LMFDB - Embedded newform 7742.2.a.bp.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7742,2,Mod(1,7742)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7742.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7742, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7742 = 2 \cdot 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7742.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,12,0,12,0,0,0,12,12,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.8201812449$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 205x^{8} - 738x^{6} + 1016x^{4} - 380x^{2} + 8 $ x^12 - 24x^10 + 205x^8 - 738x^6 + 1016x^4 - 380*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.149562$ of defining polynomial
Character	$\chi$	$=$	7742.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.149562 q^{3} +1.00000 q^{4} -1.63746 q^{5} -0.149562 q^{6} +1.00000 q^{8} -2.97763 q^{9} -1.63746 q^{10} +4.31135 q^{11} -0.149562 q^{12} -1.29177 q^{13} +0.244902 q^{15} +1.00000 q^{16} -6.73549 q^{17} -2.97763 q^{18} -4.24063 q^{19} -1.63746 q^{20} +4.31135 q^{22} -3.22991 q^{23} -0.149562 q^{24} -2.31873 q^{25} -1.29177 q^{26} +0.894028 q^{27} +3.58715 q^{29} +0.244902 q^{30} +10.4431 q^{31} +1.00000 q^{32} -0.644815 q^{33} -6.73549 q^{34} -2.97763 q^{36} +2.47930 q^{37} -4.24063 q^{38} +0.193200 q^{39} -1.63746 q^{40} +2.30405 q^{41} +12.5010 q^{43} +4.31135 q^{44} +4.87575 q^{45} -3.22991 q^{46} -4.47801 q^{47} -0.149562 q^{48} -2.31873 q^{50} +1.00737 q^{51} -1.29177 q^{52} +1.27579 q^{53} +0.894028 q^{54} -7.05966 q^{55} +0.634238 q^{57} +3.58715 q^{58} -6.37900 q^{59} +0.244902 q^{60} +1.55958 q^{61} +10.4431 q^{62} +1.00000 q^{64} +2.11522 q^{65} -0.644815 q^{66} -1.37043 q^{67} -6.73549 q^{68} +0.483072 q^{69} -0.700690 q^{71} -2.97763 q^{72} +8.02726 q^{73} +2.47930 q^{74} +0.346794 q^{75} -4.24063 q^{76} +0.193200 q^{78} +1.00000 q^{79} -1.63746 q^{80} +8.79918 q^{81} +2.30405 q^{82} +1.65710 q^{83} +11.0291 q^{85} +12.5010 q^{86} -0.536502 q^{87} +4.31135 q^{88} +4.55511 q^{89} +4.87575 q^{90} -3.22991 q^{92} -1.56190 q^{93} -4.47801 q^{94} +6.94386 q^{95} -0.149562 q^{96} +11.2404 q^{97} -12.8376 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8} + 12 q^{9} + 12 q^{11} + 8 q^{15} + 12 q^{16} + 12 q^{18} + 12 q^{22} + 16 q^{23} + 24 q^{25} + 12 q^{29} + 8 q^{30} + 12 q^{32} + 12 q^{36} + 16 q^{37} + 28 q^{39}+ \cdots + 40 q^{99}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8 + 12 * q^9 + 12 * q^11 + 8 * q^15 + 12 * q^16 + 12 * q^18 + 12 * q^22 + 16 * q^23 + 24 * q^25 + 12 * q^29 + 8 * q^30 + 12 * q^32 + 12 * q^36 + 16 * q^37 + 28 * q^39 - 4 * q^43 + 12 * q^44 + 16 * q^46 + 24 * q^50 + 24 * q^53 + 28 * q^57 + 12 * q^58 + 8 * q^60 + 12 * q^64 - 4 * q^65 + 56 * q^67 + 16 * q^71 + 12 * q^72 + 16 * q^74 + 28 * q^78 + 12 * q^79 + 8 * q^81 - 20 * q^85 - 4 * q^86 + 12 * q^88 + 16 * q^92 + 12 * q^93 + 80 * q^95 + 40 * q^99

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
$3$	−0.149562	−0.0863498	−0.0431749	−	0.999068i	$-0.513747\pi$
$3$	−0.149562	−0.0863498	−0.0431749	+	0.999068i	$0.513747\pi$
$4$	1.00000	0.500000
$5$	−1.63746	−0.732294	−0.366147	−	0.930557i	$-0.619323\pi$
$5$	−1.63746	−0.732294	−0.366147	+	0.930557i	$0.619323\pi$
$6$	−0.149562	−0.0610585
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.97763	−0.992544
$10$	−1.63746	−0.517810
$11$	4.31135	1.29992	0.649961	−	0.759968i	$-0.274785\pi$
$11$	4.31135	1.29992	0.649961	+	0.759968i	$0.274785\pi$
$12$	−0.149562	−0.0431749
$13$	−1.29177	−0.358272	−0.179136	−	0.983824i	$-0.557330\pi$
$13$	−1.29177	−0.358272	−0.179136	+	0.983824i	$0.557330\pi$
$14$	0	0
$15$	0.244902	0.0632334
$16$	1.00000	0.250000
$17$	−6.73549	−1.63360	−0.816798	−	0.576924i	$-0.804253\pi$
$17$	−6.73549	−1.63360	−0.816798	+	0.576924i	$0.804253\pi$
$18$	−2.97763	−0.701834
$19$	−4.24063	−0.972868	−0.486434	−	0.873717i	$-0.661703\pi$
$19$	−4.24063	−0.972868	−0.486434	+	0.873717i	$0.661703\pi$
$20$	−1.63746	−0.366147
$21$	0	0
$22$	4.31135	0.919184
$23$	−3.22991	−0.673482	−0.336741	−	0.941597i	$-0.609325\pi$
$23$	−3.22991	−0.673482	−0.336741	+	0.941597i	$0.609325\pi$
$24$	−0.149562	−0.0305293
$25$	−2.31873	−0.463746
$26$	−1.29177	−0.253337
$27$	0.894028	0.172056
$28$	0	0
$29$	3.58715	0.666116	0.333058	−	0.942906i	$-0.391919\pi$
$29$	3.58715	0.666116	0.333058	+	0.942906i	$0.391919\pi$
$30$	0.244902	0.0447128
$31$	10.4431	1.87564	0.937821	−	0.347118i	$-0.112840\pi$
$31$	10.4431	1.87564	0.937821	+	0.347118i	$0.112840\pi$
$32$	1.00000	0.176777
$33$	−0.644815	−0.112248
$34$	−6.73549	−1.15513
$35$	0	0
$36$	−2.97763	−0.496272
$37$	2.47930	0.407595	0.203798	−	0.979013i	$-0.434672\pi$
$37$	2.47930	0.407595	0.203798	+	0.979013i	$0.434672\pi$
$38$	−4.24063	−0.687921
$39$	0.193200	0.0309367
$40$	−1.63746	−0.258905
$41$	2.30405	0.359832	0.179916	−	0.983682i	$-0.442417\pi$
$41$	2.30405	0.359832	0.179916	+	0.983682i	$0.442417\pi$
$42$	0	0
$43$	12.5010	1.90639	0.953194	−	0.302360i	$-0.0977746\pi$
$43$	12.5010	1.90639	0.953194	+	0.302360i	$0.0977746\pi$
$44$	4.31135	0.649961
$45$	4.87575	0.726834
$46$	−3.22991	−0.476224
$47$	−4.47801	−0.653185	−0.326592	−	0.945165i	$-0.605900\pi$
$47$	−4.47801	−0.653185	−0.326592	+	0.945165i	$0.605900\pi$
$48$	−0.149562	−0.0215874
$49$	0	0
$50$	−2.31873	−0.327918
$51$	1.00737	0.141061
$52$	−1.29177	−0.179136
$53$	1.27579	0.175244	0.0876219	−	0.996154i	$-0.472073\pi$
$53$	1.27579	0.175244	0.0876219	+	0.996154i	$0.472073\pi$
$54$	0.894028	0.121662
$55$	−7.05966	−0.951925
$56$	0	0
$57$	0.634238	0.0840069
$58$	3.58715	0.471015
$59$	−6.37900	−0.830475	−0.415237	−	0.909713i	$-0.636301\pi$
$59$	−6.37900	−0.830475	−0.415237	+	0.909713i	$0.636301\pi$
$60$	0.244902	0.0316167
$61$	1.55958	0.199684	0.0998420	−	0.995003i	$-0.468166\pi$
$61$	1.55958	0.199684	0.0998420	+	0.995003i	$0.468166\pi$
$62$	10.4431	1.32628
$63$	0	0
$64$	1.00000	0.125000
$65$	2.11522	0.262360
$66$	−0.644815	−0.0793713
$67$	−1.37043	−0.167425	−0.0837124	−	0.996490i	$-0.526678\pi$
$67$	−1.37043	−0.167425	−0.0837124	+	0.996490i	$0.526678\pi$
$68$	−6.73549	−0.816798
$69$	0.483072	0.0581550
$70$	0	0
$71$	−0.700690	−0.0831566	−0.0415783	−	0.999135i	$-0.513239\pi$
$71$	−0.700690	−0.0831566	−0.0415783	+	0.999135i	$0.513239\pi$
$72$	−2.97763	−0.350917
$73$	8.02726	0.939519	0.469760	−	0.882794i	$-0.344341\pi$
$73$	8.02726	0.939519	0.469760	+	0.882794i	$0.344341\pi$
$74$	2.47930	0.288213
$75$	0.346794	0.0400443
$76$	−4.24063	−0.486434
$77$	0	0
$78$	0.193200	0.0218756
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−1.63746	−0.183073
$81$	8.79918	0.977687
$82$	2.30405	0.254440
$83$	1.65710	0.181890	0.0909449	−	0.995856i	$-0.471011\pi$
$83$	1.65710	0.181890	0.0909449	+	0.995856i	$0.471011\pi$
$84$	0	0
$85$	11.0291	1.19627
$86$	12.5010	1.34802
$87$	−0.536502	−0.0575190
$88$	4.31135	0.459592
$89$	4.55511	0.482840	0.241420	−	0.970421i	$-0.422387\pi$
$89$	4.55511	0.482840	0.241420	+	0.970421i	$0.422387\pi$
$90$	4.87575	0.513949
$91$	0	0
$92$	−3.22991	−0.336741
$93$	−1.56190	−0.161961
$94$	−4.47801	−0.461871
$95$	6.94386	0.712425
$96$	−0.149562	−0.0152646
$97$	11.2404	1.14129	0.570646	−	0.821196i	$-0.306693\pi$
$97$	11.2404	1.14129	0.570646	+	0.821196i	$0.306693\pi$
$98$	0	0
$99$	−12.8376	−1.29023
$100$	−2.31873	−0.231873
$101$	−9.13815	−0.909280	−0.454640	−	0.890675i	$-0.650232\pi$
$101$	−9.13815	−0.909280	−0.454640	+	0.890675i	$0.650232\pi$
$102$	1.00737	0.0997449
$103$	11.2019	1.10376	0.551879	−	0.833924i	$-0.313911\pi$
$103$	11.2019	1.10376	0.551879	+	0.833924i	$0.313911\pi$
$104$	−1.29177	−0.126668
$105$	0	0
$106$	1.27579	0.123916
$107$	−4.54576	−0.439455	−0.219727	−	0.975561i	$-0.570517\pi$
$107$	−4.54576	−0.439455	−0.219727	+	0.975561i	$0.570517\pi$
$108$	0.894028	0.0860278
$109$	7.50517	0.718865	0.359433	−	0.933171i	$-0.382970\pi$
$109$	7.50517	0.718865	0.359433	+	0.933171i	$0.382970\pi$
$110$	−7.05966	−0.673113
$111$	−0.370810	−0.0351957
$112$	0	0
$113$	14.2326	1.33889	0.669445	−	0.742861i	$-0.266532\pi$
$113$	14.2326	1.33889	0.669445	+	0.742861i	$0.266532\pi$
$114$	0.634238	0.0594018
$115$	5.28884	0.493187
$116$	3.58715	0.333058
$117$	3.84641	0.355601
$118$	−6.37900	−0.587234
$119$	0	0
$120$	0.244902	0.0223564
$121$	7.58777	0.689797
$122$	1.55958	0.141198
$123$	−0.344598	−0.0310714
$124$	10.4431	0.937821
$125$	11.9841	1.07189
$126$	0	0
$127$	3.00737	0.266861	0.133431	−	0.991058i	$-0.457401\pi$
$127$	3.00737	0.266861	0.133431	+	0.991058i	$0.457401\pi$
$128$	1.00000	0.0883883
$129$	−1.86968	−0.164616
$130$	2.11522	0.185517
$131$	3.25012	0.283965	0.141982	−	0.989869i	$-0.454652\pi$
$131$	3.25012	0.283965	0.141982	+	0.989869i	$0.454652\pi$
$132$	−0.644815	−0.0561240
$133$	0	0
$134$	−1.37043	−0.118387
$135$	−1.46393	−0.125995
$136$	−6.73549	−0.577563
$137$	9.17429	0.783813	0.391906	−	0.920005i	$-0.371816\pi$
$137$	9.17429	0.783813	0.391906	+	0.920005i	$0.371816\pi$
$138$	0.483072	0.0411218
$139$	0.547234	0.0464157	0.0232079	−	0.999731i	$-0.492612\pi$
$139$	0.547234	0.0464157	0.0232079	+	0.999731i	$0.492612\pi$
$140$	0	0
$141$	0.669741	0.0564023
$142$	−0.700690	−0.0588006
$143$	−5.56927	−0.465726
$144$	−2.97763	−0.248136
$145$	−5.87381	−0.487793
$146$	8.02726	0.664341
$147$	0	0
$148$	2.47930	0.203798
$149$	3.78328	0.309939	0.154969	−	0.987919i	$-0.450472\pi$
$149$	3.78328	0.309939	0.154969	+	0.987919i	$0.450472\pi$
$150$	0.346794	0.0283156
$151$	10.6845	0.869491	0.434746	−	0.900553i	$-0.356838\pi$
$151$	10.6845	0.869491	0.434746	+	0.900553i	$0.356838\pi$
$152$	−4.24063	−0.343961
$153$	20.0558	1.62142
$154$	0	0
$155$	−17.1002	−1.37352
$156$	0.193200	0.0154684
$157$	−3.15493	−0.251791	−0.125895	−	0.992044i	$-0.540180\pi$
$157$	−3.15493	−0.251791	−0.125895	+	0.992044i	$0.540180\pi$
$158$	1.00000	0.0795557
$159$	−0.190810	−0.0151323
$160$	−1.63746	−0.129452
$161$	0	0
$162$	8.79918	0.691329
$163$	−13.8450	−1.08442	−0.542212	−	0.840242i	$-0.682413\pi$
$163$	−13.8450	−1.08442	−0.542212	+	0.840242i	$0.682413\pi$
$164$	2.30405	0.179916
$165$	1.05586	0.0821985
$166$	1.65710	0.128616
$167$	−15.6896	−1.21410	−0.607050	−	0.794664i	$-0.707647\pi$
$167$	−15.6896	−1.21410	−0.607050	+	0.794664i	$0.707647\pi$
$168$	0	0
$169$	−11.3313	−0.871641
$170$	11.0291	0.845892
$171$	12.6270	0.965614
$172$	12.5010	0.953194
$173$	5.18749	0.394397	0.197199	−	0.980364i	$-0.436816\pi$
$173$	5.18749	0.394397	0.197199	+	0.980364i	$0.436816\pi$
$174$	−0.536502	−0.0406721
$175$	0	0
$176$	4.31135	0.324980
$177$	0.954057	0.0717113
$178$	4.55511	0.341420
$179$	22.3348	1.66938	0.834691	−	0.550719i	$-0.185646\pi$
$179$	22.3348	1.66938	0.834691	+	0.550719i	$0.185646\pi$
$180$	4.87575	0.363417
$181$	3.96844	0.294972	0.147486	−	0.989064i	$-0.452882\pi$
$181$	3.96844	0.294972	0.147486	+	0.989064i	$0.452882\pi$
$182$	0	0
$183$	−0.233255	−0.0172427
$184$	−3.22991	−0.238112
$185$	−4.05976	−0.298479
$186$	−1.56190	−0.114524
$187$	−29.0391	−2.12355
$188$	−4.47801	−0.326592
$189$	0	0
$190$	6.94386	0.503761
$191$	−9.93970	−0.719212	−0.359606	−	0.933104i	$-0.617089\pi$
$191$	−9.93970	−0.719212	−0.359606	+	0.933104i	$0.617089\pi$
$192$	−0.149562	−0.0107937
$193$	7.12634	0.512965	0.256483	−	0.966549i	$-0.417436\pi$
$193$	7.12634	0.512965	0.256483	+	0.966549i	$0.417436\pi$
$194$	11.2404	0.807016
$195$	−0.316356	−0.0226548
$196$	0	0
$197$	−13.9000	−0.990336	−0.495168	−	0.868797i	$-0.664893\pi$
$197$	−13.9000	−0.990336	−0.495168	+	0.868797i	$0.664893\pi$
$198$	−12.8376	−0.912330
$199$	−16.8913	−1.19740	−0.598698	−	0.800975i	$-0.704315\pi$
$199$	−16.8913	−1.19740	−0.598698	+	0.800975i	$0.704315\pi$
$200$	−2.31873	−0.163959
$201$	0.204965	0.0144571
$202$	−9.13815	−0.642958
$203$	0	0
$204$	1.00737	0.0705303
$205$	−3.77278	−0.263503
$206$	11.2019	0.780475
$207$	9.61747	0.668461
$208$	−1.29177	−0.0895680
$209$	−18.2829	−1.26465
$210$	0	0
$211$	−5.45746	−0.375707	−0.187854	−	0.982197i	$-0.560153\pi$
$211$	−5.45746	−0.375707	−0.187854	+	0.982197i	$0.560153\pi$
$212$	1.27579	0.0876219
$213$	0.104797	0.00718055
$214$	−4.54576	−0.310741
$215$	−20.4699	−1.39604
$216$	0.894028	0.0608309
$217$	0	0
$218$	7.50517	0.508315
$219$	−1.20057	−0.0811273
$220$	−7.05966	−0.475962
$221$	8.70069	0.585272
$222$	−0.370810	−0.0248872
$223$	−8.15306	−0.545969	−0.272985	−	0.962018i	$-0.588011\pi$
$223$	−8.15306	−0.545969	−0.272985	+	0.962018i	$0.588011\pi$
$224$	0	0
$225$	6.90432	0.460288
$226$	14.2326	0.946738
$227$	15.3046	1.01580	0.507902	−	0.861415i	$-0.330421\pi$
$227$	15.3046	1.01580	0.507902	+	0.861415i	$0.330421\pi$
$228$	0.634238	0.0420034
$229$	14.2977	0.944819	0.472409	−	0.881379i	$-0.343384\pi$
$229$	14.2977	0.944819	0.472409	+	0.881379i	$0.343384\pi$
$230$	5.28884	0.348736
$231$	0	0
$232$	3.58715	0.235508
$233$	16.7256	1.09573	0.547866	−	0.836566i	$-0.315440\pi$
$233$	16.7256	1.09573	0.547866	+	0.836566i	$0.315440\pi$
$234$	3.84641	0.251448
$235$	7.33256	0.478323
$236$	−6.37900	−0.415237
$237$	−0.149562	−0.00971511
$238$	0	0
$239$	−6.83839	−0.442339	−0.221169	−	0.975235i	$-0.570987\pi$
$239$	−6.83839	−0.442339	−0.221169	+	0.975235i	$0.570987\pi$
$240$	0.244902	0.0158084
$241$	6.57847	0.423757	0.211878	−	0.977296i	$-0.432042\pi$
$241$	6.57847	0.423757	0.211878	+	0.977296i	$0.432042\pi$
$242$	7.58777	0.487760
$243$	−3.99811	−0.256479
$244$	1.55958	0.0998420
$245$	0	0
$246$	−0.344598	−0.0219708
$247$	5.47791	0.348551
$248$	10.4431	0.663140
$249$	−0.247839	−0.0157061
$250$	11.9841	0.757942
$251$	−0.450755	−0.0284514	−0.0142257	−	0.999899i	$-0.504528\pi$
$251$	−0.450755	−0.0284514	−0.0142257	+	0.999899i	$0.504528\pi$
$252$	0	0
$253$	−13.9253	−0.875474
$254$	3.00737	0.188699
$255$	−1.64953	−0.103298
$256$	1.00000	0.0625000
$257$	−27.4267	−1.71083	−0.855414	−	0.517944i	$-0.826697\pi$
$257$	−27.4267	−1.71083	−0.855414	+	0.517944i	$0.826697\pi$
$258$	−1.86968	−0.116401
$259$	0	0
$260$	2.11522	0.131180
$261$	−10.6812	−0.661150
$262$	3.25012	0.200793
$263$	30.2619	1.86603	0.933015	−	0.359837i	$-0.117168\pi$
$263$	30.2619	1.86603	0.933015	+	0.359837i	$0.117168\pi$
$264$	−0.644815	−0.0396856
$265$	−2.08906	−0.128330
$266$	0	0
$267$	−0.681272	−0.0416932
$268$	−1.37043	−0.0837124
$269$	3.83647	0.233914	0.116957	−	0.993137i	$-0.462686\pi$
$269$	3.83647	0.233914	0.116957	+	0.993137i	$0.462686\pi$
$270$	−1.46393	−0.0890922
$271$	−2.19360	−0.133252	−0.0666258	−	0.997778i	$-0.521223\pi$
$271$	−2.19360	−0.133252	−0.0666258	+	0.997778i	$0.521223\pi$
$272$	−6.73549	−0.408399
$273$	0	0
$274$	9.17429	0.554239
$275$	−9.99686	−0.602833
$276$	0.483072	0.0290775
$277$	13.9553	0.838490	0.419245	−	0.907873i	$-0.362295\pi$
$277$	13.9553	0.838490	0.419245	+	0.907873i	$0.362295\pi$
$278$	0.547234	0.0328209
$279$	−31.0958	−1.86166
$280$	0	0
$281$	13.6254	0.812823	0.406412	−	0.913690i	$-0.366780\pi$
$281$	13.6254	0.812823	0.406412	+	0.913690i	$0.366780\pi$
$282$	0.669741	0.0398825
$283$	−2.20731	−0.131211	−0.0656055	−	0.997846i	$-0.520898\pi$
$283$	−2.20731	−0.131211	−0.0656055	+	0.997846i	$0.520898\pi$
$284$	−0.700690	−0.0415783
$285$	−1.03854	−0.0615177
$286$	−5.56927	−0.329318
$287$	0	0
$288$	−2.97763	−0.175459
$289$	28.3668	1.66864
$290$	−5.87381	−0.344922
$291$	−1.68114	−0.0985504
$292$	8.02726	0.469760
$293$	0.387196	0.0226203	0.0113101	−	0.999936i	$-0.496400\pi$
$293$	0.387196	0.0226203	0.0113101	+	0.999936i	$0.496400\pi$
$294$	0	0
$295$	10.4453	0.608152
$296$	2.47930	0.144107
$297$	3.85447	0.223659
$298$	3.78328	0.219160
$299$	4.17229	0.241290
$300$	0.346794	0.0200222
$301$	0	0
$302$	10.6845	0.614823
$303$	1.36672	0.0785161
$304$	−4.24063	−0.243217
$305$	−2.55375	−0.146227
$306$	20.0558	1.14651
$307$	28.7476	1.64071	0.820356	−	0.571853i	$-0.193775\pi$
$307$	28.7476	1.64071	0.820356	+	0.571853i	$0.193775\pi$
$308$	0	0
$309$	−1.67538	−0.0953093
$310$	−17.1002	−0.971227
$311$	−21.0553	−1.19394	−0.596969	−	0.802265i	$-0.703628\pi$
$311$	−21.0553	−1.19394	−0.596969	+	0.802265i	$0.703628\pi$
$312$	0.193200	0.0109378
$313$	15.1952	0.858885	0.429442	−	0.903094i	$-0.358710\pi$
$313$	15.1952	0.858885	0.429442	+	0.903094i	$0.358710\pi$
$314$	−3.15493	−0.178043
$315$	0	0
$316$	1.00000	0.0562544
$317$	11.5366	0.647959	0.323979	−	0.946064i	$-0.394979\pi$
$317$	11.5366	0.647959	0.323979	+	0.946064i	$0.394979\pi$
$318$	−0.190810	−0.0107001
$319$	15.4655	0.865899
$320$	−1.63746	−0.0915367
$321$	0.679873	0.0379468
$322$	0	0
$323$	28.5627	1.58927
$324$	8.79918	0.488843
$325$	2.99526	0.166147
$326$	−13.8450	−0.766803
$327$	−1.12249	−0.0620739
$328$	2.30405	0.127220
$329$	0	0
$330$	1.05586	0.0581231
$331$	−11.3004	−0.621129	−0.310564	−	0.950552i	$-0.600518\pi$
$331$	−11.3004	−0.621129	−0.310564	+	0.950552i	$0.600518\pi$
$332$	1.65710	0.0909449
$333$	−7.38245	−0.404556
$334$	−15.6896	−0.858498
$335$	2.24402	0.122604
$336$	0	0
$337$	−26.5410	−1.44578	−0.722890	−	0.690963i	$-0.757187\pi$
$337$	−26.5410	−1.44578	−0.722890	+	0.690963i	$0.757187\pi$
$338$	−11.3313	−0.616343
$339$	−2.12866	−0.115613
$340$	11.0291	0.598136
$341$	45.0241	2.43819
$342$	12.6270	0.682792
$343$	0	0
$344$	12.5010	0.674010
$345$	−0.791011	−0.0425866
$346$	5.18749	0.278881
$347$	2.28670	0.122757	0.0613783	−	0.998115i	$-0.480450\pi$
$347$	2.28670	0.122757	0.0613783	+	0.998115i	$0.480450\pi$
$348$	−0.536502	−0.0287595
$349$	−13.3598	−0.715131	−0.357566	−	0.933888i	$-0.616393\pi$
$349$	−13.3598	−0.715131	−0.357566	+	0.933888i	$0.616393\pi$
$350$	0	0
$351$	−1.15488	−0.0616427
$352$	4.31135	0.229796
$353$	23.6970	1.26126	0.630632	−	0.776082i	$-0.282796\pi$
$353$	23.6970	1.26126	0.630632	+	0.776082i	$0.282796\pi$
$354$	0.954057	0.0507076
$355$	1.14735	0.0608950
$356$	4.55511	0.241420
$357$	0	0
$358$	22.3348	1.18043
$359$	9.21369	0.486280	0.243140	−	0.969991i	$-0.421823\pi$
$359$	9.21369	0.486280	0.243140	+	0.969991i	$0.421823\pi$
$360$	4.87575	0.256975
$361$	−1.01705	−0.0535287
$362$	3.96844	0.208576
$363$	−1.13484	−0.0595638
$364$	0	0
$365$	−13.1443	−0.688004
$366$	−0.233255	−0.0121924
$367$	6.85127	0.357633	0.178817	−	0.983882i	$-0.442773\pi$
$367$	6.85127	0.357633	0.178817	+	0.983882i	$0.442773\pi$
$368$	−3.22991	−0.168371
$369$	−6.86061	−0.357149
$370$	−4.05976	−0.211057
$371$	0	0
$372$	−1.56190	−0.0809807
$373$	11.0110	0.570129	0.285064	−	0.958508i	$-0.407985\pi$
$373$	11.0110	0.570129	0.285064	+	0.958508i	$0.407985\pi$
$374$	−29.0391	−1.50157
$375$	−1.79237	−0.0925576
$376$	−4.47801	−0.230936
$377$	−4.63376	−0.238651
$378$	0	0
$379$	11.0021	0.565140	0.282570	−	0.959247i	$-0.408813\pi$
$379$	11.0021	0.565140	0.282570	+	0.959247i	$0.408813\pi$
$380$	6.94386	0.356213
$381$	−0.449790	−0.0230434
$382$	−9.93970	−0.508559
$383$	26.2214	1.33985	0.669927	−	0.742427i	$-0.266326\pi$
$383$	26.2214	1.33985	0.669927	+	0.742427i	$0.266326\pi$
$384$	−0.149562	−0.00763231
$385$	0	0
$386$	7.12634	0.362721
$387$	−37.2234	−1.89217
$388$	11.2404	0.570646
$389$	−4.14765	−0.210294	−0.105147	−	0.994457i	$-0.533531\pi$
$389$	−4.14765	−0.210294	−0.105147	+	0.994457i	$0.533531\pi$
$390$	−0.316356	−0.0160193
$391$	21.7550	1.10020
$392$	0	0
$393$	−0.486096	−0.0245203
$394$	−13.9000	−0.700273
$395$	−1.63746	−0.0823895
$396$	−12.8376	−0.645115
$397$	−5.18688	−0.260322	−0.130161	−	0.991493i	$-0.541549\pi$
$397$	−5.18688	−0.260322	−0.130161	+	0.991493i	$0.541549\pi$
$398$	−16.8913	−0.846686
$399$	0	0
$400$	−2.31873	−0.115936
$401$	−0.357823	−0.0178688	−0.00893441	−	0.999960i	$-0.502844\pi$
$401$	−0.357823	−0.0178688	−0.00893441	+	0.999960i	$0.502844\pi$
$402$	0.204965	0.0102227
$403$	−13.4901	−0.671990
$404$	−9.13815	−0.454640
$405$	−14.4083	−0.715954
$406$	0	0
$407$	10.6892	0.529842
$408$	1.00737	0.0498725
$409$	24.7058	1.22162	0.610810	−	0.791777i	$-0.290844\pi$
$409$	24.7058	1.22162	0.610810	+	0.791777i	$0.290844\pi$
$410$	−3.77278	−0.186325
$411$	−1.37213	−0.0676821
$412$	11.2019	0.551879
$413$	0	0
$414$	9.61747	0.472673
$415$	−2.71343	−0.133197
$416$	−1.29177	−0.0633341
$417$	−0.0818454	−0.00400799
$418$	−18.2829	−0.894244
$419$	−2.91163	−0.142243	−0.0711213	−	0.997468i	$-0.522658\pi$
$419$	−2.91163	−0.142243	−0.0711213	+	0.997468i	$0.522658\pi$
$420$	0	0
$421$	7.11481	0.346754	0.173377	−	0.984855i	$-0.444532\pi$
$421$	7.11481	0.346754	0.173377	+	0.984855i	$0.444532\pi$
$422$	−5.45746	−0.265665
$423$	13.3339	0.648314
$424$	1.27579	0.0619580
$425$	15.6178	0.757573
$426$	0.104797	0.00507742
$427$	0	0
$428$	−4.54576	−0.219727
$429$	0.832952	0.0402153
$430$	−20.4699	−0.987146
$431$	23.3100	1.12280	0.561402	−	0.827543i	$-0.310262\pi$
$431$	23.3100	1.12280	0.561402	+	0.827543i	$0.310262\pi$
$432$	0.894028	0.0430139
$433$	32.1346	1.54429	0.772145	−	0.635446i	$-0.219184\pi$
$433$	32.1346	1.54429	0.772145	+	0.635446i	$0.219184\pi$
$434$	0	0
$435$	0.878499	0.0421208
$436$	7.50517	0.359433
$437$	13.6968	0.655209
$438$	−1.20057	−0.0573657
$439$	30.4288	1.45229	0.726143	−	0.687543i	$-0.241311\pi$
$439$	30.4288	1.45229	0.726143	+	0.687543i	$0.241311\pi$
$440$	−7.05966	−0.336556
$441$	0	0
$442$	8.70069	0.413850
$443$	−18.6950	−0.888228	−0.444114	−	0.895970i	$-0.646481\pi$
$443$	−18.6950	−0.888228	−0.444114	+	0.895970i	$0.646481\pi$
$444$	−0.370810	−0.0175979
$445$	−7.45880	−0.353581
$446$	−8.15306	−0.386059
$447$	−0.565836	−0.0267631
$448$	0	0
$449$	20.5950	0.971939	0.485969	−	0.873976i	$-0.338467\pi$
$449$	20.5950	0.971939	0.485969	+	0.873976i	$0.338467\pi$
$450$	6.90432	0.325473
$451$	9.93357	0.467753
$452$	14.2326	0.669445
$453$	−1.59800	−0.0750804
$454$	15.3046	0.718282
$455$	0	0
$456$	0.634238	0.0297009
$457$	−18.5056	−0.865657	−0.432828	−	0.901476i	$-0.642484\pi$
$457$	−18.5056	−0.865657	−0.432828	+	0.901476i	$0.642484\pi$
$458$	14.2977	0.668088
$459$	−6.02171	−0.281070
$460$	5.28884	0.246594
$461$	−30.1943	−1.40629	−0.703143	−	0.711048i	$-0.748221\pi$
$461$	−30.1943	−1.40629	−0.703143	+	0.711048i	$0.748221\pi$
$462$	0	0
$463$	18.8763	0.877256	0.438628	−	0.898669i	$-0.355465\pi$
$463$	18.8763	0.877256	0.438628	+	0.898669i	$0.355465\pi$
$464$	3.58715	0.166529
$465$	2.55754	0.118603
$466$	16.7256	0.774799
$467$	38.3593	1.77506	0.887529	−	0.460753i	$-0.152420\pi$
$467$	38.3593	1.77506	0.887529	+	0.460753i	$0.152420\pi$
$468$	3.84641	0.177800
$469$	0	0
$470$	7.33256	0.338226
$471$	0.471858	0.0217421
$472$	−6.37900	−0.293617
$473$	53.8963	2.47815
$474$	−0.149562	−0.00686962
$475$	9.83287	0.451163
$476$	0	0
$477$	−3.79884	−0.173937
$478$	−6.83839	−0.312781
$479$	−40.5172	−1.85128	−0.925640	−	0.378407i	$-0.876472\pi$
$479$	−40.5172	−1.85128	−0.925640	+	0.378407i	$0.876472\pi$
$480$	0.244902	0.0111782
$481$	−3.20269	−0.146030
$482$	6.57847	0.299641
$483$	0	0
$484$	7.58777	0.344899
$485$	−18.4057	−0.835762
$486$	−3.99811	−0.181358
$487$	13.2186	0.598993	0.299496	−	0.954097i	$-0.403181\pi$
$487$	13.2186	0.598993	0.299496	+	0.954097i	$0.403181\pi$
$488$	1.55958	0.0705990
$489$	2.07069	0.0936397
$490$	0	0
$491$	−4.37529	−0.197454	−0.0987270	−	0.995115i	$-0.531477\pi$
$491$	−4.37529	−0.197454	−0.0987270	+	0.995115i	$0.531477\pi$
$492$	−0.344598	−0.0155357
$493$	−24.1612	−1.08817
$494$	5.47791	0.246463
$495$	21.0211	0.944827
$496$	10.4431	0.468911
$497$	0	0
$498$	−0.247839	−0.0111059
$499$	−14.9041	−0.667198	−0.333599	−	0.942715i	$-0.608263\pi$
$499$	−14.9041	−0.667198	−0.333599	+	0.942715i	$0.608263\pi$
$500$	11.9841	0.535946
$501$	2.34657	0.104837
$502$	−0.450755	−0.0201182
$503$	17.1623	0.765228	0.382614	−	0.923908i	$-0.375024\pi$
$503$	17.1623	0.765228	0.382614	+	0.923908i	$0.375024\pi$
$504$	0	0
$505$	14.9634	0.665860
$506$	−13.9253	−0.619054
$507$	1.69474	0.0752660
$508$	3.00737	0.133431
$509$	−24.1821	−1.07185	−0.535927	−	0.844264i	$-0.680038\pi$
$509$	−24.1821	−1.07185	−0.535927	+	0.844264i	$0.680038\pi$
$510$	−1.64953	−0.0730426
$511$	0	0
$512$	1.00000	0.0441942
$513$	−3.79124	−0.167387
$514$	−27.4267	−1.20974
$515$	−18.3427	−0.808275
$516$	−1.86968	−0.0823081
$517$	−19.3063	−0.849089
$518$	0	0
$519$	−0.775852	−0.0340561
$520$	2.11522	0.0927584
$521$	9.44885	0.413962	0.206981	−	0.978345i	$-0.433636\pi$
$521$	9.44885	0.413962	0.206981	+	0.978345i	$0.433636\pi$
$522$	−10.6812	−0.467503
$523$	−29.5378	−1.29160	−0.645798	−	0.763508i	$-0.723475\pi$
$523$	−29.5378	−1.29160	−0.645798	+	0.763508i	$0.723475\pi$
$524$	3.25012	0.141982
$525$	0	0
$526$	30.2619	1.31948
$527$	−70.3396	−3.06404
$528$	−0.644815	−0.0280620
$529$	−12.5677	−0.546422
$530$	−2.08906	−0.0907429
$531$	18.9943	0.824283
$532$	0	0
$533$	−2.97630	−0.128918
$534$	−0.681272	−0.0294815
$535$	7.44349	0.321810
$536$	−1.37043	−0.0591936
$537$	−3.34044	−0.144151
$538$	3.83647	0.165402
$539$	0	0
$540$	−1.46393	−0.0629977
$541$	−29.2741	−1.25859	−0.629296	−	0.777166i	$-0.716657\pi$
$541$	−29.2741	−1.25859	−0.629296	+	0.777166i	$0.716657\pi$
$542$	−2.19360	−0.0942231
$543$	−0.593528	−0.0254707
$544$	−6.73549	−0.288782
$545$	−12.2894	−0.526421
$546$	0	0
$547$	−44.3494	−1.89625	−0.948123	−	0.317905i	$-0.897021\pi$
$547$	−44.3494	−1.89625	−0.948123	+	0.317905i	$0.897021\pi$
$548$	9.17429	0.391906
$549$	−4.64386	−0.198195
$550$	−9.99686	−0.426267
$551$	−15.2118	−0.648043
$552$	0.483072	0.0205609
$553$	0	0
$554$	13.9553	0.592902
$555$	0.607187	0.0257736
$556$	0.547234	0.0232079
$557$	22.2992	0.944848	0.472424	−	0.881371i	$-0.343379\pi$
$557$	22.2992	0.944848	0.472424	+	0.881371i	$0.343379\pi$
$558$	−31.0958	−1.31639
$559$	−16.1484	−0.683005
$560$	0	0
$561$	4.34315	0.183368
$562$	13.6254	0.574753
$563$	22.5456	0.950183	0.475091	−	0.879936i	$-0.342415\pi$
$563$	22.5456	0.950183	0.475091	+	0.879936i	$0.342415\pi$
$564$	0.669741	0.0282012
$565$	−23.3053	−0.980461
$566$	−2.20731	−0.0927802
$567$	0	0
$568$	−0.700690	−0.0294003
$569$	−39.4077	−1.65205	−0.826027	−	0.563630i	$-0.809404\pi$
$569$	−39.4077	−1.65205	−0.826027	+	0.563630i	$0.809404\pi$
$570$	−1.03854	−0.0434996
$571$	−13.3566	−0.558958	−0.279479	−	0.960152i	$-0.590162\pi$
$571$	−13.3566	−0.558958	−0.279479	+	0.960152i	$0.590162\pi$
$572$	−5.56927	−0.232863
$573$	1.48660	0.0621038
$574$	0	0
$575$	7.48928	0.312324
$576$	−2.97763	−0.124068
$577$	34.8193	1.44955	0.724774	−	0.688987i	$-0.241944\pi$
$577$	34.8193	1.44955	0.724774	+	0.688987i	$0.241944\pi$
$578$	28.3668	1.17990
$579$	−1.06583	−0.0442944
$580$	−5.87381	−0.243897
$581$	0	0
$582$	−1.68114	−0.0696856
$583$	5.50040	0.227803
$584$	8.02726	0.332170
$585$	−6.29834	−0.260404
$586$	0.387196	0.0159949
$587$	37.5665	1.55054	0.775268	−	0.631633i	$-0.217615\pi$
$587$	37.5665	1.55054	0.775268	+	0.631633i	$0.217615\pi$
$588$	0	0
$589$	−44.2855	−1.82475
$590$	10.4453	0.430028
$591$	2.07892	0.0855153
$592$	2.47930	0.101899
$593$	−3.53217	−0.145049	−0.0725245	−	0.997367i	$-0.523106\pi$
$593$	−3.53217	−0.145049	−0.0725245	+	0.997367i	$0.523106\pi$
$594$	3.85447	0.158151
$595$	0	0
$596$	3.78328	0.154969
$597$	2.52631	0.103395
$598$	4.17229	0.170618
$599$	3.56657	0.145726	0.0728630	−	0.997342i	$-0.476786\pi$
$599$	3.56657	0.145726	0.0728630	+	0.997342i	$0.476786\pi$
$600$	0.346794	0.0141578
$601$	−29.5166	−1.20401	−0.602003	−	0.798494i	$-0.705630\pi$
$601$	−29.5166	−1.20401	−0.602003	+	0.798494i	$0.705630\pi$
$602$	0	0
$603$	4.08064	0.166176
$604$	10.6845	0.434746
$605$	−12.4247	−0.505134
$606$	1.36672	0.0555193
$607$	−15.7275	−0.638360	−0.319180	−	0.947694i	$-0.603408\pi$
$607$	−15.7275	−0.638360	−0.319180	+	0.947694i	$0.603408\pi$
$608$	−4.24063	−0.171980
$609$	0	0
$610$	−2.55375	−0.103398
$611$	5.78455	0.234018
$612$	20.0558	0.810708
$613$	18.6076	0.751556	0.375778	−	0.926710i	$-0.377376\pi$
$613$	18.6076	0.751556	0.375778	+	0.926710i	$0.377376\pi$
$614$	28.7476	1.16016
$615$	0.564266	0.0227534
$616$	0	0
$617$	−0.478721	−0.0192726	−0.00963629	−	0.999954i	$-0.503067\pi$
$617$	−0.478721	−0.0192726	−0.00963629	+	0.999954i	$0.503067\pi$
$618$	−1.67538	−0.0673938
$619$	−45.1622	−1.81522	−0.907611	−	0.419812i	$-0.862096\pi$
$619$	−45.1622	−1.81522	−0.907611	+	0.419812i	$0.862096\pi$
$620$	−17.1002	−0.686761
$621$	−2.88763	−0.115876
$622$	−21.0553	−0.844241
$623$	0	0
$624$	0.193200	0.00773418
$625$	−8.02986	−0.321194
$626$	15.1952	0.607323
$627$	2.73442	0.109202
$628$	−3.15493	−0.125895
$629$	−16.6993	−0.665846
$630$	0	0
$631$	0.723984	0.0288214	0.0144107	−	0.999896i	$-0.495413\pi$
$631$	0.723984	0.0288214	0.0144107	+	0.999896i	$0.495413\pi$
$632$	1.00000	0.0397779
$633$	0.816230	0.0324422
$634$	11.5366	0.458176
$635$	−4.92445	−0.195421
$636$	−0.190810	−0.00756613
$637$	0	0
$638$	15.4655	0.612283
$639$	2.08640	0.0825365
$640$	−1.63746	−0.0647262
$641$	−4.18974	−0.165485	−0.0827423	−	0.996571i	$-0.526368\pi$
$641$	−4.18974	−0.165485	−0.0827423	+	0.996571i	$0.526368\pi$
$642$	0.679873	0.0268325
$643$	−0.452029	−0.0178263	−0.00891315	−	0.999960i	$-0.502837\pi$
$643$	−0.452029	−0.0178263	−0.00891315	+	0.999960i	$0.502837\pi$
$644$	0	0
$645$	3.06152	0.120547
$646$	28.5627	1.12379
$647$	1.93802	0.0761914	0.0380957	−	0.999274i	$-0.487871\pi$
$647$	1.93802	0.0761914	0.0380957	+	0.999274i	$0.487871\pi$
$648$	8.79918	0.345664
$649$	−27.5021	−1.07955
$650$	2.99526	0.117484
$651$	0	0
$652$	−13.8450	−0.542212
$653$	−16.5609	−0.648079	−0.324040	−	0.946044i	$-0.605041\pi$
$653$	−16.5609	−0.648079	−0.324040	+	0.946044i	$0.605041\pi$
$654$	−1.12249	−0.0438928
$655$	−5.32195	−0.207946
$656$	2.30405	0.0899580
$657$	−23.9022	−0.932514
$658$	0	0
$659$	−43.2954	−1.68655	−0.843275	−	0.537482i	$-0.819376\pi$
$659$	−43.2954	−1.68655	−0.843275	+	0.537482i	$0.819376\pi$
$660$	1.05586	0.0410993
$661$	−36.2939	−1.41167	−0.705835	−	0.708376i	$-0.749428\pi$
$661$	−36.2939	−1.41167	−0.705835	+	0.708376i	$0.749428\pi$
$662$	−11.3004	−0.439204
$663$	−1.30129	−0.0505381
$664$	1.65710	0.0643078
$665$	0	0
$666$	−7.38245	−0.286064
$667$	−11.5862	−0.448618
$668$	−15.6896	−0.607050
$669$	1.21939	0.0471443
$670$	2.24402	0.0866942
$671$	6.72391	0.259574
$672$	0	0
$673$	14.3005	0.551244	0.275622	−	0.961266i	$-0.411116\pi$
$673$	14.3005	0.551244	0.275622	+	0.961266i	$0.411116\pi$
$674$	−26.5410	−1.02232
$675$	−2.07301	−0.0797901
$676$	−11.3313	−0.435821
$677$	34.3223	1.31911	0.659556	−	0.751655i	$-0.270744\pi$
$677$	34.3223	1.31911	0.659556	+	0.751655i	$0.270744\pi$
$678$	−2.12866	−0.0817506
$679$	0	0
$680$	11.0291	0.422946
$681$	−2.28899	−0.0877145
$682$	45.0241	1.72406
$683$	24.6089	0.941634	0.470817	−	0.882231i	$-0.343959\pi$
$683$	24.6089	0.941634	0.470817	+	0.882231i	$0.343959\pi$
$684$	12.6270	0.482807
$685$	−15.0225	−0.573981
$686$	0	0
$687$	−2.13840	−0.0815849
$688$	12.5010	0.476597
$689$	−1.64803	−0.0627849
$690$	−0.791011	−0.0301133
$691$	−6.74408	−0.256557	−0.128279	−	0.991738i	$-0.540945\pi$
$691$	−6.74408	−0.256557	−0.128279	+	0.991738i	$0.540945\pi$
$692$	5.18749	0.197199
$693$	0	0
$694$	2.28670	0.0868020
$695$	−0.896072	−0.0339900
$696$	−0.536502	−0.0203360
$697$	−15.5189	−0.587820
$698$	−13.3598	−0.505674
$699$	−2.50152	−0.0946162
$700$	0	0
$701$	−31.5348	−1.19105	−0.595526	−	0.803336i	$-0.703056\pi$
$701$	−31.5348	−1.19105	−0.595526	+	0.803336i	$0.703056\pi$
$702$	−1.15488	−0.0435880
$703$	−10.5138	−0.396536
$704$	4.31135	0.162490
$705$	−1.09667	−0.0413031
$706$	23.6970	0.891848
$707$	0	0
$708$	0.954057	0.0358557
$709$	45.1319	1.69496	0.847482	−	0.530824i	$-0.178118\pi$
$709$	45.1319	1.69496	0.847482	+	0.530824i	$0.178118\pi$
$710$	1.14735	0.0430593
$711$	−2.97763	−0.111670
$712$	4.55511	0.170710
$713$	−33.7304	−1.26321
$714$	0	0
$715$	9.11945	0.341048
$716$	22.3348	0.834691
$717$	1.02276	0.0381959
$718$	9.21369	0.343852
$719$	−38.8023	−1.44708	−0.723542	−	0.690281i	$-0.757487\pi$
$719$	−38.8023	−1.44708	−0.723542	+	0.690281i	$0.757487\pi$
$720$	4.87575	0.181708
$721$	0	0
$722$	−1.01705	−0.0378505
$723$	−0.983891	−0.0365913
$724$	3.96844	0.147486
$725$	−8.31762	−0.308909
$726$	−1.13484	−0.0421180
$727$	44.4176	1.64736	0.823678	−	0.567057i	$-0.191918\pi$
$727$	44.4176	1.64736	0.823678	+	0.567057i	$0.191918\pi$
$728$	0	0
$729$	−25.7996	−0.955540
$730$	−13.1443	−0.486493
$731$	−84.2005	−3.11427
$732$	−0.233255	−0.00862134
$733$	44.4466	1.64167	0.820837	−	0.571162i	$-0.193507\pi$
$733$	44.4466	1.64167	0.820837	+	0.571162i	$0.193507\pi$
$734$	6.85127	0.252885
$735$	0	0
$736$	−3.22991	−0.119056
$737$	−5.90841	−0.217639
$738$	−6.86061	−0.252542
$739$	−12.8750	−0.473615	−0.236807	−	0.971557i	$-0.576101\pi$
$739$	−12.8750	−0.473615	−0.236807	+	0.971557i	$0.576101\pi$
$740$	−4.05976	−0.149240
$741$	−0.819288	−0.0300973
$742$	0	0
$743$	43.3246	1.58943	0.794713	−	0.606986i	$-0.207622\pi$
$743$	43.3246	1.58943	0.794713	+	0.606986i	$0.207622\pi$
$744$	−1.56190	−0.0572620
$745$	−6.19497	−0.226966
$746$	11.0110	0.403142
$747$	−4.93422	−0.180534
$748$	−29.0391	−1.06177
$749$	0	0
$750$	−1.79237	−0.0654481
$751$	1.84223	0.0672238	0.0336119	−	0.999435i	$-0.489299\pi$
$751$	1.84223	0.0672238	0.0336119	+	0.999435i	$0.489299\pi$
$752$	−4.47801	−0.163296
$753$	0.0674158	0.00245677
$754$	−4.63376	−0.168752
$755$	−17.4954	−0.636723
$756$	0	0
$757$	2.80471	0.101939	0.0509694	−	0.998700i	$-0.483769\pi$
$757$	2.80471	0.101939	0.0509694	+	0.998700i	$0.483769\pi$
$758$	11.0021	0.399615
$759$	2.08269	0.0755970
$760$	6.94386	0.251880
$761$	51.6534	1.87243	0.936217	−	0.351424i	$-0.114302\pi$
$761$	51.6534	1.87243	0.936217	+	0.351424i	$0.114302\pi$
$762$	−0.449790	−0.0162942
$763$	0	0
$764$	−9.93970	−0.359606
$765$	−32.8406	−1.18735
$766$	26.2214	0.947419
$767$	8.24018	0.297536
$768$	−0.149562	−0.00539686
$769$	49.1469	1.77228	0.886141	−	0.463416i	$-0.153377\pi$
$769$	49.1469	1.77228	0.886141	+	0.463416i	$0.153377\pi$
$770$	0	0
$771$	4.10199	0.147730
$772$	7.12634	0.256483
$773$	−35.9481	−1.29296	−0.646482	−	0.762929i	$-0.723761\pi$
$773$	−35.9481	−1.29296	−0.646482	+	0.762929i	$0.723761\pi$
$774$	−37.2234	−1.33797
$775$	−24.2148	−0.869821
$776$	11.2404	0.403508
$777$	0	0
$778$	−4.14765	−0.148701
$779$	−9.77062	−0.350069
$780$	−0.316356	−0.0113274
$781$	−3.02092	−0.108097
$782$	21.7550	0.777958
$783$	3.20701	0.114609
$784$	0	0
$785$	5.16607	0.184385
$786$	−0.486096	−0.0173385
$787$	−10.4375	−0.372056	−0.186028	−	0.982544i	$-0.559562\pi$
$787$	−10.4375	−0.372056	−0.186028	+	0.982544i	$0.559562\pi$
$788$	−13.9000	−0.495168
$789$	−4.52604	−0.161131
$790$	−1.63746	−0.0582582
$791$	0	0
$792$	−12.8376	−0.456165
$793$	−2.01462	−0.0715412
$794$	−5.18688	−0.184075
$795$	0.312444	0.0110813
$796$	−16.8913	−0.598698
$797$	−11.9748	−0.424169	−0.212084	−	0.977251i	$-0.568025\pi$
$797$	−11.9748	−0.424169	−0.212084	+	0.977251i	$0.568025\pi$
$798$	0	0
$799$	30.1616	1.06704
$800$	−2.31873	−0.0819794
$801$	−13.5634	−0.479240
$802$	−0.357823	−0.0126352
$803$	34.6083	1.22130
$804$	0.204965	0.00722854
$805$	0	0
$806$	−13.4901	−0.475169
$807$	−0.573791	−0.0201984
$808$	−9.13815	−0.321479
$809$	−14.6377	−0.514633	−0.257316	−	0.966327i	$-0.582838\pi$
$809$	−14.6377	−0.514633	−0.257316	+	0.966327i	$0.582838\pi$
$810$	−14.4083	−0.506256
$811$	19.8307	0.696350	0.348175	−	0.937430i	$-0.386801\pi$
$811$	19.8307	0.696350	0.348175	+	0.937430i	$0.386801\pi$
$812$	0	0
$813$	0.328079	0.0115062
$814$	10.6892	0.374655
$815$	22.6706	0.794117
$816$	1.00737	0.0352652
$817$	−53.0122	−1.85466
$818$	24.7058	0.863816
$819$	0	0
$820$	−3.77278	−0.131751
$821$	2.40469	0.0839242	0.0419621	−	0.999119i	$-0.486639\pi$
$821$	2.40469	0.0839242	0.0419621	+	0.999119i	$0.486639\pi$
$822$	−1.37213	−0.0478584
$823$	−18.5096	−0.645205	−0.322602	−	0.946535i	$-0.604558\pi$
$823$	−18.5096	−0.645205	−0.322602	+	0.946535i	$0.604558\pi$
$824$	11.2019	0.390237
$825$	1.49515	0.0520545
$826$	0	0
$827$	−9.16475	−0.318690	−0.159345	−	0.987223i	$-0.550938\pi$
$827$	−9.16475	−0.318690	−0.159345	+	0.987223i	$0.550938\pi$
$828$	9.61747	0.334230
$829$	42.3085	1.46943	0.734717	−	0.678374i	$-0.237315\pi$
$829$	42.3085	1.46943	0.734717	+	0.678374i	$0.237315\pi$
$830$	−2.71343	−0.0941844
$831$	−2.08718	−0.0724035
$832$	−1.29177	−0.0447840
$833$	0	0
$834$	−0.0818454	−0.00283408
$835$	25.6911	0.889078
$836$	−18.2829	−0.632326
$837$	9.33645	0.322715
$838$	−2.91163	−0.100581
$839$	1.63763	0.0565373	0.0282686	−	0.999600i	$-0.491001\pi$
$839$	1.63763	0.0565373	0.0282686	+	0.999600i	$0.491001\pi$
$840$	0	0
$841$	−16.1324	−0.556289
$842$	7.11481	0.245192
$843$	−2.03784	−0.0701871
$844$	−5.45746	−0.187854
$845$	18.5546	0.638298
$846$	13.3339	0.458427
$847$	0	0
$848$	1.27579	0.0438109
$849$	0.330130	0.0113300
$850$	15.6178	0.535685
$851$	−8.00793	−0.274508
$852$	0.104797	0.00359027
$853$	28.2165	0.966115	0.483058	−	0.875589i	$-0.339526\pi$
$853$	28.2165	0.966115	0.483058	+	0.875589i	$0.339526\pi$
$854$	0	0
$855$	−20.6763	−0.707113
$856$	−4.54576	−0.155371
$857$	0.731170	0.0249763	0.0124881	−	0.999922i	$-0.496025\pi$
$857$	0.731170	0.0249763	0.0124881	+	0.999922i	$0.496025\pi$
$858$	0.832952	0.0284365
$859$	51.4071	1.75399	0.876994	−	0.480502i	$-0.159545\pi$
$859$	51.4071	1.75399	0.876994	+	0.480502i	$0.159545\pi$
$860$	−20.4699	−0.698018
$861$	0	0
$862$	23.3100	0.793942
$863$	−36.8177	−1.25329	−0.626644	−	0.779306i	$-0.715572\pi$
$863$	−36.8177	−1.25329	−0.626644	+	0.779306i	$0.715572\pi$
$864$	0.894028	0.0304154
$865$	−8.49429	−0.288815
$866$	32.1346	1.09198
$867$	−4.24260	−0.144086
$868$	0	0
$869$	4.31135	0.146253
$870$	0.878499	0.0297839
$871$	1.77028	0.0599836
$872$	7.50517	0.254157
$873$	−33.4699	−1.13278
$874$	13.6968	0.463303
$875$	0	0
$876$	−1.20057	−0.0405636
$877$	−45.4799	−1.53575	−0.767874	−	0.640601i	$-0.778685\pi$
$877$	−45.4799	−1.53575	−0.767874	+	0.640601i	$0.778685\pi$
$878$	30.4288	1.02692
$879$	−0.0579099	−0.00195325
$880$	−7.05966	−0.237981
$881$	47.5733	1.60278	0.801392	−	0.598139i	$-0.204093\pi$
$881$	47.5733	1.60278	0.801392	+	0.598139i	$0.204093\pi$
$882$	0	0
$883$	−2.49964	−0.0841197	−0.0420599	−	0.999115i	$-0.513392\pi$
$883$	−2.49964	−0.0841197	−0.0420599	+	0.999115i	$0.513392\pi$
$884$	8.70069	0.292636
$885$	−1.56223	−0.0525138
$886$	−18.6950	−0.628072
$887$	−40.1880	−1.34938	−0.674691	−	0.738101i	$-0.735723\pi$
$887$	−40.1880	−1.34938	−0.674691	+	0.738101i	$0.735723\pi$
$888$	−0.370810	−0.0124436
$889$	0	0
$890$	−7.45880	−0.250020
$891$	37.9364	1.27092
$892$	−8.15306	−0.272985
$893$	18.9896	0.635462
$894$	−0.565836	−0.0189244
$895$	−36.5723	−1.22248
$896$	0	0
$897$	−0.624017	−0.0208353
$898$	20.5950	0.687265
$899$	37.4611	1.24940
$900$	6.90432	0.230144
$901$	−8.59309	−0.286277
$902$	9.93357	0.330752
$903$	0	0
$904$	14.2326	0.473369
$905$	−6.49815	−0.216006
$906$	−1.59800	−0.0530899
$907$	41.7973	1.38786	0.693928	−	0.720044i	$-0.255879\pi$
$907$	41.7973	1.38786	0.693928	+	0.720044i	$0.255879\pi$
$908$	15.3046	0.507902
$909$	27.2101	0.902500
$910$	0	0
$911$	29.4658	0.976246	0.488123	−	0.872775i	$-0.337682\pi$
$911$	29.4658	0.976246	0.488123	+	0.872775i	$0.337682\pi$
$912$	0.634238	0.0210017
$913$	7.14432	0.236443
$914$	−18.5056	−0.612112
$915$	0.381945	0.0126267
$916$	14.2977	0.472409
$917$	0	0
$918$	−6.02171	−0.198746
$919$	−31.1446	−1.02736	−0.513682	−	0.857980i	$-0.671719\pi$
$919$	−31.1446	−1.02736	−0.513682	+	0.857980i	$0.671719\pi$
$920$	5.28884	0.174368
$921$	−4.29956	−0.141675
$922$	−30.1943	−0.994395
$923$	0.905128	0.0297927
$924$	0	0
$925$	−5.74883	−0.189020
$926$	18.8763	0.620314
$927$	−33.3552	−1.09553
$928$	3.58715	0.117754
$929$	−34.6550	−1.13699	−0.568497	−	0.822685i	$-0.692475\pi$
$929$	−34.6550	−1.13699	−0.568497	+	0.822685i	$0.692475\pi$
$930$	2.55754	0.0838652
$931$	0	0
$932$	16.7256	0.547866
$933$	3.14908	0.103096
$934$	38.3593	1.25516
$935$	47.5503	1.55506
$936$	3.84641	0.125724
$937$	−1.48968	−0.0486658	−0.0243329	−	0.999704i	$-0.507746\pi$
$937$	−1.48968	−0.0486658	−0.0243329	+	0.999704i	$0.507746\pi$
$938$	0	0
$939$	−2.27263	−0.0741645
$940$	7.33256	0.239162
$941$	−6.61724	−0.215716	−0.107858	−	0.994166i	$-0.534399\pi$
$941$	−6.61724	−0.215716	−0.107858	+	0.994166i	$0.534399\pi$
$942$	0.471858	0.0153740
$943$	−7.44186	−0.242340
$944$	−6.37900	−0.207619
$945$	0	0
$946$	53.8963	1.75232
$947$	2.24547	0.0729680	0.0364840	−	0.999334i	$-0.488384\pi$
$947$	2.24547	0.0729680	0.0364840	+	0.999334i	$0.488384\pi$
$948$	−0.149562	−0.00485755
$949$	−10.3694	−0.336603
$950$	9.83287	0.319020
$951$	−1.72544	−0.0559511
$952$	0	0
$953$	40.3823	1.30811	0.654056	−	0.756446i	$-0.273066\pi$
$953$	40.3823	1.30811	0.654056	+	0.756446i	$0.273066\pi$
$954$	−3.79884	−0.122992
$955$	16.2759	0.526674
$956$	−6.83839	−0.221169
$957$	−2.31305	−0.0747702
$958$	−40.5172	−1.30905
$959$	0	0
$960$	0.244902	0.00790418
$961$	78.0591	2.51804
$962$	−3.20269	−0.103259
$963$	13.5356	0.436178
$964$	6.57847	0.211878
$965$	−11.6691	−0.375641
$966$	0	0
$967$	23.2845	0.748780	0.374390	−	0.927271i	$-0.377852\pi$
$967$	23.2845	0.748780	0.374390	+	0.927271i	$0.377852\pi$
$968$	7.58777	0.243880
$969$	−4.27190	−0.137233
$970$	−18.4057	−0.590973
$971$	−44.3411	−1.42297	−0.711487	−	0.702700i	$-0.751978\pi$
$971$	−44.3411	−1.42297	−0.711487	+	0.702700i	$0.751978\pi$
$972$	−3.99811	−0.128239
$973$	0	0
$974$	13.2186	0.423552
$975$	−0.447977	−0.0143468
$976$	1.55958	0.0499210
$977$	−48.1226	−1.53958	−0.769789	−	0.638298i	$-0.779639\pi$
$977$	−48.1226	−1.53958	−0.769789	+	0.638298i	$0.779639\pi$
$978$	2.07069	0.0662133
$979$	19.6387	0.627655
$980$	0	0
$981$	−22.3476	−0.713505
$982$	−4.37529	−0.139621
$983$	−35.1556	−1.12129	−0.560644	−	0.828057i	$-0.689446\pi$
$983$	−35.1556	−1.12129	−0.560644	+	0.828057i	$0.689446\pi$
$984$	−0.344598	−0.0109854
$985$	22.7607	0.725217
$986$	−24.1612	−0.769449
$987$	0	0
$988$	5.47791	0.174276
$989$	−40.3771	−1.28392
$990$	21.0211	0.668094
$991$	6.75190	0.214481	0.107241	−	0.994233i	$-0.465798\pi$
$991$	6.75190	0.214481	0.107241	+	0.994233i	$0.465798\pi$
$992$	10.4431	0.331570
$993$	1.69012	0.0536343
$994$	0	0
$995$	27.6589	0.876845
$996$	−0.247839	−0.00785307
$997$	−10.8209	−0.342702	−0.171351	−	0.985210i	$-0.554813\pi$
$997$	−10.8209	−0.342702	−0.171351	+	0.985210i	$0.554813\pi$
$998$	−14.9041	−0.471780
$999$	2.21657	0.0701291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7742.2.a.bp.1.6	✓	12
7.6	odd	2	inner	7742.2.a.bp.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7742.2.a.bp.1.6	✓	12	1.1	even	1	trivial
7742.2.a.bp.1.7	yes	12	7.6	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7742 = 2 \cdot 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7742.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.149562 q^{3} +1.00000 q^{4} -1.63746 q^{5} -0.149562 q^{6} +1.00000 q^{8} -2.97763 q^{9} -1.63746 q^{10} +4.31135 q^{11} -0.149562 q^{12} -1.29177 q^{13} +0.244902 q^{15} +1.00000 q^{16} -6.73549 q^{17} -2.97763 q^{18} -4.24063 q^{19} -1.63746 q^{20} +4.31135 q^{22} -3.22991 q^{23} -0.149562 q^{24} -2.31873 q^{25} -1.29177 q^{26} +0.894028 q^{27} +3.58715 q^{29} +0.244902 q^{30} +10.4431 q^{31} +1.00000 q^{32} -0.644815 q^{33} -6.73549 q^{34} -2.97763 q^{36} +2.47930 q^{37} -4.24063 q^{38} +0.193200 q^{39} -1.63746 q^{40} +2.30405 q^{41} +12.5010 q^{43} +4.31135 q^{44} +4.87575 q^{45} -3.22991 q^{46} -4.47801 q^{47} -0.149562 q^{48} -2.31873 q^{50} +1.00737 q^{51} -1.29177 q^{52} +1.27579 q^{53} +0.894028 q^{54} -7.05966 q^{55} +0.634238 q^{57} +3.58715 q^{58} -6.37900 q^{59} +0.244902 q^{60} +1.55958 q^{61} +10.4431 q^{62} +1.00000 q^{64} +2.11522 q^{65} -0.644815 q^{66} -1.37043 q^{67} -6.73549 q^{68} +0.483072 q^{69} -0.700690 q^{71} -2.97763 q^{72} +8.02726 q^{73} +2.47930 q^{74} +0.346794 q^{75} -4.24063 q^{76} +0.193200 q^{78} +1.00000 q^{79} -1.63746 q^{80} +8.79918 q^{81} +2.30405 q^{82} +1.65710 q^{83} +11.0291 q^{85} +12.5010 q^{86} -0.536502 q^{87} +4.31135 q^{88} +4.55511 q^{89} +4.87575 q^{90} -3.22991 q^{92} -1.56190 q^{93} -4.47801 q^{94} +6.94386 q^{95} -0.149562 q^{96} +11.2404 q^{97} -12.8376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8} + 12 q^{9} + 12 q^{11} + 8 q^{15} + 12 q^{16} + 12 q^{18} + 12 q^{22} + 16 q^{23} + 24 q^{25} + 12 q^{29} + 8 q^{30} + 12 q^{32} + 12 q^{36} + 16 q^{37} + 28 q^{39}+ \cdots + 40 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 12 * q^8 + 12 * q^9 + 12 * q^11 + 8 * q^15 + 12 * q^16 + 12 * q^18 + 12 * q^22 + 16 * q^23 + 24 * q^25 + 12 * q^29 + 8 * q^30 + 12 * q^32 + 12 * q^36 + 16 * q^37 + 28 * q^39 - 4 * q^43 + 12 * q^44 + 16 * q^46 + 24 * q^50 + 24 * q^53 + 28 * q^57 + 12 * q^58 + 8 * q^60 + 12 * q^64 - 4 * q^65 + 56 * q^67 + 16 * q^71 + 12 * q^72 + 16 * q^74 + 28 * q^78 + 12 * q^79 + 8 * q^81 - 20 * q^85 - 4 * q^86 + 12 * q^88 + 16 * q^92 + 12 * q^93 + 80 * q^95 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more