LMFDB - Embedded newform 48.24.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,24,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, 24, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 24 $
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:	$160.897937926$
Analytic rank:	$1$
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} - 313478447x - 3858843765 $ x^3 - x^2 - 313478447*x - 3858843765
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-17698.7$ 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)

$f(q)$	$=$	$q+177147. q^{3} -1.57121e8 q^{5} -8.81163e9 q^{7} +3.13811e10 q^{9} +O(q^{10})$	$q+177147. q^{3} -1.57121e8 q^{5} -8.81163e9 q^{7} +3.13811e10 q^{9} +4.03414e10 q^{11} +7.60927e12 q^{13} -2.78335e13 q^{15} -1.79230e14 q^{17} +5.45844e14 q^{19} -1.56095e15 q^{21} +8.25114e15 q^{23} +1.27660e16 q^{25} +5.55906e15 q^{27} +8.83761e16 q^{29} +1.96640e16 q^{31} +7.14635e15 q^{33} +1.38449e18 q^{35} +2.53547e17 q^{37} +1.34796e18 q^{39} -5.07657e17 q^{41} -1.05976e19 q^{43} -4.93062e18 q^{45} +1.49978e19 q^{47} +5.02761e19 q^{49} -3.17501e19 q^{51} -6.83781e19 q^{53} -6.33847e18 q^{55} +9.66946e19 q^{57} -3.48234e19 q^{59} -4.54539e20 q^{61} -2.76518e20 q^{63} -1.19557e21 q^{65} -9.99124e20 q^{67} +1.46167e21 q^{69} -2.37111e21 q^{71} +2.48627e21 q^{73} +2.26146e21 q^{75} -3.55473e20 q^{77} +4.32369e21 q^{79} +9.84771e20 q^{81} -1.35805e22 q^{83} +2.81608e22 q^{85} +1.56556e22 q^{87} +3.27737e22 q^{89} -6.70501e22 q^{91} +3.48341e21 q^{93} -8.57634e22 q^{95} -1.23343e23 q^{97} +1.26595e21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9}+O(q^{10}) $ 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9	$ 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9} + 66881746140 q^{11} + 5138719820610 q^{13} - 13680206481402 q^{15} - 232964235839898 q^{17} + 39051643356612 q^{19} - 591782205641184 q^{21} + 43\!\cdots\!64 q^{23}+ \cdots + 20\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9 + 66881746140 * q^11 + 5138719820610 * q^13 - 13680206481402 * q^15 - 232964235839898 * q^17 + 39051643356612 * q^19 - 591782205641184 * q^21 + 4358324846210664 * q^23 - 5164619678396379 * q^25 + 16677181699666569 * q^27 - 25471683594266694 * q^29 + 125107985305907448 * q^31 + 11847900683462580 * q^33 + 1311200449040041536 * q^35 - 1292646678108250614 * q^37 + 910308800061599670 * q^39 + 144934201638163518 * q^41 - 8133920080752699684 * q^43 - 2423407537560920094 * q^45 - 10029621278906529216 * q^47 + 41823513653599245723 * q^49 - 41268915486330411006 * q^51 - 28251334125947614062 * q^53 - 85062431363722250328 * q^55 + 6917881465693745964 * q^57 - 187335386683361480100 * q^59 - 555296430938057984862 * q^61 - 104832442382718822048 * q^63 - 1392409010623622551476 * q^65 - 1602075967132026564156 * q^67 + 772064171531680495608 * q^69 - 3528346267933953055080 * q^71 + 2100403635635469857646 * q^73 - 914896882168883350713 * q^75 + 8279040903546572168064 * q^77 - 12175210588212413874744 * q^79 + 2954312706550833698643 * q^81 - 30717412717926068652348 * q^83 + 44709116429093587923108 * q^85 - 4512232333673562042018 * q^87 + 58208079477660202949598 * q^89 - 63276323944720307992512 * q^91 + 22162504272985586690856 * q^93 - 79678214445310528084968 * q^95 + 76406389646793178923846 * q^97 + 2098820062373345659260 * 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
$3$	177147.	0.577350
$3$	177147.	0.577350
$4$	0	0
$5$	−1.57121e8	−1.43906	−0.719530	−	0.694462i	$-0.755643\pi$
$5$	−1.57121e8	−1.43906	−0.719530	+	0.694462i	$0.755643\pi$
$6$	0	0
$7$	−8.81163e9	−1.68434	−0.842168	−	0.539214i	$-0.818721\pi$
$7$	−8.81163e9	−1.68434	−0.842168	+	0.539214i	$0.818721\pi$
$8$	0	0
$9$	3.13811e10	0.333333
$10$	0	0
$11$	4.03414e10	0.0426319	0.0213159	−	0.999773i	$-0.493214\pi$
$11$	4.03414e10	0.0426319	0.0213159	+	0.999773i	$0.493214\pi$
$12$	0	0
$13$	7.60927e12	1.17759	0.588795	−	0.808282i	$-0.299602\pi$
$13$	7.60927e12	1.17759	0.588795	+	0.808282i	$0.299602\pi$
$14$	0	0
$15$	−2.78335e13	−0.830841
$16$	0	0
$17$	−1.79230e14	−1.26838	−0.634188	−	0.773179i	$-0.718665\pi$
$17$	−1.79230e14	−1.26838	−0.634188	+	0.773179i	$0.718665\pi$
$18$	0	0
$19$	5.45844e14	1.07498	0.537492	−	0.843269i	$-0.319372\pi$
$19$	5.45844e14	1.07498	0.537492	+	0.843269i	$0.319372\pi$
$20$	0	0
$21$	−1.56095e15	−0.972452
$22$	0	0
$23$	8.25114e15	1.80569	0.902847	−	0.429962i	$-0.141473\pi$
$23$	8.25114e15	1.80569	0.902847	+	0.429962i	$0.141473\pi$
$24$	0	0
$25$	1.27660e16	1.07089
$26$	0	0
$27$	5.55906e15	0.192450
$28$	0	0
$29$	8.83761e16	1.34511	0.672555	−	0.740047i	$-0.265197\pi$
$29$	8.83761e16	1.34511	0.672555	+	0.740047i	$0.265197\pi$
$30$	0	0
$31$	1.96640e16	0.138999	0.0694994	−	0.997582i	$-0.477860\pi$
$31$	1.96640e16	0.138999	0.0694994	+	0.997582i	$0.477860\pi$
$32$	0	0
$33$	7.14635e15	0.0246135
$34$	0	0
$35$	1.38449e18	2.42386
$36$	0	0
$37$	2.53547e17	0.234282	0.117141	−	0.993115i	$-0.462627\pi$
$37$	2.53547e17	0.234282	0.117141	+	0.993115i	$0.462627\pi$
$38$	0	0
$39$	1.34796e18	0.679882
$40$	0	0
$41$	−5.07657e17	−0.144064	−0.0720321	−	0.997402i	$-0.522948\pi$
$41$	−5.07657e17	−0.144064	−0.0720321	+	0.997402i	$0.522948\pi$
$42$	0	0
$43$	−1.05976e19	−1.73908	−0.869538	−	0.493867i	$-0.835583\pi$
$43$	−1.05976e19	−1.73908	−0.869538	+	0.493867i	$0.835583\pi$
$44$	0	0
$45$	−4.93062e18	−0.479686
$46$	0	0
$47$	1.49978e19	0.884914	0.442457	−	0.896790i	$-0.354107\pi$
$47$	1.49978e19	0.884914	0.442457	+	0.896790i	$0.354107\pi$
$48$	0	0
$49$	5.02761e19	1.83699
$50$	0	0
$51$	−3.17501e19	−0.732297
$52$	0	0
$53$	−6.83781e19	−1.01332	−0.506658	−	0.862147i	$-0.669119\pi$
$53$	−6.83781e19	−1.01332	−0.506658	+	0.862147i	$0.669119\pi$
$54$	0	0
$55$	−6.33847e18	−0.0613498
$56$	0	0
$57$	9.66946e19	0.620642
$58$	0	0
$59$	−3.48234e19	−0.150339	−0.0751697	−	0.997171i	$-0.523950\pi$
$59$	−3.48234e19	−0.150339	−0.0751697	+	0.997171i	$0.523950\pi$
$60$	0	0
$61$	−4.54539e20	−1.33745	−0.668726	−	0.743509i	$-0.733160\pi$
$61$	−4.54539e20	−1.33745	−0.668726	+	0.743509i	$0.733160\pi$
$62$	0	0
$63$	−2.76518e20	−0.561446
$64$	0	0
$65$	−1.19557e21	−1.69462
$66$	0	0
$67$	−9.99124e20	−0.999446	−0.499723	−	0.866185i	$-0.666565\pi$
$67$	−9.99124e20	−0.999446	−0.499723	+	0.866185i	$0.666565\pi$
$68$	0	0
$69$	1.46167e21	1.04252
$70$	0	0
$71$	−2.37111e21	−1.21753	−0.608767	−	0.793349i	$-0.708336\pi$
$71$	−2.37111e21	−1.21753	−0.608767	+	0.793349i	$0.708336\pi$
$72$	0	0
$73$	2.48627e21	0.927547	0.463774	−	0.885954i	$-0.346495\pi$
$73$	2.48627e21	0.927547	0.463774	+	0.885954i	$0.346495\pi$
$74$	0	0
$75$	2.26146e21	0.618279
$76$	0	0
$77$	−3.55473e20	−0.0718065
$78$	0	0
$79$	4.32369e21	0.650344	0.325172	−	0.945655i	$-0.394578\pi$
$79$	4.32369e21	0.650344	0.325172	+	0.945655i	$0.394578\pi$
$80$	0	0
$81$	9.84771e20	0.111111
$82$	0	0
$83$	−1.35805e22	−1.15749	−0.578746	−	0.815508i	$-0.696458\pi$
$83$	−1.35805e22	−1.15749	−0.578746	+	0.815508i	$0.696458\pi$
$84$	0	0
$85$	2.81608e22	1.82527
$86$	0	0
$87$	1.56556e22	0.776600
$88$	0	0
$89$	3.27737e22	1.25182	0.625909	−	0.779896i	$-0.284728\pi$
$89$	3.27737e22	1.25182	0.625909	+	0.779896i	$0.284728\pi$
$90$	0	0
$91$	−6.70501e22	−1.98346
$92$	0	0
$93$	3.48341e21	0.0802510
$94$	0	0
$95$	−8.57634e22	−1.54697
$96$	0	0
$97$	−1.23343e23	−1.75081	−0.875404	−	0.483393i	$-0.839404\pi$
$97$	−1.23343e23	−1.75081	−0.875404	+	0.483393i	$0.839404\pi$
$98$	0	0
$99$	1.26595e21	0.0142106
$100$	0	0
$101$	4.49160e22	0.400595	0.200298	−	0.979735i	$-0.435809\pi$
$101$	4.49160e22	0.400595	0.200298	+	0.979735i	$0.435809\pi$
$102$	0	0
$103$	8.97611e22	0.638940	0.319470	−	0.947596i	$-0.396495\pi$
$103$	8.97611e22	0.638940	0.319470	+	0.947596i	$0.396495\pi$
$104$	0	0
$105$	2.45258e23	1.39942
$106$	0	0
$107$	−1.58982e23	−0.730186	−0.365093	−	0.930971i	$-0.618963\pi$
$107$	−1.58982e23	−0.730186	−0.365093	+	0.930971i	$0.618963\pi$
$108$	0	0
$109$	5.06892e23	1.88153	0.940763	−	0.339064i	$-0.110110\pi$
$109$	5.06892e23	1.88153	0.940763	+	0.339064i	$0.110110\pi$
$110$	0	0
$111$	4.49151e22	0.135263
$112$	0	0
$113$	4.68677e23	1.14940	0.574701	−	0.818364i	$-0.305118\pi$
$113$	4.68677e23	1.14940	0.574701	+	0.818364i	$0.305118\pi$
$114$	0	0
$115$	−1.29643e24	−2.59850
$116$	0	0
$117$	2.38787e23	0.392530
$118$	0	0
$119$	1.57931e24	2.13637
$120$	0	0
$121$	−8.93803e23	−0.998183
$122$	0	0
$123$	−8.99300e22	−0.0831755
$124$	0	0
$125$	−1.32781e23	−0.102017
$126$	0	0
$127$	−1.70982e24	−1.09448	−0.547240	−	0.836976i	$-0.684321\pi$
$127$	−1.70982e24	−1.09448	−0.547240	+	0.836976i	$0.684321\pi$
$128$	0	0
$129$	−1.87732e24	−1.00406
$130$	0	0
$131$	−1.33830e23	−0.0599697	−0.0299849	−	0.999550i	$-0.509546\pi$
$131$	−1.33830e23	−0.0599697	−0.0299849	+	0.999550i	$0.509546\pi$
$132$	0	0
$133$	−4.80977e24	−1.81063
$134$	0	0
$135$	−8.73444e23	−0.276947
$136$	0	0
$137$	−6.43725e22	−0.0172351	−0.00861755	−	0.999963i	$-0.502743\pi$
$137$	−6.43725e22	−0.0172351	−0.00861755	+	0.999963i	$0.502743\pi$
$138$	0	0
$139$	−1.95709e24	−0.443547	−0.221774	−	0.975098i	$-0.571185\pi$
$139$	−1.95709e24	−0.443547	−0.221774	+	0.975098i	$0.571185\pi$
$140$	0	0
$141$	2.65681e24	0.510906
$142$	0	0
$143$	3.06968e23	0.0502029
$144$	0	0
$145$	−1.38857e25	−1.93569
$146$	0	0
$147$	8.90627e24	1.06059
$148$	0	0
$149$	8.03463e24	0.819075	0.409537	−	0.912293i	$-0.365690\pi$
$149$	8.03463e24	0.819075	0.409537	+	0.912293i	$0.365690\pi$
$150$	0	0
$151$	1.81832e25	1.59014	0.795069	−	0.606519i	$-0.207435\pi$
$151$	1.81832e25	1.59014	0.795069	+	0.606519i	$0.207435\pi$
$152$	0	0
$153$	−5.62443e24	−0.422792
$154$	0	0
$155$	−3.08962e24	−0.200028
$156$	0	0
$157$	2.14734e25	1.19965	0.599826	−	0.800130i	$-0.295236\pi$
$157$	2.14734e25	1.19965	0.599826	+	0.800130i	$0.295236\pi$
$158$	0	0
$159$	−1.21130e25	−0.585038
$160$	0	0
$161$	−7.27061e25	−3.04140
$162$	0	0
$163$	1.45948e25	0.529712	0.264856	−	0.964288i	$-0.414676\pi$
$163$	1.45948e25	0.529712	0.264856	+	0.964288i	$0.414676\pi$
$164$	0	0
$165$	−1.12284e24	−0.0354203
$166$	0	0
$167$	−1.89306e25	−0.519905	−0.259953	−	0.965621i	$-0.583707\pi$
$167$	−1.89306e25	−0.519905	−0.259953	+	0.965621i	$0.583707\pi$
$168$	0	0
$169$	1.61470e25	0.386719
$170$	0	0
$171$	1.71292e25	0.358328
$172$	0	0
$173$	−5.28942e25	−0.968005	−0.484002	−	0.875067i	$-0.660817\pi$
$173$	−5.28942e25	−0.968005	−0.484002	+	0.875067i	$0.660817\pi$
$174$	0	0
$175$	−1.12489e26	−1.80374
$176$	0	0
$177$	−6.16886e24	−0.0867985
$178$	0	0
$179$	−5.38948e25	−0.666403	−0.333202	−	0.942856i	$-0.608129\pi$
$179$	−5.38948e25	−0.666403	−0.333202	+	0.942856i	$0.608129\pi$
$180$	0	0
$181$	2.13425e25	0.232242	0.116121	−	0.993235i	$-0.462954\pi$
$181$	2.13425e25	0.232242	0.116121	+	0.993235i	$0.462954\pi$
$182$	0	0
$183$	−8.05202e25	−0.772178
$184$	0	0
$185$	−3.98375e25	−0.337146
$186$	0	0
$187$	−7.23038e24	−0.0540733
$188$	0	0
$189$	−4.89844e25	−0.324151
$190$	0	0
$191$	−2.15927e26	−1.26597	−0.632985	−	0.774164i	$-0.718171\pi$
$191$	−2.15927e26	−1.26597	−0.632985	+	0.774164i	$0.718171\pi$
$192$	0	0
$193$	−1.24633e26	−0.648223	−0.324112	−	0.946019i	$-0.605065\pi$
$193$	−1.24633e26	−0.648223	−0.324112	+	0.946019i	$0.605065\pi$
$194$	0	0
$195$	−2.11792e26	−0.978391
$196$	0	0
$197$	2.17670e26	0.894205	0.447103	−	0.894483i	$-0.352456\pi$
$197$	2.17670e26	0.894205	0.447103	+	0.894483i	$0.352456\pi$
$198$	0	0
$199$	−2.30229e26	−0.842074	−0.421037	−	0.907044i	$-0.638334\pi$
$199$	−2.30229e26	−0.842074	−0.421037	+	0.907044i	$0.638334\pi$
$200$	0	0
$201$	−1.76992e26	−0.577030
$202$	0	0
$203$	−7.78738e26	−2.26562
$204$	0	0
$205$	7.97635e25	0.207317
$206$	0	0
$207$	2.58930e26	0.601898
$208$	0	0
$209$	2.20201e25	0.0458286
$210$	0	0
$211$	−6.41286e26	−1.19620	−0.598100	−	0.801421i	$-0.704077\pi$
$211$	−6.41286e26	−1.19620	−0.598100	+	0.801421i	$0.704077\pi$
$212$	0	0
$213$	−4.20035e26	−0.702944
$214$	0	0
$215$	1.66510e27	2.50263
$216$	0	0
$217$	−1.73272e26	−0.234121
$218$	0	0
$219$	4.40436e26	0.535520
$220$	0	0
$221$	−1.36381e27	−1.49363
$222$	0	0
$223$	−1.29766e27	−1.28131	−0.640655	−	0.767828i	$-0.721337\pi$
$223$	−1.29766e27	−1.28131	−0.640655	+	0.767828i	$0.721337\pi$
$224$	0	0
$225$	4.00611e26	0.356964
$226$	0	0
$227$	−5.12462e26	−0.412443	−0.206222	−	0.978505i	$-0.566117\pi$
$227$	−5.12462e26	−0.412443	−0.206222	+	0.978505i	$0.566117\pi$
$228$	0	0
$229$	−1.73308e27	−1.26099	−0.630495	−	0.776193i	$-0.717148\pi$
$229$	−1.73308e27	−1.26099	−0.630495	+	0.776193i	$0.717148\pi$
$230$	0	0
$231$	−6.29710e25	−0.0414575
$232$	0	0
$233$	5.22831e26	0.311723	0.155861	−	0.987779i	$-0.450185\pi$
$233$	5.22831e26	0.311723	0.155861	+	0.987779i	$0.450185\pi$
$234$	0	0
$235$	−2.35646e27	−1.27344
$236$	0	0
$237$	7.65929e26	0.375476
$238$	0	0
$239$	6.31381e26	0.281006	0.140503	−	0.990080i	$-0.455128\pi$
$239$	6.31381e26	0.281006	0.140503	+	0.990080i	$0.455128\pi$
$240$	0	0
$241$	3.41013e26	0.137903	0.0689516	−	0.997620i	$-0.478035\pi$
$241$	3.41013e26	0.137903	0.0689516	+	0.997620i	$0.478035\pi$
$242$	0	0
$243$	1.74449e26	0.0641500
$244$	0	0
$245$	−7.89943e27	−2.64354
$246$	0	0
$247$	4.15347e27	1.26589
$248$	0	0
$249$	−2.40575e27	−0.668278
$250$	0	0
$251$	4.07074e27	1.03140	0.515698	−	0.856770i	$-0.327533\pi$
$251$	4.07074e27	1.03140	0.515698	+	0.856770i	$0.327533\pi$
$252$	0	0
$253$	3.32862e26	0.0769801
$254$	0	0
$255$	4.98859e27	1.05382
$256$	0	0
$257$	2.34690e27	0.453173	0.226587	−	0.973991i	$-0.427243\pi$
$257$	2.34690e27	0.453173	0.226587	+	0.973991i	$0.427243\pi$
$258$	0	0
$259$	−2.23416e27	−0.394610
$260$	0	0
$261$	2.77334e27	0.448370
$262$	0	0
$263$	−4.76043e27	−0.704944	−0.352472	−	0.935822i	$-0.614659\pi$
$263$	−4.76043e27	−0.704944	−0.352472	+	0.935822i	$0.614659\pi$
$264$	0	0
$265$	1.07436e28	1.45822
$266$	0	0
$267$	5.80577e27	0.722737
$268$	0	0
$269$	5.39725e27	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$269$	5.39725e27	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$270$	0	0
$271$	−9.05206e27	−0.949731	−0.474866	−	0.880058i	$-0.657503\pi$
$271$	−9.05206e27	−0.949731	−0.474866	+	0.880058i	$0.657503\pi$
$272$	0	0
$273$	−1.18777e28	−1.14515
$274$	0	0
$275$	5.14999e26	0.0456541
$276$	0	0
$277$	2.19075e28	1.78680	0.893398	−	0.449266i	$-0.148315\pi$
$277$	2.19075e28	1.78680	0.893398	+	0.449266i	$0.148315\pi$
$278$	0	0
$279$	6.17076e26	0.0463330
$280$	0	0
$281$	−1.16506e27	−0.0805795	−0.0402897	−	0.999188i	$-0.512828\pi$
$281$	−1.16506e27	−0.0805795	−0.0402897	+	0.999188i	$0.512828\pi$
$282$	0	0
$283$	−1.77405e28	−1.13090	−0.565448	−	0.824784i	$-0.691297\pi$
$283$	−1.77405e28	−1.13090	−0.565448	+	0.824784i	$0.691297\pi$
$284$	0	0
$285$	−1.51927e28	−0.893141
$286$	0	0
$287$	4.47329e27	0.242653
$288$	0	0
$289$	1.21558e28	0.608778
$290$	0	0
$291$	−2.18498e28	−1.01083
$292$	0	0
$293$	1.09362e28	0.467617	0.233808	−	0.972283i	$-0.424881\pi$
$293$	1.09362e28	0.467617	0.233808	+	0.972283i	$0.424881\pi$
$294$	0	0
$295$	5.47148e27	0.216347
$296$	0	0
$297$	2.24260e26	0.00820451
$298$	0	0
$299$	6.27852e28	2.12637
$300$	0	0
$301$	9.33818e28	2.92919
$302$	0	0
$303$	7.95674e27	0.231284
$304$	0	0
$305$	7.14175e28	1.92467
$306$	0	0
$307$	1.26433e28	0.316059	0.158030	−	0.987434i	$-0.449486\pi$
$307$	1.26433e28	0.316059	0.158030	+	0.987434i	$0.449486\pi$
$308$	0	0
$309$	1.59009e28	0.368892
$310$	0	0
$311$	6.93363e28	1.49354	0.746769	−	0.665084i	$-0.231604\pi$
$311$	6.93363e28	1.49354	0.746769	+	0.665084i	$0.231604\pi$
$312$	0	0
$313$	−2.09418e27	−0.0419038	−0.0209519	−	0.999780i	$-0.506670\pi$
$313$	−2.09418e27	−0.0419038	−0.0209519	+	0.999780i	$0.506670\pi$
$314$	0	0
$315$	4.34468e28	0.807954
$316$	0	0
$317$	−1.09160e29	−1.88748	−0.943738	−	0.330695i	$-0.892717\pi$
$317$	−1.09160e29	−1.88748	−0.943738	+	0.330695i	$0.892717\pi$
$318$	0	0
$319$	3.56521e27	0.0573446
$320$	0	0
$321$	−2.81631e28	−0.421573
$322$	0	0
$323$	−9.78315e28	−1.36348
$324$	0	0
$325$	9.71401e28	1.26107
$326$	0	0
$327$	8.97943e28	1.08630
$328$	0	0
$329$	−1.32155e29	−1.49049
$330$	0	0
$331$	−1.30504e29	−1.37278	−0.686390	−	0.727234i	$-0.740806\pi$
$331$	−1.30504e29	−1.37278	−0.686390	+	0.727234i	$0.740806\pi$
$332$	0	0
$333$	7.95658e27	0.0780940
$334$	0	0
$335$	1.56983e29	1.43826
$336$	0	0
$337$	−9.30659e28	−0.796245	−0.398122	−	0.917332i	$-0.630338\pi$
$337$	−9.30659e28	−0.796245	−0.398122	+	0.917332i	$0.630338\pi$
$338$	0	0
$339$	8.30248e28	0.663607
$340$	0	0
$341$	7.93270e26	0.00592579
$342$	0	0
$343$	−2.01852e29	−1.40977
$344$	0	0
$345$	−2.29658e29	−1.50024
$346$	0	0
$347$	−1.72120e29	−1.05206	−0.526032	−	0.850465i	$-0.676321\pi$
$347$	−1.72120e29	−1.05206	−0.526032	+	0.850465i	$0.676321\pi$
$348$	0	0
$349$	−2.35689e29	−1.34848	−0.674242	−	0.738510i	$-0.735530\pi$
$349$	−2.35689e29	−1.34848	−0.674242	+	0.738510i	$0.735530\pi$
$350$	0	0
$351$	4.23004e28	0.226627
$352$	0	0
$353$	2.18980e29	1.09899	0.549497	−	0.835496i	$-0.314819\pi$
$353$	2.18980e29	1.09899	0.549497	+	0.835496i	$0.314819\pi$
$354$	0	0
$355$	3.72551e29	1.75210
$356$	0	0
$357$	2.79770e29	1.23344
$358$	0	0
$359$	1.51427e29	0.626060	0.313030	−	0.949743i	$-0.398656\pi$
$359$	1.51427e29	0.626060	0.313030	+	0.949743i	$0.398656\pi$
$360$	0	0
$361$	4.01157e28	0.155590
$362$	0	0
$363$	−1.58334e29	−0.576301
$364$	0	0
$365$	−3.90645e29	−1.33480
$366$	0	0
$367$	−4.04094e29	−1.29665	−0.648325	−	0.761364i	$-0.724530\pi$
$367$	−4.04094e29	−1.29665	−0.648325	+	0.761364i	$0.724530\pi$
$368$	0	0
$369$	−1.59308e28	−0.0480214
$370$	0	0
$371$	6.02523e29	1.70676
$372$	0	0
$373$	4.96896e28	0.132317	0.0661583	−	0.997809i	$-0.478926\pi$
$373$	4.96896e28	0.132317	0.0661583	+	0.997809i	$0.478926\pi$
$374$	0	0
$375$	−2.35218e28	−0.0588995
$376$	0	0
$377$	6.72478e29	1.58399
$378$	0	0
$379$	−1.38943e29	−0.307953	−0.153976	−	0.988075i	$-0.549208\pi$
$379$	−1.38943e29	−0.307953	−0.153976	+	0.988075i	$0.549208\pi$
$380$	0	0
$381$	−3.02889e29	−0.631898
$382$	0	0
$383$	−7.93381e29	−1.55846	−0.779231	−	0.626737i	$-0.784390\pi$
$383$	−7.93381e29	−1.55846	−0.779231	+	0.626737i	$0.784390\pi$
$384$	0	0
$385$	5.58522e28	0.103334
$386$	0	0
$387$	−3.32562e29	−0.579692
$388$	0	0
$389$	−1.06596e30	−1.75113	−0.875567	−	0.483097i	$-0.839512\pi$
$389$	−1.06596e30	−1.75113	−0.875567	+	0.483097i	$0.839512\pi$
$390$	0	0
$391$	−1.47885e30	−2.29030
$392$	0	0
$393$	−2.37075e28	−0.0346235
$394$	0	0
$395$	−6.79342e29	−0.935884
$396$	0	0
$397$	6.43516e29	0.836505	0.418253	−	0.908331i	$-0.362643\pi$
$397$	6.43516e29	0.836505	0.418253	+	0.908331i	$0.362643\pi$
$398$	0	0
$399$	−8.52037e29	−1.04537
$400$	0	0
$401$	1.22922e30	1.42387	0.711933	−	0.702248i	$-0.247820\pi$
$401$	1.22922e30	1.42387	0.711933	+	0.702248i	$0.247820\pi$
$402$	0	0
$403$	1.49628e29	0.163684
$404$	0	0
$405$	−1.54728e29	−0.159895
$406$	0	0
$407$	1.02284e28	0.00998789
$408$	0	0
$409$	−1.71008e30	−1.57833	−0.789166	−	0.614180i	$-0.789487\pi$
$409$	−1.71008e30	−1.57833	−0.789166	+	0.614180i	$0.789487\pi$
$410$	0	0
$411$	−1.14034e28	−0.00995069
$412$	0	0
$413$	3.06851e29	0.253222
$414$	0	0
$415$	2.13378e30	1.66570
$416$	0	0
$417$	−3.46692e29	−0.256082
$418$	0	0
$419$	−1.24474e30	−0.870198	−0.435099	−	0.900383i	$-0.643287\pi$
$419$	−1.24474e30	−0.870198	−0.435099	+	0.900383i	$0.643287\pi$
$420$	0	0
$421$	−1.35737e30	−0.898367	−0.449184	−	0.893440i	$-0.648285\pi$
$421$	−1.35737e30	−0.898367	−0.449184	+	0.893440i	$0.648285\pi$
$422$	0	0
$423$	4.70646e29	0.294971
$424$	0	0
$425$	−2.28805e30	−1.35829
$426$	0	0
$427$	4.00523e30	2.25272
$428$	0	0
$429$	5.43785e28	0.0289847
$430$	0	0
$431$	−2.49265e30	−1.25943	−0.629714	−	0.776827i	$-0.716828\pi$
$431$	−2.49265e30	−1.25943	−0.629714	+	0.776827i	$0.716828\pi$
$432$	0	0
$433$	1.11855e30	0.535853	0.267927	−	0.963439i	$-0.413661\pi$
$433$	1.11855e30	0.535853	0.267927	+	0.963439i	$0.413661\pi$
$434$	0	0
$435$	−2.45982e30	−1.11757
$436$	0	0
$437$	4.50383e30	1.94109
$438$	0	0
$439$	−4.99745e29	−0.204365	−0.102182	−	0.994766i	$-0.532583\pi$
$439$	−4.99745e29	−0.204365	−0.102182	+	0.994766i	$0.532583\pi$
$440$	0	0
$441$	1.57772e30	0.612330
$442$	0	0
$443$	1.51565e30	0.558414	0.279207	−	0.960231i	$-0.409928\pi$
$443$	1.51565e30	0.558414	0.279207	+	0.960231i	$0.409928\pi$
$444$	0	0
$445$	−5.14944e30	−1.80144
$446$	0	0
$447$	1.42331e30	0.472893
$448$	0	0
$449$	−9.14479e29	−0.288630	−0.144315	−	0.989532i	$-0.546098\pi$
$449$	−9.14479e29	−0.288630	−0.144315	+	0.989532i	$0.546098\pi$
$450$	0	0
$451$	−2.04796e28	−0.00614173
$452$	0	0
$453$	3.22109e30	0.918067
$454$	0	0
$455$	1.05350e31	2.85431
$456$	0	0
$457$	−7.10896e30	−1.83134	−0.915672	−	0.401926i	$-0.868341\pi$
$457$	−7.10896e30	−1.83134	−0.915672	+	0.401926i	$0.868341\pi$
$458$	0	0
$459$	−9.96350e29	−0.244099
$460$	0	0
$461$	1.11488e30	0.259817	0.129908	−	0.991526i	$-0.458532\pi$
$461$	1.11488e30	0.259817	0.129908	+	0.991526i	$0.458532\pi$
$462$	0	0
$463$	4.51676e29	0.100149	0.0500744	−	0.998745i	$-0.484054\pi$
$463$	4.51676e29	0.100149	0.0500744	+	0.998745i	$0.484054\pi$
$464$	0	0
$465$	−5.47316e29	−0.115486
$466$	0	0
$467$	4.73482e29	0.0950954	0.0475477	−	0.998869i	$-0.484859\pi$
$467$	4.73482e29	0.0950954	0.0475477	+	0.998869i	$0.484859\pi$
$468$	0	0
$469$	8.80392e30	1.68340
$470$	0	0
$471$	3.80395e30	0.692619
$472$	0	0
$473$	−4.27520e29	−0.0741401
$474$	0	0
$475$	6.96825e30	1.15119
$476$	0	0
$477$	−2.14578e30	−0.337772
$478$	0	0
$479$	3.90153e30	0.585298	0.292649	−	0.956220i	$-0.405463\pi$
$479$	3.90153e30	0.585298	0.292649	+	0.956220i	$0.405463\pi$
$480$	0	0
$481$	1.92931e30	0.275888
$482$	0	0
$483$	−1.28797e31	−1.75595
$484$	0	0
$485$	1.93797e31	2.51952
$486$	0	0
$487$	−1.13082e31	−1.40220	−0.701102	−	0.713061i	$-0.747308\pi$
$487$	−1.13082e31	−1.40220	−0.701102	+	0.713061i	$0.747308\pi$
$488$	0	0
$489$	2.58542e30	0.305829
$490$	0	0
$491$	3.58672e30	0.404819	0.202409	−	0.979301i	$-0.435123\pi$
$491$	3.58672e30	0.404819	0.202409	+	0.979301i	$0.435123\pi$
$492$	0	0
$493$	−1.58397e31	−1.70611
$494$	0	0
$495$	−1.98908e29	−0.0204499
$496$	0	0
$497$	2.08934e31	2.05074
$498$	0	0
$499$	1.73959e31	1.63039	0.815195	−	0.579186i	$-0.196630\pi$
$499$	1.73959e31	1.63039	0.815195	+	0.579186i	$0.196630\pi$
$500$	0	0
$501$	−3.35349e30	−0.300167
$502$	0	0
$503$	8.49237e30	0.726101	0.363051	−	0.931769i	$-0.381735\pi$
$503$	8.49237e30	0.726101	0.363051	+	0.931769i	$0.381735\pi$
$504$	0	0
$505$	−7.05725e30	−0.576480
$506$	0	0
$507$	2.86040e30	0.223272
$508$	0	0
$509$	2.01029e31	1.49970	0.749848	−	0.661610i	$-0.230127\pi$
$509$	2.01029e31	1.49970	0.749848	+	0.661610i	$0.230127\pi$
$510$	0	0
$511$	−2.19081e31	−1.56230
$512$	0	0
$513$	3.03438e30	0.206881
$514$	0	0
$515$	−1.41033e31	−0.919472
$516$	0	0
$517$	6.05030e29	0.0377256
$518$	0	0
$519$	−9.37004e30	−0.558878
$520$	0	0
$521$	−6.65578e30	−0.379809	−0.189905	−	0.981803i	$-0.560818\pi$
$521$	−6.65578e30	−0.379809	−0.189905	+	0.981803i	$0.560818\pi$
$522$	0	0
$523$	−1.15283e31	−0.629502	−0.314751	−	0.949174i	$-0.601921\pi$
$523$	−1.15283e31	−0.629502	−0.314751	+	0.949174i	$0.601921\pi$
$524$	0	0
$525$	−1.99272e31	−1.04139
$526$	0	0
$527$	−3.52437e30	−0.176303
$528$	0	0
$529$	4.72009e31	2.26053
$530$	0	0
$531$	−1.09279e30	−0.0501131
$532$	0	0
$533$	−3.86290e30	−0.169649
$534$	0	0
$535$	2.49793e31	1.05078
$536$	0	0
$537$	−9.54730e30	−0.384748
$538$	0	0
$539$	2.02821e30	0.0783144
$540$	0	0
$541$	3.39259e31	1.25534	0.627672	−	0.778478i	$-0.284008\pi$
$541$	3.39259e31	1.25534	0.627672	+	0.778478i	$0.284008\pi$
$542$	0	0
$543$	3.78075e30	0.134085
$544$	0	0
$545$	−7.96432e31	−2.70763
$546$	0	0
$547$	−2.02292e31	−0.659363	−0.329682	−	0.944092i	$-0.606941\pi$
$547$	−2.02292e31	−0.659363	−0.329682	+	0.944092i	$0.606941\pi$
$548$	0	0
$549$	−1.42639e31	−0.445817
$550$	0	0
$551$	4.82396e31	1.44597
$552$	0	0
$553$	−3.80988e31	−1.09540
$554$	0	0
$555$	−7.05710e30	−0.194651
$556$	0	0
$557$	−5.58561e31	−1.47821	−0.739105	−	0.673590i	$-0.764751\pi$
$557$	−5.58561e31	−1.47821	−0.739105	+	0.673590i	$0.764751\pi$
$558$	0	0
$559$	−8.06396e31	−2.04792
$560$	0	0
$561$	−1.28084e30	−0.0312192
$562$	0	0
$563$	−1.92115e30	−0.0449486	−0.0224743	−	0.999747i	$-0.507154\pi$
$563$	−1.92115e30	−0.0449486	−0.0224743	+	0.999747i	$0.507154\pi$
$564$	0	0
$565$	−7.36390e31	−1.65406
$566$	0	0
$567$	−8.67744e30	−0.187149
$568$	0	0
$569$	7.81735e31	1.61908	0.809541	−	0.587064i	$-0.199716\pi$
$569$	7.81735e31	1.61908	0.809541	+	0.587064i	$0.199716\pi$
$570$	0	0
$571$	4.36111e31	0.867524	0.433762	−	0.901028i	$-0.357186\pi$
$571$	4.36111e31	0.867524	0.433762	+	0.901028i	$0.357186\pi$
$572$	0	0
$573$	−3.82509e31	−0.730908
$574$	0	0
$575$	1.05334e32	1.93370
$576$	0	0
$577$	6.52282e30	0.115057	0.0575286	−	0.998344i	$-0.481678\pi$
$577$	6.52282e30	0.115057	0.0575286	+	0.998344i	$0.481678\pi$
$578$	0	0
$579$	−2.20784e31	−0.374252
$580$	0	0
$581$	1.19666e32	1.94961
$582$	0	0
$583$	−2.75847e30	−0.0431996
$584$	0	0
$585$	−3.75184e31	−0.564874
$586$	0	0
$587$	4.60923e31	0.667252	0.333626	−	0.942706i	$-0.391728\pi$
$587$	4.60923e31	0.667252	0.333626	+	0.942706i	$0.391728\pi$
$588$	0	0
$589$	1.07334e31	0.149422
$590$	0	0
$591$	3.85596e31	0.516270
$592$	0	0
$593$	−5.07532e31	−0.653633	−0.326817	−	0.945088i	$-0.605976\pi$
$593$	−5.07532e31	−0.653633	−0.326817	+	0.945088i	$0.605976\pi$
$594$	0	0
$595$	−2.48142e32	−3.07437
$596$	0	0
$597$	−4.07844e31	−0.486171
$598$	0	0
$599$	3.64679e31	0.418314	0.209157	−	0.977882i	$-0.432928\pi$
$599$	3.64679e31	0.418314	0.209157	+	0.977882i	$0.432928\pi$
$600$	0	0
$601$	−1.09549e32	−1.20936	−0.604678	−	0.796470i	$-0.706698\pi$
$601$	−1.09549e32	−1.20936	−0.604678	+	0.796470i	$0.706698\pi$
$602$	0	0
$603$	−3.13536e31	−0.333149
$604$	0	0
$605$	1.40435e32	1.43644
$606$	0	0
$607$	−1.11689e32	−1.09987	−0.549936	−	0.835207i	$-0.685348\pi$
$607$	−1.11689e32	−1.09987	−0.549936	+	0.835207i	$0.685348\pi$
$608$	0	0
$609$	−1.37951e32	−1.30806
$610$	0	0
$611$	1.14122e32	1.04207
$612$	0	0
$613$	−1.10958e31	−0.0975805	−0.0487902	−	0.998809i	$-0.515537\pi$
$613$	−1.10958e31	−0.0975805	−0.0487902	+	0.998809i	$0.515537\pi$
$614$	0	0
$615$	1.41299e31	0.119695
$616$	0	0
$617$	−2.63249e31	−0.214826	−0.107413	−	0.994214i	$-0.534257\pi$
$617$	−2.63249e31	−0.214826	−0.107413	+	0.994214i	$0.534257\pi$
$618$	0	0
$619$	−2.63435e31	−0.207124	−0.103562	−	0.994623i	$-0.533024\pi$
$619$	−2.63435e31	−0.207124	−0.103562	+	0.994623i	$0.533024\pi$
$620$	0	0
$621$	4.58686e31	0.347506
$622$	0	0
$623$	−2.88790e32	−2.10848
$624$	0	0
$625$	−1.31320e32	−0.924083
$626$	0	0
$627$	3.90079e30	0.0264591
$628$	0	0
$629$	−4.54433e31	−0.297158
$630$	0	0
$631$	−2.38295e32	−1.50237	−0.751185	−	0.660092i	$-0.770517\pi$
$631$	−2.38295e32	−1.50237	−0.751185	+	0.660092i	$0.770517\pi$
$632$	0	0
$633$	−1.13602e32	−0.690626
$634$	0	0
$635$	2.68648e32	1.57502
$636$	0	0
$637$	3.82565e32	2.16322
$638$	0	0
$639$	−7.44080e31	−0.405845
$640$	0	0
$641$	1.31206e32	0.690379	0.345190	−	0.938533i	$-0.387815\pi$
$641$	1.31206e32	0.690379	0.345190	+	0.938533i	$0.387815\pi$
$642$	0	0
$643$	−5.40757e31	−0.274521	−0.137261	−	0.990535i	$-0.543830\pi$
$643$	−5.40757e31	−0.274521	−0.137261	+	0.990535i	$0.543830\pi$
$644$	0	0
$645$	2.94967e32	1.44490
$646$	0	0
$647$	2.13738e32	1.01037	0.505187	−	0.863010i	$-0.331424\pi$
$647$	2.13738e32	1.01037	0.505187	+	0.863010i	$0.331424\pi$
$648$	0	0
$649$	−1.40482e30	−0.00640925
$650$	0	0
$651$	−3.06945e31	−0.135170
$652$	0	0
$653$	2.87849e32	1.22367	0.611834	−	0.790986i	$-0.290432\pi$
$653$	2.87849e32	1.22367	0.611834	+	0.790986i	$0.290432\pi$
$654$	0	0
$655$	2.10274e31	0.0863000
$656$	0	0
$657$	7.80219e31	0.309182
$658$	0	0
$659$	2.75005e32	1.05235	0.526173	−	0.850377i	$-0.323626\pi$
$659$	2.75005e32	1.05235	0.526173	+	0.850377i	$0.323626\pi$
$660$	0	0
$661$	4.12541e32	1.52458	0.762291	−	0.647235i	$-0.224075\pi$
$661$	4.12541e32	1.52458	0.762291	+	0.647235i	$0.224075\pi$
$662$	0	0
$663$	−2.41595e32	−0.862346
$664$	0	0
$665$	7.55716e32	2.60561
$666$	0	0
$667$	7.29204e32	2.42886
$668$	0	0
$669$	−2.29876e32	−0.739765
$670$	0	0
$671$	−1.83367e31	−0.0570181
$672$	0	0
$673$	−6.43151e32	−1.93259	−0.966295	−	0.257436i	$-0.917122\pi$
$673$	−6.43151e32	−1.93259	−0.966295	+	0.257436i	$0.917122\pi$
$674$	0	0
$675$	7.09671e31	0.206093
$676$	0	0
$677$	−4.89556e32	−1.37415	−0.687073	−	0.726589i	$-0.741105\pi$
$677$	−4.89556e32	−1.37415	−0.687073	+	0.726589i	$0.741105\pi$
$678$	0	0
$679$	1.08685e33	2.94895
$680$	0	0
$681$	−9.07811e31	−0.238124
$682$	0	0
$683$	−8.06542e31	−0.204545	−0.102273	−	0.994756i	$-0.532611\pi$
$683$	−8.06542e31	−0.204545	−0.102273	+	0.994756i	$0.532611\pi$
$684$	0	0
$685$	1.01143e31	0.0248023
$686$	0	0
$687$	−3.07011e32	−0.728033
$688$	0	0
$689$	−5.20307e32	−1.19327
$690$	0	0
$691$	−6.50392e32	−1.44271	−0.721353	−	0.692568i	$-0.756479\pi$
$691$	−6.50392e32	−1.44271	−0.721353	+	0.692568i	$0.756479\pi$
$692$	0	0
$693$	−1.11551e31	−0.0239355
$694$	0	0
$695$	3.07499e32	0.638290
$696$	0	0
$697$	9.09874e31	0.182728
$698$	0	0
$699$	9.26180e31	0.179973
$700$	0	0
$701$	7.16326e32	1.34696	0.673478	−	0.739207i	$-0.264800\pi$
$701$	7.16326e32	1.34696	0.673478	+	0.739207i	$0.264800\pi$
$702$	0	0
$703$	1.38397e32	0.251849
$704$	0	0
$705$	−4.17440e32	−0.735223
$706$	0	0
$707$	−3.95784e32	−0.674737
$708$	0	0
$709$	−6.66300e32	−1.09961	−0.549804	−	0.835294i	$-0.685298\pi$
$709$	−6.66300e32	−1.09961	−0.549804	+	0.835294i	$0.685298\pi$
$710$	0	0
$711$	1.35682e32	0.216781
$712$	0	0
$713$	1.62250e32	0.250989
$714$	0	0
$715$	−4.82311e31	−0.0722449
$716$	0	0
$717$	1.11847e32	0.162239
$718$	0	0
$719$	−9.30896e31	−0.130773	−0.0653866	−	0.997860i	$-0.520828\pi$
$719$	−9.30896e31	−0.130773	−0.0653866	+	0.997860i	$0.520828\pi$
$720$	0	0
$721$	−7.90941e32	−1.07619
$722$	0	0
$723$	6.04093e31	0.0796185
$724$	0	0
$725$	1.12821e33	1.44047
$726$	0	0
$727$	1.16189e33	1.43721	0.718604	−	0.695420i	$-0.244781\pi$
$727$	1.16189e33	1.43721	0.718604	+	0.695420i	$0.244781\pi$
$728$	0	0
$729$	3.09032e31	0.0370370
$730$	0	0
$731$	1.89940e33	2.20580
$732$	0	0
$733$	−2.79603e32	−0.314663	−0.157331	−	0.987546i	$-0.550289\pi$
$733$	−2.79603e32	−0.314663	−0.157331	+	0.987546i	$0.550289\pi$
$734$	0	0
$735$	−1.39936e33	−1.52625
$736$	0	0
$737$	−4.03060e31	−0.0426083
$738$	0	0
$739$	−2.20384e32	−0.225823	−0.112911	−	0.993605i	$-0.536018\pi$
$739$	−2.20384e32	−0.225823	−0.112911	+	0.993605i	$0.536018\pi$
$740$	0	0
$741$	7.35775e32	0.730862
$742$	0	0
$743$	4.23262e32	0.407603	0.203802	−	0.979012i	$-0.434670\pi$
$743$	4.23262e32	0.407603	0.203802	+	0.979012i	$0.434670\pi$
$744$	0	0
$745$	−1.26241e33	−1.17870
$746$	0	0
$747$	−4.26171e32	−0.385831
$748$	0	0
$749$	1.40089e33	1.22988
$750$	0	0
$751$	2.42857e32	0.206771	0.103386	−	0.994641i	$-0.467032\pi$
$751$	2.42857e32	0.206771	0.103386	+	0.994641i	$0.467032\pi$
$752$	0	0
$753$	7.21119e32	0.595477
$754$	0	0
$755$	−2.85696e33	−2.28830
$756$	0	0
$757$	−6.18178e32	−0.480298	−0.240149	−	0.970736i	$-0.577196\pi$
$757$	−6.18178e32	−0.480298	−0.240149	+	0.970736i	$0.577196\pi$
$758$	0	0
$759$	5.89656e31	0.0444445
$760$	0	0
$761$	−5.12850e32	−0.375030	−0.187515	−	0.982262i	$-0.560043\pi$
$761$	−5.12850e32	−0.375030	−0.187515	+	0.982262i	$0.560043\pi$
$762$	0	0
$763$	−4.46654e33	−3.16912
$764$	0	0
$765$	8.83714e32	0.608423
$766$	0	0
$767$	−2.64980e32	−0.177038
$768$	0	0
$769$	1.16105e33	0.752832	0.376416	−	0.926451i	$-0.377156\pi$
$769$	1.16105e33	0.752832	0.376416	+	0.926451i	$0.377156\pi$
$770$	0	0
$771$	4.15747e32	0.261640
$772$	0	0
$773$	−2.02327e33	−1.23592	−0.617960	−	0.786210i	$-0.712041\pi$
$773$	−2.02327e33	−1.23592	−0.617960	+	0.786210i	$0.712041\pi$
$774$	0	0
$775$	2.51030e32	0.148853
$776$	0	0
$777$	−3.95776e32	−0.227828
$778$	0	0
$779$	−2.77102e32	−0.154867
$780$	0	0
$781$	−9.56539e31	−0.0519058
$782$	0	0
$783$	4.91288e32	0.258867
$784$	0	0
$785$	−3.37392e33	−1.72637
$786$	0	0
$787$	2.21364e33	1.10001	0.550005	−	0.835161i	$-0.314626\pi$
$787$	2.21364e33	1.10001	0.550005	+	0.835161i	$0.314626\pi$
$788$	0	0
$789$	−8.43295e32	−0.407000
$790$	0	0
$791$	−4.12981e33	−1.93598
$792$	0	0
$793$	−3.45871e33	−1.57497
$794$	0	0
$795$	1.90320e33	0.841904
$796$	0	0
$797$	3.48203e33	1.49645	0.748224	−	0.663447i	$-0.230907\pi$
$797$	3.48203e33	1.49645	0.748224	+	0.663447i	$0.230907\pi$
$798$	0	0
$799$	−2.68805e33	−1.12240
$800$	0	0
$801$	1.02847e33	0.417272
$802$	0	0
$803$	1.00300e32	0.0395431
$804$	0	0
$805$	1.14236e34	4.37675
$806$	0	0
$807$	9.56107e32	0.356009
$808$	0	0
$809$	−4.69918e33	−1.70065	−0.850323	−	0.526262i	$-0.823593\pi$
$809$	−4.69918e33	−1.70065	−0.850323	+	0.526262i	$0.823593\pi$
$810$	0	0
$811$	2.56473e33	0.902197	0.451099	−	0.892474i	$-0.351032\pi$
$811$	2.56473e33	0.902197	0.451099	+	0.892474i	$0.351032\pi$
$812$	0	0
$813$	−1.60355e33	−0.548328
$814$	0	0
$815$	−2.29314e33	−0.762286
$816$	0	0
$817$	−5.78461e33	−1.86948
$818$	0	0
$819$	−2.10410e33	−0.661153
$820$	0	0
$821$	4.82193e33	1.47324	0.736622	−	0.676304i	$-0.236420\pi$
$821$	4.82193e33	1.47324	0.736622	+	0.676304i	$0.236420\pi$
$822$	0	0
$823$	−2.45636e33	−0.729785	−0.364892	−	0.931050i	$-0.618894\pi$
$823$	−2.45636e33	−0.729785	−0.364892	+	0.931050i	$0.618894\pi$
$824$	0	0
$825$	9.12304e31	0.0263584
$826$	0	0
$827$	−8.49118e32	−0.238591	−0.119296	−	0.992859i	$-0.538064\pi$
$827$	−8.49118e32	−0.238591	−0.119296	+	0.992859i	$0.538064\pi$
$828$	0	0
$829$	−2.56796e33	−0.701796	−0.350898	−	0.936414i	$-0.614124\pi$
$829$	−2.56796e33	−0.701796	−0.350898	+	0.936414i	$0.614124\pi$
$830$	0	0
$831$	3.88084e33	1.03161
$832$	0	0
$833$	−9.01099e33	−2.33000
$834$	0	0
$835$	2.97439e33	0.748174
$836$	0	0
$837$	1.09313e32	0.0267503
$838$	0	0
$839$	2.61032e33	0.621487	0.310743	−	0.950494i	$-0.399422\pi$
$839$	2.61032e33	0.621487	0.310743	+	0.950494i	$0.399422\pi$
$840$	0	0
$841$	3.49362e33	0.809323
$842$	0	0
$843$	−2.06386e32	−0.0465226
$844$	0	0
$845$	−2.53704e33	−0.556512
$846$	0	0
$847$	7.87586e33	1.68128
$848$	0	0
$849$	−3.14268e33	−0.652923
$850$	0	0
$851$	2.09205e33	0.423042
$852$	0	0
$853$	1.58808e33	0.312578	0.156289	−	0.987711i	$-0.450047\pi$
$853$	1.58808e33	0.312578	0.156289	+	0.987711i	$0.450047\pi$
$854$	0	0
$855$	−2.69135e33	−0.515655
$856$	0	0
$857$	−6.65431e33	−1.24115	−0.620574	−	0.784148i	$-0.713100\pi$
$857$	−6.65431e33	−1.24115	−0.620574	+	0.784148i	$0.713100\pi$
$858$	0	0
$859$	−2.19052e33	−0.397765	−0.198883	−	0.980023i	$-0.563731\pi$
$859$	−2.19052e33	−0.397765	−0.198883	+	0.980023i	$0.563731\pi$
$860$	0	0
$861$	7.92430e32	0.140096
$862$	0	0
$863$	8.59960e33	1.48032	0.740158	−	0.672433i	$-0.234751\pi$
$863$	8.59960e33	1.48032	0.740158	+	0.672433i	$0.234751\pi$
$864$	0	0
$865$	8.31077e33	1.39302
$866$	0	0
$867$	2.15337e33	0.351478
$868$	0	0
$869$	1.74424e32	0.0277254
$870$	0	0
$871$	−7.60260e33	−1.17694
$872$	0	0
$873$	−3.87062e33	−0.583602
$874$	0	0
$875$	1.17002e33	0.171831
$876$	0	0
$877$	1.88095e33	0.269081	0.134540	−	0.990908i	$-0.457044\pi$
$877$	1.88095e33	0.269081	0.134540	+	0.990908i	$0.457044\pi$
$878$	0	0
$879$	1.93732e33	0.269979
$880$	0	0
$881$	−3.11670e33	−0.423129	−0.211564	−	0.977364i	$-0.567856\pi$
$881$	−3.11670e33	−0.423129	−0.211564	+	0.977364i	$0.567856\pi$
$882$	0	0
$883$	−4.91478e33	−0.650063	−0.325032	−	0.945703i	$-0.605375\pi$
$883$	−4.91478e33	−0.650063	−0.325032	+	0.945703i	$0.605375\pi$
$884$	0	0
$885$	9.69256e32	0.124908
$886$	0	0
$887$	−2.63916e33	−0.331394	−0.165697	−	0.986177i	$-0.552987\pi$
$887$	−2.63916e33	−0.331394	−0.165697	+	0.986177i	$0.552987\pi$
$888$	0	0
$889$	1.50663e34	1.84347
$890$	0	0
$891$	3.97270e31	0.00473688
$892$	0	0
$893$	8.18644e33	0.951268
$894$	0	0
$895$	8.46800e33	0.958994
$896$	0	0
$897$	1.11222e34	1.22766
$898$	0	0
$899$	1.73782e33	0.186969
$900$	0	0
$901$	1.22554e34	1.28527
$902$	0	0
$903$	1.65423e34	1.69117
$904$	0	0
$905$	−3.35335e33	−0.334210
$906$	0	0
$907$	−5.24324e32	−0.0509467	−0.0254733	−	0.999676i	$-0.508109\pi$
$907$	−5.24324e32	−0.0509467	−0.0254733	+	0.999676i	$0.508109\pi$
$908$	0	0
$909$	1.40951e33	0.133532
$910$	0	0
$911$	−1.37016e34	−1.26564	−0.632819	−	0.774300i	$-0.718102\pi$
$911$	−1.37016e34	−1.26564	−0.632819	+	0.774300i	$0.718102\pi$
$912$	0	0
$913$	−5.47856e32	−0.0493460
$914$	0	0
$915$	1.26514e34	1.11121
$916$	0	0
$917$	1.17926e33	0.101009
$918$	0	0
$919$	−1.66584e34	−1.39157	−0.695786	−	0.718249i	$-0.744944\pi$
$919$	−1.66584e34	−1.39157	−0.695786	+	0.718249i	$0.744944\pi$
$920$	0	0
$921$	2.23972e33	0.182477
$922$	0	0
$923$	−1.80424e34	−1.43376
$924$	0	0
$925$	3.23679e33	0.250891
$926$	0	0
$927$	2.81680e33	0.212980
$928$	0	0
$929$	1.41966e34	1.04713	0.523567	−	0.851984i	$-0.324601\pi$
$929$	1.41966e34	1.04713	0.523567	+	0.851984i	$0.324601\pi$
$930$	0	0
$931$	2.74429e34	1.97474
$932$	0	0
$933$	1.22827e34	0.862295
$934$	0	0
$935$	1.13604e33	0.0778146
$936$	0	0
$937$	−2.31238e34	−1.54545	−0.772723	−	0.634743i	$-0.781106\pi$
$937$	−2.31238e34	−1.54545	−0.772723	+	0.634743i	$0.781106\pi$
$938$	0	0
$939$	−3.70977e32	−0.0241932
$940$	0	0
$941$	−2.49154e34	−1.58558	−0.792788	−	0.609497i	$-0.791371\pi$
$941$	−2.49154e34	−1.58558	−0.792788	+	0.609497i	$0.791371\pi$
$942$	0	0
$943$	−4.18875e33	−0.260136
$944$	0	0
$945$	7.69647e33	0.466472
$946$	0	0
$947$	−1.33752e34	−0.791179	−0.395590	−	0.918427i	$-0.629460\pi$
$947$	−1.33752e34	−0.791179	−0.395590	+	0.918427i	$0.629460\pi$
$948$	0	0
$949$	1.89187e34	1.09227
$950$	0	0
$951$	−1.93373e34	−1.08973
$952$	0	0
$953$	3.72509e33	0.204913	0.102456	−	0.994738i	$-0.467330\pi$
$953$	3.72509e33	0.204913	0.102456	+	0.994738i	$0.467330\pi$
$954$	0	0
$955$	3.39267e34	1.82181
$956$	0	0
$957$	6.31567e32	0.0331079
$958$	0	0
$959$	5.67227e32	0.0290297
$960$	0	0
$961$	−1.96266e34	−0.980679
$962$	0	0
$963$	−4.98901e33	−0.243395
$964$	0	0
$965$	1.95825e34	0.932832
$966$	0	0
$967$	−3.25266e34	−1.51298	−0.756492	−	0.654003i	$-0.773088\pi$
$967$	−3.25266e34	−1.51298	−0.756492	+	0.654003i	$0.773088\pi$
$968$	0	0
$969$	−1.73306e34	−0.787208
$970$	0	0
$971$	−3.07260e34	−1.36296	−0.681482	−	0.731835i	$-0.738664\pi$
$971$	−3.07260e34	−1.36296	−0.681482	+	0.731835i	$0.738664\pi$
$972$	0	0
$973$	1.72451e34	0.747083
$974$	0	0
$975$	1.72081e34	0.728080
$976$	0	0
$977$	−1.55004e34	−0.640555	−0.320278	−	0.947324i	$-0.603776\pi$
$977$	−1.55004e34	−0.640555	−0.320278	+	0.947324i	$0.603776\pi$
$978$	0	0
$979$	1.32214e33	0.0533673
$980$	0	0
$981$	1.59068e34	0.627175
$982$	0	0
$983$	3.55003e34	1.36731	0.683655	−	0.729805i	$-0.260389\pi$
$983$	3.55003e34	1.36731	0.683655	+	0.729805i	$0.260389\pi$
$984$	0	0
$985$	−3.42005e34	−1.28681
$986$	0	0
$987$	−2.34108e34	−0.860537
$988$	0	0
$989$	−8.74419e34	−3.14024
$990$	0	0
$991$	−4.18079e34	−1.46694	−0.733469	−	0.679723i	$-0.762100\pi$
$991$	−4.18079e34	−1.46694	−0.733469	+	0.679723i	$0.762100\pi$
$992$	0	0
$993$	−2.31183e34	−0.792575
$994$	0	0
$995$	3.61738e34	1.21179
$996$	0	0
$997$	−4.07343e34	−1.33342	−0.666708	−	0.745319i	$-0.732297\pi$
$997$	−4.07343e34	−1.33342	−0.666708	+	0.745319i	$0.732297\pi$
$998$	0	0
$999$	1.40948e33	0.0450876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.24.a.l.1.1		3
4.3	odd	2		24.24.a.b.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.24.a.b.1.1	✓	3	4.3	odd	2
48.24.a.l.1.1		3	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:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+177147. q^{3} -1.57121e8 q^{5} -8.81163e9 q^{7} +3.13811e10 q^{9} +O(q^{10})\)	\(q+177147. q^{3} -1.57121e8 q^{5} -8.81163e9 q^{7} +3.13811e10 q^{9} +4.03414e10 q^{11} +7.60927e12 q^{13} -2.78335e13 q^{15} -1.79230e14 q^{17} +5.45844e14 q^{19} -1.56095e15 q^{21} +8.25114e15 q^{23} +1.27660e16 q^{25} +5.55906e15 q^{27} +8.83761e16 q^{29} +1.96640e16 q^{31} +7.14635e15 q^{33} +1.38449e18 q^{35} +2.53547e17 q^{37} +1.34796e18 q^{39} -5.07657e17 q^{41} -1.05976e19 q^{43} -4.93062e18 q^{45} +1.49978e19 q^{47} +5.02761e19 q^{49} -3.17501e19 q^{51} -6.83781e19 q^{53} -6.33847e18 q^{55} +9.66946e19 q^{57} -3.48234e19 q^{59} -4.54539e20 q^{61} -2.76518e20 q^{63} -1.19557e21 q^{65} -9.99124e20 q^{67} +1.46167e21 q^{69} -2.37111e21 q^{71} +2.48627e21 q^{73} +2.26146e21 q^{75} -3.55473e20 q^{77} +4.32369e21 q^{79} +9.84771e20 q^{81} -1.35805e22 q^{83} +2.81608e22 q^{85} +1.56556e22 q^{87} +3.27737e22 q^{89} -6.70501e22 q^{91} +3.48341e21 q^{93} -8.57634e22 q^{95} -1.23343e23 q^{97} +1.26595e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9}+O(q^{10}) \) 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9	\( 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9} + 66881746140 q^{11} + 5138719820610 q^{13} - 13680206481402 q^{15} - 232964235839898 q^{17} + 39051643356612 q^{19} - 591782205641184 q^{21} + 43\!\cdots\!64 q^{23}+ \cdots + 20\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9 + 66881746140 * q^11 + 5138719820610 * q^13 - 13680206481402 * q^15 - 232964235839898 * q^17 + 39051643356612 * q^19 - 591782205641184 * q^21 + 4358324846210664 * q^23 - 5164619678396379 * q^25 + 16677181699666569 * q^27 - 25471683594266694 * q^29 + 125107985305907448 * q^31 + 11847900683462580 * q^33 + 1311200449040041536 * q^35 - 1292646678108250614 * q^37 + 910308800061599670 * q^39 + 144934201638163518 * q^41 - 8133920080752699684 * q^43 - 2423407537560920094 * q^45 - 10029621278906529216 * q^47 + 41823513653599245723 * q^49 - 41268915486330411006 * q^51 - 28251334125947614062 * q^53 - 85062431363722250328 * q^55 + 6917881465693745964 * q^57 - 187335386683361480100 * q^59 - 555296430938057984862 * q^61 - 104832442382718822048 * q^63 - 1392409010623622551476 * q^65 - 1602075967132026564156 * q^67 + 772064171531680495608 * q^69 - 3528346267933953055080 * q^71 + 2100403635635469857646 * q^73 - 914896882168883350713 * q^75 + 8279040903546572168064 * q^77 - 12175210588212413874744 * q^79 + 2954312706550833698643 * q^81 - 30717412717926068652348 * q^83 + 44709116429093587923108 * q^85 - 4512232333673562042018 * q^87 + 58208079477660202949598 * q^89 - 63276323944720307992512 * q^91 + 22162504272985586690856 * q^93 - 79678214445310528084968 * q^95 + 76406389646793178923846 * q^97 + 2098820062373345659260 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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