LMFDB - Embedded newform 4650.2.a.ck.1.2

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:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
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			1.2
Root			$-2.29240$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.255105 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.255105 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.03730 q^{11} +1.00000 q^{12} -1.70760 q^{13} +0.255105 q^{14} +1.00000 q^{16} +5.83991 q^{17} -1.00000 q^{18} +6.09501 q^{19} -0.255105 q^{21} -2.03730 q^{22} -4.83991 q^{23} -1.00000 q^{24} +1.70760 q^{26} +1.00000 q^{27} -0.255105 q^{28} -2.58480 q^{29} +1.00000 q^{31} -1.00000 q^{32} +2.03730 q^{33} -5.83991 q^{34} +1.00000 q^{36} -6.09501 q^{38} -1.70760 q^{39} +11.1696 q^{41} +0.255105 q^{42} -0.255105 q^{43} +2.03730 q^{44} +4.83991 q^{46} +9.83991 q^{47} +1.00000 q^{48} -6.93492 q^{49} +5.83991 q^{51} -1.70760 q^{52} -1.16009 q^{53} -1.00000 q^{54} +0.255105 q^{56} +6.09501 q^{57} +2.58480 q^{58} -5.09501 q^{59} +0.160092 q^{61} -1.00000 q^{62} -0.255105 q^{63} +1.00000 q^{64} -2.03730 q^{66} -7.38741 q^{67} +5.83991 q^{68} -4.83991 q^{69} -5.64252 q^{71} -1.00000 q^{72} -9.93492 q^{73} +6.09501 q^{76} -0.519725 q^{77} +1.70760 q^{78} +15.1900 q^{79} +1.00000 q^{81} -11.1696 q^{82} -8.42471 q^{83} -0.255105 q^{84} +0.255105 q^{86} -2.58480 q^{87} -2.03730 q^{88} +10.2551 q^{89} +0.435617 q^{91} -4.83991 q^{92} +1.00000 q^{93} -9.83991 q^{94} -1.00000 q^{96} +4.87720 q^{97} +6.93492 q^{98} +2.03730 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9} + q^{11} + 3 q^{12} - 12 q^{13} - q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{18} + q^{19} + q^{21} - q^{22} + q^{23} - 3 q^{24} + 12 q^{26} + 3 q^{27} + q^{28} + 6 q^{29} + 3 q^{31} - 3 q^{32} + q^{33} - 2 q^{34} + 3 q^{36} - q^{38} - 12 q^{39} + 6 q^{41} - q^{42} + q^{43} + q^{44} - q^{46} + 14 q^{47} + 3 q^{48} + 12 q^{49} + 2 q^{51} - 12 q^{52} - 19 q^{53} - 3 q^{54} - q^{56} + q^{57} - 6 q^{58} + 2 q^{59} + 16 q^{61} - 3 q^{62} + q^{63} + 3 q^{64} - q^{66} + 2 q^{67} + 2 q^{68} + q^{69} + 9 q^{71} - 3 q^{72} + 3 q^{73} + q^{76} + 45 q^{77} + 12 q^{78} + 11 q^{79} + 3 q^{81} - 6 q^{82} + 4 q^{83} + q^{84} - q^{86} + 6 q^{87} - q^{88} + 29 q^{89} + 8 q^{91} + q^{92} + 3 q^{93} - 14 q^{94} - 3 q^{96} - 6 q^{97} - 12 q^{98} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9 + q^11 + 3 * q^12 - 12 * q^13 - q^14 + 3 * q^16 + 2 * q^17 - 3 * q^18 + q^19 + q^21 - q^22 + q^23 - 3 * q^24 + 12 * q^26 + 3 * q^27 + q^28 + 6 * q^29 + 3 * q^31 - 3 * q^32 + q^33 - 2 * q^34 + 3 * q^36 - q^38 - 12 * q^39 + 6 * q^41 - q^42 + q^43 + q^44 - q^46 + 14 * q^47 + 3 * q^48 + 12 * q^49 + 2 * q^51 - 12 * q^52 - 19 * q^53 - 3 * q^54 - q^56 + q^57 - 6 * q^58 + 2 * q^59 + 16 * q^61 - 3 * q^62 + q^63 + 3 * q^64 - q^66 + 2 * q^67 + 2 * q^68 + q^69 + 9 * q^71 - 3 * q^72 + 3 * q^73 + q^76 + 45 * q^77 + 12 * q^78 + 11 * q^79 + 3 * q^81 - 6 * q^82 + 4 * q^83 + q^84 - q^86 + 6 * q^87 - q^88 + 29 * q^89 + 8 * q^91 + q^92 + 3 * q^93 - 14 * q^94 - 3 * q^96 - 6 * q^97 - 12 * q^98 + 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
$7$	−0.255105	−0.0964206	−0.0482103	−	0.998837i	$-0.515352\pi$
$7$	−0.255105	−0.0964206	−0.0482103	+	0.998837i	$0.515352\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.03730	0.614268	0.307134	−	0.951666i	$-0.400630\pi$
$11$	2.03730	0.614268	0.307134	+	0.951666i	$0.400630\pi$
$12$	1.00000	0.288675
$13$	−1.70760	−0.473603	−0.236801	−	0.971558i	$-0.576099\pi$
$13$	−1.70760	−0.473603	−0.236801	+	0.971558i	$0.576099\pi$
$14$	0.255105	0.0681797
$15$	0	0
$16$	1.00000	0.250000
$17$	5.83991	1.41639	0.708193	−	0.706019i	$-0.249511\pi$
$17$	5.83991	1.41639	0.708193	+	0.706019i	$0.249511\pi$
$18$	−1.00000	−0.235702
$19$	6.09501	1.39829	0.699146	−	0.714979i	$-0.253564\pi$
$19$	6.09501	1.39829	0.699146	+	0.714979i	$0.253564\pi$
$20$	0	0
$21$	−0.255105	−0.0556685
$22$	−2.03730	−0.434353
$23$	−4.83991	−1.00919	−0.504595	−	0.863356i	$-0.668358\pi$
$23$	−4.83991	−1.00919	−0.504595	+	0.863356i	$0.668358\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.70760	0.334888
$27$	1.00000	0.192450
$28$	−0.255105	−0.0482103
$29$	−2.58480	−0.479986	−0.239993	−	0.970775i	$-0.577145\pi$
$29$	−2.58480	−0.479986	−0.239993	+	0.970775i	$0.577145\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	2.03730	0.354648
$34$	−5.83991	−1.00154
$35$	0	0
$36$	1.00000	0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−6.09501	−0.988742
$39$	−1.70760	−0.273435
$40$	0	0
$41$	11.1696	1.74440	0.872200	−	0.489150i	$-0.162693\pi$
$41$	11.1696	1.74440	0.872200	+	0.489150i	$0.162693\pi$
$42$	0.255105	0.0393636
$43$	−0.255105	−0.0389032	−0.0194516	−	0.999811i	$-0.506192\pi$
$43$	−0.255105	−0.0389032	−0.0194516	+	0.999811i	$0.506192\pi$
$44$	2.03730	0.307134
$45$	0	0
$46$	4.83991	0.713606
$47$	9.83991	1.43530	0.717649	−	0.696405i	$-0.245218\pi$
$47$	9.83991	1.43530	0.717649	+	0.696405i	$0.245218\pi$
$48$	1.00000	0.144338
$49$	−6.93492	−0.990703
$50$	0	0
$51$	5.83991	0.817751
$52$	−1.70760	−0.236801
$53$	−1.16009	−0.159351	−0.0796754	−	0.996821i	$-0.525388\pi$
$53$	−1.16009	−0.159351	−0.0796754	+	0.996821i	$0.525388\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0.255105	0.0340898
$57$	6.09501	0.807304
$58$	2.58480	0.339401
$59$	−5.09501	−0.663314	−0.331657	−	0.943400i	$-0.607608\pi$
$59$	−5.09501	−0.663314	−0.331657	+	0.943400i	$0.607608\pi$
$60$	0	0
$61$	0.160092	0.0204977	0.0102488	−	0.999947i	$-0.496738\pi$
$61$	0.160092	0.0204977	0.0102488	+	0.999947i	$0.496738\pi$
$62$	−1.00000	−0.127000
$63$	−0.255105	−0.0321402
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.03730	−0.250774
$67$	−7.38741	−0.902516	−0.451258	−	0.892393i	$-0.649025\pi$
$67$	−7.38741	−0.902516	−0.451258	+	0.892393i	$0.649025\pi$
$68$	5.83991	0.708193
$69$	−4.83991	−0.582656
$70$	0	0
$71$	−5.64252	−0.669644	−0.334822	−	0.942281i	$-0.608676\pi$
$71$	−5.64252	−0.669644	−0.334822	+	0.942281i	$0.608676\pi$
$72$	−1.00000	−0.117851
$73$	−9.93492	−1.16279	−0.581397	−	0.813620i	$-0.697494\pi$
$73$	−9.93492	−1.16279	−0.581397	+	0.813620i	$0.697494\pi$
$74$	0	0
$75$	0	0
$76$	6.09501	0.699146
$77$	−0.519725	−0.0592281
$78$	1.70760	0.193347
$79$	15.1900	1.70901	0.854506	−	0.519442i	$-0.173860\pi$
$79$	15.1900	1.70901	0.854506	+	0.519442i	$0.173860\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−11.1696	−1.23348
$83$	−8.42471	−0.924732	−0.462366	−	0.886689i	$-0.652999\pi$
$83$	−8.42471	−0.924732	−0.462366	+	0.886689i	$0.652999\pi$
$84$	−0.255105	−0.0278342
$85$	0	0
$86$	0.255105	0.0275087
$87$	−2.58480	−0.277120
$88$	−2.03730	−0.217177
$89$	10.2551	1.08704	0.543519	−	0.839397i	$-0.317091\pi$
$89$	10.2551	1.08704	0.543519	+	0.839397i	$0.317091\pi$
$90$	0	0
$91$	0.435617	0.0456651
$92$	−4.83991	−0.504595
$93$	1.00000	0.103695
$94$	−9.83991	−1.01491
$95$	0	0
$96$	−1.00000	−0.102062
$97$	4.87720	0.495205	0.247603	−	0.968862i	$-0.420357\pi$
$97$	4.87720	0.495205	0.247603	+	0.968862i	$0.420357\pi$
$98$	6.93492	0.700533
$99$	2.03730	0.204756
$100$	0	0
$101$	0.575289	0.0572434	0.0286217	−	0.999590i	$-0.490888\pi$
$101$	0.575289	0.0572434	0.0286217	+	0.999590i	$0.490888\pi$
$102$	−5.83991	−0.578237
$103$	−5.48979	−0.540925	−0.270463	−	0.962730i	$-0.587177\pi$
$103$	−5.48979	−0.540925	−0.270463	+	0.962730i	$0.587177\pi$
$104$	1.70760	0.167444
$105$	0	0
$106$	1.16009	0.112678
$107$	13.3501	1.29060	0.645302	−	0.763927i	$-0.276731\pi$
$107$	13.3501	1.29060	0.645302	+	0.763927i	$0.276731\pi$
$108$	1.00000	0.0962250
$109$	6.26462	0.600042	0.300021	−	0.953933i	$-0.403006\pi$
$109$	6.26462	0.600042	0.300021	+	0.953933i	$0.403006\pi$
$110$	0	0
$111$	0	0
$112$	−0.255105	−0.0241052
$113$	0.255105	0.0239983	0.0119991	−	0.999928i	$-0.496180\pi$
$113$	0.255105	0.0239983	0.0119991	+	0.999928i	$0.496180\pi$
$114$	−6.09501	−0.570850
$115$	0	0
$116$	−2.58480	−0.239993
$117$	−1.70760	−0.157868
$118$	5.09501	0.469034
$119$	−1.48979	−0.136569
$120$	0	0
$121$	−6.84942	−0.622675
$122$	−0.160092	−0.0144940
$123$	11.1696	1.00713
$124$	1.00000	0.0898027
$125$	0	0
$126$	0.255105	0.0227266
$127$	−7.41520	−0.657992	−0.328996	−	0.944331i	$-0.606710\pi$
$127$	−7.41520	−0.657992	−0.328996	+	0.944331i	$0.606710\pi$
$128$	−1.00000	−0.0883883
$129$	−0.255105	−0.0224607
$130$	0	0
$131$	17.2442	1.50663	0.753316	−	0.657658i	$-0.228453\pi$
$131$	17.2442	1.50663	0.753316	+	0.657658i	$0.228453\pi$
$132$	2.03730	0.177324
$133$	−1.55487	−0.134824
$134$	7.38741	0.638175
$135$	0	0
$136$	−5.83991	−0.500768
$137$	16.1900	1.38321	0.691604	−	0.722277i	$-0.256905\pi$
$137$	16.1900	1.38321	0.691604	+	0.722277i	$0.256905\pi$
$138$	4.83991	0.412000
$139$	15.0950	1.28034	0.640171	−	0.768232i	$-0.278863\pi$
$139$	15.0950	1.28034	0.640171	+	0.768232i	$0.278863\pi$
$140$	0	0
$141$	9.83991	0.828670
$142$	5.64252	0.473510
$143$	−3.47888	−0.290919
$144$	1.00000	0.0833333
$145$	0	0
$146$	9.93492	0.822220
$147$	−6.93492	−0.571983
$148$	0	0
$149$	13.4525	1.10207	0.551036	−	0.834482i	$-0.314233\pi$
$149$	13.4525	1.10207	0.551036	+	0.834482i	$0.314233\pi$
$150$	0	0
$151$	0.934921	0.0760828	0.0380414	−	0.999276i	$-0.487888\pi$
$151$	0.934921	0.0760828	0.0380414	+	0.999276i	$0.487888\pi$
$152$	−6.09501	−0.494371
$153$	5.83991	0.472129
$154$	0.519725	0.0418806
$155$	0	0
$156$	−1.70760	−0.136717
$157$	1.35012	0.107751	0.0538756	−	0.998548i	$-0.482843\pi$
$157$	1.35012	0.107751	0.0538756	+	0.998548i	$0.482843\pi$
$158$	−15.1900	−1.20845
$159$	−1.16009	−0.0920013
$160$	0	0
$161$	1.23468	0.0973068
$162$	−1.00000	−0.0785674
$163$	20.4824	1.60431	0.802154	−	0.597117i	$-0.203687\pi$
$163$	20.4824	1.60431	0.802154	+	0.597117i	$0.203687\pi$
$164$	11.1696	0.872200
$165$	0	0
$166$	8.42471	0.653884
$167$	−9.42471	−0.729306	−0.364653	−	0.931143i	$-0.618812\pi$
$167$	−9.42471	−0.729306	−0.364653	+	0.931143i	$0.618812\pi$
$168$	0.255105	0.0196818
$169$	−10.0841	−0.775701
$170$	0	0
$171$	6.09501	0.466097
$172$	−0.255105	−0.0194516
$173$	−12.4247	−0.944633	−0.472317	−	0.881429i	$-0.656582\pi$
$173$	−12.4247	−0.944633	−0.472317	+	0.881429i	$0.656582\pi$
$174$	2.58480	0.195953
$175$	0	0
$176$	2.03730	0.153567
$177$	−5.09501	−0.382965
$178$	−10.2551	−0.768653
$179$	10.6126	0.793222	0.396611	−	0.917987i	$-0.370186\pi$
$179$	10.6126	0.793222	0.396611	+	0.917987i	$0.370186\pi$
$180$	0	0
$181$	22.0841	1.64150	0.820749	−	0.571288i	$-0.193556\pi$
$181$	22.0841	1.64150	0.820749	+	0.571288i	$0.193556\pi$
$182$	−0.435617	−0.0322901
$183$	0.160092	0.0118343
$184$	4.83991	0.356803
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	11.8976	0.870040
$188$	9.83991	0.717649
$189$	−0.255105	−0.0185562
$190$	0	0
$191$	−5.75441	−0.416374	−0.208187	−	0.978089i	$-0.566756\pi$
$191$	−5.75441	−0.416374	−0.208187	+	0.978089i	$0.566756\pi$
$192$	1.00000	0.0721688
$193$	−13.8976	−1.00037	−0.500186	−	0.865918i	$-0.666735\pi$
$193$	−13.8976	−1.00037	−0.500186	+	0.865918i	$0.666735\pi$
$194$	−4.87720	−0.350163
$195$	0	0
$196$	−6.93492	−0.495352
$197$	14.1492	1.00809	0.504044	−	0.863678i	$-0.331845\pi$
$197$	14.1492	1.00809	0.504044	+	0.863678i	$0.331845\pi$
$198$	−2.03730	−0.144784
$199$	−16.8698	−1.19587	−0.597936	−	0.801544i	$-0.704012\pi$
$199$	−16.8698	−1.19587	−0.597936	+	0.801544i	$0.704012\pi$
$200$	0	0
$201$	−7.38741	−0.521068
$202$	−0.575289	−0.0404772
$203$	0.659396	0.0462805
$204$	5.83991	0.408875
$205$	0	0
$206$	5.48979	0.382492
$207$	−4.83991	−0.336397
$208$	−1.70760	−0.118401
$209$	12.4173	0.858926
$210$	0	0
$211$	−9.23468	−0.635742	−0.317871	−	0.948134i	$-0.602968\pi$
$211$	−9.23468	−0.635742	−0.317871	+	0.948134i	$0.602968\pi$
$212$	−1.16009	−0.0796754
$213$	−5.64252	−0.386619
$214$	−13.3501	−0.912595
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−0.255105	−0.0173177
$218$	−6.26462	−0.424294
$219$	−9.93492	−0.671340
$220$	0	0
$221$	−9.97222	−0.670804
$222$	0	0
$223$	−16.9518	−1.13518	−0.567588	−	0.823313i	$-0.692123\pi$
$223$	−16.9518	−1.13518	−0.567588	+	0.823313i	$0.692123\pi$
$224$	0.255105	0.0170449
$225$	0	0
$226$	−0.255105	−0.0169693
$227$	−9.16009	−0.607977	−0.303988	−	0.952676i	$-0.598318\pi$
$227$	−9.16009	−0.607977	−0.303988	+	0.952676i	$0.598318\pi$
$228$	6.09501	0.403652
$229$	−4.60522	−0.304322	−0.152161	−	0.988356i	$-0.548623\pi$
$229$	−4.60522	−0.304322	−0.152161	+	0.988356i	$0.548623\pi$
$230$	0	0
$231$	−0.519725	−0.0341954
$232$	2.58480	0.169701
$233$	12.4451	0.815308	0.407654	−	0.913137i	$-0.366347\pi$
$233$	12.4451	0.815308	0.407654	+	0.913137i	$0.366347\pi$
$234$	1.70760	0.111629
$235$	0	0
$236$	−5.09501	−0.331657
$237$	15.1900	0.986698
$238$	1.48979	0.0965687
$239$	20.7748	1.34381	0.671906	−	0.740636i	$-0.265476\pi$
$239$	20.7748	1.34381	0.671906	+	0.740636i	$0.265476\pi$
$240$	0	0
$241$	4.58480	0.295333	0.147667	−	0.989037i	$-0.452824\pi$
$241$	4.58480	0.295333	0.147667	+	0.989037i	$0.452824\pi$
$242$	6.84942	0.440298
$243$	1.00000	0.0641500
$244$	0.160092	0.0102488
$245$	0	0
$246$	−11.1696	−0.712148
$247$	−10.4078	−0.662235
$248$	−1.00000	−0.0635001
$249$	−8.42471	−0.533894
$250$	0	0
$251$	13.6052	0.858754	0.429377	−	0.903125i	$-0.358733\pi$
$251$	13.6052	0.858754	0.429377	+	0.903125i	$0.358733\pi$
$252$	−0.255105	−0.0160701
$253$	−9.86033	−0.619914
$254$	7.41520	0.465271
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.25511	−0.390183	−0.195091	−	0.980785i	$-0.562500\pi$
$257$	−6.25511	−0.390183	−0.195091	+	0.980785i	$0.562500\pi$
$258$	0.255105	0.0158821
$259$	0	0
$260$	0	0
$261$	−2.58480	−0.159995
$262$	−17.2442	−1.06535
$263$	24.5292	1.51254	0.756269	−	0.654261i	$-0.227020\pi$
$263$	24.5292	1.51254	0.756269	+	0.654261i	$0.227020\pi$
$264$	−2.03730	−0.125387
$265$	0	0
$266$	1.55487	0.0953351
$267$	10.2551	0.627602
$268$	−7.38741	−0.451258
$269$	15.1696	0.924907	0.462454	−	0.886643i	$-0.346969\pi$
$269$	15.1696	0.924907	0.462454	+	0.886643i	$0.346969\pi$
$270$	0	0
$271$	5.51021	0.334721	0.167361	−	0.985896i	$-0.446476\pi$
$271$	5.51021	0.334721	0.167361	+	0.985896i	$0.446476\pi$
$272$	5.83991	0.354096
$273$	0.435617	0.0263647
$274$	−16.1900	−0.978075
$275$	0	0
$276$	−4.83991	−0.291328
$277$	−27.3319	−1.64221	−0.821106	−	0.570776i	$-0.806643\pi$
$277$	−27.3319	−1.64221	−0.821106	+	0.570776i	$0.806643\pi$
$278$	−15.0950	−0.905339
$279$	1.00000	0.0598684
$280$	0	0
$281$	21.8698	1.30465	0.652323	−	0.757941i	$-0.273795\pi$
$281$	21.8698	1.30465	0.652323	+	0.757941i	$0.273795\pi$
$282$	−9.83991	−0.585958
$283$	26.1622	1.55518	0.777592	−	0.628769i	$-0.216441\pi$
$283$	26.1622	1.55518	0.777592	+	0.628769i	$0.216441\pi$
$284$	−5.64252	−0.334822
$285$	0	0
$286$	3.47888	0.205711
$287$	−2.84942	−0.168196
$288$	−1.00000	−0.0589256
$289$	17.1045	1.00615
$290$	0	0
$291$	4.87720	0.285907
$292$	−9.93492	−0.581397
$293$	−13.3596	−0.780478	−0.390239	−	0.920714i	$-0.627608\pi$
$293$	−13.3596	−0.780478	−0.390239	+	0.920714i	$0.627608\pi$
$294$	6.93492	0.404453
$295$	0	0
$296$	0	0
$297$	2.03730	0.118216
$298$	−13.4525	−0.779282
$299$	8.26462	0.477955
$300$	0	0
$301$	0.0650786	0.00375107
$302$	−0.934921	−0.0537987
$303$	0.575289	0.0330495
$304$	6.09501	0.349573
$305$	0	0
$306$	−5.83991	−0.333845
$307$	−18.7748	−1.07154	−0.535768	−	0.844365i	$-0.679978\pi$
$307$	−18.7748	−1.07154	−0.535768	+	0.844365i	$0.679978\pi$
$308$	−0.519725	−0.0296141
$309$	−5.48979	−0.312303
$310$	0	0
$311$	−14.2924	−0.810448	−0.405224	−	0.914217i	$-0.632806\pi$
$311$	−14.2924	−0.810448	−0.405224	+	0.914217i	$0.632806\pi$
$312$	1.70760	0.0966737
$313$	−19.1696	−1.08353	−0.541765	−	0.840530i	$-0.682244\pi$
$313$	−19.1696	−1.08353	−0.541765	+	0.840530i	$0.682244\pi$
$314$	−1.35012	−0.0761916
$315$	0	0
$316$	15.1900	0.854506
$317$	24.4247	1.37183	0.685914	−	0.727682i	$-0.259403\pi$
$317$	24.4247	1.37183	0.685914	+	0.727682i	$0.259403\pi$
$318$	1.16009	0.0650547
$319$	−5.26601	−0.294840
$320$	0	0
$321$	13.3501	0.745131
$322$	−1.23468	−0.0688063
$323$	35.5943	1.98052
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.4824	−1.13442
$327$	6.26462	0.346434
$328$	−11.1696	−0.616738
$329$	−2.51021	−0.138392
$330$	0	0
$331$	−7.24420	−0.398177	−0.199089	−	0.979981i	$-0.563798\pi$
$331$	−7.24420	−0.398177	−0.199089	+	0.979981i	$0.563798\pi$
$332$	−8.42471	−0.462366
$333$	0	0
$334$	9.42471	0.515697
$335$	0	0
$336$	−0.255105	−0.0139171
$337$	−16.5102	−0.899368	−0.449684	−	0.893188i	$-0.648463\pi$
$337$	−16.5102	−0.899368	−0.449684	+	0.893188i	$0.648463\pi$
$338$	10.0841	0.548503
$339$	0.255105	0.0138554
$340$	0	0
$341$	2.03730	0.110326
$342$	−6.09501	−0.329581
$343$	3.55487	0.191945
$344$	0.255105	0.0137543
$345$	0	0
$346$	12.4247	0.667957
$347$	4.27553	0.229522	0.114761	−	0.993393i	$-0.463390\pi$
$347$	4.27553	0.229522	0.114761	+	0.993393i	$0.463390\pi$
$348$	−2.58480	−0.138560
$349$	20.1492	1.07856	0.539281	−	0.842126i	$-0.318696\pi$
$349$	20.1492	1.07856	0.539281	+	0.842126i	$0.318696\pi$
$350$	0	0
$351$	−1.70760	−0.0911449
$352$	−2.03730	−0.108588
$353$	35.1587	1.87131	0.935654	−	0.352918i	$-0.114810\pi$
$353$	35.1587	1.87131	0.935654	+	0.352918i	$0.114810\pi$
$354$	5.09501	0.270797
$355$	0	0
$356$	10.2551	0.543519
$357$	−1.48979	−0.0788480
$358$	−10.6126	−0.560893
$359$	−0.622100	−0.0328332	−0.0164166	−	0.999865i	$-0.505226\pi$
$359$	−0.622100	−0.0328332	−0.0164166	+	0.999865i	$0.505226\pi$
$360$	0	0
$361$	18.1492	0.955220
$362$	−22.0841	−1.16071
$363$	−6.84942	−0.359501
$364$	0.435617	0.0228325
$365$	0	0
$366$	−0.160092	−0.00836813
$367$	28.0877	1.46616	0.733082	−	0.680141i	$-0.238081\pi$
$367$	28.0877	1.46616	0.733082	+	0.680141i	$0.238081\pi$
$368$	−4.83991	−0.252298
$369$	11.1696	0.581466
$370$	0	0
$371$	0.295945	0.0153647
$372$	1.00000	0.0518476
$373$	−17.7449	−0.918796	−0.459398	−	0.888231i	$-0.651935\pi$
$373$	−17.7449	−0.918796	−0.459398	+	0.888231i	$0.651935\pi$
$374$	−11.8976	−0.615212
$375$	0	0
$376$	−9.83991	−0.507455
$377$	4.41381	0.227323
$378$	0.255105	0.0131212
$379$	−25.7748	−1.32396	−0.661982	−	0.749520i	$-0.730284\pi$
$379$	−25.7748	−1.32396	−0.661982	+	0.749520i	$0.730284\pi$
$380$	0	0
$381$	−7.41520	−0.379892
$382$	5.75441	0.294421
$383$	−19.7952	−1.01149	−0.505745	−	0.862683i	$-0.668782\pi$
$383$	−19.7952	−1.01149	−0.505745	+	0.862683i	$0.668782\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	13.8976	0.707370
$387$	−0.255105	−0.0129677
$388$	4.87720	0.247603
$389$	−23.9444	−1.21403	−0.607016	−	0.794690i	$-0.707633\pi$
$389$	−23.9444	−1.21403	−0.607016	+	0.794690i	$0.707633\pi$
$390$	0	0
$391$	−28.2646	−1.42940
$392$	6.93492	0.350266
$393$	17.2442	0.869855
$394$	−14.1492	−0.712826
$395$	0	0
$396$	2.03730	0.102378
$397$	−10.3297	−0.518433	−0.259216	−	0.965819i	$-0.583464\pi$
$397$	−10.3297	−0.518433	−0.259216	+	0.965819i	$0.583464\pi$
$398$	16.8698	0.845609
$399$	−1.55487	−0.0778408
$400$	0	0
$401$	0.398320	0.0198912	0.00994559	−	0.999951i	$-0.496834\pi$
$401$	0.398320	0.0198912	0.00994559	+	0.999951i	$0.496834\pi$
$402$	7.38741	0.368451
$403$	−1.70760	−0.0850615
$404$	0.575289	0.0286217
$405$	0	0
$406$	−0.659396	−0.0327253
$407$	0	0
$408$	−5.83991	−0.289119
$409$	2.77483	0.137206	0.0686032	−	0.997644i	$-0.478146\pi$
$409$	2.77483	0.137206	0.0686032	+	0.997644i	$0.478146\pi$
$410$	0	0
$411$	16.1900	0.798595
$412$	−5.48979	−0.270463
$413$	1.29976	0.0639572
$414$	4.83991	0.237869
$415$	0	0
$416$	1.70760	0.0837219
$417$	15.0950	0.739206
$418$	−12.4173	−0.607352
$419$	−35.1140	−1.71543	−0.857717	−	0.514123i	$-0.828118\pi$
$419$	−35.1140	−1.71543	−0.857717	+	0.514123i	$0.828118\pi$
$420$	0	0
$421$	12.3392	0.601376	0.300688	−	0.953722i	$-0.402784\pi$
$421$	12.3392	0.601376	0.300688	+	0.953722i	$0.402784\pi$
$422$	9.23468	0.449537
$423$	9.83991	0.478433
$424$	1.16009	0.0563390
$425$	0	0
$426$	5.64252	0.273381
$427$	−0.0408402	−0.00197640
$428$	13.3501	0.645302
$429$	−3.47888	−0.167962
$430$	0	0
$431$	11.9444	0.575343	0.287672	−	0.957729i	$-0.407119\pi$
$431$	11.9444	0.575343	0.287672	+	0.957729i	$0.407119\pi$
$432$	1.00000	0.0481125
$433$	−10.7653	−0.517348	−0.258674	−	0.965965i	$-0.583286\pi$
$433$	−10.7653	−0.517348	−0.258674	+	0.965965i	$0.583286\pi$
$434$	0.255105	0.0122454
$435$	0	0
$436$	6.26462	0.300021
$437$	−29.4993	−1.41114
$438$	9.93492	0.474709
$439$	−23.2442	−1.10939	−0.554693	−	0.832055i	$-0.687164\pi$
$439$	−23.2442	−1.10939	−0.554693	+	0.832055i	$0.687164\pi$
$440$	0	0
$441$	−6.93492	−0.330234
$442$	9.97222	0.474330
$443$	34.4451	1.63654	0.818269	−	0.574836i	$-0.194934\pi$
$443$	34.4451	1.63654	0.818269	+	0.574836i	$0.194934\pi$
$444$	0	0
$445$	0	0
$446$	16.9518	0.802691
$447$	13.4525	0.636281
$448$	−0.255105	−0.0120526
$449$	24.5570	1.15892	0.579459	−	0.815002i	$-0.303264\pi$
$449$	24.5570	1.15892	0.579459	+	0.815002i	$0.303264\pi$
$450$	0	0
$451$	22.7558	1.07153
$452$	0.255105	0.0119991
$453$	0.934921	0.0439264
$454$	9.16009	0.429904
$455$	0	0
$456$	−6.09501	−0.285425
$457$	30.1154	1.40874	0.704370	−	0.709833i	$-0.251229\pi$
$457$	30.1154	1.40874	0.704370	+	0.709833i	$0.251229\pi$
$458$	4.60522	0.215188
$459$	5.83991	0.272584
$460$	0	0
$461$	−22.7748	−1.06073	−0.530365	−	0.847770i	$-0.677945\pi$
$461$	−22.7748	−1.06073	−0.530365	+	0.847770i	$0.677945\pi$
$462$	0.519725	0.0241798
$463$	29.0264	1.34897	0.674485	−	0.738288i	$-0.264366\pi$
$463$	29.0264	1.34897	0.674485	+	0.738288i	$0.264366\pi$
$464$	−2.58480	−0.119996
$465$	0	0
$466$	−12.4451	−0.576510
$467$	−17.3188	−0.801418	−0.400709	−	0.916205i	$-0.631236\pi$
$467$	−17.3188	−0.801418	−0.400709	+	0.916205i	$0.631236\pi$
$468$	−1.70760	−0.0789338
$469$	1.88457	0.0870212
$470$	0	0
$471$	1.35012	0.0622102
$472$	5.09501	0.234517
$473$	−0.519725	−0.0238970
$474$	−15.1900	−0.697701
$475$	0	0
$476$	−1.48979	−0.0682844
$477$	−1.16009	−0.0531170
$478$	−20.7748	−0.950219
$479$	−36.6221	−1.67331	−0.836653	−	0.547733i	$-0.815491\pi$
$479$	−36.6221	−1.67331	−0.836653	+	0.547733i	$0.815491\pi$
$480$	0	0
$481$	0	0
$482$	−4.58480	−0.208832
$483$	1.23468	0.0561801
$484$	−6.84942	−0.311337
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	34.9371	1.58315	0.791575	−	0.611072i	$-0.209261\pi$
$487$	34.9371	1.58315	0.791575	+	0.611072i	$0.209261\pi$
$488$	−0.160092	−0.00724702
$489$	20.4824	0.926247
$490$	0	0
$491$	24.0095	1.08353	0.541767	−	0.840529i	$-0.317755\pi$
$491$	24.0095	1.08353	0.541767	+	0.840529i	$0.317755\pi$
$492$	11.1696	0.503565
$493$	−15.0950	−0.679845
$494$	10.4078	0.468271
$495$	0	0
$496$	1.00000	0.0449013
$497$	1.43944	0.0645675
$498$	8.42471	0.377520
$499$	4.65940	0.208583	0.104292	−	0.994547i	$-0.466742\pi$
$499$	4.65940	0.208583	0.104292	+	0.994547i	$0.466742\pi$
$500$	0	0
$501$	−9.42471	−0.421065
$502$	−13.6052	−0.607231
$503$	4.30928	0.192141	0.0960706	−	0.995375i	$-0.469373\pi$
$503$	4.30928	0.192141	0.0960706	+	0.995375i	$0.469373\pi$
$504$	0.255105	0.0113633
$505$	0	0
$506$	9.86033	0.438345
$507$	−10.0841	−0.447851
$508$	−7.41520	−0.328996
$509$	−33.1696	−1.47022	−0.735108	−	0.677950i	$-0.762869\pi$
$509$	−33.1696	−1.47022	−0.735108	+	0.677950i	$0.762869\pi$
$510$	0	0
$511$	2.53445	0.112117
$512$	−1.00000	−0.0441942
$513$	6.09501	0.269101
$514$	6.25511	0.275901
$515$	0	0
$516$	−0.255105	−0.0112304
$517$	20.0468	0.881658
$518$	0	0
$519$	−12.4247	−0.545384
$520$	0	0
$521$	−20.4138	−0.894345	−0.447173	−	0.894448i	$-0.647569\pi$
$521$	−20.4138	−0.894345	−0.447173	+	0.894448i	$0.647569\pi$
$522$	2.58480	0.113134
$523$	18.0841	0.790763	0.395381	−	0.918517i	$-0.370612\pi$
$523$	18.0841	0.790763	0.395381	+	0.918517i	$0.370612\pi$
$524$	17.2442	0.753316
$525$	0	0
$526$	−24.5292	−1.06953
$527$	5.83991	0.254390
$528$	2.03730	0.0886620
$529$	0.424711	0.0184657
$530$	0	0
$531$	−5.09501	−0.221105
$532$	−1.55487	−0.0674121
$533$	−19.0732	−0.826152
$534$	−10.2551	−0.443782
$535$	0	0
$536$	7.38741	0.319088
$537$	10.6126	0.457967
$538$	−15.1696	−0.654008
$539$	−14.1285	−0.608557
$540$	0	0
$541$	10.4546	0.449480	0.224740	−	0.974419i	$-0.427847\pi$
$541$	10.4546	0.449480	0.224740	+	0.974419i	$0.427847\pi$
$542$	−5.51021	−0.236684
$543$	22.0841	0.947720
$544$	−5.83991	−0.250384
$545$	0	0
$546$	−0.435617	−0.0186427
$547$	−11.9444	−0.510707	−0.255354	−	0.966848i	$-0.582192\pi$
$547$	−11.9444	−0.510707	−0.255354	+	0.966848i	$0.582192\pi$
$548$	16.1900	0.691604
$549$	0.160092	0.00683255
$550$	0	0
$551$	−15.7544	−0.671160
$552$	4.83991	0.206000
$553$	−3.87505	−0.164784
$554$	27.3319	1.16122
$555$	0	0
$556$	15.0950	0.640171
$557$	−20.0841	−0.850991	−0.425495	−	0.904961i	$-0.639900\pi$
$557$	−20.0841	−0.850991	−0.425495	+	0.904961i	$0.639900\pi$
$558$	−1.00000	−0.0423334
$559$	0.435617	0.0184246
$560$	0	0
$561$	11.8976	0.502318
$562$	−21.8698	−0.922524
$563$	28.9050	1.21820	0.609100	−	0.793093i	$-0.291531\pi$
$563$	28.9050	1.21820	0.609100	+	0.793093i	$0.291531\pi$
$564$	9.83991	0.414335
$565$	0	0
$566$	−26.1622	−1.09968
$567$	−0.255105	−0.0107134
$568$	5.64252	0.236755
$569$	−0.724475	−0.0303716	−0.0151858	−	0.999885i	$-0.504834\pi$
$569$	−0.724475	−0.0303716	−0.0151858	+	0.999885i	$0.504834\pi$
$570$	0	0
$571$	12.6594	0.529779	0.264890	−	0.964279i	$-0.414664\pi$
$571$	12.6594	0.529779	0.264890	+	0.964279i	$0.414664\pi$
$572$	−3.47888	−0.145459
$573$	−5.75441	−0.240394
$574$	2.84942	0.118933
$575$	0	0
$576$	1.00000	0.0416667
$577$	−31.0117	−1.29103	−0.645516	−	0.763746i	$-0.723358\pi$
$577$	−31.0117	−1.29103	−0.645516	+	0.763746i	$0.723358\pi$
$578$	−17.1045	−0.711455
$579$	−13.8976	−0.577566
$580$	0	0
$581$	2.14919	0.0891633
$582$	−4.87720	−0.202167
$583$	−2.36345	−0.0978841
$584$	9.93492	0.411110
$585$	0	0
$586$	13.3596	0.551881
$587$	13.2959	0.548782	0.274391	−	0.961618i	$-0.411524\pi$
$587$	13.2959	0.548782	0.274391	+	0.961618i	$0.411524\pi$
$588$	−6.93492	−0.285991
$589$	6.09501	0.251141
$590$	0	0
$591$	14.1492	0.582020
$592$	0	0
$593$	−3.05417	−0.125420	−0.0627099	−	0.998032i	$-0.519974\pi$
$593$	−3.05417	−0.125420	−0.0627099	+	0.998032i	$0.519974\pi$
$594$	−2.03730	−0.0835913
$595$	0	0
$596$	13.4525	0.551036
$597$	−16.8698	−0.690437
$598$	−8.26462	−0.337965
$599$	−0.983124	−0.0401693	−0.0200847	−	0.999798i	$-0.506394\pi$
$599$	−0.983124	−0.0401693	−0.0200847	+	0.999798i	$0.506394\pi$
$600$	0	0
$601$	43.8290	1.78782	0.893911	−	0.448244i	$-0.147950\pi$
$601$	43.8290	1.78782	0.893911	+	0.448244i	$0.147950\pi$
$602$	−0.0650786	−0.00265240
$603$	−7.38741	−0.300839
$604$	0.934921	0.0380414
$605$	0	0
$606$	−0.575289	−0.0233695
$607$	4.91450	0.199473	0.0997367	−	0.995014i	$-0.468200\pi$
$607$	4.91450	0.199473	0.0997367	+	0.995014i	$0.468200\pi$
$608$	−6.09501	−0.247185
$609$	0.659396	0.0267201
$610$	0	0
$611$	−16.8026	−0.679761
$612$	5.83991	0.236064
$613$	−28.6316	−1.15642	−0.578210	−	0.815888i	$-0.696249\pi$
$613$	−28.6316	−1.15642	−0.578210	+	0.815888i	$0.696249\pi$
$614$	18.7748	0.757690
$615$	0	0
$616$	0.519725	0.0209403
$617$	22.5943	0.909613	0.454806	−	0.890590i	$-0.349708\pi$
$617$	22.5943	0.909613	0.454806	+	0.890590i	$0.349708\pi$
$618$	5.48979	0.220832
$619$	21.0204	0.844882	0.422441	−	0.906390i	$-0.361173\pi$
$619$	21.0204	0.844882	0.422441	+	0.906390i	$0.361173\pi$
$620$	0	0
$621$	−4.83991	−0.194219
$622$	14.2924	0.573073
$623$	−2.61613	−0.104813
$624$	−1.70760	−0.0683586
$625$	0	0
$626$	19.1696	0.766172
$627$	12.4173	0.495901
$628$	1.35012	0.0538756
$629$	0	0
$630$	0	0
$631$	−46.6352	−1.85652	−0.928258	−	0.371937i	$-0.878694\pi$
$631$	−46.6352	−1.85652	−0.928258	+	0.371937i	$0.878694\pi$
$632$	−15.1900	−0.604227
$633$	−9.23468	−0.367046
$634$	−24.4247	−0.970029
$635$	0	0
$636$	−1.16009	−0.0460006
$637$	11.8421	0.469200
$638$	5.26601	0.208483
$639$	−5.64252	−0.223215
$640$	0	0
$641$	36.5015	1.44172	0.720860	−	0.693080i	$-0.243747\pi$
$641$	36.5015	1.44172	0.720860	+	0.693080i	$0.243747\pi$
$642$	−13.3501	−0.526887
$643$	4.45986	0.175880	0.0879398	−	0.996126i	$-0.471972\pi$
$643$	4.45986	0.175880	0.0879398	+	0.996126i	$0.471972\pi$
$644$	1.23468	0.0486534
$645$	0	0
$646$	−35.5943	−1.40044
$647$	10.4451	0.410640	0.205320	−	0.978695i	$-0.434176\pi$
$647$	10.4451	0.410640	0.205320	+	0.978695i	$0.434176\pi$
$648$	−1.00000	−0.0392837
$649$	−10.3801	−0.407453
$650$	0	0
$651$	−0.255105	−0.00999835
$652$	20.4824	0.802154
$653$	26.4437	1.03482	0.517412	−	0.855737i	$-0.326896\pi$
$653$	26.4437	1.03482	0.517412	+	0.855737i	$0.326896\pi$
$654$	−6.26462	−0.244966
$655$	0	0
$656$	11.1696	0.436100
$657$	−9.93492	−0.387598
$658$	2.51021	0.0978582
$659$	−17.2104	−0.670424	−0.335212	−	0.942143i	$-0.608808\pi$
$659$	−17.2104	−0.670424	−0.335212	+	0.942143i	$0.608808\pi$
$660$	0	0
$661$	−28.1154	−1.09356	−0.546782	−	0.837275i	$-0.684147\pi$
$661$	−28.1154	−1.09356	−0.546782	+	0.837275i	$0.684147\pi$
$662$	7.24420	0.281554
$663$	−9.97222	−0.387289
$664$	8.42471	0.326942
$665$	0	0
$666$	0	0
$667$	12.5102	0.484397
$668$	−9.42471	−0.364653
$669$	−16.9518	−0.655394
$670$	0	0
$671$	0.326154	0.0125911
$672$	0.255105	0.00984089
$673$	−9.56438	−0.368680	−0.184340	−	0.982863i	$-0.559015\pi$
$673$	−9.56438	−0.368680	−0.184340	+	0.982863i	$0.559015\pi$
$674$	16.5102	0.635950
$675$	0	0
$676$	−10.0841	−0.387850
$677$	−16.0841	−0.618162	−0.309081	−	0.951036i	$-0.600022\pi$
$677$	−16.0841	−0.618162	−0.309081	+	0.951036i	$0.600022\pi$
$678$	−0.255105	−0.00979725
$679$	−1.24420	−0.0477480
$680$	0	0
$681$	−9.16009	−0.351015
$682$	−2.03730	−0.0780121
$683$	31.4247	1.20243	0.601217	−	0.799086i	$-0.294683\pi$
$683$	31.4247	1.20243	0.601217	+	0.799086i	$0.294683\pi$
$684$	6.09501	0.233049
$685$	0	0
$686$	−3.55487	−0.135726
$687$	−4.60522	−0.175700
$688$	−0.255105	−0.00972579
$689$	1.98097	0.0754690
$690$	0	0
$691$	−15.4546	−0.587922	−0.293961	−	0.955817i	$-0.594974\pi$
$691$	−15.4546	−0.587922	−0.293961	+	0.955817i	$0.594974\pi$
$692$	−12.4247	−0.472317
$693$	−0.519725	−0.0197427
$694$	−4.27553	−0.162297
$695$	0	0
$696$	2.58480	0.0979767
$697$	65.2295	2.47074
$698$	−20.1492	−0.762658
$699$	12.4451	0.470718
$700$	0	0
$701$	−39.2069	−1.48082	−0.740412	−	0.672153i	$-0.765370\pi$
$701$	−39.2069	−1.48082	−0.740412	+	0.672153i	$0.765370\pi$
$702$	1.70760	0.0644491
$703$	0	0
$704$	2.03730	0.0767835
$705$	0	0
$706$	−35.1587	−1.32322
$707$	−0.146759	−0.00551944
$708$	−5.09501	−0.191482
$709$	14.0950	0.529349	0.264675	−	0.964338i	$-0.414735\pi$
$709$	14.0950	0.529349	0.264675	+	0.964338i	$0.414735\pi$
$710$	0	0
$711$	15.1900	0.569670
$712$	−10.2551	−0.384326
$713$	−4.83991	−0.181256
$714$	1.48979	0.0557540
$715$	0	0
$716$	10.6126	0.396611
$717$	20.7748	0.775850
$718$	0.622100	0.0232166
$719$	19.6052	0.731151	0.365576	−	0.930782i	$-0.380872\pi$
$719$	19.6052	0.731151	0.365576	+	0.930782i	$0.380872\pi$
$720$	0	0
$721$	1.40047	0.0521563
$722$	−18.1492	−0.675443
$723$	4.58480	0.170511
$724$	22.0841	0.820749
$725$	0	0
$726$	6.84942	0.254206
$727$	2.89547	0.107387	0.0536936	−	0.998557i	$-0.482901\pi$
$727$	2.89547	0.107387	0.0536936	+	0.998557i	$0.482901\pi$
$728$	−0.435617	−0.0161450
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.48979	−0.0551019
$732$	0.160092	0.00591716
$733$	10.2646	0.379132	0.189566	−	0.981868i	$-0.439292\pi$
$733$	10.2646	0.379132	0.189566	+	0.981868i	$0.439292\pi$
$734$	−28.0877	−1.03673
$735$	0	0
$736$	4.83991	0.178401
$737$	−15.0504	−0.554387
$738$	−11.1696	−0.411159
$739$	13.9444	0.512954	0.256477	−	0.966550i	$-0.417438\pi$
$739$	13.9444	0.512954	0.256477	+	0.966550i	$0.417438\pi$
$740$	0	0
$741$	−10.4078	−0.382341
$742$	−0.295945	−0.0108645
$743$	27.1454	0.995867	0.497933	−	0.867215i	$-0.334092\pi$
$743$	27.1454	0.995867	0.497933	+	0.867215i	$0.334092\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	17.7449	0.649687
$747$	−8.42471	−0.308244
$748$	11.8976	0.435020
$749$	−3.40568	−0.124441
$750$	0	0
$751$	7.16961	0.261623	0.130811	−	0.991407i	$-0.458242\pi$
$751$	7.16961	0.261623	0.130811	+	0.991407i	$0.458242\pi$
$752$	9.83991	0.358825
$753$	13.6052	0.495802
$754$	−4.41381	−0.160741
$755$	0	0
$756$	−0.255105	−0.00927808
$757$	39.2104	1.42513	0.712564	−	0.701607i	$-0.247534\pi$
$757$	39.2104	1.42513	0.712564	+	0.701607i	$0.247534\pi$
$758$	25.7748	0.936184
$759$	−9.86033	−0.357907
$760$	0	0
$761$	−38.7566	−1.40492	−0.702462	−	0.711721i	$-0.747916\pi$
$761$	−38.7566	−1.40492	−0.702462	+	0.711721i	$0.747916\pi$
$762$	7.41520	0.268624
$763$	−1.59814	−0.0578564
$764$	−5.75441	−0.208187
$765$	0	0
$766$	19.7952	0.715231
$767$	8.70024	0.314147
$768$	1.00000	0.0360844
$769$	15.7449	0.567775	0.283888	−	0.958858i	$-0.408376\pi$
$769$	15.7449	0.567775	0.283888	+	0.958858i	$0.408376\pi$
$770$	0	0
$771$	−6.25511	−0.225272
$772$	−13.8976	−0.500186
$773$	21.9758	0.790413	0.395207	−	0.918592i	$-0.370673\pi$
$773$	21.9758	0.790413	0.395207	+	0.918592i	$0.370673\pi$
$774$	0.255105	0.00916956
$775$	0	0
$776$	−4.87720	−0.175081
$777$	0	0
$778$	23.9444	0.858450
$779$	68.0789	2.43918
$780$	0	0
$781$	−11.4955	−0.411341
$782$	28.2646	1.01074
$783$	−2.58480	−0.0923733
$784$	−6.93492	−0.247676
$785$	0	0
$786$	−17.2442	−0.615080
$787$	−6.00951	−0.214216	−0.107108	−	0.994247i	$-0.534159\pi$
$787$	−6.00951	−0.214216	−0.107108	+	0.994247i	$0.534159\pi$
$788$	14.1492	0.504044
$789$	24.5292	0.873264
$790$	0	0
$791$	−0.0650786	−0.00231393
$792$	−2.03730	−0.0723922
$793$	−0.273373	−0.00970774
$794$	10.3297	0.366587
$795$	0	0
$796$	−16.8698	−0.597936
$797$	6.58480	0.233246	0.116623	−	0.993176i	$-0.462793\pi$
$797$	6.58480	0.233246	0.116623	+	0.993176i	$0.462793\pi$
$798$	1.55487	0.0550417
$799$	57.4642	2.03294
$800$	0	0
$801$	10.2551	0.362346
$802$	−0.398320	−0.0140652
$803$	−20.2404	−0.714268
$804$	−7.38741	−0.260534
$805$	0	0
$806$	1.70760	0.0601476
$807$	15.1696	0.533995
$808$	−0.575289	−0.0202386
$809$	−42.2273	−1.48463	−0.742317	−	0.670049i	$-0.766273\pi$
$809$	−42.2273	−1.48463	−0.742317	+	0.670049i	$0.766273\pi$
$810$	0	0
$811$	3.29455	0.115687	0.0578437	−	0.998326i	$-0.481577\pi$
$811$	3.29455	0.115687	0.0578437	+	0.998326i	$0.481577\pi$
$812$	0.659396	0.0231403
$813$	5.51021	0.193252
$814$	0	0
$815$	0	0
$816$	5.83991	0.204438
$817$	−1.55487	−0.0543980
$818$	−2.77483	−0.0970196
$819$	0.435617	0.0152217
$820$	0	0
$821$	6.52924	0.227872	0.113936	−	0.993488i	$-0.463654\pi$
$821$	6.52924	0.227872	0.113936	+	0.993488i	$0.463654\pi$
$822$	−16.1900	−0.564692
$823$	−44.6725	−1.55718	−0.778592	−	0.627531i	$-0.784066\pi$
$823$	−44.6725	−1.55718	−0.778592	+	0.627531i	$0.784066\pi$
$824$	5.48979	0.191246
$825$	0	0
$826$	−1.29976	−0.0452246
$827$	23.2551	0.808659	0.404330	−	0.914613i	$-0.367505\pi$
$827$	23.2551	0.808659	0.404330	+	0.914613i	$0.367505\pi$
$828$	−4.83991	−0.168198
$829$	−44.6352	−1.55024	−0.775122	−	0.631812i	$-0.782311\pi$
$829$	−44.6352	−1.55024	−0.775122	+	0.631812i	$0.782311\pi$
$830$	0	0
$831$	−27.3319	−0.948131
$832$	−1.70760	−0.0592003
$833$	−40.4993	−1.40322
$834$	−15.0950	−0.522698
$835$	0	0
$836$	12.4173	0.429463
$837$	1.00000	0.0345651
$838$	35.1140	1.21299
$839$	40.1587	1.38643	0.693216	−	0.720730i	$-0.256193\pi$
$839$	40.1587	1.38643	0.693216	+	0.720730i	$0.256193\pi$
$840$	0	0
$841$	−22.3188	−0.769614
$842$	−12.3392	−0.425237
$843$	21.8698	0.753237
$844$	−9.23468	−0.317871
$845$	0	0
$846$	−9.83991	−0.338303
$847$	1.74732	0.0600387
$848$	−1.16009	−0.0398377
$849$	26.1622	0.897886
$850$	0	0
$851$	0	0
$852$	−5.64252	−0.193310
$853$	−10.8589	−0.371803	−0.185901	−	0.982568i	$-0.559521\pi$
$853$	−10.8589	−0.371803	−0.185901	+	0.982568i	$0.559521\pi$
$854$	0.0408402	0.00139752
$855$	0	0
$856$	−13.3501	−0.456298
$857$	36.0190	1.23039	0.615193	−	0.788376i	$-0.289078\pi$
$857$	36.0190	1.23039	0.615193	+	0.788376i	$0.289078\pi$
$858$	3.47888	0.118767
$859$	−35.9782	−1.22756	−0.613780	−	0.789477i	$-0.710352\pi$
$859$	−35.9782	−1.22756	−0.613780	+	0.789477i	$0.710352\pi$
$860$	0	0
$861$	−2.84942	−0.0971081
$862$	−11.9444	−0.406829
$863$	−48.5388	−1.65228	−0.826139	−	0.563466i	$-0.809468\pi$
$863$	−48.5388	−1.65228	−0.826139	+	0.563466i	$0.809468\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	10.7653	0.365820
$867$	17.1045	0.580900
$868$	−0.255105	−0.00865883
$869$	30.9466	1.04979
$870$	0	0
$871$	12.6147	0.427434
$872$	−6.26462	−0.212147
$873$	4.87720	0.165068
$874$	29.4993	0.997829
$875$	0	0
$876$	−9.93492	−0.335670
$877$	−56.7748	−1.91715	−0.958575	−	0.284841i	$-0.908059\pi$
$877$	−56.7748	−1.91715	−0.958575	+	0.284841i	$0.908059\pi$
$878$	23.2442	0.784454
$879$	−13.3596	−0.450609
$880$	0	0
$881$	−1.34657	−0.0453673	−0.0226836	−	0.999743i	$-0.507221\pi$
$881$	−1.34657	−0.0453673	−0.0226836	+	0.999743i	$0.507221\pi$
$882$	6.93492	0.233511
$883$	36.3705	1.22397	0.611983	−	0.790871i	$-0.290372\pi$
$883$	36.3705	1.22397	0.611983	+	0.790871i	$0.290372\pi$
$884$	−9.97222	−0.335402
$885$	0	0
$886$	−34.4451	−1.15721
$887$	4.38005	0.147068	0.0735339	−	0.997293i	$-0.476572\pi$
$887$	4.38005	0.147068	0.0735339	+	0.997293i	$0.476572\pi$
$888$	0	0
$889$	1.89165	0.0634440
$890$	0	0
$891$	2.03730	0.0682520
$892$	−16.9518	−0.567588
$893$	59.9744	2.00697
$894$	−13.4525	−0.449919
$895$	0	0
$896$	0.255105	0.00852246
$897$	8.26462	0.275948
$898$	−24.5570	−0.819478
$899$	−2.58480	−0.0862080
$900$	0	0
$901$	−6.77483	−0.225702
$902$	−22.7558	−0.757685
$903$	0.0650786	0.00216568
$904$	−0.255105	−0.00848467
$905$	0	0
$906$	−0.934921	−0.0310607
$907$	6.29240	0.208936	0.104468	−	0.994528i	$-0.466686\pi$
$907$	6.29240	0.208936	0.104468	+	0.994528i	$0.466686\pi$
$908$	−9.16009	−0.303988
$909$	0.575289	0.0190811
$910$	0	0
$911$	−32.1682	−1.06578	−0.532890	−	0.846184i	$-0.678894\pi$
$911$	−32.1682	−1.06578	−0.532890	+	0.846184i	$0.678894\pi$
$912$	6.09501	0.201826
$913$	−17.1636	−0.568033
$914$	−30.1154	−0.996130
$915$	0	0
$916$	−4.60522	−0.152161
$917$	−4.39908	−0.145270
$918$	−5.83991	−0.192746
$919$	−32.1154	−1.05939	−0.529695	−	0.848188i	$-0.677694\pi$
$919$	−32.1154	−1.05939	−0.529695	+	0.848188i	$0.677694\pi$
$920$	0	0
$921$	−18.7748	−0.618652
$922$	22.7748	0.750049
$923$	9.63516	0.317145
$924$	−0.519725	−0.0170977
$925$	0	0
$926$	−29.0264	−0.953866
$927$	−5.48979	−0.180308
$928$	2.58480	0.0848503
$929$	−59.9540	−1.96703	−0.983513	−	0.180839i	$-0.942119\pi$
$929$	−59.9540	−1.96703	−0.983513	+	0.180839i	$0.942119\pi$
$930$	0	0
$931$	−42.2684	−1.38529
$932$	12.4451	0.407654
$933$	−14.2924	−0.467912
$934$	17.3188	0.566688
$935$	0	0
$936$	1.70760	0.0558146
$937$	−52.7661	−1.72379	−0.861896	−	0.507085i	$-0.830723\pi$
$937$	−52.7661	−1.72379	−0.861896	+	0.507085i	$0.830723\pi$
$938$	−1.88457	−0.0615333
$939$	−19.1696	−0.625576
$940$	0	0
$941$	−54.3055	−1.77031	−0.885154	−	0.465299i	$-0.845947\pi$
$941$	−54.3055	−1.77031	−0.885154	+	0.465299i	$0.845947\pi$
$942$	−1.35012	−0.0439892
$943$	−54.0599	−1.76043
$944$	−5.09501	−0.165829
$945$	0	0
$946$	0.519725	0.0168977
$947$	−49.2741	−1.60119	−0.800597	−	0.599203i	$-0.795484\pi$
$947$	−49.2741	−1.60119	−0.800597	+	0.599203i	$0.795484\pi$
$948$	15.1900	0.493349
$949$	16.9649	0.550703
$950$	0	0
$951$	24.4247	0.792026
$952$	1.48979	0.0482844
$953$	−20.9012	−0.677055	−0.338528	−	0.940956i	$-0.609929\pi$
$953$	−20.9012	−0.677055	−0.338528	+	0.940956i	$0.609929\pi$
$954$	1.16009	0.0375594
$955$	0	0
$956$	20.7748	0.671906
$957$	−5.26601	−0.170226
$958$	36.6221	1.18321
$959$	−4.13016	−0.133370
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	13.3501	0.430202
$964$	4.58480	0.147667
$965$	0	0
$966$	−1.23468	−0.0397253
$967$	7.19739	0.231452	0.115726	−	0.993281i	$-0.463080\pi$
$967$	7.19739	0.231452	0.115726	+	0.993281i	$0.463080\pi$
$968$	6.84942	0.220149
$969$	35.5943	1.14345
$970$	0	0
$971$	−42.7939	−1.37332	−0.686660	−	0.726979i	$-0.740924\pi$
$971$	−42.7939	−1.37332	−0.686660	+	0.726979i	$0.740924\pi$
$972$	1.00000	0.0320750
$973$	−3.85081	−0.123451
$974$	−34.9371	−1.11946
$975$	0	0
$976$	0.160092	0.00512441
$977$	−18.1492	−0.580644	−0.290322	−	0.956929i	$-0.593762\pi$
$977$	−18.1492	−0.580644	−0.290322	+	0.956929i	$0.593762\pi$
$978$	−20.4824	−0.654956
$979$	20.8927	0.667733
$980$	0	0
$981$	6.26462	0.200014
$982$	−24.0095	−0.766174
$983$	6.30546	0.201113	0.100556	−	0.994931i	$-0.467938\pi$
$983$	6.30546	0.201113	0.100556	+	0.994931i	$0.467938\pi$
$984$	−11.1696	−0.356074
$985$	0	0
$986$	15.0950	0.480723
$987$	−2.51021	−0.0799009
$988$	−10.4078	−0.331117
$989$	1.23468	0.0392607
$990$	0	0
$991$	−27.5549	−0.875309	−0.437655	−	0.899143i	$-0.644191\pi$
$991$	−27.5549	−0.875309	−0.437655	+	0.899143i	$0.644191\pi$
$992$	−1.00000	−0.0317500
$993$	−7.24420	−0.229888
$994$	−1.43944	−0.0456561
$995$	0	0
$996$	−8.42471	−0.266947
$997$	0.945827	0.0299546	0.0149773	−	0.999888i	$-0.495232\pi$
$997$	0.945827	0.0299546	0.0149773	+	0.999888i	$0.495232\pi$
$998$	−4.65940	−0.147491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.ck.1.2		3
5.2	odd	4		930.2.d.h.559.1	✓	6
5.3	odd	4		930.2.d.h.559.4	yes	6
5.4	even	2		4650.2.a.cn.1.2		3
15.2	even	4		2790.2.d.k.559.6		6
15.8	even	4		2790.2.d.k.559.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.d.h.559.1	✓	6	5.2	odd	4
930.2.d.h.559.4	yes	6	5.3	odd	4
2790.2.d.k.559.3		6	15.8	even	4
2790.2.d.k.559.6		6	15.2	even	4
4650.2.a.ck.1.2		3	1.1	even	1	trivial
4650.2.a.cn.1.2		3	5.4	even	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.255105 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.255105 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.03730 q^{11} +1.00000 q^{12} -1.70760 q^{13} +0.255105 q^{14} +1.00000 q^{16} +5.83991 q^{17} -1.00000 q^{18} +6.09501 q^{19} -0.255105 q^{21} -2.03730 q^{22} -4.83991 q^{23} -1.00000 q^{24} +1.70760 q^{26} +1.00000 q^{27} -0.255105 q^{28} -2.58480 q^{29} +1.00000 q^{31} -1.00000 q^{32} +2.03730 q^{33} -5.83991 q^{34} +1.00000 q^{36} -6.09501 q^{38} -1.70760 q^{39} +11.1696 q^{41} +0.255105 q^{42} -0.255105 q^{43} +2.03730 q^{44} +4.83991 q^{46} +9.83991 q^{47} +1.00000 q^{48} -6.93492 q^{49} +5.83991 q^{51} -1.70760 q^{52} -1.16009 q^{53} -1.00000 q^{54} +0.255105 q^{56} +6.09501 q^{57} +2.58480 q^{58} -5.09501 q^{59} +0.160092 q^{61} -1.00000 q^{62} -0.255105 q^{63} +1.00000 q^{64} -2.03730 q^{66} -7.38741 q^{67} +5.83991 q^{68} -4.83991 q^{69} -5.64252 q^{71} -1.00000 q^{72} -9.93492 q^{73} +6.09501 q^{76} -0.519725 q^{77} +1.70760 q^{78} +15.1900 q^{79} +1.00000 q^{81} -11.1696 q^{82} -8.42471 q^{83} -0.255105 q^{84} +0.255105 q^{86} -2.58480 q^{87} -2.03730 q^{88} +10.2551 q^{89} +0.435617 q^{91} -4.83991 q^{92} +1.00000 q^{93} -9.83991 q^{94} -1.00000 q^{96} +4.87720 q^{97} +6.93492 q^{98} +2.03730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9} + q^{11} + 3 q^{12} - 12 q^{13} - q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{18} + q^{19} + q^{21} - q^{22} + q^{23} - 3 q^{24} + 12 q^{26} + 3 q^{27} + q^{28} + 6 q^{29} + 3 q^{31} - 3 q^{32} + q^{33} - 2 q^{34} + 3 q^{36} - q^{38} - 12 q^{39} + 6 q^{41} - q^{42} + q^{43} + q^{44} - q^{46} + 14 q^{47} + 3 q^{48} + 12 q^{49} + 2 q^{51} - 12 q^{52} - 19 q^{53} - 3 q^{54} - q^{56} + q^{57} - 6 q^{58} + 2 q^{59} + 16 q^{61} - 3 q^{62} + q^{63} + 3 q^{64} - q^{66} + 2 q^{67} + 2 q^{68} + q^{69} + 9 q^{71} - 3 q^{72} + 3 q^{73} + q^{76} + 45 q^{77} + 12 q^{78} + 11 q^{79} + 3 q^{81} - 6 q^{82} + 4 q^{83} + q^{84} - q^{86} + 6 q^{87} - q^{88} + 29 q^{89} + 8 q^{91} + q^{92} + 3 q^{93} - 14 q^{94} - 3 q^{96} - 6 q^{97} - 12 q^{98} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9 + q^11 + 3 * q^12 - 12 * q^13 - q^14 + 3 * q^16 + 2 * q^17 - 3 * q^18 + q^19 + q^21 - q^22 + q^23 - 3 * q^24 + 12 * q^26 + 3 * q^27 + q^28 + 6 * q^29 + 3 * q^31 - 3 * q^32 + q^33 - 2 * q^34 + 3 * q^36 - q^38 - 12 * q^39 + 6 * q^41 - q^42 + q^43 + q^44 - q^46 + 14 * q^47 + 3 * q^48 + 12 * q^49 + 2 * q^51 - 12 * q^52 - 19 * q^53 - 3 * q^54 - q^56 + q^57 - 6 * q^58 + 2 * q^59 + 16 * q^61 - 3 * q^62 + q^63 + 3 * q^64 - q^66 + 2 * q^67 + 2 * q^68 + q^69 + 9 * q^71 - 3 * q^72 + 3 * q^73 + q^76 + 45 * q^77 + 12 * q^78 + 11 * q^79 + 3 * q^81 - 6 * q^82 + 4 * q^83 + q^84 - q^86 + 6 * q^87 - q^88 + 29 * q^89 + 8 * q^91 + q^92 + 3 * q^93 - 14 * q^94 - 3 * q^96 - 6 * q^97 - 12 * q^98 + q^99

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