LMFDB - Embedded newform 117.8.a.i.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,8,Mod(1,117)] 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("117.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	117.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$36.5490479816$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 729x^{6} + 138692x^{4} - 8145792x^{2} + 116376832 $ x^8 - 729x^6 + 138692x^4 - 8145792*x^2 + 116376832
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$8.37511$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.37511 q^{2} -57.8575 q^{4} -78.0252 q^{5} -1418.83 q^{7} -1556.58 q^{8} -653.469 q^{10} +3976.49 q^{11} -2197.00 q^{13} -11882.9 q^{14} -5630.74 q^{16} +32107.8 q^{17} +34359.9 q^{19} +4514.34 q^{20} +33303.5 q^{22} +65325.6 q^{23} -72037.1 q^{25} -18400.1 q^{26} +82090.2 q^{28} -84814.6 q^{29} -154740. q^{31} +152084. q^{32} +268906. q^{34} +110705. q^{35} +338365. q^{37} +287768. q^{38} +121452. q^{40} +736691. q^{41} +211542. q^{43} -230070. q^{44} +547109. q^{46} -1.33488e6 q^{47} +1.18955e6 q^{49} -603318. q^{50} +127113. q^{52} -1.52669e6 q^{53} -310266. q^{55} +2.20852e6 q^{56} -710331. q^{58} +1.62421e6 q^{59} +492614. q^{61} -1.29597e6 q^{62} +1.99445e6 q^{64} +171421. q^{65} +2.16287e6 q^{67} -1.85768e6 q^{68} +927164. q^{70} -3.27264e6 q^{71} +4.95434e6 q^{73} +2.83384e6 q^{74} -1.98798e6 q^{76} -5.64198e6 q^{77} +7.64247e6 q^{79} +439340. q^{80} +6.16987e6 q^{82} +4.43774e6 q^{83} -2.50521e6 q^{85} +1.77169e6 q^{86} -6.18971e6 q^{88} -2.43479e6 q^{89} +3.11718e6 q^{91} -3.77958e6 q^{92} -1.11798e7 q^{94} -2.68093e6 q^{95} -7.11530e6 q^{97} +9.96258e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 434 q^{4} - 72 q^{7} + 13924 q^{10} - 17576 q^{13} + 79314 q^{16} - 11352 q^{19} + 243952 q^{22} + 97224 q^{25} - 28372 q^{28} + 5656 q^{31} - 223920 q^{34} + 603152 q^{37} + 3411964 q^{40} + 2930960 q^{43}+ \cdots + 26916464 q^{97}+O(q^{100}) $ 8 * q + 434 * q^4 - 72 * q^7 + 13924 * q^10 - 17576 * q^13 + 79314 * q^16 - 11352 * q^19 + 243952 * q^22 + 97224 * q^25 - 28372 * q^28 + 5656 * q^31 - 223920 * q^34 + 603152 * q^37 + 3411964 * q^40 + 2930960 * q^43 + 4783304 * q^46 + 3382232 * q^49 - 953498 * q^52 + 6106032 * q^55 + 12363392 * q^58 + 10218304 * q^61 + 18433978 * q^64 + 1734584 * q^67 + 6347032 * q^70 + 9903984 * q^73 - 5818028 * q^76 + 446688 * q^79 + 9877140 * q^82 + 2396896 * q^85 + 33949728 * q^88 + 158184 * q^91 - 39327112 * q^94 + 26916464 * q^97

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 $

$2$	8.37511	0.740262	0.370131	−	0.928980i	$-0.379313\pi$
$2$	8.37511	0.740262	0.370131	+	0.928980i	$0.379313\pi$
$3$	0	0
$3$	0	0
$4$	−57.8575	−0.452012
$5$	−78.0252	−0.279151	−0.139576	−	0.990211i	$-0.544574\pi$
$5$	−78.0252	−0.279151	−0.139576	+	0.990211i	$0.544574\pi$
$6$	0	0
$7$	−1418.83	−1.56347	−0.781733	−	0.623614i	$-0.785664\pi$
$7$	−1418.83	−1.56347	−0.781733	+	0.623614i	$0.785664\pi$
$8$	−1556.58	−1.07487
$9$	0	0
$10$	−653.469	−0.206645
$11$	3976.49	0.900794	0.450397	−	0.892828i	$-0.351282\pi$
$11$	3976.49	0.900794	0.450397	+	0.892828i	$0.351282\pi$
$12$	0	0
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−11882.9	−1.15737
$15$	0	0
$16$	−5630.74	−0.343673
$17$	32107.8	1.58503	0.792517	−	0.609850i	$-0.208770\pi$
$17$	32107.8	1.58503	0.792517	+	0.609850i	$0.208770\pi$
$18$	0	0
$19$	34359.9	1.14925	0.574624	−	0.818418i	$-0.305148\pi$
$19$	34359.9	1.14925	0.574624	+	0.818418i	$0.305148\pi$
$20$	4514.34	0.126180
$21$	0	0
$22$	33303.5	0.666824
$23$	65325.6	1.11953	0.559765	−	0.828651i	$-0.310891\pi$
$23$	65325.6	1.11953	0.559765	+	0.828651i	$0.310891\pi$
$24$	0	0
$25$	−72037.1	−0.922075
$26$	−18400.1	−0.205312
$27$	0	0
$28$	82090.2	0.706705
$29$	−84814.6	−0.645769	−0.322885	−	0.946438i	$-0.604653\pi$
$29$	−84814.6	−0.645769	−0.322885	+	0.946438i	$0.604653\pi$
$30$	0	0
$31$	−154740.	−0.932904	−0.466452	−	0.884547i	$-0.654468\pi$
$31$	−154740.	−0.932904	−0.466452	+	0.884547i	$0.654468\pi$
$32$	152084.	0.820461
$33$	0	0
$34$	268906.	1.17334
$35$	110705.	0.436443
$36$	0	0
$37$	338365.	1.09819	0.549097	−	0.835758i	$-0.314972\pi$
$37$	338365.	1.09819	0.549097	+	0.835758i	$0.314972\pi$
$38$	287768.	0.850745
$39$	0	0
$40$	121452.	0.300051
$41$	736691.	1.66933	0.834664	−	0.550759i	$-0.185662\pi$
$41$	736691.	1.66933	0.834664	+	0.550759i	$0.185662\pi$
$42$	0	0
$43$	211542.	0.405749	0.202874	−	0.979205i	$-0.434972\pi$
$43$	211542.	0.405749	0.202874	+	0.979205i	$0.434972\pi$
$44$	−230070.	−0.407170
$45$	0	0
$46$	547109.	0.828746
$47$	−1.33488e6	−1.87543	−0.937715	−	0.347404i	$-0.887063\pi$
$47$	−1.33488e6	−1.87543	−0.937715	+	0.347404i	$0.887063\pi$
$48$	0	0
$49$	1.18955e6	1.44442
$50$	−603318.	−0.682577
$51$	0	0
$52$	127113.	0.125366
$53$	−1.52669e6	−1.40859	−0.704295	−	0.709908i	$-0.748737\pi$
$53$	−1.52669e6	−1.40859	−0.704295	+	0.709908i	$0.748737\pi$
$54$	0	0
$55$	−310266.	−0.251458
$56$	2.20852e6	1.68052
$57$	0	0
$58$	−710331.	−0.478039
$59$	1.62421e6	1.02958	0.514790	−	0.857316i	$-0.327870\pi$
$59$	1.62421e6	1.02958	0.514790	+	0.857316i	$0.327870\pi$
$60$	0	0
$61$	492614.	0.277877	0.138939	−	0.990301i	$-0.455631\pi$
$61$	492614.	0.277877	0.138939	+	0.990301i	$0.455631\pi$
$62$	−1.29597e6	−0.690593
$63$	0	0
$64$	1.99445e6	0.951030
$65$	171421.	0.0774226
$66$	0	0
$67$	2.16287e6	0.878553	0.439276	−	0.898352i	$-0.355235\pi$
$67$	2.16287e6	0.878553	0.439276	+	0.898352i	$0.355235\pi$
$68$	−1.85768e6	−0.716455
$69$	0	0
$70$	927164.	0.323083
$71$	−3.27264e6	−1.08516	−0.542581	−	0.840004i	$-0.682553\pi$
$71$	−3.27264e6	−1.08516	−0.542581	+	0.840004i	$0.682553\pi$
$72$	0	0
$73$	4.95434e6	1.49058	0.745291	−	0.666739i	$-0.232311\pi$
$73$	4.95434e6	1.49058	0.745291	+	0.666739i	$0.232311\pi$
$74$	2.83384e6	0.812952
$75$	0	0
$76$	−1.98798e6	−0.519474
$77$	−5.64198e6	−1.40836
$78$	0	0
$79$	7.64247e6	1.74397	0.871985	−	0.489532i	$-0.162832\pi$
$79$	7.64247e6	1.74397	0.871985	+	0.489532i	$0.162832\pi$
$80$	439340.	0.0959368
$81$	0	0
$82$	6.16987e6	1.23574
$83$	4.43774e6	0.851899	0.425950	−	0.904747i	$-0.359940\pi$
$83$	4.43774e6	0.851899	0.425950	+	0.904747i	$0.359940\pi$
$84$	0	0
$85$	−2.50521e6	−0.442464
$86$	1.77169e6	0.300361
$87$	0	0
$88$	−6.18971e6	−0.968236
$89$	−2.43479e6	−0.366097	−0.183049	−	0.983104i	$-0.558597\pi$
$89$	−2.43479e6	−0.366097	−0.183049	+	0.983104i	$0.558597\pi$
$90$	0	0
$91$	3.11718e6	0.433627
$92$	−3.77958e6	−0.506041
$93$	0	0
$94$	−1.11798e7	−1.38831
$95$	−2.68093e6	−0.320814
$96$	0	0
$97$	−7.11530e6	−0.791575	−0.395787	−	0.918342i	$-0.629528\pi$
$97$	−7.11530e6	−0.791575	−0.395787	+	0.918342i	$0.629528\pi$
$98$	9.96258e6	1.06925
$99$	0	0
$100$	4.16789e6	0.416789
$101$	1.15494e7	1.11541	0.557704	−	0.830040i	$-0.311683\pi$
$101$	1.15494e7	1.11541	0.557704	+	0.830040i	$0.311683\pi$
$102$	0	0
$103$	6.81710e6	0.614708	0.307354	−	0.951595i	$-0.400556\pi$
$103$	6.81710e6	0.614708	0.307354	+	0.951595i	$0.400556\pi$
$104$	3.41980e6	0.298115
$105$	0	0
$106$	−1.27862e7	−1.04273
$107$	1.94149e6	0.153212	0.0766059	−	0.997061i	$-0.475592\pi$
$107$	1.94149e6	0.153212	0.0766059	+	0.997061i	$0.475592\pi$
$108$	0	0
$109$	−2.62040e7	−1.93809	−0.969046	−	0.246879i	$-0.920595\pi$
$109$	−2.62040e7	−1.93809	−0.969046	+	0.246879i	$0.920595\pi$
$110$	−2.59851e6	−0.186145
$111$	0	0
$112$	7.98909e6	0.537321
$113$	2.85374e7	1.86054	0.930272	−	0.366870i	$-0.119571\pi$
$113$	2.85374e7	1.86054	0.930272	+	0.366870i	$0.119571\pi$
$114$	0	0
$115$	−5.09704e6	−0.312518
$116$	4.90716e6	0.291895
$117$	0	0
$118$	1.36029e7	0.762160
$119$	−4.55556e7	−2.47815
$120$	0	0
$121$	−3.67471e6	−0.188570
$122$	4.12570e6	0.205702
$123$	0	0
$124$	8.95288e6	0.421684
$125$	1.17164e7	0.536550
$126$	0	0
$127$	4.87796e6	0.211312	0.105656	−	0.994403i	$-0.466306\pi$
$127$	4.87796e6	0.211312	0.105656	+	0.994403i	$0.466306\pi$
$128$	−2.76296e6	−0.116450
$129$	0	0
$130$	1.43567e6	0.0573130
$131$	−1.42247e7	−0.552834	−0.276417	−	0.961038i	$-0.589147\pi$
$131$	−1.42247e7	−0.552834	−0.276417	+	0.961038i	$0.589147\pi$
$132$	0	0
$133$	−4.87509e7	−1.79681
$134$	1.81142e7	0.650359
$135$	0	0
$136$	−4.99782e7	−1.70371
$137$	−3.49760e7	−1.16211	−0.581057	−	0.813863i	$-0.697361\pi$
$137$	−3.49760e7	−1.16211	−0.581057	+	0.813863i	$0.697361\pi$
$138$	0	0
$139$	−1.29899e7	−0.410256	−0.205128	−	0.978735i	$-0.565761\pi$
$139$	−1.29899e7	−0.410256	−0.205128	+	0.978735i	$0.565761\pi$
$140$	−6.40510e6	−0.197278
$141$	0	0
$142$	−2.74087e7	−0.803304
$143$	−8.73635e6	−0.249835
$144$	0	0
$145$	6.61767e6	0.180267
$146$	4.14931e7	1.10342
$147$	0	0
$148$	−1.95770e7	−0.496397
$149$	−6.94078e6	−0.171892	−0.0859462	−	0.996300i	$-0.527391\pi$
$149$	−6.94078e6	−0.171892	−0.0859462	+	0.996300i	$0.527391\pi$
$150$	0	0
$151$	4.70900e7	1.11304	0.556518	−	0.830836i	$-0.312137\pi$
$151$	4.70900e7	1.11304	0.556518	+	0.830836i	$0.312137\pi$
$152$	−5.34838e7	−1.23529
$153$	0	0
$154$	−4.72522e7	−1.04256
$155$	1.20736e7	0.260421
$156$	0	0
$157$	4.00504e7	0.825959	0.412979	−	0.910740i	$-0.364488\pi$
$157$	4.00504e7	0.825959	0.412979	+	0.910740i	$0.364488\pi$
$158$	6.40066e7	1.29100
$159$	0	0
$160$	−1.18664e7	−0.229033
$161$	−9.26861e7	−1.75035
$162$	0	0
$163$	−3.03416e7	−0.548760	−0.274380	−	0.961621i	$-0.588473\pi$
$163$	−3.03416e7	−0.548760	−0.274380	+	0.961621i	$0.588473\pi$
$164$	−4.26231e7	−0.754556
$165$	0	0
$166$	3.71665e7	0.630629
$167$	3.89368e7	0.646922	0.323461	−	0.946241i	$-0.395154\pi$
$167$	3.89368e7	0.646922	0.323461	+	0.946241i	$0.395154\pi$
$168$	0	0
$169$	4.82681e6	0.0769231
$170$	−2.09814e7	−0.327540
$171$	0	0
$172$	−1.22393e7	−0.183403
$173$	4.27780e7	0.628144	0.314072	−	0.949399i	$-0.398307\pi$
$173$	4.27780e7	0.628144	0.314072	+	0.949399i	$0.398307\pi$
$174$	0	0
$175$	1.02209e8	1.44163
$176$	−2.23906e7	−0.309579
$177$	0	0
$178$	−2.03916e7	−0.271008
$179$	−7.29707e6	−0.0950961	−0.0475481	−	0.998869i	$-0.515141\pi$
$179$	−7.29707e6	−0.0950961	−0.0475481	+	0.998869i	$0.515141\pi$
$180$	0	0
$181$	6.55707e6	0.0821931	0.0410965	−	0.999155i	$-0.486915\pi$
$181$	6.55707e6	0.0821931	0.0410965	+	0.999155i	$0.486915\pi$
$182$	2.61067e7	0.320998
$183$	0	0
$184$	−1.01684e8	−1.20335
$185$	−2.64010e7	−0.306562
$186$	0	0
$187$	1.27676e8	1.42779
$188$	7.72331e7	0.847717
$189$	0	0
$190$	−2.24531e7	−0.237487
$191$	−5.38047e7	−0.558731	−0.279366	−	0.960185i	$-0.590124\pi$
$191$	−5.38047e7	−0.558731	−0.279366	+	0.960185i	$0.590124\pi$
$192$	0	0
$193$	1.04771e8	1.04903	0.524516	−	0.851400i	$-0.324246\pi$
$193$	1.04771e8	1.04903	0.524516	+	0.851400i	$0.324246\pi$
$194$	−5.95914e7	−0.585973
$195$	0	0
$196$	−6.88242e7	−0.652897
$197$	5.18935e6	0.0483594	0.0241797	−	0.999708i	$-0.492303\pi$
$197$	5.18935e6	0.0483594	0.0241797	+	0.999708i	$0.492303\pi$
$198$	0	0
$199$	8.24241e6	0.0741427	0.0370714	−	0.999313i	$-0.488197\pi$
$199$	8.24241e6	0.0741427	0.0370714	+	0.999313i	$0.488197\pi$
$200$	1.12131e8	0.991110
$201$	0	0
$202$	9.67273e7	0.825694
$203$	1.20338e8	1.00964
$204$	0	0
$205$	−5.74804e7	−0.465995
$206$	5.70939e7	0.455045
$207$	0	0
$208$	1.23707e7	0.0953178
$209$	1.36632e8	1.03524
$210$	0	0
$211$	2.33974e8	1.71466	0.857332	−	0.514764i	$-0.172121\pi$
$211$	2.33974e8	1.71466	0.857332	+	0.514764i	$0.172121\pi$
$212$	8.83303e7	0.636699
$213$	0	0
$214$	1.62602e7	0.113417
$215$	−1.65056e7	−0.113265
$216$	0	0
$217$	2.19550e8	1.45856
$218$	−2.19461e8	−1.43470
$219$	0	0
$220$	1.79512e7	0.113662
$221$	−7.05407e7	−0.439609
$222$	0	0
$223$	3.81145e7	0.230157	0.115078	−	0.993356i	$-0.463288\pi$
$223$	3.81145e7	0.230157	0.115078	+	0.993356i	$0.463288\pi$
$224$	−2.15782e8	−1.28276
$225$	0	0
$226$	2.39004e8	1.37729
$227$	3.40570e8	1.93249	0.966243	−	0.257632i	$-0.0829421\pi$
$227$	3.40570e8	1.93249	0.966243	+	0.257632i	$0.0829421\pi$
$228$	0	0
$229$	−8.73986e7	−0.480928	−0.240464	−	0.970658i	$-0.577300\pi$
$229$	−8.73986e7	−0.480928	−0.240464	+	0.970658i	$0.577300\pi$
$230$	−4.26882e7	−0.231345
$231$	0	0
$232$	1.32020e8	0.694118
$233$	−8.55257e7	−0.442946	−0.221473	−	0.975167i	$-0.571086\pi$
$233$	−8.55257e7	−0.442946	−0.221473	+	0.975167i	$0.571086\pi$
$234$	0	0
$235$	1.04155e8	0.523529
$236$	−9.39728e7	−0.465383
$237$	0	0
$238$	−3.81533e8	−1.83448
$239$	1.31646e8	0.623757	0.311878	−	0.950122i	$-0.399042\pi$
$239$	1.31646e8	0.623757	0.311878	+	0.950122i	$0.399042\pi$
$240$	0	0
$241$	7.89647e7	0.363390	0.181695	−	0.983355i	$-0.441842\pi$
$241$	7.89647e7	0.363390	0.181695	+	0.983355i	$0.441842\pi$
$242$	−3.07761e7	−0.139592
$243$	0	0
$244$	−2.85015e7	−0.125604
$245$	−9.28145e7	−0.403213
$246$	0	0
$247$	−7.54886e7	−0.318744
$248$	2.40865e8	1.00275
$249$	0	0
$250$	9.81263e7	0.397187
$251$	−1.42569e8	−0.569073	−0.284537	−	0.958665i	$-0.591840\pi$
$251$	−1.42569e8	−0.569073	−0.284537	+	0.958665i	$0.591840\pi$
$252$	0	0
$253$	2.59766e8	1.00847
$254$	4.08534e7	0.156427
$255$	0	0
$256$	−2.78430e8	−1.03723
$257$	6.04174e7	0.222022	0.111011	−	0.993819i	$-0.464591\pi$
$257$	6.04174e7	0.222022	0.111011	+	0.993819i	$0.464591\pi$
$258$	0	0
$259$	−4.80084e8	−1.71699
$260$	−9.91801e6	−0.0349960
$261$	0	0
$262$	−1.19134e8	−0.409242
$263$	−1.26963e8	−0.430361	−0.215180	−	0.976574i	$-0.569034\pi$
$263$	−1.26963e8	−0.430361	−0.215180	+	0.976574i	$0.569034\pi$
$264$	0	0
$265$	1.19120e8	0.393210
$266$	−4.08294e8	−1.33011
$267$	0	0
$268$	−1.25138e8	−0.397116
$269$	2.68950e8	0.842439	0.421219	−	0.906959i	$-0.361602\pi$
$269$	2.68950e8	0.842439	0.421219	+	0.906959i	$0.361602\pi$
$270$	0	0
$271$	−1.95379e8	−0.596328	−0.298164	−	0.954515i	$-0.596374\pi$
$271$	−1.95379e8	−0.596328	−0.298164	+	0.954515i	$0.596374\pi$
$272$	−1.80790e8	−0.544734
$273$	0	0
$274$	−2.92928e8	−0.860269
$275$	−2.86455e8	−0.830599
$276$	0	0
$277$	1.23578e8	0.349350	0.174675	−	0.984626i	$-0.444113\pi$
$277$	1.23578e8	0.349350	0.174675	+	0.984626i	$0.444113\pi$
$278$	−1.08792e8	−0.303697
$279$	0	0
$280$	−1.72320e8	−0.469120
$281$	6.53613e8	1.75731	0.878656	−	0.477456i	$-0.158441\pi$
$281$	6.53613e8	1.75731	0.878656	+	0.477456i	$0.158441\pi$
$282$	0	0
$283$	3.62432e8	0.950548	0.475274	−	0.879838i	$-0.342349\pi$
$283$	3.62432e8	0.950548	0.475274	+	0.879838i	$0.342349\pi$
$284$	1.89347e8	0.490506
$285$	0	0
$286$	−7.31679e7	−0.184944
$287$	−1.04524e9	−2.60994
$288$	0	0
$289$	6.20569e8	1.51233
$290$	5.54237e7	0.133445
$291$	0	0
$292$	−2.86646e8	−0.673761
$293$	−1.45058e8	−0.336903	−0.168452	−	0.985710i	$-0.553877\pi$
$293$	−1.45058e8	−0.336903	−0.168452	+	0.985710i	$0.553877\pi$
$294$	0	0
$295$	−1.26729e8	−0.287409
$296$	−5.26691e8	−1.18042
$297$	0	0
$298$	−5.81298e7	−0.127245
$299$	−1.43520e8	−0.310502
$300$	0	0
$301$	−3.00143e8	−0.634374
$302$	3.94384e8	0.823938
$303$	0	0
$304$	−1.93471e8	−0.394966
$305$	−3.84363e7	−0.0775697
$306$	0	0
$307$	4.12640e8	0.813929	0.406965	−	0.913444i	$-0.366587\pi$
$307$	4.12640e8	0.813929	0.406965	+	0.913444i	$0.366587\pi$
$308$	3.26431e8	0.636596
$309$	0	0
$310$	1.01118e8	0.192780
$311$	5.64006e8	1.06322	0.531609	−	0.846990i	$-0.321588\pi$
$311$	5.64006e8	1.06322	0.531609	+	0.846990i	$0.321588\pi$
$312$	0	0
$313$	−5.65784e8	−1.04291	−0.521453	−	0.853280i	$-0.674610\pi$
$313$	−5.65784e8	−1.04291	−0.521453	+	0.853280i	$0.674610\pi$
$314$	3.35427e8	0.611426
$315$	0	0
$316$	−4.42175e8	−0.788296
$317$	−7.79510e7	−0.137440	−0.0687201	−	0.997636i	$-0.521892\pi$
$317$	−7.79510e7	−0.137440	−0.0687201	+	0.997636i	$0.521892\pi$
$318$	0	0
$319$	−3.37264e8	−0.581705
$320$	−1.55618e8	−0.265481
$321$	0	0
$322$	−7.76256e8	−1.29572
$323$	1.10322e9	1.82160
$324$	0	0
$325$	1.58265e8	0.255737
$326$	−2.54115e8	−0.406226
$327$	0	0
$328$	−1.14672e9	−1.79431
$329$	1.89398e9	2.93217
$330$	0	0
$331$	−5.05965e8	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$331$	−5.05965e8	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$332$	−2.56756e8	−0.385069
$333$	0	0
$334$	3.26100e8	0.478892
$335$	−1.68758e8	−0.245249
$336$	0	0
$337$	4.83484e8	0.688141	0.344070	−	0.938944i	$-0.388194\pi$
$337$	4.83484e8	0.688141	0.344070	+	0.938944i	$0.388194\pi$
$338$	4.04251e7	0.0569432
$339$	0	0
$340$	1.44945e8	0.199999
$341$	−6.15322e8	−0.840354
$342$	0	0
$343$	−5.19297e8	−0.694842
$344$	−3.29282e8	−0.436127
$345$	0	0
$346$	3.58270e8	0.464991
$347$	−9.52859e8	−1.22427	−0.612133	−	0.790755i	$-0.709688\pi$
$347$	−9.52859e8	−1.22427	−0.612133	+	0.790755i	$0.709688\pi$
$348$	0	0
$349$	−5.80675e8	−0.731213	−0.365607	−	0.930769i	$-0.619138\pi$
$349$	−5.80675e8	−0.731213	−0.365607	+	0.930769i	$0.619138\pi$
$350$	8.56008e8	1.06719
$351$	0	0
$352$	6.04760e8	0.739066
$353$	−1.55889e9	−1.88627	−0.943134	−	0.332412i	$-0.892137\pi$
$353$	−1.55889e9	−1.88627	−0.943134	+	0.332412i	$0.892137\pi$
$354$	0	0
$355$	2.55348e8	0.302924
$356$	1.40871e8	0.165480
$357$	0	0
$358$	−6.11138e7	−0.0703961
$359$	−3.14671e8	−0.358944	−0.179472	−	0.983763i	$-0.557439\pi$
$359$	−3.14671e8	−0.358944	−0.179472	+	0.983763i	$0.557439\pi$
$360$	0	0
$361$	2.86728e8	0.320771
$362$	5.49162e7	0.0608444
$363$	0	0
$364$	−1.80352e8	−0.196005
$365$	−3.86563e8	−0.416098
$366$	0	0
$367$	−3.05352e8	−0.322456	−0.161228	−	0.986917i	$-0.551545\pi$
$367$	−3.05352e8	−0.322456	−0.161228	+	0.986917i	$0.551545\pi$
$368$	−3.67831e8	−0.384752
$369$	0	0
$370$	−2.21111e8	−0.226937
$371$	2.16611e9	2.20228
$372$	0	0
$373$	1.64657e9	1.64285	0.821425	−	0.570316i	$-0.193179\pi$
$373$	1.64657e9	1.64285	0.821425	+	0.570316i	$0.193179\pi$
$374$	1.06930e9	1.05694
$375$	0	0
$376$	2.07785e9	2.01584
$377$	1.86338e8	0.179104
$378$	0	0
$379$	8.88173e7	0.0838032	0.0419016	−	0.999122i	$-0.486658\pi$
$379$	8.88173e7	0.0838032	0.0419016	+	0.999122i	$0.486658\pi$
$380$	1.55112e8	0.145012
$381$	0	0
$382$	−4.50620e8	−0.413608
$383$	3.36278e8	0.305846	0.152923	−	0.988238i	$-0.451131\pi$
$383$	3.36278e8	0.305846	0.152923	+	0.988238i	$0.451131\pi$
$384$	0	0
$385$	4.40216e8	0.393146
$386$	8.77465e8	0.776559
$387$	0	0
$388$	4.11673e8	0.357801
$389$	−1.70532e9	−1.46887	−0.734434	−	0.678680i	$-0.762552\pi$
$389$	−1.70532e9	−1.46887	−0.734434	+	0.678680i	$0.762552\pi$
$390$	0	0
$391$	2.09746e9	1.77449
$392$	−1.85162e9	−1.55257
$393$	0	0
$394$	4.34614e7	0.0357986
$395$	−5.96305e8	−0.486832
$396$	0	0
$397$	−2.16516e9	−1.73670	−0.868348	−	0.495955i	$-0.834818\pi$
$397$	−2.16516e9	−1.73670	−0.868348	+	0.495955i	$0.834818\pi$
$398$	6.90311e7	0.0548850
$399$	0	0
$400$	4.05622e8	0.316892
$401$	1.40199e9	1.08578	0.542889	−	0.839804i	$-0.317330\pi$
$401$	1.40199e9	1.08578	0.542889	+	0.839804i	$0.317330\pi$
$402$	0	0
$403$	3.39964e8	0.258741
$404$	−6.68218e8	−0.504178
$405$	0	0
$406$	1.00784e9	0.747397
$407$	1.34550e9	0.989247
$408$	0	0
$409$	−5.58196e8	−0.403418	−0.201709	−	0.979445i	$-0.564650\pi$
$409$	−5.58196e8	−0.403418	−0.201709	+	0.979445i	$0.564650\pi$
$410$	−4.81405e8	−0.344959
$411$	0	0
$412$	−3.94420e8	−0.277856
$413$	−2.30449e9	−1.60971
$414$	0	0
$415$	−3.46255e8	−0.237809
$416$	−3.34128e8	−0.227555
$417$	0	0
$418$	1.14430e9	0.766346
$419$	−2.56173e9	−1.70132	−0.850658	−	0.525720i	$-0.823796\pi$
$419$	−2.56173e9	−1.70132	−0.850658	+	0.525720i	$0.823796\pi$
$420$	0	0
$421$	−1.97138e9	−1.28761	−0.643803	−	0.765191i	$-0.722645\pi$
$421$	−1.97138e9	−1.28761	−0.643803	+	0.765191i	$0.722645\pi$
$422$	1.95956e9	1.26930
$423$	0	0
$424$	2.37641e9	1.51405
$425$	−2.31295e9	−1.46152
$426$	0	0
$427$	−6.98938e8	−0.434451
$428$	−1.12330e8	−0.0692536
$429$	0	0
$430$	−1.38236e8	−0.0838460
$431$	1.14235e8	0.0687272	0.0343636	−	0.999409i	$-0.489060\pi$
$431$	1.14235e8	0.0687272	0.0343636	+	0.999409i	$0.489060\pi$
$432$	0	0
$433$	2.90458e9	1.71940	0.859698	−	0.510803i	$-0.170652\pi$
$433$	2.90458e9	1.71940	0.859698	+	0.510803i	$0.170652\pi$
$434$	1.83876e9	1.07972
$435$	0	0
$436$	1.51610e9	0.876041
$437$	2.24458e9	1.28662
$438$	0	0
$439$	1.23076e9	0.694300	0.347150	−	0.937810i	$-0.387150\pi$
$439$	1.23076e9	0.694300	0.347150	+	0.937810i	$0.387150\pi$
$440$	4.82953e8	0.270284
$441$	0	0
$442$	−5.90786e8	−0.325426
$443$	4.98361e8	0.272352	0.136176	−	0.990685i	$-0.456519\pi$
$443$	4.98361e8	0.272352	0.136176	+	0.990685i	$0.456519\pi$
$444$	0	0
$445$	1.89975e8	0.102196
$446$	3.19213e8	0.170376
$447$	0	0
$448$	−2.82980e9	−1.48690
$449$	−1.74332e9	−0.908898	−0.454449	−	0.890773i	$-0.650164\pi$
$449$	−1.74332e9	−0.908898	−0.454449	+	0.890773i	$0.650164\pi$
$450$	0	0
$451$	2.92944e9	1.50372
$452$	−1.65110e9	−0.840988
$453$	0	0
$454$	2.85231e9	1.43055
$455$	−2.43218e8	−0.121048
$456$	0	0
$457$	−2.32029e9	−1.13720	−0.568599	−	0.822615i	$-0.692514\pi$
$457$	−2.32029e9	−1.13720	−0.568599	+	0.822615i	$0.692514\pi$
$458$	−7.31973e8	−0.356013
$459$	0	0
$460$	2.94902e8	0.141262
$461$	−6.51561e8	−0.309743	−0.154872	−	0.987935i	$-0.549496\pi$
$461$	−6.51561e8	−0.309743	−0.154872	+	0.987935i	$0.549496\pi$
$462$	0	0
$463$	3.11159e9	1.45697	0.728483	−	0.685064i	$-0.240226\pi$
$463$	3.11159e9	1.45697	0.728483	+	0.685064i	$0.240226\pi$
$464$	4.77569e8	0.221934
$465$	0	0
$466$	−7.16287e8	−0.327896
$467$	−1.41396e9	−0.642431	−0.321216	−	0.947006i	$-0.604091\pi$
$467$	−1.41396e9	−0.642431	−0.321216	+	0.947006i	$0.604091\pi$
$468$	0	0
$469$	−3.06875e9	−1.37359
$470$	8.72306e8	0.387549
$471$	0	0
$472$	−2.52821e9	−1.10666
$473$	8.41195e8	0.365496
$474$	0	0
$475$	−2.47518e9	−1.05969
$476$	2.63573e9	1.12015
$477$	0	0
$478$	1.10255e9	0.461744
$479$	1.91626e9	0.796675	0.398337	−	0.917239i	$-0.369587\pi$
$479$	1.91626e9	0.796675	0.398337	+	0.917239i	$0.369587\pi$
$480$	0	0
$481$	−7.43388e8	−0.304584
$482$	6.61338e8	0.269004
$483$	0	0
$484$	2.12609e8	0.0852361
$485$	5.55172e8	0.220969
$486$	0	0
$487$	2.40543e9	0.943716	0.471858	−	0.881675i	$-0.343584\pi$
$487$	2.40543e9	0.943716	0.471858	+	0.881675i	$0.343584\pi$
$488$	−7.66792e8	−0.298682
$489$	0	0
$490$	−7.77332e8	−0.298483
$491$	−2.43715e9	−0.929175	−0.464587	−	0.885527i	$-0.653797\pi$
$491$	−2.43715e9	−0.929175	−0.464587	+	0.885527i	$0.653797\pi$
$492$	0	0
$493$	−2.72321e9	−1.02357
$494$	−6.32225e8	−0.235954
$495$	0	0
$496$	8.71301e8	0.320614
$497$	4.64333e9	1.69661
$498$	0	0
$499$	−1.07161e8	−0.0386086	−0.0193043	−	0.999814i	$-0.506145\pi$
$499$	−1.07161e8	−0.0386086	−0.0193043	+	0.999814i	$0.506145\pi$
$500$	−6.77883e8	−0.242527
$501$	0	0
$502$	−1.19403e9	−0.421263
$503$	5.20439e9	1.82340	0.911700	−	0.410857i	$-0.134771\pi$
$503$	5.20439e9	1.82340	0.911700	+	0.410857i	$0.134771\pi$
$504$	0	0
$505$	−9.01142e8	−0.311367
$506$	2.17557e9	0.746529
$507$	0	0
$508$	−2.82227e8	−0.0955158
$509$	2.30372e9	0.774316	0.387158	−	0.922013i	$-0.373457\pi$
$509$	2.30372e9	0.774316	0.387158	+	0.922013i	$0.373457\pi$
$510$	0	0
$511$	−7.02938e9	−2.33047
$512$	−1.97822e9	−0.651374
$513$	0	0
$514$	5.06003e8	0.164355
$515$	−5.31905e8	−0.171597
$516$	0	0
$517$	−5.30815e9	−1.68938
$518$	−4.02075e9	−1.27102
$519$	0	0
$520$	−2.66830e8	−0.0832192
$521$	−1.12710e9	−0.349165	−0.174582	−	0.984643i	$-0.555858\pi$
$521$	−1.12710e9	−0.349165	−0.174582	+	0.984643i	$0.555858\pi$
$522$	0	0
$523$	5.63804e9	1.72335	0.861673	−	0.507464i	$-0.169417\pi$
$523$	5.63804e9	1.72335	0.861673	+	0.507464i	$0.169417\pi$
$524$	8.23008e8	0.249888
$525$	0	0
$526$	−1.06333e9	−0.318580
$527$	−4.96836e9	−1.47868
$528$	0	0
$529$	8.62603e8	0.253347
$530$	9.97643e8	0.291078
$531$	0	0
$532$	2.82061e9	0.812179
$533$	−1.61851e9	−0.462988
$534$	0	0
$535$	−1.51485e8	−0.0427693
$536$	−3.36667e9	−0.944329
$537$	0	0
$538$	2.25248e9	0.623626
$539$	4.73022e9	1.30113
$540$	0	0
$541$	−7.75692e7	−0.0210620	−0.0105310	−	0.999945i	$-0.503352\pi$
$541$	−7.75692e7	−0.0210620	−0.0105310	+	0.999945i	$0.503352\pi$
$542$	−1.63632e9	−0.441439
$543$	0	0
$544$	4.88307e9	1.30046
$545$	2.04457e9	0.541021
$546$	0	0
$547$	−4.43025e9	−1.15737	−0.578686	−	0.815551i	$-0.696434\pi$
$547$	−4.43025e9	−1.15737	−0.578686	+	0.815551i	$0.696434\pi$
$548$	2.02363e9	0.525289
$549$	0	0
$550$	−2.39909e9	−0.614861
$551$	−2.91422e9	−0.742149
$552$	0	0
$553$	−1.08434e10	−2.72664
$554$	1.03498e9	0.258610
$555$	0	0
$556$	7.51566e8	0.185441
$557$	3.18990e9	0.782138	0.391069	−	0.920361i	$-0.372105\pi$
$557$	3.18990e9	0.782138	0.391069	+	0.920361i	$0.372105\pi$
$558$	0	0
$559$	−4.64758e8	−0.112535
$560$	−6.23350e8	−0.149994
$561$	0	0
$562$	5.47408e9	1.30087
$563$	3.82449e9	0.903221	0.451610	−	0.892215i	$-0.350850\pi$
$563$	3.82449e9	0.903221	0.451610	+	0.892215i	$0.350850\pi$
$564$	0	0
$565$	−2.22664e9	−0.519373
$566$	3.03541e9	0.703654
$567$	0	0
$568$	5.09412e9	1.16641
$569$	−3.39322e9	−0.772181	−0.386091	−	0.922461i	$-0.626175\pi$
$569$	−3.39322e9	−0.772181	−0.386091	+	0.922461i	$0.626175\pi$
$570$	0	0
$571$	−4.74973e9	−1.06768	−0.533842	−	0.845584i	$-0.679252\pi$
$571$	−4.74973e9	−1.06768	−0.533842	+	0.845584i	$0.679252\pi$
$572$	5.05463e8	0.112929
$573$	0	0
$574$	−8.75402e9	−1.93204
$575$	−4.70586e9	−1.03229
$576$	0	0
$577$	1.63294e9	0.353880	0.176940	−	0.984222i	$-0.443380\pi$
$577$	1.63294e9	0.353880	0.176940	+	0.984222i	$0.443380\pi$
$578$	5.19733e9	1.11952
$579$	0	0
$580$	−3.82882e8	−0.0814830
$581$	−6.29641e9	−1.33191
$582$	0	0
$583$	−6.07085e9	−1.26885
$584$	−7.71181e9	−1.60218
$585$	0	0
$586$	−1.21488e9	−0.249397
$587$	−8.86642e9	−1.80932	−0.904659	−	0.426137i	$-0.859874\pi$
$587$	−8.86642e9	−1.80932	−0.904659	+	0.426137i	$0.859874\pi$
$588$	0	0
$589$	−5.31685e9	−1.07214
$590$	−1.06137e9	−0.212758
$591$	0	0
$592$	−1.90525e9	−0.377420
$593$	1.84381e9	0.363098	0.181549	−	0.983382i	$-0.441889\pi$
$593$	1.84381e9	0.363098	0.181549	+	0.983382i	$0.441889\pi$
$594$	0	0
$595$	3.55448e9	0.691778
$596$	4.01577e8	0.0776974
$597$	0	0
$598$	−1.20200e9	−0.229853
$599$	1.78692e9	0.339711	0.169856	−	0.985469i	$-0.445670\pi$
$599$	1.78692e9	0.339711	0.169856	+	0.985469i	$0.445670\pi$
$600$	0	0
$601$	6.42068e9	1.20648	0.603240	−	0.797559i	$-0.293876\pi$
$601$	6.42068e9	1.20648	0.603240	+	0.797559i	$0.293876\pi$
$602$	−2.51373e9	−0.469603
$603$	0	0
$604$	−2.72451e9	−0.503105
$605$	2.86719e8	0.0526397
$606$	0	0
$607$	3.78234e9	0.686437	0.343218	−	0.939256i	$-0.388483\pi$
$607$	3.78234e9	0.686437	0.343218	+	0.939256i	$0.388483\pi$
$608$	5.22558e9	0.942913
$609$	0	0
$610$	−3.21908e8	−0.0574219
$611$	2.93274e9	0.520151
$612$	0	0
$613$	−4.75053e9	−0.832972	−0.416486	−	0.909142i	$-0.636739\pi$
$613$	−4.75053e9	−0.832972	−0.416486	+	0.909142i	$0.636739\pi$
$614$	3.45590e9	0.602521
$615$	0	0
$616$	8.78217e9	1.51380
$617$	−9.17034e9	−1.57176	−0.785882	−	0.618376i	$-0.787791\pi$
$617$	−9.17034e9	−1.57176	−0.785882	+	0.618376i	$0.787791\pi$
$618$	0	0
$619$	4.61570e9	0.782204	0.391102	−	0.920347i	$-0.372094\pi$
$619$	4.61570e9	0.782204	0.391102	+	0.920347i	$0.372094\pi$
$620$	−6.98550e8	−0.117714
$621$	0	0
$622$	4.72361e9	0.787060
$623$	3.45456e9	0.572380
$624$	0	0
$625$	4.71372e9	0.772296
$626$	−4.73850e9	−0.772024
$627$	0	0
$628$	−2.31722e9	−0.373343
$629$	1.08641e10	1.74068
$630$	0	0
$631$	3.44532e9	0.545918	0.272959	−	0.962026i	$-0.411998\pi$
$631$	3.44532e9	0.545918	0.272959	+	0.962026i	$0.411998\pi$
$632$	−1.18961e10	−1.87454
$633$	0	0
$634$	−6.52848e8	−0.101742
$635$	−3.80604e8	−0.0589882
$636$	0	0
$637$	−2.61343e9	−0.400611
$638$	−2.82462e9	−0.430614
$639$	0	0
$640$	2.15580e8	0.0325072
$641$	−9.82463e7	−0.0147337	−0.00736687	−	0.999973i	$-0.502345\pi$
$641$	−9.82463e7	−0.0147337	−0.00736687	+	0.999973i	$0.502345\pi$
$642$	0	0
$643$	4.07562e9	0.604583	0.302291	−	0.953216i	$-0.402248\pi$
$643$	4.07562e9	0.604583	0.302291	+	0.953216i	$0.402248\pi$
$644$	5.36259e9	0.791178
$645$	0	0
$646$	9.23957e9	1.34846
$647$	7.00089e9	1.01622	0.508111	−	0.861292i	$-0.330344\pi$
$647$	7.00089e9	1.01622	0.508111	+	0.861292i	$0.330344\pi$
$648$	0	0
$649$	6.45866e9	0.927440
$650$	1.32549e9	0.189313
$651$	0	0
$652$	1.75549e9	0.248046
$653$	6.97942e9	0.980897	0.490448	−	0.871470i	$-0.336833\pi$
$653$	6.97942e9	0.980897	0.490448	+	0.871470i	$0.336833\pi$
$654$	0	0
$655$	1.10989e9	0.154324
$656$	−4.14812e9	−0.573703
$657$	0	0
$658$	1.58623e10	2.17058
$659$	6.20737e9	0.844907	0.422453	−	0.906385i	$-0.361169\pi$
$659$	6.20737e9	0.844907	0.422453	+	0.906385i	$0.361169\pi$
$660$	0	0
$661$	7.75251e8	0.104409	0.0522044	−	0.998636i	$-0.483375\pi$
$661$	7.75251e8	0.104409	0.0522044	+	0.998636i	$0.483375\pi$
$662$	−4.23752e9	−0.567686
$663$	0	0
$664$	−6.90768e9	−0.915680
$665$	3.80380e9	0.501582
$666$	0	0
$667$	−5.54056e9	−0.722958
$668$	−2.25278e9	−0.292417
$669$	0	0
$670$	−1.41337e9	−0.181549
$671$	1.95888e9	0.250310
$672$	0	0
$673$	3.89012e9	0.491937	0.245969	−	0.969278i	$-0.420894\pi$
$673$	3.89012e9	0.491937	0.245969	+	0.969278i	$0.420894\pi$
$674$	4.04923e9	0.509404
$675$	0	0
$676$	−2.79267e8	−0.0347702
$677$	1.20651e9	0.149442	0.0747208	−	0.997204i	$-0.476193\pi$
$677$	1.20651e9	0.149442	0.0747208	+	0.997204i	$0.476193\pi$
$678$	0	0
$679$	1.00954e10	1.23760
$680$	3.89956e9	0.475591
$681$	0	0
$682$	−5.15339e9	−0.622082
$683$	6.45534e9	0.775259	0.387629	−	0.921815i	$-0.373294\pi$
$683$	6.45534e9	0.775259	0.387629	+	0.921815i	$0.373294\pi$
$684$	0	0
$685$	2.72901e9	0.324405
$686$	−4.34917e9	−0.514366
$687$	0	0
$688$	−1.19114e9	−0.139445
$689$	3.35413e9	0.390672
$690$	0	0
$691$	−8.05007e9	−0.928167	−0.464084	−	0.885791i	$-0.653616\pi$
$691$	−8.05007e9	−0.928167	−0.464084	+	0.885791i	$0.653616\pi$
$692$	−2.47503e9	−0.283928
$693$	0	0
$694$	−7.98030e9	−0.906277
$695$	1.01354e9	0.114524
$696$	0	0
$697$	2.36535e10	2.64594
$698$	−4.86322e9	−0.541290
$699$	0	0
$700$	−5.91354e9	−0.651635
$701$	4.44290e9	0.487140	0.243570	−	0.969883i	$-0.421681\pi$
$701$	4.44290e9	0.487140	0.243570	+	0.969883i	$0.421681\pi$
$702$	0	0
$703$	1.16262e10	1.26210
$704$	7.93092e9	0.856682
$705$	0	0
$706$	−1.30559e10	−1.39633
$707$	−1.63866e10	−1.74390
$708$	0	0
$709$	−7.64450e9	−0.805541	−0.402770	−	0.915301i	$-0.631953\pi$
$709$	−7.64450e9	−0.805541	−0.402770	+	0.915301i	$0.631953\pi$
$710$	2.13857e9	0.224243
$711$	0	0
$712$	3.78994e9	0.393507
$713$	−1.01085e10	−1.04441
$714$	0	0
$715$	6.81655e8	0.0697418
$716$	4.22190e8	0.0429846
$717$	0	0
$718$	−2.63541e9	−0.265712
$719$	−9.16074e9	−0.919135	−0.459567	−	0.888143i	$-0.651996\pi$
$719$	−9.16074e9	−0.919135	−0.459567	+	0.888143i	$0.651996\pi$
$720$	0	0
$721$	−9.67232e9	−0.961075
$722$	2.40138e9	0.237455
$723$	0	0
$724$	−3.79376e8	−0.0371522
$725$	6.10979e9	0.595448
$726$	0	0
$727$	1.39651e10	1.34795	0.673974	−	0.738755i	$-0.264586\pi$
$727$	1.39651e10	1.34795	0.673974	+	0.738755i	$0.264586\pi$
$728$	−4.85213e9	−0.466093
$729$	0	0
$730$	−3.23751e9	−0.308021
$731$	6.79214e9	0.643126
$732$	0	0
$733$	−1.22423e10	−1.14815	−0.574074	−	0.818803i	$-0.694638\pi$
$733$	−1.22423e10	−1.14815	−0.574074	+	0.818803i	$0.694638\pi$
$734$	−2.55736e9	−0.238702
$735$	0	0
$736$	9.93496e9	0.918531
$737$	8.60061e9	0.791395
$738$	0	0
$739$	−3.72415e9	−0.339447	−0.169723	−	0.985492i	$-0.554287\pi$
$739$	−3.72415e9	−0.339447	−0.169723	+	0.985492i	$0.554287\pi$
$740$	1.52750e9	0.138570
$741$	0	0
$742$	1.81414e10	1.63027
$743$	−8.70451e9	−0.778545	−0.389272	−	0.921123i	$-0.627274\pi$
$743$	−8.70451e9	−0.778545	−0.389272	+	0.921123i	$0.627274\pi$
$744$	0	0
$745$	5.41556e8	0.0479840
$746$	1.37902e10	1.21614
$747$	0	0
$748$	−7.38703e9	−0.645378
$749$	−2.75465e9	−0.239541
$750$	0	0
$751$	4.77226e9	0.411135	0.205567	−	0.978643i	$-0.434096\pi$
$751$	4.77226e9	0.411135	0.205567	+	0.978643i	$0.434096\pi$
$752$	7.51639e9	0.644535
$753$	0	0
$754$	1.56060e9	0.132584
$755$	−3.67420e9	−0.310705
$756$	0	0
$757$	−5.66163e9	−0.474358	−0.237179	−	0.971466i	$-0.576223\pi$
$757$	−5.66163e9	−0.474358	−0.237179	+	0.971466i	$0.576223\pi$
$758$	7.43855e8	0.0620363
$759$	0	0
$760$	4.17308e9	0.344833
$761$	−1.03908e10	−0.854682	−0.427341	−	0.904091i	$-0.640550\pi$
$761$	−1.03908e10	−0.854682	−0.427341	+	0.904091i	$0.640550\pi$
$762$	0	0
$763$	3.71791e10	3.03014
$764$	3.11300e9	0.252553
$765$	0	0
$766$	2.81637e9	0.226407
$767$	−3.56839e9	−0.285554
$768$	0	0
$769$	−1.61920e10	−1.28398	−0.641990	−	0.766713i	$-0.721891\pi$
$769$	−1.61920e10	−1.28398	−0.641990	+	0.766713i	$0.721891\pi$
$770$	3.68686e9	0.291031
$771$	0	0
$772$	−6.06177e9	−0.474175
$773$	−1.30852e10	−1.01894	−0.509472	−	0.860487i	$-0.670159\pi$
$773$	−1.30852e10	−1.01894	−0.509472	+	0.860487i	$0.670159\pi$
$774$	0	0
$775$	1.11470e10	0.860207
$776$	1.10755e10	0.850839
$777$	0	0
$778$	−1.42823e10	−1.08735
$779$	2.53126e10	1.91847
$780$	0	0
$781$	−1.30136e10	−0.977506
$782$	1.75664e10	1.31359
$783$	0	0
$784$	−6.69802e9	−0.496410
$785$	−3.12494e9	−0.230567
$786$	0	0
$787$	−2.43918e10	−1.78374	−0.891872	−	0.452288i	$-0.850608\pi$
$787$	−2.43918e10	−1.78374	−0.891872	+	0.452288i	$0.850608\pi$
$788$	−3.00243e8	−0.0218590
$789$	0	0
$790$	−4.99412e9	−0.360383
$791$	−4.04898e10	−2.90890
$792$	0	0
$793$	−1.08227e9	−0.0770692
$794$	−1.81335e10	−1.28561
$795$	0	0
$796$	−4.76886e8	−0.0335134
$797$	−2.78830e10	−1.95090	−0.975450	−	0.220219i	$-0.929323\pi$
$797$	−2.78830e10	−1.95090	−0.975450	+	0.220219i	$0.929323\pi$
$798$	0	0
$799$	−4.28601e10	−2.97262
$800$	−1.09557e10	−0.756526
$801$	0	0
$802$	1.17419e10	0.803760
$803$	1.97009e10	1.34271
$804$	0	0
$805$	7.23185e9	0.488611
$806$	2.84724e9	0.191536
$807$	0	0
$808$	−1.79775e10	−1.19892
$809$	−1.50237e10	−0.997600	−0.498800	−	0.866717i	$-0.666226\pi$
$809$	−1.50237e10	−0.997600	−0.498800	+	0.866717i	$0.666226\pi$
$810$	0	0
$811$	1.90447e10	1.25372	0.626861	−	0.779131i	$-0.284339\pi$
$811$	1.90447e10	1.25372	0.626861	+	0.779131i	$0.284339\pi$
$812$	−6.96245e9	−0.456369
$813$	0	0
$814$	1.12687e10	0.732302
$815$	2.36741e9	0.153187
$816$	0	0
$817$	7.26856e9	0.466306
$818$	−4.67496e9	−0.298635
$819$	0	0
$820$	3.32568e9	0.210635
$821$	2.56581e10	1.61817	0.809084	−	0.587693i	$-0.199964\pi$
$821$	2.56581e10	1.61817	0.809084	+	0.587693i	$0.199964\pi$
$822$	0	0
$823$	−9.13563e9	−0.571267	−0.285633	−	0.958339i	$-0.592204\pi$
$823$	−9.13563e9	−0.571267	−0.285633	+	0.958339i	$0.592204\pi$
$824$	−1.06113e10	−0.660731
$825$	0	0
$826$	−1.93003e10	−1.19161
$827$	−2.87638e10	−1.76839	−0.884194	−	0.467120i	$-0.845292\pi$
$827$	−2.87638e10	−1.76839	−0.884194	+	0.467120i	$0.845292\pi$
$828$	0	0
$829$	−9.59018e9	−0.584636	−0.292318	−	0.956321i	$-0.594427\pi$
$829$	−9.59018e9	−0.584636	−0.292318	+	0.956321i	$0.594427\pi$
$830$	−2.89992e9	−0.176041
$831$	0	0
$832$	−4.38181e9	−0.263768
$833$	3.81936e10	2.28946
$834$	0	0
$835$	−3.03805e9	−0.180589
$836$	−7.90517e9	−0.467939
$837$	0	0
$838$	−2.14548e10	−1.25942
$839$	2.31118e9	0.135103	0.0675517	−	0.997716i	$-0.478481\pi$
$839$	2.31118e9	0.135103	0.0675517	+	0.997716i	$0.478481\pi$
$840$	0	0
$841$	−1.00564e10	−0.582982
$842$	−1.65105e10	−0.953167
$843$	0	0
$844$	−1.35372e10	−0.775048
$845$	−3.76613e8	−0.0214732
$846$	0	0
$847$	5.21380e9	0.294823
$848$	8.59638e9	0.484094
$849$	0	0
$850$	−1.93712e10	−1.08191
$851$	2.21039e10	1.22946
$852$	0	0
$853$	2.49072e10	1.37405	0.687025	−	0.726633i	$-0.258916\pi$
$853$	2.49072e10	1.37405	0.687025	+	0.726633i	$0.258916\pi$
$854$	−5.85368e9	−0.321608
$855$	0	0
$856$	−3.02208e9	−0.164683
$857$	−2.80959e10	−1.52479	−0.762396	−	0.647110i	$-0.775977\pi$
$857$	−2.80959e10	−1.52479	−0.762396	+	0.647110i	$0.775977\pi$
$858$	0	0
$859$	1.10331e10	0.593910	0.296955	−	0.954892i	$-0.404029\pi$
$859$	1.10331e10	0.593910	0.296955	+	0.954892i	$0.404029\pi$
$860$	9.54974e8	0.0511973
$861$	0	0
$862$	9.56731e8	0.0508762
$863$	1.99198e10	1.05499	0.527494	−	0.849559i	$-0.323132\pi$
$863$	1.99198e10	1.05499	0.527494	+	0.849559i	$0.323132\pi$
$864$	0	0
$865$	−3.33776e9	−0.175347
$866$	2.43262e10	1.27280
$867$	0	0
$868$	−1.27026e10	−0.659288
$869$	3.03902e10	1.57096
$870$	0	0
$871$	−4.75182e9	−0.243667
$872$	4.07885e10	2.08320
$873$	0	0
$874$	1.87986e10	0.952434
$875$	−1.66236e10	−0.838877
$876$	0	0
$877$	1.25041e10	0.625971	0.312985	−	0.949758i	$-0.398671\pi$
$877$	1.25041e10	0.625971	0.312985	+	0.949758i	$0.398671\pi$
$878$	1.03077e10	0.513964
$879$	0	0
$880$	1.74703e9	0.0864193
$881$	−1.47457e10	−0.726521	−0.363261	−	0.931688i	$-0.618337\pi$
$881$	−1.47457e10	−0.726521	−0.363261	+	0.931688i	$0.618337\pi$
$882$	0	0
$883$	3.78289e10	1.84910	0.924552	−	0.381057i	$-0.124440\pi$
$883$	3.78289e10	1.84910	0.924552	+	0.381057i	$0.124440\pi$
$884$	4.08131e9	0.198709
$885$	0	0
$886$	4.17383e9	0.201612
$887$	−4.41598e9	−0.212469	−0.106234	−	0.994341i	$-0.533879\pi$
$887$	−4.41598e9	−0.212469	−0.106234	+	0.994341i	$0.533879\pi$
$888$	0	0
$889$	−6.92101e9	−0.330380
$890$	1.59106e9	0.0756522
$891$	0	0
$892$	−2.20521e9	−0.104034
$893$	−4.58664e10	−2.15534
$894$	0	0
$895$	5.69355e8	0.0265462
$896$	3.92018e9	0.182066
$897$	0	0
$898$	−1.46005e10	−0.672822
$899$	1.31242e10	0.602441
$900$	0	0
$901$	−4.90185e10	−2.23266
$902$	2.45344e10	1.11315
$903$	0	0
$904$	−4.44207e10	−1.99984
$905$	−5.11617e8	−0.0229443
$906$	0	0
$907$	2.68010e10	1.19269	0.596343	−	0.802730i	$-0.296620\pi$
$907$	2.68010e10	1.19269	0.596343	+	0.802730i	$0.296620\pi$
$908$	−1.97046e10	−0.873507
$909$	0	0
$910$	−2.03698e9	−0.0896070
$911$	4.22204e9	0.185015	0.0925077	−	0.995712i	$-0.470512\pi$
$911$	4.22204e9	0.185015	0.0925077	+	0.995712i	$0.470512\pi$
$912$	0	0
$913$	1.76466e10	0.767385
$914$	−1.94327e10	−0.841824
$915$	0	0
$916$	5.05667e9	0.217385
$917$	2.01825e10	0.864337
$918$	0	0
$919$	3.31907e10	1.41063	0.705313	−	0.708896i	$-0.250806\pi$
$919$	3.31907e10	1.41063	0.705313	+	0.708896i	$0.250806\pi$
$920$	7.93393e9	0.335916
$921$	0	0
$922$	−5.45690e9	−0.229291
$923$	7.18999e9	0.300970
$924$	0	0
$925$	−2.43748e10	−1.01262
$926$	2.60599e10	1.07854
$927$	0	0
$928$	−1.28989e10	−0.529829
$929$	6.90493e9	0.282556	0.141278	−	0.989970i	$-0.454879\pi$
$929$	6.90493e9	0.282556	0.141278	+	0.989970i	$0.454879\pi$
$930$	0	0
$931$	4.08726e10	1.66000
$932$	4.94830e9	0.200217
$933$	0	0
$934$	−1.18420e10	−0.475568
$935$	−9.96195e9	−0.398569
$936$	0	0
$937$	−4.46363e10	−1.77256	−0.886278	−	0.463154i	$-0.846718\pi$
$937$	−4.46363e10	−1.77256	−0.886278	+	0.463154i	$0.846718\pi$
$938$	−2.57011e10	−1.01681
$939$	0	0
$940$	−6.02612e9	−0.236641
$941$	−4.46604e10	−1.74727	−0.873633	−	0.486585i	$-0.838243\pi$
$941$	−4.46604e10	−1.74727	−0.873633	+	0.486585i	$0.838243\pi$
$942$	0	0
$943$	4.81247e10	1.86886
$944$	−9.14551e9	−0.353839
$945$	0	0
$946$	7.04510e9	0.270563
$947$	9.94621e9	0.380568	0.190284	−	0.981729i	$-0.439059\pi$
$947$	9.94621e9	0.380568	0.190284	+	0.981729i	$0.439059\pi$
$948$	0	0
$949$	−1.08847e10	−0.413413
$950$	−2.07299e10	−0.784450
$951$	0	0
$952$	7.09107e10	2.66368
$953$	2.94139e10	1.10085	0.550423	−	0.834886i	$-0.314466\pi$
$953$	2.94139e10	1.10085	0.550423	+	0.834886i	$0.314466\pi$
$954$	0	0
$955$	4.19812e9	0.155971
$956$	−7.61672e9	−0.281946
$957$	0	0
$958$	1.60489e10	0.589748
$959$	4.96252e10	1.81692
$960$	0	0
$961$	−3.56812e9	−0.129690
$962$	−6.22595e9	−0.225472
$963$	0	0
$964$	−4.56870e9	−0.164257
$965$	−8.17474e9	−0.292839
$966$	0	0
$967$	2.80836e10	0.998757	0.499379	−	0.866384i	$-0.333562\pi$
$967$	2.80836e10	0.998757	0.499379	+	0.866384i	$0.333562\pi$
$968$	5.71996e9	0.202689
$969$	0	0
$970$	4.64963e9	0.163575
$971$	−5.91936e9	−0.207495	−0.103747	−	0.994604i	$-0.533083\pi$
$971$	−5.91936e9	−0.207495	−0.103747	+	0.994604i	$0.533083\pi$
$972$	0	0
$973$	1.84306e10	0.641422
$974$	2.01457e10	0.698597
$975$	0	0
$976$	−2.77378e9	−0.0954989
$977$	3.36951e10	1.15594	0.577970	−	0.816058i	$-0.303845\pi$
$977$	3.36951e10	1.15594	0.577970	+	0.816058i	$0.303845\pi$
$978$	0	0
$979$	−9.68191e9	−0.329778
$980$	5.37002e9	0.182257
$981$	0	0
$982$	−2.04114e10	−0.687833
$983$	−2.08403e10	−0.699789	−0.349895	−	0.936789i	$-0.613783\pi$
$983$	−2.08403e10	−0.699789	−0.349895	+	0.936789i	$0.613783\pi$
$984$	0	0
$985$	−4.04900e8	−0.0134996
$986$	−2.28071e10	−0.757708
$987$	0	0
$988$	4.36758e9	0.144076
$989$	1.38191e10	0.454248
$990$	0	0
$991$	4.91301e10	1.60358	0.801788	−	0.597609i	$-0.203882\pi$
$991$	4.91301e10	1.60358	0.801788	+	0.597609i	$0.203882\pi$
$992$	−2.35335e10	−0.765411
$993$	0	0
$994$	3.88884e10	1.25594
$995$	−6.43115e8	−0.0206970
$996$	0	0
$997$	2.36829e10	0.756837	0.378419	−	0.925635i	$-0.376468\pi$
$997$	2.36829e10	0.756837	0.378419	+	0.925635i	$0.376468\pi$
$998$	−8.97484e8	−0.0285805
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.8.a.i.1.6	yes	8
3.2	odd	2	inner	117.8.a.i.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.8.a.i.1.3	✓	8	3.2	odd	2	inner
117.8.a.i.1.6	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+8.37511 q^{2} -57.8575 q^{4} -78.0252 q^{5} -1418.83 q^{7} -1556.58 q^{8} -653.469 q^{10} +3976.49 q^{11} -2197.00 q^{13} -11882.9 q^{14} -5630.74 q^{16} +32107.8 q^{17} +34359.9 q^{19} +4514.34 q^{20} +33303.5 q^{22} +65325.6 q^{23} -72037.1 q^{25} -18400.1 q^{26} +82090.2 q^{28} -84814.6 q^{29} -154740. q^{31} +152084. q^{32} +268906. q^{34} +110705. q^{35} +338365. q^{37} +287768. q^{38} +121452. q^{40} +736691. q^{41} +211542. q^{43} -230070. q^{44} +547109. q^{46} -1.33488e6 q^{47} +1.18955e6 q^{49} -603318. q^{50} +127113. q^{52} -1.52669e6 q^{53} -310266. q^{55} +2.20852e6 q^{56} -710331. q^{58} +1.62421e6 q^{59} +492614. q^{61} -1.29597e6 q^{62} +1.99445e6 q^{64} +171421. q^{65} +2.16287e6 q^{67} -1.85768e6 q^{68} +927164. q^{70} -3.27264e6 q^{71} +4.95434e6 q^{73} +2.83384e6 q^{74} -1.98798e6 q^{76} -5.64198e6 q^{77} +7.64247e6 q^{79} +439340. q^{80} +6.16987e6 q^{82} +4.43774e6 q^{83} -2.50521e6 q^{85} +1.77169e6 q^{86} -6.18971e6 q^{88} -2.43479e6 q^{89} +3.11718e6 q^{91} -3.77958e6 q^{92} -1.11798e7 q^{94} -2.68093e6 q^{95} -7.11530e6 q^{97} +9.96258e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 434 q^{4} - 72 q^{7} + 13924 q^{10} - 17576 q^{13} + 79314 q^{16} - 11352 q^{19} + 243952 q^{22} + 97224 q^{25} - 28372 q^{28} + 5656 q^{31} - 223920 q^{34} + 603152 q^{37} + 3411964 q^{40} + 2930960 q^{43}+ \cdots + 26916464 q^{97}+O(q^{100}) \) 8 * q + 434 * q^4 - 72 * q^7 + 13924 * q^10 - 17576 * q^13 + 79314 * q^16 - 11352 * q^19 + 243952 * q^22 + 97224 * q^25 - 28372 * q^28 + 5656 * q^31 - 223920 * q^34 + 603152 * q^37 + 3411964 * q^40 + 2930960 * q^43 + 4783304 * q^46 + 3382232 * q^49 - 953498 * q^52 + 6106032 * q^55 + 12363392 * q^58 + 10218304 * q^61 + 18433978 * q^64 + 1734584 * q^67 + 6347032 * q^70 + 9903984 * q^73 - 5818028 * q^76 + 446688 * q^79 + 9877140 * q^82 + 2396896 * q^85 + 33949728 * q^88 + 158184 * q^91 - 39327112 * q^94 + 26916464 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more