.36945	−0.0938809
$638$	23.0309	0.911803
$639$	−13.8207	−0.546740
$640$	0	0
$641$	3.59058	0.141819	0.0709096	−	0.997483i	$-0.477410\pi$
$641$	3.59058	0.141819	0.0709096	+	0.997483i	$0.477410\pi$
$642$	−3.03093	−0.119621
$643$	−38.6505	−1.52423	−0.762114	−	0.647443i	$-0.775839\pi$
$643$	−38.6505	−1.52423	−0.762114	+	0.647443i	$0.775839\pi$
$644$	−8.84074	−0.348374
$645$	0	0
$646$	−0.0132448	−0.000521110 0
$647$	32.9026	1.29353	0.646767	−	0.762687i	$-0.276120\pi$
$647$	32.9026	1.29353	0.646767	+	0.762687i	$0.276120\pi$
$648$	1.00000	0.0392837
$649$	−21.8317	−0.856969
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	9.82075	0.384610
$653$	25.5264	0.998926	0.499463	−	0.866335i	$-0.333531\pi$
$653$	25.5264	0.998926	0.499463	+	0.866335i	$0.333531\pi$
$654$	−8.19020	−0.320262
$655$	0	0
$656$	2.00000	0.0780869
$657$	−10.8407	−0.422937
$658$	0.230172	0.00897305
$659$	6.84074	0.266477	0.133239	−	0.991084i	$-0.457462\pi$
$659$	6.84074	0.266477	0.133239	+	0.991084i	$0.457462\pi$
$660$	0	0
$661$	−21.2611	−0.826961	−0.413481	−	0.910513i	$-0.635687\pi$
$661$	−21.2611	−0.826961	−0.413481	+	0.910513i	$0.635687\pi$
$662$	−20.8008	−0.808445
$663$	−0.0908968	−0.00353014
$664$	15.1460	0.587780
$665$	0	0
$666$	6.61056	0.256154
$667$	−19.5397	−0.756578
$668$	2.00000	0.0773823
$669$	−19.6306	−0.758961
$670$	0	0
$671$	65.1037	2.51330
$672$	2.00000	0.0771517
$673$	39.7034	1.53045	0.765226	−	0.643762i	$-0.222627\pi$
$673$	39.7034	1.53045	0.765226	+	0.643762i	$0.222627\pi$
$674$	−23.0709	−0.888658
$675$	0	0
$676$	−12.3762	−0.476007
$677$	−3.45130	−0.132644	−0.0663221	−	0.997798i	$-0.521126\pi$
$677$	−3.45130	−0.132644	−0.0663221	+	0.997798i	$0.521126\pi$
$678$	10.3804	0.398657
$679$	−18.4204	−0.706909
$680$	0	0
$681$	−6.42037	−0.246029
$682$	−5.21018	−0.199508
$683$	−20.4603	−0.782893	−0.391447	−	0.920201i	$-0.628025\pi$
$683$	−20.4603	−0.782893	−0.391447	+	0.920201i	$0.628025\pi$
$684$	0.115086	0.00440043
$685$	0	0
$686$	−20.0000	−0.763604
$687$	−29.7166	−1.13376
$688$	−8.61056	−0.328275
$689$	−0.150220	−0.00572292
$690$	0	0
$691$	−9.94678	−0.378393	−0.189197	−	0.981939i	$-0.560588\pi$
$691$	−9.94678	−0.378393	−0.189197	+	0.981939i	$0.560588\pi$
$692$	−0.535454	−0.0203549
$693$	10.4204	0.395837
$694$	−31.9868	−1.21420
$695$	0	0
$696$	4.42037	0.167554
$697$	−0.230172	−0.00871839
$698$	23.4513	0.887645
$699$	5.38944	0.203847
$700$	0	0
$701$	−14.7898	−0.558604	−0.279302	−	0.960203i	$-0.590103\pi$
$701$	−14.7898	−0.558604	−0.279302	+	0.960203i	$0.590103\pi$
$702$	0.789816	0.0298097
$703$	0.760784	0.0286935
$704$	5.21018	0.196366
$705$	0	0
$706$	3.88491	0.146211
$707$	0	0
$708$	−4.19020	−0.157477
$709$	−10.4336	−0.391843	−0.195921	−	0.980620i	$-0.562770\pi$
$709$	−10.4336	−0.391843	−0.195921	+	0.980620i	$0.562770\pi$
$710$	0	0
$711$	2.30528	0.0864548
$712$	−2.23017	−0.0835792
$713$	4.42037	0.165544
$714$	−0.230172	−0.00861398
$715$	0	0
$716$	10.4313	0.389836
$717$	7.03093	0.262575
$718$	−21.3604	−0.797163
$719$	−20.3320	−0.758256	−0.379128	−	0.925344i	$-0.623776\pi$
$719$	−20.3320	−0.758256	−0.379128	+	0.925344i	$0.623776\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	−18.9868	−0.706614
$723$	−0.190196	−0.00707347
$724$	17.4513	0.648572
$725$	0	0
$726$	16.1460	0.599235
$727$	−37.5132	−1.39129	−0.695643	−	0.718387i	$-0.744881\pi$
$727$	−37.5132	−1.39129	−0.695643	+	0.718387i	$0.744881\pi$
$728$	1.57963	0.0585450
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.990956	0.0366518
$732$	12.4955	0.461846
$733$	28.6505	1.05823	0.529116	−	0.848550i	$-0.322524\pi$
$733$	28.6505	1.05823	0.529116	+	0.848550i	$0.322524\pi$
$734$	−20.4713	−0.755609
$735$	0	0
$736$	−4.42037	−0.162937
$737$	30.3272	1.11712
$738$	2.00000	0.0736210
$739$	21.6415	0.796095	0.398048	−	0.917365i	$-0.369688\pi$
$739$	21.6415	0.796095	0.398048	+	0.917365i	$0.369688\pi$
$740$	0	0
$741$	0.0908968	0.00333918
$742$	−0.380392	−0.0139646
$743$	−5.76983	−0.211674	−0.105837	−	0.994383i	$-0.533752\pi$
$743$	−5.76983	−0.211674	−0.105837	+	0.994383i	$0.533752\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−9.76983	−0.357699
$747$	15.1460	0.554164
$748$	−0.599620	−0.0219243
$749$	−6.06187	−0.221496
$750$	0	0
$751$	−31.7698	−1.15930	−0.579649	−	0.814866i	$-0.696810\pi$
$751$	−31.7698	−1.15930	−0.579649	+	0.814866i	$0.696810\pi$
$752$	0.115086	0.00419676
$753$	14.4204	0.525507
$754$	3.49128	0.127145
$755$	0	0
$756$	2.00000	0.0727393
$757$	14.6905	0.533936	0.266968	−	0.963705i	$-0.413978\pi$
$757$	14.6905	0.533936	0.266968	+	0.963705i	$0.413978\pi$
$758$	3.65474	0.132746
$759$	−23.0309	−0.835970
$760$	0	0
$761$	44.1528	1.60054	0.800268	−	0.599642i	$-0.204690\pi$
$761$	44.1528	1.60054	0.800268	+	0.599642i	$0.204690\pi$
$762$	−10.4204	−0.377490
$763$	−16.3804	−0.593010
$764$	19.2611	0.696842
$765$	0	0
$766$	−25.4913	−0.921037
$767$	−3.30948	−0.119499
$768$	1.00000	0.0360844
$769$	2.07995	0.0750050	0.0375025	−	0.999297i	$-0.488060\pi$
$769$	2.07995	0.0750050	0.0375025	+	0.999297i	$0.488060\pi$
$770$	0	0
$771$	−1.76983	−0.0637388
$772$	14.4313	0.519394
$773$	30.1502	1.08443	0.542214	−	0.840240i	$-0.317586\pi$
$773$	30.1502	1.08443	0.542214	+	0.840240i	$0.317586\pi$
$774$	−8.61056	−0.309500
$775$	0	0
$776$	−9.21018	−0.330626
$777$	13.2211	0.474305
$778$	4.65054	0.166730
$779$	0.230172	0.00824678
$780$	0	0
$781$	−72.0086	−2.57667
$782$	0.508723	0.0181919
$783$	4.42037	0.157971
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−0.190196	−0.00678407
$787$	15.8717	0.565764	0.282882	−	0.959155i	$-0.408710\pi$
$787$	15.8717	0.565764	0.282882	+	0.959155i	$0.408710\pi$
$788$	10.3804	0.369786
$789$	14.2302	0.506608
$790$	0	0
$791$	20.7608	0.738169
$792$	5.21018	0.185136
$793$	9.86913	0.350463
$794$	26.7124	0.947988
$795$	0	0
$796$	14.5355	0.515196
$797$	−25.6415	−0.908268	−0.454134	−	0.890933i	$-0.650051\pi$
$797$	−25.6415	−0.908268	−0.454134	+	0.890933i	$0.650051\pi$
$798$	0.230172	0.00814801
$799$	−0.0132448	−0.000468568 0
$800$	0	0
$801$	−2.23017	−0.0787993
$802$	16.0509	0.566778
$803$	−56.4822	−1.99321
$804$	5.82075	0.205282
$805$	0	0
$806$	−0.789816	−0.0278201
$807$	−1.15926	−0.0408080
$808$	0	0
$809$	−37.7324	−1.32660	−0.663300	−	0.748353i	$-0.730845\pi$
$809$	−37.7324	−1.32660	−0.663300	+	0.748353i	$0.730845\pi$
$810$	0	0
$811$	9.22113	0.323798	0.161899	−	0.986807i	$-0.448238\pi$
$811$	9.22113	0.323798	0.161899	+	0.986807i	$0.448238\pi$
$812$	8.84074	0.310249
$813$	22.6857	0.795621
$814$	34.4423	1.20720
$815$	0	0
$816$	−0.115086	−0.00402882
$817$	−0.990956	−0.0346692
$818$	9.87167	0.345155
$819$	1.57963	0.0551968
$820$	0	0
$821$	−26.3804	−0.920682	−0.460341	−	0.887742i	$-0.652273\pi$
$821$	−26.3804	−0.920682	−0.460341	+	0.887742i	$0.652273\pi$
$822$	−21.4513	−0.748200
$823$	−24.0109	−0.836969	−0.418484	−	0.908224i	$-0.637439\pi$
$823$	−24.0109	−0.836969	−0.418484	+	0.908224i	$0.637439\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−8.38039	−0.291591
$827$	−7.52641	−0.261719	−0.130859	−	0.991401i	$-0.541774\pi$
$827$	−7.52641	−0.261719	−0.130859	+	0.991401i	$0.541774\pi$
$828$	−4.42037	−0.153618
$829$	7.30948	0.253869	0.126934	−	0.991911i	$-0.459486\pi$
$829$	7.30948	0.253869	0.126934	+	0.991911i	$0.459486\pi$
$830$	0	0
$831$	−9.63055	−0.334080
$832$	0.789816	0.0273819
$833$	0.345258	0.0119625
$834$	16.4204	0.568591
$835$	0	0
$836$	0.599620	0.0207383
$837$	−1.00000	−0.0345651
$838$	−0.420368	−0.0145214
$839$	−45.2430	−1.56196	−0.780981	−	0.624555i	$-0.785281\pi$
$839$	−45.2430	−1.56196	−0.780981	+	0.624555i	$0.785281\pi$
$840$	0	0
$841$	−9.46034	−0.326219
$842$	−39.2211	−1.35165
$843$	10.8407	0.373375
$844$	24.8407	0.855053
$845$	0	0
$846$	0.115086	0.00395674
$847$	32.2920	1.10957
$848$	−0.190196	−0.00653136
$849$	−19.7808	−0.678874
$850$	0	0
$851$	−29.2211	−1.00169
$852$	−13.8207	−0.473491
$853$	−9.11088	−0.311951	−0.155975	−	0.987761i	$-0.549852\pi$
$853$	−9.11088	−0.311951	−0.155975	+	0.987761i	$0.549852\pi$
$854$	24.9910	0.855173
$855$	0	0
$856$	−3.03093	−0.103595
$857$	−6.99096	−0.238807	−0.119403	−	0.992846i	$-0.538098\pi$
$857$	−6.99096	−0.238807	−0.119403	+	0.992846i	$0.538098\pi$
$858$	4.11509	0.140487
$859$	−17.6196	−0.601173	−0.300587	−	0.953755i	$-0.597182\pi$
$859$	−17.6196	−0.601173	−0.300587	+	0.953755i	$0.597182\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	−6.88071	−0.234358
$863$	45.8717	1.56149	0.780745	−	0.624850i	$-0.214840\pi$
$863$	45.8717	1.56149	0.780745	+	0.624850i	$0.214840\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−1.38944	−0.0472149
$867$	−16.9868	−0.576900
$868$	−2.00000	−0.0678844
$869$	12.0109	0.407443
$870$	0	0
$871$	4.59732	0.155774
$872$	−8.19020	−0.277355
$873$	−9.21018	−0.311718
$874$	−0.508723	−0.0172078
$875$	0	0
$876$	−10.8407	−0.366275
$877$	48.4204	1.63504	0.817520	−	0.575900i	$-0.195348\pi$
$877$	48.4204	1.63504	0.817520	+	0.575900i	$0.195348\pi$
$878$	19.8717	0.670636
$879$	31.2211	1.05306
$880$	0	0
$881$	−34.7717	−1.17149	−0.585745	−	0.810496i	$-0.699198\pi$
$881$	−34.7717	−1.17149	−0.585745	+	0.810496i	$0.699198\pi$
$882$	−3.00000	−0.101015
$883$	−35.7214	−1.20212	−0.601061	−	0.799203i	$-0.705255\pi$
$883$	−35.7214	−1.20212	−0.601061	+	0.799203i	$0.705255\pi$
$884$	−0.0908968	−0.00305719
$885$	0	0
$886$	8.61056	0.289278
$887$	24.7608	0.831386	0.415693	−	0.909505i	$-0.363539\pi$
$887$	24.7608	0.831386	0.415693	+	0.909505i	$0.363539\pi$
$888$	6.61056	0.221836
$889$	−20.8407	−0.698976
$890$	0	0
$891$	5.21018	0.174548
$892$	−19.6306	−0.657280
$893$	0.0132448	0.000443221 0
$894$	−3.63055	−0.121424
$895$	0	0
$896$	2.00000	0.0668153
$897$	−3.49128	−0.116570
$898$	−10.3694	−0.346033
$899$	−4.42037	−0.147428
$900$	0	0
$901$	0.0218889	0.000729226 0
$902$	10.4204	0.346960
$903$	−17.2211	−0.573083
$904$	10.3804	0.345247
$905$	0	0
$906$	−15.7566	−0.523477
$907$	20.5197	0.681344	0.340672	−	0.940182i	$-0.389345\pi$
$907$	20.5197	0.681344	0.340672	+	0.940182i	$0.389345\pi$
$908$	−6.42037	−0.213067
$909$	0	0
$910$	0	0
$911$	−27.1593	−0.899827	−0.449913	−	0.893072i	$-0.648545\pi$
$911$	−27.1593	−0.899827	−0.449913	+	0.893072i	$0.648545\pi$
$912$	0.115086	0.00381088
$913$	78.9135	2.61166
$914$	−21.4913	−0.710868
$915$	0	0
$916$	−29.7166	−0.981864
$917$	−0.380392	−0.0125617
$918$	−0.115086	−0.00379841
$919$	−37.5531	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$919$	−37.5531	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$920$	0	0
$921$	−13.5796	−0.447464
$922$	3.34946	0.110309
$923$	−10.9158	−0.359299
$924$	10.4204	0.342805
$925$	0	0
$926$	−28.0109	−0.920497
$927$	−6.00000	−0.197066
$928$	4.42037	0.145106
$929$	28.2121	0.925608	0.462804	−	0.886461i	$-0.346843\pi$
$929$	28.2121	0.925608	0.462804	+	0.886461i	$0.346843\pi$
$930$	0	0
$931$	−0.345258	−0.0113154
$932$	5.38944	0.176537
$933$	18.2811	0.598496
$934$	−37.6815	−1.23298
$935$	0	0
$936$	0.789816	0.0258159
$937$	−54.0728	−1.76648	−0.883241	−	0.468920i	$-0.844643\pi$
$937$	−54.0728	−1.76648	−0.883241	+	0.468920i	$0.844643\pi$
$938$	11.6415	0.380108
$939$	−14.4603	−0.471896
$940$	0	0
$941$	27.9600	0.911471	0.455735	−	0.890115i	$-0.349376\pi$
$941$	27.9600	0.911471	0.455735	+	0.890115i	$0.349376\pi$
$942$	−3.96002	−0.129025
$943$	−8.84074	−0.287894
$944$	−4.19020	−0.136379
$945$	0	0
$946$	−44.8626	−1.45861
$947$	−6.68567	−0.217255	−0.108628	−	0.994083i	$-0.534646\pi$
$947$	−6.68567	−0.217255	−0.108628	+	0.994083i	$0.534646\pi$
$948$	2.30528	0.0748721
$949$	−8.56219	−0.277940
$950$	0	0
$951$	−3.46455	−0.112346
$952$	−0.230172	−0.00745992
$953$	33.3362	1.07987	0.539933	−	0.841708i	$-0.318450\pi$
$953$	33.3362	1.07987	0.539933	+	0.841708i	$0.318450\pi$
$954$	−0.190196	−0.00615782
$955$	0	0
$956$	7.03093	0.227397
$957$	23.0309	0.744484
$958$	−10.2811	−0.332167
$959$	−42.9026	−1.38540
$960$	0	0
$961$	1.00000	0.0322581
$962$	5.22113	0.168336
$963$	−3.03093	−0.0976704
$964$	−0.190196	−0.00612580
$965$	0	0
$966$	−8.84074	−0.284446
$967$	40.1128	1.28994	0.644970	−	0.764208i	$-0.276870\pi$
$967$	40.1128	1.28994	0.644970	+	0.764208i	$0.276870\pi$
$968$	16.1460	0.518953
$969$	−0.0132448	−0.000425485 0
$970$	0	0
$971$	−53.1547	−1.70581	−0.852907	−	0.522063i	$-0.825163\pi$
$971$	−53.1547	−1.70581	−0.852907	+	0.522063i	$0.825163\pi$
$972$	1.00000	0.0320750
$973$	32.8407	1.05283
$974$	−32.9316	−1.05520
$975$	0	0
$976$	12.4955	0.399971
$977$	37.2830	1.19279	0.596394	−	0.802692i	$-0.296599\pi$
$977$	37.2830	1.19279	0.596394	+	0.802692i	$0.296599\pi$
$978$	9.82075	0.314033
$979$	−11.6196	−0.371364
$980$	0	0
$981$	−8.19020	−0.261493
$982$	−41.7034	−1.33081
$983$	45.6415	1.45574	0.727869	−	0.685716i	$-0.240511\pi$
$983$	45.6415	1.45574	0.727869	+	0.685716i	$0.240511\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−0.508723	−0.0162010
$987$	0.230172	0.00732646
$988$	0.0908968	0.00289181
$989$	38.0619	1.21030
$990$	0	0
$991$	58.9026	1.87110	0.935551	−	0.353191i	$-0.114903\pi$
$991$	58.9026	1.87110	0.935551	+	0.353191i	$0.114903\pi$
$992$	−1.00000	−0.0317500
$993$	−20.8008	−0.660092
$994$	−27.6415	−0.876735
$995$	0	0
$996$	15.1460	0.479920
$997$	4.11024	0.130173	0.0650864	−	0.997880i	$-0.479268\pi$
$997$	4.11024	0.130173	0.0650864	+	0.997880i	$0.479268\pi$
$998$	22.2302	0.703684
$999$	6.61056	0.209149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.cp.1.3		3
5.2	odd	4		930.2.d.i.559.6	yes	6
5.3	odd	4		930.2.d.i.559.3	✓	6
5.4	even	2		4650.2.a.ci.1.3		3
15.2	even	4		2790.2.d.j.559.1		6
15.8	even	4		2790.2.d.j.559.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.d.i.559.3	✓	6	5.3	odd	4
930.2.d.i.559.6	yes	6	5.2	odd	4
2790.2.d.j.559.1		6	15.2	even	4
2790.2.d.j.559.4		6	15.8	even	4
4650.2.a.ci.1.3		3	5.4	even	2
4650.2.a.cp.1.3		3	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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.21018 q^{11} +1.00000 q^{12} +0.789816 q^{13} +2.00000 q^{14} +1.00000 q^{16} -0.115086 q^{17} +1.00000 q^{18} +0.115086 q^{19} +2.00000 q^{21} +5.21018 q^{22} -4.42037 q^{23} +1.00000 q^{24} +0.789816 q^{26} +1.00000 q^{27} +2.00000 q^{28} +4.42037 q^{29} -1.00000 q^{31} +1.00000 q^{32} +5.21018 q^{33} -0.115086 q^{34} +1.00000 q^{36} +6.61056 q^{37} +0.115086 q^{38} +0.789816 q^{39} +2.00000 q^{41} +2.00000 q^{42} -8.61056 q^{43} +5.21018 q^{44} -4.42037 q^{46} +0.115086 q^{47} +1.00000 q^{48} -3.00000 q^{49} -0.115086 q^{51} +0.789816 q^{52} -0.190196 q^{53} +1.00000 q^{54} +2.00000 q^{56} +0.115086 q^{57} +4.42037 q^{58} -4.19020 q^{59} +12.4955 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +5.21018 q^{66} +5.82075 q^{67} -0.115086 q^{68} -4.42037 q^{69} -13.8207 q^{71} +1.00000 q^{72} -10.8407 q^{73} +6.61056 q^{74} +0.115086 q^{76} +10.4204 q^{77} +0.789816 q^{78} +2.30528 q^{79} +1.00000 q^{81} +2.00000 q^{82} +15.1460 q^{83} +2.00000 q^{84} -8.61056 q^{86} +4.42037 q^{87} +5.21018 q^{88} -2.23017 q^{89} +1.57963 q^{91} -4.42037 q^{92} -1.00000 q^{93} +0.115086 q^{94} +1.00000 q^{96} -9.21018 q^{97} -3.00000 q^{98} +5.21018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 8 q^{11} + 3 q^{12} + 10 q^{13} + 6 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{18} - 2 q^{19} + 6 q^{21} + 8 q^{22} + 2 q^{23} + 3 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 8 * q^11 + 3 * q^12 + 10 * q^13 + 6 * q^14 + 3 * q^16 + 2 * q^17 + 3 * q^18 - 2 * q^19 + 6 * q^21 + 8 * q^22 + 2 * q^23 + 3 * q^24 + 10 * q^26 + 3 * q^27 + 6 * q^28 - 2 * q^29 - 3 * q^31 + 3 * q^32 + 8 * q^33 + 2 * q^34 + 3 * q^36 - 6 * q^37 - 2 * q^38 + 10 * q^39 + 6 * q^41 + 6 * q^42 + 8 * q^44 + 2 * q^46 - 2 * q^47 + 3 * q^48 - 9 * q^49 + 2 * q^51 + 10 * q^52 + 10 * q^53 + 3 * q^54 + 6 * q^56 - 2 * q^57 - 2 * q^58 - 2 * q^59 + 14 * q^61 - 3 * q^62 + 6 * q^63 + 3 * q^64 + 8 * q^66 - 16 * q^67 + 2 * q^68 + 2 * q^69 - 8 * q^71 + 3 * q^72 - 2 * q^73 - 6 * q^74 - 2 * q^76 + 16 * q^77 + 10 * q^78 - 6 * q^79 + 3 * q^81 + 6 * q^82 + 2 * q^83 + 6 * q^84 - 2 * q^87 + 8 * q^88 - 2 * q^89 + 20 * q^91 + 2 * q^92 - 3 * q^93 - 2 * q^94 + 3 * q^96 - 20 * q^97 - 9 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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