LMFDB - Embedded newform 3330.2.a.bc.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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -2.12311 q^{11} -6.68466 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -3.56155 q^{19} +1.00000 q^{20} +2.12311 q^{22} +2.43845 q^{23} +1.00000 q^{25} +6.68466 q^{26} -1.00000 q^{28} -5.68466 q^{29} +0.561553 q^{31} -1.00000 q^{32} -5.00000 q^{34} -1.00000 q^{35} +1.00000 q^{37} +3.56155 q^{38} -1.00000 q^{40} +3.68466 q^{41} -0.315342 q^{43} -2.12311 q^{44} -2.43845 q^{46} +8.00000 q^{47} -6.00000 q^{49} -1.00000 q^{50} -6.68466 q^{52} +11.2462 q^{53} -2.12311 q^{55} +1.00000 q^{56} +5.68466 q^{58} +12.2462 q^{59} +6.80776 q^{61} -0.561553 q^{62} +1.00000 q^{64} -6.68466 q^{65} +4.87689 q^{67} +5.00000 q^{68} +1.00000 q^{70} +7.12311 q^{71} -2.43845 q^{73} -1.00000 q^{74} -3.56155 q^{76} +2.12311 q^{77} -2.24621 q^{79} +1.00000 q^{80} -3.68466 q^{82} +11.5616 q^{83} +5.00000 q^{85} +0.315342 q^{86} +2.12311 q^{88} +6.68466 q^{89} +6.68466 q^{91} +2.43845 q^{92} -8.00000 q^{94} -3.56155 q^{95} +15.6847 q^{97} +6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 4 q^{11} - q^{13} + 2 q^{14} + 2 q^{16} + 10 q^{17} - 3 q^{19} + 2 q^{20} - 4 q^{22} + 9 q^{23} + 2 q^{25} + q^{26} - 2 q^{28} + q^{29} - 3 q^{31} - 2 q^{32} - 10 q^{34} - 2 q^{35} + 2 q^{37} + 3 q^{38} - 2 q^{40} - 5 q^{41} - 13 q^{43} + 4 q^{44} - 9 q^{46} + 16 q^{47} - 12 q^{49} - 2 q^{50} - q^{52} + 6 q^{53} + 4 q^{55} + 2 q^{56} - q^{58} + 8 q^{59} - 7 q^{61} + 3 q^{62} + 2 q^{64} - q^{65} + 18 q^{67} + 10 q^{68} + 2 q^{70} + 6 q^{71} - 9 q^{73} - 2 q^{74} - 3 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 5 q^{82} + 19 q^{83} + 10 q^{85} + 13 q^{86} - 4 q^{88} + q^{89} + q^{91} + 9 q^{92} - 16 q^{94} - 3 q^{95} + 19 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 4 * q^11 - q^13 + 2 * q^14 + 2 * q^16 + 10 * q^17 - 3 * q^19 + 2 * q^20 - 4 * q^22 + 9 * q^23 + 2 * q^25 + q^26 - 2 * q^28 + q^29 - 3 * q^31 - 2 * q^32 - 10 * q^34 - 2 * q^35 + 2 * q^37 + 3 * q^38 - 2 * q^40 - 5 * q^41 - 13 * q^43 + 4 * q^44 - 9 * q^46 + 16 * q^47 - 12 * q^49 - 2 * q^50 - q^52 + 6 * q^53 + 4 * q^55 + 2 * q^56 - q^58 + 8 * q^59 - 7 * q^61 + 3 * q^62 + 2 * q^64 - q^65 + 18 * q^67 + 10 * q^68 + 2 * q^70 + 6 * q^71 - 9 * q^73 - 2 * q^74 - 3 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 5 * q^82 + 19 * q^83 + 10 * q^85 + 13 * q^86 - 4 * q^88 + q^89 + q^91 + 9 * q^92 - 16 * q^94 - 3 * q^95 + 19 * q^97 + 12 * 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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−2.12311	−0.640140	−0.320070	−	0.947394i	$-0.603707\pi$
$11$	−2.12311	−0.640140	−0.320070	+	0.947394i	$0.603707\pi$
$12$	0	0
$13$	−6.68466	−1.85399	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−6.68466	−1.85399	−0.926995	+	0.375073i	$0.877618\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	−3.56155	−0.817076	−0.408538	−	0.912741i	$-0.633961\pi$
$19$	−3.56155	−0.817076	−0.408538	+	0.912741i	$0.633961\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	2.12311	0.452648
$23$	2.43845	0.508451	0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845	0.508451	0.254226	+	0.967145i	$0.418179\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.68466	1.31097
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	0.561553	0.100858	0.0504289	−	0.998728i	$-0.483941\pi$
$31$	0.561553	0.100858	0.0504289	+	0.998728i	$0.483941\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.00000	−0.857493
$35$	−1.00000	−0.169031
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	3.56155	0.577760
$39$	0	0
$40$	−1.00000	−0.158114
$41$	3.68466	0.575447	0.287723	−	0.957714i	$-0.407102\pi$
$41$	3.68466	0.575447	0.287723	+	0.957714i	$0.407102\pi$
$42$	0	0
$43$	−0.315342	−0.0480891	−0.0240446	−	0.999711i	$-0.507654\pi$
$43$	−0.315342	−0.0480891	−0.0240446	+	0.999711i	$0.507654\pi$
$44$	−2.12311	−0.320070
$45$	0	0
$46$	−2.43845	−0.359529
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−6.68466	−0.926995
$53$	11.2462	1.54479	0.772393	−	0.635145i	$-0.219060\pi$
$53$	11.2462	1.54479	0.772393	+	0.635145i	$0.219060\pi$
$54$	0	0
$55$	−2.12311	−0.286280
$56$	1.00000	0.133631
$57$	0	0
$58$	5.68466	0.746432
$59$	12.2462	1.59432	0.797160	−	0.603768i	$-0.206334\pi$
$59$	12.2462	1.59432	0.797160	+	0.603768i	$0.206334\pi$
$60$	0	0
$61$	6.80776	0.871645	0.435822	−	0.900033i	$-0.356458\pi$
$61$	6.80776	0.871645	0.435822	+	0.900033i	$0.356458\pi$
$62$	−0.561553	−0.0713173
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.68466	−0.829130
$66$	0	0
$67$	4.87689	0.595807	0.297904	−	0.954596i	$-0.403713\pi$
$67$	4.87689	0.595807	0.297904	+	0.954596i	$0.403713\pi$
$68$	5.00000	0.606339
$69$	0	0
$70$	1.00000	0.119523
$71$	7.12311	0.845357	0.422679	−	0.906280i	$-0.361090\pi$
$71$	7.12311	0.845357	0.422679	+	0.906280i	$0.361090\pi$
$72$	0	0
$73$	−2.43845	−0.285399	−0.142699	−	0.989766i	$-0.545578\pi$
$73$	−2.43845	−0.285399	−0.142699	+	0.989766i	$0.545578\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−3.56155	−0.408538
$77$	2.12311	0.241950
$78$	0	0
$79$	−2.24621	−0.252719	−0.126359	−	0.991985i	$-0.540329\pi$
$79$	−2.24621	−0.252719	−0.126359	+	0.991985i	$0.540329\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−3.68466	−0.406902
$83$	11.5616	1.26905	0.634523	−	0.772904i	$-0.281197\pi$
$83$	11.5616	1.26905	0.634523	+	0.772904i	$0.281197\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0.315342	0.0340042
$87$	0	0
$88$	2.12311	0.226324
$89$	6.68466	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$89$	6.68466	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$90$	0	0
$91$	6.68466	0.700743
$92$	2.43845	0.254226
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−3.56155	−0.365408
$96$	0	0
$97$	15.6847	1.59254	0.796268	−	0.604944i	$-0.206805\pi$
$97$	15.6847	1.59254	0.796268	+	0.604944i	$0.206805\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	1.00000	0.100000
$101$	−19.1231	−1.90282	−0.951410	−	0.307927i	$-0.900365\pi$
$101$	−19.1231	−1.90282	−0.951410	+	0.307927i	$0.900365\pi$
$102$	0	0
$103$	16.4924	1.62505	0.812523	−	0.582929i	$-0.198093\pi$
$103$	16.4924	1.62505	0.812523	+	0.582929i	$0.198093\pi$
$104$	6.68466	0.655485
$105$	0	0
$106$	−11.2462	−1.09233
$107$	−9.80776	−0.948152	−0.474076	−	0.880484i	$-0.657218\pi$
$107$	−9.80776	−0.948152	−0.474076	+	0.880484i	$0.657218\pi$
$108$	0	0
$109$	6.12311	0.586487	0.293244	−	0.956038i	$-0.405265\pi$
$109$	6.12311	0.586487	0.293244	+	0.956038i	$0.405265\pi$
$110$	2.12311	0.202430
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−1.68466	−0.158479	−0.0792397	−	0.996856i	$-0.525249\pi$
$113$	−1.68466	−0.158479	−0.0792397	+	0.996856i	$0.525249\pi$
$114$	0	0
$115$	2.43845	0.227386
$116$	−5.68466	−0.527807
$117$	0	0
$118$	−12.2462	−1.12736
$119$	−5.00000	−0.458349
$120$	0	0
$121$	−6.49242	−0.590220
$122$	−6.80776	−0.616346
$123$	0	0
$124$	0.561553	0.0504289
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4384	0.926262	0.463131	−	0.886290i	$-0.346726\pi$
$127$	10.4384	0.926262	0.463131	+	0.886290i	$0.346726\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.68466	0.586283
$131$	1.12311	0.0981262	0.0490631	−	0.998796i	$-0.484376\pi$
$131$	1.12311	0.0981262	0.0490631	+	0.998796i	$0.484376\pi$
$132$	0	0
$133$	3.56155	0.308826
$134$	−4.87689	−0.421300
$135$	0	0
$136$	−5.00000	−0.428746
$137$	−12.2462	−1.04626	−0.523132	−	0.852252i	$-0.675237\pi$
$137$	−12.2462	−1.04626	−0.523132	+	0.852252i	$0.675237\pi$
$138$	0	0
$139$	−6.80776	−0.577427	−0.288714	−	0.957416i	$-0.593228\pi$
$139$	−6.80776	−0.577427	−0.288714	+	0.957416i	$0.593228\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−7.12311	−0.597758
$143$	14.1922	1.18681
$144$	0	0
$145$	−5.68466	−0.472085
$146$	2.43845	0.201807
$147$	0	0
$148$	1.00000	0.0821995
$149$	−24.2462	−1.98633	−0.993164	−	0.116731i	$-0.962758\pi$
$149$	−24.2462	−1.98633	−0.993164	+	0.116731i	$0.962758\pi$
$150$	0	0
$151$	−19.5616	−1.59190	−0.795948	−	0.605365i	$-0.793027\pi$
$151$	−19.5616	−1.59190	−0.795948	+	0.605365i	$0.793027\pi$
$152$	3.56155	0.288880
$153$	0	0
$154$	−2.12311	−0.171085
$155$	0.561553	0.0451050
$156$	0	0
$157$	19.0540	1.52067	0.760336	−	0.649530i	$-0.225034\pi$
$157$	19.0540	1.52067	0.760336	+	0.649530i	$0.225034\pi$
$158$	2.24621	0.178699
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−2.43845	−0.192177
$162$	0	0
$163$	−13.4924	−1.05681	−0.528404	−	0.848993i	$-0.677209\pi$
$163$	−13.4924	−1.05681	−0.528404	+	0.848993i	$0.677209\pi$
$164$	3.68466	0.287723
$165$	0	0
$166$	−11.5616	−0.897351
$167$	18.0540	1.39706	0.698529	−	0.715581i	$-0.253838\pi$
$167$	18.0540	1.39706	0.698529	+	0.715581i	$0.253838\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	−5.00000	−0.383482
$171$	0	0
$172$	−0.315342	−0.0240446
$173$	−6.12311	−0.465531	−0.232766	−	0.972533i	$-0.574777\pi$
$173$	−6.12311	−0.465531	−0.232766	+	0.972533i	$0.574777\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−2.12311	−0.160035
$177$	0	0
$178$	−6.68466	−0.501036
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	3.75379	0.279017	0.139508	−	0.990221i	$-0.455448\pi$
$181$	3.75379	0.279017	0.139508	+	0.990221i	$0.455448\pi$
$182$	−6.68466	−0.495500
$183$	0	0
$184$	−2.43845	−0.179765
$185$	1.00000	0.0735215
$186$	0	0
$187$	−10.6155	−0.776284
$188$	8.00000	0.583460
$189$	0	0
$190$	3.56155	0.258382
$191$	10.3693	0.750297	0.375149	−	0.926965i	$-0.377592\pi$
$191$	10.3693	0.750297	0.375149	+	0.926965i	$0.377592\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−15.6847	−1.12609
$195$	0	0
$196$	−6.00000	−0.428571
$197$	14.6847	1.04624	0.523119	−	0.852259i	$-0.324768\pi$
$197$	14.6847	1.04624	0.523119	+	0.852259i	$0.324768\pi$
$198$	0	0
$199$	16.4924	1.16912	0.584558	−	0.811352i	$-0.301268\pi$
$199$	16.4924	1.16912	0.584558	+	0.811352i	$0.301268\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	19.1231	1.34550
$203$	5.68466	0.398985
$204$	0	0
$205$	3.68466	0.257348
$206$	−16.4924	−1.14908
$207$	0	0
$208$	−6.68466	−0.463498
$209$	7.56155	0.523044
$210$	0	0
$211$	8.56155	0.589402	0.294701	−	0.955590i	$-0.404780\pi$
$211$	8.56155	0.589402	0.294701	+	0.955590i	$0.404780\pi$
$212$	11.2462	0.772393
$213$	0	0
$214$	9.80776	0.670445
$215$	−0.315342	−0.0215061
$216$	0	0
$217$	−0.561553	−0.0381207
$218$	−6.12311	−0.414709
$219$	0	0
$220$	−2.12311	−0.143140
$221$	−33.4233	−2.24829
$222$	0	0
$223$	0.807764	0.0540919	0.0270459	−	0.999634i	$-0.491390\pi$
$223$	0.807764	0.0540919	0.0270459	+	0.999634i	$0.491390\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	1.68466	0.112062
$227$	−17.9309	−1.19011	−0.595057	−	0.803684i	$-0.702870\pi$
$227$	−17.9309	−1.19011	−0.595057	+	0.803684i	$0.702870\pi$
$228$	0	0
$229$	3.12311	0.206381	0.103190	−	0.994662i	$-0.467095\pi$
$229$	3.12311	0.206381	0.103190	+	0.994662i	$0.467095\pi$
$230$	−2.43845	−0.160786
$231$	0	0
$232$	5.68466	0.373216
$233$	13.3693	0.875853	0.437927	−	0.899011i	$-0.355713\pi$
$233$	13.3693	0.875853	0.437927	+	0.899011i	$0.355713\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	12.2462	0.797160
$237$	0	0
$238$	5.00000	0.324102
$239$	12.8078	0.828465	0.414233	−	0.910171i	$-0.364050\pi$
$239$	12.8078	0.828465	0.414233	+	0.910171i	$0.364050\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	6.49242	0.417349
$243$	0	0
$244$	6.80776	0.435822
$245$	−6.00000	−0.383326
$246$	0	0
$247$	23.8078	1.51485
$248$	−0.561553	−0.0356586
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	15.6155	0.985643	0.492822	−	0.870130i	$-0.335965\pi$
$251$	15.6155	0.985643	0.492822	+	0.870130i	$0.335965\pi$
$252$	0	0
$253$	−5.17708	−0.325480
$254$	−10.4384	−0.654966
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.80776	0.611792	0.305896	−	0.952065i	$-0.401044\pi$
$257$	9.80776	0.611792	0.305896	+	0.952065i	$0.401044\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	−6.68466	−0.414565
$261$	0	0
$262$	−1.12311	−0.0693857
$263$	10.5616	0.651253	0.325627	−	0.945498i	$-0.394425\pi$
$263$	10.5616	0.651253	0.325627	+	0.945498i	$0.394425\pi$
$264$	0	0
$265$	11.2462	0.690849
$266$	−3.56155	−0.218373
$267$	0	0
$268$	4.87689	0.297904
$269$	−3.80776	−0.232163	−0.116082	−	0.993240i	$-0.537033\pi$
$269$	−3.80776	−0.232163	−0.116082	+	0.993240i	$0.537033\pi$
$270$	0	0
$271$	16.4924	1.00184	0.500922	−	0.865493i	$-0.332994\pi$
$271$	16.4924	1.00184	0.500922	+	0.865493i	$0.332994\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	12.2462	0.739821
$275$	−2.12311	−0.128028
$276$	0	0
$277$	15.3153	0.920210	0.460105	−	0.887865i	$-0.347812\pi$
$277$	15.3153	0.920210	0.460105	+	0.887865i	$0.347812\pi$
$278$	6.80776	0.408303
$279$	0	0
$280$	1.00000	0.0597614
$281$	23.1771	1.38263	0.691314	−	0.722554i	$-0.257032\pi$
$281$	23.1771	1.38263	0.691314	+	0.722554i	$0.257032\pi$
$282$	0	0
$283$	−18.9309	−1.12532	−0.562662	−	0.826687i	$-0.690223\pi$
$283$	−18.9309	−1.12532	−0.562662	+	0.826687i	$0.690223\pi$
$284$	7.12311	0.422679
$285$	0	0
$286$	−14.1922	−0.839205
$287$	−3.68466	−0.217499
$288$	0	0
$289$	8.00000	0.470588
$290$	5.68466	0.333815
$291$	0	0
$292$	−2.43845	−0.142699
$293$	−24.6155	−1.43805	−0.719027	−	0.694982i	$-0.755412\pi$
$293$	−24.6155	−1.43805	−0.719027	+	0.694982i	$0.755412\pi$
$294$	0	0
$295$	12.2462	0.713002
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	24.2462	1.40455
$299$	−16.3002	−0.942664
$300$	0	0
$301$	0.315342	0.0181760
$302$	19.5616	1.12564
$303$	0	0
$304$	−3.56155	−0.204269
$305$	6.80776	0.389811
$306$	0	0
$307$	16.2462	0.927220	0.463610	−	0.886039i	$-0.346554\pi$
$307$	16.2462	0.927220	0.463610	+	0.886039i	$0.346554\pi$
$308$	2.12311	0.120975
$309$	0	0
$310$	−0.561553	−0.0318941
$311$	−24.1771	−1.37096	−0.685478	−	0.728093i	$-0.740407\pi$
$311$	−24.1771	−1.37096	−0.685478	+	0.728093i	$0.740407\pi$
$312$	0	0
$313$	−33.1231	−1.87223	−0.936114	−	0.351696i	$-0.885605\pi$
$313$	−33.1231	−1.87223	−0.936114	+	0.351696i	$0.885605\pi$
$314$	−19.0540	−1.07528
$315$	0	0
$316$	−2.24621	−0.126359
$317$	11.9309	0.670104	0.335052	−	0.942200i	$-0.391246\pi$
$317$	11.9309	0.670104	0.335052	+	0.942200i	$0.391246\pi$
$318$	0	0
$319$	12.0691	0.675742
$320$	1.00000	0.0559017
$321$	0	0
$322$	2.43845	0.135889
$323$	−17.8078	−0.990850
$324$	0	0
$325$	−6.68466	−0.370798
$326$	13.4924	0.747276
$327$	0	0
$328$	−3.68466	−0.203451
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	11.5616	0.634523
$333$	0	0
$334$	−18.0540	−0.987870
$335$	4.87689	0.266453
$336$	0	0
$337$	0.438447	0.0238837	0.0119419	−	0.999929i	$-0.496199\pi$
$337$	0.438447	0.0238837	0.0119419	+	0.999929i	$0.496199\pi$
$338$	−31.6847	−1.72342
$339$	0	0
$340$	5.00000	0.271163
$341$	−1.19224	−0.0645632
$342$	0	0
$343$	13.0000	0.701934
$344$	0.315342	0.0170021
$345$	0	0
$346$	6.12311	0.329180
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	−2.24621	−0.120237	−0.0601185	−	0.998191i	$-0.519148\pi$
$349$	−2.24621	−0.120237	−0.0601185	+	0.998191i	$0.519148\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	2.12311	0.113162
$353$	9.19224	0.489253	0.244627	−	0.969617i	$-0.421335\pi$
$353$	9.19224	0.489253	0.244627	+	0.969617i	$0.421335\pi$
$354$	0	0
$355$	7.12311	0.378055
$356$	6.68466	0.354286
$357$	0	0
$358$	−4.00000	−0.211407
$359$	29.1231	1.53706	0.768529	−	0.639815i	$-0.220989\pi$
$359$	29.1231	1.53706	0.768529	+	0.639815i	$0.220989\pi$
$360$	0	0
$361$	−6.31534	−0.332386
$362$	−3.75379	−0.197295
$363$	0	0
$364$	6.68466	0.350371
$365$	−2.43845	−0.127634
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	2.43845	0.127113
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	−11.2462	−0.583874
$372$	0	0
$373$	2.49242	0.129053	0.0645264	−	0.997916i	$-0.479446\pi$
$373$	2.49242	0.129053	0.0645264	+	0.997916i	$0.479446\pi$
$374$	10.6155	0.548916
$375$	0	0
$376$	−8.00000	−0.412568
$377$	38.0000	1.95710
$378$	0	0
$379$	−11.1231	−0.571356	−0.285678	−	0.958326i	$-0.592219\pi$
$379$	−11.1231	−0.571356	−0.285678	+	0.958326i	$0.592219\pi$
$380$	−3.56155	−0.182704
$381$	0	0
$382$	−10.3693	−0.530540
$383$	15.3153	0.782577	0.391289	−	0.920268i	$-0.372029\pi$
$383$	15.3153	0.782577	0.391289	+	0.920268i	$0.372029\pi$
$384$	0	0
$385$	2.12311	0.108203
$386$	14.0000	0.712581
$387$	0	0
$388$	15.6847	0.796268
$389$	7.93087	0.402111	0.201056	−	0.979580i	$-0.435563\pi$
$389$	7.93087	0.402111	0.201056	+	0.979580i	$0.435563\pi$
$390$	0	0
$391$	12.1922	0.616588
$392$	6.00000	0.303046
$393$	0	0
$394$	−14.6847	−0.739802
$395$	−2.24621	−0.113019
$396$	0	0
$397$	18.8769	0.947404	0.473702	−	0.880685i	$-0.342917\pi$
$397$	18.8769	0.947404	0.473702	+	0.880685i	$0.342917\pi$
$398$	−16.4924	−0.826690
$399$	0	0
$400$	1.00000	0.0500000
$401$	−5.80776	−0.290026	−0.145013	−	0.989430i	$-0.546322\pi$
$401$	−5.80776	−0.290026	−0.145013	+	0.989430i	$0.546322\pi$
$402$	0	0
$403$	−3.75379	−0.186990
$404$	−19.1231	−0.951410
$405$	0	0
$406$	−5.68466	−0.282125
$407$	−2.12311	−0.105238
$408$	0	0
$409$	−9.12311	−0.451109	−0.225554	−	0.974231i	$-0.572419\pi$
$409$	−9.12311	−0.451109	−0.225554	+	0.974231i	$0.572419\pi$
$410$	−3.68466	−0.181972
$411$	0	0
$412$	16.4924	0.812523
$413$	−12.2462	−0.602597
$414$	0	0
$415$	11.5616	0.567534
$416$	6.68466	0.327742
$417$	0	0
$418$	−7.56155	−0.369848
$419$	6.43845	0.314539	0.157269	−	0.987556i	$-0.449731\pi$
$419$	6.43845	0.314539	0.157269	+	0.987556i	$0.449731\pi$
$420$	0	0
$421$	−34.9848	−1.70506	−0.852529	−	0.522681i	$-0.824932\pi$
$421$	−34.9848	−1.70506	−0.852529	+	0.522681i	$0.824932\pi$
$422$	−8.56155	−0.416770
$423$	0	0
$424$	−11.2462	−0.546164
$425$	5.00000	0.242536
$426$	0	0
$427$	−6.80776	−0.329451
$428$	−9.80776	−0.474076
$429$	0	0
$430$	0.315342	0.0152071
$431$	29.0000	1.39688	0.698440	−	0.715668i	$-0.253878\pi$
$431$	29.0000	1.39688	0.698440	+	0.715668i	$0.253878\pi$
$432$	0	0
$433$	31.4233	1.51011	0.755054	−	0.655663i	$-0.227611\pi$
$433$	31.4233	1.51011	0.755054	+	0.655663i	$0.227611\pi$
$434$	0.561553	0.0269554
$435$	0	0
$436$	6.12311	0.293244
$437$	−8.68466	−0.415444
$438$	0	0
$439$	23.0540	1.10031	0.550153	−	0.835064i	$-0.314569\pi$
$439$	23.0540	1.10031	0.550153	+	0.835064i	$0.314569\pi$
$440$	2.12311	0.101215
$441$	0	0
$442$	33.4233	1.58978
$443$	23.8617	1.13371	0.566853	−	0.823819i	$-0.308161\pi$
$443$	23.8617	1.13371	0.566853	+	0.823819i	$0.308161\pi$
$444$	0	0
$445$	6.68466	0.316883
$446$	−0.807764	−0.0382487
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−39.6155	−1.86957	−0.934786	−	0.355211i	$-0.884409\pi$
$449$	−39.6155	−1.86957	−0.934786	+	0.355211i	$0.884409\pi$
$450$	0	0
$451$	−7.82292	−0.368367
$452$	−1.68466	−0.0792397
$453$	0	0
$454$	17.9309	0.841537
$455$	6.68466	0.313382
$456$	0	0
$457$	−2.06913	−0.0967898	−0.0483949	−	0.998828i	$-0.515411\pi$
$457$	−2.06913	−0.0967898	−0.0483949	+	0.998828i	$0.515411\pi$
$458$	−3.12311	−0.145933
$459$	0	0
$460$	2.43845	0.113693
$461$	−4.56155	−0.212453	−0.106226	−	0.994342i	$-0.533877\pi$
$461$	−4.56155	−0.212453	−0.106226	+	0.994342i	$0.533877\pi$
$462$	0	0
$463$	13.1231	0.609882	0.304941	−	0.952371i	$-0.401363\pi$
$463$	13.1231	0.609882	0.304941	+	0.952371i	$0.401363\pi$
$464$	−5.68466	−0.263904
$465$	0	0
$466$	−13.3693	−0.619322
$467$	−6.56155	−0.303632	−0.151816	−	0.988409i	$-0.548512\pi$
$467$	−6.56155	−0.303632	−0.151816	+	0.988409i	$0.548512\pi$
$468$	0	0
$469$	−4.87689	−0.225194
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−12.2462	−0.563678
$473$	0.669503	0.0307838
$474$	0	0
$475$	−3.56155	−0.163415
$476$	−5.00000	−0.229175
$477$	0	0
$478$	−12.8078	−0.585813
$479$	2.43845	0.111415	0.0557077	−	0.998447i	$-0.482258\pi$
$479$	2.43845	0.111415	0.0557077	+	0.998447i	$0.482258\pi$
$480$	0	0
$481$	−6.68466	−0.304794
$482$	2.00000	0.0910975
$483$	0	0
$484$	−6.49242	−0.295110
$485$	15.6847	0.712204
$486$	0	0
$487$	−32.7386	−1.48353	−0.741765	−	0.670660i	$-0.766011\pi$
$487$	−32.7386	−1.48353	−0.741765	+	0.670660i	$0.766011\pi$
$488$	−6.80776	−0.308173
$489$	0	0
$490$	6.00000	0.271052
$491$	−33.1771	−1.49726	−0.748630	−	0.662988i	$-0.769288\pi$
$491$	−33.1771	−1.49726	−0.748630	+	0.662988i	$0.769288\pi$
$492$	0	0
$493$	−28.4233	−1.28012
$494$	−23.8078	−1.07116
$495$	0	0
$496$	0.561553	0.0252145
$497$	−7.12311	−0.319515
$498$	0	0
$499$	12.1922	0.545799	0.272900	−	0.962042i	$-0.412017\pi$
$499$	12.1922	0.545799	0.272900	+	0.962042i	$0.412017\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−15.6155	−0.696955
$503$	1.75379	0.0781976	0.0390988	−	0.999235i	$-0.487551\pi$
$503$	1.75379	0.0781976	0.0390988	+	0.999235i	$0.487551\pi$
$504$	0	0
$505$	−19.1231	−0.850967
$506$	5.17708	0.230149
$507$	0	0
$508$	10.4384	0.463131
$509$	14.0540	0.622932	0.311466	−	0.950257i	$-0.399180\pi$
$509$	14.0540	0.622932	0.311466	+	0.950257i	$0.399180\pi$
$510$	0	0
$511$	2.43845	0.107871
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−9.80776	−0.432602
$515$	16.4924	0.726743
$516$	0	0
$517$	−16.9848	−0.746993
$518$	1.00000	0.0439375
$519$	0	0
$520$	6.68466	0.293142
$521$	22.5616	0.988440	0.494220	−	0.869337i	$-0.335454\pi$
$521$	22.5616	0.988440	0.494220	+	0.869337i	$0.335454\pi$
$522$	0	0
$523$	−18.7386	−0.819383	−0.409692	−	0.912224i	$-0.634364\pi$
$523$	−18.7386	−0.819383	−0.409692	+	0.912224i	$0.634364\pi$
$524$	1.12311	0.0490631
$525$	0	0
$526$	−10.5616	−0.460506
$527$	2.80776	0.122308
$528$	0	0
$529$	−17.0540	−0.741477
$530$	−11.2462	−0.488504
$531$	0	0
$532$	3.56155	0.154413
$533$	−24.6307	−1.06687
$534$	0	0
$535$	−9.80776	−0.424027
$536$	−4.87689	−0.210650
$537$	0	0
$538$	3.80776	0.164164
$539$	12.7386	0.548692
$540$	0	0
$541$	41.4233	1.78093	0.890463	−	0.455055i	$-0.150380\pi$
$541$	41.4233	1.78093	0.890463	+	0.455055i	$0.150380\pi$
$542$	−16.4924	−0.708410
$543$	0	0
$544$	−5.00000	−0.214373
$545$	6.12311	0.262285
$546$	0	0
$547$	9.73863	0.416394	0.208197	−	0.978087i	$-0.433240\pi$
$547$	9.73863	0.416394	0.208197	+	0.978087i	$0.433240\pi$
$548$	−12.2462	−0.523132
$549$	0	0
$550$	2.12311	0.0905295
$551$	20.2462	0.862518
$552$	0	0
$553$	2.24621	0.0955186
$554$	−15.3153	−0.650687
$555$	0	0
$556$	−6.80776	−0.288714
$557$	33.6155	1.42434	0.712168	−	0.702009i	$-0.247714\pi$
$557$	33.6155	1.42434	0.712168	+	0.702009i	$0.247714\pi$
$558$	0	0
$559$	2.10795	0.0891568
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−23.1771	−0.977666
$563$	23.9309	1.00857	0.504283	−	0.863538i	$-0.331757\pi$
$563$	23.9309	1.00857	0.504283	+	0.863538i	$0.331757\pi$
$564$	0	0
$565$	−1.68466	−0.0708741
$566$	18.9309	0.795724
$567$	0	0
$568$	−7.12311	−0.298879
$569$	−14.6847	−0.615613	−0.307806	−	0.951449i	$-0.599595\pi$
$569$	−14.6847	−0.615613	−0.307806	+	0.951449i	$0.599595\pi$
$570$	0	0
$571$	14.8078	0.619686	0.309843	−	0.950788i	$-0.399724\pi$
$571$	14.8078	0.619686	0.309843	+	0.950788i	$0.399724\pi$
$572$	14.1922	0.593407
$573$	0	0
$574$	3.68466	0.153795
$575$	2.43845	0.101690
$576$	0	0
$577$	18.4924	0.769850	0.384925	−	0.922948i	$-0.374227\pi$
$577$	18.4924	0.769850	0.384925	+	0.922948i	$0.374227\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−5.68466	−0.236043
$581$	−11.5616	−0.479654
$582$	0	0
$583$	−23.8769	−0.988880
$584$	2.43845	0.100904
$585$	0	0
$586$	24.6155	1.01686
$587$	−41.3002	−1.70464	−0.852321	−	0.523020i	$-0.824805\pi$
$587$	−41.3002	−1.70464	−0.852321	+	0.523020i	$0.824805\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	−12.2462	−0.504169
$591$	0	0
$592$	1.00000	0.0410997
$593$	−13.1231	−0.538901	−0.269451	−	0.963014i	$-0.586842\pi$
$593$	−13.1231	−0.538901	−0.269451	+	0.963014i	$0.586842\pi$
$594$	0	0
$595$	−5.00000	−0.204980
$596$	−24.2462	−0.993164
$597$	0	0
$598$	16.3002	0.666564
$599$	22.7386	0.929075	0.464538	−	0.885553i	$-0.346221\pi$
$599$	22.7386	0.929075	0.464538	+	0.885553i	$0.346221\pi$
$600$	0	0
$601$	14.8617	0.606223	0.303111	−	0.952955i	$-0.401975\pi$
$601$	14.8617	0.606223	0.303111	+	0.952955i	$0.401975\pi$
$602$	−0.315342	−0.0128524
$603$	0	0
$604$	−19.5616	−0.795948
$605$	−6.49242	−0.263955
$606$	0	0
$607$	6.49242	0.263519	0.131760	−	0.991282i	$-0.457937\pi$
$607$	6.49242	0.263519	0.131760	+	0.991282i	$0.457937\pi$
$608$	3.56155	0.144440
$609$	0	0
$610$	−6.80776	−0.275638
$611$	−53.4773	−2.16346
$612$	0	0
$613$	12.5616	0.507356	0.253678	−	0.967289i	$-0.418360\pi$
$613$	12.5616	0.507356	0.253678	+	0.967289i	$0.418360\pi$
$614$	−16.2462	−0.655644
$615$	0	0
$616$	−2.12311	−0.0855424
$617$	18.4924	0.744477	0.372238	−	0.928137i	$-0.378590\pi$
$617$	18.4924	0.744477	0.372238	+	0.928137i	$0.378590\pi$
$618$	0	0
$619$	−15.1922	−0.610628	−0.305314	−	0.952252i	$-0.598761\pi$
$619$	−15.1922	−0.610628	−0.305314	+	0.952252i	$0.598761\pi$
$620$	0.561553	0.0225525
$621$	0	0
$622$	24.1771	0.969413
$623$	−6.68466	−0.267815
$624$	0	0
$625$	1.00000	0.0400000
$626$	33.1231	1.32387
$627$	0	0
$628$	19.0540	0.760336
$629$	5.00000	0.199363
$630$	0	0
$631$	27.6847	1.10211	0.551054	−	0.834469i	$-0.314226\pi$
$631$	27.6847	1.10211	0.551054	+	0.834469i	$0.314226\pi$
$632$	2.24621	0.0893495
$633$	0	0
$634$	−11.9309	−0.473835
$635$	10.4384	0.414237
$636$	0	0
$637$	40.1080	1.58913
$638$	−12.0691	−0.477821
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−39.0540	−1.54254	−0.771270	−	0.636509i	$-0.780378\pi$
$641$	−39.0540	−1.54254	−0.771270	+	0.636509i	$0.780378\pi$
$642$	0	0
$643$	−25.8769	−1.02049	−0.510243	−	0.860031i	$-0.670444\pi$
$643$	−25.8769	−1.02049	−0.510243	+	0.860031i	$0.670444\pi$
$644$	−2.43845	−0.0960883
$645$	0	0
$646$	17.8078	0.700637
$647$	16.1922	0.636582	0.318291	−	0.947993i	$-0.396891\pi$
$647$	16.1922	0.636582	0.318291	+	0.947993i	$0.396891\pi$
$648$	0	0
$649$	−26.0000	−1.02059
$650$	6.68466	0.262194
$651$	0	0
$652$	−13.4924	−0.528404
$653$	−4.49242	−0.175802	−0.0879010	−	0.996129i	$-0.528016\pi$
$653$	−4.49242	−0.175802	−0.0879010	+	0.996129i	$0.528016\pi$
$654$	0	0
$655$	1.12311	0.0438834
$656$	3.68466	0.143862
$657$	0	0
$658$	8.00000	0.311872
$659$	1.75379	0.0683179	0.0341590	−	0.999416i	$-0.489125\pi$
$659$	1.75379	0.0683179	0.0341590	+	0.999416i	$0.489125\pi$
$660$	0	0
$661$	−24.8617	−0.967010	−0.483505	−	0.875342i	$-0.660636\pi$
$661$	−24.8617	−0.967010	−0.483505	+	0.875342i	$0.660636\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	−11.5616	−0.448675
$665$	3.56155	0.138111
$666$	0	0
$667$	−13.8617	−0.536729
$668$	18.0540	0.698529
$669$	0	0
$670$	−4.87689	−0.188411
$671$	−14.4536	−0.557975
$672$	0	0
$673$	−14.4384	−0.556561	−0.278281	−	0.960500i	$-0.589765\pi$
$673$	−14.4384	−0.556561	−0.278281	+	0.960500i	$0.589765\pi$
$674$	−0.438447	−0.0168884
$675$	0	0
$676$	31.6847	1.21864
$677$	−7.06913	−0.271689	−0.135844	−	0.990730i	$-0.543375\pi$
$677$	−7.06913	−0.271689	−0.135844	+	0.990730i	$0.543375\pi$
$678$	0	0
$679$	−15.6847	−0.601922
$680$	−5.00000	−0.191741
$681$	0	0
$682$	1.19224	0.0456531
$683$	34.5616	1.32246	0.661231	−	0.750183i	$-0.270035\pi$
$683$	34.5616	1.32246	0.661231	+	0.750183i	$0.270035\pi$
$684$	0	0
$685$	−12.2462	−0.467904
$686$	−13.0000	−0.496342
$687$	0	0
$688$	−0.315342	−0.0120223
$689$	−75.1771	−2.86402
$690$	0	0
$691$	−17.6847	−0.672756	−0.336378	−	0.941727i	$-0.609202\pi$
$691$	−17.6847	−0.672756	−0.336378	+	0.941727i	$0.609202\pi$
$692$	−6.12311	−0.232766
$693$	0	0
$694$	−28.0000	−1.06287
$695$	−6.80776	−0.258233
$696$	0	0
$697$	18.4233	0.697832
$698$	2.24621	0.0850203
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	15.3693	0.580491	0.290246	−	0.956952i	$-0.406263\pi$
$701$	15.3693	0.580491	0.290246	+	0.956952i	$0.406263\pi$
$702$	0	0
$703$	−3.56155	−0.134327
$704$	−2.12311	−0.0800176
$705$	0	0
$706$	−9.19224	−0.345954
$707$	19.1231	0.719198
$708$	0	0
$709$	−37.4924	−1.40806	−0.704029	−	0.710171i	$-0.748617\pi$
$709$	−37.4924	−1.40806	−0.704029	+	0.710171i	$0.748617\pi$
$710$	−7.12311	−0.267325
$711$	0	0
$712$	−6.68466	−0.250518
$713$	1.36932	0.0512813
$714$	0	0
$715$	14.1922	0.530760
$716$	4.00000	0.149487
$717$	0	0
$718$	−29.1231	−1.08686
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	−16.4924	−0.614210
$722$	6.31534	0.235033
$723$	0	0
$724$	3.75379	0.139508
$725$	−5.68466	−0.211123
$726$	0	0
$727$	−14.9848	−0.555757	−0.277879	−	0.960616i	$-0.589631\pi$
$727$	−14.9848	−0.555757	−0.277879	+	0.960616i	$0.589631\pi$
$728$	−6.68466	−0.247750
$729$	0	0
$730$	2.43845	0.0902510
$731$	−1.57671	−0.0583166
$732$	0	0
$733$	31.3002	1.15610	0.578049	−	0.816002i	$-0.303814\pi$
$733$	31.3002	1.15610	0.578049	+	0.816002i	$0.303814\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	−2.43845	−0.0898824
$737$	−10.3542	−0.381400
$738$	0	0
$739$	6.17708	0.227228	0.113614	−	0.993525i	$-0.463757\pi$
$739$	6.17708	0.227228	0.113614	+	0.993525i	$0.463757\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	11.2462	0.412861
$743$	40.4233	1.48299	0.741493	−	0.670960i	$-0.234118\pi$
$743$	40.4233	1.48299	0.741493	+	0.670960i	$0.234118\pi$
$744$	0	0
$745$	−24.2462	−0.888312
$746$	−2.49242	−0.0912541
$747$	0	0
$748$	−10.6155	−0.388142
$749$	9.80776	0.358368
$750$	0	0
$751$	−26.7386	−0.975707	−0.487853	−	0.872926i	$-0.662220\pi$
$751$	−26.7386	−0.975707	−0.487853	+	0.872926i	$0.662220\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−38.0000	−1.38388
$755$	−19.5616	−0.711918
$756$	0	0
$757$	−52.0540	−1.89193	−0.945967	−	0.324263i	$-0.894884\pi$
$757$	−52.0540	−1.89193	−0.945967	+	0.324263i	$0.894884\pi$
$758$	11.1231	0.404009
$759$	0	0
$760$	3.56155	0.129191
$761$	−29.6847	−1.07607	−0.538034	−	0.842923i	$-0.680833\pi$
$761$	−29.6847	−1.07607	−0.538034	+	0.842923i	$0.680833\pi$
$762$	0	0
$763$	−6.12311	−0.221671
$764$	10.3693	0.375149
$765$	0	0
$766$	−15.3153	−0.553366
$767$	−81.8617	−2.95586
$768$	0	0
$769$	−42.4924	−1.53232	−0.766158	−	0.642652i	$-0.777834\pi$
$769$	−42.4924	−1.53232	−0.766158	+	0.642652i	$0.777834\pi$
$770$	−2.12311	−0.0765114
$771$	0	0
$772$	−14.0000	−0.503871
$773$	−29.4924	−1.06077	−0.530384	−	0.847757i	$-0.677952\pi$
$773$	−29.4924	−1.06077	−0.530384	+	0.847757i	$0.677952\pi$
$774$	0	0
$775$	0.561553	0.0201716
$776$	−15.6847	−0.563046
$777$	0	0
$778$	−7.93087	−0.284335
$779$	−13.1231	−0.470184
$780$	0	0
$781$	−15.1231	−0.541147
$782$	−12.1922	−0.435993
$783$	0	0
$784$	−6.00000	−0.214286
$785$	19.0540	0.680066
$786$	0	0
$787$	35.1231	1.25200	0.626002	−	0.779822i	$-0.284690\pi$
$787$	35.1231	1.25200	0.626002	+	0.779822i	$0.284690\pi$
$788$	14.6847	0.523119
$789$	0	0
$790$	2.24621	0.0799166
$791$	1.68466	0.0598996
$792$	0	0
$793$	−45.5076	−1.61602
$794$	−18.8769	−0.669916
$795$	0	0
$796$	16.4924	0.584558
$797$	−15.6155	−0.553130	−0.276565	−	0.960995i	$-0.589196\pi$
$797$	−15.6155	−0.553130	−0.276565	+	0.960995i	$0.589196\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	5.80776	0.205079
$803$	5.17708	0.182695
$804$	0	0
$805$	−2.43845	−0.0859440
$806$	3.75379	0.132222
$807$	0	0
$808$	19.1231	0.672749
$809$	29.5616	1.03933	0.519664	−	0.854370i	$-0.326057\pi$
$809$	29.5616	1.03933	0.519664	+	0.854370i	$0.326057\pi$
$810$	0	0
$811$	−46.7386	−1.64122	−0.820608	−	0.571492i	$-0.806365\pi$
$811$	−46.7386	−1.64122	−0.820608	+	0.571492i	$0.806365\pi$
$812$	5.68466	0.199492
$813$	0	0
$814$	2.12311	0.0744148
$815$	−13.4924	−0.472619
$816$	0	0
$817$	1.12311	0.0392925
$818$	9.12311	0.318982
$819$	0	0
$820$	3.68466	0.128674
$821$	−27.5616	−0.961905	−0.480952	−	0.876747i	$-0.659709\pi$
$821$	−27.5616	−0.961905	−0.480952	+	0.876747i	$0.659709\pi$
$822$	0	0
$823$	−38.5464	−1.34364	−0.671821	−	0.740713i	$-0.734488\pi$
$823$	−38.5464	−1.34364	−0.671821	+	0.740713i	$0.734488\pi$
$824$	−16.4924	−0.574541
$825$	0	0
$826$	12.2462	0.426100
$827$	14.5616	0.506355	0.253177	−	0.967420i	$-0.418524\pi$
$827$	14.5616	0.506355	0.253177	+	0.967420i	$0.418524\pi$
$828$	0	0
$829$	−13.8769	−0.481964	−0.240982	−	0.970530i	$-0.577470\pi$
$829$	−13.8769	−0.481964	−0.240982	+	0.970530i	$0.577470\pi$
$830$	−11.5616	−0.401307
$831$	0	0
$832$	−6.68466	−0.231749
$833$	−30.0000	−1.03944
$834$	0	0
$835$	18.0540	0.624784
$836$	7.56155	0.261522
$837$	0	0
$838$	−6.43845	−0.222412
$839$	−11.7538	−0.405786	−0.202893	−	0.979201i	$-0.565034\pi$
$839$	−11.7538	−0.405786	−0.202893	+	0.979201i	$0.565034\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	34.9848	1.20566
$843$	0	0
$844$	8.56155	0.294701
$845$	31.6847	1.08999
$846$	0	0
$847$	6.49242	0.223082
$848$	11.2462	0.386196
$849$	0	0
$850$	−5.00000	−0.171499
$851$	2.43845	0.0835889
$852$	0	0
$853$	−20.9309	−0.716659	−0.358330	−	0.933595i	$-0.616654\pi$
$853$	−20.9309	−0.716659	−0.358330	+	0.933595i	$0.616654\pi$
$854$	6.80776	0.232957
$855$	0	0
$856$	9.80776	0.335222
$857$	25.2462	0.862394	0.431197	−	0.902258i	$-0.358091\pi$
$857$	25.2462	0.862394	0.431197	+	0.902258i	$0.358091\pi$
$858$	0	0
$859$	17.8078	0.607593	0.303797	−	0.952737i	$-0.401746\pi$
$859$	17.8078	0.607593	0.303797	+	0.952737i	$0.401746\pi$
$860$	−0.315342	−0.0107531
$861$	0	0
$862$	−29.0000	−0.987744
$863$	−40.8078	−1.38911	−0.694556	−	0.719438i	$-0.744399\pi$
$863$	−40.8078	−1.38911	−0.694556	+	0.719438i	$0.744399\pi$
$864$	0	0
$865$	−6.12311	−0.208192
$866$	−31.4233	−1.06781
$867$	0	0
$868$	−0.561553	−0.0190603
$869$	4.76894	0.161775
$870$	0	0
$871$	−32.6004	−1.10462
$872$	−6.12311	−0.207355
$873$	0	0
$874$	8.68466	0.293763
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	18.0691	0.610151	0.305076	−	0.952328i	$-0.401318\pi$
$877$	18.0691	0.610151	0.305076	+	0.952328i	$0.401318\pi$
$878$	−23.0540	−0.778034
$879$	0	0
$880$	−2.12311	−0.0715699
$881$	−6.06913	−0.204474	−0.102237	−	0.994760i	$-0.532600\pi$
$881$	−6.06913	−0.204474	−0.102237	+	0.994760i	$0.532600\pi$
$882$	0	0
$883$	10.3693	0.348955	0.174478	−	0.984661i	$-0.444176\pi$
$883$	10.3693	0.348955	0.174478	+	0.984661i	$0.444176\pi$
$884$	−33.4233	−1.12415
$885$	0	0
$886$	−23.8617	−0.801651
$887$	−2.80776	−0.0942755	−0.0471377	−	0.998888i	$-0.515010\pi$
$887$	−2.80776	−0.0942755	−0.0471377	+	0.998888i	$0.515010\pi$
$888$	0	0
$889$	−10.4384	−0.350094
$890$	−6.68466	−0.224070
$891$	0	0
$892$	0.807764	0.0270459
$893$	−28.4924	−0.953463
$894$	0	0
$895$	4.00000	0.133705
$896$	1.00000	0.0334077
$897$	0	0
$898$	39.6155	1.32199
$899$	−3.19224	−0.106467
$900$	0	0
$901$	56.2311	1.87333
$902$	7.82292	0.260475
$903$	0	0
$904$	1.68466	0.0560309
$905$	3.75379	0.124780
$906$	0	0
$907$	49.5616	1.64566	0.822832	−	0.568284i	$-0.192393\pi$
$907$	49.5616	1.64566	0.822832	+	0.568284i	$0.192393\pi$
$908$	−17.9309	−0.595057
$909$	0	0
$910$	−6.68466	−0.221594
$911$	12.4924	0.413892	0.206946	−	0.978352i	$-0.433647\pi$
$911$	12.4924	0.413892	0.206946	+	0.978352i	$0.433647\pi$
$912$	0	0
$913$	−24.5464	−0.812367
$914$	2.06913	0.0684407
$915$	0	0
$916$	3.12311	0.103190
$917$	−1.12311	−0.0370882
$918$	0	0
$919$	−42.7386	−1.40982	−0.704909	−	0.709298i	$-0.749012\pi$
$919$	−42.7386	−1.40982	−0.704909	+	0.709298i	$0.749012\pi$
$920$	−2.43845	−0.0803932
$921$	0	0
$922$	4.56155	0.150227
$923$	−47.6155	−1.56728
$924$	0	0
$925$	1.00000	0.0328798
$926$	−13.1231	−0.431252
$927$	0	0
$928$	5.68466	0.186608
$929$	−14.8078	−0.485827	−0.242913	−	0.970048i	$-0.578103\pi$
$929$	−14.8078	−0.485827	−0.242913	+	0.970048i	$0.578103\pi$
$930$	0	0
$931$	21.3693	0.700351
$932$	13.3693	0.437927
$933$	0	0
$934$	6.56155	0.214701
$935$	−10.6155	−0.347165
$936$	0	0
$937$	−25.3693	−0.828779	−0.414390	−	0.910100i	$-0.636005\pi$
$937$	−25.3693	−0.828779	−0.414390	+	0.910100i	$0.636005\pi$
$938$	4.87689	0.159236
$939$	0	0
$940$	8.00000	0.260931
$941$	−8.24621	−0.268819	−0.134409	−	0.990926i	$-0.542914\pi$
$941$	−8.24621	−0.268819	−0.134409	+	0.990926i	$0.542914\pi$
$942$	0	0
$943$	8.98485	0.292587
$944$	12.2462	0.398580
$945$	0	0
$946$	−0.669503	−0.0217674
$947$	37.3002	1.21209	0.606047	−	0.795429i	$-0.292754\pi$
$947$	37.3002	1.21209	0.606047	+	0.795429i	$0.292754\pi$
$948$	0	0
$949$	16.3002	0.529126
$950$	3.56155	0.115552
$951$	0	0
$952$	5.00000	0.162051
$953$	−39.6155	−1.28327	−0.641636	−	0.767009i	$-0.721744\pi$
$953$	−39.6155	−1.28327	−0.641636	+	0.767009i	$0.721744\pi$
$954$	0	0
$955$	10.3693	0.335543
$956$	12.8078	0.414233
$957$	0	0
$958$	−2.43845	−0.0787827
$959$	12.2462	0.395451
$960$	0	0
$961$	−30.6847	−0.989828
$962$	6.68466	0.215522
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	−14.0000	−0.450676
$966$	0	0
$967$	57.3693	1.84487	0.922436	−	0.386149i	$-0.126195\pi$
$967$	57.3693	1.84487	0.922436	+	0.386149i	$0.126195\pi$
$968$	6.49242	0.208674
$969$	0	0
$970$	−15.6847	−0.503604
$971$	−4.80776	−0.154288	−0.0771442	−	0.997020i	$-0.524580\pi$
$971$	−4.80776	−0.154288	−0.0771442	+	0.997020i	$0.524580\pi$
$972$	0	0
$973$	6.80776	0.218247
$974$	32.7386	1.04901
$975$	0	0
$976$	6.80776	0.217911
$977$	45.4924	1.45543	0.727716	−	0.685879i	$-0.240582\pi$
$977$	45.4924	1.45543	0.727716	+	0.685879i	$0.240582\pi$
$978$	0	0
$979$	−14.1922	−0.453586
$980$	−6.00000	−0.191663
$981$	0	0
$982$	33.1771	1.05872
$983$	−3.93087	−0.125375	−0.0626876	−	0.998033i	$-0.519967\pi$
$983$	−3.93087	−0.125375	−0.0626876	+	0.998033i	$0.519967\pi$
$984$	0	0
$985$	14.6847	0.467892
$986$	28.4233	0.905182
$987$	0	0
$988$	23.8078	0.757426
$989$	−0.768944	−0.0244510
$990$	0	0
$991$	−58.4233	−1.85588	−0.927939	−	0.372733i	$-0.878421\pi$
$991$	−58.4233	−1.85588	−0.927939	+	0.372733i	$0.878421\pi$
$992$	−0.561553	−0.0178293
$993$	0	0
$994$	7.12311	0.225931
$995$	16.4924	0.522845
$996$	0	0
$997$	−34.3002	−1.08630	−0.543149	−	0.839636i	$-0.682768\pi$
$997$	−34.3002	−1.08630	−0.543149	+	0.839636i	$0.682768\pi$
$998$	−12.1922	−0.385938
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bc.1.1		2
3.2	odd	2		1110.2.a.r.1.2	✓	2
12.11	even	2		8880.2.a.bp.1.1		2
15.14	odd	2		5550.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.r.1.2	✓	2	3.2	odd	2
3330.2.a.bc.1.1		2	1.1	even	1	trivial
5550.2.a.bw.1.2		2	15.14	odd	2
8880.2.a.bp.1.1		2	12.11	even	2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -2.12311 q^{11} -6.68466 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -3.56155 q^{19} +1.00000 q^{20} +2.12311 q^{22} +2.43845 q^{23} +1.00000 q^{25} +6.68466 q^{26} -1.00000 q^{28} -5.68466 q^{29} +0.561553 q^{31} -1.00000 q^{32} -5.00000 q^{34} -1.00000 q^{35} +1.00000 q^{37} +3.56155 q^{38} -1.00000 q^{40} +3.68466 q^{41} -0.315342 q^{43} -2.12311 q^{44} -2.43845 q^{46} +8.00000 q^{47} -6.00000 q^{49} -1.00000 q^{50} -6.68466 q^{52} +11.2462 q^{53} -2.12311 q^{55} +1.00000 q^{56} +5.68466 q^{58} +12.2462 q^{59} +6.80776 q^{61} -0.561553 q^{62} +1.00000 q^{64} -6.68466 q^{65} +4.87689 q^{67} +5.00000 q^{68} +1.00000 q^{70} +7.12311 q^{71} -2.43845 q^{73} -1.00000 q^{74} -3.56155 q^{76} +2.12311 q^{77} -2.24621 q^{79} +1.00000 q^{80} -3.68466 q^{82} +11.5616 q^{83} +5.00000 q^{85} +0.315342 q^{86} +2.12311 q^{88} +6.68466 q^{89} +6.68466 q^{91} +2.43845 q^{92} -8.00000 q^{94} -3.56155 q^{95} +15.6847 q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 4 q^{11} - q^{13} + 2 q^{14} + 2 q^{16} + 10 q^{17} - 3 q^{19} + 2 q^{20} - 4 q^{22} + 9 q^{23} + 2 q^{25} + q^{26} - 2 q^{28} + q^{29} - 3 q^{31} - 2 q^{32} - 10 q^{34} - 2 q^{35} + 2 q^{37} + 3 q^{38} - 2 q^{40} - 5 q^{41} - 13 q^{43} + 4 q^{44} - 9 q^{46} + 16 q^{47} - 12 q^{49} - 2 q^{50} - q^{52} + 6 q^{53} + 4 q^{55} + 2 q^{56} - q^{58} + 8 q^{59} - 7 q^{61} + 3 q^{62} + 2 q^{64} - q^{65} + 18 q^{67} + 10 q^{68} + 2 q^{70} + 6 q^{71} - 9 q^{73} - 2 q^{74} - 3 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 5 q^{82} + 19 q^{83} + 10 q^{85} + 13 q^{86} - 4 q^{88} + q^{89} + q^{91} + 9 q^{92} - 16 q^{94} - 3 q^{95} + 19 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 4 * q^11 - q^13 + 2 * q^14 + 2 * q^16 + 10 * q^17 - 3 * q^19 + 2 * q^20 - 4 * q^22 + 9 * q^23 + 2 * q^25 + q^26 - 2 * q^28 + q^29 - 3 * q^31 - 2 * q^32 - 10 * q^34 - 2 * q^35 + 2 * q^37 + 3 * q^38 - 2 * q^40 - 5 * q^41 - 13 * q^43 + 4 * q^44 - 9 * q^46 + 16 * q^47 - 12 * q^49 - 2 * q^50 - q^52 + 6 * q^53 + 4 * q^55 + 2 * q^56 - q^58 + 8 * q^59 - 7 * q^61 + 3 * q^62 + 2 * q^64 - q^65 + 18 * q^67 + 10 * q^68 + 2 * q^70 + 6 * q^71 - 9 * q^73 - 2 * q^74 - 3 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 5 * q^82 + 19 * q^83 + 10 * q^85 + 13 * q^86 - 4 * q^88 + q^89 + q^91 + 9 * q^92 - 16 * q^94 - 3 * q^95 + 19 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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