LMFDB - Embedded newform 2151.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1767625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - x^{3} + 11x^{2} + x - 1 $ x^6 - 7x^4 - x^3 + 11x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.65963$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22058 q^{2} +2.93097 q^{4} +0.0570810 q^{5} -3.64374 q^{7} +2.06730 q^{8} +O(q^{10})$	\(q+2.22058 q^{2} +2.93097 q^{4} +0.0570810 q^{5} -3.64374 q^{7} +2.06730 q^{8} +0.126753 q^{10} -3.71735 q^{11} -0.736497 q^{13} -8.09121 q^{14} -1.27135 q^{16} -1.72628 q^{17} -0.753715 q^{19} +0.167303 q^{20} -8.25467 q^{22} -1.29699 q^{23} -4.99674 q^{25} -1.63545 q^{26} -10.6797 q^{28} -2.49193 q^{29} +6.31032 q^{31} -6.95772 q^{32} -3.83334 q^{34} -0.207988 q^{35} -1.11169 q^{37} -1.67368 q^{38} +0.118003 q^{40} +0.457952 q^{41} -3.49470 q^{43} -10.8955 q^{44} -2.88007 q^{46} +6.66411 q^{47} +6.27684 q^{49} -11.0957 q^{50} -2.15865 q^{52} +0.214663 q^{53} -0.212190 q^{55} -7.53269 q^{56} -5.53352 q^{58} +7.80818 q^{59} -3.44774 q^{61} +14.0126 q^{62} -12.9075 q^{64} -0.0420400 q^{65} -11.5114 q^{67} -5.05968 q^{68} -0.461855 q^{70} +9.97235 q^{71} -2.09226 q^{73} -2.46859 q^{74} -2.20912 q^{76} +13.5451 q^{77} -14.2848 q^{79} -0.0725697 q^{80} +1.01692 q^{82} +9.92940 q^{83} -0.0985378 q^{85} -7.76025 q^{86} -7.68487 q^{88} -0.440716 q^{89} +2.68360 q^{91} -3.80144 q^{92} +14.7982 q^{94} -0.0430228 q^{95} -7.68409 q^{97} +13.9382 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8} - 11 q^{10} + 13 q^{11} - q^{13} - 4 q^{16} - 11 q^{17} - 22 q^{19} + q^{20} - 2 q^{22} + 12 q^{23} - q^{25} - 12 q^{26} - 16 q^{28} - 18 q^{31} - 7 q^{32} - 3 q^{34} + 9 q^{35} - 8 q^{37} + 5 q^{38} - 11 q^{40} - 10 q^{41} - 14 q^{43} + 4 q^{44} - 18 q^{46} + 9 q^{47} + 5 q^{49} - 4 q^{50} - 16 q^{52} + 8 q^{53} - 20 q^{55} - 11 q^{56} - 15 q^{58} + 10 q^{59} - 12 q^{61} + 13 q^{62} - 31 q^{64} + 11 q^{65} - 36 q^{67} - 22 q^{68} + q^{70} + 3 q^{71} - 32 q^{73} - 9 q^{74} - 4 q^{76} - 6 q^{77} - q^{79} + 7 q^{80} + 7 q^{82} + 7 q^{83} - 14 q^{85} - 45 q^{86} - 15 q^{88} - 17 q^{89} - 23 q^{91} + 12 q^{92} + 50 q^{94} - 28 q^{97} - 13 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8 - 11 * q^10 + 13 * q^11 - q^13 - 4 * q^16 - 11 * q^17 - 22 * q^19 + q^20 - 2 * q^22 + 12 * q^23 - q^25 - 12 * q^26 - 16 * q^28 - 18 * q^31 - 7 * q^32 - 3 * q^34 + 9 * q^35 - 8 * q^37 + 5 * q^38 - 11 * q^40 - 10 * q^41 - 14 * q^43 + 4 * q^44 - 18 * q^46 + 9 * q^47 + 5 * q^49 - 4 * q^50 - 16 * q^52 + 8 * q^53 - 20 * q^55 - 11 * q^56 - 15 * q^58 + 10 * q^59 - 12 * q^61 + 13 * q^62 - 31 * q^64 + 11 * q^65 - 36 * q^67 - 22 * q^68 + q^70 + 3 * q^71 - 32 * q^73 - 9 * q^74 - 4 * q^76 - 6 * q^77 - q^79 + 7 * q^80 + 7 * q^82 + 7 * q^83 - 14 * q^85 - 45 * q^86 - 15 * q^88 - 17 * q^89 - 23 * q^91 + 12 * q^92 + 50 * q^94 - 28 * q^97 - 13 * 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 $

$2$	2.22058	1.57019	0.785093	−	0.619377i	$-0.212615\pi$
$2$	2.22058	1.57019	0.785093	+	0.619377i	$0.212615\pi$
$3$	0	0
$3$	0	0
$4$	2.93097	1.46549
$5$	0.0570810	0.0255274	0.0127637	−	0.999919i	$-0.495937\pi$
$5$	0.0570810	0.0255274	0.0127637	+	0.999919i	$0.495937\pi$
$6$	0	0
$7$	−3.64374	−1.37720	−0.688602	−	0.725139i	$-0.741775\pi$
$7$	−3.64374	−1.37720	−0.688602	+	0.725139i	$0.741775\pi$
$8$	2.06730	0.730900
$9$	0	0
$10$	0.126753	0.0400828
$11$	−3.71735	−1.12082	−0.560412	−	0.828214i	$-0.689357\pi$
$11$	−3.71735	−1.12082	−0.560412	+	0.828214i	$0.689357\pi$
$12$	0	0
$13$	−0.736497	−0.204267	−0.102134	−	0.994771i	$-0.532567\pi$
$13$	−0.736497	−0.204267	−0.102134	+	0.994771i	$0.532567\pi$
$14$	−8.09121	−2.16247
$15$	0	0
$16$	−1.27135	−0.317836
$17$	−1.72628	−0.418684	−0.209342	−	0.977842i	$-0.567132\pi$
$17$	−1.72628	−0.418684	−0.209342	+	0.977842i	$0.567132\pi$
$18$	0	0
$19$	−0.753715	−0.172914	−0.0864571	−	0.996256i	$-0.527555\pi$
$19$	−0.753715	−0.172914	−0.0864571	+	0.996256i	$0.527555\pi$
$20$	0.167303	0.0374101
$21$	0	0
$22$	−8.25467	−1.75990
$23$	−1.29699	−0.270441	−0.135221	−	0.990816i	$-0.543174\pi$
$23$	−1.29699	−0.270441	−0.135221	+	0.990816i	$0.543174\pi$
$24$	0	0
$25$	−4.99674	−0.999348
$26$	−1.63545	−0.320738
$27$	0	0
$28$	−10.6797	−2.01827
$29$	−2.49193	−0.462739	−0.231369	−	0.972866i	$-0.574321\pi$
$29$	−2.49193	−0.462739	−0.231369	+	0.972866i	$0.574321\pi$
$30$	0	0
$31$	6.31032	1.13337	0.566684	−	0.823935i	$-0.308226\pi$
$31$	6.31032	1.13337	0.566684	+	0.823935i	$0.308226\pi$
$32$	−6.95772	−1.22996
$33$	0	0
$34$	−3.83334	−0.657413
$35$	−0.207988	−0.0351565
$36$	0	0
$37$	−1.11169	−0.182761	−0.0913803	−	0.995816i	$-0.529128\pi$
$37$	−1.11169	−0.182761	−0.0913803	+	0.995816i	$0.529128\pi$
$38$	−1.67368	−0.271507
$39$	0	0
$40$	0.118003	0.0186580
$41$	0.457952	0.0715201	0.0357601	−	0.999360i	$-0.488615\pi$
$41$	0.457952	0.0715201	0.0357601	+	0.999360i	$0.488615\pi$
$42$	0	0
$43$	−3.49470	−0.532936	−0.266468	−	0.963844i	$-0.585857\pi$
$43$	−3.49470	−0.532936	−0.266468	+	0.963844i	$0.585857\pi$
$44$	−10.8955	−1.64255
$45$	0	0
$46$	−2.88007	−0.424643
$47$	6.66411	0.972061	0.486030	−	0.873942i	$-0.338444\pi$
$47$	6.66411	0.972061	0.486030	+	0.873942i	$0.338444\pi$
$48$	0	0
$49$	6.27684	0.896691
$50$	−11.0957	−1.56916
$51$	0	0
$52$	−2.15865	−0.299351
$53$	0.214663	0.0294862	0.0147431	−	0.999891i	$-0.495307\pi$
$53$	0.214663	0.0294862	0.0147431	+	0.999891i	$0.495307\pi$
$54$	0	0
$55$	−0.212190	−0.0286117
$56$	−7.53269	−1.00660
$57$	0	0
$58$	−5.53352	−0.726586
$59$	7.80818	1.01654	0.508269	−	0.861198i	$-0.330285\pi$
$59$	7.80818	1.01654	0.508269	+	0.861198i	$0.330285\pi$
$60$	0	0
$61$	−3.44774	−0.441437	−0.220719	−	0.975338i	$-0.570840\pi$
$61$	−3.44774	−0.441437	−0.220719	+	0.975338i	$0.570840\pi$
$62$	14.0126	1.77960
$63$	0	0
$64$	−12.9075	−1.61343
$65$	−0.0420400	−0.00521442
$66$	0	0
$67$	−11.5114	−1.40634	−0.703168	−	0.711024i	$-0.748232\pi$
$67$	−11.5114	−1.40634	−0.703168	+	0.711024i	$0.748232\pi$
$68$	−5.05968	−0.613576
$69$	0	0
$70$	−0.461855	−0.0552022
$71$	9.97235	1.18350	0.591750	−	0.806121i	$-0.298437\pi$
$71$	9.97235	1.18350	0.591750	+	0.806121i	$0.298437\pi$
$72$	0	0
$73$	−2.09226	−0.244881	−0.122440	−	0.992476i	$-0.539072\pi$
$73$	−2.09226	−0.244881	−0.122440	+	0.992476i	$0.539072\pi$
$74$	−2.46859	−0.286968
$75$	0	0
$76$	−2.20912	−0.253403
$77$	13.5451	1.54360
$78$	0	0
$79$	−14.2848	−1.60717	−0.803583	−	0.595193i	$-0.797076\pi$
$79$	−14.2848	−1.60717	−0.803583	+	0.595193i	$0.797076\pi$
$80$	−0.0725697	−0.00811354
$81$	0	0
$82$	1.01692	0.112300
$83$	9.92940	1.08989	0.544947	−	0.838471i	$-0.316550\pi$
$83$	9.92940	1.08989	0.544947	+	0.838471i	$0.316550\pi$
$84$	0	0
$85$	−0.0985378	−0.0106879
$86$	−7.76025	−0.836810
$87$	0	0
$88$	−7.68487	−0.819210
$89$	−0.440716	−0.0467158	−0.0233579	−	0.999727i	$-0.507436\pi$
$89$	−0.440716	−0.0467158	−0.0233579	+	0.999727i	$0.507436\pi$
$90$	0	0
$91$	2.68360	0.281318
$92$	−3.80144	−0.396328
$93$	0	0
$94$	14.7982	1.52632
$95$	−0.0430228	−0.00441405
$96$	0	0
$97$	−7.68409	−0.780201	−0.390100	−	0.920772i	$-0.627560\pi$
$97$	−7.68409	−0.780201	−0.390100	+	0.920772i	$0.627560\pi$
$98$	13.9382	1.40797
$99$	0	0
$100$	−14.6453	−1.46453
$101$	−12.8775	−1.28135	−0.640677	−	0.767810i	$-0.721346\pi$
$101$	−12.8775	−1.28135	−0.640677	+	0.767810i	$0.721346\pi$
$102$	0	0
$103$	−4.75094	−0.468124	−0.234062	−	0.972222i	$-0.575202\pi$
$103$	−4.75094	−0.468124	−0.234062	+	0.972222i	$0.575202\pi$
$104$	−1.52256	−0.149299
$105$	0	0
$106$	0.476676	0.0462989
$107$	20.4655	1.97847	0.989236	−	0.146327i	$-0.0467452\pi$
$107$	20.4655	1.97847	0.989236	+	0.146327i	$0.0467452\pi$
$108$	0	0
$109$	11.6172	1.11272	0.556361	−	0.830940i	$-0.312197\pi$
$109$	11.6172	1.11272	0.556361	+	0.830940i	$0.312197\pi$
$110$	−0.471185	−0.0449257
$111$	0	0
$112$	4.63245	0.437726
$113$	−5.69792	−0.536015	−0.268008	−	0.963417i	$-0.586365\pi$
$113$	−5.69792	−0.536015	−0.268008	+	0.963417i	$0.586365\pi$
$114$	0	0
$115$	−0.0740335	−0.00690366
$116$	−7.30376	−0.678137
$117$	0	0
$118$	17.3387	1.59616
$119$	6.29011	0.576614
$120$	0	0
$121$	2.81870	0.256246
$122$	−7.65597	−0.693139
$123$	0	0
$124$	18.4954	1.66093
$125$	−0.570624	−0.0510382
$126$	0	0
$127$	2.93084	0.260070	0.130035	−	0.991509i	$-0.458491\pi$
$127$	2.93084	0.260070	0.130035	+	0.991509i	$0.458491\pi$
$128$	−14.7466	−1.30343
$129$	0	0
$130$	−0.0933531	−0.00818761
$131$	−4.34492	−0.379617	−0.189809	−	0.981821i	$-0.560787\pi$
$131$	−4.34492	−0.379617	−0.189809	+	0.981821i	$0.560787\pi$
$132$	0	0
$133$	2.74634	0.238138
$134$	−25.5619	−2.20821
$135$	0	0
$136$	−3.56873	−0.306016
$137$	10.8025	0.922922	0.461461	−	0.887160i	$-0.347325\pi$
$137$	10.8025	0.922922	0.461461	+	0.887160i	$0.347325\pi$
$138$	0	0
$139$	1.59821	0.135558	0.0677791	−	0.997700i	$-0.478409\pi$
$139$	1.59821	0.135558	0.0677791	+	0.997700i	$0.478409\pi$
$140$	−0.609608	−0.0515213
$141$	0	0
$142$	22.1444	1.85832
$143$	2.73782	0.228948
$144$	0	0
$145$	−0.142242	−0.0118125
$146$	−4.64603	−0.384508
$147$	0	0
$148$	−3.25833	−0.267833
$149$	−8.16766	−0.669121	−0.334560	−	0.942374i	$-0.608588\pi$
$149$	−8.16766	−0.669121	−0.334560	+	0.942374i	$0.608588\pi$
$150$	0	0
$151$	−11.6281	−0.946280	−0.473140	−	0.880987i	$-0.656880\pi$
$151$	−11.6281	−0.946280	−0.473140	+	0.880987i	$0.656880\pi$
$152$	−1.55815	−0.126383
$153$	0	0
$154$	30.0779	2.42374
$155$	0.360200	0.0289319
$156$	0	0
$157$	7.31612	0.583890	0.291945	−	0.956435i	$-0.405698\pi$
$157$	7.31612	0.583890	0.291945	+	0.956435i	$0.405698\pi$
$158$	−31.7205	−2.52355
$159$	0	0
$160$	−0.397154	−0.0313978
$161$	4.72590	0.372453
$162$	0	0
$163$	17.5848	1.37735	0.688675	−	0.725070i	$-0.258193\pi$
$163$	17.5848	1.37735	0.688675	+	0.725070i	$0.258193\pi$
$164$	1.34225	0.104812
$165$	0	0
$166$	22.0490	1.71134
$167$	3.05610	0.236488	0.118244	−	0.992985i	$-0.462274\pi$
$167$	3.05610	0.236488	0.118244	+	0.992985i	$0.462274\pi$
$168$	0	0
$169$	−12.4576	−0.958275
$170$	−0.218811	−0.0167820
$171$	0	0
$172$	−10.2429	−0.781011
$173$	−25.2508	−1.91978	−0.959890	−	0.280377i	$-0.909541\pi$
$173$	−25.2508	−1.91978	−0.959890	+	0.280377i	$0.909541\pi$
$174$	0	0
$175$	18.2068	1.37631
$176$	4.72604	0.356239
$177$	0	0
$178$	−0.978645	−0.0733525
$179$	2.05516	0.153610	0.0768050	−	0.997046i	$-0.475528\pi$
$179$	2.05516	0.153610	0.0768050	+	0.997046i	$0.475528\pi$
$180$	0	0
$181$	−25.1941	−1.87266	−0.936330	−	0.351122i	$-0.885800\pi$
$181$	−25.1941	−1.87266	−0.936330	+	0.351122i	$0.885800\pi$
$182$	5.95915	0.441722
$183$	0	0
$184$	−2.68127	−0.197666
$185$	−0.0634564	−0.00466540
$186$	0	0
$187$	6.41719	0.469271
$188$	19.5323	1.42454
$189$	0	0
$190$	−0.0955356	−0.00693088
$191$	−0.594359	−0.0430063	−0.0215031	−	0.999769i	$-0.506845\pi$
$191$	−0.594359	−0.0430063	−0.0215031	+	0.999769i	$0.506845\pi$
$192$	0	0
$193$	14.3926	1.03600	0.518002	−	0.855379i	$-0.326676\pi$
$193$	14.3926	1.03600	0.518002	+	0.855379i	$0.326676\pi$
$194$	−17.0631	−1.22506
$195$	0	0
$196$	18.3972	1.31409
$197$	4.63529	0.330251	0.165125	−	0.986273i	$-0.447197\pi$
$197$	4.63529	0.330251	0.165125	+	0.986273i	$0.447197\pi$
$198$	0	0
$199$	−12.5742	−0.891363	−0.445682	−	0.895192i	$-0.647039\pi$
$199$	−12.5742	−0.891363	−0.445682	+	0.895192i	$0.647039\pi$
$200$	−10.3298	−0.730424
$201$	0	0
$202$	−28.5954	−2.01197
$203$	9.07993	0.637286
$204$	0	0
$205$	0.0261404	0.00182572
$206$	−10.5498	−0.735042
$207$	0	0
$208$	0.936342	0.0649236
$209$	2.80182	0.193806
$210$	0	0
$211$	8.38295	0.577106	0.288553	−	0.957464i	$-0.406826\pi$
$211$	8.38295	0.577106	0.288553	+	0.957464i	$0.406826\pi$
$212$	0.629172	0.0432117
$213$	0	0
$214$	45.4452	3.10657
$215$	−0.199481	−0.0136045
$216$	0	0
$217$	−22.9932	−1.56088
$218$	25.7968	1.74718
$219$	0	0
$220$	−0.621924	−0.0419301
$221$	1.27140	0.0855236
$222$	0	0
$223$	−4.69892	−0.314663	−0.157332	−	0.987546i	$-0.550289\pi$
$223$	−4.69892	−0.314663	−0.157332	+	0.987546i	$0.550289\pi$
$224$	25.3521	1.69391
$225$	0	0
$226$	−12.6527	−0.841644
$227$	11.3000	0.750008	0.375004	−	0.927023i	$-0.377641\pi$
$227$	11.3000	0.750008	0.375004	+	0.927023i	$0.377641\pi$
$228$	0	0
$229$	−6.45502	−0.426560	−0.213280	−	0.976991i	$-0.568415\pi$
$229$	−6.45502	−0.426560	−0.213280	+	0.976991i	$0.568415\pi$
$230$	−0.164397	−0.0108400
$231$	0	0
$232$	−5.15155	−0.338216
$233$	17.6315	1.15508	0.577540	−	0.816363i	$-0.304013\pi$
$233$	17.6315	1.15508	0.577540	+	0.816363i	$0.304013\pi$
$234$	0	0
$235$	0.380394	0.0248142
$236$	22.8856	1.48972
$237$	0	0
$238$	13.9677	0.905391
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−2.36031	−0.152041	−0.0760205	−	0.997106i	$-0.524221\pi$
$241$	−2.36031	−0.152041	−0.0760205	+	0.997106i	$0.524221\pi$
$242$	6.25915	0.402354
$243$	0	0
$244$	−10.1052	−0.646920
$245$	0.358288	0.0228902
$246$	0	0
$247$	0.555109	0.0353207
$248$	13.0453	0.828379
$249$	0	0
$250$	−1.26712	−0.0801395
$251$	8.31979	0.525140	0.262570	−	0.964913i	$-0.415430\pi$
$251$	8.31979	0.525140	0.262570	+	0.964913i	$0.415430\pi$
$252$	0	0
$253$	4.82137	0.303117
$254$	6.50816	0.408358
$255$	0	0
$256$	−6.93112	−0.433195
$257$	−26.9279	−1.67972	−0.839858	−	0.542806i	$-0.817362\pi$
$257$	−26.9279	−1.67972	−0.839858	+	0.542806i	$0.817362\pi$
$258$	0	0
$259$	4.05071	0.251699
$260$	−0.123218	−0.00764166
$261$	0	0
$262$	−9.64824	−0.596070
$263$	−2.32501	−0.143366	−0.0716831	−	0.997427i	$-0.522837\pi$
$263$	−2.32501	−0.143366	−0.0716831	+	0.997427i	$0.522837\pi$
$264$	0	0
$265$	0.0122532	0.000752707 0
$266$	6.09847	0.373921
$267$	0	0
$268$	−33.7395	−2.06097
$269$	−13.3580	−0.814452	−0.407226	−	0.913327i	$-0.633504\pi$
$269$	−13.3580	−0.814452	−0.407226	+	0.913327i	$0.633504\pi$
$270$	0	0
$271$	16.5887	1.00769	0.503847	−	0.863793i	$-0.331917\pi$
$271$	16.5887	1.00769	0.503847	+	0.863793i	$0.331917\pi$
$272$	2.19470	0.133073
$273$	0	0
$274$	23.9879	1.44916
$275$	18.5746	1.12009
$276$	0	0
$277$	17.5340	1.05352	0.526759	−	0.850015i	$-0.323407\pi$
$277$	17.5340	1.05352	0.526759	+	0.850015i	$0.323407\pi$
$278$	3.54895	0.212852
$279$	0	0
$280$	−0.429974	−0.0256959
$281$	−11.3858	−0.679221	−0.339610	−	0.940566i	$-0.610295\pi$
$281$	−11.3858	−0.679221	−0.339610	+	0.940566i	$0.610295\pi$
$282$	0	0
$283$	−10.7683	−0.640108	−0.320054	−	0.947399i	$-0.603701\pi$
$283$	−10.7683	−0.640108	−0.320054	+	0.947399i	$0.603701\pi$
$284$	29.2287	1.73440
$285$	0	0
$286$	6.07954	0.359491
$287$	−1.66866	−0.0984978
$288$	0	0
$289$	−14.0200	−0.824703
$290$	−0.315859	−0.0185479
$291$	0	0
$292$	−6.13236	−0.358869
$293$	−12.8250	−0.749246	−0.374623	−	0.927177i	$-0.622228\pi$
$293$	−12.8250	−0.749246	−0.374623	+	0.927177i	$0.622228\pi$
$294$	0	0
$295$	0.445699	0.0259496
$296$	−2.29819	−0.133580
$297$	0	0
$298$	−18.1369	−1.05064
$299$	0.955229	0.0552423
$300$	0	0
$301$	12.7338	0.733962
$302$	−25.8211	−1.48584
$303$	0	0
$304$	0.958233	0.0549584
$305$	−0.196800	−0.0112688
$306$	0	0
$307$	−19.6775	−1.12305	−0.561527	−	0.827458i	$-0.689786\pi$
$307$	−19.6775	−1.12305	−0.561527	+	0.827458i	$0.689786\pi$
$308$	39.7002	2.26213
$309$	0	0
$310$	0.799852	0.0454285
$311$	−25.7035	−1.45751	−0.728757	−	0.684772i	$-0.759902\pi$
$311$	−25.7035	−1.45751	−0.728757	+	0.684772i	$0.759902\pi$
$312$	0	0
$313$	−9.41285	−0.532046	−0.266023	−	0.963967i	$-0.585710\pi$
$313$	−9.41285	−0.532046	−0.266023	+	0.963967i	$0.585710\pi$
$314$	16.2460	0.916816
$315$	0	0
$316$	−41.8684	−2.35528
$317$	1.87215	0.105151	0.0525753	−	0.998617i	$-0.483257\pi$
$317$	1.87215	0.105151	0.0525753	+	0.998617i	$0.483257\pi$
$318$	0	0
$319$	9.26336	0.518649
$320$	−0.736772	−0.0411868
$321$	0	0
$322$	10.4942	0.584820
$323$	1.30112	0.0723964
$324$	0	0
$325$	3.68008	0.204134
$326$	39.0485	2.16270
$327$	0	0
$328$	0.946724	0.0522741
$329$	−24.2823	−1.33873
$330$	0	0
$331$	−16.8715	−0.927341	−0.463670	−	0.886008i	$-0.653468\pi$
$331$	−16.8715	−0.927341	−0.463670	+	0.886008i	$0.653468\pi$
$332$	29.1028	1.59722
$333$	0	0
$334$	6.78631	0.371330
$335$	−0.657080	−0.0359001
$336$	0	0
$337$	−19.1903	−1.04536	−0.522681	−	0.852528i	$-0.675068\pi$
$337$	−19.1903	−1.04536	−0.522681	+	0.852528i	$0.675068\pi$
$338$	−27.6630	−1.50467
$339$	0	0
$340$	−0.288812	−0.0156630
$341$	−23.4577	−1.27031
$342$	0	0
$343$	2.63502	0.142278
$344$	−7.22458	−0.389523
$345$	0	0
$346$	−56.0713	−3.01441
$347$	23.8362	1.27960	0.639798	−	0.768543i	$-0.279018\pi$
$347$	23.8362	1.27960	0.639798	+	0.768543i	$0.279018\pi$
$348$	0	0
$349$	−13.5199	−0.723703	−0.361852	−	0.932236i	$-0.617855\pi$
$349$	−13.5199	−0.723703	−0.361852	+	0.932236i	$0.617855\pi$
$350$	40.4297	2.16106
$351$	0	0
$352$	25.8643	1.37857
$353$	14.3286	0.762636	0.381318	−	0.924444i	$-0.375470\pi$
$353$	14.3286	0.762636	0.381318	+	0.924444i	$0.375470\pi$
$354$	0	0
$355$	0.569232	0.0302117
$356$	−1.29173	−0.0684614
$357$	0	0
$358$	4.56365	0.241196
$359$	15.1952	0.801974	0.400987	−	0.916084i	$-0.368667\pi$
$359$	15.1952	0.801974	0.400987	+	0.916084i	$0.368667\pi$
$360$	0	0
$361$	−18.4319	−0.970101
$362$	−55.9454	−2.94042
$363$	0	0
$364$	7.86556	0.412268
$365$	−0.119428	−0.00625117
$366$	0	0
$367$	31.2749	1.63253	0.816267	−	0.577675i	$-0.196040\pi$
$367$	31.2749	1.63253	0.816267	+	0.577675i	$0.196040\pi$
$368$	1.64892	0.0859561
$369$	0	0
$370$	−0.140910	−0.00732556
$371$	−0.782176	−0.0406086
$372$	0	0
$373$	13.6383	0.706166	0.353083	−	0.935592i	$-0.385133\pi$
$373$	13.6383	0.706166	0.353083	+	0.935592i	$0.385133\pi$
$374$	14.2499	0.736844
$375$	0	0
$376$	13.7767	0.710479
$377$	1.83529	0.0945225
$378$	0	0
$379$	−22.3666	−1.14890	−0.574448	−	0.818541i	$-0.694783\pi$
$379$	−22.3666	−1.14890	−0.574448	+	0.818541i	$0.694783\pi$
$380$	−0.126099	−0.00646873
$381$	0	0
$382$	−1.31982	−0.0675279
$383$	−12.1215	−0.619380	−0.309690	−	0.950838i	$-0.600225\pi$
$383$	−12.1215	−0.619380	−0.309690	+	0.950838i	$0.600225\pi$
$384$	0	0
$385$	0.773166	0.0394042
$386$	31.9600	1.62672
$387$	0	0
$388$	−22.5218	−1.14337
$389$	11.4985	0.582999	0.291500	−	0.956571i	$-0.405846\pi$
$389$	11.4985	0.582999	0.291500	+	0.956571i	$0.405846\pi$
$390$	0	0
$391$	2.23897	0.113230
$392$	12.9761	0.655392
$393$	0	0
$394$	10.2930	0.518555
$395$	−0.815391	−0.0410268
$396$	0	0
$397$	−25.2172	−1.26561	−0.632807	−	0.774309i	$-0.718098\pi$
$397$	−25.2172	−1.26561	−0.632807	+	0.774309i	$0.718098\pi$
$398$	−27.9221	−1.39961
$399$	0	0
$400$	6.35259	0.317629
$401$	29.3323	1.46479	0.732393	−	0.680883i	$-0.238404\pi$
$401$	29.3323	1.46479	0.732393	+	0.680883i	$0.238404\pi$
$402$	0	0
$403$	−4.64753	−0.231510
$404$	−37.7435	−1.87781
$405$	0	0
$406$	20.1627	1.00066
$407$	4.13254	0.204842
$408$	0	0
$409$	38.7995	1.91851	0.959256	−	0.282539i	$-0.0911767\pi$
$409$	38.7995	1.91851	0.959256	+	0.282539i	$0.0911767\pi$
$410$	0.0580468	0.00286673
$411$	0	0
$412$	−13.9249	−0.686029
$413$	−28.4510	−1.39998
$414$	0	0
$415$	0.566781	0.0278222
$416$	5.12434	0.251241
$417$	0	0
$418$	6.22167	0.304312
$419$	26.0609	1.27316	0.636578	−	0.771212i	$-0.280349\pi$
$419$	26.0609	1.27316	0.636578	+	0.771212i	$0.280349\pi$
$420$	0	0
$421$	−11.7112	−0.570767	−0.285383	−	0.958413i	$-0.592121\pi$
$421$	−11.7112	−0.570767	−0.285383	+	0.958413i	$0.592121\pi$
$422$	18.6150	0.906164
$423$	0	0
$424$	0.443773	0.0215515
$425$	8.62577	0.418412
$426$	0	0
$427$	12.5626	0.607949
$428$	59.9838	2.89942
$429$	0	0
$430$	−0.442963	−0.0213616
$431$	−19.3962	−0.934284	−0.467142	−	0.884182i	$-0.654716\pi$
$431$	−19.3962	−0.934284	−0.467142	+	0.884182i	$0.654716\pi$
$432$	0	0
$433$	−3.56849	−0.171490	−0.0857452	−	0.996317i	$-0.527327\pi$
$433$	−3.56849	−0.171490	−0.0857452	+	0.996317i	$0.527327\pi$
$434$	−51.0582	−2.45087
$435$	0	0
$436$	34.0496	1.63068
$437$	0.977561	0.0467631
$438$	0	0
$439$	31.5488	1.50574	0.752872	−	0.658167i	$-0.228668\pi$
$439$	31.5488	1.50574	0.752872	+	0.658167i	$0.228668\pi$
$440$	−0.438660	−0.0209123
$441$	0	0
$442$	2.82324	0.134288
$443$	−2.17420	−0.103299	−0.0516496	−	0.998665i	$-0.516448\pi$
$443$	−2.17420	−0.103299	−0.0516496	+	0.998665i	$0.516448\pi$
$444$	0	0
$445$	−0.0251565	−0.00119253
$446$	−10.4343	−0.494080
$447$	0	0
$448$	47.0315	2.22203
$449$	−33.2412	−1.56875	−0.784375	−	0.620287i	$-0.787016\pi$
$449$	−33.2412	−1.56875	−0.784375	+	0.620287i	$0.787016\pi$
$450$	0	0
$451$	−1.70237	−0.0801614
$452$	−16.7005	−0.785523
$453$	0	0
$454$	25.0926	1.17765
$455$	0.153183	0.00718132
$456$	0	0
$457$	−26.5058	−1.23989	−0.619944	−	0.784646i	$-0.712845\pi$
$457$	−26.5058	−1.23989	−0.619944	+	0.784646i	$0.712845\pi$
$458$	−14.3339	−0.669779
$459$	0	0
$460$	−0.216990	−0.0101172
$461$	36.1977	1.68590	0.842948	−	0.537995i	$-0.180818\pi$
$461$	36.1977	1.68590	0.842948	+	0.537995i	$0.180818\pi$
$462$	0	0
$463$	6.82777	0.317314	0.158657	−	0.987334i	$-0.449284\pi$
$463$	6.82777	0.317314	0.158657	+	0.987334i	$0.449284\pi$
$464$	3.16810	0.147075
$465$	0	0
$466$	39.1522	1.81369
$467$	−33.8542	−1.56659	−0.783293	−	0.621652i	$-0.786462\pi$
$467$	−33.8542	−1.56659	−0.783293	+	0.621652i	$0.786462\pi$
$468$	0	0
$469$	41.9444	1.93681
$470$	0.844696	0.0389629
$471$	0	0
$472$	16.1418	0.742988
$473$	12.9910	0.597328
$474$	0	0
$475$	3.76612	0.172801
$476$	18.4362	0.845020
$477$	0	0
$478$	2.22058	0.101567
$479$	−8.14302	−0.372064	−0.186032	−	0.982544i	$-0.559563\pi$
$479$	−8.14302	−0.372064	−0.186032	+	0.982544i	$0.559563\pi$
$480$	0	0
$481$	0.818755	0.0373320
$482$	−5.24126	−0.238733
$483$	0	0
$484$	8.26154	0.375524
$485$	−0.438616	−0.0199165
$486$	0	0
$487$	−14.0940	−0.638661	−0.319330	−	0.947643i	$-0.603458\pi$
$487$	−14.0940	−0.638661	−0.319330	+	0.947643i	$0.603458\pi$
$488$	−7.12750	−0.322647
$489$	0	0
$490$	0.795608	0.0359419
$491$	30.5029	1.37658	0.688289	−	0.725437i	$-0.258362\pi$
$491$	30.5029	1.37658	0.688289	+	0.725437i	$0.258362\pi$
$492$	0	0
$493$	4.30176	0.193742
$494$	1.23266	0.0554601
$495$	0	0
$496$	−8.02261	−0.360226
$497$	−36.3367	−1.62992
$498$	0	0
$499$	−15.6252	−0.699479	−0.349739	−	0.936847i	$-0.613730\pi$
$499$	−15.6252	−0.699479	−0.349739	+	0.936847i	$0.613730\pi$
$500$	−1.67248	−0.0747958
$501$	0	0
$502$	18.4747	0.824568
$503$	−10.0657	−0.448807	−0.224404	−	0.974496i	$-0.572043\pi$
$503$	−10.0657	−0.448807	−0.224404	+	0.974496i	$0.572043\pi$
$504$	0	0
$505$	−0.735058	−0.0327097
$506$	10.7062	0.475950
$507$	0	0
$508$	8.59020	0.381129
$509$	22.5797	1.00083	0.500414	−	0.865786i	$-0.333181\pi$
$509$	22.5797	1.00083	0.500414	+	0.865786i	$0.333181\pi$
$510$	0	0
$511$	7.62365	0.337250
$512$	14.1022	0.623234
$513$	0	0
$514$	−59.7955	−2.63747
$515$	−0.271188	−0.0119500
$516$	0	0
$517$	−24.7728	−1.08951
$518$	8.99491	0.395214
$519$	0	0
$520$	−0.0869092	−0.00381122
$521$	10.6301	0.465715	0.232857	−	0.972511i	$-0.425192\pi$
$521$	10.6301	0.465715	0.232857	+	0.972511i	$0.425192\pi$
$522$	0	0
$523$	12.8900	0.563642	0.281821	−	0.959467i	$-0.409062\pi$
$523$	12.8900	0.563642	0.281821	+	0.959467i	$0.409062\pi$
$524$	−12.7348	−0.556324
$525$	0	0
$526$	−5.16287	−0.225112
$527$	−10.8934	−0.474523
$528$	0	0
$529$	−21.3178	−0.926862
$530$	0.0272092	0.00118189
$531$	0	0
$532$	8.04945	0.348988
$533$	−0.337280	−0.0146092
$534$	0	0
$535$	1.16819	0.0505053
$536$	−23.7974	−1.02789
$537$	0	0
$538$	−29.6625	−1.27884
$539$	−23.3332	−1.00503
$540$	0	0
$541$	34.1904	1.46996	0.734980	−	0.678089i	$-0.237191\pi$
$541$	34.1904	1.46996	0.734980	+	0.678089i	$0.237191\pi$
$542$	36.8366	1.58227
$543$	0	0
$544$	12.0110	0.514966
$545$	0.663120	0.0284049
$546$	0	0
$547$	−12.6930	−0.542714	−0.271357	−	0.962479i	$-0.587472\pi$
$547$	−12.6930	−0.542714	−0.271357	+	0.962479i	$0.587472\pi$
$548$	31.6619	1.35253
$549$	0	0
$550$	41.2465	1.75876
$551$	1.87820	0.0800141
$552$	0	0
$553$	52.0501	2.21340
$554$	38.9357	1.65422
$555$	0	0
$556$	4.68430	0.198659
$557$	32.3710	1.37160	0.685801	−	0.727789i	$-0.259452\pi$
$557$	32.3710	1.37160	0.685801	+	0.727789i	$0.259452\pi$
$558$	0	0
$559$	2.57383	0.108862
$560$	0.264425	0.0111740
$561$	0	0
$562$	−25.2831	−1.06650
$563$	−17.5807	−0.740938	−0.370469	−	0.928845i	$-0.620803\pi$
$563$	−17.5807	−0.740938	−0.370469	+	0.928845i	$0.620803\pi$
$564$	0	0
$565$	−0.325243	−0.0136831
$566$	−23.9118	−1.00509
$567$	0	0
$568$	20.6158	0.865021
$569$	−44.7549	−1.87622	−0.938112	−	0.346332i	$-0.887427\pi$
$569$	−44.7549	−1.87622	−0.938112	+	0.346332i	$0.887427\pi$
$570$	0	0
$571$	20.9021	0.874727	0.437364	−	0.899285i	$-0.355912\pi$
$571$	20.9021	0.874727	0.437364	+	0.899285i	$0.355912\pi$
$572$	8.02447	0.335520
$573$	0	0
$574$	−3.70539	−0.154660
$575$	6.48073	0.270265
$576$	0	0
$577$	−6.77713	−0.282135	−0.141068	−	0.990000i	$-0.545054\pi$
$577$	−6.77713	−0.282135	−0.141068	+	0.990000i	$0.545054\pi$
$578$	−31.1324	−1.29494
$579$	0	0
$580$	−0.416906	−0.0173111
$581$	−36.1802	−1.50101
$582$	0	0
$583$	−0.797978	−0.0330489
$584$	−4.32533	−0.178983
$585$	0	0
$586$	−28.4790	−1.17646
$587$	−26.4488	−1.09166	−0.545830	−	0.837896i	$-0.683786\pi$
$587$	−26.4488	−1.09166	−0.545830	+	0.837896i	$0.683786\pi$
$588$	0	0
$589$	−4.75619	−0.195975
$590$	0.989710	0.0407457
$591$	0	0
$592$	1.41334	0.0580880
$593$	3.87732	0.159222	0.0796112	−	0.996826i	$-0.474632\pi$
$593$	3.87732	0.159222	0.0796112	+	0.996826i	$0.474632\pi$
$594$	0	0
$595$	0.359046	0.0147195
$596$	−23.9392	−0.980587
$597$	0	0
$598$	2.12116	0.0867408
$599$	−25.3189	−1.03450	−0.517251	−	0.855834i	$-0.673045\pi$
$599$	−25.3189	−1.03450	−0.517251	+	0.855834i	$0.673045\pi$
$600$	0	0
$601$	21.4456	0.874782	0.437391	−	0.899271i	$-0.355903\pi$
$601$	21.4456	0.874782	0.437391	+	0.899271i	$0.355903\pi$
$602$	28.2763	1.15246
$603$	0	0
$604$	−34.0816	−1.38676
$605$	0.160894	0.00654129
$606$	0	0
$607$	44.0058	1.78614	0.893070	−	0.449919i	$-0.148547\pi$
$607$	44.0058	1.78614	0.893070	+	0.449919i	$0.148547\pi$
$608$	5.24414	0.212678
$609$	0	0
$610$	−0.437011	−0.0176940
$611$	−4.90810	−0.198560
$612$	0	0
$613$	9.11434	0.368125	0.184062	−	0.982915i	$-0.441075\pi$
$613$	9.11434	0.368125	0.184062	+	0.982915i	$0.441075\pi$
$614$	−43.6955	−1.76341
$615$	0	0
$616$	28.0017	1.12822
$617$	−2.06698	−0.0832133	−0.0416067	−	0.999134i	$-0.513248\pi$
$617$	−2.06698	−0.0832133	−0.0416067	+	0.999134i	$0.513248\pi$
$618$	0	0
$619$	−22.7993	−0.916383	−0.458191	−	0.888854i	$-0.651503\pi$
$619$	−22.7993	−0.916383	−0.458191	+	0.888854i	$0.651503\pi$
$620$	1.05574	0.0423994
$621$	0	0
$622$	−57.0768	−2.28857
$623$	1.60585	0.0643372
$624$	0	0
$625$	24.9511	0.998045
$626$	−20.9020	−0.835411
$627$	0	0
$628$	21.4433	0.855682
$629$	1.91909	0.0765190
$630$	0	0
$631$	−31.4481	−1.25193	−0.625964	−	0.779852i	$-0.715294\pi$
$631$	−31.4481	−1.25193	−0.625964	+	0.779852i	$0.715294\pi$
$632$	−29.5309	−1.17468
$633$	0	0
$634$	4.15727	0.165106
$635$	0.167295	0.00663891
$636$	0	0
$637$	−4.62287	−0.183165
$638$	20.5700	0.814375
$639$	0	0
$640$	−0.841753	−0.0332732
$641$	−6.51455	−0.257309	−0.128655	−	0.991689i	$-0.541066\pi$
$641$	−6.51455	−0.257309	−0.128655	+	0.991689i	$0.541066\pi$
$642$	0	0
$643$	−2.08816	−0.0823491	−0.0411746	−	0.999152i	$-0.513110\pi$
$643$	−2.08816	−0.0823491	−0.0411746	+	0.999152i	$0.513110\pi$
$644$	13.8515	0.545824
$645$	0	0
$646$	2.88925	0.113676
$647$	11.2824	0.443555	0.221778	−	0.975097i	$-0.428814\pi$
$647$	11.2824	0.443555	0.221778	+	0.975097i	$0.428814\pi$
$648$	0	0
$649$	−29.0258	−1.13936
$650$	8.17192	0.320529
$651$	0	0
$652$	51.5406	2.01849
$653$	15.7251	0.615372	0.307686	−	0.951488i	$-0.400445\pi$
$653$	15.7251	0.615372	0.307686	+	0.951488i	$0.400445\pi$
$654$	0	0
$655$	−0.248013	−0.00969065
$656$	−0.582216	−0.0227317
$657$	0	0
$658$	−53.9207	−2.10205
$659$	1.91095	0.0744400	0.0372200	−	0.999307i	$-0.488150\pi$
$659$	1.91095	0.0744400	0.0372200	+	0.999307i	$0.488150\pi$
$660$	0	0
$661$	−45.7455	−1.77929	−0.889647	−	0.456648i	$-0.849050\pi$
$661$	−45.7455	−1.77929	−0.889647	+	0.456648i	$0.849050\pi$
$662$	−37.4645	−1.45610
$663$	0	0
$664$	20.5270	0.796604
$665$	0.156764	0.00607905
$666$	0	0
$667$	3.23200	0.125144
$668$	8.95734	0.346570
$669$	0	0
$670$	−1.45910	−0.0563699
$671$	12.8164	0.494773
$672$	0	0
$673$	36.5553	1.40910	0.704552	−	0.709653i	$-0.251148\pi$
$673$	36.5553	1.40910	0.704552	+	0.709653i	$0.251148\pi$
$674$	−42.6136	−1.64141
$675$	0	0
$676$	−36.5128	−1.40434
$677$	−26.1615	−1.00547	−0.502733	−	0.864442i	$-0.667672\pi$
$677$	−26.1615	−1.00547	−0.502733	+	0.864442i	$0.667672\pi$
$678$	0	0
$679$	27.9988	1.07450
$680$	−0.203707	−0.00781181
$681$	0	0
$682$	−52.0897	−1.99462
$683$	−32.3175	−1.23660	−0.618298	−	0.785944i	$-0.712177\pi$
$683$	−32.3175	−1.23660	−0.618298	+	0.785944i	$0.712177\pi$
$684$	0	0
$685$	0.616619	0.0235598
$686$	5.85126	0.223402
$687$	0	0
$688$	4.44297	0.169387
$689$	−0.158099	−0.00602308
$690$	0	0
$691$	−3.28734	−0.125056	−0.0625282	−	0.998043i	$-0.519916\pi$
$691$	−3.28734	−0.125056	−0.0625282	+	0.998043i	$0.519916\pi$
$692$	−74.0093	−2.81341
$693$	0	0
$694$	52.9303	2.00921
$695$	0.0912273	0.00346045
$696$	0	0
$697$	−0.790554	−0.0299444
$698$	−30.0220	−1.13635
$699$	0	0
$700$	53.3637	2.01696
$701$	11.2419	0.424602	0.212301	−	0.977204i	$-0.431904\pi$
$701$	11.2419	0.424602	0.212301	+	0.977204i	$0.431904\pi$
$702$	0	0
$703$	0.837897	0.0316019
$704$	47.9816	1.80838
$705$	0	0
$706$	31.8179	1.19748
$707$	46.9221	1.76469
$708$	0	0
$709$	4.71294	0.176998	0.0884990	−	0.996076i	$-0.471793\pi$
$709$	4.71294	0.176998	0.0884990	+	0.996076i	$0.471793\pi$
$710$	1.26402	0.0474380
$711$	0	0
$712$	−0.911091	−0.0341446
$713$	−8.18443	−0.306509
$714$	0	0
$715$	0.156277	0.00584444
$716$	6.02362	0.225113
$717$	0	0
$718$	33.7422	1.25925
$719$	−25.7339	−0.959713	−0.479856	−	0.877347i	$-0.659311\pi$
$719$	−25.7339	−0.959713	−0.479856	+	0.877347i	$0.659311\pi$
$720$	0	0
$721$	17.3112	0.644702
$722$	−40.9295	−1.52324
$723$	0	0
$724$	−73.8431	−2.74436
$725$	12.4515	0.462437
$726$	0	0
$727$	−10.9895	−0.407576	−0.203788	−	0.979015i	$-0.565325\pi$
$727$	−10.9895	−0.407576	−0.203788	+	0.979015i	$0.565325\pi$
$728$	5.54780	0.205615
$729$	0	0
$730$	−0.265200	−0.00981550
$731$	6.03283	0.223132
$732$	0	0
$733$	−27.7612	−1.02538	−0.512692	−	0.858572i	$-0.671352\pi$
$733$	−27.7612	−1.02538	−0.512692	+	0.858572i	$0.671352\pi$
$734$	69.4483	2.56338
$735$	0	0
$736$	9.02410	0.332633
$737$	42.7918	1.57625
$738$	0	0
$739$	−25.7282	−0.946426	−0.473213	−	0.880948i	$-0.656906\pi$
$739$	−25.7282	−0.946426	−0.473213	+	0.880948i	$0.656906\pi$
$740$	−0.185989	−0.00683709
$741$	0	0
$742$	−1.73688	−0.0637630
$743$	38.6081	1.41639	0.708197	−	0.706015i	$-0.249509\pi$
$743$	38.6081	1.41639	0.708197	+	0.706015i	$0.249509\pi$
$744$	0	0
$745$	−0.466218	−0.0170809
$746$	30.2850	1.10881
$747$	0	0
$748$	18.8086	0.687711
$749$	−74.5709	−2.72476
$750$	0	0
$751$	−31.2572	−1.14059	−0.570296	−	0.821439i	$-0.693171\pi$
$751$	−31.2572	−1.14059	−0.570296	+	0.821439i	$0.693171\pi$
$752$	−8.47239	−0.308956
$753$	0	0
$754$	4.07542	0.148418
$755$	−0.663743	−0.0241561
$756$	0	0
$757$	−22.2245	−0.807764	−0.403882	−	0.914811i	$-0.632339\pi$
$757$	−22.2245	−0.807764	−0.403882	+	0.914811i	$0.632339\pi$
$758$	−49.6668	−1.80398
$759$	0	0
$760$	−0.0889410	−0.00322623
$761$	−29.5640	−1.07170	−0.535848	−	0.844314i	$-0.680008\pi$
$761$	−29.5640	−1.07170	−0.535848	+	0.844314i	$0.680008\pi$
$762$	0	0
$763$	−42.3299	−1.53245
$764$	−1.74205	−0.0630251
$765$	0	0
$766$	−26.9168	−0.972543
$767$	−5.75070	−0.207646
$768$	0	0
$769$	31.5926	1.13926	0.569628	−	0.821902i	$-0.307087\pi$
$769$	31.5926	1.13926	0.569628	+	0.821902i	$0.307087\pi$
$770$	1.71688	0.0618719
$771$	0	0
$772$	42.1844	1.51825
$773$	−46.1914	−1.66139	−0.830694	−	0.556729i	$-0.812056\pi$
$773$	−46.1914	−1.66139	−0.830694	+	0.556729i	$0.812056\pi$
$774$	0	0
$775$	−31.5311	−1.13263
$776$	−15.8853	−0.570249
$777$	0	0
$778$	25.5334	0.915418
$779$	−0.345166	−0.0123668
$780$	0	0
$781$	−37.0707	−1.32650
$782$	4.97181	0.177791
$783$	0	0
$784$	−7.98003	−0.285001
$785$	0.417611	0.0149052
$786$	0	0
$787$	12.5553	0.447549	0.223775	−	0.974641i	$-0.428162\pi$
$787$	12.5553	0.447549	0.223775	+	0.974641i	$0.428162\pi$
$788$	13.5859	0.483978
$789$	0	0
$790$	−1.81064	−0.0644197
$791$	20.7617	0.738203
$792$	0	0
$793$	2.53925	0.0901713
$794$	−55.9968	−1.98725
$795$	0	0
$796$	−36.8547	−1.30628
$797$	−44.8797	−1.58972	−0.794860	−	0.606793i	$-0.792456\pi$
$797$	−44.8797	−1.58972	−0.794860	+	0.606793i	$0.792456\pi$
$798$	0	0
$799$	−11.5041	−0.406987
$800$	34.7659	1.22916
$801$	0	0
$802$	65.1347	2.29999
$803$	7.77767	0.274468
$804$	0	0
$805$	0.269759	0.00950775
$806$	−10.3202	−0.363514
$807$	0	0
$808$	−26.6215	−0.936542
$809$	−41.3567	−1.45402	−0.727011	−	0.686625i	$-0.759091\pi$
$809$	−41.3567	−1.45402	−0.727011	+	0.686625i	$0.759091\pi$
$810$	0	0
$811$	−48.3830	−1.69896	−0.849479	−	0.527623i	$-0.823083\pi$
$811$	−48.3830	−1.69896	−0.849479	+	0.527623i	$0.823083\pi$
$812$	26.6130	0.933934
$813$	0	0
$814$	9.17663	0.321641
$815$	1.00376	0.0351602
$816$	0	0
$817$	2.63401	0.0921522
$818$	86.1574	3.01242
$819$	0	0
$820$	0.0766167	0.00267557
$821$	36.0285	1.25740	0.628702	−	0.777647i	$-0.283587\pi$
$821$	36.0285	1.25740	0.628702	+	0.777647i	$0.283587\pi$
$822$	0	0
$823$	37.4334	1.30484	0.652422	−	0.757856i	$-0.273753\pi$
$823$	37.4334	1.30484	0.652422	+	0.757856i	$0.273753\pi$
$824$	−9.82161	−0.342152
$825$	0	0
$826$	−63.1777	−2.19823
$827$	−32.3428	−1.12467	−0.562335	−	0.826909i	$-0.690097\pi$
$827$	−32.3428	−1.12467	−0.562335	+	0.826909i	$0.690097\pi$
$828$	0	0
$829$	16.6445	0.578087	0.289043	−	0.957316i	$-0.406663\pi$
$829$	16.6445	0.578087	0.289043	+	0.957316i	$0.406663\pi$
$830$	1.25858	0.0436860
$831$	0	0
$832$	9.50631	0.329572
$833$	−10.8356	−0.375431
$834$	0	0
$835$	0.174445	0.00603693
$836$	8.21207	0.284020
$837$	0	0
$838$	57.8702	1.99909
$839$	37.2541	1.28615	0.643077	−	0.765802i	$-0.277658\pi$
$839$	37.2541	1.28615	0.643077	+	0.765802i	$0.277658\pi$
$840$	0	0
$841$	−22.7903	−0.785873
$842$	−26.0055	−0.896210
$843$	0	0
$844$	24.5702	0.845741
$845$	−0.711091	−0.0244623
$846$	0	0
$847$	−10.2706	−0.352903
$848$	−0.272911	−0.00937181
$849$	0	0
$850$	19.1542	0.656984
$851$	1.44185	0.0494260
$852$	0	0
$853$	−45.4165	−1.55503	−0.777515	−	0.628864i	$-0.783520\pi$
$853$	−45.4165	−1.55503	−0.777515	+	0.628864i	$0.783520\pi$
$854$	27.8964	0.954594
$855$	0	0
$856$	42.3082	1.44607
$857$	46.3601	1.58363	0.791815	−	0.610761i	$-0.209137\pi$
$857$	46.3601	1.58363	0.791815	+	0.610761i	$0.209137\pi$
$858$	0	0
$859$	−36.7731	−1.25468	−0.627341	−	0.778745i	$-0.715857\pi$
$859$	−36.7731	−1.25468	−0.627341	+	0.778745i	$0.715857\pi$
$860$	−0.584673	−0.0199372
$861$	0	0
$862$	−43.0709	−1.46700
$863$	−4.40567	−0.149971	−0.0749854	−	0.997185i	$-0.523891\pi$
$863$	−4.40567	−0.149971	−0.0749854	+	0.997185i	$0.523891\pi$
$864$	0	0
$865$	−1.44134	−0.0490070
$866$	−7.92411	−0.269272
$867$	0	0
$868$	−67.3924	−2.28745
$869$	53.1016	1.80135
$870$	0	0
$871$	8.47807	0.287269
$872$	24.0161	0.813289
$873$	0	0
$874$	2.17075	0.0734268
$875$	2.07921	0.0702900
$876$	0	0
$877$	14.8414	0.501158	0.250579	−	0.968096i	$-0.419379\pi$
$877$	14.8414	0.501158	0.250579	+	0.968096i	$0.419379\pi$
$878$	70.0567	2.36430
$879$	0	0
$880$	0.269767	0.00909385
$881$	−17.1858	−0.579003	−0.289502	−	0.957178i	$-0.593490\pi$
$881$	−17.1858	−0.579003	−0.289502	+	0.957178i	$0.593490\pi$
$882$	0	0
$883$	−13.2297	−0.445216	−0.222608	−	0.974908i	$-0.571457\pi$
$883$	−13.2297	−0.445216	−0.222608	+	0.974908i	$0.571457\pi$
$884$	3.72644	0.125334
$885$	0	0
$886$	−4.82797	−0.162199
$887$	24.0644	0.808002	0.404001	−	0.914759i	$-0.367619\pi$
$887$	24.0644	0.808002	0.404001	+	0.914759i	$0.367619\pi$
$888$	0	0
$889$	−10.6792	−0.358169
$890$	−0.0558620	−0.00187250
$891$	0	0
$892$	−13.7724	−0.461135
$893$	−5.02284	−0.168083
$894$	0	0
$895$	0.117311	0.00392127
$896$	53.7329	1.79509
$897$	0	0
$898$	−73.8147	−2.46323
$899$	−15.7249	−0.524453
$900$	0	0
$901$	−0.370569	−0.0123454
$902$	−3.78025	−0.125868
$903$	0	0
$904$	−11.7793	−0.391774
$905$	−1.43810	−0.0478041
$906$	0	0
$907$	−19.6429	−0.652233	−0.326117	−	0.945330i	$-0.605740\pi$
$907$	−19.6429	−0.652233	−0.326117	+	0.945330i	$0.605740\pi$
$908$	33.1200	1.09913
$909$	0	0
$910$	0.340154	0.0112760
$911$	−4.93178	−0.163397	−0.0816986	−	0.996657i	$-0.526034\pi$
$911$	−4.93178	−0.163397	−0.0816986	+	0.996657i	$0.526034\pi$
$912$	0	0
$913$	−36.9111	−1.22158
$914$	−58.8582	−1.94686
$915$	0	0
$916$	−18.9195	−0.625118
$917$	15.8318	0.522811
$918$	0	0
$919$	6.26183	0.206559	0.103279	−	0.994652i	$-0.467066\pi$
$919$	6.26183	0.206559	0.103279	+	0.994652i	$0.467066\pi$
$920$	−0.153049	−0.00504589
$921$	0	0
$922$	80.3799	2.64717
$923$	−7.34460	−0.241751
$924$	0	0
$925$	5.55482	0.182641
$926$	15.1616	0.498242
$927$	0	0
$928$	17.3381	0.569152
$929$	−19.8755	−0.652094	−0.326047	−	0.945353i	$-0.605717\pi$
$929$	−19.8755	−0.652094	−0.326047	+	0.945353i	$0.605717\pi$
$930$	0	0
$931$	−4.73095	−0.155051
$932$	51.6775	1.69275
$933$	0	0
$934$	−75.1760	−2.45983
$935$	0.366300	0.0119793
$936$	0	0
$937$	−16.1857	−0.528764	−0.264382	−	0.964418i	$-0.585168\pi$
$937$	−16.1857	−0.528764	−0.264382	+	0.964418i	$0.585168\pi$
$938$	93.1408	3.04116
$939$	0	0
$940$	1.11493	0.0363649
$941$	26.8508	0.875310	0.437655	−	0.899143i	$-0.355809\pi$
$941$	26.8508	0.875310	0.437655	+	0.899143i	$0.355809\pi$
$942$	0	0
$943$	−0.593960	−0.0193420
$944$	−9.92690	−0.323093
$945$	0	0
$946$	28.8476	0.937916
$947$	−18.3250	−0.595481	−0.297741	−	0.954647i	$-0.596233\pi$
$947$	−18.3250	−0.595481	−0.297741	+	0.954647i	$0.596233\pi$
$948$	0	0
$949$	1.54094	0.0500211
$950$	8.36297	0.271331
$951$	0	0
$952$	13.0035	0.421447
$953$	24.6170	0.797423	0.398711	−	0.917076i	$-0.369458\pi$
$953$	24.6170	0.797423	0.398711	+	0.917076i	$0.369458\pi$
$954$	0	0
$955$	−0.0339266	−0.00109784
$956$	2.93097	0.0947944
$957$	0	0
$958$	−18.0822	−0.584210
$959$	−39.3616	−1.27105
$960$	0	0
$961$	8.82019	0.284522
$962$	1.81811	0.0586183
$963$	0	0
$964$	−6.91800	−0.222814
$965$	0.821547	0.0264465
$966$	0	0
$967$	−57.4790	−1.84840	−0.924200	−	0.381908i	$-0.875267\pi$
$967$	−57.4790	−1.84840	−0.924200	+	0.381908i	$0.875267\pi$
$968$	5.82710	0.187290
$969$	0	0
$970$	−0.973981	−0.0312726
$971$	10.4481	0.335295	0.167648	−	0.985847i	$-0.446383\pi$
$971$	10.4481	0.335295	0.167648	+	0.985847i	$0.446383\pi$
$972$	0	0
$973$	−5.82345	−0.186691
$974$	−31.2969	−1.00282
$975$	0	0
$976$	4.38326	0.140305
$977$	5.55962	0.177868	0.0889340	−	0.996038i	$-0.471654\pi$
$977$	5.55962	0.177868	0.0889340	+	0.996038i	$0.471654\pi$
$978$	0	0
$979$	1.63830	0.0523602
$980$	1.05013	0.0335453
$981$	0	0
$982$	67.7342	2.16148
$983$	20.8116	0.663787	0.331894	−	0.943317i	$-0.392312\pi$
$983$	20.8116	0.663787	0.331894	+	0.943317i	$0.392312\pi$
$984$	0	0
$985$	0.264587	0.00843044
$986$	9.55240	0.304210
$987$	0	0
$988$	1.62701	0.0517620
$989$	4.53259	0.144128
$990$	0	0
$991$	56.1881	1.78487	0.892437	−	0.451173i	$-0.148994\pi$
$991$	56.1881	1.78487	0.892437	+	0.451173i	$0.148994\pi$
$992$	−43.9055	−1.39400
$993$	0	0
$994$	−80.6884	−2.55928
$995$	−0.717750	−0.0227542
$996$	0	0
$997$	43.1483	1.36652	0.683261	−	0.730175i	$-0.260561\pi$
$997$	43.1483	1.36652	0.683261	+	0.730175i	$0.260561\pi$
$998$	−34.6969	−1.09831
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.e.1.6		6
3.2	odd	2		717.2.a.d.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.d.1.1	✓	6	3.2	odd	2
2151.2.a.e.1.6		6	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22058 q^{2} +2.93097 q^{4} +0.0570810 q^{5} -3.64374 q^{7} +2.06730 q^{8} +O(q^{10})\)	\(q+2.22058 q^{2} +2.93097 q^{4} +0.0570810 q^{5} -3.64374 q^{7} +2.06730 q^{8} +0.126753 q^{10} -3.71735 q^{11} -0.736497 q^{13} -8.09121 q^{14} -1.27135 q^{16} -1.72628 q^{17} -0.753715 q^{19} +0.167303 q^{20} -8.25467 q^{22} -1.29699 q^{23} -4.99674 q^{25} -1.63545 q^{26} -10.6797 q^{28} -2.49193 q^{29} +6.31032 q^{31} -6.95772 q^{32} -3.83334 q^{34} -0.207988 q^{35} -1.11169 q^{37} -1.67368 q^{38} +0.118003 q^{40} +0.457952 q^{41} -3.49470 q^{43} -10.8955 q^{44} -2.88007 q^{46} +6.66411 q^{47} +6.27684 q^{49} -11.0957 q^{50} -2.15865 q^{52} +0.214663 q^{53} -0.212190 q^{55} -7.53269 q^{56} -5.53352 q^{58} +7.80818 q^{59} -3.44774 q^{61} +14.0126 q^{62} -12.9075 q^{64} -0.0420400 q^{65} -11.5114 q^{67} -5.05968 q^{68} -0.461855 q^{70} +9.97235 q^{71} -2.09226 q^{73} -2.46859 q^{74} -2.20912 q^{76} +13.5451 q^{77} -14.2848 q^{79} -0.0725697 q^{80} +1.01692 q^{82} +9.92940 q^{83} -0.0985378 q^{85} -7.76025 q^{86} -7.68487 q^{88} -0.440716 q^{89} +2.68360 q^{91} -3.80144 q^{92} +14.7982 q^{94} -0.0430228 q^{95} -7.68409 q^{97} +13.9382 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8} - 11 q^{10} + 13 q^{11} - q^{13} - 4 q^{16} - 11 q^{17} - 22 q^{19} + q^{20} - 2 q^{22} + 12 q^{23} - q^{25} - 12 q^{26} - 16 q^{28} - 18 q^{31} - 7 q^{32} - 3 q^{34} + 9 q^{35} - 8 q^{37} + 5 q^{38} - 11 q^{40} - 10 q^{41} - 14 q^{43} + 4 q^{44} - 18 q^{46} + 9 q^{47} + 5 q^{49} - 4 q^{50} - 16 q^{52} + 8 q^{53} - 20 q^{55} - 11 q^{56} - 15 q^{58} + 10 q^{59} - 12 q^{61} + 13 q^{62} - 31 q^{64} + 11 q^{65} - 36 q^{67} - 22 q^{68} + q^{70} + 3 q^{71} - 32 q^{73} - 9 q^{74} - 4 q^{76} - 6 q^{77} - q^{79} + 7 q^{80} + 7 q^{82} + 7 q^{83} - 14 q^{85} - 45 q^{86} - 15 q^{88} - 17 q^{89} - 23 q^{91} + 12 q^{92} + 50 q^{94} - 28 q^{97} - 13 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8 - 11 * q^10 + 13 * q^11 - q^13 - 4 * q^16 - 11 * q^17 - 22 * q^19 + q^20 - 2 * q^22 + 12 * q^23 - q^25 - 12 * q^26 - 16 * q^28 - 18 * q^31 - 7 * q^32 - 3 * q^34 + 9 * q^35 - 8 * q^37 + 5 * q^38 - 11 * q^40 - 10 * q^41 - 14 * q^43 + 4 * q^44 - 18 * q^46 + 9 * q^47 + 5 * q^49 - 4 * q^50 - 16 * q^52 + 8 * q^53 - 20 * q^55 - 11 * q^56 - 15 * q^58 + 10 * q^59 - 12 * q^61 + 13 * q^62 - 31 * q^64 + 11 * q^65 - 36 * q^67 - 22 * q^68 + q^70 + 3 * q^71 - 32 * q^73 - 9 * q^74 - 4 * q^76 - 6 * q^77 - q^79 + 7 * q^80 + 7 * q^82 + 7 * q^83 - 14 * q^85 - 45 * q^86 - 15 * q^88 - 17 * q^89 - 23 * q^91 + 12 * q^92 + 50 * q^94 - 28 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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