LMFDB - Embedded newform 7742.2.a.bk.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,-3,12,3,3,0,-12,17,-3,-2] 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} - 3 x^{11} - 22 x^{10} + 67 x^{9} + 148 x^{8} - 471 x^{7} - 306 x^{6} + 1117 x^{5} + 207 x^{4} + \cdots - 7 $ x^12 - 3x^11 - 22x^10 + 67x^9 + 148x^8 - 471x^7 - 306x^6 + 1117x^5 + 207x^4 - 927x^3 + 11x^2 + 200*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1106)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.540089$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +0.540089 q^{3} +1.00000 q^{4} +2.34575 q^{5} -0.540089 q^{6} -1.00000 q^{8} -2.70830 q^{9} -2.34575 q^{10} +4.81099 q^{11} +0.540089 q^{12} +0.465358 q^{13} +1.26691 q^{15} +1.00000 q^{16} -7.48660 q^{17} +2.70830 q^{18} -3.95232 q^{19} +2.34575 q^{20} -4.81099 q^{22} -7.87103 q^{23} -0.540089 q^{24} +0.502533 q^{25} -0.465358 q^{26} -3.08299 q^{27} +9.07011 q^{29} -1.26691 q^{30} +4.00334 q^{31} -1.00000 q^{32} +2.59836 q^{33} +7.48660 q^{34} -2.70830 q^{36} -6.62563 q^{37} +3.95232 q^{38} +0.251335 q^{39} -2.34575 q^{40} +1.75310 q^{41} +11.6750 q^{43} +4.81099 q^{44} -6.35300 q^{45} +7.87103 q^{46} +0.109639 q^{47} +0.540089 q^{48} -0.502533 q^{50} -4.04343 q^{51} +0.465358 q^{52} +4.14410 q^{53} +3.08299 q^{54} +11.2854 q^{55} -2.13460 q^{57} -9.07011 q^{58} -4.87503 q^{59} +1.26691 q^{60} -3.90449 q^{61} -4.00334 q^{62} +1.00000 q^{64} +1.09161 q^{65} -2.59836 q^{66} -6.40231 q^{67} -7.48660 q^{68} -4.25105 q^{69} -12.3035 q^{71} +2.70830 q^{72} -10.2392 q^{73} +6.62563 q^{74} +0.271413 q^{75} -3.95232 q^{76} -0.251335 q^{78} -1.00000 q^{79} +2.34575 q^{80} +6.45982 q^{81} -1.75310 q^{82} -5.21405 q^{83} -17.5617 q^{85} -11.6750 q^{86} +4.89866 q^{87} -4.81099 q^{88} +3.25457 q^{89} +6.35300 q^{90} -7.87103 q^{92} +2.16216 q^{93} -0.109639 q^{94} -9.27115 q^{95} -0.540089 q^{96} +1.45337 q^{97} -13.0296 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} - 12 q^{8} + 17 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 13 q^{13} - 2 q^{15} + 12 q^{16} - 5 q^{17} - 17 q^{18} - 14 q^{19} + 3 q^{20} + 2 q^{22}+ \cdots - 19 q^{99}+O(q^{100}) $ 12 * q - 12 * q^2 - 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 - 12 * q^8 + 17 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 13 * q^13 - 2 * q^15 + 12 * q^16 - 5 * q^17 - 17 * q^18 - 14 * q^19 + 3 * q^20 + 2 * q^22 + 2 * q^23 + 3 * q^24 + 21 * q^25 + 13 * q^26 - 6 * q^27 + 7 * q^29 + 2 * q^30 - 15 * q^31 - 12 * q^32 - 23 * q^33 + 5 * q^34 + 17 * q^36 - 15 * q^37 + 14 * q^38 + 18 * q^39 - 3 * q^40 - 10 * q^41 + 8 * q^43 - 2 * q^44 - 47 * q^45 - 2 * q^46 + 13 * q^47 - 3 * q^48 - 21 * q^50 - 24 * q^51 - 13 * q^52 + 8 * q^53 + 6 * q^54 + q^55 - 20 * q^57 - 7 * q^58 - 10 * q^59 - 2 * q^60 + 6 * q^61 + 15 * q^62 + 12 * q^64 - 25 * q^65 + 23 * q^66 + 18 * q^67 - 5 * q^68 + 25 * q^69 - 21 * q^71 - 17 * q^72 - 45 * q^73 + 15 * q^74 + 3 * q^75 - 14 * q^76 - 18 * q^78 - 12 * q^79 + 3 * q^80 + 76 * q^81 + 10 * q^82 + 15 * q^83 - 35 * q^85 - 8 * q^86 - 50 * q^87 + 2 * q^88 - 5 * q^89 + 47 * q^90 + 2 * q^92 - 4 * q^93 - 13 * q^94 + 49 * q^95 + 3 * q^96 - 53 * q^97 - 19 * 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
$3$	0.540089	0.311820	0.155910	−	0.987771i	$-0.450169\pi$
$3$	0.540089	0.311820	0.155910	+	0.987771i	$0.450169\pi$
$4$	1.00000	0.500000
$5$	2.34575	1.04905	0.524525	−	0.851395i	$-0.324243\pi$
$5$	2.34575	1.04905	0.524525	+	0.851395i	$0.324243\pi$
$6$	−0.540089	−0.220490
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.70830	−0.902768
$10$	−2.34575	−0.741791
$11$	4.81099	1.45057	0.725284	−	0.688450i	$-0.241709\pi$
$11$	4.81099	1.45057	0.725284	+	0.688450i	$0.241709\pi$
$12$	0.540089	0.155910
$13$	0.465358	0.129067	0.0645335	−	0.997916i	$-0.479444\pi$
$13$	0.465358	0.129067	0.0645335	+	0.997916i	$0.479444\pi$
$14$	0	0
$15$	1.26691	0.327115
$16$	1.00000	0.250000
$17$	−7.48660	−1.81577	−0.907884	−	0.419222i	$-0.862303\pi$
$17$	−7.48660	−1.81577	−0.907884	+	0.419222i	$0.862303\pi$
$18$	2.70830	0.638353
$19$	−3.95232	−0.906724	−0.453362	−	0.891326i	$-0.649776\pi$
$19$	−3.95232	−0.906724	−0.453362	+	0.891326i	$0.649776\pi$
$20$	2.34575	0.524525
$21$	0	0
$22$	−4.81099	−1.02571
$23$	−7.87103	−1.64122	−0.820611	−	0.571487i	$-0.806367\pi$
$23$	−7.87103	−1.64122	−0.820611	+	0.571487i	$0.806367\pi$
$24$	−0.540089	−0.110245
$25$	0.502533	0.100507
$26$	−0.465358	−0.0912642
$27$	−3.08299	−0.593322
$28$	0	0
$29$	9.07011	1.68428	0.842138	−	0.539261i	$-0.181297\pi$
$29$	9.07011	1.68428	0.842138	+	0.539261i	$0.181297\pi$
$30$	−1.26691	−0.231305
$31$	4.00334	0.719021	0.359511	−	0.933141i	$-0.382944\pi$
$31$	4.00334	0.719021	0.359511	+	0.933141i	$0.382944\pi$
$32$	−1.00000	−0.176777
$33$	2.59836	0.452317
$34$	7.48660	1.28394
$35$	0	0
$36$	−2.70830	−0.451384
$37$	−6.62563	−1.08925	−0.544624	−	0.838681i	$-0.683327\pi$
$37$	−6.62563	−1.08925	−0.544624	+	0.838681i	$0.683327\pi$
$38$	3.95232	0.641151
$39$	0.251335	0.0402458
$40$	−2.34575	−0.370895
$41$	1.75310	0.273789	0.136894	−	0.990586i	$-0.456288\pi$
$41$	1.75310	0.273789	0.136894	+	0.990586i	$0.456288\pi$
$42$	0	0
$43$	11.6750	1.78042	0.890210	−	0.455550i	$-0.150557\pi$
$43$	11.6750	1.78042	0.890210	+	0.455550i	$0.150557\pi$
$44$	4.81099	0.725284
$45$	−6.35300	−0.947049
$46$	7.87103	1.16052
$47$	0.109639	0.0159924	0.00799621	−	0.999968i	$-0.497455\pi$
$47$	0.109639	0.0159924	0.00799621	+	0.999968i	$0.497455\pi$
$48$	0.540089	0.0779551
$49$	0	0
$50$	−0.502533	−0.0710689
$51$	−4.04343	−0.566193
$52$	0.465358	0.0645335
$53$	4.14410	0.569236	0.284618	−	0.958641i	$-0.408133\pi$
$53$	4.14410	0.569236	0.284618	+	0.958641i	$0.408133\pi$
$54$	3.08299	0.419542
$55$	11.2854	1.52172
$56$	0	0
$57$	−2.13460	−0.282735
$58$	−9.07011	−1.19096
$59$	−4.87503	−0.634675	−0.317338	−	0.948313i	$-0.602789\pi$
$59$	−4.87503	−0.634675	−0.317338	+	0.948313i	$0.602789\pi$
$60$	1.26691	0.163558
$61$	−3.90449	−0.499919	−0.249959	−	0.968256i	$-0.580417\pi$
$61$	−3.90449	−0.499919	−0.249959	+	0.968256i	$0.580417\pi$
$62$	−4.00334	−0.508425
$63$	0	0
$64$	1.00000	0.125000
$65$	1.09161	0.135398
$66$	−2.59836	−0.319836
$67$	−6.40231	−0.782166	−0.391083	−	0.920355i	$-0.627899\pi$
$67$	−6.40231	−0.782166	−0.391083	+	0.920355i	$0.627899\pi$
$68$	−7.48660	−0.907884
$69$	−4.25105	−0.511767
$70$	0	0
$71$	−12.3035	−1.46015	−0.730076	−	0.683366i	$-0.760516\pi$
$71$	−12.3035	−1.46015	−0.730076	+	0.683366i	$0.760516\pi$
$72$	2.70830	0.319177
$73$	−10.2392	−1.19840	−0.599201	−	0.800599i	$-0.704515\pi$
$73$	−10.2392	−1.19840	−0.599201	+	0.800599i	$0.704515\pi$
$74$	6.62563	0.770214
$75$	0.271413	0.0313400
$76$	−3.95232	−0.453362
$77$	0	0
$78$	−0.251335	−0.0284580
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	2.34575	0.262263
$81$	6.45982	0.717758
$82$	−1.75310	−0.193598
$83$	−5.21405	−0.572317	−0.286158	−	0.958182i	$-0.592378\pi$
$83$	−5.21405	−0.572317	−0.286158	+	0.958182i	$0.592378\pi$
$84$	0	0
$85$	−17.5617	−1.90483
$86$	−11.6750	−1.25895
$87$	4.89866	0.525192
$88$	−4.81099	−0.512853
$89$	3.25457	0.344984	0.172492	−	0.985011i	$-0.444818\pi$
$89$	3.25457	0.344984	0.172492	+	0.985011i	$0.444818\pi$
$90$	6.35300	0.669665
$91$	0	0
$92$	−7.87103	−0.820611
$93$	2.16216	0.224206
$94$	−0.109639	−0.0113084
$95$	−9.27115	−0.951200
$96$	−0.540089	−0.0551226
$97$	1.45337	0.147567	0.0737836	−	0.997274i	$-0.476493\pi$
$97$	1.45337	0.147567	0.0737836	+	0.997274i	$0.476493\pi$
$98$	0	0
$99$	−13.0296	−1.30953
$100$	0.502533	0.0502533
$101$	18.5887	1.84965	0.924823	−	0.380397i	$-0.124213\pi$
$101$	18.5887	1.84965	0.924823	+	0.380397i	$0.124213\pi$
$102$	4.04343	0.400359
$103$	−10.7765	−1.06184	−0.530918	−	0.847423i	$-0.678153\pi$
$103$	−10.7765	−1.06184	−0.530918	+	0.847423i	$0.678153\pi$
$104$	−0.465358	−0.0456321
$105$	0	0
$106$	−4.14410	−0.402511
$107$	4.84763	0.468638	0.234319	−	0.972160i	$-0.424714\pi$
$107$	4.84763	0.468638	0.234319	+	0.972160i	$0.424714\pi$
$108$	−3.08299	−0.296661
$109$	−2.15624	−0.206530	−0.103265	−	0.994654i	$-0.532929\pi$
$109$	−2.15624	−0.206530	−0.103265	+	0.994654i	$0.532929\pi$
$110$	−11.2854	−1.07602
$111$	−3.57843	−0.339650
$112$	0	0
$113$	−14.4701	−1.36123	−0.680615	−	0.732641i	$-0.738287\pi$
$113$	−14.4701	−1.36123	−0.680615	+	0.732641i	$0.738287\pi$
$114$	2.13460	0.199924
$115$	−18.4634	−1.72172
$116$	9.07011	0.842138
$117$	−1.26033	−0.116518
$118$	4.87503	0.448783
$119$	0	0
$120$	−1.26691	−0.115653
$121$	12.1456	1.10415
$122$	3.90449	0.353496
$123$	0.946831	0.0853729
$124$	4.00334	0.359511
$125$	−10.5499	−0.943614
$126$	0	0
$127$	−9.87284	−0.876073	−0.438037	−	0.898957i	$-0.644326\pi$
$127$	−9.87284	−0.876073	−0.438037	+	0.898957i	$0.644326\pi$
$128$	−1.00000	−0.0883883
$129$	6.30554	0.555172
$130$	−1.09161	−0.0957408
$131$	−15.1901	−1.32717	−0.663583	−	0.748103i	$-0.730965\pi$
$131$	−15.1901	−1.32717	−0.663583	+	0.748103i	$0.730965\pi$
$132$	2.59836	0.226158
$133$	0	0
$134$	6.40231	0.553075
$135$	−7.23192	−0.622425
$136$	7.48660	0.641971
$137$	−21.9483	−1.87517	−0.937587	−	0.347752i	$-0.886945\pi$
$137$	−21.9483	−1.87517	−0.937587	+	0.347752i	$0.886945\pi$
$138$	4.25105	0.361874
$139$	22.3469	1.89544	0.947718	−	0.319109i	$-0.103384\pi$
$139$	22.3469	1.89544	0.947718	+	0.319109i	$0.103384\pi$
$140$	0	0
$141$	0.0592145	0.00498676
$142$	12.3035	1.03248
$143$	2.23883	0.187221
$144$	−2.70830	−0.225692
$145$	21.2762	1.76689
$146$	10.2392	0.847398
$147$	0	0
$148$	−6.62563	−0.544624
$149$	−3.20887	−0.262881	−0.131441	−	0.991324i	$-0.541960\pi$
$149$	−3.20887	−0.262881	−0.131441	+	0.991324i	$0.541960\pi$
$150$	−0.271413	−0.0221607
$151$	−11.1004	−0.903339	−0.451670	−	0.892185i	$-0.649171\pi$
$151$	−11.1004	−0.903339	−0.451670	+	0.892185i	$0.649171\pi$
$152$	3.95232	0.320576
$153$	20.2760	1.63922
$154$	0	0
$155$	9.39083	0.754290
$156$	0.251335	0.0201229
$157$	−23.7364	−1.89437	−0.947185	−	0.320687i	$-0.896086\pi$
$157$	−23.7364	−1.89437	−0.947185	+	0.320687i	$0.896086\pi$
$158$	1.00000	0.0795557
$159$	2.23818	0.177499
$160$	−2.34575	−0.185448
$161$	0	0
$162$	−6.45982	−0.507532
$163$	1.73722	0.136069	0.0680347	−	0.997683i	$-0.478327\pi$
$163$	1.73722	0.136069	0.0680347	+	0.997683i	$0.478327\pi$
$164$	1.75310	0.136894
$165$	6.09510	0.474503
$166$	5.21405	0.404689
$167$	−7.07847	−0.547749	−0.273874	−	0.961765i	$-0.588305\pi$
$167$	−7.07847	−0.547749	−0.273874	+	0.961765i	$0.588305\pi$
$168$	0	0
$169$	−12.7834	−0.983342
$170$	17.5617	1.34692
$171$	10.7041	0.818562
$172$	11.6750	0.890210
$173$	4.40152	0.334641	0.167321	−	0.985903i	$-0.446488\pi$
$173$	4.40152	0.334641	0.167321	+	0.985903i	$0.446488\pi$
$174$	−4.89866	−0.371367
$175$	0	0
$176$	4.81099	0.362642
$177$	−2.63295	−0.197905
$178$	−3.25457	−0.243941
$179$	−2.58332	−0.193087	−0.0965433	−	0.995329i	$-0.530779\pi$
$179$	−2.58332	−0.193087	−0.0965433	+	0.995329i	$0.530779\pi$
$180$	−6.35300	−0.473525
$181$	8.66649	0.644175	0.322088	−	0.946710i	$-0.395615\pi$
$181$	8.66649	0.644175	0.322088	+	0.946710i	$0.395615\pi$
$182$	0	0
$183$	−2.10877	−0.155885
$184$	7.87103	0.580260
$185$	−15.5421	−1.14268
$186$	−2.16216	−0.158537
$187$	−36.0179	−2.63389
$188$	0.109639	0.00799621
$189$	0	0
$190$	9.27115	0.672600
$191$	−10.4948	−0.759378	−0.379689	−	0.925114i	$-0.623969\pi$
$191$	−10.4948	−0.759378	−0.379689	+	0.925114i	$0.623969\pi$
$192$	0.540089	0.0389776
$193$	18.3614	1.32169	0.660843	−	0.750524i	$-0.270199\pi$
$193$	18.3614	1.32169	0.660843	+	0.750524i	$0.270199\pi$
$194$	−1.45337	−0.104346
$195$	0.589568	0.0422198
$196$	0	0
$197$	20.0932	1.43158	0.715791	−	0.698314i	$-0.246066\pi$
$197$	20.0932	1.43158	0.715791	+	0.698314i	$0.246066\pi$
$198$	13.0296	0.925975
$199$	12.8398	0.910188	0.455094	−	0.890443i	$-0.349606\pi$
$199$	12.8398	0.910188	0.455094	+	0.890443i	$0.349606\pi$
$200$	−0.502533	−0.0355345
$201$	−3.45781	−0.243895
$202$	−18.5887	−1.30790
$203$	0	0
$204$	−4.04343	−0.283097
$205$	4.11234	0.287218
$206$	10.7765	0.750832
$207$	21.3171	1.48164
$208$	0.465358	0.0322668
$209$	−19.0146	−1.31527
$210$	0	0
$211$	19.2697	1.32658	0.663290	−	0.748362i	$-0.269160\pi$
$211$	19.2697	1.32658	0.663290	+	0.748362i	$0.269160\pi$
$212$	4.14410	0.284618
$213$	−6.64496	−0.455305
$214$	−4.84763	−0.331377
$215$	27.3866	1.86775
$216$	3.08299	0.209771
$217$	0	0
$218$	2.15624	0.146039
$219$	−5.53005	−0.373686
$220$	11.2854	0.760859
$221$	−3.48395	−0.234356
$222$	3.57843	0.240169
$223$	−16.8664	−1.12946	−0.564729	−	0.825276i	$-0.691019\pi$
$223$	−16.8664	−1.12946	−0.564729	+	0.825276i	$0.691019\pi$
$224$	0	0
$225$	−1.36101	−0.0907342
$226$	14.4701	0.962535
$227$	−17.6436	−1.17104	−0.585522	−	0.810656i	$-0.699110\pi$
$227$	−17.6436	−1.17104	−0.585522	+	0.810656i	$0.699110\pi$
$228$	−2.13460	−0.141368
$229$	−8.49583	−0.561420	−0.280710	−	0.959793i	$-0.590570\pi$
$229$	−8.49583	−0.561420	−0.280710	+	0.959793i	$0.590570\pi$
$230$	18.4634	1.21744
$231$	0	0
$232$	−9.07011	−0.595482
$233$	−11.0571	−0.724373	−0.362186	−	0.932106i	$-0.617970\pi$
$233$	−11.0571	−0.724373	−0.362186	+	0.932106i	$0.617970\pi$
$234$	1.26033	0.0823904
$235$	0.257184	0.0167769
$236$	−4.87503	−0.317338
$237$	−0.540089	−0.0350825
$238$	0	0
$239$	1.10050	0.0711855	0.0355927	−	0.999366i	$-0.488668\pi$
$239$	1.10050	0.0711855	0.0355927	+	0.999366i	$0.488668\pi$
$240$	1.26691	0.0817788
$241$	−25.6900	−1.65484	−0.827420	−	0.561584i	$-0.810192\pi$
$241$	−25.6900	−1.65484	−0.827420	+	0.561584i	$0.810192\pi$
$242$	−12.1456	−0.780749
$243$	12.7379	0.817134
$244$	−3.90449	−0.249959
$245$	0	0
$246$	−0.946831	−0.0603677
$247$	−1.83924	−0.117028
$248$	−4.00334	−0.254212
$249$	−2.81605	−0.178460
$250$	10.5499	0.667236
$251$	−18.6294	−1.17588	−0.587938	−	0.808906i	$-0.700060\pi$
$251$	−18.6294	−1.17588	−0.587938	+	0.808906i	$0.700060\pi$
$252$	0	0
$253$	−37.8674	−2.38070
$254$	9.87284	0.619477
$255$	−9.48486	−0.593965
$256$	1.00000	0.0625000
$257$	1.60934	0.100388	0.0501938	−	0.998739i	$-0.484016\pi$
$257$	1.60934	0.100388	0.0501938	+	0.998739i	$0.484016\pi$
$258$	−6.30554	−0.392566
$259$	0	0
$260$	1.09161	0.0676989
$261$	−24.5646	−1.52051
$262$	15.1901	0.938448
$263$	16.6016	1.02370	0.511849	−	0.859076i	$-0.328961\pi$
$263$	16.6016	1.02370	0.511849	+	0.859076i	$0.328961\pi$
$264$	−2.59836	−0.159918
$265$	9.72102	0.597157
$266$	0	0
$267$	1.75776	0.107573
$268$	−6.40231	−0.391083
$269$	19.1920	1.17016	0.585079	−	0.810977i	$-0.301064\pi$
$269$	19.1920	1.17016	0.585079	+	0.810977i	$0.301064\pi$
$270$	7.23192	0.440121
$271$	−6.14566	−0.373322	−0.186661	−	0.982424i	$-0.559767\pi$
$271$	−6.14566	−0.373322	−0.186661	+	0.982424i	$0.559767\pi$
$272$	−7.48660	−0.453942
$273$	0	0
$274$	21.9483	1.32595
$275$	2.41768	0.145792
$276$	−4.25105	−0.255883
$277$	−9.25779	−0.556247	−0.278123	−	0.960545i	$-0.589712\pi$
$277$	−9.25779	−0.556247	−0.278123	+	0.960545i	$0.589712\pi$
$278$	−22.3469	−1.34028
$279$	−10.8423	−0.649109
$280$	0	0
$281$	−13.5357	−0.807474	−0.403737	−	0.914875i	$-0.632289\pi$
$281$	−13.5357	−0.807474	−0.403737	+	0.914875i	$0.632289\pi$
$282$	−0.0592145	−0.00352617
$283$	−7.24504	−0.430673	−0.215336	−	0.976540i	$-0.569085\pi$
$283$	−7.24504	−0.430673	−0.215336	+	0.976540i	$0.569085\pi$
$284$	−12.3035	−0.730076
$285$	−5.00724	−0.296603
$286$	−2.23883	−0.132385
$287$	0	0
$288$	2.70830	0.159588
$289$	39.0492	2.29701
$290$	−21.2762	−1.24938
$291$	0.784948	0.0460144
$292$	−10.2392	−0.599201
$293$	5.02208	0.293393	0.146697	−	0.989182i	$-0.453136\pi$
$293$	5.02208	0.293393	0.146697	+	0.989182i	$0.453136\pi$
$294$	0	0
$295$	−11.4356	−0.665806
$296$	6.62563	0.385107
$297$	−14.8322	−0.860654
$298$	3.20887	0.185885
$299$	−3.66284	−0.211828
$300$	0.271413	0.0156700
$301$	0	0
$302$	11.1004	0.638757
$303$	10.0396	0.576758
$304$	−3.95232	−0.226681
$305$	−9.15895	−0.524440
$306$	−20.2760	−1.15910
$307$	−22.1425	−1.26374	−0.631869	−	0.775075i	$-0.717712\pi$
$307$	−22.1425	−1.26374	−0.631869	+	0.775075i	$0.717712\pi$
$308$	0	0
$309$	−5.82025	−0.331102
$310$	−9.39083	−0.533363
$311$	10.1458	0.575313	0.287657	−	0.957734i	$-0.407124\pi$
$311$	10.1458	0.575313	0.287657	+	0.957734i	$0.407124\pi$
$312$	−0.251335	−0.0142290
$313$	6.02420	0.340508	0.170254	−	0.985400i	$-0.445541\pi$
$313$	6.02420	0.340508	0.170254	+	0.985400i	$0.445541\pi$
$314$	23.7364	1.33952
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	−4.98228	−0.279833	−0.139916	−	0.990163i	$-0.544683\pi$
$317$	−4.98228	−0.279833	−0.139916	+	0.990163i	$0.544683\pi$
$318$	−2.23818	−0.125511
$319$	43.6362	2.44316
$320$	2.34575	0.131131
$321$	2.61815	0.146131
$322$	0	0
$323$	29.5894	1.64640
$324$	6.45982	0.358879
$325$	0.233858	0.0129721
$326$	−1.73722	−0.0962156
$327$	−1.16456	−0.0644002
$328$	−1.75310	−0.0967989
$329$	0	0
$330$	−6.09510	−0.335524
$331$	16.9665	0.932565	0.466282	−	0.884636i	$-0.345593\pi$
$331$	16.9665	0.932565	0.466282	+	0.884636i	$0.345593\pi$
$332$	−5.21405	−0.286158
$333$	17.9442	0.983338
$334$	7.07847	0.387317
$335$	−15.0182	−0.820532
$336$	0	0
$337$	11.5669	0.630092	0.315046	−	0.949076i	$-0.397980\pi$
$337$	11.5669	0.630092	0.315046	+	0.949076i	$0.397980\pi$
$338$	12.7834	0.695328
$339$	−7.81513	−0.424459
$340$	−17.5617	−0.952416
$341$	19.2600	1.04299
$342$	−10.7041	−0.578811
$343$	0	0
$344$	−11.6750	−0.629474
$345$	−9.97190	−0.536869
$346$	−4.40152	−0.236627
$347$	5.44473	0.292288	0.146144	−	0.989263i	$-0.453314\pi$
$347$	5.44473	0.292288	0.146144	+	0.989263i	$0.453314\pi$
$348$	4.89866	0.262596
$349$	−22.2511	−1.19107	−0.595537	−	0.803328i	$-0.703061\pi$
$349$	−22.2511	−1.19107	−0.595537	+	0.803328i	$0.703061\pi$
$350$	0	0
$351$	−1.43469	−0.0765783
$352$	−4.81099	−0.256427
$353$	3.32286	0.176858	0.0884290	−	0.996082i	$-0.471815\pi$
$353$	3.32286	0.176858	0.0884290	+	0.996082i	$0.471815\pi$
$354$	2.63295	0.139940
$355$	−28.8608	−1.53177
$356$	3.25457	0.172492
$357$	0	0
$358$	2.58332	0.136533
$359$	−13.8574	−0.731364	−0.365682	−	0.930740i	$-0.619164\pi$
$359$	−13.8574	−0.731364	−0.365682	+	0.930740i	$0.619164\pi$
$360$	6.35300	0.334832
$361$	−3.37916	−0.177851
$362$	−8.66649	−0.455501
$363$	6.55971	0.344295
$364$	0	0
$365$	−24.0185	−1.25718
$366$	2.10877	0.110227
$367$	0.00898137	0.000468824 0	0.000234412	−	1.00000i	$-0.499925\pi$
$367$	0.00898137	0.000468824 0	0.000234412		1.00000i	$0.499925\pi$
$368$	−7.87103	−0.410306
$369$	−4.74793	−0.247168
$370$	15.5421	0.807993
$371$	0	0
$372$	2.16216	0.112103
$373$	25.4249	1.31645	0.658225	−	0.752822i	$-0.271308\pi$
$373$	25.4249	1.31645	0.658225	+	0.752822i	$0.271308\pi$
$374$	36.0179	1.86244
$375$	−5.69790	−0.294238
$376$	−0.109639	−0.00565418
$377$	4.22085	0.217385
$378$	0	0
$379$	3.59033	0.184423	0.0922115	−	0.995739i	$-0.470606\pi$
$379$	3.59033	0.184423	0.0922115	+	0.995739i	$0.470606\pi$
$380$	−9.27115	−0.475600
$381$	−5.33221	−0.273178
$382$	10.4948	0.536961
$383$	−26.7701	−1.36789	−0.683944	−	0.729535i	$-0.739737\pi$
$383$	−26.7701	−1.36789	−0.683944	+	0.729535i	$0.739737\pi$
$384$	−0.540089	−0.0275613
$385$	0	0
$386$	−18.3614	−0.934573
$387$	−31.6194	−1.60731
$388$	1.45337	0.0737836
$389$	3.85558	0.195485	0.0977427	−	0.995212i	$-0.468838\pi$
$389$	3.85558	0.195485	0.0977427	+	0.995212i	$0.468838\pi$
$390$	−0.589568	−0.0298539
$391$	58.9272	2.98008
$392$	0	0
$393$	−8.20401	−0.413838
$394$	−20.0932	−1.01228
$395$	−2.34575	−0.118027
$396$	−13.0296	−0.654763
$397$	−15.2820	−0.766979	−0.383490	−	0.923545i	$-0.625278\pi$
$397$	−15.2820	−0.766979	−0.383490	+	0.923545i	$0.625278\pi$
$398$	−12.8398	−0.643600
$399$	0	0
$400$	0.502533	0.0251267
$401$	34.1864	1.70719	0.853594	−	0.520939i	$-0.174418\pi$
$401$	34.1864	1.70719	0.853594	+	0.520939i	$0.174418\pi$
$402$	3.45781	0.172460
$403$	1.86299	0.0928020
$404$	18.5887	0.924823
$405$	15.1531	0.752964
$406$	0	0
$407$	−31.8758	−1.58003
$408$	4.04343	0.200180
$409$	−14.1407	−0.699215	−0.349607	−	0.936896i	$-0.613685\pi$
$409$	−14.1407	−0.699215	−0.349607	+	0.936896i	$0.613685\pi$
$410$	−4.11234	−0.203094
$411$	−11.8541	−0.584717
$412$	−10.7765	−0.530918
$413$	0	0
$414$	−21.3171	−1.04768
$415$	−12.2309	−0.600389
$416$	−0.465358	−0.0228161
$417$	12.0693	0.591036
$418$	19.0146	0.930033
$419$	9.44643	0.461488	0.230744	−	0.973014i	$-0.425884\pi$
$419$	9.44643	0.461488	0.230744	+	0.973014i	$0.425884\pi$
$420$	0	0
$421$	−1.28366	−0.0625618	−0.0312809	−	0.999511i	$-0.509959\pi$
$421$	−1.28366	−0.0625618	−0.0312809	+	0.999511i	$0.509959\pi$
$422$	−19.2697	−0.938034
$423$	−0.296934	−0.0144374
$424$	−4.14410	−0.201255
$425$	−3.76226	−0.182497
$426$	6.64496	0.321950
$427$	0	0
$428$	4.84763	0.234319
$429$	1.20917	0.0583792
$430$	−27.3866	−1.32070
$431$	−14.4976	−0.698325	−0.349162	−	0.937062i	$-0.613534\pi$
$431$	−14.4976	−0.698325	−0.349162	+	0.937062i	$0.613534\pi$
$432$	−3.08299	−0.148330
$433$	13.4127	0.644575	0.322288	−	0.946642i	$-0.395548\pi$
$433$	13.4127	0.644575	0.322288	+	0.946642i	$0.395548\pi$
$434$	0	0
$435$	11.4910	0.550953
$436$	−2.15624	−0.103265
$437$	31.1088	1.48814
$438$	5.53005	0.264236
$439$	6.95936	0.332152	0.166076	−	0.986113i	$-0.446890\pi$
$439$	6.95936	0.332152	0.166076	+	0.986113i	$0.446890\pi$
$440$	−11.2854	−0.538009
$441$	0	0
$442$	3.48395	0.165715
$443$	10.3140	0.490033	0.245016	−	0.969519i	$-0.421207\pi$
$443$	10.3140	0.490033	0.245016	+	0.969519i	$0.421207\pi$
$444$	−3.57843	−0.169825
$445$	7.63441	0.361906
$446$	16.8664	0.798648
$447$	−1.73308	−0.0819717
$448$	0	0
$449$	−6.83454	−0.322542	−0.161271	−	0.986910i	$-0.551559\pi$
$449$	−6.83454	−0.322542	−0.161271	+	0.986910i	$0.551559\pi$
$450$	1.36101	0.0641587
$451$	8.43415	0.397149
$452$	−14.4701	−0.680615
$453$	−5.99521	−0.281680
$454$	17.6436	0.828053
$455$	0	0
$456$	2.13460	0.0999620
$457$	−12.2818	−0.574519	−0.287260	−	0.957853i	$-0.592744\pi$
$457$	−12.2818	−0.574519	−0.287260	+	0.957853i	$0.592744\pi$
$458$	8.49583	0.396984
$459$	23.0811	1.07733
$460$	−18.4634	−0.860862
$461$	−39.6568	−1.84700	−0.923501	−	0.383597i	$-0.874685\pi$
$461$	−39.6568	−1.84700	−0.923501	+	0.383597i	$0.874685\pi$
$462$	0	0
$463$	25.2682	1.17431	0.587156	−	0.809474i	$-0.300248\pi$
$463$	25.2682	1.17431	0.587156	+	0.809474i	$0.300248\pi$
$464$	9.07011	0.421069
$465$	5.07188	0.235203
$466$	11.0571	0.512209
$467$	13.9880	0.647289	0.323645	−	0.946179i	$-0.395092\pi$
$467$	13.9880	0.647289	0.323645	+	0.946179i	$0.395092\pi$
$468$	−1.26033	−0.0582588
$469$	0	0
$470$	−0.257184	−0.0118630
$471$	−12.8198	−0.590703
$472$	4.87503	0.224392
$473$	56.1683	2.58262
$474$	0.540089	0.0248071
$475$	−1.98617	−0.0911318
$476$	0	0
$477$	−11.2235	−0.513888
$478$	−1.10050	−0.0503357
$479$	11.8658	0.542164	0.271082	−	0.962556i	$-0.412619\pi$
$479$	11.8658	0.542164	0.271082	+	0.962556i	$0.412619\pi$
$480$	−1.26691	−0.0578264
$481$	−3.08329	−0.140586
$482$	25.6900	1.17015
$483$	0	0
$484$	12.1456	0.552073
$485$	3.40923	0.154805
$486$	−12.7379	−0.577801
$487$	−0.971111	−0.0440052	−0.0220026	−	0.999758i	$-0.507004\pi$
$487$	−0.971111	−0.0440052	−0.0220026	+	0.999758i	$0.507004\pi$
$488$	3.90449	0.176748
$489$	0.938252	0.0424292
$490$	0	0
$491$	−15.1423	−0.683363	−0.341682	−	0.939816i	$-0.610996\pi$
$491$	−15.1423	−0.683363	−0.341682	+	0.939816i	$0.610996\pi$
$492$	0.946831	0.0426864
$493$	−67.9043	−3.05825
$494$	1.83924	0.0827515
$495$	−30.5642	−1.37376
$496$	4.00334	0.179755
$497$	0	0
$498$	2.81605	0.126190
$499$	30.3024	1.35652	0.678261	−	0.734821i	$-0.262734\pi$
$499$	30.3024	1.35652	0.678261	+	0.734821i	$0.262734\pi$
$500$	−10.5499	−0.471807
$501$	−3.82300	−0.170799
$502$	18.6294	0.831470
$503$	−27.1651	−1.21123	−0.605617	−	0.795756i	$-0.707074\pi$
$503$	−27.1651	−1.21123	−0.605617	+	0.795756i	$0.707074\pi$
$504$	0	0
$505$	43.6044	1.94037
$506$	37.8674	1.68341
$507$	−6.90419	−0.306626
$508$	−9.87284	−0.438037
$509$	20.5197	0.909520	0.454760	−	0.890614i	$-0.349725\pi$
$509$	20.5197	0.909520	0.454760	+	0.890614i	$0.349725\pi$
$510$	9.48486	0.419997
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	12.1850	0.537980
$514$	−1.60934	−0.0709848
$515$	−25.2789	−1.11392
$516$	6.30554	0.277586
$517$	0.527470	0.0231981
$518$	0	0
$519$	2.37721	0.104348
$520$	−1.09161	−0.0478704
$521$	0.120435	0.00527635	0.00263817	−	0.999997i	$-0.499160\pi$
$521$	0.120435	0.00527635	0.00263817	+	0.999997i	$0.499160\pi$
$522$	24.5646	1.07516
$523$	−14.6946	−0.642548	−0.321274	−	0.946986i	$-0.604111\pi$
$523$	−14.6946	−0.642548	−0.321274	+	0.946986i	$0.604111\pi$
$524$	−15.1901	−0.663583
$525$	0	0
$526$	−16.6016	−0.723863
$527$	−29.9714	−1.30558
$528$	2.59836	0.113079
$529$	38.9530	1.69361
$530$	−9.72102	−0.422254
$531$	13.2031	0.572964
$532$	0	0
$533$	0.815820	0.0353371
$534$	−1.75776	−0.0760656
$535$	11.3713	0.491625
$536$	6.40231	0.276537
$537$	−1.39522	−0.0602083
$538$	−19.1920	−0.827426
$539$	0	0
$540$	−7.23192	−0.311212
$541$	−36.5364	−1.57082	−0.785411	−	0.618975i	$-0.787548\pi$
$541$	−36.5364	−1.57082	−0.785411	+	0.618975i	$0.787548\pi$
$542$	6.14566	0.263979
$543$	4.68068	0.200867
$544$	7.48660	0.320985
$545$	−5.05798	−0.216660
$546$	0	0
$547$	42.4371	1.81448	0.907239	−	0.420615i	$-0.138186\pi$
$547$	42.4371	1.81448	0.907239	+	0.420615i	$0.138186\pi$
$548$	−21.9483	−0.937587
$549$	10.5745	0.451311
$550$	−2.41768	−0.103090
$551$	−35.8480	−1.52718
$552$	4.25105	0.180937
$553$	0	0
$554$	9.25779	0.393326
$555$	−8.39410	−0.356310
$556$	22.3469	0.947718
$557$	2.09096	0.0885967	0.0442983	−	0.999018i	$-0.485895\pi$
$557$	2.09096	0.0885967	0.0442983	+	0.999018i	$0.485895\pi$
$558$	10.8423	0.458990
$559$	5.43305	0.229794
$560$	0	0
$561$	−19.4529	−0.821302
$562$	13.5357	0.570971
$563$	30.1818	1.27201	0.636005	−	0.771685i	$-0.280586\pi$
$563$	30.1818	1.27201	0.636005	+	0.771685i	$0.280586\pi$
$564$	0.0592145	0.00249338
$565$	−33.9432	−1.42800
$566$	7.24504	0.304532
$567$	0	0
$568$	12.3035	0.516242
$569$	41.1818	1.72643	0.863216	−	0.504835i	$-0.168447\pi$
$569$	41.1818	1.72643	0.863216	+	0.504835i	$0.168447\pi$
$570$	5.00724	0.209730
$571$	21.4934	0.899469	0.449734	−	0.893162i	$-0.351519\pi$
$571$	21.4934	0.899469	0.449734	+	0.893162i	$0.351519\pi$
$572$	2.23883	0.0936103
$573$	−5.66813	−0.236789
$574$	0	0
$575$	−3.95545	−0.164954
$576$	−2.70830	−0.112846
$577$	38.4897	1.60235	0.801173	−	0.598433i	$-0.204210\pi$
$577$	38.4897	1.60235	0.801173	+	0.598433i	$0.204210\pi$
$578$	−39.0492	−1.62423
$579$	9.91681	0.412129
$580$	21.2762	0.883446
$581$	0	0
$582$	−0.784948	−0.0325371
$583$	19.9372	0.825715
$584$	10.2392	0.423699
$585$	−2.95642	−0.122233
$586$	−5.02208	−0.207460
$587$	−33.5481	−1.38468	−0.692339	−	0.721573i	$-0.743420\pi$
$587$	−33.5481	−1.38468	−0.692339	+	0.721573i	$0.743420\pi$
$588$	0	0
$589$	−15.8225	−0.651954
$590$	11.4356	0.470796
$591$	10.8521	0.446397
$592$	−6.62563	−0.272312
$593$	20.1738	0.828440	0.414220	−	0.910177i	$-0.364054\pi$
$593$	20.1738	0.828440	0.414220	+	0.910177i	$0.364054\pi$
$594$	14.8322	0.608574
$595$	0	0
$596$	−3.20887	−0.131441
$597$	6.93462	0.283815
$598$	3.66284	0.149785
$599$	−6.44420	−0.263303	−0.131651	−	0.991296i	$-0.542028\pi$
$599$	−6.44420	−0.263303	−0.131651	+	0.991296i	$0.542028\pi$
$600$	−0.271413	−0.0110804
$601$	−21.8404	−0.890890	−0.445445	−	0.895309i	$-0.646954\pi$
$601$	−21.8404	−0.890890	−0.445445	+	0.895309i	$0.646954\pi$
$602$	0	0
$603$	17.3394	0.706115
$604$	−11.1004	−0.451670
$605$	28.4905	1.15831
$606$	−10.0396	−0.407829
$607$	−1.37152	−0.0556683	−0.0278342	−	0.999613i	$-0.508861\pi$
$607$	−1.37152	−0.0556683	−0.0278342	+	0.999613i	$0.508861\pi$
$608$	3.95232	0.160288
$609$	0	0
$610$	9.15895	0.370835
$611$	0.0510212	0.00206410
$612$	20.2760	0.819608
$613$	−13.0154	−0.525685	−0.262843	−	0.964839i	$-0.584660\pi$
$613$	−13.0154	−0.525685	−0.262843	+	0.964839i	$0.584660\pi$
$614$	22.1425	0.893598
$615$	2.22103	0.0895604
$616$	0	0
$617$	−5.26743	−0.212059	−0.106029	−	0.994363i	$-0.533814\pi$
$617$	−5.26743	−0.212059	−0.106029	+	0.994363i	$0.533814\pi$
$618$	5.82025	0.234125
$619$	−21.2550	−0.854312	−0.427156	−	0.904178i	$-0.640484\pi$
$619$	−21.2550	−0.854312	−0.427156	+	0.904178i	$0.640484\pi$
$620$	9.39083	0.377145
$621$	24.2663	0.973773
$622$	−10.1458	−0.406808
$623$	0	0
$624$	0.251335	0.0100614
$625$	−27.2601	−1.09041
$626$	−6.02420	−0.240775
$627$	−10.2696	−0.410127
$628$	−23.7364	−0.947185
$629$	49.6035	1.97782
$630$	0	0
$631$	26.3943	1.05074	0.525371	−	0.850873i	$-0.323927\pi$
$631$	26.3943	1.05074	0.525371	+	0.850873i	$0.323927\pi$
$632$	1.00000	0.0397779
$633$	10.4073	0.413655
$634$	4.98228	0.197872
$635$	−23.1592	−0.919045
$636$	2.23818	0.0887497
$637$	0	0
$638$	−43.6362	−1.72757
$639$	33.3215	1.31818
$640$	−2.34575	−0.0927238
$641$	32.3430	1.27747	0.638736	−	0.769426i	$-0.279458\pi$
$641$	32.3430	1.27747	0.638736	+	0.769426i	$0.279458\pi$
$642$	−2.61815	−0.103330
$643$	−6.15699	−0.242808	−0.121404	−	0.992603i	$-0.538740\pi$
$643$	−6.15699	−0.242808	−0.121404	+	0.992603i	$0.538740\pi$
$644$	0	0
$645$	14.7912	0.582403
$646$	−29.5894	−1.16418
$647$	−5.73440	−0.225443	−0.112721	−	0.993627i	$-0.535957\pi$
$647$	−5.73440	−0.225443	−0.112721	+	0.993627i	$0.535957\pi$
$648$	−6.45982	−0.253766
$649$	−23.4537	−0.920639
$650$	−0.233858	−0.00917266
$651$	0	0
$652$	1.73722	0.0680347
$653$	−12.7937	−0.500657	−0.250328	−	0.968161i	$-0.580539\pi$
$653$	−12.7937	−0.500657	−0.250328	+	0.968161i	$0.580539\pi$
$654$	1.16456	0.0455378
$655$	−35.6322	−1.39226
$656$	1.75310	0.0684471
$657$	27.7307	1.08188
$658$	0	0
$659$	−37.5687	−1.46347	−0.731734	−	0.681590i	$-0.761289\pi$
$659$	−37.5687	−1.46347	−0.731734	+	0.681590i	$0.761289\pi$
$660$	6.09510	0.237251
$661$	−47.1470	−1.83381	−0.916903	−	0.399110i	$-0.869319\pi$
$661$	−47.1470	−1.83381	−0.916903	+	0.399110i	$0.869319\pi$
$662$	−16.9665	−0.659423
$663$	−1.88164	−0.0730769
$664$	5.21405	0.202345
$665$	0	0
$666$	−17.9442	−0.695325
$667$	−71.3910	−2.76427
$668$	−7.07847	−0.273874
$669$	−9.10936	−0.352188
$670$	15.0182	0.580203
$671$	−18.7845	−0.725166
$672$	0	0
$673$	−16.9161	−0.652069	−0.326034	−	0.945358i	$-0.605713\pi$
$673$	−16.9161	−0.652069	−0.326034	+	0.945358i	$0.605713\pi$
$674$	−11.5669	−0.445542
$675$	−1.54931	−0.0596328
$676$	−12.7834	−0.491671
$677$	−10.8739	−0.417920	−0.208960	−	0.977924i	$-0.567008\pi$
$677$	−10.8739	−0.417920	−0.208960	+	0.977924i	$0.567008\pi$
$678$	7.81513	0.300138
$679$	0	0
$680$	17.5617	0.673459
$681$	−9.52909	−0.365155
$682$	−19.2600	−0.737505
$683$	41.3046	1.58048	0.790239	−	0.612799i	$-0.209956\pi$
$683$	41.3046	1.58048	0.790239	+	0.612799i	$0.209956\pi$
$684$	10.7041	0.409281
$685$	−51.4853	−1.96715
$686$	0	0
$687$	−4.58850	−0.175062
$688$	11.6750	0.445105
$689$	1.92849	0.0734696
$690$	9.97190	0.379624
$691$	16.7040	0.635449	0.317724	−	0.948183i	$-0.397081\pi$
$691$	16.7040	0.635449	0.317724	+	0.948183i	$0.397081\pi$
$692$	4.40152	0.167321
$693$	0	0
$694$	−5.44473	−0.206679
$695$	52.4201	1.98841
$696$	−4.89866	−0.185683
$697$	−13.1248	−0.497136
$698$	22.2511	0.842216
$699$	−5.97180	−0.225874
$700$	0	0
$701$	16.2649	0.614317	0.307158	−	0.951658i	$-0.400622\pi$
$701$	16.2649	0.614317	0.307158	+	0.951658i	$0.400622\pi$
$702$	1.43469	0.0541491
$703$	26.1866	0.987647
$704$	4.81099	0.181321
$705$	0.138902	0.00523137
$706$	−3.32286	−0.125057
$707$	0	0
$708$	−2.63295	−0.0989523
$709$	26.1371	0.981599	0.490800	−	0.871272i	$-0.336705\pi$
$709$	26.1371	0.981599	0.490800	+	0.871272i	$0.336705\pi$
$710$	28.8608	1.08313
$711$	2.70830	0.101569
$712$	−3.25457	−0.121970
$713$	−31.5104	−1.18007
$714$	0	0
$715$	5.25174	0.196404
$716$	−2.58332	−0.0965433
$717$	0.594368	0.0221971
$718$	13.8574	0.517152
$719$	30.4268	1.13473	0.567364	−	0.823467i	$-0.307963\pi$
$719$	30.4268	1.13473	0.567364	+	0.823467i	$0.307963\pi$
$720$	−6.35300	−0.236762
$721$	0	0
$722$	3.37916	0.125759
$723$	−13.8749	−0.516013
$724$	8.66649	0.322088
$725$	4.55803	0.169281
$726$	−6.55971	−0.243454
$727$	−16.6912	−0.619041	−0.309520	−	0.950893i	$-0.600168\pi$
$727$	−16.6912	−0.619041	−0.309520	+	0.950893i	$0.600168\pi$
$728$	0	0
$729$	−12.4999	−0.462959
$730$	24.0185	0.888963
$731$	−87.4060	−3.23283
$732$	−2.10877	−0.0779425
$733$	−24.0479	−0.888228	−0.444114	−	0.895970i	$-0.646481\pi$
$733$	−24.0479	−0.888228	−0.444114	+	0.895970i	$0.646481\pi$
$734$	−0.00898137	−0.000331508 0
$735$	0	0
$736$	7.87103	0.290130
$737$	−30.8014	−1.13458
$738$	4.74793	0.174774
$739$	−28.3188	−1.04172	−0.520862	−	0.853641i	$-0.674390\pi$
$739$	−28.3188	−1.04172	−0.520862	+	0.853641i	$0.674390\pi$
$740$	−15.5421	−0.571338
$741$	−0.993355	−0.0364918
$742$	0	0
$743$	12.8287	0.470640	0.235320	−	0.971918i	$-0.424386\pi$
$743$	12.8287	0.470640	0.235320	+	0.971918i	$0.424386\pi$
$744$	−2.16216	−0.0792686
$745$	−7.52721	−0.275775
$746$	−25.4249	−0.930870
$747$	14.1212	0.516669
$748$	−36.0179	−1.31695
$749$	0	0
$750$	5.69790	0.208058
$751$	−26.3737	−0.962390	−0.481195	−	0.876613i	$-0.659797\pi$
$751$	−26.3737	−0.962390	−0.481195	+	0.876613i	$0.659797\pi$
$752$	0.109639	0.00399811
$753$	−10.0615	−0.366662
$754$	−4.22085	−0.153714
$755$	−26.0388	−0.947648
$756$	0	0
$757$	−25.1677	−0.914735	−0.457367	−	0.889278i	$-0.651208\pi$
$757$	−25.1677	−0.914735	−0.457367	+	0.889278i	$0.651208\pi$
$758$	−3.59033	−0.130407
$759$	−20.4518	−0.742352
$760$	9.27115	0.336300
$761$	−36.8884	−1.33720	−0.668601	−	0.743621i	$-0.733107\pi$
$761$	−36.8884	−1.33720	−0.668601	+	0.743621i	$0.733107\pi$
$762$	5.33221	0.193166
$763$	0	0
$764$	−10.4948	−0.379689
$765$	47.5624	1.71962
$766$	26.7701	0.967243
$767$	−2.26863	−0.0819157
$768$	0.540089	0.0194888
$769$	31.7350	1.14439	0.572197	−	0.820116i	$-0.306091\pi$
$769$	31.7350	1.14439	0.572197	+	0.820116i	$0.306091\pi$
$770$	0	0
$771$	0.869184	0.0313029
$772$	18.3614	0.660843
$773$	−15.2305	−0.547804	−0.273902	−	0.961758i	$-0.588314\pi$
$773$	−15.2305	−0.547804	−0.273902	+	0.961758i	$0.588314\pi$
$774$	31.6194	1.13654
$775$	2.01181	0.0722664
$776$	−1.45337	−0.0521728
$777$	0	0
$778$	−3.85558	−0.138229
$779$	−6.92882	−0.248251
$780$	0.589568	0.0211099
$781$	−59.1918	−2.11805
$782$	−58.9272	−2.10723
$783$	−27.9631	−0.999318
$784$	0	0
$785$	−55.6796	−1.98729
$786$	8.20401	0.292627
$787$	−46.9605	−1.67396	−0.836980	−	0.547234i	$-0.815681\pi$
$787$	−46.9605	−1.67396	−0.836980	+	0.547234i	$0.815681\pi$
$788$	20.0932	0.715791
$789$	8.96633	0.319210
$790$	2.34575	0.0834580
$791$	0	0
$792$	13.0296	0.462987
$793$	−1.81699	−0.0645231
$794$	15.2820	0.542336
$795$	5.25021	0.186206
$796$	12.8398	0.455094
$797$	32.8320	1.16297	0.581485	−	0.813557i	$-0.302472\pi$
$797$	32.8320	1.16297	0.581485	+	0.813557i	$0.302472\pi$
$798$	0	0
$799$	−0.820820	−0.0290385
$800$	−0.502533	−0.0177672
$801$	−8.81437	−0.311441
$802$	−34.1864	−1.20716
$803$	−49.2604	−1.73836
$804$	−3.45781	−0.121948
$805$	0	0
$806$	−1.86299	−0.0656209
$807$	10.3654	0.364879
$808$	−18.5887	−0.653949
$809$	−30.0404	−1.05616	−0.528081	−	0.849194i	$-0.677088\pi$
$809$	−30.0404	−1.05616	−0.528081	+	0.849194i	$0.677088\pi$
$810$	−15.1531	−0.532426
$811$	−31.7635	−1.11537	−0.557683	−	0.830054i	$-0.688310\pi$
$811$	−31.7635	−1.11537	−0.557683	+	0.830054i	$0.688310\pi$
$812$	0	0
$813$	−3.31920	−0.116409
$814$	31.8758	1.11725
$815$	4.07507	0.142744
$816$	−4.04343	−0.141548
$817$	−46.1433	−1.61435
$818$	14.1407	0.494419
$819$	0	0
$820$	4.11234	0.143609
$821$	−27.4468	−0.957900	−0.478950	−	0.877842i	$-0.658982\pi$
$821$	−27.4468	−0.957900	−0.478950	+	0.877842i	$0.658982\pi$
$822$	11.8541	0.413458
$823$	−37.0791	−1.29250	−0.646249	−	0.763127i	$-0.723663\pi$
$823$	−37.0791	−1.29250	−0.646249	+	0.763127i	$0.723663\pi$
$824$	10.7765	0.375416
$825$	1.30576	0.0454608
$826$	0	0
$827$	−1.30520	−0.0453861	−0.0226931	−	0.999742i	$-0.507224\pi$
$827$	−1.30520	−0.0453861	−0.0226931	+	0.999742i	$0.507224\pi$
$828$	21.3171	0.740821
$829$	−47.7435	−1.65820	−0.829100	−	0.559100i	$-0.811147\pi$
$829$	−47.7435	−1.65820	−0.829100	+	0.559100i	$0.811147\pi$
$830$	12.2309	0.424539
$831$	−5.00003	−0.173449
$832$	0.465358	0.0161334
$833$	0	0
$834$	−12.0693	−0.417925
$835$	−16.6043	−0.574616
$836$	−19.0146	−0.657633
$837$	−12.3423	−0.426611
$838$	−9.44643	−0.326321
$839$	45.3547	1.56582	0.782908	−	0.622137i	$-0.213735\pi$
$839$	45.3547	1.56582	0.782908	+	0.622137i	$0.213735\pi$
$840$	0	0
$841$	53.2669	1.83679
$842$	1.28366	0.0442379
$843$	−7.31050	−0.251787
$844$	19.2697	0.663290
$845$	−29.9867	−1.03157
$846$	0.296934	0.0102088
$847$	0	0
$848$	4.14410	0.142309
$849$	−3.91297	−0.134293
$850$	3.76226	0.129045
$851$	52.1505	1.78770
$852$	−6.64496	−0.227653
$853$	−15.2535	−0.522269	−0.261134	−	0.965302i	$-0.584097\pi$
$853$	−15.2535	−0.522269	−0.261134	+	0.965302i	$0.584097\pi$
$854$	0	0
$855$	25.1091	0.858713
$856$	−4.84763	−0.165689
$857$	−33.4793	−1.14363	−0.571816	−	0.820382i	$-0.693761\pi$
$857$	−33.4793	−1.14363	−0.571816	+	0.820382i	$0.693761\pi$
$858$	−1.20917	−0.0412803
$859$	−8.40664	−0.286831	−0.143415	−	0.989663i	$-0.545808\pi$
$859$	−8.40664	−0.286831	−0.143415	+	0.989663i	$0.545808\pi$
$860$	27.3866	0.933875
$861$	0	0
$862$	14.4976	0.493790
$863$	36.2355	1.23347	0.616735	−	0.787171i	$-0.288455\pi$
$863$	36.2355	1.23347	0.616735	+	0.787171i	$0.288455\pi$
$864$	3.08299	0.104885
$865$	10.3249	0.351056
$866$	−13.4127	−0.455784
$867$	21.0900	0.716255
$868$	0	0
$869$	−4.81099	−0.163202
$870$	−11.4910	−0.389582
$871$	−2.97936	−0.100952
$872$	2.15624	0.0730193
$873$	−3.93616	−0.133219
$874$	−31.1088	−1.05227
$875$	0	0
$876$	−5.53005	−0.186843
$877$	−2.33397	−0.0788126	−0.0394063	−	0.999223i	$-0.512547\pi$
$877$	−2.33397	−0.0788126	−0.0394063	+	0.999223i	$0.512547\pi$
$878$	−6.95936	−0.234867
$879$	2.71237	0.0914860
$880$	11.2854	0.380430
$881$	15.9704	0.538057	0.269029	−	0.963132i	$-0.413297\pi$
$881$	15.9704	0.538057	0.269029	+	0.963132i	$0.413297\pi$
$882$	0	0
$883$	−36.0256	−1.21236	−0.606178	−	0.795329i	$-0.707298\pi$
$883$	−36.0256	−1.21236	−0.606178	+	0.795329i	$0.707298\pi$
$884$	−3.48395	−0.117178
$885$	−6.17624	−0.207612
$886$	−10.3140	−0.346505
$887$	45.2477	1.51927	0.759634	−	0.650351i	$-0.225378\pi$
$887$	45.2477	1.51927	0.759634	+	0.650351i	$0.225378\pi$
$888$	3.57843	0.120084
$889$	0	0
$890$	−7.63441	−0.255906
$891$	31.0781	1.04116
$892$	−16.8664	−0.564729
$893$	−0.433327	−0.0145007
$894$	1.73308	0.0579627
$895$	−6.05982	−0.202558
$896$	0	0
$897$	−1.97826	−0.0660522
$898$	6.83454	0.228071
$899$	36.3107	1.21103
$900$	−1.36101	−0.0453671
$901$	−31.0252	−1.03360
$902$	−8.43415	−0.280827
$903$	0	0
$904$	14.4701	0.481268
$905$	20.3294	0.675773
$906$	5.99521	0.199178
$907$	30.4079	1.00968	0.504839	−	0.863213i	$-0.331552\pi$
$907$	30.4079	1.00968	0.504839	+	0.863213i	$0.331552\pi$
$908$	−17.6436	−0.585522
$909$	−50.3439	−1.66980
$910$	0	0
$911$	45.1611	1.49625	0.748127	−	0.663555i	$-0.230953\pi$
$911$	45.1611	1.49625	0.748127	+	0.663555i	$0.230953\pi$
$912$	−2.13460	−0.0706838
$913$	−25.0848	−0.830184
$914$	12.2818	0.406246
$915$	−4.94665	−0.163531
$916$	−8.49583	−0.280710
$917$	0	0
$918$	−23.0811	−0.761791
$919$	7.43182	0.245153	0.122576	−	0.992459i	$-0.460884\pi$
$919$	7.43182	0.245153	0.122576	+	0.992459i	$0.460884\pi$
$920$	18.4634	0.608722
$921$	−11.9589	−0.394059
$922$	39.6568	1.30603
$923$	−5.72552	−0.188458
$924$	0	0
$925$	−3.32960	−0.109477
$926$	−25.2682	−0.830364
$927$	29.1859	0.958592
$928$	−9.07011	−0.297741
$929$	49.1427	1.61232	0.806160	−	0.591698i	$-0.201542\pi$
$929$	49.1427	1.61232	0.806160	+	0.591698i	$0.201542\pi$
$930$	−5.07188	−0.166314
$931$	0	0
$932$	−11.0571	−0.362186
$933$	5.47961	0.179394
$934$	−13.9880	−0.457703
$935$	−84.4890	−2.76309
$936$	1.26033	0.0411952
$937$	−6.72824	−0.219802	−0.109901	−	0.993943i	$-0.535053\pi$
$937$	−6.72824	−0.219802	−0.109901	+	0.993943i	$0.535053\pi$
$938$	0	0
$939$	3.25360	0.106177
$940$	0.257184	0.00838843
$941$	−56.9762	−1.85737	−0.928685	−	0.370869i	$-0.879060\pi$
$941$	−56.9762	−1.85737	−0.928685	+	0.370869i	$0.879060\pi$
$942$	12.8198	0.417690
$943$	−13.7987	−0.449348
$944$	−4.87503	−0.158669
$945$	0	0
$946$	−56.1683	−1.82619
$947$	−41.2259	−1.33966	−0.669831	−	0.742514i	$-0.733633\pi$
$947$	−41.2259	−1.33966	−0.669831	+	0.742514i	$0.733633\pi$
$948$	−0.540089	−0.0175413
$949$	−4.76487	−0.154674
$950$	1.98617	0.0644399
$951$	−2.69088	−0.0872576
$952$	0	0
$953$	14.4413	0.467799	0.233899	−	0.972261i	$-0.424851\pi$
$953$	14.4413	0.467799	0.233899	+	0.972261i	$0.424851\pi$
$954$	11.2235	0.363374
$955$	−24.6182	−0.796625
$956$	1.10050	0.0355927
$957$	23.5674	0.761826
$958$	−11.8658	−0.383368
$959$	0	0
$960$	1.26691	0.0408894
$961$	−14.9733	−0.483008
$962$	3.08329	0.0994093
$963$	−13.1289	−0.423072
$964$	−25.6900	−0.827420
$965$	43.0713	1.38651
$966$	0	0
$967$	−0.104395	−0.00335712	−0.00167856	−	0.999999i	$-0.500534\pi$
$967$	−0.104395	−0.00335712	−0.00167856	+	0.999999i	$0.500534\pi$
$968$	−12.1456	−0.390375
$969$	15.9809	0.513381
$970$	−3.40923	−0.109464
$971$	56.2910	1.80646	0.903231	−	0.429154i	$-0.141188\pi$
$971$	56.2910	1.80646	0.903231	+	0.429154i	$0.141188\pi$
$972$	12.7379	0.408567
$973$	0	0
$974$	0.971111	0.0311164
$975$	0.126304	0.00404496
$976$	−3.90449	−0.124980
$977$	44.4705	1.42274	0.711369	−	0.702818i	$-0.248075\pi$
$977$	44.4705	1.42274	0.711369	+	0.702818i	$0.248075\pi$
$978$	−0.938252	−0.0300020
$979$	15.6577	0.500423
$980$	0	0
$981$	5.83974	0.186449
$982$	15.1423	0.483211
$983$	24.1578	0.770515	0.385257	−	0.922809i	$-0.374113\pi$
$983$	24.1578	0.770515	0.385257	+	0.922809i	$0.374113\pi$
$984$	−0.946831	−0.0301839
$985$	47.1336	1.50180
$986$	67.9043	2.16251
$987$	0	0
$988$	−1.83924	−0.0585141
$989$	−91.8942	−2.92207
$990$	30.5642	0.971394
$991$	2.23959	0.0711431	0.0355715	−	0.999367i	$-0.488675\pi$
$991$	2.23959	0.0711431	0.0355715	+	0.999367i	$0.488675\pi$
$992$	−4.00334	−0.127106
$993$	9.16343	0.290793
$994$	0	0
$995$	30.1189	0.954833
$996$	−2.81605	−0.0892300
$997$	−26.9161	−0.852443	−0.426221	−	0.904619i	$-0.640156\pi$
$997$	−26.9161	−0.852443	−0.426221	+	0.904619i	$0.640156\pi$
$998$	−30.3024	−0.959206
$999$	20.4268	0.646274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7742.2.a.bk.1.8		12
7.2	even	3		1106.2.f.f.949.5	yes	24
7.4	even	3		1106.2.f.f.317.5	✓	24
7.6	odd	2		7742.2.a.bn.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1106.2.f.f.317.5	✓	24	7.4	even	3
1106.2.f.f.949.5	yes	24	7.2	even	3
7742.2.a.bk.1.8		12	1.1	even	1	trivial
7742.2.a.bn.1.5		12	7.6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.540089 q^{3} +1.00000 q^{4} +2.34575 q^{5} -0.540089 q^{6} -1.00000 q^{8} -2.70830 q^{9} -2.34575 q^{10} +4.81099 q^{11} +0.540089 q^{12} +0.465358 q^{13} +1.26691 q^{15} +1.00000 q^{16} -7.48660 q^{17} +2.70830 q^{18} -3.95232 q^{19} +2.34575 q^{20} -4.81099 q^{22} -7.87103 q^{23} -0.540089 q^{24} +0.502533 q^{25} -0.465358 q^{26} -3.08299 q^{27} +9.07011 q^{29} -1.26691 q^{30} +4.00334 q^{31} -1.00000 q^{32} +2.59836 q^{33} +7.48660 q^{34} -2.70830 q^{36} -6.62563 q^{37} +3.95232 q^{38} +0.251335 q^{39} -2.34575 q^{40} +1.75310 q^{41} +11.6750 q^{43} +4.81099 q^{44} -6.35300 q^{45} +7.87103 q^{46} +0.109639 q^{47} +0.540089 q^{48} -0.502533 q^{50} -4.04343 q^{51} +0.465358 q^{52} +4.14410 q^{53} +3.08299 q^{54} +11.2854 q^{55} -2.13460 q^{57} -9.07011 q^{58} -4.87503 q^{59} +1.26691 q^{60} -3.90449 q^{61} -4.00334 q^{62} +1.00000 q^{64} +1.09161 q^{65} -2.59836 q^{66} -6.40231 q^{67} -7.48660 q^{68} -4.25105 q^{69} -12.3035 q^{71} +2.70830 q^{72} -10.2392 q^{73} +6.62563 q^{74} +0.271413 q^{75} -3.95232 q^{76} -0.251335 q^{78} -1.00000 q^{79} +2.34575 q^{80} +6.45982 q^{81} -1.75310 q^{82} -5.21405 q^{83} -17.5617 q^{85} -11.6750 q^{86} +4.89866 q^{87} -4.81099 q^{88} +3.25457 q^{89} +6.35300 q^{90} -7.87103 q^{92} +2.16216 q^{93} -0.109639 q^{94} -9.27115 q^{95} -0.540089 q^{96} +1.45337 q^{97} -13.0296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} - 12 q^{8} + 17 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 13 q^{13} - 2 q^{15} + 12 q^{16} - 5 q^{17} - 17 q^{18} - 14 q^{19} + 3 q^{20} + 2 q^{22}+ \cdots - 19 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 - 12 * q^8 + 17 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 13 * q^13 - 2 * q^15 + 12 * q^16 - 5 * q^17 - 17 * q^18 - 14 * q^19 + 3 * q^20 + 2 * q^22 + 2 * q^23 + 3 * q^24 + 21 * q^25 + 13 * q^26 - 6 * q^27 + 7 * q^29 + 2 * q^30 - 15 * q^31 - 12 * q^32 - 23 * q^33 + 5 * q^34 + 17 * q^36 - 15 * q^37 + 14 * q^38 + 18 * q^39 - 3 * q^40 - 10 * q^41 + 8 * q^43 - 2 * q^44 - 47 * q^45 - 2 * q^46 + 13 * q^47 - 3 * q^48 - 21 * q^50 - 24 * q^51 - 13 * q^52 + 8 * q^53 + 6 * q^54 + q^55 - 20 * q^57 - 7 * q^58 - 10 * q^59 - 2 * q^60 + 6 * q^61 + 15 * q^62 + 12 * q^64 - 25 * q^65 + 23 * q^66 + 18 * q^67 - 5 * q^68 + 25 * q^69 - 21 * q^71 - 17 * q^72 - 45 * q^73 + 15 * q^74 + 3 * q^75 - 14 * q^76 - 18 * q^78 - 12 * q^79 + 3 * q^80 + 76 * q^81 + 10 * q^82 + 15 * q^83 - 35 * q^85 - 8 * q^86 - 50 * q^87 + 2 * q^88 - 5 * q^89 + 47 * q^90 + 2 * q^92 - 4 * q^93 - 13 * q^94 + 49 * q^95 + 3 * q^96 - 53 * q^97 - 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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