LMFDB - Embedded newform 2790.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, 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("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.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:	$22.2782621639$
Analytic rank:	$0$
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 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.37228 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} -0.627719 q^{11} -2.00000 q^{13} -3.37228 q^{14} +1.00000 q^{16} +4.74456 q^{17} -0.627719 q^{19} -1.00000 q^{20} +0.627719 q^{22} -3.37228 q^{23} +1.00000 q^{25} +2.00000 q^{26} +3.37228 q^{28} +8.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} -4.74456 q^{34} -3.37228 q^{35} -0.744563 q^{37} +0.627719 q^{38} +1.00000 q^{40} -0.744563 q^{41} +0.627719 q^{43} -0.627719 q^{44} +3.37228 q^{46} +6.74456 q^{47} +4.37228 q^{49} -1.00000 q^{50} -2.00000 q^{52} -1.37228 q^{53} +0.627719 q^{55} -3.37228 q^{56} -8.74456 q^{58} -2.74456 q^{59} +11.4891 q^{61} +1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +10.7446 q^{67} +4.74456 q^{68} +3.37228 q^{70} -3.37228 q^{71} -8.11684 q^{73} +0.744563 q^{74} -0.627719 q^{76} -2.11684 q^{77} -4.62772 q^{79} -1.00000 q^{80} +0.744563 q^{82} +12.0000 q^{83} -4.74456 q^{85} -0.627719 q^{86} +0.627719 q^{88} +1.37228 q^{89} -6.74456 q^{91} -3.37228 q^{92} -6.74456 q^{94} +0.627719 q^{95} +2.00000 q^{97} -4.37228 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - 7 q^{11} - 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 7 q^{19} - 2 q^{20} + 7 q^{22} - q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 6 q^{29} - 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 10 q^{37} + 7 q^{38} + 2 q^{40} + 10 q^{41} + 7 q^{43} - 7 q^{44} + q^{46} + 2 q^{47} + 3 q^{49} - 2 q^{50} - 4 q^{52} + 3 q^{53} + 7 q^{55} - q^{56} - 6 q^{58} + 6 q^{59} + 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 2 q^{68} + q^{70} - q^{71} + q^{73} - 10 q^{74} - 7 q^{76} + 13 q^{77} - 15 q^{79} - 2 q^{80} - 10 q^{82} + 24 q^{83} + 2 q^{85} - 7 q^{86} + 7 q^{88} - 3 q^{89} - 2 q^{91} - q^{92} - 2 q^{94} + 7 q^{95} + 4 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - 7 * q^11 - 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 7 * q^19 - 2 * q^20 + 7 * q^22 - q^23 + 2 * q^25 + 4 * q^26 + q^28 + 6 * q^29 - 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 10 * q^37 + 7 * q^38 + 2 * q^40 + 10 * q^41 + 7 * q^43 - 7 * q^44 + q^46 + 2 * q^47 + 3 * q^49 - 2 * q^50 - 4 * q^52 + 3 * q^53 + 7 * q^55 - q^56 - 6 * q^58 + 6 * q^59 + 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 2 * q^68 + q^70 - q^71 + q^73 - 10 * q^74 - 7 * q^76 + 13 * q^77 - 15 * q^79 - 2 * q^80 - 10 * q^82 + 24 * q^83 + 2 * q^85 - 7 * q^86 + 7 * q^88 - 3 * q^89 - 2 * q^91 - q^92 - 2 * q^94 + 7 * q^95 + 4 * q^97 - 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−0.627719	−0.189264	−0.0946322	−	0.995512i	$-0.530167\pi$
$11$	−0.627719	−0.189264	−0.0946322	+	0.995512i	$0.530167\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.37228	−0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	4.74456	1.15073	0.575363	−	0.817898i	$-0.304861\pi$
$17$	4.74456	1.15073	0.575363	+	0.817898i	$0.304861\pi$
$18$	0	0
$19$	−0.627719	−0.144009	−0.0720043	−	0.997404i	$-0.522940\pi$
$19$	−0.627719	−0.144009	−0.0720043	+	0.997404i	$0.522940\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0.627719	0.133830
$23$	−3.37228	−0.703169	−0.351585	−	0.936156i	$-0.614357\pi$
$23$	−3.37228	−0.703169	−0.351585	+	0.936156i	$0.614357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	3.37228	0.637301
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.74456	−0.813686
$35$	−3.37228	−0.570020
$36$	0	0
$37$	−0.744563	−0.122405	−0.0612027	−	0.998125i	$-0.519494\pi$
$37$	−0.744563	−0.122405	−0.0612027	+	0.998125i	$0.519494\pi$
$38$	0.627719	0.101829
$39$	0	0
$40$	1.00000	0.158114
$41$	−0.744563	−0.116281	−0.0581406	−	0.998308i	$-0.518517\pi$
$41$	−0.744563	−0.116281	−0.0581406	+	0.998308i	$0.518517\pi$
$42$	0	0
$43$	0.627719	0.0957262	0.0478631	−	0.998854i	$-0.484759\pi$
$43$	0.627719	0.0957262	0.0478631	+	0.998854i	$0.484759\pi$
$44$	−0.627719	−0.0946322
$45$	0	0
$46$	3.37228	0.497216
$47$	6.74456	0.983796	0.491898	−	0.870653i	$-0.336303\pi$
$47$	6.74456	0.983796	0.491898	+	0.870653i	$0.336303\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−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
$55$	0.627719	0.0846416
$56$	−3.37228	−0.450640
$57$	0	0
$58$	−8.74456	−1.14822
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	11.4891	1.47103	0.735516	−	0.677507i	$-0.236940\pi$
$61$	11.4891	1.47103	0.735516	+	0.677507i	$0.236940\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	10.7446	1.31266	0.656329	−	0.754475i	$-0.272108\pi$
$67$	10.7446	1.31266	0.656329	+	0.754475i	$0.272108\pi$
$68$	4.74456	0.575363
$69$	0	0
$70$	3.37228	0.403065
$71$	−3.37228	−0.400216	−0.200108	−	0.979774i	$-0.564129\pi$
$71$	−3.37228	−0.400216	−0.200108	+	0.979774i	$0.564129\pi$
$72$	0	0
$73$	−8.11684	−0.950005	−0.475002	−	0.879985i	$-0.657553\pi$
$73$	−8.11684	−0.950005	−0.475002	+	0.879985i	$0.657553\pi$
$74$	0.744563	0.0865536
$75$	0	0
$76$	−0.627719	−0.0720043
$77$	−2.11684	−0.241237
$78$	0	0
$79$	−4.62772	−0.520659	−0.260330	−	0.965520i	$-0.583831\pi$
$79$	−4.62772	−0.520659	−0.260330	+	0.965520i	$0.583831\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	0.744563	0.0822232
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−4.74456	−0.514620
$86$	−0.627719	−0.0676886
$87$	0	0
$88$	0.627719	0.0669150
$89$	1.37228	0.145462	0.0727308	−	0.997352i	$-0.476829\pi$
$89$	1.37228	0.145462	0.0727308	+	0.997352i	$0.476829\pi$
$90$	0	0
$91$	−6.74456	−0.707022
$92$	−3.37228	−0.351585
$93$	0	0
$94$	−6.74456	−0.695649
$95$	0.627719	0.0644026
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−4.37228	−0.441667
$99$	0	0
$100$	1.00000	0.100000
$101$	−8.11684	−0.807656	−0.403828	−	0.914835i	$-0.632321\pi$
$101$	−8.11684	−0.807656	−0.403828	+	0.914835i	$0.632321\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	1.37228	0.133288
$107$	−0.627719	−0.0606839	−0.0303419	−	0.999540i	$-0.509660\pi$
$107$	−0.627719	−0.0606839	−0.0303419	+	0.999540i	$0.509660\pi$
$108$	0	0
$109$	4.74456	0.454447	0.227223	−	0.973843i	$-0.427035\pi$
$109$	4.74456	0.454447	0.227223	+	0.973843i	$0.427035\pi$
$110$	−0.627719	−0.0598506
$111$	0	0
$112$	3.37228	0.318651
$113$	2.62772	0.247195	0.123597	−	0.992332i	$-0.460557\pi$
$113$	2.62772	0.247195	0.123597	+	0.992332i	$0.460557\pi$
$114$	0	0
$115$	3.37228	0.314467
$116$	8.74456	0.811912
$117$	0	0
$118$	2.74456	0.252657
$119$	16.0000	1.46672
$120$	0	0
$121$	−10.6060	−0.964179
$122$	−11.4891	−1.04018
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.4891	1.19697	0.598483	−	0.801135i	$-0.295770\pi$
$127$	13.4891	1.19697	0.598483	+	0.801135i	$0.295770\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	16.2337	1.41834	0.709172	−	0.705036i	$-0.249069\pi$
$131$	16.2337	1.41834	0.709172	+	0.705036i	$0.249069\pi$
$132$	0	0
$133$	−2.11684	−0.183554
$134$	−10.7446	−0.928189
$135$	0	0
$136$	−4.74456	−0.406843
$137$	3.48913	0.298096	0.149048	−	0.988830i	$-0.452379\pi$
$137$	3.48913	0.298096	0.149048	+	0.988830i	$0.452379\pi$
$138$	0	0
$139$	−10.7446	−0.911342	−0.455671	−	0.890148i	$-0.650601\pi$
$139$	−10.7446	−0.911342	−0.455671	+	0.890148i	$0.650601\pi$
$140$	−3.37228	−0.285010
$141$	0	0
$142$	3.37228	0.282996
$143$	1.25544	0.104985
$144$	0	0
$145$	−8.74456	−0.726196
$146$	8.11684	0.671755
$147$	0	0
$148$	−0.744563	−0.0612027
$149$	12.1168	0.992651	0.496325	−	0.868137i	$-0.334682\pi$
$149$	12.1168	0.992651	0.496325	+	0.868137i	$0.334682\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0.627719	0.0509147
$153$	0	0
$154$	2.11684	0.170580
$155$	1.00000	0.0803219
$156$	0	0
$157$	9.37228	0.747989	0.373995	−	0.927431i	$-0.377988\pi$
$157$	9.37228	0.747989	0.373995	+	0.927431i	$0.377988\pi$
$158$	4.62772	0.368162
$159$	0	0
$160$	1.00000	0.0790569
$161$	−11.3723	−0.896261
$162$	0	0
$163$	24.2337	1.89813	0.949064	−	0.315082i	$-0.102032\pi$
$163$	24.2337	1.89813	0.949064	+	0.315082i	$0.102032\pi$
$164$	−0.744563	−0.0581406
$165$	0	0
$166$	−12.0000	−0.931381
$167$	12.6277	0.977162	0.488581	−	0.872518i	$-0.337515\pi$
$167$	12.6277	0.977162	0.488581	+	0.872518i	$0.337515\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	4.74456	0.363891
$171$	0	0
$172$	0.627719	0.0478631
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	3.37228	0.254921
$176$	−0.627719	−0.0473161
$177$	0	0
$178$	−1.37228	−0.102857
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.8614	1.10464	0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614	1.10464	0.552320	+	0.833632i	$0.313743\pi$
$182$	6.74456	0.499940
$183$	0	0
$184$	3.37228	0.248608
$185$	0.744563	0.0547413
$186$	0	0
$187$	−2.97825	−0.217791
$188$	6.74456	0.491898
$189$	0	0
$190$	−0.627719	−0.0455395
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	15.4891	1.11493	0.557466	−	0.830200i	$-0.311774\pi$
$193$	15.4891	1.11493	0.557466	+	0.830200i	$0.311774\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	4.37228	0.312306
$197$	−19.4891	−1.38854	−0.694271	−	0.719713i	$-0.744273\pi$
$197$	−19.4891	−1.38854	−0.694271	+	0.719713i	$0.744273\pi$
$198$	0	0
$199$	−12.6277	−0.895155	−0.447578	−	0.894245i	$-0.647713\pi$
$199$	−12.6277	−0.895155	−0.447578	+	0.894245i	$0.647713\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	8.11684	0.571099
$203$	29.4891	2.06973
$204$	0	0
$205$	0.744563	0.0520025
$206$	8.00000	0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0.394031	0.0272557
$210$	0	0
$211$	−0.627719	−0.0432139	−0.0216070	−	0.999767i	$-0.506878\pi$
$211$	−0.627719	−0.0432139	−0.0216070	+	0.999767i	$0.506878\pi$
$212$	−1.37228	−0.0942487
$213$	0	0
$214$	0.627719	0.0429100
$215$	−0.627719	−0.0428101
$216$	0	0
$217$	−3.37228	−0.228925
$218$	−4.74456	−0.321342
$219$	0	0
$220$	0.627719	0.0423208
$221$	−9.48913	−0.638308
$222$	0	0
$223$	9.25544	0.619790	0.309895	−	0.950771i	$-0.399706\pi$
$223$	9.25544	0.619790	0.309895	+	0.950771i	$0.399706\pi$
$224$	−3.37228	−0.225320
$225$	0	0
$226$	−2.62772	−0.174793
$227$	−6.11684	−0.405989	−0.202995	−	0.979180i	$-0.565067\pi$
$227$	−6.11684	−0.405989	−0.202995	+	0.979180i	$0.565067\pi$
$228$	0	0
$229$	3.88316	0.256606	0.128303	−	0.991735i	$-0.459047\pi$
$229$	3.88316	0.256606	0.128303	+	0.991735i	$0.459047\pi$
$230$	−3.37228	−0.222362
$231$	0	0
$232$	−8.74456	−0.574109
$233$	14.8614	0.973603	0.486802	−	0.873513i	$-0.338163\pi$
$233$	14.8614	0.973603	0.486802	+	0.873513i	$0.338163\pi$
$234$	0	0
$235$	−6.74456	−0.439967
$236$	−2.74456	−0.178656
$237$	0	0
$238$	−16.0000	−1.03713
$239$	29.4891	1.90749	0.953746	−	0.300612i	$-0.0971910\pi$
$239$	29.4891	1.90749	0.953746	+	0.300612i	$0.0971910\pi$
$240$	0	0
$241$	4.51087	0.290571	0.145285	−	0.989390i	$-0.453590\pi$
$241$	4.51087	0.290571	0.145285	+	0.989390i	$0.453590\pi$
$242$	10.6060	0.681778
$243$	0	0
$244$	11.4891	0.735516
$245$	−4.37228	−0.279335
$246$	0	0
$247$	1.25544	0.0798816
$248$	1.00000	0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	2.11684	0.133085
$254$	−13.4891	−0.846383
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.3723	−0.834140	−0.417070	−	0.908874i	$-0.636943\pi$
$257$	−13.3723	−0.834140	−0.417070	+	0.908874i	$0.636943\pi$
$258$	0	0
$259$	−2.51087	−0.156018
$260$	2.00000	0.124035
$261$	0	0
$262$	−16.2337	−1.00292
$263$	18.9783	1.17025	0.585125	−	0.810943i	$-0.301046\pi$
$263$	18.9783	1.17025	0.585125	+	0.810943i	$0.301046\pi$
$264$	0	0
$265$	1.37228	0.0842986
$266$	2.11684	0.129792
$267$	0	0
$268$	10.7446	0.656329
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−13.8832	−0.843342	−0.421671	−	0.906749i	$-0.638556\pi$
$271$	−13.8832	−0.843342	−0.421671	+	0.906749i	$0.638556\pi$
$272$	4.74456	0.287681
$273$	0	0
$274$	−3.48913	−0.210786
$275$	−0.627719	−0.0378529
$276$	0	0
$277$	15.2554	0.916610	0.458305	−	0.888795i	$-0.348457\pi$
$277$	15.2554	0.916610	0.458305	+	0.888795i	$0.348457\pi$
$278$	10.7446	0.644416
$279$	0	0
$280$	3.37228	0.201532
$281$	−16.7446	−0.998897	−0.499448	−	0.866344i	$-0.666464\pi$
$281$	−16.7446	−0.998897	−0.499448	+	0.866344i	$0.666464\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−3.37228	−0.200108
$285$	0	0
$286$	−1.25544	−0.0742356
$287$	−2.51087	−0.148212
$288$	0	0
$289$	5.51087	0.324169
$290$	8.74456	0.513498
$291$	0	0
$292$	−8.11684	−0.475002
$293$	7.48913	0.437519	0.218760	−	0.975779i	$-0.429799\pi$
$293$	7.48913	0.437519	0.218760	+	0.975779i	$0.429799\pi$
$294$	0	0
$295$	2.74456	0.159795
$296$	0.744563	0.0432768
$297$	0	0
$298$	−12.1168	−0.701910
$299$	6.74456	0.390048
$300$	0	0
$301$	2.11684	0.122013
$302$	8.00000	0.460348
$303$	0	0
$304$	−0.627719	−0.0360021
$305$	−11.4891	−0.657865
$306$	0	0
$307$	30.9783	1.76802	0.884011	−	0.467466i	$-0.154833\pi$
$307$	30.9783	1.76802	0.884011	+	0.467466i	$0.154833\pi$
$308$	−2.11684	−0.120618
$309$	0	0
$310$	−1.00000	−0.0567962
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−9.37228	−0.528908
$315$	0	0
$316$	−4.62772	−0.260330
$317$	24.7446	1.38979	0.694897	−	0.719110i	$-0.255450\pi$
$317$	24.7446	1.38979	0.694897	+	0.719110i	$0.255450\pi$
$318$	0	0
$319$	−5.48913	−0.307332
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	11.3723	0.633752
$323$	−2.97825	−0.165714
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−24.2337	−1.34218
$327$	0	0
$328$	0.744563	0.0411116
$329$	22.7446	1.25395
$330$	0	0
$331$	−1.48913	−0.0818497	−0.0409249	−	0.999162i	$-0.513030\pi$
$331$	−1.48913	−0.0818497	−0.0409249	+	0.999162i	$0.513030\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−12.6277	−0.690958
$335$	−10.7446	−0.587038
$336$	0	0
$337$	28.9783	1.57855	0.789273	−	0.614043i	$-0.210458\pi$
$337$	28.9783	1.57855	0.789273	+	0.614043i	$0.210458\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−4.74456	−0.257310
$341$	0.627719	0.0339929
$342$	0	0
$343$	−8.86141	−0.478471
$344$	−0.627719	−0.0338443
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−36.4674	−1.95767	−0.978836	−	0.204648i	$-0.934395\pi$
$347$	−36.4674	−1.95767	−0.978836	+	0.204648i	$0.934395\pi$
$348$	0	0
$349$	7.25544	0.388375	0.194187	−	0.980964i	$-0.437793\pi$
$349$	7.25544	0.388375	0.194187	+	0.980964i	$0.437793\pi$
$350$	−3.37228	−0.180256
$351$	0	0
$352$	0.627719	0.0334575
$353$	−15.4891	−0.824403	−0.412201	−	0.911093i	$-0.635240\pi$
$353$	−15.4891	−0.824403	−0.412201	+	0.911093i	$0.635240\pi$
$354$	0	0
$355$	3.37228	0.178982
$356$	1.37228	0.0727308
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−3.37228	−0.177982	−0.0889911	−	0.996032i	$-0.528364\pi$
$359$	−3.37228	−0.177982	−0.0889911	+	0.996032i	$0.528364\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	−14.8614	−0.781098
$363$	0	0
$364$	−6.74456	−0.353511
$365$	8.11684	0.424855
$366$	0	0
$367$	−20.2337	−1.05619	−0.528095	−	0.849185i	$-0.677093\pi$
$367$	−20.2337	−1.05619	−0.528095	+	0.849185i	$0.677093\pi$
$368$	−3.37228	−0.175792
$369$	0	0
$370$	−0.744563	−0.0387080
$371$	−4.62772	−0.240259
$372$	0	0
$373$	14.8614	0.769494	0.384747	−	0.923022i	$-0.374289\pi$
$373$	14.8614	0.769494	0.384747	+	0.923022i	$0.374289\pi$
$374$	2.97825	0.154002
$375$	0	0
$376$	−6.74456	−0.347824
$377$	−17.4891	−0.900736
$378$	0	0
$379$	−28.8614	−1.48251	−0.741255	−	0.671223i	$-0.765769\pi$
$379$	−28.8614	−1.48251	−0.741255	+	0.671223i	$0.765769\pi$
$380$	0.627719	0.0322013
$381$	0	0
$382$	16.0000	0.818631
$383$	−13.4891	−0.689262	−0.344631	−	0.938738i	$-0.611996\pi$
$383$	−13.4891	−0.689262	−0.344631	+	0.938738i	$0.611996\pi$
$384$	0	0
$385$	2.11684	0.107884
$386$	−15.4891	−0.788376
$387$	0	0
$388$	2.00000	0.101535
$389$	−16.9783	−0.860831	−0.430416	−	0.902631i	$-0.641633\pi$
$389$	−16.9783	−0.860831	−0.430416	+	0.902631i	$0.641633\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	−4.37228	−0.220834
$393$	0	0
$394$	19.4891	0.981848
$395$	4.62772	0.232846
$396$	0	0
$397$	−1.60597	−0.0806013	−0.0403006	−	0.999188i	$-0.512832\pi$
$397$	−1.60597	−0.0806013	−0.0403006	+	0.999188i	$0.512832\pi$
$398$	12.6277	0.632970
$399$	0	0
$400$	1.00000	0.0500000
$401$	−22.6277	−1.12997	−0.564987	−	0.825100i	$-0.691119\pi$
$401$	−22.6277	−1.12997	−0.564987	+	0.825100i	$0.691119\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	−8.11684	−0.403828
$405$	0	0
$406$	−29.4891	−1.46352
$407$	0.467376	0.0231670
$408$	0	0
$409$	−10.2337	−0.506023	−0.253012	−	0.967463i	$-0.581421\pi$
$409$	−10.2337	−0.506023	−0.253012	+	0.967463i	$0.581421\pi$
$410$	−0.744563	−0.0367713
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−9.25544	−0.455430
$414$	0	0
$415$	−12.0000	−0.589057
$416$	2.00000	0.0980581
$417$	0	0
$418$	−0.394031	−0.0192727
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	19.4891	0.949842	0.474921	−	0.880028i	$-0.342477\pi$
$421$	19.4891	0.949842	0.474921	+	0.880028i	$0.342477\pi$
$422$	0.627719	0.0305569
$423$	0	0
$424$	1.37228	0.0666439
$425$	4.74456	0.230145
$426$	0	0
$427$	38.7446	1.87498
$428$	−0.627719	−0.0303419
$429$	0	0
$430$	0.627719	0.0302713
$431$	26.9783	1.29950	0.649748	−	0.760149i	$-0.274874\pi$
$431$	26.9783	1.29950	0.649748	+	0.760149i	$0.274874\pi$
$432$	0	0
$433$	36.1168	1.73566	0.867832	−	0.496857i	$-0.165513\pi$
$433$	36.1168	1.73566	0.867832	+	0.496857i	$0.165513\pi$
$434$	3.37228	0.161875
$435$	0	0
$436$	4.74456	0.227223
$437$	2.11684	0.101262
$438$	0	0
$439$	14.7446	0.703720	0.351860	−	0.936053i	$-0.385549\pi$
$439$	14.7446	0.703720	0.351860	+	0.936053i	$0.385549\pi$
$440$	−0.627719	−0.0299253
$441$	0	0
$442$	9.48913	0.451352
$443$	−34.3505	−1.63204	−0.816022	−	0.578022i	$-0.803825\pi$
$443$	−34.3505	−1.63204	−0.816022	+	0.578022i	$0.803825\pi$
$444$	0	0
$445$	−1.37228	−0.0650524
$446$	−9.25544	−0.438258
$447$	0	0
$448$	3.37228	0.159325
$449$	8.97825	0.423710	0.211855	−	0.977301i	$-0.432050\pi$
$449$	8.97825	0.423710	0.211855	+	0.977301i	$0.432050\pi$
$450$	0	0
$451$	0.467376	0.0220079
$452$	2.62772	0.123597
$453$	0	0
$454$	6.11684	0.287078
$455$	6.74456	0.316190
$456$	0	0
$457$	34.4674	1.61232	0.806158	−	0.591700i	$-0.201543\pi$
$457$	34.4674	1.61232	0.806158	+	0.591700i	$0.201543\pi$
$458$	−3.88316	−0.181448
$459$	0	0
$460$	3.37228	0.157233
$461$	19.7228	0.918583	0.459291	−	0.888286i	$-0.348103\pi$
$461$	19.7228	0.918583	0.459291	+	0.888286i	$0.348103\pi$
$462$	0	0
$463$	2.51087	0.116690	0.0583451	−	0.998296i	$-0.481418\pi$
$463$	2.51087	0.116690	0.0583451	+	0.998296i	$0.481418\pi$
$464$	8.74456	0.405956
$465$	0	0
$466$	−14.8614	−0.688441
$467$	−6.51087	−0.301287	−0.150644	−	0.988588i	$-0.548135\pi$
$467$	−6.51087	−0.301287	−0.150644	+	0.988588i	$0.548135\pi$
$468$	0	0
$469$	36.2337	1.67312
$470$	6.74456	0.311103
$471$	0	0
$472$	2.74456	0.126329
$473$	−0.394031	−0.0181176
$474$	0	0
$475$	−0.627719	−0.0288017
$476$	16.0000	0.733359
$477$	0	0
$478$	−29.4891	−1.34880
$479$	−8.86141	−0.404888	−0.202444	−	0.979294i	$-0.564888\pi$
$479$	−8.86141	−0.404888	−0.202444	+	0.979294i	$0.564888\pi$
$480$	0	0
$481$	1.48913	0.0678983
$482$	−4.51087	−0.205465
$483$	0	0
$484$	−10.6060	−0.482090
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−37.4891	−1.69879	−0.849397	−	0.527754i	$-0.823034\pi$
$487$	−37.4891	−1.69879	−0.849397	+	0.527754i	$0.823034\pi$
$488$	−11.4891	−0.520088
$489$	0	0
$490$	4.37228	0.197520
$491$	−27.6060	−1.24584	−0.622920	−	0.782286i	$-0.714054\pi$
$491$	−27.6060	−1.24584	−0.622920	+	0.782286i	$0.714054\pi$
$492$	0	0
$493$	41.4891	1.86858
$494$	−1.25544	−0.0564848
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−11.3723	−0.510117
$498$	0	0
$499$	33.4891	1.49918	0.749590	−	0.661903i	$-0.230251\pi$
$499$	33.4891	1.49918	0.749590	+	0.661903i	$0.230251\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−5.48913	−0.244748	−0.122374	−	0.992484i	$-0.539051\pi$
$503$	−5.48913	−0.244748	−0.122374	+	0.992484i	$0.539051\pi$
$504$	0	0
$505$	8.11684	0.361195
$506$	−2.11684	−0.0941052
$507$	0	0
$508$	13.4891	0.598483
$509$	27.2554	1.20808	0.604038	−	0.796956i	$-0.293558\pi$
$509$	27.2554	1.20808	0.604038	+	0.796956i	$0.293558\pi$
$510$	0	0
$511$	−27.3723	−1.21088
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	13.3723	0.589826
$515$	8.00000	0.352522
$516$	0	0
$517$	−4.23369	−0.186197
$518$	2.51087	0.110322
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−36.9783	−1.62005	−0.810023	−	0.586398i	$-0.800546\pi$
$521$	−36.9783	−1.62005	−0.810023	+	0.586398i	$0.800546\pi$
$522$	0	0
$523$	−12.8614	−0.562390	−0.281195	−	0.959651i	$-0.590731\pi$
$523$	−12.8614	−0.562390	−0.281195	+	0.959651i	$0.590731\pi$
$524$	16.2337	0.709172
$525$	0	0
$526$	−18.9783	−0.827491
$527$	−4.74456	−0.206676
$528$	0	0
$529$	−11.6277	−0.505553
$530$	−1.37228	−0.0596081
$531$	0	0
$532$	−2.11684	−0.0917768
$533$	1.48913	0.0645012
$534$	0	0
$535$	0.627719	0.0271386
$536$	−10.7446	−0.464094
$537$	0	0
$538$	−2.00000	−0.0862261
$539$	−2.74456	−0.118217
$540$	0	0
$541$	−15.4891	−0.665930	−0.332965	−	0.942939i	$-0.608049\pi$
$541$	−15.4891	−0.665930	−0.332965	+	0.942939i	$0.608049\pi$
$542$	13.8832	0.596333
$543$	0	0
$544$	−4.74456	−0.203421
$545$	−4.74456	−0.203235
$546$	0	0
$547$	−2.74456	−0.117349	−0.0586745	−	0.998277i	$-0.518687\pi$
$547$	−2.74456	−0.117349	−0.0586745	+	0.998277i	$0.518687\pi$
$548$	3.48913	0.149048
$549$	0	0
$550$	0.627719	0.0267660
$551$	−5.48913	−0.233845
$552$	0	0
$553$	−15.6060	−0.663633
$554$	−15.2554	−0.648141
$555$	0	0
$556$	−10.7446	−0.455671
$557$	−11.8832	−0.503505	−0.251753	−	0.967792i	$-0.581007\pi$
$557$	−11.8832	−0.503505	−0.251753	+	0.967792i	$0.581007\pi$
$558$	0	0
$559$	−1.25544	−0.0530993
$560$	−3.37228	−0.142505
$561$	0	0
$562$	16.7446	0.706327
$563$	6.97825	0.294098	0.147049	−	0.989129i	$-0.453022\pi$
$563$	6.97825	0.294098	0.147049	+	0.989129i	$0.453022\pi$
$564$	0	0
$565$	−2.62772	−0.110549
$566$	−12.0000	−0.504398
$567$	0	0
$568$	3.37228	0.141498
$569$	1.37228	0.0575290	0.0287645	−	0.999586i	$-0.490843\pi$
$569$	1.37228	0.0575290	0.0287645	+	0.999586i	$0.490843\pi$
$570$	0	0
$571$	−26.7446	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$571$	−26.7446	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$572$	1.25544	0.0524925
$573$	0	0
$574$	2.51087	0.104802
$575$	−3.37228	−0.140634
$576$	0	0
$577$	−2.23369	−0.0929896	−0.0464948	−	0.998919i	$-0.514805\pi$
$577$	−2.23369	−0.0929896	−0.0464948	+	0.998919i	$0.514805\pi$
$578$	−5.51087	−0.229222
$579$	0	0
$580$	−8.74456	−0.363098
$581$	40.4674	1.67887
$582$	0	0
$583$	0.861407	0.0356758
$584$	8.11684	0.335877
$585$	0	0
$586$	−7.48913	−0.309373
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0.627719	0.0258647
$590$	−2.74456	−0.112992
$591$	0	0
$592$	−0.744563	−0.0306013
$593$	−44.9783	−1.84704	−0.923518	−	0.383556i	$-0.874699\pi$
$593$	−44.9783	−1.84704	−0.923518	+	0.383556i	$0.874699\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	12.1168	0.496325
$597$	0	0
$598$	−6.74456	−0.275806
$599$	−10.1168	−0.413363	−0.206682	−	0.978408i	$-0.566266\pi$
$599$	−10.1168	−0.413363	−0.206682	+	0.978408i	$0.566266\pi$
$600$	0	0
$601$	12.5109	0.510329	0.255165	−	0.966898i	$-0.417870\pi$
$601$	12.5109	0.510329	0.255165	+	0.966898i	$0.417870\pi$
$602$	−2.11684	−0.0862761
$603$	0	0
$604$	−8.00000	−0.325515
$605$	10.6060	0.431194
$606$	0	0
$607$	−4.62772	−0.187833	−0.0939167	−	0.995580i	$-0.529939\pi$
$607$	−4.62772	−0.187833	−0.0939167	+	0.995580i	$0.529939\pi$
$608$	0.627719	0.0254574
$609$	0	0
$610$	11.4891	0.465181
$611$	−13.4891	−0.545712
$612$	0	0
$613$	35.4891	1.43339	0.716696	−	0.697386i	$-0.245653\pi$
$613$	35.4891	1.43339	0.716696	+	0.697386i	$0.245653\pi$
$614$	−30.9783	−1.25018
$615$	0	0
$616$	2.11684	0.0852901
$617$	−7.88316	−0.317364	−0.158682	−	0.987330i	$-0.550724\pi$
$617$	−7.88316	−0.317364	−0.158682	+	0.987330i	$0.550724\pi$
$618$	0	0
$619$	16.2337	0.652487	0.326244	−	0.945286i	$-0.394217\pi$
$619$	16.2337	0.652487	0.326244	+	0.945286i	$0.394217\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	8.00000	0.320771
$623$	4.62772	0.185406
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	9.37228	0.373995
$629$	−3.53262	−0.140855
$630$	0	0
$631$	−46.3505	−1.84519	−0.922593	−	0.385775i	$-0.873934\pi$
$631$	−46.3505	−1.84519	−0.922593	+	0.385775i	$0.873934\pi$
$632$	4.62772	0.184081
$633$	0	0
$634$	−24.7446	−0.982732
$635$	−13.4891	−0.535300
$636$	0	0
$637$	−8.74456	−0.346472
$638$	5.48913	0.217317
$639$	0	0
$640$	1.00000	0.0395285
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−25.0951	−0.989654	−0.494827	−	0.868992i	$-0.664769\pi$
$643$	−25.0951	−0.989654	−0.494827	+	0.868992i	$0.664769\pi$
$644$	−11.3723	−0.448131
$645$	0	0
$646$	2.97825	0.117178
$647$	−50.5842	−1.98867	−0.994335	−	0.106287i	$-0.966104\pi$
$647$	−50.5842	−1.98867	−0.994335	+	0.106287i	$0.966104\pi$
$648$	0	0
$649$	1.72281	0.0676263
$650$	2.00000	0.0784465
$651$	0	0
$652$	24.2337	0.949064
$653$	−40.9783	−1.60360	−0.801801	−	0.597591i	$-0.796125\pi$
$653$	−40.9783	−1.60360	−0.801801	+	0.597591i	$0.796125\pi$
$654$	0	0
$655$	−16.2337	−0.634303
$656$	−0.744563	−0.0290703
$657$	0	0
$658$	−22.7446	−0.886675
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−4.97825	−0.193632	−0.0968158	−	0.995302i	$-0.530866\pi$
$661$	−4.97825	−0.193632	−0.0968158	+	0.995302i	$0.530866\pi$
$662$	1.48913	0.0578765
$663$	0	0
$664$	−12.0000	−0.465690
$665$	2.11684	0.0820877
$666$	0	0
$667$	−29.4891	−1.14182
$668$	12.6277	0.488581
$669$	0	0
$670$	10.7446	0.415099
$671$	−7.21194	−0.278414
$672$	0	0
$673$	−0.510875	−0.0196928	−0.00984639	−	0.999952i	$-0.503134\pi$
$673$	−0.510875	−0.0196928	−0.00984639	+	0.999952i	$0.503134\pi$
$674$	−28.9783	−1.11620
$675$	0	0
$676$	−9.00000	−0.346154
$677$	41.6060	1.59905	0.799524	−	0.600635i	$-0.205085\pi$
$677$	41.6060	1.59905	0.799524	+	0.600635i	$0.205085\pi$
$678$	0	0
$679$	6.74456	0.258833
$680$	4.74456	0.181946
$681$	0	0
$682$	−0.627719	−0.0240366
$683$	12.8614	0.492128	0.246064	−	0.969254i	$-0.420863\pi$
$683$	12.8614	0.492128	0.246064	+	0.969254i	$0.420863\pi$
$684$	0	0
$685$	−3.48913	−0.133313
$686$	8.86141	0.338330
$687$	0	0
$688$	0.627719	0.0239316
$689$	2.74456	0.104560
$690$	0	0
$691$	−19.1386	−0.728066	−0.364033	−	0.931386i	$-0.618601\pi$
$691$	−19.1386	−0.728066	−0.364033	+	0.931386i	$0.618601\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	36.4674	1.38428
$695$	10.7446	0.407564
$696$	0	0
$697$	−3.53262	−0.133808
$698$	−7.25544	−0.274622
$699$	0	0
$700$	3.37228	0.127460
$701$	−45.6060	−1.72251	−0.861257	−	0.508170i	$-0.830322\pi$
$701$	−45.6060	−1.72251	−0.861257	+	0.508170i	$0.830322\pi$
$702$	0	0
$703$	0.467376	0.0176274
$704$	−0.627719	−0.0236580
$705$	0	0
$706$	15.4891	0.582941
$707$	−27.3723	−1.02944
$708$	0	0
$709$	40.1168	1.50662	0.753310	−	0.657666i	$-0.228456\pi$
$709$	40.1168	1.50662	0.753310	+	0.657666i	$0.228456\pi$
$710$	−3.37228	−0.126559
$711$	0	0
$712$	−1.37228	−0.0514284
$713$	3.37228	0.126293
$714$	0	0
$715$	−1.25544	−0.0469507
$716$	12.0000	0.448461
$717$	0	0
$718$	3.37228	0.125852
$719$	−38.7446	−1.44493	−0.722464	−	0.691408i	$-0.756991\pi$
$719$	−38.7446	−1.44493	−0.722464	+	0.691408i	$0.756991\pi$
$720$	0	0
$721$	−26.9783	−1.00472
$722$	18.6060	0.692442
$723$	0	0
$724$	14.8614	0.552320
$725$	8.74456	0.324765
$726$	0	0
$727$	19.3723	0.718478	0.359239	−	0.933246i	$-0.383036\pi$
$727$	19.3723	0.718478	0.359239	+	0.933246i	$0.383036\pi$
$728$	6.74456	0.249970
$729$	0	0
$730$	−8.11684	−0.300418
$731$	2.97825	0.110155
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	20.2337	0.746839
$735$	0	0
$736$	3.37228	0.124304
$737$	−6.74456	−0.248439
$738$	0	0
$739$	−22.9783	−0.845269	−0.422634	−	0.906300i	$-0.638895\pi$
$739$	−22.9783	−0.845269	−0.422634	+	0.906300i	$0.638895\pi$
$740$	0.744563	0.0273707
$741$	0	0
$742$	4.62772	0.169889
$743$	35.3723	1.29768	0.648842	−	0.760924i	$-0.275254\pi$
$743$	35.3723	1.29768	0.648842	+	0.760924i	$0.275254\pi$
$744$	0	0
$745$	−12.1168	−0.443927
$746$	−14.8614	−0.544115
$747$	0	0
$748$	−2.97825	−0.108896
$749$	−2.11684	−0.0773478
$750$	0	0
$751$	−38.7446	−1.41381	−0.706905	−	0.707309i	$-0.749909\pi$
$751$	−38.7446	−1.41381	−0.706905	+	0.707309i	$0.749909\pi$
$752$	6.74456	0.245949
$753$	0	0
$754$	17.4891	0.636916
$755$	8.00000	0.291150
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	28.8614	1.04829
$759$	0	0
$760$	−0.627719	−0.0227697
$761$	46.8614	1.69872	0.849362	−	0.527810i	$-0.176987\pi$
$761$	46.8614	1.69872	0.849362	+	0.527810i	$0.176987\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	−16.0000	−0.578860
$765$	0	0
$766$	13.4891	0.487382
$767$	5.48913	0.198201
$768$	0	0
$769$	−9.37228	−0.337973	−0.168987	−	0.985618i	$-0.554049\pi$
$769$	−9.37228	−0.337973	−0.168987	+	0.985618i	$0.554049\pi$
$770$	−2.11684	−0.0762858
$771$	0	0
$772$	15.4891	0.557466
$773$	−21.6060	−0.777113	−0.388556	−	0.921425i	$-0.627026\pi$
$773$	−21.6060	−0.777113	−0.388556	+	0.921425i	$0.627026\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	16.9783	0.608700
$779$	0.467376	0.0167455
$780$	0	0
$781$	2.11684	0.0757466
$782$	16.0000	0.572159
$783$	0	0
$784$	4.37228	0.156153
$785$	−9.37228	−0.334511
$786$	0	0
$787$	−9.88316	−0.352296	−0.176148	−	0.984364i	$-0.556364\pi$
$787$	−9.88316	−0.352296	−0.176148	+	0.984364i	$0.556364\pi$
$788$	−19.4891	−0.694271
$789$	0	0
$790$	−4.62772	−0.164647
$791$	8.86141	0.315075
$792$	0	0
$793$	−22.9783	−0.815982
$794$	1.60597	0.0569937
$795$	0	0
$796$	−12.6277	−0.447578
$797$	20.5109	0.726532	0.363266	−	0.931685i	$-0.381662\pi$
$797$	20.5109	0.726532	0.363266	+	0.931685i	$0.381662\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	22.6277	0.799013
$803$	5.09509	0.179802
$804$	0	0
$805$	11.3723	0.400820
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	8.11684	0.285550
$809$	−9.60597	−0.337728	−0.168864	−	0.985639i	$-0.554010\pi$
$809$	−9.60597	−0.337728	−0.168864	+	0.985639i	$0.554010\pi$
$810$	0	0
$811$	−40.6277	−1.42663	−0.713316	−	0.700842i	$-0.752808\pi$
$811$	−40.6277	−1.42663	−0.713316	+	0.700842i	$0.752808\pi$
$812$	29.4891	1.03487
$813$	0	0
$814$	−0.467376	−0.0163815
$815$	−24.2337	−0.848869
$816$	0	0
$817$	−0.394031	−0.0137854
$818$	10.2337	0.357813
$819$	0	0
$820$	0.744563	0.0260013
$821$	−15.2554	−0.532418	−0.266209	−	0.963915i	$-0.585771\pi$
$821$	−15.2554	−0.532418	−0.266209	+	0.963915i	$0.585771\pi$
$822$	0	0
$823$	−17.2554	−0.601487	−0.300743	−	0.953705i	$-0.597235\pi$
$823$	−17.2554	−0.601487	−0.300743	+	0.953705i	$0.597235\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	9.25544	0.322038
$827$	17.4891	0.608156	0.304078	−	0.952647i	$-0.401652\pi$
$827$	17.4891	0.608156	0.304078	+	0.952647i	$0.401652\pi$
$828$	0	0
$829$	−31.0951	−1.07998	−0.539989	−	0.841672i	$-0.681571\pi$
$829$	−31.0951	−1.07998	−0.539989	+	0.841672i	$0.681571\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	20.7446	0.718756
$834$	0	0
$835$	−12.6277	−0.437000
$836$	0.394031	0.0136278
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−16.8614	−0.582120	−0.291060	−	0.956705i	$-0.594008\pi$
$839$	−16.8614	−0.582120	−0.291060	+	0.956705i	$0.594008\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	−19.4891	−0.671640
$843$	0	0
$844$	−0.627719	−0.0216070
$845$	9.00000	0.309609
$846$	0	0
$847$	−35.7663	−1.22895
$848$	−1.37228	−0.0471243
$849$	0	0
$850$	−4.74456	−0.162737
$851$	2.51087	0.0860717
$852$	0	0
$853$	−39.0951	−1.33859	−0.669295	−	0.742997i	$-0.733404\pi$
$853$	−39.0951	−1.33859	−0.669295	+	0.742997i	$0.733404\pi$
$854$	−38.7446	−1.32581
$855$	0	0
$856$	0.627719	0.0214550
$857$	−7.48913	−0.255824	−0.127912	−	0.991786i	$-0.540827\pi$
$857$	−7.48913	−0.255824	−0.127912	+	0.991786i	$0.540827\pi$
$858$	0	0
$859$	−17.4891	−0.596721	−0.298361	−	0.954453i	$-0.596440\pi$
$859$	−17.4891	−0.596721	−0.298361	+	0.954453i	$0.596440\pi$
$860$	−0.627719	−0.0214050
$861$	0	0
$862$	−26.9783	−0.918883
$863$	6.35053	0.216175	0.108087	−	0.994141i	$-0.465527\pi$
$863$	6.35053	0.216175	0.108087	+	0.994141i	$0.465527\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−36.1168	−1.22730
$867$	0	0
$868$	−3.37228	−0.114463
$869$	2.90491	0.0985422
$870$	0	0
$871$	−21.4891	−0.728131
$872$	−4.74456	−0.160671
$873$	0	0
$874$	−2.11684	−0.0716033
$875$	−3.37228	−0.114004
$876$	0	0
$877$	−44.9783	−1.51881	−0.759404	−	0.650620i	$-0.774509\pi$
$877$	−44.9783	−1.51881	−0.759404	+	0.650620i	$0.774509\pi$
$878$	−14.7446	−0.497605
$879$	0	0
$880$	0.627719	0.0211604
$881$	3.02175	0.101805	0.0509027	−	0.998704i	$-0.483790\pi$
$881$	3.02175	0.101805	0.0509027	+	0.998704i	$0.483790\pi$
$882$	0	0
$883$	44.8614	1.50971	0.754853	−	0.655894i	$-0.227708\pi$
$883$	44.8614	1.50971	0.754853	+	0.655894i	$0.227708\pi$
$884$	−9.48913	−0.319154
$885$	0	0
$886$	34.3505	1.15403
$887$	30.7446	1.03230	0.516151	−	0.856498i	$-0.327364\pi$
$887$	30.7446	1.03230	0.516151	+	0.856498i	$0.327364\pi$
$888$	0	0
$889$	45.4891	1.52566
$890$	1.37228	0.0459990
$891$	0	0
$892$	9.25544	0.309895
$893$	−4.23369	−0.141675
$894$	0	0
$895$	−12.0000	−0.401116
$896$	−3.37228	−0.112660
$897$	0	0
$898$	−8.97825	−0.299608
$899$	−8.74456	−0.291647
$900$	0	0
$901$	−6.51087	−0.216909
$902$	−0.467376	−0.0155619
$903$	0	0
$904$	−2.62772	−0.0873966
$905$	−14.8614	−0.494010
$906$	0	0
$907$	−17.4891	−0.580717	−0.290358	−	0.956918i	$-0.593775\pi$
$907$	−17.4891	−0.580717	−0.290358	+	0.956918i	$0.593775\pi$
$908$	−6.11684	−0.202995
$909$	0	0
$910$	−6.74456	−0.223580
$911$	2.51087	0.0831890	0.0415945	−	0.999135i	$-0.486756\pi$
$911$	2.51087	0.0831890	0.0415945	+	0.999135i	$0.486756\pi$
$912$	0	0
$913$	−7.53262	−0.249293
$914$	−34.4674	−1.14008
$915$	0	0
$916$	3.88316	0.128303
$917$	54.7446	1.80782
$918$	0	0
$919$	−12.2337	−0.403552	−0.201776	−	0.979432i	$-0.564671\pi$
$919$	−12.2337	−0.403552	−0.201776	+	0.979432i	$0.564671\pi$
$920$	−3.37228	−0.111181
$921$	0	0
$922$	−19.7228	−0.649536
$923$	6.74456	0.222000
$924$	0	0
$925$	−0.744563	−0.0244811
$926$	−2.51087	−0.0825125
$927$	0	0
$928$	−8.74456	−0.287054
$929$	16.1168	0.528776	0.264388	−	0.964416i	$-0.414830\pi$
$929$	16.1168	0.528776	0.264388	+	0.964416i	$0.414830\pi$
$930$	0	0
$931$	−2.74456	−0.0899494
$932$	14.8614	0.486802
$933$	0	0
$934$	6.51087	0.213042
$935$	2.97825	0.0973992
$936$	0	0
$937$	−31.2554	−1.02107	−0.510535	−	0.859857i	$-0.670553\pi$
$937$	−31.2554	−1.02107	−0.510535	+	0.859857i	$0.670553\pi$
$938$	−36.2337	−1.18307
$939$	0	0
$940$	−6.74456	−0.219983
$941$	19.7228	0.642945	0.321473	−	0.946919i	$-0.395822\pi$
$941$	19.7228	0.642945	0.321473	+	0.946919i	$0.395822\pi$
$942$	0	0
$943$	2.51087	0.0817653
$944$	−2.74456	−0.0893279
$945$	0	0
$946$	0.394031	0.0128110
$947$	2.74456	0.0891863	0.0445932	−	0.999005i	$-0.485801\pi$
$947$	2.74456	0.0891863	0.0445932	+	0.999005i	$0.485801\pi$
$948$	0	0
$949$	16.2337	0.526968
$950$	0.627719	0.0203659
$951$	0	0
$952$	−16.0000	−0.518563
$953$	−11.7228	−0.379739	−0.189870	−	0.981809i	$-0.560807\pi$
$953$	−11.7228	−0.379739	−0.189870	+	0.981809i	$0.560807\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	29.4891	0.953746
$957$	0	0
$958$	8.86141	0.286299
$959$	11.7663	0.379954
$960$	0	0
$961$	1.00000	0.0322581
$962$	−1.48913	−0.0480113
$963$	0	0
$964$	4.51087	0.145285
$965$	−15.4891	−0.498613
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	10.6060	0.340889
$969$	0	0
$970$	2.00000	0.0642161
$971$	48.7011	1.56289	0.781446	−	0.623973i	$-0.214483\pi$
$971$	48.7011	1.56289	0.781446	+	0.623973i	$0.214483\pi$
$972$	0	0
$973$	−36.2337	−1.16160
$974$	37.4891	1.20123
$975$	0	0
$976$	11.4891	0.367758
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	0	0
$979$	−0.861407	−0.0275307
$980$	−4.37228	−0.139667
$981$	0	0
$982$	27.6060	0.880942
$983$	2.97825	0.0949914	0.0474957	−	0.998871i	$-0.484876\pi$
$983$	2.97825	0.0949914	0.0474957	+	0.998871i	$0.484876\pi$
$984$	0	0
$985$	19.4891	0.620975
$986$	−41.4891	−1.32128
$987$	0	0
$988$	1.25544	0.0399408
$989$	−2.11684	−0.0673117
$990$	0	0
$991$	−47.6060	−1.51225	−0.756127	−	0.654425i	$-0.772911\pi$
$991$	−47.6060	−1.51225	−0.756127	+	0.654425i	$0.772911\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	11.3723	0.360707
$995$	12.6277	0.400326
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−33.4891	−1.06008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bd.1.2		2
3.2	odd	2		930.2.a.r.1.2	✓	2
12.11	even	2		7440.2.a.bg.1.1		2
15.2	even	4		4650.2.d.bh.3349.4		4
15.8	even	4		4650.2.d.bh.3349.1		4
15.14	odd	2		4650.2.a.by.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.r.1.2	✓	2	3.2	odd	2
2790.2.a.bd.1.2		2	1.1	even	1	trivial
4650.2.a.by.1.1		2	15.14	odd	2
4650.2.d.bh.3349.1		4	15.8	even	4
4650.2.d.bh.3349.4		4	15.2	even	4
7440.2.a.bg.1.1		2	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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.37228 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} -0.627719 q^{11} -2.00000 q^{13} -3.37228 q^{14} +1.00000 q^{16} +4.74456 q^{17} -0.627719 q^{19} -1.00000 q^{20} +0.627719 q^{22} -3.37228 q^{23} +1.00000 q^{25} +2.00000 q^{26} +3.37228 q^{28} +8.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} -4.74456 q^{34} -3.37228 q^{35} -0.744563 q^{37} +0.627719 q^{38} +1.00000 q^{40} -0.744563 q^{41} +0.627719 q^{43} -0.627719 q^{44} +3.37228 q^{46} +6.74456 q^{47} +4.37228 q^{49} -1.00000 q^{50} -2.00000 q^{52} -1.37228 q^{53} +0.627719 q^{55} -3.37228 q^{56} -8.74456 q^{58} -2.74456 q^{59} +11.4891 q^{61} +1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +10.7446 q^{67} +4.74456 q^{68} +3.37228 q^{70} -3.37228 q^{71} -8.11684 q^{73} +0.744563 q^{74} -0.627719 q^{76} -2.11684 q^{77} -4.62772 q^{79} -1.00000 q^{80} +0.744563 q^{82} +12.0000 q^{83} -4.74456 q^{85} -0.627719 q^{86} +0.627719 q^{88} +1.37228 q^{89} -6.74456 q^{91} -3.37228 q^{92} -6.74456 q^{94} +0.627719 q^{95} +2.00000 q^{97} -4.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - 7 q^{11} - 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 7 q^{19} - 2 q^{20} + 7 q^{22} - q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 6 q^{29} - 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 10 q^{37} + 7 q^{38} + 2 q^{40} + 10 q^{41} + 7 q^{43} - 7 q^{44} + q^{46} + 2 q^{47} + 3 q^{49} - 2 q^{50} - 4 q^{52} + 3 q^{53} + 7 q^{55} - q^{56} - 6 q^{58} + 6 q^{59} + 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 2 q^{68} + q^{70} - q^{71} + q^{73} - 10 q^{74} - 7 q^{76} + 13 q^{77} - 15 q^{79} - 2 q^{80} - 10 q^{82} + 24 q^{83} + 2 q^{85} - 7 q^{86} + 7 q^{88} - 3 q^{89} - 2 q^{91} - q^{92} - 2 q^{94} + 7 q^{95} + 4 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - 7 * q^11 - 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 7 * q^19 - 2 * q^20 + 7 * q^22 - q^23 + 2 * q^25 + 4 * q^26 + q^28 + 6 * q^29 - 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 10 * q^37 + 7 * q^38 + 2 * q^40 + 10 * q^41 + 7 * q^43 - 7 * q^44 + q^46 + 2 * q^47 + 3 * q^49 - 2 * q^50 - 4 * q^52 + 3 * q^53 + 7 * q^55 - q^56 - 6 * q^58 + 6 * q^59 + 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 2 * q^68 + q^70 - q^71 + q^73 - 10 * q^74 - 7 * q^76 + 13 * q^77 - 15 * q^79 - 2 * q^80 - 10 * q^82 + 24 * q^83 + 2 * q^85 - 7 * q^86 + 7 * q^88 - 3 * q^89 - 2 * q^91 - q^92 - 2 * q^94 + 7 * q^95 + 4 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more