LMFDB - Embedded newform 3330.2.a.bb.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.37228 q^{11} -4.74456 q^{13} -1.37228 q^{14} +1.00000 q^{16} +5.37228 q^{17} -2.00000 q^{19} -1.00000 q^{20} -3.37228 q^{22} -6.74456 q^{23} +1.00000 q^{25} +4.74456 q^{26} +1.37228 q^{28} -8.11684 q^{29} -2.62772 q^{31} -1.00000 q^{32} -5.37228 q^{34} -1.37228 q^{35} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} -5.37228 q^{41} +7.37228 q^{43} +3.37228 q^{44} +6.74456 q^{46} +8.74456 q^{47} -5.11684 q^{49} -1.00000 q^{50} -4.74456 q^{52} -1.37228 q^{53} -3.37228 q^{55} -1.37228 q^{56} +8.11684 q^{58} -12.7446 q^{59} -5.37228 q^{61} +2.62772 q^{62} +1.00000 q^{64} +4.74456 q^{65} +4.74456 q^{67} +5.37228 q^{68} +1.37228 q^{70} +6.74456 q^{71} +8.74456 q^{73} -1.00000 q^{74} -2.00000 q^{76} +4.62772 q^{77} -4.74456 q^{79} -1.00000 q^{80} +5.37228 q^{82} +0.744563 q^{83} -5.37228 q^{85} -7.37228 q^{86} -3.37228 q^{88} -10.0000 q^{89} -6.51087 q^{91} -6.74456 q^{92} -8.74456 q^{94} +2.00000 q^{95} -0.116844 q^{97} +5.11684 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 3 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8} + 2 q^{10} + q^{11} + 2 q^{13} + 3 q^{14} + 2 q^{16} + 5 q^{17} - 4 q^{19} - 2 q^{20} - q^{22} - 2 q^{23} + 2 q^{25} - 2 q^{26} - 3 q^{28} + q^{29} - 11 q^{31} - 2 q^{32} - 5 q^{34} + 3 q^{35} + 2 q^{37} + 4 q^{38} + 2 q^{40} - 5 q^{41} + 9 q^{43} + q^{44} + 2 q^{46} + 6 q^{47} + 7 q^{49} - 2 q^{50} + 2 q^{52} + 3 q^{53} - q^{55} + 3 q^{56} - q^{58} - 14 q^{59} - 5 q^{61} + 11 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{67} + 5 q^{68} - 3 q^{70} + 2 q^{71} + 6 q^{73} - 2 q^{74} - 4 q^{76} + 15 q^{77} + 2 q^{79} - 2 q^{80} + 5 q^{82} - 10 q^{83} - 5 q^{85} - 9 q^{86} - q^{88} - 20 q^{89} - 36 q^{91} - 2 q^{92} - 6 q^{94} + 4 q^{95} + 17 q^{97} - 7 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 3 * q^7 - 2 * q^8 + 2 * q^10 + q^11 + 2 * q^13 + 3 * q^14 + 2 * q^16 + 5 * q^17 - 4 * q^19 - 2 * q^20 - q^22 - 2 * q^23 + 2 * q^25 - 2 * q^26 - 3 * q^28 + q^29 - 11 * q^31 - 2 * q^32 - 5 * q^34 + 3 * q^35 + 2 * q^37 + 4 * q^38 + 2 * q^40 - 5 * q^41 + 9 * q^43 + q^44 + 2 * q^46 + 6 * q^47 + 7 * q^49 - 2 * q^50 + 2 * q^52 + 3 * q^53 - q^55 + 3 * q^56 - q^58 - 14 * q^59 - 5 * q^61 + 11 * q^62 + 2 * q^64 - 2 * q^65 - 2 * q^67 + 5 * q^68 - 3 * q^70 + 2 * q^71 + 6 * q^73 - 2 * q^74 - 4 * q^76 + 15 * q^77 + 2 * q^79 - 2 * q^80 + 5 * q^82 - 10 * q^83 - 5 * q^85 - 9 * q^86 - q^88 - 20 * q^89 - 36 * q^91 - 2 * q^92 - 6 * q^94 + 4 * q^95 + 17 * q^97 - 7 * 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
$7$	1.37228	0.518674	0.259337	−	0.965787i	$-0.416496\pi$
$7$	1.37228	0.518674	0.259337	+	0.965787i	$0.416496\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	3.37228	1.01678	0.508391	−	0.861127i	$-0.330241\pi$
$11$	3.37228	1.01678	0.508391	+	0.861127i	$0.330241\pi$
$12$	0	0
$13$	−4.74456	−1.31590	−0.657952	−	0.753059i	$-0.728577\pi$
$13$	−4.74456	−1.31590	−0.657952	+	0.753059i	$0.728577\pi$
$14$	−1.37228	−0.366758
$15$	0	0
$16$	1.00000	0.250000
$17$	5.37228	1.30297	0.651485	−	0.758662i	$-0.274146\pi$
$17$	5.37228	1.30297	0.651485	+	0.758662i	$0.274146\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−3.37228	−0.718973
$23$	−6.74456	−1.40634	−0.703169	−	0.711022i	$-0.748232\pi$
$23$	−6.74456	−1.40634	−0.703169	+	0.711022i	$0.748232\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.74456	0.930485
$27$	0	0
$28$	1.37228	0.259337
$29$	−8.11684	−1.50726	−0.753630	−	0.657299i	$-0.771699\pi$
$29$	−8.11684	−1.50726	−0.753630	+	0.657299i	$0.771699\pi$
$30$	0	0
$31$	−2.62772	−0.471952	−0.235976	−	0.971759i	$-0.575829\pi$
$31$	−2.62772	−0.471952	−0.235976	+	0.971759i	$0.575829\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.37228	−0.921339
$35$	−1.37228	−0.231958
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	2.00000	0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	−5.37228	−0.839009	−0.419505	−	0.907753i	$-0.637796\pi$
$41$	−5.37228	−0.839009	−0.419505	+	0.907753i	$0.637796\pi$
$42$	0	0
$43$	7.37228	1.12426	0.562131	−	0.827048i	$-0.309982\pi$
$43$	7.37228	1.12426	0.562131	+	0.827048i	$0.309982\pi$
$44$	3.37228	0.508391
$45$	0	0
$46$	6.74456	0.994432
$47$	8.74456	1.27553	0.637763	−	0.770233i	$-0.279860\pi$
$47$	8.74456	1.27553	0.637763	+	0.770233i	$0.279860\pi$
$48$	0	0
$49$	−5.11684	−0.730978
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−4.74456	−0.657952
$53$	−1.37228	−0.188497	−0.0942487	−	0.995549i	$-0.530045\pi$
$53$	−1.37228	−0.188497	−0.0942487	+	0.995549i	$0.530045\pi$
$54$	0	0
$55$	−3.37228	−0.454718
$56$	−1.37228	−0.183379
$57$	0	0
$58$	8.11684	1.06579
$59$	−12.7446	−1.65920	−0.829600	−	0.558358i	$-0.811432\pi$
$59$	−12.7446	−1.65920	−0.829600	+	0.558358i	$0.811432\pi$
$60$	0	0
$61$	−5.37228	−0.687850	−0.343925	−	0.938997i	$-0.611757\pi$
$61$	−5.37228	−0.687850	−0.343925	+	0.938997i	$0.611757\pi$
$62$	2.62772	0.333721
$63$	0	0
$64$	1.00000	0.125000
$65$	4.74456	0.588491
$66$	0	0
$67$	4.74456	0.579641	0.289820	−	0.957081i	$-0.406404\pi$
$67$	4.74456	0.579641	0.289820	+	0.957081i	$0.406404\pi$
$68$	5.37228	0.651485
$69$	0	0
$70$	1.37228	0.164019
$71$	6.74456	0.800432	0.400216	−	0.916421i	$-0.368935\pi$
$71$	6.74456	0.800432	0.400216	+	0.916421i	$0.368935\pi$
$72$	0	0
$73$	8.74456	1.02347	0.511737	−	0.859142i	$-0.329002\pi$
$73$	8.74456	1.02347	0.511737	+	0.859142i	$0.329002\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−2.00000	−0.229416
$77$	4.62772	0.527377
$78$	0	0
$79$	−4.74456	−0.533805	−0.266903	−	0.963724i	$-0.586000\pi$
$79$	−4.74456	−0.533805	−0.266903	+	0.963724i	$0.586000\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	5.37228	0.593269
$83$	0.744563	0.0817264	0.0408632	−	0.999165i	$-0.486989\pi$
$83$	0.744563	0.0817264	0.0408632	+	0.999165i	$0.486989\pi$
$84$	0	0
$85$	−5.37228	−0.582706
$86$	−7.37228	−0.794974
$87$	0	0
$88$	−3.37228	−0.359486
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−6.51087	−0.682525
$92$	−6.74456	−0.703169
$93$	0	0
$94$	−8.74456	−0.901933
$95$	2.00000	0.205196
$96$	0	0
$97$	−0.116844	−0.0118637	−0.00593185	−	0.999982i	$-0.501888\pi$
$97$	−0.116844	−0.0118637	−0.00593185	+	0.999982i	$0.501888\pi$
$98$	5.11684	0.516879
$99$	0	0
$100$	1.00000	0.100000
$101$	11.4891	1.14321	0.571605	−	0.820529i	$-0.306321\pi$
$101$	11.4891	1.14321	0.571605	+	0.820529i	$0.306321\pi$
$102$	0	0
$103$	−9.48913	−0.934991	−0.467496	−	0.883995i	$-0.654844\pi$
$103$	−9.48913	−0.934991	−0.467496	+	0.883995i	$0.654844\pi$
$104$	4.74456	0.465243
$105$	0	0
$106$	1.37228	0.133288
$107$	3.48913	0.337306	0.168653	−	0.985675i	$-0.446058\pi$
$107$	3.48913	0.337306	0.168653	+	0.985675i	$0.446058\pi$
$108$	0	0
$109$	0.116844	0.0111916	0.00559581	−	0.999984i	$-0.498219\pi$
$109$	0.116844	0.0111916	0.00559581	+	0.999984i	$0.498219\pi$
$110$	3.37228	0.321534
$111$	0	0
$112$	1.37228	0.129668
$113$	−17.3723	−1.63425	−0.817123	−	0.576463i	$-0.804433\pi$
$113$	−17.3723	−1.63425	−0.817123	+	0.576463i	$0.804433\pi$
$114$	0	0
$115$	6.74456	0.628934
$116$	−8.11684	−0.753630
$117$	0	0
$118$	12.7446	1.17323
$119$	7.37228	0.675816
$120$	0	0
$121$	0.372281	0.0338438
$122$	5.37228	0.486383
$123$	0	0
$124$	−2.62772	−0.235976
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.7446	−1.48584	−0.742920	−	0.669380i	$-0.766560\pi$
$127$	−16.7446	−1.48584	−0.742920	+	0.669380i	$0.766560\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.74456	−0.416126
$131$	3.25544	0.284429	0.142214	−	0.989836i	$-0.454578\pi$
$131$	3.25544	0.284429	0.142214	+	0.989836i	$0.454578\pi$
$132$	0	0
$133$	−2.74456	−0.237984
$134$	−4.74456	−0.409868
$135$	0	0
$136$	−5.37228	−0.460669
$137$	−16.7446	−1.43058	−0.715292	−	0.698825i	$-0.753706\pi$
$137$	−16.7446	−1.43058	−0.715292	+	0.698825i	$0.753706\pi$
$138$	0	0
$139$	−1.88316	−0.159727	−0.0798636	−	0.996806i	$-0.525448\pi$
$139$	−1.88316	−0.159727	−0.0798636	+	0.996806i	$0.525448\pi$
$140$	−1.37228	−0.115979
$141$	0	0
$142$	−6.74456	−0.565991
$143$	−16.0000	−1.33799
$144$	0	0
$145$	8.11684	0.674067
$146$	−8.74456	−0.723705
$147$	0	0
$148$	1.00000	0.0821995
$149$	11.4891	0.941226	0.470613	−	0.882340i	$-0.344033\pi$
$149$	11.4891	0.941226	0.470613	+	0.882340i	$0.344033\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−4.62772	−0.372912
$155$	2.62772	0.211063
$156$	0	0
$157$	17.3723	1.38646	0.693229	−	0.720717i	$-0.256187\pi$
$157$	17.3723	1.38646	0.693229	+	0.720717i	$0.256187\pi$
$158$	4.74456	0.377457
$159$	0	0
$160$	1.00000	0.0790569
$161$	−9.25544	−0.729431
$162$	0	0
$163$	19.3723	1.51735	0.758677	−	0.651467i	$-0.225846\pi$
$163$	19.3723	1.51735	0.758677	+	0.651467i	$0.225846\pi$
$164$	−5.37228	−0.419505
$165$	0	0
$166$	−0.744563	−0.0577893
$167$	−1.48913	−0.115232	−0.0576160	−	0.998339i	$-0.518350\pi$
$167$	−1.48913	−0.115232	−0.0576160	+	0.998339i	$0.518350\pi$
$168$	0	0
$169$	9.51087	0.731606
$170$	5.37228	0.412035
$171$	0	0
$172$	7.37228	0.562131
$173$	−22.8614	−1.73812	−0.869060	−	0.494706i	$-0.835276\pi$
$173$	−22.8614	−1.73812	−0.869060	+	0.494706i	$0.835276\pi$
$174$	0	0
$175$	1.37228	0.103735
$176$	3.37228	0.254195
$177$	0	0
$178$	10.0000	0.749532
$179$	26.2337	1.96080	0.980399	−	0.197023i	$-0.0631272\pi$
$179$	26.2337	1.96080	0.980399	+	0.197023i	$0.0631272\pi$
$180$	0	0
$181$	7.48913	0.556662	0.278331	−	0.960485i	$-0.410219\pi$
$181$	7.48913	0.556662	0.278331	+	0.960485i	$0.410219\pi$
$182$	6.51087	0.482618
$183$	0	0
$184$	6.74456	0.497216
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	18.1168	1.32483
$188$	8.74456	0.637763
$189$	0	0
$190$	−2.00000	−0.145095
$191$	−18.6277	−1.34785	−0.673927	−	0.738798i	$-0.735394\pi$
$191$	−18.6277	−1.34785	−0.673927	+	0.738798i	$0.735394\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0.116844	0.00838891
$195$	0	0
$196$	−5.11684	−0.365489
$197$	−27.4891	−1.95852	−0.979260	−	0.202610i	$-0.935058\pi$
$197$	−27.4891	−1.95852	−0.979260	+	0.202610i	$0.935058\pi$
$198$	0	0
$199$	−11.2554	−0.797877	−0.398938	−	0.916978i	$-0.630621\pi$
$199$	−11.2554	−0.797877	−0.398938	+	0.916978i	$0.630621\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−11.4891	−0.808372
$203$	−11.1386	−0.781776
$204$	0	0
$205$	5.37228	0.375216
$206$	9.48913	0.661139
$207$	0	0
$208$	−4.74456	−0.328976
$209$	−6.74456	−0.466531
$210$	0	0
$211$	−14.1168	−0.971844	−0.485922	−	0.874002i	$-0.661516\pi$
$211$	−14.1168	−0.971844	−0.485922	+	0.874002i	$0.661516\pi$
$212$	−1.37228	−0.0942487
$213$	0	0
$214$	−3.48913	−0.238512
$215$	−7.37228	−0.502785
$216$	0	0
$217$	−3.60597	−0.244789
$218$	−0.116844	−0.00791367
$219$	0	0
$220$	−3.37228	−0.227359
$221$	−25.4891	−1.71458
$222$	0	0
$223$	−1.37228	−0.0918948	−0.0459474	−	0.998944i	$-0.514631\pi$
$223$	−1.37228	−0.0918948	−0.0459474	+	0.998944i	$0.514631\pi$
$224$	−1.37228	−0.0916894
$225$	0	0
$226$	17.3723	1.15559
$227$	11.3723	0.754805	0.377402	−	0.926049i	$-0.376817\pi$
$227$	11.3723	0.754805	0.377402	+	0.926049i	$0.376817\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−6.74456	−0.444723
$231$	0	0
$232$	8.11684	0.532897
$233$	−6.23369	−0.408382	−0.204191	−	0.978931i	$-0.565456\pi$
$233$	−6.23369	−0.408382	−0.204191	+	0.978931i	$0.565456\pi$
$234$	0	0
$235$	−8.74456	−0.570432
$236$	−12.7446	−0.829600
$237$	0	0
$238$	−7.37228	−0.477874
$239$	−17.6060	−1.13884	−0.569418	−	0.822048i	$-0.692831\pi$
$239$	−17.6060	−1.13884	−0.569418	+	0.822048i	$0.692831\pi$
$240$	0	0
$241$	−10.2337	−0.659210	−0.329605	−	0.944119i	$-0.606916\pi$
$241$	−10.2337	−0.659210	−0.329605	+	0.944119i	$0.606916\pi$
$242$	−0.372281	−0.0239311
$243$	0	0
$244$	−5.37228	−0.343925
$245$	5.11684	0.326903
$246$	0	0
$247$	9.48913	0.603779
$248$	2.62772	0.166860
$249$	0	0
$250$	1.00000	0.0632456
$251$	−11.4891	−0.725187	−0.362594	−	0.931947i	$-0.618109\pi$
$251$	−11.4891	−0.725187	−0.362594	+	0.931947i	$0.618109\pi$
$252$	0	0
$253$	−22.7446	−1.42994
$254$	16.7446	1.05065
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.9783	1.30859	0.654294	−	0.756241i	$-0.272966\pi$
$257$	20.9783	1.30859	0.654294	+	0.756241i	$0.272966\pi$
$258$	0	0
$259$	1.37228	0.0852694
$260$	4.74456	0.294245
$261$	0	0
$262$	−3.25544	−0.201122
$263$	−0.116844	−0.00720491	−0.00360245	−	0.999994i	$-0.501147\pi$
$263$	−0.116844	−0.00720491	−0.00360245	+	0.999994i	$0.501147\pi$
$264$	0	0
$265$	1.37228	0.0842986
$266$	2.74456	0.168280
$267$	0	0
$268$	4.74456	0.289820
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−6.74456	−0.409703	−0.204852	−	0.978793i	$-0.565671\pi$
$271$	−6.74456	−0.409703	−0.204852	+	0.978793i	$0.565671\pi$
$272$	5.37228	0.325742
$273$	0	0
$274$	16.7446	1.01158
$275$	3.37228	0.203356
$276$	0	0
$277$	0.744563	0.0447364	0.0223682	−	0.999750i	$-0.492879\pi$
$277$	0.744563	0.0447364	0.0223682	+	0.999750i	$0.492879\pi$
$278$	1.88316	0.112944
$279$	0	0
$280$	1.37228	0.0820095
$281$	−24.7446	−1.47614	−0.738068	−	0.674726i	$-0.764262\pi$
$281$	−24.7446	−1.47614	−0.738068	+	0.674726i	$0.764262\pi$
$282$	0	0
$283$	5.48913	0.326295	0.163147	−	0.986602i	$-0.447835\pi$
$283$	5.48913	0.326295	0.163147	+	0.986602i	$0.447835\pi$
$284$	6.74456	0.400216
$285$	0	0
$286$	16.0000	0.946100
$287$	−7.37228	−0.435172
$288$	0	0
$289$	11.8614	0.697730
$290$	−8.11684	−0.476637
$291$	0	0
$292$	8.74456	0.511737
$293$	18.8614	1.10190	0.550948	−	0.834540i	$-0.314266\pi$
$293$	18.8614	1.10190	0.550948	+	0.834540i	$0.314266\pi$
$294$	0	0
$295$	12.7446	0.742017
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−11.4891	−0.665547
$299$	32.0000	1.85061
$300$	0	0
$301$	10.1168	0.583125
$302$	20.0000	1.15087
$303$	0	0
$304$	−2.00000	−0.114708
$305$	5.37228	0.307616
$306$	0	0
$307$	−23.4891	−1.34060	−0.670298	−	0.742092i	$-0.733834\pi$
$307$	−23.4891	−1.34060	−0.670298	+	0.742092i	$0.733834\pi$
$308$	4.62772	0.263689
$309$	0	0
$310$	−2.62772	−0.149244
$311$	1.37228	0.0778149	0.0389075	−	0.999243i	$-0.487612\pi$
$311$	1.37228	0.0778149	0.0389075	+	0.999243i	$0.487612\pi$
$312$	0	0
$313$	−3.48913	−0.197217	−0.0986085	−	0.995126i	$-0.531439\pi$
$313$	−3.48913	−0.197217	−0.0986085	+	0.995126i	$0.531439\pi$
$314$	−17.3723	−0.980375
$315$	0	0
$316$	−4.74456	−0.266903
$317$	−25.3723	−1.42505	−0.712525	−	0.701647i	$-0.752448\pi$
$317$	−25.3723	−1.42505	−0.712525	+	0.701647i	$0.752448\pi$
$318$	0	0
$319$	−27.3723	−1.53255
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	9.25544	0.515785
$323$	−10.7446	−0.597843
$324$	0	0
$325$	−4.74456	−0.263181
$326$	−19.3723	−1.07293
$327$	0	0
$328$	5.37228	0.296635
$329$	12.0000	0.661581
$330$	0	0
$331$	23.4891	1.29108	0.645540	−	0.763727i	$-0.276633\pi$
$331$	23.4891	1.29108	0.645540	+	0.763727i	$0.276633\pi$
$332$	0.744563	0.0408632
$333$	0	0
$334$	1.48913	0.0814813
$335$	−4.74456	−0.259223
$336$	0	0
$337$	−7.25544	−0.395229	−0.197614	−	0.980280i	$-0.563319\pi$
$337$	−7.25544	−0.395229	−0.197614	+	0.980280i	$0.563319\pi$
$338$	−9.51087	−0.517323
$339$	0	0
$340$	−5.37228	−0.291353
$341$	−8.86141	−0.479872
$342$	0	0
$343$	−16.6277	−0.897812
$344$	−7.37228	−0.397487
$345$	0	0
$346$	22.8614	1.22904
$347$	22.9783	1.23354	0.616769	−	0.787145i	$-0.288441\pi$
$347$	22.9783	1.23354	0.616769	+	0.787145i	$0.288441\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−1.37228	−0.0733515
$351$	0	0
$352$	−3.37228	−0.179743
$353$	22.8614	1.21679	0.608395	−	0.793634i	$-0.291814\pi$
$353$	22.8614	1.21679	0.608395	+	0.793634i	$0.291814\pi$
$354$	0	0
$355$	−6.74456	−0.357964
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−26.2337	−1.38649
$359$	30.9783	1.63497	0.817485	−	0.575950i	$-0.195368\pi$
$359$	30.9783	1.63497	0.817485	+	0.575950i	$0.195368\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−7.48913	−0.393620
$363$	0	0
$364$	−6.51087	−0.341263
$365$	−8.74456	−0.457711
$366$	0	0
$367$	3.88316	0.202699	0.101350	−	0.994851i	$-0.467684\pi$
$367$	3.88316	0.202699	0.101350	+	0.994851i	$0.467684\pi$
$368$	−6.74456	−0.351585
$369$	0	0
$370$	1.00000	0.0519875
$371$	−1.88316	−0.0977686
$372$	0	0
$373$	−31.4891	−1.63045	−0.815223	−	0.579148i	$-0.803385\pi$
$373$	−31.4891	−1.63045	−0.815223	+	0.579148i	$0.803385\pi$
$374$	−18.1168	−0.936800
$375$	0	0
$376$	−8.74456	−0.450966
$377$	38.5109	1.98341
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	18.6277	0.953077
$383$	13.4891	0.689262	0.344631	−	0.938738i	$-0.388004\pi$
$383$	13.4891	0.689262	0.344631	+	0.938738i	$0.388004\pi$
$384$	0	0
$385$	−4.62772	−0.235850
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−0.116844	−0.00593185
$389$	−14.8614	−0.753503	−0.376752	−	0.926314i	$-0.622959\pi$
$389$	−14.8614	−0.753503	−0.376752	+	0.926314i	$0.622959\pi$
$390$	0	0
$391$	−36.2337	−1.83242
$392$	5.11684	0.258440
$393$	0	0
$394$	27.4891	1.38488
$395$	4.74456	0.238725
$396$	0	0
$397$	24.9783	1.25362	0.626811	−	0.779171i	$-0.284360\pi$
$397$	24.9783	1.25362	0.626811	+	0.779171i	$0.284360\pi$
$398$	11.2554	0.564184
$399$	0	0
$400$	1.00000	0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	12.4674	0.621044
$404$	11.4891	0.571605
$405$	0	0
$406$	11.1386	0.552799
$407$	3.37228	0.167158
$408$	0	0
$409$	6.23369	0.308236	0.154118	−	0.988052i	$-0.450746\pi$
$409$	6.23369	0.308236	0.154118	+	0.988052i	$0.450746\pi$
$410$	−5.37228	−0.265318
$411$	0	0
$412$	−9.48913	−0.467496
$413$	−17.4891	−0.860584
$414$	0	0
$415$	−0.744563	−0.0365491
$416$	4.74456	0.232621
$417$	0	0
$418$	6.74456	0.329887
$419$	−13.4891	−0.658987	−0.329493	−	0.944158i	$-0.606878\pi$
$419$	−13.4891	−0.658987	−0.329493	+	0.944158i	$0.606878\pi$
$420$	0	0
$421$	−7.48913	−0.364998	−0.182499	−	0.983206i	$-0.558419\pi$
$421$	−7.48913	−0.364998	−0.182499	+	0.983206i	$0.558419\pi$
$422$	14.1168	0.687197
$423$	0	0
$424$	1.37228	0.0666439
$425$	5.37228	0.260594
$426$	0	0
$427$	−7.37228	−0.356770
$428$	3.48913	0.168653
$429$	0	0
$430$	7.37228	0.355523
$431$	1.37228	0.0661005	0.0330502	−	0.999454i	$-0.489478\pi$
$431$	1.37228	0.0661005	0.0330502	+	0.999454i	$0.489478\pi$
$432$	0	0
$433$	12.9783	0.623695	0.311847	−	0.950132i	$-0.399052\pi$
$433$	12.9783	0.623695	0.311847	+	0.950132i	$0.399052\pi$
$434$	3.60597	0.173092
$435$	0	0
$436$	0.116844	0.00559581
$437$	13.4891	0.645272
$438$	0	0
$439$	−13.3723	−0.638224	−0.319112	−	0.947717i	$-0.603385\pi$
$439$	−13.3723	−0.638224	−0.319112	+	0.947717i	$0.603385\pi$
$440$	3.37228	0.160767
$441$	0	0
$442$	25.4891	1.21239
$443$	16.9783	0.806661	0.403331	−	0.915054i	$-0.367852\pi$
$443$	16.9783	0.806661	0.403331	+	0.915054i	$0.367852\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	1.37228	0.0649794
$447$	0	0
$448$	1.37228	0.0648342
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−18.1168	−0.853089
$452$	−17.3723	−0.817123
$453$	0	0
$454$	−11.3723	−0.533728
$455$	6.51087	0.305235
$456$	0	0
$457$	18.8614	0.882299	0.441150	−	0.897434i	$-0.354571\pi$
$457$	18.8614	0.882299	0.441150	+	0.897434i	$0.354571\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	6.74456	0.314467
$461$	39.0951	1.82084	0.910420	−	0.413685i	$-0.135759\pi$
$461$	39.0951	1.82084	0.910420	+	0.413685i	$0.135759\pi$
$462$	0	0
$463$	−5.48913	−0.255101	−0.127551	−	0.991832i	$-0.540712\pi$
$463$	−5.48913	−0.255101	−0.127551	+	0.991832i	$0.540712\pi$
$464$	−8.11684	−0.376815
$465$	0	0
$466$	6.23369	0.288770
$467$	−30.3505	−1.40446	−0.702228	−	0.711953i	$-0.747811\pi$
$467$	−30.3505	−1.40446	−0.702228	+	0.711953i	$0.747811\pi$
$468$	0	0
$469$	6.51087	0.300644
$470$	8.74456	0.403357
$471$	0	0
$472$	12.7446	0.586616
$473$	24.8614	1.14313
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	7.37228	0.337908
$477$	0	0
$478$	17.6060	0.805278
$479$	−3.25544	−0.148745	−0.0743724	−	0.997231i	$-0.523695\pi$
$479$	−3.25544	−0.148745	−0.0743724	+	0.997231i	$0.523695\pi$
$480$	0	0
$481$	−4.74456	−0.216333
$482$	10.2337	0.466132
$483$	0	0
$484$	0.372281	0.0169219
$485$	0.116844	0.00530561
$486$	0	0
$487$	−37.7228	−1.70938	−0.854692	−	0.519136i	$-0.826254\pi$
$487$	−37.7228	−1.70938	−0.854692	+	0.519136i	$0.826254\pi$
$488$	5.37228	0.243192
$489$	0	0
$490$	−5.11684	−0.231155
$491$	−14.9783	−0.675959	−0.337979	−	0.941153i	$-0.609743\pi$
$491$	−14.9783	−0.675959	−0.337979	+	0.941153i	$0.609743\pi$
$492$	0	0
$493$	−43.6060	−1.96391
$494$	−9.48913	−0.426936
$495$	0	0
$496$	−2.62772	−0.117988
$497$	9.25544	0.415163
$498$	0	0
$499$	−24.9783	−1.11818	−0.559090	−	0.829107i	$-0.688849\pi$
$499$	−24.9783	−1.11818	−0.559090	+	0.829107i	$0.688849\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	11.4891	0.512785
$503$	−29.4891	−1.31486	−0.657428	−	0.753518i	$-0.728355\pi$
$503$	−29.4891	−1.31486	−0.657428	+	0.753518i	$0.728355\pi$
$504$	0	0
$505$	−11.4891	−0.511259
$506$	22.7446	1.01112
$507$	0	0
$508$	−16.7446	−0.742920
$509$	11.2554	0.498888	0.249444	−	0.968389i	$-0.419752\pi$
$509$	11.2554	0.498888	0.249444	+	0.968389i	$0.419752\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−20.9783	−0.925311
$515$	9.48913	0.418141
$516$	0	0
$517$	29.4891	1.29693
$518$	−1.37228	−0.0602946
$519$	0	0
$520$	−4.74456	−0.208063
$521$	−27.0951	−1.18706	−0.593529	−	0.804813i	$-0.702266\pi$
$521$	−27.0951	−1.18706	−0.593529	+	0.804813i	$0.702266\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	3.25544	0.142214
$525$	0	0
$526$	0.116844	0.00509464
$527$	−14.1168	−0.614939
$528$	0	0
$529$	22.4891	0.977788
$530$	−1.37228	−0.0596081
$531$	0	0
$532$	−2.74456	−0.118992
$533$	25.4891	1.10406
$534$	0	0
$535$	−3.48913	−0.150848
$536$	−4.74456	−0.204934
$537$	0	0
$538$	10.0000	0.431131
$539$	−17.2554	−0.743244
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	6.74456	0.289704
$543$	0	0
$544$	−5.37228	−0.230335
$545$	−0.116844	−0.00500505
$546$	0	0
$547$	39.3723	1.68344	0.841719	−	0.539916i	$-0.181544\pi$
$547$	39.3723	1.68344	0.841719	+	0.539916i	$0.181544\pi$
$548$	−16.7446	−0.715292
$549$	0	0
$550$	−3.37228	−0.143795
$551$	16.2337	0.691578
$552$	0	0
$553$	−6.51087	−0.276871
$554$	−0.744563	−0.0316334
$555$	0	0
$556$	−1.88316	−0.0798636
$557$	−8.74456	−0.370519	−0.185260	−	0.982690i	$-0.559313\pi$
$557$	−8.74456	−0.370519	−0.185260	+	0.982690i	$0.559313\pi$
$558$	0	0
$559$	−34.9783	−1.47942
$560$	−1.37228	−0.0579895
$561$	0	0
$562$	24.7446	1.04379
$563$	−33.0951	−1.39479	−0.697396	−	0.716686i	$-0.745658\pi$
$563$	−33.0951	−1.39479	−0.697396	+	0.716686i	$0.745658\pi$
$564$	0	0
$565$	17.3723	0.730857
$566$	−5.48913	−0.230725
$567$	0	0
$568$	−6.74456	−0.282996
$569$	32.9783	1.38252	0.691260	−	0.722606i	$-0.257056\pi$
$569$	32.9783	1.38252	0.691260	+	0.722606i	$0.257056\pi$
$570$	0	0
$571$	−27.6060	−1.15527	−0.577637	−	0.816294i	$-0.696025\pi$
$571$	−27.6060	−1.15527	−0.577637	+	0.816294i	$0.696025\pi$
$572$	−16.0000	−0.668994
$573$	0	0
$574$	7.37228	0.307713
$575$	−6.74456	−0.281268
$576$	0	0
$577$	−7.48913	−0.311776	−0.155888	−	0.987775i	$-0.549824\pi$
$577$	−7.48913	−0.311776	−0.155888	+	0.987775i	$0.549824\pi$
$578$	−11.8614	−0.493369
$579$	0	0
$580$	8.11684	0.337034
$581$	1.02175	0.0423893
$582$	0	0
$583$	−4.62772	−0.191661
$584$	−8.74456	−0.361853
$585$	0	0
$586$	−18.8614	−0.779158
$587$	−47.8397	−1.97455	−0.987277	−	0.159010i	$-0.949170\pi$
$587$	−47.8397	−1.97455	−0.987277	+	0.159010i	$0.949170\pi$
$588$	0	0
$589$	5.25544	0.216547
$590$	−12.7446	−0.524685
$591$	0	0
$592$	1.00000	0.0410997
$593$	10.2337	0.420247	0.210124	−	0.977675i	$-0.432613\pi$
$593$	10.2337	0.420247	0.210124	+	0.977675i	$0.432613\pi$
$594$	0	0
$595$	−7.37228	−0.302234
$596$	11.4891	0.470613
$597$	0	0
$598$	−32.0000	−1.30858
$599$	−17.4891	−0.714586	−0.357293	−	0.933992i	$-0.616300\pi$
$599$	−17.4891	−0.714586	−0.357293	+	0.933992i	$0.616300\pi$
$600$	0	0
$601$	−5.37228	−0.219140	−0.109570	−	0.993979i	$-0.534947\pi$
$601$	−5.37228	−0.219140	−0.109570	+	0.993979i	$0.534947\pi$
$602$	−10.1168	−0.412332
$603$	0	0
$604$	−20.0000	−0.813788
$605$	−0.372281	−0.0151354
$606$	0	0
$607$	17.7228	0.719347	0.359673	−	0.933078i	$-0.382888\pi$
$607$	17.7228	0.719347	0.359673	+	0.933078i	$0.382888\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−5.37228	−0.217517
$611$	−41.4891	−1.67847
$612$	0	0
$613$	43.0951	1.74059	0.870297	−	0.492527i	$-0.163927\pi$
$613$	43.0951	1.74059	0.870297	+	0.492527i	$0.163927\pi$
$614$	23.4891	0.947944
$615$	0	0
$616$	−4.62772	−0.186456
$617$	31.7228	1.27711	0.638556	−	0.769575i	$-0.279532\pi$
$617$	31.7228	1.27711	0.638556	+	0.769575i	$0.279532\pi$
$618$	0	0
$619$	30.1168	1.21050	0.605249	−	0.796036i	$-0.293074\pi$
$619$	30.1168	1.21050	0.605249	+	0.796036i	$0.293074\pi$
$620$	2.62772	0.105532
$621$	0	0
$622$	−1.37228	−0.0550235
$623$	−13.7228	−0.549793
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.48913	0.139453
$627$	0	0
$628$	17.3723	0.693229
$629$	5.37228	0.214207
$630$	0	0
$631$	−19.0951	−0.760164	−0.380082	−	0.924953i	$-0.624104\pi$
$631$	−19.0951	−0.760164	−0.380082	+	0.924953i	$0.624104\pi$
$632$	4.74456	0.188729
$633$	0	0
$634$	25.3723	1.00766
$635$	16.7446	0.664488
$636$	0	0
$637$	24.2772	0.961897
$638$	27.3723	1.08368
$639$	0	0
$640$	1.00000	0.0395285
$641$	49.6060	1.95932	0.979659	−	0.200670i	$-0.0643117\pi$
$641$	49.6060	1.95932	0.979659	+	0.200670i	$0.0643117\pi$
$642$	0	0
$643$	3.37228	0.132990	0.0664949	−	0.997787i	$-0.478818\pi$
$643$	3.37228	0.132990	0.0664949	+	0.997787i	$0.478818\pi$
$644$	−9.25544	−0.364715
$645$	0	0
$646$	10.7446	0.422739
$647$	13.4891	0.530312	0.265156	−	0.964205i	$-0.414577\pi$
$647$	13.4891	0.530312	0.265156	+	0.964205i	$0.414577\pi$
$648$	0	0
$649$	−42.9783	−1.68704
$650$	4.74456	0.186097
$651$	0	0
$652$	19.3723	0.758677
$653$	−0.510875	−0.0199921	−0.00999604	−	0.999950i	$-0.503182\pi$
$653$	−0.510875	−0.0199921	−0.00999604	+	0.999950i	$0.503182\pi$
$654$	0	0
$655$	−3.25544	−0.127200
$656$	−5.37228	−0.209752
$657$	0	0
$658$	−12.0000	−0.467809
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	4.35053	0.169216	0.0846080	−	0.996414i	$-0.473036\pi$
$661$	4.35053	0.169216	0.0846080	+	0.996414i	$0.473036\pi$
$662$	−23.4891	−0.912931
$663$	0	0
$664$	−0.744563	−0.0288946
$665$	2.74456	0.106430
$666$	0	0
$667$	54.7446	2.11972
$668$	−1.48913	−0.0576160
$669$	0	0
$670$	4.74456	0.183298
$671$	−18.1168	−0.699393
$672$	0	0
$673$	−8.51087	−0.328070	−0.164035	−	0.986455i	$-0.552451\pi$
$673$	−8.51087	−0.328070	−0.164035	+	0.986455i	$0.552451\pi$
$674$	7.25544	0.279469
$675$	0	0
$676$	9.51087	0.365803
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−0.160343	−0.00615339
$680$	5.37228	0.206018
$681$	0	0
$682$	8.86141	0.339321
$683$	−8.62772	−0.330130	−0.165065	−	0.986283i	$-0.552783\pi$
$683$	−8.62772	−0.330130	−0.165065	+	0.986283i	$0.552783\pi$
$684$	0	0
$685$	16.7446	0.639777
$686$	16.6277	0.634849
$687$	0	0
$688$	7.37228	0.281066
$689$	6.51087	0.248045
$690$	0	0
$691$	22.3505	0.850254	0.425127	−	0.905134i	$-0.360229\pi$
$691$	22.3505	0.850254	0.425127	+	0.905134i	$0.360229\pi$
$692$	−22.8614	−0.869060
$693$	0	0
$694$	−22.9783	−0.872242
$695$	1.88316	0.0714322
$696$	0	0
$697$	−28.8614	−1.09320
$698$	22.0000	0.832712
$699$	0	0
$700$	1.37228	0.0518674
$701$	26.4674	0.999659	0.499829	−	0.866124i	$-0.333396\pi$
$701$	26.4674	0.999659	0.499829	+	0.866124i	$0.333396\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	3.37228	0.127098
$705$	0	0
$706$	−22.8614	−0.860400
$707$	15.7663	0.592953
$708$	0	0
$709$	40.1168	1.50662	0.753310	−	0.657666i	$-0.228456\pi$
$709$	40.1168	1.50662	0.753310	+	0.657666i	$0.228456\pi$
$710$	6.74456	0.253119
$711$	0	0
$712$	10.0000	0.374766
$713$	17.7228	0.663725
$714$	0	0
$715$	16.0000	0.598366
$716$	26.2337	0.980399
$717$	0	0
$718$	−30.9783	−1.15610
$719$	34.7446	1.29575	0.647877	−	0.761745i	$-0.275657\pi$
$719$	34.7446	1.29575	0.647877	+	0.761745i	$0.275657\pi$
$720$	0	0
$721$	−13.0217	−0.484955
$722$	15.0000	0.558242
$723$	0	0
$724$	7.48913	0.278331
$725$	−8.11684	−0.301452
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	6.51087	0.241309
$729$	0	0
$730$	8.74456	0.323651
$731$	39.6060	1.46488
$732$	0	0
$733$	−29.3723	−1.08489	−0.542445	−	0.840091i	$-0.682501\pi$
$733$	−29.3723	−1.08489	−0.542445	+	0.840091i	$0.682501\pi$
$734$	−3.88316	−0.143330
$735$	0	0
$736$	6.74456	0.248608
$737$	16.0000	0.589368
$738$	0	0
$739$	−36.8614	−1.35597	−0.677984	−	0.735076i	$-0.737146\pi$
$739$	−36.8614	−1.35597	−0.677984	+	0.735076i	$0.737146\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	1.88316	0.0691328
$743$	5.37228	0.197090	0.0985449	−	0.995133i	$-0.468581\pi$
$743$	5.37228	0.197090	0.0985449	+	0.995133i	$0.468581\pi$
$744$	0	0
$745$	−11.4891	−0.420929
$746$	31.4891	1.15290
$747$	0	0
$748$	18.1168	0.662417
$749$	4.78806	0.174952
$750$	0	0
$751$	10.7446	0.392075	0.196037	−	0.980596i	$-0.437193\pi$
$751$	10.7446	0.392075	0.196037	+	0.980596i	$0.437193\pi$
$752$	8.74456	0.318881
$753$	0	0
$754$	−38.5109	−1.40248
$755$	20.0000	0.727875
$756$	0	0
$757$	43.9565	1.59763	0.798813	−	0.601579i	$-0.205462\pi$
$757$	43.9565	1.59763	0.798813	+	0.601579i	$0.205462\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	−2.00000	−0.0725476
$761$	−5.37228	−0.194745	−0.0973725	−	0.995248i	$-0.531044\pi$
$761$	−5.37228	−0.194745	−0.0973725	+	0.995248i	$0.531044\pi$
$762$	0	0
$763$	0.160343	0.00580480
$764$	−18.6277	−0.673927
$765$	0	0
$766$	−13.4891	−0.487382
$767$	60.4674	2.18335
$768$	0	0
$769$	46.2337	1.66723	0.833615	−	0.552346i	$-0.186267\pi$
$769$	46.2337	1.66723	0.833615	+	0.552346i	$0.186267\pi$
$770$	4.62772	0.166771
$771$	0	0
$772$	2.00000	0.0719816
$773$	24.3505	0.875828	0.437914	−	0.899017i	$-0.355717\pi$
$773$	24.3505	0.875828	0.437914	+	0.899017i	$0.355717\pi$
$774$	0	0
$775$	−2.62772	−0.0943904
$776$	0.116844	0.00419445
$777$	0	0
$778$	14.8614	0.532807
$779$	10.7446	0.384964
$780$	0	0
$781$	22.7446	0.813864
$782$	36.2337	1.29571
$783$	0	0
$784$	−5.11684	−0.182744
$785$	−17.3723	−0.620043
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−27.4891	−0.979260
$789$	0	0
$790$	−4.74456	−0.168804
$791$	−23.8397	−0.847641
$792$	0	0
$793$	25.4891	0.905145
$794$	−24.9783	−0.886445
$795$	0	0
$796$	−11.2554	−0.398938
$797$	27.4891	0.973715	0.486857	−	0.873481i	$-0.338143\pi$
$797$	27.4891	0.973715	0.486857	+	0.873481i	$0.338143\pi$
$798$	0	0
$799$	46.9783	1.66197
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	10.0000	0.353112
$803$	29.4891	1.04065
$804$	0	0
$805$	9.25544	0.326211
$806$	−12.4674	−0.439145
$807$	0	0
$808$	−11.4891	−0.404186
$809$	51.4891	1.81026	0.905131	−	0.425134i	$-0.139773\pi$
$809$	51.4891	1.81026	0.905131	+	0.425134i	$0.139773\pi$
$810$	0	0
$811$	49.4891	1.73780	0.868899	−	0.494989i	$-0.164828\pi$
$811$	49.4891	1.73780	0.868899	+	0.494989i	$0.164828\pi$
$812$	−11.1386	−0.390888
$813$	0	0
$814$	−3.37228	−0.118198
$815$	−19.3723	−0.678581
$816$	0	0
$817$	−14.7446	−0.515847
$818$	−6.23369	−0.217956
$819$	0	0
$820$	5.37228	0.187608
$821$	32.7446	1.14279	0.571397	−	0.820674i	$-0.306402\pi$
$821$	32.7446	1.14279	0.571397	+	0.820674i	$0.306402\pi$
$822$	0	0
$823$	−39.7228	−1.38465	−0.692325	−	0.721586i	$-0.743414\pi$
$823$	−39.7228	−1.38465	−0.692325	+	0.721586i	$0.743414\pi$
$824$	9.48913	0.330569
$825$	0	0
$826$	17.4891	0.608524
$827$	54.3505	1.88995	0.944977	−	0.327138i	$-0.106084\pi$
$827$	54.3505	1.88995	0.944977	+	0.327138i	$0.106084\pi$
$828$	0	0
$829$	−40.3505	−1.40143	−0.700716	−	0.713440i	$-0.747136\pi$
$829$	−40.3505	−1.40143	−0.700716	+	0.713440i	$0.747136\pi$
$830$	0.744563	0.0258441
$831$	0	0
$832$	−4.74456	−0.164488
$833$	−27.4891	−0.952442
$834$	0	0
$835$	1.48913	0.0515333
$836$	−6.74456	−0.233266
$837$	0	0
$838$	13.4891	0.465974
$839$	29.4891	1.01808	0.509039	−	0.860744i	$-0.330001\pi$
$839$	29.4891	1.01808	0.509039	+	0.860744i	$0.330001\pi$
$840$	0	0
$841$	36.8832	1.27183
$842$	7.48913	0.258092
$843$	0	0
$844$	−14.1168	−0.485922
$845$	−9.51087	−0.327184
$846$	0	0
$847$	0.510875	0.0175539
$848$	−1.37228	−0.0471243
$849$	0	0
$850$	−5.37228	−0.184268
$851$	−6.74456	−0.231201
$852$	0	0
$853$	35.4891	1.21512	0.607562	−	0.794272i	$-0.292148\pi$
$853$	35.4891	1.21512	0.607562	+	0.794272i	$0.292148\pi$
$854$	7.37228	0.252274
$855$	0	0
$856$	−3.48913	−0.119256
$857$	29.1386	0.995355	0.497678	−	0.867362i	$-0.334186\pi$
$857$	29.1386	0.995355	0.497678	+	0.867362i	$0.334186\pi$
$858$	0	0
$859$	−19.7228	−0.672934	−0.336467	−	0.941695i	$-0.609232\pi$
$859$	−19.7228	−0.672934	−0.336467	+	0.941695i	$0.609232\pi$
$860$	−7.37228	−0.251393
$861$	0	0
$862$	−1.37228	−0.0467401
$863$	−53.8397	−1.83272	−0.916362	−	0.400352i	$-0.868888\pi$
$863$	−53.8397	−1.83272	−0.916362	+	0.400352i	$0.868888\pi$
$864$	0	0
$865$	22.8614	0.777311
$866$	−12.9783	−0.441019
$867$	0	0
$868$	−3.60597	−0.122395
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−22.5109	−0.762752
$872$	−0.116844	−0.00395684
$873$	0	0
$874$	−13.4891	−0.456276
$875$	−1.37228	−0.0463916
$876$	0	0
$877$	49.3723	1.66718	0.833592	−	0.552381i	$-0.186281\pi$
$877$	49.3723	1.66718	0.833592	+	0.552381i	$0.186281\pi$
$878$	13.3723	0.451293
$879$	0	0
$880$	−3.37228	−0.113680
$881$	−1.37228	−0.0462333	−0.0231167	−	0.999733i	$-0.507359\pi$
$881$	−1.37228	−0.0462333	−0.0231167	+	0.999733i	$0.507359\pi$
$882$	0	0
$883$	−11.3723	−0.382708	−0.191354	−	0.981521i	$-0.561288\pi$
$883$	−11.3723	−0.382708	−0.191354	+	0.981521i	$0.561288\pi$
$884$	−25.4891	−0.857292
$885$	0	0
$886$	−16.9783	−0.570395
$887$	25.3723	0.851918	0.425959	−	0.904743i	$-0.359937\pi$
$887$	25.3723	0.851918	0.425959	+	0.904743i	$0.359937\pi$
$888$	0	0
$889$	−22.9783	−0.770666
$890$	−10.0000	−0.335201
$891$	0	0
$892$	−1.37228	−0.0459474
$893$	−17.4891	−0.585251
$894$	0	0
$895$	−26.2337	−0.876895
$896$	−1.37228	−0.0458447
$897$	0	0
$898$	18.0000	0.600668
$899$	21.3288	0.711355
$900$	0	0
$901$	−7.37228	−0.245606
$902$	18.1168	0.603225
$903$	0	0
$904$	17.3723	0.577793
$905$	−7.48913	−0.248947
$906$	0	0
$907$	40.4674	1.34370	0.671849	−	0.740689i	$-0.265501\pi$
$907$	40.4674	1.34370	0.671849	+	0.740689i	$0.265501\pi$
$908$	11.3723	0.377402
$909$	0	0
$910$	−6.51087	−0.215833
$911$	−17.7663	−0.588624	−0.294312	−	0.955709i	$-0.595091\pi$
$911$	−17.7663	−0.588624	−0.294312	+	0.955709i	$0.595091\pi$
$912$	0	0
$913$	2.51087	0.0830978
$914$	−18.8614	−0.623880
$915$	0	0
$916$	−10.0000	−0.330409
$917$	4.46738	0.147526
$918$	0	0
$919$	7.72281	0.254752	0.127376	−	0.991854i	$-0.459344\pi$
$919$	7.72281	0.254752	0.127376	+	0.991854i	$0.459344\pi$
$920$	−6.74456	−0.222362
$921$	0	0
$922$	−39.0951	−1.28753
$923$	−32.0000	−1.05329
$924$	0	0
$925$	1.00000	0.0328798
$926$	5.48913	0.180384
$927$	0	0
$928$	8.11684	0.266448
$929$	−31.0951	−1.02020	−0.510098	−	0.860116i	$-0.670391\pi$
$929$	−31.0951	−1.02020	−0.510098	+	0.860116i	$0.670391\pi$
$930$	0	0
$931$	10.2337	0.335396
$932$	−6.23369	−0.204191
$933$	0	0
$934$	30.3505	0.993100
$935$	−18.1168	−0.592484
$936$	0	0
$937$	36.9783	1.20803	0.604013	−	0.796974i	$-0.293567\pi$
$937$	36.9783	1.20803	0.604013	+	0.796974i	$0.293567\pi$
$938$	−6.51087	−0.212588
$939$	0	0
$940$	−8.74456	−0.285216
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	36.2337	1.17993
$944$	−12.7446	−0.414800
$945$	0	0
$946$	−24.8614	−0.808314
$947$	−37.0951	−1.20543	−0.602714	−	0.797957i	$-0.705914\pi$
$947$	−37.0951	−1.20543	−0.602714	+	0.797957i	$0.705914\pi$
$948$	0	0
$949$	−41.4891	−1.34679
$950$	2.00000	0.0648886
$951$	0	0
$952$	−7.37228	−0.238937
$953$	−46.2337	−1.49766	−0.748828	−	0.662764i	$-0.769383\pi$
$953$	−46.2337	−1.49766	−0.748828	+	0.662764i	$0.769383\pi$
$954$	0	0
$955$	18.6277	0.602779
$956$	−17.6060	−0.569418
$957$	0	0
$958$	3.25544	0.105178
$959$	−22.9783	−0.742006
$960$	0	0
$961$	−24.0951	−0.777261
$962$	4.74456	0.152971
$963$	0	0
$964$	−10.2337	−0.329605
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	−0.372281	−0.0119656
$969$	0	0
$970$	−0.116844	−0.00375163
$971$	19.6060	0.629185	0.314593	−	0.949227i	$-0.398132\pi$
$971$	19.6060	0.629185	0.314593	+	0.949227i	$0.398132\pi$
$972$	0	0
$973$	−2.58422	−0.0828463
$974$	37.7228	1.20872
$975$	0	0
$976$	−5.37228	−0.171963
$977$	11.8832	0.380176	0.190088	−	0.981767i	$-0.439123\pi$
$977$	11.8832	0.380176	0.190088	+	0.981767i	$0.439123\pi$
$978$	0	0
$979$	−33.7228	−1.07779
$980$	5.11684	0.163452
$981$	0	0
$982$	14.9783	0.477975
$983$	−30.8614	−0.984326	−0.492163	−	0.870503i	$-0.663794\pi$
$983$	−30.8614	−0.984326	−0.492163	+	0.870503i	$0.663794\pi$
$984$	0	0
$985$	27.4891	0.875876
$986$	43.6060	1.38870
$987$	0	0
$988$	9.48913	0.301889
$989$	−49.7228	−1.58109
$990$	0	0
$991$	25.6060	0.813400	0.406700	−	0.913562i	$-0.366679\pi$
$991$	25.6060	0.813400	0.406700	+	0.913562i	$0.366679\pi$
$992$	2.62772	0.0834302
$993$	0	0
$994$	−9.25544	−0.293565
$995$	11.2554	0.356821
$996$	0	0
$997$	26.2337	0.830829	0.415415	−	0.909632i	$-0.363637\pi$
$997$	26.2337	0.830829	0.415415	+	0.909632i	$0.363637\pi$
$998$	24.9783	0.790673
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bb.1.2		2
3.2	odd	2		370.2.a.f.1.2	✓	2
12.11	even	2		2960.2.a.o.1.1		2
15.2	even	4		1850.2.b.m.149.4		4
15.8	even	4		1850.2.b.m.149.1		4
15.14	odd	2		1850.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.f.1.2	✓	2	3.2	odd	2
1850.2.a.q.1.1		2	15.14	odd	2
1850.2.b.m.149.1		4	15.8	even	4
1850.2.b.m.149.4		4	15.2	even	4
2960.2.a.o.1.1		2	12.11	even	2
3330.2.a.bb.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.37228 q^{11} -4.74456 q^{13} -1.37228 q^{14} +1.00000 q^{16} +5.37228 q^{17} -2.00000 q^{19} -1.00000 q^{20} -3.37228 q^{22} -6.74456 q^{23} +1.00000 q^{25} +4.74456 q^{26} +1.37228 q^{28} -8.11684 q^{29} -2.62772 q^{31} -1.00000 q^{32} -5.37228 q^{34} -1.37228 q^{35} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} -5.37228 q^{41} +7.37228 q^{43} +3.37228 q^{44} +6.74456 q^{46} +8.74456 q^{47} -5.11684 q^{49} -1.00000 q^{50} -4.74456 q^{52} -1.37228 q^{53} -3.37228 q^{55} -1.37228 q^{56} +8.11684 q^{58} -12.7446 q^{59} -5.37228 q^{61} +2.62772 q^{62} +1.00000 q^{64} +4.74456 q^{65} +4.74456 q^{67} +5.37228 q^{68} +1.37228 q^{70} +6.74456 q^{71} +8.74456 q^{73} -1.00000 q^{74} -2.00000 q^{76} +4.62772 q^{77} -4.74456 q^{79} -1.00000 q^{80} +5.37228 q^{82} +0.744563 q^{83} -5.37228 q^{85} -7.37228 q^{86} -3.37228 q^{88} -10.0000 q^{89} -6.51087 q^{91} -6.74456 q^{92} -8.74456 q^{94} +2.00000 q^{95} -0.116844 q^{97} +5.11684 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 3 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8} + 2 q^{10} + q^{11} + 2 q^{13} + 3 q^{14} + 2 q^{16} + 5 q^{17} - 4 q^{19} - 2 q^{20} - q^{22} - 2 q^{23} + 2 q^{25} - 2 q^{26} - 3 q^{28} + q^{29} - 11 q^{31} - 2 q^{32} - 5 q^{34} + 3 q^{35} + 2 q^{37} + 4 q^{38} + 2 q^{40} - 5 q^{41} + 9 q^{43} + q^{44} + 2 q^{46} + 6 q^{47} + 7 q^{49} - 2 q^{50} + 2 q^{52} + 3 q^{53} - q^{55} + 3 q^{56} - q^{58} - 14 q^{59} - 5 q^{61} + 11 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{67} + 5 q^{68} - 3 q^{70} + 2 q^{71} + 6 q^{73} - 2 q^{74} - 4 q^{76} + 15 q^{77} + 2 q^{79} - 2 q^{80} + 5 q^{82} - 10 q^{83} - 5 q^{85} - 9 q^{86} - q^{88} - 20 q^{89} - 36 q^{91} - 2 q^{92} - 6 q^{94} + 4 q^{95} + 17 q^{97} - 7 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 3 * q^7 - 2 * q^8 + 2 * q^10 + q^11 + 2 * q^13 + 3 * q^14 + 2 * q^16 + 5 * q^17 - 4 * q^19 - 2 * q^20 - q^22 - 2 * q^23 + 2 * q^25 - 2 * q^26 - 3 * q^28 + q^29 - 11 * q^31 - 2 * q^32 - 5 * q^34 + 3 * q^35 + 2 * q^37 + 4 * q^38 + 2 * q^40 - 5 * q^41 + 9 * q^43 + q^44 + 2 * q^46 + 6 * q^47 + 7 * q^49 - 2 * q^50 + 2 * q^52 + 3 * q^53 - q^55 + 3 * q^56 - q^58 - 14 * q^59 - 5 * q^61 + 11 * q^62 + 2 * q^64 - 2 * q^65 - 2 * q^67 + 5 * q^68 - 3 * q^70 + 2 * q^71 + 6 * q^73 - 2 * q^74 - 4 * q^76 + 15 * q^77 + 2 * q^79 - 2 * q^80 + 5 * q^82 - 10 * q^83 - 5 * q^85 - 9 * q^86 - q^88 - 20 * q^89 - 36 * q^91 - 2 * q^92 - 6 * q^94 + 4 * q^95 + 17 * q^97 - 7 * q^98

Introduction

L-functions

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Database

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