LMFDB - Embedded newform 7742.2.a.bl.1.8

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,-12,12,0,4] 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:	$1$
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} - 20x^{10} + 144x^{8} - 456x^{6} + 612x^{4} - 248x^{2} + 16 $ x^12 - 20x^10 + 144x^8 - 456x^6 + 612x^4 - 248*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.68362$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +0.824076 q^{3} +1.00000 q^{4} -1.68362 q^{5} -0.824076 q^{6} -1.00000 q^{8} -2.32090 q^{9} +1.68362 q^{10} -3.87375 q^{11} +0.824076 q^{12} -0.453480 q^{13} -1.38743 q^{15} +1.00000 q^{16} -1.08584 q^{17} +2.32090 q^{18} +3.91515 q^{19} -1.68362 q^{20} +3.87375 q^{22} +7.28311 q^{23} -0.824076 q^{24} -2.16542 q^{25} +0.453480 q^{26} -4.38483 q^{27} +2.84539 q^{29} +1.38743 q^{30} +2.04446 q^{31} -1.00000 q^{32} -3.19227 q^{33} +1.08584 q^{34} -2.32090 q^{36} -2.27568 q^{37} -3.91515 q^{38} -0.373702 q^{39} +1.68362 q^{40} +9.09968 q^{41} +10.8295 q^{43} -3.87375 q^{44} +3.90751 q^{45} -7.28311 q^{46} -2.11107 q^{47} +0.824076 q^{48} +2.16542 q^{50} -0.894817 q^{51} -0.453480 q^{52} +9.66060 q^{53} +4.38483 q^{54} +6.52193 q^{55} +3.22639 q^{57} -2.84539 q^{58} +3.07898 q^{59} -1.38743 q^{60} -14.1835 q^{61} -2.04446 q^{62} +1.00000 q^{64} +0.763489 q^{65} +3.19227 q^{66} -2.35782 q^{67} -1.08584 q^{68} +6.00184 q^{69} +2.42535 q^{71} +2.32090 q^{72} -8.61161 q^{73} +2.27568 q^{74} -1.78447 q^{75} +3.91515 q^{76} +0.373702 q^{78} -1.00000 q^{79} -1.68362 q^{80} +3.34926 q^{81} -9.09968 q^{82} -8.54786 q^{83} +1.82815 q^{85} -10.8295 q^{86} +2.34482 q^{87} +3.87375 q^{88} +2.34290 q^{89} -3.90751 q^{90} +7.28311 q^{92} +1.68479 q^{93} +2.11107 q^{94} -6.59164 q^{95} -0.824076 q^{96} +12.8858 q^{97} +8.99058 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} + 12 q^{9} + 4 q^{11} - 12 q^{15} + 12 q^{16} - 12 q^{18} - 4 q^{22} - 4 q^{23} - 20 q^{25} - 4 q^{29} + 12 q^{30} - 12 q^{32} + 12 q^{36} - 16 q^{37} - 24 q^{39} + 4 q^{43}+ \cdots + 44 q^{99}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 12 * q^8 + 12 * q^9 + 4 * q^11 - 12 * q^15 + 12 * q^16 - 12 * q^18 - 4 * q^22 - 4 * q^23 - 20 * q^25 - 4 * q^29 + 12 * q^30 - 12 * q^32 + 12 * q^36 - 16 * q^37 - 24 * q^39 + 4 * q^43 + 4 * q^44 + 4 * q^46 + 20 * q^50 + 4 * q^51 - 8 * q^53 - 28 * q^57 + 4 * q^58 - 12 * q^60 + 12 * q^64 - 36 * q^65 - 112 * q^67 + 44 * q^71 - 12 * q^72 + 16 * q^74 + 24 * q^78 - 12 * q^79 - 12 * q^81 + 4 * q^85 - 4 * q^86 - 4 * q^88 - 4 * q^92 - 60 * q^93 - 16 * q^95 + 44 * 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
$3$	0.824076	0.475781	0.237890	−	0.971292i	$-0.423544\pi$
$3$	0.824076	0.475781	0.237890	+	0.971292i	$0.423544\pi$
$4$	1.00000	0.500000
$5$	−1.68362	−0.752938	−0.376469	−	0.926429i	$-0.622862\pi$
$5$	−1.68362	−0.752938	−0.376469	+	0.926429i	$0.622862\pi$
$6$	−0.824076	−0.336428
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.32090	−0.773633
$10$	1.68362	0.532408
$11$	−3.87375	−1.16798	−0.583990	−	0.811761i	$-0.698509\pi$
$11$	−3.87375	−1.16798	−0.583990	+	0.811761i	$0.698509\pi$
$12$	0.824076	0.237890
$13$	−0.453480	−0.125773	−0.0628864	−	0.998021i	$-0.520031\pi$
$13$	−0.453480	−0.125773	−0.0628864	+	0.998021i	$0.520031\pi$
$14$	0	0
$15$	−1.38743	−0.358234
$16$	1.00000	0.250000
$17$	−1.08584	−0.263355	−0.131678	−	0.991293i	$-0.542036\pi$
$17$	−1.08584	−0.263355	−0.131678	+	0.991293i	$0.542036\pi$
$18$	2.32090	0.547041
$19$	3.91515	0.898198	0.449099	−	0.893482i	$-0.351745\pi$
$19$	3.91515	0.898198	0.449099	+	0.893482i	$0.351745\pi$
$20$	−1.68362	−0.376469
$21$	0	0
$22$	3.87375	0.825886
$23$	7.28311	1.51863	0.759317	−	0.650721i	$-0.225533\pi$
$23$	7.28311	1.51863	0.759317	+	0.650721i	$0.225533\pi$
$24$	−0.824076	−0.168214
$25$	−2.16542	−0.433084
$26$	0.453480	0.0889348
$27$	−4.38483	−0.843860
$28$	0	0
$29$	2.84539	0.528375	0.264188	−	0.964471i	$-0.414896\pi$
$29$	2.84539	0.528375	0.264188	+	0.964471i	$0.414896\pi$
$30$	1.38743	0.253309
$31$	2.04446	0.367196	0.183598	−	0.983001i	$-0.441225\pi$
$31$	2.04446	0.367196	0.183598	+	0.983001i	$0.441225\pi$
$32$	−1.00000	−0.176777
$33$	−3.19227	−0.555702
$34$	1.08584	0.186220
$35$	0	0
$36$	−2.32090	−0.386816
$37$	−2.27568	−0.374120	−0.187060	−	0.982349i	$-0.559896\pi$
$37$	−2.27568	−0.374120	−0.187060	+	0.982349i	$0.559896\pi$
$38$	−3.91515	−0.635122
$39$	−0.373702	−0.0598403
$40$	1.68362	0.266204
$41$	9.09968	1.42113	0.710565	−	0.703631i	$-0.248439\pi$
$41$	9.09968	1.42113	0.710565	+	0.703631i	$0.248439\pi$
$42$	0	0
$43$	10.8295	1.65149	0.825744	−	0.564045i	$-0.190755\pi$
$43$	10.8295	1.65149	0.825744	+	0.564045i	$0.190755\pi$
$44$	−3.87375	−0.583990
$45$	3.90751	0.582498
$46$	−7.28311	−1.07384
$47$	−2.11107	−0.307931	−0.153965	−	0.988076i	$-0.549204\pi$
$47$	−2.11107	−0.307931	−0.153965	+	0.988076i	$0.549204\pi$
$48$	0.824076	0.118945
$49$	0	0
$50$	2.16542	0.306237
$51$	−0.894817	−0.125299
$52$	−0.453480	−0.0628864
$53$	9.66060	1.32699	0.663493	−	0.748182i	$-0.269073\pi$
$53$	9.66060	1.32699	0.663493	+	0.748182i	$0.269073\pi$
$54$	4.38483	0.596699
$55$	6.52193	0.879417
$56$	0	0
$57$	3.22639	0.427345
$58$	−2.84539	−0.373618
$59$	3.07898	0.400850	0.200425	−	0.979709i	$-0.435768\pi$
$59$	3.07898	0.400850	0.200425	+	0.979709i	$0.435768\pi$
$60$	−1.38743	−0.179117
$61$	−14.1835	−1.81601	−0.908004	−	0.418962i	$-0.862394\pi$
$61$	−14.1835	−1.81601	−0.908004	+	0.418962i	$0.862394\pi$
$62$	−2.04446	−0.259647
$63$	0	0
$64$	1.00000	0.125000
$65$	0.763489	0.0946992
$66$	3.19227	0.392941
$67$	−2.35782	−0.288053	−0.144027	−	0.989574i	$-0.546005\pi$
$67$	−2.35782	−0.288053	−0.144027	+	0.989574i	$0.546005\pi$
$68$	−1.08584	−0.131678
$69$	6.00184	0.722537
$70$	0	0
$71$	2.42535	0.287836	0.143918	−	0.989590i	$-0.454030\pi$
$71$	2.42535	0.287836	0.143918	+	0.989590i	$0.454030\pi$
$72$	2.32090	0.273520
$73$	−8.61161	−1.00791	−0.503957	−	0.863729i	$-0.668123\pi$
$73$	−8.61161	−1.00791	−0.503957	+	0.863729i	$0.668123\pi$
$74$	2.27568	0.264543
$75$	−1.78447	−0.206053
$76$	3.91515	0.449099
$77$	0	0
$78$	0.373702	0.0423135
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	−1.68362	−0.188235
$81$	3.34926	0.372140
$82$	−9.09968	−1.00489
$83$	−8.54786	−0.938249	−0.469125	−	0.883132i	$-0.655431\pi$
$83$	−8.54786	−0.938249	−0.469125	+	0.883132i	$0.655431\pi$
$84$	0	0
$85$	1.82815	0.198290
$86$	−10.8295	−1.16778
$87$	2.34482	0.251391
$88$	3.87375	0.412943
$89$	2.34290	0.248347	0.124174	−	0.992261i	$-0.460372\pi$
$89$	2.34290	0.248347	0.124174	+	0.992261i	$0.460372\pi$
$90$	−3.90751	−0.411888
$91$	0	0
$92$	7.28311	0.759317
$93$	1.68479	0.174705
$94$	2.11107	0.217740
$95$	−6.59164	−0.676288
$96$	−0.824076	−0.0841069
$97$	12.8858	1.30836	0.654178	−	0.756340i	$-0.273015\pi$
$97$	12.8858	1.30836	0.654178	+	0.756340i	$0.273015\pi$
$98$	0	0
$99$	8.99058	0.903587
$100$	−2.16542	−0.216542
$101$	18.1323	1.80423	0.902116	−	0.431494i	$-0.142014\pi$
$101$	18.1323	1.80423	0.902116	+	0.431494i	$0.142014\pi$
$102$	0.894817	0.0886001
$103$	−16.9795	−1.67304	−0.836521	−	0.547935i	$-0.815414\pi$
$103$	−16.9795	−1.67304	−0.836521	+	0.547935i	$0.815414\pi$
$104$	0.453480	0.0444674
$105$	0	0
$106$	−9.66060	−0.938321
$107$	−16.6598	−1.61056	−0.805281	−	0.592894i	$-0.797985\pi$
$107$	−16.6598	−1.61056	−0.805281	+	0.592894i	$0.797985\pi$
$108$	−4.38483	−0.421930
$109$	−10.3447	−0.990843	−0.495422	−	0.868653i	$-0.664986\pi$
$109$	−10.3447	−0.990843	−0.495422	+	0.868653i	$0.664986\pi$
$110$	−6.52193	−0.621841
$111$	−1.87534	−0.177999
$112$	0	0
$113$	14.8905	1.40078	0.700390	−	0.713760i	$-0.253009\pi$
$113$	14.8905	1.40078	0.700390	+	0.713760i	$0.253009\pi$
$114$	−3.22639	−0.302179
$115$	−12.2620	−1.14344
$116$	2.84539	0.264188
$117$	1.05248	0.0973020
$118$	−3.07898	−0.283443
$119$	0	0
$120$	1.38743	0.126655
$121$	4.00594	0.364177
$122$	14.1835	1.28411
$123$	7.49883	0.676147
$124$	2.04446	0.183598
$125$	12.0639	1.07902
$126$	0	0
$127$	−15.3006	−1.35771	−0.678853	−	0.734274i	$-0.737523\pi$
$127$	−15.3006	−1.35771	−0.678853	+	0.734274i	$0.737523\pi$
$128$	−1.00000	−0.0883883
$129$	8.92436	0.785746
$130$	−0.763489	−0.0669624
$131$	17.2633	1.50830	0.754152	−	0.656699i	$-0.228048\pi$
$131$	17.2633	1.50830	0.754152	+	0.656699i	$0.228048\pi$
$132$	−3.19227	−0.277851
$133$	0	0
$134$	2.35782	0.203684
$135$	7.38239	0.635375
$136$	1.08584	0.0931102
$137$	−23.1413	−1.97709	−0.988546	−	0.150921i	$-0.951776\pi$
$137$	−23.1413	−1.97709	−0.988546	+	0.150921i	$0.951776\pi$
$138$	−6.00184	−0.510911
$139$	−9.77888	−0.829434	−0.414717	−	0.909950i	$-0.636119\pi$
$139$	−9.77888	−0.829434	−0.414717	+	0.909950i	$0.636119\pi$
$140$	0	0
$141$	−1.73968	−0.146508
$142$	−2.42535	−0.203531
$143$	1.75667	0.146900
$144$	−2.32090	−0.193408
$145$	−4.79055	−0.397834
$146$	8.61161	0.712702
$147$	0	0
$148$	−2.27568	−0.187060
$149$	−21.4966	−1.76107	−0.880534	−	0.473983i	$-0.842816\pi$
$149$	−21.4966	−1.76107	−0.880534	+	0.473983i	$0.842816\pi$
$150$	1.78447	0.145701
$151$	−10.7081	−0.871411	−0.435706	−	0.900089i	$-0.643501\pi$
$151$	−10.7081	−0.871411	−0.435706	+	0.900089i	$0.643501\pi$
$152$	−3.91515	−0.317561
$153$	2.52013	0.203740
$154$	0	0
$155$	−3.44210	−0.276476
$156$	−0.373702	−0.0299201
$157$	−1.81460	−0.144821	−0.0724103	−	0.997375i	$-0.523069\pi$
$157$	−1.81460	−0.144821	−0.0724103	+	0.997375i	$0.523069\pi$
$158$	1.00000	0.0795557
$159$	7.96108	0.631354
$160$	1.68362	0.133102
$161$	0	0
$162$	−3.34926	−0.263143
$163$	7.40489	0.579996	0.289998	−	0.957027i	$-0.406345\pi$
$163$	7.40489	0.579996	0.289998	+	0.957027i	$0.406345\pi$
$164$	9.09968	0.710565
$165$	5.37457	0.418409
$166$	8.54786	0.663443
$167$	−8.17952	−0.632950	−0.316475	−	0.948601i	$-0.602499\pi$
$167$	−8.17952	−0.632950	−0.316475	+	0.948601i	$0.602499\pi$
$168$	0	0
$169$	−12.7944	−0.984181
$170$	−1.82815	−0.140212
$171$	−9.08667	−0.694875
$172$	10.8295	0.825744
$173$	−3.70505	−0.281690	−0.140845	−	0.990032i	$-0.544982\pi$
$173$	−3.70505	−0.281690	−0.140845	+	0.990032i	$0.544982\pi$
$174$	−2.34482	−0.177760
$175$	0	0
$176$	−3.87375	−0.291995
$177$	2.53732	0.190716
$178$	−2.34290	−0.175608
$179$	−25.3448	−1.89436	−0.947181	−	0.320699i	$-0.896082\pi$
$179$	−25.3448	−1.89436	−0.947181	+	0.320699i	$0.896082\pi$
$180$	3.90751	0.291249
$181$	9.60410	0.713867	0.356934	−	0.934130i	$-0.383822\pi$
$181$	9.60410	0.713867	0.356934	+	0.934130i	$0.383822\pi$
$182$	0	0
$183$	−11.6883	−0.864021
$184$	−7.28311	−0.536918
$185$	3.83139	0.281689
$186$	−1.68479	−0.123535
$187$	4.20628	0.307594
$188$	−2.11107	−0.153965
$189$	0	0
$190$	6.59164	0.478208
$191$	4.81241	0.348214	0.174107	−	0.984727i	$-0.444296\pi$
$191$	4.81241	0.348214	0.174107	+	0.984727i	$0.444296\pi$
$192$	0.824076	0.0594726
$193$	−20.0451	−1.44288	−0.721439	−	0.692478i	$-0.756519\pi$
$193$	−20.0451	−1.44288	−0.721439	+	0.692478i	$0.756519\pi$
$194$	−12.8858	−0.925148
$195$	0.629173	0.0450560
$196$	0	0
$197$	10.1512	0.723243	0.361622	−	0.932325i	$-0.382223\pi$
$197$	10.1512	0.723243	0.361622	+	0.932325i	$0.382223\pi$
$198$	−8.99058	−0.638933
$199$	−1.81283	−0.128508	−0.0642540	−	0.997934i	$-0.520467\pi$
$199$	−1.81283	−0.128508	−0.0642540	+	0.997934i	$0.520467\pi$
$200$	2.16542	0.153118
$201$	−1.94302	−0.137050
$202$	−18.1323	−1.27578
$203$	0	0
$204$	−0.894817	−0.0626497
$205$	−15.3204	−1.07002
$206$	16.9795	1.18302
$207$	−16.9034	−1.17487
$208$	−0.453480	−0.0314432
$209$	−15.1663	−1.04908
$210$	0	0
$211$	−25.9414	−1.78588	−0.892939	−	0.450178i	$-0.851360\pi$
$211$	−25.9414	−1.78588	−0.892939	+	0.450178i	$0.851360\pi$
$212$	9.66060	0.663493
$213$	1.99868	0.136947
$214$	16.6598	1.13884
$215$	−18.2328	−1.24347
$216$	4.38483	0.298350
$217$	0	0
$218$	10.3447	0.700632
$219$	−7.09663	−0.479546
$220$	6.52193	0.439708
$221$	0.492408	0.0331230
$222$	1.87534	0.125864
$223$	4.06874	0.272463	0.136232	−	0.990677i	$-0.456501\pi$
$223$	4.06874	0.272463	0.136232	+	0.990677i	$0.456501\pi$
$224$	0	0
$225$	5.02572	0.335048
$226$	−14.8905	−0.990501
$227$	−5.39652	−0.358180	−0.179090	−	0.983833i	$-0.557315\pi$
$227$	−5.39652	−0.358180	−0.179090	+	0.983833i	$0.557315\pi$
$228$	3.22639	0.213673
$229$	−26.2002	−1.73136	−0.865680	−	0.500598i	$-0.833113\pi$
$229$	−26.2002	−1.73136	−0.865680	+	0.500598i	$0.833113\pi$
$230$	12.2620	0.808533
$231$	0	0
$232$	−2.84539	−0.186809
$233$	17.0742	1.11857	0.559284	−	0.828976i	$-0.311076\pi$
$233$	17.0742	1.11857	0.559284	+	0.828976i	$0.311076\pi$
$234$	−1.05248	−0.0688029
$235$	3.55424	0.231853
$236$	3.07898	0.200425
$237$	−0.824076	−0.0535295
$238$	0	0
$239$	13.4401	0.869370	0.434685	−	0.900583i	$-0.356860\pi$
$239$	13.4401	0.869370	0.434685	+	0.900583i	$0.356860\pi$
$240$	−1.38743	−0.0895584
$241$	11.4319	0.736396	0.368198	−	0.929747i	$-0.379975\pi$
$241$	11.4319	0.736396	0.368198	+	0.929747i	$0.379975\pi$
$242$	−4.00594	−0.257512
$243$	15.9145	1.02092
$244$	−14.1835	−0.908004
$245$	0	0
$246$	−7.49883	−0.478108
$247$	−1.77545	−0.112969
$248$	−2.04446	−0.129824
$249$	−7.04409	−0.446401
$250$	−12.0639	−0.762985
$251$	4.03097	0.254433	0.127216	−	0.991875i	$-0.459396\pi$
$251$	4.03097	0.254433	0.127216	+	0.991875i	$0.459396\pi$
$252$	0	0
$253$	−28.2130	−1.77373
$254$	15.3006	0.960043
$255$	1.50653	0.0943428
$256$	1.00000	0.0625000
$257$	−28.3361	−1.76756	−0.883779	−	0.467904i	$-0.845009\pi$
$257$	−28.3361	−1.76756	−0.883779	+	0.467904i	$0.845009\pi$
$258$	−8.92436	−0.555607
$259$	0	0
$260$	0.763489	0.0473496
$261$	−6.60385	−0.408768
$262$	−17.2633	−1.06653
$263$	6.38103	0.393471	0.196735	−	0.980457i	$-0.436966\pi$
$263$	6.38103	0.393471	0.196735	+	0.980457i	$0.436966\pi$
$264$	3.19227	0.196470
$265$	−16.2648	−0.999139
$266$	0	0
$267$	1.93073	0.118159
$268$	−2.35782	−0.144027
$269$	15.9075	0.969898	0.484949	−	0.874542i	$-0.338838\pi$
$269$	15.9075	0.969898	0.484949	+	0.874542i	$0.338838\pi$
$270$	−7.38239	−0.449278
$271$	−26.4747	−1.60822	−0.804112	−	0.594477i	$-0.797359\pi$
$271$	−26.4747	−1.60822	−0.804112	+	0.594477i	$0.797359\pi$
$272$	−1.08584	−0.0658389
$273$	0	0
$274$	23.1413	1.39801
$275$	8.38830	0.505833
$276$	6.00184	0.361268
$277$	−15.3441	−0.921936	−0.460968	−	0.887417i	$-0.652498\pi$
$277$	−15.3441	−0.921936	−0.460968	+	0.887417i	$0.652498\pi$
$278$	9.77888	0.586499
$279$	−4.74499	−0.284075
$280$	0	0
$281$	9.57774	0.571360	0.285680	−	0.958325i	$-0.407781\pi$
$281$	9.57774	0.571360	0.285680	+	0.958325i	$0.407781\pi$
$282$	1.73968	0.103597
$283$	19.7850	1.17609	0.588047	−	0.808826i	$-0.299897\pi$
$283$	19.7850	1.17609	0.588047	+	0.808826i	$0.299897\pi$
$284$	2.42535	0.143918
$285$	−5.43201	−0.321765
$286$	−1.75667	−0.103874
$287$	0	0
$288$	2.32090	0.136760
$289$	−15.8209	−0.930644
$290$	4.79055	0.281311
$291$	10.6189	0.622491
$292$	−8.61161	−0.503957
$293$	0.438727	0.0256307	0.0128154	−	0.999918i	$-0.495921\pi$
$293$	0.438727	0.0256307	0.0128154	+	0.999918i	$0.495921\pi$
$294$	0	0
$295$	−5.18384	−0.301815
$296$	2.27568	0.132271
$297$	16.9857	0.985612
$298$	21.4966	1.24526
$299$	−3.30275	−0.191003
$300$	−1.78447	−0.103027
$301$	0	0
$302$	10.7081	0.616181
$303$	14.9424	0.858418
$304$	3.91515	0.224550
$305$	23.8796	1.36734
$306$	−2.52013	−0.144066
$307$	−2.19659	−0.125366	−0.0626831	−	0.998033i	$-0.519966\pi$
$307$	−2.19659	−0.125366	−0.0626831	+	0.998033i	$0.519966\pi$
$308$	0	0
$309$	−13.9924	−0.796001
$310$	3.44210	0.195498
$311$	−17.5465	−0.994970	−0.497485	−	0.867473i	$-0.665743\pi$
$311$	−17.5465	−0.994970	−0.497485	+	0.867473i	$0.665743\pi$
$312$	0.373702	0.0211567
$313$	26.4011	1.49228	0.746140	−	0.665789i	$-0.231905\pi$
$313$	26.4011	1.49228	0.746140	+	0.665789i	$0.231905\pi$
$314$	1.81460	0.102404
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	5.49375	0.308560	0.154280	−	0.988027i	$-0.450694\pi$
$317$	5.49375	0.308560	0.154280	+	0.988027i	$0.450694\pi$
$318$	−7.96108	−0.446435
$319$	−11.0223	−0.617131
$320$	−1.68362	−0.0941173
$321$	−13.7289	−0.766274
$322$	0	0
$323$	−4.25124	−0.236545
$324$	3.34926	0.186070
$325$	0.981976	0.0544702
$326$	−7.40489	−0.410119
$327$	−8.52483	−0.471424
$328$	−9.09968	−0.502446
$329$	0	0
$330$	−5.37457	−0.295860
$331$	−0.449521	−0.0247079	−0.0123540	−	0.999924i	$-0.503932\pi$
$331$	−0.449521	−0.0247079	−0.0123540	+	0.999924i	$0.503932\pi$
$332$	−8.54786	−0.469125
$333$	5.28162	0.289431
$334$	8.17952	0.447563
$335$	3.96967	0.216886
$336$	0	0
$337$	−5.62376	−0.306346	−0.153173	−	0.988199i	$-0.548949\pi$
$337$	−5.62376	−0.306346	−0.153173	+	0.988199i	$0.548949\pi$
$338$	12.7944	0.695921
$339$	12.2709	0.666464
$340$	1.82815	0.0991452
$341$	−7.91974	−0.428878
$342$	9.08667	0.491351
$343$	0	0
$344$	−10.8295	−0.583889
$345$	−10.1048	−0.544026
$346$	3.70505	0.199185
$347$	27.2048	1.46043	0.730214	−	0.683218i	$-0.239420\pi$
$347$	27.2048	1.46043	0.730214	+	0.683218i	$0.239420\pi$
$348$	2.34482	0.125695
$349$	3.71469	0.198843	0.0994215	−	0.995045i	$-0.468301\pi$
$349$	3.71469	0.198843	0.0994215	+	0.995045i	$0.468301\pi$
$350$	0	0
$351$	1.98843	0.106135
$352$	3.87375	0.206472
$353$	11.5710	0.615861	0.307930	−	0.951409i	$-0.400364\pi$
$353$	11.5710	0.615861	0.307930	+	0.951409i	$0.400364\pi$
$354$	−2.53732	−0.134857
$355$	−4.08337	−0.216723
$356$	2.34290	0.124174
$357$	0	0
$358$	25.3448	1.33952
$359$	−23.8032	−1.25629	−0.628143	−	0.778098i	$-0.716185\pi$
$359$	−23.8032	−1.25629	−0.628143	+	0.778098i	$0.716185\pi$
$360$	−3.90751	−0.205944
$361$	−3.67157	−0.193240
$362$	−9.60410	−0.504780
$363$	3.30120	0.173268
$364$	0	0
$365$	14.4987	0.758896
$366$	11.6883	0.610955
$367$	29.1639	1.52235	0.761173	−	0.648549i	$-0.224624\pi$
$367$	29.1639	1.52235	0.761173	+	0.648549i	$0.224624\pi$
$368$	7.28311	0.379659
$369$	−21.1194	−1.09943
$370$	−3.83139	−0.199184
$371$	0	0
$372$	1.68479	0.0873525
$373$	−11.8178	−0.611904	−0.305952	−	0.952047i	$-0.598975\pi$
$373$	−11.8178	−0.611904	−0.305952	+	0.952047i	$0.598975\pi$
$374$	−4.20628	−0.217502
$375$	9.94154	0.513379
$376$	2.11107	0.108870
$377$	−1.29033	−0.0664552
$378$	0	0
$379$	24.7462	1.27113	0.635563	−	0.772049i	$-0.280768\pi$
$379$	24.7462	1.27113	0.635563	+	0.772049i	$0.280768\pi$
$380$	−6.59164	−0.338144
$381$	−12.6088	−0.645970
$382$	−4.81241	−0.246224
$383$	−1.20077	−0.0613564	−0.0306782	−	0.999529i	$-0.509767\pi$
$383$	−1.20077	−0.0613564	−0.0306782	+	0.999529i	$0.509767\pi$
$384$	−0.824076	−0.0420535
$385$	0	0
$386$	20.0451	1.02027
$387$	−25.1342	−1.27765
$388$	12.8858	0.654178
$389$	−26.0638	−1.32149	−0.660743	−	0.750612i	$-0.729759\pi$
$389$	−26.0638	−1.32149	−0.660743	+	0.750612i	$0.729759\pi$
$390$	−0.629173	−0.0318594
$391$	−7.90831	−0.399941
$392$	0	0
$393$	14.2263	0.717622
$394$	−10.1512	−0.511410
$395$	1.68362	0.0847122
$396$	8.99058	0.451794
$397$	24.3057	1.21987	0.609934	−	0.792452i	$-0.291196\pi$
$397$	24.3057	1.21987	0.609934	+	0.792452i	$0.291196\pi$
$398$	1.81283	0.0908689
$399$	0	0
$400$	−2.16542	−0.108271
$401$	4.02693	0.201095	0.100548	−	0.994932i	$-0.467941\pi$
$401$	4.02693	0.201095	0.100548	+	0.994932i	$0.467941\pi$
$402$	1.94302	0.0969091
$403$	−0.927124	−0.0461833
$404$	18.1323	0.902116
$405$	−5.63889	−0.280199
$406$	0	0
$407$	8.81542	0.436964
$408$	0.894817	0.0443001
$409$	−1.80916	−0.0894570	−0.0447285	−	0.998999i	$-0.514242\pi$
$409$	−1.80916	−0.0894570	−0.0447285	+	0.998999i	$0.514242\pi$
$410$	15.3204	0.756621
$411$	−19.0702	−0.940662
$412$	−16.9795	−0.836521
$413$	0	0
$414$	16.9034	0.830755
$415$	14.3914	0.706444
$416$	0.453480	0.0222337
$417$	−8.05855	−0.394629
$418$	15.1663	0.741810
$419$	−32.1832	−1.57225	−0.786127	−	0.618065i	$-0.787917\pi$
$419$	−32.1832	−1.57225	−0.786127	+	0.618065i	$0.787917\pi$
$420$	0	0
$421$	−0.858901	−0.0418603	−0.0209301	−	0.999781i	$-0.506663\pi$
$421$	−0.858901	−0.0418603	−0.0209301	+	0.999781i	$0.506663\pi$
$422$	25.9414	1.26281
$423$	4.89957	0.238225
$424$	−9.66060	−0.469160
$425$	2.35131	0.114055
$426$	−1.99868	−0.0968361
$427$	0	0
$428$	−16.6598	−0.805281
$429$	1.44763	0.0698922
$430$	18.2328	0.879265
$431$	−22.0754	−1.06333	−0.531667	−	0.846953i	$-0.678434\pi$
$431$	−22.0754	−1.06333	−0.531667	+	0.846953i	$0.678434\pi$
$432$	−4.38483	−0.210965
$433$	−8.15515	−0.391911	−0.195956	−	0.980613i	$-0.562781\pi$
$433$	−8.15515	−0.391911	−0.195956	+	0.980613i	$0.562781\pi$
$434$	0	0
$435$	−3.94778	−0.189282
$436$	−10.3447	−0.495422
$437$	28.5145	1.36403
$438$	7.09663	0.339090
$439$	26.3289	1.25661	0.628304	−	0.777968i	$-0.283749\pi$
$439$	26.3289	1.25661	0.628304	+	0.777968i	$0.283749\pi$
$440$	−6.52193	−0.310921
$441$	0	0
$442$	−0.492408	−0.0234215
$443$	15.9048	0.755661	0.377830	−	0.925875i	$-0.376670\pi$
$443$	15.9048	0.755661	0.377830	+	0.925875i	$0.376670\pi$
$444$	−1.87534	−0.0889995
$445$	−3.94456	−0.186990
$446$	−4.06874	−0.192660
$447$	−17.7148	−0.837882
$448$	0	0
$449$	−37.4697	−1.76831	−0.884153	−	0.467198i	$-0.845263\pi$
$449$	−37.4697	−1.76831	−0.884153	+	0.467198i	$0.845263\pi$
$450$	−5.02572	−0.236915
$451$	−35.2499	−1.65985
$452$	14.8905	0.700390
$453$	−8.82428	−0.414601
$454$	5.39652	0.253271
$455$	0	0
$456$	−3.22639	−0.151089
$457$	−8.81695	−0.412440	−0.206220	−	0.978506i	$-0.566116\pi$
$457$	−8.81695	−0.412440	−0.206220	+	0.978506i	$0.566116\pi$
$458$	26.2002	1.22426
$459$	4.76123	0.222235
$460$	−12.2620	−0.571719
$461$	25.7035	1.19713	0.598565	−	0.801074i	$-0.295738\pi$
$461$	25.7035	1.19713	0.598565	+	0.801074i	$0.295738\pi$
$462$	0	0
$463$	−2.56657	−0.119278	−0.0596392	−	0.998220i	$-0.518995\pi$
$463$	−2.56657	−0.119278	−0.0596392	+	0.998220i	$0.518995\pi$
$464$	2.84539	0.132094
$465$	−2.83655	−0.131542
$466$	−17.0742	−0.790947
$467$	−29.8036	−1.37915	−0.689574	−	0.724216i	$-0.742202\pi$
$467$	−29.8036	−1.37915	−0.689574	+	0.724216i	$0.742202\pi$
$468$	1.05248	0.0486510
$469$	0	0
$470$	−3.55424	−0.163945
$471$	−1.49537	−0.0689029
$472$	−3.07898	−0.141722
$473$	−41.9509	−1.92891
$474$	0.824076	0.0378511
$475$	−8.47795	−0.388995
$476$	0	0
$477$	−22.4213	−1.02660
$478$	−13.4401	−0.614737
$479$	10.2725	0.469364	0.234682	−	0.972072i	$-0.424595\pi$
$479$	10.2725	0.469364	0.234682	+	0.972072i	$0.424595\pi$
$480$	1.38743	0.0633273
$481$	1.03198	0.0470541
$482$	−11.4319	−0.520711
$483$	0	0
$484$	4.00594	0.182088
$485$	−21.6948	−0.985112
$486$	−15.9145	−0.721898
$487$	−13.3276	−0.603932	−0.301966	−	0.953319i	$-0.597643\pi$
$487$	−13.3276	−0.603932	−0.301966	+	0.953319i	$0.597643\pi$
$488$	14.1835	0.642056
$489$	6.10219	0.275951
$490$	0	0
$491$	16.1567	0.729141	0.364570	−	0.931176i	$-0.381216\pi$
$491$	16.1567	0.729141	0.364570	+	0.931176i	$0.381216\pi$
$492$	7.49883	0.338073
$493$	−3.08964	−0.139150
$494$	1.77545	0.0798811
$495$	−15.1367	−0.680345
$496$	2.04446	0.0917991
$497$	0	0
$498$	7.04409	0.315653
$499$	−3.57847	−0.160194	−0.0800971	−	0.996787i	$-0.525523\pi$
$499$	−3.57847	−0.160194	−0.0800971	+	0.996787i	$0.525523\pi$
$500$	12.0639	0.539512
$501$	−6.74055	−0.301145
$502$	−4.03097	−0.179911
$503$	−12.6373	−0.563468	−0.281734	−	0.959493i	$-0.590910\pi$
$503$	−12.6373	−0.563468	−0.281734	+	0.959493i	$0.590910\pi$
$504$	0	0
$505$	−30.5279	−1.35847
$506$	28.2130	1.25422
$507$	−10.5435	−0.468254
$508$	−15.3006	−0.678853
$509$	13.9002	0.616115	0.308058	−	0.951368i	$-0.400321\pi$
$509$	13.9002	0.616115	0.308058	+	0.951368i	$0.400321\pi$
$510$	−1.50653	−0.0667104
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−17.1673	−0.757954
$514$	28.3361	1.24985
$515$	28.5871	1.25970
$516$	8.92436	0.392873
$517$	8.17775	0.359657
$518$	0	0
$519$	−3.05324	−0.134022
$520$	−0.763489	−0.0334812
$521$	34.0296	1.49086	0.745431	−	0.666582i	$-0.232243\pi$
$521$	34.0296	1.49086	0.745431	+	0.666582i	$0.232243\pi$
$522$	6.60385	0.289043
$523$	3.49346	0.152758	0.0763792	−	0.997079i	$-0.475664\pi$
$523$	3.49346	0.152758	0.0763792	+	0.997079i	$0.475664\pi$
$524$	17.2633	0.754152
$525$	0	0
$526$	−6.38103	−0.278226
$527$	−2.21997	−0.0967032
$528$	−3.19227	−0.138926
$529$	30.0438	1.30625
$530$	16.2648	0.706498
$531$	−7.14601	−0.310110
$532$	0	0
$533$	−4.12653	−0.178740
$534$	−1.93073	−0.0835508
$535$	28.0487	1.21265
$536$	2.35782	0.101842
$537$	−20.8861	−0.901301
$538$	−15.9075	−0.685822
$539$	0	0
$540$	7.38239	0.317687
$541$	12.5398	0.539127	0.269564	−	0.962983i	$-0.413121\pi$
$541$	12.5398	0.539127	0.269564	+	0.962983i	$0.413121\pi$
$542$	26.4747	1.13719
$543$	7.91451	0.339644
$544$	1.08584	0.0465551
$545$	17.4166	0.746044
$546$	0	0
$547$	−3.47324	−0.148505	−0.0742525	−	0.997239i	$-0.523657\pi$
$547$	−3.47324	−0.148505	−0.0742525	+	0.997239i	$0.523657\pi$
$548$	−23.1413	−0.988546
$549$	32.9184	1.40492
$550$	−8.38830	−0.357678
$551$	11.1401	0.474585
$552$	−6.00184	−0.255455
$553$	0	0
$554$	15.3441	0.651908
$555$	3.15735	0.134022
$556$	−9.77888	−0.414717
$557$	13.2035	0.559451	0.279725	−	0.960080i	$-0.409757\pi$
$557$	13.2035	0.559451	0.279725	+	0.960080i	$0.409757\pi$
$558$	4.74499	0.200871
$559$	−4.91098	−0.207712
$560$	0	0
$561$	3.46630	0.146347
$562$	−9.57774	−0.404013
$563$	−43.4658	−1.83186	−0.915932	−	0.401333i	$-0.868547\pi$
$563$	−43.4658	−1.83186	−0.915932	+	0.401333i	$0.868547\pi$
$564$	−1.73968	−0.0732538
$565$	−25.0700	−1.05470
$566$	−19.7850	−0.831625
$567$	0	0
$568$	−2.42535	−0.101765
$569$	−0.180875	−0.00758266	−0.00379133	−	0.999993i	$-0.501207\pi$
$569$	−0.180875	−0.00758266	−0.00379133	+	0.999993i	$0.501207\pi$
$570$	5.43201	0.227522
$571$	4.49568	0.188138	0.0940692	−	0.995566i	$-0.470013\pi$
$571$	4.49568	0.188138	0.0940692	+	0.995566i	$0.470013\pi$
$572$	1.75667	0.0734501
$573$	3.96579	0.165673
$574$	0	0
$575$	−15.7710	−0.657696
$576$	−2.32090	−0.0967041
$577$	−44.3300	−1.84548	−0.922740	−	0.385422i	$-0.874056\pi$
$577$	−44.3300	−1.84548	−0.922740	+	0.385422i	$0.874056\pi$
$578$	15.8209	0.658065
$579$	−16.5187	−0.686494
$580$	−4.79055	−0.198917
$581$	0	0
$582$	−10.6189	−0.440167
$583$	−37.4228	−1.54989
$584$	8.61161	0.356351
$585$	−1.77198	−0.0732624
$586$	−0.438727	−0.0181237
$587$	−1.87636	−0.0774455	−0.0387228	−	0.999250i	$-0.512329\pi$
$587$	−1.87636	−0.0774455	−0.0387228	+	0.999250i	$0.512329\pi$
$588$	0	0
$589$	8.00439	0.329815
$590$	5.18384	0.213415
$591$	8.36537	0.344105
$592$	−2.27568	−0.0935299
$593$	34.2085	1.40478	0.702388	−	0.711794i	$-0.252117\pi$
$593$	34.2085	1.40478	0.702388	+	0.711794i	$0.252117\pi$
$594$	−16.9857	−0.696933
$595$	0	0
$596$	−21.4966	−0.880534
$597$	−1.49391	−0.0611416
$598$	3.30275	0.135059
$599$	15.6146	0.637993	0.318997	−	0.947756i	$-0.396654\pi$
$599$	15.6146	0.637993	0.318997	+	0.947756i	$0.396654\pi$
$600$	1.78447	0.0728507
$601$	22.9408	0.935774	0.467887	−	0.883788i	$-0.345015\pi$
$601$	22.9408	0.935774	0.467887	+	0.883788i	$0.345015\pi$
$602$	0	0
$603$	5.47225	0.222847
$604$	−10.7081	−0.435706
$605$	−6.74449	−0.274203
$606$	−14.9424	−0.606994
$607$	−6.17361	−0.250579	−0.125289	−	0.992120i	$-0.539986\pi$
$607$	−6.17361	−0.250579	−0.125289	+	0.992120i	$0.539986\pi$
$608$	−3.91515	−0.158780
$609$	0	0
$610$	−23.8796	−0.966856
$611$	0.957328	0.0387293
$612$	2.52013	0.101870
$613$	−33.9589	−1.37159	−0.685794	−	0.727795i	$-0.740545\pi$
$613$	−33.9589	−1.37159	−0.685794	+	0.727795i	$0.740545\pi$
$614$	2.19659	0.0886473
$615$	−12.6252	−0.509097
$616$	0	0
$617$	14.0655	0.566255	0.283128	−	0.959082i	$-0.408628\pi$
$617$	14.0655	0.566255	0.283128	+	0.959082i	$0.408628\pi$
$618$	13.9924	0.562858
$619$	10.5281	0.423159	0.211579	−	0.977361i	$-0.432139\pi$
$619$	10.5281	0.423159	0.211579	+	0.977361i	$0.432139\pi$
$620$	−3.44210	−0.138238
$621$	−31.9352	−1.28152
$622$	17.5465	0.703550
$623$	0	0
$624$	−0.373702	−0.0149601
$625$	−9.48386	−0.379354
$626$	−26.4011	−1.05520
$627$	−12.4982	−0.499131
$628$	−1.81460	−0.0724103
$629$	2.47103	0.0985265
$630$	0	0
$631$	−43.6073	−1.73598	−0.867989	−	0.496583i	$-0.834588\pi$
$631$	−43.6073	−1.73598	−0.867989	+	0.496583i	$0.834588\pi$
$632$	1.00000	0.0397779
$633$	−21.3777	−0.849686
$634$	−5.49375	−0.218185
$635$	25.7604	1.02227
$636$	7.96108	0.315677
$637$	0	0
$638$	11.0223	0.436378
$639$	−5.62899	−0.222680
$640$	1.68362	0.0665510
$641$	27.7079	1.09440	0.547198	−	0.837003i	$-0.315694\pi$
$641$	27.7079	1.09440	0.547198	+	0.837003i	$0.315694\pi$
$642$	13.7289	0.541837
$643$	19.9150	0.785371	0.392686	−	0.919673i	$-0.371546\pi$
$643$	19.9150	0.785371	0.392686	+	0.919673i	$0.371546\pi$
$644$	0	0
$645$	−15.0252	−0.591619
$646$	4.25124	0.167263
$647$	−17.0180	−0.669046	−0.334523	−	0.942388i	$-0.608575\pi$
$647$	−17.0180	−0.669046	−0.334523	+	0.942388i	$0.608575\pi$
$648$	−3.34926	−0.131571
$649$	−11.9272	−0.468184
$650$	−0.981976	−0.0385162
$651$	0	0
$652$	7.40489	0.289998
$653$	9.10756	0.356406	0.178203	−	0.983994i	$-0.442972\pi$
$653$	9.10756	0.356406	0.178203	+	0.983994i	$0.442972\pi$
$654$	8.52483	0.333347
$655$	−29.0649	−1.13566
$656$	9.09968	0.355283
$657$	19.9867	0.779755
$658$	0	0
$659$	−9.25137	−0.360382	−0.180191	−	0.983632i	$-0.557672\pi$
$659$	−9.25137	−0.360382	−0.180191	+	0.983632i	$0.557672\pi$
$660$	5.37457	0.209205
$661$	−14.4968	−0.563859	−0.281929	−	0.959435i	$-0.590974\pi$
$661$	−14.4968	−0.563859	−0.281929	+	0.959435i	$0.590974\pi$
$662$	0.449521	0.0174711
$663$	0.405782	0.0157593
$664$	8.54786	0.331721
$665$	0	0
$666$	−5.28162	−0.204659
$667$	20.7233	0.802408
$668$	−8.17952	−0.316475
$669$	3.35295	0.129633
$670$	−3.96967	−0.153362
$671$	54.9432	2.12106
$672$	0	0
$673$	12.0066	0.462819	0.231409	−	0.972856i	$-0.425666\pi$
$673$	12.0066	0.462819	0.231409	+	0.972856i	$0.425666\pi$
$674$	5.62376	0.216619
$675$	9.49499	0.365462
$676$	−12.7944	−0.492091
$677$	42.0166	1.61483	0.807415	−	0.589984i	$-0.200866\pi$
$677$	42.0166	1.61483	0.807415	+	0.589984i	$0.200866\pi$
$678$	−12.2709	−0.471261
$679$	0	0
$680$	−1.82815	−0.0701062
$681$	−4.44714	−0.170415
$682$	7.91974	0.303263
$683$	−50.7091	−1.94033	−0.970165	−	0.242445i	$-0.922051\pi$
$683$	−50.7091	−1.94033	−0.970165	+	0.242445i	$0.922051\pi$
$684$	−9.08667	−0.347438
$685$	38.9611	1.48863
$686$	0	0
$687$	−21.5910	−0.823747
$688$	10.8295	0.412872
$689$	−4.38089	−0.166899
$690$	10.1048	0.384684
$691$	10.6774	0.406186	0.203093	−	0.979159i	$-0.434901\pi$
$691$	10.6774	0.406186	0.203093	+	0.979159i	$0.434901\pi$
$692$	−3.70505	−0.140845
$693$	0	0
$694$	−27.2048	−1.03268
$695$	16.4639	0.624513
$696$	−2.34482	−0.0888800
$697$	−9.88082	−0.374263
$698$	−3.71469	−0.140603
$699$	14.0705	0.532193
$700$	0	0
$701$	3.72516	0.140697	0.0703487	−	0.997522i	$-0.477589\pi$
$701$	3.72516	0.140697	0.0703487	+	0.997522i	$0.477589\pi$
$702$	−1.98843	−0.0750486
$703$	−8.90964	−0.336034
$704$	−3.87375	−0.145997
$705$	2.92896	0.110311
$706$	−11.5710	−0.435479
$707$	0	0
$708$	2.53732	0.0953582
$709$	−16.5287	−0.620749	−0.310374	−	0.950614i	$-0.600454\pi$
$709$	−16.5287	−0.620749	−0.310374	+	0.950614i	$0.600454\pi$
$710$	4.08337	0.153246
$711$	2.32090	0.0870405
$712$	−2.34290	−0.0878039
$713$	14.8901	0.557637
$714$	0	0
$715$	−2.95757	−0.110607
$716$	−25.3448	−0.947181
$717$	11.0757	0.413630
$718$	23.8032	0.888328
$719$	−33.1328	−1.23565	−0.617823	−	0.786317i	$-0.711985\pi$
$719$	−33.1328	−1.23565	−0.617823	+	0.786317i	$0.711985\pi$
$720$	3.90751	0.145624
$721$	0	0
$722$	3.67157	0.136642
$723$	9.42079	0.350363
$724$	9.60410	0.356934
$725$	−6.16146	−0.228831
$726$	−3.30120	−0.122519
$727$	45.9498	1.70418	0.852091	−	0.523393i	$-0.175334\pi$
$727$	45.9498	1.70418	0.852091	+	0.523393i	$0.175334\pi$
$728$	0	0
$729$	3.06700	0.113593
$730$	−14.4987	−0.536621
$731$	−11.7592	−0.434929
$732$	−11.6883	−0.432011
$733$	29.4271	1.08692	0.543458	−	0.839436i	$-0.317115\pi$
$733$	29.4271	1.08692	0.543458	+	0.839436i	$0.317115\pi$
$734$	−29.1639	−1.07646
$735$	0	0
$736$	−7.28311	−0.268459
$737$	9.13360	0.336440
$738$	21.1194	0.777417
$739$	23.7340	0.873069	0.436534	−	0.899688i	$-0.356206\pi$
$739$	23.7340	0.873069	0.436534	+	0.899688i	$0.356206\pi$
$740$	3.83139	0.140845
$741$	−1.46310	−0.0537484
$742$	0	0
$743$	35.7622	1.31199	0.655993	−	0.754767i	$-0.272250\pi$
$743$	35.7622	1.31199	0.655993	+	0.754767i	$0.272250\pi$
$744$	−1.68479	−0.0617675
$745$	36.1921	1.32598
$746$	11.8178	0.432681
$747$	19.8387	0.725860
$748$	4.20628	0.153797
$749$	0	0
$750$	−9.94154	−0.363014
$751$	−7.91184	−0.288707	−0.144354	−	0.989526i	$-0.546110\pi$
$751$	−7.91184	−0.288707	−0.144354	+	0.989526i	$0.546110\pi$
$752$	−2.11107	−0.0769827
$753$	3.32183	0.121054
$754$	1.29033	0.0469909
$755$	18.0283	0.656119
$756$	0	0
$757$	−17.3642	−0.631112	−0.315556	−	0.948907i	$-0.602191\pi$
$757$	−17.3642	−0.631112	−0.315556	+	0.948907i	$0.602191\pi$
$758$	−24.7462	−0.898822
$759$	−23.2496	−0.843909
$760$	6.59164	0.239104
$761$	−0.655566	−0.0237642	−0.0118821	−	0.999929i	$-0.503782\pi$
$761$	−0.655566	−0.0237642	−0.0118821	+	0.999929i	$0.503782\pi$
$762$	12.6088	0.456770
$763$	0	0
$764$	4.81241	0.174107
$765$	−4.24294	−0.153404
$766$	1.20077	0.0433855
$767$	−1.39626	−0.0504160
$768$	0.824076	0.0297363
$769$	−11.8682	−0.427980	−0.213990	−	0.976836i	$-0.568646\pi$
$769$	−11.8682	−0.427980	−0.213990	+	0.976836i	$0.568646\pi$
$770$	0	0
$771$	−23.3511	−0.840970
$772$	−20.0451	−0.721439
$773$	−8.44453	−0.303729	−0.151864	−	0.988401i	$-0.548528\pi$
$773$	−8.44453	−0.303729	−0.151864	+	0.988401i	$0.548528\pi$
$774$	25.1342	0.903432
$775$	−4.42712	−0.159027
$776$	−12.8858	−0.462574
$777$	0	0
$778$	26.0638	0.934431
$779$	35.6266	1.27646
$780$	0.629173	0.0225280
$781$	−9.39521	−0.336187
$782$	7.90831	0.282801
$783$	−12.4765	−0.445875
$784$	0	0
$785$	3.05509	0.109041
$786$	−14.2263	−0.507436
$787$	11.2818	0.402152	0.201076	−	0.979576i	$-0.435556\pi$
$787$	11.2818	0.402152	0.201076	+	0.979576i	$0.435556\pi$
$788$	10.1512	0.361622
$789$	5.25845	0.187206
$790$	−1.68362	−0.0599006
$791$	0	0
$792$	−8.99058	−0.319466
$793$	6.43193	0.228404
$794$	−24.3057	−0.862577
$795$	−13.4034	−0.475371
$796$	−1.81283	−0.0642540
$797$	−50.3744	−1.78435	−0.892176	−	0.451687i	$-0.850822\pi$
$797$	−50.3744	−1.78435	−0.892176	+	0.451687i	$0.850822\pi$
$798$	0	0
$799$	2.29229	0.0810953
$800$	2.16542	0.0765592
$801$	−5.43763	−0.192129
$802$	−4.02693	−0.142196
$803$	33.3592	1.17722
$804$	−1.94302	−0.0685251
$805$	0	0
$806$	0.927124	0.0326566
$807$	13.1090	0.461459
$808$	−18.1323	−0.637892
$809$	18.2376	0.641200	0.320600	−	0.947215i	$-0.396115\pi$
$809$	18.2376	0.641200	0.320600	+	0.947215i	$0.396115\pi$
$810$	5.63889	0.198130
$811$	45.1798	1.58648	0.793239	−	0.608911i	$-0.208393\pi$
$811$	45.1798	1.58648	0.793239	+	0.608911i	$0.208393\pi$
$812$	0	0
$813$	−21.8172	−0.765162
$814$	−8.81542	−0.308980
$815$	−12.4670	−0.436701
$816$	−0.894817	−0.0313249
$817$	42.3993	1.48336
$818$	1.80916	0.0632557
$819$	0	0
$820$	−15.3204	−0.535012
$821$	−35.3415	−1.23343	−0.616714	−	0.787187i	$-0.711537\pi$
$821$	−35.3415	−1.23343	−0.616714	+	0.787187i	$0.711537\pi$
$822$	19.0702	0.665149
$823$	−3.92969	−0.136981	−0.0684903	−	0.997652i	$-0.521818\pi$
$823$	−3.92969	−0.136981	−0.0684903	+	0.997652i	$0.521818\pi$
$824$	16.9795	0.591509
$825$	6.91260	0.240666
$826$	0	0
$827$	−4.83566	−0.168152	−0.0840762	−	0.996459i	$-0.526794\pi$
$827$	−4.83566	−0.168152	−0.0840762	+	0.996459i	$0.526794\pi$
$828$	−16.9034	−0.587433
$829$	−1.44732	−0.0502675	−0.0251337	−	0.999684i	$-0.508001\pi$
$829$	−1.44732	−0.0502675	−0.0251337	+	0.999684i	$0.508001\pi$
$830$	−14.3914	−0.499531
$831$	−12.6447	−0.438640
$832$	−0.453480	−0.0157216
$833$	0	0
$834$	8.05855	0.279045
$835$	13.7712	0.476572
$836$	−15.1663	−0.524539
$837$	−8.96462	−0.309863
$838$	32.1832	1.11175
$839$	−8.16265	−0.281806	−0.140903	−	0.990023i	$-0.545001\pi$
$839$	−8.16265	−0.281806	−0.140903	+	0.990023i	$0.545001\pi$
$840$	0	0
$841$	−20.9038	−0.720820
$842$	0.858901	0.0295997
$843$	7.89279	0.271842
$844$	−25.9414	−0.892939
$845$	21.5408	0.741028
$846$	−4.89957	−0.168451
$847$	0	0
$848$	9.66060	0.331747
$849$	16.3043	0.559563
$850$	−2.35131	−0.0806491
$851$	−16.5740	−0.568151
$852$	1.99868	0.0684735
$853$	−47.6689	−1.63215	−0.816075	−	0.577946i	$-0.803855\pi$
$853$	−47.6689	−1.63215	−0.816075	+	0.577946i	$0.803855\pi$
$854$	0	0
$855$	15.2985	0.523198
$856$	16.6598	0.569419
$857$	−40.5448	−1.38498	−0.692491	−	0.721426i	$-0.743487\pi$
$857$	−40.5448	−1.38498	−0.692491	+	0.721426i	$0.743487\pi$
$858$	−1.44763	−0.0494213
$859$	−44.1672	−1.50696	−0.753482	−	0.657468i	$-0.771627\pi$
$859$	−44.1672	−1.50696	−0.753482	+	0.657468i	$0.771627\pi$
$860$	−18.2328	−0.621734
$861$	0	0
$862$	22.0754	0.751891
$863$	−48.6021	−1.65443	−0.827217	−	0.561883i	$-0.810077\pi$
$863$	−48.6021	−1.65443	−0.827217	+	0.561883i	$0.810077\pi$
$864$	4.38483	0.149175
$865$	6.23790	0.212095
$866$	8.15515	0.277123
$867$	−13.0377	−0.442782
$868$	0	0
$869$	3.87375	0.131408
$870$	3.94778	0.133842
$871$	1.06922	0.0362293
$872$	10.3447	0.350316
$873$	−29.9067	−1.01219
$874$	−28.5145	−0.964518
$875$	0	0
$876$	−7.09663	−0.239773
$877$	−57.5125	−1.94206	−0.971029	−	0.238963i	$-0.923193\pi$
$877$	−57.5125	−1.94206	−0.971029	+	0.238963i	$0.923193\pi$
$878$	−26.3289	−0.888556
$879$	0.361545	0.0121946
$880$	6.52193	0.219854
$881$	−26.1540	−0.881151	−0.440576	−	0.897716i	$-0.645226\pi$
$881$	−26.1540	−0.881151	−0.440576	+	0.897716i	$0.645226\pi$
$882$	0	0
$883$	−4.89433	−0.164707	−0.0823536	−	0.996603i	$-0.526244\pi$
$883$	−4.89433	−0.164707	−0.0823536	+	0.996603i	$0.526244\pi$
$884$	0.492408	0.0165615
$885$	−4.27188	−0.143598
$886$	−15.9048	−0.534333
$887$	−5.75592	−0.193265	−0.0966324	−	0.995320i	$-0.530807\pi$
$887$	−5.75592	−0.193265	−0.0966324	+	0.995320i	$0.530807\pi$
$888$	1.87534	0.0629321
$889$	0	0
$890$	3.94456	0.132222
$891$	−12.9742	−0.434652
$892$	4.06874	0.136232
$893$	−8.26516	−0.276583
$894$	17.7148	0.592472
$895$	42.6711	1.42634
$896$	0	0
$897$	−2.72172	−0.0908755
$898$	37.4697	1.25038
$899$	5.81729	0.194017
$900$	5.02572	0.167524
$901$	−10.4899	−0.349469
$902$	35.2499	1.17369
$903$	0	0
$904$	−14.8905	−0.495251
$905$	−16.1697	−0.537498
$906$	8.82428	0.293167
$907$	43.4100	1.44141	0.720703	−	0.693244i	$-0.243819\pi$
$907$	43.4100	1.44141	0.720703	+	0.693244i	$0.243819\pi$
$908$	−5.39652	−0.179090
$909$	−42.0832	−1.39581
$910$	0	0
$911$	−14.9509	−0.495345	−0.247672	−	0.968844i	$-0.579666\pi$
$911$	−14.9509	−0.495345	−0.247672	+	0.968844i	$0.579666\pi$
$912$	3.22639	0.106836
$913$	33.1123	1.09586
$914$	8.81695	0.291639
$915$	19.6786	0.650555
$916$	−26.2002	−0.865680
$917$	0	0
$918$	−4.76123	−0.157144
$919$	−1.31010	−0.0432163	−0.0216082	−	0.999767i	$-0.506879\pi$
$919$	−1.31010	−0.0432163	−0.0216082	+	0.999767i	$0.506879\pi$
$920$	12.2620	0.404266
$921$	−1.81016	−0.0596469
$922$	−25.7035	−0.846499
$923$	−1.09985	−0.0362020
$924$	0	0
$925$	4.92781	0.162025
$926$	2.56657	0.0843426
$927$	39.4077	1.29432
$928$	−2.84539	−0.0934044
$929$	−7.51439	−0.246539	−0.123270	−	0.992373i	$-0.539338\pi$
$929$	−7.51439	−0.246539	−0.123270	+	0.992373i	$0.539338\pi$
$930$	2.83655	0.0930143
$931$	0	0
$932$	17.0742	0.559284
$933$	−14.4596	−0.473388
$934$	29.8036	0.975204
$935$	−7.08179	−0.231599
$936$	−1.05248	−0.0344014
$937$	5.34963	0.174765	0.0873824	−	0.996175i	$-0.472150\pi$
$937$	5.34963	0.174765	0.0873824	+	0.996175i	$0.472150\pi$
$938$	0	0
$939$	21.7566	0.709998
$940$	3.55424	0.115927
$941$	49.4200	1.61105	0.805524	−	0.592564i	$-0.201884\pi$
$941$	49.4200	1.61105	0.805524	+	0.592564i	$0.201884\pi$
$942$	1.49537	0.0487217
$943$	66.2740	2.15818
$944$	3.07898	0.100212
$945$	0	0
$946$	41.9509	1.36394
$947$	−46.6275	−1.51519	−0.757595	−	0.652725i	$-0.773626\pi$
$947$	−46.6275	−1.51519	−0.757595	+	0.652725i	$0.773626\pi$
$948$	−0.824076	−0.0267648
$949$	3.90520	0.126768
$950$	8.47795	0.275061
$951$	4.52727	0.146807
$952$	0	0
$953$	−33.8697	−1.09715	−0.548574	−	0.836102i	$-0.684829\pi$
$953$	−33.8697	−1.09715	−0.548574	+	0.836102i	$0.684829\pi$
$954$	22.4213	0.725916
$955$	−8.10228	−0.262183
$956$	13.4401	0.434685
$957$	−9.08323	−0.293619
$958$	−10.2725	−0.331890
$959$	0	0
$960$	−1.38743	−0.0447792
$961$	−26.8202	−0.865167
$962$	−1.03198	−0.0332723
$963$	38.6656	1.24598
$964$	11.4319	0.368198
$965$	33.7484	1.08640
$966$	0	0
$967$	−19.5621	−0.629076	−0.314538	−	0.949245i	$-0.601850\pi$
$967$	−19.5621	−0.629076	−0.314538	+	0.949245i	$0.601850\pi$
$968$	−4.00594	−0.128756
$969$	−3.50335	−0.112544
$970$	21.6948	0.696579
$971$	44.4282	1.42577	0.712885	−	0.701281i	$-0.247388\pi$
$971$	44.4282	1.42577	0.712885	+	0.701281i	$0.247388\pi$
$972$	15.9145	0.510459
$973$	0	0
$974$	13.3276	0.427045
$975$	0.809223	0.0259159
$976$	−14.1835	−0.454002
$977$	−12.0738	−0.386276	−0.193138	−	0.981172i	$-0.561867\pi$
$977$	−12.0738	−0.386276	−0.193138	+	0.981172i	$0.561867\pi$
$978$	−6.10219	−0.195127
$979$	−9.07581	−0.290064
$980$	0	0
$981$	24.0090	0.766549
$982$	−16.1567	−0.515580
$983$	−35.1363	−1.12067	−0.560337	−	0.828264i	$-0.689329\pi$
$983$	−35.1363	−1.12067	−0.560337	+	0.828264i	$0.689329\pi$
$984$	−7.49883	−0.239054
$985$	−17.0908	−0.544558
$986$	3.08964	0.0983942
$987$	0	0
$988$	−1.77545	−0.0564845
$989$	78.8727	2.50801
$990$	15.1367	0.481077
$991$	1.60609	0.0510190	0.0255095	−	0.999675i	$-0.491879\pi$
$991$	1.60609	0.0510190	0.0255095	+	0.999675i	$0.491879\pi$
$992$	−2.04446	−0.0649118
$993$	−0.370440	−0.0117556
$994$	0	0
$995$	3.05212	0.0967586
$996$	−7.04409	−0.223201
$997$	−58.9518	−1.86702	−0.933512	−	0.358546i	$-0.883272\pi$
$997$	−58.9518	−1.86702	−0.933512	+	0.358546i	$0.883272\pi$
$998$	3.57847	0.113274
$999$	9.97847	0.315705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7742.2.a.bl.1.8	yes	12
7.6	odd	2	inner	7742.2.a.bl.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7742.2.a.bl.1.5	✓	12	7.6	odd	2	inner
7742.2.a.bl.1.8	yes	12	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.824076 q^{3} +1.00000 q^{4} -1.68362 q^{5} -0.824076 q^{6} -1.00000 q^{8} -2.32090 q^{9} +1.68362 q^{10} -3.87375 q^{11} +0.824076 q^{12} -0.453480 q^{13} -1.38743 q^{15} +1.00000 q^{16} -1.08584 q^{17} +2.32090 q^{18} +3.91515 q^{19} -1.68362 q^{20} +3.87375 q^{22} +7.28311 q^{23} -0.824076 q^{24} -2.16542 q^{25} +0.453480 q^{26} -4.38483 q^{27} +2.84539 q^{29} +1.38743 q^{30} +2.04446 q^{31} -1.00000 q^{32} -3.19227 q^{33} +1.08584 q^{34} -2.32090 q^{36} -2.27568 q^{37} -3.91515 q^{38} -0.373702 q^{39} +1.68362 q^{40} +9.09968 q^{41} +10.8295 q^{43} -3.87375 q^{44} +3.90751 q^{45} -7.28311 q^{46} -2.11107 q^{47} +0.824076 q^{48} +2.16542 q^{50} -0.894817 q^{51} -0.453480 q^{52} +9.66060 q^{53} +4.38483 q^{54} +6.52193 q^{55} +3.22639 q^{57} -2.84539 q^{58} +3.07898 q^{59} -1.38743 q^{60} -14.1835 q^{61} -2.04446 q^{62} +1.00000 q^{64} +0.763489 q^{65} +3.19227 q^{66} -2.35782 q^{67} -1.08584 q^{68} +6.00184 q^{69} +2.42535 q^{71} +2.32090 q^{72} -8.61161 q^{73} +2.27568 q^{74} -1.78447 q^{75} +3.91515 q^{76} +0.373702 q^{78} -1.00000 q^{79} -1.68362 q^{80} +3.34926 q^{81} -9.09968 q^{82} -8.54786 q^{83} +1.82815 q^{85} -10.8295 q^{86} +2.34482 q^{87} +3.87375 q^{88} +2.34290 q^{89} -3.90751 q^{90} +7.28311 q^{92} +1.68479 q^{93} +2.11107 q^{94} -6.59164 q^{95} -0.824076 q^{96} +12.8858 q^{97} +8.99058 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} + 12 q^{9} + 4 q^{11} - 12 q^{15} + 12 q^{16} - 12 q^{18} - 4 q^{22} - 4 q^{23} - 20 q^{25} - 4 q^{29} + 12 q^{30} - 12 q^{32} + 12 q^{36} - 16 q^{37} - 24 q^{39} + 4 q^{43}+ \cdots + 44 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 12 * q^8 + 12 * q^9 + 4 * q^11 - 12 * q^15 + 12 * q^16 - 12 * q^18 - 4 * q^22 - 4 * q^23 - 20 * q^25 - 4 * q^29 + 12 * q^30 - 12 * q^32 + 12 * q^36 - 16 * q^37 - 24 * q^39 + 4 * q^43 + 4 * q^44 + 4 * q^46 + 20 * q^50 + 4 * q^51 - 8 * q^53 - 28 * q^57 + 4 * q^58 - 12 * q^60 + 12 * q^64 - 36 * q^65 - 112 * q^67 + 44 * q^71 - 12 * q^72 + 16 * q^74 + 24 * q^78 - 12 * q^79 - 12 * q^81 + 4 * q^85 - 4 * q^86 - 4 * q^88 - 4 * q^92 - 60 * q^93 - 16 * q^95 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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