LMFDB - Embedded newform 3330.2.a.bj.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:	$0$
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			1.2
Root			$3.36007$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.487359 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.487359 q^{7} -1.00000 q^{8} +1.00000 q^{10} +5.20750 q^{11} +4.12492 q^{13} -0.487359 q^{14} +1.00000 q^{16} -3.67781 q^{17} +0.430057 q^{19} -1.00000 q^{20} -5.20750 q^{22} -7.31537 q^{23} +1.00000 q^{25} -4.12492 q^{26} +0.487359 q^{28} +6.77745 q^{29} +7.80273 q^{31} -1.00000 q^{32} +3.67781 q^{34} -0.487359 q^{35} -1.00000 q^{37} -0.430057 q^{38} +1.00000 q^{40} +9.80273 q^{41} +4.77745 q^{43} +5.20750 q^{44} +7.31537 q^{46} -3.52969 q^{47} -6.76248 q^{49} -1.00000 q^{50} +4.12492 q^{52} +7.67781 q^{53} -5.20750 q^{55} -0.487359 q^{56} -6.77745 q^{58} +0.974718 q^{59} -14.5229 q^{61} -7.80273 q^{62} +1.00000 q^{64} -4.12492 q^{65} -1.19045 q^{67} -3.67781 q^{68} +0.487359 q^{70} -8.58018 q^{71} -7.15020 q^{73} +1.00000 q^{74} +0.430057 q^{76} +2.53792 q^{77} +4.00000 q^{79} -1.00000 q^{80} -9.80273 q^{82} +16.5399 q^{83} +3.67781 q^{85} -4.77745 q^{86} -5.20750 q^{88} -16.7051 q^{89} +2.01032 q^{91} -7.31537 q^{92} +3.52969 q^{94} -0.430057 q^{95} -7.16726 q^{97} +6.76248 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	0.487359	0.184204	0.0921022	−	0.995750i	$-0.470641\pi$
$7$	0.487359	0.184204	0.0921022	+	0.995750i	$0.470641\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	5.20750	1.57012	0.785061	−	0.619419i	$-0.212632\pi$
$11$	5.20750	1.57012	0.785061	+	0.619419i	$0.212632\pi$
$12$	0	0
$13$	4.12492	1.14405	0.572023	−	0.820237i	$-0.306159\pi$
$13$	4.12492	1.14405	0.572023	+	0.820237i	$0.306159\pi$
$14$	−0.487359	−0.130252
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.67781	−0.892000	−0.446000	−	0.895033i	$-0.647152\pi$
$17$	−3.67781	−0.892000	−0.446000	+	0.895033i	$0.647152\pi$
$18$	0	0
$19$	0.430057	0.0986618	0.0493309	−	0.998782i	$-0.484291\pi$
$19$	0.430057	0.0986618	0.0493309	+	0.998782i	$0.484291\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−5.20750	−1.11024
$23$	−7.31537	−1.52536	−0.762680	−	0.646776i	$-0.776117\pi$
$23$	−7.31537	−1.52536	−0.762680	+	0.646776i	$0.776117\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.12492	−0.808963
$27$	0	0
$28$	0.487359	0.0921022
$29$	6.77745	1.25854	0.629270	−	0.777187i	$-0.283354\pi$
$29$	6.77745	1.25854	0.629270	+	0.777187i	$0.283354\pi$
$30$	0	0
$31$	7.80273	1.40141	0.700706	−	0.713450i	$-0.252869\pi$
$31$	7.80273	1.40141	0.700706	+	0.713450i	$0.252869\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.67781	0.630739
$35$	−0.487359	−0.0823787
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.430057	−0.0697645
$39$	0	0
$40$	1.00000	0.158114
$41$	9.80273	1.53093	0.765465	−	0.643478i	$-0.222509\pi$
$41$	9.80273	1.53093	0.765465	+	0.643478i	$0.222509\pi$
$42$	0	0
$43$	4.77745	0.728554	0.364277	−	0.931291i	$-0.381316\pi$
$43$	4.77745	0.728554	0.364277	+	0.931291i	$0.381316\pi$
$44$	5.20750	0.785061
$45$	0	0
$46$	7.31537	1.07859
$47$	−3.52969	−0.514859	−0.257429	−	0.966297i	$-0.582875\pi$
$47$	−3.52969	−0.514859	−0.257429	+	0.966297i	$0.582875\pi$
$48$	0	0
$49$	−6.76248	−0.966069
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.12492	0.572023
$53$	7.67781	1.05463	0.527314	−	0.849670i	$-0.323199\pi$
$53$	7.67781	1.05463	0.527314	+	0.849670i	$0.323199\pi$
$54$	0	0
$55$	−5.20750	−0.702180
$56$	−0.487359	−0.0651261
$57$	0	0
$58$	−6.77745	−0.889922
$59$	0.974718	0.126897	0.0634487	−	0.997985i	$-0.479790\pi$
$59$	0.974718	0.126897	0.0634487	+	0.997985i	$0.479790\pi$
$60$	0	0
$61$	−14.5229	−1.85946	−0.929732	−	0.368237i	$-0.879961\pi$
$61$	−14.5229	−1.85946	−0.929732	+	0.368237i	$0.879961\pi$
$62$	−7.80273	−0.990948
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.12492	−0.511633
$66$	0	0
$67$	−1.19045	−0.145437	−0.0727183	−	0.997353i	$-0.523167\pi$
$67$	−1.19045	−0.145437	−0.0727183	+	0.997353i	$0.523167\pi$
$68$	−3.67781	−0.446000
$69$	0	0
$70$	0.487359	0.0582505
$71$	−8.58018	−1.01828	−0.509140	−	0.860684i	$-0.670036\pi$
$71$	−8.58018	−1.01828	−0.509140	+	0.860684i	$0.670036\pi$
$72$	0	0
$73$	−7.15020	−0.836868	−0.418434	−	0.908247i	$-0.637421\pi$
$73$	−7.15020	−0.836868	−0.418434	+	0.908247i	$0.637421\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	0.430057	0.0493309
$77$	2.53792	0.289223
$78$	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
$82$	−9.80273	−1.08253
$83$	16.5399	1.81549	0.907747	−	0.419519i	$-0.137801\pi$
$83$	16.5399	1.81549	0.907747	+	0.419519i	$0.137801\pi$
$84$	0	0
$85$	3.67781	0.398914
$86$	−4.77745	−0.515165
$87$	0	0
$88$	−5.20750	−0.555122
$89$	−16.7051	−1.77074	−0.885368	−	0.464890i	$-0.846094\pi$
$89$	−16.7051	−1.77074	−0.885368	+	0.464890i	$0.846094\pi$
$90$	0	0
$91$	2.01032	0.210738
$92$	−7.31537	−0.762680
$93$	0	0
$94$	3.52969	0.364060
$95$	−0.430057	−0.0441229
$96$	0	0
$97$	−7.16726	−0.727725	−0.363862	−	0.931453i	$-0.618542\pi$
$97$	−7.16726	−0.727725	−0.363862	+	0.931453i	$0.618542\pi$
$98$	6.76248	0.683114
$99$	0	0
$100$	1.00000	0.100000
$101$	13.7454	1.36772	0.683861	−	0.729613i	$-0.260300\pi$
$101$	13.7454	1.36772	0.683861	+	0.729613i	$0.260300\pi$
$102$	0	0
$103$	−3.52969	−0.347791	−0.173896	−	0.984764i	$-0.555636\pi$
$103$	−3.52969	−0.347791	−0.173896	+	0.984764i	$0.555636\pi$
$104$	−4.12492	−0.404482
$105$	0	0
$106$	−7.67781	−0.745735
$107$	−7.09964	−0.686348	−0.343174	−	0.939272i	$-0.611502\pi$
$107$	−7.09964	−0.686348	−0.343174	+	0.939272i	$0.611502\pi$
$108$	0	0
$109$	16.4321	1.57391	0.786953	−	0.617013i	$-0.211657\pi$
$109$	16.4321	1.57391	0.786953	+	0.617013i	$0.211657\pi$
$110$	5.20750	0.496516
$111$	0	0
$112$	0.487359	0.0460511
$113$	1.22255	0.115008	0.0575040	−	0.998345i	$-0.481686\pi$
$113$	1.22255	0.115008	0.0575040	+	0.998345i	$0.481686\pi$
$114$	0	0
$115$	7.31537	0.682162
$116$	6.77745	0.629270
$117$	0	0
$118$	−0.974718	−0.0897300
$119$	−1.79241	−0.164310
$120$	0	0
$121$	16.1181	1.46528
$122$	14.5229	1.31484
$123$	0	0
$124$	7.80273	0.700706
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.5652	1.38119	0.690595	−	0.723242i	$-0.257349\pi$
$127$	15.5652	1.38119	0.690595	+	0.723242i	$0.257349\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.12492	0.361779
$131$	17.5549	1.53378	0.766889	−	0.641780i	$-0.221804\pi$
$131$	17.5549	1.53378	0.766889	+	0.641780i	$0.221804\pi$
$132$	0	0
$133$	0.209592	0.0181739
$134$	1.19045	0.102839
$135$	0	0
$136$	3.67781	0.315370
$137$	21.2498	1.81549	0.907745	−	0.419523i	$-0.137803\pi$
$137$	21.2498	1.81549	0.907745	+	0.419523i	$0.137803\pi$
$138$	0	0
$139$	−15.4082	−1.30691	−0.653453	−	0.756967i	$-0.726680\pi$
$139$	−15.4082	−1.30691	−0.653453	+	0.756967i	$0.726680\pi$
$140$	−0.487359	−0.0411893
$141$	0	0
$142$	8.58018	0.720032
$143$	21.4805	1.79629
$144$	0	0
$145$	−6.77745	−0.562836
$146$	7.15020	0.591755
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	3.47913	0.285021	0.142511	−	0.989793i	$-0.454482\pi$
$149$	3.47913	0.285021	0.142511	+	0.989793i	$0.454482\pi$
$150$	0	0
$151$	−17.7304	−1.44288	−0.721439	−	0.692478i	$-0.756519\pi$
$151$	−17.7304	−1.44288	−0.721439	+	0.692478i	$0.756519\pi$
$152$	−0.430057	−0.0348822
$153$	0	0
$154$	−2.53792	−0.204512
$155$	−7.80273	−0.626730
$156$	0	0
$157$	11.8533	0.945996	0.472998	−	0.881064i	$-0.343172\pi$
$157$	11.8533	0.945996	0.472998	+	0.881064i	$0.343172\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	−3.56521	−0.280978
$162$	0	0
$163$	5.89354	0.461618	0.230809	−	0.972999i	$-0.425863\pi$
$163$	5.89354	0.461618	0.230809	+	0.972999i	$0.425863\pi$
$164$	9.80273	0.765465
$165$	0	0
$166$	−16.5399	−1.28375
$167$	−24.8362	−1.92188	−0.960940	−	0.276758i	$-0.910740\pi$
$167$	−24.8362	−1.92188	−0.960940	+	0.276758i	$0.910740\pi$
$168$	0	0
$169$	4.01497	0.308844
$170$	−3.67781	−0.282075
$171$	0	0
$172$	4.77745	0.364277
$173$	19.9276	1.51507	0.757536	−	0.652794i	$-0.226403\pi$
$173$	19.9276	1.51507	0.757536	+	0.652794i	$0.226403\pi$
$174$	0	0
$175$	0.487359	0.0368409
$176$	5.20750	0.392530
$177$	0	0
$178$	16.7051	1.25210
$179$	4.33034	0.323665	0.161832	−	0.986818i	$-0.448260\pi$
$179$	4.33034	0.323665	0.161832	+	0.986818i	$0.448260\pi$
$180$	0	0
$181$	−0.0505645	−0.00375843	−0.00187922	−	0.999998i	$-0.500598\pi$
$181$	−0.0505645	−0.00375843	−0.00187922	+	0.999998i	$0.500598\pi$
$182$	−2.01032	−0.149015
$183$	0	0
$184$	7.31537	0.539296
$185$	1.00000	0.0735215
$186$	0	0
$187$	−19.1522	−1.40055
$188$	−3.52969	−0.257429
$189$	0	0
$190$	0.430057	0.0311996
$191$	4.37276	0.316401	0.158201	−	0.987407i	$-0.449431\pi$
$191$	4.37276	0.316401	0.158201	+	0.987407i	$0.449431\pi$
$192$	0	0
$193$	−2.47031	−0.177816	−0.0889082	−	0.996040i	$-0.528338\pi$
$193$	−2.47031	−0.177816	−0.0889082	+	0.996040i	$0.528338\pi$
$194$	7.16726	0.514579
$195$	0	0
$196$	−6.76248	−0.483034
$197$	13.3154	0.948681	0.474340	−	0.880341i	$-0.342687\pi$
$197$	13.3154	0.948681	0.474340	+	0.880341i	$0.342687\pi$
$198$	0	0
$199$	21.4403	1.51986	0.759931	−	0.650004i	$-0.225233\pi$
$199$	21.4403	1.51986	0.759931	+	0.650004i	$0.225233\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−13.7454	−0.967125
$203$	3.30305	0.231829
$204$	0	0
$205$	−9.80273	−0.684652
$206$	3.52969	0.245925
$207$	0	0
$208$	4.12492	0.286012
$209$	2.23952	0.154911
$210$	0	0
$211$	13.9679	0.961590	0.480795	−	0.876833i	$-0.340348\pi$
$211$	13.9679	0.961590	0.480795	+	0.876833i	$0.340348\pi$
$212$	7.67781	0.527314
$213$	0	0
$214$	7.09964	0.485321
$215$	−4.77745	−0.325819
$216$	0	0
$217$	3.80273	0.258146
$218$	−16.4321	−1.11292
$219$	0	0
$220$	−5.20750	−0.351090
$221$	−15.1707	−1.02049
$222$	0	0
$223$	−22.4335	−1.50226	−0.751128	−	0.660156i	$-0.770490\pi$
$223$	−22.4335	−1.50226	−0.751128	+	0.660156i	$0.770490\pi$
$224$	−0.487359	−0.0325630
$225$	0	0
$226$	−1.22255	−0.0813230
$227$	−23.8368	−1.58211	−0.791053	−	0.611747i	$-0.790467\pi$
$227$	−23.8368	−1.58211	−0.791053	+	0.611747i	$0.790467\pi$
$228$	0	0
$229$	10.2157	0.675075	0.337537	−	0.941312i	$-0.390406\pi$
$229$	10.2157	0.675075	0.337537	+	0.941312i	$0.390406\pi$
$230$	−7.31537	−0.482361
$231$	0	0
$232$	−6.77745	−0.444961
$233$	−2.07576	−0.135988	−0.0679939	−	0.997686i	$-0.521660\pi$
$233$	−2.07576	−0.135988	−0.0679939	+	0.997686i	$0.521660\pi$
$234$	0	0
$235$	3.52969	0.230252
$236$	0.974718	0.0634487
$237$	0	0
$238$	1.79241	0.116185
$239$	28.2683	1.82852	0.914262	−	0.405123i	$-0.132771\pi$
$239$	28.2683	1.82852	0.914262	+	0.405123i	$0.132771\pi$
$240$	0	0
$241$	20.4491	1.31724	0.658622	−	0.752474i	$-0.271140\pi$
$241$	20.4491	1.31724	0.658622	+	0.752474i	$0.271140\pi$
$242$	−16.1181	−1.03611
$243$	0	0
$244$	−14.5229	−0.929732
$245$	6.76248	0.432039
$246$	0	0
$247$	1.77395	0.112874
$248$	−7.80273	−0.495474
$249$	0	0
$250$	1.00000	0.0632456
$251$	19.3092	1.21879	0.609394	−	0.792868i	$-0.291413\pi$
$251$	19.3092	1.21879	0.609394	+	0.792868i	$0.291413\pi$
$252$	0	0
$253$	−38.0948	−2.39500
$254$	−15.5652	−0.976648
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.735194	−0.0458601	−0.0229301	−	0.999737i	$-0.507300\pi$
$257$	−0.735194	−0.0458601	−0.0229301	+	0.999737i	$0.507300\pi$
$258$	0	0
$259$	−0.487359	−0.0302830
$260$	−4.12492	−0.255817
$261$	0	0
$262$	−17.5549	−1.08454
$263$	−6.47231	−0.399100	−0.199550	−	0.979888i	$-0.563948\pi$
$263$	−6.47231	−0.399100	−0.199550	+	0.979888i	$0.563948\pi$
$264$	0	0
$265$	−7.67781	−0.471644
$266$	−0.209592	−0.0128509
$267$	0	0
$268$	−1.19045	−0.0727183
$269$	0.810958	0.0494450	0.0247225	−	0.999694i	$-0.492130\pi$
$269$	0.810958	0.0494450	0.0247225	+	0.999694i	$0.492130\pi$
$270$	0	0
$271$	17.4403	1.05942	0.529711	−	0.848178i	$-0.322300\pi$
$271$	17.4403	1.05942	0.529711	+	0.848178i	$0.322300\pi$
$272$	−3.67781	−0.223000
$273$	0	0
$274$	−21.2498	−1.28374
$275$	5.20750	0.314024
$276$	0	0
$277$	−1.46007	−0.0877272	−0.0438636	−	0.999038i	$-0.513967\pi$
$277$	−1.46007	−0.0877272	−0.0438636	+	0.999038i	$0.513967\pi$
$278$	15.4082	0.924122
$279$	0	0
$280$	0.487359	0.0291253
$281$	12.1249	0.723312	0.361656	−	0.932312i	$-0.382211\pi$
$281$	12.1249	0.723312	0.361656	+	0.932312i	$0.382211\pi$
$282$	0	0
$283$	19.5652	1.16303	0.581516	−	0.813535i	$-0.302460\pi$
$283$	19.5652	1.16303	0.581516	+	0.813535i	$0.302460\pi$
$284$	−8.58018	−0.509140
$285$	0	0
$286$	−21.4805	−1.27017
$287$	4.77745	0.282004
$288$	0	0
$289$	−3.47372	−0.204336
$290$	6.77745	0.397985
$291$	0	0
$292$	−7.15020	−0.418434
$293$	17.5126	1.02310	0.511550	−	0.859254i	$-0.329072\pi$
$293$	17.5126	1.02310	0.511550	+	0.859254i	$0.329072\pi$
$294$	0	0
$295$	−0.974718	−0.0567503
$296$	1.00000	0.0581238
$297$	0	0
$298$	−3.47913	−0.201541
$299$	−30.1753	−1.74508
$300$	0	0
$301$	2.32833	0.134203
$302$	17.7304	1.02027
$303$	0	0
$304$	0.430057	0.0246655
$305$	14.5229	0.831577
$306$	0	0
$307$	−0.364444	−0.0207999	−0.0104000	−	0.999946i	$-0.503310\pi$
$307$	−0.364444	−0.0207999	−0.0104000	+	0.999946i	$0.503310\pi$
$308$	2.53792	0.144612
$309$	0	0
$310$	7.80273	0.443165
$311$	15.1583	0.859551	0.429776	−	0.902936i	$-0.358593\pi$
$311$	15.1583	0.859551	0.429776	+	0.902936i	$0.358593\pi$
$312$	0	0
$313$	−14.7201	−0.832032	−0.416016	−	0.909357i	$-0.636574\pi$
$313$	−14.7201	−0.832032	−0.416016	+	0.909357i	$0.636574\pi$
$314$	−11.8533	−0.668920
$315$	0	0
$316$	4.00000	0.225018
$317$	0.942615	0.0529426	0.0264713	−	0.999650i	$-0.491573\pi$
$317$	0.942615	0.0529426	0.0264713	+	0.999650i	$0.491573\pi$
$318$	0	0
$319$	35.2936	1.97606
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	3.56521	0.198681
$323$	−1.58167	−0.0880063
$324$	0	0
$325$	4.12492	0.228809
$326$	−5.89354	−0.326413
$327$	0	0
$328$	−9.80273	−0.541265
$329$	−1.72023	−0.0948392
$330$	0	0
$331$	15.8600	0.871746	0.435873	−	0.900008i	$-0.356440\pi$
$331$	15.8600	0.871746	0.435873	+	0.900008i	$0.356440\pi$
$332$	16.5399	0.907747
$333$	0	0
$334$	24.8362	1.35897
$335$	1.19045	0.0650413
$336$	0	0
$337$	16.9549	0.923594	0.461797	−	0.886986i	$-0.347205\pi$
$337$	16.9549	0.923594	0.461797	+	0.886986i	$0.347205\pi$
$338$	−4.01497	−0.218385
$339$	0	0
$340$	3.67781	0.199457
$341$	40.6327	2.20039
$342$	0	0
$343$	−6.70727	−0.362158
$344$	−4.77745	−0.257583
$345$	0	0
$346$	−19.9276	−1.07132
$347$	5.44029	0.292050	0.146025	−	0.989281i	$-0.453352\pi$
$347$	5.44029	0.292050	0.146025	+	0.989281i	$0.453352\pi$
$348$	0	0
$349$	23.1262	1.23792	0.618960	−	0.785423i	$-0.287554\pi$
$349$	23.1262	1.23792	0.618960	+	0.785423i	$0.287554\pi$
$350$	−0.487359	−0.0260504
$351$	0	0
$352$	−5.20750	−0.277561
$353$	−24.7638	−1.31804	−0.659022	−	0.752123i	$-0.729030\pi$
$353$	−24.7638	−1.31804	−0.659022	+	0.752123i	$0.729030\pi$
$354$	0	0
$355$	8.58018	0.455388
$356$	−16.7051	−0.885368
$357$	0	0
$358$	−4.33034	−0.228865
$359$	17.5549	0.926512	0.463256	−	0.886225i	$-0.346681\pi$
$359$	17.5549	0.926512	0.463256	+	0.886225i	$0.346681\pi$
$360$	0	0
$361$	−18.8151	−0.990266
$362$	0.0505645	0.00265761
$363$	0	0
$364$	2.01032	0.105369
$365$	7.15020	0.374259
$366$	0	0
$367$	−8.17331	−0.426644	−0.213322	−	0.976982i	$-0.568428\pi$
$367$	−8.17331	−0.426644	−0.213322	+	0.976982i	$0.568428\pi$
$368$	−7.31537	−0.381340
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	3.74185	0.194267
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	19.1522	0.990337
$375$	0	0
$376$	3.52969	0.182030
$377$	27.9564	1.43983
$378$	0	0
$379$	26.1993	1.34577	0.672883	−	0.739749i	$-0.265056\pi$
$379$	26.1993	1.34577	0.672883	+	0.739749i	$0.265056\pi$
$380$	−0.430057	−0.0220615
$381$	0	0
$382$	−4.37276	−0.223730
$383$	5.87508	0.300203	0.150101	−	0.988671i	$-0.452040\pi$
$383$	5.87508	0.300203	0.150101	+	0.988671i	$0.452040\pi$
$384$	0	0
$385$	−2.53792	−0.129345
$386$	2.47031	0.125735
$387$	0	0
$388$	−7.16726	−0.363862
$389$	−33.8368	−1.71560	−0.857798	−	0.513987i	$-0.828168\pi$
$389$	−33.8368	−1.71560	−0.857798	+	0.513987i	$0.828168\pi$
$390$	0	0
$391$	26.9045	1.36062
$392$	6.76248	0.341557
$393$	0	0
$394$	−13.3154	−0.670819
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−1.75016	−0.0878380	−0.0439190	−	0.999035i	$-0.513984\pi$
$397$	−1.75016	−0.0878380	−0.0439190	+	0.999035i	$0.513984\pi$
$398$	−21.4403	−1.07470
$399$	0	0
$400$	1.00000	0.0500000
$401$	−24.7051	−1.23371	−0.616857	−	0.787075i	$-0.711594\pi$
$401$	−24.7051	−1.23371	−0.616857	+	0.787075i	$0.711594\pi$
$402$	0	0
$403$	32.1856	1.60328
$404$	13.7454	0.683861
$405$	0	0
$406$	−3.30305	−0.163928
$407$	−5.20750	−0.258126
$408$	0	0
$409$	6.75899	0.334210	0.167105	−	0.985939i	$-0.446558\pi$
$409$	6.75899	0.334210	0.167105	+	0.985939i	$0.446558\pi$
$410$	9.80273	0.484122
$411$	0	0
$412$	−3.52969	−0.173896
$413$	0.475037	0.0233751
$414$	0	0
$415$	−16.5399	−0.811913
$416$	−4.12492	−0.202241
$417$	0	0
$418$	−2.23952	−0.109539
$419$	−12.2143	−0.596709	−0.298354	−	0.954455i	$-0.596438\pi$
$419$	−12.2143	−0.596709	−0.298354	+	0.954455i	$0.596438\pi$
$420$	0	0
$421$	−8.91942	−0.434706	−0.217353	−	0.976093i	$-0.569742\pi$
$421$	−8.91942	−0.434706	−0.217353	+	0.976093i	$0.569742\pi$
$422$	−13.9679	−0.679947
$423$	0	0
$424$	−7.67781	−0.372867
$425$	−3.67781	−0.178400
$426$	0	0
$427$	−7.07785	−0.342521
$428$	−7.09964	−0.343174
$429$	0	0
$430$	4.77745	0.230389
$431$	−4.98704	−0.240217	−0.120109	−	0.992761i	$-0.538324\pi$
$431$	−4.98704	−0.240217	−0.120109	+	0.992761i	$0.538324\pi$
$432$	0	0
$433$	15.4805	0.743947	0.371974	−	0.928243i	$-0.378681\pi$
$433$	15.4805	0.743947	0.371974	+	0.928243i	$0.378681\pi$
$434$	−3.80273	−0.182537
$435$	0	0
$436$	16.4321	0.786953
$437$	−3.14603	−0.150495
$438$	0	0
$439$	−21.4929	−1.02580	−0.512899	−	0.858449i	$-0.671429\pi$
$439$	−21.4929	−1.02580	−0.512899	+	0.858449i	$0.671429\pi$
$440$	5.20750	0.248258
$441$	0	0
$442$	15.1707	0.721595
$443$	32.4355	1.54106	0.770528	−	0.637406i	$-0.219993\pi$
$443$	32.4355	1.54106	0.770528	+	0.637406i	$0.219993\pi$
$444$	0	0
$445$	16.7051	0.791898
$446$	22.4335	1.06226
$447$	0	0
$448$	0.487359	0.0230255
$449$	−15.8348	−0.747292	−0.373646	−	0.927571i	$-0.621892\pi$
$449$	−15.8348	−0.747292	−0.373646	+	0.927571i	$0.621892\pi$
$450$	0	0
$451$	51.0478	2.40374
$452$	1.22255	0.0575040
$453$	0	0
$454$	23.8368	1.11872
$455$	−2.01032	−0.0942451
$456$	0	0
$457$	−29.4470	−1.37747	−0.688737	−	0.725011i	$-0.741834\pi$
$457$	−29.4470	−1.37747	−0.688737	+	0.725011i	$0.741834\pi$
$458$	−10.2157	−0.477350
$459$	0	0
$460$	7.31537	0.341081
$461$	−18.4975	−0.861515	−0.430757	−	0.902468i	$-0.641754\pi$
$461$	−18.4975	−0.861515	−0.430757	+	0.902468i	$0.641754\pi$
$462$	0	0
$463$	−7.46549	−0.346951	−0.173475	−	0.984838i	$-0.555500\pi$
$463$	−7.46549	−0.346951	−0.173475	+	0.984838i	$0.555500\pi$
$464$	6.77745	0.314635
$465$	0	0
$466$	2.07576	0.0961579
$467$	−33.3917	−1.54519	−0.772593	−	0.634902i	$-0.781040\pi$
$467$	−33.3917	−1.54519	−0.772593	+	0.634902i	$0.781040\pi$
$468$	0	0
$469$	−0.580177	−0.0267901
$470$	−3.52969	−0.162813
$471$	0	0
$472$	−0.974718	−0.0448650
$473$	24.8786	1.14392
$474$	0	0
$475$	0.430057	0.0197324
$476$	−1.79241	−0.0821551
$477$	0	0
$478$	−28.2683	−1.29296
$479$	−26.7051	−1.22019	−0.610094	−	0.792329i	$-0.708868\pi$
$479$	−26.7051	−1.22019	−0.610094	+	0.792329i	$0.708868\pi$
$480$	0	0
$481$	−4.12492	−0.188080
$482$	−20.4491	−0.931432
$483$	0	0
$484$	16.1181	0.732641
$485$	7.16726	0.325448
$486$	0	0
$487$	−1.61437	−0.0731539	−0.0365770	−	0.999331i	$-0.511645\pi$
$487$	−1.61437	−0.0731539	−0.0365770	+	0.999331i	$0.511645\pi$
$488$	14.5229	0.657420
$489$	0	0
$490$	−6.76248	−0.305498
$491$	−1.40477	−0.0633966	−0.0316983	−	0.999497i	$-0.510092\pi$
$491$	−1.40477	−0.0633966	−0.0316983	+	0.999497i	$0.510092\pi$
$492$	0	0
$493$	−24.9262	−1.12262
$494$	−1.77395	−0.0798138
$495$	0	0
$496$	7.80273	0.350353
$497$	−4.18163	−0.187572
$498$	0	0
$499$	−9.98495	−0.446988	−0.223494	−	0.974705i	$-0.571746\pi$
$499$	−9.98495	−0.446988	−0.223494	+	0.974705i	$0.571746\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−19.3092	−0.861813
$503$	20.8300	0.928765	0.464382	−	0.885635i	$-0.346276\pi$
$503$	20.8300	0.928765	0.464382	+	0.885635i	$0.346276\pi$
$504$	0	0
$505$	−13.7454	−0.611663
$506$	38.0948	1.69352
$507$	0	0
$508$	15.5652	0.690595
$509$	−26.5816	−1.17821	−0.589104	−	0.808057i	$-0.700519\pi$
$509$	−26.5816	−1.17821	−0.589104	+	0.808057i	$0.700519\pi$
$510$	0	0
$511$	−3.48471	−0.154155
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0.735194	0.0324280
$515$	3.52969	0.155537
$516$	0	0
$517$	−18.3809	−0.808391
$518$	0.487359	0.0214133
$519$	0	0
$520$	4.12492	0.180890
$521$	9.91733	0.434486	0.217243	−	0.976118i	$-0.430294\pi$
$521$	9.91733	0.434486	0.217243	+	0.976118i	$0.430294\pi$
$522$	0	0
$523$	5.61910	0.245706	0.122853	−	0.992425i	$-0.460796\pi$
$523$	5.61910	0.245706	0.122853	+	0.992425i	$0.460796\pi$
$524$	17.5549	0.766889
$525$	0	0
$526$	6.47231	0.282206
$527$	−28.6970	−1.25006
$528$	0	0
$529$	30.5146	1.32672
$530$	7.67781	0.333503
$531$	0	0
$532$	0.209592	0.00908697
$533$	40.4355	1.75145
$534$	0	0
$535$	7.09964	0.306944
$536$	1.19045	0.0514196
$537$	0	0
$538$	−0.810958	−0.0349629
$539$	−35.2156	−1.51685
$540$	0	0
$541$	−0.315288	−0.0135553	−0.00677764	−	0.999977i	$-0.502157\pi$
$541$	−0.315288	−0.0135553	−0.00677764	+	0.999977i	$0.502157\pi$
$542$	−17.4403	−0.749125
$543$	0	0
$544$	3.67781	0.157685
$545$	−16.4321	−0.703872
$546$	0	0
$547$	22.1133	0.945496	0.472748	−	0.881198i	$-0.343262\pi$
$547$	22.1133	0.945496	0.472748	+	0.881198i	$0.343262\pi$
$548$	21.2498	0.907745
$549$	0	0
$550$	−5.20750	−0.222049
$551$	2.91469	0.124170
$552$	0	0
$553$	1.94944	0.0828984
$554$	1.46007	0.0620325
$555$	0	0
$556$	−15.4082	−0.653453
$557$	−24.9611	−1.05763	−0.528817	−	0.848736i	$-0.677364\pi$
$557$	−24.9611	−1.05763	−0.528817	+	0.848736i	$0.677364\pi$
$558$	0	0
$559$	19.7066	0.833500
$560$	−0.487359	−0.0205947
$561$	0	0
$562$	−12.1249	−0.511459
$563$	29.0642	1.22491	0.612455	−	0.790505i	$-0.290182\pi$
$563$	29.0642	1.22491	0.612455	+	0.790505i	$0.290182\pi$
$564$	0	0
$565$	−1.22255	−0.0514332
$566$	−19.5652	−0.822387
$567$	0	0
$568$	8.58018	0.360016
$569$	−34.5740	−1.44942	−0.724709	−	0.689055i	$-0.758026\pi$
$569$	−34.5740	−1.44942	−0.724709	+	0.689055i	$0.758026\pi$
$570$	0	0
$571$	−40.4539	−1.69294	−0.846472	−	0.532433i	$-0.821278\pi$
$571$	−40.4539	−1.69294	−0.846472	+	0.532433i	$0.821278\pi$
$572$	21.4805	0.898146
$573$	0	0
$574$	−4.77745	−0.199407
$575$	−7.31537	−0.305072
$576$	0	0
$577$	−15.5803	−0.648615	−0.324307	−	0.945952i	$-0.605131\pi$
$577$	−15.5803	−0.648615	−0.324307	+	0.945952i	$0.605131\pi$
$578$	3.47372	0.144488
$579$	0	0
$580$	−6.77745	−0.281418
$581$	8.06088	0.334422
$582$	0	0
$583$	39.9822	1.65589
$584$	7.15020	0.295877
$585$	0	0
$586$	−17.5126	−0.723441
$587$	−25.6717	−1.05958	−0.529792	−	0.848128i	$-0.677730\pi$
$587$	−25.6717	−1.05958	−0.529792	+	0.848128i	$0.677730\pi$
$588$	0	0
$589$	3.35562	0.138266
$590$	0.974718	0.0401285
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−22.9399	−0.942028	−0.471014	−	0.882126i	$-0.656112\pi$
$593$	−22.9399	−0.942028	−0.471014	+	0.882126i	$0.656112\pi$
$594$	0	0
$595$	1.79241	0.0734818
$596$	3.47913	0.142511
$597$	0	0
$598$	30.1753	1.23396
$599$	10.9583	0.447742	0.223871	−	0.974619i	$-0.428131\pi$
$599$	10.9583	0.447742	0.223871	+	0.974619i	$0.428131\pi$
$600$	0	0
$601$	38.0123	1.55055	0.775277	−	0.631621i	$-0.217610\pi$
$601$	38.0123	1.55055	0.775277	+	0.631621i	$0.217610\pi$
$602$	−2.32833	−0.0948957
$603$	0	0
$604$	−17.7304	−0.721439
$605$	−16.1181	−0.655294
$606$	0	0
$607$	−5.16509	−0.209644	−0.104822	−	0.994491i	$-0.533427\pi$
$607$	−5.16509	−0.209644	−0.104822	+	0.994491i	$0.533427\pi$
$608$	−0.430057	−0.0174411
$609$	0	0
$610$	−14.5229	−0.588014
$611$	−14.5597	−0.589023
$612$	0	0
$613$	35.6075	1.43817	0.719086	−	0.694921i	$-0.244561\pi$
$613$	35.6075	1.43817	0.719086	+	0.694921i	$0.244561\pi$
$614$	0.364444	0.0147078
$615$	0	0
$616$	−2.53792	−0.102256
$617$	−32.0799	−1.29149	−0.645745	−	0.763553i	$-0.723453\pi$
$617$	−32.0799	−1.29149	−0.645745	+	0.763553i	$0.723453\pi$
$618$	0	0
$619$	24.6628	0.991283	0.495642	−	0.868527i	$-0.334933\pi$
$619$	24.6628	0.991283	0.495642	+	0.868527i	$0.334933\pi$
$620$	−7.80273	−0.313365
$621$	0	0
$622$	−15.1583	−0.607794
$623$	−8.14138	−0.326177
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.7201	0.588335
$627$	0	0
$628$	11.8533	0.472998
$629$	3.67781	0.146644
$630$	0	0
$631$	8.89205	0.353987	0.176993	−	0.984212i	$-0.443363\pi$
$631$	8.89205	0.353987	0.176993	+	0.984212i	$0.443363\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	−0.942615	−0.0374360
$635$	−15.5652	−0.617687
$636$	0	0
$637$	−27.8947	−1.10523
$638$	−35.2936	−1.39729
$639$	0	0
$640$	1.00000	0.0395285
$641$	−41.5775	−1.64221	−0.821107	−	0.570774i	$-0.806643\pi$
$641$	−41.5775	−1.64221	−0.821107	+	0.570774i	$0.806643\pi$
$642$	0	0
$643$	43.1522	1.70176	0.850878	−	0.525363i	$-0.176070\pi$
$643$	43.1522	1.70176	0.850878	+	0.525363i	$0.176070\pi$
$644$	−3.56521	−0.140489
$645$	0	0
$646$	1.58167	0.0622299
$647$	15.4641	0.607956	0.303978	−	0.952679i	$-0.401685\pi$
$647$	15.4641	0.607956	0.303978	+	0.952679i	$0.401685\pi$
$648$	0	0
$649$	5.07585	0.199244
$650$	−4.12492	−0.161793
$651$	0	0
$652$	5.89354	0.230809
$653$	−17.3556	−0.679178	−0.339589	−	0.940574i	$-0.610288\pi$
$653$	−17.3556	−0.679178	−0.339589	+	0.940574i	$0.610288\pi$
$654$	0	0
$655$	−17.5549	−0.685926
$656$	9.80273	0.382732
$657$	0	0
$658$	1.72023	0.0670615
$659$	15.3509	0.597986	0.298993	−	0.954255i	$-0.403349\pi$
$659$	15.3509	0.597986	0.298993	+	0.954255i	$0.403349\pi$
$660$	0	0
$661$	−23.8218	−0.926560	−0.463280	−	0.886212i	$-0.653328\pi$
$661$	−23.8218	−0.926560	−0.463280	+	0.886212i	$0.653328\pi$
$662$	−15.8600	−0.616418
$663$	0	0
$664$	−16.5399	−0.641874
$665$	−0.209592	−0.00812763
$666$	0	0
$667$	−49.5795	−1.91973
$668$	−24.8362	−0.960940
$669$	0	0
$670$	−1.19045	−0.0459911
$671$	−75.6279	−2.91958
$672$	0	0
$673$	−37.3523	−1.43983	−0.719913	−	0.694065i	$-0.755818\pi$
$673$	−37.3523	−1.43983	−0.719913	+	0.694065i	$0.755818\pi$
$674$	−16.9549	−0.653080
$675$	0	0
$676$	4.01497	0.154422
$677$	−40.0143	−1.53788	−0.768938	−	0.639324i	$-0.779214\pi$
$677$	−40.0143	−1.53788	−0.768938	+	0.639324i	$0.779214\pi$
$678$	0	0
$679$	−3.49303	−0.134050
$680$	−3.67781	−0.141038
$681$	0	0
$682$	−40.6327	−1.55591
$683$	−7.25948	−0.277776	−0.138888	−	0.990308i	$-0.544353\pi$
$683$	−7.25948	−0.277776	−0.138888	+	0.990308i	$0.544353\pi$
$684$	0	0
$685$	−21.2498	−0.811911
$686$	6.70727	0.256085
$687$	0	0
$688$	4.77745	0.182138
$689$	31.6703	1.20654
$690$	0	0
$691$	7.91733	0.301190	0.150595	−	0.988596i	$-0.451881\pi$
$691$	7.91733	0.301190	0.150595	+	0.988596i	$0.451881\pi$
$692$	19.9276	0.757536
$693$	0	0
$694$	−5.44029	−0.206511
$695$	15.4082	0.584466
$696$	0	0
$697$	−36.0526	−1.36559
$698$	−23.1262	−0.875341
$699$	0	0
$700$	0.487359	0.0184204
$701$	39.1303	1.47793	0.738965	−	0.673744i	$-0.235315\pi$
$701$	39.1303	1.47793	0.738965	+	0.673744i	$0.235315\pi$
$702$	0	0
$703$	−0.430057	−0.0162199
$704$	5.20750	0.196265
$705$	0	0
$706$	24.7638	0.931998
$707$	6.69896	0.251940
$708$	0	0
$709$	−40.6478	−1.52656	−0.763280	−	0.646068i	$-0.776412\pi$
$709$	−40.6478	−1.52656	−0.763280	+	0.646068i	$0.776412\pi$
$710$	−8.58018	−0.322008
$711$	0	0
$712$	16.7051	0.626050
$713$	−57.0799	−2.13766
$714$	0	0
$715$	−21.4805	−0.803327
$716$	4.33034	0.161832
$717$	0	0
$718$	−17.5549	−0.655143
$719$	34.9147	1.30210	0.651049	−	0.759036i	$-0.274329\pi$
$719$	34.9147	1.30210	0.651049	+	0.759036i	$0.274329\pi$
$720$	0	0
$721$	−1.72023	−0.0640646
$722$	18.8151	0.700224
$723$	0	0
$724$	−0.0505645	−0.00187922
$725$	6.77745	0.251708
$726$	0	0
$727$	−16.1741	−0.599863	−0.299932	−	0.953961i	$-0.596964\pi$
$727$	−16.1741	−0.599863	−0.299932	+	0.953961i	$0.596964\pi$
$728$	−2.01032	−0.0745073
$729$	0	0
$730$	−7.15020	−0.264641
$731$	−17.5705	−0.649870
$732$	0	0
$733$	50.0895	1.85010	0.925049	−	0.379848i	$-0.124024\pi$
$733$	50.0895	1.85010	0.925049	+	0.379848i	$0.124024\pi$
$734$	8.17331	0.301683
$735$	0	0
$736$	7.31537	0.269648
$737$	−6.19928	−0.228353
$738$	0	0
$739$	−4.01846	−0.147822	−0.0739108	−	0.997265i	$-0.523548\pi$
$739$	−4.01846	−0.147822	−0.0739108	+	0.997265i	$0.523548\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−3.74185	−0.137368
$743$	14.4382	0.529686	0.264843	−	0.964292i	$-0.414680\pi$
$743$	14.4382	0.529686	0.264843	+	0.964292i	$0.414680\pi$
$744$	0	0
$745$	−3.47913	−0.127465
$746$	6.00000	0.219676
$747$	0	0
$748$	−19.1522	−0.700274
$749$	−3.46007	−0.126428
$750$	0	0
$751$	25.0088	0.912585	0.456292	−	0.889830i	$-0.349177\pi$
$751$	25.0088	0.912585	0.456292	+	0.889830i	$0.349177\pi$
$752$	−3.52969	−0.128715
$753$	0	0
$754$	−27.9564	−1.01811
$755$	17.7304	0.645275
$756$	0	0
$757$	−16.4049	−0.596245	−0.298122	−	0.954528i	$-0.596360\pi$
$757$	−16.4049	−0.596245	−0.298122	+	0.954528i	$0.596360\pi$
$758$	−26.1993	−0.951601
$759$	0	0
$760$	0.430057	0.0155998
$761$	−19.3877	−0.702804	−0.351402	−	0.936225i	$-0.614295\pi$
$761$	−19.3877	−0.702804	−0.351402	+	0.936225i	$0.614295\pi$
$762$	0	0
$763$	8.00831	0.289920
$764$	4.37276	0.158201
$765$	0	0
$766$	−5.87508	−0.212275
$767$	4.02063	0.145177
$768$	0	0
$769$	14.1693	0.510960	0.255480	−	0.966814i	$-0.417767\pi$
$769$	14.1693	0.510960	0.255480	+	0.966814i	$0.417767\pi$
$770$	2.53792	0.0914604
$771$	0	0
$772$	−2.47031	−0.0889082
$773$	9.58367	0.344701	0.172350	−	0.985036i	$-0.444864\pi$
$773$	9.58367	0.344701	0.172350	+	0.985036i	$0.444864\pi$
$774$	0	0
$775$	7.80273	0.280282
$776$	7.16726	0.257289
$777$	0	0
$778$	33.8368	1.21311
$779$	4.21573	0.151044
$780$	0	0
$781$	−44.6813	−1.59882
$782$	−26.9045	−0.962104
$783$	0	0
$784$	−6.76248	−0.241517
$785$	−11.8533	−0.423062
$786$	0	0
$787$	2.47921	0.0883744	0.0441872	−	0.999023i	$-0.485930\pi$
$787$	2.47921	0.0883744	0.0441872	+	0.999023i	$0.485930\pi$
$788$	13.3154	0.474340
$789$	0	0
$790$	4.00000	0.142314
$791$	0.595822	0.0211850
$792$	0	0
$793$	−59.9057	−2.12731
$794$	1.75016	0.0621108
$795$	0	0
$796$	21.4403	0.759931
$797$	47.0961	1.66823	0.834116	−	0.551590i	$-0.185979\pi$
$797$	47.0961	1.66823	0.834116	+	0.551590i	$0.185979\pi$
$798$	0	0
$799$	12.9815	0.459254
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	24.7051	0.872367
$803$	−37.2347	−1.31398
$804$	0	0
$805$	3.56521	0.125657
$806$	−32.1856	−1.13369
$807$	0	0
$808$	−13.7454	−0.483562
$809$	−36.6409	−1.28823	−0.644113	−	0.764931i	$-0.722773\pi$
$809$	−36.6409	−1.28823	−0.644113	+	0.764931i	$0.722773\pi$
$810$	0	0
$811$	12.4997	0.438923	0.219462	−	0.975621i	$-0.429570\pi$
$811$	12.4997	0.438923	0.219462	+	0.975621i	$0.429570\pi$
$812$	3.30305	0.115914
$813$	0	0
$814$	5.20750	0.182523
$815$	−5.89354	−0.206442
$816$	0	0
$817$	2.05457	0.0718805
$818$	−6.75899	−0.236322
$819$	0	0
$820$	−9.80273	−0.342326
$821$	11.7351	0.409558	0.204779	−	0.978808i	$-0.434352\pi$
$821$	11.7351	0.409558	0.204779	+	0.978808i	$0.434352\pi$
$822$	0	0
$823$	40.5058	1.41194	0.705972	−	0.708240i	$-0.250510\pi$
$823$	40.5058	1.41194	0.705972	+	0.708240i	$0.250510\pi$
$824$	3.52969	0.122963
$825$	0	0
$826$	−0.475037	−0.0165287
$827$	6.51115	0.226415	0.113207	−	0.993571i	$-0.463888\pi$
$827$	6.51115	0.226415	0.113207	+	0.993571i	$0.463888\pi$
$828$	0	0
$829$	11.0422	0.383510	0.191755	−	0.981443i	$-0.438582\pi$
$829$	11.0422	0.383510	0.191755	+	0.981443i	$0.438582\pi$
$830$	16.5399	0.574109
$831$	0	0
$832$	4.12492	0.143006
$833$	24.8711	0.861733
$834$	0	0
$835$	24.8362	0.859491
$836$	2.23952	0.0774555
$837$	0	0
$838$	12.2143	0.421937
$839$	−3.12641	−0.107936	−0.0539679	−	0.998543i	$-0.517187\pi$
$839$	−3.12641	−0.107936	−0.0539679	+	0.998543i	$0.517187\pi$
$840$	0	0
$841$	16.9338	0.583924
$842$	8.91942	0.307384
$843$	0	0
$844$	13.9679	0.480795
$845$	−4.01497	−0.138119
$846$	0	0
$847$	7.85530	0.269911
$848$	7.67781	0.263657
$849$	0	0
$850$	3.67781	0.126148
$851$	7.31537	0.250768
$852$	0	0
$853$	−8.98503	−0.307642	−0.153821	−	0.988099i	$-0.549158\pi$
$853$	−8.98503	−0.307642	−0.153821	+	0.988099i	$0.549158\pi$
$854$	7.07785	0.242199
$855$	0	0
$856$	7.09964	0.242661
$857$	−5.59314	−0.191058	−0.0955290	−	0.995427i	$-0.530454\pi$
$857$	−5.59314	−0.191058	−0.0955290	+	0.995427i	$0.530454\pi$
$858$	0	0
$859$	−18.8110	−0.641822	−0.320911	−	0.947109i	$-0.603989\pi$
$859$	−18.8110	−0.641822	−0.320911	+	0.947109i	$0.603989\pi$
$860$	−4.77745	−0.162910
$861$	0	0
$862$	4.98704	0.169859
$863$	29.8116	1.01480	0.507400	−	0.861711i	$-0.330607\pi$
$863$	29.8116	1.01480	0.507400	+	0.861711i	$0.330607\pi$
$864$	0	0
$865$	−19.9276	−0.677560
$866$	−15.4805	−0.526050
$867$	0	0
$868$	3.80273	0.129073
$869$	20.8300	0.706610
$870$	0	0
$871$	−4.91051	−0.166386
$872$	−16.4321	−0.556460
$873$	0	0
$874$	3.14603	0.106416
$875$	−0.487359	−0.0164757
$876$	0	0
$877$	−9.96806	−0.336598	−0.168299	−	0.985736i	$-0.553827\pi$
$877$	−9.96806	−0.336598	−0.168299	+	0.985736i	$0.553827\pi$
$878$	21.4929	0.725349
$879$	0	0
$880$	−5.20750	−0.175545
$881$	−27.7387	−0.934540	−0.467270	−	0.884115i	$-0.654762\pi$
$881$	−27.7387	−0.934540	−0.467270	+	0.884115i	$0.654762\pi$
$882$	0	0
$883$	7.79659	0.262376	0.131188	−	0.991358i	$-0.458121\pi$
$883$	7.79659	0.262376	0.131188	+	0.991358i	$0.458121\pi$
$884$	−15.1707	−0.510245
$885$	0	0
$886$	−32.4355	−1.08969
$887$	−19.1536	−0.643115	−0.321558	−	0.946890i	$-0.604206\pi$
$887$	−19.1536	−0.643115	−0.321558	+	0.946890i	$0.604206\pi$
$888$	0	0
$889$	7.58584	0.254421
$890$	−16.7051	−0.559956
$891$	0	0
$892$	−22.4335	−0.751128
$893$	−1.51797	−0.0507969
$894$	0	0
$895$	−4.33034	−0.144747
$896$	−0.487359	−0.0162815
$897$	0	0
$898$	15.8348	0.528415
$899$	52.8826	1.76373
$900$	0	0
$901$	−28.2375	−0.940728
$902$	−51.0478	−1.69970
$903$	0	0
$904$	−1.22255	−0.0406615
$905$	0.0505645	0.00168082
$906$	0	0
$907$	1.34530	0.0446700	0.0223350	−	0.999751i	$-0.492890\pi$
$907$	1.34530	0.0446700	0.0223350	+	0.999751i	$0.492890\pi$
$908$	−23.8368	−0.791053
$909$	0	0
$910$	2.01032	0.0666413
$911$	−2.95826	−0.0980116	−0.0490058	−	0.998798i	$-0.515605\pi$
$911$	−2.95826	−0.0980116	−0.0490058	+	0.998798i	$0.515605\pi$
$912$	0	0
$913$	86.1317	2.85054
$914$	29.4470	0.974021
$915$	0	0
$916$	10.2157	0.337537
$917$	8.55553	0.282529
$918$	0	0
$919$	16.0504	0.529454	0.264727	−	0.964323i	$-0.414718\pi$
$919$	16.0504	0.529454	0.264727	+	0.964323i	$0.414718\pi$
$920$	−7.31537	−0.241181
$921$	0	0
$922$	18.4975	0.609183
$923$	−35.3925	−1.16496
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	7.46549	0.245331
$927$	0	0
$928$	−6.77745	−0.222481
$929$	2.20009	0.0721825	0.0360913	−	0.999348i	$-0.488509\pi$
$929$	2.20009	0.0721825	0.0360913	+	0.999348i	$0.488509\pi$
$930$	0	0
$931$	−2.90825	−0.0953141
$932$	−2.07576	−0.0679939
$933$	0	0
$934$	33.3917	1.09261
$935$	19.1522	0.626344
$936$	0	0
$937$	−15.9030	−0.519530	−0.259765	−	0.965672i	$-0.583645\pi$
$937$	−15.9030	−0.519530	−0.259765	+	0.965672i	$0.583645\pi$
$938$	0.580177	0.0189434
$939$	0	0
$940$	3.52969	0.115126
$941$	−50.0021	−1.63002	−0.815011	−	0.579446i	$-0.803269\pi$
$941$	−50.0021	−1.63002	−0.815011	+	0.579446i	$0.803269\pi$
$942$	0	0
$943$	−71.7106	−2.33522
$944$	0.974718	0.0317244
$945$	0	0
$946$	−24.8786	−0.808872
$947$	−8.11677	−0.263760	−0.131880	−	0.991266i	$-0.542101\pi$
$947$	−8.11677	−0.263760	−0.131880	+	0.991266i	$0.542101\pi$
$948$	0	0
$949$	−29.4940	−0.957416
$950$	−0.430057	−0.0139529
$951$	0	0
$952$	1.79241	0.0580924
$953$	34.2451	1.10931	0.554654	−	0.832081i	$-0.312851\pi$
$953$	34.2451	1.10931	0.554654	+	0.832081i	$0.312851\pi$
$954$	0	0
$955$	−4.37276	−0.141499
$956$	28.2683	0.914262
$957$	0	0
$958$	26.7051	0.862803
$959$	10.3563	0.334421
$960$	0	0
$961$	29.8826	0.963954
$962$	4.12492	0.132993
$963$	0	0
$964$	20.4491	0.658622
$965$	2.47031	0.0795219
$966$	0	0
$967$	−7.49960	−0.241171	−0.120585	−	0.992703i	$-0.538477\pi$
$967$	−7.49960	−0.241171	−0.120585	+	0.992703i	$0.538477\pi$
$968$	−16.1181	−0.518055
$969$	0	0
$970$	−7.16726	−0.230127
$971$	−55.6327	−1.78534	−0.892669	−	0.450714i	$-0.851170\pi$
$971$	−55.6327	−1.78534	−0.892669	+	0.450714i	$0.851170\pi$
$972$	0	0
$973$	−7.50932	−0.240738
$974$	1.61437	0.0517277
$975$	0	0
$976$	−14.5229	−0.464866
$977$	−33.1685	−1.06115	−0.530577	−	0.847637i	$-0.678025\pi$
$977$	−33.1685	−1.06115	−0.530577	+	0.847637i	$0.678025\pi$
$978$	0	0
$979$	−86.9919	−2.78027
$980$	6.76248	0.216020
$981$	0	0
$982$	1.40477	0.0448282
$983$	−55.2178	−1.76117	−0.880587	−	0.473884i	$-0.842852\pi$
$983$	−55.2178	−1.76117	−0.880587	+	0.473884i	$0.842852\pi$
$984$	0	0
$985$	−13.3154	−0.424263
$986$	24.9262	0.793811
$987$	0	0
$988$	1.77395	0.0564369
$989$	−34.9488	−1.11131
$990$	0	0
$991$	−36.7815	−1.16840	−0.584201	−	0.811609i	$-0.698592\pi$
$991$	−36.7815	−1.16840	−0.584201	+	0.811609i	$0.698592\pi$
$992$	−7.80273	−0.247737
$993$	0	0
$994$	4.18163	0.132633
$995$	−21.4403	−0.679703
$996$	0	0
$997$	−42.5058	−1.34617	−0.673086	−	0.739564i	$-0.735032\pi$
$997$	−42.5058	−1.34617	−0.673086	+	0.739564i	$0.735032\pi$
$998$	9.98495	0.316068
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bj.1.2		4
3.2	odd	2		1110.2.a.s.1.2	✓	4
12.11	even	2		8880.2.a.cg.1.3		4
15.14	odd	2		5550.2.a.cj.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.s.1.2	✓	4	3.2	odd	2
3330.2.a.bj.1.2		4	1.1	even	1	trivial
5550.2.a.cj.1.3		4	15.14	odd	2
8880.2.a.cg.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.487359 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.487359 q^{7} -1.00000 q^{8} +1.00000 q^{10} +5.20750 q^{11} +4.12492 q^{13} -0.487359 q^{14} +1.00000 q^{16} -3.67781 q^{17} +0.430057 q^{19} -1.00000 q^{20} -5.20750 q^{22} -7.31537 q^{23} +1.00000 q^{25} -4.12492 q^{26} +0.487359 q^{28} +6.77745 q^{29} +7.80273 q^{31} -1.00000 q^{32} +3.67781 q^{34} -0.487359 q^{35} -1.00000 q^{37} -0.430057 q^{38} +1.00000 q^{40} +9.80273 q^{41} +4.77745 q^{43} +5.20750 q^{44} +7.31537 q^{46} -3.52969 q^{47} -6.76248 q^{49} -1.00000 q^{50} +4.12492 q^{52} +7.67781 q^{53} -5.20750 q^{55} -0.487359 q^{56} -6.77745 q^{58} +0.974718 q^{59} -14.5229 q^{61} -7.80273 q^{62} +1.00000 q^{64} -4.12492 q^{65} -1.19045 q^{67} -3.67781 q^{68} +0.487359 q^{70} -8.58018 q^{71} -7.15020 q^{73} +1.00000 q^{74} +0.430057 q^{76} +2.53792 q^{77} +4.00000 q^{79} -1.00000 q^{80} -9.80273 q^{82} +16.5399 q^{83} +3.67781 q^{85} -4.77745 q^{86} -5.20750 q^{88} -16.7051 q^{89} +2.01032 q^{91} -7.31537 q^{92} +3.52969 q^{94} -0.430057 q^{95} -7.16726 q^{97} +6.76248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) 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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Database

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