LMFDB - Embedded newform 117.10.a.g.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,10,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, 10); 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, 10, names="a")

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
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:	$60.2591928312$
Analytic rank:	$1$
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} - 2613x^{6} + 2010020x^{4} - 399948096x^{2} + 6898597888 $ x^8 - 2613x^6 + 2010020x^4 - 399948096*x^2 + 6898597888
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{4}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$31.9131$ 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)

$f(q)$	$=$	\(q+31.9131 q^{2} +506.444 q^{4} -1635.60 q^{5} +9377.46 q^{7} -177.316 q^{8} -52197.1 q^{10} -74.3848 q^{11} -28561.0 q^{13} +299264. q^{14} -264958. q^{16} -21083.6 q^{17} -967016. q^{19} -828340. q^{20} -2373.85 q^{22} +1.49197e6 q^{23} +722067. q^{25} -911469. q^{26} +4.74916e6 q^{28} +744914. q^{29} -5.33293e6 q^{31} -8.36483e6 q^{32} -672841. q^{34} -1.53378e7 q^{35} -1.21137e7 q^{37} -3.08605e7 q^{38} +290017. q^{40} +1.06424e7 q^{41} -3.31733e7 q^{43} -37671.7 q^{44} +4.76134e7 q^{46} -5.10256e7 q^{47} +4.75832e7 q^{49} +2.30434e7 q^{50} -1.44645e7 q^{52} -8.32040e7 q^{53} +121664. q^{55} -1.66277e6 q^{56} +2.37725e7 q^{58} +9.70896e7 q^{59} -1.05052e8 q^{61} -1.70190e8 q^{62} -1.31289e8 q^{64} +4.67144e7 q^{65} -2.17098e8 q^{67} -1.06776e7 q^{68} -4.89476e8 q^{70} +3.05805e8 q^{71} +8.93888e7 q^{73} -3.86586e8 q^{74} -4.89739e8 q^{76} -697540. q^{77} -8.48632e7 q^{79} +4.33366e8 q^{80} +3.39631e8 q^{82} +4.41256e8 q^{83} +3.44843e7 q^{85} -1.05866e9 q^{86} +13189.6 q^{88} +2.29385e8 q^{89} -2.67830e8 q^{91} +7.55600e8 q^{92} -1.62838e9 q^{94} +1.58165e9 q^{95} +4.66565e8 q^{97} +1.51852e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1130 q^{4} + 9756 q^{7} - 38516 q^{10} - 228488 q^{13} - 314526 q^{16} - 884340 q^{19} - 1422584 q^{22} + 1535856 q^{25} - 1630900 q^{28} - 12449540 q^{31} - 6019872 q^{34} + 6397904 q^{37} - 13411388 q^{40}+ \cdots - 3297114688 q^{97}+O(q^{100}) $ 8 * q + 1130 * q^4 + 9756 * q^7 - 38516 * q^10 - 228488 * q^13 - 314526 * q^16 - 884340 * q^19 - 1422584 * q^22 + 1535856 * q^25 - 1630900 * q^28 - 12449540 * q^31 - 6019872 * q^34 + 6397904 * q^37 - 13411388 * q^40 - 53044624 * q^43 + 7474280 * q^46 - 60752704 * q^49 - 32273930 * q^52 - 182669760 * q^55 - 228402352 * q^58 - 298501160 * q^61 - 661011230 * q^64 - 713390956 * q^67 - 1215462584 * q^70 - 953259360 * q^73 - 1790107484 * q^76 - 1270033632 * q^79 - 1406396292 * q^82 - 2048639264 * q^85 - 2044743576 * q^88 - 278641116 * q^91 - 3755823328 * q^94 - 3297114688 * 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 $

$2$	31.9131	1.41037	0.705186	−	0.709023i	$-0.250864\pi$
$2$	31.9131	1.41037	0.705186	+	0.709023i	$0.250864\pi$
$3$	0	0
$3$	0	0
$4$	506.444	0.989148
$5$	−1635.60	−1.17034	−0.585171	−	0.810910i	$-0.698972\pi$
$5$	−1635.60	−1.17034	−0.585171	+	0.810910i	$0.698972\pi$
$6$	0	0
$7$	9377.46	1.47620	0.738098	−	0.674693i	$-0.235724\pi$
$7$	9377.46	1.47620	0.738098	+	0.674693i	$0.235724\pi$
$8$	−177.316	−0.0153053
$9$	0	0
$10$	−52197.1	−1.65062
$11$	−74.3848	−0.00153185	−0.000765926	−	1.00000i	$-0.500244\pi$
$11$	−74.3848	−0.00153185	−0.000765926		1.00000i	$0.500244\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	299264.	2.08199
$15$	0	0
$16$	−264958.	−1.01073
$17$	−21083.6	−0.0612243	−0.0306121	−	0.999531i	$-0.509746\pi$
$17$	−21083.6	−0.0612243	−0.0306121	+	0.999531i	$0.509746\pi$
$18$	0	0
$19$	−967016.	−1.70233	−0.851163	−	0.524902i	$-0.824102\pi$
$19$	−967016.	−1.70233	−0.851163	+	0.524902i	$0.824102\pi$
$20$	−828340.	−1.15764
$21$	0	0
$22$	−2373.85	−0.00216048
$23$	1.49197e6	1.11169	0.555847	−	0.831285i	$-0.312394\pi$
$23$	1.49197e6	1.11169	0.555847	+	0.831285i	$0.312394\pi$
$24$	0	0
$25$	722067.	0.369698
$26$	−911469.	−0.391167
$27$	0	0
$28$	4.74916e6	1.46018
$29$	744914.	0.195576	0.0977879	−	0.995207i	$-0.468823\pi$
$29$	744914.	0.195576	0.0977879	+	0.995207i	$0.468823\pi$
$30$	0	0
$31$	−5.33293e6	−1.03714	−0.518571	−	0.855035i	$-0.673536\pi$
$31$	−5.33293e6	−1.03714	−0.518571	+	0.855035i	$0.673536\pi$
$32$	−8.36483e6	−1.41021
$33$	0	0
$34$	−672841.	−0.0863490
$35$	−1.53378e7	−1.72765
$36$	0	0
$37$	−1.21137e7	−1.06260	−0.531300	−	0.847184i	$-0.678296\pi$
$37$	−1.21137e7	−1.06260	−0.531300	+	0.847184i	$0.678296\pi$
$38$	−3.08605e7	−2.40091
$39$	0	0
$40$	290017.	0.0179124
$41$	1.06424e7	0.588181	0.294091	−	0.955778i	$-0.404983\pi$
$41$	1.06424e7	0.588181	0.294091	+	0.955778i	$0.404983\pi$
$42$	0	0
$43$	−3.31733e7	−1.47972	−0.739861	−	0.672759i	$-0.765109\pi$
$43$	−3.31733e7	−1.47972	−0.739861	+	0.672759i	$0.765109\pi$
$44$	−37671.7	−0.00151523
$45$	0	0
$46$	4.76134e7	1.56790
$47$	−5.10256e7	−1.52527	−0.762637	−	0.646827i	$-0.776096\pi$
$47$	−5.10256e7	−1.52527	−0.762637	+	0.646827i	$0.776096\pi$
$48$	0	0
$49$	4.75832e7	1.17916
$50$	2.30434e7	0.521412
$51$	0	0
$52$	−1.44645e7	−0.274340
$53$	−8.32040e7	−1.44845	−0.724223	−	0.689565i	$-0.757802\pi$
$53$	−8.32040e7	−1.44845	−0.724223	+	0.689565i	$0.757802\pi$
$54$	0	0
$55$	121664.	0.00179279
$56$	−1.66277e6	−0.0225936
$57$	0	0
$58$	2.37725e7	0.275835
$59$	9.70896e7	1.04313	0.521566	−	0.853211i	$-0.325348\pi$
$59$	9.70896e7	1.04313	0.521566	+	0.853211i	$0.325348\pi$
$60$	0	0
$61$	−1.05052e8	−0.971447	−0.485724	−	0.874112i	$-0.661444\pi$
$61$	−1.05052e8	−0.971447	−0.485724	+	0.874112i	$0.661444\pi$
$62$	−1.70190e8	−1.46275
$63$	0	0
$64$	−1.31289e8	−0.978180
$65$	4.67144e7	0.324594
$66$	0	0
$67$	−2.17098e8	−1.31619	−0.658097	−	0.752933i	$-0.728638\pi$
$67$	−2.17098e8	−1.31619	−0.658097	+	0.752933i	$0.728638\pi$
$68$	−1.06776e7	−0.0605599
$69$	0	0
$70$	−4.89476e8	−2.43663
$71$	3.05805e8	1.42818	0.714088	−	0.700056i	$-0.246842\pi$
$71$	3.05805e8	1.42818	0.714088	+	0.700056i	$0.246842\pi$
$72$	0	0
$73$	8.93888e7	0.368409	0.184205	−	0.982888i	$-0.441029\pi$
$73$	8.93888e7	0.368409	0.184205	+	0.982888i	$0.441029\pi$
$74$	−3.86586e8	−1.49866
$75$	0	0
$76$	−4.89739e8	−1.68385
$77$	−697540.	−0.00226131
$78$	0	0
$79$	−8.48632e7	−0.245131	−0.122565	−	0.992460i	$-0.539112\pi$
$79$	−8.48632e7	−0.245131	−0.122565	+	0.992460i	$0.539112\pi$
$80$	4.33366e8	1.18290
$81$	0	0
$82$	3.39631e8	0.829554
$83$	4.41256e8	1.02056	0.510280	−	0.860008i	$-0.329542\pi$
$83$	4.41256e8	1.02056	0.510280	+	0.860008i	$0.329542\pi$
$84$	0	0
$85$	3.44843e7	0.0716533
$86$	−1.05866e9	−2.08696
$87$	0	0
$88$	13189.6	2.34455e−5 0
$89$	2.29385e8	0.387535	0.193767	−	0.981048i	$-0.437929\pi$
$89$	2.29385e8	0.387535	0.193767	+	0.981048i	$0.437929\pi$
$90$	0	0
$91$	−2.67830e8	−0.409423
$92$	7.55600e8	1.09963
$93$	0	0
$94$	−1.62838e9	−2.15120
$95$	1.58165e9	1.99230
$96$	0	0
$97$	4.66565e8	0.535106	0.267553	−	0.963543i	$-0.413785\pi$
$97$	4.66565e8	0.535106	0.267553	+	0.963543i	$0.413785\pi$
$98$	1.51852e9	1.66305
$99$	0	0
$100$	3.65686e8	0.365686
$101$	1.55744e9	1.48924	0.744620	−	0.667488i	$-0.232631\pi$
$101$	1.55744e9	1.48924	0.744620	+	0.667488i	$0.232631\pi$
$102$	0	0
$103$	1.43030e9	1.25216	0.626082	−	0.779757i	$-0.284658\pi$
$103$	1.43030e9	1.25216	0.626082	+	0.779757i	$0.284658\pi$
$104$	5.06431e6	0.00424493
$105$	0	0
$106$	−2.65529e9	−2.04285
$107$	−2.52831e9	−1.86468	−0.932339	−	0.361585i	$-0.882236\pi$
$107$	−2.52831e9	−1.86468	−0.932339	+	0.361585i	$0.882236\pi$
$108$	0	0
$109$	−1.44244e8	−0.0978762	−0.0489381	−	0.998802i	$-0.515584\pi$
$109$	−1.44244e8	−0.0978762	−0.0489381	+	0.998802i	$0.515584\pi$
$110$	3.88267e6	0.00252850
$111$	0	0
$112$	−2.48463e9	−1.49204
$113$	−6.66656e8	−0.384635	−0.192318	−	0.981333i	$-0.561600\pi$
$113$	−6.66656e8	−0.384635	−0.192318	+	0.981333i	$0.561600\pi$
$114$	0	0
$115$	−2.44027e9	−1.30106
$116$	3.77257e8	0.193453
$117$	0	0
$118$	3.09843e9	1.47120
$119$	−1.97710e8	−0.0903791
$120$	0	0
$121$	−2.35794e9	−0.999998
$122$	−3.35253e9	−1.37010
$123$	0	0
$124$	−2.70083e9	−1.02589
$125$	2.01352e9	0.737668
$126$	0	0
$127$	2.39963e9	0.818515	0.409258	−	0.912419i	$-0.365788\pi$
$127$	2.39963e9	0.818515	0.409258	+	0.912419i	$0.365788\pi$
$128$	9.29589e7	0.0306088
$129$	0	0
$130$	1.49080e9	0.457798
$131$	−1.80325e9	−0.534978	−0.267489	−	0.963561i	$-0.586194\pi$
$131$	−1.80325e9	−0.534978	−0.267489	+	0.963561i	$0.586194\pi$
$132$	0	0
$133$	−9.06816e9	−2.51297
$134$	−6.92828e9	−1.85632
$135$	0	0
$136$	3.73844e6	0.000937056 0
$137$	−2.25443e9	−0.546758	−0.273379	−	0.961906i	$-0.588141\pi$
$137$	−2.25443e9	−0.546758	−0.273379	+	0.961906i	$0.588141\pi$
$138$	0	0
$139$	3.86030e9	0.877111	0.438556	−	0.898704i	$-0.355490\pi$
$139$	3.86030e9	0.877111	0.438556	+	0.898704i	$0.355490\pi$
$140$	−7.76773e9	−1.70890
$141$	0	0
$142$	9.75916e9	2.01426
$143$	2.12450e6	0.000424859 0
$144$	0	0
$145$	−1.21838e9	−0.228890
$146$	2.85267e9	0.519594
$147$	0	0
$148$	−6.13492e9	−1.05107
$149$	−7.79151e9	−1.29504	−0.647520	−	0.762048i	$-0.724194\pi$
$149$	−7.79151e9	−1.29504	−0.647520	+	0.762048i	$0.724194\pi$
$150$	0	0
$151$	7.53509e9	1.17949	0.589743	−	0.807591i	$-0.299229\pi$
$151$	7.53509e9	1.17949	0.589743	+	0.807591i	$0.299229\pi$
$152$	1.71467e8	0.0260546
$153$	0	0
$154$	−2.22606e7	−0.00318929
$155$	8.72254e9	1.21381
$156$	0	0
$157$	8.46506e9	1.11194	0.555970	−	0.831202i	$-0.312347\pi$
$157$	8.46506e9	1.11194	0.555970	+	0.831202i	$0.312347\pi$
$158$	−2.70824e9	−0.345725
$159$	0	0
$160$	1.36815e10	1.65042
$161$	1.39909e10	1.64108
$162$	0	0
$163$	−7.61688e8	−0.0845149	−0.0422574	−	0.999107i	$-0.513455\pi$
$163$	−7.61688e8	−0.0845149	−0.0422574	+	0.999107i	$0.513455\pi$
$164$	5.38976e9	0.581798
$165$	0	0
$166$	1.40818e10	1.43937
$167$	1.74042e9	0.173153	0.0865763	−	0.996245i	$-0.472407\pi$
$167$	1.74042e9	0.173153	0.0865763	+	0.996245i	$0.472407\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	1.10050e9	0.101058
$171$	0	0
$172$	−1.68004e10	−1.46366
$173$	−1.12865e10	−0.957969	−0.478985	−	0.877823i	$-0.658995\pi$
$173$	−1.12865e10	−0.957969	−0.478985	+	0.877823i	$0.658995\pi$
$174$	0	0
$175$	6.77115e9	0.545747
$176$	1.97088e7	0.00154830
$177$	0	0
$178$	7.32039e9	0.546568
$179$	1.60955e10	1.17183	0.585916	−	0.810372i	$-0.300735\pi$
$179$	1.60955e10	1.17183	0.585916	+	0.810372i	$0.300735\pi$
$180$	0	0
$181$	−1.86841e10	−1.29395	−0.646977	−	0.762509i	$-0.723967\pi$
$181$	−1.86841e10	−1.29395	−0.646977	+	0.762509i	$0.723967\pi$
$182$	−8.54727e9	−0.577439
$183$	0	0
$184$	−2.64550e8	−0.0170148
$185$	1.98132e10	1.24360
$186$	0	0
$187$	1.56830e6	9.37866e−5 0
$188$	−2.58416e10	−1.50872
$189$	0	0
$190$	5.04754e10	2.80989
$191$	1.59100e10	0.865010	0.432505	−	0.901631i	$-0.357630\pi$
$191$	1.59100e10	0.865010	0.432505	+	0.901631i	$0.357630\pi$
$192$	0	0
$193$	2.10791e9	0.109357	0.0546783	−	0.998504i	$-0.482587\pi$
$193$	2.10791e9	0.109357	0.0546783	+	0.998504i	$0.482587\pi$
$194$	1.48895e10	0.754698
$195$	0	0
$196$	2.40982e10	1.16636
$197$	−2.17603e9	−0.102936	−0.0514679	−	0.998675i	$-0.516390\pi$
$197$	−2.17603e9	−0.102936	−0.0514679	+	0.998675i	$0.516390\pi$
$198$	0	0
$199$	−2.82660e9	−0.127769	−0.0638844	−	0.997957i	$-0.520349\pi$
$199$	−2.82660e9	−0.127769	−0.0638844	+	0.997957i	$0.520349\pi$
$200$	−1.28034e8	−0.00565834
$201$	0	0
$202$	4.97026e10	2.10038
$203$	6.98540e9	0.288708
$204$	0	0
$205$	−1.74067e10	−0.688373
$206$	4.56454e10	1.76602
$207$	0	0
$208$	7.56746e9	0.280327
$209$	7.19313e7	0.00260771
$210$	0	0
$211$	2.75512e9	0.0956906	0.0478453	−	0.998855i	$-0.484765\pi$
$211$	2.75512e9	0.0956906	0.0478453	+	0.998855i	$0.484765\pi$
$212$	−4.21381e10	−1.43273
$213$	0	0
$214$	−8.06862e10	−2.62989
$215$	5.42583e10	1.73178
$216$	0	0
$217$	−5.00093e10	−1.53102
$218$	−4.60325e9	−0.138042
$219$	0	0
$220$	6.16159e7	0.00177333
$221$	6.02167e8	0.0169806
$222$	0	0
$223$	3.45788e9	0.0936350	0.0468175	−	0.998903i	$-0.485092\pi$
$223$	3.45788e9	0.0936350	0.0468175	+	0.998903i	$0.485092\pi$
$224$	−7.84409e10	−2.08174
$225$	0	0
$226$	−2.12750e10	−0.542478
$227$	6.29759e10	1.57419	0.787096	−	0.616830i	$-0.211583\pi$
$227$	6.29759e10	1.57419	0.787096	+	0.616830i	$0.211583\pi$
$228$	0	0
$229$	2.76028e10	0.663274	0.331637	−	0.943407i	$-0.392399\pi$
$229$	2.76028e10	0.663274	0.331637	+	0.943407i	$0.392399\pi$
$230$	−7.78765e10	−1.83498
$231$	0	0
$232$	−1.32085e8	−0.00299335
$233$	7.86651e10	1.74856	0.874280	−	0.485423i	$-0.161334\pi$
$233$	7.86651e10	1.74856	0.874280	+	0.485423i	$0.161334\pi$
$234$	0	0
$235$	8.34575e10	1.78509
$236$	4.91705e10	1.03181
$237$	0	0
$238$	−6.30954e9	−0.127468
$239$	−1.98319e10	−0.393164	−0.196582	−	0.980487i	$-0.562984\pi$
$239$	−1.98319e10	−0.393164	−0.196582	+	0.980487i	$0.562984\pi$
$240$	0	0
$241$	−2.09497e10	−0.400038	−0.200019	−	0.979792i	$-0.564100\pi$
$241$	−2.09497e10	−0.400038	−0.200019	+	0.979792i	$0.564100\pi$
$242$	−7.52492e10	−1.41037
$243$	0	0
$244$	−5.32028e10	−0.960905
$245$	−7.78271e10	−1.38001
$246$	0	0
$247$	2.76190e10	0.472140
$248$	9.45610e8	0.0158738
$249$	0	0
$250$	6.42576e10	1.04039
$251$	6.38611e9	0.101556	0.0507779	−	0.998710i	$-0.483830\pi$
$251$	6.38611e9	0.101556	0.0507779	+	0.998710i	$0.483830\pi$
$252$	0	0
$253$	−1.10980e8	−0.00170295
$254$	7.65794e10	1.15441
$255$	0	0
$256$	7.01866e10	1.02135
$257$	−2.49862e10	−0.357274	−0.178637	−	0.983915i	$-0.557169\pi$
$257$	−2.49862e10	−0.357274	−0.178637	+	0.983915i	$0.557169\pi$
$258$	0	0
$259$	−1.13596e11	−1.56861
$260$	2.36582e10	0.321072
$261$	0	0
$262$	−5.75473e10	−0.754518
$263$	−1.23695e11	−1.59423	−0.797116	−	0.603827i	$-0.793642\pi$
$263$	−1.23695e11	−1.59423	−0.797116	+	0.603827i	$0.793642\pi$
$264$	0	0
$265$	1.36088e11	1.69518
$266$	−2.89393e11	−3.54422
$267$	0	0
$268$	−1.09948e11	−1.30191
$269$	−9.78701e10	−1.13963	−0.569816	−	0.821772i	$-0.692986\pi$
$269$	−9.78701e10	−1.13963	−0.569816	+	0.821772i	$0.692986\pi$
$270$	0	0
$271$	−3.83068e10	−0.431433	−0.215717	−	0.976456i	$-0.569209\pi$
$271$	−3.83068e10	−0.431433	−0.215717	+	0.976456i	$0.569209\pi$
$272$	5.58626e9	0.0618815
$273$	0	0
$274$	−7.19459e10	−0.771132
$275$	−5.37108e7	−0.000566323 0
$276$	0	0
$277$	−1.74121e11	−1.77702	−0.888509	−	0.458859i	$-0.848258\pi$
$277$	−1.74121e11	−1.77702	−0.888509	+	0.458859i	$0.848258\pi$
$278$	1.23194e11	1.23705
$279$	0	0
$280$	2.71963e9	0.0264422
$281$	−1.39295e11	−1.33277	−0.666386	−	0.745607i	$-0.732160\pi$
$281$	−1.39295e11	−1.33277	−0.666386	+	0.745607i	$0.732160\pi$
$282$	0	0
$283$	−2.05354e11	−1.90311	−0.951555	−	0.307477i	$-0.900515\pi$
$283$	−2.05354e11	−1.90311	−0.951555	+	0.307477i	$0.900515\pi$
$284$	1.54873e11	1.41268
$285$	0	0
$286$	6.77994e7	0.000599210 0
$287$	9.97984e10	0.868271
$288$	0	0
$289$	−1.18143e11	−0.996252
$290$	−3.88823e10	−0.322821
$291$	0	0
$292$	4.52704e10	0.364411
$293$	5.52246e10	0.437752	0.218876	−	0.975753i	$-0.429761\pi$
$293$	5.52246e10	0.437752	0.218876	+	0.975753i	$0.429761\pi$
$294$	0	0
$295$	−1.58800e11	−1.22082
$296$	2.14795e9	0.0162634
$297$	0	0
$298$	−2.48651e11	−1.82649
$299$	−4.26122e10	−0.308328
$300$	0	0
$301$	−3.11081e11	−2.18436
$302$	2.40468e11	1.66351
$303$	0	0
$304$	2.56219e11	1.72060
$305$	1.71823e11	1.13692
$306$	0	0
$307$	1.58590e11	1.01895	0.509476	−	0.860485i	$-0.329839\pi$
$307$	1.58590e11	1.01895	0.509476	+	0.860485i	$0.329839\pi$
$308$	−3.53265e8	−0.00223677
$309$	0	0
$310$	2.78363e11	1.71192
$311$	4.54685e10	0.275606	0.137803	−	0.990460i	$-0.455996\pi$
$311$	4.54685e10	0.275606	0.137803	+	0.990460i	$0.455996\pi$
$312$	0	0
$313$	−4.85320e10	−0.285811	−0.142906	−	0.989736i	$-0.545645\pi$
$313$	−4.85320e10	−0.285811	−0.142906	+	0.989736i	$0.545645\pi$
$314$	2.70146e11	1.56825
$315$	0	0
$316$	−4.29784e10	−0.242470
$317$	−1.20808e11	−0.671940	−0.335970	−	0.941873i	$-0.609064\pi$
$317$	−1.20808e11	−0.671940	−0.335970	+	0.941873i	$0.609064\pi$
$318$	0	0
$319$	−5.54102e7	−0.000299593 0
$320$	2.14737e11	1.14480
$321$	0	0
$322$	4.46493e11	2.31453
$323$	2.03881e10	0.104224
$324$	0	0
$325$	−2.06229e10	−0.102536
$326$	−2.43078e10	−0.119197
$327$	0	0
$328$	−1.88706e9	−0.00900229
$329$	−4.78490e11	−2.25160
$330$	0	0
$331$	3.42530e11	1.56846	0.784228	−	0.620473i	$-0.213059\pi$
$331$	3.42530e11	1.56846	0.784228	+	0.620473i	$0.213059\pi$
$332$	2.23471e11	1.00949
$333$	0	0
$334$	5.55420e10	0.244209
$335$	3.55086e11	1.54040
$336$	0	0
$337$	4.51920e9	0.0190865	0.00954327	−	0.999954i	$-0.496962\pi$
$337$	4.51920e9	0.0190865	0.00954327	+	0.999954i	$0.496962\pi$
$338$	2.60325e10	0.108490
$339$	0	0
$340$	1.74644e10	0.0708757
$341$	3.96688e8	0.00158875
$342$	0	0
$343$	6.77949e10	0.264468
$344$	5.88214e9	0.0226476
$345$	0	0
$346$	−3.60187e11	−1.35109
$347$	3.96340e11	1.46752	0.733762	−	0.679407i	$-0.237763\pi$
$347$	3.96340e11	1.46752	0.733762	+	0.679407i	$0.237763\pi$
$348$	0	0
$349$	−2.94824e11	−1.06377	−0.531885	−	0.846817i	$-0.678516\pi$
$349$	−2.94824e11	−1.06377	−0.531885	+	0.846817i	$0.678516\pi$
$350$	2.16088e11	0.769706
$351$	0	0
$352$	6.22216e8	0.00216023
$353$	−2.48932e11	−0.853286	−0.426643	−	0.904420i	$-0.640304\pi$
$353$	−2.48932e11	−0.853286	−0.426643	+	0.904420i	$0.640304\pi$
$354$	0	0
$355$	−5.00174e11	−1.67145
$356$	1.16171e11	0.383329
$357$	0	0
$358$	5.13656e11	1.65272
$359$	−9.97869e10	−0.317065	−0.158533	−	0.987354i	$-0.550676\pi$
$359$	−9.97869e10	−0.317065	−0.158533	+	0.987354i	$0.550676\pi$
$360$	0	0
$361$	6.12433e11	1.89791
$362$	−5.96268e11	−1.82496
$363$	0	0
$364$	−1.35641e11	−0.404980
$365$	−1.46204e11	−0.431164
$366$	0	0
$367$	1.78985e11	0.515015	0.257507	−	0.966276i	$-0.417099\pi$
$367$	1.78985e11	0.515015	0.257507	+	0.966276i	$0.417099\pi$
$368$	−3.95310e11	−1.12363
$369$	0	0
$370$	6.32301e11	1.75394
$371$	−7.80242e11	−2.13819
$372$	0	0
$373$	−1.55535e11	−0.416044	−0.208022	−	0.978124i	$-0.566703\pi$
$373$	−1.55535e11	−0.416044	−0.208022	+	0.978124i	$0.566703\pi$
$374$	5.00491e7	0.000132274 0
$375$	0	0
$376$	9.04763e9	0.0233448
$377$	−2.12755e10	−0.0542430
$378$	0	0
$379$	−5.02913e11	−1.25203	−0.626017	−	0.779809i	$-0.715316\pi$
$379$	−5.02913e11	−1.25203	−0.626017	+	0.779809i	$0.715316\pi$
$380$	8.01018e11	1.97068
$381$	0	0
$382$	5.07738e11	1.21999
$383$	−5.65974e11	−1.34401	−0.672004	−	0.740548i	$-0.734566\pi$
$383$	−5.65974e11	−1.34401	−0.672004	+	0.740548i	$0.734566\pi$
$384$	0	0
$385$	1.14090e9	0.00264651
$386$	6.72700e10	0.154233
$387$	0	0
$388$	2.36289e11	0.529299
$389$	1.93917e11	0.429380	0.214690	−	0.976682i	$-0.431126\pi$
$389$	1.93917e11	0.429380	0.214690	+	0.976682i	$0.431126\pi$
$390$	0	0
$391$	−3.14561e10	−0.0680627
$392$	−8.43723e9	−0.0180473
$393$	0	0
$394$	−6.94438e10	−0.145178
$395$	1.38802e11	0.286886
$396$	0	0
$397$	2.39924e11	0.484748	0.242374	−	0.970183i	$-0.422074\pi$
$397$	2.39924e11	0.484748	0.242374	+	0.970183i	$0.422074\pi$
$398$	−9.02053e10	−0.180202
$399$	0	0
$400$	−1.91317e11	−0.373667
$401$	−9.60416e11	−1.85485	−0.927427	−	0.374004i	$-0.877984\pi$
$401$	−9.60416e11	−1.85485	−0.927427	+	0.374004i	$0.877984\pi$
$402$	0	0
$403$	1.52314e11	0.287651
$404$	7.88755e11	1.47308
$405$	0	0
$406$	2.22926e11	0.407186
$407$	9.01076e8	0.00162775
$408$	0	0
$409$	4.33056e11	0.765225	0.382613	−	0.923909i	$-0.375024\pi$
$409$	4.33056e11	0.765225	0.382613	+	0.923909i	$0.375024\pi$
$410$	−5.55501e11	−0.970861
$411$	0	0
$412$	7.24369e11	1.23857
$413$	9.10454e11	1.53987
$414$	0	0
$415$	−7.21718e11	−1.19440
$416$	2.38908e11	0.391121
$417$	0	0
$418$	2.29555e9	0.00367784
$419$	−6.61052e11	−1.04779	−0.523893	−	0.851784i	$-0.675521\pi$
$419$	−6.61052e11	−1.04779	−0.523893	+	0.851784i	$0.675521\pi$
$420$	0	0
$421$	1.84053e11	0.285544	0.142772	−	0.989756i	$-0.454398\pi$
$421$	1.84053e11	0.285544	0.142772	+	0.989756i	$0.454398\pi$
$422$	8.79243e10	0.134959
$423$	0	0
$424$	1.47534e10	0.0221689
$425$	−1.52237e10	−0.0226345
$426$	0	0
$427$	−9.85119e11	−1.43405
$428$	−1.28045e12	−1.84444
$429$	0	0
$430$	1.73155e12	2.44245
$431$	9.41302e11	1.31396	0.656979	−	0.753909i	$-0.271834\pi$
$431$	9.41302e11	1.31396	0.656979	+	0.753909i	$0.271834\pi$
$432$	0	0
$433$	1.30198e12	1.77996	0.889978	−	0.456004i	$-0.150720\pi$
$433$	1.30198e12	1.77996	0.889978	+	0.456004i	$0.150720\pi$
$434$	−1.59595e12	−2.15931
$435$	0	0
$436$	−7.30513e10	−0.0968141
$437$	−1.44276e12	−1.89247
$438$	0	0
$439$	−1.39753e12	−1.79585	−0.897927	−	0.440144i	$-0.854927\pi$
$439$	−1.39753e12	−1.79585	−0.897927	+	0.440144i	$0.854927\pi$
$440$	−2.15729e7	−2.74392e−5 0
$441$	0	0
$442$	1.92170e10	0.0239489
$443$	1.11492e12	1.37539	0.687695	−	0.725999i	$-0.258623\pi$
$443$	1.11492e12	1.37539	0.687695	+	0.725999i	$0.258623\pi$
$444$	0	0
$445$	−3.75183e11	−0.453548
$446$	1.10352e11	0.132060
$447$	0	0
$448$	−1.23116e12	−1.44399
$449$	1.12477e12	1.30604	0.653018	−	0.757343i	$-0.273503\pi$
$449$	1.12477e12	1.30604	0.653018	+	0.757343i	$0.273503\pi$
$450$	0	0
$451$	−7.91630e8	−0.000901007 0
$452$	−3.37624e11	−0.380461
$453$	0	0
$454$	2.00975e12	2.22020
$455$	4.38063e11	0.479165
$456$	0	0
$457$	1.53500e11	0.164621	0.0823105	−	0.996607i	$-0.473770\pi$
$457$	1.53500e11	0.164621	0.0823105	+	0.996607i	$0.473770\pi$
$458$	8.80889e11	0.935463
$459$	0	0
$460$	−1.23586e12	−1.28694
$461$	8.04982e11	0.830103	0.415051	−	0.909798i	$-0.363764\pi$
$461$	8.04982e11	0.830103	0.415051	+	0.909798i	$0.363764\pi$
$462$	0	0
$463$	9.38936e10	0.0949558	0.0474779	−	0.998872i	$-0.484882\pi$
$463$	9.38936e10	0.0949558	0.0474779	+	0.998872i	$0.484882\pi$
$464$	−1.97371e11	−0.197675
$465$	0	0
$466$	2.51044e12	2.46612
$467$	−8.94027e11	−0.869810	−0.434905	−	0.900476i	$-0.643218\pi$
$467$	−8.94027e11	−0.869810	−0.434905	+	0.900476i	$0.643218\pi$
$468$	0	0
$469$	−2.03583e12	−1.94296
$470$	2.66339e12	2.51764
$471$	0	0
$472$	−1.72155e10	−0.0159654
$473$	2.46759e9	0.00226672
$474$	0	0
$475$	−6.98250e11	−0.629347
$476$	−1.00129e11	−0.0893983
$477$	0	0
$478$	−6.32896e11	−0.554507
$479$	−1.84695e12	−1.60304	−0.801520	−	0.597968i	$-0.795975\pi$
$479$	−1.84695e12	−1.60304	−0.801520	+	0.597968i	$0.795975\pi$
$480$	0	0
$481$	3.45980e11	0.294712
$482$	−6.68569e11	−0.564202
$483$	0	0
$484$	−1.19417e12	−0.989146
$485$	−7.63115e11	−0.626256
$486$	0	0
$487$	−1.40809e12	−1.13436	−0.567178	−	0.823595i	$-0.691965\pi$
$487$	−1.40809e12	−1.13436	−0.567178	+	0.823595i	$0.691965\pi$
$488$	1.86273e10	0.0148683
$489$	0	0
$490$	−2.48370e12	−1.94633
$491$	2.38257e12	1.85003	0.925015	−	0.379931i	$-0.124052\pi$
$491$	2.38257e12	1.85003	0.925015	+	0.379931i	$0.124052\pi$
$492$	0	0
$493$	−1.57054e10	−0.0119740
$494$	8.81405e11	0.665893
$495$	0	0
$496$	1.41300e12	1.04827
$497$	2.86767e12	2.10827
$498$	0	0
$499$	−4.28780e11	−0.309586	−0.154793	−	0.987947i	$-0.549471\pi$
$499$	−4.28780e11	−0.309586	−0.154793	+	0.987947i	$0.549471\pi$
$500$	1.01973e12	0.729663
$501$	0	0
$502$	2.03800e11	0.143231
$503$	4.32750e11	0.301426	0.150713	−	0.988578i	$-0.451843\pi$
$503$	4.32750e11	0.301426	0.150713	+	0.988578i	$0.451843\pi$
$504$	0	0
$505$	−2.54735e12	−1.74292
$506$	−3.54171e9	−0.00240179
$507$	0	0
$508$	1.21528e12	0.809633
$509$	8.12557e11	0.536567	0.268284	−	0.963340i	$-0.413544\pi$
$509$	8.12557e11	0.536567	0.268284	+	0.963340i	$0.413544\pi$
$510$	0	0
$511$	8.38240e11	0.543844
$512$	2.19227e12	1.40987
$513$	0	0
$514$	−7.97386e11	−0.503889
$515$	−2.33941e12	−1.46546
$516$	0	0
$517$	3.79553e9	0.00233649
$518$	−3.62519e12	−2.21232
$519$	0	0
$520$	−8.28319e9	−0.00496801
$521$	1.15899e12	0.689145	0.344572	−	0.938760i	$-0.388024\pi$
$521$	1.15899e12	0.689145	0.344572	+	0.938760i	$0.388024\pi$
$522$	0	0
$523$	5.57641e11	0.325910	0.162955	−	0.986634i	$-0.447898\pi$
$523$	5.57641e11	0.325910	0.162955	+	0.986634i	$0.447898\pi$
$524$	−9.13246e11	−0.529172
$525$	0	0
$526$	−3.94749e12	−2.24846
$527$	1.12437e11	0.0634982
$528$	0	0
$529$	4.24826e11	0.235864
$530$	4.34300e12	2.39083
$531$	0	0
$532$	−4.59251e12	−2.48570
$533$	−3.03957e11	−0.163132
$534$	0	0
$535$	4.13531e12	2.18231
$536$	3.84949e10	0.0201447
$537$	0	0
$538$	−3.12334e12	−1.60731
$539$	−3.53946e9	−0.00180629
$540$	0	0
$541$	1.59082e12	0.798425	0.399213	−	0.916858i	$-0.369283\pi$
$541$	1.59082e12	0.798425	0.399213	+	0.916858i	$0.369283\pi$
$542$	−1.22249e12	−0.608481
$543$	0	0
$544$	1.76360e11	0.0863388
$545$	2.35925e11	0.114549
$546$	0	0
$547$	−2.29550e12	−1.09631	−0.548155	−	0.836377i	$-0.684670\pi$
$547$	−2.29550e12	−1.09631	−0.548155	+	0.836377i	$0.684670\pi$
$548$	−1.14174e12	−0.540824
$549$	0	0
$550$	−1.71408e9	−0.000798726 0
$551$	−7.20344e11	−0.332934
$552$	0	0
$553$	−7.95801e11	−0.361861
$554$	−5.55673e12	−2.50626
$555$	0	0
$556$	1.95503e12	0.867593
$557$	−1.43761e12	−0.632840	−0.316420	−	0.948619i	$-0.602481\pi$
$557$	−1.43761e12	−0.632840	−0.316420	+	0.948619i	$0.602481\pi$
$558$	0	0
$559$	9.47462e11	0.410401
$560$	4.06387e12	1.74620
$561$	0	0
$562$	−4.44532e12	−1.87970
$563$	−4.09132e12	−1.71623	−0.858115	−	0.513458i	$-0.828364\pi$
$563$	−4.09132e12	−1.71623	−0.858115	+	0.513458i	$0.828364\pi$
$564$	0	0
$565$	1.09038e12	0.450154
$566$	−6.55347e12	−2.68409
$567$	0	0
$568$	−5.42239e10	−0.0218586
$569$	−3.59636e12	−1.43833	−0.719164	−	0.694840i	$-0.755475\pi$
$569$	−3.59636e12	−1.43833	−0.719164	+	0.694840i	$0.755475\pi$
$570$	0	0
$571$	4.80155e12	1.89025	0.945125	−	0.326709i	$-0.105940\pi$
$571$	4.80155e12	1.89025	0.945125	+	0.326709i	$0.105940\pi$
$572$	1.07594e9	0.000420249 0
$573$	0	0
$574$	3.18487e12	1.22458
$575$	1.07730e12	0.410991
$576$	0	0
$577$	−3.56080e12	−1.33738	−0.668692	−	0.743540i	$-0.733145\pi$
$577$	−3.56080e12	−1.33738	−0.668692	+	0.743540i	$0.733145\pi$
$578$	−3.77032e12	−1.40508
$579$	0	0
$580$	−6.17042e11	−0.226407
$581$	4.13786e12	1.50655
$582$	0	0
$583$	6.18911e9	0.00221881
$584$	−1.58500e10	−0.00563861
$585$	0	0
$586$	1.76239e12	0.617393
$587$	4.79964e12	1.66854	0.834272	−	0.551354i	$-0.185888\pi$
$587$	4.79964e12	1.66854	0.834272	+	0.551354i	$0.185888\pi$
$588$	0	0
$589$	5.15703e12	1.76555
$590$	−5.06779e12	−1.72181
$591$	0	0
$592$	3.20963e12	1.07401
$593$	4.85669e12	1.61285	0.806425	−	0.591336i	$-0.201399\pi$
$593$	4.85669e12	1.61285	0.806425	+	0.591336i	$0.201399\pi$
$594$	0	0
$595$	3.23375e11	0.105774
$596$	−3.94596e12	−1.28099
$597$	0	0
$598$	−1.35989e12	−0.434858
$599$	−6.93558e10	−0.0220121	−0.0110061	−	0.999939i	$-0.503503\pi$
$599$	−6.93558e10	−0.0220121	−0.0110061	+	0.999939i	$0.503503\pi$
$600$	0	0
$601$	3.19131e12	0.997779	0.498889	−	0.866666i	$-0.333741\pi$
$601$	3.19131e12	0.997779	0.498889	+	0.866666i	$0.333741\pi$
$602$	−9.92755e12	−3.08076
$603$	0	0
$604$	3.81610e12	1.16669
$605$	3.85665e12	1.17034
$606$	0	0
$607$	−8.42635e11	−0.251936	−0.125968	−	0.992034i	$-0.540204\pi$
$607$	−8.42635e11	−0.251936	−0.125968	+	0.992034i	$0.540204\pi$
$608$	8.08893e12	2.40063
$609$	0	0
$610$	5.48340e12	1.60349
$611$	1.45734e12	0.423035
$612$	0	0
$613$	−3.11060e12	−0.889757	−0.444879	−	0.895591i	$-0.646753\pi$
$613$	−3.11060e12	−0.889757	−0.444879	+	0.895591i	$0.646753\pi$
$614$	5.06110e12	1.43710
$615$	0	0
$616$	1.23685e8	3.46101e−5 0
$617$	−5.41157e12	−1.50328	−0.751640	−	0.659574i	$-0.770737\pi$
$617$	−5.41157e12	−1.50328	−0.751640	+	0.659574i	$0.770737\pi$
$618$	0	0
$619$	−3.45632e12	−0.946249	−0.473125	−	0.880996i	$-0.656874\pi$
$619$	−3.45632e12	−0.946249	−0.473125	+	0.880996i	$0.656874\pi$
$620$	4.41748e12	1.20064
$621$	0	0
$622$	1.45104e12	0.388707
$623$	2.15105e12	0.572077
$624$	0	0
$625$	−4.70360e12	−1.23302
$626$	−1.54881e12	−0.403100
$627$	0	0
$628$	4.28708e12	1.09987
$629$	2.55400e11	0.0650569
$630$	0	0
$631$	−5.21983e12	−1.31076	−0.655382	−	0.755298i	$-0.727492\pi$
$631$	−5.21983e12	−1.31076	−0.655382	+	0.755298i	$0.727492\pi$
$632$	1.50476e10	0.00375180
$633$	0	0
$634$	−3.85537e12	−0.947686
$635$	−3.92483e12	−0.957942
$636$	0	0
$637$	−1.35902e12	−0.327039
$638$	−1.76831e9	−0.000422538 0
$639$	0	0
$640$	−1.52044e11	−0.0358227
$641$	−6.61129e12	−1.54677	−0.773384	−	0.633938i	$-0.781438\pi$
$641$	−6.61129e12	−1.54677	−0.773384	+	0.633938i	$0.781438\pi$
$642$	0	0
$643$	7.87205e12	1.81609	0.908047	−	0.418867i	$-0.137573\pi$
$643$	7.87205e12	1.81609	0.908047	+	0.418867i	$0.137573\pi$
$644$	7.08561e12	1.62327
$645$	0	0
$646$	6.50648e11	0.146994
$647$	4.51241e12	1.01237	0.506185	−	0.862425i	$-0.331055\pi$
$647$	4.51241e12	1.01237	0.506185	+	0.862425i	$0.331055\pi$
$648$	0	0
$649$	−7.22199e9	−0.00159792
$650$	−6.58142e11	−0.144614
$651$	0	0
$652$	−3.85752e11	−0.0835977
$653$	3.44581e12	0.741621	0.370811	−	0.928708i	$-0.379080\pi$
$653$	3.44581e12	0.741621	0.370811	+	0.928708i	$0.379080\pi$
$654$	0	0
$655$	2.94940e12	0.626107
$656$	−2.81978e12	−0.594495
$657$	0	0
$658$	−1.52701e13	−3.17560
$659$	−6.13892e12	−1.26797	−0.633983	−	0.773347i	$-0.718581\pi$
$659$	−6.13892e12	−1.26797	−0.633983	+	0.773347i	$0.718581\pi$
$660$	0	0
$661$	−8.59895e11	−0.175202	−0.0876009	−	0.996156i	$-0.527920\pi$
$661$	−8.59895e11	−0.175202	−0.0876009	+	0.996156i	$0.527920\pi$
$662$	1.09312e13	2.21211
$663$	0	0
$664$	−7.82415e10	−0.0156200
$665$	1.48319e13	2.94103
$666$	0	0
$667$	1.11139e12	0.217420
$668$	8.81423e11	0.171274
$669$	0	0
$670$	1.13319e13	2.17253
$671$	7.81425e9	0.00148811
$672$	0	0
$673$	4.15663e12	0.781041	0.390520	−	0.920594i	$-0.372295\pi$
$673$	4.15663e12	0.781041	0.390520	+	0.920594i	$0.372295\pi$
$674$	1.44222e11	0.0269191
$675$	0	0
$676$	4.13122e11	0.0760883
$677$	8.56370e12	1.56680	0.783398	−	0.621521i	$-0.213485\pi$
$677$	8.56370e12	1.56680	0.783398	+	0.621521i	$0.213485\pi$
$678$	0	0
$679$	4.37520e12	0.789921
$680$	−6.11460e9	−0.00109668
$681$	0	0
$682$	1.26595e10	0.00224072
$683$	−7.71959e12	−1.35738	−0.678689	−	0.734425i	$-0.737452\pi$
$683$	−7.71959e12	−1.35738	−0.678689	+	0.734425i	$0.737452\pi$
$684$	0	0
$685$	3.68736e12	0.639893
$686$	2.16354e12	0.372999
$687$	0	0
$688$	8.78952e12	1.49561
$689$	2.37639e12	0.401727
$690$	0	0
$691$	4.52966e12	0.755814	0.377907	−	0.925844i	$-0.376644\pi$
$691$	4.52966e12	0.755814	0.377907	+	0.925844i	$0.376644\pi$
$692$	−5.71597e12	−0.947573
$693$	0	0
$694$	1.26484e13	2.06975
$695$	−6.31392e12	−1.02652
$696$	0	0
$697$	−2.24379e11	−0.0360110
$698$	−9.40872e12	−1.50031
$699$	0	0
$700$	3.42921e12	0.539825
$701$	−3.97479e12	−0.621703	−0.310852	−	0.950458i	$-0.600614\pi$
$701$	−3.97479e12	−0.621703	−0.310852	+	0.950458i	$0.600614\pi$
$702$	0	0
$703$	1.17142e13	1.80889
$704$	9.76590e9	0.00149843
$705$	0	0
$706$	−7.94419e12	−1.20345
$707$	1.46048e13	2.19841
$708$	0	0
$709$	−3.67322e11	−0.0545932	−0.0272966	−	0.999627i	$-0.508690\pi$
$709$	−3.67322e11	−0.0545932	−0.0272966	+	0.999627i	$0.508690\pi$
$710$	−1.59621e13	−2.35737
$711$	0	0
$712$	−4.06736e10	−0.00593133
$713$	−7.95657e12	−1.15298
$714$	0	0
$715$	−3.47484e9	−0.000497230 0
$716$	8.15146e12	1.15912
$717$	0	0
$718$	−3.18451e12	−0.447180
$719$	−6.50487e12	−0.907734	−0.453867	−	0.891070i	$-0.649956\pi$
$719$	−6.50487e12	−0.907734	−0.453867	+	0.891070i	$0.649956\pi$
$720$	0	0
$721$	1.34126e13	1.84844
$722$	1.95446e13	2.67676
$723$	0	0
$724$	−9.46246e12	−1.27991
$725$	5.37878e11	0.0723040
$726$	0	0
$727$	−1.03562e13	−1.37497	−0.687486	−	0.726198i	$-0.741286\pi$
$727$	−1.03562e13	−1.37497	−0.687486	+	0.726198i	$0.741286\pi$
$728$	4.74904e10	0.00626634
$729$	0	0
$730$	−4.66583e12	−0.608102
$731$	6.99411e11	0.0905950
$732$	0	0
$733$	−7.33031e12	−0.937896	−0.468948	−	0.883226i	$-0.655367\pi$
$733$	−7.33031e12	−0.937896	−0.468948	+	0.883226i	$0.655367\pi$
$734$	5.71196e12	0.726362
$735$	0	0
$736$	−1.24801e13	−1.56772
$737$	1.61488e10	0.00201622
$738$	0	0
$739$	−1.10338e13	−1.36090	−0.680448	−	0.732796i	$-0.738215\pi$
$739$	−1.10338e13	−1.36090	−0.680448	+	0.732796i	$0.738215\pi$
$740$	1.00343e13	1.23011
$741$	0	0
$742$	−2.48999e13	−3.01564
$743$	9.14558e12	1.10093	0.550467	−	0.834857i	$-0.314450\pi$
$743$	9.14558e12	1.10093	0.550467	+	0.834857i	$0.314450\pi$
$744$	0	0
$745$	1.27438e13	1.51564
$746$	−4.96361e12	−0.586776
$747$	0	0
$748$	7.94253e8	9.27688e−5 0
$749$	−2.37092e13	−2.75263
$750$	0	0
$751$	6.13184e12	0.703414	0.351707	−	0.936110i	$-0.385601\pi$
$751$	6.13184e12	0.703414	0.351707	+	0.936110i	$0.385601\pi$
$752$	1.35196e13	1.54165
$753$	0	0
$754$	−6.78966e11	−0.0765028
$755$	−1.23244e13	−1.38040
$756$	0	0
$757$	−6.68988e11	−0.0740435	−0.0370218	−	0.999314i	$-0.511787\pi$
$757$	−6.68988e11	−0.0740435	−0.0370218	+	0.999314i	$0.511787\pi$
$758$	−1.60495e13	−1.76583
$759$	0	0
$760$	−2.80452e11	−0.0304928
$761$	6.21021e12	0.671236	0.335618	−	0.941998i	$-0.391055\pi$
$761$	6.21021e12	0.671236	0.335618	+	0.941998i	$0.391055\pi$
$762$	0	0
$763$	−1.35264e12	−0.144485
$764$	8.05754e12	0.855623
$765$	0	0
$766$	−1.80620e13	−1.89555
$767$	−2.77298e12	−0.289313
$768$	0	0
$769$	1.60156e13	1.65149	0.825744	−	0.564046i	$-0.190756\pi$
$769$	1.60156e13	1.65149	0.825744	+	0.564046i	$0.190756\pi$
$770$	3.64095e10	0.00373256
$771$	0	0
$772$	1.06754e12	0.108170
$773$	−5.39801e12	−0.543783	−0.271892	−	0.962328i	$-0.587649\pi$
$773$	−5.39801e12	−0.543783	−0.271892	+	0.962328i	$0.587649\pi$
$774$	0	0
$775$	−3.85073e12	−0.383429
$776$	−8.27293e10	−0.00818996
$777$	0	0
$778$	6.18848e12	0.605586
$779$	−1.02913e13	−1.00128
$780$	0	0
$781$	−2.27472e10	−0.00218775
$782$	−1.00386e12	−0.0959937
$783$	0	0
$784$	−1.26075e13	−1.19181
$785$	−1.38455e13	−1.30135
$786$	0	0
$787$	−4.49004e12	−0.417219	−0.208609	−	0.977999i	$-0.566894\pi$
$787$	−4.49004e12	−0.417219	−0.208609	+	0.977999i	$0.566894\pi$
$788$	−1.10204e12	−0.101819
$789$	0	0
$790$	4.42961e12	0.404616
$791$	−6.25154e12	−0.567797
$792$	0	0
$793$	3.00039e12	0.269431
$794$	7.65671e12	0.683675
$795$	0	0
$796$	−1.43151e12	−0.126382
$797$	−1.09919e13	−0.964965	−0.482482	−	0.875906i	$-0.660265\pi$
$797$	−1.09919e13	−0.964965	−0.482482	+	0.875906i	$0.660265\pi$
$798$	0	0
$799$	1.07580e12	0.0933838
$800$	−6.03997e12	−0.521350
$801$	0	0
$802$	−3.06498e13	−2.61603
$803$	−6.64917e9	−0.000564348 0
$804$	0	0
$805$	−2.28835e13	−1.92062
$806$	4.86080e12	0.405695
$807$	0	0
$808$	−2.76158e11	−0.0227933
$809$	−1.66900e13	−1.36989	−0.684947	−	0.728593i	$-0.740175\pi$
$809$	−1.66900e13	−1.36989	−0.684947	+	0.728593i	$0.740175\pi$
$810$	0	0
$811$	6.11644e12	0.496484	0.248242	−	0.968698i	$-0.420147\pi$
$811$	6.11644e12	0.496484	0.248242	+	0.968698i	$0.420147\pi$
$812$	3.53771e12	0.285575
$813$	0	0
$814$	2.87561e10	0.00229573
$815$	1.24582e12	0.0989112
$816$	0	0
$817$	3.20791e13	2.51897
$818$	1.38201e13	1.07925
$819$	0	0
$820$	−8.81550e12	−0.680902
$821$	−7.83927e12	−0.602187	−0.301094	−	0.953595i	$-0.597352\pi$
$821$	−7.83927e12	−0.602187	−0.301094	+	0.953595i	$0.597352\pi$
$822$	0	0
$823$	−7.11436e12	−0.540551	−0.270276	−	0.962783i	$-0.587115\pi$
$823$	−7.11436e12	−0.540551	−0.270276	+	0.962783i	$0.587115\pi$
$824$	−2.53615e11	−0.0191647
$825$	0	0
$826$	2.90554e13	2.17178
$827$	3.05948e12	0.227443	0.113722	−	0.993513i	$-0.463723\pi$
$827$	3.05948e12	0.227443	0.113722	+	0.993513i	$0.463723\pi$
$828$	0	0
$829$	−2.38849e13	−1.75642	−0.878209	−	0.478277i	$-0.841261\pi$
$829$	−2.38849e13	−1.75642	−0.878209	+	0.478277i	$0.841261\pi$
$830$	−2.30322e13	−1.68455
$831$	0	0
$832$	3.74975e12	0.271298
$833$	−1.00322e12	−0.0721929
$834$	0	0
$835$	−2.84663e12	−0.202648
$836$	3.64291e10	0.00257941
$837$	0	0
$838$	−2.10962e13	−1.47777
$839$	2.10384e13	1.46583	0.732916	−	0.680319i	$-0.238159\pi$
$839$	2.10384e13	1.46583	0.732916	+	0.680319i	$0.238159\pi$
$840$	0	0
$841$	−1.39522e13	−0.961750
$842$	5.87369e12	0.402723
$843$	0	0
$844$	1.39531e12	0.0946522
$845$	−1.33421e12	−0.0900262
$846$	0	0
$847$	−2.21115e13	−1.47619
$848$	2.20455e13	1.46399
$849$	0	0
$850$	−4.85836e11	−0.0319231
$851$	−1.80733e13	−1.18129
$852$	0	0
$853$	1.43288e13	0.926700	0.463350	−	0.886175i	$-0.346647\pi$
$853$	1.43288e13	0.926700	0.463350	+	0.886175i	$0.346647\pi$
$854$	−3.14382e13	−2.02254
$855$	0	0
$856$	4.48309e11	0.0285395
$857$	5.42361e10	0.00343459	0.00171729	−	0.999999i	$-0.499453\pi$
$857$	5.42361e10	0.00343459	0.00171729	+	0.999999i	$0.499453\pi$
$858$	0	0
$859$	2.29372e13	1.43738	0.718689	−	0.695332i	$-0.244743\pi$
$859$	2.29372e13	1.43738	0.718689	+	0.695332i	$0.244743\pi$
$860$	2.74788e13	1.71299
$861$	0	0
$862$	3.00398e13	1.85317
$863$	1.78347e13	1.09451	0.547253	−	0.836967i	$-0.315674\pi$
$863$	1.78347e13	1.09451	0.547253	+	0.836967i	$0.315674\pi$
$864$	0	0
$865$	1.84602e13	1.12115
$866$	4.15502e13	2.51040
$867$	0	0
$868$	−2.53269e13	−1.51441
$869$	6.31253e9	0.000375504 0
$870$	0	0
$871$	6.20055e12	0.365047
$872$	2.55766e10	0.00149802
$873$	0	0
$874$	−4.60429e13	−2.66908
$875$	1.88817e13	1.08894
$876$	0	0
$877$	−8.23771e11	−0.0470228	−0.0235114	−	0.999724i	$-0.507485\pi$
$877$	−8.23771e11	−0.0470228	−0.0235114	+	0.999724i	$0.507485\pi$
$878$	−4.45995e13	−2.53282
$879$	0	0
$880$	−3.22358e10	−0.00181203
$881$	−4.37716e12	−0.244794	−0.122397	−	0.992481i	$-0.539058\pi$
$881$	−4.37716e12	−0.244794	−0.122397	+	0.992481i	$0.539058\pi$
$882$	0	0
$883$	−2.67063e13	−1.47840	−0.739199	−	0.673487i	$-0.764796\pi$
$883$	−2.67063e13	−1.47840	−0.739199	+	0.673487i	$0.764796\pi$
$884$	3.04964e11	0.0167963
$885$	0	0
$886$	3.55804e13	1.93981
$887$	1.61428e13	0.875634	0.437817	−	0.899064i	$-0.355752\pi$
$887$	1.61428e13	0.875634	0.437817	+	0.899064i	$0.355752\pi$
$888$	0	0
$889$	2.25024e13	1.20829
$890$	−1.19732e13	−0.639671
$891$	0	0
$892$	1.75122e12	0.0926189
$893$	4.93426e13	2.59651
$894$	0	0
$895$	−2.63258e13	−1.37144
$896$	8.71719e11	0.0451846
$897$	0	0
$898$	3.58948e13	1.84200
$899$	−3.97257e12	−0.202840
$900$	0	0
$901$	1.75424e12	0.0886801
$902$	−2.52633e10	−0.00127075
$903$	0	0
$904$	1.18208e11	0.00588696
$905$	3.05598e13	1.51437
$906$	0	0
$907$	1.34941e13	0.662080	0.331040	−	0.943617i	$-0.392601\pi$
$907$	1.34941e13	0.662080	0.331040	+	0.943617i	$0.392601\pi$
$908$	3.18937e13	1.55711
$909$	0	0
$910$	1.39799e13	0.675800
$911$	2.94258e13	1.41545	0.707726	−	0.706487i	$-0.249721\pi$
$911$	2.94258e13	1.41545	0.707726	+	0.706487i	$0.249721\pi$
$912$	0	0
$913$	−3.28227e10	−0.00156335
$914$	4.89866e12	0.232177
$915$	0	0
$916$	1.39792e13	0.656076
$917$	−1.69099e13	−0.789732
$918$	0	0
$919$	1.98741e13	0.919108	0.459554	−	0.888150i	$-0.348009\pi$
$919$	1.98741e13	0.919108	0.459554	+	0.888150i	$0.348009\pi$
$920$	4.32698e11	0.0199131
$921$	0	0
$922$	2.56894e13	1.17075
$923$	−8.73409e12	−0.396105
$924$	0	0
$925$	−8.74692e12	−0.392841
$926$	2.99643e12	0.133923
$927$	0	0
$928$	−6.23108e12	−0.275802
$929$	2.44152e13	1.07545	0.537725	−	0.843120i	$-0.319284\pi$
$929$	2.44152e13	1.07545	0.537725	+	0.843120i	$0.319284\pi$
$930$	0	0
$931$	−4.60137e13	−2.00731
$932$	3.98394e13	1.72958
$933$	0	0
$934$	−2.85311e13	−1.22676
$935$	−2.56511e9	−0.000109762 0
$936$	0	0
$937$	4.13081e13	1.75068	0.875341	−	0.483505i	$-0.160637\pi$
$937$	4.13081e13	1.75068	0.875341	+	0.483505i	$0.160637\pi$
$938$	−6.49696e13	−2.74030
$939$	0	0
$940$	4.22665e13	1.76572
$941$	−1.03027e13	−0.428350	−0.214175	−	0.976795i	$-0.568706\pi$
$941$	−1.03027e13	−0.428350	−0.214175	+	0.976795i	$0.568706\pi$
$942$	0	0
$943$	1.58781e13	0.653877
$944$	−2.57247e13	−1.05433
$945$	0	0
$946$	7.87482e10	0.00319691
$947$	−1.32108e13	−0.533771	−0.266886	−	0.963728i	$-0.585995\pi$
$947$	−1.32108e13	−0.533771	−0.266886	+	0.963728i	$0.585995\pi$
$948$	0	0
$949$	−2.55303e12	−0.102178
$950$	−2.22833e13	−0.887613
$951$	0	0
$952$	3.50571e10	0.00138328
$953$	−6.34581e12	−0.249212	−0.124606	−	0.992206i	$-0.539767\pi$
$953$	−6.34581e12	−0.249212	−0.124606	+	0.992206i	$0.539767\pi$
$954$	0	0
$955$	−2.60225e13	−1.01236
$956$	−1.00437e13	−0.388897
$957$	0	0
$958$	−5.89417e13	−2.26088
$959$	−2.11409e13	−0.807122
$960$	0	0
$961$	2.00048e12	0.0756621
$962$	1.10413e13	0.415654
$963$	0	0
$964$	−1.06098e13	−0.395697
$965$	−3.44771e12	−0.127985
$966$	0	0
$967$	−2.33250e13	−0.857832	−0.428916	−	0.903344i	$-0.641104\pi$
$967$	−2.33250e13	−0.857832	−0.428916	+	0.903344i	$0.641104\pi$
$968$	4.18100e11	0.0153053
$969$	0	0
$970$	−2.43533e13	−0.883254
$971$	2.48234e13	0.896139	0.448069	−	0.893999i	$-0.352112\pi$
$971$	2.48234e13	0.896139	0.448069	+	0.893999i	$0.352112\pi$
$972$	0	0
$973$	3.61998e13	1.29479
$974$	−4.49364e13	−1.59986
$975$	0	0
$976$	2.78343e13	0.981875
$977$	3.52755e12	0.123865	0.0619323	−	0.998080i	$-0.480274\pi$
$977$	3.52755e12	0.123865	0.0619323	+	0.998080i	$0.480274\pi$
$978$	0	0
$979$	−1.70628e10	−0.000593646 0
$980$	−3.94150e13	−1.36504
$981$	0	0
$982$	7.60351e13	2.60923
$983$	−2.74635e13	−0.938133	−0.469067	−	0.883163i	$-0.655410\pi$
$983$	−2.74635e13	−0.938133	−0.469067	+	0.883163i	$0.655410\pi$
$984$	0	0
$985$	3.55912e12	0.120470
$986$	−5.01209e11	−0.0168878
$987$	0	0
$988$	1.39874e13	0.467016
$989$	−4.94936e13	−1.64500
$990$	0	0
$991$	−4.47851e13	−1.47503	−0.737517	−	0.675329i	$-0.764002\pi$
$991$	−4.47851e13	−1.47503	−0.737517	+	0.675329i	$0.764002\pi$
$992$	4.46090e13	1.46258
$993$	0	0
$994$	9.15162e13	2.97344
$995$	4.62318e12	0.149533
$996$	0	0
$997$	2.79118e13	0.894663	0.447332	−	0.894368i	$-0.352374\pi$
$997$	2.79118e13	0.894663	0.447332	+	0.894368i	$0.352374\pi$
$998$	−1.36837e13	−0.436632
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.10.a.g.1.7	yes	8
3.2	odd	2	inner	117.10.a.g.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.10.a.g.1.2	✓	8	3.2	odd	2	inner
117.10.a.g.1.7	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+31.9131 q^{2} +506.444 q^{4} -1635.60 q^{5} +9377.46 q^{7} -177.316 q^{8} -52197.1 q^{10} -74.3848 q^{11} -28561.0 q^{13} +299264. q^{14} -264958. q^{16} -21083.6 q^{17} -967016. q^{19} -828340. q^{20} -2373.85 q^{22} +1.49197e6 q^{23} +722067. q^{25} -911469. q^{26} +4.74916e6 q^{28} +744914. q^{29} -5.33293e6 q^{31} -8.36483e6 q^{32} -672841. q^{34} -1.53378e7 q^{35} -1.21137e7 q^{37} -3.08605e7 q^{38} +290017. q^{40} +1.06424e7 q^{41} -3.31733e7 q^{43} -37671.7 q^{44} +4.76134e7 q^{46} -5.10256e7 q^{47} +4.75832e7 q^{49} +2.30434e7 q^{50} -1.44645e7 q^{52} -8.32040e7 q^{53} +121664. q^{55} -1.66277e6 q^{56} +2.37725e7 q^{58} +9.70896e7 q^{59} -1.05052e8 q^{61} -1.70190e8 q^{62} -1.31289e8 q^{64} +4.67144e7 q^{65} -2.17098e8 q^{67} -1.06776e7 q^{68} -4.89476e8 q^{70} +3.05805e8 q^{71} +8.93888e7 q^{73} -3.86586e8 q^{74} -4.89739e8 q^{76} -697540. q^{77} -8.48632e7 q^{79} +4.33366e8 q^{80} +3.39631e8 q^{82} +4.41256e8 q^{83} +3.44843e7 q^{85} -1.05866e9 q^{86} +13189.6 q^{88} +2.29385e8 q^{89} -2.67830e8 q^{91} +7.55600e8 q^{92} -1.62838e9 q^{94} +1.58165e9 q^{95} +4.66565e8 q^{97} +1.51852e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1130 q^{4} + 9756 q^{7} - 38516 q^{10} - 228488 q^{13} - 314526 q^{16} - 884340 q^{19} - 1422584 q^{22} + 1535856 q^{25} - 1630900 q^{28} - 12449540 q^{31} - 6019872 q^{34} + 6397904 q^{37} - 13411388 q^{40}+ \cdots - 3297114688 q^{97}+O(q^{100}) \) 8 * q + 1130 * q^4 + 9756 * q^7 - 38516 * q^10 - 228488 * q^13 - 314526 * q^16 - 884340 * q^19 - 1422584 * q^22 + 1535856 * q^25 - 1630900 * q^28 - 12449540 * q^31 - 6019872 * q^34 + 6397904 * q^37 - 13411388 * q^40 - 53044624 * q^43 + 7474280 * q^46 - 60752704 * q^49 - 32273930 * q^52 - 182669760 * q^55 - 228402352 * q^58 - 298501160 * q^61 - 661011230 * q^64 - 713390956 * q^67 - 1215462584 * q^70 - 953259360 * q^73 - 1790107484 * q^76 - 1270033632 * q^79 - 1406396292 * q^82 - 2048639264 * q^85 - 2044743576 * q^88 - 278641116 * q^91 - 3755823328 * q^94 - 3297114688 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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