LMFDB - Embedded newform 48.25.g.c.31.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,25,Mod(31,48)]

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([1, 0, 0]))

N = Newforms(chi, 25, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 25);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 25 $
Character orbit:	$[\chi]$	$=$	48.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$175.184233084$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} + 2573102906805 x^{6} + \cdots + 38\!\cdots\!76 $ x^8 - 2x^7 + 2573102906805x^6 + 1783629383693579998x^5 + 5999817840484201711703029x^4 + 2294726839216023950878194166302x^3 + 2393325969444049306469752166042088180x^2 - 553850057255288932043196968402428127074848*x + 385689371056814708765718482430825246402445598976
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{52}\cdot 3^{15}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.3
Root			$-447343. + 774820. i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.25.g.c.31.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-306828. i q^{3} +2.72537e8 q^{5} +2.40937e10i q^{7} -9.41432e10 q^{9} +O(q^{10})$	$q-306828. i q^{3} +2.72537e8 q^{5} +2.40937e10i q^{7} -9.41432e10 q^{9} +3.29168e12i q^{11} -6.11450e12 q^{13} -8.36217e13i q^{15} +4.02211e14 q^{17} +2.49861e15i q^{19} +7.39260e15 q^{21} +2.45325e16i q^{23} +1.46715e16 q^{25} +2.88857e16i q^{27} +4.38191e16 q^{29} +9.03616e17i q^{31} +1.00998e18 q^{33} +6.56640e18i q^{35} -5.07183e18 q^{37} +1.87610e18i q^{39} -3.14455e19 q^{41} -2.74548e19i q^{43} -2.56575e19 q^{45} -7.90113e19i q^{47} -3.88923e20 q^{49} -1.23409e20i q^{51} +3.47052e20 q^{53} +8.97102e20i q^{55} +7.66641e20 q^{57} +1.27751e20i q^{59} +1.61163e21 q^{61} -2.26825e21i q^{63} -1.66643e21 q^{65} -1.28097e22i q^{67} +7.52724e21 q^{69} -1.55644e22i q^{71} -1.51996e22 q^{73} -4.50163e21i q^{75} -7.93085e22 q^{77} -3.33576e22i q^{79} +8.86294e21 q^{81} +1.53208e23i q^{83} +1.09617e23 q^{85} -1.34449e22i q^{87} -3.74743e23 q^{89} -1.47321e23i q^{91} +2.77254e23 q^{93} +6.80962e23i q^{95} +1.14440e24 q^{97} -3.09889e23i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 462495600 q^{5} - 753145430616 q^{9}+O(q^{10}) $ 8 * q + 462495600 * q^5 - 753145430616 * q^9	$ 8 q + 462495600 q^{5} - 753145430616 q^{9} - 1530027465904 q^{13} - 35559567490608 q^{17} + 84\!\cdots\!56 q^{21}+ \cdots + 12\!\cdots\!80 q^{97}+O(q^{100}) $ 8 * q + 462495600 * q^5 - 753145430616 * q^9 - 1530027465904 * q^13 - 35559567490608 * q^17 + 8439296905969056 * q^21 + 142743524026591000 * q^25 + 476759032205115312 * q^29 - 2160858161263465440 * q^33 - 15760822588656347888 * q^37 - 45389784103575648048 * q^41 - 43540805977500661200 * q^45 + 30951519803487811208 * q^49 - 1729329636487575669840 * q^53 - 1269107008528107899808 * q^57 + 2694636617324641569424 * q^61 - 3149721076444705130400 * q^65 - 6624622265621256652032 * q^69 - 54562892980695920861680 * q^73 - 268036140084799319975808 * q^77 + 70903504957220008767432 * q^81 - 328621744624984652887200 * q^85 + 223962296701000391132688 * q^89 + 645901021207453779145440 * q^93 + 1281098591387557493688080 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 306828.i	− 0.577350i
$3$	− 306828.i	− 0.577350i
$4$	0	0
$5$	2.72537e8	1.11631	0.558155	−	0.829737i	$-0.311509\pi$
$5$	2.72537e8	1.11631	0.558155	+	0.829737i	$0.311509\pi$
$6$	0	0
$7$	2.40937e10i	1.74071i	0.492426	+	0.870354i	$0.336110\pi$
$7$	2.40937e10i	1.74071i	−0.492426	+	0.870354i	$0.663890\pi$
$8$	0	0
$9$	−9.41432e10	−0.333333
$10$	0	0
$11$	3.29168e12i	1.04883i	0.851463	+	0.524415i	$0.175716\pi$
$11$	3.29168e12i	1.04883i	−0.851463	+	0.524415i	$0.824284\pi$
$12$	0	0
$13$	−6.11450e12	−0.262447	−0.131223	−	0.991353i	$-0.541890\pi$
$13$	−6.11450e12	−0.262447	−0.131223	+	0.991353i	$0.541890\pi$
$14$	0	0
$15$	− 8.36217e13i	− 0.644502i
$16$	0	0
$17$	4.02211e14	0.690346	0.345173	−	0.938539i	$-0.387820\pi$
$17$	4.02211e14	0.690346	0.345173	+	0.938539i	$0.387820\pi$
$18$	0	0
$19$	2.49861e15i	1.12890i	0.825468	+	0.564449i	$0.190911\pi$
$19$	2.49861e15i	1.12890i	−0.825468	+	0.564449i	$0.809089\pi$
$20$	0	0
$21$	7.39260e15	1.00500
$22$	0	0
$23$	2.45325e16i	1.11946i	0.828676	+	0.559728i	$0.189094\pi$
$23$	2.45325e16i	1.11946i	−0.828676	+	0.559728i	$0.810906\pi$
$24$	0	0
$25$	1.46715e16	0.246148
$26$	0	0
$27$	2.88857e16i	0.192450i
$28$	0	0
$29$	4.38191e16	0.123848	0.0619238	−	0.998081i	$-0.480276\pi$
$29$	4.38191e16	0.123848	0.0619238	+	0.998081i	$0.480276\pi$
$30$	0	0
$31$	9.03616e17i	1.14721i	0.819132	+	0.573606i	$0.194456\pi$
$31$	9.03616e17i	1.14721i	−0.819132	+	0.573606i	$0.805544\pi$
$32$	0	0
$33$	1.00998e18	0.605542
$34$	0	0
$35$	6.56640e18i	1.94317i
$36$	0	0
$37$	−5.07183e18	−0.770449	−0.385224	−	0.922823i	$-0.625876\pi$
$37$	−5.07183e18	−0.770449	−0.385224	+	0.922823i	$0.625876\pi$
$38$	0	0
$39$	1.87610e18i	0.151524i
$40$	0	0
$41$	−3.14455e19	−1.39365	−0.696824	−	0.717243i	$-0.745404\pi$
$41$	−3.14455e19	−1.39365	−0.696824	+	0.717243i	$0.745404\pi$
$42$	0	0
$43$	− 2.74548e19i	− 0.687064i	−0.939141	−	0.343532i	$-0.888377\pi$
$43$	− 2.74548e19i	− 0.687064i	0.939141	−	0.343532i	$-0.111623\pi$
$44$	0	0
$45$	−2.56575e19	−0.372103
$46$	0	0
$47$	− 7.90113e19i	− 0.680010i	−0.940424	−	0.340005i	$-0.889571\pi$
$47$	− 7.90113e19i	− 0.680010i	0.940424	−	0.340005i	$-0.110429\pi$
$48$	0	0
$49$	−3.88923e20	−2.03007
$50$	0	0
$51$	− 1.23409e20i	− 0.398572i
$52$	0	0
$53$	3.47052e20	0.706455	0.353227	−	0.935537i	$-0.385084\pi$
$53$	3.47052e20	0.706455	0.353227	+	0.935537i	$0.385084\pi$
$54$	0	0
$55$	8.97102e20i	1.17082i
$56$	0	0
$57$	7.66641e20	0.651769
$58$	0	0
$59$	1.27751e20i	0.0718028i	0.999355	+	0.0359014i	$0.0114302\pi$
$59$	1.27751e20i	0.0718028i	−0.999355	+	0.0359014i	$0.988570\pi$
$60$	0	0
$61$	1.61163e21	0.607168	0.303584	−	0.952805i	$-0.401817\pi$
$61$	1.61163e21	0.607168	0.303584	+	0.952805i	$0.401817\pi$
$62$	0	0
$63$	− 2.26825e21i	− 0.580236i
$64$	0	0
$65$	−1.66643e21	−0.292972
$66$	0	0
$67$	− 1.28097e22i	− 1.56545i	−0.622366	−	0.782726i	$-0.713828\pi$
$67$	− 1.28097e22i	− 1.56545i	0.622366	−	0.782726i	$-0.286172\pi$
$68$	0	0
$69$	7.52724e21	0.646318
$70$	0	0
$71$	− 1.55644e22i	− 0.948491i	−0.880393	−	0.474245i	$-0.842721\pi$
$71$	− 1.55644e22i	− 0.948491i	0.880393	−	0.474245i	$-0.157279\pi$
$72$	0	0
$73$	−1.51996e22	−0.663680	−0.331840	−	0.943336i	$-0.607669\pi$
$73$	−1.51996e22	−0.663680	−0.331840	+	0.943336i	$0.607669\pi$
$74$	0	0
$75$	− 4.50163e21i	− 0.142113i
$76$	0	0
$77$	−7.93085e22	−1.82571
$78$	0	0
$79$	− 3.33576e22i	− 0.564507i	−0.959340	−	0.282253i	$-0.908918\pi$
$79$	− 3.33576e22i	− 0.564507i	0.959340	−	0.282253i	$-0.0910819\pi$
$80$	0	0
$81$	8.86294e21	0.111111
$82$	0	0
$83$	1.53208e23i	1.43332i	0.697422	+	0.716661i	$0.254330\pi$
$83$	1.53208e23i	1.43332i	−0.697422	+	0.716661i	$0.745670\pi$
$84$	0	0
$85$	1.09617e23	0.770640
$86$	0	0
$87$	− 1.34449e22i	− 0.0715034i
$88$	0	0
$89$	−3.74743e23	−1.51724	−0.758618	−	0.651536i	$-0.774125\pi$
$89$	−3.74743e23	−1.51724	−0.758618	+	0.651536i	$0.774125\pi$
$90$	0	0
$91$	− 1.47321e23i	− 0.456843i
$92$	0	0
$93$	2.77254e23	0.662343
$94$	0	0
$95$	6.80962e23i	1.26020i
$96$	0	0
$97$	1.14440e24	1.64936	0.824681	−	0.565598i	$-0.191355\pi$
$97$	1.14440e24	1.64936	0.824681	+	0.565598i	$0.191355\pi$
$98$	0	0
$99$	− 3.09889e23i	− 0.349610i
$100$	0	0
$101$	−1.09154e24	−0.968685	−0.484343	−	0.874878i	$-0.660941\pi$
$101$	−1.09154e24	−0.968685	−0.484343	+	0.874878i	$0.660941\pi$
$102$	0	0
$103$	− 2.51194e24i	− 1.76182i	−0.473281	−	0.880912i	$-0.656930\pi$
$103$	− 2.51194e24i	− 1.76182i	0.473281	−	0.880912i	$-0.343070\pi$
$104$	0	0
$105$	2.01475e24	1.12189
$106$	0	0
$107$	8.36229e23i	0.371296i	0.982616	+	0.185648i	$0.0594384\pi$
$107$	8.36229e23i	0.371296i	−0.982616	+	0.185648i	$0.940562\pi$
$108$	0	0
$109$	−4.44797e24	−1.58141	−0.790704	−	0.612199i	$-0.790285\pi$
$109$	−4.44797e24	−1.58141	−0.790704	+	0.612199i	$0.790285\pi$
$110$	0	0
$111$	1.55618e24i	0.444819i
$112$	0	0
$113$	6.41500e24	1.47998	0.739989	−	0.672619i	$-0.234831\pi$
$113$	6.41500e24	1.47998	0.739989	+	0.672619i	$0.234831\pi$
$114$	0	0
$115$	6.68599e24i	1.24966i
$116$	0	0
$117$	5.75639e23	0.0874822
$118$	0	0
$119$	9.69073e24i	1.20169i
$120$	0	0
$121$	−9.85407e23	−0.100044
$122$	0	0
$123$	9.64836e24i	0.804623i
$124$	0	0
$125$	−1.22459e25	−0.841533
$126$	0	0
$127$	− 2.87030e25i	− 1.63036i	−0.579211	−	0.815178i	$-0.696639\pi$
$127$	− 2.87030e25i	− 1.63036i	0.579211	−	0.815178i	$-0.303361\pi$
$128$	0	0
$129$	−8.42390e24	−0.396677
$130$	0	0
$131$	− 2.36778e25i	− 0.927014i	−0.886093	−	0.463507i	$-0.846591\pi$
$131$	− 2.36778e25i	− 0.927014i	0.886093	−	0.463507i	$-0.153409\pi$
$132$	0	0
$133$	−6.02006e25	−1.96508
$134$	0	0
$135$	7.87242e24i	0.214834i
$136$	0	0
$137$	6.78571e25	1.55220	0.776102	−	0.630608i	$-0.217194\pi$
$137$	6.78571e25	1.55220	0.776102	+	0.630608i	$0.217194\pi$
$138$	0	0
$139$	− 6.01862e25i	− 1.15696i	−0.815696	−	0.578481i	$-0.803646\pi$
$139$	− 6.01862e25i	− 1.15696i	0.815696	−	0.578481i	$-0.196354\pi$
$140$	0	0
$141$	−2.42429e25	−0.392604
$142$	0	0
$143$	− 2.01270e25i	− 0.275262i
$144$	0	0
$145$	1.19423e25	0.138252
$146$	0	0
$147$	1.19332e26i	1.17206i
$148$	0	0
$149$	1.22220e26	1.02072	0.510362	−	0.859960i	$-0.329511\pi$
$149$	1.22220e26	1.02072	0.510362	+	0.859960i	$0.329511\pi$
$150$	0	0
$151$	− 1.13830e25i	− 0.0810091i	−0.999179	−	0.0405046i	$-0.987103\pi$
$151$	− 1.13830e25i	− 0.0810091i	0.999179	−	0.0405046i	$-0.0128965\pi$
$152$	0	0
$153$	−3.78654e25	−0.230115
$154$	0	0
$155$	2.46268e26i	1.28064i
$156$	0	0
$157$	−1.95691e26	−0.872520	−0.436260	−	0.899821i	$-0.643697\pi$
$157$	−1.95691e26	−0.872520	−0.436260	+	0.899821i	$0.643697\pi$
$158$	0	0
$159$	− 1.06485e26i	− 0.407872i
$160$	0	0
$161$	−5.91076e26	−1.94865
$162$	0	0
$163$	4.53501e26i	1.28922i	0.764512	+	0.644610i	$0.222980\pi$
$163$	4.53501e26i	1.28922i	−0.764512	+	0.644610i	$0.777020\pi$
$164$	0	0
$165$	2.75256e26	0.675973
$166$	0	0
$167$	− 1.36015e26i	− 0.289061i	−0.989500	−	0.144531i	$-0.953833\pi$
$167$	− 1.36015e26i	− 0.289061i	0.989500	−	0.144531i	$-0.0461672\pi$
$168$	0	0
$169$	−5.05414e26	−0.931122
$170$	0	0
$171$	− 2.35227e26i	− 0.376299i
$172$	0	0
$173$	−9.51269e26	−1.32358	−0.661790	−	0.749689i	$-0.730203\pi$
$173$	−9.51269e26	−1.32358	−0.661790	+	0.749689i	$0.730203\pi$
$174$	0	0
$175$	3.53491e26i	0.428471i
$176$	0	0
$177$	3.91976e25	0.0414554
$178$	0	0
$179$	1.83854e27i	1.69917i	0.527455	+	0.849583i	$0.323146\pi$
$179$	1.83854e27i	1.69917i	−0.527455	+	0.849583i	$0.676854\pi$
$180$	0	0
$181$	1.39884e27	1.13142	0.565711	−	0.824604i	$-0.308602\pi$
$181$	1.39884e27	1.13142	0.565711	+	0.824604i	$0.308602\pi$
$182$	0	0
$183$	− 4.94494e26i	− 0.350548i
$184$	0	0
$185$	−1.38226e27	−0.860060
$186$	0	0
$187$	1.32395e27i	0.724056i
$188$	0	0
$189$	−6.95963e26	−0.335000
$190$	0	0
$191$	− 9.15803e26i	− 0.388509i	−0.980951	−	0.194255i	$-0.937771\pi$
$191$	− 9.15803e26i	− 0.388509i	0.980951	−	0.194255i	$-0.0622288\pi$
$192$	0	0
$193$	−2.44418e26	−0.0915051	−0.0457526	−	0.998953i	$-0.514569\pi$
$193$	−2.44418e26	−0.0915051	−0.0457526	+	0.998953i	$0.514569\pi$
$194$	0	0
$195$	5.11305e26i	0.169147i
$196$	0	0
$197$	−4.44070e27	−1.29974	−0.649870	−	0.760045i	$-0.725177\pi$
$197$	−4.44070e27	−1.29974	−0.649870	+	0.760045i	$0.725177\pi$
$198$	0	0
$199$	3.19449e27i	0.828255i	0.910219	+	0.414128i	$0.135913\pi$
$199$	3.19449e27i	0.828255i	−0.910219	+	0.414128i	$0.864087\pi$
$200$	0	0
$201$	−3.93036e27	−0.903815
$202$	0	0
$203$	1.05576e27i	0.215582i
$204$	0	0
$205$	−8.57006e27	−1.55574
$206$	0	0
$207$	− 2.30956e27i	− 0.373152i
$208$	0	0
$209$	−8.22461e27	−1.18402
$210$	0	0
$211$	4.39031e27i	0.563775i	0.959447	+	0.281888i	$0.0909606\pi$
$211$	4.39031e27i	0.563775i	−0.959447	+	0.281888i	$0.909039\pi$
$212$	0	0
$213$	−4.77560e27	−0.547611
$214$	0	0
$215$	− 7.48245e27i	− 0.766977i
$216$	0	0
$217$	−2.17714e28	−1.99696
$218$	0	0
$219$	4.66367e27i	0.383176i
$220$	0	0
$221$	−2.45932e27	−0.181179
$222$	0	0
$223$	2.91173e28i	1.92527i	0.270795	+	0.962637i	$0.412714\pi$
$223$	2.91173e28i	1.92527i	−0.270795	+	0.962637i	$0.587286\pi$
$224$	0	0
$225$	−1.38123e27	−0.0820492
$226$	0	0
$227$	2.15103e28i	1.14904i	0.818490	+	0.574521i	$0.194812\pi$
$227$	2.15103e28i	1.14904i	−0.818490	+	0.574521i	$0.805188\pi$
$228$	0	0
$229$	−2.32405e28	−1.11743	−0.558714	−	0.829360i	$-0.688705\pi$
$229$	−2.32405e28	−1.11743	−0.558714	+	0.829360i	$0.688705\pi$
$230$	0	0
$231$	2.43340e28i	1.05407i
$232$	0	0
$233$	4.42862e28	1.72980	0.864902	−	0.501940i	$-0.167380\pi$
$233$	4.42862e28	1.72980	0.864902	+	0.501940i	$0.167380\pi$
$234$	0	0
$235$	− 2.15335e28i	− 0.759101i
$236$	0	0
$237$	−1.02350e28	−0.325918
$238$	0	0
$239$	3.82773e28i	1.10196i	0.834518	+	0.550980i	$0.185746\pi$
$239$	3.82773e28i	1.10196i	−0.834518	+	0.550980i	$0.814254\pi$
$240$	0	0
$241$	5.68054e27	0.147974	0.0739869	−	0.997259i	$-0.476428\pi$
$241$	5.68054e27	0.147974	0.0739869	+	0.997259i	$0.476428\pi$
$242$	0	0
$243$	− 2.71939e27i	− 0.0641500i
$244$	0	0
$245$	−1.05996e29	−2.26618
$246$	0	0
$247$	− 1.52777e28i	− 0.296275i
$248$	0	0
$249$	4.70084e28	0.827528
$250$	0	0
$251$	1.65332e28i	0.264407i	0.991223	+	0.132203i	$0.0422052\pi$
$251$	1.65332e28i	0.264407i	−0.991223	+	0.132203i	$0.957795\pi$
$252$	0	0
$253$	−8.07529e28	−1.17412
$254$	0	0
$255$	− 3.36336e28i	− 0.444929i
$256$	0	0
$257$	−1.32291e29	−1.59344	−0.796718	−	0.604352i	$-0.793432\pi$
$257$	−1.32291e29	−1.59344	−0.796718	+	0.604352i	$0.793432\pi$
$258$	0	0
$259$	− 1.22199e29i	− 1.34113i
$260$	0	0
$261$	−4.12527e27	−0.0412825
$262$	0	0
$263$	1.37865e29i	1.25888i	0.777049	+	0.629441i	$0.216716\pi$
$263$	1.37865e29i	1.25888i	−0.777049	+	0.629441i	$0.783284\pi$
$264$	0	0
$265$	9.45844e28	0.788623
$266$	0	0
$267$	1.14981e29i	0.875976i
$268$	0	0
$269$	−4.79825e28	−0.334238	−0.167119	−	0.985937i	$-0.553446\pi$
$269$	−4.79825e28	−0.334238	−0.167119	+	0.985937i	$0.553446\pi$
$270$	0	0
$271$	1.95433e29i	1.24556i	0.782396	+	0.622781i	$0.213997\pi$
$271$	1.95433e29i	1.24556i	−0.782396	+	0.622781i	$0.786003\pi$
$272$	0	0
$273$	−4.52021e28	−0.263759
$274$	0	0
$275$	4.82940e28i	0.258167i
$276$	0	0
$277$	2.20252e28	0.107935	0.0539675	−	0.998543i	$-0.482813\pi$
$277$	2.20252e28	0.107935	0.0539675	+	0.998543i	$0.482813\pi$
$278$	0	0
$279$	− 8.50693e28i	− 0.382404i
$280$	0	0
$281$	1.56666e29	0.646398	0.323199	−	0.946331i	$-0.395242\pi$
$281$	1.56666e29	0.646398	0.323199	+	0.946331i	$0.395242\pi$
$282$	0	0
$283$	− 2.50495e29i	− 0.949211i	−0.880199	−	0.474605i	$-0.842591\pi$
$283$	− 2.50495e29i	− 0.949211i	0.880199	−	0.474605i	$-0.157409\pi$
$284$	0	0
$285$	2.08938e29	0.727577
$286$	0	0
$287$	− 7.57638e29i	− 2.42593i
$288$	0	0
$289$	−1.77675e29	−0.523422
$290$	0	0
$291$	− 3.51133e29i	− 0.952260i
$292$	0	0
$293$	6.76701e29	1.69038	0.845192	−	0.534463i	$-0.179486\pi$
$293$	6.76701e29	1.69038	0.845192	+	0.534463i	$0.179486\pi$
$294$	0	0
$295$	3.48169e28i	0.0801542i
$296$	0	0
$297$	−9.50825e28	−0.201847
$298$	0	0
$299$	− 1.50004e29i	− 0.293797i
$300$	0	0
$301$	6.61487e29	1.19598
$302$	0	0
$303$	3.34914e29i	0.559271i
$304$	0	0
$305$	4.39229e29	0.677787
$306$	0	0
$307$	9.50153e29i	1.35560i	0.735244	+	0.677802i	$0.237067\pi$
$307$	9.50153e29i	1.35560i	−0.735244	+	0.677802i	$0.762933\pi$
$308$	0	0
$309$	−7.70732e29	−1.01719
$310$	0	0
$311$	− 5.84388e29i	− 0.713800i	−0.934143	−	0.356900i	$-0.883834\pi$
$311$	− 5.84388e29i	− 0.713800i	0.934143	−	0.356900i	$-0.116166\pi$
$312$	0	0
$313$	1.50346e30	1.70043	0.850217	−	0.526433i	$-0.176471\pi$
$313$	1.50346e30	1.70043	0.850217	+	0.526433i	$0.176471\pi$
$314$	0	0
$315$	− 6.18182e29i	− 0.647723i
$316$	0	0
$317$	6.44575e29	0.625983	0.312991	−	0.949756i	$-0.398669\pi$
$317$	6.44575e29	0.625983	0.312991	+	0.949756i	$0.398669\pi$
$318$	0	0
$319$	1.44238e29i	0.129895i
$320$	0	0
$321$	2.56578e29	0.214368
$322$	0	0
$323$	1.00497e30i	0.779330i
$324$	0	0
$325$	−8.97092e28	−0.0646006
$326$	0	0
$327$	1.36476e30i	0.913026i
$328$	0	0
$329$	1.90367e30	1.18370
$330$	0	0
$331$	− 7.49003e29i	− 0.433060i	−0.976276	−	0.216530i	$-0.930526\pi$
$331$	− 7.49003e29i	− 0.433060i	0.976276	−	0.216530i	$-0.0694739\pi$
$332$	0	0
$333$	4.77478e29	0.256816
$334$	0	0
$335$	− 3.49110e30i	− 1.74753i
$336$	0	0
$337$	2.08197e30	0.970323	0.485162	−	0.874425i	$-0.338761\pi$
$337$	2.08197e30	0.970323	0.485162	+	0.874425i	$0.338761\pi$
$338$	0	0
$339$	− 1.96830e30i	− 0.854465i
$340$	0	0
$341$	−2.97441e30	−1.20323
$342$	0	0
$343$	− 4.75468e30i	− 1.79305i
$344$	0	0
$345$	2.05145e30	0.721491
$346$	0	0
$347$	2.71606e30i	0.891223i	0.895227	+	0.445611i	$0.147014\pi$
$347$	2.71606e30i	0.891223i	−0.895227	+	0.445611i	$0.852986\pi$
$348$	0	0
$349$	5.82965e29	0.178541	0.0892704	−	0.996007i	$-0.471546\pi$
$349$	5.82965e29	0.178541	0.0892704	+	0.996007i	$0.471546\pi$
$350$	0	0
$351$	− 1.76622e29i	− 0.0505079i
$352$	0	0
$353$	1.66237e30	0.444050	0.222025	−	0.975041i	$-0.428733\pi$
$353$	1.66237e30	0.444050	0.222025	+	0.975041i	$0.428733\pi$
$354$	0	0
$355$	− 4.24188e30i	− 1.05881i
$356$	0	0
$357$	2.97338e30	0.693797
$358$	0	0
$359$	− 2.36100e30i	− 0.515185i	−0.966254	−	0.257593i	$-0.917071\pi$
$359$	− 2.36100e30i	− 0.515185i	0.966254	−	0.257593i	$-0.0829292\pi$
$360$	0	0
$361$	−1.34427e30	−0.274410
$362$	0	0
$363$	3.02350e29i	0.0577605i
$364$	0	0
$365$	−4.14246e30	−0.740873
$366$	0	0
$367$	4.52024e30i	0.757126i	0.925575	+	0.378563i	$0.123582\pi$
$367$	4.52024e30i	0.757126i	−0.925575	+	0.378563i	$0.876418\pi$
$368$	0	0
$369$	2.96038e30	0.464549
$370$	0	0
$371$	8.36176e30i	1.22973i
$372$	0	0
$373$	3.60863e29	0.0497550	0.0248775	−	0.999691i	$-0.492080\pi$
$373$	3.60863e29	0.0497550	0.0248775	+	0.999691i	$0.492080\pi$
$374$	0	0
$375$	3.75738e30i	0.485859i
$376$	0	0
$377$	−2.67932e29	−0.0325034
$378$	0	0
$379$	4.92353e29i	0.0560540i	0.999607	+	0.0280270i	$0.00892243\pi$
$379$	4.92353e29i	0.0560540i	−0.999607	+	0.0280270i	$0.991078\pi$
$380$	0	0
$381$	−8.80687e30	−0.941286
$382$	0	0
$383$	4.67483e30i	0.469224i	0.972089	+	0.234612i	$0.0753820\pi$
$383$	4.67483e30i	0.469224i	−0.972089	+	0.234612i	$0.924618\pi$
$384$	0	0
$385$	−2.16145e31	−2.03805
$386$	0	0
$387$	2.58469e30i	0.229021i
$388$	0	0
$389$	9.93029e30	0.827116	0.413558	−	0.910478i	$-0.364286\pi$
$389$	9.93029e30	0.827116	0.413558	+	0.910478i	$0.364286\pi$
$390$	0	0
$391$	9.86722e30i	0.772812i
$392$	0	0
$393$	−7.26501e30	−0.535212
$394$	0	0
$395$	− 9.09115e30i	− 0.630164i
$396$	0	0
$397$	2.12579e30	0.138687	0.0693433	−	0.997593i	$-0.477910\pi$
$397$	2.12579e30	0.138687	0.0693433	+	0.997593i	$0.477910\pi$
$398$	0	0
$399$	1.84712e31i	1.13454i
$400$	0	0
$401$	−2.65209e31	−1.53411	−0.767053	−	0.641584i	$-0.778278\pi$
$401$	−2.65209e31	−1.53411	−0.767053	+	0.641584i	$0.778278\pi$
$402$	0	0
$403$	− 5.52516e30i	− 0.301082i
$404$	0	0
$405$	2.41547e30	0.124034
$406$	0	0
$407$	− 1.66948e31i	− 0.808070i
$408$	0	0
$409$	−1.10889e31	−0.506066	−0.253033	−	0.967458i	$-0.581428\pi$
$409$	−1.10889e31	−0.506066	−0.253033	+	0.967458i	$0.581428\pi$
$410$	0	0
$411$	− 2.08204e31i	− 0.896165i
$412$	0	0
$413$	−3.07800e30	−0.124988
$414$	0	0
$415$	4.17547e31i	1.60003i
$416$	0	0
$417$	−1.84668e31	−0.667972
$418$	0	0
$419$	2.80108e31i	0.956660i	0.878180	+	0.478330i	$0.158758\pi$
$419$	2.80108e31i	0.956660i	−0.878180	+	0.478330i	$0.841242\pi$
$420$	0	0
$421$	4.06158e31	1.31012	0.655058	−	0.755579i	$-0.272644\pi$
$421$	4.06158e31	1.31012	0.655058	+	0.755579i	$0.272644\pi$
$422$	0	0
$423$	7.43838e30i	0.226670i
$424$	0	0
$425$	5.90105e30	0.169927
$426$	0	0
$427$	3.88302e31i	1.05690i
$428$	0	0
$429$	−6.17551e30	−0.158923
$430$	0	0
$431$	− 2.70034e31i	− 0.657190i	−0.944471	−	0.328595i	$-0.893425\pi$
$431$	− 2.70034e31i	− 0.657190i	0.944471	−	0.328595i	$-0.106575\pi$
$432$	0	0
$433$	−5.44781e31	−1.25420	−0.627100	−	0.778939i	$-0.715758\pi$
$433$	−5.44781e31	−1.25420	−0.627100	+	0.778939i	$0.715758\pi$
$434$	0	0
$435$	− 3.66423e30i	− 0.0798200i
$436$	0	0
$437$	−6.12970e31	−1.26375
$438$	0	0
$439$	6.53308e31i	1.27510i	0.770410	+	0.637549i	$0.220052\pi$
$439$	6.53308e31i	1.27510i	−0.770410	+	0.637549i	$0.779948\pi$
$440$	0	0
$441$	3.66144e31	0.676689
$442$	0	0
$443$	− 2.01782e31i	− 0.353214i	−0.984281	−	0.176607i	$-0.943488\pi$
$443$	− 2.01782e31i	− 0.353214i	0.984281	−	0.176607i	$-0.0565122\pi$
$444$	0	0
$445$	−1.02131e32	−1.69370
$446$	0	0
$447$	− 3.75006e31i	− 0.589315i
$448$	0	0
$449$	−1.02071e32	−1.52035	−0.760176	−	0.649717i	$-0.774887\pi$
$449$	−1.02071e32	−1.52035	−0.760176	+	0.649717i	$0.774887\pi$
$450$	0	0
$451$	− 1.03509e32i	− 1.46170i
$452$	0	0
$453$	−3.49262e30	−0.0467706
$454$	0	0
$455$	− 4.01503e31i	− 0.509978i
$456$	0	0
$457$	5.85178e31	0.705169	0.352585	−	0.935780i	$-0.385303\pi$
$457$	5.85178e31	0.705169	0.352585	+	0.935780i	$0.385303\pi$
$458$	0	0
$459$	1.16182e31i	0.132857i
$460$	0	0
$461$	−1.25648e32	−1.36378	−0.681890	−	0.731454i	$-0.738842\pi$
$461$	−1.25648e32	−1.36378	−0.681890	+	0.731454i	$0.738842\pi$
$462$	0	0
$463$	− 4.92378e31i	− 0.507372i	−0.967287	−	0.253686i	$-0.918357\pi$
$463$	− 4.92378e31i	− 0.507372i	0.967287	−	0.253686i	$-0.0816431\pi$
$464$	0	0
$465$	7.55619e31	0.739380
$466$	0	0
$467$	1.40069e32i	1.30178i	0.759170	+	0.650892i	$0.225605\pi$
$467$	1.40069e32i	1.30178i	−0.759170	+	0.650892i	$0.774395\pi$
$468$	0	0
$469$	3.08631e32	2.72500
$470$	0	0
$471$	6.00435e31i	0.503750i
$472$	0	0
$473$	9.03725e31	0.720614
$474$	0	0
$475$	3.66584e31i	0.277875i
$476$	0	0
$477$	−3.26726e31	−0.235485
$478$	0	0
$479$	− 7.16474e31i	− 0.491105i	−0.969383	−	0.245552i	$-0.921031\pi$
$479$	− 7.16474e31i	− 0.491105i	0.969383	−	0.245552i	$-0.0789693\pi$
$480$	0	0
$481$	3.10117e31	0.202202
$482$	0	0
$483$	1.81359e32i	1.12505i
$484$	0	0
$485$	3.11890e32	1.84120
$486$	0	0
$487$	− 5.30086e31i	− 0.297851i	−0.988848	−	0.148925i	$-0.952419\pi$
$487$	− 5.30086e31i	− 0.297851i	0.988848	−	0.148925i	$-0.0475814\pi$
$488$	0	0
$489$	1.39147e32	0.744332
$490$	0	0
$491$	− 2.57107e32i	− 1.30959i	−0.755805	−	0.654797i	$-0.772754\pi$
$491$	− 2.57107e32i	− 1.30959i	0.755805	−	0.654797i	$-0.227246\pi$
$492$	0	0
$493$	1.76245e31	0.0854977
$494$	0	0
$495$	− 8.44561e31i	− 0.390273i
$496$	0	0
$497$	3.75004e32	1.65105
$498$	0	0
$499$	8.80155e31i	0.369277i	0.982806	+	0.184639i	$0.0591115\pi$
$499$	8.80155e31i	0.369277i	−0.982806	+	0.184639i	$0.940889\pi$
$500$	0	0
$501$	−4.17332e31	−0.166890
$502$	0	0
$503$	− 2.28213e32i	− 0.870010i	−0.900428	−	0.435005i	$-0.856747\pi$
$503$	− 2.28213e32i	− 0.870010i	0.900428	−	0.435005i	$-0.143253\pi$
$504$	0	0
$505$	−2.97484e32	−1.08135
$506$	0	0
$507$	1.55075e32i	0.537583i
$508$	0	0
$509$	−3.38923e32	−1.12069	−0.560347	−	0.828258i	$-0.689332\pi$
$509$	−3.38923e32	−1.12069	−0.560347	+	0.828258i	$0.689332\pi$
$510$	0	0
$511$	− 3.66215e32i	− 1.15527i
$512$	0	0
$513$	−7.21741e31	−0.217256
$514$	0	0
$515$	− 6.84595e32i	− 1.96674i
$516$	0	0
$517$	2.60080e32	0.713214
$518$	0	0
$519$	2.91876e32i	0.764169i
$520$	0	0
$521$	4.62918e32	1.15731	0.578657	−	0.815571i	$-0.303577\pi$
$521$	4.62918e32	1.15731	0.578657	+	0.815571i	$0.303577\pi$
$522$	0	0
$523$	8.68164e31i	0.207292i	0.994614	+	0.103646i	$0.0330508\pi$
$523$	8.68164e31i	0.207292i	−0.994614	+	0.103646i	$0.966949\pi$
$524$	0	0
$525$	1.08461e32	0.247378
$526$	0	0
$527$	3.63444e32i	0.791973i
$528$	0	0
$529$	−1.21591e32	−0.253182
$530$	0	0
$531$	− 1.20269e31i	− 0.0239343i
$532$	0	0
$533$	1.92274e32	0.365758
$534$	0	0
$535$	2.27903e32i	0.414481i
$536$	0	0
$537$	5.64114e32	0.981014
$538$	0	0
$539$	− 1.28021e33i	− 2.12920i
$540$	0	0
$541$	8.02582e32	1.27680	0.638399	−	0.769706i	$-0.279597\pi$
$541$	8.02582e32	1.27680	0.638399	+	0.769706i	$0.279597\pi$
$542$	0	0
$543$	− 4.29202e32i	− 0.653227i
$544$	0	0
$545$	−1.21223e33	−1.76534
$546$	0	0
$547$	7.75251e32i	1.08042i	0.841529	+	0.540212i	$0.181656\pi$
$547$	7.75251e32i	1.08042i	−0.841529	+	0.540212i	$0.818344\pi$
$548$	0	0
$549$	−1.51724e32	−0.202389
$550$	0	0
$551$	1.09487e32i	0.139811i
$552$	0	0
$553$	8.03705e32	0.982642
$554$	0	0
$555$	4.24115e32i	0.496556i
$556$	0	0
$557$	−1.05854e33	−1.18698	−0.593492	−	0.804840i	$-0.702251\pi$
$557$	−1.05854e33	−1.18698	−0.593492	+	0.804840i	$0.702251\pi$
$558$	0	0
$559$	1.67873e32i	0.180318i
$560$	0	0
$561$	4.06224e32	0.418034
$562$	0	0
$563$	− 1.33471e33i	− 1.31609i	−0.752977	−	0.658047i	$-0.771383\pi$
$563$	− 1.33471e33i	− 1.31609i	0.752977	−	0.658047i	$-0.228617\pi$
$564$	0	0
$565$	1.74832e33	1.65211
$566$	0	0
$567$	2.13541e32i	0.193412i
$568$	0	0
$569$	−6.14180e31	−0.0533272	−0.0266636	−	0.999644i	$-0.508488\pi$
$569$	−6.14180e31	−0.0533272	−0.0266636	+	0.999644i	$0.508488\pi$
$570$	0	0
$571$	6.36810e32i	0.530123i	0.964232	+	0.265061i	$0.0853922\pi$
$571$	6.36810e32i	0.530123i	−0.964232	+	0.265061i	$0.914608\pi$
$572$	0	0
$573$	−2.80993e32	−0.224306
$574$	0	0
$575$	3.59929e32i	0.275551i
$576$	0	0
$577$	−5.82495e32	−0.427742	−0.213871	−	0.976862i	$-0.568607\pi$
$577$	−5.82495e32	−0.427742	−0.213871	+	0.976862i	$0.568607\pi$
$578$	0	0
$579$	7.49942e31i	0.0528305i
$580$	0	0
$581$	−3.69133e33	−2.49499
$582$	0	0
$583$	1.14238e33i	0.740951i
$584$	0	0
$585$	1.56883e32	0.0976572
$586$	0	0
$587$	− 2.11187e33i	− 1.26186i	−0.775841	−	0.630928i	$-0.782674\pi$
$587$	− 2.11187e33i	− 1.26186i	0.775841	−	0.630928i	$-0.217326\pi$
$588$	0	0
$589$	−2.25778e33	−1.29508
$590$	0	0
$591$	1.36253e33i	0.750406i
$592$	0	0
$593$	−1.89349e33	−1.00140	−0.500699	−	0.865622i	$-0.666924\pi$
$593$	−1.89349e33	−1.00140	−0.500699	+	0.865622i	$0.666924\pi$
$594$	0	0
$595$	2.64108e33i	1.34146i
$596$	0	0
$597$	9.80157e32	0.478193
$598$	0	0
$599$	2.29989e33i	1.07791i	0.842333	+	0.538957i	$0.181182\pi$
$599$	2.29989e33i	1.07791i	−0.842333	+	0.538957i	$0.818818\pi$
$600$	0	0
$601$	5.38759e31	0.0242606	0.0121303	−	0.999926i	$-0.496139\pi$
$601$	5.38759e31	0.0242606	0.0121303	+	0.999926i	$0.496139\pi$
$602$	0	0
$603$	1.20594e33i	0.521818i
$604$	0	0
$605$	−2.68560e32	−0.111680
$606$	0	0
$607$	3.19997e33i	1.27903i	0.768777	+	0.639517i	$0.220866\pi$
$607$	3.19997e33i	1.27903i	−0.768777	+	0.639517i	$0.779134\pi$
$608$	0	0
$609$	3.23937e32	0.124467
$610$	0	0
$611$	4.83115e32i	0.178466i
$612$	0	0
$613$	−3.33396e33	−1.18423	−0.592113	−	0.805855i	$-0.701706\pi$
$613$	−3.33396e33	−1.18423	−0.592113	+	0.805855i	$0.701706\pi$
$614$	0	0
$615$	2.62953e33i	0.898208i
$616$	0	0
$617$	1.93281e33	0.634991	0.317496	−	0.948260i	$-0.397158\pi$
$617$	1.93281e33	0.634991	0.317496	+	0.948260i	$0.397158\pi$
$618$	0	0
$619$	− 1.71270e33i	− 0.541245i	−0.962686	−	0.270623i	$-0.912770\pi$
$619$	− 1.71270e33i	− 0.541245i	0.962686	−	0.270623i	$-0.0872296\pi$
$620$	0	0
$621$	−7.08638e32	−0.215439
$622$	0	0
$623$	− 9.02892e33i	− 2.64106i
$624$	0	0
$625$	−4.21195e33	−1.18556
$626$	0	0
$627$	2.52354e33i	0.683595i
$628$	0	0
$629$	−2.03994e33	−0.531876
$630$	0	0
$631$	4.32549e33i	1.08563i	0.839851	+	0.542817i	$0.182642\pi$
$631$	4.32549e33i	1.08563i	−0.839851	+	0.542817i	$0.817358\pi$
$632$	0	0
$633$	1.34707e33	0.325496
$634$	0	0
$635$	− 7.82261e33i	− 1.81998i
$636$	0	0
$637$	2.37807e33	0.532784
$638$	0	0
$639$	1.46528e33i	0.316164i
$640$	0	0
$641$	−1.63564e33	−0.339931	−0.169966	−	0.985450i	$-0.554366\pi$
$641$	−1.63564e33	−0.339931	−0.169966	+	0.985450i	$0.554366\pi$
$642$	0	0
$643$	− 3.75046e33i	− 0.750849i	−0.926853	−	0.375425i	$-0.877497\pi$
$643$	− 3.75046e33i	− 0.750849i	0.926853	−	0.375425i	$-0.122503\pi$
$644$	0	0
$645$	−2.29582e33	−0.442814
$646$	0	0
$647$	− 6.64765e33i	− 1.23543i	−0.786404	−	0.617713i	$-0.788059\pi$
$647$	− 6.64765e33i	− 1.23543i	0.786404	−	0.617713i	$-0.211941\pi$
$648$	0	0
$649$	−4.20516e32	−0.0753089
$650$	0	0
$651$	6.68007e33i	1.15295i
$652$	0	0
$653$	−2.39119e33	−0.397792	−0.198896	−	0.980021i	$-0.563736\pi$
$653$	−2.39119e33	−0.397792	−0.198896	+	0.980021i	$0.563736\pi$
$654$	0	0
$655$	− 6.45307e33i	− 1.03483i
$656$	0	0
$657$	1.43094e33	0.221227
$658$	0	0
$659$	7.34105e33i	1.09429i	0.837037	+	0.547146i	$0.184286\pi$
$659$	7.34105e33i	1.09429i	−0.837037	+	0.547146i	$0.815714\pi$
$660$	0	0
$661$	5.22304e33	0.750768	0.375384	−	0.926869i	$-0.377511\pi$
$661$	5.22304e33	0.750768	0.375384	+	0.926869i	$0.377511\pi$
$662$	0	0
$663$	7.54587e32i	0.104604i
$664$	0	0
$665$	−1.64069e34	−2.19364
$666$	0	0
$667$	1.07499e33i	0.138642i
$668$	0	0
$669$	8.93399e33	1.11156
$670$	0	0
$671$	5.30498e33i	0.636816i
$672$	0	0
$673$	9.13511e33	1.05812	0.529058	−	0.848586i	$-0.322545\pi$
$673$	9.13511e33	1.05812	0.529058	+	0.848586i	$0.322545\pi$
$674$	0	0
$675$	4.23798e32i	0.0473711i
$676$	0	0
$677$	−2.71674e33	−0.293079	−0.146539	−	0.989205i	$-0.546814\pi$
$677$	−2.71674e33	−0.293079	−0.146539	+	0.989205i	$0.546814\pi$
$678$	0	0
$679$	2.75727e34i	2.87106i
$680$	0	0
$681$	6.59995e33	0.663400
$682$	0	0
$683$	− 1.41999e34i	− 1.37796i	−0.724780	−	0.688981i	$-0.758058\pi$
$683$	− 1.41999e34i	− 1.37796i	0.724780	−	0.688981i	$-0.241942\pi$
$684$	0	0
$685$	1.84935e34	1.73274
$686$	0	0
$687$	7.13083e33i	0.645147i
$688$	0	0
$689$	−2.12205e33	−0.185407
$690$	0	0
$691$	3.14372e33i	0.265282i	0.991164	+	0.132641i	$0.0423457\pi$
$691$	3.14372e33i	0.265282i	−0.991164	+	0.132641i	$0.957654\pi$
$692$	0	0
$693$	7.46636e33	0.608569
$694$	0	0
$695$	− 1.64029e34i	− 1.29153i
$696$	0	0
$697$	−1.26477e34	−0.962099
$698$	0	0
$699$	− 1.35882e34i	− 0.998703i
$700$	0	0
$701$	2.08107e34	1.47798	0.738992	−	0.673714i	$-0.235302\pi$
$701$	2.08107e34	1.47798	0.738992	+	0.673714i	$0.235302\pi$
$702$	0	0
$703$	− 1.26725e34i	− 0.869758i
$704$	0	0
$705$	−6.60707e33	−0.438267
$706$	0	0
$707$	− 2.62992e34i	− 1.68620i
$708$	0	0
$709$	2.10847e34	1.30681	0.653406	−	0.757008i	$-0.273340\pi$
$709$	2.10847e34	1.30681	0.653406	+	0.757008i	$0.273340\pi$
$710$	0	0
$711$	3.14039e33i	0.188169i
$712$	0	0
$713$	−2.21679e34	−1.28425
$714$	0	0
$715$	− 5.48534e33i	− 0.307278i
$716$	0	0
$717$	1.17445e34	0.636217
$718$	0	0
$719$	− 3.54814e34i	− 1.85889i	−0.368959	−	0.929446i	$-0.620286\pi$
$719$	− 3.54814e34i	− 1.85889i	0.368959	−	0.929446i	$-0.379714\pi$
$720$	0	0
$721$	6.05218e34	3.06682
$722$	0	0
$723$	− 1.74295e33i	− 0.0854327i
$724$	0	0
$725$	6.42893e32	0.0304848
$726$	0	0
$727$	3.61639e34i	1.65906i	0.558460	+	0.829531i	$0.311392\pi$
$727$	3.61639e34i	1.65906i	−0.558460	+	0.829531i	$0.688608\pi$
$728$	0	0
$729$	−8.34385e32	−0.0370370
$730$	0	0
$731$	− 1.10426e34i	− 0.474312i
$732$	0	0
$733$	−4.65419e34	−1.93462	−0.967311	−	0.253594i	$-0.918387\pi$
$733$	−4.65419e34	−1.93462	−0.967311	+	0.253594i	$0.918387\pi$
$734$	0	0
$735$	3.25224e34i	1.30838i
$736$	0	0
$737$	4.21653e34	1.64189
$738$	0	0
$739$	− 1.00455e34i	− 0.378649i	−0.981915	−	0.189325i	$-0.939370\pi$
$739$	− 1.00455e34i	− 0.378649i	0.981915	−	0.189325i	$-0.0606298\pi$
$740$	0	0
$741$	−4.68763e33	−0.171055
$742$	0	0
$743$	− 1.89824e34i	− 0.670632i	−0.942106	−	0.335316i	$-0.891157\pi$
$743$	− 1.89824e34i	− 0.670632i	0.942106	−	0.335316i	$-0.108843\pi$
$744$	0	0
$745$	3.33095e34	1.13944
$746$	0	0
$747$	− 1.44235e34i	− 0.477774i
$748$	0	0
$749$	−2.01478e34	−0.646318
$750$	0	0
$751$	5.84776e34i	1.81681i	0.418088	+	0.908407i	$0.362700\pi$
$751$	5.84776e34i	1.81681i	−0.418088	+	0.908407i	$0.637300\pi$
$752$	0	0
$753$	5.07285e33	0.152655
$754$	0	0
$755$	− 3.10229e33i	− 0.0904313i
$756$	0	0
$757$	−4.76935e34	−1.34682	−0.673408	−	0.739271i	$-0.735170\pi$
$757$	−4.76935e34	−1.34682	−0.673408	+	0.739271i	$0.735170\pi$
$758$	0	0
$759$	2.47772e34i	0.677878i
$760$	0	0
$761$	1.97029e34	0.522294	0.261147	−	0.965299i	$-0.415899\pi$
$761$	1.97029e34	0.522294	0.261147	+	0.965299i	$0.415899\pi$
$762$	0	0
$763$	− 1.07168e35i	− 2.75277i
$764$	0	0
$765$	−1.03197e34	−0.256880
$766$	0	0
$767$	− 7.81136e32i	− 0.0188444i
$768$	0	0
$769$	−2.92438e34	−0.683781	−0.341890	−	0.939740i	$-0.611067\pi$
$769$	−2.92438e34	−0.683781	−0.341890	+	0.939740i	$0.611067\pi$
$770$	0	0
$771$	4.05907e34i	0.919970i
$772$	0	0
$773$	−5.64083e34	−1.23934	−0.619668	−	0.784864i	$-0.712733\pi$
$773$	−5.64083e34	−1.23934	−0.619668	+	0.784864i	$0.712733\pi$
$774$	0	0
$775$	1.32574e34i	0.282383i
$776$	0	0
$777$	−3.74940e34	−0.774300
$778$	0	0
$779$	− 7.85700e34i	− 1.57329i
$780$	0	0
$781$	5.12331e34	0.994805
$782$	0	0
$783$	1.26575e33i	0.0238345i
$784$	0	0
$785$	−5.33330e34	−0.974003
$786$	0	0
$787$	4.40454e34i	0.780196i	0.920773	+	0.390098i	$0.127559\pi$
$787$	4.40454e34i	0.780196i	−0.920773	+	0.390098i	$0.872441\pi$
$788$	0	0
$789$	4.23008e34	0.726816
$790$	0	0
$791$	1.54561e35i	2.57621i
$792$	0	0
$793$	−9.85435e33	−0.159349
$794$	0	0
$795$	− 2.90211e34i	− 0.455311i
$796$	0	0
$797$	9.07728e34	1.38183	0.690917	−	0.722934i	$-0.257207\pi$
$797$	9.07728e34	1.38183	0.690917	+	0.722934i	$0.257207\pi$
$798$	0	0
$799$	− 3.17792e34i	− 0.469442i
$800$	0	0
$801$	3.52795e34	0.505745
$802$	0	0
$803$	− 5.00323e34i	− 0.696088i
$804$	0	0
$805$	−1.61090e35	−2.17529
$806$	0	0
$807$	1.47224e34i	0.192972i
$808$	0	0
$809$	−7.87745e34	−1.00231	−0.501157	−	0.865357i	$-0.667092\pi$
$809$	−7.87745e34	−1.00231	−0.501157	+	0.865357i	$0.667092\pi$
$810$	0	0
$811$	7.69790e34i	0.950872i	0.879750	+	0.475436i	$0.157710\pi$
$811$	7.69790e34i	0.950872i	−0.879750	+	0.475436i	$0.842290\pi$
$812$	0	0
$813$	5.99641e34	0.719126
$814$	0	0
$815$	1.23596e35i	1.43917i
$816$	0	0
$817$	6.85989e34	0.775626
$818$	0	0
$819$	1.38692e34i	0.152281i
$820$	0	0
$821$	−6.45455e34	−0.688253	−0.344127	−	0.938923i	$-0.611825\pi$
$821$	−6.45455e34	−0.688253	−0.344127	+	0.938923i	$0.611825\pi$
$822$	0	0
$823$	1.40111e34i	0.145102i	0.997365	+	0.0725512i	$0.0231141\pi$
$823$	1.40111e34i	0.145102i	−0.997365	+	0.0725512i	$0.976886\pi$
$824$	0	0
$825$	1.48179e34	0.149053
$826$	0	0
$827$	7.28750e34i	0.712053i	0.934476	+	0.356027i	$0.115869\pi$
$827$	7.28750e34i	0.712053i	−0.934476	+	0.356027i	$0.884131\pi$
$828$	0	0
$829$	1.98331e34	0.188250	0.0941252	−	0.995560i	$-0.469995\pi$
$829$	1.98331e34	0.188250	0.0941252	+	0.995560i	$0.469995\pi$
$830$	0	0
$831$	− 6.75794e33i	− 0.0623163i
$832$	0	0
$833$	−1.56429e35	−1.40145
$834$	0	0
$835$	− 3.70691e34i	− 0.322682i
$836$	0	0
$837$	−2.61016e34	−0.220781
$838$	0	0
$839$	1.36310e35i	1.12043i	0.828347	+	0.560215i	$0.189281\pi$
$839$	1.36310e35i	1.12043i	−0.828347	+	0.560215i	$0.810719\pi$
$840$	0	0
$841$	−1.23265e35	−0.984662
$842$	0	0
$843$	− 4.80696e34i	− 0.373198i
$844$	0	0
$845$	−1.37744e35	−1.03942
$846$	0	0
$847$	− 2.37421e34i	− 0.174148i
$848$	0	0
$849$	−7.68589e34	−0.548027
$850$	0	0
$851$	− 1.24424e35i	− 0.862484i
$852$	0	0
$853$	7.10685e34	0.478949	0.239474	−	0.970903i	$-0.423025\pi$
$853$	7.10685e34	0.478949	0.239474	+	0.970903i	$0.423025\pi$
$854$	0	0
$855$	− 6.41079e34i	− 0.420067i
$856$	0	0
$857$	4.84289e34	0.308557	0.154278	−	0.988027i	$-0.450695\pi$
$857$	4.84289e34	0.308557	0.154278	+	0.988027i	$0.450695\pi$
$858$	0	0
$859$	8.81626e34i	0.546219i	0.961983	+	0.273109i	$0.0880521\pi$
$859$	8.81626e34i	0.546219i	−0.961983	+	0.273109i	$0.911948\pi$
$860$	0	0
$861$	−2.32464e35	−1.40061
$862$	0	0
$863$	4.79792e34i	0.281140i	0.990071	+	0.140570i	$0.0448936\pi$
$863$	4.79792e34i	0.281140i	−0.990071	+	0.140570i	$0.955106\pi$
$864$	0	0
$865$	−2.59256e35	−1.47752
$866$	0	0
$867$	5.45156e34i	0.302198i
$868$	0	0
$869$	1.09802e35	0.592072
$870$	0	0
$871$	7.83247e34i	0.410848i
$872$	0	0
$873$	−1.07737e35	−0.549787
$874$	0	0
$875$	− 2.95049e35i	− 1.46486i
$876$	0	0
$877$	−1.66045e35	−0.802104	−0.401052	−	0.916055i	$-0.631355\pi$
$877$	−1.66045e35	−0.802104	−0.401052	+	0.916055i	$0.631355\pi$
$878$	0	0
$879$	− 2.07630e35i	− 0.975943i
$880$	0	0
$881$	3.74419e35	1.71257	0.856283	−	0.516507i	$-0.172768\pi$
$881$	3.74419e35	1.71257	0.856283	+	0.516507i	$0.172768\pi$
$882$	0	0
$883$	− 3.83674e35i	− 1.70779i	−0.520448	−	0.853893i	$-0.674235\pi$
$883$	− 3.83674e35i	− 1.70779i	0.520448	−	0.853893i	$-0.325765\pi$
$884$	0	0
$885$	1.06828e34	0.0462770
$886$	0	0
$887$	1.86724e35i	0.787257i	0.919270	+	0.393629i	$0.128780\pi$
$887$	1.86724e35i	0.787257i	−0.919270	+	0.393629i	$0.871220\pi$
$888$	0	0
$889$	6.91559e35	2.83797
$890$	0	0
$891$	2.91739e34i	0.116537i
$892$	0	0
$893$	1.97418e35	0.767661
$894$	0	0
$895$	5.01068e35i	1.89680i
$896$	0	0
$897$	−4.60253e34	−0.169624
$898$	0	0
$899$	3.95956e34i	0.142079i
$900$	0	0
$901$	1.39588e35	0.487698
$902$	0	0
$903$	− 2.02963e35i	− 0.690499i
$904$	0	0
$905$	3.81235e35	1.26302
$906$	0	0
$907$	− 2.56347e35i	− 0.827068i	−0.910489	−	0.413534i	$-0.864294\pi$
$907$	− 2.56347e35i	− 0.827068i	0.910489	−	0.413534i	$-0.135706\pi$
$908$	0	0
$909$	1.02761e35	0.322895
$910$	0	0
$911$	4.34114e35i	1.32857i	0.747481	+	0.664283i	$0.231263\pi$
$911$	4.34114e35i	1.32857i	−0.747481	+	0.664283i	$0.768737\pi$
$912$	0	0
$913$	−5.04310e35	−1.50331
$914$	0	0
$915$	− 1.34768e35i	− 0.391321i
$916$	0	0
$917$	5.70485e35	1.61366
$918$	0	0
$919$	1.46245e35i	0.402990i	0.979489	+	0.201495i	$0.0645801\pi$
$919$	1.46245e35i	0.402990i	−0.979489	+	0.201495i	$0.935420\pi$
$920$	0	0
$921$	2.91533e35	0.782658
$922$	0	0
$923$	9.51688e34i	0.248928i
$924$	0	0
$925$	−7.44115e34	−0.189644
$926$	0	0
$927$	2.36482e35i	0.587274i
$928$	0	0
$929$	−1.57123e35	−0.380234	−0.190117	−	0.981761i	$-0.560887\pi$
$929$	−1.57123e35	−0.380234	−0.190117	+	0.981761i	$0.560887\pi$
$930$	0	0
$931$	− 9.71765e35i	− 2.29174i
$932$	0	0
$933$	−1.79306e35	−0.412113
$934$	0	0
$935$	3.60824e35i	0.808270i
$936$	0	0
$937$	2.00029e35	0.436735	0.218368	−	0.975867i	$-0.429927\pi$
$937$	2.00029e35	0.436735	0.218368	+	0.975867i	$0.429927\pi$
$938$	0	0
$939$	− 4.61303e35i	− 0.981746i
$940$	0	0
$941$	−9.90987e34	−0.205585	−0.102793	−	0.994703i	$-0.532778\pi$
$941$	−9.90987e34	−0.205585	−0.102793	+	0.994703i	$0.532778\pi$
$942$	0	0
$943$	− 7.71436e35i	− 1.56013i
$944$	0	0
$945$	−1.89675e35	−0.373963
$946$	0	0
$947$	− 4.01235e35i	− 0.771257i	−0.922654	−	0.385629i	$-0.873985\pi$
$947$	− 4.01235e35i	− 0.771257i	0.922654	−	0.385629i	$-0.126015\pi$
$948$	0	0
$949$	9.29382e34	0.174181
$950$	0	0
$951$	− 1.97773e35i	− 0.361411i
$952$	0	0
$953$	5.13762e35	0.915476	0.457738	−	0.889087i	$-0.348660\pi$
$953$	5.13762e35	0.915476	0.457738	+	0.889087i	$0.348660\pi$
$954$	0	0
$955$	− 2.49590e35i	− 0.433697i
$956$	0	0
$957$	4.42563e34	0.0749949
$958$	0	0
$959$	1.63493e36i	2.70193i
$960$	0	0
$961$	−1.96109e35	−0.316094
$962$	0	0
$963$	− 7.87253e34i	− 0.123765i
$964$	0	0
$965$	−6.66128e34	−0.102148
$966$	0	0
$967$	3.88986e35i	0.581856i	0.956745	+	0.290928i	$0.0939641\pi$
$967$	3.88986e35i	0.581856i	−0.956745	+	0.290928i	$0.906036\pi$
$968$	0	0
$969$	3.08352e35	0.449947
$970$	0	0
$971$	6.00316e35i	0.854574i	0.904116	+	0.427287i	$0.140531\pi$
$971$	6.00316e35i	0.854574i	−0.904116	+	0.427287i	$0.859469\pi$
$972$	0	0
$973$	1.45010e36	2.01393
$974$	0	0
$975$	2.75252e34i	0.0372972i
$976$	0	0
$977$	1.63760e35	0.216507	0.108254	−	0.994123i	$-0.465474\pi$
$977$	1.63760e35	0.216507	0.108254	+	0.994123i	$0.465474\pi$
$978$	0	0
$979$	− 1.23353e36i	− 1.59132i
$980$	0	0
$981$	4.18746e35	0.527136
$982$	0	0
$983$	− 1.15950e35i	− 0.142439i	−0.997461	−	0.0712197i	$-0.977311\pi$
$983$	− 1.15950e35i	− 0.142439i	0.997461	−	0.0712197i	$-0.0226891\pi$
$984$	0	0
$985$	−1.21025e36	−1.45091
$986$	0	0
$987$	− 5.84099e35i	− 0.683409i
$988$	0	0
$989$	6.73535e35	0.769138
$990$	0	0
$991$	− 1.13176e36i	− 1.26145i	−0.776006	−	0.630726i	$-0.782757\pi$
$991$	− 1.13176e36i	− 1.26145i	0.776006	−	0.630726i	$-0.217243\pi$
$992$	0	0
$993$	−2.29815e35	−0.250027
$994$	0	0
$995$	8.70614e35i	0.924589i
$996$	0	0
$997$	8.65498e35	0.897272	0.448636	−	0.893715i	$-0.351910\pi$
$997$	8.65498e35	0.897272	0.448636	+	0.893715i	$0.351910\pi$
$998$	0	0
$999$	− 1.46503e35i	− 0.148273i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.25.g.c.31.3	✓	8
4.3	odd	2	inner	48.25.g.c.31.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.25.g.c.31.3	✓	8	1.1	even	1	trivial
48.25.g.c.31.7	yes	8	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 25 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-306828. i q^{3} +2.72537e8 q^{5} +2.40937e10i q^{7} -9.41432e10 q^{9} +O(q^{10})\)	\(q-306828. i q^{3} +2.72537e8 q^{5} +2.40937e10i q^{7} -9.41432e10 q^{9} +3.29168e12i q^{11} -6.11450e12 q^{13} -8.36217e13i q^{15} +4.02211e14 q^{17} +2.49861e15i q^{19} +7.39260e15 q^{21} +2.45325e16i q^{23} +1.46715e16 q^{25} +2.88857e16i q^{27} +4.38191e16 q^{29} +9.03616e17i q^{31} +1.00998e18 q^{33} +6.56640e18i q^{35} -5.07183e18 q^{37} +1.87610e18i q^{39} -3.14455e19 q^{41} -2.74548e19i q^{43} -2.56575e19 q^{45} -7.90113e19i q^{47} -3.88923e20 q^{49} -1.23409e20i q^{51} +3.47052e20 q^{53} +8.97102e20i q^{55} +7.66641e20 q^{57} +1.27751e20i q^{59} +1.61163e21 q^{61} -2.26825e21i q^{63} -1.66643e21 q^{65} -1.28097e22i q^{67} +7.52724e21 q^{69} -1.55644e22i q^{71} -1.51996e22 q^{73} -4.50163e21i q^{75} -7.93085e22 q^{77} -3.33576e22i q^{79} +8.86294e21 q^{81} +1.53208e23i q^{83} +1.09617e23 q^{85} -1.34449e22i q^{87} -3.74743e23 q^{89} -1.47321e23i q^{91} +2.77254e23 q^{93} +6.80962e23i q^{95} +1.14440e24 q^{97} -3.09889e23i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 462495600 q^{5} - 753145430616 q^{9}+O(q^{10}) \) 8 * q + 462495600 * q^5 - 753145430616 * q^9	\( 8 q + 462495600 q^{5} - 753145430616 q^{9} - 1530027465904 q^{13} - 35559567490608 q^{17} + 84\!\cdots\!56 q^{21}+ \cdots + 12\!\cdots\!80 q^{97}+O(q^{100}) \) 8 * q + 462495600 * q^5 - 753145430616 * q^9 - 1530027465904 * q^13 - 35559567490608 * q^17 + 8439296905969056 * q^21 + 142743524026591000 * q^25 + 476759032205115312 * q^29 - 2160858161263465440 * q^33 - 15760822588656347888 * q^37 - 45389784103575648048 * q^41 - 43540805977500661200 * q^45 + 30951519803487811208 * q^49 - 1729329636487575669840 * q^53 - 1269107008528107899808 * q^57 + 2694636617324641569424 * q^61 - 3149721076444705130400 * q^65 - 6624622265621256652032 * q^69 - 54562892980695920861680 * q^73 - 268036140084799319975808 * q^77 + 70903504957220008767432 * q^81 - 328621744624984652887200 * q^85 + 223962296701000391132688 * q^89 + 645901021207453779145440 * q^93 + 1281098591387557493688080 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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