LMFDB - Embedded newform 48.22.a.l.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} - x^{2} - 2295485x - 828958533 $ x^3 - x^2 - 2295485*x - 828958533
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$-1283.97$ 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)

$f(q)$	$=$	$q+59049.0 q^{3} -8.90852e6 q^{5} -5.87822e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$	$q+59049.0 q^{3} -8.90852e6 q^{5} -5.87822e8 q^{7} +3.48678e9 q^{9} +1.42332e11 q^{11} +1.49035e11 q^{13} -5.26039e11 q^{15} +9.95199e12 q^{17} +5.02926e12 q^{19} -3.47103e13 q^{21} -3.25343e14 q^{23} -3.97475e14 q^{25} +2.05891e14 q^{27} +1.07038e15 q^{29} -3.98818e15 q^{31} +8.40458e15 q^{33} +5.23663e15 q^{35} -3.35659e16 q^{37} +8.80039e15 q^{39} +4.54703e16 q^{41} +8.38924e16 q^{43} -3.10621e16 q^{45} +5.34085e16 q^{47} -2.13011e17 q^{49} +5.87655e17 q^{51} +1.14205e18 q^{53} -1.26797e18 q^{55} +2.96973e17 q^{57} +5.97497e18 q^{59} +4.12289e17 q^{61} -2.04961e18 q^{63} -1.32769e18 q^{65} +2.42251e19 q^{67} -1.92112e19 q^{69} -7.25973e16 q^{71} -2.97771e19 q^{73} -2.34705e19 q^{75} -8.36661e19 q^{77} +5.78794e19 q^{79} +1.21577e19 q^{81} +2.48286e20 q^{83} -8.86575e19 q^{85} +6.32049e19 q^{87} -5.56792e19 q^{89} -8.76063e19 q^{91} -2.35498e20 q^{93} -4.48032e19 q^{95} -8.39559e20 q^{97} +4.96282e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10}) $ 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9	$ 3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9} - 13839247500 q^{11} + 718855551690 q^{13} + 122823455274 q^{15} + 2135189843046 q^{17} + 40122324686988 q^{19} + 71170700597136 q^{21} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 48\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9 - 13839247500 * q^11 + 718855551690 * q^13 + 122823455274 * q^15 + 2135189843046 * q^17 + 40122324686988 * q^19 + 71170700597136 * q^21 - 278424417682632 * q^23 + 1348043260553901 * q^25 + 617673396283947 * q^27 + 442708167991794 * q^29 - 8016070162990152 * q^31 - 817193725627500 * q^33 - 125384157242400 * q^35 + 27729341388737058 * q^37 + 42447701471742810 * q^39 - 125648125186340562 * q^41 + 229052541499074612 * q^43 + 7252602210474426 * q^45 + 448613782068047712 * q^47 + 365221903446092427 * q^49 + 126080825042023254 * q^51 + 1406206217208267066 * q^53 + 3437829264920292504 * q^55 + 2369183150441954412 * q^57 + 1844638981471622100 * q^59 - 3294066300350351382 * q^61 + 4202558699560283664 * q^63 - 19537666262756991444 * q^65 + 33491023693155020652 * q^67 - 16440683439741736968 * q^69 + 79431018431598881160 * q^71 - 46612822906958319618 * q^73 + 79600606492447300149 * q^75 - 255759806110876888128 * q^77 + 197919973704661098024 * q^79 + 36472996377170786403 * q^81 - 111848528551886940276 * q^83 - 531596971435705186956 * q^85 + 26141474611747443906 * q^87 - 731008175840243483058 * q^89 + 546304658547739915488 * q^91 - 473340927054405485448 * q^93 + 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97 - 48254472304578247500 * 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
$5$	−8.90852e6	−0.407963	−0.203981	−	0.978975i	$-0.565388\pi$
$5$	−8.90852e6	−0.407963	−0.203981	+	0.978975i	$0.565388\pi$
$6$	0	0
$7$	−5.87822e8	−0.786532	−0.393266	−	0.919425i	$-0.628655\pi$
$7$	−5.87822e8	−0.786532	−0.393266	+	0.919425i	$0.628655\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.42332e11	1.65455	0.827276	−	0.561796i	$-0.189889\pi$
$11$	1.42332e11	1.65455	0.827276	+	0.561796i	$0.189889\pi$
$12$	0	0
$13$	1.49035e11	0.299836	0.149918	−	0.988698i	$-0.452099\pi$
$13$	1.49035e11	0.299836	0.149918	+	0.988698i	$0.452099\pi$
$14$	0	0
$15$	−5.26039e11	−0.235537
$16$	0	0
$17$	9.95199e12	1.19728	0.598641	−	0.801018i	$-0.295708\pi$
$17$	9.95199e12	1.19728	0.598641	+	0.801018i	$0.295708\pi$
$18$	0	0
$19$	5.02926e12	0.188188	0.0940938	−	0.995563i	$-0.470005\pi$
$19$	5.02926e12	0.188188	0.0940938	+	0.995563i	$0.470005\pi$
$20$	0	0
$21$	−3.47103e13	−0.454105
$22$	0	0
$23$	−3.25343e14	−1.63757	−0.818783	−	0.574103i	$-0.805351\pi$
$23$	−3.25343e14	−1.63757	−0.818783	+	0.574103i	$0.805351\pi$
$24$	0	0
$25$	−3.97475e14	−0.833566
$26$	0	0
$27$	2.05891e14	0.192450
$28$	0	0
$29$	1.07038e15	0.472454	0.236227	−	0.971698i	$-0.424089\pi$
$29$	1.07038e15	0.472454	0.236227	+	0.971698i	$0.424089\pi$
$30$	0	0
$31$	−3.98818e15	−0.873932	−0.436966	−	0.899478i	$-0.643947\pi$
$31$	−3.98818e15	−0.873932	−0.436966	+	0.899478i	$0.643947\pi$
$32$	0	0
$33$	8.40458e15	0.955256
$34$	0	0
$35$	5.23663e15	0.320876
$36$	0	0
$37$	−3.35659e16	−1.14757	−0.573787	−	0.819004i	$-0.694526\pi$
$37$	−3.35659e16	−1.14757	−0.573787	+	0.819004i	$0.694526\pi$
$38$	0	0
$39$	8.80039e15	0.173111
$40$	0	0
$41$	4.54703e16	0.529051	0.264526	−	0.964379i	$-0.414785\pi$
$41$	4.54703e16	0.529051	0.264526	+	0.964379i	$0.414785\pi$
$42$	0	0
$43$	8.38924e16	0.591976	0.295988	−	0.955192i	$-0.404351\pi$
$43$	8.38924e16	0.591976	0.295988	+	0.955192i	$0.404351\pi$
$44$	0	0
$45$	−3.10621e16	−0.135988
$46$	0	0
$47$	5.34085e16	0.148110	0.0740548	−	0.997254i	$-0.476406\pi$
$47$	5.34085e16	0.148110	0.0740548	+	0.997254i	$0.476406\pi$
$48$	0	0
$49$	−2.13011e17	−0.381367
$50$	0	0
$51$	5.87655e17	0.691251
$52$	0	0
$53$	1.14205e18	0.896988	0.448494	−	0.893786i	$-0.351961\pi$
$53$	1.14205e18	0.896988	0.448494	+	0.893786i	$0.351961\pi$
$54$	0	0
$55$	−1.26797e18	−0.674996
$56$	0	0
$57$	2.96973e17	0.108650
$58$	0	0
$59$	5.97497e18	1.52191	0.760956	−	0.648803i	$-0.224730\pi$
$59$	5.97497e18	1.52191	0.760956	+	0.648803i	$0.224730\pi$
$60$	0	0
$61$	4.12289e17	0.0740011	0.0370006	−	0.999315i	$-0.488220\pi$
$61$	4.12289e17	0.0740011	0.0370006	+	0.999315i	$0.488220\pi$
$62$	0	0
$63$	−2.04961e18	−0.262177
$64$	0	0
$65$	−1.32769e18	−0.122322
$66$	0	0
$67$	2.42251e19	1.62360	0.811802	−	0.583933i	$-0.198487\pi$
$67$	2.42251e19	1.62360	0.811802	+	0.583933i	$0.198487\pi$
$68$	0	0
$69$	−1.92112e19	−0.945449
$70$	0	0
$71$	−7.25973e16	−0.00264672	−0.00132336	−	0.999999i	$-0.500421\pi$
$71$	−7.25973e16	−0.00264672	−0.00132336	+	0.999999i	$0.500421\pi$
$72$	0	0
$73$	−2.97771e19	−0.810947	−0.405473	−	0.914107i	$-0.632893\pi$
$73$	−2.97771e19	−0.810947	−0.405473	+	0.914107i	$0.632893\pi$
$74$	0	0
$75$	−2.34705e19	−0.481260
$76$	0	0
$77$	−8.36661e19	−1.30136
$78$	0	0
$79$	5.78794e19	0.687764	0.343882	−	0.939013i	$-0.388258\pi$
$79$	5.78794e19	0.687764	0.343882	+	0.939013i	$0.388258\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	2.48286e20	1.75643	0.878217	−	0.478262i	$-0.158733\pi$
$83$	2.48286e20	1.75643	0.878217	+	0.478262i	$0.158733\pi$
$84$	0	0
$85$	−8.86575e19	−0.488446
$86$	0	0
$87$	6.32049e19	0.272771
$88$	0	0
$89$	−5.56792e19	−0.189277	−0.0946385	−	0.995512i	$-0.530170\pi$
$89$	−5.56792e19	−0.189277	−0.0946385	+	0.995512i	$0.530170\pi$
$90$	0	0
$91$	−8.76063e19	−0.235831
$92$	0	0
$93$	−2.35498e20	−0.504565
$94$	0	0
$95$	−4.48032e19	−0.0767735
$96$	0	0
$97$	−8.39559e20	−1.15597	−0.577987	−	0.816046i	$-0.696161\pi$
$97$	−8.39559e20	−1.15597	−0.577987	+	0.816046i	$0.696161\pi$
$98$	0	0
$99$	4.96282e20	0.551517
$100$	0	0
$101$	−7.25262e20	−0.653312	−0.326656	−	0.945143i	$-0.605922\pi$
$101$	−7.25262e20	−0.653312	−0.326656	+	0.945143i	$0.605922\pi$
$102$	0	0
$103$	1.79494e21	1.31601	0.658004	−	0.753015i	$-0.271401\pi$
$103$	1.79494e21	1.31601	0.658004	+	0.753015i	$0.271401\pi$
$104$	0	0
$105$	3.09217e20	0.185258
$106$	0	0
$107$	1.59231e20	0.0782522	0.0391261	−	0.999234i	$-0.487543\pi$
$107$	1.59231e20	0.0782522	0.0391261	+	0.999234i	$0.487543\pi$
$108$	0	0
$109$	2.97325e21	1.20297	0.601484	−	0.798885i	$-0.294577\pi$
$109$	2.97325e21	1.20297	0.601484	+	0.798885i	$0.294577\pi$
$110$	0	0
$111$	−1.98204e21	−0.662552
$112$	0	0
$113$	1.38076e21	0.382644	0.191322	−	0.981527i	$-0.438723\pi$
$113$	1.38076e21	0.382644	0.191322	+	0.981527i	$0.438723\pi$
$114$	0	0
$115$	2.89832e21	0.668066
$116$	0	0
$117$	5.19654e20	0.0999455
$118$	0	0
$119$	−5.85000e21	−0.941700
$120$	0	0
$121$	1.28582e22	1.73754
$122$	0	0
$123$	2.68498e21	0.305448
$124$	0	0
$125$	7.78883e21	0.748027
$126$	0	0
$127$	−4.56871e21	−0.371411	−0.185706	−	0.982605i	$-0.559457\pi$
$127$	−4.56871e21	−0.371411	−0.185706	+	0.982605i	$0.559457\pi$
$128$	0	0
$129$	4.95376e21	0.341777
$130$	0	0
$131$	8.64013e21	0.507191	0.253595	−	0.967310i	$-0.418387\pi$
$131$	8.64013e21	0.507191	0.253595	+	0.967310i	$0.418387\pi$
$132$	0	0
$133$	−2.95631e21	−0.148016
$134$	0	0
$135$	−1.83419e21	−0.0785125
$136$	0	0
$137$	2.68277e22	0.984052	0.492026	−	0.870581i	$-0.336257\pi$
$137$	2.68277e22	0.984052	0.492026	+	0.870581i	$0.336257\pi$
$138$	0	0
$139$	−3.90524e22	−1.23025	−0.615123	−	0.788431i	$-0.710894\pi$
$139$	−3.90524e22	−1.23025	−0.615123	+	0.788431i	$0.710894\pi$
$140$	0	0
$141$	3.15372e21	0.0855111
$142$	0	0
$143$	2.12126e22	0.496095
$144$	0	0
$145$	−9.53551e21	−0.192744
$146$	0	0
$147$	−1.25781e22	−0.220182
$148$	0	0
$149$	3.86793e22	0.587521	0.293760	−	0.955879i	$-0.405093\pi$
$149$	3.86793e22	0.587521	0.293760	+	0.955879i	$0.405093\pi$
$150$	0	0
$151$	1.07575e23	1.42054	0.710269	−	0.703930i	$-0.248573\pi$
$151$	1.07575e23	1.42054	0.710269	+	0.703930i	$0.248573\pi$
$152$	0	0
$153$	3.47004e22	0.399094
$154$	0	0
$155$	3.55288e22	0.356532
$156$	0	0
$157$	6.91463e22	0.606489	0.303244	−	0.952913i	$-0.401930\pi$
$157$	6.91463e22	0.606489	0.303244	+	0.952913i	$0.401930\pi$
$158$	0	0
$159$	6.74366e22	0.517876
$160$	0	0
$161$	1.91244e23	1.28800
$162$	0	0
$163$	2.70783e23	1.60196	0.800980	−	0.598691i	$-0.204312\pi$
$163$	2.70783e23	1.60196	0.800980	+	0.598691i	$0.204312\pi$
$164$	0	0
$165$	−7.48724e22	−0.389709
$166$	0	0
$167$	2.30715e23	1.05816	0.529081	−	0.848571i	$-0.322537\pi$
$167$	2.30715e23	1.05816	0.529081	+	0.848571i	$0.322537\pi$
$168$	0	0
$169$	−2.24853e23	−0.910098
$170$	0	0
$171$	1.75359e22	0.0627292
$172$	0	0
$173$	−3.68393e23	−1.16635	−0.583173	−	0.812348i	$-0.698189\pi$
$173$	−3.68393e23	−1.16635	−0.583173	+	0.812348i	$0.698189\pi$
$174$	0	0
$175$	2.33645e23	0.655627
$176$	0	0
$177$	3.52816e23	0.878677
$178$	0	0
$179$	6.78508e23	1.50175	0.750876	−	0.660443i	$-0.229631\pi$
$179$	6.78508e23	1.50175	0.750876	+	0.660443i	$0.229631\pi$
$180$	0	0
$181$	9.41228e23	1.85383	0.926915	−	0.375272i	$-0.122451\pi$
$181$	9.41228e23	1.85383	0.926915	+	0.375272i	$0.122451\pi$
$182$	0	0
$183$	2.43452e22	0.0427246
$184$	0	0
$185$	2.99023e23	0.468168
$186$	0	0
$187$	1.41649e24	1.98096
$188$	0	0
$189$	−1.21027e23	−0.151368
$190$	0	0
$191$	−1.00355e24	−1.12380	−0.561899	−	0.827206i	$-0.689929\pi$
$191$	−1.00355e24	−1.12380	−0.561899	+	0.827206i	$0.689929\pi$
$192$	0	0
$193$	−1.40841e24	−1.41377	−0.706885	−	0.707328i	$-0.749900\pi$
$193$	−1.40841e24	−1.41377	−0.706885	+	0.707328i	$0.749900\pi$
$194$	0	0
$195$	−7.83985e22	−0.0706227
$196$	0	0
$197$	1.96986e24	1.59419	0.797096	−	0.603853i	$-0.206369\pi$
$197$	1.96986e24	1.59419	0.797096	+	0.603853i	$0.206369\pi$
$198$	0	0
$199$	−1.33492e24	−0.971620	−0.485810	−	0.874064i	$-0.661475\pi$
$199$	−1.33492e24	−0.971620	−0.485810	+	0.874064i	$0.661475\pi$
$200$	0	0
$201$	1.43047e24	0.937388
$202$	0	0
$203$	−6.29193e23	−0.371600
$204$	0	0
$205$	−4.05073e23	−0.215833
$206$	0	0
$207$	−1.13440e24	−0.545855
$208$	0	0
$209$	7.15826e23	0.311366
$210$	0	0
$211$	1.21610e24	0.478633	0.239316	−	0.970942i	$-0.423077\pi$
$211$	1.21610e24	0.478633	0.239316	+	0.970942i	$0.423077\pi$
$212$	0	0
$213$	−4.28680e21	−0.00152808
$214$	0	0
$215$	−7.47357e23	−0.241504
$216$	0	0
$217$	2.34434e24	0.687375
$218$	0	0
$219$	−1.75831e24	−0.468200
$220$	0	0
$221$	1.48320e24	0.358989
$222$	0	0
$223$	5.72126e24	1.25977	0.629884	−	0.776689i	$-0.283102\pi$
$223$	5.72126e24	1.25977	0.629884	+	0.776689i	$0.283102\pi$
$224$	0	0
$225$	−1.38591e24	−0.277855
$226$	0	0
$227$	−5.60839e24	−1.02463	−0.512314	−	0.858798i	$-0.671212\pi$
$227$	−5.60839e24	−1.02463	−0.512314	+	0.858798i	$0.671212\pi$
$228$	0	0
$229$	−2.51872e24	−0.419670	−0.209835	−	0.977737i	$-0.567293\pi$
$229$	−2.51872e24	−0.419670	−0.209835	+	0.977737i	$0.567293\pi$
$230$	0	0
$231$	−4.94040e24	−0.751340
$232$	0	0
$233$	1.98939e24	0.276365	0.138182	−	0.990407i	$-0.455874\pi$
$233$	1.98939e24	0.276365	0.138182	+	0.990407i	$0.455874\pi$
$234$	0	0
$235$	−4.75791e23	−0.0604232
$236$	0	0
$237$	3.41772e24	0.397081
$238$	0	0
$239$	−7.32699e24	−0.779378	−0.389689	−	0.920947i	$-0.627417\pi$
$239$	−7.32699e24	−0.779378	−0.389689	+	0.920947i	$0.627417\pi$
$240$	0	0
$241$	−1.88926e25	−1.84125	−0.920625	−	0.390448i	$-0.872320\pi$
$241$	−1.88926e25	−1.84125	−0.920625	+	0.390448i	$0.872320\pi$
$242$	0	0
$243$	7.17898e23	0.0641500
$244$	0	0
$245$	1.89761e24	0.155584
$246$	0	0
$247$	7.49537e23	0.0564255
$248$	0	0
$249$	1.46610e25	1.01408
$250$	0	0
$251$	1.65459e25	1.05224	0.526122	−	0.850409i	$-0.323645\pi$
$251$	1.65459e25	1.05224	0.526122	+	0.850409i	$0.323645\pi$
$252$	0	0
$253$	−4.63068e25	−2.70944
$254$	0	0
$255$	−5.23514e24	−0.282005
$256$	0	0
$257$	−2.93804e25	−1.45801	−0.729004	−	0.684509i	$-0.760017\pi$
$257$	−2.93804e25	−1.45801	−0.729004	+	0.684509i	$0.760017\pi$
$258$	0	0
$259$	1.97308e25	0.902604
$260$	0	0
$261$	3.73219e24	0.157485
$262$	0	0
$263$	−3.29044e25	−1.28150	−0.640749	−	0.767750i	$-0.721376\pi$
$263$	−3.29044e25	−1.28150	−0.640749	+	0.767750i	$0.721376\pi$
$264$	0	0
$265$	−1.01739e25	−0.365938
$266$	0	0
$267$	−3.28780e24	−0.109279
$268$	0	0
$269$	−4.30191e24	−0.132209	−0.0661047	−	0.997813i	$-0.521057\pi$
$269$	−4.30191e24	−0.132209	−0.0661047	+	0.997813i	$0.521057\pi$
$270$	0	0
$271$	2.72610e25	0.775110	0.387555	−	0.921846i	$-0.373320\pi$
$271$	2.72610e25	0.775110	0.387555	+	0.921846i	$0.373320\pi$
$272$	0	0
$273$	−5.17307e24	−0.136157
$274$	0	0
$275$	−5.65736e25	−1.37918
$276$	0	0
$277$	−4.61030e25	−1.04158	−0.520788	−	0.853686i	$-0.674362\pi$
$277$	−4.61030e25	−1.04158	−0.520788	+	0.853686i	$0.674362\pi$
$278$	0	0
$279$	−1.39059e25	−0.291311
$280$	0	0
$281$	−7.74125e24	−0.150451	−0.0752254	−	0.997167i	$-0.523968\pi$
$281$	−7.74125e24	−0.150451	−0.0752254	+	0.997167i	$0.523968\pi$
$282$	0	0
$283$	1.41856e25	0.255911	0.127955	−	0.991780i	$-0.459159\pi$
$283$	1.41856e25	0.255911	0.127955	+	0.991780i	$0.459159\pi$
$284$	0	0
$285$	−2.64559e24	−0.0443252
$286$	0	0
$287$	−2.67285e25	−0.416116
$288$	0	0
$289$	2.99502e25	0.433483
$290$	0	0
$291$	−4.95751e25	−0.667401
$292$	0	0
$293$	8.61121e25	1.07883	0.539416	−	0.842039i	$-0.318645\pi$
$293$	8.61121e25	1.07883	0.539416	+	0.842039i	$0.318645\pi$
$294$	0	0
$295$	−5.32282e25	−0.620884
$296$	0	0
$297$	2.93050e25	0.318419
$298$	0	0
$299$	−4.84876e25	−0.491002
$300$	0	0
$301$	−4.93138e25	−0.465608
$302$	0	0
$303$	−4.28260e25	−0.377190
$304$	0	0
$305$	−3.67288e24	−0.0301897
$306$	0	0
$307$	9.91709e25	0.761082	0.380541	−	0.924764i	$-0.375738\pi$
$307$	9.91709e25	0.761082	0.380541	+	0.924764i	$0.375738\pi$
$308$	0	0
$309$	1.05989e26	0.759797
$310$	0	0
$311$	1.15943e26	0.776709	0.388355	−	0.921510i	$-0.373044\pi$
$311$	1.15943e26	0.776709	0.388355	+	0.921510i	$0.373044\pi$
$312$	0	0
$313$	2.06648e26	1.29424	0.647122	−	0.762386i	$-0.275972\pi$
$313$	2.06648e26	1.29424	0.647122	+	0.762386i	$0.275972\pi$
$314$	0	0
$315$	1.82590e25	0.106959
$316$	0	0
$317$	2.90072e26	1.58995	0.794976	−	0.606641i	$-0.207483\pi$
$317$	2.90072e26	1.58995	0.794976	+	0.606641i	$0.207483\pi$
$318$	0	0
$319$	1.52350e26	0.781699
$320$	0	0
$321$	9.40241e24	0.0451790
$322$	0	0
$323$	5.00511e25	0.225314
$324$	0	0
$325$	−5.92379e25	−0.249934
$326$	0	0
$327$	1.75568e26	0.694533
$328$	0	0
$329$	−3.13947e25	−0.116493
$330$	0	0
$331$	−2.56275e26	−0.892303	−0.446151	−	0.894958i	$-0.647206\pi$
$331$	−2.56275e26	−0.892303	−0.446151	+	0.894958i	$0.647206\pi$
$332$	0	0
$333$	−1.17037e26	−0.382525
$334$	0	0
$335$	−2.15810e26	−0.662370
$336$	0	0
$337$	1.42732e26	0.411535	0.205768	−	0.978601i	$-0.434031\pi$
$337$	1.42732e26	0.411535	0.205768	+	0.978601i	$0.434031\pi$
$338$	0	0
$339$	8.15325e25	0.220919
$340$	0	0
$341$	−5.67648e26	−1.44597
$342$	0	0
$343$	4.53538e26	1.08649
$344$	0	0
$345$	1.71143e26	0.385708
$346$	0	0
$347$	4.74021e26	1.00540	0.502699	−	0.864462i	$-0.332341\pi$
$347$	4.74021e26	1.00540	0.502699	+	0.864462i	$0.332341\pi$
$348$	0	0
$349$	8.88859e26	1.77487	0.887434	−	0.460935i	$-0.152486\pi$
$349$	8.88859e26	1.77487	0.887434	+	0.460935i	$0.152486\pi$
$350$	0	0
$351$	3.06851e25	0.0577035
$352$	0	0
$353$	−4.86433e26	−0.861764	−0.430882	−	0.902408i	$-0.641798\pi$
$353$	−4.86433e26	−0.861764	−0.430882	+	0.902408i	$0.641798\pi$
$354$	0	0
$355$	6.46734e23	0.00107976
$356$	0	0
$357$	−3.45437e26	−0.543691
$358$	0	0
$359$	4.36541e26	0.647938	0.323969	−	0.946068i	$-0.394983\pi$
$359$	4.36541e26	0.647938	0.323969	+	0.946068i	$0.394983\pi$
$360$	0	0
$361$	−6.88916e26	−0.964585
$362$	0	0
$363$	7.59267e26	1.00317
$364$	0	0
$365$	2.65270e26	0.330836
$366$	0	0
$367$	1.13289e27	1.33412	0.667060	−	0.745004i	$-0.267553\pi$
$367$	1.13289e27	1.33412	0.667060	+	0.745004i	$0.267553\pi$
$368$	0	0
$369$	1.58545e26	0.176350
$370$	0	0
$371$	−6.71319e26	−0.705510
$372$	0	0
$373$	6.51576e26	0.647176	0.323588	−	0.946198i	$-0.395111\pi$
$373$	6.51576e26	0.647176	0.323588	+	0.946198i	$0.395111\pi$
$374$	0	0
$375$	4.59923e26	0.431874
$376$	0	0
$377$	1.59525e26	0.141659
$378$	0	0
$379$	−1.71451e27	−1.44022	−0.720109	−	0.693861i	$-0.755908\pi$
$379$	−1.71451e27	−1.44022	−0.720109	+	0.693861i	$0.755908\pi$
$380$	0	0
$381$	−2.69778e26	−0.214434
$382$	0	0
$383$	1.63419e27	1.22946	0.614731	−	0.788737i	$-0.289265\pi$
$383$	1.63419e27	1.22946	0.614731	+	0.788737i	$0.289265\pi$
$384$	0	0
$385$	7.45341e26	0.530906
$386$	0	0
$387$	2.92515e26	0.197325
$388$	0	0
$389$	−2.64030e27	−1.68726	−0.843632	−	0.536922i	$-0.819587\pi$
$389$	−2.64030e27	−1.68726	−0.843632	+	0.536922i	$0.819587\pi$
$390$	0	0
$391$	−3.23781e27	−1.96063
$392$	0	0
$393$	5.10191e26	0.292827
$394$	0	0
$395$	−5.15620e26	−0.280582
$396$	0	0
$397$	2.94709e27	1.52087	0.760437	−	0.649411i	$-0.224985\pi$
$397$	2.94709e27	1.52087	0.760437	+	0.649411i	$0.224985\pi$
$398$	0	0
$399$	−1.74567e26	−0.0854568
$400$	0	0
$401$	1.81255e27	0.841927	0.420963	−	0.907078i	$-0.361692\pi$
$401$	1.81255e27	0.841927	0.420963	+	0.907078i	$0.361692\pi$
$402$	0	0
$403$	−5.94381e26	−0.262037
$404$	0	0
$405$	−1.08307e26	−0.0453292
$406$	0	0
$407$	−4.77752e27	−1.89872
$408$	0	0
$409$	2.59753e27	0.980542	0.490271	−	0.871570i	$-0.336898\pi$
$409$	2.59753e27	0.980542	0.490271	+	0.871570i	$0.336898\pi$
$410$	0	0
$411$	1.58415e27	0.568143
$412$	0	0
$413$	−3.51222e27	−1.19703
$414$	0	0
$415$	−2.21186e27	−0.716560
$416$	0	0
$417$	−2.30600e27	−0.710283
$418$	0	0
$419$	3.40142e26	0.0996353	0.0498176	−	0.998758i	$-0.484136\pi$
$419$	3.40142e26	0.0996353	0.0498176	+	0.998758i	$0.484136\pi$
$420$	0	0
$421$	−4.22891e27	−1.17833	−0.589164	−	0.808014i	$-0.700543\pi$
$421$	−4.22891e27	−1.17833	−0.589164	+	0.808014i	$0.700543\pi$
$422$	0	0
$423$	1.86224e26	0.0493698
$424$	0	0
$425$	−3.95567e27	−0.998014
$426$	0	0
$427$	−2.42353e26	−0.0582043
$428$	0	0
$429$	1.25258e27	0.286421
$430$	0	0
$431$	−7.72212e27	−1.68161	−0.840805	−	0.541339i	$-0.817918\pi$
$431$	−7.72212e27	−1.68161	−0.840805	+	0.541339i	$0.817918\pi$
$432$	0	0
$433$	3.14232e27	0.651819	0.325910	−	0.945401i	$-0.394330\pi$
$433$	3.14232e27	0.651819	0.325910	+	0.945401i	$0.394330\pi$
$434$	0	0
$435$	−5.63062e26	−0.111281
$436$	0	0
$437$	−1.63623e27	−0.308170
$438$	0	0
$439$	3.34873e27	0.601177	0.300588	−	0.953754i	$-0.402817\pi$
$439$	3.34873e27	0.601177	0.300588	+	0.953754i	$0.402817\pi$
$440$	0	0
$441$	−7.42724e26	−0.127122
$442$	0	0
$443$	−5.82708e27	−0.951068	−0.475534	−	0.879697i	$-0.657745\pi$
$443$	−5.82708e27	−0.951068	−0.475534	+	0.879697i	$0.657745\pi$
$444$	0	0
$445$	4.96019e26	0.0772179
$446$	0	0
$447$	2.28398e27	0.339205
$448$	0	0
$449$	−1.14216e28	−1.61860	−0.809300	−	0.587396i	$-0.800153\pi$
$449$	−1.14216e28	−1.61860	−0.809300	+	0.587396i	$0.800153\pi$
$450$	0	0
$451$	6.47190e27	0.875343
$452$	0	0
$453$	6.35219e27	0.820148
$454$	0	0
$455$	7.80443e26	0.0962103
$456$	0	0
$457$	1.70411e27	0.200622	0.100311	−	0.994956i	$-0.468016\pi$
$457$	1.70411e27	0.200622	0.100311	+	0.994956i	$0.468016\pi$
$458$	0	0
$459$	2.04903e27	0.230417
$460$	0	0
$461$	−1.57190e28	−1.68875	−0.844374	−	0.535753i	$-0.820028\pi$
$461$	−1.57190e28	−1.68875	−0.844374	+	0.535753i	$0.820028\pi$
$462$	0	0
$463$	2.45032e27	0.251549	0.125774	−	0.992059i	$-0.459858\pi$
$463$	2.45032e27	0.251549	0.125774	+	0.992059i	$0.459858\pi$
$464$	0	0
$465$	2.09794e27	0.205844
$466$	0	0
$467$	−4.79077e26	−0.0449344	−0.0224672	−	0.999748i	$-0.507152\pi$
$467$	−4.79077e26	−0.0449344	−0.0224672	+	0.999748i	$0.507152\pi$
$468$	0	0
$469$	−1.42400e28	−1.27702
$470$	0	0
$471$	4.08302e27	0.350156
$472$	0	0
$473$	1.19406e28	0.979455
$474$	0	0
$475$	−1.99901e27	−0.156867
$476$	0	0
$477$	3.98207e27	0.298996
$478$	0	0
$479$	5.29156e27	0.380243	0.190121	−	0.981761i	$-0.439112\pi$
$479$	5.29156e27	0.380243	0.190121	+	0.981761i	$0.439112\pi$
$480$	0	0
$481$	−5.00252e27	−0.344085
$482$	0	0
$483$	1.12927e28	0.743626
$484$	0	0
$485$	7.47922e27	0.471594
$486$	0	0
$487$	2.99405e28	1.80803	0.904013	−	0.427506i	$-0.140608\pi$
$487$	2.99405e28	1.80803	0.904013	+	0.427506i	$0.140608\pi$
$488$	0	0
$489$	1.59895e28	0.924893
$490$	0	0
$491$	−1.72464e27	−0.0955749	−0.0477875	−	0.998858i	$-0.515217\pi$
$491$	−1.72464e27	−0.0955749	−0.0477875	+	0.998858i	$0.515217\pi$
$492$	0	0
$493$	1.06524e28	0.565660
$494$	0	0
$495$	−4.42114e27	−0.224999
$496$	0	0
$497$	4.26743e25	0.00208173
$498$	0	0
$499$	1.82237e28	0.852276	0.426138	−	0.904658i	$-0.359874\pi$
$499$	1.82237e28	0.852276	0.426138	+	0.904658i	$0.359874\pi$
$500$	0	0
$501$	1.36235e28	0.610930
$502$	0	0
$503$	−3.98628e28	−1.71437	−0.857183	−	0.515011i	$-0.827788\pi$
$503$	−3.98628e28	−1.71437	−0.857183	+	0.515011i	$0.827788\pi$
$504$	0	0
$505$	6.46101e27	0.266527
$506$	0	0
$507$	−1.32773e28	−0.525445
$508$	0	0
$509$	−1.48470e28	−0.563771	−0.281886	−	0.959448i	$-0.590960\pi$
$509$	−1.48470e28	−0.563771	−0.281886	+	0.959448i	$0.590960\pi$
$510$	0	0
$511$	1.75036e28	0.637836
$512$	0	0
$513$	1.03548e27	0.0362167
$514$	0	0
$515$	−1.59902e28	−0.536882
$516$	0	0
$517$	7.60176e27	0.245055
$518$	0	0
$519$	−2.17532e28	−0.673390
$520$	0	0
$521$	4.40264e28	1.30893	0.654465	−	0.756092i	$-0.272894\pi$
$521$	4.40264e28	1.30893	0.654465	+	0.756092i	$0.272894\pi$
$522$	0	0
$523$	3.71320e28	1.06043	0.530214	−	0.847864i	$-0.322112\pi$
$523$	3.71320e28	1.06043	0.530214	+	0.847864i	$0.322112\pi$
$524$	0	0
$525$	1.37965e28	0.378526
$526$	0	0
$527$	−3.96904e28	−1.04634
$528$	0	0
$529$	6.63763e28	1.68162
$530$	0	0
$531$	2.08334e28	0.507304
$532$	0	0
$533$	6.77669e27	0.158629
$534$	0	0
$535$	−1.41851e27	−0.0319240
$536$	0	0
$537$	4.00652e28	0.867037
$538$	0	0
$539$	−3.03184e28	−0.630992
$540$	0	0
$541$	−4.12076e28	−0.824909	−0.412454	−	0.910978i	$-0.635328\pi$
$541$	−4.12076e28	−0.824909	−0.412454	+	0.910978i	$0.635328\pi$
$542$	0	0
$543$	5.55786e28	1.07031
$544$	0	0
$545$	−2.64873e28	−0.490766
$546$	0	0
$547$	2.13981e28	0.381512	0.190756	−	0.981637i	$-0.438906\pi$
$547$	2.13981e28	0.381512	0.190756	+	0.981637i	$0.438906\pi$
$548$	0	0
$549$	1.43756e27	0.0246670
$550$	0	0
$551$	5.38322e27	0.0889099
$552$	0	0
$553$	−3.40228e28	−0.540948
$554$	0	0
$555$	1.76570e28	0.270297
$556$	0	0
$557$	−9.11122e28	−1.34307	−0.671533	−	0.740975i	$-0.734364\pi$
$557$	−9.11122e28	−1.34307	−0.671533	+	0.740975i	$0.734364\pi$
$558$	0	0
$559$	1.25029e28	0.177496
$560$	0	0
$561$	8.36423e28	1.14371
$562$	0	0
$563$	−7.72184e28	−1.01714	−0.508572	−	0.861019i	$-0.669826\pi$
$563$	−7.72184e28	−1.01714	−0.508572	+	0.861019i	$0.669826\pi$
$564$	0	0
$565$	−1.23005e28	−0.156104
$566$	0	0
$567$	−7.14654e27	−0.0873925
$568$	0	0
$569$	−8.18518e28	−0.964605	−0.482303	−	0.876005i	$-0.660199\pi$
$569$	−8.18518e28	−0.964605	−0.482303	+	0.876005i	$0.660199\pi$
$570$	0	0
$571$	−3.83570e27	−0.0435678	−0.0217839	−	0.999763i	$-0.506935\pi$
$571$	−3.83570e27	−0.0435678	−0.0217839	+	0.999763i	$0.506935\pi$
$572$	0	0
$573$	−5.92586e28	−0.648825
$574$	0	0
$575$	1.29316e29	1.36502
$576$	0	0
$577$	1.15551e29	1.17606	0.588028	−	0.808841i	$-0.299905\pi$
$577$	1.15551e29	1.17606	0.588028	+	0.808841i	$0.299905\pi$
$578$	0	0
$579$	−8.31655e28	−0.816241
$580$	0	0
$581$	−1.45948e29	−1.38149
$582$	0	0
$583$	1.62550e29	1.48411
$584$	0	0
$585$	−4.62935e27	−0.0407740
$586$	0	0
$587$	−1.04413e29	−0.887271	−0.443636	−	0.896207i	$-0.646312\pi$
$587$	−1.04413e29	−0.887271	−0.443636	+	0.896207i	$0.646312\pi$
$588$	0	0
$589$	−2.00576e28	−0.164463
$590$	0	0
$591$	1.16318e29	0.920407
$592$	0	0
$593$	1.30270e29	0.994879	0.497439	−	0.867499i	$-0.334274\pi$
$593$	1.30270e29	0.994879	0.497439	+	0.867499i	$0.334274\pi$
$594$	0	0
$595$	5.21148e28	0.384179
$596$	0	0
$597$	−7.88255e28	−0.560965
$598$	0	0
$599$	−1.35818e29	−0.933205	−0.466603	−	0.884467i	$-0.654522\pi$
$599$	−1.35818e29	−0.933205	−0.466603	+	0.884467i	$0.654522\pi$
$600$	0	0
$601$	−5.67794e28	−0.376712	−0.188356	−	0.982101i	$-0.560316\pi$
$601$	−5.67794e28	−0.376712	−0.188356	+	0.982101i	$0.560316\pi$
$602$	0	0
$603$	8.44677e28	0.541201
$604$	0	0
$605$	−1.14548e29	−0.708853
$606$	0	0
$607$	−2.93166e29	−1.75239	−0.876197	−	0.481953i	$-0.839927\pi$
$607$	−2.93166e29	−1.75239	−0.876197	+	0.481953i	$0.839927\pi$
$608$	0	0
$609$	−3.71532e28	−0.214543
$610$	0	0
$611$	7.95976e27	0.0444086
$612$	0	0
$613$	−1.03374e29	−0.557282	−0.278641	−	0.960395i	$-0.589884\pi$
$613$	−1.03374e29	−0.557282	−0.278641	+	0.960395i	$0.589884\pi$
$614$	0	0
$615$	−2.39192e28	−0.124611
$616$	0	0
$617$	−2.61665e29	−1.31750	−0.658750	−	0.752362i	$-0.728915\pi$
$617$	−2.61665e29	−1.31750	−0.658750	+	0.752362i	$0.728915\pi$
$618$	0	0
$619$	−2.52208e29	−1.22746	−0.613729	−	0.789517i	$-0.710331\pi$
$619$	−2.52208e29	−1.22746	−0.613729	+	0.789517i	$0.710331\pi$
$620$	0	0
$621$	−6.69852e28	−0.315150
$622$	0	0
$623$	3.27294e28	0.148872
$624$	0	0
$625$	1.20144e29	0.528399
$626$	0	0
$627$	4.22688e28	0.179767
$628$	0	0
$629$	−3.34048e29	−1.37397
$630$	0	0
$631$	−4.18556e29	−1.66512	−0.832559	−	0.553936i	$-0.813125\pi$
$631$	−4.18556e29	−1.66512	−0.832559	+	0.553936i	$0.813125\pi$
$632$	0	0
$633$	7.18093e28	0.276339
$634$	0	0
$635$	4.07005e28	0.151522
$636$	0	0
$637$	−3.17462e28	−0.114348
$638$	0	0
$639$	−2.53131e26	−0.000882239 0
$640$	0	0
$641$	4.16106e29	1.40344	0.701721	−	0.712451i	$-0.252415\pi$
$641$	4.16106e29	1.40344	0.701721	+	0.712451i	$0.252415\pi$
$642$	0	0
$643$	−1.16723e29	−0.381015	−0.190508	−	0.981686i	$-0.561013\pi$
$643$	−1.16723e29	−0.381015	−0.190508	+	0.981686i	$0.561013\pi$
$644$	0	0
$645$	−4.41307e28	−0.139432
$646$	0	0
$647$	−2.12199e29	−0.649007	−0.324503	−	0.945885i	$-0.605197\pi$
$647$	−2.12199e29	−0.649007	−0.324503	+	0.945885i	$0.605197\pi$
$648$	0	0
$649$	8.50432e29	2.51808
$650$	0	0
$651$	1.38431e29	0.396856
$652$	0	0
$653$	2.19143e29	0.608329	0.304165	−	0.952619i	$-0.401623\pi$
$653$	2.19143e29	0.608329	0.304165	+	0.952619i	$0.401623\pi$
$654$	0	0
$655$	−7.69707e28	−0.206915
$656$	0	0
$657$	−1.03826e29	−0.270316
$658$	0	0
$659$	7.81698e28	0.197125	0.0985627	−	0.995131i	$-0.468576\pi$
$659$	7.81698e28	0.197125	0.0985627	+	0.995131i	$0.468576\pi$
$660$	0	0
$661$	−4.98165e29	−1.21691	−0.608454	−	0.793589i	$-0.708210\pi$
$661$	−4.98165e29	−1.21691	−0.608454	+	0.793589i	$0.708210\pi$
$662$	0	0
$663$	8.75814e28	0.207262
$664$	0	0
$665$	2.63363e28	0.0603849
$666$	0	0
$667$	−3.48241e29	−0.773674
$668$	0	0
$669$	3.37835e29	0.727328
$670$	0	0
$671$	5.86820e28	0.122439
$672$	0	0
$673$	7.45598e29	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$673$	7.45598e29	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$674$	0	0
$675$	−8.18367e28	−0.160420
$676$	0	0
$677$	1.07199e29	0.203708	0.101854	−	0.994799i	$-0.467523\pi$
$677$	1.07199e29	0.203708	0.101854	+	0.994799i	$0.467523\pi$
$678$	0	0
$679$	4.93511e29	0.909210
$680$	0	0
$681$	−3.31170e29	−0.591569
$682$	0	0
$683$	6.20112e29	1.07412	0.537060	−	0.843544i	$-0.319535\pi$
$683$	6.20112e29	1.07412	0.537060	+	0.843544i	$0.319535\pi$
$684$	0	0
$685$	−2.38995e29	−0.401457
$686$	0	0
$687$	−1.48728e29	−0.242297
$688$	0	0
$689$	1.70205e29	0.268950
$690$	0	0
$691$	1.11755e30	1.71296	0.856482	−	0.516177i	$-0.172645\pi$
$691$	1.11755e30	1.71296	0.856482	+	0.516177i	$0.172645\pi$
$692$	0	0
$693$	−2.91726e29	−0.433786
$694$	0	0
$695$	3.47899e29	0.501895
$696$	0	0
$697$	4.52520e29	0.633423
$698$	0	0
$699$	1.17471e29	0.159559
$700$	0	0
$701$	−9.02930e29	−1.19019	−0.595093	−	0.803657i	$-0.702885\pi$
$701$	−9.02930e29	−1.19019	−0.595093	+	0.803657i	$0.702885\pi$
$702$	0	0
$703$	−1.68812e29	−0.215959
$704$	0	0
$705$	−2.80950e28	−0.0348853
$706$	0	0
$707$	4.26325e29	0.513851
$708$	0	0
$709$	1.25054e29	0.146322	0.0731612	−	0.997320i	$-0.476691\pi$
$709$	1.25054e29	0.146322	0.0731612	+	0.997320i	$0.476691\pi$
$710$	0	0
$711$	2.01813e29	0.229255
$712$	0	0
$713$	1.29753e30	1.43112
$714$	0	0
$715$	−1.88973e29	−0.202388
$716$	0	0
$717$	−4.32652e29	−0.449974
$718$	0	0
$719$	7.25984e29	0.733287	0.366643	−	0.930362i	$-0.380507\pi$
$719$	7.25984e29	0.733287	0.366643	+	0.930362i	$0.380507\pi$
$720$	0	0
$721$	−1.05510e30	−1.03508
$722$	0	0
$723$	−1.11559e30	−1.06305
$724$	0	0
$725$	−4.25450e29	−0.393822
$726$	0	0
$727$	3.73992e29	0.336318	0.168159	−	0.985760i	$-0.446218\pi$
$727$	3.73992e29	0.336318	0.168159	+	0.985760i	$0.446218\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	8.34897e29	0.708762
$732$	0	0
$733$	−1.77085e30	−1.46080	−0.730401	−	0.683019i	$-0.760667\pi$
$733$	−1.77085e30	−1.46080	−0.730401	+	0.683019i	$0.760667\pi$
$734$	0	0
$735$	1.12052e29	0.0898263
$736$	0	0
$737$	3.44802e30	2.68634
$738$	0	0
$739$	1.88567e30	1.42791	0.713953	−	0.700194i	$-0.246903\pi$
$739$	1.88567e30	1.42791	0.713953	+	0.700194i	$0.246903\pi$
$740$	0	0
$741$	4.42594e28	0.0325773
$742$	0	0
$743$	8.39029e29	0.600336	0.300168	−	0.953886i	$-0.402957\pi$
$743$	8.39029e29	0.600336	0.300168	+	0.953886i	$0.402957\pi$
$744$	0	0
$745$	−3.44576e29	−0.239687
$746$	0	0
$747$	8.65719e29	0.585478
$748$	0	0
$749$	−9.35992e28	−0.0615479
$750$	0	0
$751$	−1.32480e30	−0.847095	−0.423547	−	0.905874i	$-0.639215\pi$
$751$	−1.32480e30	−0.847095	−0.423547	+	0.905874i	$0.639215\pi$
$752$	0	0
$753$	9.77020e29	0.607514
$754$	0	0
$755$	−9.58333e29	−0.579527
$756$	0	0
$757$	−8.91190e29	−0.524159	−0.262080	−	0.965046i	$-0.584408\pi$
$757$	−8.91190e29	−0.524159	−0.262080	+	0.965046i	$0.584408\pi$
$758$	0	0
$759$	−2.73437e30	−1.56429
$760$	0	0
$761$	2.22977e30	1.24086	0.620429	−	0.784263i	$-0.286959\pi$
$761$	2.22977e30	1.24086	0.620429	+	0.784263i	$0.286959\pi$
$762$	0	0
$763$	−1.74774e30	−0.946172
$764$	0	0
$765$	−3.09130e29	−0.162815
$766$	0	0
$767$	8.90482e29	0.456325
$768$	0	0
$769$	−1.65220e30	−0.823827	−0.411913	−	0.911223i	$-0.635139\pi$
$769$	−1.65220e30	−0.823827	−0.411913	+	0.911223i	$0.635139\pi$
$770$	0	0
$771$	−1.73488e30	−0.841782
$772$	0	0
$773$	−7.95379e29	−0.375569	−0.187784	−	0.982210i	$-0.560131\pi$
$773$	−7.95379e29	−0.375569	−0.187784	+	0.982210i	$0.560131\pi$
$774$	0	0
$775$	1.58521e30	0.728480
$776$	0	0
$777$	1.16508e30	0.521119
$778$	0	0
$779$	2.28682e29	0.0995609
$780$	0	0
$781$	−1.03329e28	−0.00437913
$782$	0	0
$783$	2.20382e29	0.0909238
$784$	0	0
$785$	−6.15992e29	−0.247425
$786$	0	0
$787$	−2.00720e30	−0.784977	−0.392489	−	0.919757i	$-0.628386\pi$
$787$	−2.00720e30	−0.784977	−0.392489	+	0.919757i	$0.628386\pi$
$788$	0	0
$789$	−1.94297e30	−0.739873
$790$	0	0
$791$	−8.11641e29	−0.300962
$792$	0	0
$793$	6.14457e28	0.0221882
$794$	0	0
$795$	−6.00761e29	−0.211274
$796$	0	0
$797$	3.69070e30	1.26414	0.632071	−	0.774910i	$-0.282205\pi$
$797$	3.69070e30	1.26414	0.632071	+	0.774910i	$0.282205\pi$
$798$	0	0
$799$	5.31521e29	0.177329
$800$	0	0
$801$	−1.94141e29	−0.0630923
$802$	0	0
$803$	−4.23824e30	−1.34175
$804$	0	0
$805$	−1.70370e30	−0.525455
$806$	0	0
$807$	−2.54024e29	−0.0763311
$808$	0	0
$809$	4.72614e30	1.38372	0.691859	−	0.722033i	$-0.256792\pi$
$809$	4.72614e30	1.38372	0.691859	+	0.722033i	$0.256792\pi$
$810$	0	0
$811$	−2.65583e30	−0.757673	−0.378837	−	0.925464i	$-0.623676\pi$
$811$	−2.65583e30	−0.757673	−0.378837	+	0.925464i	$0.623676\pi$
$812$	0	0
$813$	1.60973e30	0.447510
$814$	0	0
$815$	−2.41228e30	−0.653540
$816$	0	0
$817$	4.21917e29	0.111403
$818$	0	0
$819$	−3.05464e29	−0.0786103
$820$	0	0
$821$	−5.10610e30	−1.28082	−0.640408	−	0.768035i	$-0.721235\pi$
$821$	−5.10610e30	−1.28082	−0.640408	+	0.768035i	$0.721235\pi$
$822$	0	0
$823$	4.35473e30	1.06479	0.532394	−	0.846497i	$-0.321293\pi$
$823$	4.35473e30	1.06479	0.532394	+	0.846497i	$0.321293\pi$
$824$	0	0
$825$	−3.34061e30	−0.796269
$826$	0	0
$827$	4.07345e30	0.946573	0.473287	−	0.880909i	$-0.343068\pi$
$827$	4.07345e30	0.946573	0.473287	+	0.880909i	$0.343068\pi$
$828$	0	0
$829$	−2.77876e30	−0.629548	−0.314774	−	0.949167i	$-0.601929\pi$
$829$	−2.77876e30	−0.629548	−0.314774	+	0.949167i	$0.601929\pi$
$830$	0	0
$831$	−2.72233e30	−0.601354
$832$	0	0
$833$	−2.11988e30	−0.456604
$834$	0	0
$835$	−2.05533e30	−0.431691
$836$	0	0
$837$	−8.21132e29	−0.168188
$838$	0	0
$839$	−3.15950e30	−0.631128	−0.315564	−	0.948904i	$-0.602194\pi$
$839$	−3.15950e30	−0.631128	−0.315564	+	0.948904i	$0.602194\pi$
$840$	0	0
$841$	−3.98713e30	−0.776787
$842$	0	0
$843$	−4.57113e29	−0.0868628
$844$	0	0
$845$	2.00311e30	0.371286
$846$	0	0
$847$	−7.55836e30	−1.36663
$848$	0	0
$849$	8.37643e29	0.147750
$850$	0	0
$851$	1.09204e31	1.87923
$852$	0	0
$853$	−1.12969e31	−1.89668	−0.948338	−	0.317260i	$-0.897237\pi$
$853$	−1.12969e31	−1.89668	−0.948338	+	0.317260i	$0.897237\pi$
$854$	0	0
$855$	−1.56219e29	−0.0255912
$856$	0	0
$857$	−1.09526e31	−1.75073	−0.875366	−	0.483462i	$-0.839379\pi$
$857$	−1.09526e31	−1.75073	−0.875366	+	0.483462i	$0.839379\pi$
$858$	0	0
$859$	8.09790e29	0.126312	0.0631559	−	0.998004i	$-0.479883\pi$
$859$	8.09790e29	0.126312	0.0631559	+	0.998004i	$0.479883\pi$
$860$	0	0
$861$	−1.57829e30	−0.240245
$862$	0	0
$863$	9.41039e30	1.39796	0.698979	−	0.715142i	$-0.253638\pi$
$863$	9.41039e30	1.39796	0.698979	+	0.715142i	$0.253638\pi$
$864$	0	0
$865$	3.28184e30	0.475825
$866$	0	0
$867$	1.76853e30	0.250272
$868$	0	0
$869$	8.23811e30	1.13794
$870$	0	0
$871$	3.61040e30	0.486816
$872$	0	0
$873$	−2.92736e30	−0.385324
$874$	0	0
$875$	−4.57845e30	−0.588347
$876$	0	0
$877$	9.07878e30	1.13902	0.569510	−	0.821984i	$-0.307133\pi$
$877$	9.07878e30	1.13902	0.569510	+	0.821984i	$0.307133\pi$
$878$	0	0
$879$	5.08483e30	0.622864
$880$	0	0
$881$	3.03572e30	0.363090	0.181545	−	0.983383i	$-0.441890\pi$
$881$	3.03572e30	0.363090	0.181545	+	0.983383i	$0.441890\pi$
$882$	0	0
$883$	−6.61886e28	−0.00773030	−0.00386515	−	0.999993i	$-0.501230\pi$
$883$	−6.61886e28	−0.00773030	−0.00386515	+	0.999993i	$0.501230\pi$
$884$	0	0
$885$	−3.14307e30	−0.358467
$886$	0	0
$887$	3.50101e30	0.389937	0.194969	−	0.980809i	$-0.437539\pi$
$887$	3.50101e30	0.389937	0.194969	+	0.980809i	$0.437539\pi$
$888$	0	0
$889$	2.68559e30	0.292127
$890$	0	0
$891$	1.73043e30	0.183839
$892$	0	0
$893$	2.68605e29	0.0278724
$894$	0	0
$895$	−6.04451e30	−0.612659
$896$	0	0
$897$	−2.86314e30	−0.283480
$898$	0	0
$899$	−4.26888e30	−0.412892
$900$	0	0
$901$	1.13656e31	1.07395
$902$	0	0
$903$	−2.91193e30	−0.268819
$904$	0	0
$905$	−8.38495e30	−0.756293
$906$	0	0
$907$	−5.24604e29	−0.0462333	−0.0231167	−	0.999733i	$-0.507359\pi$
$907$	−5.24604e29	−0.0462333	−0.0231167	+	0.999733i	$0.507359\pi$
$908$	0	0
$909$	−2.52883e30	−0.217771
$910$	0	0
$911$	1.16565e31	0.980902	0.490451	−	0.871469i	$-0.336832\pi$
$911$	1.16565e31	0.980902	0.490451	+	0.871469i	$0.336832\pi$
$912$	0	0
$913$	3.53391e31	2.90611
$914$	0	0
$915$	−2.16880e29	−0.0174300
$916$	0	0
$917$	−5.07886e30	−0.398922
$918$	0	0
$919$	7.15232e30	0.549078	0.274539	−	0.961576i	$-0.411475\pi$
$919$	7.15232e30	0.549078	0.274539	+	0.961576i	$0.411475\pi$
$920$	0	0
$921$	5.85594e30	0.439411
$922$	0	0
$923$	−1.08196e28	−0.000793582 0
$924$	0	0
$925$	1.33416e31	0.956579
$926$	0	0
$927$	6.25856e30	0.438669
$928$	0	0
$929$	9.72648e30	0.666486	0.333243	−	0.942841i	$-0.391857\pi$
$929$	9.72648e30	0.666486	0.333243	+	0.942841i	$0.391857\pi$
$930$	0	0
$931$	−1.07129e30	−0.0717686
$932$	0	0
$933$	6.84629e30	0.448433
$934$	0	0
$935$	−1.26188e31	−0.808160
$936$	0	0
$937$	−1.38653e31	−0.868288	−0.434144	−	0.900844i	$-0.642949\pi$
$937$	−1.38653e31	−0.868288	−0.434144	+	0.900844i	$0.642949\pi$
$938$	0	0
$939$	1.22024e31	0.747233
$940$	0	0
$941$	2.03835e31	1.22064	0.610320	−	0.792155i	$-0.291041\pi$
$941$	2.03835e31	1.22064	0.610320	+	0.792155i	$0.291041\pi$
$942$	0	0
$943$	−1.47934e31	−0.866356
$944$	0	0
$945$	1.07817e30	0.0617526
$946$	0	0
$947$	3.05609e30	0.171195	0.0855975	−	0.996330i	$-0.472720\pi$
$947$	3.05609e30	0.171195	0.0855975	+	0.996330i	$0.472720\pi$
$948$	0	0
$949$	−4.43784e30	−0.243151
$950$	0	0
$951$	1.71285e31	0.917960
$952$	0	0
$953$	−1.54521e31	−0.810053	−0.405026	−	0.914305i	$-0.632738\pi$
$953$	−1.54521e31	−0.810053	−0.405026	+	0.914305i	$0.632738\pi$
$954$	0	0
$955$	8.94014e30	0.458468
$956$	0	0
$957$	8.99610e30	0.451314
$958$	0	0
$959$	−1.57699e31	−0.773988
$960$	0	0
$961$	−4.91989e30	−0.236243
$962$	0	0
$963$	5.55203e29	0.0260841
$964$	0	0
$965$	1.25469e31	0.576766
$966$	0	0
$967$	−2.30504e31	−1.03681	−0.518406	−	0.855135i	$-0.673474\pi$
$967$	−2.30504e31	−1.03681	−0.518406	+	0.855135i	$0.673474\pi$
$968$	0	0
$969$	2.95547e30	0.130085
$970$	0	0
$971$	−5.03493e29	−0.0216866	−0.0108433	−	0.999941i	$-0.503452\pi$
$971$	−5.03493e29	−0.0216866	−0.0108433	+	0.999941i	$0.503452\pi$
$972$	0	0
$973$	2.29558e31	0.967628
$974$	0	0
$975$	−3.49794e30	−0.144299
$976$	0	0
$977$	−3.24415e31	−1.30981	−0.654905	−	0.755711i	$-0.727291\pi$
$977$	−3.24415e31	−1.30981	−0.654905	+	0.755711i	$0.727291\pi$
$978$	0	0
$979$	−7.92495e30	−0.313169
$980$	0	0
$981$	1.03671e31	0.400989
$982$	0	0
$983$	−1.25577e31	−0.475444	−0.237722	−	0.971333i	$-0.576401\pi$
$983$	−1.25577e31	−0.475444	−0.237722	+	0.971333i	$0.576401\pi$
$984$	0	0
$985$	−1.75486e31	−0.650371
$986$	0	0
$987$	−1.85383e30	−0.0672572
$988$	0	0
$989$	−2.72938e31	−0.969399
$990$	0	0
$991$	1.56643e31	0.544676	0.272338	−	0.962202i	$-0.412203\pi$
$991$	1.56643e31	0.544676	0.272338	+	0.962202i	$0.412203\pi$
$992$	0	0
$993$	−1.51328e31	−0.515171
$994$	0	0
$995$	1.18921e31	0.396385
$996$	0	0
$997$	4.16202e31	1.35833	0.679165	−	0.733986i	$-0.262342\pi$
$997$	4.16202e31	1.35833	0.679165	+	0.733986i	$0.262342\pi$
$998$	0	0
$999$	−6.91093e30	−0.220851

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.c.1.2	✓	3	4.3	odd	2
48.22.a.l.1.2		3	1.1	even	1	trivial
72.22.a.d.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)

\(f(q)\)	\(=\)	\(q+59049.0 q^{3} -8.90852e6 q^{5} -5.87822e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\)	\(q+59049.0 q^{3} -8.90852e6 q^{5} -5.87822e8 q^{7} +3.48678e9 q^{9} +1.42332e11 q^{11} +1.49035e11 q^{13} -5.26039e11 q^{15} +9.95199e12 q^{17} +5.02926e12 q^{19} -3.47103e13 q^{21} -3.25343e14 q^{23} -3.97475e14 q^{25} +2.05891e14 q^{27} +1.07038e15 q^{29} -3.98818e15 q^{31} +8.40458e15 q^{33} +5.23663e15 q^{35} -3.35659e16 q^{37} +8.80039e15 q^{39} +4.54703e16 q^{41} +8.38924e16 q^{43} -3.10621e16 q^{45} +5.34085e16 q^{47} -2.13011e17 q^{49} +5.87655e17 q^{51} +1.14205e18 q^{53} -1.26797e18 q^{55} +2.96973e17 q^{57} +5.97497e18 q^{59} +4.12289e17 q^{61} -2.04961e18 q^{63} -1.32769e18 q^{65} +2.42251e19 q^{67} -1.92112e19 q^{69} -7.25973e16 q^{71} -2.97771e19 q^{73} -2.34705e19 q^{75} -8.36661e19 q^{77} +5.78794e19 q^{79} +1.21577e19 q^{81} +2.48286e20 q^{83} -8.86575e19 q^{85} +6.32049e19 q^{87} -5.56792e19 q^{89} -8.76063e19 q^{91} -2.35498e20 q^{93} -4.48032e19 q^{95} -8.39559e20 q^{97} +4.96282e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10}) \) 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9	\( 3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9} - 13839247500 q^{11} + 718855551690 q^{13} + 122823455274 q^{15} + 2135189843046 q^{17} + 40122324686988 q^{19} + 71170700597136 q^{21} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 48\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9 - 13839247500 * q^11 + 718855551690 * q^13 + 122823455274 * q^15 + 2135189843046 * q^17 + 40122324686988 * q^19 + 71170700597136 * q^21 - 278424417682632 * q^23 + 1348043260553901 * q^25 + 617673396283947 * q^27 + 442708167991794 * q^29 - 8016070162990152 * q^31 - 817193725627500 * q^33 - 125384157242400 * q^35 + 27729341388737058 * q^37 + 42447701471742810 * q^39 - 125648125186340562 * q^41 + 229052541499074612 * q^43 + 7252602210474426 * q^45 + 448613782068047712 * q^47 + 365221903446092427 * q^49 + 126080825042023254 * q^51 + 1406206217208267066 * q^53 + 3437829264920292504 * q^55 + 2369183150441954412 * q^57 + 1844638981471622100 * q^59 - 3294066300350351382 * q^61 + 4202558699560283664 * q^63 - 19537666262756991444 * q^65 + 33491023693155020652 * q^67 - 16440683439741736968 * q^69 + 79431018431598881160 * q^71 - 46612822906958319618 * q^73 + 79600606492447300149 * q^75 - 255759806110876888128 * q^77 + 197919973704661098024 * q^79 + 36472996377170786403 * q^81 - 111848528551886940276 * q^83 - 531596971435705186956 * q^85 + 26141474611747443906 * q^87 - 731008175840243483058 * q^89 + 546304658547739915488 * q^91 - 473340927054405485448 * q^93 + 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97 - 48254472304578247500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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