LMFDB - Embedded newform 50.26.a.l.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,26,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 26, names="a")

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 26);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 26 $
Character orbit:	$[\chi]$	$=$	50.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:	$197.998389976$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2 x^{5} - 27590779188 x^{4} + 26487255863952 x^{3} + \cdots - 30\!\cdots\!08 $ x^6 - 2x^5 - 27590779188x^4 + 26487255863952x^3 + 180405425727179733780x^2 - 1479960352440847053525960*x - 3018239644098422083113677808
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{4}\cdot 5^{18} $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-122054.$ of defining polynomial
Character	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4096.00 q^{2} +1.08697e6 q^{3} +1.67772e7 q^{4} +4.45223e9 q^{6} -4.89841e10 q^{7} +6.87195e10 q^{8} +3.34213e11 q^{9} +1.14584e13 q^{11} +1.82363e13 q^{12} +1.28077e14 q^{13} -2.00639e14 q^{14} +2.81475e14 q^{16} -3.78822e15 q^{17} +1.36894e15 q^{18} -1.02229e16 q^{19} -5.32442e16 q^{21} +4.69335e16 q^{22} -1.81999e17 q^{23} +7.46959e16 q^{24} +5.24604e17 q^{26} -5.57697e17 q^{27} -8.21816e17 q^{28} +2.19149e18 q^{29} +2.89689e18 q^{31} +1.15292e18 q^{32} +1.24549e19 q^{33} -1.55166e19 q^{34} +5.60717e18 q^{36} -4.56681e19 q^{37} -4.18730e19 q^{38} +1.39216e20 q^{39} -4.22120e18 q^{41} -2.18088e20 q^{42} +1.22773e19 q^{43} +1.92240e20 q^{44} -7.45466e20 q^{46} -4.60132e20 q^{47} +3.05955e20 q^{48} +1.05837e21 q^{49} -4.11768e21 q^{51} +2.14878e21 q^{52} -2.54574e21 q^{53} -2.28433e21 q^{54} -3.36616e21 q^{56} -1.11120e22 q^{57} +8.97634e21 q^{58} -2.20173e21 q^{59} +7.41676e21 q^{61} +1.18656e22 q^{62} -1.63711e22 q^{63} +4.72237e21 q^{64} +5.10153e22 q^{66} +6.01134e22 q^{67} -6.35558e22 q^{68} -1.97827e23 q^{69} +3.59036e22 q^{71} +2.29670e22 q^{72} -1.98356e23 q^{73} -1.87057e23 q^{74} -1.71512e23 q^{76} -5.61278e23 q^{77} +5.70229e23 q^{78} -7.84538e23 q^{79} -8.89375e23 q^{81} -1.72901e22 q^{82} -4.60438e23 q^{83} -8.93289e23 q^{84} +5.02877e22 q^{86} +2.38208e24 q^{87} +7.87413e23 q^{88} +5.94026e23 q^{89} -6.27374e24 q^{91} -3.05343e24 q^{92} +3.14882e24 q^{93} -1.88470e24 q^{94} +1.25319e24 q^{96} +8.71039e24 q^{97} +4.33508e24 q^{98} +3.82954e24 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 24576 q^{2} - 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} - 34007705352 q^{7} + 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} - 13445529337856 q^{12} - 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) $ 6 * q + 24576 * q^2 - 801416 * q^3 + 100663296 * q^4 - 3282599936 * q^6 - 34007705352 * q^7 + 412316860416 * q^8 + 541468782118 * q^9 + 9861544614312 * q^11 - 13445529337856 * q^12 - 30787386783696 * q^13 - 139295561121792 * q^14 + 1688849860263936 * q^16 - 3432836939805312 * q^17 + 2217856131555328 * q^18 + 2007656488916520 * q^19 - 28181710291014728 * q^21 + 40392886740221952 * q^22 - 135711886432707576 * q^23 - 55072888167858176 * q^24 - 126105136266018816 * q^26 - 788054364552235760 * q^27 - 570554618354860032 * q^28 - 2926032031655504820 * q^29 + 79292935777014912 * q^31 + 6917529027641081856 * q^32 + 6844843638395003168 * q^33 - 14060900105442557952 * q^34 + 9084338714850623488 * q^36 - 3345885298190974032 * q^37 + 8223360978602065920 * q^38 + 124726636622561793456 * q^39 + 103229301719876244012 * q^41 - 115432285351996325888 * q^42 - 166390524239317503336 * q^43 + 165449264087949115392 * q^44 - 555875886828370231296 * q^46 - 405906243391034677992 * q^47 - 225578549935547088896 * q^48 + 1716280748139027694542 * q^49 - 1516225893661962013568 * q^51 - 516526638145613070336 * q^52 - 9772760301640799789616 * q^53 - 3227870677205957672960 * q^54 - 2336991716781506691072 * q^56 - 26174725126915157341920 * q^57 - 11985027201660947742720 * q^58 - 12499075130880780168840 * q^59 + 3687285610311118296612 * q^61 + 324783864942653079552 * q^62 - 93007818034972665339656 * q^63 + 28334198897217871282176 * q^64 + 28036479542865932976128 * q^66 - 16759935062266648886712 * q^67 - 57593446831892717371392 * q^68 + 185125325982378999854536 * q^69 + 74568876221253661649712 * q^71 + 37209451376028153806848 * q^72 - 381797761857106829638176 * q^73 - 13704746181390229635072 * q^74 + 33682886568354062008320 * q^76 + 80271541631956375339296 * q^77 + 510880303606013105995776 * q^78 - 400850316873811624799520 * q^79 - 1432110118616113828343714 * q^81 + 422827219844613095473152 * q^82 + 2423159733509985740001144 * q^83 - 472810640801776950837248 * q^84 - 681535587284244493664256 * q^86 + 1231328665433636473947120 * q^87 + 677680185704239576645632 * q^88 + 2982172528060341024616740 * q^89 - 7001711719121019241314768 * q^91 - 2276867632449004467388416 * q^92 + 8806417055581711481018368 * q^93 - 1662591972929678041055232 * q^94 - 923969740536000876118016 * q^96 + 4960469410756873985567808 * q^97 + 7029885944377457436844032 * q^98 - 2053720289282411723573464 * 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 $

$2$	4096.00	0.707107
$2$	4096.00	0.707107
$3$	1.08697e6	1.18087	0.590434	−	0.807086i	$-0.298957\pi$
$3$	1.08697e6	1.18087	0.590434	+	0.807086i	$0.298957\pi$
$4$	1.67772e7	0.500000
$5$	0	0
$5$	0	0
$6$	4.45223e9	0.835000
$7$	−4.89841e10	−1.33761	−0.668805	−	0.743438i	$-0.733194\pi$
$7$	−4.89841e10	−1.33761	−0.668805	+	0.743438i	$0.733194\pi$
$8$	6.87195e10	0.353553
$9$	3.34213e11	0.394450
$10$	0	0
$11$	1.14584e13	1.10082	0.550408	−	0.834896i	$-0.314472\pi$
$11$	1.14584e13	1.10082	0.550408	+	0.834896i	$0.314472\pi$
$12$	1.82363e13	0.590434
$13$	1.28077e14	1.52468	0.762342	−	0.647174i	$-0.224049\pi$
$13$	1.28077e14	1.52468	0.762342	+	0.647174i	$0.224049\pi$
$14$	−2.00639e14	−0.945833
$15$	0	0
$16$	2.81475e14	0.250000
$17$	−3.78822e15	−1.57697	−0.788486	−	0.615053i	$-0.789134\pi$
$17$	−3.78822e15	−1.57697	−0.788486	+	0.615053i	$0.789134\pi$
$18$	1.36894e15	0.278918
$19$	−1.02229e16	−1.05963	−0.529815	−	0.848113i	$-0.677739\pi$
$19$	−1.02229e16	−1.05963	−0.529815	+	0.848113i	$0.677739\pi$
$20$	0	0
$21$	−5.32442e16	−1.57954
$22$	4.69335e16	0.778394
$23$	−1.81999e17	−1.73169	−0.865845	−	0.500312i	$-0.833218\pi$
$23$	−1.81999e17	−1.73169	−0.865845	+	0.500312i	$0.833218\pi$
$24$	7.46959e16	0.417500
$25$	0	0
$26$	5.24604e17	1.07811
$27$	−5.57697e17	−0.715075
$28$	−8.21816e17	−0.668805
$29$	2.19149e18	1.15018	0.575088	−	0.818092i	$-0.304968\pi$
$29$	2.19149e18	1.15018	0.575088	+	0.818092i	$0.304968\pi$
$30$	0	0
$31$	2.89689e18	0.660557	0.330278	−	0.943884i	$-0.392857\pi$
$31$	2.89689e18	0.660557	0.330278	+	0.943884i	$0.392857\pi$
$32$	1.15292e18	0.176777
$33$	1.24549e19	1.29992
$34$	−1.55166e19	−1.11509
$35$	0	0
$36$	5.60717e18	0.197225
$37$	−4.56681e19	−1.14049	−0.570245	−	0.821475i	$-0.693152\pi$
$37$	−4.56681e19	−1.14049	−0.570245	+	0.821475i	$0.693152\pi$
$38$	−4.18730e19	−0.749271
$39$	1.39216e20	1.80045
$40$	0	0
$41$	−4.22120e18	−0.0292172	−0.0146086	−	0.999893i	$-0.504650\pi$
$41$	−4.22120e18	−0.0292172	−0.0146086	+	0.999893i	$0.504650\pi$
$42$	−2.18088e20	−1.11690
$43$	1.22773e19	0.0468540	0.0234270	−	0.999726i	$-0.492542\pi$
$43$	1.22773e19	0.0468540	0.0234270	+	0.999726i	$0.492542\pi$
$44$	1.92240e20	0.550408
$45$	0	0
$46$	−7.45466e20	−1.22449
$47$	−4.60132e20	−0.577642	−0.288821	−	0.957383i	$-0.593263\pi$
$47$	−4.60132e20	−0.577642	−0.288821	+	0.957383i	$0.593263\pi$
$48$	3.05955e20	0.295217
$49$	1.05837e21	0.789199
$50$	0	0
$51$	−4.11768e21	−1.86220
$52$	2.14878e21	0.762342
$53$	−2.54574e21	−0.711814	−0.355907	−	0.934521i	$-0.615828\pi$
$53$	−2.54574e21	−0.711814	−0.355907	+	0.934521i	$0.615828\pi$
$54$	−2.28433e21	−0.505634
$55$	0	0
$56$	−3.36616e21	−0.472916
$57$	−1.11120e22	−1.25128
$58$	8.97634e21	0.813297
$59$	−2.20173e21	−0.161107	−0.0805536	−	0.996750i	$-0.525669\pi$
$59$	−2.20173e21	−0.161107	−0.0805536	+	0.996750i	$0.525669\pi$
$60$	0	0
$61$	7.41676e21	0.357760	0.178880	−	0.983871i	$-0.442753\pi$
$61$	7.41676e21	0.357760	0.178880	+	0.983871i	$0.442753\pi$
$62$	1.18656e22	0.467084
$63$	−1.63711e22	−0.527620
$64$	4.72237e21	0.125000
$65$	0	0
$66$	5.10153e22	0.919181
$67$	6.01134e22	0.897503	0.448752	−	0.893656i	$-0.351869\pi$
$67$	6.01134e22	0.897503	0.448752	+	0.893656i	$0.351869\pi$
$68$	−6.35558e22	−0.788486
$69$	−1.97827e23	−2.04490
$70$	0	0
$71$	3.59036e22	0.259662	0.129831	−	0.991536i	$-0.458557\pi$
$71$	3.59036e22	0.259662	0.129831	+	0.991536i	$0.458557\pi$
$72$	2.29670e22	0.139459
$73$	−1.98356e23	−1.01370	−0.506851	−	0.862034i	$-0.669190\pi$
$73$	−1.98356e23	−1.01370	−0.506851	+	0.862034i	$0.669190\pi$
$74$	−1.87057e23	−0.806448
$75$	0	0
$76$	−1.71512e23	−0.529815
$77$	−5.61278e23	−1.47246
$78$	5.70229e23	1.27311
$79$	−7.84538e23	−1.49374	−0.746870	−	0.664970i	$-0.768444\pi$
$79$	−7.84538e23	−1.49374	−0.746870	+	0.664970i	$0.768444\pi$
$80$	0	0
$81$	−8.89375e23	−1.23886
$82$	−1.72901e22	−0.0206597
$83$	−4.60438e23	−0.472819	−0.236409	−	0.971654i	$-0.575971\pi$
$83$	−4.60438e23	−0.472819	−0.236409	+	0.971654i	$0.575971\pi$
$84$	−8.93289e23	−0.789770
$85$	0	0
$86$	5.02877e22	0.0331308
$87$	2.38208e24	1.35821
$88$	7.87413e23	0.389197
$89$	5.94026e23	0.254936	0.127468	−	0.991843i	$-0.459315\pi$
$89$	5.94026e23	0.254936	0.127468	+	0.991843i	$0.459315\pi$
$90$	0	0
$91$	−6.27374e24	−2.03943
$92$	−3.05343e24	−0.865845
$93$	3.14882e24	0.780031
$94$	−1.88470e24	−0.408455
$95$	0	0
$96$	1.25319e24	0.208750
$97$	8.71039e24	1.27465	0.637325	−	0.770595i	$-0.280041\pi$
$97$	8.71039e24	1.27465	0.637325	+	0.770595i	$0.280041\pi$
$98$	4.33508e24	0.558048
$99$	3.82954e24	0.434217
$100$	0	0
$101$	7.81456e24	0.690061	0.345031	−	0.938591i	$-0.387869\pi$
$101$	7.81456e24	0.690061	0.345031	+	0.938591i	$0.387869\pi$
$102$	−1.68660e25	−1.31677
$103$	−1.05677e25	−0.730321	−0.365161	−	0.930945i	$-0.618986\pi$
$103$	−1.05677e25	−0.730321	−0.365161	+	0.930945i	$0.618986\pi$
$104$	8.80140e24	0.539057
$105$	0	0
$106$	−1.04274e25	−0.503328
$107$	4.46639e24	0.191716	0.0958582	−	0.995395i	$-0.469440\pi$
$107$	4.46639e24	0.191716	0.0958582	+	0.995395i	$0.469440\pi$
$108$	−9.35660e24	−0.357537
$109$	−4.49825e25	−1.53184	−0.765918	−	0.642939i	$-0.777715\pi$
$109$	−4.49825e25	−1.53184	−0.765918	+	0.642939i	$0.777715\pi$
$110$	0	0
$111$	−4.96398e25	−1.34677
$112$	−1.37878e25	−0.334402
$113$	−2.79039e25	−0.605598	−0.302799	−	0.953054i	$-0.597921\pi$
$113$	−2.79039e25	−0.605598	−0.302799	+	0.953054i	$0.597921\pi$
$114$	−4.55146e25	−0.884791
$115$	0	0
$116$	3.67671e25	0.575088
$117$	4.28051e25	0.601412
$118$	−9.01830e24	−0.113920
$119$	1.85562e26	2.10937
$120$	0	0
$121$	2.29472e25	0.211794
$122$	3.03790e25	0.252974
$123$	−4.58832e24	−0.0345016
$124$	4.86017e25	0.330278
$125$	0	0
$126$	−6.70561e25	−0.373084
$127$	−1.91643e26	−0.965930	−0.482965	−	0.875640i	$-0.660440\pi$
$127$	−1.91643e26	−0.965930	−0.482965	+	0.875640i	$0.660440\pi$
$128$	1.93428e25	0.0883883
$129$	1.33450e25	0.0553284
$130$	0	0
$131$	−2.22743e26	−0.761925	−0.380963	−	0.924590i	$-0.624407\pi$
$131$	−2.22743e26	−0.761925	−0.380963	+	0.924590i	$0.624407\pi$
$132$	2.08958e26	0.649959
$133$	5.00759e26	1.41737
$134$	2.46224e26	0.634631
$135$	0	0
$136$	−2.60325e26	−0.557543
$137$	−3.59939e26	−0.703432	−0.351716	−	0.936107i	$-0.614402\pi$
$137$	−3.59939e26	−0.703432	−0.351716	+	0.936107i	$0.614402\pi$
$138$	−8.10299e26	−1.44596
$139$	−5.18148e26	−0.844829	−0.422414	−	0.906403i	$-0.638817\pi$
$139$	−5.18148e26	−0.844829	−0.422414	+	0.906403i	$0.638817\pi$
$140$	0	0
$141$	−5.00149e26	−0.682119
$142$	1.47061e26	0.183609
$143$	1.46756e27	1.67840
$144$	9.40727e25	0.0986126
$145$	0	0
$146$	−8.12467e26	−0.716795
$147$	1.15042e27	0.931940
$148$	−7.66184e26	−0.570245
$149$	−1.91990e27	−1.31356	−0.656780	−	0.754082i	$-0.728082\pi$
$149$	−1.91990e27	−1.31356	−0.656780	+	0.754082i	$0.728082\pi$
$150$	0	0
$151$	−1.20411e27	−0.697356	−0.348678	−	0.937243i	$-0.613369\pi$
$151$	−1.20411e27	−0.697356	−0.348678	+	0.937243i	$0.613369\pi$
$152$	−7.02512e26	−0.374636
$153$	−1.26607e27	−0.622037
$154$	−2.29899e27	−1.04119
$155$	0	0
$156$	2.33566e27	0.900226
$157$	−4.32983e27	−1.54073	−0.770363	−	0.637606i	$-0.779925\pi$
$157$	−4.32983e27	−1.54073	−0.770363	+	0.637606i	$0.779925\pi$
$158$	−3.21347e27	−1.05623
$159$	−2.76715e27	−0.840559
$160$	0	0
$161$	8.91503e27	2.31632
$162$	−3.64288e27	−0.876006
$163$	5.02220e27	1.11828	0.559138	−	0.829075i	$-0.311132\pi$
$163$	5.02220e27	1.11828	0.559138	+	0.829075i	$0.311132\pi$
$164$	−7.08201e25	−0.0146086
$165$	0	0
$166$	−1.88595e27	−0.334333
$167$	1.18728e28	1.95253	0.976265	−	0.216580i	$-0.0694903\pi$
$167$	1.18728e28	1.95253	0.976265	+	0.216580i	$0.0694903\pi$
$168$	−3.65891e27	−0.558452
$169$	9.34736e27	1.32466
$170$	0	0
$171$	−3.41663e27	−0.417971
$172$	2.05979e26	0.0234270
$173$	−4.16222e27	−0.440300	−0.220150	−	0.975466i	$-0.570655\pi$
$173$	−4.16222e27	−0.440300	−0.220150	+	0.975466i	$0.570655\pi$
$174$	9.75700e27	0.960397
$175$	0	0
$176$	3.22524e27	0.275204
$177$	−2.39322e27	−0.190246
$178$	2.43313e27	0.180267
$179$	1.09350e28	0.755363	0.377681	−	0.925936i	$-0.376721\pi$
$179$	1.09350e28	0.755363	0.377681	+	0.925936i	$0.376721\pi$
$180$	0	0
$181$	−8.64913e27	−0.519984	−0.259992	−	0.965611i	$-0.583720\pi$
$181$	−8.64913e27	−0.519984	−0.259992	+	0.965611i	$0.583720\pi$
$182$	−2.56972e28	−1.44210
$183$	8.06179e27	0.422467
$184$	−1.25068e28	−0.612245
$185$	0	0
$186$	1.28976e28	0.551565
$187$	−4.34068e28	−1.73595
$188$	−7.71973e27	−0.288821
$189$	2.73183e28	0.956490
$190$	0	0
$191$	1.76410e28	0.541511	0.270755	−	0.962648i	$-0.412727\pi$
$191$	1.76410e28	0.541511	0.270755	+	0.962648i	$0.412727\pi$
$192$	5.13307e27	0.147609
$193$	−5.06807e27	−0.136577	−0.0682883	−	0.997666i	$-0.521754\pi$
$193$	−5.06807e27	−0.136577	−0.0682883	+	0.997666i	$0.521754\pi$
$194$	3.56778e28	0.901314
$195$	0	0
$196$	1.77565e28	0.394599
$197$	−5.65172e27	−0.117856	−0.0589282	−	0.998262i	$-0.518768\pi$
$197$	−5.65172e27	−0.117856	−0.0589282	+	0.998262i	$0.518768\pi$
$198$	1.56858e28	0.307038
$199$	2.19231e28	0.402938	0.201469	−	0.979495i	$-0.435428\pi$
$199$	2.19231e28	0.402938	0.201469	+	0.979495i	$0.435428\pi$
$200$	0	0
$201$	6.53414e28	1.05983
$202$	3.20085e28	0.487947
$203$	−1.07348e29	−1.53849
$204$	−6.90832e28	−0.931098
$205$	0	0
$206$	−4.32852e28	−0.516415
$207$	−6.08263e28	−0.683066
$208$	3.60505e28	0.381171
$209$	−1.17138e29	−1.16646
$210$	0	0
$211$	9.10994e28	0.805350	0.402675	−	0.915343i	$-0.368080\pi$
$211$	9.10994e28	0.805350	0.402675	+	0.915343i	$0.368080\pi$
$212$	−4.27105e28	−0.355907
$213$	3.90261e28	0.306627
$214$	1.82943e28	0.135564
$215$	0	0
$216$	−3.83247e28	−0.252817
$217$	−1.41901e29	−0.883567
$218$	−1.84248e29	−1.08317
$219$	−2.15607e29	−1.19705
$220$	0	0
$221$	−4.85185e29	−2.40438
$222$	−2.03325e29	−0.952309
$223$	1.06006e29	0.469374	0.234687	−	0.972071i	$-0.424594\pi$
$223$	1.06006e29	0.469374	0.234687	+	0.972071i	$0.424594\pi$
$224$	−5.64748e28	−0.236458
$225$	0	0
$226$	−1.14294e29	−0.428223
$227$	−5.01045e29	−1.77645	−0.888226	−	0.459407i	$-0.848062\pi$
$227$	−5.01045e29	−1.77645	−0.888226	+	0.459407i	$0.848062\pi$
$228$	−1.86428e29	−0.625641
$229$	−3.20108e28	−0.101708	−0.0508538	−	0.998706i	$-0.516194\pi$
$229$	−3.20108e28	−0.101708	−0.0508538	+	0.998706i	$0.516194\pi$
$230$	0	0
$231$	−6.10091e29	−1.73878
$232$	1.50598e29	0.406648
$233$	−6.19627e29	−1.58556	−0.792778	−	0.609510i	$-0.791366\pi$
$233$	−6.19627e29	−1.58556	−0.792778	+	0.609510i	$0.791366\pi$
$234$	1.75330e29	0.425263
$235$	0	0
$236$	−3.69390e28	−0.0805536
$237$	−8.52768e29	−1.76391
$238$	7.60064e29	1.49155
$239$	2.16736e29	0.403605	0.201803	−	0.979426i	$-0.435320\pi$
$239$	2.16736e29	0.403605	0.201803	+	0.979426i	$0.435320\pi$
$240$	0	0
$241$	7.46346e29	1.25236	0.626178	−	0.779681i	$-0.284619\pi$
$241$	7.46346e29	1.25236	0.626178	+	0.779681i	$0.284619\pi$
$242$	9.39918e28	0.149761
$243$	−4.94192e29	−0.747855
$244$	1.24433e29	0.178880
$245$	0	0
$246$	−1.87938e28	−0.0243963
$247$	−1.30932e30	−1.61560
$248$	1.99072e29	0.233542
$249$	−5.00481e29	−0.558337
$250$	0	0
$251$	7.46681e29	0.753726	0.376863	−	0.926269i	$-0.377003\pi$
$251$	7.46681e29	0.753726	0.376863	+	0.926269i	$0.377003\pi$
$252$	−2.74662e29	−0.263810
$253$	−2.08541e30	−1.90627
$254$	−7.84968e29	−0.683016
$255$	0	0
$256$	7.92282e28	0.0625000
$257$	−2.03309e29	−0.152754	−0.0763769	−	0.997079i	$-0.524335\pi$
$257$	−2.03309e29	−0.152754	−0.0763769	+	0.997079i	$0.524335\pi$
$258$	5.46612e28	0.0391231
$259$	2.23701e30	1.52553
$260$	0	0
$261$	7.32424e29	0.453687
$262$	−9.12354e29	−0.538763
$263$	3.18718e30	1.79457	0.897284	−	0.441454i	$-0.145537\pi$
$263$	3.18718e30	1.79457	0.897284	+	0.441454i	$0.145537\pi$
$264$	8.55894e29	0.459590
$265$	0	0
$266$	2.05111e30	1.00223
$267$	6.45688e29	0.301046
$268$	1.00854e30	0.448752
$269$	1.08398e29	0.0460382	0.0230191	−	0.999735i	$-0.492672\pi$
$269$	1.08398e29	0.0460382	0.0230191	+	0.999735i	$0.492672\pi$
$270$	0	0
$271$	8.40525e29	0.325413	0.162706	−	0.986675i	$-0.447978\pi$
$271$	8.40525e29	0.325413	0.162706	+	0.986675i	$0.447978\pi$
$272$	−1.06629e30	−0.394243
$273$	−6.81936e30	−2.40830
$274$	−1.47431e30	−0.497402
$275$	0	0
$276$	−3.31898e30	−1.02245
$277$	−5.28289e30	−1.55551	−0.777757	−	0.628564i	$-0.783643\pi$
$277$	−5.28289e30	−1.55551	−0.777757	+	0.628564i	$0.783643\pi$
$278$	−2.12233e30	−0.597384
$279$	9.68177e29	0.260557
$280$	0	0
$281$	−5.89709e30	−1.45147	−0.725736	−	0.687973i	$-0.758501\pi$
$281$	−5.89709e30	−1.45147	−0.725736	+	0.687973i	$0.758501\pi$
$282$	−2.04861e30	−0.482331
$283$	1.58325e30	0.356631	0.178315	−	0.983973i	$-0.442935\pi$
$283$	1.58325e30	0.356631	0.178315	+	0.983973i	$0.442935\pi$
$284$	6.02362e29	0.129831
$285$	0	0
$286$	6.01111e30	1.18680
$287$	2.06772e29	0.0390812
$288$	3.85322e29	0.0697296
$289$	8.57999e30	1.48684
$290$	0	0
$291$	9.46793e30	1.50519
$292$	−3.32787e30	−0.506851
$293$	4.96361e30	0.724357	0.362179	−	0.932109i	$-0.382033\pi$
$293$	4.96361e30	0.724357	0.362179	+	0.932109i	$0.382033\pi$
$294$	4.71210e30	0.658981
$295$	0	0
$296$	−3.13829e30	−0.403224
$297$	−6.39030e30	−0.787165
$298$	−7.86391e30	−0.928828
$299$	−2.33099e31	−2.64028
$300$	0	0
$301$	−6.01391e29	−0.0626723
$302$	−4.93203e30	−0.493105
$303$	8.49419e30	0.814871
$304$	−2.87749e30	−0.264907
$305$	0	0
$306$	−5.18584e30	−0.439846
$307$	1.29347e31	1.05323	0.526617	−	0.850102i	$-0.323460\pi$
$307$	1.29347e31	1.05323	0.526617	+	0.850102i	$0.323460\pi$
$308$	−9.41668e30	−0.736230
$309$	−1.14867e31	−0.862413
$310$	0	0
$311$	−1.68637e30	−0.116801	−0.0584007	−	0.998293i	$-0.518600\pi$
$311$	−1.68637e30	−0.116801	−0.0584007	+	0.998293i	$0.518600\pi$
$312$	9.56685e30	0.636556
$313$	2.70375e30	0.172847	0.0864236	−	0.996258i	$-0.472456\pi$
$313$	2.70375e30	0.172847	0.0864236	+	0.996258i	$0.472456\pi$
$314$	−1.77350e31	−1.08946
$315$	0	0
$316$	−1.31624e31	−0.746870
$317$	3.29106e31	1.79513	0.897563	−	0.440887i	$-0.145336\pi$
$317$	3.29106e31	1.79513	0.897563	+	0.440887i	$0.145336\pi$
$318$	−1.13342e31	−0.594365
$319$	2.51109e31	1.26613
$320$	0	0
$321$	4.85483e30	0.226392
$322$	3.65160e31	1.63789
$323$	3.87266e31	1.67100
$324$	−1.49212e31	−0.619430
$325$	0	0
$326$	2.05709e31	0.790740
$327$	−4.88946e31	−1.80890
$328$	−2.90079e29	−0.0103298
$329$	2.25391e31	0.772659
$330$	0	0
$331$	−3.10367e31	−0.986337	−0.493168	−	0.869934i	$-0.664161\pi$
$331$	−3.10367e31	−0.986337	−0.493168	+	0.869934i	$0.664161\pi$
$332$	−7.72486e30	−0.236409
$333$	−1.52629e31	−0.449867
$334$	4.86310e31	1.38065
$335$	0	0
$336$	−1.49869e31	−0.394885
$337$	3.19922e31	0.812214	0.406107	−	0.913825i	$-0.366886\pi$
$337$	3.19922e31	0.812214	0.406107	+	0.913825i	$0.366886\pi$
$338$	3.82868e31	0.936678
$339$	−3.03307e31	−0.715132
$340$	0	0
$341$	3.31936e31	0.727151
$342$	−1.39945e31	−0.295550
$343$	1.38477e31	0.281970
$344$	8.43688e29	0.0165654
$345$	0	0
$346$	−1.70484e31	−0.311339
$347$	4.34215e31	0.764867	0.382434	−	0.923983i	$-0.375086\pi$
$347$	4.34215e31	0.764867	0.382434	+	0.923983i	$0.375086\pi$
$348$	3.99647e31	0.679103
$349$	9.72186e31	1.59379	0.796896	−	0.604117i	$-0.206474\pi$
$349$	9.72186e31	1.59379	0.796896	+	0.604117i	$0.206474\pi$
$350$	0	0
$351$	−7.14283e31	−1.09026
$352$	1.32106e31	0.194598
$353$	−1.03434e32	−1.47055	−0.735277	−	0.677767i	$-0.762948\pi$
$353$	−1.03434e32	−1.47055	−0.735277	+	0.677767i	$0.762948\pi$
$354$	−9.80262e30	−0.134524
$355$	0	0
$356$	9.96611e30	0.127468
$357$	2.01701e32	2.49089
$358$	4.47897e31	0.534122
$359$	4.33845e31	0.499636	0.249818	−	0.968293i	$-0.419629\pi$
$359$	4.33845e31	0.499636	0.249818	+	0.968293i	$0.419629\pi$
$360$	0	0
$361$	1.14311e31	0.122814
$362$	−3.54268e31	−0.367684
$363$	2.49429e31	0.250100
$364$	−1.05256e32	−1.01972
$365$	0	0
$366$	3.30211e31	0.298729
$367$	−4.71466e31	−0.412216	−0.206108	−	0.978529i	$-0.566080\pi$
$367$	−4.71466e31	−0.412216	−0.206108	+	0.978529i	$0.566080\pi$
$368$	−5.12281e31	−0.432922
$369$	−1.41078e30	−0.0115247
$370$	0	0
$371$	1.24701e32	0.952129
$372$	5.28285e31	0.390015
$373$	9.92589e31	0.708613	0.354306	−	0.935129i	$-0.384717\pi$
$373$	9.92589e31	0.708613	0.354306	+	0.935129i	$0.384717\pi$
$374$	−1.77794e32	−1.22750
$375$	0	0
$376$	−3.16200e31	−0.204227
$377$	2.80680e32	1.75365
$378$	1.11896e32	0.676341
$379$	−6.28833e31	−0.367743	−0.183872	−	0.982950i	$-0.558863\pi$
$379$	−6.28833e31	−0.367743	−0.183872	+	0.982950i	$0.558863\pi$
$380$	0	0
$381$	−2.08310e32	−1.14064
$382$	7.22577e31	0.382906
$383$	−4.95371e31	−0.254065	−0.127033	−	0.991899i	$-0.540545\pi$
$383$	−4.95371e31	−0.254065	−0.127033	+	0.991899i	$0.540545\pi$
$384$	2.10250e31	0.104375
$385$	0	0
$386$	−2.07588e31	−0.0965742
$387$	4.10323e30	0.0184816
$388$	1.46136e32	0.637325
$389$	−4.10071e32	−1.73177	−0.865883	−	0.500246i	$-0.833243\pi$
$389$	−4.10071e32	−1.73177	−0.865883	+	0.500246i	$0.833243\pi$
$390$	0	0
$391$	6.89451e32	2.73082
$392$	7.27306e31	0.279024
$393$	−2.42114e32	−0.899734
$394$	−2.31495e31	−0.0833370
$395$	0	0
$396$	6.42490e31	0.217108
$397$	−1.81115e32	−0.593026	−0.296513	−	0.955029i	$-0.595824\pi$
$397$	−1.81115e32	−0.593026	−0.296513	+	0.955029i	$0.595824\pi$
$398$	8.97970e31	0.284920
$399$	5.44310e32	1.67373
$400$	0	0
$401$	1.30258e32	0.376271	0.188136	−	0.982143i	$-0.439756\pi$
$401$	1.30258e32	0.376271	0.188136	+	0.982143i	$0.439756\pi$
$402$	2.67638e32	0.749415
$403$	3.71025e32	1.00714
$404$	1.31107e32	0.345031
$405$	0	0
$406$	−4.39697e32	−1.08787
$407$	−5.23282e32	−1.25547
$408$	−2.82965e32	−0.658385
$409$	8.59252e32	1.93901	0.969503	−	0.245081i	$-0.0788144\pi$
$409$	8.59252e32	1.93901	0.969503	+	0.245081i	$0.0788144\pi$
$410$	0	0
$411$	−3.91243e32	−0.830661
$412$	−1.77296e32	−0.365161
$413$	1.07850e32	0.215498
$414$	−2.49145e32	−0.483000
$415$	0	0
$416$	1.47663e32	0.269529
$417$	−5.63211e32	−0.997631
$418$	−4.79796e32	−0.824809
$419$	−4.67029e32	−0.779235	−0.389618	−	0.920977i	$-0.627393\pi$
$419$	−4.67029e32	−0.779235	−0.389618	+	0.920977i	$0.627393\pi$
$420$	0	0
$421$	−7.42072e32	−1.16659	−0.583297	−	0.812259i	$-0.698238\pi$
$421$	−7.42072e32	−1.16659	−0.583297	+	0.812259i	$0.698238\pi$
$422$	3.73143e32	0.569468
$423$	−1.53782e32	−0.227851
$424$	−1.74942e32	−0.251664
$425$	0	0
$426$	1.59851e32	0.216818
$427$	−3.63303e32	−0.478543
$428$	7.49336e31	0.0958582
$429$	1.59519e33	1.98196
$430$	0	0
$431$	1.06420e33	1.24755	0.623774	−	0.781605i	$-0.285599\pi$
$431$	1.06420e33	1.24755	0.623774	+	0.781605i	$0.285599\pi$
$432$	−1.56978e32	−0.178769
$433$	9.20340e32	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$433$	9.20340e32	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$434$	−5.81227e32	−0.624776
$435$	0	0
$436$	−7.54681e32	−0.765918
$437$	1.86055e33	1.83495
$438$	−8.83127e32	−0.846441
$439$	−3.84917e32	−0.358559	−0.179279	−	0.983798i	$-0.557377\pi$
$439$	−3.84917e32	−0.358559	−0.179279	+	0.983798i	$0.557377\pi$
$440$	0	0
$441$	3.53721e32	0.311300
$442$	−1.98732e33	−1.70016
$443$	−8.53492e32	−0.709827	−0.354914	−	0.934899i	$-0.615490\pi$
$443$	−8.53492e32	−0.709827	−0.354914	+	0.934899i	$0.615490\pi$
$444$	−8.32818e32	−0.673384
$445$	0	0
$446$	4.34200e32	0.331897
$447$	−2.08687e33	−1.55114
$448$	−2.31321e32	−0.167201
$449$	3.42652e32	0.240865	0.120432	−	0.992722i	$-0.461572\pi$
$449$	3.42652e32	0.240865	0.120432	+	0.992722i	$0.461572\pi$
$450$	0	0
$451$	−4.83681e31	−0.0321627
$452$	−4.68150e32	−0.302799
$453$	−1.30883e33	−0.823485
$454$	−2.05228e33	−1.25614
$455$	0	0
$456$	−7.63609e32	−0.442395
$457$	4.52087e32	0.254841	0.127421	−	0.991849i	$-0.459330\pi$
$457$	4.52087e32	0.254841	0.127421	+	0.991849i	$0.459330\pi$
$458$	−1.31116e32	−0.0719181
$459$	2.11268e33	1.12765
$460$	0	0
$461$	2.47159e33	1.24944	0.624718	−	0.780850i	$-0.285214\pi$
$461$	2.47159e33	1.24944	0.624718	+	0.780850i	$0.285214\pi$
$462$	−2.49893e33	−1.22950
$463$	−1.72628e33	−0.826703	−0.413351	−	0.910572i	$-0.635642\pi$
$463$	−1.72628e33	−0.826703	−0.413351	+	0.910572i	$0.635642\pi$
$464$	6.16849e32	0.287544
$465$	0	0
$466$	−2.53799e33	−1.12116
$467$	−1.88292e33	−0.809787	−0.404893	−	0.914364i	$-0.632691\pi$
$467$	−1.88292e33	−0.809787	−0.404893	+	0.914364i	$0.632691\pi$
$468$	7.18150e32	0.300706
$469$	−2.94460e33	−1.20051
$470$	0	0
$471$	−4.70639e33	−1.81939
$472$	−1.51302e32	−0.0569600
$473$	1.40678e32	0.0515775
$474$	−3.49294e33	−1.24727
$475$	0	0
$476$	3.11322e33	1.05469
$477$	−8.50822e32	−0.280775
$478$	8.87749e32	0.285392
$479$	1.07303e33	0.336061	0.168030	−	0.985782i	$-0.446259\pi$
$479$	1.07303e33	0.336061	0.168030	+	0.985782i	$0.446259\pi$
$480$	0	0
$481$	−5.84904e33	−1.73889
$482$	3.05704e33	0.885549
$483$	9.69036e33	2.73527
$484$	3.84991e32	0.105897
$485$	0	0
$486$	−2.02421e33	−0.528814
$487$	−6.26292e33	−1.59465	−0.797324	−	0.603552i	$-0.793752\pi$
$487$	−6.26292e33	−1.59465	−0.797324	+	0.603552i	$0.793752\pi$
$488$	5.09676e32	0.126487
$489$	5.45897e33	1.32054
$490$	0	0
$491$	3.10431e33	0.713586	0.356793	−	0.934183i	$-0.383870\pi$
$491$	3.10431e33	0.713586	0.356793	+	0.934183i	$0.383870\pi$
$492$	−7.69792e31	−0.0172508
$493$	−8.30184e33	−1.81379
$494$	−5.36298e33	−1.14240
$495$	0	0
$496$	8.15401e32	0.165139
$497$	−1.75870e33	−0.347326
$498$	−2.04997e33	−0.394804
$499$	7.12443e33	1.33811	0.669057	−	0.743211i	$-0.266698\pi$
$499$	7.12443e33	1.33811	0.669057	+	0.743211i	$0.266698\pi$
$500$	0	0
$501$	1.29054e34	2.30568
$502$	3.05840e33	0.532965
$503$	−1.03191e34	−1.75406	−0.877028	−	0.480439i	$-0.840477\pi$
$503$	−1.03191e34	−1.75406	−0.877028	+	0.480439i	$0.840477\pi$
$504$	−1.12501e33	−0.186542
$505$	0	0
$506$	−8.54183e33	−1.34794
$507$	1.01603e34	1.56425
$508$	−3.21523e33	−0.482965
$509$	−4.04421e32	−0.0592736	−0.0296368	−	0.999561i	$-0.509435\pi$
$509$	−4.04421e32	−0.0592736	−0.0296368	+	0.999561i	$0.509435\pi$
$510$	0	0
$511$	9.71629e33	1.35594
$512$	3.24519e32	0.0441942
$513$	5.70128e33	0.757714
$514$	−8.32753e32	−0.108013
$515$	0	0
$516$	2.23892e32	0.0276642
$517$	−5.27236e33	−0.635877
$518$	9.16279e33	1.07871
$519$	−4.52420e33	−0.519936
$520$	0	0
$521$	9.14369e33	1.00150	0.500749	−	0.865593i	$-0.333058\pi$
$521$	9.14369e33	1.00150	0.500749	+	0.865593i	$0.333058\pi$
$522$	3.00001e33	0.320805
$523$	1.50595e33	0.157231	0.0786156	−	0.996905i	$-0.474950\pi$
$523$	1.50595e33	0.157231	0.0786156	+	0.996905i	$0.474950\pi$
$524$	−3.73700e33	−0.380963
$525$	0	0
$526$	1.30547e34	1.26895
$527$	−1.09740e34	−1.04168
$528$	3.50574e33	0.324979
$529$	2.20777e34	1.99875
$530$	0	0
$531$	−7.35849e32	−0.0635487
$532$	8.40134e33	0.708685
$533$	−5.40640e32	−0.0445470
$534$	2.64474e33	0.212871
$535$	0	0
$536$	4.13096e33	0.317315
$537$	1.18860e34	0.891984
$538$	4.43999e32	0.0325539
$539$	1.21272e34	0.868762
$540$	0	0
$541$	7.92779e32	0.0542234	0.0271117	−	0.999632i	$-0.491369\pi$
$541$	7.92779e32	0.0542234	0.0271117	+	0.999632i	$0.491369\pi$
$542$	3.44279e33	0.230102
$543$	−9.40134e33	−0.614033
$544$	−4.36752e33	−0.278772
$545$	0	0
$546$	−2.79321e34	−1.70293
$547$	2.56023e34	1.52559	0.762794	−	0.646641i	$-0.223827\pi$
$547$	2.56023e34	1.52559	0.762794	+	0.646641i	$0.223827\pi$
$548$	−6.03878e33	−0.351716
$549$	2.47878e33	0.141118
$550$	0	0
$551$	−2.24034e34	−1.21876
$552$	−1.35946e34	−0.722981
$553$	3.84298e34	1.99804
$554$	−2.16387e34	−1.09992
$555$	0	0
$556$	−8.69308e33	−0.422414
$557$	2.77017e34	1.31618	0.658091	−	0.752938i	$-0.271364\pi$
$557$	2.77017e34	1.31618	0.658091	+	0.752938i	$0.271364\pi$
$558$	3.96565e33	0.184241
$559$	1.57244e33	0.0714375
$560$	0	0
$561$	−4.71819e34	−2.04993
$562$	−2.41545e34	−1.02635
$563$	−1.80222e33	−0.0748951	−0.0374476	−	0.999299i	$-0.511923\pi$
$563$	−1.80222e33	−0.0748951	−0.0374476	+	0.999299i	$0.511923\pi$
$564$	−8.39110e33	−0.341060
$565$	0	0
$566$	6.48499e33	0.252176
$567$	4.35652e34	1.65711
$568$	2.46728e33	0.0918044
$569$	−2.00804e34	−0.730920	−0.365460	−	0.930827i	$-0.619088\pi$
$569$	−2.00804e34	−0.730920	−0.365460	+	0.930827i	$0.619088\pi$
$570$	0	0
$571$	3.27237e34	1.14002	0.570008	−	0.821639i	$-0.306940\pi$
$571$	3.27237e34	1.14002	0.570008	+	0.821639i	$0.306940\pi$
$572$	2.46215e34	0.839198
$573$	1.91753e34	0.639453
$574$	8.46937e32	0.0276346
$575$	0	0
$576$	1.57828e33	0.0493063
$577$	−1.86866e34	−0.571257	−0.285629	−	0.958340i	$-0.592202\pi$
$577$	−1.86866e34	−0.571257	−0.285629	+	0.958340i	$0.592202\pi$
$578$	3.51436e34	1.05135
$579$	−5.50883e33	−0.161279
$580$	0	0
$581$	2.25541e34	0.632447
$582$	3.87806e34	1.06433
$583$	−2.91701e34	−0.783575
$584$	−1.36309e34	−0.358398
$585$	0	0
$586$	2.03310e34	0.512198
$587$	2.44688e33	0.0603443	0.0301721	−	0.999545i	$-0.490394\pi$
$587$	2.44688e33	0.0603443	0.0301721	+	0.999545i	$0.490394\pi$
$588$	1.93008e34	0.465970
$589$	−2.96146e34	−0.699945
$590$	0	0
$591$	−6.14325e33	−0.139173
$592$	−1.28544e34	−0.285122
$593$	−3.48844e34	−0.757613	−0.378806	−	0.925476i	$-0.623665\pi$
$593$	−3.48844e34	−0.757613	−0.378806	+	0.925476i	$0.623665\pi$
$594$	−2.61747e34	−0.556610
$595$	0	0
$596$	−3.22106e34	−0.656780
$597$	2.38297e34	0.475817
$598$	−9.54772e34	−1.86696
$599$	6.48903e34	1.24264	0.621319	−	0.783558i	$-0.286597\pi$
$599$	6.48903e34	1.24264	0.621319	+	0.783558i	$0.286597\pi$
$600$	0	0
$601$	−2.18034e34	−0.400492	−0.200246	−	0.979746i	$-0.564174\pi$
$601$	−2.18034e34	−0.400492	−0.200246	+	0.979746i	$0.564174\pi$
$602$	−2.46330e33	−0.0443160
$603$	2.00907e34	0.354020
$604$	−2.02016e34	−0.348678
$605$	0	0
$606$	3.47922e34	0.576201
$607$	−7.37348e34	−1.19623	−0.598114	−	0.801411i	$-0.704083\pi$
$607$	−7.37348e34	−1.19623	−0.598114	+	0.801411i	$0.704083\pi$
$608$	−1.17862e34	−0.187318
$609$	−1.16684e35	−1.81675
$610$	0	0
$611$	−5.89324e34	−0.880722
$612$	−2.12412e34	−0.311018
$613$	1.05162e35	1.50870	0.754349	−	0.656473i	$-0.227953\pi$
$613$	1.05162e35	1.50870	0.754349	+	0.656473i	$0.227953\pi$
$614$	5.29804e34	0.744750
$615$	0	0
$616$	−3.85707e34	−0.520593
$617$	8.12769e34	1.07499	0.537493	−	0.843268i	$-0.319372\pi$
$617$	8.12769e34	1.07499	0.537493	+	0.843268i	$0.319372\pi$
$618$	−4.70496e34	−0.609818
$619$	4.81305e34	0.611346	0.305673	−	0.952137i	$-0.401119\pi$
$619$	4.81305e34	0.611346	0.305673	+	0.952137i	$0.401119\pi$
$620$	0	0
$621$	1.01500e35	1.23829
$622$	−6.90737e33	−0.0825910
$623$	−2.90978e34	−0.341004
$624$	3.91858e34	0.450113
$625$	0	0
$626$	1.10746e34	0.122221
$627$	−1.27325e35	−1.37743
$628$	−7.26425e34	−0.770363
$629$	1.73001e35	1.79852
$630$	0	0
$631$	6.11675e34	0.611158	0.305579	−	0.952167i	$-0.401150\pi$
$631$	6.11675e34	0.611158	0.305579	+	0.952167i	$0.401150\pi$
$632$	−5.39130e34	−0.528117
$633$	9.90222e34	0.951012
$634$	1.34802e35	1.26935
$635$	0	0
$636$	−4.64250e34	−0.420279
$637$	1.35553e35	1.20328
$638$	1.02854e35	0.895289
$639$	1.19995e34	0.102424
$640$	0	0
$641$	9.15866e34	0.751807	0.375904	−	0.926659i	$-0.377332\pi$
$641$	9.15866e34	0.751807	0.375904	+	0.926659i	$0.377332\pi$
$642$	1.98854e34	0.160083
$643$	−5.09730e34	−0.402442	−0.201221	−	0.979546i	$-0.564491\pi$
$643$	−5.09730e34	−0.402442	−0.201221	+	0.979546i	$0.564491\pi$
$644$	1.49569e35	1.15816
$645$	0	0
$646$	1.58624e35	1.18158
$647$	−2.28658e35	−1.67064	−0.835320	−	0.549765i	$-0.814717\pi$
$647$	−2.28658e35	−1.67064	−0.835320	+	0.549765i	$0.814717\pi$
$648$	−6.11174e34	−0.438003
$649$	−2.52283e34	−0.177349
$650$	0	0
$651$	−1.54242e35	−1.04338
$652$	8.42585e34	0.559138
$653$	1.93736e35	1.26124	0.630618	−	0.776093i	$-0.282801\pi$
$653$	1.93736e35	1.26124	0.630618	+	0.776093i	$0.282801\pi$
$654$	−2.00272e35	−1.27908
$655$	0	0
$656$	−1.18816e33	−0.00730429
$657$	−6.62933e34	−0.399855
$658$	9.23202e34	0.546353
$659$	−2.14078e35	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$659$	−2.14078e35	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$660$	0	0
$661$	1.52863e35	0.854641	0.427320	−	0.904100i	$-0.359458\pi$
$661$	1.52863e35	0.854641	0.427320	+	0.904100i	$0.359458\pi$
$662$	−1.27126e35	−0.697445
$663$	−5.27381e35	−2.83926
$664$	−3.16410e34	−0.167167
$665$	0	0
$666$	−6.25168e34	−0.318104
$667$	−3.98848e35	−1.99175
$668$	1.99193e35	0.976265
$669$	1.15225e35	0.554268
$670$	0	0
$671$	8.49840e34	0.393827
$672$	−6.13863e34	−0.279226
$673$	−2.04386e35	−0.912564	−0.456282	−	0.889835i	$-0.650819\pi$
$673$	−2.04386e35	−0.912564	−0.456282	+	0.889835i	$0.650819\pi$
$674$	1.31040e35	0.574322
$675$	0	0
$676$	1.56823e35	0.662331
$677$	4.21908e34	0.174928	0.0874640	−	0.996168i	$-0.472124\pi$
$677$	4.21908e34	0.174928	0.0874640	+	0.996168i	$0.472124\pi$
$678$	−1.24235e35	−0.505675
$679$	−4.26671e35	−1.70498
$680$	0	0
$681$	−5.44620e35	−2.09776
$682$	1.35961e35	0.514173
$683$	−2.99380e35	−1.11164	−0.555820	−	0.831303i	$-0.687595\pi$
$683$	−2.99380e35	−1.11164	−0.555820	+	0.831303i	$0.687595\pi$
$684$	−5.73215e34	−0.208986
$685$	0	0
$686$	5.67203e34	0.199383
$687$	−3.47948e34	−0.120103
$688$	3.45575e33	0.0117135
$689$	−3.26052e35	−1.08529
$690$	0	0
$691$	8.03613e34	0.257971	0.128986	−	0.991646i	$-0.458828\pi$
$691$	8.03613e34	0.257971	0.128986	+	0.991646i	$0.458828\pi$
$692$	−6.98304e34	−0.220150
$693$	−1.87586e35	−0.580812
$694$	1.77854e35	0.540843
$695$	0	0
$696$	1.63695e35	0.480198
$697$	1.59909e34	0.0460746
$698$	3.98207e35	1.12698
$699$	−6.73516e35	−1.87233
$700$	0	0
$701$	−4.53710e35	−1.21704	−0.608518	−	0.793540i	$-0.708235\pi$
$701$	−4.53710e35	−1.21704	−0.608518	+	0.793540i	$0.708235\pi$
$702$	−2.92570e35	−0.770932
$703$	4.66860e35	1.20850
$704$	5.41106e34	0.137602
$705$	0	0
$706$	−4.23667e35	−1.03984
$707$	−3.82789e35	−0.923032
$708$	−4.01515e34	−0.0951232
$709$	−1.72814e35	−0.402255	−0.201128	−	0.979565i	$-0.564461\pi$
$709$	−1.72814e35	−0.402255	−0.201128	+	0.979565i	$0.564461\pi$
$710$	0	0
$711$	−2.62203e35	−0.589206
$712$	4.08212e34	0.0901334
$713$	−5.27229e35	−1.14388
$714$	8.26166e35	1.76133
$715$	0	0
$716$	1.83459e35	0.377681
$717$	2.35585e35	0.476604
$718$	1.77703e35	0.353296
$719$	9.50282e35	1.85670	0.928349	−	0.371709i	$-0.121228\pi$
$719$	9.50282e35	1.85670	0.928349	+	0.371709i	$0.121228\pi$
$720$	0	0
$721$	5.17647e35	0.976884
$722$	4.68219e34	0.0868429
$723$	8.11256e35	1.47887
$724$	−1.45108e35	−0.259992
$725$	0	0
$726$	1.02166e35	0.176848
$727$	1.71833e35	0.292367	0.146183	−	0.989258i	$-0.453301\pi$
$727$	1.71833e35	0.292367	0.146183	+	0.989258i	$0.453301\pi$
$728$	−4.31128e35	−0.721048
$729$	2.16385e35	0.355741
$730$	0	0
$731$	−4.65090e34	−0.0738873
$732$	1.35254e35	0.211233
$733$	−5.57098e35	−0.855327	−0.427664	−	0.903938i	$-0.640663\pi$
$733$	−5.57098e35	−0.855327	−0.427664	+	0.903938i	$0.640663\pi$
$734$	−1.93113e35	−0.291481
$735$	0	0
$736$	−2.09830e35	−0.306122
$737$	6.88801e35	0.987985
$738$	−5.77856e33	−0.00814921
$739$	6.81418e35	0.944839	0.472420	−	0.881374i	$-0.343381\pi$
$739$	6.81418e35	0.944839	0.472420	+	0.881374i	$0.343381\pi$
$740$	0	0
$741$	−1.42319e36	−1.90781
$742$	5.10775e35	0.673257
$743$	7.94423e35	1.02965	0.514827	−	0.857294i	$-0.327856\pi$
$743$	7.94423e35	1.02965	0.514827	+	0.857294i	$0.327856\pi$
$744$	2.16386e35	0.275782
$745$	0	0
$746$	4.06564e35	0.501065
$747$	−1.53884e35	−0.186503
$748$	−7.28246e35	−0.867977
$749$	−2.18782e35	−0.256442
$750$	0	0
$751$	9.09984e35	1.03166	0.515828	−	0.856692i	$-0.327484\pi$
$751$	9.09984e35	1.03166	0.515828	+	0.856692i	$0.327484\pi$
$752$	−1.29516e35	−0.144411
$753$	8.11619e35	0.890051
$754$	1.14966e36	1.24002
$755$	0	0
$756$	4.58325e35	0.478245
$757$	9.30502e35	0.955034	0.477517	−	0.878623i	$-0.341537\pi$
$757$	9.30502e35	0.955034	0.477517	+	0.878623i	$0.341537\pi$
$758$	−2.57570e35	−0.260034
$759$	−2.26677e36	−2.25105
$760$	0	0
$761$	1.13441e36	1.09009	0.545045	−	0.838407i	$-0.316513\pi$
$761$	1.13441e36	1.09009	0.545045	+	0.838407i	$0.316513\pi$
$762$	−8.53236e35	−0.806552
$763$	2.20342e36	2.04900
$764$	2.95967e35	0.270755
$765$	0	0
$766$	−2.02904e35	−0.179651
$767$	−2.81992e35	−0.245637
$768$	8.61186e34	0.0738043
$769$	−2.29001e36	−1.93090	−0.965448	−	0.260597i	$-0.916081\pi$
$769$	−2.29001e36	−1.93090	−0.965448	+	0.260597i	$0.916081\pi$
$770$	0	0
$771$	−2.20990e35	−0.180382
$772$	−8.50281e34	−0.0682883
$773$	−3.21550e35	−0.254100	−0.127050	−	0.991896i	$-0.540551\pi$
$773$	−3.21550e35	−0.254100	−0.127050	+	0.991896i	$0.540551\pi$
$774$	1.68068e34	0.0130684
$775$	0	0
$776$	5.98574e35	0.450657
$777$	2.43156e36	1.80145
$778$	−1.67965e36	−1.22454
$779$	4.31529e34	0.0309594
$780$	0	0
$781$	4.11397e35	0.285840
$782$	2.82399e36	1.93098
$783$	−1.22219e36	−0.822461
$784$	2.97905e35	0.197300
$785$	0	0
$786$	−9.91701e35	−0.636208
$787$	−3.25977e35	−0.205827	−0.102914	−	0.994690i	$-0.532817\pi$
$787$	−3.25977e35	−0.205827	−0.102914	+	0.994690i	$0.532817\pi$
$788$	−9.48202e34	−0.0589282
$789$	3.46437e36	2.11915
$790$	0	0
$791$	1.36685e36	0.810054
$792$	2.63164e35	0.153519
$793$	9.49918e35	0.545470
$794$	−7.41849e35	−0.419333
$795$	0	0
$796$	3.67809e35	0.201469
$797$	8.54851e35	0.460958	0.230479	−	0.973077i	$-0.425971\pi$
$797$	8.54851e35	0.460958	0.230479	+	0.973077i	$0.425971\pi$
$798$	2.22949e36	1.18350
$799$	1.74308e36	0.910925
$800$	0	0
$801$	1.98532e35	0.100559
$802$	5.33538e35	0.266064
$803$	−2.27284e36	−1.11590
$804$	1.09625e36	0.529917
$805$	0	0
$806$	1.51972e36	0.712156
$807$	1.17825e35	0.0543651
$808$	5.37013e35	0.243973
$809$	−3.81958e36	−1.70867	−0.854336	−	0.519721i	$-0.826036\pi$
$809$	−3.81958e36	−1.70867	−0.854336	+	0.519721i	$0.826036\pi$
$810$	0	0
$811$	−2.53614e36	−1.10005	−0.550025	−	0.835148i	$-0.685382\pi$
$811$	−2.53614e36	−1.10005	−0.550025	+	0.835148i	$0.685382\pi$
$812$	−1.80100e36	−0.769243
$813$	9.13625e35	0.384270
$814$	−2.14336e36	−0.887750
$815$	0	0
$816$	−1.15902e36	−0.465549
$817$	−1.25509e35	−0.0496478
$818$	3.51949e36	1.37108
$819$	−2.09677e36	−0.804455
$820$	0	0
$821$	−6.03536e35	−0.224602	−0.112301	−	0.993674i	$-0.535822\pi$
$821$	−6.03536e35	−0.224602	−0.112301	+	0.993674i	$0.535822\pi$
$822$	−1.60253e36	−0.587366
$823$	3.78844e36	1.36761	0.683804	−	0.729665i	$-0.260324\pi$
$823$	3.78844e36	1.36761	0.683804	+	0.729665i	$0.260324\pi$
$824$	−7.26205e35	−0.258208
$825$	0	0
$826$	4.41753e35	0.152380
$827$	3.11766e36	1.05928	0.529639	−	0.848223i	$-0.322328\pi$
$827$	3.11766e36	1.05928	0.529639	+	0.848223i	$0.322328\pi$
$828$	−1.02050e36	−0.341533
$829$	5.42365e36	1.78797	0.893984	−	0.448099i	$-0.147899\pi$
$829$	5.42365e36	1.78797	0.893984	+	0.448099i	$0.147899\pi$
$830$	0	0
$831$	−5.74233e36	−1.83686
$832$	6.04827e35	0.190586
$833$	−4.00934e36	−1.24454
$834$	−2.30691e36	−0.705432
$835$	0	0
$836$	−1.96525e36	−0.583228
$837$	−1.61558e36	−0.472347
$838$	−1.91295e36	−0.551003
$839$	−3.97800e35	−0.112886	−0.0564431	−	0.998406i	$-0.517976\pi$
$839$	−3.97800e35	−0.112886	−0.0564431	+	0.998406i	$0.517976\pi$
$840$	0	0
$841$	1.17226e36	0.322904
$842$	−3.03953e36	−0.824907
$843$	−6.40996e36	−1.71400
$844$	1.52839e36	0.402675
$845$	0	0
$846$	−6.29891e35	−0.161115
$847$	−1.12405e36	−0.283297
$848$	−7.16563e35	−0.177953
$849$	1.72094e36	0.421134
$850$	0	0
$851$	8.31153e36	1.97497
$852$	6.54749e35	0.153313
$853$	2.49788e36	0.576379	0.288189	−	0.957573i	$-0.406947\pi$
$853$	2.49788e36	0.576379	0.288189	+	0.957573i	$0.406947\pi$
$854$	−1.48809e36	−0.338381
$855$	0	0
$856$	3.06928e35	0.0677820
$857$	−5.58799e36	−1.21617	−0.608086	−	0.793871i	$-0.708062\pi$
$857$	−5.58799e36	−1.21617	−0.608086	+	0.793871i	$0.708062\pi$
$858$	6.53389e36	1.40146
$859$	4.00214e36	0.846015	0.423007	−	0.906126i	$-0.360974\pi$
$859$	4.00214e36	0.846015	0.423007	+	0.906126i	$0.360974\pi$
$860$	0	0
$861$	2.24755e35	0.0461497
$862$	4.35896e36	0.882149
$863$	1.01320e36	0.202097	0.101048	−	0.994882i	$-0.467780\pi$
$863$	1.01320e36	0.202097	0.101048	+	0.994882i	$0.467780\pi$
$864$	−6.42981e35	−0.126409
$865$	0	0
$866$	3.76971e36	0.720003
$867$	9.32618e36	1.75576
$868$	−2.38071e36	−0.441783
$869$	−8.98952e36	−1.64433
$870$	0	0
$871$	7.69915e36	1.36841
$872$	−3.09117e36	−0.541586
$873$	2.91113e36	0.502786
$874$	7.62083e36	1.29751
$875$	0	0
$876$	−3.61729e36	−0.598524
$877$	−1.04808e37	−1.70962	−0.854808	−	0.518944i	$-0.826325\pi$
$877$	−1.04808e37	−1.70962	−0.854808	+	0.518944i	$0.826325\pi$
$878$	−1.57662e36	−0.253539
$879$	5.39529e36	0.855370
$880$	0	0
$881$	−7.82632e36	−1.20603	−0.603016	−	0.797729i	$-0.706035\pi$
$881$	−7.82632e36	−1.20603	−0.603016	+	0.797729i	$0.706035\pi$
$882$	1.44884e36	0.220122
$883$	−1.14075e37	−1.70876	−0.854379	−	0.519651i	$-0.826062\pi$
$883$	−1.14075e37	−1.70876	−0.854379	+	0.519651i	$0.826062\pi$
$884$	−8.14005e36	−1.20219
$885$	0	0
$886$	−3.49590e36	−0.501924
$887$	−2.08825e35	−0.0295622	−0.0147811	−	0.999891i	$-0.504705\pi$
$887$	−2.08825e35	−0.0295622	−0.0147811	+	0.999891i	$0.504705\pi$
$888$	−3.41122e36	−0.476155
$889$	9.38744e36	1.29204
$890$	0	0
$891$	−1.01908e37	−1.36375
$892$	1.77848e36	0.234687
$893$	4.70388e36	0.612086
$894$	−8.54783e36	−1.09682
$895$	0	0
$896$	−9.47490e35	−0.118229
$897$	−2.53371e37	−3.11782
$898$	1.40350e36	0.170317
$899$	6.34849e36	0.759756
$900$	0	0
$901$	9.64384e36	1.12251
$902$	−1.98116e35	−0.0227425
$903$	−6.53693e35	−0.0740077
$904$	−1.91754e36	−0.214111
$905$	0	0
$906$	−5.36097e36	−0.582292
$907$	−1.26800e37	−1.35841	−0.679203	−	0.733950i	$-0.737674\pi$
$907$	−1.26800e37	−1.35841	−0.679203	+	0.733950i	$0.737674\pi$
$908$	−8.40614e36	−0.888226
$909$	2.61173e36	0.272195
$910$	0	0
$911$	−9.78070e36	−0.991722	−0.495861	−	0.868402i	$-0.665148\pi$
$911$	−9.78070e36	−0.991722	−0.495861	+	0.868402i	$0.665148\pi$
$912$	−3.12774e36	−0.312821
$913$	−5.27586e36	−0.520486
$914$	1.85175e36	0.180200
$915$	0	0
$916$	−5.37053e35	−0.0508538
$917$	1.09108e37	1.01916
$918$	8.65354e36	0.797370
$919$	−7.03823e36	−0.639764	−0.319882	−	0.947457i	$-0.603643\pi$
$919$	−7.03823e36	−0.639764	−0.319882	+	0.947457i	$0.603643\pi$
$920$	0	0
$921$	1.40596e37	1.24373
$922$	1.01236e37	0.883485
$923$	4.59843e36	0.395903
$924$	−1.02356e37	−0.869391
$925$	0	0
$926$	−7.07085e36	−0.584567
$927$	−3.53185e36	−0.288075
$928$	2.52661e36	0.203324
$929$	2.91517e36	0.231455	0.115728	−	0.993281i	$-0.463080\pi$
$929$	2.91517e36	0.231455	0.115728	+	0.993281i	$0.463080\pi$
$930$	0	0
$931$	−1.08196e37	−0.836258
$932$	−1.03956e37	−0.792778
$933$	−1.83303e36	−0.137927
$934$	−7.71244e36	−0.572606
$935$	0	0
$936$	2.94154e36	0.212631
$937$	−1.15106e37	−0.821016	−0.410508	−	0.911857i	$-0.634649\pi$
$937$	−1.15106e37	−0.821016	−0.410508	+	0.911857i	$0.634649\pi$
$938$	−1.20611e37	−0.848888
$939$	2.93889e36	0.204110
$940$	0	0
$941$	1.44936e37	0.980179	0.490090	−	0.871672i	$-0.336964\pi$
$941$	1.44936e37	0.980179	0.490090	+	0.871672i	$0.336964\pi$
$942$	−1.92774e37	−1.28651
$943$	7.68253e35	0.0505951
$944$	−6.19733e35	−0.0402768
$945$	0	0
$946$	5.76215e35	0.0364708
$947$	2.53109e37	1.58100	0.790502	−	0.612459i	$-0.209820\pi$
$947$	2.53109e37	1.58100	0.790502	+	0.612459i	$0.209820\pi$
$948$	−1.43071e37	−0.881956
$949$	−2.54049e37	−1.54557
$950$	0	0
$951$	3.57728e37	2.11981
$952$	1.27518e37	0.745775
$953$	8.20660e36	0.473698	0.236849	−	0.971546i	$-0.423885\pi$
$953$	8.20660e36	0.473698	0.236849	+	0.971546i	$0.423885\pi$
$954$	−3.48496e36	−0.198538
$955$	0	0
$956$	3.63622e36	0.201803
$957$	2.72948e37	1.49513
$958$	4.39513e36	0.237631
$959$	1.76313e37	0.940917
$960$	0	0
$961$	−1.08408e37	−0.563665
$962$	−2.39577e37	−1.22958
$963$	1.49273e36	0.0756226
$964$	1.25216e37	0.626178
$965$	0	0
$966$	3.96917e37	1.93413
$967$	2.92106e37	1.40511	0.702555	−	0.711630i	$-0.252043\pi$
$967$	2.92106e37	1.40511	0.702555	+	0.711630i	$0.252043\pi$
$968$	1.57692e36	0.0748804
$969$	4.20946e37	1.97324
$970$	0	0
$971$	9.97325e36	0.455613	0.227807	−	0.973706i	$-0.426845\pi$
$971$	9.97325e36	0.455613	0.227807	+	0.973706i	$0.426845\pi$
$972$	−8.29117e36	−0.373928
$973$	2.53810e37	1.13005
$974$	−2.56529e37	−1.12759
$975$	0	0
$976$	2.08763e36	0.0894399
$977$	−3.07257e37	−1.29963	−0.649816	−	0.760092i	$-0.725154\pi$
$977$	−3.07257e37	−1.29963	−0.649816	+	0.760092i	$0.725154\pi$
$978$	2.23600e37	0.933760
$979$	6.80658e36	0.280637
$980$	0	0
$981$	−1.50337e37	−0.604233
$982$	1.27153e37	0.504582
$983$	5.34152e36	0.209289	0.104644	−	0.994510i	$-0.466630\pi$
$983$	5.34152e36	0.209289	0.104644	+	0.994510i	$0.466630\pi$
$984$	−3.15307e35	−0.0121982
$985$	0	0
$986$	−3.40043e37	−1.28255
$987$	2.44993e37	0.912409
$988$	−2.19667e37	−0.807800
$989$	−2.23445e36	−0.0811365
$990$	0	0
$991$	−4.37784e36	−0.155003	−0.0775013	−	0.996992i	$-0.524694\pi$
$991$	−4.37784e36	−0.155003	−0.0775013	+	0.996992i	$0.524694\pi$
$992$	3.33988e36	0.116771
$993$	−3.37359e37	−1.16473
$994$	−7.20365e36	−0.245597
$995$	0	0
$996$	−8.39668e36	−0.279168
$997$	2.23560e37	0.734015	0.367008	−	0.930218i	$-0.380382\pi$
$997$	2.23560e37	0.734015	0.367008	+	0.930218i	$0.380382\pi$
$998$	2.91817e37	0.946190
$999$	2.54690e37	0.815535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.26.a.l.1.6		6
5.2	odd	4		10.26.b.a.9.7	yes	12
5.3	odd	4		10.26.b.a.9.6	✓	12
5.4	even	2		50.26.a.k.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.26.b.a.9.6	✓	12	5.3	odd	4
10.26.b.a.9.7	yes	12	5.2	odd	4
50.26.a.k.1.1		6	5.4	even	2
50.26.a.l.1.6		6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

\(f(q)\)	\(=\)	\(q+4096.00 q^{2} +1.08697e6 q^{3} +1.67772e7 q^{4} +4.45223e9 q^{6} -4.89841e10 q^{7} +6.87195e10 q^{8} +3.34213e11 q^{9} +1.14584e13 q^{11} +1.82363e13 q^{12} +1.28077e14 q^{13} -2.00639e14 q^{14} +2.81475e14 q^{16} -3.78822e15 q^{17} +1.36894e15 q^{18} -1.02229e16 q^{19} -5.32442e16 q^{21} +4.69335e16 q^{22} -1.81999e17 q^{23} +7.46959e16 q^{24} +5.24604e17 q^{26} -5.57697e17 q^{27} -8.21816e17 q^{28} +2.19149e18 q^{29} +2.89689e18 q^{31} +1.15292e18 q^{32} +1.24549e19 q^{33} -1.55166e19 q^{34} +5.60717e18 q^{36} -4.56681e19 q^{37} -4.18730e19 q^{38} +1.39216e20 q^{39} -4.22120e18 q^{41} -2.18088e20 q^{42} +1.22773e19 q^{43} +1.92240e20 q^{44} -7.45466e20 q^{46} -4.60132e20 q^{47} +3.05955e20 q^{48} +1.05837e21 q^{49} -4.11768e21 q^{51} +2.14878e21 q^{52} -2.54574e21 q^{53} -2.28433e21 q^{54} -3.36616e21 q^{56} -1.11120e22 q^{57} +8.97634e21 q^{58} -2.20173e21 q^{59} +7.41676e21 q^{61} +1.18656e22 q^{62} -1.63711e22 q^{63} +4.72237e21 q^{64} +5.10153e22 q^{66} +6.01134e22 q^{67} -6.35558e22 q^{68} -1.97827e23 q^{69} +3.59036e22 q^{71} +2.29670e22 q^{72} -1.98356e23 q^{73} -1.87057e23 q^{74} -1.71512e23 q^{76} -5.61278e23 q^{77} +5.70229e23 q^{78} -7.84538e23 q^{79} -8.89375e23 q^{81} -1.72901e22 q^{82} -4.60438e23 q^{83} -8.93289e23 q^{84} +5.02877e22 q^{86} +2.38208e24 q^{87} +7.87413e23 q^{88} +5.94026e23 q^{89} -6.27374e24 q^{91} -3.05343e24 q^{92} +3.14882e24 q^{93} -1.88470e24 q^{94} +1.25319e24 q^{96} +8.71039e24 q^{97} +4.33508e24 q^{98} +3.82954e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24576 q^{2} - 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} - 34007705352 q^{7} + 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} - 13445529337856 q^{12} - 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) 6 * q + 24576 * q^2 - 801416 * q^3 + 100663296 * q^4 - 3282599936 * q^6 - 34007705352 * q^7 + 412316860416 * q^8 + 541468782118 * q^9 + 9861544614312 * q^11 - 13445529337856 * q^12 - 30787386783696 * q^13 - 139295561121792 * q^14 + 1688849860263936 * q^16 - 3432836939805312 * q^17 + 2217856131555328 * q^18 + 2007656488916520 * q^19 - 28181710291014728 * q^21 + 40392886740221952 * q^22 - 135711886432707576 * q^23 - 55072888167858176 * q^24 - 126105136266018816 * q^26 - 788054364552235760 * q^27 - 570554618354860032 * q^28 - 2926032031655504820 * q^29 + 79292935777014912 * q^31 + 6917529027641081856 * q^32 + 6844843638395003168 * q^33 - 14060900105442557952 * q^34 + 9084338714850623488 * q^36 - 3345885298190974032 * q^37 + 8223360978602065920 * q^38 + 124726636622561793456 * q^39 + 103229301719876244012 * q^41 - 115432285351996325888 * q^42 - 166390524239317503336 * q^43 + 165449264087949115392 * q^44 - 555875886828370231296 * q^46 - 405906243391034677992 * q^47 - 225578549935547088896 * q^48 + 1716280748139027694542 * q^49 - 1516225893661962013568 * q^51 - 516526638145613070336 * q^52 - 9772760301640799789616 * q^53 - 3227870677205957672960 * q^54 - 2336991716781506691072 * q^56 - 26174725126915157341920 * q^57 - 11985027201660947742720 * q^58 - 12499075130880780168840 * q^59 + 3687285610311118296612 * q^61 + 324783864942653079552 * q^62 - 93007818034972665339656 * q^63 + 28334198897217871282176 * q^64 + 28036479542865932976128 * q^66 - 16759935062266648886712 * q^67 - 57593446831892717371392 * q^68 + 185125325982378999854536 * q^69 + 74568876221253661649712 * q^71 + 37209451376028153806848 * q^72 - 381797761857106829638176 * q^73 - 13704746181390229635072 * q^74 + 33682886568354062008320 * q^76 + 80271541631956375339296 * q^77 + 510880303606013105995776 * q^78 - 400850316873811624799520 * q^79 - 1432110118616113828343714 * q^81 + 422827219844613095473152 * q^82 + 2423159733509985740001144 * q^83 - 472810640801776950837248 * q^84 - 681535587284244493664256 * q^86 + 1231328665433636473947120 * q^87 + 677680185704239576645632 * q^88 + 2982172528060341024616740 * q^89 - 7001711719121019241314768 * q^91 - 2276867632449004467388416 * q^92 + 8806417055581711481018368 * q^93 - 1662591972929678041055232 * q^94 - 923969740536000876118016 * q^96 + 4960469410756873985567808 * q^97 + 7029885944377457436844032 * q^98 - 2053720289282411723573464 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more