LMFDB - Embedded newform 48.22.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,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, 22, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
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:	$134.149125258$
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} - 53560x - 70812 $ x^3 - 53560*x - 70812
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.32215$ 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)

$f(q)$	$=$	$q+59049.0 q^{3} +995860. q^{5} +1.18014e9 q^{7} +3.48678e9 q^{9} +O(q^{10})$	$q+59049.0 q^{3} +995860. q^{5} +1.18014e9 q^{7} +3.48678e9 q^{9} +3.52053e10 q^{11} -8.44606e11 q^{13} +5.88045e10 q^{15} +1.17645e13 q^{17} +1.89418e13 q^{19} +6.96862e13 q^{21} +2.72308e14 q^{23} -4.75845e14 q^{25} +2.05891e14 q^{27} +1.38288e15 q^{29} +1.93945e15 q^{31} +2.07884e15 q^{33} +1.17526e15 q^{35} -5.29854e16 q^{37} -4.98732e16 q^{39} +1.37432e17 q^{41} +1.40042e16 q^{43} +3.47235e15 q^{45} +1.53143e17 q^{47} +8.34188e17 q^{49} +6.94683e17 q^{51} -1.04072e18 q^{53} +3.50595e16 q^{55} +1.11850e18 q^{57} -4.07509e18 q^{59} +3.59286e18 q^{61} +4.11490e18 q^{63} -8.41109e17 q^{65} +6.22840e16 q^{67} +1.60795e19 q^{69} -3.80609e19 q^{71} +5.06568e19 q^{73} -2.80982e19 q^{75} +4.15472e19 q^{77} +2.16125e19 q^{79} +1.21577e19 q^{81} -1.97785e20 q^{83} +1.17158e19 q^{85} +8.16575e19 q^{87} -9.11760e19 q^{89} -9.96755e20 q^{91} +1.14523e20 q^{93} +1.88634e19 q^{95} +9.35999e20 q^{97} +1.22753e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 177147 q^{3} - 4833126 q^{5} - 271431024 q^{7} + 10460353203 q^{9}+O(q^{10}) $ 3 * q + 177147 * q^3 - 4833126 * q^5 - 271431024 * q^7 + 10460353203 * q^9	$ 3 q + 177147 q^{3} - 4833126 q^{5} - 271431024 q^{7} + 10460353203 q^{9} - 61194658188 q^{11} - 594486202422 q^{13} - 285391257174 q^{15} + 1424819519334 q^{17} + 35094825681804 q^{19} - 16027730536176 q^{21} + 94911043073592 q^{23} - 613780025295699 q^{25} + 617673396283947 q^{27} - 32\!\cdots\!34 q^{29}+ \cdots - 21\!\cdots\!88 q^{99}+O(q^{100}) $ 3 * q + 177147 * q^3 - 4833126 * q^5 - 271431024 * q^7 + 10460353203 * q^9 - 61194658188 * q^11 - 594486202422 * q^13 - 285391257174 * q^15 + 1424819519334 * q^17 + 35094825681804 * q^19 - 16027730536176 * q^21 + 94911043073592 * q^23 - 613780025295699 * q^25 + 617673396283947 * q^27 - 3292734981070734 * q^29 + 1928658139449144 * q^31 - 3613483371343212 * q^33 + 18129892754606304 * q^35 + 22185275061682722 * q^37 - 35103815766816678 * q^39 + 167394191883971118 * q^41 - 69547615491362508 * q^43 - 16852068344867526 * q^45 + 49110618973894752 * q^47 + 973320764343712395 * q^49 + 84134167797153366 * q^51 + 488391341947078074 * q^53 + 3110835549535695000 * q^55 + 2072314361684844396 * q^57 + 1846144806145488852 * q^59 + 8183707455377473002 * q^61 - 946421460430656624 * q^63 + 28951429805381293356 * q^65 - 37438935583939030164 * q^67 + 5604402182452534008 * q^69 - 65865243803437791096 * q^71 + 88415465494171854846 * q^73 - 36243096713685730251 * q^75 + 156034186748470726080 * q^77 - 221381156829825400920 * q^79 + 36472996377170786403 * q^81 - 305200914127129955700 * q^83 + 184695330643817576052 * q^85 - 194432707897245771966 * q^87 + 327556458098783035470 * q^89 - 692083854852141366816 * q^91 + 113885334476332504056 * q^93 - 1486593794603974227096 * q^95 + 1980043166774365090758 * q^97 - 213372579594445325388 * 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
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	0	0
$5$	995860.	0.0456051	0.0228025	−	0.999740i	$-0.492741\pi$
$5$	995860.	0.0456051	0.0228025	+	0.999740i	$0.492741\pi$
$6$	0	0
$7$	1.18014e9	1.57908	0.789541	−	0.613698i	$-0.210319\pi$
$7$	1.18014e9	1.57908	0.789541	+	0.613698i	$0.210319\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	3.52053e10	0.409246	0.204623	−	0.978841i	$-0.434403\pi$
$11$	3.52053e10	0.409246	0.204623	+	0.978841i	$0.434403\pi$
$12$	0	0
$13$	−8.44606e11	−1.69922	−0.849609	−	0.527413i	$-0.823162\pi$
$13$	−8.44606e11	−1.69922	−0.849609	+	0.527413i	$0.823162\pi$
$14$	0	0
$15$	5.88045e10	0.0263301
$16$	0	0
$17$	1.17645e13	1.41534	0.707670	−	0.706544i	$-0.249747\pi$
$17$	1.17645e13	1.41534	0.707670	+	0.706544i	$0.249747\pi$
$18$	0	0
$19$	1.89418e13	0.708776	0.354388	−	0.935098i	$-0.384689\pi$
$19$	1.89418e13	0.708776	0.354388	+	0.935098i	$0.384689\pi$
$20$	0	0
$21$	6.96862e13	0.911683
$22$	0	0
$23$	2.72308e14	1.37062	0.685312	−	0.728250i	$-0.259666\pi$
$23$	2.72308e14	1.37062	0.685312	+	0.728250i	$0.259666\pi$
$24$	0	0
$25$	−4.75845e14	−0.997920
$26$	0	0
$27$	2.05891e14	0.192450
$28$	0	0
$29$	1.38288e15	0.610386	0.305193	−	0.952290i	$-0.401279\pi$
$29$	1.38288e15	0.610386	0.305193	+	0.952290i	$0.401279\pi$
$30$	0	0
$31$	1.93945e15	0.424992	0.212496	−	0.977162i	$-0.431841\pi$
$31$	1.93945e15	0.424992	0.212496	+	0.977162i	$0.431841\pi$
$32$	0	0
$33$	2.07884e15	0.236278
$34$	0	0
$35$	1.17526e15	0.0720141
$36$	0	0
$37$	−5.29854e16	−1.81150	−0.905750	−	0.423812i	$-0.860692\pi$
$37$	−5.29854e16	−1.81150	−0.905750	+	0.423812i	$0.860692\pi$
$38$	0	0
$39$	−4.98732e16	−0.981044
$40$	0	0
$41$	1.37432e17	1.59903	0.799514	−	0.600647i	$-0.205090\pi$
$41$	1.37432e17	1.59903	0.799514	+	0.600647i	$0.205090\pi$
$42$	0	0
$43$	1.40042e16	0.0988189	0.0494095	−	0.998779i	$-0.484266\pi$
$43$	1.40042e16	0.0988189	0.0494095	+	0.998779i	$0.484266\pi$
$44$	0	0
$45$	3.47235e15	0.0152017
$46$	0	0
$47$	1.53143e17	0.424688	0.212344	−	0.977195i	$-0.431890\pi$
$47$	1.53143e17	0.424688	0.212344	+	0.977195i	$0.431890\pi$
$48$	0	0
$49$	8.34188e17	1.49350
$50$	0	0
$51$	6.94683e17	0.817146
$52$	0	0
$53$	−1.04072e18	−0.817403	−0.408702	−	0.912668i	$-0.634018\pi$
$53$	−1.04072e18	−0.817403	−0.408702	+	0.912668i	$0.634018\pi$
$54$	0	0
$55$	3.50595e16	0.0186637
$56$	0	0
$57$	1.11850e18	0.409212
$58$	0	0
$59$	−4.07509e18	−1.03799	−0.518993	−	0.854779i	$-0.673693\pi$
$59$	−4.07509e18	−1.03799	−0.518993	+	0.854779i	$0.673693\pi$
$60$	0	0
$61$	3.59286e18	0.644877	0.322439	−	0.946590i	$-0.395497\pi$
$61$	3.59286e18	0.644877	0.322439	+	0.946590i	$0.395497\pi$
$62$	0	0
$63$	4.11490e18	0.526361
$64$	0	0
$65$	−8.41109e17	−0.0774930
$66$	0	0
$67$	6.22840e16	0.00417437	0.00208719	−	0.999998i	$-0.499336\pi$
$67$	6.22840e16	0.00417437	0.00208719	+	0.999998i	$0.499336\pi$
$68$	0	0
$69$	1.60795e19	0.791330
$70$	0	0
$71$	−3.80609e19	−1.38761	−0.693803	−	0.720165i	$-0.744066\pi$
$71$	−3.80609e19	−1.38761	−0.693803	+	0.720165i	$0.744066\pi$
$72$	0	0
$73$	5.06568e19	1.37958	0.689792	−	0.724008i	$-0.257702\pi$
$73$	5.06568e19	1.37958	0.689792	+	0.724008i	$0.257702\pi$
$74$	0	0
$75$	−2.80982e19	−0.576149
$76$	0	0
$77$	4.15472e19	0.646233
$78$	0	0
$79$	2.16125e19	0.256815	0.128407	−	0.991722i	$-0.459014\pi$
$79$	2.16125e19	0.256815	0.128407	+	0.991722i	$0.459014\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	−1.97785e20	−1.39918	−0.699591	−	0.714544i	$-0.746634\pi$
$83$	−1.97785e20	−1.39918	−0.699591	+	0.714544i	$0.746634\pi$
$84$	0	0
$85$	1.17158e19	0.0645466
$86$	0	0
$87$	8.16575e19	0.352407
$88$	0	0
$89$	−9.11760e19	−0.309945	−0.154973	−	0.987919i	$-0.549529\pi$
$89$	−9.11760e19	−0.309945	−0.154973	+	0.987919i	$0.549529\pi$
$90$	0	0
$91$	−9.96755e20	−2.68321
$92$	0	0
$93$	1.14523e20	0.245369
$94$	0	0
$95$	1.88634e19	0.0323238
$96$	0	0
$97$	9.35999e20	1.28876	0.644380	−	0.764705i	$-0.277115\pi$
$97$	9.35999e20	1.28876	0.644380	+	0.764705i	$0.277115\pi$
$98$	0	0
$99$	1.22753e20	0.136415
$100$	0	0
$101$	1.46129e21	1.31632	0.658159	−	0.752879i	$-0.271335\pi$
$101$	1.46129e21	1.31632	0.658159	+	0.752879i	$0.271335\pi$
$102$	0	0
$103$	1.98468e21	1.45512	0.727560	−	0.686044i	$-0.240654\pi$
$103$	1.98468e21	1.45512	0.727560	+	0.686044i	$0.240654\pi$
$104$	0	0
$105$	6.93976e19	0.0415774
$106$	0	0
$107$	−2.24032e20	−0.110098	−0.0550492	−	0.998484i	$-0.517532\pi$
$107$	−2.24032e20	−0.110098	−0.0550492	+	0.998484i	$0.517532\pi$
$108$	0	0
$109$	−1.34317e21	−0.543441	−0.271720	−	0.962376i	$-0.587593\pi$
$109$	−1.34317e21	−0.543441	−0.271720	+	0.962376i	$0.587593\pi$
$110$	0	0
$111$	−3.12874e21	−1.04587
$112$	0	0
$113$	1.93517e21	0.536283	0.268142	−	0.963379i	$-0.413590\pi$
$113$	1.93517e21	0.536283	0.268142	+	0.963379i	$0.413590\pi$
$114$	0	0
$115$	2.71181e20	0.0625074
$116$	0	0
$117$	−2.94496e21	−0.566406
$118$	0	0
$119$	1.38838e22	2.23494
$120$	0	0
$121$	−6.16084e21	−0.832518
$122$	0	0
$123$	8.11520e21	0.923200
$124$	0	0
$125$	−9.48738e20	−0.0911153
$126$	0	0
$127$	2.29611e21	0.186661	0.0933305	−	0.995635i	$-0.470249\pi$
$127$	2.29611e21	0.186661	0.0933305	+	0.995635i	$0.470249\pi$
$128$	0	0
$129$	8.26935e20	0.0570531
$130$	0	0
$131$	1.02004e22	0.598783	0.299391	−	0.954130i	$-0.403216\pi$
$131$	1.02004e22	0.598783	0.299391	+	0.954130i	$0.403216\pi$
$132$	0	0
$133$	2.23540e22	1.11922
$134$	0	0
$135$	2.05039e20	0.00877670
$136$	0	0
$137$	−2.02671e22	−0.743407	−0.371703	−	0.928352i	$-0.621226\pi$
$137$	−2.02671e22	−0.743407	−0.371703	+	0.928352i	$0.621226\pi$
$138$	0	0
$139$	4.87536e22	1.53586	0.767929	−	0.640535i	$-0.221287\pi$
$139$	4.87536e22	1.53586	0.767929	+	0.640535i	$0.221287\pi$
$140$	0	0
$141$	9.04296e21	0.245194
$142$	0	0
$143$	−2.97346e22	−0.695398
$144$	0	0
$145$	1.37715e21	0.0278367
$146$	0	0
$147$	4.92580e22	0.862273
$148$	0	0
$149$	−4.22945e22	−0.642433	−0.321216	−	0.947006i	$-0.604092\pi$
$149$	−4.22945e22	−0.642433	−0.321216	+	0.947006i	$0.604092\pi$
$150$	0	0
$151$	−3.24829e21	−0.0428940	−0.0214470	−	0.999770i	$-0.506827\pi$
$151$	−3.24829e21	−0.0428940	−0.0214470	+	0.999770i	$0.506827\pi$
$152$	0	0
$153$	4.10203e22	0.471780
$154$	0	0
$155$	1.93142e21	0.0193818
$156$	0	0
$157$	7.93683e22	0.696146	0.348073	−	0.937467i	$-0.386836\pi$
$157$	7.93683e22	0.696146	0.348073	+	0.937467i	$0.386836\pi$
$158$	0	0
$159$	−6.14534e22	−0.471928
$160$	0	0
$161$	3.21362e23	2.16433
$162$	0	0
$163$	1.28319e23	0.759139	0.379569	−	0.925163i	$-0.376072\pi$
$163$	1.28319e23	0.759139	0.379569	+	0.925163i	$0.376072\pi$
$164$	0	0
$165$	2.07023e21	0.0107755
$166$	0	0
$167$	3.95828e23	1.81545	0.907724	−	0.419569i	$-0.137819\pi$
$167$	3.95828e23	1.81545	0.907724	+	0.419569i	$0.137819\pi$
$168$	0	0
$169$	4.66296e23	1.88734
$170$	0	0
$171$	6.60461e22	0.236259
$172$	0	0
$173$	4.16073e23	1.31730	0.658651	−	0.752449i	$-0.271127\pi$
$173$	4.16073e23	1.31730	0.658651	+	0.752449i	$0.271127\pi$
$174$	0	0
$175$	−5.61565e23	−1.57580
$176$	0	0
$177$	−2.40630e23	−0.599281
$178$	0	0
$179$	−1.27122e23	−0.281360	−0.140680	−	0.990055i	$-0.544929\pi$
$179$	−1.27122e23	−0.281360	−0.140680	+	0.990055i	$0.544929\pi$
$180$	0	0
$181$	8.28267e23	1.63134	0.815671	−	0.578516i	$-0.196368\pi$
$181$	8.28267e23	1.63134	0.815671	+	0.578516i	$0.196368\pi$
$182$	0	0
$183$	2.12155e23	0.372320
$184$	0	0
$185$	−5.27661e22	−0.0826136
$186$	0	0
$187$	4.14173e23	0.579222
$188$	0	0
$189$	2.42981e23	0.303894
$190$	0	0
$191$	8.25338e23	0.924233	0.462117	−	0.886819i	$-0.347090\pi$
$191$	8.25338e23	0.924233	0.462117	+	0.886819i	$0.347090\pi$
$192$	0	0
$193$	−6.79233e23	−0.681815	−0.340908	−	0.940097i	$-0.610734\pi$
$193$	−6.79233e23	−0.681815	−0.340908	+	0.940097i	$0.610734\pi$
$194$	0	0
$195$	−4.96667e22	−0.0447406
$196$	0	0
$197$	−3.06756e23	−0.248254	−0.124127	−	0.992266i	$-0.539613\pi$
$197$	−3.06756e23	−0.248254	−0.124127	+	0.992266i	$0.539613\pi$
$198$	0	0
$199$	8.09278e23	0.589034	0.294517	−	0.955646i	$-0.404841\pi$
$199$	8.09278e23	0.589034	0.294517	+	0.955646i	$0.404841\pi$
$200$	0	0
$201$	3.67781e21	0.00241008
$202$	0	0
$203$	1.63199e24	0.963850
$204$	0	0
$205$	1.36863e23	0.0729238
$206$	0	0
$207$	9.49480e23	0.456875
$208$	0	0
$209$	6.66852e23	0.290064
$210$	0	0
$211$	−1.58081e24	−0.622176	−0.311088	−	0.950381i	$-0.600693\pi$
$211$	−1.58081e24	−0.622176	−0.311088	+	0.950381i	$0.600693\pi$
$212$	0	0
$213$	−2.24746e24	−0.801135
$214$	0	0
$215$	1.39462e22	0.00450664
$216$	0	0
$217$	2.28883e24	0.671097
$218$	0	0
$219$	2.99123e24	0.796503
$220$	0	0
$221$	−9.93639e24	−2.40497
$222$	0	0
$223$	−4.04934e24	−0.891626	−0.445813	−	0.895126i	$-0.647085\pi$
$223$	−4.04934e24	−0.891626	−0.445813	+	0.895126i	$0.647085\pi$
$224$	0	0
$225$	−1.65917e24	−0.332640
$226$	0	0
$227$	4.16746e24	0.761377	0.380688	−	0.924703i	$-0.375687\pi$
$227$	4.16746e24	0.761377	0.380688	+	0.924703i	$0.375687\pi$
$228$	0	0
$229$	7.85951e24	1.30955	0.654776	−	0.755823i	$-0.272763\pi$
$229$	7.85951e24	1.30955	0.654776	+	0.755823i	$0.272763\pi$
$230$	0	0
$231$	2.45332e24	0.373103
$232$	0	0
$233$	−4.71187e24	−0.654570	−0.327285	−	0.944926i	$-0.606134\pi$
$233$	−4.71187e24	−0.654570	−0.327285	+	0.944926i	$0.606134\pi$
$234$	0	0
$235$	1.52509e23	0.0193679
$236$	0	0
$237$	1.27619e24	0.148272
$238$	0	0
$239$	−1.76020e25	−1.87234	−0.936169	−	0.351550i	$-0.885655\pi$
$239$	−1.76020e25	−1.87234	−0.936169	+	0.351550i	$0.885655\pi$
$240$	0	0
$241$	−1.92639e23	−0.0187744	−0.00938718	−	0.999956i	$-0.502988\pi$
$241$	−1.92639e23	−0.0187744	−0.00938718	+	0.999956i	$0.502988\pi$
$242$	0	0
$243$	7.17898e23	0.0641500
$244$	0	0
$245$	8.30734e23	0.0681112
$246$	0	0
$247$	−1.59984e25	−1.20437
$248$	0	0
$249$	−1.16790e25	−0.807818
$250$	0	0
$251$	2.97473e25	1.89179	0.945895	−	0.324472i	$-0.105187\pi$
$251$	2.97473e25	1.89179	0.945895	+	0.324472i	$0.105187\pi$
$252$	0	0
$253$	9.58668e24	0.560922
$254$	0	0
$255$	6.91807e23	0.0372660
$256$	0	0
$257$	1.69961e25	0.843436	0.421718	−	0.906727i	$-0.361427\pi$
$257$	1.69961e25	0.843436	0.421718	+	0.906727i	$0.361427\pi$
$258$	0	0
$259$	−6.25303e25	−2.86051
$260$	0	0
$261$	4.82180e24	0.203462
$262$	0	0
$263$	−4.05987e25	−1.58116	−0.790582	−	0.612356i	$-0.790222\pi$
$263$	−4.05987e25	−1.58116	−0.790582	+	0.612356i	$0.790222\pi$
$264$	0	0
$265$	−1.03641e24	−0.0372777
$266$	0	0
$267$	−5.38385e24	−0.178947
$268$	0	0
$269$	−4.80880e25	−1.47787	−0.738937	−	0.673775i	$-0.764672\pi$
$269$	−4.80880e25	−1.47787	−0.738937	+	0.673775i	$0.764672\pi$
$270$	0	0
$271$	−4.82122e25	−1.37082	−0.685408	−	0.728159i	$-0.740376\pi$
$271$	−4.82122e25	−1.37082	−0.685408	+	0.728159i	$0.740376\pi$
$272$	0	0
$273$	−5.88574e25	−1.54915
$274$	0	0
$275$	−1.67523e25	−0.408395
$276$	0	0
$277$	2.25627e25	0.509747	0.254873	−	0.966974i	$-0.417966\pi$
$277$	2.25627e25	0.509747	0.254873	+	0.966974i	$0.417966\pi$
$278$	0	0
$279$	6.76244e24	0.141664
$280$	0	0
$281$	2.49516e24	0.0484932	0.0242466	−	0.999706i	$-0.492281\pi$
$281$	2.49516e24	0.0484932	0.0242466	+	0.999706i	$0.492281\pi$
$282$	0	0
$283$	1.36006e25	0.245359	0.122679	−	0.992446i	$-0.460851\pi$
$283$	1.36006e25	0.245359	0.122679	+	0.992446i	$0.460851\pi$
$284$	0	0
$285$	1.11387e24	0.0186621
$286$	0	0
$287$	1.62189e26	2.52500
$288$	0	0
$289$	6.93120e25	1.00318
$290$	0	0
$291$	5.52698e25	0.744066
$292$	0	0
$293$	−1.56999e26	−1.96693	−0.983463	−	0.181110i	$-0.942031\pi$
$293$	−1.56999e26	−1.96693	−0.983463	+	0.181110i	$0.942031\pi$
$294$	0	0
$295$	−4.05822e24	−0.0473374
$296$	0	0
$297$	7.24845e24	0.0787594
$298$	0	0
$299$	−2.29993e26	−2.32899
$300$	0	0
$301$	1.65270e25	0.156043
$302$	0	0
$303$	8.62875e25	0.759977
$304$	0	0
$305$	3.57798e24	0.0294097
$306$	0	0
$307$	1.74656e26	1.34038	0.670192	−	0.742187i	$-0.266212\pi$
$307$	1.74656e26	1.34038	0.670192	+	0.742187i	$0.266212\pi$
$308$	0	0
$309$	1.17193e26	0.840114
$310$	0	0
$311$	1.24531e26	0.834246	0.417123	−	0.908850i	$-0.363039\pi$
$311$	1.24531e26	0.834246	0.417123	+	0.908850i	$0.363039\pi$
$312$	0	0
$313$	−2.42614e26	−1.51950	−0.759749	−	0.650216i	$-0.774678\pi$
$313$	−2.42614e26	−1.51950	−0.759749	+	0.650216i	$0.774678\pi$
$314$	0	0
$315$	4.09786e24	0.0240047
$316$	0	0
$317$	−1.12301e25	−0.0615548	−0.0307774	−	0.999526i	$-0.509798\pi$
$317$	−1.12301e25	−0.0615548	−0.0307774	+	0.999526i	$0.509798\pi$
$318$	0	0
$319$	4.86846e25	0.249798
$320$	0	0
$321$	−1.32289e25	−0.0635654
$322$	0	0
$323$	2.22842e26	1.00316
$324$	0	0
$325$	4.01902e26	1.69568
$326$	0	0
$327$	−7.93127e25	−0.313756
$328$	0	0
$329$	1.80731e26	0.670618
$330$	0	0
$331$	−1.06388e26	−0.370422	−0.185211	−	0.982699i	$-0.559297\pi$
$331$	−1.06388e26	−0.370422	−0.185211	+	0.982699i	$0.559297\pi$
$332$	0	0
$333$	−1.84749e26	−0.603833
$334$	0	0
$335$	6.20261e22	0.000190373 0
$336$	0	0
$337$	4.17700e26	1.20434	0.602172	−	0.798366i	$-0.294302\pi$
$337$	4.17700e26	1.20434	0.602172	+	0.798366i	$0.294302\pi$
$338$	0	0
$339$	1.14270e26	0.309623
$340$	0	0
$341$	6.82789e25	0.173926
$342$	0	0
$343$	3.25297e26	0.779277
$344$	0	0
$345$	1.60129e25	0.0360887
$346$	0	0
$347$	2.09387e26	0.444109	0.222054	−	0.975034i	$-0.428724\pi$
$347$	2.09387e26	0.444109	0.222054	+	0.975034i	$0.428724\pi$
$348$	0	0
$349$	−4.26717e26	−0.852066	−0.426033	−	0.904708i	$-0.640089\pi$
$349$	−4.26717e26	−0.852066	−0.426033	+	0.904708i	$0.640089\pi$
$350$	0	0
$351$	−1.73897e26	−0.327015
$352$	0	0
$353$	1.08602e27	1.92399	0.961995	−	0.273066i	$-0.0880377\pi$
$353$	1.08602e27	1.92399	0.961995	+	0.273066i	$0.0880377\pi$
$354$	0	0
$355$	−3.79033e25	−0.0632818
$356$	0	0
$357$	8.19824e26	1.29034
$358$	0	0
$359$	−9.95627e26	−1.47776	−0.738882	−	0.673835i	$-0.764646\pi$
$359$	−9.95627e26	−1.47776	−0.738882	+	0.673835i	$0.764646\pi$
$360$	0	0
$361$	−3.55417e26	−0.497636
$362$	0	0
$363$	−3.63791e26	−0.480654
$364$	0	0
$365$	5.04471e25	0.0629160
$366$	0	0
$367$	3.79742e26	0.447193	0.223596	−	0.974682i	$-0.428220\pi$
$367$	3.79742e26	0.447193	0.223596	+	0.974682i	$0.428220\pi$
$368$	0	0
$369$	4.79195e26	0.533010
$370$	0	0
$371$	−1.22819e27	−1.29075
$372$	0	0
$373$	2.69702e26	0.267880	0.133940	−	0.990989i	$-0.457237\pi$
$373$	2.69702e26	0.267880	0.133940	+	0.990989i	$0.457237\pi$
$374$	0	0
$375$	−5.60220e25	−0.0526054
$376$	0	0
$377$	−1.16799e27	−1.03718
$378$	0	0
$379$	1.52196e27	1.27848	0.639238	−	0.769009i	$-0.279250\pi$
$379$	1.52196e27	1.27848	0.639238	+	0.769009i	$0.279250\pi$
$380$	0	0
$381$	1.35583e26	0.107769
$382$	0	0
$383$	2.90339e26	0.218433	0.109217	−	0.994018i	$-0.465166\pi$
$383$	2.90339e26	0.218433	0.109217	+	0.994018i	$0.465166\pi$
$384$	0	0
$385$	4.13752e25	0.0294715
$386$	0	0
$387$	4.88297e25	0.0329396
$388$	0	0
$389$	3.69707e26	0.236258	0.118129	−	0.992998i	$-0.462310\pi$
$389$	3.69707e26	0.236258	0.118129	+	0.992998i	$0.462310\pi$
$390$	0	0
$391$	3.20357e27	1.93990
$392$	0	0
$393$	6.02324e26	0.345707
$394$	0	0
$395$	2.15230e25	0.0117120
$396$	0	0
$397$	−1.41599e27	−0.730736	−0.365368	−	0.930863i	$-0.619057\pi$
$397$	−1.41599e27	−0.730736	−0.365368	+	0.930863i	$0.619057\pi$
$398$	0	0
$399$	1.31998e27	0.646180
$400$	0	0
$401$	−1.31866e27	−0.612515	−0.306257	−	0.951949i	$-0.599077\pi$
$401$	−1.31866e27	−0.612515	−0.306257	+	0.951949i	$0.599077\pi$
$402$	0	0
$403$	−1.63807e27	−0.722154
$404$	0	0
$405$	1.21073e25	0.00506723
$406$	0	0
$407$	−1.86537e27	−0.741349
$408$	0	0
$409$	−3.20193e27	−1.20870	−0.604348	−	0.796721i	$-0.706566\pi$
$409$	−3.20193e27	−1.20870	−0.604348	+	0.796721i	$0.706566\pi$
$410$	0	0
$411$	−1.19675e27	−0.429206
$412$	0	0
$413$	−4.80919e27	−1.63906
$414$	0	0
$415$	−1.96966e26	−0.0638097
$416$	0	0
$417$	2.87885e27	0.886728
$418$	0	0
$419$	−3.51711e27	−1.03024	−0.515120	−	0.857118i	$-0.672253\pi$
$419$	−3.51711e27	−1.03024	−0.515120	+	0.857118i	$0.672253\pi$
$420$	0	0
$421$	−5.09209e27	−1.41884	−0.709420	−	0.704786i	$-0.751043\pi$
$421$	−5.09209e27	−1.41884	−0.709420	+	0.704786i	$0.751043\pi$
$422$	0	0
$423$	5.33978e26	0.141563
$424$	0	0
$425$	−5.59809e27	−1.41240
$426$	0	0
$427$	4.24008e27	1.01831
$428$	0	0
$429$	−1.75580e27	−0.401488
$430$	0	0
$431$	−7.67930e27	−1.67228	−0.836142	−	0.548513i	$-0.815194\pi$
$431$	−7.67930e27	−1.67228	−0.836142	+	0.548513i	$0.815194\pi$
$432$	0	0
$433$	4.55461e27	0.944774	0.472387	−	0.881391i	$-0.343392\pi$
$433$	4.55461e27	0.944774	0.472387	+	0.881391i	$0.343392\pi$
$434$	0	0
$435$	8.13194e25	0.0160715
$436$	0	0
$437$	5.15801e27	0.971466
$438$	0	0
$439$	−2.37092e27	−0.425637	−0.212819	−	0.977092i	$-0.568264\pi$
$439$	−2.37092e27	−0.425637	−0.212819	+	0.977092i	$0.568264\pi$
$440$	0	0
$441$	2.90863e27	0.497833
$442$	0	0
$443$	−1.41566e27	−0.231058	−0.115529	−	0.993304i	$-0.536856\pi$
$443$	−1.41566e27	−0.231058	−0.115529	+	0.993304i	$0.536856\pi$
$444$	0	0
$445$	−9.07985e25	−0.0141351
$446$	0	0
$447$	−2.49745e27	−0.370909
$448$	0	0
$449$	−4.28806e27	−0.607679	−0.303839	−	0.952723i	$-0.598269\pi$
$449$	−4.28806e27	−0.607679	−0.303839	+	0.952723i	$0.598269\pi$
$450$	0	0
$451$	4.83832e27	0.654396
$452$	0	0
$453$	−1.91808e26	−0.0247649
$454$	0	0
$455$	−9.92628e26	−0.122368
$456$	0	0
$457$	2.53901e27	0.298912	0.149456	−	0.988768i	$-0.452248\pi$
$457$	2.53901e27	0.298912	0.149456	+	0.988768i	$0.452248\pi$
$458$	0	0
$459$	2.42221e27	0.272382
$460$	0	0
$461$	−1.34137e27	−0.144108	−0.0720541	−	0.997401i	$-0.522955\pi$
$461$	−1.34137e27	−0.144108	−0.0720541	+	0.997401i	$0.522955\pi$
$462$	0	0
$463$	−1.82493e28	−1.87347	−0.936735	−	0.350038i	$-0.886169\pi$
$463$	−1.82493e28	−1.87347	−0.936735	+	0.350038i	$0.886169\pi$
$464$	0	0
$465$	1.14048e26	0.0111901
$466$	0	0
$467$	−3.72909e27	−0.349764	−0.174882	−	0.984589i	$-0.555954\pi$
$467$	−3.72909e27	−0.349764	−0.174882	+	0.984589i	$0.555954\pi$
$468$	0	0
$469$	7.35040e25	0.00659168
$470$	0	0
$471$	4.68662e27	0.401920
$472$	0	0
$473$	4.93022e26	0.0404413
$474$	0	0
$475$	−9.01338e27	−0.707302
$476$	0	0
$477$	−3.62876e27	−0.272468
$478$	0	0
$479$	3.98246e27	0.286173	0.143086	−	0.989710i	$-0.454297\pi$
$479$	3.98246e27	0.286173	0.143086	+	0.989710i	$0.454297\pi$
$480$	0	0
$481$	4.47518e28	3.07813
$482$	0	0
$483$	1.89761e28	1.24957
$484$	0	0
$485$	9.32124e26	0.0587740
$486$	0	0
$487$	9.44835e26	0.0570561	0.0285280	−	0.999593i	$-0.490918\pi$
$487$	9.44835e26	0.0570561	0.0285280	+	0.999593i	$0.490918\pi$
$488$	0	0
$489$	7.57711e27	0.438289
$490$	0	0
$491$	−1.33942e28	−0.742268	−0.371134	−	0.928579i	$-0.621031\pi$
$491$	−1.33942e28	−0.742268	−0.371134	+	0.928579i	$0.621031\pi$
$492$	0	0
$493$	1.62689e28	0.863903
$494$	0	0
$495$	1.22245e26	0.00622123
$496$	0	0
$497$	−4.49172e28	−2.19114
$498$	0	0
$499$	3.84837e28	1.79978	0.899892	−	0.436112i	$-0.143645\pi$
$499$	3.84837e28	1.79978	0.899892	+	0.436112i	$0.143645\pi$
$500$	0	0
$501$	2.33733e28	1.04815
$502$	0	0
$503$	−1.67092e28	−0.718610	−0.359305	−	0.933220i	$-0.616986\pi$
$503$	−1.67092e28	−0.718610	−0.359305	+	0.933220i	$0.616986\pi$
$504$	0	0
$505$	1.45524e27	0.0600308
$506$	0	0
$507$	2.75343e28	1.08966
$508$	0	0
$509$	−4.22522e28	−1.60440	−0.802200	−	0.597055i	$-0.796337\pi$
$509$	−4.22522e28	−1.60440	−0.802200	+	0.597055i	$0.796337\pi$
$510$	0	0
$511$	5.97822e28	2.17847
$512$	0	0
$513$	3.89996e27	0.136404
$514$	0	0
$515$	1.97646e27	0.0663608
$516$	0	0
$517$	5.39145e27	0.173802
$518$	0	0
$519$	2.45687e28	0.760544
$520$	0	0
$521$	5.08622e28	1.51216	0.756082	−	0.654477i	$-0.227111\pi$
$521$	5.08622e28	1.51216	0.756082	+	0.654477i	$0.227111\pi$
$522$	0	0
$523$	−5.12940e28	−1.46487	−0.732434	−	0.680838i	$-0.761616\pi$
$523$	−5.12940e28	−1.46487	−0.732434	+	0.680838i	$0.761616\pi$
$524$	0	0
$525$	−3.31598e28	−0.909787
$526$	0	0
$527$	2.28167e28	0.601508
$528$	0	0
$529$	3.46801e28	0.878609
$530$	0	0
$531$	−1.42090e28	−0.345995
$532$	0	0
$533$	−1.16076e29	−2.71710
$534$	0	0
$535$	−2.23105e26	−0.00502105
$536$	0	0
$537$	−7.50640e27	−0.162443
$538$	0	0
$539$	2.93678e28	0.611209
$540$	0	0
$541$	2.98076e28	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$541$	2.98076e28	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$542$	0	0
$543$	4.89083e28	0.941856
$544$	0	0
$545$	−1.33761e27	−0.0247836
$546$	0	0
$547$	−5.19133e28	−0.925575	−0.462788	−	0.886469i	$-0.653151\pi$
$547$	−5.19133e28	−0.925575	−0.462788	+	0.886469i	$0.653151\pi$
$548$	0	0
$549$	1.25275e28	0.214959
$550$	0	0
$551$	2.61942e28	0.432627
$552$	0	0
$553$	2.55057e28	0.405531
$554$	0	0
$555$	−3.11578e27	−0.0476970
$556$	0	0
$557$	5.37882e28	0.792881	0.396440	−	0.918060i	$-0.370245\pi$
$557$	5.37882e28	0.792881	0.396440	+	0.918060i	$0.370245\pi$
$558$	0	0
$559$	−1.18281e28	−0.167915
$560$	0	0
$561$	2.44565e28	0.334414
$562$	0	0
$563$	−5.50087e28	−0.724592	−0.362296	−	0.932063i	$-0.618007\pi$
$563$	−5.50087e28	−0.724592	−0.362296	+	0.932063i	$0.618007\pi$
$564$	0	0
$565$	1.92715e27	0.0244572
$566$	0	0
$567$	1.43478e28	0.175454
$568$	0	0
$569$	1.68810e29	1.98939	0.994695	−	0.102871i	$-0.0328029\pi$
$569$	1.68810e29	1.98939	0.994695	+	0.102871i	$0.0328029\pi$
$570$	0	0
$571$	1.01286e29	1.15046	0.575230	−	0.817992i	$-0.304913\pi$
$571$	1.01286e29	1.15046	0.575230	+	0.817992i	$0.304913\pi$
$572$	0	0
$573$	4.87354e28	0.533606
$574$	0	0
$575$	−1.29577e29	−1.36777
$576$	0	0
$577$	−1.45726e28	−0.148317	−0.0741585	−	0.997246i	$-0.523627\pi$
$577$	−1.45726e28	−0.148317	−0.0741585	+	0.997246i	$0.523627\pi$
$578$	0	0
$579$	−4.01080e28	−0.393646
$580$	0	0
$581$	−2.33415e29	−2.20942
$582$	0	0
$583$	−3.66388e28	−0.334519
$584$	0	0
$585$	−2.93277e27	−0.0258310
$586$	0	0
$587$	−9.93629e28	−0.844353	−0.422177	−	0.906514i	$-0.638734\pi$
$587$	−9.93629e28	−0.844353	−0.422177	+	0.906514i	$0.638734\pi$
$588$	0	0
$589$	3.67367e28	0.301224
$590$	0	0
$591$	−1.81136e28	−0.143330
$592$	0	0
$593$	−1.83739e29	−1.40323	−0.701614	−	0.712557i	$-0.747537\pi$
$593$	−1.83739e29	−1.40323	−0.701614	+	0.712557i	$0.747537\pi$
$594$	0	0
$595$	1.38263e28	0.101924
$596$	0	0
$597$	4.77871e28	0.340079
$598$	0	0
$599$	−2.52254e29	−1.73323	−0.866614	−	0.498979i	$-0.833708\pi$
$599$	−2.52254e29	−1.73323	−0.866614	+	0.498979i	$0.833708\pi$
$600$	0	0
$601$	1.14887e28	0.0762234	0.0381117	−	0.999273i	$-0.487866\pi$
$601$	1.14887e28	0.0762234	0.0381117	+	0.999273i	$0.487866\pi$
$602$	0	0
$603$	2.17171e26	0.00139146
$604$	0	0
$605$	−6.13533e27	−0.0379670
$606$	0	0
$607$	6.52264e28	0.389890	0.194945	−	0.980814i	$-0.437547\pi$
$607$	6.52264e28	0.389890	0.194945	+	0.980814i	$0.437547\pi$
$608$	0	0
$609$	9.63674e28	0.556479
$610$	0	0
$611$	−1.29346e29	−0.721638
$612$	0	0
$613$	−3.10672e29	−1.67482	−0.837409	−	0.546577i	$-0.815930\pi$
$613$	−3.10672e29	−1.67482	−0.837409	+	0.546577i	$0.815930\pi$
$614$	0	0
$615$	8.08160e27	0.0421026
$616$	0	0
$617$	2.33223e29	1.17429	0.587147	−	0.809481i	$-0.300251\pi$
$617$	2.33223e29	1.17429	0.587147	+	0.809481i	$0.300251\pi$
$618$	0	0
$619$	−8.01970e28	−0.390307	−0.195154	−	0.980773i	$-0.562521\pi$
$619$	−8.01970e28	−0.390307	−0.195154	+	0.980773i	$0.562521\pi$
$620$	0	0
$621$	5.60658e28	0.263777
$622$	0	0
$623$	−1.07601e29	−0.489429
$624$	0	0
$625$	2.25956e29	0.993765
$626$	0	0
$627$	3.93770e28	0.167468
$628$	0	0
$629$	−6.23348e29	−2.56389
$630$	0	0
$631$	5.48800e28	0.218326	0.109163	−	0.994024i	$-0.465183\pi$
$631$	5.48800e28	0.218326	0.109163	+	0.994024i	$0.465183\pi$
$632$	0	0
$633$	−9.33451e28	−0.359213
$634$	0	0
$635$	2.28660e27	0.00851268
$636$	0	0
$637$	−7.04561e29	−2.53778
$638$	0	0
$639$	−1.32710e29	−0.462535
$640$	0	0
$641$	3.70840e29	1.25077	0.625385	−	0.780316i	$-0.284942\pi$
$641$	3.70840e29	1.25077	0.625385	+	0.780316i	$0.284942\pi$
$642$	0	0
$643$	3.51390e29	1.14703	0.573514	−	0.819196i	$-0.305580\pi$
$643$	3.51390e29	1.14703	0.573514	+	0.819196i	$0.305580\pi$
$644$	0	0
$645$	8.23511e26	0.00260191
$646$	0	0
$647$	5.82634e29	1.78197	0.890986	−	0.454031i	$-0.150014\pi$
$647$	5.82634e29	1.78197	0.890986	+	0.454031i	$0.150014\pi$
$648$	0	0
$649$	−1.43465e29	−0.424792
$650$	0	0
$651$	1.35153e29	0.387458
$652$	0	0
$653$	3.13916e28	0.0871416	0.0435708	−	0.999050i	$-0.486127\pi$
$653$	3.13916e28	0.0871416	0.0435708	+	0.999050i	$0.486127\pi$
$654$	0	0
$655$	1.01582e28	0.0273075
$656$	0	0
$657$	1.76629e29	0.459861
$658$	0	0
$659$	−3.33583e29	−0.841215	−0.420607	−	0.907243i	$-0.638183\pi$
$659$	−3.33583e29	−0.841215	−0.420607	+	0.907243i	$0.638183\pi$
$660$	0	0
$661$	5.89161e28	0.143919	0.0719596	−	0.997408i	$-0.477075\pi$
$661$	5.89161e28	0.143919	0.0719596	+	0.997408i	$0.477075\pi$
$662$	0	0
$663$	−5.86734e29	−1.38851
$664$	0	0
$665$	2.22615e28	0.0510419
$666$	0	0
$667$	3.76569e29	0.836610
$668$	0	0
$669$	−2.39109e29	−0.514781
$670$	0	0
$671$	1.26488e29	0.263913
$672$	0	0
$673$	−5.66425e29	−1.14547	−0.572735	−	0.819740i	$-0.694118\pi$
$673$	−5.66425e29	−1.14547	−0.572735	+	0.819740i	$0.694118\pi$
$674$	0	0
$675$	−9.79724e28	−0.192050
$676$	0	0
$677$	1.74529e29	0.331654	0.165827	−	0.986155i	$-0.446971\pi$
$677$	1.74529e29	0.331654	0.165827	+	0.986155i	$0.446971\pi$
$678$	0	0
$679$	1.10461e30	2.03506
$680$	0	0
$681$	2.46084e29	0.439581
$682$	0	0
$683$	2.06449e29	0.357598	0.178799	−	0.983886i	$-0.442779\pi$
$683$	2.06449e29	0.357598	0.178799	+	0.983886i	$0.442779\pi$
$684$	0	0
$685$	−2.01832e28	−0.0339031
$686$	0	0
$687$	4.64096e29	0.756070
$688$	0	0
$689$	8.78997e29	1.38895
$690$	0	0
$691$	−9.21370e29	−1.41226	−0.706130	−	0.708082i	$-0.749561\pi$
$691$	−9.21370e29	−1.41226	−0.706130	+	0.708082i	$0.749561\pi$
$692$	0	0
$693$	1.44866e29	0.215411
$694$	0	0
$695$	4.85517e28	0.0700429
$696$	0	0
$697$	1.61682e30	2.26317
$698$	0	0
$699$	−2.78231e29	−0.377916
$700$	0	0
$701$	1.27540e30	1.68115	0.840576	−	0.541694i	$-0.182217\pi$
$701$	1.27540e30	1.68115	0.840576	+	0.541694i	$0.182217\pi$
$702$	0	0
$703$	−1.00364e30	−1.28395
$704$	0	0
$705$	9.00552e27	0.0111821
$706$	0	0
$707$	1.72453e30	2.07858
$708$	0	0
$709$	−5.15684e29	−0.603390	−0.301695	−	0.953404i	$-0.597552\pi$
$709$	−5.15684e29	−0.603390	−0.301695	+	0.953404i	$0.597552\pi$
$710$	0	0
$711$	7.53580e28	0.0856048
$712$	0	0
$713$	5.28128e29	0.582504
$714$	0	0
$715$	−2.96115e28	−0.0317137
$716$	0	0
$717$	−1.03938e30	−1.08100
$718$	0	0
$719$	−3.68507e29	−0.372214	−0.186107	−	0.982530i	$-0.559587\pi$
$719$	−3.68507e29	−0.372214	−0.186107	+	0.982530i	$0.559587\pi$
$720$	0	0
$721$	2.34220e30	2.29775
$722$	0	0
$723$	−1.13751e28	−0.0108394
$724$	0	0
$725$	−6.58036e29	−0.609117
$726$	0	0
$727$	−2.00029e30	−1.79880	−0.899398	−	0.437130i	$-0.855995\pi$
$727$	−2.00029e30	−1.79880	−0.899398	+	0.437130i	$0.855995\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	1.64753e29	0.139862
$732$	0	0
$733$	−1.63425e30	−1.34812	−0.674059	−	0.738678i	$-0.735451\pi$
$733$	−1.63425e30	−1.34812	−0.674059	+	0.738678i	$0.735451\pi$
$734$	0	0
$735$	4.90540e28	0.0393240
$736$	0	0
$737$	2.19273e27	0.00170835
$738$	0	0
$739$	2.17982e30	1.65064	0.825322	−	0.564662i	$-0.190993\pi$
$739$	2.17982e30	1.65064	0.825322	+	0.564662i	$0.190993\pi$
$740$	0	0
$741$	−9.44689e29	−0.695341
$742$	0	0
$743$	−8.10599e29	−0.579994	−0.289997	−	0.957028i	$-0.593654\pi$
$743$	−8.10599e29	−0.579994	−0.289997	+	0.957028i	$0.593654\pi$
$744$	0	0
$745$	−4.21193e28	−0.0292982
$746$	0	0
$747$	−6.89635e29	−0.466394
$748$	0	0
$749$	−2.64390e29	−0.173854
$750$	0	0
$751$	−4.66500e27	−0.00298286	−0.00149143	−	0.999999i	$-0.500475\pi$
$751$	−4.66500e27	−0.00298286	−0.00149143	+	0.999999i	$0.500475\pi$
$752$	0	0
$753$	1.75655e30	1.09223
$754$	0	0
$755$	−3.23484e27	−0.00195619
$756$	0	0
$757$	2.77260e29	0.163072	0.0815362	−	0.996670i	$-0.474017\pi$
$757$	2.77260e29	0.163072	0.0815362	+	0.996670i	$0.474017\pi$
$758$	0	0
$759$	5.66084e29	0.323849
$760$	0	0
$761$	−2.32433e29	−0.129348	−0.0646738	−	0.997906i	$-0.520601\pi$
$761$	−2.32433e29	−0.129348	−0.0646738	+	0.997906i	$0.520601\pi$
$762$	0	0
$763$	−1.58513e30	−0.858137
$764$	0	0
$765$	4.08505e28	0.0215155
$766$	0	0
$767$	3.44185e30	1.76376
$768$	0	0
$769$	1.70610e30	0.850705	0.425352	−	0.905028i	$-0.360150\pi$
$769$	1.70610e30	0.850705	0.425352	+	0.905028i	$0.360150\pi$
$770$	0	0
$771$	1.00360e30	0.486958
$772$	0	0
$773$	1.50011e30	0.708335	0.354168	−	0.935182i	$-0.384764\pi$
$773$	1.50011e30	0.708335	0.354168	+	0.935182i	$0.384764\pi$
$774$	0	0
$775$	−9.22878e29	−0.424108
$776$	0	0
$777$	−3.69235e30	−1.65151
$778$	0	0
$779$	2.60321e30	1.13335
$780$	0	0
$781$	−1.33994e30	−0.567872
$782$	0	0
$783$	2.84722e29	0.117469
$784$	0	0
$785$	7.90396e28	0.0317478
$786$	0	0
$787$	−4.40356e30	−1.72214	−0.861071	−	0.508485i	$-0.830206\pi$
$787$	−4.40356e30	−1.72214	−0.861071	+	0.508485i	$0.830206\pi$
$788$	0	0
$789$	−2.39731e30	−0.912885
$790$	0	0
$791$	2.28377e30	0.846836
$792$	0	0
$793$	−3.03455e30	−1.09579
$794$	0	0
$795$	−6.11989e28	−0.0215223
$796$	0	0
$797$	−2.64344e30	−0.905434	−0.452717	−	0.891654i	$-0.649545\pi$
$797$	−2.64344e30	−0.905434	−0.452717	+	0.891654i	$0.649545\pi$
$798$	0	0
$799$	1.80166e30	0.601078
$800$	0	0
$801$	−3.17911e29	−0.103315
$802$	0	0
$803$	1.78339e30	0.564589
$804$	0	0
$805$	3.20031e29	0.0987043
$806$	0	0
$807$	−2.83955e30	−0.853251
$808$	0	0
$809$	1.79810e30	0.526447	0.263224	−	0.964735i	$-0.415214\pi$
$809$	1.79810e30	0.526447	0.263224	+	0.964735i	$0.415214\pi$
$810$	0	0
$811$	−6.41630e30	−1.83048	−0.915241	−	0.402906i	$-0.868000\pi$
$811$	−6.41630e30	−1.83048	−0.915241	+	0.402906i	$0.868000\pi$
$812$	0	0
$813$	−2.84688e30	−0.791441
$814$	0	0
$815$	1.27788e29	0.0346206
$816$	0	0
$817$	2.65266e29	0.0700405
$818$	0	0
$819$	−3.47547e30	−0.894402
$820$	0	0
$821$	−4.18469e30	−1.04969	−0.524844	−	0.851198i	$-0.675877\pi$
$821$	−4.18469e30	−1.04969	−0.524844	+	0.851198i	$0.675877\pi$
$822$	0	0
$823$	1.69536e30	0.414537	0.207269	−	0.978284i	$-0.433543\pi$
$823$	1.69536e30	0.414537	0.207269	+	0.978284i	$0.433543\pi$
$824$	0	0
$825$	−9.89205e29	−0.235787
$826$	0	0
$827$	3.73546e29	0.0868032	0.0434016	−	0.999058i	$-0.486180\pi$
$827$	3.73546e29	0.0868032	0.0434016	+	0.999058i	$0.486180\pi$
$828$	0	0
$829$	−3.37284e30	−0.764141	−0.382071	−	0.924133i	$-0.624789\pi$
$829$	−3.37284e30	−0.764141	−0.382071	+	0.924133i	$0.624789\pi$
$830$	0	0
$831$	1.33231e30	0.294302
$832$	0	0
$833$	9.81382e30	2.11381
$834$	0	0
$835$	3.94189e29	0.0827936
$836$	0	0
$837$	3.99316e29	0.0817898
$838$	0	0
$839$	2.40344e30	0.480102	0.240051	−	0.970760i	$-0.422836\pi$
$839$	2.40344e30	0.480102	0.240051	+	0.970760i	$0.422836\pi$
$840$	0	0
$841$	−3.22049e30	−0.627429
$842$	0	0
$843$	1.47336e29	0.0279976
$844$	0	0
$845$	4.64365e29	0.0860724
$846$	0	0
$847$	−7.27066e30	−1.31461
$848$	0	0
$849$	8.03103e29	0.141658
$850$	0	0
$851$	−1.44284e31	−2.48289
$852$	0	0
$853$	1.29292e30	0.217074	0.108537	−	0.994092i	$-0.465383\pi$
$853$	1.29292e30	0.217074	0.108537	+	0.994092i	$0.465383\pi$
$854$	0	0
$855$	6.57726e28	0.0107746
$856$	0	0
$857$	−2.56383e30	−0.409817	−0.204909	−	0.978781i	$-0.565690\pi$
$857$	−2.56383e30	−0.409817	−0.204909	+	0.978781i	$0.565690\pi$
$858$	0	0
$859$	1.40233e30	0.218737	0.109369	−	0.994001i	$-0.465117\pi$
$859$	1.40233e30	0.218737	0.109369	+	0.994001i	$0.465117\pi$
$860$	0	0
$861$	9.57709e30	1.45781
$862$	0	0
$863$	−5.48238e30	−0.814434	−0.407217	−	0.913331i	$-0.633501\pi$
$863$	−5.48238e30	−0.814434	−0.407217	+	0.913331i	$0.633501\pi$
$864$	0	0
$865$	4.14350e29	0.0600756
$866$	0	0
$867$	4.09280e30	0.579189
$868$	0	0
$869$	7.60872e29	0.105100
$870$	0	0
$871$	−5.26055e28	−0.00709317
$872$	0	0
$873$	3.26363e30	0.429587
$874$	0	0
$875$	−1.11965e30	−0.143878
$876$	0	0
$877$	5.89368e30	0.739419	0.369709	−	0.929147i	$-0.379457\pi$
$877$	5.89368e30	0.739419	0.369709	+	0.929147i	$0.379457\pi$
$878$	0	0
$879$	−9.27066e30	−1.13561
$880$	0	0
$881$	−1.21940e30	−0.145848	−0.0729238	−	0.997338i	$-0.523233\pi$
$881$	−1.21940e30	−0.145848	−0.0729238	+	0.997338i	$0.523233\pi$
$882$	0	0
$883$	1.36507e31	1.59429	0.797144	−	0.603790i	$-0.206343\pi$
$883$	1.36507e31	1.59429	0.797144	+	0.603790i	$0.206343\pi$
$884$	0	0
$885$	−2.39634e29	−0.0273303
$886$	0	0
$887$	6.50423e30	0.724433	0.362216	−	0.932094i	$-0.382020\pi$
$887$	6.50423e30	0.724433	0.362216	+	0.932094i	$0.382020\pi$
$888$	0	0
$889$	2.70973e30	0.294753
$890$	0	0
$891$	4.28014e29	0.0454718
$892$	0	0
$893$	2.90081e30	0.301009
$894$	0	0
$895$	−1.26595e29	−0.0128314
$896$	0	0
$897$	−1.35809e31	−1.34464
$898$	0	0
$899$	2.68202e30	0.259409
$900$	0	0
$901$	−1.22435e31	−1.15690
$902$	0	0
$903$	9.75901e29	0.0900916
$904$	0	0
$905$	8.24837e29	0.0743975
$906$	0	0
$907$	1.08087e31	0.952571	0.476286	−	0.879291i	$-0.341983\pi$
$907$	1.08087e31	0.952571	0.476286	+	0.879291i	$0.341983\pi$
$908$	0	0
$909$	5.09519e30	0.438773
$910$	0	0
$911$	−3.63526e30	−0.305909	−0.152954	−	0.988233i	$-0.548879\pi$
$911$	−3.63526e30	−0.305909	−0.152954	+	0.988233i	$0.548879\pi$
$912$	0	0
$913$	−6.96308e30	−0.572609
$914$	0	0
$915$	2.11276e29	0.0169797
$916$	0	0
$917$	1.20379e31	0.945527
$918$	0	0
$919$	5.58006e30	0.428377	0.214188	−	0.976792i	$-0.431289\pi$
$919$	5.58006e30	0.428377	0.214188	+	0.976792i	$0.431289\pi$
$920$	0	0
$921$	1.03132e31	0.773872
$922$	0	0
$923$	3.21465e31	2.35785
$924$	0	0
$925$	2.52129e31	1.80773
$926$	0	0
$927$	6.92014e30	0.485040
$928$	0	0
$929$	2.15700e31	1.47804	0.739020	−	0.673684i	$-0.235289\pi$
$929$	2.15700e31	1.47804	0.739020	+	0.673684i	$0.235289\pi$
$930$	0	0
$931$	1.58011e31	1.05856
$932$	0	0
$933$	7.35344e30	0.481652
$934$	0	0
$935$	4.12458e29	0.0264155
$936$	0	0
$937$	−1.41281e31	−0.884743	−0.442371	−	0.896832i	$-0.645863\pi$
$937$	−1.41281e31	−0.884743	−0.442371	+	0.896832i	$0.645863\pi$
$938$	0	0
$939$	−1.43261e31	−0.877283
$940$	0	0
$941$	−6.25041e30	−0.374298	−0.187149	−	0.982332i	$-0.559925\pi$
$941$	−6.25041e30	−0.374298	−0.187149	+	0.982332i	$0.559925\pi$
$942$	0	0
$943$	3.74237e31	2.19167
$944$	0	0
$945$	2.41975e29	0.0138591
$946$	0	0
$947$	−1.56241e31	−0.875225	−0.437612	−	0.899164i	$-0.644176\pi$
$947$	−1.56241e31	−0.875225	−0.437612	+	0.899164i	$0.644176\pi$
$948$	0	0
$949$	−4.27851e31	−2.34421
$950$	0	0
$951$	−6.63127e29	−0.0355387
$952$	0	0
$953$	−2.51938e31	−1.32074	−0.660371	−	0.750939i	$-0.729601\pi$
$953$	−2.51938e31	−1.32074	−0.660371	+	0.750939i	$0.729601\pi$
$954$	0	0
$955$	8.21921e29	0.0421497
$956$	0	0
$957$	2.87478e30	0.144221
$958$	0	0
$959$	−2.39181e31	−1.17390
$960$	0	0
$961$	−1.70640e31	−0.819382
$962$	0	0
$963$	−7.81153e29	−0.0366995
$964$	0	0
$965$	−6.76420e29	−0.0310942
$966$	0	0
$967$	−2.95777e31	−1.33041	−0.665206	−	0.746660i	$-0.731656\pi$
$967$	−2.95777e31	−1.33041	−0.665206	+	0.746660i	$0.731656\pi$
$968$	0	0
$969$	1.31586e31	0.579174
$970$	0	0
$971$	−6.84548e30	−0.294851	−0.147425	−	0.989073i	$-0.547099\pi$
$971$	−6.84548e30	−0.294851	−0.147425	+	0.989073i	$0.547099\pi$
$972$	0	0
$973$	5.75361e31	2.42525
$974$	0	0
$975$	2.37319e31	0.979004
$976$	0	0
$977$	−2.28485e31	−0.922498	−0.461249	−	0.887271i	$-0.652598\pi$
$977$	−2.28485e31	−0.922498	−0.461249	+	0.887271i	$0.652598\pi$
$978$	0	0
$979$	−3.20987e30	−0.126844
$980$	0	0
$981$	−4.68334e30	−0.181147
$982$	0	0
$983$	−3.53478e30	−0.133829	−0.0669146	−	0.997759i	$-0.521315\pi$
$983$	−3.53478e30	−0.133829	−0.0669146	+	0.997759i	$0.521315\pi$
$984$	0	0
$985$	−3.05485e29	−0.0113217
$986$	0	0
$987$	1.06720e31	0.387181
$988$	0	0
$989$	3.81346e30	0.135444
$990$	0	0
$991$	2.61702e31	0.909982	0.454991	−	0.890496i	$-0.349642\pi$
$991$	2.61702e31	0.909982	0.454991	+	0.890496i	$0.349642\pi$
$992$	0	0
$993$	−6.28208e30	−0.213863
$994$	0	0
$995$	8.05927e29	0.0268629
$996$	0	0
$997$	4.15801e31	1.35702	0.678511	−	0.734590i	$-0.262626\pi$
$997$	4.15801e31	1.35702	0.678511	+	0.734590i	$0.262626\pi$
$998$	0	0
$999$	−1.09092e31	−0.348623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.k.1.2		3
4.3	odd	2		24.22.a.b.1.2	✓	3
12.11	even	2		72.22.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.b.1.2	✓	3	4.3	odd	2
48.22.a.k.1.2		3	1.1	even	1	trivial
72.22.a.e.1.2		3	12.11	even	2

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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+59049.0 q^{3} +995860. q^{5} +1.18014e9 q^{7} +3.48678e9 q^{9} +O(q^{10})\)	\(q+59049.0 q^{3} +995860. q^{5} +1.18014e9 q^{7} +3.48678e9 q^{9} +3.52053e10 q^{11} -8.44606e11 q^{13} +5.88045e10 q^{15} +1.17645e13 q^{17} +1.89418e13 q^{19} +6.96862e13 q^{21} +2.72308e14 q^{23} -4.75845e14 q^{25} +2.05891e14 q^{27} +1.38288e15 q^{29} +1.93945e15 q^{31} +2.07884e15 q^{33} +1.17526e15 q^{35} -5.29854e16 q^{37} -4.98732e16 q^{39} +1.37432e17 q^{41} +1.40042e16 q^{43} +3.47235e15 q^{45} +1.53143e17 q^{47} +8.34188e17 q^{49} +6.94683e17 q^{51} -1.04072e18 q^{53} +3.50595e16 q^{55} +1.11850e18 q^{57} -4.07509e18 q^{59} +3.59286e18 q^{61} +4.11490e18 q^{63} -8.41109e17 q^{65} +6.22840e16 q^{67} +1.60795e19 q^{69} -3.80609e19 q^{71} +5.06568e19 q^{73} -2.80982e19 q^{75} +4.15472e19 q^{77} +2.16125e19 q^{79} +1.21577e19 q^{81} -1.97785e20 q^{83} +1.17158e19 q^{85} +8.16575e19 q^{87} -9.11760e19 q^{89} -9.96755e20 q^{91} +1.14523e20 q^{93} +1.88634e19 q^{95} +9.35999e20 q^{97} +1.22753e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 177147 q^{3} - 4833126 q^{5} - 271431024 q^{7} + 10460353203 q^{9}+O(q^{10}) \) 3 * q + 177147 * q^3 - 4833126 * q^5 - 271431024 * q^7 + 10460353203 * q^9	\( 3 q + 177147 q^{3} - 4833126 q^{5} - 271431024 q^{7} + 10460353203 q^{9} - 61194658188 q^{11} - 594486202422 q^{13} - 285391257174 q^{15} + 1424819519334 q^{17} + 35094825681804 q^{19} - 16027730536176 q^{21} + 94911043073592 q^{23} - 613780025295699 q^{25} + 617673396283947 q^{27} - 32\!\cdots\!34 q^{29}+ \cdots - 21\!\cdots\!88 q^{99}+O(q^{100}) \) 3 * q + 177147 * q^3 - 4833126 * q^5 - 271431024 * q^7 + 10460353203 * q^9 - 61194658188 * q^11 - 594486202422 * q^13 - 285391257174 * q^15 + 1424819519334 * q^17 + 35094825681804 * q^19 - 16027730536176 * q^21 + 94911043073592 * q^23 - 613780025295699 * q^25 + 617673396283947 * q^27 - 3292734981070734 * q^29 + 1928658139449144 * q^31 - 3613483371343212 * q^33 + 18129892754606304 * q^35 + 22185275061682722 * q^37 - 35103815766816678 * q^39 + 167394191883971118 * q^41 - 69547615491362508 * q^43 - 16852068344867526 * q^45 + 49110618973894752 * q^47 + 973320764343712395 * q^49 + 84134167797153366 * q^51 + 488391341947078074 * q^53 + 3110835549535695000 * q^55 + 2072314361684844396 * q^57 + 1846144806145488852 * q^59 + 8183707455377473002 * q^61 - 946421460430656624 * q^63 + 28951429805381293356 * q^65 - 37438935583939030164 * q^67 + 5604402182452534008 * q^69 - 65865243803437791096 * q^71 + 88415465494171854846 * q^73 - 36243096713685730251 * q^75 + 156034186748470726080 * q^77 - 221381156829825400920 * q^79 + 36472996377170786403 * q^81 - 305200914127129955700 * q^83 + 184695330643817576052 * q^85 - 194432707897245771966 * q^87 + 327556458098783035470 * q^89 - 692083854852141366816 * q^91 + 113885334476332504056 * q^93 - 1486593794603974227096 * q^95 + 1980043166774365090758 * q^97 - 213372579594445325388 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more