LMFDB - Embedded newform 150.12.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,12,Mod(1,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.1"); S:= CuspForms(chi, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-96,729,3072,0,-23328,-67384] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$115.251477084$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 175039x - 13178910 $ x^3 - 175039*x - 13178910
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-78.0027$ of defining polynomial
Character	$\chi$	$=$	150.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-32.0000 q^{2} +243.000 q^{3} +1024.00 q^{4} -7776.00 q^{6} +52907.3 q^{7} -32768.0 q^{8} +59049.0 q^{9} +531122. q^{11} +248832. q^{12} +373795. q^{13} -1.69303e6 q^{14} +1.04858e6 q^{16} +4.81399e6 q^{17} -1.88957e6 q^{18} +5.80222e6 q^{19} +1.28565e7 q^{21} -1.69959e7 q^{22} +3.32148e7 q^{23} -7.96262e6 q^{24} -1.19614e7 q^{26} +1.43489e7 q^{27} +5.41770e7 q^{28} -1.30307e8 q^{29} +6.18143e7 q^{31} -3.35544e7 q^{32} +1.29063e8 q^{33} -1.54048e8 q^{34} +6.04662e7 q^{36} +1.37358e8 q^{37} -1.85671e8 q^{38} +9.08322e7 q^{39} +1.46886e9 q^{41} -4.11407e8 q^{42} -5.30160e8 q^{43} +5.43869e8 q^{44} -1.06287e9 q^{46} +1.34319e9 q^{47} +2.54804e8 q^{48} +8.21851e8 q^{49} +1.16980e9 q^{51} +3.82766e8 q^{52} -4.74535e9 q^{53} -4.59165e8 q^{54} -1.73366e9 q^{56} +1.40994e9 q^{57} +4.16982e9 q^{58} +4.37861e9 q^{59} -7.78414e9 q^{61} -1.97806e9 q^{62} +3.12412e9 q^{63} +1.07374e9 q^{64} -4.13000e9 q^{66} +4.98481e9 q^{67} +4.92953e9 q^{68} +8.07119e9 q^{69} +9.09417e9 q^{71} -1.93492e9 q^{72} -1.88613e10 q^{73} -4.39545e9 q^{74} +5.94148e9 q^{76} +2.81002e10 q^{77} -2.90663e9 q^{78} -4.50312e10 q^{79} +3.48678e9 q^{81} -4.70035e10 q^{82} -6.88856e10 q^{83} +1.31650e10 q^{84} +1.69651e10 q^{86} -3.16645e10 q^{87} -1.74038e10 q^{88} -3.49597e9 q^{89} +1.97765e10 q^{91} +3.40119e10 q^{92} +1.50209e10 q^{93} -4.29820e10 q^{94} -8.15373e9 q^{96} -1.62848e11 q^{97} -2.62992e10 q^{98} +3.13622e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 96 q^{2} + 729 q^{3} + 3072 q^{4} - 23328 q^{6} - 67384 q^{7} - 98304 q^{8} + 177147 q^{9} + 876700 q^{11} + 746496 q^{12} - 315192 q^{13} + 2156288 q^{14} + 3145728 q^{16} - 2874324 q^{17} - 5668704 q^{18}+ \cdots + 51768258300 q^{99}+O(q^{100}) $ 3 * q - 96 * q^2 + 729 * q^3 + 3072 * q^4 - 23328 * q^6 - 67384 * q^7 - 98304 * q^8 + 177147 * q^9 + 876700 * q^11 + 746496 * q^12 - 315192 * q^13 + 2156288 * q^14 + 3145728 * q^16 - 2874324 * q^17 - 5668704 * q^18 + 2034912 * q^19 - 16374312 * q^21 - 28054400 * q^22 - 49866300 * q^23 - 23887872 * q^24 + 10086144 * q^26 + 43046721 * q^27 - 69001216 * q^28 - 35857782 * q^29 - 61121676 * q^31 - 100663296 * q^32 + 213038100 * q^33 + 91978368 * q^34 + 181398528 * q^36 + 935590592 * q^37 - 65117184 * q^38 - 76591656 * q^39 + 1003384282 * q^41 + 523977984 * q^42 + 980338484 * q^43 + 897740800 * q^44 + 1595721600 * q^46 + 2190634500 * q^47 + 764411904 * q^48 + 4795452723 * q^49 - 698460732 * q^51 - 322756608 * q^52 - 6993816924 * q^53 - 1377495072 * q^54 + 2208038912 * q^56 + 494483616 * q^57 + 1147449024 * q^58 + 2034730100 * q^59 - 6193233330 * q^61 + 1955893632 * q^62 - 3978957816 * q^63 + 3221225472 * q^64 - 6817219200 * q^66 - 5079955484 * q^67 - 2943307776 * q^68 - 12117510900 * q^69 + 6851752800 * q^71 - 5804752896 * q^72 - 54889949992 * q^73 - 29938898944 * q^74 + 2083749888 * q^76 + 32141607600 * q^77 + 2450932992 * q^78 + 6719163876 * q^79 + 10460353203 * q^81 - 32108297024 * q^82 - 57462212652 * q^83 - 16767295488 * q^84 - 31370831488 * q^86 - 8713441026 * q^87 - 28727705600 * q^88 + 63718294546 * q^89 + 135921691776 * q^91 - 51063091200 * q^92 - 14852567268 * q^93 - 70100304000 * q^94 - 24461180928 * q^96 - 315259594808 * q^97 - 153454487136 * q^98 + 51768258300 * 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 $

$2$	−32.0000	−0.707107
$2$	−32.0000	−0.707107
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	1024.00	0.500000
$5$	0	0
$5$	0	0
$6$	−7776.00	−0.408248
$7$	52907.3	1.18981	0.594903	−	0.803798i	$-0.297191\pi$
$7$	52907.3	1.18981	0.594903	+	0.803798i	$0.297191\pi$
$8$	−32768.0	−0.353553
$9$	59049.0	0.333333
$10$	0	0
$11$	531122.	0.994339	0.497169	−	0.867654i	$-0.334373\pi$
$11$	531122.	0.994339	0.497169	+	0.867654i	$0.334373\pi$
$12$	248832.	0.288675
$13$	373795.	0.279219	0.139610	−	0.990207i	$-0.455415\pi$
$13$	373795.	0.279219	0.139610	+	0.990207i	$0.455415\pi$
$14$	−1.69303e6	−0.841320
$15$	0	0
$16$	1.04858e6	0.250000
$17$	4.81399e6	0.822311	0.411156	−	0.911565i	$-0.365125\pi$
$17$	4.81399e6	0.822311	0.411156	+	0.911565i	$0.365125\pi$
$18$	−1.88957e6	−0.235702
$19$	5.80222e6	0.537588	0.268794	−	0.963198i	$-0.413375\pi$
$19$	5.80222e6	0.537588	0.268794	+	0.963198i	$0.413375\pi$
$20$	0	0
$21$	1.28565e7	0.686935
$22$	−1.69959e7	−0.703104
$23$	3.32148e7	1.07604	0.538020	−	0.842932i	$-0.319173\pi$
$23$	3.32148e7	1.07604	0.538020	+	0.842932i	$0.319173\pi$
$24$	−7.96262e6	−0.204124
$25$	0	0
$26$	−1.19614e7	−0.197438
$27$	1.43489e7	0.192450
$28$	5.41770e7	0.594903
$29$	−1.30307e8	−1.17972	−0.589859	−	0.807507i	$-0.700816\pi$
$29$	−1.30307e8	−1.17972	−0.589859	+	0.807507i	$0.700816\pi$
$30$	0	0
$31$	6.18143e7	0.387793	0.193896	−	0.981022i	$-0.437887\pi$
$31$	6.18143e7	0.387793	0.193896	+	0.981022i	$0.437887\pi$
$32$	−3.35544e7	−0.176777
$33$	1.29063e8	0.574082
$34$	−1.54048e8	−0.581462
$35$	0	0
$36$	6.04662e7	0.166667
$37$	1.37358e8	0.325645	0.162822	−	0.986655i	$-0.447940\pi$
$37$	1.37358e8	0.325645	0.162822	+	0.986655i	$0.447940\pi$
$38$	−1.85671e8	−0.380132
$39$	9.08322e7	0.161207
$40$	0	0
$41$	1.46886e9	1.98002	0.990009	−	0.141003i	$-0.0450326\pi$
$41$	1.46886e9	1.98002	0.990009	+	0.141003i	$0.0450326\pi$
$42$	−4.11407e8	−0.485736
$43$	−5.30160e8	−0.549960	−0.274980	−	0.961450i	$-0.588671\pi$
$43$	−5.30160e8	−0.549960	−0.274980	+	0.961450i	$0.588671\pi$
$44$	5.43869e8	0.497169
$45$	0	0
$46$	−1.06287e9	−0.760875
$47$	1.34319e9	0.854277	0.427138	−	0.904186i	$-0.359522\pi$
$47$	1.34319e9	0.854277	0.427138	+	0.904186i	$0.359522\pi$
$48$	2.54804e8	0.144338
$49$	8.21851e8	0.415637
$50$	0	0
$51$	1.16980e9	0.474762
$52$	3.82766e8	0.139610
$53$	−4.74535e9	−1.55866	−0.779329	−	0.626615i	$-0.784440\pi$
$53$	−4.74535e9	−1.55866	−0.779329	+	0.626615i	$0.784440\pi$
$54$	−4.59165e8	−0.136083
$55$	0	0
$56$	−1.73366e9	−0.420660
$57$	1.40994e9	0.310377
$58$	4.16982e9	0.834186
$59$	4.37861e9	0.797352	0.398676	−	0.917092i	$-0.369470\pi$
$59$	4.37861e9	0.797352	0.398676	+	0.917092i	$0.369470\pi$
$60$	0	0
$61$	−7.78414e9	−1.18004	−0.590020	−	0.807389i	$-0.700880\pi$
$61$	−7.78414e9	−1.18004	−0.590020	+	0.807389i	$0.700880\pi$
$62$	−1.97806e9	−0.274211
$63$	3.12412e9	0.396602
$64$	1.07374e9	0.125000
$65$	0	0
$66$	−4.13000e9	−0.405937
$67$	4.98481e9	0.451063	0.225531	−	0.974236i	$-0.427588\pi$
$67$	4.98481e9	0.451063	0.225531	+	0.974236i	$0.427588\pi$
$68$	4.92953e9	0.411156
$69$	8.07119e9	0.621252
$70$	0	0
$71$	9.09417e9	0.598194	0.299097	−	0.954223i	$-0.403315\pi$
$71$	9.09417e9	0.598194	0.299097	+	0.954223i	$0.403315\pi$
$72$	−1.93492e9	−0.117851
$73$	−1.88613e10	−1.06487	−0.532435	−	0.846471i	$-0.678723\pi$
$73$	−1.88613e10	−1.06487	−0.532435	+	0.846471i	$0.678723\pi$
$74$	−4.39545e9	−0.230266
$75$	0	0
$76$	5.94148e9	0.268794
$77$	2.81002e10	1.18307
$78$	−2.90663e9	−0.113991
$79$	−4.50312e10	−1.64651	−0.823255	−	0.567672i	$-0.807844\pi$
$79$	−4.50312e10	−1.64651	−0.823255	+	0.567672i	$0.807844\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	−4.70035e10	−1.40008
$83$	−6.88856e10	−1.91955	−0.959774	−	0.280773i	$-0.909409\pi$
$83$	−6.88856e10	−1.91955	−0.959774	+	0.280773i	$0.909409\pi$
$84$	1.31650e10	0.343467
$85$	0	0
$86$	1.69651e10	0.388880
$87$	−3.16645e10	−0.681110
$88$	−1.74038e10	−0.351552
$89$	−3.49597e9	−0.0663624	−0.0331812	−	0.999449i	$-0.510564\pi$
$89$	−3.49597e9	−0.0663624	−0.0331812	+	0.999449i	$0.510564\pi$
$90$	0	0
$91$	1.97765e10	0.332217
$92$	3.40119e10	0.538020
$93$	1.50209e10	0.223892
$94$	−4.29820e10	−0.604065
$95$	0	0
$96$	−8.15373e9	−0.102062
$97$	−1.62848e11	−1.92548	−0.962738	−	0.270437i	$-0.912832\pi$
$97$	−1.62848e11	−1.92548	−0.962738	+	0.270437i	$0.912832\pi$
$98$	−2.62992e10	−0.293900
$99$	3.13622e10	0.331446
$100$	0	0
$101$	1.33281e11	1.26183	0.630915	−	0.775852i	$-0.282680\pi$
$101$	1.33281e11	1.26183	0.630915	+	0.775852i	$0.282680\pi$
$102$	−3.74336e10	−0.335707
$103$	8.49052e10	0.721655	0.360827	−	0.932633i	$-0.382494\pi$
$103$	8.49052e10	0.721655	0.360827	+	0.932633i	$0.382494\pi$
$104$	−1.22485e10	−0.0987189
$105$	0	0
$106$	1.51851e11	1.10214
$107$	4.19151e10	0.288908	0.144454	−	0.989512i	$-0.453857\pi$
$107$	4.19151e10	0.288908	0.144454	+	0.989512i	$0.453857\pi$
$108$	1.46933e10	0.0962250
$109$	2.67212e11	1.66345	0.831727	−	0.555185i	$-0.187353\pi$
$109$	2.67212e11	1.66345	0.831727	+	0.555185i	$0.187353\pi$
$110$	0	0
$111$	3.33780e10	0.188011
$112$	5.54773e10	0.297451
$113$	−2.37749e11	−1.21391	−0.606957	−	0.794735i	$-0.707610\pi$
$113$	−2.37749e11	−1.21391	−0.606957	+	0.794735i	$0.707610\pi$
$114$	−4.51181e10	−0.219469
$115$	0	0
$116$	−1.33434e11	−0.589859
$117$	2.20722e10	0.0930731
$118$	−1.40116e11	−0.563813
$119$	2.54695e11	0.978390
$120$	0	0
$121$	−3.22132e9	−0.0112905
$122$	2.49093e11	0.834414
$123$	3.56933e11	1.14316
$124$	6.32978e10	0.193896
$125$	0	0
$126$	−9.99719e10	−0.280440
$127$	−7.15504e10	−0.192173	−0.0960863	−	0.995373i	$-0.530632\pi$
$127$	−7.15504e10	−0.192173	−0.0960863	+	0.995373i	$0.530632\pi$
$128$	−3.43597e10	−0.0883883
$129$	−1.28829e11	−0.317519
$130$	0	0
$131$	4.56783e11	1.03447	0.517235	−	0.855844i	$-0.326961\pi$
$131$	4.56783e11	1.03447	0.517235	+	0.855844i	$0.326961\pi$
$132$	1.32160e11	0.287041
$133$	3.06980e11	0.639626
$134$	−1.59514e11	−0.318949
$135$	0	0
$136$	−1.57745e11	−0.290731
$137$	2.86655e10	0.0507454	0.0253727	−	0.999678i	$-0.491923\pi$
$137$	2.86655e10	0.0507454	0.0253727	+	0.999678i	$0.491923\pi$
$138$	−2.58278e11	−0.439291
$139$	1.20093e12	1.96307	0.981536	−	0.191280i	$-0.0612638\pi$
$139$	1.20093e12	1.96307	0.981536	+	0.191280i	$0.0612638\pi$
$140$	0	0
$141$	3.26395e11	0.493217
$142$	−2.91014e11	−0.422987
$143$	1.98531e11	0.277638
$144$	6.19174e10	0.0833333
$145$	0	0
$146$	6.03563e11	0.752977
$147$	1.99710e11	0.239968
$148$	1.40655e11	0.162822
$149$	−6.64637e11	−0.741412	−0.370706	−	0.928750i	$-0.620884\pi$
$149$	−6.64637e11	−0.741412	−0.370706	+	0.928750i	$0.620884\pi$
$150$	0	0
$151$	1.70963e12	1.77226	0.886130	−	0.463436i	$-0.153384\pi$
$151$	1.70963e12	1.77226	0.886130	+	0.463436i	$0.153384\pi$
$152$	−1.90127e11	−0.190066
$153$	2.84261e11	0.274104
$154$	−8.99206e11	−0.836557
$155$	0	0
$156$	9.30122e10	0.0806036
$157$	9.96231e11	0.833512	0.416756	−	0.909018i	$-0.363167\pi$
$157$	9.96231e11	0.833512	0.416756	+	0.909018i	$0.363167\pi$
$158$	1.44100e12	1.16426
$159$	−1.15312e12	−0.899892
$160$	0	0
$161$	1.75730e12	1.28028
$162$	−1.11577e11	−0.0785674
$163$	−2.56482e11	−0.174592	−0.0872960	−	0.996182i	$-0.527823\pi$
$163$	−2.56482e11	−0.174592	−0.0872960	+	0.996182i	$0.527823\pi$
$164$	1.50411e12	0.990009
$165$	0	0
$166$	2.20434e12	1.35733
$167$	1.02735e12	0.612040	0.306020	−	0.952025i	$-0.401003\pi$
$167$	1.02735e12	0.612040	0.306020	+	0.952025i	$0.401003\pi$
$168$	−4.21281e11	−0.242868
$169$	−1.65244e12	−0.922037
$170$	0	0
$171$	3.42616e11	0.179196
$172$	−5.42884e11	−0.274980
$173$	3.51424e12	1.72416	0.862082	−	0.506769i	$-0.169160\pi$
$173$	3.51424e12	1.72416	0.862082	+	0.506769i	$0.169160\pi$
$174$	1.01327e12	0.481617
$175$	0	0
$176$	5.56922e11	0.248585
$177$	1.06400e12	0.460352
$178$	1.11871e11	0.0469253
$179$	4.59480e12	1.86885	0.934427	−	0.356156i	$-0.115913\pi$
$179$	4.59480e12	1.86885	0.934427	+	0.356156i	$0.115913\pi$
$180$	0	0
$181$	−9.54953e11	−0.365384	−0.182692	−	0.983170i	$-0.558481\pi$
$181$	−9.54953e11	−0.365384	−0.182692	+	0.983170i	$0.558481\pi$
$182$	−6.32847e11	−0.234913
$183$	−1.89155e12	−0.681297
$184$	−1.08838e12	−0.380437
$185$	0	0
$186$	−4.80668e11	−0.158316
$187$	2.55682e12	0.817656
$188$	1.37542e12	0.427138
$189$	7.59161e11	0.228978
$190$	0	0
$191$	4.18247e12	1.19055	0.595277	−	0.803520i	$-0.297042\pi$
$191$	4.18247e12	1.19055	0.595277	+	0.803520i	$0.297042\pi$
$192$	2.60919e11	0.0721688
$193$	−3.35128e12	−0.900836	−0.450418	−	0.892818i	$-0.648725\pi$
$193$	−3.35128e12	−0.900836	−0.450418	+	0.892818i	$0.648725\pi$
$194$	5.21114e12	1.36152
$195$	0	0
$196$	8.41575e11	0.207819
$197$	−7.22875e12	−1.73580	−0.867899	−	0.496742i	$-0.834530\pi$
$197$	−7.22875e12	−1.73580	−0.867899	+	0.496742i	$0.834530\pi$
$198$	−1.00359e12	−0.234368
$199$	1.05138e12	0.238818	0.119409	−	0.992845i	$-0.461900\pi$
$199$	1.05138e12	0.238818	0.119409	+	0.992845i	$0.461900\pi$
$200$	0	0
$201$	1.21131e12	0.260421
$202$	−4.26499e12	−0.892248
$203$	−6.89417e12	−1.40363
$204$	1.19787e12	0.237381
$205$	0	0
$206$	−2.71697e12	−0.510287
$207$	1.96130e12	0.358680
$208$	3.91953e11	0.0698048
$209$	3.08169e12	0.534545
$210$	0	0
$211$	3.80994e11	0.0627141	0.0313570	−	0.999508i	$-0.490017\pi$
$211$	3.80994e11	0.0627141	0.0313570	+	0.999508i	$0.490017\pi$
$212$	−4.85924e12	−0.779329
$213$	2.20988e12	0.345368
$214$	−1.34128e12	−0.204289
$215$	0	0
$216$	−4.70185e11	−0.0680414
$217$	3.27042e12	0.461398
$218$	−8.55079e12	−1.17624
$219$	−4.58331e12	−0.614803
$220$	0	0
$221$	1.79945e12	0.229605
$222$	−1.06810e12	−0.132944
$223$	−3.50569e12	−0.425693	−0.212846	−	0.977086i	$-0.568273\pi$
$223$	−3.50569e12	−0.425693	−0.212846	+	0.977086i	$0.568273\pi$
$224$	−1.77527e12	−0.210330
$225$	0	0
$226$	7.60798e12	0.858367
$227$	4.84938e12	0.534003	0.267002	−	0.963696i	$-0.413967\pi$
$227$	4.84938e12	0.534003	0.267002	+	0.963696i	$0.413967\pi$
$228$	1.44378e12	0.155188
$229$	1.41260e12	0.148225	0.0741127	−	0.997250i	$-0.476388\pi$
$229$	1.41260e12	0.148225	0.0741127	+	0.997250i	$0.476388\pi$
$230$	0	0
$231$	6.82835e12	0.683046
$232$	4.26989e12	0.417093
$233$	9.88438e12	0.942957	0.471478	−	0.881878i	$-0.343721\pi$
$233$	9.88438e12	0.942957	0.471478	+	0.881878i	$0.343721\pi$
$234$	−7.06311e11	−0.0658126
$235$	0	0
$236$	4.48370e12	0.398676
$237$	−1.09426e13	−0.950613
$238$	−8.15024e12	−0.691826
$239$	−5.87747e12	−0.487531	−0.243765	−	0.969834i	$-0.578383\pi$
$239$	−5.87747e12	−0.487531	−0.243765	+	0.969834i	$0.578383\pi$
$240$	0	0
$241$	−9.32982e12	−0.739230	−0.369615	−	0.929185i	$-0.620510\pi$
$241$	−9.32982e12	−0.739230	−0.369615	+	0.929185i	$0.620510\pi$
$242$	1.03082e11	0.00798360
$243$	8.47289e11	0.0641500
$244$	−7.97096e12	−0.590020
$245$	0	0
$246$	−1.14219e13	−0.808339
$247$	2.16884e12	0.150105
$248$	−2.02553e12	−0.137105
$249$	−1.67392e13	−1.10825
$250$	0	0
$251$	1.76150e13	1.11603	0.558016	−	0.829830i	$-0.311563\pi$
$251$	1.76150e13	1.11603	0.558016	+	0.829830i	$0.311563\pi$
$252$	3.19910e12	0.198301
$253$	1.76411e13	1.06995
$254$	2.28961e12	0.135887
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	1.61047e13	0.896023	0.448012	−	0.894028i	$-0.352132\pi$
$257$	1.61047e13	0.896023	0.448012	+	0.894028i	$0.352132\pi$
$258$	4.12253e12	0.224520
$259$	7.26723e12	0.387454
$260$	0	0
$261$	−7.69448e12	−0.393239
$262$	−1.46171e13	−0.731480
$263$	8.60423e12	0.421653	0.210826	−	0.977524i	$-0.432384\pi$
$263$	8.60423e12	0.421653	0.210826	+	0.977524i	$0.432384\pi$
$264$	−4.22912e12	−0.202969
$265$	0	0
$266$	−9.82335e12	−0.452284
$267$	−8.49520e11	−0.0383144
$268$	5.10444e12	0.225531
$269$	1.03462e13	0.447861	0.223931	−	0.974605i	$-0.428111\pi$
$269$	1.03462e13	0.447861	0.223931	+	0.974605i	$0.428111\pi$
$270$	0	0
$271$	−3.19603e13	−1.32825	−0.664124	−	0.747622i	$-0.731196\pi$
$271$	−3.19603e13	−1.32825	−0.664124	+	0.747622i	$0.731196\pi$
$272$	5.04783e12	0.205578
$273$	4.80568e12	0.191805
$274$	−9.17297e11	−0.0358824
$275$	0	0
$276$	8.26490e12	0.310626
$277$	−7.21606e12	−0.265865	−0.132933	−	0.991125i	$-0.542439\pi$
$277$	−7.21606e12	−0.265865	−0.132933	+	0.991125i	$0.542439\pi$
$278$	−3.84297e13	−1.38810
$279$	3.65007e12	0.129264
$280$	0	0
$281$	−4.86472e13	−1.65643	−0.828216	−	0.560409i	$-0.810644\pi$
$281$	−4.86472e13	−1.65643	−0.828216	+	0.560409i	$0.810644\pi$
$282$	−1.04446e13	−0.348757
$283$	−4.50061e13	−1.47382	−0.736912	−	0.675988i	$-0.763717\pi$
$283$	−4.50061e13	−1.47382	−0.736912	+	0.675988i	$0.763717\pi$
$284$	9.31243e12	0.299097
$285$	0	0
$286$	−6.35298e12	−0.196320
$287$	7.77134e13	2.35584
$288$	−1.98136e12	−0.0589256
$289$	−1.10974e13	−0.323804
$290$	0	0
$291$	−3.95721e13	−1.11167
$292$	−1.93140e13	−0.532435
$293$	1.55228e13	0.419949	0.209975	−	0.977707i	$-0.432662\pi$
$293$	1.55228e13	0.419949	0.209975	+	0.977707i	$0.432662\pi$
$294$	−6.39071e12	−0.169683
$295$	0	0
$296$	−4.50094e12	−0.115133
$297$	7.62102e12	0.191361
$298$	2.12684e13	0.524258
$299$	1.24155e13	0.300451
$300$	0	0
$301$	−2.80493e13	−0.654345
$302$	−5.47080e13	−1.25318
$303$	3.23873e13	0.728518
$304$	6.08407e12	0.134397
$305$	0	0
$306$	−9.09636e12	−0.193821
$307$	5.15314e13	1.07848	0.539239	−	0.842153i	$-0.318712\pi$
$307$	5.15314e13	1.07848	0.539239	+	0.842153i	$0.318712\pi$
$308$	2.87746e13	0.591535
$309$	2.06320e13	0.416648
$310$	0	0
$311$	−1.21704e13	−0.237204	−0.118602	−	0.992942i	$-0.537841\pi$
$311$	−1.21704e13	−0.237204	−0.118602	+	0.992942i	$0.537841\pi$
$312$	−2.97639e12	−0.0569954
$313$	4.90592e13	0.923053	0.461527	−	0.887126i	$-0.347302\pi$
$313$	4.90592e13	0.923053	0.461527	+	0.887126i	$0.347302\pi$
$314$	−3.18794e13	−0.589382
$315$	0	0
$316$	−4.61119e13	−0.823255
$317$	−2.65586e12	−0.0465992	−0.0232996	−	0.999729i	$-0.507417\pi$
$317$	−2.65586e12	−0.0465992	−0.0232996	+	0.999729i	$0.507417\pi$
$318$	3.68998e13	0.636320
$319$	−6.92087e13	−1.17304
$320$	0	0
$321$	1.01854e13	0.166801
$322$	−5.62337e13	−0.905293
$323$	2.79319e13	0.442065
$324$	3.57047e12	0.0555556
$325$	0	0
$326$	8.20741e12	0.123455
$327$	6.49326e13	0.960395
$328$	−4.81316e13	−0.700042
$329$	7.10644e13	1.01642
$330$	0	0
$331$	8.86699e13	1.22665	0.613327	−	0.789829i	$-0.289831\pi$
$331$	8.86699e13	1.22665	0.613327	+	0.789829i	$0.289831\pi$
$332$	−7.05389e13	−0.959774
$333$	8.11085e12	0.108548
$334$	−3.28753e13	−0.432777
$335$	0	0
$336$	1.34810e13	0.171734
$337$	−5.28648e13	−0.662525	−0.331262	−	0.943539i	$-0.607475\pi$
$337$	−5.28648e13	−0.662525	−0.331262	+	0.943539i	$0.607475\pi$
$338$	5.28780e13	0.651978
$339$	−5.77731e13	−0.700853
$340$	0	0
$341$	3.28309e13	0.385597
$342$	−1.09637e13	−0.126711
$343$	−6.11331e13	−0.695278
$344$	1.73723e13	0.194440
$345$	0	0
$346$	−1.12456e14	−1.21917
$347$	2.14116e13	0.228474	0.114237	−	0.993454i	$-0.463558\pi$
$347$	2.14116e13	0.228474	0.114237	+	0.993454i	$0.463558\pi$
$348$	−3.24245e13	−0.340555
$349$	−1.29282e14	−1.33659	−0.668295	−	0.743896i	$-0.732976\pi$
$349$	−1.29282e14	−1.33659	−0.668295	+	0.743896i	$0.732976\pi$
$350$	0	0
$351$	5.36355e12	0.0537358
$352$	−1.78215e13	−0.175776
$353$	3.24173e13	0.314786	0.157393	−	0.987536i	$-0.449691\pi$
$353$	3.24173e13	0.314786	0.157393	+	0.987536i	$0.449691\pi$
$354$	−3.40481e13	−0.325518
$355$	0	0
$356$	−3.57987e12	−0.0331812
$357$	6.18909e13	0.564874
$358$	−1.47034e14	−1.32148
$359$	−1.01176e14	−0.895483	−0.447741	−	0.894163i	$-0.647771\pi$
$359$	−1.01176e14	−0.895483	−0.447741	+	0.894163i	$0.647771\pi$
$360$	0	0
$361$	−8.28244e13	−0.710999
$362$	3.05585e13	0.258366
$363$	−7.82780e11	−0.00651858
$364$	2.02511e13	0.166108
$365$	0	0
$366$	6.05295e13	0.481749
$367$	−6.19096e12	−0.0485395	−0.0242697	−	0.999705i	$-0.507726\pi$
$367$	−6.19096e12	−0.0485395	−0.0242697	+	0.999705i	$0.507726\pi$
$368$	3.48282e13	0.269010
$369$	8.67347e13	0.660006
$370$	0	0
$371$	−2.51063e14	−1.85450
$372$	1.53814e13	0.111946
$373$	−2.17159e14	−1.55733	−0.778664	−	0.627442i	$-0.784102\pi$
$373$	−2.17159e14	−1.55733	−0.778664	+	0.627442i	$0.784102\pi$
$374$	−8.18181e13	−0.578170
$375$	0	0
$376$	−4.40136e13	−0.302033
$377$	−4.87080e13	−0.329400
$378$	−2.42932e13	−0.161912
$379$	1.35854e14	0.892397	0.446199	−	0.894934i	$-0.352778\pi$
$379$	1.35854e14	0.892397	0.446199	+	0.894934i	$0.352778\pi$
$380$	0	0
$381$	−1.73867e13	−0.110951
$382$	−1.33839e14	−0.841849
$383$	1.53210e14	0.949932	0.474966	−	0.880004i	$-0.342460\pi$
$383$	1.53210e14	0.949932	0.474966	+	0.880004i	$0.342460\pi$
$384$	−8.34942e12	−0.0510310
$385$	0	0
$386$	1.07241e14	0.636988
$387$	−3.13054e13	−0.183320
$388$	−1.66756e14	−0.962738
$389$	6.34461e13	0.361145	0.180573	−	0.983562i	$-0.442205\pi$
$389$	6.34461e13	0.361145	0.180573	+	0.983562i	$0.442205\pi$
$390$	0	0
$391$	1.59896e14	0.884839
$392$	−2.69304e13	−0.146950
$393$	1.10998e14	0.597251
$394$	2.31320e14	1.22739
$395$	0	0
$396$	3.21149e13	0.165723
$397$	2.82212e14	1.43624	0.718121	−	0.695918i	$-0.245002\pi$
$397$	2.82212e14	1.43624	0.718121	+	0.695918i	$0.245002\pi$
$398$	−3.36442e13	−0.168870
$399$	7.45961e13	0.369288
$400$	0	0
$401$	−4.11588e13	−0.198230	−0.0991149	−	0.995076i	$-0.531601\pi$
$401$	−4.11588e13	−0.198230	−0.0991149	+	0.995076i	$0.531601\pi$
$402$	−3.87619e13	−0.184146
$403$	2.31059e13	0.108279
$404$	1.36480e14	0.630915
$405$	0	0
$406$	2.20613e14	0.992519
$407$	7.29538e13	0.323801
$408$	−3.83320e13	−0.167854
$409$	7.28620e13	0.314792	0.157396	−	0.987536i	$-0.449690\pi$
$409$	7.28620e13	0.314792	0.157396	+	0.987536i	$0.449690\pi$
$410$	0	0
$411$	6.96573e12	0.0292979
$412$	8.69429e13	0.360827
$413$	2.31660e14	0.948694
$414$	−6.27616e13	−0.253625
$415$	0	0
$416$	−1.25425e13	−0.0493594
$417$	2.91826e14	1.13338
$418$	−9.86140e13	−0.377980
$419$	−4.44807e13	−0.168265	−0.0841326	−	0.996455i	$-0.526812\pi$
$419$	−4.44807e13	−0.168265	−0.0841326	+	0.996455i	$0.526812\pi$
$420$	0	0
$421$	4.08282e13	0.150456	0.0752279	−	0.997166i	$-0.476032\pi$
$421$	4.08282e13	0.150456	0.0752279	+	0.997166i	$0.476032\pi$
$422$	−1.21918e13	−0.0443455
$423$	7.93139e13	0.284759
$424$	1.55496e14	0.551069
$425$	0	0
$426$	−7.07163e13	−0.244212
$427$	−4.11838e14	−1.40402
$428$	4.29210e13	0.144454
$429$	4.82430e13	0.160295
$430$	0	0
$431$	5.01831e14	1.62530	0.812648	−	0.582754i	$-0.198025\pi$
$431$	5.01831e14	1.62530	0.812648	+	0.582754i	$0.198025\pi$
$432$	1.50459e13	0.0481125
$433$	−3.09744e14	−0.977957	−0.488979	−	0.872296i	$-0.662630\pi$
$433$	−3.09744e14	−0.977957	−0.488979	+	0.872296i	$0.662630\pi$
$434$	−1.04654e14	−0.326258
$435$	0	0
$436$	2.73625e14	0.831727
$437$	1.92720e14	0.578466
$438$	1.46666e14	0.434732
$439$	−1.08027e13	−0.0316213	−0.0158106	−	0.999875i	$-0.505033\pi$
$439$	−1.08027e13	−0.0316213	−0.0158106	+	0.999875i	$0.505033\pi$
$440$	0	0
$441$	4.85295e13	0.138546
$442$	−5.75823e13	−0.162355
$443$	−3.45097e14	−0.960993	−0.480497	−	0.876997i	$-0.659544\pi$
$443$	−3.45097e14	−0.960993	−0.480497	+	0.876997i	$0.659544\pi$
$444$	3.41790e13	0.0940056
$445$	0	0
$446$	1.12182e14	0.301010
$447$	−1.61507e14	−0.428055
$448$	5.68087e13	0.148726
$449$	−5.12768e14	−1.32607	−0.663035	−	0.748589i	$-0.730732\pi$
$449$	−5.12768e14	−1.32607	−0.663035	+	0.748589i	$0.730732\pi$
$450$	0	0
$451$	7.80144e14	1.96881
$452$	−2.43455e14	−0.606957
$453$	4.15439e14	1.02322
$454$	−1.55180e14	−0.377597
$455$	0	0
$456$	−4.62009e13	−0.109735
$457$	−2.33364e14	−0.547640	−0.273820	−	0.961781i	$-0.588287\pi$
$457$	−2.33364e14	−0.547640	−0.273820	+	0.961781i	$0.588287\pi$
$458$	−4.52030e13	−0.104811
$459$	6.90755e13	0.158254
$460$	0	0
$461$	1.62814e14	0.364197	0.182098	−	0.983280i	$-0.441711\pi$
$461$	1.62814e14	0.364197	0.182098	+	0.983280i	$0.441711\pi$
$462$	−2.18507e14	−0.482986
$463$	−2.48926e14	−0.543718	−0.271859	−	0.962337i	$-0.587639\pi$
$463$	−2.48926e14	−0.543718	−0.271859	+	0.962337i	$0.587639\pi$
$464$	−1.36637e14	−0.294929
$465$	0	0
$466$	−3.16300e14	−0.666771
$467$	3.50158e14	0.729494	0.364747	−	0.931107i	$-0.381155\pi$
$467$	3.50158e14	0.729494	0.364747	+	0.931107i	$0.381155\pi$
$468$	2.26020e13	0.0465365
$469$	2.63732e14	0.536677
$470$	0	0
$471$	2.42084e14	0.481228
$472$	−1.43478e14	−0.281907
$473$	−2.81580e14	−0.546846
$474$	3.50162e14	0.672185
$475$	0	0
$476$	2.60808e14	0.489195
$477$	−2.80208e14	−0.519553
$478$	1.88079e14	0.344736
$479$	4.41157e14	0.799370	0.399685	−	0.916653i	$-0.369119\pi$
$479$	4.41157e14	0.799370	0.399685	+	0.916653i	$0.369119\pi$
$480$	0	0
$481$	5.13437e13	0.0909263
$482$	2.98554e14	0.522715
$483$	4.27024e14	0.739169
$484$	−3.29863e12	−0.00564526
$485$	0	0
$486$	−2.71132e13	−0.0453609
$487$	−4.35803e12	−0.00720909	−0.00360455	−	0.999994i	$-0.501147\pi$
$487$	−4.35803e12	−0.00720909	−0.00360455	+	0.999994i	$0.501147\pi$
$488$	2.55071e14	0.417207
$489$	−6.23250e13	−0.100801
$490$	0	0
$491$	−6.73844e13	−0.106564	−0.0532821	−	0.998580i	$-0.516968\pi$
$491$	−6.73844e13	−0.106564	−0.0532821	+	0.998580i	$0.516968\pi$
$492$	3.65499e14	0.571582
$493$	−6.27295e14	−0.970094
$494$	−6.94030e13	−0.106140
$495$	0	0
$496$	6.48170e13	0.0969481
$497$	4.81148e14	0.711735
$498$	5.35655e14	0.783652
$499$	8.06287e14	1.16664	0.583320	−	0.812243i	$-0.301753\pi$
$499$	8.06287e14	1.16664	0.583320	+	0.812243i	$0.301753\pi$
$500$	0	0
$501$	2.49647e14	0.353361
$502$	−5.63680e14	−0.789154
$503$	1.30287e15	1.80416	0.902082	−	0.431565i	$-0.142038\pi$
$503$	1.30287e15	1.80416	0.902082	+	0.431565i	$0.142038\pi$
$504$	−1.02371e14	−0.140220
$505$	0	0
$506$	−5.64515e14	−0.756567
$507$	−4.01542e14	−0.532338
$508$	−7.32676e13	−0.0960863
$509$	−1.09216e15	−1.41690	−0.708450	−	0.705761i	$-0.750605\pi$
$509$	−1.09216e15	−1.41690	−0.708450	+	0.705761i	$0.750605\pi$
$510$	0	0
$511$	−9.97902e14	−1.26699
$512$	−3.51844e13	−0.0441942
$513$	8.32556e13	0.103459
$514$	−5.15349e14	−0.633584
$515$	0	0
$516$	−1.31921e14	−0.158760
$517$	7.13397e14	0.849441
$518$	−2.32551e14	−0.273971
$519$	8.53961e14	0.995446
$520$	0	0
$521$	5.50995e14	0.628840	0.314420	−	0.949284i	$-0.398190\pi$
$521$	5.50995e14	0.628840	0.314420	+	0.949284i	$0.398190\pi$
$522$	2.46223e14	0.278062
$523$	−4.43612e14	−0.495729	−0.247865	−	0.968795i	$-0.579729\pi$
$523$	−4.43612e14	−0.495729	−0.247865	+	0.968795i	$0.579729\pi$
$524$	4.67746e14	0.517235
$525$	0	0
$526$	−2.75335e14	−0.298154
$527$	2.97573e14	0.318886
$528$	1.35332e14	0.143520
$529$	1.50411e14	0.157861
$530$	0	0
$531$	2.58553e14	0.265784
$532$	3.14347e14	0.319813
$533$	5.49053e14	0.552859
$534$	2.71846e13	0.0270923
$535$	0	0
$536$	−1.63342e14	−0.159475
$537$	1.11654e15	1.07898
$538$	−3.31078e14	−0.316686
$539$	4.36503e14	0.413284
$540$	0	0
$541$	−7.37013e14	−0.683739	−0.341869	−	0.939747i	$-0.611060\pi$
$541$	−7.37013e14	−0.683739	−0.341869	+	0.939747i	$0.611060\pi$
$542$	1.02273e15	0.939214
$543$	−2.32054e14	−0.210955
$544$	−1.61531e14	−0.145365
$545$	0	0
$546$	−1.53782e14	−0.135627
$547$	−2.04180e15	−1.78272	−0.891358	−	0.453299i	$-0.850247\pi$
$547$	−2.04180e15	−1.78272	−0.891358	+	0.453299i	$0.850247\pi$
$548$	2.93535e13	0.0253727
$549$	−4.59646e14	−0.393347
$550$	0	0
$551$	−7.56069e14	−0.634202
$552$	−2.64477e14	−0.219646
$553$	−2.38248e15	−1.95903
$554$	2.30914e14	0.187995
$555$	0	0
$556$	1.22975e15	0.981536
$557$	−2.16374e15	−1.71002	−0.855011	−	0.518611i	$-0.826449\pi$
$557$	−2.16374e15	−1.71002	−0.855011	+	0.518611i	$0.826449\pi$
$558$	−1.16802e14	−0.0914036
$559$	−1.98171e14	−0.153559
$560$	0	0
$561$	6.21306e14	0.472074
$562$	1.55671e15	1.17127
$563$	2.38895e15	1.77996	0.889982	−	0.455995i	$-0.150717\pi$
$563$	2.38895e15	1.77996	0.889982	+	0.455995i	$0.150717\pi$
$564$	3.34228e14	0.246609
$565$	0	0
$566$	1.44019e15	1.04215
$567$	1.84476e14	0.132201
$568$	−2.97998e14	−0.211494
$569$	8.76463e13	0.0616050	0.0308025	−	0.999525i	$-0.490194\pi$
$569$	8.76463e13	0.0616050	0.0308025	+	0.999525i	$0.490194\pi$
$570$	0	0
$571$	1.74097e15	1.20031	0.600155	−	0.799884i	$-0.295106\pi$
$571$	1.74097e15	1.20031	0.600155	+	0.799884i	$0.295106\pi$
$572$	2.03296e14	0.138819
$573$	1.01634e15	0.687367
$574$	−2.48683e15	−1.66583
$575$	0	0
$576$	6.34034e13	0.0416667
$577$	−1.81953e15	−1.18438	−0.592192	−	0.805797i	$-0.701737\pi$
$577$	−1.81953e15	−1.18438	−0.592192	+	0.805797i	$0.701737\pi$
$578$	3.55117e14	0.228964
$579$	−8.14362e14	−0.520098
$580$	0	0
$581$	−3.64455e15	−2.28389
$582$	1.26631e15	0.786072
$583$	−2.52036e15	−1.54983
$584$	6.18048e14	0.376489
$585$	0	0
$586$	−4.96728e14	−0.296949
$587$	−7.68456e14	−0.455103	−0.227551	−	0.973766i	$-0.573072\pi$
$587$	−7.68456e14	−0.455103	−0.227551	+	0.973766i	$0.573072\pi$
$588$	2.04503e14	0.119984
$589$	3.58660e14	0.208473
$590$	0	0
$591$	−1.75659e15	−1.00216
$592$	1.44030e14	0.0814112
$593$	−1.21221e15	−0.678855	−0.339427	−	0.940632i	$-0.610233\pi$
$593$	−1.21221e15	−0.678855	−0.339427	+	0.940632i	$0.610233\pi$
$594$	−2.43873e14	−0.135312
$595$	0	0
$596$	−6.80588e14	−0.370706
$597$	2.55485e14	0.137882
$598$	−3.97297e14	−0.212451
$599$	−1.29651e15	−0.686954	−0.343477	−	0.939161i	$-0.611605\pi$
$599$	−1.29651e15	−0.686954	−0.343477	+	0.939161i	$0.611605\pi$
$600$	0	0
$601$	−1.83742e15	−0.955871	−0.477935	−	0.878395i	$-0.658615\pi$
$601$	−1.83742e15	−0.955871	−0.477935	+	0.878395i	$0.658615\pi$
$602$	8.97579e14	0.462692
$603$	2.94348e14	0.150354
$604$	1.75066e15	0.886130
$605$	0	0
$606$	−1.03639e15	−0.515140
$607$	−9.38310e14	−0.462177	−0.231089	−	0.972933i	$-0.574229\pi$
$607$	−9.38310e14	−0.462177	−0.231089	+	0.972933i	$0.574229\pi$
$608$	−1.94690e14	−0.0950331
$609$	−1.67528e15	−0.810388
$610$	0	0
$611$	5.02077e14	0.238531
$612$	2.91084e14	0.137052
$613$	8.45726e13	0.0394636	0.0197318	−	0.999805i	$-0.493719\pi$
$613$	8.45726e13	0.0394636	0.0197318	+	0.999805i	$0.493719\pi$
$614$	−1.64901e15	−0.762599
$615$	0	0
$616$	−9.20787e14	−0.418278
$617$	−1.30490e15	−0.587501	−0.293750	−	0.955882i	$-0.594903\pi$
$617$	−1.30490e15	−0.587501	−0.293750	+	0.955882i	$0.594903\pi$
$618$	−6.60223e14	−0.294614
$619$	−1.79372e15	−0.793335	−0.396668	−	0.917962i	$-0.629833\pi$
$619$	−1.79372e15	−0.793335	−0.396668	+	0.917962i	$0.629833\pi$
$620$	0	0
$621$	4.76596e14	0.207084
$622$	3.89453e14	0.167729
$623$	−1.84962e14	−0.0789584
$624$	9.52445e13	0.0403018
$625$	0	0
$626$	−1.56990e15	−0.652697
$627$	7.48850e14	0.308620
$628$	1.02014e15	0.416756
$629$	6.61240e14	0.267781
$630$	0	0
$631$	−1.89462e15	−0.753981	−0.376991	−	0.926217i	$-0.623041\pi$
$631$	−1.89462e15	−0.753981	−0.376991	+	0.926217i	$0.623041\pi$
$632$	1.47558e15	0.582129
$633$	9.25816e13	0.0362080
$634$	8.49874e13	0.0329506
$635$	0	0
$636$	−1.18079e15	−0.449946
$637$	3.07204e14	0.116054
$638$	2.21468e15	0.829463
$639$	5.37002e14	0.199398
$640$	0	0
$641$	−2.48785e14	−0.0908040	−0.0454020	−	0.998969i	$-0.514457\pi$
$641$	−2.48785e14	−0.0908040	−0.0454020	+	0.998969i	$0.514457\pi$
$642$	−3.25931e14	−0.117946
$643$	−4.34116e15	−1.55756	−0.778780	−	0.627297i	$-0.784161\pi$
$643$	−4.34116e15	−1.55756	−0.778780	+	0.627297i	$0.784161\pi$
$644$	1.79948e15	0.640139
$645$	0	0
$646$	−8.93819e14	−0.312587
$647$	1.76476e15	0.611944	0.305972	−	0.952041i	$-0.401019\pi$
$647$	1.76476e15	0.611944	0.305972	+	0.952041i	$0.401019\pi$
$648$	−1.14255e14	−0.0392837
$649$	2.32558e15	0.792838
$650$	0	0
$651$	7.94713e14	0.266388
$652$	−2.62637e14	−0.0872960
$653$	8.57218e14	0.282533	0.141266	−	0.989972i	$-0.454883\pi$
$653$	8.57218e14	0.282533	0.141266	+	0.989972i	$0.454883\pi$
$654$	−2.07784e15	−0.679102
$655$	0	0
$656$	1.54021e15	0.495005
$657$	−1.11374e15	−0.354957
$658$	−2.27406e15	−0.718720
$659$	2.32789e15	0.729613	0.364807	−	0.931083i	$-0.381135\pi$
$659$	2.32789e15	0.729613	0.364807	+	0.931083i	$0.381135\pi$
$660$	0	0
$661$	−3.06684e15	−0.945328	−0.472664	−	0.881243i	$-0.656708\pi$
$661$	−3.06684e15	−0.945328	−0.472664	+	0.881243i	$0.656708\pi$
$662$	−2.83744e15	−0.867375
$663$	4.37266e14	0.132563
$664$	2.25724e15	0.678663
$665$	0	0
$666$	−2.59547e14	−0.0767552
$667$	−4.32811e15	−1.26942
$668$	1.05201e15	0.306020
$669$	−8.51882e14	−0.245774
$670$	0	0
$671$	−4.13433e15	−1.17336
$672$	−4.31391e14	−0.121434
$673$	2.69617e15	0.752775	0.376387	−	0.926462i	$-0.377166\pi$
$673$	2.69617e15	0.752775	0.376387	+	0.926462i	$0.377166\pi$
$674$	1.69167e15	0.468476
$675$	0	0
$676$	−1.69210e15	−0.461018
$677$	−5.36073e15	−1.44872	−0.724362	−	0.689419i	$-0.757866\pi$
$677$	−5.36073e15	−1.44872	−0.724362	+	0.689419i	$0.757866\pi$
$678$	1.84874e15	0.495578
$679$	−8.61584e15	−2.29094
$680$	0	0
$681$	1.17840e15	0.308307
$682$	−1.05059e15	−0.272658
$683$	−7.26562e15	−1.87050	−0.935252	−	0.353981i	$-0.884828\pi$
$683$	−7.26562e15	−1.87050	−0.935252	+	0.353981i	$0.884828\pi$
$684$	3.50838e14	0.0895980
$685$	0	0
$686$	1.95626e15	0.491636
$687$	3.43261e14	0.0855780
$688$	−5.55913e14	−0.137490
$689$	−1.77379e15	−0.435207
$690$	0	0
$691$	2.83285e15	0.684061	0.342030	−	0.939689i	$-0.388885\pi$
$691$	2.83285e15	0.684061	0.342030	+	0.939689i	$0.388885\pi$
$692$	3.59859e15	0.862082
$693$	1.65929e15	0.394357
$694$	−6.85171e14	−0.161556
$695$	0	0
$696$	1.03758e15	0.240809
$697$	7.07108e15	1.62819
$698$	4.13703e15	0.945112
$699$	2.40190e15	0.544416
$700$	0	0
$701$	6.35487e15	1.41794	0.708969	−	0.705239i	$-0.249161\pi$
$701$	6.35487e15	1.41794	0.708969	+	0.705239i	$0.249161\pi$
$702$	−1.71634e14	−0.0379969
$703$	7.96982e14	0.175063
$704$	5.70288e14	0.124292
$705$	0	0
$706$	−1.03735e15	−0.222587
$707$	7.05153e15	1.50133
$708$	1.08954e15	0.230176
$709$	5.55781e15	1.16506	0.582531	−	0.812808i	$-0.302062\pi$
$709$	5.55781e15	1.16506	0.582531	+	0.812808i	$0.302062\pi$
$710$	0	0
$711$	−2.65905e15	−0.548837
$712$	1.14556e14	0.0234627
$713$	2.05315e15	0.417280
$714$	−1.98051e15	−0.399426
$715$	0	0
$716$	4.70508e15	0.934427
$717$	−1.42823e15	−0.281476
$718$	3.23763e15	0.633202
$719$	1.05266e15	0.204305	0.102153	−	0.994769i	$-0.467427\pi$
$719$	1.05266e15	0.204305	0.102153	+	0.994769i	$0.467427\pi$
$720$	0	0
$721$	4.49210e15	0.858629
$722$	2.65038e15	0.502752
$723$	−2.26715e15	−0.426795
$724$	−9.77872e14	−0.182692
$725$	0	0
$726$	2.50490e13	0.00460934
$727$	−3.83121e15	−0.699675	−0.349838	−	0.936810i	$-0.613763\pi$
$727$	−3.83121e15	−0.699675	−0.349838	+	0.936810i	$0.613763\pi$
$728$	−6.48036e14	−0.117456
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−2.55219e15	−0.452238
$732$	−1.93694e15	−0.340648
$733$	7.31689e15	1.27719	0.638594	−	0.769544i	$-0.279517\pi$
$733$	7.31689e15	1.27719	0.638594	+	0.769544i	$0.279517\pi$
$734$	1.98111e14	0.0343226
$735$	0	0
$736$	−1.11450e15	−0.190219
$737$	2.64754e15	0.448509
$738$	−2.77551e15	−0.466695
$739$	−1.04805e16	−1.74920	−0.874598	−	0.484848i	$-0.838875\pi$
$739$	−1.04805e16	−1.74920	−0.874598	+	0.484848i	$0.838875\pi$
$740$	0	0
$741$	5.27029e14	0.0866631
$742$	8.03403e15	1.31133
$743$	−3.28089e15	−0.531561	−0.265780	−	0.964034i	$-0.585630\pi$
$743$	−3.28089e15	−0.531561	−0.265780	+	0.964034i	$0.585630\pi$
$744$	−4.92204e14	−0.0791578
$745$	0	0
$746$	6.94910e15	1.10120
$747$	−4.06763e15	−0.639850
$748$	2.61818e15	0.408828
$749$	2.21761e15	0.343744
$750$	0	0
$751$	9.30704e15	1.42165	0.710824	−	0.703370i	$-0.248322\pi$
$751$	9.30704e15	1.42165	0.710824	+	0.703370i	$0.248322\pi$
$752$	1.40843e15	0.213569
$753$	4.28044e15	0.644342
$754$	1.55866e15	0.232921
$755$	0	0
$756$	7.77381e14	0.114489
$757$	−6.20592e15	−0.907359	−0.453680	−	0.891165i	$-0.649889\pi$
$757$	−6.20592e15	−0.907359	−0.453680	+	0.891165i	$0.649889\pi$
$758$	−4.34734e15	−0.631020
$759$	4.28678e15	0.617735
$760$	0	0
$761$	−6.29521e14	−0.0894118	−0.0447059	−	0.999000i	$-0.514235\pi$
$761$	−6.29521e14	−0.0894118	−0.0447059	+	0.999000i	$0.514235\pi$
$762$	5.56376e14	0.0784542
$763$	1.41375e16	1.97919
$764$	4.28285e15	0.595277
$765$	0	0
$766$	−4.90271e15	−0.671703
$767$	1.63670e15	0.222636
$768$	2.67181e14	0.0360844
$769$	4.37850e15	0.587126	0.293563	−	0.955940i	$-0.405159\pi$
$769$	4.37850e15	0.587126	0.293563	+	0.955940i	$0.405159\pi$
$770$	0	0
$771$	3.91343e15	0.517319
$772$	−3.43171e15	−0.450418
$773$	6.24169e15	0.813420	0.406710	−	0.913557i	$-0.366676\pi$
$773$	6.24169e15	0.813420	0.406710	+	0.913557i	$0.366676\pi$
$774$	1.00177e15	0.129627
$775$	0	0
$776$	5.33620e15	0.680758
$777$	1.76594e15	0.223697
$778$	−2.03027e15	−0.255368
$779$	8.52266e15	1.06443
$780$	0	0
$781$	4.83011e15	0.594808
$782$	−5.11666e15	−0.625676
$783$	−1.86976e15	−0.227037
$784$	8.61773e14	0.103909
$785$	0	0
$786$	−3.55194e15	−0.422320
$787$	1.67198e16	1.97411	0.987053	−	0.160397i	$-0.0512776\pi$
$787$	1.67198e16	1.97411	0.987053	+	0.160397i	$0.0512776\pi$
$788$	−7.40224e15	−0.867899
$789$	2.09083e15	0.243441
$790$	0	0
$791$	−1.25787e16	−1.44432
$792$	−1.02768e15	−0.117184
$793$	−2.90968e15	−0.329490
$794$	−9.03078e15	−1.01558
$795$	0	0
$796$	1.07661e15	0.119409
$797$	−4.36301e15	−0.480580	−0.240290	−	0.970701i	$-0.577242\pi$
$797$	−4.36301e15	−0.480580	−0.240290	+	0.970701i	$0.577242\pi$
$798$	−2.38707e15	−0.261126
$799$	6.46610e15	0.702481
$800$	0	0
$801$	−2.06433e14	−0.0221208
$802$	1.31708e15	0.140170
$803$	−1.00177e16	−1.05884
$804$	1.24038e15	0.130211
$805$	0	0
$806$	−7.39388e14	−0.0765649
$807$	2.51413e15	0.258573
$808$	−4.36735e15	−0.446124
$809$	3.55238e15	0.360415	0.180208	−	0.983629i	$-0.442323\pi$
$809$	3.55238e15	0.360415	0.180208	+	0.983629i	$0.442323\pi$
$810$	0	0
$811$	1.38794e16	1.38917	0.694585	−	0.719411i	$-0.255588\pi$
$811$	1.38794e16	1.38917	0.694585	+	0.719411i	$0.255588\pi$
$812$	−7.05963e15	−0.701817
$813$	−7.76635e15	−0.766865
$814$	−2.33452e15	−0.228962
$815$	0	0
$816$	1.22662e15	0.118690
$817$	−3.07611e15	−0.295652
$818$	−2.33158e15	−0.222591
$819$	1.16778e15	0.110739
$820$	0	0
$821$	9.65670e15	0.903528	0.451764	−	0.892138i	$-0.350795\pi$
$821$	9.65670e15	0.903528	0.451764	+	0.892138i	$0.350795\pi$
$822$	−2.22903e14	−0.0207167
$823$	−1.62599e16	−1.50114	−0.750568	−	0.660793i	$-0.770220\pi$
$823$	−1.62599e16	−1.50114	−0.750568	+	0.660793i	$0.770220\pi$
$824$	−2.78217e15	−0.255144
$825$	0	0
$826$	−7.41313e15	−0.670828
$827$	−9.52899e15	−0.856578	−0.428289	−	0.903642i	$-0.640883\pi$
$827$	−9.52899e15	−0.856578	−0.428289	+	0.903642i	$0.640883\pi$
$828$	2.00837e15	0.179340
$829$	1.66053e16	1.47298	0.736492	−	0.676446i	$-0.236481\pi$
$829$	1.66053e16	1.47298	0.736492	+	0.676446i	$0.236481\pi$
$830$	0	0
$831$	−1.75350e15	−0.153497
$832$	4.01360e14	0.0349024
$833$	3.95638e15	0.341783
$834$	−9.33843e15	−0.801420
$835$	0	0
$836$	3.15565e15	0.267272
$837$	8.86967e14	0.0746307
$838$	1.42338e15	0.118981
$839$	−1.01371e16	−0.841827	−0.420913	−	0.907101i	$-0.638290\pi$
$839$	−1.01371e16	−0.841827	−0.420913	+	0.907101i	$0.638290\pi$
$840$	0	0
$841$	4.77933e15	0.391732
$842$	−1.30650e15	−0.106388
$843$	−1.18213e16	−0.956341
$844$	3.90138e14	0.0313570
$845$	0	0
$846$	−2.53805e15	−0.201355
$847$	−1.70431e14	−0.0134335
$848$	−4.97586e15	−0.389665
$849$	−1.09365e16	−0.850913
$850$	0	0
$851$	4.56231e15	0.350407
$852$	2.26292e15	0.172684
$853$	2.03075e16	1.53970	0.769851	−	0.638224i	$-0.220331\pi$
$853$	2.03075e16	1.53970	0.769851	+	0.638224i	$0.220331\pi$
$854$	1.31788e16	0.992791
$855$	0	0
$856$	−1.37347e15	−0.102144
$857$	4.99342e15	0.368981	0.184490	−	0.982834i	$-0.440937\pi$
$857$	4.99342e15	0.368981	0.184490	+	0.982834i	$0.440937\pi$
$858$	−1.54378e15	−0.113345
$859$	5.27939e15	0.385142	0.192571	−	0.981283i	$-0.438317\pi$
$859$	5.27939e15	0.385142	0.192571	+	0.981283i	$0.438317\pi$
$860$	0	0
$861$	1.88843e16	1.36014
$862$	−1.60586e16	−1.14926
$863$	6.94499e15	0.493869	0.246935	−	0.969032i	$-0.420577\pi$
$863$	6.94499e15	0.493869	0.246935	+	0.969032i	$0.420577\pi$
$864$	−4.81469e14	−0.0340207
$865$	0	0
$866$	9.91182e15	0.691520
$867$	−2.69667e15	−0.186949
$868$	3.34891e15	0.230699
$869$	−2.39170e16	−1.63719
$870$	0	0
$871$	1.86330e15	0.125945
$872$	−8.75601e15	−0.588120
$873$	−9.61601e15	−0.641825
$874$	−6.16703e15	−0.409037
$875$	0	0
$876$	−4.69330e15	−0.307402
$877$	1.66689e16	1.08495	0.542473	−	0.840073i	$-0.317488\pi$
$877$	1.66689e16	1.08495	0.542473	+	0.840073i	$0.317488\pi$
$878$	3.45688e14	0.0223596
$879$	3.77203e15	0.242458
$880$	0	0
$881$	−7.30716e15	−0.463854	−0.231927	−	0.972733i	$-0.574503\pi$
$881$	−7.30716e15	−0.463854	−0.231927	+	0.972733i	$0.574503\pi$
$882$	−1.55294e15	−0.0979667
$883$	1.41660e16	0.888100	0.444050	−	0.896002i	$-0.353541\pi$
$883$	1.41660e16	0.888100	0.444050	+	0.896002i	$0.353541\pi$
$884$	1.84263e15	0.114803
$885$	0	0
$886$	1.10431e16	0.679525
$887$	−1.48432e16	−0.907713	−0.453857	−	0.891075i	$-0.649952\pi$
$887$	−1.48432e16	−0.907713	−0.453857	+	0.891075i	$0.649952\pi$
$888$	−1.09373e15	−0.0664720
$889$	−3.78554e15	−0.228648
$890$	0	0
$891$	1.85191e15	0.110482
$892$	−3.58982e15	−0.212846
$893$	7.79348e15	0.459249
$894$	5.16822e15	0.302680
$895$	0	0
$896$	−1.81788e15	−0.105165
$897$	3.01697e15	0.173465
$898$	1.64086e16	0.937673
$899$	−8.05482e15	−0.457486
$900$	0	0
$901$	−2.28441e16	−1.28170
$902$	−2.49646e16	−1.39216
$903$	−6.81599e15	−0.377786
$904$	7.79057e15	0.429183
$905$	0	0
$906$	−1.32940e16	−0.723522
$907$	1.13330e16	0.613061	0.306530	−	0.951861i	$-0.400832\pi$
$907$	1.13330e16	0.613061	0.306530	+	0.951861i	$0.400832\pi$
$908$	4.96577e15	0.267002
$909$	7.87011e15	0.420610
$910$	0	0
$911$	−1.91130e16	−1.00920	−0.504602	−	0.863352i	$-0.668361\pi$
$911$	−1.91130e16	−1.00920	−0.504602	+	0.863352i	$0.668361\pi$
$912$	1.47843e15	0.0775942
$913$	−3.65867e16	−1.90868
$914$	7.46766e15	0.387240
$915$	0	0
$916$	1.44650e15	0.0741127
$917$	2.41671e16	1.23082
$918$	−2.21042e15	−0.111902
$919$	7.09706e15	0.357144	0.178572	−	0.983927i	$-0.442852\pi$
$919$	7.09706e15	0.357144	0.178572	+	0.983927i	$0.442852\pi$
$920$	0	0
$921$	1.25221e16	0.622660
$922$	−5.21004e15	−0.257526
$923$	3.39936e15	0.167027
$924$	6.99223e15	0.341523
$925$	0	0
$926$	7.96562e15	0.384467
$927$	5.01357e15	0.240552
$928$	4.37237e15	0.208546
$929$	−1.87622e16	−0.889604	−0.444802	−	0.895629i	$-0.646726\pi$
$929$	−1.87622e16	−0.889604	−0.444802	+	0.895629i	$0.646726\pi$
$930$	0	0
$931$	4.76856e15	0.223442
$932$	1.01216e16	0.471478
$933$	−2.95741e15	−0.136950
$934$	−1.12051e16	−0.515830
$935$	0	0
$936$	−7.23263e14	−0.0329063
$937$	3.08165e16	1.39385	0.696924	−	0.717145i	$-0.254552\pi$
$937$	3.08165e16	1.39385	0.696924	+	0.717145i	$0.254552\pi$
$938$	−8.43944e15	−0.379488
$939$	1.19214e16	0.532925
$940$	0	0
$941$	−4.26974e16	−1.88651	−0.943255	−	0.332070i	$-0.892253\pi$
$941$	−4.26974e16	−1.88651	−0.943255	+	0.332070i	$0.892253\pi$
$942$	−7.74669e15	−0.340280
$943$	4.87879e16	2.13058
$944$	4.59131e15	0.199338
$945$	0	0
$946$	9.01055e15	0.386679
$947$	1.63317e16	0.696799	0.348399	−	0.937346i	$-0.386725\pi$
$947$	1.63317e16	0.696799	0.348399	+	0.937346i	$0.386725\pi$
$948$	−1.12052e16	−0.475306
$949$	−7.05028e15	−0.297332
$950$	0	0
$951$	−6.45373e14	−0.0269041
$952$	−8.34585e15	−0.345913
$953$	−2.78821e16	−1.14899	−0.574494	−	0.818509i	$-0.694801\pi$
$953$	−2.78821e16	−1.14899	−0.574494	+	0.818509i	$0.694801\pi$
$954$	8.96666e15	0.367379
$955$	0	0
$956$	−6.01853e15	−0.243765
$957$	−1.68177e16	−0.677254
$958$	−1.41170e16	−0.565240
$959$	1.51662e15	0.0603772
$960$	0	0
$961$	−2.15875e16	−0.849617
$962$	−1.64300e15	−0.0642946
$963$	2.47504e15	0.0963026
$964$	−9.55374e15	−0.369615
$965$	0	0
$966$	−1.36648e16	−0.522671
$967$	−7.21160e15	−0.274275	−0.137137	−	0.990552i	$-0.543790\pi$
$967$	−7.21160e15	−0.274275	−0.137137	+	0.990552i	$0.543790\pi$
$968$	1.05556e14	0.00399180
$969$	6.78744e15	0.255226
$970$	0	0
$971$	−4.56053e16	−1.69555	−0.847773	−	0.530359i	$-0.822057\pi$
$971$	−4.56053e16	−1.69555	−0.847773	+	0.530359i	$0.822057\pi$
$972$	8.67624e14	0.0320750
$973$	6.35379e16	2.33567
$974$	1.39457e14	0.00509760
$975$	0	0
$976$	−8.16227e15	−0.295010
$977$	4.85178e16	1.74374	0.871868	−	0.489740i	$-0.162908\pi$
$977$	4.85178e16	1.74374	0.871868	+	0.489740i	$0.162908\pi$
$978$	1.99440e15	0.0712769
$979$	−1.85678e15	−0.0659867
$980$	0	0
$981$	1.57786e16	0.554485
$982$	2.15630e15	0.0753522
$983$	−4.37954e16	−1.52189	−0.760947	−	0.648814i	$-0.775265\pi$
$983$	−4.37954e16	−1.52189	−0.760947	+	0.648814i	$0.775265\pi$
$984$	−1.16960e16	−0.404170
$985$	0	0
$986$	2.00735e16	0.685960
$987$	1.72687e16	0.586832
$988$	2.22090e15	0.0750525
$989$	−1.76092e16	−0.591778
$990$	0	0
$991$	−2.55075e16	−0.847740	−0.423870	−	0.905723i	$-0.639329\pi$
$991$	−2.55075e16	−0.847740	−0.423870	+	0.905723i	$0.639329\pi$
$992$	−2.07414e15	−0.0685527
$993$	2.15468e16	0.708209
$994$	−1.53967e16	−0.503273
$995$	0	0
$996$	−1.71409e16	−0.554126
$997$	−8.08618e15	−0.259968	−0.129984	−	0.991516i	$-0.541493\pi$
$997$	−8.08618e15	−0.259968	−0.129984	+	0.991516i	$0.541493\pi$
$998$	−2.58012e16	−0.824939
$999$	1.97094e15	0.0626704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.12.a.t.1.3		3
5.2	odd	4		30.12.c.b.19.1	✓	6
5.3	odd	4		30.12.c.b.19.4	yes	6
5.4	even	2		150.12.a.u.1.1		3
15.2	even	4		90.12.c.c.19.6		6
15.8	even	4		90.12.c.c.19.3		6
20.3	even	4		240.12.f.b.49.1		6
20.7	even	4		240.12.f.b.49.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.12.c.b.19.1	✓	6	5.2	odd	4
30.12.c.b.19.4	yes	6	5.3	odd	4
90.12.c.c.19.3		6	15.8	even	4
90.12.c.c.19.6		6	15.2	even	4
150.12.a.t.1.3		3	1.1	even	1	trivial
150.12.a.u.1.1		3	5.4	even	2
240.12.f.b.49.1		6	20.3	even	4
240.12.f.b.49.4		6	20.7	even	4

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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

\(f(q)\)	\(=\)	\(q-32.0000 q^{2} +243.000 q^{3} +1024.00 q^{4} -7776.00 q^{6} +52907.3 q^{7} -32768.0 q^{8} +59049.0 q^{9} +531122. q^{11} +248832. q^{12} +373795. q^{13} -1.69303e6 q^{14} +1.04858e6 q^{16} +4.81399e6 q^{17} -1.88957e6 q^{18} +5.80222e6 q^{19} +1.28565e7 q^{21} -1.69959e7 q^{22} +3.32148e7 q^{23} -7.96262e6 q^{24} -1.19614e7 q^{26} +1.43489e7 q^{27} +5.41770e7 q^{28} -1.30307e8 q^{29} +6.18143e7 q^{31} -3.35544e7 q^{32} +1.29063e8 q^{33} -1.54048e8 q^{34} +6.04662e7 q^{36} +1.37358e8 q^{37} -1.85671e8 q^{38} +9.08322e7 q^{39} +1.46886e9 q^{41} -4.11407e8 q^{42} -5.30160e8 q^{43} +5.43869e8 q^{44} -1.06287e9 q^{46} +1.34319e9 q^{47} +2.54804e8 q^{48} +8.21851e8 q^{49} +1.16980e9 q^{51} +3.82766e8 q^{52} -4.74535e9 q^{53} -4.59165e8 q^{54} -1.73366e9 q^{56} +1.40994e9 q^{57} +4.16982e9 q^{58} +4.37861e9 q^{59} -7.78414e9 q^{61} -1.97806e9 q^{62} +3.12412e9 q^{63} +1.07374e9 q^{64} -4.13000e9 q^{66} +4.98481e9 q^{67} +4.92953e9 q^{68} +8.07119e9 q^{69} +9.09417e9 q^{71} -1.93492e9 q^{72} -1.88613e10 q^{73} -4.39545e9 q^{74} +5.94148e9 q^{76} +2.81002e10 q^{77} -2.90663e9 q^{78} -4.50312e10 q^{79} +3.48678e9 q^{81} -4.70035e10 q^{82} -6.88856e10 q^{83} +1.31650e10 q^{84} +1.69651e10 q^{86} -3.16645e10 q^{87} -1.74038e10 q^{88} -3.49597e9 q^{89} +1.97765e10 q^{91} +3.40119e10 q^{92} +1.50209e10 q^{93} -4.29820e10 q^{94} -8.15373e9 q^{96} -1.62848e11 q^{97} -2.62992e10 q^{98} +3.13622e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 96 q^{2} + 729 q^{3} + 3072 q^{4} - 23328 q^{6} - 67384 q^{7} - 98304 q^{8} + 177147 q^{9} + 876700 q^{11} + 746496 q^{12} - 315192 q^{13} + 2156288 q^{14} + 3145728 q^{16} - 2874324 q^{17} - 5668704 q^{18}+ \cdots + 51768258300 q^{99}+O(q^{100}) \) 3 * q - 96 * q^2 + 729 * q^3 + 3072 * q^4 - 23328 * q^6 - 67384 * q^7 - 98304 * q^8 + 177147 * q^9 + 876700 * q^11 + 746496 * q^12 - 315192 * q^13 + 2156288 * q^14 + 3145728 * q^16 - 2874324 * q^17 - 5668704 * q^18 + 2034912 * q^19 - 16374312 * q^21 - 28054400 * q^22 - 49866300 * q^23 - 23887872 * q^24 + 10086144 * q^26 + 43046721 * q^27 - 69001216 * q^28 - 35857782 * q^29 - 61121676 * q^31 - 100663296 * q^32 + 213038100 * q^33 + 91978368 * q^34 + 181398528 * q^36 + 935590592 * q^37 - 65117184 * q^38 - 76591656 * q^39 + 1003384282 * q^41 + 523977984 * q^42 + 980338484 * q^43 + 897740800 * q^44 + 1595721600 * q^46 + 2190634500 * q^47 + 764411904 * q^48 + 4795452723 * q^49 - 698460732 * q^51 - 322756608 * q^52 - 6993816924 * q^53 - 1377495072 * q^54 + 2208038912 * q^56 + 494483616 * q^57 + 1147449024 * q^58 + 2034730100 * q^59 - 6193233330 * q^61 + 1955893632 * q^62 - 3978957816 * q^63 + 3221225472 * q^64 - 6817219200 * q^66 - 5079955484 * q^67 - 2943307776 * q^68 - 12117510900 * q^69 + 6851752800 * q^71 - 5804752896 * q^72 - 54889949992 * q^73 - 29938898944 * q^74 + 2083749888 * q^76 + 32141607600 * q^77 + 2450932992 * q^78 + 6719163876 * q^79 + 10460353203 * q^81 - 32108297024 * q^82 - 57462212652 * q^83 - 16767295488 * q^84 - 31370831488 * q^86 - 8713441026 * q^87 - 28727705600 * q^88 + 63718294546 * q^89 + 135921691776 * q^91 - 51063091200 * q^92 - 14852567268 * q^93 - 70100304000 * q^94 - 24461180928 * q^96 - 315259594808 * q^97 - 153454487136 * q^98 + 51768258300 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more