LMFDB - Embedded newform 54.12.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [54,12,Mod(1,54)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("54.1"); S:= CuspForms(chi, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	54.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,64,0,2048,-3720] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.4905317502$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.72015$ of defining polynomial
Character	$\chi$	$=$	54.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+32.0000 q^{2} +1024.00 q^{4} +3777.77 q^{5} -53009.3 q^{7} +32768.0 q^{8} +120888. q^{10} +191435. q^{11} -1.32530e6 q^{13} -1.69630e6 q^{14} +1.04858e6 q^{16} -7.39161e6 q^{17} -6.06542e6 q^{19} +3.86843e6 q^{20} +6.12592e6 q^{22} +4.65331e7 q^{23} -3.45566e7 q^{25} -4.24096e7 q^{26} -5.42816e7 q^{28} -6.50061e7 q^{29} -5.59140e7 q^{31} +3.35544e7 q^{32} -2.36531e8 q^{34} -2.00257e8 q^{35} -2.00878e8 q^{37} -1.94093e8 q^{38} +1.23790e8 q^{40} -5.72523e8 q^{41} +9.67573e7 q^{43} +1.96030e8 q^{44} +1.48906e9 q^{46} -2.39673e9 q^{47} +8.32663e8 q^{49} -1.10581e9 q^{50} -1.35711e9 q^{52} -3.14068e9 q^{53} +7.23197e8 q^{55} -1.73701e9 q^{56} -2.08020e9 q^{58} -2.20467e9 q^{59} +1.09589e10 q^{61} -1.78925e9 q^{62} +1.07374e9 q^{64} -5.00667e9 q^{65} -1.06335e10 q^{67} -7.56901e9 q^{68} -6.40822e9 q^{70} -2.95131e10 q^{71} +2.30487e10 q^{73} -6.42809e9 q^{74} -6.21099e9 q^{76} -1.01478e10 q^{77} -3.32219e10 q^{79} +3.96127e9 q^{80} -1.83207e10 q^{82} +2.34153e10 q^{83} -2.79238e10 q^{85} +3.09623e9 q^{86} +6.27295e9 q^{88} +4.47401e10 q^{89} +7.02532e10 q^{91} +4.76499e10 q^{92} -7.66954e10 q^{94} -2.29137e10 q^{95} +2.46368e10 q^{97} +2.66452e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 64 q^{2} + 2048 q^{4} - 3720 q^{5} - 6794 q^{7} + 65536 q^{8} - 119040 q^{10} - 832632 q^{11} + 984634 q^{13} - 217408 q^{14} + 2097152 q^{16} - 5111064 q^{17} - 22919270 q^{19} - 3809280 q^{20} - 26644224 q^{22}+ \cdots + 31718195328 q^{98}+O(q^{100}) $ 2 * q + 64 * q^2 + 2048 * q^4 - 3720 * q^5 - 6794 * q^7 + 65536 * q^8 - 119040 * q^10 - 832632 * q^11 + 984634 * q^13 - 217408 * q^14 + 2097152 * q^16 - 5111064 * q^17 - 22919270 * q^19 - 3809280 * q^20 - 26644224 * q^22 + 5263656 * q^23 - 27168250 * q^25 + 31508288 * q^26 - 6957056 * q^28 - 37534848 * q^29 - 328859480 * q^31 + 67108864 * q^32 - 163554048 * q^34 - 546768600 * q^35 - 96567386 * q^37 - 733416640 * q^38 - 121896960 * q^40 - 478653312 * q^41 + 133709104 * q^43 - 852615168 * q^44 + 168436992 * q^46 - 4431713208 * q^47 + 991193604 * q^49 - 869384000 * q^50 + 1008265216 * q^52 - 4754972016 * q^53 + 8401412160 * q^55 - 222625792 * q^56 - 1201115136 * q^58 - 9940537848 * q^59 + 6058984510 * q^61 - 10523503360 * q^62 + 2147483648 * q^64 - 22326000360 * q^65 - 6772264670 * q^67 - 5233729536 * q^68 - 17496595200 * q^70 - 25581365424 * q^71 + 15807624466 * q^73 - 3090156352 * q^74 - 23469332480 * q^76 - 57475455528 * q^77 - 28139021846 * q^79 - 3900702720 * q^80 - 15316905984 * q^82 + 77207454480 * q^83 - 45022737600 * q^85 + 4278691328 * q^86 - 27283685376 * q^88 + 38295584136 * q^89 + 177007512158 * q^91 + 5389983744 * q^92 - 141814822656 * q^94 + 103452470040 * q^95 + 109290600874 * q^97 + 31718195328 * q^98

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
$3$	0	0
$3$	0	0
$4$	1024.00	0.500000
$5$	3777.77	0.540630	0.270315	−	0.962772i	$-0.412872\pi$
$5$	3777.77	0.540630	0.270315	+	0.962772i	$0.412872\pi$
$6$	0	0
$7$	−53009.3	−1.19210	−0.596051	−	0.802947i	$-0.703264\pi$
$7$	−53009.3	−1.19210	−0.596051	+	0.802947i	$0.703264\pi$
$8$	32768.0	0.353553
$9$	0	0
$10$	120888.	0.382283
$11$	191435.	0.358395	0.179197	−	0.983813i	$-0.442650\pi$
$11$	191435.	0.358395	0.179197	+	0.983813i	$0.442650\pi$
$12$	0	0
$13$	−1.32530e6	−0.989977	−0.494989	−	0.868899i	$-0.664828\pi$
$13$	−1.32530e6	−0.989977	−0.494989	+	0.868899i	$0.664828\pi$
$14$	−1.69630e6	−0.842943
$15$	0	0
$16$	1.04858e6	0.250000
$17$	−7.39161e6	−1.26261	−0.631306	−	0.775534i	$-0.717481\pi$
$17$	−7.39161e6	−1.26261	−0.631306	+	0.775534i	$0.717481\pi$
$18$	0	0
$19$	−6.06542e6	−0.561974	−0.280987	−	0.959712i	$-0.590662\pi$
$19$	−6.06542e6	−0.561974	−0.280987	+	0.959712i	$0.590662\pi$
$20$	3.86843e6	0.270315
$21$	0	0
$22$	6.12592e6	0.253423
$23$	4.65331e7	1.50751	0.753753	−	0.657158i	$-0.228242\pi$
$23$	4.65331e7	1.50751	0.753753	+	0.657158i	$0.228242\pi$
$24$	0	0
$25$	−3.45566e7	−0.707719
$26$	−4.24096e7	−0.700020
$27$	0	0
$28$	−5.42816e7	−0.596051
$29$	−6.50061e7	−0.588525	−0.294263	−	0.955725i	$-0.595074\pi$
$29$	−6.50061e7	−0.588525	−0.294263	+	0.955725i	$0.595074\pi$
$30$	0	0
$31$	−5.59140e7	−0.350777	−0.175389	−	0.984499i	$-0.556118\pi$
$31$	−5.59140e7	−0.350777	−0.175389	+	0.984499i	$0.556118\pi$
$32$	3.35544e7	0.176777
$33$	0	0
$34$	−2.36531e8	−0.892801
$35$	−2.00257e8	−0.644485
$36$	0	0
$37$	−2.00878e8	−0.476236	−0.238118	−	0.971236i	$-0.576531\pi$
$37$	−2.00878e8	−0.476236	−0.238118	+	0.971236i	$0.576531\pi$
$38$	−1.94093e8	−0.397376
$39$	0	0
$40$	1.23790e8	0.191141
$41$	−5.72523e8	−0.771759	−0.385880	−	0.922549i	$-0.626102\pi$
$41$	−5.72523e8	−0.771759	−0.385880	+	0.922549i	$0.626102\pi$
$42$	0	0
$43$	9.67573e7	0.100371	0.0501854	−	0.998740i	$-0.484019\pi$
$43$	9.67573e7	0.100371	0.0501854	+	0.998740i	$0.484019\pi$
$44$	1.96030e8	0.179197
$45$	0	0
$46$	1.48906e9	1.06597
$47$	−2.39673e9	−1.52434	−0.762169	−	0.647379i	$-0.775865\pi$
$47$	−2.39673e9	−1.52434	−0.762169	+	0.647379i	$0.775865\pi$
$48$	0	0
$49$	8.32663e8	0.421105
$50$	−1.10581e9	−0.500433
$51$	0	0
$52$	−1.35711e9	−0.494989
$53$	−3.14068e9	−1.03159	−0.515794	−	0.856713i	$-0.672503\pi$
$53$	−3.14068e9	−1.03159	−0.515794	+	0.856713i	$0.672503\pi$
$54$	0	0
$55$	7.23197e8	0.193759
$56$	−1.73701e9	−0.421471
$57$	0	0
$58$	−2.08020e9	−0.416150
$59$	−2.20467e9	−0.401473	−0.200737	−	0.979645i	$-0.564334\pi$
$59$	−2.20467e9	−0.401473	−0.200737	+	0.979645i	$0.564334\pi$
$60$	0	0
$61$	1.09589e10	1.66131	0.830657	−	0.556785i	$-0.187965\pi$
$61$	1.09589e10	1.66131	0.830657	+	0.556785i	$0.187965\pi$
$62$	−1.78925e9	−0.248037
$63$	0	0
$64$	1.07374e9	0.125000
$65$	−5.00667e9	−0.535211
$66$	0	0
$67$	−1.06335e10	−0.962196	−0.481098	−	0.876667i	$-0.659762\pi$
$67$	−1.06335e10	−0.962196	−0.481098	+	0.876667i	$0.659762\pi$
$68$	−7.56901e9	−0.631306
$69$	0	0
$70$	−6.40822e9	−0.455720
$71$	−2.95131e10	−1.94131	−0.970653	−	0.240486i	$-0.922693\pi$
$71$	−2.95131e10	−1.94131	−0.970653	+	0.240486i	$0.922693\pi$
$72$	0	0
$73$	2.30487e10	1.30128	0.650641	−	0.759386i	$-0.274500\pi$
$73$	2.30487e10	1.30128	0.650641	+	0.759386i	$0.274500\pi$
$74$	−6.42809e9	−0.336750
$75$	0	0
$76$	−6.21099e9	−0.280987
$77$	−1.01478e10	−0.427243
$78$	0	0
$79$	−3.32219e10	−1.21472	−0.607358	−	0.794428i	$-0.707771\pi$
$79$	−3.32219e10	−1.21472	−0.607358	+	0.794428i	$0.707771\pi$
$80$	3.96127e9	0.135157
$81$	0	0
$82$	−1.83207e10	−0.545716
$83$	2.34153e10	0.652484	0.326242	−	0.945286i	$-0.394218\pi$
$83$	2.34153e10	0.652484	0.326242	+	0.945286i	$0.394218\pi$
$84$	0	0
$85$	−2.79238e10	−0.682606
$86$	3.09623e9	0.0709728
$87$	0	0
$88$	6.27295e9	0.126712
$89$	4.47401e10	0.849281	0.424641	−	0.905362i	$-0.360400\pi$
$89$	4.47401e10	0.849281	0.424641	+	0.905362i	$0.360400\pi$
$90$	0	0
$91$	7.02532e10	1.18015
$92$	4.76499e10	0.753753
$93$	0	0
$94$	−7.66954e10	−1.07787
$95$	−2.29137e10	−0.303820
$96$	0	0
$97$	2.46368e10	0.291300	0.145650	−	0.989336i	$-0.453473\pi$
$97$	2.46368e10	0.291300	0.145650	+	0.989336i	$0.453473\pi$
$98$	2.66452e10	0.297767
$99$	0	0
$100$	−3.53860e10	−0.353860
$101$	8.35085e10	0.790611	0.395306	−	0.918550i	$-0.370639\pi$
$101$	8.35085e10	0.790611	0.395306	+	0.918550i	$0.370639\pi$
$102$	0	0
$103$	1.75475e11	1.49146	0.745728	−	0.666250i	$-0.232102\pi$
$103$	1.75475e11	1.49146	0.745728	+	0.666250i	$0.232102\pi$
$104$	−4.34274e10	−0.350010
$105$	0	0
$106$	−1.00502e11	−0.729443
$107$	2.96445e10	0.204330	0.102165	−	0.994767i	$-0.467423\pi$
$107$	2.96445e10	0.204330	0.102165	+	0.994767i	$0.467423\pi$
$108$	0	0
$109$	2.52394e11	1.57120	0.785602	−	0.618732i	$-0.212353\pi$
$109$	2.52394e11	1.57120	0.785602	+	0.618732i	$0.212353\pi$
$110$	2.31423e10	0.137008
$111$	0	0
$112$	−5.55843e10	−0.298025
$113$	3.50530e11	1.78975	0.894877	−	0.446313i	$-0.147263\pi$
$113$	3.50530e11	1.78975	0.894877	+	0.446313i	$0.147263\pi$
$114$	0	0
$115$	1.75791e11	0.815002
$116$	−6.65663e10	−0.294263
$117$	0	0
$118$	−7.05493e10	−0.283885
$119$	3.91824e11	1.50516
$120$	0	0
$121$	−2.48664e11	−0.871553
$122$	3.50684e11	1.17473
$123$	0	0
$124$	−5.72560e10	−0.175389
$125$	−3.15008e11	−0.923244
$126$	0	0
$127$	8.86417e9	0.0238077	0.0119039	−	0.999929i	$-0.496211\pi$
$127$	8.86417e9	0.0238077	0.0119039	+	0.999929i	$0.496211\pi$
$128$	3.43597e10	0.0883883
$129$	0	0
$130$	−1.60213e11	−0.378452
$131$	−2.45331e11	−0.555598	−0.277799	−	0.960639i	$-0.589605\pi$
$131$	−2.45331e11	−0.555598	−0.277799	+	0.960639i	$0.589605\pi$
$132$	0	0
$133$	3.21524e11	0.669930
$134$	−3.40271e11	−0.680375
$135$	0	0
$136$	−2.42208e11	−0.446401
$137$	4.37981e11	0.775339	0.387670	−	0.921798i	$-0.373280\pi$
$137$	4.37981e11	0.775339	0.387670	+	0.921798i	$0.373280\pi$
$138$	0	0
$139$	4.16402e11	0.680661	0.340331	−	0.940306i	$-0.389461\pi$
$139$	4.16402e11	0.680661	0.340331	+	0.940306i	$0.389461\pi$
$140$	−2.05063e11	−0.322243
$141$	0	0
$142$	−9.44419e11	−1.37271
$143$	−2.53709e11	−0.354803
$144$	0	0
$145$	−2.45578e11	−0.318174
$146$	7.37560e11	0.920145
$147$	0	0
$148$	−2.05699e11	−0.238118
$149$	−1.31657e12	−1.46865	−0.734327	−	0.678796i	$-0.762502\pi$
$149$	−1.31657e12	−1.46865	−0.734327	+	0.678796i	$0.762502\pi$
$150$	0	0
$151$	6.20176e11	0.642898	0.321449	−	0.946927i	$-0.395830\pi$
$151$	6.20176e11	0.642898	0.321449	+	0.946927i	$0.395830\pi$
$152$	−1.98752e11	−0.198688
$153$	0	0
$154$	−3.24731e11	−0.302106
$155$	−2.11230e11	−0.189641
$156$	0	0
$157$	5.97589e11	0.499982	0.249991	−	0.968248i	$-0.419572\pi$
$157$	5.97589e11	0.499982	0.249991	+	0.968248i	$0.419572\pi$
$158$	−1.06310e12	−0.858934
$159$	0	0
$160$	1.26761e11	0.0955707
$161$	−2.46669e12	−1.79710
$162$	0	0
$163$	−1.93116e12	−1.31458	−0.657290	−	0.753638i	$-0.728297\pi$
$163$	−1.93116e12	−1.31458	−0.657290	+	0.753638i	$0.728297\pi$
$164$	−5.86264e11	−0.385880
$165$	0	0
$166$	7.49289e11	0.461376
$167$	−9.00420e11	−0.536419	−0.268210	−	0.963361i	$-0.586432\pi$
$167$	−9.00420e11	−0.536419	−0.268210	+	0.963361i	$0.586432\pi$
$168$	0	0
$169$	−3.57440e10	−0.0199446
$170$	−8.93560e11	−0.482675
$171$	0	0
$172$	9.90794e10	0.0501854
$173$	3.54309e12	1.73832	0.869158	−	0.494535i	$-0.164662\pi$
$173$	3.54309e12	1.73832	0.869158	+	0.494535i	$0.164662\pi$
$174$	0	0
$175$	1.83182e12	0.843673
$176$	2.00734e11	0.0895987
$177$	0	0
$178$	1.43168e12	0.600533
$179$	−3.54131e12	−1.44036	−0.720182	−	0.693785i	$-0.755942\pi$
$179$	−3.54131e12	−1.44036	−0.720182	+	0.693785i	$0.755942\pi$
$180$	0	0
$181$	2.51189e12	0.961098	0.480549	−	0.876968i	$-0.340437\pi$
$181$	2.51189e12	0.961098	0.480549	+	0.876968i	$0.340437\pi$
$182$	2.24810e12	0.834494
$183$	0	0
$184$	1.52480e12	0.532984
$185$	−7.58869e11	−0.257467
$186$	0	0
$187$	−1.41501e12	−0.452514
$188$	−2.45425e12	−0.762169
$189$	0	0
$190$	−7.33240e11	−0.214833
$191$	6.31326e12	1.79709	0.898546	−	0.438879i	$-0.144624\pi$
$191$	6.31326e12	1.79709	0.898546	+	0.438879i	$0.144624\pi$
$192$	0	0
$193$	−1.08992e12	−0.292975	−0.146488	−	0.989213i	$-0.546797\pi$
$193$	−1.08992e12	−0.292975	−0.146488	+	0.989213i	$0.546797\pi$
$194$	7.88378e11	0.205980
$195$	0	0
$196$	8.52647e11	0.210553
$197$	−4.77834e12	−1.14740	−0.573698	−	0.819067i	$-0.694492\pi$
$197$	−4.77834e12	−1.14740	−0.573698	+	0.819067i	$0.694492\pi$
$198$	0	0
$199$	−3.39102e12	−0.770261	−0.385130	−	0.922862i	$-0.625844\pi$
$199$	−3.39102e12	−0.770261	−0.385130	+	0.922862i	$0.625844\pi$
$200$	−1.13235e12	−0.250217
$201$	0	0
$202$	2.67227e12	0.559047
$203$	3.44593e12	0.701582
$204$	0	0
$205$	−2.16286e12	−0.417236
$206$	5.61520e12	1.05462
$207$	0	0
$208$	−1.38968e12	−0.247494
$209$	−1.16113e12	−0.201409
$210$	0	0
$211$	−1.07177e13	−1.76420	−0.882102	−	0.471058i	$-0.843872\pi$
$211$	−1.07177e13	−1.76420	−0.882102	+	0.471058i	$0.843872\pi$
$212$	−3.21606e12	−0.515794
$213$	0	0
$214$	9.48623e11	0.144483
$215$	3.65526e11	0.0542634
$216$	0	0
$217$	2.96397e12	0.418162
$218$	8.07660e12	1.11101
$219$	0	0
$220$	7.40554e11	0.0968795
$221$	9.79609e12	1.24996
$222$	0	0
$223$	1.97543e12	0.239875	0.119938	−	0.992781i	$-0.461731\pi$
$223$	1.97543e12	0.239875	0.119938	+	0.992781i	$0.461731\pi$
$224$	−1.77870e12	−0.210736
$225$	0	0
$226$	1.12169e13	1.26555
$227$	1.64216e13	1.80831	0.904154	−	0.427207i	$-0.140502\pi$
$227$	1.64216e13	1.80831	0.904154	+	0.427207i	$0.140502\pi$
$228$	0	0
$229$	−1.66067e13	−1.74257	−0.871283	−	0.490780i	$-0.836712\pi$
$229$	−1.66067e13	−1.74257	−0.871283	+	0.490780i	$0.836712\pi$
$230$	5.62532e12	0.576294
$231$	0	0
$232$	−2.13012e12	−0.208075
$233$	1.39122e13	1.32721	0.663605	−	0.748083i	$-0.269026\pi$
$233$	1.39122e13	1.32721	0.663605	+	0.748083i	$0.269026\pi$
$234$	0	0
$235$	−9.05429e12	−0.824102
$236$	−2.25758e12	−0.200737
$237$	0	0
$238$	1.25384e13	1.06431
$239$	−1.57281e13	−1.30463	−0.652314	−	0.757949i	$-0.726202\pi$
$239$	−1.57281e13	−1.30463	−0.652314	+	0.757949i	$0.726202\pi$
$240$	0	0
$241$	1.58009e12	0.125196	0.0625978	−	0.998039i	$-0.480061\pi$
$241$	1.58009e12	0.125196	0.0625978	+	0.998039i	$0.480061\pi$
$242$	−7.95726e12	−0.616281
$243$	0	0
$244$	1.12219e13	0.830657
$245$	3.14561e12	0.227662
$246$	0	0
$247$	8.03849e12	0.556341
$248$	−1.83219e12	−0.124019
$249$	0	0
$250$	−1.00803e13	−0.652832
$251$	−1.45264e13	−0.920351	−0.460176	−	0.887828i	$-0.652214\pi$
$251$	−1.45264e13	−0.920351	−0.460176	+	0.887828i	$0.652214\pi$
$252$	0	0
$253$	8.90807e12	0.540282
$254$	2.83653e11	0.0168346
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	4.87500e12	0.271233	0.135616	−	0.990761i	$-0.456699\pi$
$257$	4.87500e12	0.271233	0.135616	+	0.990761i	$0.456699\pi$
$258$	0	0
$259$	1.06484e13	0.567722
$260$	−5.12683e12	−0.267606
$261$	0	0
$262$	−7.85060e12	−0.392867
$263$	−1.42657e13	−0.699095	−0.349547	−	0.936919i	$-0.613665\pi$
$263$	−1.42657e13	−0.699095	−0.349547	+	0.936919i	$0.613665\pi$
$264$	0	0
$265$	−1.18648e13	−0.557707
$266$	1.02888e13	0.473712
$267$	0	0
$268$	−1.08887e13	−0.481098
$269$	−1.72215e13	−0.745475	−0.372737	−	0.927937i	$-0.621581\pi$
$269$	−1.72215e13	−0.745475	−0.372737	+	0.927937i	$0.621581\pi$
$270$	0	0
$271$	2.46849e13	1.02589	0.512944	−	0.858422i	$-0.328555\pi$
$271$	2.46849e13	1.02589	0.512944	+	0.858422i	$0.328555\pi$
$272$	−7.75066e12	−0.315653
$273$	0	0
$274$	1.40154e13	0.548248
$275$	−6.61535e12	−0.253643
$276$	0	0
$277$	9.61890e12	0.354394	0.177197	−	0.984175i	$-0.443297\pi$
$277$	9.61890e12	0.354394	0.177197	+	0.984175i	$0.443297\pi$
$278$	1.33249e13	0.481300
$279$	0	0
$280$	−6.56202e12	−0.227860
$281$	−3.32447e13	−1.13198	−0.565988	−	0.824413i	$-0.691505\pi$
$281$	−3.32447e13	−1.13198	−0.565988	+	0.824413i	$0.691505\pi$
$282$	0	0
$283$	−2.93742e13	−0.961925	−0.480962	−	0.876741i	$-0.659713\pi$
$283$	−2.93742e13	−0.961925	−0.480962	+	0.876741i	$0.659713\pi$
$284$	−3.02214e13	−0.970653
$285$	0	0
$286$	−8.11868e12	−0.250884
$287$	3.03491e13	0.920015
$288$	0	0
$289$	2.03640e13	0.594188
$290$	−7.85849e12	−0.224983
$291$	0	0
$292$	2.36019e13	0.650641
$293$	−4.74926e13	−1.28486	−0.642428	−	0.766346i	$-0.722073\pi$
$293$	−4.74926e13	−1.28486	−0.642428	+	0.766346i	$0.722073\pi$
$294$	0	0
$295$	−8.32871e12	−0.217049
$296$	−6.58236e12	−0.168375
$297$	0	0
$298$	−4.21302e13	−1.03850
$299$	−6.16703e13	−1.49240
$300$	0	0
$301$	−5.12904e12	−0.119652
$302$	1.98456e13	0.454597
$303$	0	0
$304$	−6.36005e12	−0.140493
$305$	4.14000e13	0.898155
$306$	0	0
$307$	−6.85550e13	−1.43476	−0.717378	−	0.696684i	$-0.754658\pi$
$307$	−6.85550e13	−1.43476	−0.717378	+	0.696684i	$0.754658\pi$
$308$	−1.03914e13	−0.213622
$309$	0	0
$310$	−6.75936e12	−0.134096
$311$	5.64700e12	0.110062	0.0550308	−	0.998485i	$-0.482474\pi$
$311$	5.64700e12	0.110062	0.0550308	+	0.998485i	$0.482474\pi$
$312$	0	0
$313$	9.59682e13	1.80565	0.902825	−	0.430008i	$-0.141489\pi$
$313$	9.59682e13	1.80565	0.902825	+	0.430008i	$0.141489\pi$
$314$	1.91228e13	0.353541
$315$	0	0
$316$	−3.40192e13	−0.607358
$317$	−2.74425e13	−0.481502	−0.240751	−	0.970587i	$-0.577394\pi$
$317$	−2.74425e13	−0.481502	−0.240751	+	0.970587i	$0.577394\pi$
$318$	0	0
$319$	−1.24445e13	−0.210925
$320$	4.05634e12	0.0675787
$321$	0	0
$322$	−7.89341e13	−1.27074
$323$	4.48332e13	0.709555
$324$	0	0
$325$	4.57978e13	0.700626
$326$	−6.17972e13	−0.929549
$327$	0	0
$328$	−1.87604e13	−0.272858
$329$	1.27049e14	1.81716
$330$	0	0
$331$	−1.19196e13	−0.164895	−0.0824474	−	0.996595i	$-0.526274\pi$
$331$	−1.19196e13	−0.164895	−0.0824474	+	0.996595i	$0.526274\pi$
$332$	2.39772e13	0.326242
$333$	0	0
$334$	−2.88134e13	−0.379306
$335$	−4.01707e13	−0.520192
$336$	0	0
$337$	1.12070e14	1.40450	0.702252	−	0.711928i	$-0.252178\pi$
$337$	1.12070e14	1.40450	0.702252	+	0.711928i	$0.252178\pi$
$338$	−1.14381e12	−0.0141030
$339$	0	0
$340$	−2.85939e13	−0.341303
$341$	−1.07039e13	−0.125717
$342$	0	0
$343$	6.06779e13	0.690101
$344$	3.17054e12	0.0354864
$345$	0	0
$346$	1.13379e14	1.22917
$347$	8.10239e13	0.864572	0.432286	−	0.901737i	$-0.357707\pi$
$347$	8.10239e13	0.864572	0.432286	+	0.901737i	$0.357707\pi$
$348$	0	0
$349$	1.79822e14	1.85910	0.929551	−	0.368693i	$-0.120195\pi$
$349$	1.79822e14	1.85910	0.929551	+	0.368693i	$0.120195\pi$
$350$	5.86183e13	0.596567
$351$	0	0
$352$	6.42350e12	0.0633559
$353$	−7.83466e13	−0.760780	−0.380390	−	0.924826i	$-0.624210\pi$
$353$	−7.83466e13	−0.760780	−0.380390	+	0.924826i	$0.624210\pi$
$354$	0	0
$355$	−1.11494e14	−1.04953
$356$	4.58138e13	0.424641
$357$	0	0
$358$	−1.13322e14	−1.01849
$359$	1.87927e14	1.66329	0.831646	−	0.555306i	$-0.187399\pi$
$359$	1.87927e14	1.66329	0.831646	+	0.555306i	$0.187399\pi$
$360$	0	0
$361$	−7.97009e13	−0.684185
$362$	8.03803e13	0.679599
$363$	0	0
$364$	7.19393e13	0.590077
$365$	8.70727e13	0.703512
$366$	0	0
$367$	5.93042e13	0.464967	0.232483	−	0.972600i	$-0.425315\pi$
$367$	5.93042e13	0.464967	0.232483	+	0.972600i	$0.425315\pi$
$368$	4.87935e13	0.376876
$369$	0	0
$370$	−2.42838e13	−0.182057
$371$	1.66485e14	1.22976
$372$	0	0
$373$	−1.23055e12	−0.00882474	−0.00441237	−	0.999990i	$-0.501405\pi$
$373$	−1.23055e12	−0.00882474	−0.00441237	+	0.999990i	$0.501405\pi$
$374$	−4.52804e13	−0.319975
$375$	0	0
$376$	−7.85361e13	−0.538935
$377$	8.61525e13	0.582627
$378$	0	0
$379$	1.21423e13	0.0797600	0.0398800	−	0.999204i	$-0.487302\pi$
$379$	1.21423e13	0.0797600	0.0398800	+	0.999204i	$0.487302\pi$
$380$	−2.34637e13	−0.151910
$381$	0	0
$382$	2.02024e14	1.27074
$383$	−2.12398e14	−1.31691	−0.658457	−	0.752619i	$-0.728790\pi$
$383$	−2.12398e14	−1.31691	−0.658457	+	0.752619i	$0.728790\pi$
$384$	0	0
$385$	−3.83362e13	−0.230980
$386$	−3.48776e13	−0.207165
$387$	0	0
$388$	2.52281e13	0.145650
$389$	−1.68175e14	−0.957282	−0.478641	−	0.878011i	$-0.658870\pi$
$389$	−1.68175e14	−0.957282	−0.478641	+	0.878011i	$0.658870\pi$
$390$	0	0
$391$	−3.43954e14	−1.90339
$392$	2.72847e13	0.148883
$393$	0	0
$394$	−1.52907e14	−0.811331
$395$	−1.25504e14	−0.656712
$396$	0	0
$397$	−1.43772e13	−0.0731691	−0.0365845	−	0.999331i	$-0.511648\pi$
$397$	−1.43772e13	−0.0731691	−0.0365845	+	0.999331i	$0.511648\pi$
$398$	−1.08512e14	−0.544657
$399$	0	0
$400$	−3.62352e13	−0.176930
$401$	−1.26246e14	−0.608026	−0.304013	−	0.952668i	$-0.598327\pi$
$401$	−1.26246e14	−0.608026	−0.304013	+	0.952668i	$0.598327\pi$
$402$	0	0
$403$	7.41028e13	0.347262
$404$	8.55127e13	0.395306
$405$	0	0
$406$	1.10270e14	0.496093
$407$	−3.84550e13	−0.170681
$408$	0	0
$409$	−1.04201e14	−0.450189	−0.225095	−	0.974337i	$-0.572269\pi$
$409$	−1.04201e14	−0.450189	−0.225095	+	0.974337i	$0.572269\pi$
$410$	−6.92115e13	−0.295030
$411$	0	0
$412$	1.79686e14	0.745728
$413$	1.16868e14	0.478597
$414$	0	0
$415$	8.84574e13	0.352752
$416$	−4.44696e13	−0.175005
$417$	0	0
$418$	−3.71563e13	−0.142417
$419$	4.49811e12	0.0170158	0.00850791	−	0.999964i	$-0.497292\pi$
$419$	4.49811e12	0.0170158	0.00850791	+	0.999964i	$0.497292\pi$
$420$	0	0
$421$	1.38510e14	0.510423	0.255212	−	0.966885i	$-0.417855\pi$
$421$	1.38510e14	0.510423	0.255212	+	0.966885i	$0.417855\pi$
$422$	−3.42967e14	−1.24748
$423$	0	0
$424$	−1.02914e14	−0.364721
$425$	2.55429e14	0.893575
$426$	0	0
$427$	−5.80922e14	−1.98045
$428$	3.03559e13	0.102165
$429$	0	0
$430$	1.16968e13	0.0383700
$431$	−3.59012e14	−1.16275	−0.581373	−	0.813637i	$-0.697484\pi$
$431$	−3.59012e14	−1.16275	−0.581373	+	0.813637i	$0.697484\pi$
$432$	0	0
$433$	2.15274e14	0.679684	0.339842	−	0.940482i	$-0.389626\pi$
$433$	2.15274e14	0.679684	0.339842	+	0.940482i	$0.389626\pi$
$434$	9.48469e13	0.295685
$435$	0	0
$436$	2.58451e14	0.785602
$437$	−2.82243e14	−0.847179
$438$	0	0
$439$	−5.57910e14	−1.63309	−0.816543	−	0.577285i	$-0.804112\pi$
$439$	−5.57910e14	−1.63309	−0.816543	+	0.577285i	$0.804112\pi$
$440$	2.36977e13	0.0685041
$441$	0	0
$442$	3.13475e14	0.883853
$443$	−2.77748e14	−0.773446	−0.386723	−	0.922196i	$-0.626393\pi$
$443$	−2.77748e14	−0.773446	−0.386723	+	0.922196i	$0.626393\pi$
$444$	0	0
$445$	1.69017e14	0.459147
$446$	6.32139e13	0.169617
$447$	0	0
$448$	−5.69183e13	−0.149013
$449$	−5.80134e14	−1.50028	−0.750142	−	0.661277i	$-0.770015\pi$
$449$	−5.80134e14	−1.50028	−0.750142	+	0.661277i	$0.770015\pi$
$450$	0	0
$451$	−1.09601e14	−0.276595
$452$	3.58942e14	0.894877
$453$	0	0
$454$	5.25490e14	1.27867
$455$	2.65400e14	0.638026
$456$	0	0
$457$	1.62773e14	0.381982	0.190991	−	0.981592i	$-0.438830\pi$
$457$	1.62773e14	0.381982	0.190991	+	0.981592i	$0.438830\pi$
$458$	−5.31416e14	−1.23218
$459$	0	0
$460$	1.80010e14	0.407501
$461$	1.13044e14	0.252867	0.126433	−	0.991975i	$-0.459647\pi$
$461$	1.13044e14	0.252867	0.126433	+	0.991975i	$0.459647\pi$
$462$	0	0
$463$	−5.51011e14	−1.20355	−0.601776	−	0.798665i	$-0.705540\pi$
$463$	−5.51011e14	−1.20355	−0.601776	+	0.798665i	$0.705540\pi$
$464$	−6.81639e13	−0.147131
$465$	0	0
$466$	4.45192e14	0.938479
$467$	−4.19636e14	−0.874238	−0.437119	−	0.899404i	$-0.644001\pi$
$467$	−4.19636e14	−0.874238	−0.437119	+	0.899404i	$0.644001\pi$
$468$	0	0
$469$	5.63673e14	1.14703
$470$	−2.89737e14	−0.582728
$471$	0	0
$472$	−7.22425e13	−0.141942
$473$	1.85227e13	0.0359724
$474$	0	0
$475$	2.09600e14	0.397720
$476$	4.01228e14	0.752581
$477$	0	0
$478$	−5.03298e14	−0.922511
$479$	4.77698e14	0.865582	0.432791	−	0.901494i	$-0.357529\pi$
$479$	4.77698e14	0.865582	0.432791	+	0.901494i	$0.357529\pi$
$480$	0	0
$481$	2.66223e14	0.471463
$482$	5.05630e13	0.0885267
$483$	0	0
$484$	−2.54632e14	−0.435777
$485$	9.30721e13	0.157485
$486$	0	0
$487$	−9.07190e14	−1.50068	−0.750342	−	0.661050i	$-0.770111\pi$
$487$	−9.07190e14	−1.50068	−0.750342	+	0.661050i	$0.770111\pi$
$488$	3.59100e14	0.587363
$489$	0	0
$490$	1.00659e14	0.160981
$491$	−6.05713e13	−0.0957897	−0.0478948	−	0.998852i	$-0.515251\pi$
$491$	−6.05713e13	−0.0957897	−0.0478948	+	0.998852i	$0.515251\pi$
$492$	0	0
$493$	4.80500e14	0.743079
$494$	2.57232e14	0.393393
$495$	0	0
$496$	−5.86301e13	−0.0876943
$497$	1.56447e15	2.31423
$498$	0	0
$499$	7.99650e14	1.15704	0.578518	−	0.815670i	$-0.303631\pi$
$499$	7.99650e14	1.15704	0.578518	+	0.815670i	$0.303631\pi$
$500$	−3.22568e14	−0.461622
$501$	0	0
$502$	−4.64846e14	−0.650787
$503$	3.09482e14	0.428560	0.214280	−	0.976772i	$-0.431259\pi$
$503$	3.09482e14	0.428560	0.214280	+	0.976772i	$0.431259\pi$
$504$	0	0
$505$	3.15475e14	0.427428
$506$	2.85058e14	0.382037
$507$	0	0
$508$	9.07691e12	0.0119039
$509$	−3.20476e14	−0.415765	−0.207882	−	0.978154i	$-0.566657\pi$
$509$	−3.20476e14	−0.415765	−0.207882	+	0.978154i	$0.566657\pi$
$510$	0	0
$511$	−1.22180e15	−1.55126
$512$	3.51844e13	0.0441942
$513$	0	0
$514$	1.56000e14	0.191791
$515$	6.62904e14	0.806326
$516$	0	0
$517$	−4.58818e14	−0.546315
$518$	3.40749e14	0.401440
$519$	0	0
$520$	−1.64058e14	−0.189226
$521$	−1.59747e14	−0.182317	−0.0911583	−	0.995836i	$-0.529057\pi$
$521$	−1.59747e14	−0.182317	−0.0911583	+	0.995836i	$0.529057\pi$
$522$	0	0
$523$	−1.60413e15	−1.79259	−0.896293	−	0.443462i	$-0.853750\pi$
$523$	−1.60413e15	−1.79259	−0.896293	+	0.443462i	$0.853750\pi$
$524$	−2.51219e14	−0.277799
$525$	0	0
$526$	−4.56502e14	−0.494335
$527$	4.13295e14	0.442896
$528$	0	0
$529$	1.21252e15	1.27257
$530$	−3.79672e14	−0.394359
$531$	0	0
$532$	3.29241e14	0.334965
$533$	7.58764e14	0.764024
$534$	0	0
$535$	1.11990e14	0.110467
$536$	−3.48437e14	−0.340188
$537$	0	0
$538$	−5.51087e14	−0.527130
$539$	1.59401e14	0.150922
$540$	0	0
$541$	1.42984e14	0.132649	0.0663243	−	0.997798i	$-0.478873\pi$
$541$	1.42984e14	0.132649	0.0663243	+	0.997798i	$0.478873\pi$
$542$	7.89916e14	0.725413
$543$	0	0
$544$	−2.48021e14	−0.223200
$545$	9.53484e14	0.849440
$546$	0	0
$547$	2.16822e14	0.189310	0.0946550	−	0.995510i	$-0.469825\pi$
$547$	2.16822e14	0.189310	0.0946550	+	0.995510i	$0.469825\pi$
$548$	4.48492e14	0.387670
$549$	0	0
$550$	−2.11691e14	−0.179353
$551$	3.94289e14	0.330736
$552$	0	0
$553$	1.76107e15	1.44807
$554$	3.07805e14	0.250595
$555$	0	0
$556$	4.26395e14	0.340331
$557$	−1.11565e15	−0.881707	−0.440853	−	0.897579i	$-0.645324\pi$
$557$	−1.11565e15	−0.881707	−0.440853	+	0.897579i	$0.645324\pi$
$558$	0	0
$559$	−1.28232e14	−0.0993648
$560$	−2.09985e14	−0.161121
$561$	0	0
$562$	−1.06383e15	−0.800428
$563$	1.46937e14	0.109480	0.0547402	−	0.998501i	$-0.482567\pi$
$563$	1.46937e14	0.109480	0.0547402	+	0.998501i	$0.482567\pi$
$564$	0	0
$565$	1.32422e15	0.967594
$566$	−9.39975e14	−0.680184
$567$	0	0
$568$	−9.67085e14	−0.686355
$569$	−2.15324e15	−1.51348	−0.756738	−	0.653718i	$-0.773208\pi$
$569$	−2.15324e15	−1.51348	−0.756738	+	0.653718i	$0.773208\pi$
$570$	0	0
$571$	1.11728e15	0.770309	0.385155	−	0.922852i	$-0.374148\pi$
$571$	1.11728e15	0.770309	0.385155	+	0.922852i	$0.374148\pi$
$572$	−2.59798e14	−0.177401
$573$	0	0
$574$	9.71170e14	0.650549
$575$	−1.60803e15	−1.06689
$576$	0	0
$577$	1.96428e15	1.27860	0.639301	−	0.768956i	$-0.279224\pi$
$577$	1.96428e15	1.27860	0.639301	+	0.768956i	$0.279224\pi$
$578$	6.51647e14	0.420155
$579$	0	0
$580$	−2.51472e14	−0.159087
$581$	−1.24123e15	−0.777827
$582$	0	0
$583$	−6.01236e14	−0.369716
$584$	7.55261e14	0.460073
$585$	0	0
$586$	−1.51976e15	−0.908530
$587$	−1.25268e15	−0.741877	−0.370939	−	0.928657i	$-0.620964\pi$
$587$	−1.25268e15	−0.741877	−0.370939	+	0.928657i	$0.620964\pi$
$588$	0	0
$589$	3.39142e14	0.197128
$590$	−2.66519e14	−0.153476
$591$	0	0
$592$	−2.10636e14	−0.119059
$593$	1.44789e15	0.810837	0.405418	−	0.914131i	$-0.367126\pi$
$593$	1.44789e15	0.810837	0.405418	+	0.914131i	$0.367126\pi$
$594$	0	0
$595$	1.48022e15	0.813735
$596$	−1.34817e15	−0.734327
$597$	0	0
$598$	−1.97345e15	−1.05528
$599$	−3.42579e14	−0.181516	−0.0907578	−	0.995873i	$-0.528929\pi$
$599$	−3.42579e14	−0.181516	−0.0907578	+	0.995873i	$0.528929\pi$
$600$	0	0
$601$	−2.45834e15	−1.27889	−0.639443	−	0.768839i	$-0.720835\pi$
$601$	−2.45834e15	−1.27889	−0.639443	+	0.768839i	$0.720835\pi$
$602$	−1.64129e14	−0.0846068
$603$	0	0
$604$	6.35061e14	0.321449
$605$	−9.39395e14	−0.471188
$606$	0	0
$607$	2.00862e15	0.989372	0.494686	−	0.869072i	$-0.335283\pi$
$607$	2.00862e15	0.989372	0.494686	+	0.869072i	$0.335283\pi$
$608$	−2.03522e14	−0.0993439
$609$	0	0
$610$	1.32480e15	0.635092
$611$	3.17638e15	1.50906
$612$	0	0
$613$	1.54595e14	0.0721379	0.0360689	−	0.999349i	$-0.488516\pi$
$613$	1.54595e14	0.0721379	0.0360689	+	0.999349i	$0.488516\pi$
$614$	−2.19376e15	−1.01453
$615$	0	0
$616$	−3.32525e14	−0.151053
$617$	−2.63672e15	−1.18712	−0.593562	−	0.804789i	$-0.702279\pi$
$617$	−2.63672e15	−1.18712	−0.593562	+	0.804789i	$0.702279\pi$
$618$	0	0
$619$	−1.54864e15	−0.684941	−0.342470	−	0.939529i	$-0.611264\pi$
$619$	−1.54864e15	−0.684941	−0.342470	+	0.939529i	$0.611264\pi$
$620$	−2.16300e14	−0.0948203
$621$	0	0
$622$	1.80704e14	0.0778253
$623$	−2.37164e15	−1.01243
$624$	0	0
$625$	4.97308e14	0.208586
$626$	3.07098e15	1.27679
$627$	0	0
$628$	6.11931e14	0.249991
$629$	1.48481e15	0.601301
$630$	0	0
$631$	9.61087e12	0.00382473	0.00191237	−	0.999998i	$-0.499391\pi$
$631$	9.61087e12	0.00382473	0.00191237	+	0.999998i	$0.499391\pi$
$632$	−1.08861e15	−0.429467
$633$	0	0
$634$	−8.78161e14	−0.340473
$635$	3.34867e13	0.0128712
$636$	0	0
$637$	−1.10353e15	−0.416885
$638$	−3.98223e14	−0.149146
$639$	0	0
$640$	1.29803e14	0.0477854
$641$	−3.95925e15	−1.44509	−0.722544	−	0.691325i	$-0.757027\pi$
$641$	−3.95925e15	−1.44509	−0.722544	+	0.691325i	$0.757027\pi$
$642$	0	0
$643$	−3.23339e15	−1.16011	−0.580054	−	0.814578i	$-0.696968\pi$
$643$	−3.23339e15	−1.16011	−0.580054	+	0.814578i	$0.696968\pi$
$644$	−2.52589e15	−0.898550
$645$	0	0
$646$	1.43466e15	0.501731
$647$	−1.30706e15	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$647$	−1.30706e15	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$648$	0	0
$649$	−4.22051e14	−0.143886
$650$	1.46553e15	0.495418
$651$	0	0
$652$	−1.97751e15	−0.657290
$653$	2.98261e15	0.983045	0.491523	−	0.870865i	$-0.336441\pi$
$653$	2.98261e15	0.983045	0.491523	+	0.870865i	$0.336441\pi$
$654$	0	0
$655$	−9.26804e14	−0.300373
$656$	−6.00334e14	−0.192940
$657$	0	0
$658$	4.06557e15	1.28493
$659$	−2.71150e15	−0.849846	−0.424923	−	0.905230i	$-0.639699\pi$
$659$	−2.71150e15	−0.849846	−0.424923	+	0.905230i	$0.639699\pi$
$660$	0	0
$661$	1.59365e15	0.491229	0.245615	−	0.969368i	$-0.421010\pi$
$661$	1.59365e15	0.491229	0.245615	+	0.969368i	$0.421010\pi$
$662$	−3.81427e14	−0.116598
$663$	0	0
$664$	7.67272e14	0.230688
$665$	1.21464e15	0.362184
$666$	0	0
$667$	−3.02494e15	−0.887206
$668$	−9.22030e14	−0.268210
$669$	0	0
$670$	−1.28546e15	−0.367831
$671$	2.09791e15	0.595406
$672$	0	0
$673$	−4.10494e15	−1.14610	−0.573052	−	0.819519i	$-0.694241\pi$
$673$	−4.10494e15	−1.14610	−0.573052	+	0.819519i	$0.694241\pi$
$674$	3.58623e15	0.993135
$675$	0	0
$676$	−3.66019e13	−0.00997232
$677$	4.62400e15	1.24963	0.624814	−	0.780774i	$-0.285175\pi$
$677$	4.62400e15	1.24963	0.624814	+	0.780774i	$0.285175\pi$
$678$	0	0
$679$	−1.30598e15	−0.347259
$680$	−9.15006e14	−0.241337
$681$	0	0
$682$	−3.42525e14	−0.0888952
$683$	2.30592e15	0.593651	0.296825	−	0.954932i	$-0.404072\pi$
$683$	2.30592e15	0.593651	0.296825	+	0.954932i	$0.404072\pi$
$684$	0	0
$685$	1.65459e15	0.419171
$686$	1.94169e15	0.487975
$687$	0	0
$688$	1.01457e14	0.0250927
$689$	4.16234e15	1.02125
$690$	0	0
$691$	−1.21214e15	−0.292700	−0.146350	−	0.989233i	$-0.546753\pi$
$691$	−1.21214e15	−0.292700	−0.146350	+	0.989233i	$0.546753\pi$
$692$	3.62812e15	0.869158
$693$	0	0
$694$	2.59277e15	0.611345
$695$	1.57307e15	0.367986
$696$	0	0
$697$	4.23187e15	0.974432
$698$	5.75431e15	1.31458
$699$	0	0
$700$	1.87579e15	0.421837
$701$	4.25099e15	0.948507	0.474254	−	0.880388i	$-0.342718\pi$
$701$	4.25099e15	0.948507	0.474254	+	0.880388i	$0.342718\pi$
$702$	0	0
$703$	1.21841e15	0.267632
$704$	2.05552e14	0.0447994
$705$	0	0
$706$	−2.50709e15	−0.537953
$707$	−4.42673e15	−0.942489
$708$	0	0
$709$	8.15467e15	1.70943	0.854717	−	0.519095i	$-0.173731\pi$
$709$	8.15467e15	1.70943	0.854717	+	0.519095i	$0.173731\pi$
$710$	−3.56779e15	−0.742128
$711$	0	0
$712$	1.46604e15	0.300266
$713$	−2.60185e15	−0.528799
$714$	0	0
$715$	−9.58452e14	−0.191817
$716$	−3.62630e15	−0.720182
$717$	0	0
$718$	6.01365e15	1.17613
$719$	6.49021e15	1.25965	0.629825	−	0.776737i	$-0.283127\pi$
$719$	6.49021e15	1.25965	0.629825	+	0.776737i	$0.283127\pi$
$720$	0	0
$721$	−9.30182e15	−1.77797
$722$	−2.55043e15	−0.483792
$723$	0	0
$724$	2.57217e15	0.480549
$725$	2.24639e15	0.416511
$726$	0	0
$727$	−2.22828e15	−0.406939	−0.203470	−	0.979081i	$-0.565222\pi$
$727$	−2.22828e15	−0.406939	−0.203470	+	0.979081i	$0.565222\pi$
$728$	2.30206e15	0.417247
$729$	0	0
$730$	2.78633e15	0.497458
$731$	−7.15192e14	−0.126729
$732$	0	0
$733$	−5.80037e15	−1.01247	−0.506237	−	0.862394i	$-0.668964\pi$
$733$	−5.80037e15	−1.01247	−0.506237	+	0.862394i	$0.668964\pi$
$734$	1.89773e15	0.328781
$735$	0	0
$736$	1.56139e15	0.266492
$737$	−2.03562e15	−0.344846
$738$	0	0
$739$	5.26802e15	0.879231	0.439615	−	0.898186i	$-0.355115\pi$
$739$	5.26802e15	0.879231	0.439615	+	0.898186i	$0.355115\pi$
$740$	−7.77082e14	−0.128734
$741$	0	0
$742$	5.32753e15	0.869570
$743$	1.35946e15	0.220256	0.110128	−	0.993917i	$-0.464874\pi$
$743$	1.35946e15	0.220256	0.110128	+	0.993917i	$0.464874\pi$
$744$	0	0
$745$	−4.97369e15	−0.793998
$746$	−3.93777e13	−0.00624003
$747$	0	0
$748$	−1.44897e15	−0.226257
$749$	−1.57143e15	−0.243582
$750$	0	0
$751$	6.48313e15	0.990296	0.495148	−	0.868809i	$-0.335114\pi$
$751$	6.48313e15	0.990296	0.495148	+	0.868809i	$0.335114\pi$
$752$	−2.51315e15	−0.381084
$753$	0	0
$754$	2.75688e15	0.411979
$755$	2.34288e15	0.347570
$756$	0	0
$757$	−7.16904e15	−1.04817	−0.524087	−	0.851664i	$-0.675594\pi$
$757$	−7.16904e15	−1.04817	−0.524087	+	0.851664i	$0.675594\pi$
$758$	3.88553e14	0.0563988
$759$	0	0
$760$	−7.50837e14	−0.107417
$761$	−2.20591e15	−0.313309	−0.156655	−	0.987653i	$-0.550071\pi$
$761$	−2.20591e15	−0.313309	−0.156655	+	0.987653i	$0.550071\pi$
$762$	0	0
$763$	−1.33792e16	−1.87303
$764$	6.46478e15	0.898546
$765$	0	0
$766$	−6.79674e15	−0.931198
$767$	2.92184e15	0.397450
$768$	0	0
$769$	1.02342e16	1.37233	0.686166	−	0.727445i	$-0.259292\pi$
$769$	1.02342e16	1.37233	0.686166	+	0.727445i	$0.259292\pi$
$770$	−1.22676e15	−0.163328
$771$	0	0
$772$	−1.11608e15	−0.146488
$773$	−9.26990e15	−1.20806	−0.604029	−	0.796962i	$-0.706439\pi$
$773$	−9.26990e15	−1.20806	−0.604029	+	0.796962i	$0.706439\pi$
$774$	0	0
$775$	1.93220e15	0.248252
$776$	8.07299e14	0.102990
$777$	0	0
$778$	−5.38161e15	−0.676900
$779$	3.47259e15	0.433709
$780$	0	0
$781$	−5.64984e15	−0.695754
$782$	−1.10065e16	−1.34590
$783$	0	0
$784$	8.73110e14	0.105276
$785$	2.25755e15	0.270305
$786$	0	0
$787$	7.27178e15	0.858578	0.429289	−	0.903167i	$-0.358764\pi$
$787$	7.27178e15	0.858578	0.429289	+	0.903167i	$0.358764\pi$
$788$	−4.89302e15	−0.573698
$789$	0	0
$790$	−4.01614e15	−0.464365
$791$	−1.85813e16	−2.13357
$792$	0	0
$793$	−1.45238e16	−1.64466
$794$	−4.60072e14	−0.0517384
$795$	0	0
$796$	−3.47240e15	−0.385130
$797$	−2.05946e15	−0.226847	−0.113424	−	0.993547i	$-0.536182\pi$
$797$	−2.05946e15	−0.226847	−0.113424	+	0.993547i	$0.536182\pi$
$798$	0	0
$799$	1.77157e16	1.92465
$800$	−1.15953e15	−0.125108
$801$	0	0
$802$	−4.03986e15	−0.429939
$803$	4.41234e15	0.466373
$804$	0	0
$805$	−9.31857e15	−0.971566
$806$	2.37129e15	0.245551
$807$	0	0
$808$	2.73641e15	0.279523
$809$	−4.11963e14	−0.0417966	−0.0208983	−	0.999782i	$-0.506653\pi$
$809$	−4.11963e14	−0.0417966	−0.0208983	+	0.999782i	$0.506653\pi$
$810$	0	0
$811$	1.59926e16	1.60068	0.800341	−	0.599545i	$-0.204652\pi$
$811$	1.59926e16	1.60068	0.800341	+	0.599545i	$0.204652\pi$
$812$	3.52863e15	0.350791
$813$	0	0
$814$	−1.23056e15	−0.120689
$815$	−7.29548e15	−0.710701
$816$	0	0
$817$	−5.86874e14	−0.0564057
$818$	−3.33444e15	−0.318332
$819$	0	0
$820$	−2.21477e15	−0.208618
$821$	−9.16232e15	−0.857271	−0.428635	−	0.903478i	$-0.641006\pi$
$821$	−9.16232e15	−0.857271	−0.428635	+	0.903478i	$0.641006\pi$
$822$	0	0
$823$	5.35862e15	0.494714	0.247357	−	0.968924i	$-0.420438\pi$
$823$	5.35862e15	0.494714	0.247357	+	0.968924i	$0.420438\pi$
$824$	5.74997e15	0.527310
$825$	0	0
$826$	3.73977e15	0.338419
$827$	−1.52149e16	−1.36770	−0.683849	−	0.729624i	$-0.739695\pi$
$827$	−1.52149e16	−1.36770	−0.683849	+	0.729624i	$0.739695\pi$
$828$	0	0
$829$	−7.51363e15	−0.666500	−0.333250	−	0.942839i	$-0.608145\pi$
$829$	−7.51363e15	−0.666500	−0.333250	+	0.942839i	$0.608145\pi$
$830$	2.83064e15	0.249433
$831$	0	0
$832$	−1.42303e15	−0.123747
$833$	−6.15472e15	−0.531693
$834$	0	0
$835$	−3.40157e15	−0.290004
$836$	−1.18900e15	−0.100704
$837$	0	0
$838$	1.43939e14	0.0120320
$839$	6.23751e15	0.517989	0.258994	−	0.965879i	$-0.416609\pi$
$839$	6.23751e15	0.517989	0.258994	+	0.965879i	$0.416609\pi$
$840$	0	0
$841$	−7.97471e15	−0.653638
$842$	4.43233e15	0.360924
$843$	0	0
$844$	−1.09749e16	−0.882102
$845$	−1.35032e14	−0.0107827
$846$	0	0
$847$	1.31815e16	1.03898
$848$	−3.29324e15	−0.257897
$849$	0	0
$850$	8.17373e15	0.631853
$851$	−9.34746e15	−0.717929
$852$	0	0
$853$	1.85638e16	1.40750	0.703749	−	0.710448i	$-0.251508\pi$
$853$	1.85638e16	1.40750	0.703749	+	0.710448i	$0.251508\pi$
$854$	−1.85895e16	−1.40039
$855$	0	0
$856$	9.71390e14	0.0722417
$857$	1.73724e16	1.28370	0.641851	−	0.766829i	$-0.278167\pi$
$857$	1.73724e16	1.28370	0.641851	+	0.766829i	$0.278167\pi$
$858$	0	0
$859$	1.16681e16	0.851214	0.425607	−	0.904908i	$-0.360061\pi$
$859$	1.16681e16	0.851214	0.425607	+	0.904908i	$0.360061\pi$
$860$	3.74299e14	0.0271317
$861$	0	0
$862$	−1.14884e16	−0.822185
$863$	1.75417e16	1.24742	0.623709	−	0.781657i	$-0.285625\pi$
$863$	1.75417e16	1.24742	0.623709	+	0.781657i	$0.285625\pi$
$864$	0	0
$865$	1.33850e16	0.939785
$866$	6.88875e15	0.480609
$867$	0	0
$868$	3.03510e15	0.209081
$869$	−6.35983e15	−0.435348
$870$	0	0
$871$	1.40925e16	0.952552
$872$	8.27043e15	0.555505
$873$	0	0
$874$	−9.03177e15	−0.599046
$875$	1.66984e16	1.10060
$876$	0	0
$877$	−2.13988e16	−1.39281	−0.696403	−	0.717651i	$-0.745217\pi$
$877$	−2.13988e16	−1.39281	−0.696403	+	0.717651i	$0.745217\pi$
$878$	−1.78531e16	−1.15477
$879$	0	0
$880$	7.58327e14	0.0484397
$881$	−1.47188e16	−0.934338	−0.467169	−	0.884168i	$-0.654726\pi$
$881$	−1.47188e16	−0.934338	−0.467169	+	0.884168i	$0.654726\pi$
$882$	0	0
$883$	−4.13136e15	−0.259006	−0.129503	−	0.991579i	$-0.541338\pi$
$883$	−4.13136e15	−0.259006	−0.129503	+	0.991579i	$0.541338\pi$
$884$	1.00312e16	0.624979
$885$	0	0
$886$	−8.88793e15	−0.546909
$887$	−5.95457e15	−0.364142	−0.182071	−	0.983285i	$-0.558280\pi$
$887$	−5.95457e15	−0.364142	−0.182071	+	0.983285i	$0.558280\pi$
$888$	0	0
$889$	−4.69884e14	−0.0283812
$890$	5.40856e15	0.324666
$891$	0	0
$892$	2.02284e15	0.119938
$893$	1.45372e16	0.856638
$894$	0	0
$895$	−1.33782e16	−0.778704
$896$	−1.82139e15	−0.105368
$897$	0	0
$898$	−1.85643e16	−1.06086
$899$	3.63475e15	0.206441
$900$	0	0
$901$	2.32147e16	1.30250
$902$	−3.50723e15	−0.195582
$903$	0	0
$904$	1.14862e16	0.632774
$905$	9.48931e15	0.519598
$906$	0	0
$907$	2.21430e16	1.19784	0.598918	−	0.800810i	$-0.295598\pi$
$907$	2.21430e16	1.19784	0.598918	+	0.800810i	$0.295598\pi$
$908$	1.68157e16	0.904154
$909$	0	0
$910$	8.49280e15	0.451153
$911$	2.06900e16	1.09247	0.546235	−	0.837632i	$-0.316061\pi$
$911$	2.06900e16	1.09247	0.546235	+	0.837632i	$0.316061\pi$
$912$	0	0
$913$	4.48250e15	0.233847
$914$	5.20873e15	0.270102
$915$	0	0
$916$	−1.70053e16	−0.871283
$917$	1.30048e16	0.662329
$918$	0	0
$919$	9.82530e15	0.494436	0.247218	−	0.968960i	$-0.420484\pi$
$919$	9.82530e15	0.494436	0.247218	+	0.968960i	$0.420484\pi$
$920$	5.76033e15	0.288147
$921$	0	0
$922$	3.61740e15	0.178804
$923$	3.91137e16	1.92185
$924$	0	0
$925$	6.94165e15	0.337042
$926$	−1.76324e16	−0.851040
$927$	0	0
$928$	−2.18124e15	−0.104038
$929$	1.69779e14	0.00805003	0.00402502	−	0.999992i	$-0.498719\pi$
$929$	1.69779e14	0.00805003	0.00402502	+	0.999992i	$0.498719\pi$
$930$	0	0
$931$	−5.05045e15	−0.236650
$932$	1.42461e16	0.663605
$933$	0	0
$934$	−1.34283e16	−0.618180
$935$	−5.34559e15	−0.244642
$936$	0	0
$937$	−3.07118e16	−1.38911	−0.694556	−	0.719439i	$-0.744399\pi$
$937$	−3.07118e16	−1.38911	−0.694556	+	0.719439i	$0.744399\pi$
$938$	1.80375e16	0.811076
$939$	0	0
$940$	−9.27159e15	−0.412051
$941$	2.05209e16	0.906679	0.453340	−	0.891338i	$-0.350232\pi$
$941$	2.05209e16	0.906679	0.453340	+	0.891338i	$0.350232\pi$
$942$	0	0
$943$	−2.66413e16	−1.16343
$944$	−2.31176e15	−0.100368
$945$	0	0
$946$	5.92728e14	0.0254363
$947$	−2.61902e16	−1.11741	−0.558707	−	0.829365i	$-0.688702\pi$
$947$	−2.61902e16	−1.11741	−0.558707	+	0.829365i	$0.688702\pi$
$948$	0	0
$949$	−3.05465e16	−1.28824
$950$	6.70721e15	0.281230
$951$	0	0
$952$	1.28393e16	0.532155
$953$	−5.86183e15	−0.241558	−0.120779	−	0.992679i	$-0.538539\pi$
$953$	−5.86183e15	−0.241558	−0.120779	+	0.992679i	$0.538539\pi$
$954$	0	0
$955$	2.38500e16	0.971562
$956$	−1.61055e16	−0.652314
$957$	0	0
$958$	1.52863e16	0.612059
$959$	−2.32171e16	−0.924283
$960$	0	0
$961$	−2.22821e16	−0.876955
$962$	8.51913e15	0.333375
$963$	0	0
$964$	1.61802e15	0.0625978
$965$	−4.11748e15	−0.158391
$966$	0	0
$967$	−3.14672e15	−0.119677	−0.0598387	−	0.998208i	$-0.519059\pi$
$967$	−3.14672e15	−0.119677	−0.0598387	+	0.998208i	$0.519059\pi$
$968$	−8.14823e15	−0.308141
$969$	0	0
$970$	2.97831e15	0.111359
$971$	−2.12016e16	−0.788249	−0.394124	−	0.919057i	$-0.628952\pi$
$971$	−2.12016e16	−0.788249	−0.394124	+	0.919057i	$0.628952\pi$
$972$	0	0
$973$	−2.20732e16	−0.811417
$974$	−2.90301e16	−1.06114
$975$	0	0
$976$	1.14912e16	0.415328
$977$	9.83306e15	0.353402	0.176701	−	0.984265i	$-0.443457\pi$
$977$	9.83306e15	0.353402	0.176701	+	0.984265i	$0.443457\pi$
$978$	0	0
$979$	8.56482e15	0.304378
$980$	3.22110e15	0.113831
$981$	0	0
$982$	−1.93828e15	−0.0677335
$983$	−9.42009e15	−0.327349	−0.163674	−	0.986514i	$-0.552335\pi$
$983$	−9.42009e15	−0.327349	−0.163674	+	0.986514i	$0.552335\pi$
$984$	0	0
$985$	−1.80515e16	−0.620316
$986$	1.53760e16	0.525436
$987$	0	0
$988$	8.23142e15	0.278171
$989$	4.50242e15	0.151309
$990$	0	0
$991$	−1.54334e16	−0.512928	−0.256464	−	0.966554i	$-0.582557\pi$
$991$	−1.54334e16	−0.512928	−0.256464	+	0.966554i	$0.582557\pi$
$992$	−1.87616e15	−0.0620093
$993$	0	0
$994$	5.00630e16	1.63641
$995$	−1.28105e16	−0.416426
$996$	0	0
$997$	2.73270e16	0.878553	0.439277	−	0.898352i	$-0.355235\pi$
$997$	2.73270e16	0.878553	0.439277	+	0.898352i	$0.355235\pi$
$998$	2.55888e16	0.818148
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.12.a.g.1.2	yes	2
3.2	odd	2		54.12.a.d.1.1	✓	2
9.2	odd	6		162.12.c.q.109.2		4
9.4	even	3		162.12.c.l.55.1		4
9.5	odd	6		162.12.c.q.55.2		4
9.7	even	3		162.12.c.l.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.12.a.d.1.1	✓	2	3.2	odd	2
54.12.a.g.1.2	yes	2	1.1	even	1	trivial
162.12.c.l.55.1		4	9.4	even	3
162.12.c.l.109.1		4	9.7	even	3
162.12.c.q.55.2		4	9.5	odd	6
162.12.c.q.109.2		4	9.2	odd	6

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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

\(f(q)\)	\(=\)	\(q+32.0000 q^{2} +1024.00 q^{4} +3777.77 q^{5} -53009.3 q^{7} +32768.0 q^{8} +120888. q^{10} +191435. q^{11} -1.32530e6 q^{13} -1.69630e6 q^{14} +1.04858e6 q^{16} -7.39161e6 q^{17} -6.06542e6 q^{19} +3.86843e6 q^{20} +6.12592e6 q^{22} +4.65331e7 q^{23} -3.45566e7 q^{25} -4.24096e7 q^{26} -5.42816e7 q^{28} -6.50061e7 q^{29} -5.59140e7 q^{31} +3.35544e7 q^{32} -2.36531e8 q^{34} -2.00257e8 q^{35} -2.00878e8 q^{37} -1.94093e8 q^{38} +1.23790e8 q^{40} -5.72523e8 q^{41} +9.67573e7 q^{43} +1.96030e8 q^{44} +1.48906e9 q^{46} -2.39673e9 q^{47} +8.32663e8 q^{49} -1.10581e9 q^{50} -1.35711e9 q^{52} -3.14068e9 q^{53} +7.23197e8 q^{55} -1.73701e9 q^{56} -2.08020e9 q^{58} -2.20467e9 q^{59} +1.09589e10 q^{61} -1.78925e9 q^{62} +1.07374e9 q^{64} -5.00667e9 q^{65} -1.06335e10 q^{67} -7.56901e9 q^{68} -6.40822e9 q^{70} -2.95131e10 q^{71} +2.30487e10 q^{73} -6.42809e9 q^{74} -6.21099e9 q^{76} -1.01478e10 q^{77} -3.32219e10 q^{79} +3.96127e9 q^{80} -1.83207e10 q^{82} +2.34153e10 q^{83} -2.79238e10 q^{85} +3.09623e9 q^{86} +6.27295e9 q^{88} +4.47401e10 q^{89} +7.02532e10 q^{91} +4.76499e10 q^{92} -7.66954e10 q^{94} -2.29137e10 q^{95} +2.46368e10 q^{97} +2.66452e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 64 q^{2} + 2048 q^{4} - 3720 q^{5} - 6794 q^{7} + 65536 q^{8} - 119040 q^{10} - 832632 q^{11} + 984634 q^{13} - 217408 q^{14} + 2097152 q^{16} - 5111064 q^{17} - 22919270 q^{19} - 3809280 q^{20} - 26644224 q^{22}+ \cdots + 31718195328 q^{98}+O(q^{100}) \) 2 * q + 64 * q^2 + 2048 * q^4 - 3720 * q^5 - 6794 * q^7 + 65536 * q^8 - 119040 * q^10 - 832632 * q^11 + 984634 * q^13 - 217408 * q^14 + 2097152 * q^16 - 5111064 * q^17 - 22919270 * q^19 - 3809280 * q^20 - 26644224 * q^22 + 5263656 * q^23 - 27168250 * q^25 + 31508288 * q^26 - 6957056 * q^28 - 37534848 * q^29 - 328859480 * q^31 + 67108864 * q^32 - 163554048 * q^34 - 546768600 * q^35 - 96567386 * q^37 - 733416640 * q^38 - 121896960 * q^40 - 478653312 * q^41 + 133709104 * q^43 - 852615168 * q^44 + 168436992 * q^46 - 4431713208 * q^47 + 991193604 * q^49 - 869384000 * q^50 + 1008265216 * q^52 - 4754972016 * q^53 + 8401412160 * q^55 - 222625792 * q^56 - 1201115136 * q^58 - 9940537848 * q^59 + 6058984510 * q^61 - 10523503360 * q^62 + 2147483648 * q^64 - 22326000360 * q^65 - 6772264670 * q^67 - 5233729536 * q^68 - 17496595200 * q^70 - 25581365424 * q^71 + 15807624466 * q^73 - 3090156352 * q^74 - 23469332480 * q^76 - 57475455528 * q^77 - 28139021846 * q^79 - 3900702720 * q^80 - 15316905984 * q^82 + 77207454480 * q^83 - 45022737600 * q^85 + 4278691328 * q^86 - 27283685376 * q^88 + 38295584136 * q^89 + 177007512158 * q^91 + 5389983744 * q^92 - 141814822656 * q^94 + 103452470040 * q^95 + 109290600874 * q^97 + 31718195328 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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