LMFDB - Embedded newform 1078.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$8.60787333789$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.64575 q^{3} +1.00000 q^{4} -3.64575 q^{5} -2.64575 q^{6} -1.00000 q^{8} +4.00000 q^{9} +3.64575 q^{10} +1.00000 q^{11} +2.64575 q^{12} -5.00000 q^{13} -9.64575 q^{15} +1.00000 q^{16} -6.00000 q^{17} -4.00000 q^{18} +0.354249 q^{19} -3.64575 q^{20} -1.00000 q^{22} -3.64575 q^{23} -2.64575 q^{24} +8.29150 q^{25} +5.00000 q^{26} +2.64575 q^{27} -4.29150 q^{29} +9.64575 q^{30} +4.00000 q^{31} -1.00000 q^{32} +2.64575 q^{33} +6.00000 q^{34} +4.00000 q^{36} -1.64575 q^{37} -0.354249 q^{38} -13.2288 q^{39} +3.64575 q^{40} +4.93725 q^{41} -4.00000 q^{43} +1.00000 q^{44} -14.5830 q^{45} +3.64575 q^{46} -13.2915 q^{47} +2.64575 q^{48} -8.29150 q^{50} -15.8745 q^{51} -5.00000 q^{52} -3.64575 q^{53} -2.64575 q^{54} -3.64575 q^{55} +0.937254 q^{57} +4.29150 q^{58} +0.645751 q^{59} -9.64575 q^{60} -3.70850 q^{61} -4.00000 q^{62} +1.00000 q^{64} +18.2288 q^{65} -2.64575 q^{66} +3.93725 q^{67} -6.00000 q^{68} -9.64575 q^{69} +9.64575 q^{71} -4.00000 q^{72} -5.64575 q^{73} +1.64575 q^{74} +21.9373 q^{75} +0.354249 q^{76} +13.2288 q^{78} +2.64575 q^{79} -3.64575 q^{80} -5.00000 q^{81} -4.93725 q^{82} -13.2915 q^{83} +21.8745 q^{85} +4.00000 q^{86} -11.3542 q^{87} -1.00000 q^{88} +14.5830 q^{89} +14.5830 q^{90} -3.64575 q^{92} +10.5830 q^{93} +13.2915 q^{94} -1.29150 q^{95} -2.64575 q^{96} +5.70850 q^{97} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 8 q^{9} + 2 q^{10} + 2 q^{11} - 10 q^{13} - 14 q^{15} + 2 q^{16} - 12 q^{17} - 8 q^{18} + 6 q^{19} - 2 q^{20} - 2 q^{22} - 2 q^{23} + 6 q^{25} + 10 q^{26} + 2 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 8 * q^9 + 2 * q^10 + 2 * q^11 - 10 * q^13 - 14 * q^15 + 2 * q^16 - 12 * q^17 - 8 * q^18 + 6 * q^19 - 2 * q^20 - 2 * q^22 - 2 * q^23 + 6 * q^25 + 10 * q^26 + 2 * q^29 + 14 * q^30 + 8 * q^31 - 2 * q^32 + 12 * q^34 + 8 * q^36 + 2 * q^37 - 6 * q^38 + 2 * q^40 - 6 * q^41 - 8 * q^43 + 2 * q^44 - 8 * q^45 + 2 * q^46 - 16 * q^47 - 6 * q^50 - 10 * q^52 - 2 * q^53 - 2 * q^55 - 14 * q^57 - 2 * q^58 - 4 * q^59 - 14 * q^60 - 18 * q^61 - 8 * q^62 + 2 * q^64 + 10 * q^65 - 8 * q^67 - 12 * q^68 - 14 * q^69 + 14 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 + 28 * q^75 + 6 * q^76 - 2 * q^80 - 10 * q^81 + 6 * q^82 - 16 * q^83 + 12 * q^85 + 8 * q^86 - 28 * q^87 - 2 * q^88 + 8 * q^89 + 8 * q^90 - 2 * q^92 + 16 * q^94 + 8 * q^95 + 22 * q^97 + 8 * 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
$3$	2.64575	1.52753	0.763763	−	0.645497i	$-0.223350\pi$
$3$	2.64575	1.52753	0.763763	+	0.645497i	$0.223350\pi$
$4$	1.00000	0.500000
$5$	−3.64575	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$5$	−3.64575	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$6$	−2.64575	−1.08012
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	4.00000	1.33333
$10$	3.64575	1.15289
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.64575	0.763763
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	−9.64575	−2.49052
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−4.00000	−0.942809
$19$	0.354249	0.0812702	0.0406351	−	0.999174i	$-0.487062\pi$
$19$	0.354249	0.0812702	0.0406351	+	0.999174i	$0.487062\pi$
$20$	−3.64575	−0.815215
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−3.64575	−0.760192	−0.380096	−	0.924947i	$-0.624109\pi$
$23$	−3.64575	−0.760192	−0.380096	+	0.924947i	$0.624109\pi$
$24$	−2.64575	−0.540062
$25$	8.29150	1.65830
$26$	5.00000	0.980581
$27$	2.64575	0.509175
$28$	0	0
$29$	−4.29150	−0.796912	−0.398456	−	0.917187i	$-0.630454\pi$
$29$	−4.29150	−0.796912	−0.398456	+	0.917187i	$0.630454\pi$
$30$	9.64575	1.76107
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	2.64575	0.460566
$34$	6.00000	1.02899
$35$	0	0
$36$	4.00000	0.666667
$37$	−1.64575	−0.270560	−0.135280	−	0.990807i	$-0.543193\pi$
$37$	−1.64575	−0.270560	−0.135280	+	0.990807i	$0.543193\pi$
$38$	−0.354249	−0.0574667
$39$	−13.2288	−2.11830
$40$	3.64575	0.576444
$41$	4.93725	0.771070	0.385535	−	0.922693i	$-0.374017\pi$
$41$	4.93725	0.771070	0.385535	+	0.922693i	$0.374017\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	−14.5830	−2.17391
$46$	3.64575	0.537537
$47$	−13.2915	−1.93876	−0.969382	−	0.245556i	$-0.921030\pi$
$47$	−13.2915	−1.93876	−0.969382	+	0.245556i	$0.921030\pi$
$48$	2.64575	0.381881
$49$	0	0
$50$	−8.29150	−1.17260
$51$	−15.8745	−2.22288
$52$	−5.00000	−0.693375
$53$	−3.64575	−0.500782	−0.250391	−	0.968145i	$-0.580559\pi$
$53$	−3.64575	−0.500782	−0.250391	+	0.968145i	$0.580559\pi$
$54$	−2.64575	−0.360041
$55$	−3.64575	−0.491593
$56$	0	0
$57$	0.937254	0.124142
$58$	4.29150	0.563502
$59$	0.645751	0.0840697	0.0420348	−	0.999116i	$-0.486616\pi$
$59$	0.645751	0.0840697	0.0420348	+	0.999116i	$0.486616\pi$
$60$	−9.64575	−1.24526
$61$	−3.70850	−0.474824	−0.237412	−	0.971409i	$-0.576299\pi$
$61$	−3.70850	−0.474824	−0.237412	+	0.971409i	$0.576299\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	18.2288	2.26100
$66$	−2.64575	−0.325669
$67$	3.93725	0.481012	0.240506	−	0.970648i	$-0.422687\pi$
$67$	3.93725	0.481012	0.240506	+	0.970648i	$0.422687\pi$
$68$	−6.00000	−0.727607
$69$	−9.64575	−1.16121
$70$	0	0
$71$	9.64575	1.14474	0.572370	−	0.819995i	$-0.306024\pi$
$71$	9.64575	1.14474	0.572370	+	0.819995i	$0.306024\pi$
$72$	−4.00000	−0.471405
$73$	−5.64575	−0.660785	−0.330393	−	0.943844i	$-0.607181\pi$
$73$	−5.64575	−0.660785	−0.330393	+	0.943844i	$0.607181\pi$
$74$	1.64575	0.191315
$75$	21.9373	2.53310
$76$	0.354249	0.0406351
$77$	0	0
$78$	13.2288	1.49786
$79$	2.64575	0.297670	0.148835	−	0.988862i	$-0.452448\pi$
$79$	2.64575	0.297670	0.148835	+	0.988862i	$0.452448\pi$
$80$	−3.64575	−0.407607
$81$	−5.00000	−0.555556
$82$	−4.93725	−0.545228
$83$	−13.2915	−1.45893	−0.729466	−	0.684017i	$-0.760231\pi$
$83$	−13.2915	−1.45893	−0.729466	+	0.684017i	$0.760231\pi$
$84$	0	0
$85$	21.8745	2.37262
$86$	4.00000	0.431331
$87$	−11.3542	−1.21730
$88$	−1.00000	−0.106600
$89$	14.5830	1.54580	0.772898	−	0.634531i	$-0.218807\pi$
$89$	14.5830	1.54580	0.772898	+	0.634531i	$0.218807\pi$
$90$	14.5830	1.53718
$91$	0	0
$92$	−3.64575	−0.380096
$93$	10.5830	1.09741
$94$	13.2915	1.37091
$95$	−1.29150	−0.132505
$96$	−2.64575	−0.270031
$97$	5.70850	0.579610	0.289805	−	0.957086i	$-0.406410\pi$
$97$	5.70850	0.579610	0.289805	+	0.957086i	$0.406410\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	8.29150	0.829150
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	15.8745	1.57181
$103$	−12.9373	−1.27475	−0.637373	−	0.770556i	$-0.719979\pi$
$103$	−12.9373	−1.27475	−0.637373	+	0.770556i	$0.719979\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	3.64575	0.354107
$107$	4.93725	0.477302	0.238651	−	0.971105i	$-0.423295\pi$
$107$	4.93725	0.477302	0.238651	+	0.971105i	$0.423295\pi$
$108$	2.64575	0.254588
$109$	10.5830	1.01367	0.506834	−	0.862044i	$-0.330816\pi$
$109$	10.5830	1.01367	0.506834	+	0.862044i	$0.330816\pi$
$110$	3.64575	0.347609
$111$	−4.35425	−0.413287
$112$	0	0
$113$	−7.70850	−0.725154	−0.362577	−	0.931954i	$-0.618103\pi$
$113$	−7.70850	−0.725154	−0.362577	+	0.931954i	$0.618103\pi$
$114$	−0.937254	−0.0877819
$115$	13.2915	1.23944
$116$	−4.29150	−0.398456
$117$	−20.0000	−1.84900
$118$	−0.645751	−0.0594462
$119$	0	0
$120$	9.64575	0.880533
$121$	1.00000	0.0909091
$122$	3.70850	0.335752
$123$	13.0627	1.17783
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	0.0627461	0.00556781	0.00278391	−	0.999996i	$-0.499114\pi$
$127$	0.0627461	0.00556781	0.00278391	+	0.999996i	$0.499114\pi$
$128$	−1.00000	−0.0883883
$129$	−10.5830	−0.931782
$130$	−18.2288	−1.59877
$131$	−15.6458	−1.36698	−0.683488	−	0.729962i	$-0.739538\pi$
$131$	−15.6458	−1.36698	−0.683488	+	0.729962i	$0.739538\pi$
$132$	2.64575	0.230283
$133$	0	0
$134$	−3.93725	−0.340127
$135$	−9.64575	−0.830174
$136$	6.00000	0.514496
$137$	18.8745	1.61256	0.806279	−	0.591535i	$-0.201478\pi$
$137$	18.8745	1.61256	0.806279	+	0.591535i	$0.201478\pi$
$138$	9.64575	0.821101
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−35.1660	−2.96151
$142$	−9.64575	−0.809453
$143$	−5.00000	−0.418121
$144$	4.00000	0.333333
$145$	15.6458	1.29931
$146$	5.64575	0.467246
$147$	0	0
$148$	−1.64575	−0.135280
$149$	−4.70850	−0.385735	−0.192868	−	0.981225i	$-0.561779\pi$
$149$	−4.70850	−0.385735	−0.192868	+	0.981225i	$0.561779\pi$
$150$	−21.9373	−1.79117
$151$	−3.35425	−0.272965	−0.136482	−	0.990642i	$-0.543580\pi$
$151$	−3.35425	−0.272965	−0.136482	+	0.990642i	$0.543580\pi$
$152$	−0.354249	−0.0287334
$153$	−24.0000	−1.94029
$154$	0	0
$155$	−14.5830	−1.17134
$156$	−13.2288	−1.05915
$157$	21.1660	1.68923	0.844616	−	0.535373i	$-0.179829\pi$
$157$	21.1660	1.68923	0.844616	+	0.535373i	$0.179829\pi$
$158$	−2.64575	−0.210485
$159$	−9.64575	−0.764958
$160$	3.64575	0.288222
$161$	0	0
$162$	5.00000	0.392837
$163$	−4.64575	−0.363883	−0.181942	−	0.983309i	$-0.558238\pi$
$163$	−4.64575	−0.363883	−0.181942	+	0.983309i	$0.558238\pi$
$164$	4.93725	0.385535
$165$	−9.64575	−0.750921
$166$	13.2915	1.03162
$167$	−15.2288	−1.17844	−0.589218	−	0.807974i	$-0.700564\pi$
$167$	−15.2288	−1.17844	−0.589218	+	0.807974i	$0.700564\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−21.8745	−1.67770
$171$	1.41699	0.108360
$172$	−4.00000	−0.304997
$173$	−10.2915	−0.782448	−0.391224	−	0.920295i	$-0.627948\pi$
$173$	−10.2915	−0.782448	−0.391224	+	0.920295i	$0.627948\pi$
$174$	11.3542	0.860763
$175$	0	0
$176$	1.00000	0.0753778
$177$	1.70850	0.128419
$178$	−14.5830	−1.09304
$179$	−4.06275	−0.303664	−0.151832	−	0.988406i	$-0.548517\pi$
$179$	−4.06275	−0.303664	−0.151832	+	0.988406i	$0.548517\pi$
$180$	−14.5830	−1.08695
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−9.81176	−0.725306
$184$	3.64575	0.268768
$185$	6.00000	0.441129
$186$	−10.5830	−0.775984
$187$	−6.00000	−0.438763
$188$	−13.2915	−0.969382
$189$	0	0
$190$	1.29150	0.0936954
$191$	−13.2915	−0.961739	−0.480870	−	0.876792i	$-0.659679\pi$
$191$	−13.2915	−0.961739	−0.480870	+	0.876792i	$0.659679\pi$
$192$	2.64575	0.190941
$193$	−11.5203	−0.829246	−0.414623	−	0.909993i	$-0.636087\pi$
$193$	−11.5203	−0.829246	−0.414623	+	0.909993i	$0.636087\pi$
$194$	−5.70850	−0.409846
$195$	48.2288	3.45373
$196$	0	0
$197$	18.8745	1.34475	0.672377	−	0.740209i	$-0.265274\pi$
$197$	18.8745	1.34475	0.672377	+	0.740209i	$0.265274\pi$
$198$	−4.00000	−0.284268
$199$	22.2288	1.57575	0.787877	−	0.615832i	$-0.211180\pi$
$199$	22.2288	1.57575	0.787877	+	0.615832i	$0.211180\pi$
$200$	−8.29150	−0.586298
$201$	10.4170	0.734758
$202$	3.00000	0.211079
$203$	0	0
$204$	−15.8745	−1.11144
$205$	−18.0000	−1.25717
$206$	12.9373	0.901381
$207$	−14.5830	−1.01359
$208$	−5.00000	−0.346688
$209$	0.354249	0.0245039
$210$	0	0
$211$	−14.9373	−1.02832	−0.514161	−	0.857693i	$-0.671897\pi$
$211$	−14.9373	−1.02832	−0.514161	+	0.857693i	$0.671897\pi$
$212$	−3.64575	−0.250391
$213$	25.5203	1.74862
$214$	−4.93725	−0.337504
$215$	14.5830	0.994553
$216$	−2.64575	−0.180021
$217$	0	0
$218$	−10.5830	−0.716772
$219$	−14.9373	−1.00937
$220$	−3.64575	−0.245797
$221$	30.0000	2.01802
$222$	4.35425	0.292238
$223$	12.3542	0.827302	0.413651	−	0.910436i	$-0.364253\pi$
$223$	12.3542	0.827302	0.413651	+	0.910436i	$0.364253\pi$
$224$	0	0
$225$	33.1660	2.21107
$226$	7.70850	0.512762
$227$	13.2915	0.882188	0.441094	−	0.897461i	$-0.354591\pi$
$227$	13.2915	0.882188	0.441094	+	0.897461i	$0.354591\pi$
$228$	0.937254	0.0620712
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	−13.2915	−0.876416
$231$	0	0
$232$	4.29150	0.281751
$233$	16.9373	1.10960	0.554798	−	0.831985i	$-0.312795\pi$
$233$	16.9373	1.10960	0.554798	+	0.831985i	$0.312795\pi$
$234$	20.0000	1.30744
$235$	48.4575	3.16102
$236$	0.645751	0.0420348
$237$	7.00000	0.454699
$238$	0	0
$239$	9.22876	0.596959	0.298479	−	0.954416i	$-0.403521\pi$
$239$	9.22876	0.596959	0.298479	+	0.954416i	$0.403521\pi$
$240$	−9.64575	−0.622631
$241$	−22.8118	−1.46943	−0.734717	−	0.678373i	$-0.762685\pi$
$241$	−22.8118	−1.46943	−0.734717	+	0.678373i	$0.762685\pi$
$242$	−1.00000	−0.0642824
$243$	−21.1660	−1.35780
$244$	−3.70850	−0.237412
$245$	0	0
$246$	−13.0627	−0.832850
$247$	−1.77124	−0.112702
$248$	−4.00000	−0.254000
$249$	−35.1660	−2.22856
$250$	12.0000	0.758947
$251$	−7.29150	−0.460236	−0.230118	−	0.973163i	$-0.573911\pi$
$251$	−7.29150	−0.460236	−0.230118	+	0.973163i	$0.573911\pi$
$252$	0	0
$253$	−3.64575	−0.229206
$254$	−0.0627461	−0.00393704
$255$	57.8745	3.62424
$256$	1.00000	0.0625000
$257$	0.416995	0.0260114	0.0130057	−	0.999915i	$-0.495860\pi$
$257$	0.416995	0.0260114	0.0130057	+	0.999915i	$0.495860\pi$
$258$	10.5830	0.658869
$259$	0	0
$260$	18.2288	1.13050
$261$	−17.1660	−1.06255
$262$	15.6458	0.966598
$263$	4.06275	0.250520	0.125260	−	0.992124i	$-0.460024\pi$
$263$	4.06275	0.250520	0.125260	+	0.992124i	$0.460024\pi$
$264$	−2.64575	−0.162835
$265$	13.2915	0.816491
$266$	0	0
$267$	38.5830	2.36124
$268$	3.93725	0.240506
$269$	−26.5830	−1.62079	−0.810397	−	0.585881i	$-0.800749\pi$
$269$	−26.5830	−1.62079	−0.810397	+	0.585881i	$0.800749\pi$
$270$	9.64575	0.587022
$271$	17.9373	1.08961	0.544805	−	0.838563i	$-0.316604\pi$
$271$	17.9373	1.08961	0.544805	+	0.838563i	$0.316604\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−18.8745	−1.14025
$275$	8.29150	0.499996
$276$	−9.64575	−0.580606
$277$	−11.7085	−0.703495	−0.351748	−	0.936095i	$-0.614413\pi$
$277$	−11.7085	−0.703495	−0.351748	+	0.936095i	$0.614413\pi$
$278$	−4.00000	−0.239904
$279$	16.0000	0.957895
$280$	0	0
$281$	−7.06275	−0.421328	−0.210664	−	0.977559i	$-0.567563\pi$
$281$	−7.06275	−0.421328	−0.210664	+	0.977559i	$0.567563\pi$
$282$	35.1660	2.09411
$283$	7.64575	0.454493	0.227246	−	0.973837i	$-0.427028\pi$
$283$	7.64575	0.454493	0.227246	+	0.973837i	$0.427028\pi$
$284$	9.64575	0.572370
$285$	−3.41699	−0.202405
$286$	5.00000	0.295656
$287$	0	0
$288$	−4.00000	−0.235702
$289$	19.0000	1.11765
$290$	−15.6458	−0.918750
$291$	15.1033	0.885369
$292$	−5.64575	−0.330393
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	−2.35425	−0.137070
$296$	1.64575	0.0956574
$297$	2.64575	0.153522
$298$	4.70850	0.272756
$299$	18.2288	1.05420
$300$	21.9373	1.26655
$301$	0	0
$302$	3.35425	0.193015
$303$	−7.93725	−0.455983
$304$	0.354249	0.0203176
$305$	13.5203	0.774168
$306$	24.0000	1.37199
$307$	4.22876	0.241348	0.120674	−	0.992692i	$-0.461494\pi$
$307$	4.22876	0.241348	0.120674	+	0.992692i	$0.461494\pi$
$308$	0	0
$309$	−34.2288	−1.94721
$310$	14.5830	0.828259
$311$	−16.9373	−0.960424	−0.480212	−	0.877153i	$-0.659440\pi$
$311$	−16.9373	−0.960424	−0.480212	+	0.877153i	$0.659440\pi$
$312$	13.2288	0.748931
$313$	−2.41699	−0.136617	−0.0683083	−	0.997664i	$-0.521760\pi$
$313$	−2.41699	−0.136617	−0.0683083	+	0.997664i	$0.521760\pi$
$314$	−21.1660	−1.19447
$315$	0	0
$316$	2.64575	0.148835
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	9.64575	0.540907
$319$	−4.29150	−0.240278
$320$	−3.64575	−0.203804
$321$	13.0627	0.729091
$322$	0	0
$323$	−2.12549	−0.118266
$324$	−5.00000	−0.277778
$325$	−41.4575	−2.29965
$326$	4.64575	0.257304
$327$	28.0000	1.54840
$328$	−4.93725	−0.272614
$329$	0	0
$330$	9.64575	0.530981
$331$	−15.3542	−0.843946	−0.421973	−	0.906608i	$-0.638662\pi$
$331$	−15.3542	−0.843946	−0.421973	+	0.906608i	$0.638662\pi$
$332$	−13.2915	−0.729466
$333$	−6.58301	−0.360746
$334$	15.2288	0.833280
$335$	−14.3542	−0.784256
$336$	0	0
$337$	24.9373	1.35842	0.679209	−	0.733945i	$-0.262323\pi$
$337$	24.9373	1.35842	0.679209	+	0.733945i	$0.262323\pi$
$338$	−12.0000	−0.652714
$339$	−20.3948	−1.10769
$340$	21.8745	1.18631
$341$	4.00000	0.216612
$342$	−1.41699	−0.0766223
$343$	0	0
$344$	4.00000	0.215666
$345$	35.1660	1.89327
$346$	10.2915	0.553275
$347$	26.8118	1.43933	0.719665	−	0.694321i	$-0.244295\pi$
$347$	26.8118	1.43933	0.719665	+	0.694321i	$0.244295\pi$
$348$	−11.3542	−0.608652
$349$	−29.8745	−1.59915	−0.799573	−	0.600569i	$-0.794941\pi$
$349$	−29.8745	−1.59915	−0.799573	+	0.600569i	$0.794941\pi$
$350$	0	0
$351$	−13.2288	−0.706099
$352$	−1.00000	−0.0533002
$353$	35.1660	1.87170	0.935849	−	0.352401i	$-0.114635\pi$
$353$	35.1660	1.87170	0.935849	+	0.352401i	$0.114635\pi$
$354$	−1.70850	−0.0908056
$355$	−35.1660	−1.86642
$356$	14.5830	0.772898
$357$	0	0
$358$	4.06275	0.214723
$359$	−10.0627	−0.531091	−0.265546	−	0.964098i	$-0.585552\pi$
$359$	−10.0627	−0.531091	−0.265546	+	0.964098i	$0.585552\pi$
$360$	14.5830	0.768592
$361$	−18.8745	−0.993395
$362$	−10.0000	−0.525588
$363$	2.64575	0.138866
$364$	0	0
$365$	20.5830	1.07736
$366$	9.81176	0.512869
$367$	9.77124	0.510055	0.255027	−	0.966934i	$-0.417916\pi$
$367$	9.77124	0.510055	0.255027	+	0.966934i	$0.417916\pi$
$368$	−3.64575	−0.190048
$369$	19.7490	1.02809
$370$	−6.00000	−0.311925
$371$	0	0
$372$	10.5830	0.548703
$373$	−10.8745	−0.563061	−0.281530	−	0.959552i	$-0.590842\pi$
$373$	−10.8745	−0.563061	−0.281530	+	0.959552i	$0.590842\pi$
$374$	6.00000	0.310253
$375$	−31.7490	−1.63951
$376$	13.2915	0.685457
$377$	21.4575	1.10512
$378$	0	0
$379$	21.9373	1.12684	0.563421	−	0.826170i	$-0.309485\pi$
$379$	21.9373	1.12684	0.563421	+	0.826170i	$0.309485\pi$
$380$	−1.29150	−0.0662527
$381$	0.166010	0.00850497
$382$	13.2915	0.680052
$383$	34.1033	1.74260	0.871298	−	0.490755i	$-0.163279\pi$
$383$	34.1033	1.74260	0.871298	+	0.490755i	$0.163279\pi$
$384$	−2.64575	−0.135015
$385$	0	0
$386$	11.5203	0.586366
$387$	−16.0000	−0.813326
$388$	5.70850	0.289805
$389$	−20.8118	−1.05520	−0.527599	−	0.849493i	$-0.676908\pi$
$389$	−20.8118	−1.05520	−0.527599	+	0.849493i	$0.676908\pi$
$390$	−48.2288	−2.44216
$391$	21.8745	1.10624
$392$	0	0
$393$	−41.3948	−2.08809
$394$	−18.8745	−0.950884
$395$	−9.64575	−0.485330
$396$	4.00000	0.201008
$397$	−31.1660	−1.56418	−0.782089	−	0.623167i	$-0.785846\pi$
$397$	−31.1660	−1.56418	−0.782089	+	0.623167i	$0.785846\pi$
$398$	−22.2288	−1.11423
$399$	0	0
$400$	8.29150	0.414575
$401$	0.416995	0.0208237	0.0104119	−	0.999946i	$-0.496686\pi$
$401$	0.416995	0.0208237	0.0104119	+	0.999946i	$0.496686\pi$
$402$	−10.4170	−0.519552
$403$	−20.0000	−0.996271
$404$	−3.00000	−0.149256
$405$	18.2288	0.905794
$406$	0	0
$407$	−1.64575	−0.0815769
$408$	15.8745	0.785905
$409$	−18.9373	−0.936387	−0.468193	−	0.883626i	$-0.655095\pi$
$409$	−18.9373	−0.936387	−0.468193	+	0.883626i	$0.655095\pi$
$410$	18.0000	0.888957
$411$	49.9373	2.46322
$412$	−12.9373	−0.637373
$413$	0	0
$414$	14.5830	0.716716
$415$	48.4575	2.37869
$416$	5.00000	0.245145
$417$	10.5830	0.518252
$418$	−0.354249	−0.0173269
$419$	−21.8745	−1.06864	−0.534320	−	0.845282i	$-0.679432\pi$
$419$	−21.8745	−1.06864	−0.534320	+	0.845282i	$0.679432\pi$
$420$	0	0
$421$	−33.1660	−1.61641	−0.808206	−	0.588900i	$-0.799561\pi$
$421$	−33.1660	−1.61641	−0.808206	+	0.588900i	$0.799561\pi$
$422$	14.9373	0.727134
$423$	−53.1660	−2.58502
$424$	3.64575	0.177053
$425$	−49.7490	−2.41318
$426$	−25.5203	−1.23646
$427$	0	0
$428$	4.93725	0.238651
$429$	−13.2288	−0.638690
$430$	−14.5830	−0.703255
$431$	2.77124	0.133486	0.0667430	−	0.997770i	$-0.478739\pi$
$431$	2.77124	0.133486	0.0667430	+	0.997770i	$0.478739\pi$
$432$	2.64575	0.127294
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	41.3948	1.98473
$436$	10.5830	0.506834
$437$	−1.29150	−0.0617809
$438$	14.9373	0.713730
$439$	11.9373	0.569734	0.284867	−	0.958567i	$-0.408051\pi$
$439$	11.9373	0.569734	0.284867	+	0.958567i	$0.408051\pi$
$440$	3.64575	0.173804
$441$	0	0
$442$	−30.0000	−1.42695
$443$	−18.4575	−0.876943	−0.438471	−	0.898745i	$-0.644480\pi$
$443$	−18.4575	−0.876943	−0.438471	+	0.898745i	$0.644480\pi$
$444$	−4.35425	−0.206643
$445$	−53.1660	−2.52031
$446$	−12.3542	−0.584991
$447$	−12.4575	−0.589220
$448$	0	0
$449$	9.87451	0.466007	0.233003	−	0.972476i	$-0.425145\pi$
$449$	9.87451	0.466007	0.233003	+	0.972476i	$0.425145\pi$
$450$	−33.1660	−1.56346
$451$	4.93725	0.232486
$452$	−7.70850	−0.362577
$453$	−8.87451	−0.416961
$454$	−13.2915	−0.623801
$455$	0	0
$456$	−0.937254	−0.0438909
$457$	−39.1660	−1.83211	−0.916054	−	0.401054i	$-0.868644\pi$
$457$	−39.1660	−1.83211	−0.916054	+	0.401054i	$0.868644\pi$
$458$	−16.0000	−0.747631
$459$	−15.8745	−0.740959
$460$	13.2915	0.619720
$461$	32.1660	1.49812	0.749060	−	0.662502i	$-0.230505\pi$
$461$	32.1660	1.49812	0.749060	+	0.662502i	$0.230505\pi$
$462$	0	0
$463$	−22.4575	−1.04369	−0.521845	−	0.853041i	$-0.674756\pi$
$463$	−22.4575	−1.04369	−0.521845	+	0.853041i	$0.674756\pi$
$464$	−4.29150	−0.199228
$465$	−38.5830	−1.78924
$466$	−16.9373	−0.784603
$467$	−10.7085	−0.495530	−0.247765	−	0.968820i	$-0.579696\pi$
$467$	−10.7085	−0.495530	−0.247765	+	0.968820i	$0.579696\pi$
$468$	−20.0000	−0.924500
$469$	0	0
$470$	−48.4575	−2.23518
$471$	56.0000	2.58034
$472$	−0.645751	−0.0297231
$473$	−4.00000	−0.183920
$474$	−7.00000	−0.321521
$475$	2.93725	0.134770
$476$	0	0
$477$	−14.5830	−0.667710
$478$	−9.22876	−0.422113
$479$	25.9373	1.18510	0.592552	−	0.805532i	$-0.298121\pi$
$479$	25.9373	1.18510	0.592552	+	0.805532i	$0.298121\pi$
$480$	9.64575	0.440266
$481$	8.22876	0.375199
$482$	22.8118	1.03905
$483$	0	0
$484$	1.00000	0.0454545
$485$	−20.8118	−0.945013
$486$	21.1660	0.960110
$487$	−30.5830	−1.38585	−0.692924	−	0.721011i	$-0.743678\pi$
$487$	−30.5830	−1.38585	−0.692924	+	0.721011i	$0.743678\pi$
$488$	3.70850	0.167876
$489$	−12.2915	−0.555841
$490$	0	0
$491$	10.7085	0.483268	0.241634	−	0.970367i	$-0.422317\pi$
$491$	10.7085	0.483268	0.241634	+	0.970367i	$0.422317\pi$
$492$	13.0627	0.588914
$493$	25.7490	1.15968
$494$	1.77124	0.0796920
$495$	−14.5830	−0.655457
$496$	4.00000	0.179605
$497$	0	0
$498$	35.1660	1.57583
$499$	17.8745	0.800173	0.400086	−	0.916477i	$-0.368980\pi$
$499$	17.8745	0.800173	0.400086	+	0.916477i	$0.368980\pi$
$500$	−12.0000	−0.536656
$501$	−40.2915	−1.80009
$502$	7.29150	0.325436
$503$	19.9373	0.888958	0.444479	−	0.895789i	$-0.353389\pi$
$503$	19.9373	0.888958	0.444479	+	0.895789i	$0.353389\pi$
$504$	0	0
$505$	10.9373	0.486701
$506$	3.64575	0.162073
$507$	31.7490	1.41002
$508$	0.0627461	0.00278391
$509$	−20.5830	−0.912326	−0.456163	−	0.889896i	$-0.650777\pi$
$509$	−20.5830	−0.912326	−0.456163	+	0.889896i	$0.650777\pi$
$510$	−57.8745	−2.56273
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0.937254	0.0413808
$514$	−0.416995	−0.0183929
$515$	47.1660	2.07838
$516$	−10.5830	−0.465891
$517$	−13.2915	−0.584560
$518$	0	0
$519$	−27.2288	−1.19521
$520$	−18.2288	−0.799384
$521$	2.12549	0.0931195	0.0465598	−	0.998916i	$-0.485174\pi$
$521$	2.12549	0.0931195	0.0465598	+	0.998916i	$0.485174\pi$
$522$	17.1660	0.751336
$523$	−15.5203	−0.678654	−0.339327	−	0.940669i	$-0.610199\pi$
$523$	−15.5203	−0.678654	−0.339327	+	0.940669i	$0.610199\pi$
$524$	−15.6458	−0.683488
$525$	0	0
$526$	−4.06275	−0.177144
$527$	−24.0000	−1.04546
$528$	2.64575	0.115142
$529$	−9.70850	−0.422109
$530$	−13.2915	−0.577346
$531$	2.58301	0.112093
$532$	0	0
$533$	−24.6863	−1.06928
$534$	−38.5830	−1.66965
$535$	−18.0000	−0.778208
$536$	−3.93725	−0.170063
$537$	−10.7490	−0.463854
$538$	26.5830	1.14607
$539$	0	0
$540$	−9.64575	−0.415087
$541$	−8.29150	−0.356480	−0.178240	−	0.983987i	$-0.557040\pi$
$541$	−8.29150	−0.356480	−0.178240	+	0.983987i	$0.557040\pi$
$542$	−17.9373	−0.770471
$543$	26.4575	1.13540
$544$	6.00000	0.257248
$545$	−38.5830	−1.65271
$546$	0	0
$547$	27.5203	1.17668	0.588341	−	0.808613i	$-0.299781\pi$
$547$	27.5203	1.17668	0.588341	+	0.808613i	$0.299781\pi$
$548$	18.8745	0.806279
$549$	−14.8340	−0.633099
$550$	−8.29150	−0.353551
$551$	−1.52026	−0.0647652
$552$	9.64575	0.410550
$553$	0	0
$554$	11.7085	0.497446
$555$	15.8745	0.673835
$556$	4.00000	0.169638
$557$	31.7490	1.34525	0.672624	−	0.739984i	$-0.265167\pi$
$557$	31.7490	1.34525	0.672624	+	0.739984i	$0.265167\pi$
$558$	−16.0000	−0.677334
$559$	20.0000	0.845910
$560$	0	0
$561$	−15.8745	−0.670222
$562$	7.06275	0.297924
$563$	−13.0627	−0.550529	−0.275265	−	0.961369i	$-0.588765\pi$
$563$	−13.0627	−0.550529	−0.275265	+	0.961369i	$0.588765\pi$
$564$	−35.1660	−1.48076
$565$	28.1033	1.18231
$566$	−7.64575	−0.321375
$567$	0	0
$568$	−9.64575	−0.404727
$569$	−41.1660	−1.72577	−0.862884	−	0.505401i	$-0.831344\pi$
$569$	−41.1660	−1.72577	−0.862884	+	0.505401i	$0.831344\pi$
$570$	3.41699	0.143122
$571$	−44.9373	−1.88057	−0.940283	−	0.340394i	$-0.889439\pi$
$571$	−44.9373	−1.88057	−0.940283	+	0.340394i	$0.889439\pi$
$572$	−5.00000	−0.209061
$573$	−35.1660	−1.46908
$574$	0	0
$575$	−30.2288	−1.26063
$576$	4.00000	0.166667
$577$	25.4575	1.05981	0.529905	−	0.848057i	$-0.322228\pi$
$577$	25.4575	1.05981	0.529905	+	0.848057i	$0.322228\pi$
$578$	−19.0000	−0.790296
$579$	−30.4797	−1.26669
$580$	15.6458	0.649654
$581$	0	0
$582$	−15.1033	−0.626050
$583$	−3.64575	−0.150992
$584$	5.64575	0.233623
$585$	72.9150	3.01467
$586$	12.0000	0.495715
$587$	7.93725	0.327606	0.163803	−	0.986493i	$-0.447624\pi$
$587$	7.93725	0.327606	0.163803	+	0.986493i	$0.447624\pi$
$588$	0	0
$589$	1.41699	0.0583863
$590$	2.35425	0.0969229
$591$	49.9373	2.05414
$592$	−1.64575	−0.0676400
$593$	−22.9373	−0.941920	−0.470960	−	0.882155i	$-0.656092\pi$
$593$	−22.9373	−0.941920	−0.470960	+	0.882155i	$0.656092\pi$
$594$	−2.64575	−0.108556
$595$	0	0
$596$	−4.70850	−0.192868
$597$	58.8118	2.40701
$598$	−18.2288	−0.745429
$599$	−19.7490	−0.806923	−0.403461	−	0.914997i	$-0.632193\pi$
$599$	−19.7490	−0.806923	−0.403461	+	0.914997i	$0.632193\pi$
$600$	−21.9373	−0.895585
$601$	24.5830	1.00276	0.501381	−	0.865227i	$-0.332826\pi$
$601$	24.5830	1.00276	0.501381	+	0.865227i	$0.332826\pi$
$602$	0	0
$603$	15.7490	0.641350
$604$	−3.35425	−0.136482
$605$	−3.64575	−0.148221
$606$	7.93725	0.322429
$607$	−21.2915	−0.864195	−0.432098	−	0.901827i	$-0.642226\pi$
$607$	−21.2915	−0.864195	−0.432098	+	0.901827i	$0.642226\pi$
$608$	−0.354249	−0.0143667
$609$	0	0
$610$	−13.5203	−0.547419
$611$	66.4575	2.68858
$612$	−24.0000	−0.970143
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−4.22876	−0.170659
$615$	−47.6235	−1.92037
$616$	0	0
$617$	−16.2915	−0.655871	−0.327936	−	0.944700i	$-0.606353\pi$
$617$	−16.2915	−0.655871	−0.327936	+	0.944700i	$0.606353\pi$
$618$	34.2288	1.37688
$619$	−34.5830	−1.39001	−0.695004	−	0.719006i	$-0.744597\pi$
$619$	−34.5830	−1.39001	−0.695004	+	0.719006i	$0.744597\pi$
$620$	−14.5830	−0.585668
$621$	−9.64575	−0.387071
$622$	16.9373	0.679122
$623$	0	0
$624$	−13.2288	−0.529574
$625$	2.29150	0.0916601
$626$	2.41699	0.0966025
$627$	0.937254	0.0374303
$628$	21.1660	0.844616
$629$	9.87451	0.393722
$630$	0	0
$631$	34.8118	1.38583	0.692917	−	0.721017i	$-0.256325\pi$
$631$	34.8118	1.38583	0.692917	+	0.721017i	$0.256325\pi$
$632$	−2.64575	−0.105242
$633$	−39.5203	−1.57079
$634$	−12.0000	−0.476581
$635$	−0.228757	−0.00907793
$636$	−9.64575	−0.382479
$637$	0	0
$638$	4.29150	0.169902
$639$	38.5830	1.52632
$640$	3.64575	0.144111
$641$	−18.8745	−0.745498	−0.372749	−	0.927932i	$-0.621585\pi$
$641$	−18.8745	−0.745498	−0.372749	+	0.927932i	$0.621585\pi$
$642$	−13.0627	−0.515545
$643$	−6.52026	−0.257134	−0.128567	−	0.991701i	$-0.541038\pi$
$643$	−6.52026	−0.257134	−0.128567	+	0.991701i	$0.541038\pi$
$644$	0	0
$645$	38.5830	1.51920
$646$	2.12549	0.0836264
$647$	−38.8118	−1.52585	−0.762924	−	0.646488i	$-0.776237\pi$
$647$	−38.8118	−1.52585	−0.762924	+	0.646488i	$0.776237\pi$
$648$	5.00000	0.196419
$649$	0.645751	0.0253480
$650$	41.4575	1.62610
$651$	0	0
$652$	−4.64575	−0.181942
$653$	2.35425	0.0921289	0.0460644	−	0.998938i	$-0.485332\pi$
$653$	2.35425	0.0921289	0.0460644	+	0.998938i	$0.485332\pi$
$654$	−28.0000	−1.09489
$655$	57.0405	2.22876
$656$	4.93725	0.192767
$657$	−22.5830	−0.881047
$658$	0	0
$659$	14.5830	0.568073	0.284037	−	0.958813i	$-0.408326\pi$
$659$	14.5830	0.568073	0.284037	+	0.958813i	$0.408326\pi$
$660$	−9.64575	−0.375460
$661$	−28.5830	−1.11175	−0.555875	−	0.831266i	$-0.687617\pi$
$661$	−28.5830	−1.11175	−0.555875	+	0.831266i	$0.687617\pi$
$662$	15.3542	0.596760
$663$	79.3725	3.08257
$664$	13.2915	0.515810
$665$	0	0
$666$	6.58301	0.255086
$667$	15.6458	0.605806
$668$	−15.2288	−0.589218
$669$	32.6863	1.26372
$670$	14.3542	0.554553
$671$	−3.70850	−0.143165
$672$	0	0
$673$	−14.9373	−0.575789	−0.287894	−	0.957662i	$-0.592955\pi$
$673$	−14.9373	−0.575789	−0.287894	+	0.957662i	$0.592955\pi$
$674$	−24.9373	−0.960547
$675$	21.9373	0.844365
$676$	12.0000	0.461538
$677$	2.12549	0.0816893	0.0408446	−	0.999166i	$-0.486995\pi$
$677$	2.12549	0.0816893	0.0408446	+	0.999166i	$0.486995\pi$
$678$	20.3948	0.783256
$679$	0	0
$680$	−21.8745	−0.838849
$681$	35.1660	1.34756
$682$	−4.00000	−0.153168
$683$	−13.9373	−0.533294	−0.266647	−	0.963794i	$-0.585916\pi$
$683$	−13.9373	−0.533294	−0.266647	+	0.963794i	$0.585916\pi$
$684$	1.41699	0.0541801
$685$	−68.8118	−2.62916
$686$	0	0
$687$	42.3320	1.61507
$688$	−4.00000	−0.152499
$689$	18.2288	0.694460
$690$	−35.1660	−1.33875
$691$	18.7712	0.714092	0.357046	−	0.934087i	$-0.383784\pi$
$691$	18.7712	0.714092	0.357046	+	0.934087i	$0.383784\pi$
$692$	−10.2915	−0.391224
$693$	0	0
$694$	−26.8118	−1.01776
$695$	−14.5830	−0.553165
$696$	11.3542	0.430382
$697$	−29.6235	−1.12207
$698$	29.8745	1.13077
$699$	44.8118	1.69494
$700$	0	0
$701$	−6.87451	−0.259647	−0.129823	−	0.991537i	$-0.541441\pi$
$701$	−6.87451	−0.259647	−0.129823	+	0.991537i	$0.541441\pi$
$702$	13.2288	0.499287
$703$	−0.583005	−0.0219885
$704$	1.00000	0.0376889
$705$	128.207	4.82854
$706$	−35.1660	−1.32349
$707$	0	0
$708$	1.70850	0.0642093
$709$	−6.81176	−0.255821	−0.127911	−	0.991786i	$-0.540827\pi$
$709$	−6.81176	−0.255821	−0.127911	+	0.991786i	$0.540827\pi$
$710$	35.1660	1.31976
$711$	10.5830	0.396894
$712$	−14.5830	−0.546521
$713$	−14.5830	−0.546138
$714$	0	0
$715$	18.2288	0.681717
$716$	−4.06275	−0.151832
$717$	24.4170	0.911869
$718$	10.0627	0.375538
$719$	3.87451	0.144495	0.0722474	−	0.997387i	$-0.476983\pi$
$719$	3.87451	0.144495	0.0722474	+	0.997387i	$0.476983\pi$
$720$	−14.5830	−0.543477
$721$	0	0
$722$	18.8745	0.702436
$723$	−60.3542	−2.24460
$724$	10.0000	0.371647
$725$	−35.5830	−1.32152
$726$	−2.64575	−0.0981930
$727$	17.2915	0.641306	0.320653	−	0.947197i	$-0.396098\pi$
$727$	17.2915	0.641306	0.320653	+	0.947197i	$0.396098\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	−20.5830	−0.761811
$731$	24.0000	0.887672
$732$	−9.81176	−0.362653
$733$	−41.4575	−1.53127	−0.765634	−	0.643276i	$-0.777575\pi$
$733$	−41.4575	−1.53127	−0.765634	+	0.643276i	$0.777575\pi$
$734$	−9.77124	−0.360663
$735$	0	0
$736$	3.64575	0.134384
$737$	3.93725	0.145031
$738$	−19.7490	−0.726971
$739$	−7.87451	−0.289668	−0.144834	−	0.989456i	$-0.546265\pi$
$739$	−7.87451	−0.289668	−0.144834	+	0.989456i	$0.546265\pi$
$740$	6.00000	0.220564
$741$	−4.68627	−0.172154
$742$	0	0
$743$	−34.7085	−1.27333	−0.636666	−	0.771140i	$-0.719687\pi$
$743$	−34.7085	−1.27333	−0.636666	+	0.771140i	$0.719687\pi$
$744$	−10.5830	−0.387992
$745$	17.1660	0.628914
$746$	10.8745	0.398144
$747$	−53.1660	−1.94524
$748$	−6.00000	−0.219382
$749$	0	0
$750$	31.7490	1.15931
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−13.2915	−0.484691
$753$	−19.2915	−0.703021
$754$	−21.4575	−0.781437
$755$	12.2288	0.445050
$756$	0	0
$757$	19.1660	0.696600	0.348300	−	0.937383i	$-0.386759\pi$
$757$	19.1660	0.696600	0.348300	+	0.937383i	$0.386759\pi$
$758$	−21.9373	−0.796797
$759$	−9.64575	−0.350119
$760$	1.29150	0.0468477
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−0.166010	−0.00601393
$763$	0	0
$764$	−13.2915	−0.480870
$765$	87.4980	3.16350
$766$	−34.1033	−1.23220
$767$	−3.22876	−0.116584
$768$	2.64575	0.0954703
$769$	15.1660	0.546900	0.273450	−	0.961886i	$-0.411835\pi$
$769$	15.1660	0.546900	0.273450	+	0.961886i	$0.411835\pi$
$770$	0	0
$771$	1.10326	0.0397331
$772$	−11.5203	−0.414623
$773$	−2.58301	−0.0929042	−0.0464521	−	0.998921i	$-0.514791\pi$
$773$	−2.58301	−0.0929042	−0.0464521	+	0.998921i	$0.514791\pi$
$774$	16.0000	0.575108
$775$	33.1660	1.19136
$776$	−5.70850	−0.204923
$777$	0	0
$778$	20.8118	0.746138
$779$	1.74902	0.0626650
$780$	48.2288	1.72687
$781$	9.64575	0.345152
$782$	−21.8745	−0.782231
$783$	−11.3542	−0.405768
$784$	0	0
$785$	−77.1660	−2.75417
$786$	41.3948	1.47650
$787$	0.811762	0.0289362	0.0144681	−	0.999895i	$-0.495395\pi$
$787$	0.811762	0.0289362	0.0144681	+	0.999895i	$0.495395\pi$
$788$	18.8745	0.672377
$789$	10.7490	0.382675
$790$	9.64575	0.343180
$791$	0	0
$792$	−4.00000	−0.142134
$793$	18.5425	0.658463
$794$	31.1660	1.10604
$795$	35.1660	1.24721
$796$	22.2288	0.787877
$797$	35.1660	1.24564	0.622822	−	0.782364i	$-0.285986\pi$
$797$	35.1660	1.24564	0.622822	+	0.782364i	$0.285986\pi$
$798$	0	0
$799$	79.7490	2.82132
$800$	−8.29150	−0.293149
$801$	58.3320	2.06106
$802$	−0.416995	−0.0147246
$803$	−5.64575	−0.199234
$804$	10.4170	0.367379
$805$	0	0
$806$	20.0000	0.704470
$807$	−70.3320	−2.47580
$808$	3.00000	0.105540
$809$	50.5830	1.77840	0.889202	−	0.457515i	$-0.151260\pi$
$809$	50.5830	1.77840	0.889202	+	0.457515i	$0.151260\pi$
$810$	−18.2288	−0.640493
$811$	−27.7490	−0.974400	−0.487200	−	0.873290i	$-0.661982\pi$
$811$	−27.7490	−0.974400	−0.487200	+	0.873290i	$0.661982\pi$
$812$	0	0
$813$	47.4575	1.66441
$814$	1.64575	0.0576836
$815$	16.9373	0.593286
$816$	−15.8745	−0.555719
$817$	−1.41699	−0.0495744
$818$	18.9373	0.662126
$819$	0	0
$820$	−18.0000	−0.628587
$821$	−7.70850	−0.269028	−0.134514	−	0.990912i	$-0.542947\pi$
$821$	−7.70850	−0.269028	−0.134514	+	0.990912i	$0.542947\pi$
$822$	−49.9373	−1.74176
$823$	−43.8745	−1.52937	−0.764685	−	0.644405i	$-0.777105\pi$
$823$	−43.8745	−1.52937	−0.764685	+	0.644405i	$0.777105\pi$
$824$	12.9373	0.450691
$825$	21.9373	0.763757
$826$	0	0
$827$	−35.3948	−1.23080	−0.615398	−	0.788216i	$-0.711005\pi$
$827$	−35.3948	−1.23080	−0.615398	+	0.788216i	$0.711005\pi$
$828$	−14.5830	−0.506794
$829$	−43.3948	−1.50716	−0.753581	−	0.657355i	$-0.771675\pi$
$829$	−43.3948	−1.50716	−0.753581	+	0.657355i	$0.771675\pi$
$830$	−48.4575	−1.68198
$831$	−30.9778	−1.07461
$832$	−5.00000	−0.173344
$833$	0	0
$834$	−10.5830	−0.366460
$835$	55.5203	1.92136
$836$	0.354249	0.0122519
$837$	10.5830	0.365802
$838$	21.8745	0.755642
$839$	−27.8745	−0.962335	−0.481167	−	0.876629i	$-0.659787\pi$
$839$	−27.8745	−0.962335	−0.481167	+	0.876629i	$0.659787\pi$
$840$	0	0
$841$	−10.5830	−0.364931
$842$	33.1660	1.14298
$843$	−18.6863	−0.643589
$844$	−14.9373	−0.514161
$845$	−43.7490	−1.50501
$846$	53.1660	1.82789
$847$	0	0
$848$	−3.64575	−0.125196
$849$	20.2288	0.694249
$850$	49.7490	1.70638
$851$	6.00000	0.205677
$852$	25.5203	0.874310
$853$	3.16601	0.108402	0.0542011	−	0.998530i	$-0.482739\pi$
$853$	3.16601	0.108402	0.0542011	+	0.998530i	$0.482739\pi$
$854$	0	0
$855$	−5.16601	−0.176674
$856$	−4.93725	−0.168752
$857$	−36.0000	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$857$	−36.0000	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$858$	13.2288	0.451622
$859$	−37.8118	−1.29012	−0.645060	−	0.764132i	$-0.723168\pi$
$859$	−37.8118	−1.29012	−0.645060	+	0.764132i	$0.723168\pi$
$860$	14.5830	0.497276
$861$	0	0
$862$	−2.77124	−0.0943889
$863$	−49.5203	−1.68569	−0.842845	−	0.538157i	$-0.819121\pi$
$863$	−49.5203	−1.68569	−0.842845	+	0.538157i	$0.819121\pi$
$864$	−2.64575	−0.0900103
$865$	37.5203	1.27573
$866$	−16.0000	−0.543702
$867$	50.2693	1.70723
$868$	0	0
$869$	2.64575	0.0897510
$870$	−41.3948	−1.40341
$871$	−19.6863	−0.667044
$872$	−10.5830	−0.358386
$873$	22.8340	0.772813
$874$	1.29150	0.0436857
$875$	0	0
$876$	−14.9373	−0.504683
$877$	8.87451	0.299671	0.149835	−	0.988711i	$-0.452126\pi$
$877$	8.87451	0.299671	0.149835	+	0.988711i	$0.452126\pi$
$878$	−11.9373	−0.402863
$879$	−31.7490	−1.07087
$880$	−3.64575	−0.122898
$881$	−6.87451	−0.231608	−0.115804	−	0.993272i	$-0.536944\pi$
$881$	−6.87451	−0.231608	−0.115804	+	0.993272i	$0.536944\pi$
$882$	0	0
$883$	6.06275	0.204028	0.102014	−	0.994783i	$-0.467471\pi$
$883$	6.06275	0.204028	0.102014	+	0.994783i	$0.467471\pi$
$884$	30.0000	1.00901
$885$	−6.22876	−0.209377
$886$	18.4575	0.620092
$887$	13.1033	0.439965	0.219982	−	0.975504i	$-0.429400\pi$
$887$	13.1033	0.439965	0.219982	+	0.975504i	$0.429400\pi$
$888$	4.35425	0.146119
$889$	0	0
$890$	53.1660	1.78213
$891$	−5.00000	−0.167506
$892$	12.3542	0.413651
$893$	−4.70850	−0.157564
$894$	12.4575	0.416642
$895$	14.8118	0.495103
$896$	0	0
$897$	48.2288	1.61031
$898$	−9.87451	−0.329517
$899$	−17.1660	−0.572519
$900$	33.1660	1.10553
$901$	21.8745	0.728746
$902$	−4.93725	−0.164393
$903$	0	0
$904$	7.70850	0.256381
$905$	−36.4575	−1.21189
$906$	8.87451	0.294836
$907$	−10.4575	−0.347236	−0.173618	−	0.984813i	$-0.555546\pi$
$907$	−10.4575	−0.347236	−0.173618	+	0.984813i	$0.555546\pi$
$908$	13.2915	0.441094
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−1.29150	−0.0427894	−0.0213947	−	0.999771i	$-0.506811\pi$
$911$	−1.29150	−0.0427894	−0.0213947	+	0.999771i	$0.506811\pi$
$912$	0.937254	0.0310356
$913$	−13.2915	−0.439885
$914$	39.1660	1.29550
$915$	35.7712	1.18256
$916$	16.0000	0.528655
$917$	0	0
$918$	15.8745	0.523937
$919$	−35.2915	−1.16416	−0.582080	−	0.813132i	$-0.697761\pi$
$919$	−35.2915	−1.16416	−0.582080	+	0.813132i	$0.697761\pi$
$920$	−13.2915	−0.438208
$921$	11.1882	0.368665
$922$	−32.1660	−1.05933
$923$	−48.2288	−1.58747
$924$	0	0
$925$	−13.6458	−0.448670
$926$	22.4575	0.738000
$927$	−51.7490	−1.69966
$928$	4.29150	0.140875
$929$	−35.5830	−1.16744	−0.583720	−	0.811955i	$-0.698404\pi$
$929$	−35.5830	−1.16744	−0.583720	+	0.811955i	$0.698404\pi$
$930$	38.5830	1.26519
$931$	0	0
$932$	16.9373	0.554798
$933$	−44.8118	−1.46707
$934$	10.7085	0.350393
$935$	21.8745	0.715373
$936$	20.0000	0.653720
$937$	46.6863	1.52517	0.762587	−	0.646886i	$-0.223929\pi$
$937$	46.6863	1.52517	0.762587	+	0.646886i	$0.223929\pi$
$938$	0	0
$939$	−6.39477	−0.208685
$940$	48.4575	1.58051
$941$	44.6235	1.45469	0.727343	−	0.686274i	$-0.240755\pi$
$941$	44.6235	1.45469	0.727343	+	0.686274i	$0.240755\pi$
$942$	−56.0000	−1.82458
$943$	−18.0000	−0.586161
$944$	0.645751	0.0210174
$945$	0	0
$946$	4.00000	0.130051
$947$	−33.8745	−1.10077	−0.550387	−	0.834910i	$-0.685520\pi$
$947$	−33.8745	−1.10077	−0.550387	+	0.834910i	$0.685520\pi$
$948$	7.00000	0.227349
$949$	28.2288	0.916344
$950$	−2.93725	−0.0952971
$951$	31.7490	1.02953
$952$	0	0
$953$	−25.5203	−0.826682	−0.413341	−	0.910576i	$-0.635638\pi$
$953$	−25.5203	−0.826682	−0.413341	+	0.910576i	$0.635638\pi$
$954$	14.5830	0.472142
$955$	48.4575	1.56805
$956$	9.22876	0.298479
$957$	−11.3542	−0.367031
$958$	−25.9373	−0.837995
$959$	0	0
$960$	−9.64575	−0.311315
$961$	−15.0000	−0.483871
$962$	−8.22876	−0.265306
$963$	19.7490	0.636403
$964$	−22.8118	−0.734717
$965$	42.0000	1.35203
$966$	0	0
$967$	−26.3320	−0.846781	−0.423390	−	0.905947i	$-0.639160\pi$
$967$	−26.3320	−0.846781	−0.423390	+	0.905947i	$0.639160\pi$
$968$	−1.00000	−0.0321412
$969$	−5.62352	−0.180654
$970$	20.8118	0.668225
$971$	25.9373	0.832366	0.416183	−	0.909281i	$-0.363368\pi$
$971$	25.9373	0.832366	0.416183	+	0.909281i	$0.363368\pi$
$972$	−21.1660	−0.678900
$973$	0	0
$974$	30.5830	0.979943
$975$	−109.686	−3.51277
$976$	−3.70850	−0.118706
$977$	36.4575	1.16638	0.583190	−	0.812336i	$-0.301804\pi$
$977$	36.4575	1.16638	0.583190	+	0.812336i	$0.301804\pi$
$978$	12.2915	0.393039
$979$	14.5830	0.466075
$980$	0	0
$981$	42.3320	1.35156
$982$	−10.7085	−0.341722
$983$	37.2915	1.18941	0.594707	−	0.803942i	$-0.297268\pi$
$983$	37.2915	1.18941	0.594707	+	0.803942i	$0.297268\pi$
$984$	−13.0627	−0.416425
$985$	−68.8118	−2.19253
$986$	−25.7490	−0.820016
$987$	0	0
$988$	−1.77124	−0.0563508
$989$	14.5830	0.463713
$990$	14.5830	0.463478
$991$	56.6863	1.80070	0.900349	−	0.435168i	$-0.143311\pi$
$991$	56.6863	1.80070	0.900349	+	0.435168i	$0.143311\pi$
$992$	−4.00000	−0.127000
$993$	−40.6235	−1.28915
$994$	0	0
$995$	−81.0405	−2.56916
$996$	−35.1660	−1.11428
$997$	42.5830	1.34862	0.674309	−	0.738450i	$-0.264442\pi$
$997$	42.5830	1.34862	0.674309	+	0.738450i	$0.264442\pi$
$998$	−17.8745	−0.565808
$999$	−4.35425	−0.137762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.n.1.2		2
3.2	odd	2		9702.2.a.dr.1.2		2
4.3	odd	2		8624.2.a.bk.1.1		2
7.2	even	3		1078.2.e.v.67.1		4
7.3	odd	6		154.2.e.f.23.2	✓	4
7.4	even	3		1078.2.e.v.177.1		4
7.5	odd	6		154.2.e.f.67.2	yes	4
7.6	odd	2		1078.2.a.s.1.1		2
21.5	even	6		1386.2.k.s.991.2		4
21.17	even	6		1386.2.k.s.793.2		4
21.20	even	2		9702.2.a.cz.1.1		2
28.3	even	6		1232.2.q.g.177.1		4
28.19	even	6		1232.2.q.g.529.1		4
28.27	even	2		8624.2.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.f.23.2	✓	4	7.3	odd	6
154.2.e.f.67.2	yes	4	7.5	odd	6
1078.2.a.n.1.2		2	1.1	even	1	trivial
1078.2.a.s.1.1		2	7.6	odd	2
1078.2.e.v.67.1		4	7.2	even	3
1078.2.e.v.177.1		4	7.4	even	3
1232.2.q.g.177.1		4	28.3	even	6
1232.2.q.g.529.1		4	28.19	even	6
1386.2.k.s.793.2		4	21.17	even	6
1386.2.k.s.991.2		4	21.5	even	6
8624.2.a.bk.1.1		2	4.3	odd	2
8624.2.a.ca.1.2		2	28.27	even	2
9702.2.a.cz.1.1		2	21.20	even	2
9702.2.a.dr.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.64575 q^{3} +1.00000 q^{4} -3.64575 q^{5} -2.64575 q^{6} -1.00000 q^{8} +4.00000 q^{9} +3.64575 q^{10} +1.00000 q^{11} +2.64575 q^{12} -5.00000 q^{13} -9.64575 q^{15} +1.00000 q^{16} -6.00000 q^{17} -4.00000 q^{18} +0.354249 q^{19} -3.64575 q^{20} -1.00000 q^{22} -3.64575 q^{23} -2.64575 q^{24} +8.29150 q^{25} +5.00000 q^{26} +2.64575 q^{27} -4.29150 q^{29} +9.64575 q^{30} +4.00000 q^{31} -1.00000 q^{32} +2.64575 q^{33} +6.00000 q^{34} +4.00000 q^{36} -1.64575 q^{37} -0.354249 q^{38} -13.2288 q^{39} +3.64575 q^{40} +4.93725 q^{41} -4.00000 q^{43} +1.00000 q^{44} -14.5830 q^{45} +3.64575 q^{46} -13.2915 q^{47} +2.64575 q^{48} -8.29150 q^{50} -15.8745 q^{51} -5.00000 q^{52} -3.64575 q^{53} -2.64575 q^{54} -3.64575 q^{55} +0.937254 q^{57} +4.29150 q^{58} +0.645751 q^{59} -9.64575 q^{60} -3.70850 q^{61} -4.00000 q^{62} +1.00000 q^{64} +18.2288 q^{65} -2.64575 q^{66} +3.93725 q^{67} -6.00000 q^{68} -9.64575 q^{69} +9.64575 q^{71} -4.00000 q^{72} -5.64575 q^{73} +1.64575 q^{74} +21.9373 q^{75} +0.354249 q^{76} +13.2288 q^{78} +2.64575 q^{79} -3.64575 q^{80} -5.00000 q^{81} -4.93725 q^{82} -13.2915 q^{83} +21.8745 q^{85} +4.00000 q^{86} -11.3542 q^{87} -1.00000 q^{88} +14.5830 q^{89} +14.5830 q^{90} -3.64575 q^{92} +10.5830 q^{93} +13.2915 q^{94} -1.29150 q^{95} -2.64575 q^{96} +5.70850 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 8 q^{9} + 2 q^{10} + 2 q^{11} - 10 q^{13} - 14 q^{15} + 2 q^{16} - 12 q^{17} - 8 q^{18} + 6 q^{19} - 2 q^{20} - 2 q^{22} - 2 q^{23} + 6 q^{25} + 10 q^{26} + 2 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 8 * q^9 + 2 * q^10 + 2 * q^11 - 10 * q^13 - 14 * q^15 + 2 * q^16 - 12 * q^17 - 8 * q^18 + 6 * q^19 - 2 * q^20 - 2 * q^22 - 2 * q^23 + 6 * q^25 + 10 * q^26 + 2 * q^29 + 14 * q^30 + 8 * q^31 - 2 * q^32 + 12 * q^34 + 8 * q^36 + 2 * q^37 - 6 * q^38 + 2 * q^40 - 6 * q^41 - 8 * q^43 + 2 * q^44 - 8 * q^45 + 2 * q^46 - 16 * q^47 - 6 * q^50 - 10 * q^52 - 2 * q^53 - 2 * q^55 - 14 * q^57 - 2 * q^58 - 4 * q^59 - 14 * q^60 - 18 * q^61 - 8 * q^62 + 2 * q^64 + 10 * q^65 - 8 * q^67 - 12 * q^68 - 14 * q^69 + 14 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 + 28 * q^75 + 6 * q^76 - 2 * q^80 - 10 * q^81 + 6 * q^82 - 16 * q^83 + 12 * q^85 + 8 * q^86 - 28 * q^87 - 2 * q^88 + 8 * q^89 + 8 * q^90 - 2 * q^92 + 16 * q^94 + 8 * q^95 + 22 * q^97 + 8 * q^99

Introduction

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