LMFDB - Embedded newform 639.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,2,Mod(1,639)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, 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("639.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	639.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:	$5.10244068916$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 213)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.75660$ of defining polynomial
Character	$\chi$	$=$	639.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75660 q^{2} +5.59883 q^{4} +1.08564 q^{5} -0.842236 q^{7} -9.92054 q^{8} +O(q^{10})$	\(q-2.75660 q^{2} +5.59883 q^{4} +1.08564 q^{5} -0.842236 q^{7} -9.92054 q^{8} -2.99267 q^{10} +4.07830 q^{11} -2.99267 q^{13} +2.32171 q^{14} +16.1493 q^{16} +3.88330 q^{17} +7.03724 q^{19} +6.07830 q^{20} -11.2422 q^{22} -4.20500 q^{23} -3.82139 q^{25} +8.24958 q^{26} -4.71554 q^{28} +4.82139 q^{29} -7.09181 q^{31} -24.6760 q^{32} -10.7047 q^{34} -0.914363 q^{35} +4.52053 q^{37} -19.3989 q^{38} -10.7701 q^{40} +2.63723 q^{41} +11.1639 q^{43} +22.8337 q^{44} +11.5915 q^{46} -2.29797 q^{47} -6.29064 q^{49} +10.5340 q^{50} -16.7554 q^{52} -1.10937 q^{53} +4.42756 q^{55} +8.35543 q^{56} -13.2906 q^{58} -3.52405 q^{59} +13.1047 q^{61} +19.5493 q^{62} +35.7232 q^{64} -3.24895 q^{65} +15.4411 q^{67} +21.7419 q^{68} +2.52053 q^{70} -1.00000 q^{71} +4.72553 q^{73} -12.4613 q^{74} +39.4004 q^{76} -3.43489 q^{77} -10.2012 q^{79} +17.5323 q^{80} -7.26979 q^{82} +12.7892 q^{83} +4.21585 q^{85} -30.7745 q^{86} -40.4590 q^{88} +6.77681 q^{89} +2.52053 q^{91} -23.5431 q^{92} +6.33459 q^{94} +7.63989 q^{95} +9.24224 q^{97} +17.3408 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8	$ 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 2 q^{11} + 5 q^{13} - q^{14} + 11 q^{16} + 8 q^{17} + 8 q^{19} + 6 q^{20} - 7 q^{22} + q^{23} - q^{25} + 12 q^{26} - 9 q^{28} + 5 q^{29} + 2 q^{31} - 17 q^{32} - 21 q^{34} - 5 q^{35} + 19 q^{37} - 3 q^{38} - 23 q^{40} + 19 q^{41} + 25 q^{43} + 19 q^{44} + 12 q^{46} - 7 q^{47} - 6 q^{49} + 31 q^{50} - 13 q^{52} + 5 q^{53} + 3 q^{55} + 8 q^{56} - 34 q^{58} - 10 q^{59} + 2 q^{61} - 4 q^{62} + 34 q^{64} + 16 q^{65} + 35 q^{67} + 45 q^{68} + 11 q^{70} - 4 q^{71} + 2 q^{73} - 20 q^{74} + 13 q^{76} - 16 q^{77} - q^{79} + 5 q^{80} - 5 q^{82} - 18 q^{83} + 11 q^{85} - 20 q^{86} - 40 q^{88} + 16 q^{89} + 11 q^{91} - 41 q^{92} - 5 q^{94} + 15 q^{95} - q^{97} + 10 q^{98}+O(q^{100}) $ 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 2 * q^11 + 5 * q^13 - q^14 + 11 * q^16 + 8 * q^17 + 8 * q^19 + 6 * q^20 - 7 * q^22 + q^23 - q^25 + 12 * q^26 - 9 * q^28 + 5 * q^29 + 2 * q^31 - 17 * q^32 - 21 * q^34 - 5 * q^35 + 19 * q^37 - 3 * q^38 - 23 * q^40 + 19 * q^41 + 25 * q^43 + 19 * q^44 + 12 * q^46 - 7 * q^47 - 6 * q^49 + 31 * q^50 - 13 * q^52 + 5 * q^53 + 3 * q^55 + 8 * q^56 - 34 * q^58 - 10 * q^59 + 2 * q^61 - 4 * q^62 + 34 * q^64 + 16 * q^65 + 35 * q^67 + 45 * q^68 + 11 * q^70 - 4 * q^71 + 2 * q^73 - 20 * q^74 + 13 * q^76 - 16 * q^77 - q^79 + 5 * q^80 - 5 * q^82 - 18 * q^83 + 11 * q^85 - 20 * q^86 - 40 * q^88 + 16 * q^89 + 11 * q^91 - 41 * q^92 - 5 * q^94 + 15 * q^95 - q^97 + 10 * 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 $

$2$	−2.75660	−1.94921	−0.974605	−	0.223932i	$-0.928110\pi$
$2$	−2.75660	−1.94921	−0.974605	+	0.223932i	$0.928110\pi$
$3$	0	0
$3$	0	0
$4$	5.59883	2.79942
$5$	1.08564	0.485512	0.242756	−	0.970087i	$-0.421949\pi$
$5$	1.08564	0.485512	0.242756	+	0.970087i	$0.421949\pi$
$6$	0	0
$7$	−0.842236	−0.318335	−0.159168	−	0.987252i	$-0.550881\pi$
$7$	−0.842236	−0.318335	−0.159168	+	0.987252i	$0.550881\pi$
$8$	−9.92054	−3.50744
$9$	0	0
$10$	−2.99267	−0.946364
$11$	4.07830	1.22965	0.614827	−	0.788662i	$-0.289226\pi$
$11$	4.07830	1.22965	0.614827	+	0.788662i	$0.289226\pi$
$12$	0	0
$13$	−2.99267	−0.830016	−0.415008	−	0.909818i	$-0.636221\pi$
$13$	−2.99267	−0.830016	−0.415008	+	0.909818i	$0.636221\pi$
$14$	2.32171	0.620502
$15$	0	0
$16$	16.1493	4.03732
$17$	3.88330	0.941838	0.470919	−	0.882177i	$-0.343922\pi$
$17$	3.88330	0.941838	0.470919	+	0.882177i	$0.343922\pi$
$18$	0	0
$19$	7.03724	1.61445	0.807227	−	0.590241i	$-0.200967\pi$
$19$	7.03724	1.61445	0.807227	+	0.590241i	$0.200967\pi$
$20$	6.07830	1.35915
$21$	0	0
$22$	−11.2422	−2.39685
$23$	−4.20500	−0.876803	−0.438402	−	0.898779i	$-0.644455\pi$
$23$	−4.20500	−0.876803	−0.438402	+	0.898779i	$0.644455\pi$
$24$	0	0
$25$	−3.82139	−0.764278
$26$	8.24958	1.61788
$27$	0	0
$28$	−4.71554	−0.891153
$29$	4.82139	0.895310	0.447655	−	0.894206i	$-0.352259\pi$
$29$	4.82139	0.895310	0.447655	+	0.894206i	$0.352259\pi$
$30$	0	0
$31$	−7.09181	−1.27373	−0.636864	−	0.770976i	$-0.719769\pi$
$31$	−7.09181	−1.27373	−0.636864	+	0.770976i	$0.719769\pi$
$32$	−24.6760	−4.36214
$33$	0	0
$34$	−10.7047	−1.83584
$35$	−0.914363	−0.154555
$36$	0	0
$37$	4.52053	0.743171	0.371585	−	0.928399i	$-0.378814\pi$
$37$	4.52053	0.743171	0.371585	+	0.928399i	$0.378814\pi$
$38$	−19.3989	−3.14691
$39$	0	0
$40$	−10.7701	−1.70290
$41$	2.63723	0.411867	0.205933	−	0.978566i	$-0.433977\pi$
$41$	2.63723	0.411867	0.205933	+	0.978566i	$0.433977\pi$
$42$	0	0
$43$	11.1639	1.70249	0.851243	−	0.524772i	$-0.175850\pi$
$43$	11.1639	1.70249	0.851243	+	0.524772i	$0.175850\pi$
$44$	22.8337	3.44232
$45$	0	0
$46$	11.5915	1.70907
$47$	−2.29797	−0.335194	−0.167597	−	0.985856i	$-0.553601\pi$
$47$	−2.29797	−0.335194	−0.167597	+	0.985856i	$0.553601\pi$
$48$	0	0
$49$	−6.29064	−0.898663
$50$	10.5340	1.48974
$51$	0	0
$52$	−16.7554	−2.32356
$53$	−1.10937	−0.152384	−0.0761918	−	0.997093i	$-0.524276\pi$
$53$	−1.10937	−0.152384	−0.0761918	+	0.997093i	$0.524276\pi$
$54$	0	0
$55$	4.42756	0.597012
$56$	8.35543	1.11654
$57$	0	0
$58$	−13.2906	−1.74515
$59$	−3.52405	−0.458792	−0.229396	−	0.973333i	$-0.573675\pi$
$59$	−3.52405	−0.458792	−0.229396	+	0.973333i	$0.573675\pi$
$60$	0	0
$61$	13.1047	1.67788	0.838942	−	0.544220i	$-0.183174\pi$
$61$	13.1047	1.67788	0.838942	+	0.544220i	$0.183174\pi$
$62$	19.5493	2.48276
$63$	0	0
$64$	35.7232	4.46540
$65$	−3.24895	−0.402983
$66$	0	0
$67$	15.4411	1.88643	0.943213	−	0.332187i	$-0.107787\pi$
$67$	15.4411	1.88643	0.943213	+	0.332187i	$0.107787\pi$
$68$	21.7419	2.63660
$69$	0	0
$70$	2.52053	0.301261
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	0	0
$73$	4.72553	0.553082	0.276541	−	0.961002i	$-0.410812\pi$
$73$	4.72553	0.553082	0.276541	+	0.961002i	$0.410812\pi$
$74$	−12.4613	−1.44860
$75$	0	0
$76$	39.4004	4.51953
$77$	−3.43489	−0.391442
$78$	0	0
$79$	−10.2012	−1.14772	−0.573861	−	0.818952i	$-0.694555\pi$
$79$	−10.2012	−1.14772	−0.573861	+	0.818952i	$0.694555\pi$
$80$	17.5323	1.96017
$81$	0	0
$82$	−7.26979	−0.802815
$83$	12.7892	1.40379	0.701897	−	0.712279i	$-0.252337\pi$
$83$	12.7892	1.40379	0.701897	+	0.712279i	$0.252337\pi$
$84$	0	0
$85$	4.21585	0.457273
$86$	−30.7745	−3.31850
$87$	0	0
$88$	−40.4590	−4.31294
$89$	6.77681	0.718341	0.359170	−	0.933272i	$-0.383060\pi$
$89$	6.77681	0.718341	0.359170	+	0.933272i	$0.383060\pi$
$90$	0	0
$91$	2.52053	0.264223
$92$	−23.5431	−2.45454
$93$	0	0
$94$	6.33459	0.653363
$95$	7.63989	0.783837
$96$	0	0
$97$	9.24224	0.938408	0.469204	−	0.883090i	$-0.344541\pi$
$97$	9.24224	0.938408	0.469204	+	0.883090i	$0.344541\pi$
$98$	17.3408	1.75168
$99$	0	0
$100$	−21.3953	−2.13953
$101$	6.86597	0.683189	0.341595	−	0.939847i	$-0.389033\pi$
$101$	6.86597	0.683189	0.341595	+	0.939847i	$0.389033\pi$
$102$	0	0
$103$	1.60617	0.158260	0.0791302	−	0.996864i	$-0.474786\pi$
$103$	1.60617	0.158260	0.0791302	+	0.996864i	$0.474786\pi$
$104$	29.6889	2.91123
$105$	0	0
$106$	3.05809	0.297028
$107$	−5.17861	−0.500635	−0.250317	−	0.968164i	$-0.580535\pi$
$107$	−5.17861	−0.500635	−0.250317	+	0.968164i	$0.580535\pi$
$108$	0	0
$109$	−0.182125	−0.0174444	−0.00872220	−	0.999962i	$-0.502776\pi$
$109$	−0.182125	−0.0174444	−0.00872220	+	0.999962i	$0.502776\pi$
$110$	−12.2050	−1.16370
$111$	0	0
$112$	−13.6015	−1.28522
$113$	5.23607	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$113$	5.23607	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$114$	0	0
$115$	−4.56511	−0.425698
$116$	26.9942	2.50635
$117$	0	0
$118$	9.71438	0.894281
$119$	−3.27065	−0.299820
$120$	0	0
$121$	5.63256	0.512051
$122$	−36.1244	−3.27055
$123$	0	0
$124$	−39.7059	−3.56569
$125$	−9.57683	−0.856578
$126$	0	0
$127$	−2.22585	−0.197512	−0.0987559	−	0.995112i	$-0.531486\pi$
$127$	−2.22585	−0.197512	−0.0987559	+	0.995112i	$0.531486\pi$
$128$	−49.1226	−4.34187
$129$	0	0
$130$	8.95605	0.785498
$131$	−9.10851	−0.795815	−0.397907	−	0.917426i	$-0.630263\pi$
$131$	−9.10851	−0.795815	−0.397907	+	0.917426i	$0.630263\pi$
$132$	0	0
$133$	−5.92702	−0.513938
$134$	−42.5648	−3.67704
$135$	0	0
$136$	−38.5244	−3.30344
$137$	20.4191	1.74452	0.872259	−	0.489044i	$-0.162654\pi$
$137$	20.4191	1.74452	0.872259	+	0.489044i	$0.162654\pi$
$138$	0	0
$139$	5.69065	0.482674	0.241337	−	0.970441i	$-0.422414\pi$
$139$	5.69065	0.482674	0.241337	+	0.970441i	$0.422414\pi$
$140$	−5.11936	−0.432665
$141$	0	0
$142$	2.75660	0.231329
$143$	−12.2050	−1.02063
$144$	0	0
$145$	5.23428	0.434684
$146$	−13.0264	−1.07807
$147$	0	0
$148$	25.3097	2.08044
$149$	14.8721	1.21837	0.609187	−	0.793027i	$-0.291496\pi$
$149$	14.8721	1.21837	0.609187	+	0.793027i	$0.291496\pi$
$150$	0	0
$151$	−15.7965	−1.28550	−0.642751	−	0.766076i	$-0.722207\pi$
$151$	−15.7965	−1.28550	−0.642751	+	0.766076i	$0.722207\pi$
$152$	−69.8132	−5.66260
$153$	0	0
$154$	9.46862	0.763003
$155$	−7.69914	−0.618410
$156$	0	0
$157$	−19.9085	−1.58887	−0.794437	−	0.607347i	$-0.792234\pi$
$157$	−19.9085	−1.58887	−0.794437	+	0.607347i	$0.792234\pi$
$158$	28.1206	2.23715
$159$	0	0
$160$	−26.7892	−2.11787
$161$	3.54160	0.279117
$162$	0	0
$163$	−5.44396	−0.426404	−0.213202	−	0.977008i	$-0.568389\pi$
$163$	−5.44396	−0.426404	−0.213202	+	0.977008i	$0.568389\pi$
$164$	14.7654	1.15299
$165$	0	0
$166$	−35.2546	−2.73629
$167$	−12.9458	−1.00177	−0.500887	−	0.865513i	$-0.666993\pi$
$167$	−12.9458	−1.00177	−0.500887	+	0.865513i	$0.666993\pi$
$168$	0	0
$169$	−4.04395	−0.311073
$170$	−11.6214	−0.891321
$171$	0	0
$172$	62.5051	4.76597
$173$	9.99947	0.760246	0.380123	−	0.924936i	$-0.375882\pi$
$173$	9.99947	0.760246	0.380123	+	0.924936i	$0.375882\pi$
$174$	0	0
$175$	3.21851	0.243297
$176$	65.8616	4.96451
$177$	0	0
$178$	−18.6810	−1.40020
$179$	0.205002	0.0153225	0.00766127	−	0.999971i	$-0.497561\pi$
$179$	0.205002	0.0153225	0.00766127	+	0.999971i	$0.497561\pi$
$180$	0	0
$181$	9.88566	0.734795	0.367397	−	0.930064i	$-0.380249\pi$
$181$	9.88566	0.734795	0.367397	+	0.930064i	$0.380249\pi$
$182$	−6.94809	−0.515027
$183$	0	0
$184$	41.7159	3.07534
$185$	4.90766	0.360818
$186$	0	0
$187$	15.8373	1.15814
$188$	−12.8660	−0.938347
$189$	0	0
$190$	−21.0601	−1.52786
$191$	−4.37916	−0.316865	−0.158433	−	0.987370i	$-0.550644\pi$
$191$	−4.37916	−0.316865	−0.158433	+	0.987370i	$0.550644\pi$
$192$	0	0
$193$	15.8302	1.13948	0.569742	−	0.821823i	$-0.307043\pi$
$193$	15.8302	1.13948	0.569742	+	0.821823i	$0.307043\pi$
$194$	−25.4772	−1.82915
$195$	0	0
$196$	−35.2202	−2.51573
$197$	5.71617	0.407260	0.203630	−	0.979048i	$-0.434726\pi$
$197$	5.71617	0.407260	0.203630	+	0.979048i	$0.434726\pi$
$198$	0	0
$199$	7.85575	0.556879	0.278440	−	0.960454i	$-0.410183\pi$
$199$	7.85575	0.556879	0.278440	+	0.960454i	$0.410183\pi$
$200$	37.9103	2.68066
$201$	0	0
$202$	−18.9267	−1.33168
$203$	−4.06075	−0.285009
$204$	0	0
$205$	2.86308	0.199966
$206$	−4.42756	−0.308483
$207$	0	0
$208$	−48.3294	−3.35104
$209$	28.7000	1.98522
$210$	0	0
$211$	−18.4191	−1.26802	−0.634010	−	0.773324i	$-0.718592\pi$
$211$	−18.4191	−1.26802	−0.634010	+	0.773324i	$0.718592\pi$
$212$	−6.21118	−0.426585
$213$	0	0
$214$	14.2753	0.975842
$215$	12.1200	0.826577
$216$	0	0
$217$	5.97298	0.405472
$218$	0.502045	0.0340028
$219$	0	0
$220$	24.7892	1.67129
$221$	−11.6214	−0.781741
$222$	0	0
$223$	−20.5775	−1.37797	−0.688984	−	0.724776i	$-0.741943\pi$
$223$	−20.5775	−1.37797	−0.688984	+	0.724776i	$0.741943\pi$
$224$	20.7830	1.38862
$225$	0	0
$226$	−14.4337	−0.960118
$227$	−10.3938	−0.689863	−0.344931	−	0.938628i	$-0.612098\pi$
$227$	−10.3938	−0.689863	−0.344931	+	0.938628i	$0.612098\pi$
$228$	0	0
$229$	−9.68065	−0.639716	−0.319858	−	0.947466i	$-0.603635\pi$
$229$	−9.68065	−0.639716	−0.319858	+	0.947466i	$0.603635\pi$
$230$	12.5842	0.829775
$231$	0	0
$232$	−47.8308	−3.14025
$233$	−1.43904	−0.0942745	−0.0471372	−	0.998888i	$-0.515010\pi$
$233$	−1.43904	−0.0942745	−0.0471372	+	0.998888i	$0.515010\pi$
$234$	0	0
$235$	−2.49477	−0.162741
$236$	−19.7306	−1.28435
$237$	0	0
$238$	9.01587	0.584412
$239$	25.3434	1.63933	0.819665	−	0.572843i	$-0.194160\pi$
$239$	25.3434	1.63933	0.819665	+	0.572843i	$0.194160\pi$
$240$	0	0
$241$	−3.20437	−0.206412	−0.103206	−	0.994660i	$-0.532910\pi$
$241$	−3.20437	−0.206412	−0.103206	+	0.994660i	$0.532910\pi$
$242$	−15.5267	−0.998095
$243$	0	0
$244$	73.3710	4.69710
$245$	−6.82935	−0.436311
$246$	0	0
$247$	−21.0601	−1.34002
$248$	70.3546	4.46752
$249$	0	0
$250$	26.3995	1.66965
$251$	−16.2065	−1.02295	−0.511473	−	0.859299i	$-0.670900\pi$
$251$	−16.2065	−1.02295	−0.511473	+	0.859299i	$0.670900\pi$
$252$	0	0
$253$	−17.1493	−1.07817
$254$	6.13576	0.384992
$255$	0	0
$256$	63.9648	3.99780
$257$	−20.9533	−1.30703	−0.653517	−	0.756912i	$-0.726707\pi$
$257$	−20.9533	−1.30703	−0.653517	+	0.756912i	$0.726707\pi$
$258$	0	0
$259$	−3.80735	−0.236577
$260$	−18.1903	−1.12812
$261$	0	0
$262$	25.1085	1.55121
$263$	−19.7073	−1.21521	−0.607604	−	0.794240i	$-0.707869\pi$
$263$	−19.7073	−1.21521	−0.607604	+	0.794240i	$0.707869\pi$
$264$	0	0
$265$	−1.20437	−0.0739841
$266$	16.3384	1.00177
$267$	0	0
$268$	86.4520	5.28090
$269$	6.18034	0.376822	0.188411	−	0.982090i	$-0.439666\pi$
$269$	6.18034	0.376822	0.188411	+	0.982090i	$0.439666\pi$
$270$	0	0
$271$	−0.144254	−0.00876279	−0.00438140	−	0.999990i	$-0.501395\pi$
$271$	−0.144254	−0.00876279	−0.00438140	+	0.999990i	$0.501395\pi$
$272$	62.7124	3.80250
$273$	0	0
$274$	−56.2872	−3.40043
$275$	−15.5848	−0.939798
$276$	0	0
$277$	−8.84459	−0.531420	−0.265710	−	0.964053i	$-0.585606\pi$
$277$	−8.84459	−0.531420	−0.265710	+	0.964053i	$0.585606\pi$
$278$	−15.6868	−0.940834
$279$	0	0
$280$	9.07097	0.542094
$281$	−22.0551	−1.31570	−0.657850	−	0.753149i	$-0.728534\pi$
$281$	−22.0551	−1.31570	−0.657850	+	0.753149i	$0.728534\pi$
$282$	0	0
$283$	−0.00203240	−0.000120813 0	−6.04067e−5	−	1.00000i	$-0.500019\pi$
$283$	−0.00203240	−0.000120813 0	−6.04067e−5		1.00000i	$0.500019\pi$
$284$	−5.59883	−0.332230
$285$	0	0
$286$	33.6443	1.98943
$287$	−2.22117	−0.131112
$288$	0	0
$289$	−1.92001	−0.112942
$290$	−14.4288	−0.847289
$291$	0	0
$292$	26.4575	1.54831
$293$	−14.5167	−0.848075	−0.424038	−	0.905645i	$-0.639388\pi$
$293$	−14.5167	−0.848075	−0.424038	+	0.905645i	$0.639388\pi$
$294$	0	0
$295$	−3.82584	−0.222749
$296$	−44.8461	−2.60663
$297$	0	0
$298$	−40.9965	−2.37487
$299$	12.5842	0.727761
$300$	0	0
$301$	−9.40267	−0.541961
$302$	43.5446	2.50571
$303$	0	0
$304$	113.646	6.51807
$305$	14.2269	0.814633
$306$	0	0
$307$	15.6419	0.892732	0.446366	−	0.894851i	$-0.352718\pi$
$307$	15.6419	0.892732	0.446366	+	0.894851i	$0.352718\pi$
$308$	−19.2314	−1.09581
$309$	0	0
$310$	21.2234	1.20541
$311$	−10.5680	−0.599256	−0.299628	−	0.954056i	$-0.596863\pi$
$311$	−10.5680	−0.599256	−0.299628	+	0.954056i	$0.596863\pi$
$312$	0	0
$313$	−2.34255	−0.132409	−0.0662044	−	0.997806i	$-0.521089\pi$
$313$	−2.34255	−0.132409	−0.0662044	+	0.997806i	$0.521089\pi$
$314$	54.8798	3.09705
$315$	0	0
$316$	−57.1147	−3.21295
$317$	21.0743	1.18365	0.591824	−	0.806067i	$-0.298408\pi$
$317$	21.0743	1.18365	0.591824	+	0.806067i	$0.298408\pi$
$318$	0	0
$319$	19.6631	1.10092
$320$	38.7825	2.16801
$321$	0	0
$322$	−9.76278	−0.544058
$323$	27.3277	1.52055
$324$	0	0
$325$	11.4361	0.634363
$326$	15.0068	0.831150
$327$	0	0
$328$	−26.1628	−1.44460
$329$	1.93543	0.106704
$330$	0	0
$331$	−31.7985	−1.74781	−0.873903	−	0.486101i	$-0.838419\pi$
$331$	−31.7985	−1.74781	−0.873903	+	0.486101i	$0.838419\pi$
$332$	71.6044	3.92980
$333$	0	0
$334$	35.6863	1.95267
$335$	16.7634	0.915883
$336$	0	0
$337$	−16.7270	−0.911180	−0.455590	−	0.890190i	$-0.650572\pi$
$337$	−16.7270	−0.911180	−0.455590	+	0.890190i	$0.650572\pi$
$338$	11.1475	0.606346
$339$	0	0
$340$	23.6039	1.28010
$341$	−28.9226	−1.56625
$342$	0	0
$343$	11.1939	0.604411
$344$	−110.752	−5.97136
$345$	0	0
$346$	−27.5645	−1.48188
$347$	−14.3830	−0.772119	−0.386059	−	0.922474i	$-0.626164\pi$
$347$	−14.3830	−0.772119	−0.386059	+	0.922474i	$0.626164\pi$
$348$	0	0
$349$	16.7270	0.895378	0.447689	−	0.894189i	$-0.352247\pi$
$349$	16.7270	0.895378	0.447689	+	0.894189i	$0.352247\pi$
$350$	−8.87215	−0.474236
$351$	0	0
$352$	−100.636	−5.36392
$353$	−26.0968	−1.38899	−0.694497	−	0.719495i	$-0.744373\pi$
$353$	−26.0968	−1.38899	−0.694497	+	0.719495i	$0.744373\pi$
$354$	0	0
$355$	−1.08564	−0.0576197
$356$	37.9423	2.01094
$357$	0	0
$358$	−0.565107	−0.0298668
$359$	−3.68829	−0.194660	−0.0973302	−	0.995252i	$-0.531030\pi$
$359$	−3.68829	−0.194660	−0.0973302	+	0.995252i	$0.531030\pi$
$360$	0	0
$361$	30.5228	1.60646
$362$	−27.2508	−1.43227
$363$	0	0
$364$	14.1120	0.739671
$365$	5.13021	0.268528
$366$	0	0
$367$	−12.3801	−0.646236	−0.323118	−	0.946359i	$-0.604731\pi$
$367$	−12.3801	−0.646236	−0.323118	+	0.946359i	$0.604731\pi$
$368$	−67.9077	−3.53993
$369$	0	0
$370$	−13.5284	−0.703310
$371$	0.934351	0.0485091
$372$	0	0
$373$	−2.57181	−0.133163	−0.0665817	−	0.997781i	$-0.521209\pi$
$373$	−2.57181	−0.133163	−0.0665817	+	0.997781i	$0.521209\pi$
$374$	−43.6570	−2.25745
$375$	0	0
$376$	22.7971	1.17567
$377$	−14.4288	−0.743122
$378$	0	0
$379$	−0.434265	−0.0223067	−0.0111533	−	0.999938i	$-0.503550\pi$
$379$	−0.434265	−0.0223067	−0.0111533	+	0.999938i	$0.503550\pi$
$380$	42.7745	2.19429
$381$	0	0
$382$	12.0716	0.617637
$383$	−6.75637	−0.345234	−0.172617	−	0.984989i	$-0.555222\pi$
$383$	−6.75637	−0.345234	−0.172617	+	0.984989i	$0.555222\pi$
$384$	0	0
$385$	−3.72905	−0.190050
$386$	−43.6376	−2.22109
$387$	0	0
$388$	51.7458	2.62699
$389$	−7.90650	−0.400875	−0.200438	−	0.979706i	$-0.564236\pi$
$389$	−7.90650	−0.400875	−0.200438	+	0.979706i	$0.564236\pi$
$390$	0	0
$391$	−16.3293	−0.825807
$392$	62.4065	3.15201
$393$	0	0
$394$	−15.7572	−0.793835
$395$	−11.0748	−0.557233
$396$	0	0
$397$	5.97159	0.299706	0.149853	−	0.988708i	$-0.452120\pi$
$397$	5.97159	0.299706	0.149853	+	0.988708i	$0.452120\pi$
$398$	−21.6551	−1.08547
$399$	0	0
$400$	−61.7127	−3.08563
$401$	0.742559	0.0370816	0.0185408	−	0.999828i	$-0.494098\pi$
$401$	0.742559	0.0370816	0.0185408	+	0.999828i	$0.494098\pi$
$402$	0	0
$403$	21.2234	1.05721
$404$	38.4414	1.91253
$405$	0	0
$406$	11.1939	0.555541
$407$	18.4361	0.913843
$408$	0	0
$409$	−5.67073	−0.280400	−0.140200	−	0.990123i	$-0.544775\pi$
$409$	−5.67073	−0.280400	−0.140200	+	0.990123i	$0.544775\pi$
$410$	−7.89236	−0.389776
$411$	0	0
$412$	8.99267	0.443037
$413$	2.96808	0.146050
$414$	0	0
$415$	13.8844	0.681558
$416$	73.8470	3.62065
$417$	0	0
$418$	−79.1144	−3.86961
$419$	23.5739	1.15166	0.575831	−	0.817569i	$-0.304679\pi$
$419$	23.5739	1.15166	0.575831	+	0.817569i	$0.304679\pi$
$420$	0	0
$421$	23.5094	1.14578	0.572889	−	0.819633i	$-0.305823\pi$
$421$	23.5094	1.14578	0.572889	+	0.819633i	$0.305823\pi$
$422$	50.7740	2.47164
$423$	0	0
$424$	11.0055	0.534477
$425$	−14.8396	−0.719826
$426$	0	0
$427$	−11.0372	−0.534130
$428$	−28.9942	−1.40149
$429$	0	0
$430$	−33.4099	−1.61117
$431$	−38.0085	−1.83081	−0.915403	−	0.402538i	$-0.868128\pi$
$431$	−38.0085	−1.83081	−0.915403	+	0.402538i	$0.868128\pi$
$432$	0	0
$433$	17.4772	0.839899	0.419949	−	0.907548i	$-0.362048\pi$
$433$	17.4772	0.839899	0.419949	+	0.907548i	$0.362048\pi$
$434$	−16.4651	−0.790350
$435$	0	0
$436$	−1.01969	−0.0488342
$437$	−29.5916	−1.41556
$438$	0	0
$439$	3.25362	0.155287	0.0776435	−	0.996981i	$-0.475260\pi$
$439$	3.25362	0.155287	0.0776435	+	0.996981i	$0.475260\pi$
$440$	−43.9238	−2.09398
$441$	0	0
$442$	32.0356	1.52378
$443$	−31.3572	−1.48982	−0.744912	−	0.667163i	$-0.767509\pi$
$443$	−31.3572	−1.48982	−0.744912	+	0.667163i	$0.767509\pi$
$444$	0	0
$445$	7.35716	0.348763
$446$	56.7238	2.68595
$447$	0	0
$448$	−30.0874	−1.42149
$449$	−24.9733	−1.17856	−0.589282	−	0.807927i	$-0.700589\pi$
$449$	−24.9733	−1.17856	−0.589282	+	0.807927i	$0.700589\pi$
$450$	0	0
$451$	10.7554	0.506454
$452$	29.3159	1.37890
$453$	0	0
$454$	28.6516	1.34469
$455$	2.73638	0.128284
$456$	0	0
$457$	10.6068	0.496165	0.248082	−	0.968739i	$-0.420200\pi$
$457$	10.6068	0.496165	0.248082	+	0.968739i	$0.420200\pi$
$458$	26.6857	1.24694
$459$	0	0
$460$	−25.5593	−1.19171
$461$	0.425200	0.0198035	0.00990176	−	0.999951i	$-0.496848\pi$
$461$	0.425200	0.0198035	0.00990176	+	0.999951i	$0.496848\pi$
$462$	0	0
$463$	22.5746	1.04913	0.524565	−	0.851371i	$-0.324228\pi$
$463$	22.5746	1.04913	0.524565	+	0.851371i	$0.324228\pi$
$464$	77.8620	3.61465
$465$	0	0
$466$	3.96685	0.183761
$467$	−23.2413	−1.07548	−0.537740	−	0.843111i	$-0.680722\pi$
$467$	−23.2413	−1.07548	−0.537740	+	0.843111i	$0.680722\pi$
$468$	0	0
$469$	−13.0050	−0.600516
$470$	6.87707	0.317215
$471$	0	0
$472$	34.9604	1.60918
$473$	45.5299	2.09347
$474$	0	0
$475$	−26.8921	−1.23389
$476$	−18.3118	−0.839321
$477$	0	0
$478$	−69.8616	−3.19540
$479$	39.6472	1.81153	0.905763	−	0.423785i	$-0.139299\pi$
$479$	39.6472	1.81153	0.905763	+	0.423785i	$0.139299\pi$
$480$	0	0
$481$	−13.5284	−0.616844
$482$	8.83317	0.402340
$483$	0	0
$484$	31.5358	1.43344
$485$	10.0337	0.455608
$486$	0	0
$487$	−28.8196	−1.30594	−0.652971	−	0.757383i	$-0.726477\pi$
$487$	−28.8196	−1.30594	−0.652971	+	0.757383i	$0.726477\pi$
$488$	−130.006	−5.88508
$489$	0	0
$490$	18.8258	0.850462
$491$	−11.7373	−0.529695	−0.264848	−	0.964290i	$-0.585322\pi$
$491$	−11.7373	−0.529695	−0.264848	+	0.964290i	$0.585322\pi$
$492$	0	0
$493$	18.7229	0.843237
$494$	58.0543	2.61199
$495$	0	0
$496$	−114.528	−5.14244
$497$	0.842236	0.0377794
$498$	0	0
$499$	−7.86338	−0.352013	−0.176007	−	0.984389i	$-0.556318\pi$
$499$	−7.86338	−0.352013	−0.176007	+	0.984389i	$0.556318\pi$
$500$	−53.6191	−2.39792
$501$	0	0
$502$	44.6748	1.99393
$503$	14.8095	0.660322	0.330161	−	0.943925i	$-0.392897\pi$
$503$	14.8095	0.660322	0.330161	+	0.943925i	$0.392897\pi$
$504$	0	0
$505$	7.45395	0.331697
$506$	47.2737	2.10157
$507$	0	0
$508$	−12.4621	−0.552918
$509$	35.3249	1.56575	0.782875	−	0.622179i	$-0.213752\pi$
$509$	35.3249	1.56575	0.782875	+	0.622179i	$0.213752\pi$
$510$	0	0
$511$	−3.98001	−0.176065
$512$	−78.0802	−3.45069
$513$	0	0
$514$	57.7599	2.54768
$515$	1.74372	0.0768373
$516$	0	0
$517$	−9.37183	−0.412173
$518$	10.4953	0.461139
$519$	0	0
$520$	32.2313	1.41344
$521$	−4.33552	−0.189942	−0.0949712	−	0.995480i	$-0.530276\pi$
$521$	−4.33552	−0.189942	−0.0949712	+	0.995480i	$0.530276\pi$
$522$	0	0
$523$	−18.0498	−0.789264	−0.394632	−	0.918839i	$-0.629128\pi$
$523$	−18.0498	−0.789264	−0.394632	+	0.918839i	$0.629128\pi$
$524$	−50.9971	−2.22782
$525$	0	0
$526$	54.3252	2.36869
$527$	−27.5396	−1.19964
$528$	0	0
$529$	−5.31796	−0.231216
$530$	3.31997	0.144210
$531$	0	0
$532$	−33.1844	−1.43873
$533$	−7.89236	−0.341856
$534$	0	0
$535$	−5.62209	−0.243064
$536$	−153.184	−6.61653
$537$	0	0
$538$	−17.0367	−0.734505
$539$	−25.6551	−1.10504
$540$	0	0
$541$	1.83148	0.0787417	0.0393708	−	0.999225i	$-0.487465\pi$
$541$	1.83148	0.0787417	0.0393708	+	0.999225i	$0.487465\pi$
$542$	0.397650	0.0170805
$543$	0	0
$544$	−95.8241	−4.10843
$545$	−0.197722	−0.00846946
$546$	0	0
$547$	3.39991	0.145370	0.0726848	−	0.997355i	$-0.476843\pi$
$547$	3.39991	0.145370	0.0726848	+	0.997355i	$0.476843\pi$
$548$	114.323	4.88364
$549$	0	0
$550$	42.9610	1.83186
$551$	33.9293	1.44544
$552$	0	0
$553$	8.59180	0.365361
$554$	24.3810	1.03585
$555$	0	0
$556$	31.8610	1.35121
$557$	−28.9956	−1.22858	−0.614291	−	0.789080i	$-0.710558\pi$
$557$	−28.9956	−1.22858	−0.614291	+	0.789080i	$0.710558\pi$
$558$	0	0
$559$	−33.4099	−1.41309
$560$	−14.7663	−0.623990
$561$	0	0
$562$	60.7972	2.56457
$563$	−18.1120	−0.763331	−0.381666	−	0.924301i	$-0.624649\pi$
$563$	−18.1120	−0.763331	−0.381666	+	0.924301i	$0.624649\pi$
$564$	0	0
$565$	5.68447	0.239148
$566$	0.00560250	0.000235491 0
$567$	0	0
$568$	9.92054	0.416257
$569$	−8.40619	−0.352406	−0.176203	−	0.984354i	$-0.556381\pi$
$569$	−8.40619	−0.352406	−0.176203	+	0.984354i	$0.556381\pi$
$570$	0	0
$571$	10.8352	0.453441	0.226721	−	0.973960i	$-0.427200\pi$
$571$	10.8352	0.453441	0.226721	+	0.973960i	$0.427200\pi$
$572$	−68.3338	−2.85718
$573$	0	0
$574$	6.12288	0.255564
$575$	16.0690	0.670122
$576$	0	0
$577$	33.5982	1.39871	0.699356	−	0.714774i	$-0.253470\pi$
$577$	33.5982	1.39871	0.699356	+	0.714774i	$0.253470\pi$
$578$	5.29270	0.220147
$579$	0	0
$580$	29.3059	1.21686
$581$	−10.7715	−0.446877
$582$	0	0
$583$	−4.52435	−0.187379
$584$	−46.8798	−1.93990
$585$	0	0
$586$	40.0167	1.65308
$587$	−3.09385	−0.127697	−0.0638484	−	0.997960i	$-0.520337\pi$
$587$	−3.09385	−0.127697	−0.0638484	+	0.997960i	$0.520337\pi$
$588$	0	0
$589$	−49.9068	−2.05637
$590$	10.5463	0.434184
$591$	0	0
$592$	73.0033	3.00042
$593$	14.6666	0.602286	0.301143	−	0.953579i	$-0.402632\pi$
$593$	14.6666	0.602286	0.301143	+	0.953579i	$0.402632\pi$
$594$	0	0
$595$	−3.55074	−0.145566
$596$	83.2667	3.41074
$597$	0	0
$598$	−34.6895	−1.41856
$599$	35.2789	1.44146	0.720728	−	0.693218i	$-0.243808\pi$
$599$	35.2789	1.44146	0.720728	+	0.693218i	$0.243808\pi$
$600$	0	0
$601$	−47.2182	−1.92607	−0.963036	−	0.269373i	$-0.913184\pi$
$601$	−47.2182	−1.92607	−0.963036	+	0.269373i	$0.913184\pi$
$602$	25.9194	1.05640
$603$	0	0
$604$	−88.4420	−3.59865
$605$	6.11492	0.248607
$606$	0	0
$607$	10.0950	0.409744	0.204872	−	0.978789i	$-0.434322\pi$
$607$	10.0950	0.409744	0.204872	+	0.978789i	$0.434322\pi$
$608$	−173.651	−7.04247
$609$	0	0
$610$	−39.2180	−1.58789
$611$	6.87707	0.278216
$612$	0	0
$613$	4.32788	0.174802	0.0874008	−	0.996173i	$-0.472144\pi$
$613$	4.32788	0.174802	0.0874008	+	0.996173i	$0.472144\pi$
$614$	−43.1185	−1.74012
$615$	0	0
$616$	34.0760	1.37296
$617$	7.96363	0.320604	0.160302	−	0.987068i	$-0.448753\pi$
$617$	7.96363	0.320604	0.160302	+	0.987068i	$0.448753\pi$
$618$	0	0
$619$	−6.76426	−0.271879	−0.135939	−	0.990717i	$-0.543405\pi$
$619$	−6.76426	−0.271879	−0.135939	+	0.990717i	$0.543405\pi$
$620$	−43.1062	−1.73119
$621$	0	0
$622$	29.1317	1.16808
$623$	−5.70767	−0.228673
$624$	0	0
$625$	8.70999	0.348400
$626$	6.45747	0.258092
$627$	0	0
$628$	−111.465	−4.44792
$629$	17.5546	0.699946
$630$	0	0
$631$	−22.7642	−0.906228	−0.453114	−	0.891453i	$-0.649687\pi$
$631$	−22.7642	−0.906228	−0.453114	+	0.891453i	$0.649687\pi$
$632$	101.201	4.02557
$633$	0	0
$634$	−58.0933	−2.30718
$635$	−2.41646	−0.0958944
$636$	0	0
$637$	18.8258	0.745905
$638$	−54.2033	−2.14593
$639$	0	0
$640$	−53.3293	−2.10803
$641$	−16.1713	−0.638727	−0.319363	−	0.947632i	$-0.603469\pi$
$641$	−16.1713	−0.638727	−0.319363	+	0.947632i	$0.603469\pi$
$642$	0	0
$643$	2.63138	0.103772	0.0518858	−	0.998653i	$-0.483477\pi$
$643$	2.63138	0.103772	0.0518858	+	0.998653i	$0.483477\pi$
$644$	19.8288	0.781366
$645$	0	0
$646$	−75.3315	−2.96388
$647$	45.0982	1.77299	0.886497	−	0.462734i	$-0.153131\pi$
$647$	45.0982	1.77299	0.886497	+	0.462734i	$0.153131\pi$
$648$	0	0
$649$	−14.3721	−0.564156
$650$	−31.5249	−1.23651
$651$	0	0
$652$	−30.4798	−1.19368
$653$	41.8083	1.63609	0.818043	−	0.575157i	$-0.195059\pi$
$653$	41.8083	1.63609	0.818043	+	0.575157i	$0.195059\pi$
$654$	0	0
$655$	−9.88854	−0.386377
$656$	42.5894	1.66284
$657$	0	0
$658$	−5.33522	−0.207988
$659$	15.2813	0.595275	0.297637	−	0.954679i	$-0.403801\pi$
$659$	15.2813	0.595275	0.297637	+	0.954679i	$0.403801\pi$
$660$	0	0
$661$	34.9700	1.36018	0.680088	−	0.733130i	$-0.261941\pi$
$661$	34.9700	1.36018	0.680088	+	0.733130i	$0.261941\pi$
$662$	87.6558	3.40684
$663$	0	0
$664$	−126.875	−4.92372
$665$	−6.43459	−0.249523
$666$	0	0
$667$	−20.2740	−0.785011
$668$	−72.4812	−2.80438
$669$	0	0
$670$	−46.2100	−1.78525
$671$	53.4449	2.06322
$672$	0	0
$673$	10.8318	0.417535	0.208767	−	0.977965i	$-0.433055\pi$
$673$	10.8318	0.417535	0.208767	+	0.977965i	$0.433055\pi$
$674$	46.1097	1.77608
$675$	0	0
$676$	−22.6414	−0.870823
$677$	−12.0461	−0.462968	−0.231484	−	0.972839i	$-0.574358\pi$
$677$	−12.0461	−0.462968	−0.231484	+	0.972839i	$0.574358\pi$
$678$	0	0
$679$	−7.78415	−0.298728
$680$	−41.8235	−1.60386
$681$	0	0
$682$	79.7279	3.05294
$683$	16.2549	0.621976	0.310988	−	0.950414i	$-0.399340\pi$
$683$	16.2549	0.621976	0.310988	+	0.950414i	$0.399340\pi$
$684$	0	0
$685$	22.1677	0.846984
$686$	−30.8569	−1.17812
$687$	0	0
$688$	180.290	6.87347
$689$	3.31997	0.126481
$690$	0	0
$691$	50.1713	1.90861	0.954303	−	0.298839i	$-0.0965994\pi$
$691$	50.1713	1.90861	0.954303	+	0.298839i	$0.0965994\pi$
$692$	55.9854	2.12824
$693$	0	0
$694$	39.6481	1.50502
$695$	6.17798	0.234344
$696$	0	0
$697$	10.2412	0.387912
$698$	−46.1097	−1.74528
$699$	0	0
$700$	18.0199	0.681089
$701$	20.0631	0.757773	0.378887	−	0.925443i	$-0.376307\pi$
$701$	20.0631	0.757773	0.378887	+	0.925443i	$0.376307\pi$
$702$	0	0
$703$	31.8121	1.19981
$704$	145.690	5.49090
$705$	0	0
$706$	71.9385	2.70744
$707$	−5.78276	−0.217483
$708$	0	0
$709$	0.871289	0.0327220	0.0163610	−	0.999866i	$-0.494792\pi$
$709$	0.871289	0.0327220	0.0163610	+	0.999866i	$0.494792\pi$
$710$	2.99267	0.112313
$711$	0	0
$712$	−67.2297	−2.51954
$713$	29.8211	1.11681
$714$	0	0
$715$	−13.2502	−0.495530
$716$	1.14777	0.0428942
$717$	0	0
$718$	10.1671	0.379434
$719$	−46.0599	−1.71775	−0.858873	−	0.512189i	$-0.828835\pi$
$719$	−46.0599	−1.71775	−0.858873	+	0.512189i	$0.828835\pi$
$720$	0	0
$721$	−1.35277	−0.0503799
$722$	−84.1391	−3.13133
$723$	0	0
$724$	55.3481	2.05700
$725$	−18.4244	−0.684266
$726$	0	0
$727$	−47.8311	−1.77396	−0.886978	−	0.461811i	$-0.847200\pi$
$727$	−47.8311	−1.77396	−0.886978	+	0.461811i	$0.847200\pi$
$728$	−25.0050	−0.926748
$729$	0	0
$730$	−14.1419	−0.523417
$731$	43.3529	1.60346
$732$	0	0
$733$	−17.9340	−0.662409	−0.331205	−	0.943559i	$-0.607455\pi$
$733$	−17.9340	−0.662409	−0.331205	+	0.943559i	$0.607455\pi$
$734$	34.1269	1.25965
$735$	0	0
$736$	103.763	3.82474
$737$	62.9734	2.31965
$738$	0	0
$739$	42.0367	1.54635	0.773173	−	0.634195i	$-0.218668\pi$
$739$	42.0367	1.54635	0.773173	+	0.634195i	$0.218668\pi$
$740$	27.4772	1.01008
$741$	0	0
$742$	−2.57563	−0.0945544
$743$	−45.2481	−1.65999	−0.829997	−	0.557768i	$-0.811658\pi$
$743$	−45.2481	−1.65999	−0.829997	+	0.557768i	$0.811658\pi$
$744$	0	0
$745$	16.1458	0.591535
$746$	7.08945	0.259563
$747$	0	0
$748$	88.6702	3.24210
$749$	4.36161	0.159370
$750$	0	0
$751$	−42.1505	−1.53810	−0.769048	−	0.639191i	$-0.779269\pi$
$751$	−42.1505	−1.53810	−0.769048	+	0.639191i	$0.779269\pi$
$752$	−37.1106	−1.35328
$753$	0	0
$754$	39.7744	1.44850
$755$	−17.1493	−0.624126
$756$	0	0
$757$	−6.07923	−0.220953	−0.110477	−	0.993879i	$-0.535238\pi$
$757$	−6.07923	−0.220953	−0.110477	+	0.993879i	$0.535238\pi$
$758$	1.19709	0.0434804
$759$	0	0
$760$	−75.7919	−2.74926
$761$	−39.7211	−1.43989	−0.719945	−	0.694031i	$-0.755833\pi$
$761$	−39.7211	−1.43989	−0.719945	+	0.694031i	$0.755833\pi$
$762$	0	0
$763$	0.153392	0.00555317
$764$	−24.5182	−0.887038
$765$	0	0
$766$	18.6246	0.672934
$767$	10.5463	0.380805
$768$	0	0
$769$	−12.4754	−0.449873	−0.224936	−	0.974373i	$-0.572217\pi$
$769$	−12.4754	−0.449873	−0.224936	+	0.974373i	$0.572217\pi$
$770$	10.2795	0.370447
$771$	0	0
$772$	88.6308	3.18989
$773$	18.7712	0.675154	0.337577	−	0.941298i	$-0.390393\pi$
$773$	18.7712	0.675154	0.337577	+	0.941298i	$0.390393\pi$
$774$	0	0
$775$	27.1006	0.973482
$776$	−91.6881	−3.29141
$777$	0	0
$778$	21.7950	0.781390
$779$	18.5589	0.664940
$780$	0	0
$781$	−4.07830	−0.145933
$782$	45.0132	1.60967
$783$	0	0
$784$	−101.589	−3.62819
$785$	−21.6134	−0.771417
$786$	0	0
$787$	27.0948	0.965824	0.482912	−	0.875669i	$-0.339579\pi$
$787$	27.0948	0.965824	0.482912	+	0.875669i	$0.339579\pi$
$788$	32.0039	1.14009
$789$	0	0
$790$	30.5287	1.08616
$791$	−4.41000	−0.156802
$792$	0	0
$793$	−39.2180	−1.39267
$794$	−16.4613	−0.584189
$795$	0	0
$796$	43.9830	1.55894
$797$	22.1391	0.784205	0.392103	−	0.919921i	$-0.371748\pi$
$797$	22.1391	0.784205	0.392103	+	0.919921i	$0.371748\pi$
$798$	0	0
$799$	−8.92371	−0.315698
$800$	94.2966	3.33389
$801$	0	0
$802$	−2.04694	−0.0722798
$803$	19.2722	0.680100
$804$	0	0
$805$	3.84490	0.135515
$806$	−58.5045	−2.06073
$807$	0	0
$808$	−68.1141	−2.39625
$809$	31.9156	1.12209	0.561047	−	0.827784i	$-0.310399\pi$
$809$	31.9156	1.12209	0.561047	+	0.827784i	$0.310399\pi$
$810$	0	0
$811$	−30.6587	−1.07657	−0.538285	−	0.842763i	$-0.680928\pi$
$811$	−30.6587	−1.07657	−0.538285	+	0.842763i	$0.680928\pi$
$812$	−22.7355	−0.797858
$813$	0	0
$814$	−50.8209	−1.78127
$815$	−5.91016	−0.207024
$816$	0	0
$817$	78.5634	2.74858
$818$	15.6319	0.546558
$819$	0	0
$820$	16.0299	0.559789
$821$	−2.46832	−0.0861449	−0.0430725	−	0.999072i	$-0.513715\pi$
$821$	−2.46832	−0.0861449	−0.0430725	+	0.999072i	$0.513715\pi$
$822$	0	0
$823$	−7.56719	−0.263776	−0.131888	−	0.991265i	$-0.542104\pi$
$823$	−7.56719	−0.263776	−0.131888	+	0.991265i	$0.542104\pi$
$824$	−15.9340	−0.555089
$825$	0	0
$826$	−8.18180	−0.284681
$827$	−32.9376	−1.14535	−0.572676	−	0.819782i	$-0.694095\pi$
$827$	−32.9376	−1.14535	−0.572676	+	0.819782i	$0.694095\pi$
$828$	0	0
$829$	17.0830	0.593317	0.296658	−	0.954984i	$-0.404128\pi$
$829$	17.0830	0.593317	0.296658	+	0.954984i	$0.404128\pi$
$830$	−38.2737	−1.32850
$831$	0	0
$832$	−106.908	−3.70636
$833$	−24.4284	−0.846394
$834$	0	0
$835$	−14.0544	−0.486373
$836$	160.687	5.55746
$837$	0	0
$838$	−64.9839	−2.24483
$839$	46.4628	1.60407	0.802037	−	0.597275i	$-0.203750\pi$
$839$	46.4628	1.60407	0.802037	+	0.597275i	$0.203750\pi$
$840$	0	0
$841$	−5.75419	−0.198420
$842$	−64.8059	−2.23336
$843$	0	0
$844$	−103.125	−3.54972
$845$	−4.39026	−0.151030
$846$	0	0
$847$	−4.74394	−0.163004
$848$	−17.9155	−0.615221
$849$	0	0
$850$	40.9068	1.40309
$851$	−19.0088	−0.651615
$852$	0	0
$853$	40.9410	1.40179	0.700897	−	0.713262i	$-0.252783\pi$
$853$	40.9410	1.40179	0.700897	+	0.713262i	$0.252783\pi$
$854$	30.4252	1.04113
$855$	0	0
$856$	51.3746	1.75595
$857$	8.83882	0.301928	0.150964	−	0.988539i	$-0.451762\pi$
$857$	8.83882	0.301928	0.150964	+	0.988539i	$0.451762\pi$
$858$	0	0
$859$	−38.4699	−1.31258	−0.656288	−	0.754511i	$-0.727874\pi$
$859$	−38.4699	−1.31258	−0.656288	+	0.754511i	$0.727874\pi$
$860$	67.8578	2.31393
$861$	0	0
$862$	104.774	3.56863
$863$	−11.7414	−0.399682	−0.199841	−	0.979828i	$-0.564043\pi$
$863$	−11.7414	−0.399682	−0.199841	+	0.979828i	$0.564043\pi$
$864$	0	0
$865$	10.8558	0.369108
$866$	−48.1775	−1.63714
$867$	0	0
$868$	33.4417	1.13509
$869$	−41.6035	−1.41130
$870$	0	0
$871$	−46.2100	−1.56577
$872$	1.80678	0.0611852
$873$	0	0
$874$	81.5722	2.75922
$875$	8.06595	0.272679
$876$	0	0
$877$	−11.9255	−0.402696	−0.201348	−	0.979520i	$-0.564532\pi$
$877$	−11.9255	−0.402696	−0.201348	+	0.979520i	$0.564532\pi$
$878$	−8.96893	−0.302687
$879$	0	0
$880$	71.5019	2.41033
$881$	9.77651	0.329379	0.164690	−	0.986345i	$-0.447338\pi$
$881$	9.77651	0.329379	0.164690	+	0.986345i	$0.447338\pi$
$882$	0	0
$883$	18.7859	0.632195	0.316098	−	0.948727i	$-0.397627\pi$
$883$	18.7859	0.632195	0.316098	+	0.948727i	$0.397627\pi$
$884$	−65.0663	−2.18842
$885$	0	0
$886$	86.4391	2.90398
$887$	2.94553	0.0989012	0.0494506	−	0.998777i	$-0.484253\pi$
$887$	2.94553	0.0989012	0.0494506	+	0.998777i	$0.484253\pi$
$888$	0	0
$889$	1.87469	0.0628750
$890$	−20.2807	−0.679812
$891$	0	0
$892$	−115.210	−3.85751
$893$	−16.1714	−0.541155
$894$	0	0
$895$	0.222557	0.00743928
$896$	41.3728	1.38217
$897$	0	0
$898$	68.8414	2.29727
$899$	−34.1924	−1.14038
$900$	0	0
$901$	−4.30801	−0.143521
$902$	−29.6484	−0.987185
$903$	0	0
$904$	−51.9446	−1.72765
$905$	10.7322	0.356752
$906$	0	0
$907$	−10.9945	−0.365065	−0.182532	−	0.983200i	$-0.558429\pi$
$907$	−10.9945	−0.365065	−0.182532	+	0.983200i	$0.558429\pi$
$908$	−58.1933	−1.93121
$909$	0	0
$910$	−7.54311	−0.250052
$911$	−3.98651	−0.132079	−0.0660395	−	0.997817i	$-0.521036\pi$
$911$	−3.98651	−0.132079	−0.0660395	+	0.997817i	$0.521036\pi$
$912$	0	0
$913$	52.1581	1.72618
$914$	−29.2387	−0.967129
$915$	0	0
$916$	−54.2004	−1.79083
$917$	7.67152	0.253336
$918$	0	0
$919$	−24.0118	−0.792077	−0.396039	−	0.918234i	$-0.629615\pi$
$919$	−24.0118	−0.792077	−0.396039	+	0.918234i	$0.629615\pi$
$920$	45.2883	1.49311
$921$	0	0
$922$	−1.17210	−0.0386012
$923$	2.99267	0.0985048
$924$	0	0
$925$	−17.2747	−0.567989
$926$	−62.2290	−2.04497
$927$	0	0
$928$	−118.973	−3.90547
$929$	−43.7824	−1.43645	−0.718227	−	0.695809i	$-0.755046\pi$
$929$	−43.7824	−1.43645	−0.718227	+	0.695809i	$0.755046\pi$
$930$	0	0
$931$	−44.2688	−1.45085
$932$	−8.05693	−0.263914
$933$	0	0
$934$	64.0670	2.09634
$935$	17.1935	0.562288
$936$	0	0
$937$	0.591199	0.0193136	0.00965682	−	0.999953i	$-0.496926\pi$
$937$	0.591199	0.0193136	0.00965682	+	0.999953i	$0.496926\pi$
$938$	35.8496	1.17053
$939$	0	0
$940$	−13.9678	−0.455579
$941$	35.0992	1.14420	0.572100	−	0.820184i	$-0.306129\pi$
$941$	35.0992	1.14420	0.572100	+	0.820184i	$0.306129\pi$
$942$	0	0
$943$	−11.0896	−0.361126
$944$	−56.9108	−1.85229
$945$	0	0
$946$	−125.508	−4.08061
$947$	−36.6341	−1.19045	−0.595224	−	0.803560i	$-0.702937\pi$
$947$	−36.6341	−1.19045	−0.595224	+	0.803560i	$0.702937\pi$
$948$	0	0
$949$	−14.1419	−0.459067
$950$	74.1306	2.40511
$951$	0	0
$952$	32.4466	1.05160
$953$	43.1915	1.39911	0.699556	−	0.714578i	$-0.253381\pi$
$953$	43.1915	1.39911	0.699556	+	0.714578i	$0.253381\pi$
$954$	0	0
$955$	−4.75419	−0.153842
$956$	141.894	4.58917
$957$	0	0
$958$	−109.291	−3.53104
$959$	−17.1977	−0.555342
$960$	0	0
$961$	19.2938	0.622382
$962$	37.2925	1.20236
$963$	0	0
$964$	−17.9408	−0.577833
$965$	17.1859	0.553233
$966$	0	0
$967$	9.54639	0.306991	0.153496	−	0.988149i	$-0.450947\pi$
$967$	9.54639	0.306991	0.153496	+	0.988149i	$0.450947\pi$
$968$	−55.8780	−1.79599
$969$	0	0
$970$	−27.6590	−0.888075
$971$	−20.4085	−0.654940	−0.327470	−	0.944862i	$-0.606196\pi$
$971$	−20.4085	−0.654940	−0.327470	+	0.944862i	$0.606196\pi$
$972$	0	0
$973$	−4.79287	−0.153652
$974$	79.4441	2.54555
$975$	0	0
$976$	211.631	6.77415
$977$	−34.3698	−1.09959	−0.549794	−	0.835300i	$-0.685294\pi$
$977$	−34.3698	−1.09959	−0.549794	+	0.835300i	$0.685294\pi$
$978$	0	0
$979$	27.6379	0.883311
$980$	−38.2364	−1.22142
$981$	0	0
$982$	32.3549	1.03249
$983$	15.8548	0.505690	0.252845	−	0.967507i	$-0.418634\pi$
$983$	15.8548	0.505690	0.252845	+	0.967507i	$0.418634\pi$
$984$	0	0
$985$	6.20568	0.197730
$986$	−51.6115	−1.64364
$987$	0	0
$988$	−117.912	−3.75128
$989$	−46.9444	−1.49274
$990$	0	0
$991$	35.8202	1.13787	0.568934	−	0.822383i	$-0.307356\pi$
$991$	35.8202	1.13787	0.568934	+	0.822383i	$0.307356\pi$
$992$	174.997	5.55618
$993$	0	0
$994$	−2.32171	−0.0736400
$995$	8.52849	0.270371
$996$	0	0
$997$	−40.7951	−1.29199	−0.645996	−	0.763341i	$-0.723558\pi$
$997$	−40.7951	−1.29199	−0.645996	+	0.763341i	$0.723558\pi$
$998$	21.6762	0.686148
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.i.1.1		4
3.2	odd	2		213.2.a.e.1.4	✓	4
12.11	even	2		3408.2.a.w.1.2		4
15.14	odd	2		5325.2.a.w.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
213.2.a.e.1.4	✓	4	3.2	odd	2
639.2.a.i.1.1		4	1.1	even	1	trivial
3408.2.a.w.1.2		4	12.11	even	2
5325.2.a.w.1.1		4	15.14	odd	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:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	639.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75660 q^{2} +5.59883 q^{4} +1.08564 q^{5} -0.842236 q^{7} -9.92054 q^{8} +O(q^{10})\)	\(q-2.75660 q^{2} +5.59883 q^{4} +1.08564 q^{5} -0.842236 q^{7} -9.92054 q^{8} -2.99267 q^{10} +4.07830 q^{11} -2.99267 q^{13} +2.32171 q^{14} +16.1493 q^{16} +3.88330 q^{17} +7.03724 q^{19} +6.07830 q^{20} -11.2422 q^{22} -4.20500 q^{23} -3.82139 q^{25} +8.24958 q^{26} -4.71554 q^{28} +4.82139 q^{29} -7.09181 q^{31} -24.6760 q^{32} -10.7047 q^{34} -0.914363 q^{35} +4.52053 q^{37} -19.3989 q^{38} -10.7701 q^{40} +2.63723 q^{41} +11.1639 q^{43} +22.8337 q^{44} +11.5915 q^{46} -2.29797 q^{47} -6.29064 q^{49} +10.5340 q^{50} -16.7554 q^{52} -1.10937 q^{53} +4.42756 q^{55} +8.35543 q^{56} -13.2906 q^{58} -3.52405 q^{59} +13.1047 q^{61} +19.5493 q^{62} +35.7232 q^{64} -3.24895 q^{65} +15.4411 q^{67} +21.7419 q^{68} +2.52053 q^{70} -1.00000 q^{71} +4.72553 q^{73} -12.4613 q^{74} +39.4004 q^{76} -3.43489 q^{77} -10.2012 q^{79} +17.5323 q^{80} -7.26979 q^{82} +12.7892 q^{83} +4.21585 q^{85} -30.7745 q^{86} -40.4590 q^{88} +6.77681 q^{89} +2.52053 q^{91} -23.5431 q^{92} +6.33459 q^{94} +7.63989 q^{95} +9.24224 q^{97} +17.3408 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8	\( 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 2 q^{11} + 5 q^{13} - q^{14} + 11 q^{16} + 8 q^{17} + 8 q^{19} + 6 q^{20} - 7 q^{22} + q^{23} - q^{25} + 12 q^{26} - 9 q^{28} + 5 q^{29} + 2 q^{31} - 17 q^{32} - 21 q^{34} - 5 q^{35} + 19 q^{37} - 3 q^{38} - 23 q^{40} + 19 q^{41} + 25 q^{43} + 19 q^{44} + 12 q^{46} - 7 q^{47} - 6 q^{49} + 31 q^{50} - 13 q^{52} + 5 q^{53} + 3 q^{55} + 8 q^{56} - 34 q^{58} - 10 q^{59} + 2 q^{61} - 4 q^{62} + 34 q^{64} + 16 q^{65} + 35 q^{67} + 45 q^{68} + 11 q^{70} - 4 q^{71} + 2 q^{73} - 20 q^{74} + 13 q^{76} - 16 q^{77} - q^{79} + 5 q^{80} - 5 q^{82} - 18 q^{83} + 11 q^{85} - 20 q^{86} - 40 q^{88} + 16 q^{89} + 11 q^{91} - 41 q^{92} - 5 q^{94} + 15 q^{95} - q^{97} + 10 q^{98}+O(q^{100}) \) 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 2 * q^11 + 5 * q^13 - q^14 + 11 * q^16 + 8 * q^17 + 8 * q^19 + 6 * q^20 - 7 * q^22 + q^23 - q^25 + 12 * q^26 - 9 * q^28 + 5 * q^29 + 2 * q^31 - 17 * q^32 - 21 * q^34 - 5 * q^35 + 19 * q^37 - 3 * q^38 - 23 * q^40 + 19 * q^41 + 25 * q^43 + 19 * q^44 + 12 * q^46 - 7 * q^47 - 6 * q^49 + 31 * q^50 - 13 * q^52 + 5 * q^53 + 3 * q^55 + 8 * q^56 - 34 * q^58 - 10 * q^59 + 2 * q^61 - 4 * q^62 + 34 * q^64 + 16 * q^65 + 35 * q^67 + 45 * q^68 + 11 * q^70 - 4 * q^71 + 2 * q^73 - 20 * q^74 + 13 * q^76 - 16 * q^77 - q^79 + 5 * q^80 - 5 * q^82 - 18 * q^83 + 11 * q^85 - 20 * q^86 - 40 * q^88 + 16 * q^89 + 11 * q^91 - 41 * q^92 - 5 * q^94 + 15 * q^95 - q^97 + 10 * q^98

Introduction

L-functions

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