LMFDB - Embedded newform 4650.2.a.co.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.36147$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.93800 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.93800 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.78493 q^{11} -1.00000 q^{12} -0.784934 q^{13} +3.93800 q^{14} +1.00000 q^{16} -0.938003 q^{17} +1.00000 q^{18} +7.87601 q^{19} -3.93800 q^{21} -2.78493 q^{22} +2.00000 q^{23} -1.00000 q^{24} -0.784934 q^{26} -1.00000 q^{27} +3.93800 q^{28} -0.938003 q^{29} -1.00000 q^{31} +1.00000 q^{32} +2.78493 q^{33} -0.938003 q^{34} +1.00000 q^{36} -4.78493 q^{37} +7.87601 q^{38} +0.784934 q^{39} +3.50787 q^{41} -3.93800 q^{42} +7.72294 q^{43} -2.78493 q^{44} +2.00000 q^{46} +4.56987 q^{47} -1.00000 q^{48} +8.50787 q^{49} +0.938003 q^{51} -0.784934 q^{52} -2.66094 q^{53} -1.00000 q^{54} +3.93800 q^{56} -7.87601 q^{57} -0.938003 q^{58} -1.56987 q^{59} -0.784934 q^{61} -1.00000 q^{62} +3.93800 q^{63} +1.00000 q^{64} +2.78493 q^{66} +3.56987 q^{67} -0.938003 q^{68} -2.00000 q^{69} +4.44588 q^{71} +1.00000 q^{72} +10.0000 q^{73} -4.78493 q^{74} +7.87601 q^{76} -10.9671 q^{77} +0.784934 q^{78} +2.93800 q^{79} +1.00000 q^{81} +3.50787 q^{82} -0.660941 q^{83} -3.93800 q^{84} +7.72294 q^{86} +0.938003 q^{87} -2.78493 q^{88} -11.2441 q^{89} -3.09107 q^{91} +2.00000 q^{92} +1.00000 q^{93} +4.56987 q^{94} -1.00000 q^{96} +1.36814 q^{97} +8.50787 q^{98} -2.78493 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{11} - 3 q^{12} + 3 q^{16} + 9 q^{17} + 3 q^{18} - 6 q^{22} + 6 q^{23} - 3 q^{24} - 3 q^{27} + 9 q^{29} - 3 q^{31} + 3 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{36} - 12 q^{37} - 6 q^{41} + 9 q^{43} - 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{48} + 9 q^{49} - 9 q^{51} + 18 q^{53} - 3 q^{54} + 9 q^{58} - 3 q^{62} + 3 q^{64} + 6 q^{66} + 6 q^{67} + 9 q^{68} - 6 q^{69} - 15 q^{71} + 3 q^{72} + 30 q^{73} - 12 q^{74} + 12 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{82} + 24 q^{83} + 9 q^{86} - 9 q^{87} - 6 q^{88} - 3 q^{89} + 12 q^{91} + 6 q^{92} + 3 q^{93} + 9 q^{94} - 3 q^{96} - 3 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 6 * q^11 - 3 * q^12 + 3 * q^16 + 9 * q^17 + 3 * q^18 - 6 * q^22 + 6 * q^23 - 3 * q^24 - 3 * q^27 + 9 * q^29 - 3 * q^31 + 3 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^36 - 12 * q^37 - 6 * q^41 + 9 * q^43 - 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^48 + 9 * q^49 - 9 * q^51 + 18 * q^53 - 3 * q^54 + 9 * q^58 - 3 * q^62 + 3 * q^64 + 6 * q^66 + 6 * q^67 + 9 * q^68 - 6 * q^69 - 15 * q^71 + 3 * q^72 + 30 * q^73 - 12 * q^74 + 12 * q^77 - 3 * q^79 + 3 * q^81 - 6 * q^82 + 24 * q^83 + 9 * q^86 - 9 * q^87 - 6 * q^88 - 3 * q^89 + 12 * q^91 + 6 * q^92 + 3 * q^93 + 9 * q^94 - 3 * q^96 - 3 * q^97 + 9 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	3.93800	1.48843	0.744213	−	0.667943i	$-0.232825\pi$
$7$	3.93800	1.48843	0.744213	+	0.667943i	$0.232825\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.78493	−0.839689	−0.419845	−	0.907596i	$-0.637915\pi$
$11$	−2.78493	−0.839689	−0.419845	+	0.907596i	$0.637915\pi$
$12$	−1.00000	−0.288675
$13$	−0.784934	−0.217702	−0.108851	−	0.994058i	$-0.534717\pi$
$13$	−0.784934	−0.217702	−0.108851	+	0.994058i	$0.534717\pi$
$14$	3.93800	1.05248
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.938003	−0.227499	−0.113750	−	0.993509i	$-0.536286\pi$
$17$	−0.938003	−0.227499	−0.113750	+	0.993509i	$0.536286\pi$
$18$	1.00000	0.235702
$19$	7.87601	1.80688	0.903440	−	0.428715i	$-0.141033\pi$
$19$	7.87601	1.80688	0.903440	+	0.428715i	$0.141033\pi$
$20$	0	0
$21$	−3.93800	−0.859343
$22$	−2.78493	−0.593750
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−0.784934	−0.153938
$27$	−1.00000	−0.192450
$28$	3.93800	0.744213
$29$	−0.938003	−0.174183	−0.0870914	−	0.996200i	$-0.527757\pi$
$29$	−0.938003	−0.174183	−0.0870914	+	0.996200i	$0.527757\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	2.78493	0.484795
$34$	−0.938003	−0.160866
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.78493	−0.786638	−0.393319	−	0.919402i	$-0.628673\pi$
$37$	−4.78493	−0.786638	−0.393319	+	0.919402i	$0.628673\pi$
$38$	7.87601	1.27766
$39$	0.784934	0.125690
$40$	0	0
$41$	3.50787	0.547837	0.273919	−	0.961753i	$-0.411680\pi$
$41$	3.50787	0.547837	0.273919	+	0.961753i	$0.411680\pi$
$42$	−3.93800	−0.607647
$43$	7.72294	1.17774	0.588868	−	0.808229i	$-0.299574\pi$
$43$	7.72294	1.17774	0.588868	+	0.808229i	$0.299574\pi$
$44$	−2.78493	−0.419845
$45$	0	0
$46$	2.00000	0.294884
$47$	4.56987	0.666584	0.333292	−	0.942824i	$-0.391841\pi$
$47$	4.56987	0.666584	0.333292	+	0.942824i	$0.391841\pi$
$48$	−1.00000	−0.144338
$49$	8.50787	1.21541
$50$	0	0
$51$	0.938003	0.131347
$52$	−0.784934	−0.108851
$53$	−2.66094	−0.365508	−0.182754	−	0.983159i	$-0.558501\pi$
$53$	−2.66094	−0.365508	−0.182754	+	0.983159i	$0.558501\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.93800	0.526238
$57$	−7.87601	−1.04320
$58$	−0.938003	−0.123166
$59$	−1.56987	−0.204379	−0.102190	−	0.994765i	$-0.532585\pi$
$59$	−1.56987	−0.204379	−0.102190	+	0.994765i	$0.532585\pi$
$60$	0	0
$61$	−0.784934	−0.100501	−0.0502503	−	0.998737i	$-0.516002\pi$
$61$	−0.784934	−0.100501	−0.0502503	+	0.998737i	$0.516002\pi$
$62$	−1.00000	−0.127000
$63$	3.93800	0.496142
$64$	1.00000	0.125000
$65$	0	0
$66$	2.78493	0.342802
$67$	3.56987	0.436129	0.218064	−	0.975934i	$-0.430026\pi$
$67$	3.56987	0.436129	0.218064	+	0.975934i	$0.430026\pi$
$68$	−0.938003	−0.113750
$69$	−2.00000	−0.240772
$70$	0	0
$71$	4.44588	0.527628	0.263814	−	0.964574i	$-0.415019\pi$
$71$	4.44588	0.527628	0.263814	+	0.964574i	$0.415019\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−4.78493	−0.556237
$75$	0	0
$76$	7.87601	0.903440
$77$	−10.9671	−1.24981
$78$	0.784934	0.0888763
$79$	2.93800	0.330551	0.165276	−	0.986247i	$-0.447149\pi$
$79$	2.93800	0.330551	0.165276	+	0.986247i	$0.447149\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.50787	0.387380
$83$	−0.660941	−0.0725477	−0.0362739	−	0.999342i	$-0.511549\pi$
$83$	−0.660941	−0.0725477	−0.0362739	+	0.999342i	$0.511549\pi$
$84$	−3.93800	−0.429671
$85$	0	0
$86$	7.72294	0.832786
$87$	0.938003	0.100565
$88$	−2.78493	−0.296875
$89$	−11.2441	−1.19188	−0.595938	−	0.803030i	$-0.703220\pi$
$89$	−11.2441	−1.19188	−0.595938	+	0.803030i	$0.703220\pi$
$90$	0	0
$91$	−3.09107	−0.324032
$92$	2.00000	0.208514
$93$	1.00000	0.103695
$94$	4.56987	0.471346
$95$	0	0
$96$	−1.00000	−0.102062
$97$	1.36814	0.138913	0.0694566	−	0.997585i	$-0.477873\pi$
$97$	1.36814	0.138913	0.0694566	+	0.997585i	$0.477873\pi$
$98$	8.50787	0.859425
$99$	−2.78493	−0.279896
$100$	0	0
$101$	−7.44588	−0.740892	−0.370446	−	0.928854i	$-0.620795\pi$
$101$	−7.44588	−0.740892	−0.370446	+	0.928854i	$0.620795\pi$
$102$	0.938003	0.0928762
$103$	6.06200	0.597306	0.298653	−	0.954362i	$-0.403463\pi$
$103$	6.06200	0.597306	0.298653	+	0.954362i	$0.403463\pi$
$104$	−0.784934	−0.0769691
$105$	0	0
$106$	−2.66094	−0.258453
$107$	−0.631865	−0.0610847	−0.0305423	−	0.999533i	$-0.509723\pi$
$107$	−0.631865	−0.0610847	−0.0305423	+	0.999533i	$0.509723\pi$
$108$	−1.00000	−0.0962250
$109$	7.36814	0.705739	0.352870	−	0.935672i	$-0.385206\pi$
$109$	7.36814	0.705739	0.352870	+	0.935672i	$0.385206\pi$
$110$	0	0
$111$	4.78493	0.454166
$112$	3.93800	0.372106
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−7.87601	−0.737656
$115$	0	0
$116$	−0.938003	−0.0870914
$117$	−0.784934	−0.0725672
$118$	−1.56987	−0.144518
$119$	−3.69386	−0.338616
$120$	0	0
$121$	−3.24414	−0.294922
$122$	−0.784934	−0.0710646
$123$	−3.50787	−0.316294
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	3.93800	0.350825
$127$	4.50787	0.400009	0.200004	−	0.979795i	$-0.435904\pi$
$127$	4.50787	0.400009	0.200004	+	0.979795i	$0.435904\pi$
$128$	1.00000	0.0883883
$129$	−7.72294	−0.679967
$130$	0	0
$131$	−20.3839	−1.78095	−0.890474	−	0.455034i	$-0.849627\pi$
$131$	−20.3839	−1.78095	−0.890474	+	0.455034i	$0.849627\pi$
$132$	2.78493	0.242397
$133$	31.0157	2.68941
$134$	3.56987	0.308390
$135$	0	0
$136$	−0.938003	−0.0804331
$137$	18.6900	1.59680	0.798398	−	0.602130i	$-0.205681\pi$
$137$	18.6900	1.59680	0.798398	+	0.602130i	$0.205681\pi$
$138$	−2.00000	−0.170251
$139$	−9.29281	−0.788205	−0.394103	−	0.919066i	$-0.628945\pi$
$139$	−9.29281	−0.788205	−0.394103	+	0.919066i	$0.628945\pi$
$140$	0	0
$141$	−4.56987	−0.384852
$142$	4.44588	0.373090
$143$	2.18599	0.182802
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	−8.50787	−0.701717
$148$	−4.78493	−0.393319
$149$	11.0157	0.902445	0.451222	−	0.892412i	$-0.350988\pi$
$149$	11.0157	0.902445	0.451222	+	0.892412i	$0.350988\pi$
$150$	0	0
$151$	−4.38388	−0.356755	−0.178377	−	0.983962i	$-0.557085\pi$
$151$	−4.38388	−0.356755	−0.178377	+	0.983962i	$0.557085\pi$
$152$	7.87601	0.638829
$153$	−0.938003	−0.0758331
$154$	−10.9671	−0.883752
$155$	0	0
$156$	0.784934	0.0628450
$157$	−10.8140	−0.863052	−0.431526	−	0.902101i	$-0.642025\pi$
$157$	−10.8140	−0.863052	−0.431526	+	0.902101i	$0.642025\pi$
$158$	2.93800	0.233735
$159$	2.66094	0.211026
$160$	0	0
$161$	7.87601	0.620716
$162$	1.00000	0.0785674
$163$	13.4459	1.05316	0.526581	−	0.850125i	$-0.323474\pi$
$163$	13.4459	1.05316	0.526581	+	0.850125i	$0.323474\pi$
$164$	3.50787	0.273919
$165$	0	0
$166$	−0.660941	−0.0512990
$167$	15.7520	1.21893	0.609464	−	0.792814i	$-0.291385\pi$
$167$	15.7520	1.21893	0.609464	+	0.792814i	$0.291385\pi$
$168$	−3.93800	−0.303824
$169$	−12.3839	−0.952606
$170$	0	0
$171$	7.87601	0.602293
$172$	7.72294	0.588868
$173$	15.4459	1.17433	0.587164	−	0.809468i	$-0.300244\pi$
$173$	15.4459	1.17433	0.587164	+	0.809468i	$0.300244\pi$
$174$	0.938003	0.0711099
$175$	0	0
$176$	−2.78493	−0.209922
$177$	1.56987	0.117999
$178$	−11.2441	−0.842784
$179$	−24.1068	−1.80183	−0.900914	−	0.433998i	$-0.857103\pi$
$179$	−24.1068	−1.80183	−0.900914	+	0.433998i	$0.857103\pi$
$180$	0	0
$181$	22.1068	1.64319	0.821593	−	0.570074i	$-0.193085\pi$
$181$	22.1068	1.64319	0.821593	+	0.570074i	$0.193085\pi$
$182$	−3.09107	−0.229126
$183$	0.784934	0.0580240
$184$	2.00000	0.147442
$185$	0	0
$186$	1.00000	0.0733236
$187$	2.61228	0.191029
$188$	4.56987	0.333292
$189$	−3.93800	−0.286448
$190$	0	0
$191$	−7.36814	−0.533140	−0.266570	−	0.963816i	$-0.585890\pi$
$191$	−7.36814	−0.533140	−0.266570	+	0.963816i	$0.585890\pi$
$192$	−1.00000	−0.0721688
$193$	2.49213	0.179387	0.0896937	−	0.995969i	$-0.471411\pi$
$193$	2.49213	0.179387	0.0896937	+	0.995969i	$0.471411\pi$
$194$	1.36814	0.0982264
$195$	0	0
$196$	8.50787	0.607705
$197$	−1.41680	−0.100943	−0.0504714	−	0.998726i	$-0.516072\pi$
$197$	−1.41680	−0.100943	−0.0504714	+	0.998726i	$0.516072\pi$
$198$	−2.78493	−0.197917
$199$	18.8918	1.33920	0.669600	−	0.742722i	$-0.266465\pi$
$199$	18.8918	1.33920	0.669600	+	0.742722i	$0.266465\pi$
$200$	0	0
$201$	−3.56987	−0.251799
$202$	−7.44588	−0.523890
$203$	−3.69386	−0.259258
$204$	0.938003	0.0656734
$205$	0	0
$206$	6.06200	0.422359
$207$	2.00000	0.139010
$208$	−0.784934	−0.0544254
$209$	−21.9342	−1.51722
$210$	0	0
$211$	15.8760	1.09295	0.546475	−	0.837476i	$-0.315969\pi$
$211$	15.8760	1.09295	0.546475	+	0.837476i	$0.315969\pi$
$212$	−2.66094	−0.182754
$213$	−4.44588	−0.304626
$214$	−0.631865	−0.0431934
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−3.93800	−0.267329
$218$	7.36814	0.499033
$219$	−10.0000	−0.675737
$220$	0	0
$221$	0.736271	0.0495269
$222$	4.78493	0.321144
$223$	3.06200	0.205046	0.102523	−	0.994731i	$-0.467308\pi$
$223$	3.06200	0.205046	0.102523	+	0.994731i	$0.467308\pi$
$224$	3.93800	0.263119
$225$	0	0
$226$	6.00000	0.399114
$227$	−0.814010	−0.0540278	−0.0270139	−	0.999635i	$-0.508600\pi$
$227$	−0.814010	−0.0540278	−0.0270139	+	0.999635i	$0.508600\pi$
$228$	−7.87601	−0.521601
$229$	−5.56987	−0.368067	−0.184034	−	0.982920i	$-0.558916\pi$
$229$	−5.56987	−0.368067	−0.184034	+	0.982920i	$0.558916\pi$
$230$	0	0
$231$	10.9671	0.721581
$232$	−0.938003	−0.0615829
$233$	14.3681	0.941288	0.470644	−	0.882323i	$-0.344022\pi$
$233$	14.3681	0.941288	0.470644	+	0.882323i	$0.344022\pi$
$234$	−0.784934	−0.0513127
$235$	0	0
$236$	−1.56987	−0.102190
$237$	−2.93800	−0.190844
$238$	−3.69386	−0.239437
$239$	16.8918	1.09264	0.546318	−	0.837578i	$-0.316029\pi$
$239$	16.8918	1.09264	0.546318	+	0.837578i	$0.316029\pi$
$240$	0	0
$241$	−3.13974	−0.202248	−0.101124	−	0.994874i	$-0.532244\pi$
$241$	−3.13974	−0.202248	−0.101124	+	0.994874i	$0.532244\pi$
$242$	−3.24414	−0.208541
$243$	−1.00000	−0.0641500
$244$	−0.784934	−0.0502503
$245$	0	0
$246$	−3.50787	−0.223654
$247$	−6.18215	−0.393361
$248$	−1.00000	−0.0635001
$249$	0.660941	0.0418854
$250$	0	0
$251$	−27.3972	−1.72930	−0.864648	−	0.502378i	$-0.832459\pi$
$251$	−27.3972	−1.72930	−0.864648	+	0.502378i	$0.832459\pi$
$252$	3.93800	0.248071
$253$	−5.56987	−0.350175
$254$	4.50787	0.282849
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.38388	−0.585350	−0.292675	−	0.956212i	$-0.594545\pi$
$257$	−9.38388	−0.585350	−0.292675	+	0.956212i	$0.594545\pi$
$258$	−7.72294	−0.480809
$259$	−18.8431	−1.17085
$260$	0	0
$261$	−0.938003	−0.0580610
$262$	−20.3839	−1.25932
$263$	1.44588	0.0891565	0.0445782	−	0.999006i	$-0.485806\pi$
$263$	1.44588	0.0891565	0.0445782	+	0.999006i	$0.485806\pi$
$264$	2.78493	0.171401
$265$	0	0
$266$	31.0157	1.90170
$267$	11.2441	0.688130
$268$	3.56987	0.218064
$269$	4.93800	0.301075	0.150538	−	0.988604i	$-0.451900\pi$
$269$	4.93800	0.301075	0.150538	+	0.988604i	$0.451900\pi$
$270$	0	0
$271$	9.67427	0.587670	0.293835	−	0.955856i	$-0.405068\pi$
$271$	9.67427	0.587670	0.293835	+	0.955856i	$0.405068\pi$
$272$	−0.938003	−0.0568748
$273$	3.09107	0.187080
$274$	18.6900	1.12911
$275$	0	0
$276$	−2.00000	−0.120386
$277$	−30.1821	−1.81347	−0.906735	−	0.421702i	$-0.861433\pi$
$277$	−30.1821	−1.81347	−0.906735	+	0.421702i	$0.861433\pi$
$278$	−9.29281	−0.557345
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	21.3839	1.27566	0.637828	−	0.770179i	$-0.279833\pi$
$281$	21.3839	1.27566	0.637828	+	0.770179i	$0.279833\pi$
$282$	−4.56987	−0.272132
$283$	15.3219	0.910791	0.455396	−	0.890289i	$-0.349498\pi$
$283$	15.3219	0.910791	0.455396	+	0.890289i	$0.349498\pi$
$284$	4.44588	0.263814
$285$	0	0
$286$	2.18599	0.129260
$287$	13.8140	0.815415
$288$	1.00000	0.0589256
$289$	−16.1201	−0.948244
$290$	0	0
$291$	−1.36814	−0.0802015
$292$	10.0000	0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−8.50787	−0.496189
$295$	0	0
$296$	−4.78493	−0.278119
$297$	2.78493	0.161598
$298$	11.0157	0.638125
$299$	−1.56987	−0.0907878
$300$	0	0
$301$	30.4130	1.75297
$302$	−4.38388	−0.252264
$303$	7.44588	0.427754
$304$	7.87601	0.451720
$305$	0	0
$306$	−0.938003	−0.0536221
$307$	17.1979	0.981535	0.490768	−	0.871290i	$-0.336716\pi$
$307$	17.1979	0.981535	0.490768	+	0.871290i	$0.336716\pi$
$308$	−10.9671	−0.624907
$309$	−6.06200	−0.344855
$310$	0	0
$311$	−28.0739	−1.59192	−0.795962	−	0.605346i	$-0.793035\pi$
$311$	−28.0739	−1.59192	−0.795962	+	0.605346i	$0.793035\pi$
$312$	0.784934	0.0444381
$313$	−5.19789	−0.293802	−0.146901	−	0.989151i	$-0.546930\pi$
$313$	−5.19789	−0.293802	−0.146901	+	0.989151i	$0.546930\pi$
$314$	−10.8140	−0.610270
$315$	0	0
$316$	2.93800	0.165276
$317$	24.8918	1.39806	0.699030	−	0.715092i	$-0.253615\pi$
$317$	24.8918	1.39806	0.699030	+	0.715092i	$0.253615\pi$
$318$	2.66094	0.149218
$319$	2.61228	0.146259
$320$	0	0
$321$	0.631865	0.0352672
$322$	7.87601	0.438913
$323$	−7.38772	−0.411064
$324$	1.00000	0.0555556
$325$	0	0
$326$	13.4459	0.744698
$327$	−7.36814	−0.407459
$328$	3.50787	0.193690
$329$	17.9962	0.992160
$330$	0	0
$331$	8.66094	0.476048	0.238024	−	0.971259i	$-0.423500\pi$
$331$	8.66094	0.476048	0.238024	+	0.971259i	$0.423500\pi$
$332$	−0.660941	−0.0362739
$333$	−4.78493	−0.262213
$334$	15.7520	0.861912
$335$	0	0
$336$	−3.93800	−0.214836
$337$	−4.30614	−0.234570	−0.117285	−	0.993098i	$-0.537419\pi$
$337$	−4.30614	−0.234570	−0.117285	+	0.993098i	$0.537419\pi$
$338$	−12.3839	−0.673594
$339$	−6.00000	−0.325875
$340$	0	0
$341$	2.78493	0.150813
$342$	7.87601	0.425886
$343$	5.93800	0.320622
$344$	7.72294	0.416393
$345$	0	0
$346$	15.4459	0.830375
$347$	−14.5369	−0.780384	−0.390192	−	0.920733i	$-0.627591\pi$
$347$	−14.5369	−0.780384	−0.390192	+	0.920733i	$0.627591\pi$
$348$	0.938003	0.0502823
$349$	24.6900	1.32163	0.660813	−	0.750550i	$-0.270212\pi$
$349$	24.6900	1.32163	0.660813	+	0.750550i	$0.270212\pi$
$350$	0	0
$351$	0.784934	0.0418967
$352$	−2.78493	−0.148437
$353$	10.5856	0.563415	0.281708	−	0.959500i	$-0.409099\pi$
$353$	10.5856	0.563415	0.281708	+	0.959500i	$0.409099\pi$
$354$	1.56987	0.0834375
$355$	0	0
$356$	−11.2441	−0.595938
$357$	3.69386	0.195500
$358$	−24.1068	−1.27408
$359$	−10.8918	−0.574845	−0.287422	−	0.957804i	$-0.592798\pi$
$359$	−10.8918	−0.574845	−0.287422	+	0.957804i	$0.592798\pi$
$360$	0	0
$361$	43.0315	2.26482
$362$	22.1068	1.16191
$363$	3.24414	0.170273
$364$	−3.09107	−0.162016
$365$	0	0
$366$	0.784934	0.0410292
$367$	6.63186	0.346181	0.173090	−	0.984906i	$-0.444625\pi$
$367$	6.63186	0.346181	0.173090	+	0.984906i	$0.444625\pi$
$368$	2.00000	0.104257
$369$	3.50787	0.182612
$370$	0	0
$371$	−10.4788	−0.544032
$372$	1.00000	0.0518476
$373$	−4.38388	−0.226989	−0.113494	−	0.993539i	$-0.536204\pi$
$373$	−4.38388	−0.226989	−0.113494	+	0.993539i	$0.536204\pi$
$374$	2.61228	0.135078
$375$	0	0
$376$	4.56987	0.235673
$377$	0.736271	0.0379199
$378$	−3.93800	−0.202549
$379$	−2.30614	−0.118458	−0.0592292	−	0.998244i	$-0.518864\pi$
$379$	−2.30614	−0.118458	−0.0592292	+	0.998244i	$0.518864\pi$
$380$	0	0
$381$	−4.50787	−0.230945
$382$	−7.36814	−0.376987
$383$	1.26373	0.0645735	0.0322868	−	0.999479i	$-0.489721\pi$
$383$	1.26373	0.0645735	0.0322868	+	0.999479i	$0.489721\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.49213	0.126846
$387$	7.72294	0.392579
$388$	1.36814	0.0694566
$389$	−8.55412	−0.433711	−0.216856	−	0.976204i	$-0.569580\pi$
$389$	−8.55412	−0.433711	−0.216856	+	0.976204i	$0.569580\pi$
$390$	0	0
$391$	−1.87601	−0.0948738
$392$	8.50787	0.429712
$393$	20.3839	1.02823
$394$	−1.41680	−0.0713773
$395$	0	0
$396$	−2.78493	−0.139948
$397$	−37.0157	−1.85777	−0.928883	−	0.370372i	$-0.879230\pi$
$397$	−37.0157	−1.85777	−0.928883	+	0.370372i	$0.879230\pi$
$398$	18.8918	0.946958
$399$	−31.0157	−1.55273
$400$	0	0
$401$	5.75201	0.287242	0.143621	−	0.989633i	$-0.454125\pi$
$401$	5.75201	0.287242	0.143621	+	0.989633i	$0.454125\pi$
$402$	−3.56987	−0.178049
$403$	0.784934	0.0391003
$404$	−7.44588	−0.370446
$405$	0	0
$406$	−3.69386	−0.183323
$407$	13.3257	0.660532
$408$	0.938003	0.0464381
$409$	10.5541	0.521868	0.260934	−	0.965357i	$-0.415970\pi$
$409$	10.5541	0.521868	0.260934	+	0.965357i	$0.415970\pi$
$410$	0	0
$411$	−18.6900	−0.921911
$412$	6.06200	0.298653
$413$	−6.18215	−0.304204
$414$	2.00000	0.0982946
$415$	0	0
$416$	−0.784934	−0.0384846
$417$	9.29281	0.455071
$418$	−21.9342	−1.07283
$419$	14.0777	0.687743	0.343871	−	0.939017i	$-0.388262\pi$
$419$	14.0777	0.687743	0.343871	+	0.939017i	$0.388262\pi$
$420$	0	0
$421$	−14.2599	−0.694984	−0.347492	−	0.937683i	$-0.612967\pi$
$421$	−14.2599	−0.694984	−0.347492	+	0.937683i	$0.612967\pi$
$422$	15.8760	0.772832
$423$	4.56987	0.222195
$424$	−2.66094	−0.129227
$425$	0	0
$426$	−4.44588	−0.215403
$427$	−3.09107	−0.149588
$428$	−0.631865	−0.0305423
$429$	−2.18599	−0.105541
$430$	0	0
$431$	−37.5660	−1.80949	−0.904746	−	0.425952i	$-0.859939\pi$
$431$	−37.5660	−1.80949	−0.904746	+	0.425952i	$0.859939\pi$
$432$	−1.00000	−0.0481125
$433$	28.0582	1.34839	0.674194	−	0.738554i	$-0.264491\pi$
$433$	28.0582	1.34839	0.674194	+	0.738554i	$0.264491\pi$
$434$	−3.93800	−0.189030
$435$	0	0
$436$	7.36814	0.352870
$437$	15.7520	0.753521
$438$	−10.0000	−0.477818
$439$	−21.6280	−1.03225	−0.516125	−	0.856514i	$-0.672626\pi$
$439$	−21.6280	−1.03225	−0.516125	+	0.856514i	$0.672626\pi$
$440$	0	0
$441$	8.50787	0.405137
$442$	0.736271	0.0350208
$443$	36.5660	1.73730	0.868652	−	0.495423i	$-0.164987\pi$
$443$	36.5660	1.73730	0.868652	+	0.495423i	$0.164987\pi$
$444$	4.78493	0.227083
$445$	0	0
$446$	3.06200	0.144990
$447$	−11.0157	−0.521027
$448$	3.93800	0.186053
$449$	−3.36814	−0.158952	−0.0794761	−	0.996837i	$-0.525325\pi$
$449$	−3.36814	−0.158952	−0.0794761	+	0.996837i	$0.525325\pi$
$450$	0	0
$451$	−9.76919	−0.460013
$452$	6.00000	0.282216
$453$	4.38388	0.205973
$454$	−0.814010	−0.0382034
$455$	0	0
$456$	−7.87601	−0.368828
$457$	10.1821	0.476301	0.238150	−	0.971228i	$-0.423459\pi$
$457$	10.1821	0.476301	0.238150	+	0.971228i	$0.423459\pi$
$458$	−5.56987	−0.260263
$459$	0.938003	0.0437823
$460$	0	0
$461$	35.1226	1.63582	0.817910	−	0.575346i	$-0.195132\pi$
$461$	35.1226	1.63582	0.817910	+	0.575346i	$0.195132\pi$
$462$	10.9671	0.510235
$463$	−8.20173	−0.381167	−0.190583	−	0.981671i	$-0.561038\pi$
$463$	−8.20173	−0.381167	−0.190583	+	0.981671i	$0.561038\pi$
$464$	−0.938003	−0.0435457
$465$	0	0
$466$	14.3681	0.665591
$467$	6.75586	0.312624	0.156312	−	0.987708i	$-0.450039\pi$
$467$	6.75586	0.312624	0.156312	+	0.987708i	$0.450039\pi$
$468$	−0.784934	−0.0362836
$469$	14.0582	0.649145
$470$	0	0
$471$	10.8140	0.498283
$472$	−1.56987	−0.0722590
$473$	−21.5079	−0.988933
$474$	−2.93800	−0.134947
$475$	0	0
$476$	−3.69386	−0.169308
$477$	−2.66094	−0.121836
$478$	16.8918	0.772611
$479$	15.5236	0.709292	0.354646	−	0.935001i	$-0.384601\pi$
$479$	15.5236	0.709292	0.354646	+	0.935001i	$0.384601\pi$
$480$	0	0
$481$	3.75586	0.171252
$482$	−3.13974	−0.143011
$483$	−7.87601	−0.358371
$484$	−3.24414	−0.147461
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−33.1201	−1.50082	−0.750409	−	0.660974i	$-0.770143\pi$
$487$	−33.1201	−1.50082	−0.750409	+	0.660974i	$0.770143\pi$
$488$	−0.784934	−0.0355323
$489$	−13.4459	−0.608043
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−3.50787	−0.158147
$493$	0.879851	0.0396265
$494$	−6.18215	−0.278148
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	17.5079	0.785335
$498$	0.660941	0.0296175
$499$	26.6634	1.19362	0.596808	−	0.802384i	$-0.296435\pi$
$499$	26.6634	1.19362	0.596808	+	0.802384i	$0.296435\pi$
$500$	0	0
$501$	−15.7520	−0.703748
$502$	−27.3972	−1.22280
$503$	−19.1821	−0.855290	−0.427645	−	0.903947i	$-0.640657\pi$
$503$	−19.1821	−0.855290	−0.427645	+	0.903947i	$0.640657\pi$
$504$	3.93800	0.175413
$505$	0	0
$506$	−5.56987	−0.247611
$507$	12.3839	0.549987
$508$	4.50787	0.200004
$509$	28.8431	1.27845	0.639224	−	0.769021i	$-0.279256\pi$
$509$	28.8431	1.27845	0.639224	+	0.769021i	$0.279256\pi$
$510$	0	0
$511$	39.3800	1.74207
$512$	1.00000	0.0441942
$513$	−7.87601	−0.347734
$514$	−9.38388	−0.413905
$515$	0	0
$516$	−7.72294	−0.339983
$517$	−12.7268	−0.559723
$518$	−18.8431	−0.827918
$519$	−15.4459	−0.677999
$520$	0	0
$521$	−19.6319	−0.860088	−0.430044	−	0.902808i	$-0.641502\pi$
$521$	−19.6319	−0.860088	−0.430044	+	0.902808i	$0.641502\pi$
$522$	−0.938003	−0.0410553
$523$	−10.2599	−0.448633	−0.224317	−	0.974516i	$-0.572015\pi$
$523$	−10.2599	−0.448633	−0.224317	+	0.974516i	$0.572015\pi$
$524$	−20.3839	−0.890474
$525$	0	0
$526$	1.44588	0.0630431
$527$	0.938003	0.0408601
$528$	2.78493	0.121199
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−1.56987	−0.0681265
$532$	31.0157	1.34470
$533$	−2.75345	−0.119265
$534$	11.2441	0.486582
$535$	0	0
$536$	3.56987	0.154195
$537$	24.1068	1.04029
$538$	4.93800	0.212892
$539$	−23.6939	−1.02057
$540$	0	0
$541$	−30.6634	−1.31832	−0.659160	−	0.752003i	$-0.729088\pi$
$541$	−30.6634	−1.31832	−0.659160	+	0.752003i	$0.729088\pi$
$542$	9.67427	0.415546
$543$	−22.1068	−0.948694
$544$	−0.938003	−0.0402166
$545$	0	0
$546$	3.09107	0.132286
$547$	−22.8918	−0.978781	−0.489390	−	0.872065i	$-0.662781\pi$
$547$	−22.8918	−0.978781	−0.489390	+	0.872065i	$0.662781\pi$
$548$	18.6900	0.798398
$549$	−0.784934	−0.0335002
$550$	0	0
$551$	−7.38772	−0.314728
$552$	−2.00000	−0.0851257
$553$	11.5699	0.492001
$554$	−30.1821	−1.28232
$555$	0	0
$556$	−9.29281	−0.394103
$557$	−12.0777	−0.511750	−0.255875	−	0.966710i	$-0.582364\pi$
$557$	−12.0777	−0.511750	−0.255875	+	0.966710i	$0.582364\pi$
$558$	−1.00000	−0.0423334
$559$	−6.06200	−0.256395
$560$	0	0
$561$	−2.61228	−0.110290
$562$	21.3839	0.902024
$563$	−36.6900	−1.54630	−0.773150	−	0.634223i	$-0.781320\pi$
$563$	−36.6900	−1.54630	−0.773150	+	0.634223i	$0.781320\pi$
$564$	−4.56987	−0.192426
$565$	0	0
$566$	15.3219	0.644027
$567$	3.93800	0.165381
$568$	4.44588	0.186545
$569$	−40.7678	−1.70907	−0.854537	−	0.519391i	$-0.826159\pi$
$569$	−40.7678	−1.70907	−0.854537	+	0.519391i	$0.826159\pi$
$570$	0	0
$571$	−38.5951	−1.61515	−0.807577	−	0.589762i	$-0.799222\pi$
$571$	−38.5951	−1.61515	−0.807577	+	0.589762i	$0.799222\pi$
$572$	2.18599	0.0914008
$573$	7.36814	0.307808
$574$	13.8140	0.576586
$575$	0	0
$576$	1.00000	0.0416667
$577$	−33.8879	−1.41077	−0.705386	−	0.708823i	$-0.749226\pi$
$577$	−33.8879	−1.41077	−0.705386	+	0.708823i	$0.749226\pi$
$578$	−16.1201	−0.670510
$579$	−2.49213	−0.103569
$580$	0	0
$581$	−2.60279	−0.107982
$582$	−1.36814	−0.0567110
$583$	7.41055	0.306913
$584$	10.0000	0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	−7.49454	−0.309333	−0.154666	−	0.987967i	$-0.549430\pi$
$587$	−7.49454	−0.309333	−0.154666	+	0.987967i	$0.549430\pi$
$588$	−8.50787	−0.350859
$589$	−7.87601	−0.324525
$590$	0	0
$591$	1.41680	0.0582793
$592$	−4.78493	−0.196660
$593$	−24.6476	−1.01216	−0.506078	−	0.862488i	$-0.668905\pi$
$593$	−24.6476	−1.01216	−0.506078	+	0.862488i	$0.668905\pi$
$594$	2.78493	0.114267
$595$	0	0
$596$	11.0157	0.451222
$597$	−18.8918	−0.773188
$598$	−1.56987	−0.0641967
$599$	−41.2136	−1.68394	−0.841972	−	0.539522i	$-0.818605\pi$
$599$	−41.2136	−1.68394	−0.841972	+	0.539522i	$0.818605\pi$
$600$	0	0
$601$	−18.7096	−0.763181	−0.381590	−	0.924332i	$-0.624623\pi$
$601$	−18.7096	−0.763181	−0.381590	+	0.924332i	$0.624623\pi$
$602$	30.4130	1.23954
$603$	3.56987	0.145376
$604$	−4.38388	−0.178377
$605$	0	0
$606$	7.44588	0.302468
$607$	13.8140	0.560693	0.280347	−	0.959899i	$-0.409551\pi$
$607$	13.8140	0.560693	0.280347	+	0.959899i	$0.409551\pi$
$608$	7.87601	0.319414
$609$	3.69386	0.149683
$610$	0	0
$611$	−3.58704	−0.145116
$612$	−0.938003	−0.0379165
$613$	−25.3219	−1.02274	−0.511371	−	0.859360i	$-0.670862\pi$
$613$	−25.3219	−1.02274	−0.511371	+	0.859360i	$0.670862\pi$
$614$	17.1979	0.694050
$615$	0	0
$616$	−10.9671	−0.441876
$617$	45.5394	1.83335	0.916673	−	0.399639i	$-0.130864\pi$
$617$	45.5394	1.83335	0.916673	+	0.399639i	$0.130864\pi$
$618$	−6.06200	−0.243849
$619$	−21.9051	−0.880440	−0.440220	−	0.897890i	$-0.645099\pi$
$619$	−21.9051	−0.880440	−0.440220	+	0.897890i	$0.645099\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	−28.0739	−1.12566
$623$	−44.2795	−1.77402
$624$	0.784934	0.0314225
$625$	0	0
$626$	−5.19789	−0.207749
$627$	21.9342	0.875966
$628$	−10.8140	−0.431526
$629$	4.48828	0.178960
$630$	0	0
$631$	−31.2756	−1.24506	−0.622532	−	0.782595i	$-0.713896\pi$
$631$	−31.2756	−1.24506	−0.622532	+	0.782595i	$0.713896\pi$
$632$	2.93800	0.116868
$633$	−15.8760	−0.631015
$634$	24.8918	0.988578
$635$	0	0
$636$	2.66094	0.105513
$637$	−6.67812	−0.264597
$638$	2.61228	0.103421
$639$	4.44588	0.175876
$640$	0	0
$641$	−24.8918	−0.983165	−0.491583	−	0.870831i	$-0.663581\pi$
$641$	−24.8918	−0.983165	−0.491583	+	0.870831i	$0.663581\pi$
$642$	0.631865	0.0249377
$643$	−25.3510	−0.999744	−0.499872	−	0.866099i	$-0.666620\pi$
$643$	−25.3510	−0.999744	−0.499872	+	0.866099i	$0.666620\pi$
$644$	7.87601	0.310358
$645$	0	0
$646$	−7.38772	−0.290666
$647$	31.5040	1.23855	0.619276	−	0.785174i	$-0.287426\pi$
$647$	31.5040	1.23855	0.619276	+	0.785174i	$0.287426\pi$
$648$	1.00000	0.0392837
$649$	4.37198	0.171615
$650$	0	0
$651$	3.93800	0.154343
$652$	13.4459	0.526581
$653$	−23.6280	−0.924636	−0.462318	−	0.886714i	$-0.652982\pi$
$653$	−23.6280	−0.924636	−0.462318	+	0.886714i	$0.652982\pi$
$654$	−7.36814	−0.288117
$655$	0	0
$656$	3.50787	0.136959
$657$	10.0000	0.390137
$658$	17.9962	0.701563
$659$	−0.709604	−0.0276423	−0.0138211	−	0.999904i	$-0.504400\pi$
$659$	−0.709604	−0.0276423	−0.0138211	+	0.999904i	$0.504400\pi$
$660$	0	0
$661$	−38.9962	−1.51677	−0.758387	−	0.651804i	$-0.774012\pi$
$661$	−38.9962	−1.51677	−0.758387	+	0.651804i	$0.774012\pi$
$662$	8.66094	0.336617
$663$	−0.736271	−0.0285944
$664$	−0.660941	−0.0256495
$665$	0	0
$666$	−4.78493	−0.185412
$667$	−1.87601	−0.0726393
$668$	15.7520	0.609464
$669$	−3.06200	−0.118384
$670$	0	0
$671$	2.18599	0.0843892
$672$	−3.93800	−0.151912
$673$	38.9499	1.50141	0.750704	−	0.660638i	$-0.229714\pi$
$673$	38.9499	1.50141	0.750704	+	0.660638i	$0.229714\pi$
$674$	−4.30614	−0.165866
$675$	0	0
$676$	−12.3839	−0.476303
$677$	−51.6571	−1.98534	−0.992672	−	0.120842i	$-0.961441\pi$
$677$	−51.6571	−1.98534	−0.992672	+	0.120842i	$0.961441\pi$
$678$	−6.00000	−0.230429
$679$	5.38772	0.206762
$680$	0	0
$681$	0.814010	0.0311930
$682$	2.78493	0.106641
$683$	−8.93800	−0.342003	−0.171002	−	0.985271i	$-0.554700\pi$
$683$	−8.93800	−0.342003	−0.171002	+	0.985271i	$0.554700\pi$
$684$	7.87601	0.301147
$685$	0	0
$686$	5.93800	0.226714
$687$	5.56987	0.212504
$688$	7.72294	0.294434
$689$	2.08866	0.0795717
$690$	0	0
$691$	−36.1821	−1.37643	−0.688217	−	0.725505i	$-0.741606\pi$
$691$	−36.1821	−1.37643	−0.688217	+	0.725505i	$0.741606\pi$
$692$	15.4459	0.587164
$693$	−10.9671	−0.416605
$694$	−14.5369	−0.551815
$695$	0	0
$696$	0.938003	0.0355549
$697$	−3.29040	−0.124633
$698$	24.6900	0.934531
$699$	−14.3681	−0.543453
$700$	0	0
$701$	−14.4883	−0.547215	−0.273607	−	0.961841i	$-0.588217\pi$
$701$	−14.4883	−0.547215	−0.273607	+	0.961841i	$0.588217\pi$
$702$	0.784934	0.0296254
$703$	−37.6862	−1.42136
$704$	−2.78493	−0.104961
$705$	0	0
$706$	10.5856	0.398395
$707$	−29.3219	−1.10276
$708$	1.56987	0.0589993
$709$	−20.0582	−0.753300	−0.376650	−	0.926356i	$-0.622924\pi$
$709$	−20.0582	−0.753300	−0.376650	+	0.926356i	$0.622924\pi$
$710$	0	0
$711$	2.93800	0.110184
$712$	−11.2441	−0.421392
$713$	−2.00000	−0.0749006
$714$	3.69386	0.138239
$715$	0	0
$716$	−24.1068	−0.900914
$717$	−16.8918	−0.630834
$718$	−10.8918	−0.406477
$719$	34.3376	1.28058	0.640289	−	0.768134i	$-0.278815\pi$
$719$	34.3376	1.28058	0.640289	+	0.768134i	$0.278815\pi$
$720$	0	0
$721$	23.8722	0.889046
$722$	43.0315	1.60147
$723$	3.13974	0.116768
$724$	22.1068	0.821593
$725$	0	0
$726$	3.24414	0.120401
$727$	−41.7215	−1.54736	−0.773682	−	0.633574i	$-0.781587\pi$
$727$	−41.7215	−1.54736	−0.773682	+	0.633574i	$0.781587\pi$
$728$	−3.09107	−0.114563
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.24414	−0.267934
$732$	0.784934	0.0290120
$733$	43.8298	1.61889	0.809444	−	0.587196i	$-0.199768\pi$
$733$	43.8298	1.61889	0.809444	+	0.587196i	$0.199768\pi$
$734$	6.63186	0.244787
$735$	0	0
$736$	2.00000	0.0737210
$737$	−9.94185	−0.366213
$738$	3.50787	0.129127
$739$	38.0119	1.39829	0.699145	−	0.714980i	$-0.253564\pi$
$739$	38.0119	1.39829	0.699145	+	0.714980i	$0.253564\pi$
$740$	0	0
$741$	6.18215	0.227107
$742$	−10.4788	−0.384689
$743$	−19.3219	−0.708851	−0.354426	−	0.935084i	$-0.615324\pi$
$743$	−19.3219	−0.708851	−0.354426	+	0.935084i	$0.615324\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−4.38388	−0.160505
$747$	−0.660941	−0.0241826
$748$	2.61228	0.0955143
$749$	−2.48828	−0.0909200
$750$	0	0
$751$	−26.7983	−0.977883	−0.488941	−	0.872317i	$-0.662617\pi$
$751$	−26.7983	−0.977883	−0.488941	+	0.872317i	$0.662617\pi$
$752$	4.56987	0.166646
$753$	27.3972	0.998410
$754$	0.736271	0.0268134
$755$	0	0
$756$	−3.93800	−0.143224
$757$	15.2732	0.555115	0.277557	−	0.960709i	$-0.410475\pi$
$757$	15.2732	0.555115	0.277557	+	0.960709i	$0.410475\pi$
$758$	−2.30614	−0.0837627
$759$	5.56987	0.202173
$760$	0	0
$761$	−12.1044	−0.438784	−0.219392	−	0.975637i	$-0.570407\pi$
$761$	−12.1044	−0.438784	−0.219392	+	0.975637i	$0.570407\pi$
$762$	−4.50787	−0.163303
$763$	29.0157	1.05044
$764$	−7.36814	−0.266570
$765$	0	0
$766$	1.26373	0.0456604
$767$	1.23224	0.0444937
$768$	−1.00000	−0.0360844
$769$	47.7678	1.72255	0.861275	−	0.508140i	$-0.169667\pi$
$769$	47.7678	1.72255	0.861275	+	0.508140i	$0.169667\pi$
$770$	0	0
$771$	9.38388	0.337952
$772$	2.49213	0.0896937
$773$	−8.69002	−0.312558	−0.156279	−	0.987713i	$-0.549950\pi$
$773$	−8.69002	−0.312558	−0.156279	+	0.987713i	$0.549950\pi$
$774$	7.72294	0.277595
$775$	0	0
$776$	1.36814	0.0491132
$777$	18.8431	0.675992
$778$	−8.55412	−0.306680
$779$	27.6280	0.989877
$780$	0	0
$781$	−12.3815	−0.443044
$782$	−1.87601	−0.0670859
$783$	0.938003	0.0335215
$784$	8.50787	0.303853
$785$	0	0
$786$	20.3839	0.727069
$787$	0.412955	0.0147203	0.00736013	−	0.999973i	$-0.497657\pi$
$787$	0.412955	0.0147203	0.00736013	+	0.999973i	$0.497657\pi$
$788$	−1.41680	−0.0504714
$789$	−1.44588	−0.0514745
$790$	0	0
$791$	23.6280	0.840116
$792$	−2.78493	−0.0989583
$793$	0.616121	0.0218791
$794$	−37.0157	−1.31364
$795$	0	0
$796$	18.8918	0.669600
$797$	6.83360	0.242058	0.121029	−	0.992649i	$-0.461381\pi$
$797$	6.83360	0.242058	0.121029	+	0.992649i	$0.461381\pi$
$798$	−31.0157	−1.09795
$799$	−4.28655	−0.151647
$800$	0	0
$801$	−11.2441	−0.397292
$802$	5.75201	0.203111
$803$	−27.8493	−0.982782
$804$	−3.56987	−0.125900
$805$	0	0
$806$	0.784934	0.0276481
$807$	−4.93800	−0.173826
$808$	−7.44588	−0.261945
$809$	−13.0935	−0.460342	−0.230171	−	0.973150i	$-0.573929\pi$
$809$	−13.0935	−0.460342	−0.230171	+	0.973150i	$0.573929\pi$
$810$	0	0
$811$	33.0739	1.16138	0.580691	−	0.814124i	$-0.302783\pi$
$811$	33.0739	1.16138	0.580691	+	0.814124i	$0.302783\pi$
$812$	−3.69386	−0.129629
$813$	−9.67427	−0.339292
$814$	13.3257	0.467066
$815$	0	0
$816$	0.938003	0.0328367
$817$	60.8259	2.12803
$818$	10.5541	0.369016
$819$	−3.09107	−0.108011
$820$	0	0
$821$	−23.9828	−0.837006	−0.418503	−	0.908215i	$-0.637445\pi$
$821$	−23.9828	−0.837006	−0.418503	+	0.908215i	$0.637445\pi$
$822$	−18.6900	−0.651889
$823$	−20.2480	−0.705800	−0.352900	−	0.935661i	$-0.614804\pi$
$823$	−20.2480	−0.705800	−0.352900	+	0.935661i	$0.614804\pi$
$824$	6.06200	0.211180
$825$	0	0
$826$	−6.18215	−0.215104
$827$	−38.7363	−1.34699	−0.673496	−	0.739191i	$-0.735208\pi$
$827$	−38.7363	−1.34699	−0.673496	+	0.739191i	$0.735208\pi$
$828$	2.00000	0.0695048
$829$	0.935594	0.0324945	0.0162473	−	0.999868i	$-0.494828\pi$
$829$	0.935594	0.0324945	0.0162473	+	0.999868i	$0.494828\pi$
$830$	0	0
$831$	30.1821	1.04701
$832$	−0.784934	−0.0272127
$833$	−7.98041	−0.276505
$834$	9.29281	0.321784
$835$	0	0
$836$	−21.9342	−0.758609
$837$	1.00000	0.0345651
$838$	14.0777	0.486307
$839$	−49.1821	−1.69796	−0.848978	−	0.528428i	$-0.822782\pi$
$839$	−49.1821	−1.69796	−0.848978	+	0.528428i	$0.822782\pi$
$840$	0	0
$841$	−28.1201	−0.969660
$842$	−14.2599	−0.491428
$843$	−21.3839	−0.736500
$844$	15.8760	0.546475
$845$	0	0
$846$	4.56987	0.157115
$847$	−12.7754	−0.438969
$848$	−2.66094	−0.0913771
$849$	−15.3219	−0.525846
$850$	0	0
$851$	−9.56987	−0.328051
$852$	−4.44588	−0.152313
$853$	49.9537	1.71038	0.855192	−	0.518312i	$-0.173439\pi$
$853$	49.9537	1.71038	0.855192	+	0.518312i	$0.173439\pi$
$854$	−3.09107	−0.105774
$855$	0	0
$856$	−0.631865	−0.0215967
$857$	3.25989	0.111356	0.0556778	−	0.998449i	$-0.482268\pi$
$857$	3.25989	0.111356	0.0556778	+	0.998449i	$0.482268\pi$
$858$	−2.18599	−0.0746285
$859$	−12.1846	−0.415732	−0.207866	−	0.978157i	$-0.566652\pi$
$859$	−12.1846	−0.415732	−0.207866	+	0.978157i	$0.566652\pi$
$860$	0	0
$861$	−13.8140	−0.470780
$862$	−37.5660	−1.27950
$863$	24.6438	0.838883	0.419442	−	0.907782i	$-0.362226\pi$
$863$	24.6438	0.838883	0.419442	+	0.907782i	$0.362226\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	28.0582	0.953455
$867$	16.1201	0.547469
$868$	−3.93800	−0.133665
$869$	−8.18215	−0.277560
$870$	0	0
$871$	−2.80211	−0.0949459
$872$	7.36814	0.249517
$873$	1.36814	0.0463044
$874$	15.7520	0.532820
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−2.78252	−0.0939592	−0.0469796	−	0.998896i	$-0.514960\pi$
$877$	−2.78252	−0.0939592	−0.0469796	+	0.998896i	$0.514960\pi$
$878$	−21.6280	−0.729910
$879$	−6.00000	−0.202375
$880$	0	0
$881$	33.7324	1.13647	0.568237	−	0.822865i	$-0.307626\pi$
$881$	33.7324	1.13647	0.568237	+	0.822865i	$0.307626\pi$
$882$	8.50787	0.286475
$883$	2.94041	0.0989528	0.0494764	−	0.998775i	$-0.484245\pi$
$883$	2.94041	0.0989528	0.0494764	+	0.998775i	$0.484245\pi$
$884$	0.736271	0.0247635
$885$	0	0
$886$	36.5660	1.22846
$887$	10.6939	0.359065	0.179532	−	0.983752i	$-0.442542\pi$
$887$	10.6939	0.359065	0.179532	+	0.983752i	$0.442542\pi$
$888$	4.78493	0.160572
$889$	17.7520	0.595383
$890$	0	0
$891$	−2.78493	−0.0932988
$892$	3.06200	0.102523
$893$	35.9923	1.20444
$894$	−11.0157	−0.368422
$895$	0	0
$896$	3.93800	0.131559
$897$	1.56987	0.0524164
$898$	−3.36814	−0.112396
$899$	0.938003	0.0312842
$900$	0	0
$901$	2.49597	0.0831529
$902$	−9.76919	−0.325278
$903$	−30.4130	−1.01208
$904$	6.00000	0.199557
$905$	0	0
$906$	4.38388	0.145645
$907$	−0.860264	−0.0285646	−0.0142823	−	0.999898i	$-0.504546\pi$
$907$	−0.860264	−0.0285646	−0.0142823	+	0.999898i	$0.504546\pi$
$908$	−0.814010	−0.0270139
$909$	−7.44588	−0.246964
$910$	0	0
$911$	−17.3219	−0.573900	−0.286950	−	0.957946i	$-0.592641\pi$
$911$	−17.3219	−0.573900	−0.286950	+	0.957946i	$0.592641\pi$
$912$	−7.87601	−0.260801
$913$	1.84068	0.0609175
$914$	10.1821	0.336795
$915$	0	0
$916$	−5.56987	−0.184034
$917$	−80.2718	−2.65081
$918$	0.938003	0.0309587
$919$	13.5345	0.446463	0.223232	−	0.974765i	$-0.428339\pi$
$919$	13.5345	0.446463	0.223232	+	0.974765i	$0.428339\pi$
$920$	0	0
$921$	−17.1979	−0.566690
$922$	35.1226	1.15670
$923$	−3.48972	−0.114865
$924$	10.9671	0.360790
$925$	0	0
$926$	−8.20173	−0.269526
$927$	6.06200	0.199102
$928$	−0.938003	−0.0307915
$929$	47.6595	1.56366	0.781829	−	0.623493i	$-0.214287\pi$
$929$	47.6595	1.56366	0.781829	+	0.623493i	$0.214287\pi$
$930$	0	0
$931$	67.0081	2.19610
$932$	14.3681	0.470644
$933$	28.0739	0.919098
$934$	6.75586	0.221058
$935$	0	0
$936$	−0.784934	−0.0256564
$937$	−22.6123	−0.738711	−0.369355	−	0.929288i	$-0.620422\pi$
$937$	−22.6123	−0.738711	−0.369355	+	0.929288i	$0.620422\pi$
$938$	14.0582	0.459015
$939$	5.19789	0.169627
$940$	0	0
$941$	12.0777	0.393723	0.196862	−	0.980431i	$-0.436925\pi$
$941$	12.0777	0.393723	0.196862	+	0.980431i	$0.436925\pi$
$942$	10.8140	0.352340
$943$	7.01574	0.228464
$944$	−1.56987	−0.0510949
$945$	0	0
$946$	−21.5079	−0.699281
$947$	−54.5951	−1.77410	−0.887051	−	0.461671i	$-0.847250\pi$
$947$	−54.5951	−1.77410	−0.887051	+	0.461671i	$0.847250\pi$
$948$	−2.93800	−0.0954219
$949$	−7.84934	−0.254800
$950$	0	0
$951$	−24.8918	−0.807170
$952$	−3.69386	−0.119719
$953$	1.01574	0.0329031	0.0164516	−	0.999865i	$-0.494763\pi$
$953$	1.01574	0.0329031	0.0164516	+	0.999865i	$0.494763\pi$
$954$	−2.66094	−0.0861511
$955$	0	0
$956$	16.8918	0.546318
$957$	−2.61228	−0.0844430
$958$	15.5236	0.501545
$959$	73.6014	2.37671
$960$	0	0
$961$	1.00000	0.0322581
$962$	3.75586	0.121094
$963$	−0.631865	−0.0203616
$964$	−3.13974	−0.101124
$965$	0	0
$966$	−7.87601	−0.253406
$967$	−3.87601	−0.124644	−0.0623220	−	0.998056i	$-0.519851\pi$
$967$	−3.87601	−0.124644	−0.0623220	+	0.998056i	$0.519851\pi$
$968$	−3.24414	−0.104271
$969$	7.38772	0.237328
$970$	0	0
$971$	−27.3681	−0.878285	−0.439143	−	0.898417i	$-0.644718\pi$
$971$	−27.3681	−0.878285	−0.439143	+	0.898417i	$0.644718\pi$
$972$	−1.00000	−0.0320750
$973$	−36.5951	−1.17319
$974$	−33.1201	−1.06124
$975$	0	0
$976$	−0.784934	−0.0251251
$977$	−49.1359	−1.57200	−0.785998	−	0.618229i	$-0.787851\pi$
$977$	−49.1359	−1.57200	−0.785998	+	0.618229i	$0.787851\pi$
$978$	−13.4459	−0.429952
$979$	31.3142	1.00081
$980$	0	0
$981$	7.36814	0.235246
$982$	−20.0000	−0.638226
$983$	3.50403	0.111761	0.0558806	−	0.998437i	$-0.482203\pi$
$983$	3.50403	0.111761	0.0558806	+	0.998437i	$0.482203\pi$
$984$	−3.50787	−0.111827
$985$	0	0
$986$	0.879851	0.0280202
$987$	−17.9962	−0.572824
$988$	−6.18215	−0.196680
$989$	15.4459	0.491150
$990$	0	0
$991$	11.6743	0.370846	0.185423	−	0.982659i	$-0.440635\pi$
$991$	11.6743	0.370846	0.185423	+	0.982659i	$0.440635\pi$
$992$	−1.00000	−0.0317500
$993$	−8.66094	−0.274847
$994$	17.5079	0.555316
$995$	0	0
$996$	0.660941	0.0209427
$997$	9.41439	0.298157	0.149078	−	0.988825i	$-0.452369\pi$
$997$	9.41439	0.298157	0.149078	+	0.988825i	$0.452369\pi$
$998$	26.6634	0.844014
$999$	4.78493	0.151389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.co.1.3	yes	3
5.2	odd	4		4650.2.d.bi.3349.6		6
5.3	odd	4		4650.2.d.bi.3349.1		6
5.4	even	2		4650.2.a.cj.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.cj.1.1	✓	3	5.4	even	2
4650.2.a.co.1.3	yes	3	1.1	even	1	trivial
4650.2.d.bi.3349.1		6	5.3	odd	4
4650.2.d.bi.3349.6		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.93800 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.93800 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.78493 q^{11} -1.00000 q^{12} -0.784934 q^{13} +3.93800 q^{14} +1.00000 q^{16} -0.938003 q^{17} +1.00000 q^{18} +7.87601 q^{19} -3.93800 q^{21} -2.78493 q^{22} +2.00000 q^{23} -1.00000 q^{24} -0.784934 q^{26} -1.00000 q^{27} +3.93800 q^{28} -0.938003 q^{29} -1.00000 q^{31} +1.00000 q^{32} +2.78493 q^{33} -0.938003 q^{34} +1.00000 q^{36} -4.78493 q^{37} +7.87601 q^{38} +0.784934 q^{39} +3.50787 q^{41} -3.93800 q^{42} +7.72294 q^{43} -2.78493 q^{44} +2.00000 q^{46} +4.56987 q^{47} -1.00000 q^{48} +8.50787 q^{49} +0.938003 q^{51} -0.784934 q^{52} -2.66094 q^{53} -1.00000 q^{54} +3.93800 q^{56} -7.87601 q^{57} -0.938003 q^{58} -1.56987 q^{59} -0.784934 q^{61} -1.00000 q^{62} +3.93800 q^{63} +1.00000 q^{64} +2.78493 q^{66} +3.56987 q^{67} -0.938003 q^{68} -2.00000 q^{69} +4.44588 q^{71} +1.00000 q^{72} +10.0000 q^{73} -4.78493 q^{74} +7.87601 q^{76} -10.9671 q^{77} +0.784934 q^{78} +2.93800 q^{79} +1.00000 q^{81} +3.50787 q^{82} -0.660941 q^{83} -3.93800 q^{84} +7.72294 q^{86} +0.938003 q^{87} -2.78493 q^{88} -11.2441 q^{89} -3.09107 q^{91} +2.00000 q^{92} +1.00000 q^{93} +4.56987 q^{94} -1.00000 q^{96} +1.36814 q^{97} +8.50787 q^{98} -2.78493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{11} - 3 q^{12} + 3 q^{16} + 9 q^{17} + 3 q^{18} - 6 q^{22} + 6 q^{23} - 3 q^{24} - 3 q^{27} + 9 q^{29} - 3 q^{31} + 3 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{36} - 12 q^{37} - 6 q^{41} + 9 q^{43} - 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{48} + 9 q^{49} - 9 q^{51} + 18 q^{53} - 3 q^{54} + 9 q^{58} - 3 q^{62} + 3 q^{64} + 6 q^{66} + 6 q^{67} + 9 q^{68} - 6 q^{69} - 15 q^{71} + 3 q^{72} + 30 q^{73} - 12 q^{74} + 12 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{82} + 24 q^{83} + 9 q^{86} - 9 q^{87} - 6 q^{88} - 3 q^{89} + 12 q^{91} + 6 q^{92} + 3 q^{93} + 9 q^{94} - 3 q^{96} - 3 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 6 * q^11 - 3 * q^12 + 3 * q^16 + 9 * q^17 + 3 * q^18 - 6 * q^22 + 6 * q^23 - 3 * q^24 - 3 * q^27 + 9 * q^29 - 3 * q^31 + 3 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^36 - 12 * q^37 - 6 * q^41 + 9 * q^43 - 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^48 + 9 * q^49 - 9 * q^51 + 18 * q^53 - 3 * q^54 + 9 * q^58 - 3 * q^62 + 3 * q^64 + 6 * q^66 + 6 * q^67 + 9 * q^68 - 6 * q^69 - 15 * q^71 + 3 * q^72 + 30 * q^73 - 12 * q^74 + 12 * q^77 - 3 * q^79 + 3 * q^81 - 6 * q^82 + 24 * q^83 + 9 * q^86 - 9 * q^87 - 6 * q^88 - 3 * q^89 + 12 * q^91 + 6 * q^92 + 3 * q^93 + 9 * q^94 - 3 * q^96 - 3 * q^97 + 9 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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