LMFDB - Embedded newform 48.22.a.g.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:	$2$
Coefficient field:	$\Q(\sqrt{649}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 162 $ x^2 - x - 162
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$13.2377$ 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)

$f(q)$	$=$	$q-59049.0 q^{3} +2.22745e7 q^{5} -4.78205e7 q^{7} +3.48678e9 q^{9} +O(q^{10})$	$q-59049.0 q^{3} +2.22745e7 q^{5} -4.78205e7 q^{7} +3.48678e9 q^{9} -1.60111e11 q^{11} -7.86968e11 q^{13} -1.31529e12 q^{15} -2.97477e12 q^{17} +2.99456e13 q^{19} +2.82376e12 q^{21} +1.91401e14 q^{23} +1.93148e13 q^{25} -2.05891e14 q^{27} +9.68170e14 q^{29} -2.80459e15 q^{31} +9.45437e15 q^{33} -1.06518e15 q^{35} +3.05038e16 q^{37} +4.64697e16 q^{39} -2.22806e16 q^{41} -1.63711e17 q^{43} +7.76663e16 q^{45} -4.08678e17 q^{47} -5.56259e17 q^{49} +1.75657e17 q^{51} +4.34009e17 q^{53} -3.56638e18 q^{55} -1.76826e18 q^{57} +5.14341e18 q^{59} +1.98980e18 q^{61} -1.66740e17 q^{63} -1.75293e19 q^{65} +1.36361e19 q^{67} -1.13020e19 q^{69} -7.35641e18 q^{71} +6.81650e19 q^{73} -1.14052e18 q^{75} +7.65658e18 q^{77} +2.12282e19 q^{79} +1.21577e19 q^{81} -1.10803e20 q^{83} -6.62613e19 q^{85} -5.71695e19 q^{87} -7.67205e18 q^{89} +3.76333e19 q^{91} +1.65608e20 q^{93} +6.67021e20 q^{95} +4.63755e20 q^{97} -5.58271e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9}+O(q^{10}) $ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9	$ 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9} - 219869122968 q^{11} - 48468909956 q^{13} - 58864530924 q^{15} - 11333529041436 q^{17} - 11960585011624 q^{19} + 40147185517488 q^{21} + 146508390063504 q^{23} - 4786354247074 q^{25} - 411782264189298 q^{27} - 17\!\cdots\!52 q^{29}+ \cdots - 76\!\cdots\!68 q^{99}+O(q^{100}) $ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 - 219869122968 * q^11 - 48468909956 * q^13 - 58864530924 * q^15 - 11333529041436 * q^17 - 11960585011624 * q^19 + 40147185517488 * q^21 + 146508390063504 * q^23 - 4786354247074 * q^25 - 411782264189298 * q^27 - 1798520043674052 * q^29 - 11169107526944992 * q^31 + 12983051842137432 * q^33 + 12383869767948000 * q^35 + 12736264858660012 * q^37 + 2862040663991844 * q^39 + 122972020616468052 * q^41 - 288455418162270040 * q^43 + 3475891686531276 * q^45 - 837243745741596960 * q^47 - 715285396941470670 * q^49 + 669233556367754364 * q^51 - 43007964012775764 * q^53 - 2294863118895313296 * q^55 + 706260584351385576 * q^57 + 3523823330903857224 * q^59 - 1779023128451013860 * q^61 - 2370651157622148912 * q^63 - 33242791066177513752 * q^65 + 16454068667621610296 * q^67 - 8651173924859847696 * q^69 - 17379227131150420944 * q^71 + 50891146268473989076 * q^73 + 282629431935472626 * q^75 + 45428450983816025664 * q^77 + 54055785594190591040 * q^79 + 24315330918113857602 * q^81 - 111108429277666677288 * q^83 + 111593012278262968152 * q^85 + 106200810058909096548 * q^87 + 226920767965448065524 * q^89 - 429154174685185000352 * q^91 + 659524630358574832608 * q^93 + 1558683223531725831696 * q^95 - 128331469252795746236 * q^97 - 766636228226373222168 * 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
$5$	2.22745e7	1.02005	0.510026	−	0.860159i	$-0.329636\pi$
$5$	2.22745e7	1.02005	0.510026	+	0.860159i	$0.329636\pi$
$6$	0	0
$7$	−4.78205e7	−0.0639860	−0.0319930	−	0.999488i	$-0.510185\pi$
$7$	−4.78205e7	−0.0639860	−0.0319930	+	0.999488i	$0.510185\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−1.60111e11	−1.86122	−0.930609	−	0.366016i	$-0.880722\pi$
$11$	−1.60111e11	−1.86122	−0.930609	+	0.366016i	$0.880722\pi$
$12$	0	0
$13$	−7.86968e11	−1.58326	−0.791630	−	0.611001i	$-0.790767\pi$
$13$	−7.86968e11	−1.58326	−0.791630	+	0.611001i	$0.790767\pi$
$14$	0	0
$15$	−1.31529e12	−0.588927
$16$	0	0
$17$	−2.97477e12	−0.357881	−0.178941	−	0.983860i	$-0.557267\pi$
$17$	−2.97477e12	−0.357881	−0.178941	+	0.983860i	$0.557267\pi$
$18$	0	0
$19$	2.99456e13	1.12052	0.560260	−	0.828317i	$-0.310701\pi$
$19$	2.99456e13	1.12052	0.560260	+	0.828317i	$0.310701\pi$
$20$	0	0
$21$	2.82376e12	0.0369423
$22$	0	0
$23$	1.91401e14	0.963390	0.481695	−	0.876339i	$-0.340021\pi$
$23$	1.91401e14	0.963390	0.481695	+	0.876339i	$0.340021\pi$
$24$	0	0
$25$	1.93148e13	0.0405061
$26$	0	0
$27$	−2.05891e14	−0.192450
$28$	0	0
$29$	9.68170e14	0.427339	0.213670	−	0.976906i	$-0.431458\pi$
$29$	9.68170e14	0.427339	0.213670	+	0.976906i	$0.431458\pi$
$30$	0	0
$31$	−2.80459e15	−0.614570	−0.307285	−	0.951618i	$-0.599420\pi$
$31$	−2.80459e15	−0.614570	−0.307285	+	0.951618i	$0.599420\pi$
$32$	0	0
$33$	9.45437e15	1.07457
$34$	0	0
$35$	−1.06518e15	−0.0652691
$36$	0	0
$37$	3.05038e16	1.04288	0.521441	−	0.853287i	$-0.325395\pi$
$37$	3.05038e16	1.04288	0.521441	+	0.853287i	$0.325395\pi$
$38$	0	0
$39$	4.64697e16	0.914095
$40$	0	0
$41$	−2.22806e16	−0.259237	−0.129618	−	0.991564i	$-0.541375\pi$
$41$	−2.22806e16	−0.259237	−0.129618	+	0.991564i	$0.541375\pi$
$42$	0	0
$43$	−1.63711e17	−1.15521	−0.577604	−	0.816317i	$-0.696012\pi$
$43$	−1.63711e17	−1.15521	−0.577604	+	0.816317i	$0.696012\pi$
$44$	0	0
$45$	7.76663e16	0.340017
$46$	0	0
$47$	−4.08678e17	−1.13332	−0.566662	−	0.823951i	$-0.691765\pi$
$47$	−4.08678e17	−1.13332	−0.566662	+	0.823951i	$0.691765\pi$
$48$	0	0
$49$	−5.56259e17	−0.995906
$50$	0	0
$51$	1.75657e17	0.206623
$52$	0	0
$53$	4.34009e17	0.340881	0.170440	−	0.985368i	$-0.445481\pi$
$53$	4.34009e17	0.340881	0.170440	+	0.985368i	$0.445481\pi$
$54$	0	0
$55$	−3.56638e18	−1.89854
$56$	0	0
$57$	−1.76826e18	−0.646932
$58$	0	0
$59$	5.14341e18	1.31010	0.655051	−	0.755584i	$-0.272647\pi$
$59$	5.14341e18	1.31010	0.655051	+	0.755584i	$0.272647\pi$
$60$	0	0
$61$	1.98980e18	0.357147	0.178573	−	0.983927i	$-0.442852\pi$
$61$	1.98980e18	0.357147	0.178573	+	0.983927i	$0.442852\pi$
$62$	0	0
$63$	−1.66740e17	−0.0213287
$64$	0	0
$65$	−1.75293e19	−1.61501
$66$	0	0
$67$	1.36361e19	0.913913	0.456956	−	0.889489i	$-0.348940\pi$
$67$	1.36361e19	0.913913	0.456956	+	0.889489i	$0.348940\pi$
$68$	0	0
$69$	−1.13020e19	−0.556213
$70$	0	0
$71$	−7.35641e18	−0.268196	−0.134098	−	0.990968i	$-0.542814\pi$
$71$	−7.35641e18	−0.268196	−0.134098	+	0.990968i	$0.542814\pi$
$72$	0	0
$73$	6.81650e19	1.85640	0.928200	−	0.372081i	$-0.121356\pi$
$73$	6.81650e19	1.85640	0.928200	+	0.372081i	$0.121356\pi$
$74$	0	0
$75$	−1.14052e18	−0.0233862
$76$	0	0
$77$	7.65658e18	0.119092
$78$	0	0
$79$	2.12282e19	0.252249	0.126124	−	0.992014i	$-0.459746\pi$
$79$	2.12282e19	0.252249	0.126124	+	0.992014i	$0.459746\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	−1.10803e20	−0.783849	−0.391924	−	0.919997i	$-0.628191\pi$
$83$	−1.10803e20	−0.783849	−0.391924	+	0.919997i	$0.628191\pi$
$84$	0	0
$85$	−6.62613e19	−0.365058
$86$	0	0
$87$	−5.71695e19	−0.246724
$88$	0	0
$89$	−7.67205e18	−0.0260805	−0.0130403	−	0.999915i	$-0.504151\pi$
$89$	−7.67205e18	−0.0260805	−0.0130403	+	0.999915i	$0.504151\pi$
$90$	0	0
$91$	3.76333e19	0.101306
$92$	0	0
$93$	1.65608e20	0.354822
$94$	0	0
$95$	6.67021e20	1.14299
$96$	0	0
$97$	4.63755e20	0.638535	0.319268	−	0.947665i	$-0.396563\pi$
$97$	4.63755e20	0.638535	0.319268	+	0.947665i	$0.396563\pi$
$98$	0	0
$99$	−5.58271e20	−0.620406
$100$	0	0
$101$	1.75642e20	0.158217	0.0791086	−	0.996866i	$-0.474793\pi$
$101$	1.75642e20	0.158217	0.0791086	+	0.996866i	$0.474793\pi$
$102$	0	0
$103$	1.29018e21	0.945933	0.472967	−	0.881080i	$-0.343183\pi$
$103$	1.29018e21	0.945933	0.472967	+	0.881080i	$0.343183\pi$
$104$	0	0
$105$	6.28976e19	0.0376831
$106$	0	0
$107$	−5.76913e19	−0.0283518	−0.0141759	−	0.999900i	$-0.504512\pi$
$107$	−5.76913e19	−0.0283518	−0.0141759	+	0.999900i	$0.504512\pi$
$108$	0	0
$109$	−5.74941e20	−0.232619	−0.116309	−	0.993213i	$-0.537106\pi$
$109$	−5.74941e20	−0.232619	−0.116309	+	0.993213i	$0.537106\pi$
$110$	0	0
$111$	−1.80122e21	−0.602109
$112$	0	0
$113$	−4.56755e21	−1.26578	−0.632892	−	0.774240i	$-0.718132\pi$
$113$	−4.56755e21	−1.26578	−0.632892	+	0.774240i	$0.718132\pi$
$114$	0	0
$115$	4.26336e21	0.982708
$116$	0	0
$117$	−2.74399e21	−0.527753
$118$	0	0
$119$	1.42255e20	0.0228994
$120$	0	0
$121$	1.82352e22	2.46413
$122$	0	0
$123$	1.31565e21	0.149671
$124$	0	0
$125$	−1.01911e22	−0.978734
$126$	0	0
$127$	−1.84938e22	−1.50344	−0.751722	−	0.659480i	$-0.770776\pi$
$127$	−1.84938e22	−1.50344	−0.751722	+	0.659480i	$0.770776\pi$
$128$	0	0
$129$	9.66700e21	0.666960
$130$	0	0
$131$	1.25296e22	0.735509	0.367754	−	0.929923i	$-0.380127\pi$
$131$	1.25296e22	0.735509	0.367754	+	0.929923i	$0.380127\pi$
$132$	0	0
$133$	−1.43201e21	−0.0716976
$134$	0	0
$135$	−4.58612e21	−0.196309
$136$	0	0
$137$	3.84075e22	1.40880	0.704402	−	0.709801i	$-0.251215\pi$
$137$	3.84075e22	1.40880	0.704402	+	0.709801i	$0.251215\pi$
$138$	0	0
$139$	9.58856e21	0.302063	0.151032	−	0.988529i	$-0.451740\pi$
$139$	9.58856e21	0.302063	0.151032	+	0.988529i	$0.451740\pi$
$140$	0	0
$141$	2.41320e22	0.654324
$142$	0	0
$143$	1.26002e23	2.94679
$144$	0	0
$145$	2.15655e22	0.435908
$146$	0	0
$147$	3.28465e22	0.574986
$148$	0	0
$149$	3.44428e22	0.523170	0.261585	−	0.965180i	$-0.415755\pi$
$149$	3.44428e22	0.523170	0.261585	+	0.965180i	$0.415755\pi$
$150$	0	0
$151$	3.63152e22	0.479546	0.239773	−	0.970829i	$-0.422927\pi$
$151$	3.63152e22	0.479546	0.239773	+	0.970829i	$0.422927\pi$
$152$	0	0
$153$	−1.03724e22	−0.119294
$154$	0	0
$155$	−6.24707e22	−0.626893
$156$	0	0
$157$	−9.41986e22	−0.826225	−0.413112	−	0.910680i	$-0.635558\pi$
$157$	−9.41986e22	−0.826225	−0.413112	+	0.910680i	$0.635558\pi$
$158$	0	0
$159$	−2.56278e22	−0.196808
$160$	0	0
$161$	−9.15290e21	−0.0616435
$162$	0	0
$163$	1.80771e23	1.06945	0.534723	−	0.845027i	$-0.320416\pi$
$163$	1.80771e23	1.06945	0.534723	+	0.845027i	$0.320416\pi$
$164$	0	0
$165$	2.10591e23	1.09612
$166$	0	0
$167$	−3.71934e22	−0.170586	−0.0852929	−	0.996356i	$-0.527183\pi$
$167$	−3.71934e22	−0.170586	−0.0852929	+	0.996356i	$0.527183\pi$
$168$	0	0
$169$	3.72255e23	1.50671
$170$	0	0
$171$	1.04414e23	0.373507
$172$	0	0
$173$	2.58398e23	0.818096	0.409048	−	0.912513i	$-0.365861\pi$
$173$	2.58398e23	0.818096	0.409048	+	0.912513i	$0.365861\pi$
$174$	0	0
$175$	−9.23645e20	−0.00259183
$176$	0	0
$177$	−3.03713e23	−0.756388
$178$	0	0
$179$	3.35184e23	0.741868	0.370934	−	0.928659i	$-0.379038\pi$
$179$	3.35184e23	0.741868	0.370934	+	0.928659i	$0.379038\pi$
$180$	0	0
$181$	−1.92267e23	−0.378687	−0.189343	−	0.981911i	$-0.560636\pi$
$181$	−1.92267e23	−0.378687	−0.189343	+	0.981911i	$0.560636\pi$
$182$	0	0
$183$	−1.17496e23	−0.206199
$184$	0	0
$185$	6.79455e23	1.06379
$186$	0	0
$187$	4.76292e23	0.666095
$188$	0	0
$189$	9.84583e21	0.0123141
$190$	0	0
$191$	4.60821e23	0.516038	0.258019	−	0.966140i	$-0.416930\pi$
$191$	4.60821e23	0.516038	0.258019	+	0.966140i	$0.416930\pi$
$192$	0	0
$193$	1.29600e24	1.30093	0.650465	−	0.759536i	$-0.274574\pi$
$193$	1.29600e24	1.30093	0.650465	+	0.759536i	$0.274574\pi$
$194$	0	0
$195$	1.03509e24	0.932425
$196$	0	0
$197$	2.05711e24	1.66480	0.832402	−	0.554172i	$-0.186965\pi$
$197$	2.05711e24	1.66480	0.832402	+	0.554172i	$0.186965\pi$
$198$	0	0
$199$	−5.34557e22	−0.0389078	−0.0194539	−	0.999811i	$-0.506193\pi$
$199$	−5.34557e22	−0.0389078	−0.0194539	+	0.999811i	$0.506193\pi$
$200$	0	0
$201$	−8.05198e23	−0.527648
$202$	0	0
$203$	−4.62984e22	−0.0273437
$204$	0	0
$205$	−4.96289e23	−0.264435
$206$	0	0
$207$	6.67374e23	0.321130
$208$	0	0
$209$	−4.79460e24	−2.08553
$210$	0	0
$211$	3.16142e24	1.24427	0.622137	−	0.782908i	$-0.286265\pi$
$211$	3.16142e24	1.24427	0.622137	+	0.782908i	$0.286265\pi$
$212$	0	0
$213$	4.34388e23	0.154843
$214$	0	0
$215$	−3.64659e24	−1.17837
$216$	0	0
$217$	1.34117e23	0.0393239
$218$	0	0
$219$	−4.02508e24	−1.07179
$220$	0	0
$221$	2.34105e24	0.566619
$222$	0	0
$223$	3.10623e24	0.683963	0.341981	−	0.939707i	$-0.388902\pi$
$223$	3.10623e24	0.683963	0.341981	+	0.939707i	$0.388902\pi$
$224$	0	0
$225$	6.73466e22	0.0135020
$226$	0	0
$227$	2.04882e24	0.374310	0.187155	−	0.982330i	$-0.440073\pi$
$227$	2.04882e24	0.374310	0.187155	+	0.982330i	$0.440073\pi$
$228$	0	0
$229$	4.12622e24	0.687512	0.343756	−	0.939059i	$-0.388301\pi$
$229$	4.12622e24	0.687512	0.343756	+	0.939059i	$0.388301\pi$
$230$	0	0
$231$	−4.52113e23	−0.0687577
$232$	0	0
$233$	3.43270e24	0.476869	0.238434	−	0.971159i	$-0.423366\pi$
$233$	3.43270e24	0.476869	0.238434	+	0.971159i	$0.423366\pi$
$234$	0	0
$235$	−9.10308e24	−1.15605
$236$	0	0
$237$	−1.25351e24	−0.145636
$238$	0	0
$239$	−8.29149e24	−0.881972	−0.440986	−	0.897514i	$-0.645371\pi$
$239$	−8.29149e24	−0.881972	−0.440986	+	0.897514i	$0.645371\pi$
$240$	0	0
$241$	−1.30526e25	−1.27209	−0.636045	−	0.771652i	$-0.719431\pi$
$241$	−1.30526e25	−1.27209	−0.636045	+	0.771652i	$0.719431\pi$
$242$	0	0
$243$	−7.17898e23	−0.0641500
$244$	0	0
$245$	−1.23904e25	−1.01588
$246$	0	0
$247$	−2.35662e25	−1.77407
$248$	0	0
$249$	6.54282e24	0.452555
$250$	0	0
$251$	1.77630e25	1.12965	0.564823	−	0.825212i	$-0.308944\pi$
$251$	1.77630e25	1.12965	0.564823	+	0.825212i	$0.308944\pi$
$252$	0	0
$253$	−3.06453e25	−1.79308
$254$	0	0
$255$	3.91267e24	0.210766
$256$	0	0
$257$	−5.70058e24	−0.282893	−0.141446	−	0.989946i	$-0.545175\pi$
$257$	−5.70058e24	−0.282893	−0.141446	+	0.989946i	$0.545175\pi$
$258$	0	0
$259$	−1.45871e24	−0.0667299
$260$	0	0
$261$	3.37580e24	0.142446
$262$	0	0
$263$	−9.73572e23	−0.0379169	−0.0189584	−	0.999820i	$-0.506035\pi$
$263$	−9.73572e23	−0.0379169	−0.0189584	+	0.999820i	$0.506035\pi$
$264$	0	0
$265$	9.66733e24	0.347716
$266$	0	0
$267$	4.53027e23	0.0150576
$268$	0	0
$269$	−5.78587e25	−1.77816	−0.889078	−	0.457756i	$-0.848653\pi$
$269$	−5.78587e25	−1.77816	−0.889078	+	0.457756i	$0.848653\pi$
$270$	0	0
$271$	−4.29033e25	−1.21987	−0.609934	−	0.792452i	$-0.708804\pi$
$271$	−4.29033e25	−1.21987	−0.609934	+	0.792452i	$0.708804\pi$
$272$	0	0
$273$	−2.22221e24	−0.0584893
$274$	0	0
$275$	−3.09251e24	−0.0753907
$276$	0	0
$277$	6.39000e25	1.44366	0.721828	−	0.692073i	$-0.243302\pi$
$277$	6.39000e25	1.44366	0.721828	+	0.692073i	$0.243302\pi$
$278$	0	0
$279$	−9.77899e24	−0.204857
$280$	0	0
$281$	3.23104e25	0.627951	0.313976	−	0.949431i	$-0.398339\pi$
$281$	3.23104e25	0.627951	0.313976	+	0.949431i	$0.398339\pi$
$282$	0	0
$283$	1.00627e25	0.181534	0.0907668	−	0.995872i	$-0.471068\pi$
$283$	1.00627e25	0.181534	0.0907668	+	0.995872i	$0.471068\pi$
$284$	0	0
$285$	−3.93869e25	−0.659905
$286$	0	0
$287$	1.06547e24	0.0165875
$288$	0	0
$289$	−6.02427e25	−0.871921
$290$	0	0
$291$	−2.73842e25	−0.368659
$292$	0	0
$293$	4.93011e25	0.617656	0.308828	−	0.951118i	$-0.400063\pi$
$293$	4.93011e25	0.617656	0.308828	+	0.951118i	$0.400063\pi$
$294$	0	0
$295$	1.14567e26	1.33637
$296$	0	0
$297$	3.29654e25	0.358191
$298$	0	0
$299$	−1.50627e26	−1.52530
$300$	0	0
$301$	7.82877e24	0.0739172
$302$	0	0
$303$	−1.03715e25	−0.0913467
$304$	0	0
$305$	4.43218e25	0.364308
$306$	0	0
$307$	1.49828e25	0.114985	0.0574925	−	0.998346i	$-0.481689\pi$
$307$	1.49828e25	0.114985	0.0574925	+	0.998346i	$0.481689\pi$
$308$	0	0
$309$	−7.61840e25	−0.546135
$310$	0	0
$311$	1.48794e26	0.996786	0.498393	−	0.866951i	$-0.333924\pi$
$311$	1.48794e26	0.996786	0.498393	+	0.866951i	$0.333924\pi$
$312$	0	0
$313$	−1.10961e26	−0.694951	−0.347476	−	0.937689i	$-0.612961\pi$
$313$	−1.10961e26	−0.694951	−0.347476	+	0.937689i	$0.612961\pi$
$314$	0	0
$315$	−3.71404e24	−0.0217564
$316$	0	0
$317$	7.14805e25	0.391801	0.195900	−	0.980624i	$-0.437237\pi$
$317$	7.14805e25	0.391801	0.195900	+	0.980624i	$0.437237\pi$
$318$	0	0
$319$	−1.55014e26	−0.795371
$320$	0	0
$321$	3.40661e24	0.0163689
$322$	0	0
$323$	−8.90810e25	−0.401013
$324$	0	0
$325$	−1.52002e25	−0.0641317
$326$	0	0
$327$	3.39497e25	0.134302
$328$	0	0
$329$	1.95432e25	0.0725168
$330$	0	0
$331$	5.05668e26	1.76065	0.880323	−	0.474375i	$-0.157326\pi$
$331$	5.05668e26	1.76065	0.880323	+	0.474375i	$0.157326\pi$
$332$	0	0
$333$	1.06360e26	0.347628
$334$	0	0
$335$	3.03737e26	0.932238
$336$	0	0
$337$	3.50546e26	1.01072	0.505360	−	0.862909i	$-0.331360\pi$
$337$	3.50546e26	1.01072	0.505360	+	0.862909i	$0.331360\pi$
$338$	0	0
$339$	2.69709e26	0.730801
$340$	0	0
$341$	4.49044e26	1.14385
$342$	0	0
$343$	5.33106e25	0.127710
$344$	0	0
$345$	−2.51747e26	−0.567366
$346$	0	0
$347$	3.51664e26	0.745879	0.372940	−	0.927856i	$-0.378350\pi$
$347$	3.51664e26	0.745879	0.372940	+	0.927856i	$0.378350\pi$
$348$	0	0
$349$	5.74285e26	1.14673	0.573364	−	0.819300i	$-0.305638\pi$
$349$	5.74285e26	1.14673	0.573364	+	0.819300i	$0.305638\pi$
$350$	0	0
$351$	1.62030e26	0.304698
$352$	0	0
$353$	1.85373e25	0.0328406	0.0164203	−	0.999865i	$-0.494773\pi$
$353$	1.85373e25	0.0328406	0.0164203	+	0.999865i	$0.494773\pi$
$354$	0	0
$355$	−1.63860e26	−0.273574
$356$	0	0
$357$	−8.40001e24	−0.0132210
$358$	0	0
$359$	3.41606e26	0.507030	0.253515	−	0.967331i	$-0.418413\pi$
$359$	3.41606e26	0.507030	0.253515	+	0.967331i	$0.418413\pi$
$360$	0	0
$361$	1.82527e26	0.255565
$362$	0	0
$363$	−1.07677e27	−1.42267
$364$	0	0
$365$	1.51834e27	1.89362
$366$	0	0
$367$	−5.79162e26	−0.682034	−0.341017	−	0.940057i	$-0.610771\pi$
$367$	−5.79162e26	−0.682034	−0.341017	+	0.940057i	$0.610771\pi$
$368$	0	0
$369$	−7.76878e25	−0.0864123
$370$	0	0
$371$	−2.07546e25	−0.0218116
$372$	0	0
$373$	−1.74513e27	−1.73335	−0.866673	−	0.498877i	$-0.833746\pi$
$373$	−1.74513e27	−1.73335	−0.866673	+	0.498877i	$0.833746\pi$
$374$	0	0
$375$	6.01772e26	0.565072
$376$	0	0
$377$	−7.61919e26	−0.676588
$378$	0	0
$379$	−1.08818e27	−0.914093	−0.457046	−	0.889443i	$-0.651093\pi$
$379$	−1.08818e27	−0.914093	−0.457046	+	0.889443i	$0.651093\pi$
$380$	0	0
$381$	1.09204e27	0.868014
$382$	0	0
$383$	2.35974e27	1.77532	0.887661	−	0.460498i	$-0.152329\pi$
$383$	2.35974e27	1.77532	0.887661	+	0.460498i	$0.152329\pi$
$384$	0	0
$385$	1.70546e26	0.121480
$386$	0	0
$387$	−5.70827e26	−0.385070
$388$	0	0
$389$	−2.72162e27	−1.73923	−0.869616	−	0.493729i	$-0.835633\pi$
$389$	−2.72162e27	−1.73923	−0.869616	+	0.493729i	$0.835633\pi$
$390$	0	0
$391$	−5.69373e26	−0.344779
$392$	0	0
$393$	−7.39859e26	−0.424646
$394$	0	0
$395$	4.72847e26	0.257307
$396$	0	0
$397$	−2.20086e27	−1.13577	−0.567887	−	0.823106i	$-0.692239\pi$
$397$	−2.20086e27	−1.13577	−0.567887	+	0.823106i	$0.692239\pi$
$398$	0	0
$399$	8.45589e25	0.0413946
$400$	0	0
$401$	2.61699e27	1.21559	0.607793	−	0.794096i	$-0.292055\pi$
$401$	2.61699e27	1.21559	0.607793	+	0.794096i	$0.292055\pi$
$402$	0	0
$403$	2.20712e27	0.973023
$404$	0	0
$405$	2.70806e26	0.113339
$406$	0	0
$407$	−4.88398e27	−1.94103
$408$	0	0
$409$	−8.92474e26	−0.336900	−0.168450	−	0.985710i	$-0.553876\pi$
$409$	−8.92474e26	−0.336900	−0.168450	+	0.985710i	$0.553876\pi$
$410$	0	0
$411$	−2.26793e27	−0.813373
$412$	0	0
$413$	−2.45961e26	−0.0838282
$414$	0	0
$415$	−2.46808e27	−0.799567
$416$	0	0
$417$	−5.66195e26	−0.174396
$418$	0	0
$419$	2.48238e27	0.727144	0.363572	−	0.931566i	$-0.381557\pi$
$419$	2.48238e27	0.727144	0.363572	+	0.931566i	$0.381557\pi$
$420$	0	0
$421$	4.09797e27	1.14184	0.570922	−	0.821004i	$-0.306586\pi$
$421$	4.09797e27	1.14184	0.570922	+	0.821004i	$0.306586\pi$
$422$	0	0
$423$	−1.42497e27	−0.377774
$424$	0	0
$425$	−5.74571e25	−0.0144964
$426$	0	0
$427$	−9.51534e25	−0.0228524
$428$	0	0
$429$	−7.44029e27	−1.70133
$430$	0	0
$431$	4.89793e27	1.06660	0.533299	−	0.845927i	$-0.320952\pi$
$431$	4.89793e27	1.06660	0.533299	+	0.845927i	$0.320952\pi$
$432$	0	0
$433$	−2.06118e27	−0.427557	−0.213778	−	0.976882i	$-0.568577\pi$
$433$	−2.06118e27	−0.427557	−0.213778	+	0.976882i	$0.568577\pi$
$434$	0	0
$435$	−1.27342e27	−0.251672
$436$	0	0
$437$	5.73161e27	1.07950
$438$	0	0
$439$	−1.00358e28	−1.80167	−0.900835	−	0.434161i	$-0.857045\pi$
$439$	−1.00358e28	−1.80167	−0.900835	+	0.434161i	$0.857045\pi$
$440$	0	0
$441$	−1.93956e27	−0.331969
$442$	0	0
$443$	−4.23345e27	−0.690964	−0.345482	−	0.938425i	$-0.612285\pi$
$443$	−4.23345e27	−0.690964	−0.345482	+	0.938425i	$0.612285\pi$
$444$	0	0
$445$	−1.70891e26	−0.0266035
$446$	0	0
$447$	−2.03382e27	−0.302053
$448$	0	0
$449$	−6.74432e27	−0.955766	−0.477883	−	0.878424i	$-0.658596\pi$
$449$	−6.74432e27	−0.955766	−0.477883	+	0.878424i	$0.658596\pi$
$450$	0	0
$451$	3.56737e27	0.482496
$452$	0	0
$453$	−2.14438e27	−0.276866
$454$	0	0
$455$	8.38261e26	0.103338
$456$	0	0
$457$	−1.13821e28	−1.34000	−0.669998	−	0.742362i	$-0.733705\pi$
$457$	−1.13821e28	−1.34000	−0.669998	+	0.742362i	$0.733705\pi$
$458$	0	0
$459$	6.12478e26	0.0688743
$460$	0	0
$461$	6.24175e27	0.670574	0.335287	−	0.942116i	$-0.391167\pi$
$461$	6.24175e27	0.670574	0.335287	+	0.942116i	$0.391167\pi$
$462$	0	0
$463$	2.31855e27	0.238021	0.119011	−	0.992893i	$-0.462028\pi$
$463$	2.31855e27	0.238021	0.119011	+	0.992893i	$0.462028\pi$
$464$	0	0
$465$	3.68883e27	0.361937
$466$	0	0
$467$	9.47846e26	0.0889018	0.0444509	−	0.999012i	$-0.485846\pi$
$467$	9.47846e26	0.0889018	0.0444509	+	0.999012i	$0.485846\pi$
$468$	0	0
$469$	−6.52086e26	−0.0584776
$470$	0	0
$471$	5.56234e27	0.477021
$472$	0	0
$473$	2.62120e28	2.15009
$474$	0	0
$475$	5.78393e26	0.0453879
$476$	0	0
$477$	1.51330e27	0.113627
$478$	0	0
$479$	1.76996e28	1.27186	0.635932	−	0.771745i	$-0.280616\pi$
$479$	1.76996e28	1.27186	0.635932	+	0.771745i	$0.280616\pi$
$480$	0	0
$481$	−2.40055e28	−1.65115
$482$	0	0
$483$	5.40470e26	0.0355899
$484$	0	0
$485$	1.03299e28	0.651339
$486$	0	0
$487$	−1.99209e28	−1.20297	−0.601484	−	0.798885i	$-0.705424\pi$
$487$	−1.99209e28	−1.20297	−0.601484	+	0.798885i	$0.705424\pi$
$488$	0	0
$489$	−1.06743e28	−0.617445
$490$	0	0
$491$	5.71237e27	0.316563	0.158282	−	0.987394i	$-0.449405\pi$
$491$	5.71237e27	0.316563	0.158282	+	0.987394i	$0.449405\pi$
$492$	0	0
$493$	−2.88008e27	−0.152937
$494$	0	0
$495$	−1.24352e28	−0.632846
$496$	0	0
$497$	3.51787e26	0.0171608
$498$	0	0
$499$	1.17032e28	0.547332	0.273666	−	0.961825i	$-0.411764\pi$
$499$	1.17032e28	0.547332	0.273666	+	0.961825i	$0.411764\pi$
$500$	0	0
$501$	2.19623e27	0.0984878
$502$	0	0
$503$	3.15184e28	1.35550	0.677750	−	0.735292i	$-0.262955\pi$
$503$	3.15184e28	1.35550	0.677750	+	0.735292i	$0.262955\pi$
$504$	0	0
$505$	3.91233e27	0.161390
$506$	0	0
$507$	−2.19813e28	−0.869899
$508$	0	0
$509$	4.58029e28	1.73923	0.869613	−	0.493733i	$-0.164368\pi$
$509$	4.58029e28	1.73923	0.869613	+	0.493733i	$0.164368\pi$
$510$	0	0
$511$	−3.25969e27	−0.118784
$512$	0	0
$513$	−6.16552e27	−0.215644
$514$	0	0
$515$	2.87382e28	0.964901
$516$	0	0
$517$	6.54337e28	2.10936
$518$	0	0
$519$	−1.52581e28	−0.472328
$520$	0	0
$521$	−5.19044e27	−0.154315	−0.0771575	−	0.997019i	$-0.524584\pi$
$521$	−5.19044e27	−0.154315	−0.0771575	+	0.997019i	$0.524584\pi$
$522$	0	0
$523$	2.08282e27	0.0594817	0.0297408	−	0.999558i	$-0.490532\pi$
$523$	2.08282e27	0.0594817	0.0297408	+	0.999558i	$0.490532\pi$
$524$	0	0
$525$	5.45403e25	0.00149639
$526$	0	0
$527$	8.34299e27	0.219943
$528$	0	0
$529$	−2.83724e27	−0.0718805
$530$	0	0
$531$	1.79340e28	0.436701
$532$	0	0
$533$	1.75342e28	0.410439
$534$	0	0
$535$	−1.28504e27	−0.0289203
$536$	0	0
$537$	−1.97923e28	−0.428318
$538$	0	0
$539$	8.90630e28	1.85360
$540$	0	0
$541$	−1.25240e28	−0.250710	−0.125355	−	0.992112i	$-0.540007\pi$
$541$	−1.25240e28	−0.250710	−0.125355	+	0.992112i	$0.540007\pi$
$542$	0	0
$543$	1.13532e28	0.218635
$544$	0	0
$545$	−1.28065e28	−0.237283
$546$	0	0
$547$	−5.63104e28	−1.00397	−0.501987	−	0.864875i	$-0.667397\pi$
$547$	−5.63104e28	−1.00397	−0.501987	+	0.864875i	$0.667397\pi$
$548$	0	0
$549$	6.93801e27	0.119049
$550$	0	0
$551$	2.89924e28	0.478842
$552$	0	0
$553$	−1.01514e27	−0.0161404
$554$	0	0
$555$	−4.01212e28	−0.614182
$556$	0	0
$557$	4.80914e28	0.708904	0.354452	−	0.935074i	$-0.384667\pi$
$557$	4.80914e28	0.708904	0.354452	+	0.935074i	$0.384667\pi$
$558$	0	0
$559$	1.28836e29	1.82899
$560$	0	0
$561$	−2.81246e28	−0.384570
$562$	0	0
$563$	1.81812e28	0.239488	0.119744	−	0.992805i	$-0.461793\pi$
$563$	1.81812e28	0.239488	0.119744	+	0.992805i	$0.461793\pi$
$564$	0	0
$565$	−1.01740e29	−1.29117
$566$	0	0
$567$	−5.81386e26	−0.00710956
$568$	0	0
$569$	1.75144e28	0.206403	0.103201	−	0.994660i	$-0.467091\pi$
$569$	1.75144e28	0.206403	0.103201	+	0.994660i	$0.467091\pi$
$570$	0	0
$571$	−1.67794e29	−1.90589	−0.952946	−	0.303140i	$-0.901965\pi$
$571$	−1.67794e29	−1.90589	−0.952946	+	0.303140i	$0.901965\pi$
$572$	0	0
$573$	−2.72110e28	−0.297935
$574$	0	0
$575$	3.69688e27	0.0390232
$576$	0	0
$577$	7.49346e28	0.762669	0.381335	−	0.924437i	$-0.375465\pi$
$577$	7.49346e28	0.762669	0.381335	+	0.924437i	$0.375465\pi$
$578$	0	0
$579$	−7.65276e28	−0.751092
$580$	0	0
$581$	5.29867e27	0.0501554
$582$	0	0
$583$	−6.94895e28	−0.634453
$584$	0	0
$585$	−6.11209e28	−0.538336
$586$	0	0
$587$	−1.69908e29	−1.44382	−0.721912	−	0.691985i	$-0.756736\pi$
$587$	−1.69908e29	−1.44382	−0.721912	+	0.691985i	$0.756736\pi$
$588$	0	0
$589$	−8.39849e28	−0.688638
$590$	0	0
$591$	−1.21471e29	−0.961175
$592$	0	0
$593$	−1.26592e29	−0.966787	−0.483394	−	0.875403i	$-0.660596\pi$
$593$	−1.26592e29	−0.966787	−0.483394	+	0.875403i	$0.660596\pi$
$594$	0	0
$595$	3.16865e27	0.0233586
$596$	0	0
$597$	3.15650e27	0.0224634
$598$	0	0
$599$	−1.73983e29	−1.19544	−0.597718	−	0.801706i	$-0.703926\pi$
$599$	−1.73983e29	−1.19544	−0.597718	+	0.801706i	$0.703926\pi$
$600$	0	0
$601$	−2.58691e29	−1.71633	−0.858164	−	0.513376i	$-0.828395\pi$
$601$	−2.58691e29	−1.71633	−0.858164	+	0.513376i	$0.828395\pi$
$602$	0	0
$603$	4.75461e28	0.304638
$604$	0	0
$605$	4.06179e29	2.51354
$606$	0	0
$607$	1.61646e29	0.966238	0.483119	−	0.875555i	$-0.339504\pi$
$607$	1.61646e29	0.966238	0.483119	+	0.875555i	$0.339504\pi$
$608$	0	0
$609$	2.73387e27	0.0157869
$610$	0	0
$611$	3.21617e29	1.79434
$612$	0	0
$613$	8.86426e28	0.477868	0.238934	−	0.971036i	$-0.423202\pi$
$613$	8.86426e28	0.477868	0.238934	+	0.971036i	$0.423202\pi$
$614$	0	0
$615$	2.93054e28	0.152672
$616$	0	0
$617$	−7.14976e28	−0.359996	−0.179998	−	0.983667i	$-0.557609\pi$
$617$	−7.14976e28	−0.359996	−0.179998	+	0.983667i	$0.557609\pi$
$618$	0	0
$619$	3.79378e28	0.184638	0.0923188	−	0.995730i	$-0.470572\pi$
$619$	3.79378e28	0.184638	0.0923188	+	0.995730i	$0.470572\pi$
$620$	0	0
$621$	−3.94078e28	−0.185404
$622$	0	0
$623$	3.66882e26	0.00166879
$624$	0	0
$625$	−2.36211e29	−1.03887
$626$	0	0
$627$	2.83116e29	1.20408
$628$	0	0
$629$	−9.07416e28	−0.373228
$630$	0	0
$631$	−4.22002e29	−1.67883	−0.839415	−	0.543491i	$-0.817102\pi$
$631$	−4.22002e29	−1.67883	−0.839415	+	0.543491i	$0.817102\pi$
$632$	0	0
$633$	−1.86679e29	−0.718382
$634$	0	0
$635$	−4.11940e29	−1.53359
$636$	0	0
$637$	4.37758e29	1.57678
$638$	0	0
$639$	−2.56502e28	−0.0893988
$640$	0	0
$641$	−2.45898e29	−0.829364	−0.414682	−	0.909966i	$-0.636107\pi$
$641$	−2.45898e29	−0.829364	−0.414682	+	0.909966i	$0.636107\pi$
$642$	0	0
$643$	−3.81795e29	−1.24628	−0.623139	−	0.782111i	$-0.714143\pi$
$643$	−3.81795e29	−1.24628	−0.623139	+	0.782111i	$0.714143\pi$
$644$	0	0
$645$	2.15327e29	0.680334
$646$	0	0
$647$	3.01566e29	0.922332	0.461166	−	0.887314i	$-0.347431\pi$
$647$	3.01566e29	0.922332	0.461166	+	0.887314i	$0.347431\pi$
$648$	0	0
$649$	−8.23515e29	−2.43839
$650$	0	0
$651$	−7.91947e27	−0.0227037
$652$	0	0
$653$	2.27685e29	0.632042	0.316021	−	0.948752i	$-0.397653\pi$
$653$	2.27685e29	0.632042	0.316021	+	0.948752i	$0.397653\pi$
$654$	0	0
$655$	2.79090e29	0.750257
$656$	0	0
$657$	2.37677e29	0.618800
$658$	0	0
$659$	4.45804e29	1.12421	0.562104	−	0.827066i	$-0.309992\pi$
$659$	4.45804e29	1.12421	0.562104	+	0.827066i	$0.309992\pi$
$660$	0	0
$661$	4.54039e29	1.10912	0.554559	−	0.832144i	$-0.312887\pi$
$661$	4.54039e29	1.10912	0.554559	+	0.832144i	$0.312887\pi$
$662$	0	0
$663$	−1.38236e29	−0.327138
$664$	0	0
$665$	−3.18973e28	−0.0731353
$666$	0	0
$667$	1.85309e29	0.411694
$668$	0	0
$669$	−1.83420e29	−0.394886
$670$	0	0
$671$	−3.18588e29	−0.664727
$672$	0	0
$673$	5.05512e28	0.102229	0.0511144	−	0.998693i	$-0.483723\pi$
$673$	5.05512e28	0.102229	0.0511144	+	0.998693i	$0.483723\pi$
$674$	0	0
$675$	−3.97675e27	−0.00779541
$676$	0	0
$677$	−3.44673e29	−0.654977	−0.327489	−	0.944855i	$-0.606202\pi$
$677$	−3.44673e29	−0.654977	−0.327489	+	0.944855i	$0.606202\pi$
$678$	0	0
$679$	−2.21770e28	−0.0408573
$680$	0	0
$681$	−1.20981e29	−0.216108
$682$	0	0
$683$	−3.07293e29	−0.532273	−0.266137	−	0.963935i	$-0.585747\pi$
$683$	−3.07293e29	−0.532273	−0.266137	+	0.963935i	$0.585747\pi$
$684$	0	0
$685$	8.55508e29	1.43705
$686$	0	0
$687$	−2.43649e29	−0.396935
$688$	0	0
$689$	−3.41552e29	−0.539703
$690$	0	0
$691$	−7.85532e28	−0.120405	−0.0602025	−	0.998186i	$-0.519175\pi$
$691$	−7.85532e28	−0.120405	−0.0602025	+	0.998186i	$0.519175\pi$
$692$	0	0
$693$	2.66968e28	0.0396973
$694$	0	0
$695$	2.13580e29	0.308120
$696$	0	0
$697$	6.62797e28	0.0927761
$698$	0	0
$699$	−2.02698e29	−0.275320
$700$	0	0
$701$	7.75849e29	1.02268	0.511338	−	0.859380i	$-0.329150\pi$
$701$	7.75849e29	1.02268	0.511338	+	0.859380i	$0.329150\pi$
$702$	0	0
$703$	9.13453e29	1.16857
$704$	0	0
$705$	5.37528e29	0.667445
$706$	0	0
$707$	−8.39929e27	−0.0101237
$708$	0	0
$709$	1.53428e30	1.79523	0.897613	−	0.440785i	$-0.145300\pi$
$709$	1.53428e30	1.79523	0.897613	+	0.440785i	$0.145300\pi$
$710$	0	0
$711$	7.40182e28	0.0840829
$712$	0	0
$713$	−5.36801e29	−0.592070
$714$	0	0
$715$	2.80663e30	3.00588
$716$	0	0
$717$	4.89604e29	0.509206
$718$	0	0
$719$	−5.56901e29	−0.562503	−0.281252	−	0.959634i	$-0.590750\pi$
$719$	−5.56901e29	−0.562503	−0.281252	+	0.959634i	$0.590750\pi$
$720$	0	0
$721$	−6.16973e28	−0.0605265
$722$	0	0
$723$	7.70743e29	0.734442
$724$	0	0
$725$	1.87000e28	0.0173098
$726$	0	0
$727$	1.43096e29	0.128681	0.0643407	−	0.997928i	$-0.479506\pi$
$727$	1.43096e29	0.128681	0.0643407	+	0.997928i	$0.479506\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	4.87003e29	0.413428
$732$	0	0
$733$	−6.81141e29	−0.561882	−0.280941	−	0.959725i	$-0.590647\pi$
$733$	−6.81141e29	−0.561882	−0.280941	+	0.959725i	$0.590647\pi$
$734$	0	0
$735$	7.31639e29	0.586516
$736$	0	0
$737$	−2.18328e30	−1.70099
$738$	0	0
$739$	−2.16993e30	−1.64315	−0.821577	−	0.570097i	$-0.806906\pi$
$739$	−2.16993e30	−1.64315	−0.821577	+	0.570097i	$0.806906\pi$
$740$	0	0
$741$	1.39156e30	1.02426
$742$	0	0
$743$	1.40097e29	0.100241	0.0501206	−	0.998743i	$-0.484039\pi$
$743$	1.40097e29	0.100241	0.0501206	+	0.998743i	$0.484039\pi$
$744$	0	0
$745$	7.67196e29	0.533661
$746$	0	0
$747$	−3.86347e29	−0.261283
$748$	0	0
$749$	2.75883e27	0.00181412
$750$	0	0
$751$	−8.06099e29	−0.515429	−0.257715	−	0.966221i	$-0.582969\pi$
$751$	−8.06099e29	−0.515429	−0.257715	+	0.966221i	$0.582969\pi$
$752$	0	0
$753$	−1.04889e30	−0.652202
$754$	0	0
$755$	8.08901e29	0.489162
$756$	0	0
$757$	−1.54959e30	−0.911401	−0.455700	−	0.890133i	$-0.650611\pi$
$757$	−1.54959e30	−0.911401	−0.455700	+	0.890133i	$0.650611\pi$
$758$	0	0
$759$	1.80958e30	1.03523
$760$	0	0
$761$	2.04061e29	0.113559	0.0567793	−	0.998387i	$-0.481917\pi$
$761$	2.04061e29	0.113559	0.0567793	+	0.998387i	$0.481917\pi$
$762$	0	0
$763$	2.74940e28	0.0148843
$764$	0	0
$765$	−2.31039e29	−0.121686
$766$	0	0
$767$	−4.04770e30	−2.07423
$768$	0	0
$769$	1.27880e30	0.637641	0.318821	−	0.947815i	$-0.396713\pi$
$769$	1.27880e30	0.637641	0.318821	+	0.947815i	$0.396713\pi$
$770$	0	0
$771$	3.36614e29	0.163328
$772$	0	0
$773$	3.43774e30	1.62326	0.811630	−	0.584172i	$-0.198581\pi$
$773$	3.43774e30	1.62326	0.811630	+	0.584172i	$0.198581\pi$
$774$	0	0
$775$	−5.41701e28	−0.0248938
$776$	0	0
$777$	8.61352e28	0.0385265
$778$	0	0
$779$	−6.67206e29	−0.290480
$780$	0	0
$781$	1.17784e30	0.499172
$782$	0	0
$783$	−1.99338e29	−0.0822414
$784$	0	0
$785$	−2.09822e30	−0.842792
$786$	0	0
$787$	−1.06317e30	−0.415784	−0.207892	−	0.978152i	$-0.566660\pi$
$787$	−1.06317e30	−0.415784	−0.207892	+	0.978152i	$0.566660\pi$
$788$	0	0
$789$	5.74885e28	0.0218913
$790$	0	0
$791$	2.18423e29	0.0809925
$792$	0	0
$793$	−1.56591e30	−0.565456
$794$	0	0
$795$	−5.70846e29	−0.200754
$796$	0	0
$797$	−2.59470e30	−0.888742	−0.444371	−	0.895843i	$-0.646573\pi$
$797$	−2.59470e30	−0.888742	−0.444371	+	0.895843i	$0.646573\pi$
$798$	0	0
$799$	1.21572e30	0.405595
$800$	0	0
$801$	−2.67508e28	−0.00869351
$802$	0	0
$803$	−1.09139e31	−3.45516
$804$	0	0
$805$	−2.03876e29	−0.0628795
$806$	0	0
$807$	3.41650e30	1.02662
$808$	0	0
$809$	−4.54632e30	−1.33107	−0.665534	−	0.746367i	$-0.731796\pi$
$809$	−4.54632e30	−1.33107	−0.665534	+	0.746367i	$0.731796\pi$
$810$	0	0
$811$	5.33885e30	1.52310	0.761550	−	0.648106i	$-0.224439\pi$
$811$	5.33885e30	1.52310	0.761550	+	0.648106i	$0.224439\pi$
$812$	0	0
$813$	2.53339e30	0.704291
$814$	0	0
$815$	4.02658e30	1.09089
$816$	0	0
$817$	−4.90243e30	−1.29443
$818$	0	0
$819$	1.31219e29	0.0337688
$820$	0	0
$821$	−6.77652e30	−1.69982	−0.849911	−	0.526926i	$-0.823344\pi$
$821$	−6.77652e30	−1.69982	−0.849911	+	0.526926i	$0.823344\pi$
$822$	0	0
$823$	−2.32096e30	−0.567506	−0.283753	−	0.958897i	$-0.591580\pi$
$823$	−2.32096e30	−0.567506	−0.283753	+	0.958897i	$0.591580\pi$
$824$	0	0
$825$	1.82610e29	0.0435268
$826$	0	0
$827$	−3.93890e30	−0.915307	−0.457654	−	0.889131i	$-0.651310\pi$
$827$	−3.93890e30	−0.915307	−0.457654	+	0.889131i	$0.651310\pi$
$828$	0	0
$829$	3.15777e30	0.715415	0.357708	−	0.933834i	$-0.383558\pi$
$829$	3.15777e30	0.715415	0.357708	+	0.933834i	$0.383558\pi$
$830$	0	0
$831$	−3.77323e30	−0.833495
$832$	0	0
$833$	1.65474e30	0.356416
$834$	0	0
$835$	−8.28463e29	−0.174006
$836$	0	0
$837$	5.77440e29	0.118274
$838$	0	0
$839$	1.71221e28	0.00342024	0.00171012	−	0.999999i	$-0.499456\pi$
$839$	1.71221e28	0.00342024	0.00171012	+	0.999999i	$0.499456\pi$
$840$	0	0
$841$	−4.19549e30	−0.817381
$842$	0	0
$843$	−1.90790e30	−0.362548
$844$	0	0
$845$	8.29177e30	1.53692
$846$	0	0
$847$	−8.72016e29	−0.157670
$848$	0	0
$849$	−5.94193e29	−0.104809
$850$	0	0
$851$	5.83845e30	1.00470
$852$	0	0
$853$	2.27930e30	0.382680	0.191340	−	0.981524i	$-0.438717\pi$
$853$	2.27930e30	0.382680	0.191340	+	0.981524i	$0.438717\pi$
$854$	0	0
$855$	2.32576e30	0.380996
$856$	0	0
$857$	−1.06629e31	−1.70441	−0.852207	−	0.523204i	$-0.824737\pi$
$857$	−1.06629e31	−1.70441	−0.852207	+	0.523204i	$0.824737\pi$
$858$	0	0
$859$	8.93547e29	0.139376	0.0696882	−	0.997569i	$-0.477800\pi$
$859$	8.93547e29	0.139376	0.0696882	+	0.997569i	$0.477800\pi$
$860$	0	0
$861$	−6.29150e28	−0.00957682
$862$	0	0
$863$	−2.82417e30	−0.419543	−0.209772	−	0.977750i	$-0.567272\pi$
$863$	−2.82417e30	−0.419543	−0.209772	+	0.977750i	$0.567272\pi$
$864$	0	0
$865$	5.75567e30	0.834501
$866$	0	0
$867$	3.55727e30	0.503404
$868$	0	0
$869$	−3.39886e30	−0.469490
$870$	0	0
$871$	−1.07312e31	−1.44696
$872$	0	0
$873$	1.61701e30	0.212845
$874$	0	0
$875$	4.87342e29	0.0626253
$876$	0	0
$877$	9.06575e30	1.13739	0.568693	−	0.822550i	$-0.307449\pi$
$877$	9.06575e30	1.13739	0.568693	+	0.822550i	$0.307449\pi$
$878$	0	0
$879$	−2.91118e30	−0.356604
$880$	0	0
$881$	1.51630e31	1.81359	0.906794	−	0.421574i	$-0.138522\pi$
$881$	1.51630e31	1.81359	0.906794	+	0.421574i	$0.138522\pi$
$882$	0	0
$883$	1.16543e31	1.36113	0.680565	−	0.732688i	$-0.261735\pi$
$883$	1.16543e31	1.36113	0.680565	+	0.732688i	$0.261735\pi$
$884$	0	0
$885$	−6.76505e30	−0.771555
$886$	0	0
$887$	7.76908e30	0.865309	0.432655	−	0.901560i	$-0.357577\pi$
$887$	7.76908e30	0.865309	0.432655	+	0.901560i	$0.357577\pi$
$888$	0	0
$889$	8.84384e29	0.0961994
$890$	0	0
$891$	−1.94657e30	−0.206802
$892$	0	0
$893$	−1.22381e31	−1.26991
$894$	0	0
$895$	7.46605e30	0.756744
$896$	0	0
$897$	8.89435e30	0.880630
$898$	0	0
$899$	−2.71532e30	−0.262630
$900$	0	0
$901$	−1.29108e30	−0.121995
$902$	0	0
$903$	−4.62281e29	−0.0426761
$904$	0	0
$905$	−4.28265e30	−0.386280
$906$	0	0
$907$	−5.71533e30	−0.503692	−0.251846	−	0.967767i	$-0.581038\pi$
$907$	−5.71533e30	−0.503692	−0.251846	+	0.967767i	$0.581038\pi$
$908$	0	0
$909$	6.12425e29	0.0527390
$910$	0	0
$911$	−2.22259e30	−0.187032	−0.0935160	−	0.995618i	$-0.529811\pi$
$911$	−2.22259e30	−0.187032	−0.0935160	+	0.995618i	$0.529811\pi$
$912$	0	0
$913$	1.77408e31	1.45891
$914$	0	0
$915$	−2.61716e30	−0.210333
$916$	0	0
$917$	−5.99171e29	−0.0470623
$918$	0	0
$919$	9.94638e30	0.763576	0.381788	−	0.924250i	$-0.375308\pi$
$919$	9.94638e30	0.763576	0.381788	+	0.924250i	$0.375308\pi$
$920$	0	0
$921$	−8.84722e29	−0.0663866
$922$	0	0
$923$	5.78926e30	0.424624
$924$	0	0
$925$	5.89175e29	0.0422431
$926$	0	0
$927$	4.49859e30	0.315311
$928$	0	0
$929$	6.57341e30	0.450429	0.225214	−	0.974309i	$-0.427692\pi$
$929$	6.57341e30	0.450429	0.225214	+	0.974309i	$0.427692\pi$
$930$	0	0
$931$	−1.66575e31	−1.11593
$932$	0	0
$933$	−8.78615e30	−0.575495
$934$	0	0
$935$	1.06091e31	0.679452
$936$	0	0
$937$	−9.88672e30	−0.619137	−0.309568	−	0.950877i	$-0.600185\pi$
$937$	−9.88672e30	−0.619137	−0.309568	+	0.950877i	$0.600185\pi$
$938$	0	0
$939$	6.55213e30	0.401230
$940$	0	0
$941$	1.35253e31	0.809947	0.404973	−	0.914328i	$-0.367281\pi$
$941$	1.35253e31	0.809947	0.404973	+	0.914328i	$0.367281\pi$
$942$	0	0
$943$	−4.26454e30	−0.249746
$944$	0	0
$945$	2.19311e29	0.0125610
$946$	0	0
$947$	2.26830e31	1.27065	0.635325	−	0.772245i	$-0.280866\pi$
$947$	2.26830e31	1.27065	0.635325	+	0.772245i	$0.280866\pi$
$948$	0	0
$949$	−5.36437e31	−2.93916
$950$	0	0
$951$	−4.22085e30	−0.226206
$952$	0	0
$953$	3.45875e31	1.81319	0.906595	−	0.422002i	$-0.138673\pi$
$953$	3.45875e31	1.81319	0.906595	+	0.422002i	$0.138673\pi$
$954$	0	0
$955$	1.02645e31	0.526386
$956$	0	0
$957$	9.15344e30	0.459207
$958$	0	0
$959$	−1.83667e30	−0.0901438
$960$	0	0
$961$	−1.29598e31	−0.622304
$962$	0	0
$963$	−2.01157e29	−0.00945059
$964$	0	0
$965$	2.88678e31	1.32702
$966$	0	0
$967$	1.92820e31	0.867311	0.433655	−	0.901079i	$-0.357224\pi$
$967$	1.92820e31	0.867311	0.433655	+	0.901079i	$0.357224\pi$
$968$	0	0
$969$	5.26015e30	0.231525
$970$	0	0
$971$	2.95793e31	1.27405	0.637024	−	0.770844i	$-0.280165\pi$
$971$	2.95793e31	1.27405	0.637024	+	0.770844i	$0.280165\pi$
$972$	0	0
$973$	−4.58530e29	−0.0193278
$974$	0	0
$975$	8.97554e29	0.0370265
$976$	0	0
$977$	4.86581e31	1.96455	0.982274	−	0.187453i	$-0.0600233\pi$
$977$	4.86581e31	1.96455	0.982274	+	0.187453i	$0.0600233\pi$
$978$	0	0
$979$	1.22838e30	0.0485415
$980$	0	0
$981$	−2.00469e30	−0.0775396
$982$	0	0
$983$	−4.31738e31	−1.63459	−0.817294	−	0.576220i	$-0.804527\pi$
$983$	−4.31738e31	−1.63459	−0.817294	+	0.576220i	$0.804527\pi$
$984$	0	0
$985$	4.58211e31	1.69819
$986$	0	0
$987$	−1.15401e30	−0.0418676
$988$	0	0
$989$	−3.13345e31	−1.11292
$990$	0	0
$991$	−3.44591e31	−1.19820	−0.599101	−	0.800673i	$-0.704475\pi$
$991$	−3.44591e31	−1.19820	−0.599101	+	0.800673i	$0.704475\pi$
$992$	0	0
$993$	−2.98592e31	−1.01651
$994$	0	0
$995$	−1.19070e30	−0.0396880
$996$	0	0
$997$	3.62353e31	1.18259	0.591293	−	0.806457i	$-0.298618\pi$
$997$	3.62353e31	1.18259	0.591293	+	0.806457i	$0.298618\pi$
$998$	0	0
$999$	−6.28046e30	−0.200703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.g.1.2		2
4.3	odd	2		3.22.a.c.1.2	✓	2
12.11	even	2		9.22.a.e.1.1		2
20.3	even	4		75.22.b.d.49.1		4
20.7	even	4		75.22.b.d.49.4		4
20.19	odd	2		75.22.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.22.a.c.1.2	✓	2	4.3	odd	2
9.22.a.e.1.1		2	12.11	even	2
48.22.a.g.1.2		2	1.1	even	1	trivial
75.22.a.d.1.1		2	20.19	odd	2
75.22.b.d.49.1		4	20.3	even	4
75.22.b.d.49.4		4	20.7	even	4

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)

\(f(q)\)	\(=\)	\(q-59049.0 q^{3} +2.22745e7 q^{5} -4.78205e7 q^{7} +3.48678e9 q^{9} +O(q^{10})\)	\(q-59049.0 q^{3} +2.22745e7 q^{5} -4.78205e7 q^{7} +3.48678e9 q^{9} -1.60111e11 q^{11} -7.86968e11 q^{13} -1.31529e12 q^{15} -2.97477e12 q^{17} +2.99456e13 q^{19} +2.82376e12 q^{21} +1.91401e14 q^{23} +1.93148e13 q^{25} -2.05891e14 q^{27} +9.68170e14 q^{29} -2.80459e15 q^{31} +9.45437e15 q^{33} -1.06518e15 q^{35} +3.05038e16 q^{37} +4.64697e16 q^{39} -2.22806e16 q^{41} -1.63711e17 q^{43} +7.76663e16 q^{45} -4.08678e17 q^{47} -5.56259e17 q^{49} +1.75657e17 q^{51} +4.34009e17 q^{53} -3.56638e18 q^{55} -1.76826e18 q^{57} +5.14341e18 q^{59} +1.98980e18 q^{61} -1.66740e17 q^{63} -1.75293e19 q^{65} +1.36361e19 q^{67} -1.13020e19 q^{69} -7.35641e18 q^{71} +6.81650e19 q^{73} -1.14052e18 q^{75} +7.65658e18 q^{77} +2.12282e19 q^{79} +1.21577e19 q^{81} -1.10803e20 q^{83} -6.62613e19 q^{85} -5.71695e19 q^{87} -7.67205e18 q^{89} +3.76333e19 q^{91} +1.65608e20 q^{93} +6.67021e20 q^{95} +4.63755e20 q^{97} -5.58271e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9}+O(q^{10}) \) 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9	\( 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9} - 219869122968 q^{11} - 48468909956 q^{13} - 58864530924 q^{15} - 11333529041436 q^{17} - 11960585011624 q^{19} + 40147185517488 q^{21} + 146508390063504 q^{23} - 4786354247074 q^{25} - 411782264189298 q^{27} - 17\!\cdots\!52 q^{29}+ \cdots - 76\!\cdots\!68 q^{99}+O(q^{100}) \) 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 - 219869122968 * q^11 - 48468909956 * q^13 - 58864530924 * q^15 - 11333529041436 * q^17 - 11960585011624 * q^19 + 40147185517488 * q^21 + 146508390063504 * q^23 - 4786354247074 * q^25 - 411782264189298 * q^27 - 1798520043674052 * q^29 - 11169107526944992 * q^31 + 12983051842137432 * q^33 + 12383869767948000 * q^35 + 12736264858660012 * q^37 + 2862040663991844 * q^39 + 122972020616468052 * q^41 - 288455418162270040 * q^43 + 3475891686531276 * q^45 - 837243745741596960 * q^47 - 715285396941470670 * q^49 + 669233556367754364 * q^51 - 43007964012775764 * q^53 - 2294863118895313296 * q^55 + 706260584351385576 * q^57 + 3523823330903857224 * q^59 - 1779023128451013860 * q^61 - 2370651157622148912 * q^63 - 33242791066177513752 * q^65 + 16454068667621610296 * q^67 - 8651173924859847696 * q^69 - 17379227131150420944 * q^71 + 50891146268473989076 * q^73 + 282629431935472626 * q^75 + 45428450983816025664 * q^77 + 54055785594190591040 * q^79 + 24315330918113857602 * q^81 - 111108429277666677288 * q^83 + 111593012278262968152 * q^85 + 106200810058909096548 * q^87 + 226920767965448065524 * q^89 - 429154174685185000352 * q^91 + 659524630358574832608 * q^93 + 1558683223531725831696 * q^95 - 128331469252795746236 * q^97 - 766636228226373222168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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