LMFDB - Embedded newform 48.14.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,14,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, 14, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 14 $
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:	$51.4708458969$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-729.000 q^{3} -24570.0 q^{5} +173704. q^{7} +531441. q^{9} +O(q^{10})$

$q-729.000 q^{3} -24570.0 q^{5} +173704. q^{7} +531441. q^{9} +970164. q^{11} -2.41494e7 q^{13} +1.79115e7 q^{15} -1.57098e8 q^{17} +1.19525e8 q^{19} -1.26630e8 q^{21} +9.49750e7 q^{23} -6.17018e8 q^{25} -3.87420e8 q^{27} +4.97957e9 q^{29} -5.63827e9 q^{31} -7.07250e8 q^{33} -4.26791e9 q^{35} -5.88141e9 q^{37} +1.76049e10 q^{39} +2.57538e10 q^{41} +6.84564e10 q^{43} -1.30575e10 q^{45} -2.96176e9 q^{47} -6.67159e10 q^{49} +1.14524e11 q^{51} +3.12743e11 q^{53} -2.38369e10 q^{55} -8.71336e10 q^{57} -4.61474e11 q^{59} +2.83119e11 q^{61} +9.23134e10 q^{63} +5.93351e11 q^{65} +1.30344e12 q^{67} -6.92368e10 q^{69} +1.26398e12 q^{71} +5.94014e11 q^{73} +4.49806e11 q^{75} +1.68521e11 q^{77} +1.15379e12 q^{79} +2.82430e11 q^{81} +4.82038e12 q^{83} +3.85990e12 q^{85} -3.63011e12 q^{87} +7.28549e11 q^{89} -4.19485e12 q^{91} +4.11030e12 q^{93} -2.93672e12 q^{95} +2.58874e12 q^{97} +5.15585e11 q^{99} +O(q^{100})$

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
$3$	−729.000	−0.577350
$3$	−729.000	−0.577350
$4$	0	0
$5$	−24570.0	−0.703234	−0.351617	−	0.936144i	$-0.614368\pi$
$5$	−24570.0	−0.703234	−0.351617	+	0.936144i	$0.614368\pi$
$6$	0	0
$7$	173704.	0.558049	0.279025	−	0.960284i	$-0.409989\pi$
$7$	173704.	0.558049	0.279025	+	0.960284i	$0.409989\pi$
$8$	0	0
$9$	531441.	0.333333
$10$	0	0
$11$	970164.	0.165117	0.0825587	−	0.996586i	$-0.473691\pi$
$11$	970164.	0.165117	0.0825587	+	0.996586i	$0.473691\pi$
$12$	0	0
$13$	−2.41494e7	−1.38763	−0.693817	−	0.720152i	$-0.744072\pi$
$13$	−2.41494e7	−1.38763	−0.693817	+	0.720152i	$0.744072\pi$
$14$	0	0
$15$	1.79115e7	0.406013
$16$	0	0
$17$	−1.57098e8	−1.57853	−0.789264	−	0.614054i	$-0.789538\pi$
$17$	−1.57098e8	−1.57853	−0.789264	+	0.614054i	$0.789538\pi$
$18$	0	0
$19$	1.19525e8	0.582854	0.291427	−	0.956593i	$-0.405870\pi$
$19$	1.19525e8	0.582854	0.291427	+	0.956593i	$0.405870\pi$
$20$	0	0
$21$	−1.26630e8	−0.322190
$22$	0	0
$23$	9.49750e7	0.133776	0.0668880	−	0.997760i	$-0.478693\pi$
$23$	9.49750e7	0.133776	0.0668880	+	0.997760i	$0.478693\pi$
$24$	0	0
$25$	−6.17018e8	−0.505461
$26$	0	0
$27$	−3.87420e8	−0.192450
$28$	0	0
$29$	4.97957e9	1.55455	0.777276	−	0.629160i	$-0.216601\pi$
$29$	4.97957e9	1.55455	0.777276	+	0.629160i	$0.216601\pi$
$30$	0	0
$31$	−5.63827e9	−1.14103	−0.570513	−	0.821289i	$-0.693255\pi$
$31$	−5.63827e9	−1.14103	−0.570513	+	0.821289i	$0.693255\pi$
$32$	0	0
$33$	−7.07250e8	−0.0953305
$34$	0	0
$35$	−4.26791e9	−0.392439
$36$	0	0
$37$	−5.88141e9	−0.376852	−0.188426	−	0.982087i	$-0.560338\pi$
$37$	−5.88141e9	−0.376852	−0.188426	+	0.982087i	$0.560338\pi$
$38$	0	0
$39$	1.76049e10	0.801150
$40$	0	0
$41$	2.57538e10	0.846734	0.423367	−	0.905958i	$-0.360848\pi$
$41$	2.57538e10	0.846734	0.423367	+	0.905958i	$0.360848\pi$
$42$	0	0
$43$	6.84564e10	1.65146	0.825732	−	0.564063i	$-0.190763\pi$
$43$	6.84564e10	1.65146	0.825732	+	0.564063i	$0.190763\pi$
$44$	0	0
$45$	−1.30575e10	−0.234411
$46$	0	0
$47$	−2.96176e9	−0.0400787	−0.0200394	−	0.999799i	$-0.506379\pi$
$47$	−2.96176e9	−0.0400787	−0.0200394	+	0.999799i	$0.506379\pi$
$48$	0	0
$49$	−6.67159e10	−0.688581
$50$	0	0
$51$	1.14524e11	0.911364
$52$	0	0
$53$	3.12743e11	1.93818	0.969090	−	0.246708i	$-0.0793488\pi$
$53$	3.12743e11	1.93818	0.969090	+	0.246708i	$0.0793488\pi$
$54$	0	0
$55$	−2.38369e10	−0.116116
$56$	0	0
$57$	−8.71336e10	−0.336511
$58$	0	0
$59$	−4.61474e11	−1.42433	−0.712163	−	0.702014i	$-0.752284\pi$
$59$	−4.61474e11	−1.42433	−0.712163	+	0.702014i	$0.752284\pi$
$60$	0	0
$61$	2.83119e11	0.703599	0.351800	−	0.936075i	$-0.385570\pi$
$61$	2.83119e11	0.703599	0.351800	+	0.936075i	$0.385570\pi$
$62$	0	0
$63$	9.23134e10	0.186016
$64$	0	0
$65$	5.93351e11	0.975832
$66$	0	0
$67$	1.30344e12	1.76037	0.880186	−	0.474629i	$-0.157418\pi$
$67$	1.30344e12	1.76037	0.880186	+	0.474629i	$0.157418\pi$
$68$	0	0
$69$	−6.92368e10	−0.0772356
$70$	0	0
$71$	1.26398e12	1.17101	0.585507	−	0.810667i	$-0.300895\pi$
$71$	1.26398e12	1.17101	0.585507	+	0.810667i	$0.300895\pi$
$72$	0	0
$73$	5.94014e11	0.459408	0.229704	−	0.973261i	$-0.426224\pi$
$73$	5.94014e11	0.459408	0.229704	+	0.973261i	$0.426224\pi$
$74$	0	0
$75$	4.49806e11	0.291828
$76$	0	0
$77$	1.68521e11	0.0921436
$78$	0	0
$79$	1.15379e12	0.534013	0.267007	−	0.963695i	$-0.413965\pi$
$79$	1.15379e12	0.534013	0.267007	+	0.963695i	$0.413965\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	0	0
$83$	4.82038e12	1.61835	0.809177	−	0.587565i	$-0.199913\pi$
$83$	4.82038e12	1.61835	0.809177	+	0.587565i	$0.199913\pi$
$84$	0	0
$85$	3.85990e12	1.11008
$86$	0	0
$87$	−3.63011e12	−0.897521
$88$	0	0
$89$	7.28549e11	0.155390	0.0776951	−	0.996977i	$-0.475244\pi$
$89$	7.28549e11	0.155390	0.0776951	+	0.996977i	$0.475244\pi$
$90$	0	0
$91$	−4.19485e12	−0.774368
$92$	0	0
$93$	4.11030e12	0.658771
$94$	0	0
$95$	−2.93672e12	−0.409883
$96$	0	0
$97$	2.58874e12	0.315552	0.157776	−	0.987475i	$-0.449568\pi$
$97$	2.58874e12	0.315552	0.157776	+	0.987475i	$0.449568\pi$
$98$	0	0
$99$	5.15585e11	0.0550391
$100$	0	0
$101$	−1.51503e13	−1.42014	−0.710072	−	0.704129i	$-0.751338\pi$
$101$	−1.51503e13	−1.42014	−0.710072	+	0.704129i	$0.751338\pi$
$102$	0	0
$103$	−6.94411e12	−0.573026	−0.286513	−	0.958076i	$-0.592496\pi$
$103$	−6.94411e12	−0.573026	−0.286513	+	0.958076i	$0.592496\pi$
$104$	0	0
$105$	3.11130e12	0.226575
$106$	0	0
$107$	1.70894e13	1.10086	0.550430	−	0.834881i	$-0.314464\pi$
$107$	1.70894e13	1.10086	0.550430	+	0.834881i	$0.314464\pi$
$108$	0	0
$109$	−1.87878e13	−1.07301	−0.536506	−	0.843896i	$-0.680256\pi$
$109$	−1.87878e13	−1.07301	−0.536506	+	0.843896i	$0.680256\pi$
$110$	0	0
$111$	4.28755e12	0.217575
$112$	0	0
$113$	1.00165e12	0.0452590	0.0226295	−	0.999744i	$-0.492796\pi$
$113$	1.00165e12	0.0452590	0.0226295	+	0.999744i	$0.492796\pi$
$114$	0	0
$115$	−2.33354e12	−0.0940759
$116$	0	0
$117$	−1.28340e13	−0.462544
$118$	0	0
$119$	−2.72885e13	−0.880897
$120$	0	0
$121$	−3.35815e13	−0.972736
$122$	0	0
$123$	−1.87745e13	−0.488862
$124$	0	0
$125$	4.51528e13	1.05869
$126$	0	0
$127$	−1.99216e13	−0.421308	−0.210654	−	0.977561i	$-0.567559\pi$
$127$	−1.99216e13	−0.421308	−0.210654	+	0.977561i	$0.567559\pi$
$128$	0	0
$129$	−4.99047e13	−0.953473
$130$	0	0
$131$	−6.83274e13	−1.18122	−0.590611	−	0.806957i	$-0.701113\pi$
$131$	−6.83274e13	−1.18122	−0.590611	+	0.806957i	$0.701113\pi$
$132$	0	0
$133$	2.07619e13	0.325261
$134$	0	0
$135$	9.51892e12	0.135338
$136$	0	0
$137$	4.81065e13	0.621613	0.310806	−	0.950473i	$-0.399401\pi$
$137$	4.81065e13	0.621613	0.310806	+	0.950473i	$0.399401\pi$
$138$	0	0
$139$	−4.00767e13	−0.471298	−0.235649	−	0.971838i	$-0.575722\pi$
$139$	−4.00767e13	−0.471298	−0.235649	+	0.971838i	$0.575722\pi$
$140$	0	0
$141$	2.15912e12	0.0231395
$142$	0	0
$143$	−2.34289e13	−0.229122
$144$	0	0
$145$	−1.22348e14	−1.09321
$146$	0	0
$147$	4.86359e13	0.397552
$148$	0	0
$149$	−4.13554e13	−0.309615	−0.154807	−	0.987945i	$-0.549476\pi$
$149$	−4.13554e13	−0.309615	−0.154807	+	0.987945i	$0.549476\pi$
$150$	0	0
$151$	1.96580e14	1.34955	0.674776	−	0.738023i	$-0.264240\pi$
$151$	1.96580e14	1.34955	0.674776	+	0.738023i	$0.264240\pi$
$152$	0	0
$153$	−8.34883e13	−0.526176
$154$	0	0
$155$	1.38532e14	0.802408
$156$	0	0
$157$	3.07441e12	0.0163838	0.00819188	−	0.999966i	$-0.497392\pi$
$157$	3.07441e12	0.0163838	0.00819188	+	0.999966i	$0.497392\pi$
$158$	0	0
$159$	−2.27989e14	−1.11901
$160$	0	0
$161$	1.64975e13	0.0746536
$162$	0	0
$163$	−2.71130e14	−1.13229	−0.566145	−	0.824306i	$-0.691566\pi$
$163$	−2.71130e14	−1.13229	−0.566145	+	0.824306i	$0.691566\pi$
$164$	0	0
$165$	1.73771e13	0.0670397
$166$	0	0
$167$	5.00523e14	1.78553	0.892765	−	0.450523i	$-0.148763\pi$
$167$	5.00523e14	1.78553	0.892765	+	0.450523i	$0.148763\pi$
$168$	0	0
$169$	2.80319e14	0.925526
$170$	0	0
$171$	6.35204e13	0.194285
$172$	0	0
$173$	5.31777e14	1.50810	0.754050	−	0.656817i	$-0.228098\pi$
$173$	5.31777e14	1.50810	0.754050	+	0.656817i	$0.228098\pi$
$174$	0	0
$175$	−1.07179e14	−0.282072
$176$	0	0
$177$	3.36415e14	0.822335
$178$	0	0
$179$	5.47920e14	1.24501	0.622504	−	0.782617i	$-0.286115\pi$
$179$	5.47920e14	1.24501	0.622504	+	0.782617i	$0.286115\pi$
$180$	0	0
$181$	3.26858e14	0.690952	0.345476	−	0.938428i	$-0.387718\pi$
$181$	3.26858e14	0.690952	0.345476	+	0.938428i	$0.387718\pi$
$182$	0	0
$183$	−2.06394e14	−0.406223
$184$	0	0
$185$	1.44506e14	0.265015
$186$	0	0
$187$	−1.52411e14	−0.260642
$188$	0	0
$189$	−6.72965e13	−0.107397
$190$	0	0
$191$	7.77910e14	1.15934	0.579672	−	0.814850i	$-0.303181\pi$
$191$	7.77910e14	1.15934	0.579672	+	0.814850i	$0.303181\pi$
$192$	0	0
$193$	−7.39265e14	−1.02962	−0.514811	−	0.857304i	$-0.672138\pi$
$193$	−7.39265e14	−1.02962	−0.514811	+	0.857304i	$0.672138\pi$
$194$	0	0
$195$	−4.32553e14	−0.563397
$196$	0	0
$197$	3.43888e14	0.419167	0.209583	−	0.977791i	$-0.432789\pi$
$197$	3.43888e14	0.419167	0.209583	+	0.977791i	$0.432789\pi$
$198$	0	0
$199$	2.05463e14	0.234525	0.117262	−	0.993101i	$-0.462588\pi$
$199$	2.05463e14	0.234525	0.117262	+	0.993101i	$0.462588\pi$
$200$	0	0
$201$	−9.50207e14	−1.01635
$202$	0	0
$203$	8.64972e14	0.867516
$204$	0	0
$205$	−6.32772e14	−0.595452
$206$	0	0
$207$	5.04736e13	0.0445920
$208$	0	0
$209$	1.15959e14	0.0962393
$210$	0	0
$211$	−6.67820e14	−0.520983	−0.260491	−	0.965476i	$-0.583885\pi$
$211$	−6.67820e14	−0.520983	−0.260491	+	0.965476i	$0.583885\pi$
$212$	0	0
$213$	−9.21444e14	−0.676086
$214$	0	0
$215$	−1.68197e15	−1.16137
$216$	0	0
$217$	−9.79391e14	−0.636748
$218$	0	0
$219$	−4.33036e14	−0.265239
$220$	0	0
$221$	3.79382e15	2.19042
$222$	0	0
$223$	−3.47790e14	−0.189381	−0.0946904	−	0.995507i	$-0.530186\pi$
$223$	−3.47790e14	−0.189381	−0.0946904	+	0.995507i	$0.530186\pi$
$224$	0	0
$225$	−3.27909e14	−0.168487
$226$	0	0
$227$	−8.79540e14	−0.426665	−0.213333	−	0.976980i	$-0.568432\pi$
$227$	−8.79540e14	−0.426665	−0.213333	+	0.976980i	$0.568432\pi$
$228$	0	0
$229$	8.75474e14	0.401155	0.200578	−	0.979678i	$-0.435718\pi$
$229$	8.75474e14	0.401155	0.200578	+	0.979678i	$0.435718\pi$
$230$	0	0
$231$	−1.22852e14	−0.0531991
$232$	0	0
$233$	−8.19733e14	−0.335629	−0.167814	−	0.985819i	$-0.553671\pi$
$233$	−8.19733e14	−0.335629	−0.167814	+	0.985819i	$0.553671\pi$
$234$	0	0
$235$	7.27705e13	0.0281847
$236$	0	0
$237$	−8.41115e14	−0.308313
$238$	0	0
$239$	−1.57547e15	−0.546794	−0.273397	−	0.961901i	$-0.588147\pi$
$239$	−1.57547e15	−0.546794	−0.273397	+	0.961901i	$0.588147\pi$
$240$	0	0
$241$	−4.41425e15	−1.45126	−0.725632	−	0.688083i	$-0.758452\pi$
$241$	−4.41425e15	−1.45126	−0.725632	+	0.688083i	$0.758452\pi$
$242$	0	0
$243$	−2.05891e14	−0.0641500
$244$	0	0
$245$	1.63921e15	0.484234
$246$	0	0
$247$	−2.88645e15	−0.808787
$248$	0	0
$249$	−3.51406e15	−0.934357
$250$	0	0
$251$	4.56770e15	1.15297	0.576486	−	0.817107i	$-0.304424\pi$
$251$	4.56770e15	1.15297	0.576486	+	0.817107i	$0.304424\pi$
$252$	0	0
$253$	9.21413e13	0.0220887
$254$	0	0
$255$	−2.81386e15	−0.640903
$256$	0	0
$257$	−4.82358e15	−1.04425	−0.522124	−	0.852869i	$-0.674860\pi$
$257$	−4.82358e15	−1.04425	−0.522124	+	0.852869i	$0.674860\pi$
$258$	0	0
$259$	−1.02162e15	−0.210302
$260$	0	0
$261$	2.64635e15	0.518184
$262$	0	0
$263$	9.79439e15	1.82501	0.912504	−	0.409067i	$-0.134146\pi$
$263$	9.79439e15	1.82501	0.912504	+	0.409067i	$0.134146\pi$
$264$	0	0
$265$	−7.68409e15	−1.36299
$266$	0	0
$267$	−5.31112e14	−0.0897146
$268$	0	0
$269$	6.00430e15	0.966212	0.483106	−	0.875562i	$-0.339509\pi$
$269$	6.00430e15	0.966212	0.483106	+	0.875562i	$0.339509\pi$
$270$	0	0
$271$	−7.10683e15	−1.08987	−0.544936	−	0.838478i	$-0.683446\pi$
$271$	−7.10683e15	−1.08987	−0.544936	+	0.838478i	$0.683446\pi$
$272$	0	0
$273$	3.05805e15	0.447081
$274$	0	0
$275$	−5.98609e14	−0.0834604
$276$	0	0
$277$	8.69369e15	1.15634	0.578170	−	0.815916i	$-0.303767\pi$
$277$	8.69369e15	1.15634	0.578170	+	0.815916i	$0.303767\pi$
$278$	0	0
$279$	−2.99641e15	−0.380342
$280$	0	0
$281$	3.09984e14	0.0375619	0.0187810	−	0.999824i	$-0.494021\pi$
$281$	3.09984e14	0.0375619	0.0187810	+	0.999824i	$0.494021\pi$
$282$	0	0
$283$	−4.34269e14	−0.0502512	−0.0251256	−	0.999684i	$-0.507999\pi$
$283$	−4.34269e14	−0.0502512	−0.0251256	+	0.999684i	$0.507999\pi$
$284$	0	0
$285$	2.14087e15	0.236646
$286$	0	0
$287$	4.47354e15	0.472519
$288$	0	0
$289$	1.47752e16	1.49175
$290$	0	0
$291$	−1.88719e15	−0.182184
$292$	0	0
$293$	−2.51640e15	−0.232349	−0.116174	−	0.993229i	$-0.537063\pi$
$293$	−2.51640e15	−0.232349	−0.116174	+	0.993229i	$0.537063\pi$
$294$	0	0
$295$	1.13384e16	1.00163
$296$	0	0
$297$	−3.75861e14	−0.0317768
$298$	0	0
$299$	−2.29359e15	−0.185632
$300$	0	0
$301$	1.18911e16	0.921598
$302$	0	0
$303$	1.10446e16	0.819920
$304$	0	0
$305$	−6.95624e15	−0.494795
$306$	0	0
$307$	1.46476e16	0.998541	0.499271	−	0.866446i	$-0.333601\pi$
$307$	1.46476e16	0.998541	0.499271	+	0.866446i	$0.333601\pi$
$308$	0	0
$309$	5.06226e15	0.330837
$310$	0	0
$311$	1.06220e16	0.665680	0.332840	−	0.942983i	$-0.391993\pi$
$311$	1.06220e16	0.665680	0.332840	+	0.942983i	$0.391993\pi$
$312$	0	0
$313$	−3.05368e16	−1.83563	−0.917814	−	0.397011i	$-0.870048\pi$
$313$	−3.05368e16	−1.83563	−0.917814	+	0.397011i	$0.870048\pi$
$314$	0	0
$315$	−2.26814e15	−0.130813
$316$	0	0
$317$	2.11397e16	1.17007	0.585037	−	0.811006i	$-0.301080\pi$
$317$	2.11397e16	1.17007	0.585037	+	0.811006i	$0.301080\pi$
$318$	0	0
$319$	4.83100e15	0.256683
$320$	0	0
$321$	−1.24582e16	−0.635582
$322$	0	0
$323$	−1.87771e16	−0.920051
$324$	0	0
$325$	1.49006e16	0.701395
$326$	0	0
$327$	1.36963e16	0.619504
$328$	0	0
$329$	−5.14470e14	−0.0223659
$330$	0	0
$331$	2.23899e16	0.935774	0.467887	−	0.883788i	$-0.345015\pi$
$331$	2.23899e16	0.935774	0.467887	+	0.883788i	$0.345015\pi$
$332$	0	0
$333$	−3.12562e15	−0.125617
$334$	0	0
$335$	−3.20255e16	−1.23795
$336$	0	0
$337$	−1.40570e15	−0.0522754	−0.0261377	−	0.999658i	$-0.508321\pi$
$337$	−1.40570e15	−0.0522754	−0.0261377	+	0.999658i	$0.508321\pi$
$338$	0	0
$339$	−7.30202e14	−0.0261303
$340$	0	0
$341$	−5.47005e15	−0.188403
$342$	0	0
$343$	−2.84188e16	−0.942311
$344$	0	0
$345$	1.70115e15	0.0543148
$346$	0	0
$347$	−3.62428e16	−1.11450	−0.557250	−	0.830345i	$-0.688144\pi$
$347$	−3.62428e16	−1.11450	−0.557250	+	0.830345i	$0.688144\pi$
$348$	0	0
$349$	1.81663e16	0.538148	0.269074	−	0.963120i	$-0.413282\pi$
$349$	1.81663e16	0.538148	0.269074	+	0.963120i	$0.413282\pi$
$350$	0	0
$351$	9.35598e15	0.267050
$352$	0	0
$353$	−2.57806e16	−0.709181	−0.354590	−	0.935022i	$-0.615380\pi$
$353$	−2.57806e16	−0.709181	−0.354590	+	0.935022i	$0.615380\pi$
$354$	0	0
$355$	−3.10561e16	−0.823498
$356$	0	0
$357$	1.98933e16	0.508586
$358$	0	0
$359$	−6.61675e16	−1.63129	−0.815645	−	0.578553i	$-0.803618\pi$
$359$	−6.61675e16	−1.63129	−0.815645	+	0.578553i	$0.803618\pi$
$360$	0	0
$361$	−2.77668e16	−0.660282
$362$	0	0
$363$	2.44809e16	0.561610
$364$	0	0
$365$	−1.45949e16	−0.323071
$366$	0	0
$367$	6.14049e16	1.31182	0.655909	−	0.754840i	$-0.272286\pi$
$367$	6.14049e16	1.31182	0.655909	+	0.754840i	$0.272286\pi$
$368$	0	0
$369$	1.36866e16	0.282245
$370$	0	0
$371$	5.43247e16	1.08160
$372$	0	0
$373$	−4.06128e16	−0.780828	−0.390414	−	0.920639i	$-0.627668\pi$
$373$	−4.06128e16	−0.780828	−0.390414	+	0.920639i	$0.627668\pi$
$374$	0	0
$375$	−3.29164e16	−0.611236
$376$	0	0
$377$	−1.20254e17	−2.15715
$378$	0	0
$379$	−7.30226e16	−1.26562	−0.632809	−	0.774308i	$-0.718098\pi$
$379$	−7.30226e16	−1.26562	−0.632809	+	0.774308i	$0.718098\pi$
$380$	0	0
$381$	1.45229e16	0.243243
$382$	0	0
$383$	4.07574e16	0.659803	0.329901	−	0.944015i	$-0.392985\pi$
$383$	4.07574e16	0.659803	0.329901	+	0.944015i	$0.392985\pi$
$384$	0	0
$385$	−4.14057e15	−0.0647986
$386$	0	0
$387$	3.63805e16	0.550488
$388$	0	0
$389$	−3.50389e16	−0.512717	−0.256359	−	0.966582i	$-0.582523\pi$
$389$	−3.50389e16	−0.512717	−0.256359	+	0.966582i	$0.582523\pi$
$390$	0	0
$391$	−1.49204e16	−0.211169
$392$	0	0
$393$	4.98107e16	0.681978
$394$	0	0
$395$	−2.83487e16	−0.375536
$396$	0	0
$397$	1.15353e17	1.47874	0.739368	−	0.673301i	$-0.235124\pi$
$397$	1.15353e17	1.47874	0.739368	+	0.673301i	$0.235124\pi$
$398$	0	0
$399$	−1.51354e16	−0.187790
$400$	0	0
$401$	−3.14508e16	−0.377740	−0.188870	−	0.982002i	$-0.560482\pi$
$401$	−3.14508e16	−0.377740	−0.188870	+	0.982002i	$0.560482\pi$
$402$	0	0
$403$	1.36161e17	1.58332
$404$	0	0
$405$	−6.93929e15	−0.0781372
$406$	0	0
$407$	−5.70593e15	−0.0622247
$408$	0	0
$409$	1.65160e17	1.74463	0.872314	−	0.488946i	$-0.162619\pi$
$409$	1.65160e17	1.74463	0.872314	+	0.488946i	$0.162619\pi$
$410$	0	0
$411$	−3.50696e16	−0.358888
$412$	0	0
$413$	−8.01599e16	−0.794844
$414$	0	0
$415$	−1.18437e17	−1.13808
$416$	0	0
$417$	2.92159e16	0.272104
$418$	0	0
$419$	−1.84207e17	−1.66309	−0.831545	−	0.555458i	$-0.812543\pi$
$419$	−1.84207e17	−1.66309	−0.831545	+	0.555458i	$0.812543\pi$
$420$	0	0
$421$	−1.79908e17	−1.57477	−0.787385	−	0.616461i	$-0.788566\pi$
$421$	−1.79908e17	−1.57477	−0.787385	+	0.616461i	$0.788566\pi$
$422$	0	0
$423$	−1.57400e15	−0.0133596
$424$	0	0
$425$	9.69323e16	0.797885
$426$	0	0
$427$	4.91789e16	0.392643
$428$	0	0
$429$	1.70797e16	0.132284
$430$	0	0
$431$	2.73823e16	0.205763	0.102882	−	0.994694i	$-0.467194\pi$
$431$	2.73823e16	0.205763	0.102882	+	0.994694i	$0.467194\pi$
$432$	0	0
$433$	1.44184e17	1.05135	0.525674	−	0.850686i	$-0.323813\pi$
$433$	1.44184e17	1.05135	0.525674	+	0.850686i	$0.323813\pi$
$434$	0	0
$435$	8.91918e16	0.631167
$436$	0	0
$437$	1.13519e16	0.0779719
$438$	0	0
$439$	1.58195e17	1.05481	0.527405	−	0.849614i	$-0.323165\pi$
$439$	1.58195e17	1.05481	0.527405	+	0.849614i	$0.323165\pi$
$440$	0	0
$441$	−3.54556e16	−0.229527
$442$	0	0
$443$	2.24157e17	1.40905	0.704527	−	0.709678i	$-0.251159\pi$
$443$	2.24157e17	1.40905	0.704527	+	0.709678i	$0.251159\pi$
$444$	0	0
$445$	−1.79004e16	−0.109276
$446$	0	0
$447$	3.01481e16	0.178756
$448$	0	0
$449$	3.09213e17	1.78097	0.890485	−	0.455014i	$-0.150366\pi$
$449$	3.09213e17	1.78097	0.890485	+	0.455014i	$0.150366\pi$
$450$	0	0
$451$	2.49854e16	0.139810
$452$	0	0
$453$	−1.43307e17	−0.779164
$454$	0	0
$455$	1.03067e17	0.544562
$456$	0	0
$457$	−2.53985e17	−1.30423	−0.652113	−	0.758122i	$-0.726117\pi$
$457$	−2.53985e17	−1.30423	−0.652113	+	0.758122i	$0.726117\pi$
$458$	0	0
$459$	6.08630e16	0.303788
$460$	0	0
$461$	1.30741e17	0.634389	0.317194	−	0.948361i	$-0.397259\pi$
$461$	1.30741e17	0.634389	0.317194	+	0.948361i	$0.397259\pi$
$462$	0	0
$463$	2.96869e17	1.40052	0.700258	−	0.713890i	$-0.253068\pi$
$463$	2.96869e17	1.40052	0.700258	+	0.713890i	$0.253068\pi$
$464$	0	0
$465$	−1.00990e17	−0.463271
$466$	0	0
$467$	−5.86244e16	−0.261528	−0.130764	−	0.991414i	$-0.541743\pi$
$467$	−5.86244e16	−0.261528	−0.130764	+	0.991414i	$0.541743\pi$
$468$	0	0
$469$	2.26413e17	0.982375
$470$	0	0
$471$	−2.24124e15	−0.00945917
$472$	0	0
$473$	6.64139e16	0.272685
$474$	0	0
$475$	−7.37490e16	−0.294610
$476$	0	0
$477$	1.66204e17	0.646060
$478$	0	0
$479$	−3.37179e17	−1.27550	−0.637749	−	0.770244i	$-0.720134\pi$
$479$	−3.37179e17	−1.27550	−0.637749	+	0.770244i	$0.720134\pi$
$480$	0	0
$481$	1.42033e17	0.522932
$482$	0	0
$483$	−1.20267e16	−0.0431013
$484$	0	0
$485$	−6.36053e16	−0.221907
$486$	0	0
$487$	−3.80927e17	−1.29391	−0.646955	−	0.762529i	$-0.723958\pi$
$487$	−3.80927e17	−1.29391	−0.646955	+	0.762529i	$0.723958\pi$
$488$	0	0
$489$	1.97654e17	0.653728
$490$	0	0
$491$	−3.84092e16	−0.123710	−0.0618552	−	0.998085i	$-0.519702\pi$
$491$	−3.84092e16	−0.123710	−0.0618552	+	0.998085i	$0.519702\pi$
$492$	0	0
$493$	−7.82281e17	−2.45390
$494$	0	0
$495$	−1.26679e16	−0.0387054
$496$	0	0
$497$	2.19559e17	0.653484
$498$	0	0
$499$	−5.93211e17	−1.72011	−0.860053	−	0.510204i	$-0.829570\pi$
$499$	−5.93211e17	−1.72011	−0.860053	+	0.510204i	$0.829570\pi$
$500$	0	0
$501$	−3.64881e17	−1.03088
$502$	0	0
$503$	3.66052e17	1.00775	0.503873	−	0.863778i	$-0.331908\pi$
$503$	3.66052e17	1.00775	0.503873	+	0.863778i	$0.331908\pi$
$504$	0	0
$505$	3.72243e17	0.998694
$506$	0	0
$507$	−2.04352e17	−0.534353
$508$	0	0
$509$	−3.78155e17	−0.963838	−0.481919	−	0.876216i	$-0.660060\pi$
$509$	−3.78155e17	−0.963838	−0.481919	+	0.876216i	$0.660060\pi$
$510$	0	0
$511$	1.03183e17	0.256372
$512$	0	0
$513$	−4.63063e16	−0.112170
$514$	0	0
$515$	1.70617e17	0.402972
$516$	0	0
$517$	−2.87339e15	−0.00661769
$518$	0	0
$519$	−3.87666e17	−0.870702
$520$	0	0
$521$	−4.53382e16	−0.0993159	−0.0496579	−	0.998766i	$-0.515813\pi$
$521$	−4.53382e16	−0.0993159	−0.0496579	+	0.998766i	$0.515813\pi$
$522$	0	0
$523$	−4.87192e16	−0.104097	−0.0520486	−	0.998645i	$-0.516575\pi$
$523$	−4.87192e16	−0.104097	−0.0520486	+	0.998645i	$0.516575\pi$
$524$	0	0
$525$	7.81332e16	0.162855
$526$	0	0
$527$	8.85761e17	1.80114
$528$	0	0
$529$	−4.95016e17	−0.982104
$530$	0	0
$531$	−2.45246e17	−0.474775
$532$	0	0
$533$	−6.21940e17	−1.17496
$534$	0	0
$535$	−4.19887e17	−0.774163
$536$	0	0
$537$	−3.99433e17	−0.718806
$538$	0	0
$539$	−6.47254e16	−0.113697
$540$	0	0
$541$	−1.79753e17	−0.308244	−0.154122	−	0.988052i	$-0.549255\pi$
$541$	−1.79753e17	−0.308244	−0.154122	+	0.988052i	$0.549255\pi$
$542$	0	0
$543$	−2.38279e17	−0.398921
$544$	0	0
$545$	4.61617e17	0.754579
$546$	0	0
$547$	8.41798e17	1.34366	0.671831	−	0.740704i	$-0.265508\pi$
$547$	8.41798e17	1.34366	0.671831	+	0.740704i	$0.265508\pi$
$548$	0	0
$549$	1.50461e17	0.234533
$550$	0	0
$551$	5.95182e17	0.906076
$552$	0	0
$553$	2.00419e17	0.298006
$554$	0	0
$555$	−1.05345e17	−0.153006
$556$	0	0
$557$	1.44215e17	0.204622	0.102311	−	0.994752i	$-0.467376\pi$
$557$	1.44215e17	0.204622	0.102311	+	0.994752i	$0.467376\pi$
$558$	0	0
$559$	−1.65318e18	−2.29163
$560$	0	0
$561$	1.11107e17	0.150482
$562$	0	0
$563$	−8.35920e17	−1.10627	−0.553134	−	0.833092i	$-0.686568\pi$
$563$	−8.35920e17	−1.10627	−0.553134	+	0.833092i	$0.686568\pi$
$564$	0	0
$565$	−2.46105e16	−0.0318277
$566$	0	0
$567$	4.90591e16	0.0620055
$568$	0	0
$569$	8.18085e17	1.01058	0.505288	−	0.862951i	$-0.331386\pi$
$569$	8.18085e17	1.01058	0.505288	+	0.862951i	$0.331386\pi$
$570$	0	0
$571$	5.30569e17	0.640630	0.320315	−	0.947311i	$-0.396211\pi$
$571$	5.30569e17	0.640630	0.320315	+	0.947311i	$0.396211\pi$
$572$	0	0
$573$	−5.67096e17	−0.669348
$574$	0	0
$575$	−5.86013e16	−0.0676186
$576$	0	0
$577$	1.09433e18	1.23454	0.617269	−	0.786752i	$-0.288239\pi$
$577$	1.09433e18	1.23454	0.617269	+	0.786752i	$0.288239\pi$
$578$	0	0
$579$	5.38924e17	0.594453
$580$	0	0
$581$	8.37319e17	0.903121
$582$	0	0
$583$	3.03412e17	0.320027
$584$	0	0
$585$	3.15331e17	0.325277
$586$	0	0
$587$	−1.17212e17	−0.118256	−0.0591282	−	0.998250i	$-0.518832\pi$
$587$	−1.17212e17	−0.118256	−0.0591282	+	0.998250i	$0.518832\pi$
$588$	0	0
$589$	−6.73914e17	−0.665051
$590$	0	0
$591$	−2.50695e17	−0.242006
$592$	0	0
$593$	1.52628e16	0.0144138	0.00720689	−	0.999974i	$-0.497706\pi$
$593$	1.52628e16	0.0144138	0.00720689	+	0.999974i	$0.497706\pi$
$594$	0	0
$595$	6.70479e17	0.619477
$596$	0	0
$597$	−1.49782e17	−0.135403
$598$	0	0
$599$	3.57493e17	0.316223	0.158112	−	0.987421i	$-0.449459\pi$
$599$	3.57493e17	0.316223	0.158112	+	0.987421i	$0.449459\pi$
$600$	0	0
$601$	−1.03886e18	−0.899232	−0.449616	−	0.893222i	$-0.648439\pi$
$601$	−1.03886e18	−0.899232	−0.449616	+	0.893222i	$0.648439\pi$
$602$	0	0
$603$	6.92701e17	0.586791
$604$	0	0
$605$	8.25097e17	0.684062
$606$	0	0
$607$	8.25642e17	0.669985	0.334993	−	0.942221i	$-0.391266\pi$
$607$	8.25642e17	0.669985	0.334993	+	0.942221i	$0.391266\pi$
$608$	0	0
$609$	−6.30564e17	−0.500861
$610$	0	0
$611$	7.15248e16	0.0556146
$612$	0	0
$613$	−1.95639e17	−0.148923	−0.0744617	−	0.997224i	$-0.523724\pi$
$613$	−1.95639e17	−0.148923	−0.0744617	+	0.997224i	$0.523724\pi$
$614$	0	0
$615$	4.61291e17	0.343785
$616$	0	0
$617$	8.23477e17	0.600894	0.300447	−	0.953798i	$-0.402864\pi$
$617$	8.23477e17	0.600894	0.300447	+	0.953798i	$0.402864\pi$
$618$	0	0
$619$	1.49824e18	1.07051	0.535256	−	0.844690i	$-0.320215\pi$
$619$	1.49824e18	1.07051	0.535256	+	0.844690i	$0.320215\pi$
$620$	0	0
$621$	−3.67953e16	−0.0257452
$622$	0	0
$623$	1.26552e17	0.0867154
$624$	0	0
$625$	−3.56209e17	−0.239048
$626$	0	0
$627$	−8.45338e16	−0.0555638
$628$	0	0
$629$	9.23957e17	0.594871
$630$	0	0
$631$	−1.02665e18	−0.647486	−0.323743	−	0.946145i	$-0.604941\pi$
$631$	−1.02665e18	−0.647486	−0.323743	+	0.946145i	$0.604941\pi$
$632$	0	0
$633$	4.86841e17	0.300789
$634$	0	0
$635$	4.89474e17	0.296279
$636$	0	0
$637$	1.61115e18	0.955498
$638$	0	0
$639$	6.71733e17	0.390338
$640$	0	0
$641$	−3.12189e18	−1.77763	−0.888813	−	0.458269i	$-0.848470\pi$
$641$	−3.12189e18	−1.77763	−0.888813	+	0.458269i	$0.848470\pi$
$642$	0	0
$643$	−2.23410e18	−1.24661	−0.623307	−	0.781977i	$-0.714211\pi$
$643$	−2.23410e18	−1.24661	−0.623307	+	0.781977i	$0.714211\pi$
$644$	0	0
$645$	1.22616e18	0.670515
$646$	0	0
$647$	4.99323e17	0.267611	0.133805	−	0.991008i	$-0.457280\pi$
$647$	4.99323e17	0.267611	0.133805	+	0.991008i	$0.457280\pi$
$648$	0	0
$649$	−4.47706e17	−0.235181
$650$	0	0
$651$	7.13976e17	0.367627
$652$	0	0
$653$	9.79475e17	0.494376	0.247188	−	0.968967i	$-0.420493\pi$
$653$	9.79475e17	0.494376	0.247188	+	0.968967i	$0.420493\pi$
$654$	0	0
$655$	1.67880e18	0.830676
$656$	0	0
$657$	3.15684e17	0.153136
$658$	0	0
$659$	−3.17243e18	−1.50882	−0.754408	−	0.656406i	$-0.772076\pi$
$659$	−3.17243e18	−1.50882	−0.754408	+	0.656406i	$0.772076\pi$
$660$	0	0
$661$	3.59409e18	1.67602	0.838010	−	0.545655i	$-0.183719\pi$
$661$	3.59409e18	1.67602	0.838010	+	0.545655i	$0.183719\pi$
$662$	0	0
$663$	−2.76570e18	−1.26464
$664$	0	0
$665$	−5.10121e17	−0.228735
$666$	0	0
$667$	4.72935e17	0.207962
$668$	0	0
$669$	2.53539e17	0.109339
$670$	0	0
$671$	2.74672e17	0.116176
$672$	0	0
$673$	1.77130e18	0.734844	0.367422	−	0.930054i	$-0.380241\pi$
$673$	1.77130e18	0.734844	0.367422	+	0.930054i	$0.380241\pi$
$674$	0	0
$675$	2.39046e17	0.0972761
$676$	0	0
$677$	−2.33881e18	−0.933615	−0.466808	−	0.884359i	$-0.654596\pi$
$677$	−2.33881e18	−0.933615	−0.466808	+	0.884359i	$0.654596\pi$
$678$	0	0
$679$	4.49674e17	0.176094
$680$	0	0
$681$	6.41184e17	0.246335
$682$	0	0
$683$	−1.44984e17	−0.0546493	−0.0273247	−	0.999627i	$-0.508699\pi$
$683$	−1.44984e17	−0.0546493	−0.0273247	+	0.999627i	$0.508699\pi$
$684$	0	0
$685$	−1.18198e18	−0.437140
$686$	0	0
$687$	−6.38220e17	−0.231607
$688$	0	0
$689$	−7.55255e18	−2.68948
$690$	0	0
$691$	−1.31745e18	−0.460392	−0.230196	−	0.973144i	$-0.573937\pi$
$691$	−1.31745e18	−0.460392	−0.230196	+	0.973144i	$0.573937\pi$
$692$	0	0
$693$	8.95592e16	0.0307145
$694$	0	0
$695$	9.84683e17	0.331433
$696$	0	0
$697$	−4.04587e18	−1.33659
$698$	0	0
$699$	5.97585e17	0.193775
$700$	0	0
$701$	−1.56284e18	−0.497448	−0.248724	−	0.968574i	$-0.580011\pi$
$701$	−1.56284e18	−0.497448	−0.248724	+	0.968574i	$0.580011\pi$
$702$	0	0
$703$	−7.02974e17	−0.219649
$704$	0	0
$705$	−5.30497e16	−0.0162725
$706$	0	0
$707$	−2.63167e18	−0.792510
$708$	0	0
$709$	1.21994e18	0.360692	0.180346	−	0.983603i	$-0.442278\pi$
$709$	1.21994e18	0.360692	0.180346	+	0.983603i	$0.442278\pi$
$710$	0	0
$711$	6.13173e17	0.178004
$712$	0	0
$713$	−5.35495e17	−0.152642
$714$	0	0
$715$	5.75648e17	0.161127
$716$	0	0
$717$	1.14852e18	0.315692
$718$	0	0
$719$	2.63729e18	0.711901	0.355950	−	0.934505i	$-0.384157\pi$
$719$	2.63729e18	0.711901	0.355950	+	0.934505i	$0.384157\pi$
$720$	0	0
$721$	−1.20622e18	−0.319777
$722$	0	0
$723$	3.21799e18	0.837887
$724$	0	0
$725$	−3.07249e18	−0.785766
$726$	0	0
$727$	−5.57685e18	−1.40093	−0.700463	−	0.713688i	$-0.747023\pi$
$727$	−5.57685e18	−1.40093	−0.700463	+	0.713688i	$0.747023\pi$
$728$	0	0
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	−1.07544e19	−2.60688
$732$	0	0
$733$	−2.28658e17	−0.0544517	−0.0272259	−	0.999629i	$-0.508667\pi$
$733$	−2.28658e17	−0.0544517	−0.0272259	+	0.999629i	$0.508667\pi$
$734$	0	0
$735$	−1.19498e18	−0.279573
$736$	0	0
$737$	1.26455e18	0.290668
$738$	0	0
$739$	8.54445e18	1.92973	0.964863	−	0.262755i	$-0.0846311\pi$
$739$	8.54445e18	1.92973	0.964863	+	0.262755i	$0.0846311\pi$
$740$	0	0
$741$	2.10422e18	0.466954
$742$	0	0
$743$	−5.21598e18	−1.13739	−0.568694	−	0.822549i	$-0.692551\pi$
$743$	−5.21598e18	−1.13739	−0.568694	+	0.822549i	$0.692551\pi$
$744$	0	0
$745$	1.01610e18	0.217732
$746$	0	0
$747$	2.56175e18	0.539451
$748$	0	0
$749$	2.96850e18	0.614335
$750$	0	0
$751$	1.45323e18	0.295580	0.147790	−	0.989019i	$-0.452784\pi$
$751$	1.45323e18	0.295580	0.147790	+	0.989019i	$0.452784\pi$
$752$	0	0
$753$	−3.32986e18	−0.665669
$754$	0	0
$755$	−4.82997e18	−0.949051
$756$	0	0
$757$	4.69618e18	0.907030	0.453515	−	0.891249i	$-0.350170\pi$
$757$	4.69618e18	0.907030	0.453515	+	0.891249i	$0.350170\pi$
$758$	0	0
$759$	−6.71710e16	−0.0127529
$760$	0	0
$761$	5.68223e18	1.06052	0.530260	−	0.847835i	$-0.322094\pi$
$761$	5.68223e18	1.06052	0.530260	+	0.847835i	$0.322094\pi$
$762$	0	0
$763$	−3.26352e18	−0.598794
$764$	0	0
$765$	2.05131e18	0.370025
$766$	0	0
$767$	1.11443e19	1.97644
$768$	0	0
$769$	−2.37228e18	−0.413662	−0.206831	−	0.978377i	$-0.566315\pi$
$769$	−2.37228e18	−0.413662	−0.206831	+	0.978377i	$0.566315\pi$
$770$	0	0
$771$	3.51639e18	0.602897
$772$	0	0
$773$	8.98369e18	1.51457	0.757283	−	0.653087i	$-0.226527\pi$
$773$	8.98369e18	1.51457	0.757283	+	0.653087i	$0.226527\pi$
$774$	0	0
$775$	3.47892e18	0.576744
$776$	0	0
$777$	7.44764e17	0.121418
$778$	0	0
$779$	3.07822e18	0.493522
$780$	0	0
$781$	1.22627e18	0.193355
$782$	0	0
$783$	−1.92919e18	−0.299174
$784$	0	0
$785$	−7.55382e16	−0.0115216
$786$	0	0
$787$	1.60782e18	0.241214	0.120607	−	0.992700i	$-0.461516\pi$
$787$	1.60782e18	0.241214	0.120607	+	0.992700i	$0.461516\pi$
$788$	0	0
$789$	−7.14011e18	−1.05367
$790$	0	0
$791$	1.73990e17	0.0252568
$792$	0	0
$793$	−6.83716e18	−0.976338
$794$	0	0
$795$	5.60170e18	0.786925
$796$	0	0
$797$	2.63037e18	0.363528	0.181764	−	0.983342i	$-0.441819\pi$
$797$	2.63037e18	0.363528	0.181764	+	0.983342i	$0.441819\pi$
$798$	0	0
$799$	4.65286e17	0.0632654
$800$	0	0
$801$	3.87181e17	0.0517967
$802$	0	0
$803$	5.76291e17	0.0758562
$804$	0	0
$805$	−4.05344e17	−0.0524990
$806$	0	0
$807$	−4.37713e18	−0.557843
$808$	0	0
$809$	−1.26684e19	−1.58876	−0.794379	−	0.607423i	$-0.792203\pi$
$809$	−1.26684e19	−1.58876	−0.794379	+	0.607423i	$0.792203\pi$
$810$	0	0
$811$	1.02640e19	1.26672	0.633362	−	0.773856i	$-0.281675\pi$
$811$	1.02640e19	1.26672	0.633362	+	0.773856i	$0.281675\pi$
$812$	0	0
$813$	5.18088e18	0.629238
$814$	0	0
$815$	6.66166e18	0.796265
$816$	0	0
$817$	8.18223e18	0.962561
$818$	0	0
$819$	−2.22931e18	−0.258123
$820$	0	0
$821$	−3.18813e18	−0.363334	−0.181667	−	0.983360i	$-0.558149\pi$
$821$	−3.18813e18	−0.363334	−0.181667	+	0.983360i	$0.558149\pi$
$822$	0	0
$823$	−1.56274e18	−0.175303	−0.0876514	−	0.996151i	$-0.527936\pi$
$823$	−1.56274e18	−0.175303	−0.0876514	+	0.996151i	$0.527936\pi$
$824$	0	0
$825$	4.36386e17	0.0481859
$826$	0	0
$827$	−3.32727e17	−0.0361661	−0.0180831	−	0.999836i	$-0.505756\pi$
$827$	−3.32727e17	−0.0361661	−0.0180831	+	0.999836i	$0.505756\pi$
$828$	0	0
$829$	−3.79136e18	−0.405687	−0.202843	−	0.979211i	$-0.565018\pi$
$829$	−3.79136e18	−0.405687	−0.202843	+	0.979211i	$0.565018\pi$
$830$	0	0
$831$	−6.33770e18	−0.667614
$832$	0	0
$833$	1.04809e19	1.08694
$834$	0	0
$835$	−1.22979e19	−1.25565
$836$	0	0
$837$	2.18438e18	0.219590
$838$	0	0
$839$	3.63059e16	0.00359355	0.00179678	−	0.999998i	$-0.499428\pi$
$839$	3.63059e16	0.00359355	0.00179678	+	0.999998i	$0.499428\pi$
$840$	0	0
$841$	1.45355e19	1.41663
$842$	0	0
$843$	−2.25978e17	−0.0216864
$844$	0	0
$845$	−6.88744e18	−0.650862
$846$	0	0
$847$	−5.83324e18	−0.542835
$848$	0	0
$849$	3.16582e17	0.0290126
$850$	0	0
$851$	−5.58587e17	−0.0504137
$852$	0	0
$853$	1.73396e19	1.54124	0.770619	−	0.637296i	$-0.219947\pi$
$853$	1.73396e19	1.54124	0.770619	+	0.637296i	$0.219947\pi$
$854$	0	0
$855$	−1.56070e18	−0.136628
$856$	0	0
$857$	1.16333e18	0.100306	0.0501531	−	0.998742i	$-0.484029\pi$
$857$	1.16333e18	0.100306	0.0501531	+	0.998742i	$0.484029\pi$
$858$	0	0
$859$	1.93144e19	1.64031	0.820153	−	0.572144i	$-0.193888\pi$
$859$	1.93144e19	1.64031	0.820153	+	0.572144i	$0.193888\pi$
$860$	0	0
$861$	−3.26121e18	−0.272809
$862$	0	0
$863$	1.70021e19	1.40098	0.700492	−	0.713660i	$-0.252964\pi$
$863$	1.70021e19	1.40098	0.700492	+	0.713660i	$0.252964\pi$
$864$	0	0
$865$	−1.30658e19	−1.06055
$866$	0	0
$867$	−1.07711e19	−0.861264
$868$	0	0
$869$	1.11937e18	0.0881748
$870$	0	0
$871$	−3.14773e19	−2.44275
$872$	0	0
$873$	1.37576e18	0.105184
$874$	0	0
$875$	7.84322e18	0.590802
$876$	0	0
$877$	−1.41383e19	−1.04930	−0.524651	−	0.851318i	$-0.675804\pi$
$877$	−1.41383e19	−1.04930	−0.524651	+	0.851318i	$0.675804\pi$
$878$	0	0
$879$	1.83445e18	0.134146
$880$	0	0
$881$	1.32768e19	0.956646	0.478323	−	0.878184i	$-0.341245\pi$
$881$	1.32768e19	0.956646	0.478323	+	0.878184i	$0.341245\pi$
$882$	0	0
$883$	−2.45626e18	−0.174393	−0.0871965	−	0.996191i	$-0.527791\pi$
$883$	−2.45626e18	−0.174393	−0.0871965	+	0.996191i	$0.527791\pi$
$884$	0	0
$885$	−8.26571e18	−0.578294
$886$	0	0
$887$	2.22255e19	1.53232	0.766159	−	0.642651i	$-0.222166\pi$
$887$	2.22255e19	1.53232	0.766159	+	0.642651i	$0.222166\pi$
$888$	0	0
$889$	−3.46046e18	−0.235111
$890$	0	0
$891$	2.74003e17	0.0183464
$892$	0	0
$893$	−3.54004e17	−0.0233600
$894$	0	0
$895$	−1.34624e19	−0.875532
$896$	0	0
$897$	1.67203e18	0.107175
$898$	0	0
$899$	−2.80762e19	−1.77378
$900$	0	0
$901$	−4.91312e19	−3.05947
$902$	0	0
$903$	−8.66864e18	−0.532085
$904$	0	0
$905$	−8.03089e18	−0.485901
$906$	0	0
$907$	1.33549e19	0.796511	0.398256	−	0.917274i	$-0.369616\pi$
$907$	1.33549e19	0.796511	0.398256	+	0.917274i	$0.369616\pi$
$908$	0	0
$909$	−8.05149e18	−0.473381
$910$	0	0
$911$	1.96969e19	1.14164	0.570819	−	0.821076i	$-0.306626\pi$
$911$	1.96969e19	1.14164	0.570819	+	0.821076i	$0.306626\pi$
$912$	0	0
$913$	4.67656e18	0.267218
$914$	0	0
$915$	5.07110e18	0.285670
$916$	0	0
$917$	−1.18687e19	−0.659180
$918$	0	0
$919$	8.19722e17	0.0448865	0.0224433	−	0.999748i	$-0.492855\pi$
$919$	8.19722e17	0.0448865	0.0224433	+	0.999748i	$0.492855\pi$
$920$	0	0
$921$	−1.06781e19	−0.576508
$922$	0	0
$923$	−3.05245e19	−1.62494
$924$	0	0
$925$	3.62894e18	0.190484
$926$	0	0
$927$	−3.69039e18	−0.191009
$928$	0	0
$929$	3.34031e19	1.70484	0.852422	−	0.522855i	$-0.175133\pi$
$929$	3.34031e19	1.70484	0.852422	+	0.522855i	$0.175133\pi$
$930$	0	0
$931$	−7.97421e18	−0.401342
$932$	0	0
$933$	−7.74347e18	−0.384331
$934$	0	0
$935$	3.74473e18	0.183293
$936$	0	0
$937$	1.16220e19	0.561014	0.280507	−	0.959852i	$-0.409497\pi$
$937$	1.16220e19	0.561014	0.280507	+	0.959852i	$0.409497\pi$
$938$	0	0
$939$	2.22613e19	1.05980
$940$	0	0
$941$	6.46781e18	0.303686	0.151843	−	0.988405i	$-0.451479\pi$
$941$	6.46781e18	0.303686	0.151843	+	0.988405i	$0.451479\pi$
$942$	0	0
$943$	2.44597e18	0.113273
$944$	0	0
$945$	1.65347e18	0.0755250
$946$	0	0
$947$	−2.87995e19	−1.29751	−0.648753	−	0.760999i	$-0.724709\pi$
$947$	−2.87995e19	−1.29751	−0.648753	+	0.760999i	$0.724709\pi$
$948$	0	0
$949$	−1.43451e19	−0.637490
$950$	0	0
$951$	−1.54108e19	−0.675543
$952$	0	0
$953$	2.73267e19	1.18163	0.590817	−	0.806805i	$-0.298805\pi$
$953$	2.73267e19	1.18163	0.590817	+	0.806805i	$0.298805\pi$
$954$	0	0
$955$	−1.91132e19	−0.815291
$956$	0	0
$957$	−3.52180e18	−0.148196
$958$	0	0
$959$	8.35629e18	0.346891
$960$	0	0
$961$	7.37259e18	0.301938
$962$	0	0
$963$	9.08201e18	0.366954
$964$	0	0
$965$	1.81637e19	0.724066
$966$	0	0
$967$	−8.14927e18	−0.320514	−0.160257	−	0.987075i	$-0.551232\pi$
$967$	−8.14927e18	−0.320514	−0.160257	+	0.987075i	$0.551232\pi$
$968$	0	0
$969$	1.36885e19	0.531192
$970$	0	0
$971$	9.30946e18	0.356451	0.178225	−	0.983990i	$-0.442964\pi$
$971$	9.30946e18	0.356451	0.178225	+	0.983990i	$0.442964\pi$
$972$	0	0
$973$	−6.96148e18	−0.263007
$974$	0	0
$975$	−1.08626e19	−0.404951
$976$	0	0
$977$	−2.16013e19	−0.794629	−0.397314	−	0.917683i	$-0.630058\pi$
$977$	−2.16013e19	−0.794629	−0.397314	+	0.917683i	$0.630058\pi$
$978$	0	0
$979$	7.06812e17	0.0256576
$980$	0	0
$981$	−9.98463e18	−0.357671
$982$	0	0
$983$	2.78261e18	0.0983681	0.0491841	−	0.998790i	$-0.484338\pi$
$983$	2.78261e18	0.0983681	0.0491841	+	0.998790i	$0.484338\pi$
$984$	0	0
$985$	−8.44934e18	−0.294773
$986$	0	0
$987$	3.75048e17	0.0129130
$988$	0	0
$989$	6.50164e18	0.220926
$990$	0	0
$991$	4.97916e19	1.66985	0.834925	−	0.550364i	$-0.185511\pi$
$991$	4.97916e19	1.66985	0.834925	+	0.550364i	$0.185511\pi$
$992$	0	0
$993$	−1.63223e19	−0.540269
$994$	0	0
$995$	−5.04822e18	−0.164926
$996$	0	0
$997$	1.86583e19	0.601664	0.300832	−	0.953677i	$-0.402736\pi$
$997$	1.86583e19	0.601664	0.300832	+	0.953677i	$0.402736\pi$
$998$	0	0
$999$	2.27858e18	0.0725251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.14.a.a.1.1		1
3.2	odd	2		144.14.a.j.1.1		1
4.3	odd	2		12.14.a.b.1.1	✓	1
8.3	odd	2		192.14.a.d.1.1		1
8.5	even	2		192.14.a.i.1.1		1
12.11	even	2		36.14.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.14.a.b.1.1	✓	1	4.3	odd	2
36.14.a.d.1.1		1	12.11	even	2
48.14.a.a.1.1		1	1.1	even	1	trivial
144.14.a.j.1.1		1	3.2	odd	2
192.14.a.d.1.1		1	8.3	odd	2
192.14.a.i.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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