LMFDB - Embedded newform 3330.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.655762$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.46569 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.46569 q^{7} -1.00000 q^{8} +1.00000 q^{10} -4.15417 q^{11} -3.73836 q^{13} +3.46569 q^{14} +1.00000 q^{16} -4.63408 q^{17} +5.88150 q^{19} -1.00000 q^{20} +4.15417 q^{22} -4.36141 q^{23} +1.00000 q^{25} +3.73836 q^{26} -3.46569 q^{28} -8.03567 q^{29} +0.895717 q^{31} -1.00000 q^{32} +4.63408 q^{34} +3.46569 q^{35} -1.00000 q^{37} -5.88150 q^{38} +1.00000 q^{40} +2.89572 q^{41} -10.0357 q^{43} -4.15417 q^{44} +4.36141 q^{46} +6.78825 q^{47} +5.01103 q^{49} -1.00000 q^{50} -3.73836 q^{52} +8.63408 q^{53} +4.15417 q^{55} +3.46569 q^{56} +8.03567 q^{58} -6.93139 q^{59} -2.20724 q^{61} -0.895717 q^{62} +1.00000 q^{64} +3.73836 q^{65} -6.09978 q^{67} -4.63408 q^{68} -3.46569 q^{70} +13.1400 q^{71} -7.19302 q^{73} +1.00000 q^{74} +5.88150 q^{76} +14.3971 q^{77} +4.00000 q^{79} -1.00000 q^{80} -2.89572 q^{82} -10.0467 q^{83} +4.63408 q^{85} +10.0357 q^{86} +4.15417 q^{88} +12.8783 q^{89} +12.9560 q^{91} -4.36141 q^{92} -6.78825 q^{94} -5.88150 q^{95} +7.06092 q^{97} -5.01103 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 2 q^{11} - 3 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} + 3 q^{19} - 4 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} + 3 q^{26} + 4 q^{28} + 3 q^{29} + 3 q^{31} - 4 q^{32} + 6 q^{34} - 4 q^{35} - 4 q^{37} - 3 q^{38} + 4 q^{40} + 11 q^{41} - 5 q^{43} - 2 q^{44} - q^{46} + 14 q^{49} - 4 q^{50} - 3 q^{52} + 22 q^{53} + 2 q^{55} - 4 q^{56} - 3 q^{58} + 8 q^{59} - 5 q^{61} - 3 q^{62} + 4 q^{64} + 3 q^{65} + 6 q^{67} - 6 q^{68} + 4 q^{70} + 18 q^{71} - 5 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 16 q^{79} - 4 q^{80} - 11 q^{82} + q^{83} + 6 q^{85} + 5 q^{86} + 2 q^{88} + 5 q^{89} - 13 q^{91} + q^{92} - 3 q^{95} + 7 q^{97} - 14 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 2 * q^11 - 3 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 + 3 * q^19 - 4 * q^20 + 2 * q^22 + q^23 + 4 * q^25 + 3 * q^26 + 4 * q^28 + 3 * q^29 + 3 * q^31 - 4 * q^32 + 6 * q^34 - 4 * q^35 - 4 * q^37 - 3 * q^38 + 4 * q^40 + 11 * q^41 - 5 * q^43 - 2 * q^44 - q^46 + 14 * q^49 - 4 * q^50 - 3 * q^52 + 22 * q^53 + 2 * q^55 - 4 * q^56 - 3 * q^58 + 8 * q^59 - 5 * q^61 - 3 * q^62 + 4 * q^64 + 3 * q^65 + 6 * q^67 - 6 * q^68 + 4 * q^70 + 18 * q^71 - 5 * q^73 + 4 * q^74 + 3 * q^76 + 4 * q^77 + 16 * q^79 - 4 * q^80 - 11 * q^82 + q^83 + 6 * q^85 + 5 * q^86 + 2 * q^88 + 5 * q^89 - 13 * q^91 + q^92 - 3 * q^95 + 7 * q^97 - 14 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.46569	−1.30991	−0.654955	−	0.755668i	$-0.727312\pi$
$7$	−3.46569	−1.30991	−0.654955	+	0.755668i	$0.727312\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−4.15417	−1.25253	−0.626265	−	0.779610i	$-0.715417\pi$
$11$	−4.15417	−1.25253	−0.626265	+	0.779610i	$0.715417\pi$
$12$	0	0
$13$	−3.73836	−1.03684	−0.518418	−	0.855127i	$-0.673479\pi$
$13$	−3.73836	−1.03684	−0.518418	+	0.855127i	$0.673479\pi$
$14$	3.46569	0.926246
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.63408	−1.12393	−0.561965	−	0.827161i	$-0.689954\pi$
$17$	−4.63408	−1.12393	−0.561965	+	0.827161i	$0.689954\pi$
$18$	0	0
$19$	5.88150	1.34931	0.674654	−	0.738134i	$-0.264293\pi$
$19$	5.88150	1.34931	0.674654	+	0.738134i	$0.264293\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	4.15417	0.885672
$23$	−4.36141	−0.909417	−0.454709	−	0.890640i	$-0.650257\pi$
$23$	−4.36141	−0.909417	−0.454709	+	0.890640i	$0.650257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.73836	0.733153
$27$	0	0
$28$	−3.46569	−0.654955
$29$	−8.03567	−1.49219	−0.746093	−	0.665841i	$-0.768073\pi$
$29$	−8.03567	−1.49219	−0.746093	+	0.665841i	$0.768073\pi$
$30$	0	0
$31$	0.895717	0.160876	0.0804378	−	0.996760i	$-0.474368\pi$
$31$	0.895717	0.160876	0.0804378	+	0.996760i	$0.474368\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.63408	0.794738
$35$	3.46569	0.585809
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−5.88150	−0.954105
$39$	0	0
$40$	1.00000	0.158114
$41$	2.89572	0.452235	0.226118	−	0.974100i	$-0.427397\pi$
$41$	2.89572	0.452235	0.226118	+	0.974100i	$0.427397\pi$
$42$	0	0
$43$	−10.0357	−1.53043	−0.765213	−	0.643778i	$-0.777366\pi$
$43$	−10.0357	−1.53043	−0.765213	+	0.643778i	$0.777366\pi$
$44$	−4.15417	−0.626265
$45$	0	0
$46$	4.36141	0.643055
$47$	6.78825	0.990168	0.495084	−	0.868845i	$-0.335137\pi$
$47$	6.78825	0.990168	0.495084	+	0.868845i	$0.335137\pi$
$48$	0	0
$49$	5.01103	0.715862
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.73836	−0.518418
$53$	8.63408	1.18598	0.592991	−	0.805209i	$-0.297947\pi$
$53$	8.63408	1.18598	0.592991	+	0.805209i	$0.297947\pi$
$54$	0	0
$55$	4.15417	0.560148
$56$	3.46569	0.463123
$57$	0	0
$58$	8.03567	1.05514
$59$	−6.93139	−0.902390	−0.451195	−	0.892425i	$-0.649002\pi$
$59$	−6.93139	−0.902390	−0.451195	+	0.892425i	$0.649002\pi$
$60$	0	0
$61$	−2.20724	−0.282608	−0.141304	−	0.989966i	$-0.545130\pi$
$61$	−2.20724	−0.282608	−0.141304	+	0.989966i	$0.545130\pi$
$62$	−0.895717	−0.113756
$63$	0	0
$64$	1.00000	0.125000
$65$	3.73836	0.463687
$66$	0	0
$67$	−6.09978	−0.745206	−0.372603	−	0.927991i	$-0.621535\pi$
$67$	−6.09978	−0.745206	−0.372603	+	0.927991i	$0.621535\pi$
$68$	−4.63408	−0.561965
$69$	0	0
$70$	−3.46569	−0.414230
$71$	13.1400	1.55943	0.779713	−	0.626137i	$-0.215365\pi$
$71$	13.1400	1.55943	0.779713	+	0.626137i	$0.215365\pi$
$72$	0	0
$73$	−7.19302	−0.841880	−0.420940	−	0.907089i	$-0.638300\pi$
$73$	−7.19302	−0.841880	−0.420940	+	0.907089i	$0.638300\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	5.88150	0.674654
$77$	14.3971	1.64070
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−2.89572	−0.319778
$83$	−10.0467	−1.10277	−0.551385	−	0.834251i	$-0.685900\pi$
$83$	−10.0467	−1.10277	−0.551385	+	0.834251i	$0.685900\pi$
$84$	0	0
$85$	4.63408	0.502637
$86$	10.0357	1.08217
$87$	0	0
$88$	4.15417	0.442836
$89$	12.8783	1.36510	0.682549	−	0.730839i	$-0.260871\pi$
$89$	12.8783	1.36510	0.682549	+	0.730839i	$0.260871\pi$
$90$	0	0
$91$	12.9560	1.35816
$92$	−4.36141	−0.454709
$93$	0	0
$94$	−6.78825	−0.700155
$95$	−5.88150	−0.603429
$96$	0	0
$97$	7.06092	0.716928	0.358464	−	0.933544i	$-0.383301\pi$
$97$	7.06092	0.716928	0.358464	+	0.933544i	$0.383301\pi$
$98$	−5.01103	−0.506191
$99$	0	0
$100$	1.00000	0.100000
$101$	16.2429	1.61623	0.808115	−	0.589025i	$-0.200488\pi$
$101$	16.2429	1.61623	0.808115	+	0.589025i	$0.200488\pi$
$102$	0	0
$103$	6.78825	0.668866	0.334433	−	0.942419i	$-0.391455\pi$
$103$	6.78825	0.668866	0.334433	+	0.942419i	$0.391455\pi$
$104$	3.73836	0.366577
$105$	0	0
$106$	−8.63408	−0.838616
$107$	8.66975	0.838137	0.419068	−	0.907955i	$-0.362357\pi$
$107$	8.66975	0.838137	0.419068	+	0.907955i	$0.362357\pi$
$108$	0	0
$109$	−16.5623	−1.58638	−0.793190	−	0.608975i	$-0.791581\pi$
$109$	−16.5623	−1.58638	−0.793190	+	0.608975i	$0.791581\pi$
$110$	−4.15417	−0.396085
$111$	0	0
$112$	−3.46569	−0.327477
$113$	16.0357	1.50851	0.754254	−	0.656582i	$-0.227998\pi$
$113$	16.0357	1.50851	0.754254	+	0.656582i	$0.227998\pi$
$114$	0	0
$115$	4.36141	0.406704
$116$	−8.03567	−0.746093
$117$	0	0
$118$	6.93139	0.638086
$119$	16.0603	1.47225
$120$	0	0
$121$	6.25713	0.568830
$122$	2.20724	0.199834
$123$	0	0
$124$	0.895717	0.0804378
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.11532	−0.276440	−0.138220	−	0.990402i	$-0.544138\pi$
$127$	−3.11532	−0.276440	−0.138220	+	0.990402i	$0.544138\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.73836	−0.327876
$131$	−12.0713	−1.05468	−0.527339	−	0.849655i	$-0.676810\pi$
$131$	−12.0713	−1.05468	−0.527339	+	0.849655i	$0.676810\pi$
$132$	0	0
$133$	−20.3835	−1.76747
$134$	6.09978	0.526940
$135$	0	0
$136$	4.63408	0.397369
$137$	−21.6912	−1.85320	−0.926602	−	0.376043i	$-0.877285\pi$
$137$	−21.6912	−1.85320	−0.926602	+	0.376043i	$0.877285\pi$
$138$	0	0
$139$	5.31285	0.450630	0.225315	−	0.974286i	$-0.427659\pi$
$139$	5.31285	0.450630	0.225315	+	0.974286i	$0.427659\pi$
$140$	3.46569	0.292905
$141$	0	0
$142$	−13.1400	−1.10268
$143$	15.5298	1.29867
$144$	0	0
$145$	8.03567	0.667326
$146$	7.19302	0.595299
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−22.6510	−1.85564	−0.927822	−	0.373023i	$-0.878321\pi$
$149$	−22.6510	−1.85564	−0.927822	+	0.373023i	$0.878321\pi$
$150$	0	0
$151$	3.94693	0.321197	0.160598	−	0.987020i	$-0.448658\pi$
$151$	3.94693	0.321197	0.160598	+	0.987020i	$0.448658\pi$
$152$	−5.88150	−0.477053
$153$	0	0
$154$	−14.3971	−1.16015
$155$	−0.895717	−0.0719458
$156$	0	0
$157$	20.7585	1.65671	0.828354	−	0.560205i	$-0.189278\pi$
$157$	20.7585	1.65671	0.828354	+	0.560205i	$0.189278\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	15.1153	1.19125
$162$	0	0
$163$	19.6652	1.54030	0.770150	−	0.637862i	$-0.220181\pi$
$163$	19.6652	1.54030	0.770150	+	0.637862i	$0.220181\pi$
$164$	2.89572	0.226118
$165$	0	0
$166$	10.0467	0.779775
$167$	−20.7980	−1.60939	−0.804697	−	0.593685i	$-0.797672\pi$
$167$	−20.7980	−1.60939	−0.804697	+	0.593685i	$0.797672\pi$
$168$	0	0
$169$	0.975364	0.0750280
$170$	−4.63408	−0.355418
$171$	0	0
$172$	−10.0357	−0.765213
$173$	5.15735	0.392106	0.196053	−	0.980593i	$-0.437187\pi$
$173$	5.15735	0.392106	0.196053	+	0.980593i	$0.437187\pi$
$174$	0	0
$175$	−3.46569	−0.261982
$176$	−4.15417	−0.313132
$177$	0	0
$178$	−12.8783	−0.965271
$179$	−1.66323	−0.124315	−0.0621576	−	0.998066i	$-0.519798\pi$
$179$	−1.66323	−0.124315	−0.0621576	+	0.998066i	$0.519798\pi$
$180$	0	0
$181$	−15.8628	−1.17907	−0.589535	−	0.807743i	$-0.700689\pi$
$181$	−15.8628	−1.17907	−0.589535	+	0.807743i	$0.700689\pi$
$182$	−12.9560	−0.960364
$183$	0	0
$184$	4.36141	0.321528
$185$	1.00000	0.0735215
$186$	0	0
$187$	19.2508	1.40776
$188$	6.78825	0.495084
$189$	0	0
$190$	5.88150	0.426689
$191$	19.2287	1.39134	0.695670	−	0.718362i	$-0.255108\pi$
$191$	19.2287	1.39134	0.695670	+	0.718362i	$0.255108\pi$
$192$	0	0
$193$	−12.7883	−0.920518	−0.460259	−	0.887785i	$-0.652244\pi$
$193$	−12.7883	−0.920518	−0.460259	+	0.887785i	$0.652244\pi$
$194$	−7.06092	−0.506945
$195$	0	0
$196$	5.01103	0.357931
$197$	10.3614	0.738220	0.369110	−	0.929386i	$-0.379663\pi$
$197$	10.3614	0.738220	0.369110	+	0.929386i	$0.379663\pi$
$198$	0	0
$199$	10.6230	0.753048	0.376524	−	0.926407i	$-0.377119\pi$
$199$	10.6230	0.753048	0.376524	+	0.926407i	$0.377119\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−16.2429	−1.14285
$203$	27.8492	1.95463
$204$	0	0
$205$	−2.89572	−0.202246
$206$	−6.78825	−0.472960
$207$	0	0
$208$	−3.73836	−0.259209
$209$	−24.4328	−1.69005
$210$	0	0
$211$	4.06410	0.279785	0.139892	−	0.990167i	$-0.455324\pi$
$211$	4.06410	0.279785	0.139892	+	0.990167i	$0.455324\pi$
$212$	8.63408	0.592991
$213$	0	0
$214$	−8.66975	−0.592652
$215$	10.0357	0.684427
$216$	0	0
$217$	−3.10428	−0.210732
$218$	16.5623	1.12174
$219$	0	0
$220$	4.15417	0.280074
$221$	17.3239	1.16533
$222$	0	0
$223$	−9.61854	−0.644105	−0.322053	−	0.946722i	$-0.604373\pi$
$223$	−9.61854	−0.644105	−0.322053	+	0.946722i	$0.604373\pi$
$224$	3.46569	0.231561
$225$	0	0
$226$	−16.0357	−1.06668
$227$	11.6122	0.770727	0.385363	−	0.922765i	$-0.374076\pi$
$227$	11.6122	0.770727	0.385363	+	0.922765i	$0.374076\pi$
$228$	0	0
$229$	23.0312	1.52194	0.760971	−	0.648786i	$-0.224723\pi$
$229$	23.0312	1.52194	0.760971	+	0.648786i	$0.224723\pi$
$230$	−4.36141	−0.287583
$231$	0	0
$232$	8.03567	0.527568
$233$	1.42031	0.0930478	0.0465239	−	0.998917i	$-0.485186\pi$
$233$	1.42031	0.0930478	0.0465239	+	0.998917i	$0.485186\pi$
$234$	0	0
$235$	−6.78825	−0.442817
$236$	−6.93139	−0.451195
$237$	0	0
$238$	−16.0603	−1.04104
$239$	18.4502	1.19344	0.596721	−	0.802449i	$-0.296470\pi$
$239$	18.4502	1.19344	0.596721	+	0.802449i	$0.296470\pi$
$240$	0	0
$241$	−26.8162	−1.72739	−0.863693	−	0.504019i	$-0.831854\pi$
$241$	−26.8162	−1.72739	−0.863693	+	0.504019i	$0.831854\pi$
$242$	−6.25713	−0.402223
$243$	0	0
$244$	−2.20724	−0.141304
$245$	−5.01103	−0.320143
$246$	0	0
$247$	−21.9872	−1.39901
$248$	−0.895717	−0.0568781
$249$	0	0
$250$	1.00000	0.0632456
$251$	−17.0532	−1.07639	−0.538195	−	0.842820i	$-0.680894\pi$
$251$	−17.0532	−1.07639	−0.538195	+	0.842820i	$0.680894\pi$
$252$	0	0
$253$	18.1180	1.13907
$254$	3.11532	0.195472
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.5014	−1.21646	−0.608231	−	0.793760i	$-0.708121\pi$
$257$	−19.5014	−1.21646	−0.608231	+	0.793760i	$0.708121\pi$
$258$	0	0
$259$	3.46569	0.215348
$260$	3.73836	0.231843
$261$	0	0
$262$	12.0713	0.745770
$263$	21.6555	1.33534	0.667669	−	0.744459i	$-0.267292\pi$
$263$	21.6555	1.33534	0.667669	+	0.744459i	$0.267292\pi$
$264$	0	0
$265$	−8.63408	−0.530387
$266$	20.3835	1.24979
$267$	0	0
$268$	−6.09978	−0.372603
$269$	16.0811	0.980479	0.490239	−	0.871588i	$-0.336909\pi$
$269$	16.0811	0.980479	0.490239	+	0.871588i	$0.336909\pi$
$270$	0	0
$271$	6.62305	0.402321	0.201161	−	0.979558i	$-0.435529\pi$
$271$	6.62305	0.402321	0.201161	+	0.979558i	$0.435529\pi$
$272$	−4.63408	−0.280982
$273$	0	0
$274$	21.6912	1.31041
$275$	−4.15417	−0.250506
$276$	0	0
$277$	−28.0467	−1.68516	−0.842582	−	0.538569i	$-0.818965\pi$
$277$	−28.0467	−1.68516	−0.842582	+	0.538569i	$0.818965\pi$
$278$	−5.31285	−0.318643
$279$	0	0
$280$	−3.46569	−0.207115
$281$	4.26164	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$281$	4.26164	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$282$	0	0
$283$	0.884684	0.0525890	0.0262945	−	0.999654i	$-0.491629\pi$
$283$	0.884684	0.0525890	0.0262945	+	0.999654i	$0.491629\pi$
$284$	13.1400	0.779713
$285$	0	0
$286$	−15.5298	−0.918296
$287$	−10.0357	−0.592387
$288$	0	0
$289$	4.47471	0.263218
$290$	−8.03567	−0.471871
$291$	0	0
$292$	−7.19302	−0.420940
$293$	21.4657	1.25404	0.627020	−	0.779003i	$-0.284275\pi$
$293$	21.4657	1.25404	0.627020	+	0.779003i	$0.284275\pi$
$294$	0	0
$295$	6.93139	0.403561
$296$	1.00000	0.0581238
$297$	0	0
$298$	22.6510	1.31214
$299$	16.3045	0.942916
$300$	0	0
$301$	34.7806	2.00472
$302$	−3.94693	−0.227120
$303$	0	0
$304$	5.88150	0.337327
$305$	2.20724	0.126386
$306$	0	0
$307$	34.1711	1.95025	0.975124	−	0.221659i	$-0.0711471\pi$
$307$	34.1711	1.95025	0.975124	+	0.221659i	$0.0711471\pi$
$308$	14.3971	0.820350
$309$	0	0
$310$	0.895717	0.0508733
$311$	10.1639	0.576341	0.288170	−	0.957579i	$-0.406953\pi$
$311$	10.1639	0.576341	0.288170	+	0.957579i	$0.406953\pi$
$312$	0	0
$313$	−9.31152	−0.526318	−0.263159	−	0.964752i	$-0.584764\pi$
$313$	−9.31152	−0.526318	−0.263159	+	0.964752i	$0.584764\pi$
$314$	−20.7585	−1.17147
$315$	0	0
$316$	4.00000	0.225018
$317$	−16.8673	−0.947361	−0.473680	−	0.880697i	$-0.657075\pi$
$317$	−16.8673	−0.947361	−0.473680	+	0.880697i	$0.657075\pi$
$318$	0	0
$319$	33.3815	1.86901
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−15.1153	−0.842344
$323$	−27.2553	−1.51653
$324$	0	0
$325$	−3.73836	−0.207367
$326$	−19.6652	−1.08916
$327$	0	0
$328$	−2.89572	−0.159889
$329$	−23.5260	−1.29703
$330$	0	0
$331$	−0.451476	−0.0248154	−0.0124077	−	0.999923i	$-0.503950\pi$
$331$	−0.451476	−0.0248154	−0.0124077	+	0.999923i	$0.503950\pi$
$332$	−10.0467	−0.551385
$333$	0	0
$334$	20.7980	1.13801
$335$	6.09978	0.333266
$336$	0	0
$337$	−28.3550	−1.54460	−0.772299	−	0.635259i	$-0.780893\pi$
$337$	−28.3550	−1.54460	−0.772299	+	0.635259i	$0.780893\pi$
$338$	−0.975364	−0.0530528
$339$	0	0
$340$	4.63408	0.251318
$341$	−3.72096	−0.201501
$342$	0	0
$343$	6.89315	0.372195
$344$	10.0357	0.541087
$345$	0	0
$346$	−5.15735	−0.277261
$347$	−5.37695	−0.288650	−0.144325	−	0.989530i	$-0.546101\pi$
$347$	−5.37695	−0.288650	−0.144325	+	0.989530i	$0.546101\pi$
$348$	0	0
$349$	8.22798	0.440434	0.220217	−	0.975451i	$-0.429323\pi$
$349$	8.22798	0.440434	0.220217	+	0.975451i	$0.429323\pi$
$350$	3.46569	0.185249
$351$	0	0
$352$	4.15417	0.221418
$353$	−5.95531	−0.316969	−0.158485	−	0.987361i	$-0.550661\pi$
$353$	−5.95531	−0.316969	−0.158485	+	0.987361i	$0.550661\pi$
$354$	0	0
$355$	−13.1400	−0.697396
$356$	12.8783	0.682549
$357$	0	0
$358$	1.66323	0.0879042
$359$	−12.0713	−0.637101	−0.318550	−	0.947906i	$-0.603196\pi$
$359$	−12.0713	−0.637101	−0.318550	+	0.947906i	$0.603196\pi$
$360$	0	0
$361$	15.5920	0.820634
$362$	15.8628	0.833729
$363$	0	0
$364$	12.9560	0.679080
$365$	7.19302	0.376500
$366$	0	0
$367$	−0.139243	−0.00726845	−0.00363422	−	0.999993i	$-0.501157\pi$
$367$	−0.139243	−0.00726845	−0.00363422	+	0.999993i	$0.501157\pi$
$368$	−4.36141	−0.227354
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	−29.9231	−1.55353
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−19.2508	−0.995433
$375$	0	0
$376$	−6.78825	−0.350077
$377$	30.0403	1.54715
$378$	0	0
$379$	−5.33950	−0.274272	−0.137136	−	0.990552i	$-0.543790\pi$
$379$	−5.33950	−0.274272	−0.137136	+	0.990552i	$0.543790\pi$
$380$	−5.88150	−0.301715
$381$	0	0
$382$	−19.2287	−0.983826
$383$	13.7384	0.701998	0.350999	−	0.936376i	$-0.385842\pi$
$383$	13.7384	0.701998	0.350999	+	0.936376i	$0.385842\pi$
$384$	0	0
$385$	−14.3971	−0.733743
$386$	12.7883	0.650905
$387$	0	0
$388$	7.06092	0.358464
$389$	1.61217	0.0817404	0.0408702	−	0.999164i	$-0.486987\pi$
$389$	1.61217	0.0817404	0.0408702	+	0.999164i	$0.486987\pi$
$390$	0	0
$391$	20.2111	1.02212
$392$	−5.01103	−0.253095
$393$	0	0
$394$	−10.3614	−0.522000
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−17.4767	−0.877132	−0.438566	−	0.898699i	$-0.644513\pi$
$397$	−17.4767	−0.877132	−0.438566	+	0.898699i	$0.644513\pi$
$398$	−10.6230	−0.532485
$399$	0	0
$400$	1.00000	0.0500000
$401$	4.87832	0.243611	0.121806	−	0.992554i	$-0.461132\pi$
$401$	4.87832	0.243611	0.121806	+	0.992554i	$0.461132\pi$
$402$	0	0
$403$	−3.34852	−0.166802
$404$	16.2429	0.808115
$405$	0	0
$406$	−27.8492	−1.38213
$407$	4.15417	0.205915
$408$	0	0
$409$	−13.9626	−0.690404	−0.345202	−	0.938528i	$-0.612190\pi$
$409$	−13.9626	−0.690404	−0.345202	+	0.938528i	$0.612190\pi$
$410$	2.89572	0.143009
$411$	0	0
$412$	6.78825	0.334433
$413$	24.0221	1.18205
$414$	0	0
$415$	10.0467	0.493173
$416$	3.73836	0.183288
$417$	0	0
$418$	24.4328	1.19504
$419$	−4.85034	−0.236954	−0.118477	−	0.992957i	$-0.537801\pi$
$419$	−4.85034	−0.236954	−0.118477	+	0.992957i	$0.537801\pi$
$420$	0	0
$421$	28.0280	1.36600	0.683000	−	0.730418i	$-0.260675\pi$
$421$	28.0280	1.36600	0.683000	+	0.730418i	$0.260675\pi$
$422$	−4.06410	−0.197838
$423$	0	0
$424$	−8.63408	−0.419308
$425$	−4.63408	−0.224786
$426$	0	0
$427$	7.64962	0.370191
$428$	8.66975	0.419068
$429$	0	0
$430$	−10.0357	−0.483963
$431$	30.4191	1.46524	0.732619	−	0.680639i	$-0.238298\pi$
$431$	30.4191	1.46524	0.732619	+	0.680639i	$0.238298\pi$
$432$	0	0
$433$	9.52980	0.457973	0.228986	−	0.973430i	$-0.426459\pi$
$433$	9.52980	0.457973	0.228986	+	0.973430i	$0.426459\pi$
$434$	3.10428	0.149010
$435$	0	0
$436$	−16.5623	−0.793190
$437$	−25.6516	−1.22708
$438$	0	0
$439$	11.9580	0.570722	0.285361	−	0.958420i	$-0.407886\pi$
$439$	11.9580	0.570722	0.285361	+	0.958420i	$0.407886\pi$
$440$	−4.15417	−0.198042
$441$	0	0
$442$	−17.3239	−0.824013
$443$	−18.8252	−0.894414	−0.447207	−	0.894430i	$-0.647581\pi$
$443$	−18.8252	−0.894414	−0.447207	+	0.894430i	$0.647581\pi$
$444$	0	0
$445$	−12.8783	−0.610491
$446$	9.61854	0.455451
$447$	0	0
$448$	−3.46569	−0.163739
$449$	−18.8316	−0.888719	−0.444359	−	0.895849i	$-0.646569\pi$
$449$	−18.8316	−0.888719	−0.444359	+	0.895849i	$0.646569\pi$
$450$	0	0
$451$	−12.0293	−0.566438
$452$	16.0357	0.754254
$453$	0	0
$454$	−11.6122	−0.544986
$455$	−12.9560	−0.607388
$456$	0	0
$457$	6.58692	0.308123	0.154062	−	0.988061i	$-0.450765\pi$
$457$	6.58692	0.308123	0.154062	+	0.988061i	$0.450765\pi$
$458$	−23.0312	−1.07618
$459$	0	0
$460$	4.36141	0.203352
$461$	28.9386	1.34781	0.673903	−	0.738820i	$-0.264617\pi$
$461$	28.9386	1.34781	0.673903	+	0.738820i	$0.264617\pi$
$462$	0	0
$463$	22.6600	1.05310	0.526551	−	0.850144i	$-0.323485\pi$
$463$	22.6600	1.05310	0.526551	+	0.850144i	$0.323485\pi$
$464$	−8.03567	−0.373047
$465$	0	0
$466$	−1.42031	−0.0657948
$467$	31.6835	1.46614	0.733069	−	0.680154i	$-0.238087\pi$
$467$	31.6835	1.46614	0.733069	+	0.680154i	$0.238087\pi$
$468$	0	0
$469$	21.1400	0.976152
$470$	6.78825	0.313119
$471$	0	0
$472$	6.93139	0.319043
$473$	41.6899	1.91690
$474$	0	0
$475$	5.88150	0.269862
$476$	16.0603	0.736123
$477$	0	0
$478$	−18.4502	−0.843890
$479$	2.87832	0.131514	0.0657568	−	0.997836i	$-0.479054\pi$
$479$	2.87832	0.131514	0.0657568	+	0.997836i	$0.479054\pi$
$480$	0	0
$481$	3.73836	0.170455
$482$	26.8162	1.22145
$483$	0	0
$484$	6.25713	0.284415
$485$	−7.06092	−0.320620
$486$	0	0
$487$	21.4334	0.971239	0.485619	−	0.874170i	$-0.338594\pi$
$487$	21.4334	0.971239	0.485619	+	0.874170i	$0.338594\pi$
$488$	2.20724	0.0999171
$489$	0	0
$490$	5.01103	0.226375
$491$	1.04989	0.0473808	0.0236904	−	0.999719i	$-0.492458\pi$
$491$	1.04989	0.0473808	0.0236904	+	0.999719i	$0.492458\pi$
$492$	0	0
$493$	37.2379	1.67711
$494$	21.9872	0.989250
$495$	0	0
$496$	0.895717	0.0402189
$497$	−45.5391	−2.04271
$498$	0	0
$499$	14.1898	0.635224	0.317612	−	0.948221i	$-0.397119\pi$
$499$	14.1898	0.635224	0.317612	+	0.948221i	$0.397119\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	17.0532	0.761123
$503$	−16.6167	−0.740901	−0.370451	−	0.928852i	$-0.620797\pi$
$503$	−16.6167	−0.740901	−0.370451	+	0.928852i	$0.620797\pi$
$504$	0	0
$505$	−16.2429	−0.722800
$506$	−18.1180	−0.805445
$507$	0	0
$508$	−3.11532	−0.138220
$509$	−25.0409	−1.10992	−0.554959	−	0.831878i	$-0.687266\pi$
$509$	−25.0409	−1.10992	−0.554959	+	0.831878i	$0.687266\pi$
$510$	0	0
$511$	24.9288	1.10279
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	19.5014	0.860168
$515$	−6.78825	−0.299126
$516$	0	0
$517$	−28.1996	−1.24021
$518$	−3.46569	−0.152274
$519$	0	0
$520$	−3.73836	−0.163938
$521$	−15.7987	−0.692152	−0.346076	−	0.938206i	$-0.612486\pi$
$521$	−15.7987	−0.692152	−0.346076	+	0.938206i	$0.612486\pi$
$522$	0	0
$523$	−4.19955	−0.183634	−0.0918168	−	0.995776i	$-0.529267\pi$
$523$	−4.19955	−0.183634	−0.0918168	+	0.995776i	$0.529267\pi$
$524$	−12.0713	−0.527339
$525$	0	0
$526$	−21.6555	−0.944226
$527$	−4.15083	−0.180813
$528$	0	0
$529$	−3.97809	−0.172960
$530$	8.63408	0.375041
$531$	0	0
$532$	−20.3835	−0.883736
$533$	−10.8252	−0.468893
$534$	0	0
$535$	−8.66975	−0.374826
$536$	6.09978	0.263470
$537$	0	0
$538$	−16.0811	−0.693303
$539$	−20.8167	−0.896638
$540$	0	0
$541$	29.8531	1.28348	0.641742	−	0.766921i	$-0.278212\pi$
$541$	29.8531	1.28348	0.641742	+	0.766921i	$0.278212\pi$
$542$	−6.62305	−0.284484
$543$	0	0
$544$	4.63408	0.198685
$545$	16.5623	0.709450
$546$	0	0
$547$	−28.1912	−1.20537	−0.602684	−	0.797980i	$-0.705902\pi$
$547$	−28.1912	−1.20537	−0.602684	+	0.797980i	$0.705902\pi$
$548$	−21.6912	−0.926602
$549$	0	0
$550$	4.15417	0.177134
$551$	−47.2618	−2.01342
$552$	0	0
$553$	−13.8628	−0.589505
$554$	28.0467	1.19159
$555$	0	0
$556$	5.31285	0.225315
$557$	−13.0596	−0.553353	−0.276676	−	0.960963i	$-0.589233\pi$
$557$	−13.0596	−0.553353	−0.276676	+	0.960963i	$0.589233\pi$
$558$	0	0
$559$	37.5170	1.58680
$560$	3.46569	0.146452
$561$	0	0
$562$	−4.26164	−0.179766
$563$	10.3414	0.435836	0.217918	−	0.975967i	$-0.430073\pi$
$563$	10.3414	0.435836	0.217918	+	0.975967i	$0.430073\pi$
$564$	0	0
$565$	−16.0357	−0.674626
$566$	−0.884684	−0.0371860
$567$	0	0
$568$	−13.1400	−0.551340
$569$	20.5546	0.861693	0.430847	−	0.902425i	$-0.358215\pi$
$569$	20.5546	0.861693	0.430847	+	0.902425i	$0.358215\pi$
$570$	0	0
$571$	4.89837	0.204990	0.102495	−	0.994734i	$-0.467317\pi$
$571$	4.89837	0.204990	0.102495	+	0.994734i	$0.467317\pi$
$572$	15.5298	0.649334
$573$	0	0
$574$	10.0357	0.418881
$575$	−4.36141	−0.181883
$576$	0	0
$577$	−21.0745	−0.877344	−0.438672	−	0.898647i	$-0.644551\pi$
$577$	−21.0745	−0.877344	−0.438672	+	0.898647i	$0.644551\pi$
$578$	−4.47471	−0.186123
$579$	0	0
$580$	8.03567	0.333663
$581$	34.8188	1.44453
$582$	0	0
$583$	−35.8674	−1.48548
$584$	7.19302	0.297649
$585$	0	0
$586$	−21.4657	−0.886740
$587$	6.78056	0.279864	0.139932	−	0.990161i	$-0.455312\pi$
$587$	6.78056	0.279864	0.139932	+	0.990161i	$0.455312\pi$
$588$	0	0
$589$	5.26816	0.217071
$590$	−6.93139	−0.285361
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	46.5449	1.91137	0.955685	−	0.294392i	$-0.0951172\pi$
$593$	46.5449	1.91137	0.955685	+	0.294392i	$0.0951172\pi$
$594$	0	0
$595$	−16.0603	−0.658408
$596$	−22.6510	−0.927822
$597$	0	0
$598$	−16.3045	−0.666742
$599$	−41.3021	−1.68756	−0.843778	−	0.536692i	$-0.819674\pi$
$599$	−41.3021	−1.68756	−0.843778	+	0.536692i	$0.819674\pi$
$600$	0	0
$601$	10.5122	0.428803	0.214402	−	0.976746i	$-0.431220\pi$
$601$	10.5122	0.428803	0.214402	+	0.976746i	$0.431220\pi$
$602$	−34.7806	−1.41755
$603$	0	0
$604$	3.94693	0.160598
$605$	−6.25713	−0.254388
$606$	0	0
$607$	25.0461	1.01659	0.508295	−	0.861183i	$-0.330276\pi$
$607$	25.0461	1.01659	0.508295	+	0.861183i	$0.330276\pi$
$608$	−5.88150	−0.238526
$609$	0	0
$610$	−2.20724	−0.0893686
$611$	−25.3770	−1.02664
$612$	0	0
$613$	−16.6524	−0.672582	−0.336291	−	0.941758i	$-0.609173\pi$
$613$	−16.6524	−0.672582	−0.336291	+	0.941758i	$0.609173\pi$
$614$	−34.1711	−1.37903
$615$	0	0
$616$	−14.3971	−0.580075
$617$	−6.12107	−0.246425	−0.123212	−	0.992380i	$-0.539320\pi$
$617$	−6.12107	−0.246425	−0.123212	+	0.992380i	$0.539320\pi$
$618$	0	0
$619$	28.6587	1.15189	0.575946	−	0.817488i	$-0.304634\pi$
$619$	28.6587	1.15189	0.575946	+	0.817488i	$0.304634\pi$
$620$	−0.895717	−0.0359729
$621$	0	0
$622$	−10.1639	−0.407534
$623$	−44.6323	−1.78816
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.31152	0.372163
$627$	0	0
$628$	20.7585	0.828354
$629$	4.63408	0.184773
$630$	0	0
$631$	−24.7301	−0.984488	−0.492244	−	0.870457i	$-0.663823\pi$
$631$	−24.7301	−0.984488	−0.492244	+	0.870457i	$0.663823\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	16.8673	0.669885
$635$	3.11532	0.123628
$636$	0	0
$637$	−18.7331	−0.742231
$638$	−33.3815	−1.32159
$639$	0	0
$640$	1.00000	0.0395285
$641$	4.60308	0.181811	0.0909053	−	0.995860i	$-0.471024\pi$
$641$	4.60308	0.181811	0.0909053	+	0.995860i	$0.471024\pi$
$642$	0	0
$643$	4.74924	0.187292	0.0936458	−	0.995606i	$-0.470148\pi$
$643$	4.74924	0.187292	0.0936458	+	0.995606i	$0.470148\pi$
$644$	15.1153	0.595627
$645$	0	0
$646$	27.2553	1.07235
$647$	−34.8409	−1.36974	−0.684868	−	0.728667i	$-0.740140\pi$
$647$	−34.8409	−1.36974	−0.684868	+	0.728667i	$0.740140\pi$
$648$	0	0
$649$	28.7942	1.13027
$650$	3.73836	0.146631
$651$	0	0
$652$	19.6652	0.770150
$653$	−19.2682	−0.754021	−0.377011	−	0.926209i	$-0.623048\pi$
$653$	−19.2682	−0.754021	−0.377011	+	0.926209i	$0.623048\pi$
$654$	0	0
$655$	12.0713	0.471666
$656$	2.89572	0.113059
$657$	0	0
$658$	23.5260	0.917139
$659$	4.03435	0.157156	0.0785779	−	0.996908i	$-0.474962\pi$
$659$	4.03435	0.157156	0.0785779	+	0.996908i	$0.474962\pi$
$660$	0	0
$661$	35.8020	1.39254	0.696268	−	0.717781i	$-0.254842\pi$
$661$	35.8020	1.39254	0.696268	+	0.717781i	$0.254842\pi$
$662$	0.451476	0.0175471
$663$	0	0
$664$	10.0467	0.389888
$665$	20.3835	0.790437
$666$	0	0
$667$	35.0469	1.35702
$668$	−20.7980	−0.804697
$669$	0	0
$670$	−6.09978	−0.235655
$671$	9.16926	0.353975
$672$	0	0
$673$	−46.2152	−1.78146	−0.890732	−	0.454529i	$-0.849808\pi$
$673$	−46.2152	−1.78146	−0.890732	+	0.454529i	$0.849808\pi$
$674$	28.3550	1.09220
$675$	0	0
$676$	0.975364	0.0375140
$677$	25.9315	0.996630	0.498315	−	0.866996i	$-0.333952\pi$
$677$	25.9315	0.996630	0.498315	+	0.866996i	$0.333952\pi$
$678$	0	0
$679$	−24.4710	−0.939110
$680$	−4.63408	−0.177709
$681$	0	0
$682$	3.72096	0.142483
$683$	−33.8894	−1.29674	−0.648372	−	0.761324i	$-0.724550\pi$
$683$	−33.8894	−1.29674	−0.648372	+	0.761324i	$0.724550\pi$
$684$	0	0
$685$	21.6912	0.828778
$686$	−6.89315	−0.263182
$687$	0	0
$688$	−10.0357	−0.382606
$689$	−32.2773	−1.22967
$690$	0	0
$691$	−17.7987	−0.677093	−0.338547	−	0.940950i	$-0.609935\pi$
$691$	−17.7987	−0.677093	−0.338547	+	0.940950i	$0.609935\pi$
$692$	5.15735	0.196053
$693$	0	0
$694$	5.37695	0.204106
$695$	−5.31285	−0.201528
$696$	0	0
$697$	−13.4190	−0.508280
$698$	−8.22798	−0.311434
$699$	0	0
$700$	−3.46569	−0.130991
$701$	−52.6596	−1.98893	−0.994463	−	0.105092i	$-0.966486\pi$
$701$	−52.6596	−1.98893	−0.994463	+	0.105092i	$0.966486\pi$
$702$	0	0
$703$	−5.88150	−0.221825
$704$	−4.15417	−0.156566
$705$	0	0
$706$	5.95531	0.224131
$707$	−56.2930	−2.11711
$708$	0	0
$709$	−20.4689	−0.768725	−0.384362	−	0.923182i	$-0.625579\pi$
$709$	−20.4689	−0.768725	−0.384362	+	0.923182i	$0.625579\pi$
$710$	13.1400	0.493134
$711$	0	0
$712$	−12.8783	−0.482635
$713$	−3.90659	−0.146303
$714$	0	0
$715$	−15.5298	−0.580782
$716$	−1.66323	−0.0621576
$717$	0	0
$718$	12.0713	0.450498
$719$	−15.2618	−0.569169	−0.284584	−	0.958651i	$-0.591856\pi$
$719$	−15.2618	−0.569169	−0.284584	+	0.958651i	$0.591856\pi$
$720$	0	0
$721$	−23.5260	−0.876154
$722$	−15.5920	−0.580276
$723$	0	0
$724$	−15.8628	−0.589535
$725$	−8.03567	−0.298437
$726$	0	0
$727$	−3.94359	−0.146260	−0.0731298	−	0.997322i	$-0.523299\pi$
$727$	−3.94359	−0.146260	−0.0731298	+	0.997322i	$0.523299\pi$
$728$	−12.9560	−0.480182
$729$	0	0
$730$	−7.19302	−0.266226
$731$	46.5061	1.72009
$732$	0	0
$733$	39.2728	1.45057	0.725286	−	0.688448i	$-0.241707\pi$
$733$	39.2728	1.45057	0.725286	+	0.688448i	$0.241707\pi$
$734$	0.139243	0.00513957
$735$	0	0
$736$	4.36141	0.160764
$737$	25.3395	0.933393
$738$	0	0
$739$	−9.92688	−0.365166	−0.182583	−	0.983190i	$-0.558446\pi$
$739$	−9.92688	−0.365166	−0.182583	+	0.983190i	$0.558446\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	29.9231	1.09851
$743$	14.8524	0.544880	0.272440	−	0.962173i	$-0.412169\pi$
$743$	14.8524	0.544880	0.272440	+	0.962173i	$0.412169\pi$
$744$	0	0
$745$	22.6510	0.829869
$746$	6.00000	0.219676
$747$	0	0
$748$	19.2508	0.703878
$749$	−30.0467	−1.09788
$750$	0	0
$751$	−11.4393	−0.417425	−0.208713	−	0.977977i	$-0.566927\pi$
$751$	−11.4393	−0.417425	−0.208713	+	0.977977i	$0.566927\pi$
$752$	6.78825	0.247542
$753$	0	0
$754$	−30.0403	−1.09400
$755$	−3.94693	−0.143643
$756$	0	0
$757$	−41.1646	−1.49615	−0.748076	−	0.663613i	$-0.769022\pi$
$757$	−41.1646	−1.49615	−0.748076	+	0.663613i	$0.769022\pi$
$758$	5.33950	0.193939
$759$	0	0
$760$	5.88150	0.213344
$761$	−31.2041	−1.13115	−0.565573	−	0.824698i	$-0.691345\pi$
$761$	−31.2041	−1.13115	−0.565573	+	0.824698i	$0.691345\pi$
$762$	0	0
$763$	57.3998	2.07801
$764$	19.2287	0.695670
$765$	0	0
$766$	−13.7384	−0.496387
$767$	25.9120	0.935630
$768$	0	0
$769$	−11.2902	−0.407136	−0.203568	−	0.979061i	$-0.565254\pi$
$769$	−11.2902	−0.407136	−0.203568	+	0.979061i	$0.565254\pi$
$770$	14.3971	0.518835
$771$	0	0
$772$	−12.7883	−0.460259
$773$	−3.18844	−0.114680	−0.0573400	−	0.998355i	$-0.518262\pi$
$773$	−3.18844	−0.114680	−0.0573400	+	0.998355i	$0.518262\pi$
$774$	0	0
$775$	0.895717	0.0321751
$776$	−7.06092	−0.253472
$777$	0	0
$778$	−1.61217	−0.0577992
$779$	17.0312	0.610205
$780$	0	0
$781$	−54.5856	−1.95323
$782$	−20.2111	−0.722749
$783$	0	0
$784$	5.01103	0.178965
$785$	−20.7585	−0.740902
$786$	0	0
$787$	3.56345	0.127023	0.0635116	−	0.997981i	$-0.479770\pi$
$787$	3.56345	0.127023	0.0635116	+	0.997981i	$0.479770\pi$
$788$	10.3614	0.369110
$789$	0	0
$790$	4.00000	0.142314
$791$	−55.5747	−1.97601
$792$	0	0
$793$	8.25147	0.293018
$794$	17.4767	0.620226
$795$	0	0
$796$	10.6230	0.376524
$797$	−16.1517	−0.572122	−0.286061	−	0.958211i	$-0.592346\pi$
$797$	−16.1517	−0.572122	−0.286061	+	0.958211i	$0.592346\pi$
$798$	0	0
$799$	−31.4573	−1.11288
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−4.87832	−0.172259
$803$	29.8810	1.05448
$804$	0	0
$805$	−15.1153	−0.532745
$806$	3.34852	0.117947
$807$	0	0
$808$	−16.2429	−0.571424
$809$	12.7501	0.448270	0.224135	−	0.974558i	$-0.428044\pi$
$809$	12.7501	0.448270	0.224135	+	0.974558i	$0.428044\pi$
$810$	0	0
$811$	−18.9535	−0.665546	−0.332773	−	0.943007i	$-0.607984\pi$
$811$	−18.9535	−0.665546	−0.332773	+	0.943007i	$0.607984\pi$
$812$	27.8492	0.977314
$813$	0	0
$814$	−4.15417	−0.145604
$815$	−19.6652	−0.688843
$816$	0	0
$817$	−59.0248	−2.06502
$818$	13.9626	0.488189
$819$	0	0
$820$	−2.89572	−0.101123
$821$	3.28689	0.114713	0.0573566	−	0.998354i	$-0.481733\pi$
$821$	3.28689	0.114713	0.0573566	+	0.998354i	$0.481733\pi$
$822$	0	0
$823$	42.4612	1.48010	0.740052	−	0.672550i	$-0.234801\pi$
$823$	42.4612	1.48010	0.740052	+	0.672550i	$0.234801\pi$
$824$	−6.78825	−0.236480
$825$	0	0
$826$	−24.0221	−0.835835
$827$	−36.9296	−1.28417	−0.642084	−	0.766634i	$-0.721930\pi$
$827$	−36.9296	−1.28417	−0.642084	+	0.766634i	$0.721930\pi$
$828$	0	0
$829$	−49.7515	−1.72794	−0.863971	−	0.503542i	$-0.832030\pi$
$829$	−49.7515	−1.72794	−0.863971	+	0.503542i	$0.832030\pi$
$830$	−10.0467	−0.348726
$831$	0	0
$832$	−3.73836	−0.129604
$833$	−23.2215	−0.804579
$834$	0	0
$835$	20.7980	0.719743
$836$	−24.4328	−0.845024
$837$	0	0
$838$	4.85034	0.167552
$839$	−42.6569	−1.47268	−0.736341	−	0.676611i	$-0.763448\pi$
$839$	−42.6569	−1.47268	−0.736341	+	0.676611i	$0.763448\pi$
$840$	0	0
$841$	35.5720	1.22662
$842$	−28.0280	−0.965908
$843$	0	0
$844$	4.06410	0.139892
$845$	−0.975364	−0.0335535
$846$	0	0
$847$	−21.6853	−0.745115
$848$	8.63408	0.296496
$849$	0	0
$850$	4.63408	0.158948
$851$	4.36141	0.149507
$852$	0	0
$853$	−12.0246	−0.411716	−0.205858	−	0.978582i	$-0.565998\pi$
$853$	−12.0246	−0.411716	−0.205858	+	0.978582i	$0.565998\pi$
$854$	−7.64962	−0.261765
$855$	0	0
$856$	−8.66975	−0.296326
$857$	−19.2792	−0.658565	−0.329282	−	0.944231i	$-0.606807\pi$
$857$	−19.2792	−0.658565	−0.329282	+	0.944231i	$0.606807\pi$
$858$	0	0
$859$	−34.0811	−1.16283	−0.581415	−	0.813607i	$-0.697501\pi$
$859$	−34.0811	−1.16283	−0.581415	+	0.813607i	$0.697501\pi$
$860$	10.0357	0.342214
$861$	0	0
$862$	−30.4191	−1.03608
$863$	13.6709	0.465363	0.232682	−	0.972553i	$-0.425250\pi$
$863$	13.6709	0.465363	0.232682	+	0.972553i	$0.425250\pi$
$864$	0	0
$865$	−5.15735	−0.175355
$866$	−9.52980	−0.323836
$867$	0	0
$868$	−3.10428	−0.105366
$869$	−16.6167	−0.563682
$870$	0	0
$871$	22.8032	0.772656
$872$	16.5623	0.560870
$873$	0	0
$874$	25.6516	0.867680
$875$	3.46569	0.117162
$876$	0	0
$877$	−54.4931	−1.84010	−0.920050	−	0.391801i	$-0.871852\pi$
$877$	−54.4931	−1.84010	−0.920050	+	0.391801i	$0.871852\pi$
$878$	−11.9580	−0.403562
$879$	0	0
$880$	4.15417	0.140037
$881$	−55.4529	−1.86826	−0.934128	−	0.356939i	$-0.883820\pi$
$881$	−55.4529	−1.86826	−0.934128	+	0.356939i	$0.883820\pi$
$882$	0	0
$883$	−32.5189	−1.09435	−0.547174	−	0.837019i	$-0.684297\pi$
$883$	−32.5189	−1.09435	−0.547174	+	0.837019i	$0.684297\pi$
$884$	17.3239	0.582665
$885$	0	0
$886$	18.8252	0.632446
$887$	−0.930064	−0.0312285	−0.0156142	−	0.999878i	$-0.504970\pi$
$887$	−0.930064	−0.0312285	−0.0156142	+	0.999878i	$0.504970\pi$
$888$	0	0
$889$	10.7967	0.362111
$890$	12.8783	0.431682
$891$	0	0
$892$	−9.61854	−0.322053
$893$	39.9251	1.33604
$894$	0	0
$895$	1.66323	0.0555955
$896$	3.46569	0.115781
$897$	0	0
$898$	18.8316	0.628419
$899$	−7.19769	−0.240056
$900$	0	0
$901$	−40.0110	−1.33296
$902$	12.0293	0.400532
$903$	0	0
$904$	−16.0357	−0.533338
$905$	15.8628	0.527296
$906$	0	0
$907$	−7.68786	−0.255271	−0.127636	−	0.991821i	$-0.540739\pi$
$907$	−7.68786	−0.255271	−0.127636	+	0.991821i	$0.540739\pi$
$908$	11.6122	0.385363
$909$	0	0
$910$	12.9560	0.429488
$911$	49.3021	1.63345	0.816725	−	0.577027i	$-0.195787\pi$
$911$	49.3021	1.63345	0.816725	+	0.577027i	$0.195787\pi$
$912$	0	0
$913$	41.7357	1.38125
$914$	−6.58692	−0.217876
$915$	0	0
$916$	23.0312	0.760971
$917$	41.8356	1.38153
$918$	0	0
$919$	−22.5662	−0.744390	−0.372195	−	0.928155i	$-0.621395\pi$
$919$	−22.5662	−0.744390	−0.372195	+	0.928155i	$0.621395\pi$
$920$	−4.36141	−0.143791
$921$	0	0
$922$	−28.9386	−0.953043
$923$	−49.1219	−1.61687
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−22.6600	−0.744655
$927$	0	0
$928$	8.03567	0.263784
$929$	49.4659	1.62292	0.811462	−	0.584405i	$-0.198672\pi$
$929$	49.4659	1.62292	0.811462	+	0.584405i	$0.198672\pi$
$930$	0	0
$931$	29.4724	0.965919
$932$	1.42031	0.0465239
$933$	0	0
$934$	−31.6835	−1.03672
$935$	−19.2508	−0.629567
$936$	0	0
$937$	38.1842	1.24742	0.623711	−	0.781655i	$-0.285624\pi$
$937$	38.1842	1.24742	0.623711	+	0.781655i	$0.285624\pi$
$938$	−21.1400	−0.690244
$939$	0	0
$940$	−6.78825	−0.221408
$941$	−0.240264	−0.00783238	−0.00391619	−	0.999992i	$-0.501247\pi$
$941$	−0.240264	−0.00783238	−0.00391619	+	0.999992i	$0.501247\pi$
$942$	0	0
$943$	−12.6294	−0.411270
$944$	−6.93139	−0.225597
$945$	0	0
$946$	−41.6899	−1.35545
$947$	−5.29078	−0.171927	−0.0859636	−	0.996298i	$-0.527397\pi$
$947$	−5.29078	−0.171927	−0.0859636	+	0.996298i	$0.527397\pi$
$948$	0	0
$949$	26.8901	0.872891
$950$	−5.88150	−0.190821
$951$	0	0
$952$	−16.0603	−0.520518
$953$	5.28946	0.171342	0.0856712	−	0.996323i	$-0.472697\pi$
$953$	5.28946	0.171342	0.0856712	+	0.996323i	$0.472697\pi$
$954$	0	0
$955$	−19.2287	−0.622226
$956$	18.4502	0.596721
$957$	0	0
$958$	−2.87832	−0.0929942
$959$	75.1751	2.42753
$960$	0	0
$961$	−30.1977	−0.974119
$962$	−3.73836	−0.120530
$963$	0	0
$964$	−26.8162	−0.863693
$965$	12.7883	0.411668
$966$	0	0
$967$	51.1679	1.64545	0.822725	−	0.568440i	$-0.192453\pi$
$967$	51.1679	1.64545	0.822725	+	0.568440i	$0.192453\pi$
$968$	−6.25713	−0.201112
$969$	0	0
$970$	7.06092	0.226713
$971$	15.9354	0.511393	0.255696	−	0.966757i	$-0.417695\pi$
$971$	15.9354	0.511393	0.255696	+	0.966757i	$0.417695\pi$
$972$	0	0
$973$	−18.4127	−0.590284
$974$	−21.4334	−0.686769
$975$	0	0
$976$	−2.20724	−0.0706521
$977$	15.3090	0.489780	0.244890	−	0.969551i	$-0.421248\pi$
$977$	15.3090	0.489780	0.244890	+	0.969551i	$0.421248\pi$
$978$	0	0
$979$	−53.4987	−1.70983
$980$	−5.01103	−0.160072
$981$	0	0
$982$	−1.04989	−0.0335033
$983$	−56.8019	−1.81170	−0.905849	−	0.423601i	$-0.860766\pi$
$983$	−56.8019	−1.81170	−0.905849	+	0.423601i	$0.860766\pi$
$984$	0	0
$985$	−10.3614	−0.330142
$986$	−37.2379	−1.18590
$987$	0	0
$988$	−21.9872	−0.699506
$989$	43.7697	1.39180
$990$	0	0
$991$	54.9232	1.74469	0.872347	−	0.488887i	$-0.162597\pi$
$991$	54.9232	1.74469	0.872347	+	0.488887i	$0.162597\pi$
$992$	−0.895717	−0.0284391
$993$	0	0
$994$	45.5391	1.44441
$995$	−10.6230	−0.336773
$996$	0	0
$997$	−44.4612	−1.40810	−0.704050	−	0.710150i	$-0.748627\pi$
$997$	−44.4612	−1.40810	−0.704050	+	0.710150i	$0.748627\pi$
$998$	−14.1898	−0.449172
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bj.1.1		4
3.2	odd	2		1110.2.a.s.1.1	✓	4
12.11	even	2		8880.2.a.cg.1.4		4
15.14	odd	2		5550.2.a.cj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.s.1.1	✓	4	3.2	odd	2
3330.2.a.bj.1.1		4	1.1	even	1	trivial
5550.2.a.cj.1.4		4	15.14	odd	2
8880.2.a.cg.1.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.46569 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.46569 q^{7} -1.00000 q^{8} +1.00000 q^{10} -4.15417 q^{11} -3.73836 q^{13} +3.46569 q^{14} +1.00000 q^{16} -4.63408 q^{17} +5.88150 q^{19} -1.00000 q^{20} +4.15417 q^{22} -4.36141 q^{23} +1.00000 q^{25} +3.73836 q^{26} -3.46569 q^{28} -8.03567 q^{29} +0.895717 q^{31} -1.00000 q^{32} +4.63408 q^{34} +3.46569 q^{35} -1.00000 q^{37} -5.88150 q^{38} +1.00000 q^{40} +2.89572 q^{41} -10.0357 q^{43} -4.15417 q^{44} +4.36141 q^{46} +6.78825 q^{47} +5.01103 q^{49} -1.00000 q^{50} -3.73836 q^{52} +8.63408 q^{53} +4.15417 q^{55} +3.46569 q^{56} +8.03567 q^{58} -6.93139 q^{59} -2.20724 q^{61} -0.895717 q^{62} +1.00000 q^{64} +3.73836 q^{65} -6.09978 q^{67} -4.63408 q^{68} -3.46569 q^{70} +13.1400 q^{71} -7.19302 q^{73} +1.00000 q^{74} +5.88150 q^{76} +14.3971 q^{77} +4.00000 q^{79} -1.00000 q^{80} -2.89572 q^{82} -10.0467 q^{83} +4.63408 q^{85} +10.0357 q^{86} +4.15417 q^{88} +12.8783 q^{89} +12.9560 q^{91} -4.36141 q^{92} -6.78825 q^{94} -5.88150 q^{95} +7.06092 q^{97} -5.01103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 2 q^{11} - 3 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} + 3 q^{19} - 4 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} + 3 q^{26} + 4 q^{28} + 3 q^{29} + 3 q^{31} - 4 q^{32} + 6 q^{34} - 4 q^{35} - 4 q^{37} - 3 q^{38} + 4 q^{40} + 11 q^{41} - 5 q^{43} - 2 q^{44} - q^{46} + 14 q^{49} - 4 q^{50} - 3 q^{52} + 22 q^{53} + 2 q^{55} - 4 q^{56} - 3 q^{58} + 8 q^{59} - 5 q^{61} - 3 q^{62} + 4 q^{64} + 3 q^{65} + 6 q^{67} - 6 q^{68} + 4 q^{70} + 18 q^{71} - 5 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 16 q^{79} - 4 q^{80} - 11 q^{82} + q^{83} + 6 q^{85} + 5 q^{86} + 2 q^{88} + 5 q^{89} - 13 q^{91} + q^{92} - 3 q^{95} + 7 q^{97} - 14 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 2 * q^11 - 3 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 + 3 * q^19 - 4 * q^20 + 2 * q^22 + q^23 + 4 * q^25 + 3 * q^26 + 4 * q^28 + 3 * q^29 + 3 * q^31 - 4 * q^32 + 6 * q^34 - 4 * q^35 - 4 * q^37 - 3 * q^38 + 4 * q^40 + 11 * q^41 - 5 * q^43 - 2 * q^44 - q^46 + 14 * q^49 - 4 * q^50 - 3 * q^52 + 22 * q^53 + 2 * q^55 - 4 * q^56 - 3 * q^58 + 8 * q^59 - 5 * q^61 - 3 * q^62 + 4 * q^64 + 3 * q^65 + 6 * q^67 - 6 * q^68 + 4 * q^70 + 18 * q^71 - 5 * q^73 + 4 * q^74 + 3 * q^76 + 4 * q^77 + 16 * q^79 - 4 * q^80 - 11 * q^82 + q^83 + 6 * q^85 + 5 * q^86 + 2 * q^88 + 5 * q^89 - 13 * q^91 + q^92 - 3 * q^95 + 7 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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