LMFDB - Embedded newform 48.26.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,26,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 26);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 26 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$190.078454377$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 1647414391x^{2} - 16173027965094x + 235708219135253151 $ x^4 - 1647414391x^2 - 16173027965094x + 235708219135253151
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$43526.6$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+531441. q^{3} +1.03572e9 q^{5} +3.32173e10 q^{7} +2.82430e11 q^{9} +O(q^{10})$	$q+531441. q^{3} +1.03572e9 q^{5} +3.32173e10 q^{7} +2.82430e11 q^{9} +7.60614e12 q^{11} +5.42918e13 q^{13} +5.50424e14 q^{15} +2.53686e15 q^{17} +1.35588e15 q^{19} +1.76530e16 q^{21} -1.50819e17 q^{23} +7.74693e17 q^{25} +1.50095e17 q^{27} -2.24955e18 q^{29} +6.09626e18 q^{31} +4.04221e18 q^{33} +3.44038e19 q^{35} -2.76915e19 q^{37} +2.88529e19 q^{39} +2.52183e20 q^{41} +2.51287e20 q^{43} +2.92518e20 q^{45} -4.65208e19 q^{47} -2.37682e20 q^{49} +1.34819e21 q^{51} +3.32933e21 q^{53} +7.87783e21 q^{55} +7.20570e20 q^{57} -1.91949e22 q^{59} -1.45149e21 q^{61} +9.38154e21 q^{63} +5.62311e22 q^{65} -3.78805e22 q^{67} -8.01512e22 q^{69} -1.33486e23 q^{71} -2.87808e23 q^{73} +4.11703e23 q^{75} +2.52655e23 q^{77} +2.03381e23 q^{79} +7.97664e22 q^{81} -4.22924e23 q^{83} +2.62747e24 q^{85} -1.19550e24 q^{87} +3.33118e24 q^{89} +1.80342e24 q^{91} +3.23980e24 q^{93} +1.40431e24 q^{95} +2.95556e24 q^{97} +2.14820e24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9}+O(q^{10}) $ 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9	$ 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9} - 4052955557392 q^{11} + 82663381481080 q^{13} + 141595061042808 q^{15} + 10\!\cdots\!28 q^{17}+ \cdots - 11\!\cdots\!52 q^{99}+O(q^{100}) $ 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9 - 4052955557392 * q^11 + 82663381481080 * q^13 + 141595061042808 * q^15 + 1091839531710728 * q^17 + 5291818285650448 * q^19 - 257528111654016 * q^21 - 22957555441633504 * q^23 + 329573428119320444 * q^25 + 600378541187996484 * q^27 - 2255995699601091048 * q^29 + 6422134828063294816 * q^31 - 2153906754375961872 * q^33 + 26450723987305847040 * q^35 - 84572383178406307176 * q^37 + 43930710117686636280 * q^39 - 191923587571359535128 * q^41 + 393512700764425221104 * q^43 + 75249420835650926328 * q^45 + 426139906889905851648 * q^47 + 1697461944491957688548 * q^49 + 580248292571880999048 * q^51 + 4165797514523716100344 * q^53 - 1377411551242297580512 * q^55 + 2812289201544359735568 * q^57 - 22289642820361283440912 * q^59 + 31420833466214461354360 * q^61 - 136860997185521917056 * q^63 + 49904798993160775849744 * q^65 + 105958367596085546226448 * q^67 - 12200586221457150999264 * q^69 + 260836089844500944794336 * q^71 + 96585608913237168931624 * q^73 + 175148832213159776079804 * q^75 + 609668306798881251586560 * q^77 + 228314222637752365466528 * q^79 + 319065772307490039453444 * q^81 - 508041934949405113277936 * q^83 + 3527355809877820452506096 * q^85 - 1198928610591703427640168 * q^87 + 9334569288453070817495784 * q^89 - 6414440882259952713889536 * q^91 + 3412985755160785460309856 * q^93 - 9916967691165637498776608 * q^95 + 17898782631688573069195784 * q^97 - 1144674359452315553217552 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	531441.	0.577350
$3$	531441.	0.577350
$4$	0	0
$5$	1.03572e9	1.89722	0.948609	−	0.316450i	$-0.102491\pi$
$5$	1.03572e9	1.89722	0.948609	+	0.316450i	$0.102491\pi$
$6$	0	0
$7$	3.32173e10	0.907065	0.453532	−	0.891240i	$-0.350164\pi$
$7$	3.32173e10	0.907065	0.453532	+	0.891240i	$0.350164\pi$
$8$	0	0
$9$	2.82430e11	0.333333
$10$	0	0
$11$	7.60614e12	0.730728	0.365364	−	0.930865i	$-0.380945\pi$
$11$	7.60614e12	0.730728	0.365364	+	0.930865i	$0.380945\pi$
$12$	0	0
$13$	5.42918e13	0.646312	0.323156	−	0.946346i	$-0.395256\pi$
$13$	5.42918e13	0.646312	0.323156	+	0.946346i	$0.395256\pi$
$14$	0	0
$15$	5.50424e14	1.09536
$16$	0	0
$17$	2.53686e15	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$17$	2.53686e15	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$18$	0	0
$19$	1.35588e15	0.140540	0.0702702	−	0.997528i	$-0.477614\pi$
$19$	1.35588e15	0.140540	0.0702702	+	0.997528i	$0.477614\pi$
$20$	0	0
$21$	1.76530e16	0.523694
$22$	0	0
$23$	−1.50819e17	−1.43502	−0.717509	−	0.696549i	$-0.754718\pi$
$23$	−1.50819e17	−1.43502	−0.717509	+	0.696549i	$0.754718\pi$
$24$	0	0
$25$	7.74693e17	2.59944
$26$	0	0
$27$	1.50095e17	0.192450
$28$	0	0
$29$	−2.24955e18	−1.18065	−0.590323	−	0.807167i	$-0.701000\pi$
$29$	−2.24955e18	−1.18065	−0.590323	+	0.807167i	$0.701000\pi$
$30$	0	0
$31$	6.09626e18	1.39009	0.695044	−	0.718967i	$-0.255385\pi$
$31$	6.09626e18	1.39009	0.695044	+	0.718967i	$0.255385\pi$
$32$	0	0
$33$	4.04221e18	0.421886
$34$	0	0
$35$	3.44038e19	1.72090
$36$	0	0
$37$	−2.76915e19	−0.691552	−0.345776	−	0.938317i	$-0.612384\pi$
$37$	−2.76915e19	−0.691552	−0.345776	+	0.938317i	$0.612384\pi$
$38$	0	0
$39$	2.88529e19	0.373148
$40$	0	0
$41$	2.52183e20	1.74549	0.872744	−	0.488178i	$-0.162338\pi$
$41$	2.52183e20	1.74549	0.872744	+	0.488178i	$0.162338\pi$
$42$	0	0
$43$	2.51287e20	0.958992	0.479496	−	0.877544i	$-0.340820\pi$
$43$	2.51287e20	0.958992	0.479496	+	0.877544i	$0.340820\pi$
$44$	0	0
$45$	2.92518e20	0.632406
$46$	0	0
$47$	−4.65208e19	−0.0584014	−0.0292007	−	0.999574i	$-0.509296\pi$
$47$	−4.65208e19	−0.0584014	−0.0292007	+	0.999574i	$0.509296\pi$
$48$	0	0
$49$	−2.37682e20	−0.177233
$50$	0	0
$51$	1.34819e21	0.609710
$52$	0	0
$53$	3.32933e21	0.930911	0.465456	−	0.885071i	$-0.345891\pi$
$53$	3.32933e21	0.930911	0.465456	+	0.885071i	$0.345891\pi$
$54$	0	0
$55$	7.87783e21	1.38635
$56$	0	0
$57$	7.20570e20	0.0811410
$58$	0	0
$59$	−1.91949e22	−1.40455	−0.702273	−	0.711907i	$-0.747832\pi$
$59$	−1.91949e22	−1.40455	−0.702273	+	0.711907i	$0.747832\pi$
$60$	0	0
$61$	−1.45149e21	−0.0700152	−0.0350076	−	0.999387i	$-0.511146\pi$
$61$	−1.45149e21	−0.0700152	−0.0350076	+	0.999387i	$0.511146\pi$
$62$	0	0
$63$	9.38154e21	0.302355
$64$	0	0
$65$	5.62311e22	1.22619
$66$	0	0
$67$	−3.78805e22	−0.565563	−0.282782	−	0.959184i	$-0.591257\pi$
$67$	−3.78805e22	−0.565563	−0.282782	+	0.959184i	$0.591257\pi$
$68$	0	0
$69$	−8.01512e22	−0.828508
$70$	0	0
$71$	−1.33486e23	−0.965399	−0.482699	−	0.875786i	$-0.660344\pi$
$71$	−1.33486e23	−0.965399	−0.482699	+	0.875786i	$0.660344\pi$
$72$	0	0
$73$	−2.87808e23	−1.47085	−0.735423	−	0.677608i	$-0.763017\pi$
$73$	−2.87808e23	−1.47085	−0.735423	+	0.677608i	$0.763017\pi$
$74$	0	0
$75$	4.11703e23	1.50079
$76$	0	0
$77$	2.52655e23	0.662818
$78$	0	0
$79$	2.03381e23	0.387232	0.193616	−	0.981077i	$-0.437978\pi$
$79$	2.03381e23	0.387232	0.193616	+	0.981077i	$0.437978\pi$
$80$	0	0
$81$	7.97664e22	0.111111
$82$	0	0
$83$	−4.22924e23	−0.434296	−0.217148	−	0.976139i	$-0.569675\pi$
$83$	−4.22924e23	−0.434296	−0.217148	+	0.976139i	$0.569675\pi$
$84$	0	0
$85$	2.62747e24	2.00356
$86$	0	0
$87$	−1.19550e24	−0.681646
$88$	0	0
$89$	3.33118e24	1.42963	0.714814	−	0.699315i	$-0.246511\pi$
$89$	3.33118e24	1.42963	0.714814	+	0.699315i	$0.246511\pi$
$90$	0	0
$91$	1.80342e24	0.586247
$92$	0	0
$93$	3.23980e24	0.802567
$94$	0	0
$95$	1.40431e24	0.266636
$96$	0	0
$97$	2.95556e24	0.432507	0.216254	−	0.976337i	$-0.430616\pi$
$97$	2.95556e24	0.432507	0.216254	+	0.976337i	$0.430616\pi$
$98$	0	0
$99$	2.14820e24	0.243576
$100$	0	0
$101$	−2.24506e25	−1.98249	−0.991247	−	0.132023i	$-0.957853\pi$
$101$	−2.24506e25	−1.98249	−0.991247	+	0.132023i	$0.957853\pi$
$102$	0	0
$103$	−2.28109e25	−1.57644	−0.788219	−	0.615395i	$-0.788996\pi$
$103$	−2.28109e25	−1.57644	−0.788219	+	0.615395i	$0.788996\pi$
$104$	0	0
$105$	1.82836e25	0.993562
$106$	0	0
$107$	−4.38483e25	−1.88216	−0.941079	−	0.338187i	$-0.890186\pi$
$107$	−4.38483e25	−1.88216	−0.941079	+	0.338187i	$0.890186\pi$
$108$	0	0
$109$	8.47757e24	0.288695	0.144348	−	0.989527i	$-0.453892\pi$
$109$	8.47757e24	0.288695	0.144348	+	0.989527i	$0.453892\pi$
$110$	0	0
$111$	−1.47164e25	−0.399268
$112$	0	0
$113$	1.22205e25	0.265221	0.132611	−	0.991168i	$-0.457664\pi$
$113$	1.22205e25	0.265221	0.132611	+	0.991168i	$0.457664\pi$
$114$	0	0
$115$	−1.56206e26	−2.72254
$116$	0	0
$117$	1.53336e25	0.215437
$118$	0	0
$119$	8.42674e25	0.957905
$120$	0	0
$121$	−5.04937e25	−0.466037
$122$	0	0
$123$	1.34020e26	1.00776
$124$	0	0
$125$	4.93696e26	3.03448
$126$	0	0
$127$	−2.47670e25	−0.124832	−0.0624162	−	0.998050i	$-0.519881\pi$
$127$	−2.47670e25	−0.124832	−0.0624162	+	0.998050i	$0.519881\pi$
$128$	0	0
$129$	1.33544e26	0.553674
$130$	0	0
$131$	−4.22730e26	−1.44601	−0.723006	−	0.690842i	$-0.757240\pi$
$131$	−4.22730e26	−1.44601	−0.723006	+	0.690842i	$0.757240\pi$
$132$	0	0
$133$	4.50386e25	0.127479
$134$	0	0
$135$	1.55456e26	0.365120
$136$	0	0
$137$	5.43358e26	1.06189	0.530945	−	0.847407i	$-0.321837\pi$
$137$	5.43358e26	1.06189	0.530945	+	0.847407i	$0.321837\pi$
$138$	0	0
$139$	−4.53658e26	−0.739679	−0.369840	−	0.929096i	$-0.620587\pi$
$139$	−4.53658e26	−0.739679	−0.369840	+	0.929096i	$0.620587\pi$
$140$	0	0
$141$	−2.47230e25	−0.0337181
$142$	0	0
$143$	4.12951e26	0.472278
$144$	0	0
$145$	−2.32990e27	−2.23994
$146$	0	0
$147$	−1.26314e26	−0.102326
$148$	0	0
$149$	3.88473e26	0.265786	0.132893	−	0.991130i	$-0.457573\pi$
$149$	3.88473e26	0.265786	0.132893	+	0.991130i	$0.457573\pi$
$150$	0	0
$151$	−9.38128e26	−0.543313	−0.271657	−	0.962394i	$-0.587571\pi$
$151$	−9.38128e26	−0.543313	−0.271657	+	0.962394i	$0.587571\pi$
$152$	0	0
$153$	7.16483e26	0.352017
$154$	0	0
$155$	6.31402e27	2.63730
$156$	0	0
$157$	1.91372e27	0.680977	0.340488	−	0.940249i	$-0.389408\pi$
$157$	1.91372e27	0.680977	0.340488	+	0.940249i	$0.389408\pi$
$158$	0	0
$159$	1.76934e27	0.537462
$160$	0	0
$161$	−5.00978e27	−1.30165
$162$	0	0
$163$	−5.50273e27	−1.22527	−0.612637	−	0.790364i	$-0.709891\pi$
$163$	−5.50273e27	−1.22527	−0.612637	+	0.790364i	$0.709891\pi$
$164$	0	0
$165$	4.18660e27	0.800410
$166$	0	0
$167$	1.05820e27	0.174024	0.0870122	−	0.996207i	$-0.472268\pi$
$167$	1.05820e27	0.174024	0.0870122	+	0.996207i	$0.472268\pi$
$168$	0	0
$169$	−4.10881e27	−0.582281
$170$	0	0
$171$	3.82940e26	0.0468468
$172$	0	0
$173$	8.13045e27	0.860079	0.430039	−	0.902810i	$-0.358500\pi$
$173$	8.13045e27	0.860079	0.430039	+	0.902810i	$0.358500\pi$
$174$	0	0
$175$	2.57332e28	2.35786
$176$	0	0
$177$	−1.02010e28	−0.810915
$178$	0	0
$179$	2.14299e28	1.48032	0.740162	−	0.672428i	$-0.234749\pi$
$179$	2.14299e28	1.48032	0.740162	+	0.672428i	$0.234749\pi$
$180$	0	0
$181$	−8.55613e26	−0.0514393	−0.0257197	−	0.999669i	$-0.508188\pi$
$181$	−8.55613e26	−0.0514393	−0.0257197	+	0.999669i	$0.508188\pi$
$182$	0	0
$183$	−7.71383e26	−0.0404233
$184$	0	0
$185$	−2.86806e28	−1.31203
$186$	0	0
$187$	1.92957e28	0.771685
$188$	0	0
$189$	4.98573e27	0.174565
$190$	0	0
$191$	−6.08648e28	−1.86831	−0.934156	−	0.356865i	$-0.883846\pi$
$191$	−6.08648e28	−1.86831	−0.934156	+	0.356865i	$0.883846\pi$
$192$	0	0
$193$	−9.37995e27	−0.252775	−0.126388	−	0.991981i	$-0.540338\pi$
$193$	−9.37995e27	−0.252775	−0.126388	+	0.991981i	$0.540338\pi$
$194$	0	0
$195$	2.98835e28	0.707944
$196$	0	0
$197$	5.79367e28	1.20816	0.604082	−	0.796922i	$-0.293540\pi$
$197$	5.79367e28	1.20816	0.604082	+	0.796922i	$0.293540\pi$
$198$	0	0
$199$	1.93145e28	0.354993	0.177497	−	0.984121i	$-0.443200\pi$
$199$	1.93145e28	0.354993	0.177497	+	0.984121i	$0.443200\pi$
$200$	0	0
$201$	−2.01313e28	−0.326528
$202$	0	0
$203$	−7.47237e28	−1.07092
$204$	0	0
$205$	2.61191e29	3.31157
$206$	0	0
$207$	−4.25957e28	−0.478339
$208$	0	0
$209$	1.03130e28	0.102697
$210$	0	0
$211$	1.94531e29	1.71972	0.859862	−	0.510527i	$-0.170550\pi$
$211$	1.94531e29	1.71972	0.859862	+	0.510527i	$0.170550\pi$
$212$	0	0
$213$	−7.09400e28	−0.557373
$214$	0	0
$215$	2.60263e29	1.81942
$216$	0	0
$217$	2.02501e29	1.26090
$218$	0	0
$219$	−1.52953e29	−0.849193
$220$	0	0
$221$	1.37730e29	0.682537
$222$	0	0
$223$	−1.71983e29	−0.761508	−0.380754	−	0.924676i	$-0.624336\pi$
$223$	−1.71983e29	−0.761508	−0.380754	+	0.924676i	$0.624336\pi$
$224$	0	0
$225$	2.18796e29	0.866479
$226$	0	0
$227$	3.90202e29	1.38346	0.691730	−	0.722156i	$-0.256849\pi$
$227$	3.90202e29	1.38346	0.691730	+	0.722156i	$0.256849\pi$
$228$	0	0
$229$	6.24500e29	1.98422	0.992108	−	0.125389i	$-0.0400179\pi$
$229$	6.24500e29	1.98422	0.992108	+	0.125389i	$0.0400179\pi$
$230$	0	0
$231$	1.34271e29	0.382678
$232$	0	0
$233$	5.50811e29	1.40946	0.704731	−	0.709475i	$-0.251068\pi$
$233$	5.50811e29	1.40946	0.704731	+	0.709475i	$0.251068\pi$
$234$	0	0
$235$	−4.81825e28	−0.110800
$236$	0	0
$237$	1.08085e29	0.223569
$238$	0	0
$239$	−9.90750e28	−0.184498	−0.0922488	−	0.995736i	$-0.529405\pi$
$239$	−9.90750e28	−0.184498	−0.0922488	+	0.995736i	$0.529405\pi$
$240$	0	0
$241$	−7.84330e29	−1.31609	−0.658045	−	0.752979i	$-0.728616\pi$
$241$	−7.84330e29	−1.31609	−0.658045	+	0.752979i	$0.728616\pi$
$242$	0	0
$243$	4.23912e28	0.0641500
$244$	0	0
$245$	−2.46172e29	−0.336250
$246$	0	0
$247$	7.36131e28	0.0908329
$248$	0	0
$249$	−2.24759e29	−0.250741
$250$	0	0
$251$	6.69272e29	0.675586	0.337793	−	0.941220i	$-0.390320\pi$
$251$	6.69272e29	0.675586	0.337793	+	0.941220i	$0.390320\pi$
$252$	0	0
$253$	−1.14715e30	−1.04861
$254$	0	0
$255$	1.39635e30	1.15675
$256$	0	0
$257$	1.46988e30	1.10438	0.552189	−	0.833719i	$-0.313793\pi$
$257$	1.46988e30	1.10438	0.552189	+	0.833719i	$0.313793\pi$
$258$	0	0
$259$	−9.19836e29	−0.627283
$260$	0	0
$261$	−6.35338e29	−0.393549
$262$	0	0
$263$	−2.95239e29	−0.166237	−0.0831184	−	0.996540i	$-0.526488\pi$
$263$	−2.95239e29	−0.166237	−0.0831184	+	0.996540i	$0.526488\pi$
$264$	0	0
$265$	3.44825e30	1.76614
$266$	0	0
$267$	1.77032e30	0.825396
$268$	0	0
$269$	−2.01088e30	−0.854050	−0.427025	−	0.904240i	$-0.640438\pi$
$269$	−2.01088e30	−0.854050	−0.427025	+	0.904240i	$0.640438\pi$
$270$	0	0
$271$	2.45837e30	0.951770	0.475885	−	0.879508i	$-0.342128\pi$
$271$	2.45837e30	0.951770	0.475885	+	0.879508i	$0.342128\pi$
$272$	0	0
$273$	9.58413e29	0.338470
$274$	0	0
$275$	5.89242e30	1.89948
$276$	0	0
$277$	−1.37534e30	−0.404959	−0.202480	−	0.979286i	$-0.564900\pi$
$277$	−1.37534e30	−0.404959	−0.202480	+	0.979286i	$0.564900\pi$
$278$	0	0
$279$	1.72176e30	0.463362
$280$	0	0
$281$	6.55349e29	0.161304	0.0806518	−	0.996742i	$-0.474300\pi$
$281$	6.55349e29	0.161304	0.0806518	+	0.996742i	$0.474300\pi$
$282$	0	0
$283$	3.35779e30	0.756349	0.378175	−	0.925734i	$-0.376552\pi$
$283$	3.35779e30	0.756349	0.378175	+	0.925734i	$0.376552\pi$
$284$	0	0
$285$	7.46309e29	0.153942
$286$	0	0
$287$	8.37682e30	1.58327
$288$	0	0
$289$	6.65011e29	0.115241
$290$	0	0
$291$	1.57071e30	0.249708
$292$	0	0
$293$	−6.43876e30	−0.939631	−0.469815	−	0.882765i	$-0.655680\pi$
$293$	−6.43876e30	−0.939631	−0.469815	+	0.882765i	$0.655680\pi$
$294$	0	0
$295$	−1.98806e31	−2.66473
$296$	0	0
$297$	1.14164e30	0.140629
$298$	0	0
$299$	−8.18821e30	−0.927469
$300$	0	0
$301$	8.34708e30	0.869868
$302$	0	0
$303$	−1.19312e31	−1.14459
$304$	0	0
$305$	−1.50334e30	−0.132834
$306$	0	0
$307$	−7.73680e30	−0.629987	−0.314994	−	0.949094i	$-0.602002\pi$
$307$	−7.73680e30	−0.629987	−0.314994	+	0.949094i	$0.602002\pi$
$308$	0	0
$309$	−1.21226e31	−0.910157
$310$	0	0
$311$	−1.69403e31	−1.17332	−0.586659	−	0.809834i	$-0.699557\pi$
$311$	−1.69403e31	−1.17332	−0.586659	+	0.809834i	$0.699557\pi$
$312$	0	0
$313$	1.94689e31	1.24462	0.622310	−	0.782771i	$-0.286194\pi$
$313$	1.94689e31	1.24462	0.622310	+	0.782771i	$0.286194\pi$
$314$	0	0
$315$	9.71664e30	0.573633
$316$	0	0
$317$	−2.01752e31	−1.10047	−0.550233	−	0.835011i	$-0.685461\pi$
$317$	−2.01752e31	−1.10047	−0.550233	+	0.835011i	$0.685461\pi$
$318$	0	0
$319$	−1.71104e31	−0.862731
$320$	0	0
$321$	−2.33028e31	−1.08666
$322$	0	0
$323$	3.43967e30	0.148418
$324$	0	0
$325$	4.20594e31	1.68005
$326$	0	0
$327$	4.50533e30	0.166678
$328$	0	0
$329$	−1.54529e30	−0.0529739
$330$	0	0
$331$	7.83114e30	0.248871	0.124436	−	0.992228i	$-0.460288\pi$
$331$	7.83114e30	0.248871	0.124436	+	0.992228i	$0.460288\pi$
$332$	0	0
$333$	−7.82090e30	−0.230517
$334$	0	0
$335$	−3.92336e31	−1.07300
$336$	0	0
$337$	−2.66493e31	−0.676569	−0.338285	−	0.941044i	$-0.609847\pi$
$337$	−2.66493e31	−0.676569	−0.338285	+	0.941044i	$0.609847\pi$
$338$	0	0
$339$	6.49448e30	0.153126
$340$	0	0
$341$	4.63690e31	1.01578
$342$	0	0
$343$	−5.24418e31	−1.06783
$344$	0	0
$345$	−8.30142e31	−1.57186
$346$	0	0
$347$	−1.54993e31	−0.273020	−0.136510	−	0.990639i	$-0.543589\pi$
$347$	−1.54993e31	−0.273020	−0.136510	+	0.990639i	$0.543589\pi$
$348$	0	0
$349$	8.05996e31	1.32134	0.660671	−	0.750676i	$-0.270272\pi$
$349$	8.05996e31	1.32134	0.660671	+	0.750676i	$0.270272\pi$
$350$	0	0
$351$	8.14890e30	0.124383
$352$	0	0
$353$	−1.18306e32	−1.68199	−0.840997	−	0.541040i	$-0.818031\pi$
$353$	−1.18306e32	−1.68199	−0.840997	+	0.541040i	$0.818031\pi$
$354$	0	0
$355$	−1.38254e32	−1.83157
$356$	0	0
$357$	4.47832e31	0.553047
$358$	0	0
$359$	6.16966e31	0.710527	0.355263	−	0.934766i	$-0.384391\pi$
$359$	6.16966e31	0.710527	0.355263	+	0.934766i	$0.384391\pi$
$360$	0	0
$361$	−9.12381e31	−0.980248
$362$	0	0
$363$	−2.68344e31	−0.269066
$364$	0	0
$365$	−2.98089e32	−2.79052
$366$	0	0
$367$	−5.91930e31	−0.517541	−0.258771	−	0.965939i	$-0.583317\pi$
$367$	−5.91930e31	−0.517541	−0.258771	+	0.965939i	$0.583317\pi$
$368$	0	0
$369$	7.12238e31	0.581830
$370$	0	0
$371$	1.10591e32	0.844397
$372$	0	0
$373$	−8.03433e31	−0.573573	−0.286787	−	0.957994i	$-0.592587\pi$
$373$	−8.03433e31	−0.573573	−0.286787	+	0.957994i	$0.592587\pi$
$374$	0	0
$375$	2.62370e32	1.75196
$376$	0	0
$377$	−1.22132e32	−0.763066
$378$	0	0
$379$	−3.92687e31	−0.229645	−0.114822	−	0.993386i	$-0.536630\pi$
$379$	−3.92687e31	−0.229645	−0.114822	+	0.993386i	$0.536630\pi$
$380$	0	0
$381$	−1.31622e31	−0.0720720
$382$	0	0
$383$	2.61593e32	1.34166	0.670828	−	0.741613i	$-0.265939\pi$
$383$	2.61593e32	1.34166	0.670828	+	0.741613i	$0.265939\pi$
$384$	0	0
$385$	2.61680e32	1.25751
$386$	0	0
$387$	7.09710e31	0.319664
$388$	0	0
$389$	3.22209e32	1.36072	0.680358	−	0.732880i	$-0.261824\pi$
$389$	3.22209e32	1.36072	0.680358	+	0.732880i	$0.261824\pi$
$390$	0	0
$391$	−3.82605e32	−1.51545
$392$	0	0
$393$	−2.24656e32	−0.834855
$394$	0	0
$395$	2.10646e32	0.734664
$396$	0	0
$397$	−1.46124e31	−0.0478455	−0.0239227	−	0.999714i	$-0.507616\pi$
$397$	−1.46124e31	−0.0478455	−0.0239227	+	0.999714i	$0.507616\pi$
$398$	0	0
$399$	2.39354e31	0.0736002
$400$	0	0
$401$	2.58797e32	0.747574	0.373787	−	0.927515i	$-0.378059\pi$
$401$	2.58797e32	0.747574	0.373787	+	0.927515i	$0.378059\pi$
$402$	0	0
$403$	3.30977e32	0.898430
$404$	0	0
$405$	8.26157e31	0.210802
$406$	0	0
$407$	−2.10625e32	−0.505336
$408$	0	0
$409$	1.49624e32	0.337644	0.168822	−	0.985647i	$-0.446004\pi$
$409$	1.49624e32	0.337644	0.168822	+	0.985647i	$0.446004\pi$
$410$	0	0
$411$	2.88763e32	0.613082
$412$	0	0
$413$	−6.37603e32	−1.27401
$414$	0	0
$415$	−4.38031e32	−0.823955
$416$	0	0
$417$	−2.41092e32	−0.427054
$418$	0	0
$419$	−8.92181e32	−1.48860	−0.744300	−	0.667846i	$-0.767217\pi$
$419$	−8.92181e32	−1.48860	−0.744300	+	0.667846i	$0.767217\pi$
$420$	0	0
$421$	−2.14602e32	−0.337371	−0.168686	−	0.985670i	$-0.553952\pi$
$421$	−2.14602e32	−0.337371	−0.168686	+	0.985670i	$0.553952\pi$
$422$	0	0
$423$	−1.31388e31	−0.0194671
$424$	0	0
$425$	1.96528e33	2.74513
$426$	0	0
$427$	−4.82147e31	−0.0635083
$428$	0	0
$429$	2.19459e32	0.272670
$430$	0	0
$431$	4.74900e32	0.556720	0.278360	−	0.960477i	$-0.410209\pi$
$431$	4.74900e32	0.556720	0.278360	+	0.960477i	$0.410209\pi$
$432$	0	0
$433$	1.00740e33	1.11456	0.557278	−	0.830326i	$-0.311846\pi$
$433$	1.00740e33	1.11456	0.557278	+	0.830326i	$0.311846\pi$
$434$	0	0
$435$	−1.23820e33	−1.29323
$436$	0	0
$437$	−2.04492e32	−0.201678
$438$	0	0
$439$	7.83160e32	0.729531	0.364766	−	0.931099i	$-0.381149\pi$
$439$	7.83160e32	0.729531	0.364766	+	0.931099i	$0.381149\pi$
$440$	0	0
$441$	−6.71284e31	−0.0590778
$442$	0	0
$443$	2.35773e32	0.196086	0.0980432	−	0.995182i	$-0.468742\pi$
$443$	2.35773e32	0.196086	0.0980432	+	0.995182i	$0.468742\pi$
$444$	0	0
$445$	3.45017e33	2.71231
$446$	0	0
$447$	2.06451e32	0.153452
$448$	0	0
$449$	−6.19556e32	−0.435513	−0.217757	−	0.976003i	$-0.569874\pi$
$449$	−6.19556e32	−0.435513	−0.217757	+	0.976003i	$0.569874\pi$
$450$	0	0
$451$	1.91814e33	1.27548
$452$	0	0
$453$	−4.98560e32	−0.313682
$454$	0	0
$455$	1.86784e33	1.11224
$456$	0	0
$457$	4.74565e32	0.267512	0.133756	−	0.991014i	$-0.457296\pi$
$457$	4.74565e32	0.267512	0.133756	+	0.991014i	$0.457296\pi$
$458$	0	0
$459$	3.80768e32	0.203237
$460$	0	0
$461$	−2.80426e33	−1.41761	−0.708805	−	0.705404i	$-0.750765\pi$
$461$	−2.80426e33	−1.41761	−0.708805	+	0.705404i	$0.750765\pi$
$462$	0	0
$463$	2.29467e33	1.09890	0.549449	−	0.835527i	$-0.314838\pi$
$463$	2.29467e33	1.09890	0.549449	+	0.835527i	$0.314838\pi$
$464$	0	0
$465$	3.35553e33	1.52265
$466$	0	0
$467$	−1.70515e32	−0.0733333	−0.0366666	−	0.999328i	$-0.511674\pi$
$467$	−1.70515e32	−0.0733333	−0.0366666	+	0.999328i	$0.511674\pi$
$468$	0	0
$469$	−1.25829e33	−0.513003
$470$	0	0
$471$	1.01703e33	0.393162
$472$	0	0
$473$	1.91133e33	0.700762
$474$	0	0
$475$	1.05039e33	0.365326
$476$	0	0
$477$	9.40301e32	0.310304
$478$	0	0
$479$	−3.95233e33	−1.23782	−0.618912	−	0.785460i	$-0.712426\pi$
$479$	−3.95233e33	−1.23782	−0.618912	+	0.785460i	$0.712426\pi$
$480$	0	0
$481$	−1.50342e33	−0.446958
$482$	0	0
$483$	−2.66240e33	−0.751510
$484$	0	0
$485$	3.06113e33	0.820560
$486$	0	0
$487$	2.29925e33	0.585429	0.292714	−	0.956200i	$-0.405441\pi$
$487$	2.29925e33	0.585429	0.292714	+	0.956200i	$0.405441\pi$
$488$	0	0
$489$	−2.92438e33	−0.707412
$490$	0	0
$491$	−3.08153e33	−0.708350	−0.354175	−	0.935179i	$-0.615238\pi$
$491$	−3.08153e33	−0.708350	−0.354175	+	0.935179i	$0.615238\pi$
$492$	0	0
$493$	−5.70677e33	−1.24682
$494$	0	0
$495$	2.22493e33	0.462117
$496$	0	0
$497$	−4.43404e33	−0.875679
$498$	0	0
$499$	9.20524e33	1.72893	0.864467	−	0.502690i	$-0.167656\pi$
$499$	9.20524e33	1.72893	0.864467	+	0.502690i	$0.167656\pi$
$500$	0	0
$501$	5.62369e32	0.100473
$502$	0	0
$503$	4.38047e33	0.744597	0.372298	−	0.928113i	$-0.378570\pi$
$503$	4.38047e33	0.744597	0.372298	+	0.928113i	$0.378570\pi$
$504$	0	0
$505$	−2.32526e34	−3.76122
$506$	0	0
$507$	−2.18359e33	−0.336180
$508$	0	0
$509$	1.85572e33	0.271983	0.135991	−	0.990710i	$-0.456578\pi$
$509$	1.85572e33	0.271983	0.135991	+	0.990710i	$0.456578\pi$
$510$	0	0
$511$	−9.56020e33	−1.33415
$512$	0	0
$513$	2.03510e32	0.0270470
$514$	0	0
$515$	−2.36257e34	−2.99085
$516$	0	0
$517$	−3.53843e32	−0.0426756
$518$	0	0
$519$	4.32085e33	0.496567
$520$	0	0
$521$	−9.84209e33	−1.07799	−0.538996	−	0.842308i	$-0.681196\pi$
$521$	−9.84209e33	−1.07799	−0.538996	+	0.842308i	$0.681196\pi$
$522$	0	0
$523$	1.34833e34	1.40774	0.703872	−	0.710327i	$-0.251453\pi$
$523$	1.34833e34	1.40774	0.703872	+	0.710327i	$0.251453\pi$
$524$	0	0
$525$	1.36757e34	1.36131
$526$	0	0
$527$	1.54653e34	1.46800
$528$	0	0
$529$	1.17005e34	1.05928
$530$	0	0
$531$	−5.42121e33	−0.468182
$532$	0	0
$533$	1.36914e34	1.12813
$534$	0	0
$535$	−4.54146e34	−3.57086
$536$	0	0
$537$	1.13887e34	0.854666
$538$	0	0
$539$	−1.80784e33	−0.129509
$540$	0	0
$541$	−1.68646e34	−1.15348	−0.576739	−	0.816928i	$-0.695675\pi$
$541$	−1.68646e34	−1.15348	−0.576739	+	0.816928i	$0.695675\pi$
$542$	0	0
$543$	−4.54708e32	−0.0296985
$544$	0	0
$545$	8.78038e33	0.547718
$546$	0	0
$547$	1.52523e34	0.908855	0.454427	−	0.890784i	$-0.349844\pi$
$547$	1.52523e34	0.908855	0.454427	+	0.890784i	$0.349844\pi$
$548$	0	0
$549$	−4.09945e32	−0.0233384
$550$	0	0
$551$	−3.05011e33	−0.165928
$552$	0	0
$553$	6.75576e33	0.351245
$554$	0	0
$555$	−1.52421e34	−0.757498
$556$	0	0
$557$	−2.15303e34	−1.02296	−0.511482	−	0.859294i	$-0.670903\pi$
$557$	−2.15303e34	−1.02296	−0.511482	+	0.859294i	$0.670903\pi$
$558$	0	0
$559$	1.36428e34	0.619808
$560$	0	0
$561$	1.02545e34	0.445533
$562$	0	0
$563$	−2.64941e34	−1.10102	−0.550509	−	0.834829i	$-0.685566\pi$
$563$	−2.64941e34	−1.10102	−0.550509	+	0.834829i	$0.685566\pi$
$564$	0	0
$565$	1.26570e34	0.503183
$566$	0	0
$567$	2.64962e33	0.100785
$568$	0	0
$569$	−3.44517e34	−1.25403	−0.627014	−	0.779008i	$-0.715723\pi$
$569$	−3.44517e34	−1.25403	−0.627014	+	0.779008i	$0.715723\pi$
$570$	0	0
$571$	2.21254e34	0.770796	0.385398	−	0.922750i	$-0.374064\pi$
$571$	2.21254e34	0.770796	0.385398	+	0.922750i	$0.374064\pi$
$572$	0	0
$573$	−3.23461e34	−1.07867
$574$	0	0
$575$	−1.16838e35	−3.73024
$576$	0	0
$577$	5.37387e33	0.164282	0.0821409	−	0.996621i	$-0.473824\pi$
$577$	5.37387e33	0.164282	0.0821409	+	0.996621i	$0.473824\pi$
$578$	0	0
$579$	−4.98489e33	−0.145940
$580$	0	0
$581$	−1.40484e34	−0.393935
$582$	0	0
$583$	2.53233e34	0.680243
$584$	0	0
$585$	1.58813e34	0.408731
$586$	0	0
$587$	1.13800e34	0.280650	0.140325	−	0.990105i	$-0.455185\pi$
$587$	1.13800e34	0.280650	0.140325	+	0.990105i	$0.455185\pi$
$588$	0	0
$589$	8.26579e33	0.195363
$590$	0	0
$591$	3.07899e34	0.697533
$592$	0	0
$593$	2.25279e33	0.0489258	0.0244629	−	0.999701i	$-0.492212\pi$
$593$	2.25279e33	0.0489258	0.0244629	+	0.999701i	$0.492212\pi$
$594$	0	0
$595$	8.72774e34	1.81736
$596$	0	0
$597$	1.02645e34	0.204956
$598$	0	0
$599$	2.80123e34	0.536431	0.268216	−	0.963359i	$-0.413566\pi$
$599$	2.80123e34	0.536431	0.268216	+	0.963359i	$0.413566\pi$
$600$	0	0
$601$	8.28210e34	1.52128	0.760641	−	0.649173i	$-0.224885\pi$
$601$	8.28210e34	1.52128	0.760641	+	0.649173i	$0.224885\pi$
$602$	0	0
$603$	−1.06986e34	−0.188521
$604$	0	0
$605$	−5.22973e34	−0.884173
$606$	0	0
$607$	−9.39877e34	−1.52480	−0.762399	−	0.647107i	$-0.775978\pi$
$607$	−9.39877e34	−1.52480	−0.762399	+	0.647107i	$0.775978\pi$
$608$	0	0
$609$	−3.97113e34	−0.618298
$610$	0	0
$611$	−2.52569e33	−0.0377455
$612$	0	0
$613$	−8.24805e34	−1.18330	−0.591651	−	0.806194i	$-0.701524\pi$
$613$	−8.24805e34	−1.18330	−0.591651	+	0.806194i	$0.701524\pi$
$614$	0	0
$615$	1.38807e35	1.91194
$616$	0	0
$617$	−3.21186e34	−0.424807	−0.212404	−	0.977182i	$-0.568129\pi$
$617$	−3.21186e34	−0.424807	−0.212404	+	0.977182i	$0.568129\pi$
$618$	0	0
$619$	1.06631e35	1.35441	0.677206	−	0.735793i	$-0.263190\pi$
$619$	1.06631e35	1.35441	0.677206	+	0.735793i	$0.263190\pi$
$620$	0	0
$621$	−2.26371e34	−0.276169
$622$	0	0
$623$	1.10653e35	1.29676
$624$	0	0
$625$	2.80454e35	3.15763
$626$	0	0
$627$	5.48076e33	0.0592920
$628$	0	0
$629$	−7.02493e34	−0.730313
$630$	0	0
$631$	1.09014e35	1.08922	0.544610	−	0.838689i	$-0.316678\pi$
$631$	1.09014e35	1.08922	0.544610	+	0.838689i	$0.316678\pi$
$632$	0	0
$633$	1.03382e35	0.992883
$634$	0	0
$635$	−2.56517e34	−0.236834
$636$	0	0
$637$	−1.29042e34	−0.114548
$638$	0	0
$639$	−3.77004e34	−0.321800
$640$	0	0
$641$	−1.12583e35	−0.924157	−0.462079	−	0.886839i	$-0.652896\pi$
$641$	−1.12583e35	−0.924157	−0.462079	+	0.886839i	$0.652896\pi$
$642$	0	0
$643$	−1.81383e35	−1.43206	−0.716029	−	0.698071i	$-0.754042\pi$
$643$	−1.81383e35	−1.43206	−0.716029	+	0.698071i	$0.754042\pi$
$644$	0	0
$645$	1.38315e35	1.05044
$646$	0	0
$647$	2.31376e35	1.69050	0.845249	−	0.534372i	$-0.179452\pi$
$647$	2.31376e35	1.69050	0.845249	+	0.534372i	$0.179452\pi$
$648$	0	0
$649$	−1.45999e35	−1.02634
$650$	0	0
$651$	1.07617e35	0.727981
$652$	0	0
$653$	−2.26159e34	−0.147231	−0.0736156	−	0.997287i	$-0.523454\pi$
$653$	−2.26159e34	−0.147231	−0.0736156	+	0.997287i	$0.523454\pi$
$654$	0	0
$655$	−4.37830e35	−2.74340
$656$	0	0
$657$	−8.12855e34	−0.490282
$658$	0	0
$659$	−1.11214e35	−0.645792	−0.322896	−	0.946434i	$-0.604656\pi$
$659$	−1.11214e35	−0.645792	−0.322896	+	0.946434i	$0.604656\pi$
$660$	0	0
$661$	3.00401e35	1.67951	0.839756	−	0.542963i	$-0.182698\pi$
$661$	3.00401e35	1.67951	0.839756	+	0.542963i	$0.182698\pi$
$662$	0	0
$663$	7.31956e34	0.394063
$664$	0	0
$665$	4.66474e34	0.241856
$666$	0	0
$667$	3.39274e35	1.69425
$668$	0	0
$669$	−9.13988e34	−0.439657
$670$	0	0
$671$	−1.10403e34	−0.0511621
$672$	0	0
$673$	−9.46703e34	−0.422693	−0.211347	−	0.977411i	$-0.567785\pi$
$673$	−9.46703e34	−0.422693	−0.211347	+	0.977411i	$0.567785\pi$
$674$	0	0
$675$	1.16277e35	0.500262
$676$	0	0
$677$	9.46740e34	0.392530	0.196265	−	0.980551i	$-0.437119\pi$
$677$	9.46740e34	0.392530	0.196265	+	0.980551i	$0.437119\pi$
$678$	0	0
$679$	9.81757e34	0.392312
$680$	0	0
$681$	2.07369e35	0.798741
$682$	0	0
$683$	−1.74667e35	−0.648562	−0.324281	−	0.945961i	$-0.605122\pi$
$683$	−1.74667e35	−0.648562	−0.324281	+	0.945961i	$0.605122\pi$
$684$	0	0
$685$	5.62767e35	2.01464
$686$	0	0
$687$	3.31885e35	1.14559
$688$	0	0
$689$	1.80755e35	0.601659
$690$	0	0
$691$	−4.95830e35	−1.59168	−0.795842	−	0.605504i	$-0.792972\pi$
$691$	−4.95830e35	−1.59168	−0.795842	+	0.605504i	$0.792972\pi$
$692$	0	0
$693$	7.13573e34	0.220939
$694$	0	0
$695$	−4.69862e35	−1.40333
$696$	0	0
$697$	6.39751e35	1.84332
$698$	0	0
$699$	2.92723e35	0.813753
$700$	0	0
$701$	−4.07843e34	−0.109400	−0.0547001	−	0.998503i	$-0.517420\pi$
$701$	−4.07843e34	−0.109400	−0.0547001	+	0.998503i	$0.517420\pi$
$702$	0	0
$703$	−3.75463e34	−0.0971910
$704$	0	0
$705$	−2.56061e34	−0.0639706
$706$	0	0
$707$	−7.45749e35	−1.79825
$708$	0	0
$709$	−1.10573e35	−0.257377	−0.128688	−	0.991685i	$-0.541077\pi$
$709$	−1.10573e35	−0.257377	−0.128688	+	0.991685i	$0.541077\pi$
$710$	0	0
$711$	5.74408e34	0.129077
$712$	0	0
$713$	−9.19430e35	−1.99480
$714$	0	0
$715$	4.27701e35	0.896015
$716$	0	0
$717$	−5.26525e34	−0.106520
$718$	0	0
$719$	−7.53230e35	−1.47169	−0.735845	−	0.677150i	$-0.763215\pi$
$719$	−7.53230e35	−1.47169	−0.735845	+	0.677150i	$0.763215\pi$
$720$	0	0
$721$	−7.57715e35	−1.42993
$722$	0	0
$723$	−4.16825e35	−0.759845
$724$	0	0
$725$	−1.74271e36	−3.06902
$726$	0	0
$727$	6.13362e35	1.04361	0.521804	−	0.853066i	$-0.325259\pi$
$727$	6.13362e35	1.04361	0.521804	+	0.853066i	$0.325259\pi$
$728$	0	0
$729$	2.25284e34	0.0370370
$730$	0	0
$731$	6.37480e35	1.01274
$732$	0	0
$733$	1.08593e35	0.166725	0.0833626	−	0.996519i	$-0.473434\pi$
$733$	1.08593e35	0.166725	0.0833626	+	0.996519i	$0.473434\pi$
$734$	0	0
$735$	−1.30826e35	−0.194134
$736$	0	0
$737$	−2.88125e35	−0.413273
$738$	0	0
$739$	−3.86171e35	−0.535457	−0.267729	−	0.963494i	$-0.586273\pi$
$739$	−3.86171e35	−0.535457	−0.267729	+	0.963494i	$0.586273\pi$
$740$	0	0
$741$	3.91210e34	0.0524424
$742$	0	0
$743$	2.49794e34	0.0323759	0.0161880	−	0.999869i	$-0.494847\pi$
$743$	2.49794e34	0.0323759	0.0161880	+	0.999869i	$0.494847\pi$
$744$	0	0
$745$	4.02349e35	0.504255
$746$	0	0
$747$	−1.19446e35	−0.144765
$748$	0	0
$749$	−1.45652e36	−1.70724
$750$	0	0
$751$	6.72396e34	0.0762300	0.0381150	−	0.999273i	$-0.487865\pi$
$751$	6.72396e34	0.0762300	0.0381150	+	0.999273i	$0.487865\pi$
$752$	0	0
$753$	3.55678e35	0.390050
$754$	0	0
$755$	−9.71638e35	−1.03078
$756$	0	0
$757$	−1.33814e36	−1.37342	−0.686711	−	0.726930i	$-0.740946\pi$
$757$	−1.33814e36	−1.37342	−0.686711	+	0.726930i	$0.740946\pi$
$758$	0	0
$759$	−6.09642e35	−0.605414
$760$	0	0
$761$	1.27906e36	1.22908	0.614542	−	0.788884i	$-0.289341\pi$
$761$	1.27906e36	1.22908	0.614542	+	0.788884i	$0.289341\pi$
$762$	0	0
$763$	2.81602e35	0.261866
$764$	0	0
$765$	7.42076e35	0.667852
$766$	0	0
$767$	−1.04213e36	−0.907775
$768$	0	0
$769$	1.63070e36	1.37497	0.687487	−	0.726197i	$-0.258714\pi$
$769$	1.63070e36	1.37497	0.687487	+	0.726197i	$0.258714\pi$
$770$	0	0
$771$	7.81154e35	0.637613
$772$	0	0
$773$	−1.40061e36	−1.10681	−0.553405	−	0.832912i	$-0.686672\pi$
$773$	−1.40061e36	−1.10681	−0.553405	+	0.832912i	$0.686672\pi$
$774$	0	0
$775$	4.72272e36	3.61344
$776$	0	0
$777$	−4.88838e35	−0.362162
$778$	0	0
$779$	3.41929e35	0.245312
$780$	0	0
$781$	−1.01531e36	−0.705444
$782$	0	0
$783$	−3.37645e35	−0.227215
$784$	0	0
$785$	1.98208e36	1.29196
$786$	0	0
$787$	−1.08027e36	−0.682103	−0.341052	−	0.940045i	$-0.610783\pi$
$787$	−1.08027e36	−0.682103	−0.341052	+	0.940045i	$0.610783\pi$
$788$	0	0
$789$	−1.56902e35	−0.0959768
$790$	0	0
$791$	4.05932e35	0.240573
$792$	0	0
$793$	−7.88042e34	−0.0452516
$794$	0	0
$795$	1.83254e36	1.01968
$796$	0	0
$797$	−2.06774e36	−1.11498	−0.557489	−	0.830185i	$-0.688235\pi$
$797$	−2.06774e36	−1.11498	−0.557489	+	0.830185i	$0.688235\pi$
$798$	0	0
$799$	−1.18016e35	−0.0616748
$800$	0	0
$801$	9.40823e35	0.476542
$802$	0	0
$803$	−2.18911e36	−1.07479
$804$	0	0
$805$	−5.18873e36	−2.46952
$806$	0	0
$807$	−1.06867e36	−0.493086
$808$	0	0
$809$	1.84174e36	0.823896	0.411948	−	0.911207i	$-0.364848\pi$
$809$	1.84174e36	0.823896	0.411948	+	0.911207i	$0.364848\pi$
$810$	0	0
$811$	−3.86140e36	−1.67488	−0.837440	−	0.546529i	$-0.815949\pi$
$811$	−3.86140e36	−1.67488	−0.837440	+	0.546529i	$0.815949\pi$
$812$	0	0
$813$	1.30648e36	0.549505
$814$	0	0
$815$	−5.69929e36	−2.32461
$816$	0	0
$817$	3.40715e35	0.134777
$818$	0	0
$819$	5.09340e35	0.195416
$820$	0	0
$821$	5.13456e36	1.91079	0.955396	−	0.295327i	$-0.0954285\pi$
$821$	5.13456e36	1.91079	0.955396	+	0.295327i	$0.0954285\pi$
$822$	0	0
$823$	2.52635e36	0.912002	0.456001	−	0.889979i	$-0.349281\pi$
$823$	2.52635e36	0.912002	0.456001	+	0.889979i	$0.349281\pi$
$824$	0	0
$825$	3.13147e36	1.09667
$826$	0	0
$827$	9.63067e35	0.327218	0.163609	−	0.986525i	$-0.447686\pi$
$827$	9.63067e35	0.327218	0.163609	+	0.986525i	$0.447686\pi$
$828$	0	0
$829$	4.93563e35	0.162709	0.0813544	−	0.996685i	$-0.474075\pi$
$829$	4.93563e35	0.162709	0.0813544	+	0.996685i	$0.474075\pi$
$830$	0	0
$831$	−7.30910e35	−0.233803
$832$	0	0
$833$	−6.02965e35	−0.187167
$834$	0	0
$835$	1.09599e36	0.330162
$836$	0	0
$837$	9.15015e35	0.267522
$838$	0	0
$839$	−1.85620e36	−0.526745	−0.263372	−	0.964694i	$-0.584835\pi$
$839$	−1.85620e36	−0.526745	−0.263372	+	0.964694i	$0.584835\pi$
$840$	0	0
$841$	1.43009e36	0.393926
$842$	0	0
$843$	3.48279e35	0.0931286
$844$	0	0
$845$	−4.25558e36	−1.10471
$846$	0	0
$847$	−1.67726e36	−0.422725
$848$	0	0
$849$	1.78446e36	0.436679
$850$	0	0
$851$	4.17640e36	0.992389
$852$	0	0
$853$	−1.78562e36	−0.412027	−0.206013	−	0.978549i	$-0.566049\pi$
$853$	−1.78562e36	−0.412027	−0.206013	+	0.978549i	$0.566049\pi$
$854$	0	0
$855$	3.96619e35	0.0888786
$856$	0	0
$857$	4.04214e36	0.879734	0.439867	−	0.898063i	$-0.355026\pi$
$857$	4.04214e36	0.879734	0.439867	+	0.898063i	$0.355026\pi$
$858$	0	0
$859$	7.87170e36	1.66400	0.832002	−	0.554773i	$-0.187195\pi$
$859$	7.87170e36	1.66400	0.832002	+	0.554773i	$0.187195\pi$
$860$	0	0
$861$	4.45178e36	0.914102
$862$	0	0
$863$	1.11698e35	0.0222798	0.0111399	−	0.999938i	$-0.496454\pi$
$863$	1.11698e35	0.0222798	0.0111399	+	0.999938i	$0.496454\pi$
$864$	0	0
$865$	8.42087e36	1.63176
$866$	0	0
$867$	3.53414e35	0.0665342
$868$	0	0
$869$	1.54694e36	0.282961
$870$	0	0
$871$	−2.05660e36	−0.365530
$872$	0	0
$873$	8.34738e35	0.144169
$874$	0	0
$875$	1.63992e37	2.75247
$876$	0	0
$877$	4.11140e36	0.670649	0.335324	−	0.942103i	$-0.391154\pi$
$877$	4.11140e36	0.670649	0.335324	+	0.942103i	$0.391154\pi$
$878$	0	0
$879$	−3.42182e36	−0.542496
$880$	0	0
$881$	−1.91002e36	−0.294333	−0.147167	−	0.989112i	$-0.547015\pi$
$881$	−1.91002e36	−0.294333	−0.147167	+	0.989112i	$0.547015\pi$
$882$	0	0
$883$	−6.74401e36	−1.01020	−0.505102	−	0.863060i	$-0.668545\pi$
$883$	−6.74401e36	−1.01020	−0.505102	+	0.863060i	$0.668545\pi$
$884$	0	0
$885$	−1.05653e37	−1.53848
$886$	0	0
$887$	3.00804e36	0.425832	0.212916	−	0.977071i	$-0.431704\pi$
$887$	3.00804e36	0.425832	0.212916	+	0.977071i	$0.431704\pi$
$888$	0	0
$889$	−8.22692e35	−0.113231
$890$	0	0
$891$	6.06715e35	0.0811920
$892$	0	0
$893$	−6.30765e34	−0.00820776
$894$	0	0
$895$	2.21954e37	2.80850
$896$	0	0
$897$	−4.35155e36	−0.535474
$898$	0	0
$899$	−1.37138e37	−1.64120
$900$	0	0
$901$	8.44603e36	0.983088
$902$	0	0
$903$	4.43598e36	0.502218
$904$	0	0
$905$	−8.86176e35	−0.0975916
$906$	0	0
$907$	9.46925e36	1.01444	0.507218	−	0.861818i	$-0.330674\pi$
$907$	9.46925e36	1.01444	0.507218	+	0.861818i	$0.330674\pi$
$908$	0	0
$909$	−6.34073e36	−0.660831
$910$	0	0
$911$	−2.55801e36	−0.259372	−0.129686	−	0.991555i	$-0.541397\pi$
$911$	−2.55801e36	−0.259372	−0.129686	+	0.991555i	$0.541397\pi$
$912$	0	0
$913$	−3.21682e36	−0.317353
$914$	0	0
$915$	−7.98937e35	−0.0766918
$916$	0	0
$917$	−1.40419e37	−1.31163
$918$	0	0
$919$	−3.05037e36	−0.277274	−0.138637	−	0.990343i	$-0.544272\pi$
$919$	−3.05037e36	−0.277274	−0.138637	+	0.990343i	$0.544272\pi$
$920$	0	0
$921$	−4.11166e36	−0.363723
$922$	0	0
$923$	−7.24720e36	−0.623949
$924$	0	0
$925$	−2.14524e37	−1.79765
$926$	0	0
$927$	−6.44247e36	−0.525479
$928$	0	0
$929$	1.54110e37	1.22359	0.611795	−	0.791017i	$-0.290448\pi$
$929$	1.54110e37	1.22359	0.611795	+	0.791017i	$0.290448\pi$
$930$	0	0
$931$	−3.22268e35	−0.0249084
$932$	0	0
$933$	−9.00276e36	−0.677416
$934$	0	0
$935$	1.99849e37	1.46405
$936$	0	0
$937$	1.52436e37	1.08728	0.543641	−	0.839318i	$-0.317045\pi$
$937$	1.52436e37	1.08728	0.543641	+	0.839318i	$0.317045\pi$
$938$	0	0
$939$	1.03466e37	0.718582
$940$	0	0
$941$	7.49284e36	0.506729	0.253365	−	0.967371i	$-0.418463\pi$
$941$	7.49284e36	0.506729	0.253365	+	0.967371i	$0.418463\pi$
$942$	0	0
$943$	−3.80339e37	−2.50481
$944$	0	0
$945$	5.16382e36	0.331187
$946$	0	0
$947$	1.46968e37	0.918012	0.459006	−	0.888433i	$-0.348206\pi$
$947$	1.46968e37	0.918012	0.459006	+	0.888433i	$0.348206\pi$
$948$	0	0
$949$	−1.56256e37	−0.950625
$950$	0	0
$951$	−1.07219e37	−0.635354
$952$	0	0
$953$	7.55270e35	0.0435955	0.0217977	−	0.999762i	$-0.493061\pi$
$953$	7.55270e35	0.0435955	0.0217977	+	0.999762i	$0.493061\pi$
$954$	0	0
$955$	−6.30389e37	−3.54460
$956$	0	0
$957$	−9.09315e36	−0.498098
$958$	0	0
$959$	1.80489e37	0.963202
$960$	0	0
$961$	1.79316e37	0.932343
$962$	0	0
$963$	−1.23841e37	−0.627386
$964$	0	0
$965$	−9.71500e36	−0.479569
$966$	0	0
$967$	−7.02769e35	−0.0338051	−0.0169025	−	0.999857i	$-0.505381\pi$
$967$	−7.02769e35	−0.0338051	−0.0169025	+	0.999857i	$0.505381\pi$
$968$	0	0
$969$	1.82798e36	0.0856889
$970$	0	0
$971$	−1.57684e37	−0.720354	−0.360177	−	0.932884i	$-0.617284\pi$
$971$	−1.57684e37	−0.720354	−0.360177	+	0.932884i	$0.617284\pi$
$972$	0	0
$973$	−1.50693e37	−0.670937
$974$	0	0
$975$	2.23521e37	0.969975
$976$	0	0
$977$	7.04013e35	0.0297782	0.0148891	−	0.999889i	$-0.495260\pi$
$977$	7.04013e35	0.0297782	0.0148891	+	0.999889i	$0.495260\pi$
$978$	0	0
$979$	2.53374e37	1.04467
$980$	0	0
$981$	2.39432e36	0.0962318
$982$	0	0
$983$	3.42880e35	0.0134346	0.00671728	−	0.999977i	$-0.497862\pi$
$983$	3.42880e35	0.0134346	0.00671728	+	0.999977i	$0.497862\pi$
$984$	0	0
$985$	6.00061e37	2.29215
$986$	0	0
$987$	−8.21232e35	−0.0305845
$988$	0	0
$989$	−3.78988e37	−1.37617
$990$	0	0
$991$	−3.39269e37	−1.20122	−0.600611	−	0.799541i	$-0.705076\pi$
$991$	−3.39269e37	−1.20122	−0.600611	+	0.799541i	$0.705076\pi$
$992$	0	0
$993$	4.16179e36	0.143686
$994$	0	0
$995$	2.00044e37	0.673500
$996$	0	0
$997$	−1.87351e37	−0.615129	−0.307564	−	0.951527i	$-0.599514\pi$
$997$	−1.87351e37	−0.615129	−0.307564	+	0.951527i	$0.599514\pi$
$998$	0	0
$999$	−4.15635e36	−0.133089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.26.a.k.1.4		4
4.3	odd	2		24.26.a.d.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.26.a.d.1.4	✓	4	4.3	odd	2
48.26.a.k.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+531441. q^{3} +1.03572e9 q^{5} +3.32173e10 q^{7} +2.82430e11 q^{9} +O(q^{10})\)	\(q+531441. q^{3} +1.03572e9 q^{5} +3.32173e10 q^{7} +2.82430e11 q^{9} +7.60614e12 q^{11} +5.42918e13 q^{13} +5.50424e14 q^{15} +2.53686e15 q^{17} +1.35588e15 q^{19} +1.76530e16 q^{21} -1.50819e17 q^{23} +7.74693e17 q^{25} +1.50095e17 q^{27} -2.24955e18 q^{29} +6.09626e18 q^{31} +4.04221e18 q^{33} +3.44038e19 q^{35} -2.76915e19 q^{37} +2.88529e19 q^{39} +2.52183e20 q^{41} +2.51287e20 q^{43} +2.92518e20 q^{45} -4.65208e19 q^{47} -2.37682e20 q^{49} +1.34819e21 q^{51} +3.32933e21 q^{53} +7.87783e21 q^{55} +7.20570e20 q^{57} -1.91949e22 q^{59} -1.45149e21 q^{61} +9.38154e21 q^{63} +5.62311e22 q^{65} -3.78805e22 q^{67} -8.01512e22 q^{69} -1.33486e23 q^{71} -2.87808e23 q^{73} +4.11703e23 q^{75} +2.52655e23 q^{77} +2.03381e23 q^{79} +7.97664e22 q^{81} -4.22924e23 q^{83} +2.62747e24 q^{85} -1.19550e24 q^{87} +3.33118e24 q^{89} +1.80342e24 q^{91} +3.23980e24 q^{93} +1.40431e24 q^{95} +2.95556e24 q^{97} +2.14820e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9}+O(q^{10}) \) 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9	\( 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9} - 4052955557392 q^{11} + 82663381481080 q^{13} + 141595061042808 q^{15} + 10\!\cdots\!28 q^{17}+ \cdots - 11\!\cdots\!52 q^{99}+O(q^{100}) \) 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9 - 4052955557392 * q^11 + 82663381481080 * q^13 + 141595061042808 * q^15 + 1091839531710728 * q^17 + 5291818285650448 * q^19 - 257528111654016 * q^21 - 22957555441633504 * q^23 + 329573428119320444 * q^25 + 600378541187996484 * q^27 - 2255995699601091048 * q^29 + 6422134828063294816 * q^31 - 2153906754375961872 * q^33 + 26450723987305847040 * q^35 - 84572383178406307176 * q^37 + 43930710117686636280 * q^39 - 191923587571359535128 * q^41 + 393512700764425221104 * q^43 + 75249420835650926328 * q^45 + 426139906889905851648 * q^47 + 1697461944491957688548 * q^49 + 580248292571880999048 * q^51 + 4165797514523716100344 * q^53 - 1377411551242297580512 * q^55 + 2812289201544359735568 * q^57 - 22289642820361283440912 * q^59 + 31420833466214461354360 * q^61 - 136860997185521917056 * q^63 + 49904798993160775849744 * q^65 + 105958367596085546226448 * q^67 - 12200586221457150999264 * q^69 + 260836089844500944794336 * q^71 + 96585608913237168931624 * q^73 + 175148832213159776079804 * q^75 + 609668306798881251586560 * q^77 + 228314222637752365466528 * q^79 + 319065772307490039453444 * q^81 - 508041934949405113277936 * q^83 + 3527355809877820452506096 * q^85 - 1198928610591703427640168 * q^87 + 9334569288453070817495784 * q^89 - 6414440882259952713889536 * q^91 + 3412985755160785460309856 * q^93 - 9916967691165637498776608 * q^95 + 17898782631688573069195784 * q^97 - 1144674359452315553217552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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