LMFDB - Embedded newform 48.12.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,12,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, 12, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
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:	$36.8804726669$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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)

$f(q)$

$=$

$q+243.000 q^{3} +5766.00 q^{5} -72464.0 q^{7} +59049.0 q^{9} +O(q^{10})$

$q+243.000 q^{3} +5766.00 q^{5} -72464.0 q^{7} +59049.0 q^{9} +408948. q^{11} +1.36756e6 q^{13} +1.40114e6 q^{15} +5.42291e6 q^{17} -1.51661e7 q^{19} -1.76088e7 q^{21} +5.21941e7 q^{23} -1.55814e7 q^{25} +1.43489e7 q^{27} +1.18581e8 q^{29} +5.76524e7 q^{31} +9.93744e7 q^{33} -4.17827e8 q^{35} -3.75985e8 q^{37} +3.32317e8 q^{39} +8.56316e8 q^{41} +1.24519e9 q^{43} +3.40477e8 q^{45} +1.30676e9 q^{47} +3.27370e9 q^{49} +1.31777e9 q^{51} +4.09556e8 q^{53} +2.35799e9 q^{55} -3.68536e9 q^{57} +2.88287e9 q^{59} +5.73177e9 q^{61} -4.27893e9 q^{63} +7.88534e9 q^{65} -3.89327e9 q^{67} +1.26832e10 q^{69} +9.07589e9 q^{71} -1.55718e10 q^{73} -3.78627e9 q^{75} -2.96340e10 q^{77} +3.01968e10 q^{79} +3.48678e9 q^{81} -2.31353e10 q^{83} +3.12685e10 q^{85} +2.88152e10 q^{87} -2.56148e10 q^{89} -9.90987e10 q^{91} +1.40095e10 q^{93} -8.74477e10 q^{95} -6.19376e10 q^{97} +2.41480e10 q^{99} +O(q^{100})$

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
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	0	0
$5$	5766.00	0.825163	0.412581	−	0.910921i	$-0.364627\pi$
$5$	5766.00	0.825163	0.412581	+	0.910921i	$0.364627\pi$
$6$	0	0
$7$	−72464.0	−1.62961	−0.814804	−	0.579737i	$-0.803155\pi$
$7$	−72464.0	−1.62961	−0.814804	+	0.579737i	$0.803155\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	408948.	0.765611	0.382806	−	0.923829i	$-0.374958\pi$
$11$	408948.	0.765611	0.382806	+	0.923829i	$0.374958\pi$
$12$	0	0
$13$	1.36756e6	1.02154	0.510772	−	0.859716i	$-0.329360\pi$
$13$	1.36756e6	1.02154	0.510772	+	0.859716i	$0.329360\pi$
$14$	0	0
$15$	1.40114e6	0.476408
$16$	0	0
$17$	5.42291e6	0.926326	0.463163	−	0.886273i	$-0.346715\pi$
$17$	5.42291e6	0.926326	0.463163	+	0.886273i	$0.346715\pi$
$18$	0	0
$19$	−1.51661e7	−1.40517	−0.702585	−	0.711599i	$-0.747971\pi$
$19$	−1.51661e7	−1.40517	−0.702585	+	0.711599i	$0.747971\pi$
$20$	0	0
$21$	−1.76088e7	−0.940854
$22$	0	0
$23$	5.21941e7	1.69090	0.845450	−	0.534054i	$-0.179332\pi$
$23$	5.21941e7	1.69090	0.845450	+	0.534054i	$0.179332\pi$
$24$	0	0
$25$	−1.55814e7	−0.319106
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	1.18581e8	1.07356	0.536780	−	0.843722i	$-0.319640\pi$
$29$	1.18581e8	1.07356	0.536780	+	0.843722i	$0.319640\pi$
$30$	0	0
$31$	5.76524e7	0.361683	0.180842	−	0.983512i	$-0.442118\pi$
$31$	5.76524e7	0.361683	0.180842	+	0.983512i	$0.442118\pi$
$32$	0	0
$33$	9.93744e7	0.442026
$34$	0	0
$35$	−4.17827e8	−1.34469
$36$	0	0
$37$	−3.75985e8	−0.891377	−0.445688	−	0.895188i	$-0.647041\pi$
$37$	−3.75985e8	−0.891377	−0.445688	+	0.895188i	$0.647041\pi$
$38$	0	0
$39$	3.32317e8	0.589789
$40$	0	0
$41$	8.56316e8	1.15431	0.577156	−	0.816634i	$-0.304163\pi$
$41$	8.56316e8	1.15431	0.577156	+	0.816634i	$0.304163\pi$
$42$	0	0
$43$	1.24519e9	1.29169	0.645846	−	0.763468i	$-0.276505\pi$
$43$	1.24519e9	1.29169	0.645846	+	0.763468i	$0.276505\pi$
$44$	0	0
$45$	3.40477e8	0.275054
$46$	0	0
$47$	1.30676e9	0.831110	0.415555	−	0.909568i	$-0.363587\pi$
$47$	1.30676e9	0.831110	0.415555	+	0.909568i	$0.363587\pi$
$48$	0	0
$49$	3.27370e9	1.65562
$50$	0	0
$51$	1.31777e9	0.534814
$52$	0	0
$53$	4.09556e8	0.134523	0.0672615	−	0.997735i	$-0.478574\pi$
$53$	4.09556e8	0.134523	0.0672615	+	0.997735i	$0.478574\pi$
$54$	0	0
$55$	2.35799e9	0.631754
$56$	0	0
$57$	−3.68536e9	−0.811276
$58$	0	0
$59$	2.88287e9	0.524975	0.262487	−	0.964935i	$-0.415457\pi$
$59$	2.88287e9	0.524975	0.262487	+	0.964935i	$0.415457\pi$
$60$	0	0
$61$	5.73177e9	0.868909	0.434455	−	0.900694i	$-0.356941\pi$
$61$	5.73177e9	0.868909	0.434455	+	0.900694i	$0.356941\pi$
$62$	0	0
$63$	−4.27893e9	−0.543203
$64$	0	0
$65$	7.88534e9	0.842941
$66$	0	0
$67$	−3.89327e9	−0.352292	−0.176146	−	0.984364i	$-0.556363\pi$
$67$	−3.89327e9	−0.352292	−0.176146	+	0.984364i	$0.556363\pi$
$68$	0	0
$69$	1.26832e10	0.976242
$70$	0	0
$71$	9.07589e9	0.596992	0.298496	−	0.954411i	$-0.403515\pi$
$71$	9.07589e9	0.596992	0.298496	+	0.954411i	$0.403515\pi$
$72$	0	0
$73$	−1.55718e10	−0.879152	−0.439576	−	0.898206i	$-0.644871\pi$
$73$	−1.55718e10	−0.879152	−0.439576	+	0.898206i	$0.644871\pi$
$74$	0	0
$75$	−3.78627e9	−0.184236
$76$	0	0
$77$	−2.96340e10	−1.24765
$78$	0	0
$79$	3.01968e10	1.10411	0.552054	−	0.833809i	$-0.313844\pi$
$79$	3.01968e10	1.10411	0.552054	+	0.833809i	$0.313844\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	−2.31353e10	−0.644681	−0.322340	−	0.946624i	$-0.604470\pi$
$83$	−2.31353e10	−0.644681	−0.322340	+	0.946624i	$0.604470\pi$
$84$	0	0
$85$	3.12685e10	0.764369
$86$	0	0
$87$	2.88152e10	0.619821
$88$	0	0
$89$	−2.56148e10	−0.486235	−0.243118	−	0.969997i	$-0.578170\pi$
$89$	−2.56148e10	−0.486235	−0.243118	+	0.969997i	$0.578170\pi$
$90$	0	0
$91$	−9.90987e10	−1.66472
$92$	0	0
$93$	1.40095e10	0.208818
$94$	0	0
$95$	−8.74477e10	−1.15949
$96$	0	0
$97$	−6.19376e10	−0.732335	−0.366167	−	0.930549i	$-0.619330\pi$
$97$	−6.19376e10	−0.732335	−0.366167	+	0.930549i	$0.619330\pi$
$98$	0	0
$99$	2.41480e10	0.255204
$100$	0	0
$101$	−9.49642e10	−0.899067	−0.449534	−	0.893263i	$-0.648410\pi$
$101$	−9.49642e10	−0.899067	−0.449534	+	0.893263i	$0.648410\pi$
$102$	0	0
$103$	−5.43719e10	−0.462136	−0.231068	−	0.972938i	$-0.574222\pi$
$103$	−5.43719e10	−0.462136	−0.231068	+	0.972938i	$0.574222\pi$
$104$	0	0
$105$	−1.01532e11	−0.776358
$106$	0	0
$107$	−5.71348e9	−0.0393813	−0.0196906	−	0.999806i	$-0.506268\pi$
$107$	−5.71348e9	−0.0393813	−0.0196906	+	0.999806i	$0.506268\pi$
$108$	0	0
$109$	−4.96724e10	−0.309221	−0.154611	−	0.987975i	$-0.549412\pi$
$109$	−4.96724e10	−0.309221	−0.154611	+	0.987975i	$0.549412\pi$
$110$	0	0
$111$	−9.13644e10	−0.514637
$112$	0	0
$113$	1.77803e11	0.907834	0.453917	−	0.891044i	$-0.350026\pi$
$113$	1.77803e11	0.907834	0.453917	+	0.891044i	$0.350026\pi$
$114$	0	0
$115$	3.00951e11	1.39527
$116$	0	0
$117$	8.07529e10	0.340515
$118$	0	0
$119$	−3.92966e11	−1.50955
$120$	0	0
$121$	−1.18073e11	−0.413839
$122$	0	0
$123$	2.08085e11	0.666442
$124$	0	0
$125$	−3.71385e11	−1.08848
$126$	0	0
$127$	4.73708e11	1.27230	0.636150	−	0.771565i	$-0.280526\pi$
$127$	4.73708e11	1.27230	0.636150	+	0.771565i	$0.280526\pi$
$128$	0	0
$129$	3.02581e11	0.745758
$130$	0	0
$131$	−1.68574e11	−0.381767	−0.190883	−	0.981613i	$-0.561135\pi$
$131$	−1.68574e11	−0.381767	−0.190883	+	0.981613i	$0.561135\pi$
$132$	0	0
$133$	1.09900e12	2.28988
$134$	0	0
$135$	8.27358e10	0.158803
$136$	0	0
$137$	−8.01479e11	−1.41883	−0.709413	−	0.704793i	$-0.751040\pi$
$137$	−8.01479e11	−1.41883	−0.709413	+	0.704793i	$0.751040\pi$
$138$	0	0
$139$	−5.24839e11	−0.857916	−0.428958	−	0.903324i	$-0.641119\pi$
$139$	−5.24839e11	−0.857916	−0.428958	+	0.903324i	$0.641119\pi$
$140$	0	0
$141$	3.17543e11	0.479842
$142$	0	0
$143$	5.59260e11	0.782106
$144$	0	0
$145$	6.83739e11	0.885862
$146$	0	0
$147$	7.95510e11	0.955873
$148$	0	0
$149$	1.25702e12	1.40223	0.701113	−	0.713051i	$-0.252687\pi$
$149$	1.25702e12	1.40223	0.701113	+	0.713051i	$0.252687\pi$
$150$	0	0
$151$	4.92117e11	0.510147	0.255074	−	0.966922i	$-0.417900\pi$
$151$	4.92117e11	0.510147	0.255074	+	0.966922i	$0.417900\pi$
$152$	0	0
$153$	3.20218e11	0.308775
$154$	0	0
$155$	3.32424e11	0.298447
$156$	0	0
$157$	−9.97092e11	−0.834232	−0.417116	−	0.908853i	$-0.636959\pi$
$157$	−9.97092e11	−0.834232	−0.417116	+	0.908853i	$0.636959\pi$
$158$	0	0
$159$	9.95222e10	0.0776669
$160$	0	0
$161$	−3.78219e12	−2.75550
$162$	0	0
$163$	4.84142e11	0.329565	0.164782	−	0.986330i	$-0.447308\pi$
$163$	4.84142e11	0.329565	0.164782	+	0.986330i	$0.447308\pi$
$164$	0	0
$165$	5.72993e11	0.364743
$166$	0	0
$167$	−2.31537e12	−1.37937	−0.689685	−	0.724110i	$-0.742251\pi$
$167$	−2.31537e12	−1.37937	−0.689685	+	0.724110i	$0.742251\pi$
$168$	0	0
$169$	7.80545e10	0.0435533
$170$	0	0
$171$	−8.95543e11	−0.468390
$172$	0	0
$173$	3.37157e12	1.65416	0.827081	−	0.562083i	$-0.190000\pi$
$173$	3.37157e12	1.65416	0.827081	+	0.562083i	$0.190000\pi$
$174$	0	0
$175$	1.12909e12	0.520018
$176$	0	0
$177$	7.00537e11	0.303094
$178$	0	0
$179$	−1.64598e12	−0.669472	−0.334736	−	0.942312i	$-0.608647\pi$
$179$	−1.64598e12	−0.669472	−0.334736	+	0.942312i	$0.608647\pi$
$180$	0	0
$181$	2.74217e12	1.04921	0.524605	−	0.851346i	$-0.324213\pi$
$181$	2.74217e12	1.04921	0.524605	+	0.851346i	$0.324213\pi$
$182$	0	0
$183$	1.39282e12	0.501665
$184$	0	0
$185$	−2.16793e12	−0.735531
$186$	0	0
$187$	2.21769e12	0.709205
$188$	0	0
$189$	−1.03978e12	−0.313618
$190$	0	0
$191$	−4.08409e12	−1.16255	−0.581275	−	0.813707i	$-0.697446\pi$
$191$	−4.08409e12	−1.16255	−0.581275	+	0.813707i	$0.697446\pi$
$192$	0	0
$193$	−4.47239e12	−1.20219	−0.601097	−	0.799176i	$-0.705269\pi$
$193$	−4.47239e12	−1.20219	−0.601097	+	0.799176i	$0.705269\pi$
$194$	0	0
$195$	1.91614e12	0.486672
$196$	0	0
$197$	−7.37025e12	−1.76977	−0.884887	−	0.465805i	$-0.845765\pi$
$197$	−7.37025e12	−1.76977	−0.884887	+	0.465805i	$0.845765\pi$
$198$	0	0
$199$	−2.69308e12	−0.611728	−0.305864	−	0.952075i	$-0.598945\pi$
$199$	−2.69308e12	−0.611728	−0.305864	+	0.952075i	$0.598945\pi$
$200$	0	0
$201$	−9.46065e11	−0.203396
$202$	0	0
$203$	−8.59286e12	−1.74948
$204$	0	0
$205$	4.93752e12	0.952495
$206$	0	0
$207$	3.08201e12	0.563634
$208$	0	0
$209$	−6.20215e12	−1.07581
$210$	0	0
$211$	−6.63458e12	−1.09209	−0.546047	−	0.837755i	$-0.683868\pi$
$211$	−6.63458e12	−1.09209	−0.546047	+	0.837755i	$0.683868\pi$
$212$	0	0
$213$	2.20544e12	0.344673
$214$	0	0
$215$	7.17976e12	1.06586
$216$	0	0
$217$	−4.17772e12	−0.589401
$218$	0	0
$219$	−3.78395e12	−0.507578
$220$	0	0
$221$	7.41615e12	0.946283
$222$	0	0
$223$	5.99559e12	0.728040	0.364020	−	0.931391i	$-0.381404\pi$
$223$	5.99559e12	0.728040	0.364020	+	0.931391i	$0.381404\pi$
$224$	0	0
$225$	−9.20064e11	−0.106369
$226$	0	0
$227$	7.74930e12	0.853337	0.426668	−	0.904408i	$-0.359687\pi$
$227$	7.74930e12	0.853337	0.426668	+	0.904408i	$0.359687\pi$
$228$	0	0
$229$	3.75804e12	0.394336	0.197168	−	0.980370i	$-0.436826\pi$
$229$	3.75804e12	0.394336	0.197168	+	0.980370i	$0.436826\pi$
$230$	0	0
$231$	−7.20106e12	−0.720329
$232$	0	0
$233$	9.02676e12	0.861141	0.430570	−	0.902557i	$-0.358312\pi$
$233$	9.02676e12	0.861141	0.430570	+	0.902557i	$0.358312\pi$
$234$	0	0
$235$	7.53479e12	0.685801
$236$	0	0
$237$	7.33781e12	0.637457
$238$	0	0
$239$	4.02269e12	0.333679	0.166839	−	0.985984i	$-0.446644\pi$
$239$	4.02269e12	0.333679	0.166839	+	0.985984i	$0.446644\pi$
$240$	0	0
$241$	−1.49997e13	−1.18847	−0.594235	−	0.804292i	$-0.702545\pi$
$241$	−1.49997e13	−1.18847	−0.594235	+	0.804292i	$0.702545\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	1.88762e13	1.36616
$246$	0	0
$247$	−2.07405e13	−1.43544
$248$	0	0
$249$	−5.62187e12	−0.372207
$250$	0	0
$251$	1.46817e13	0.930187	0.465094	−	0.885262i	$-0.346021\pi$
$251$	1.46817e13	0.930187	0.465094	+	0.885262i	$0.346021\pi$
$252$	0	0
$253$	2.13447e13	1.29457
$254$	0	0
$255$	7.59825e12	0.441309
$256$	0	0
$257$	−1.64476e13	−0.915106	−0.457553	−	0.889182i	$-0.651274\pi$
$257$	−1.64476e13	−0.915106	−0.457553	+	0.889182i	$0.651274\pi$
$258$	0	0
$259$	2.72454e13	1.45259
$260$	0	0
$261$	7.00210e12	0.357854
$262$	0	0
$263$	1.96818e13	0.964514	0.482257	−	0.876030i	$-0.339817\pi$
$263$	1.96818e13	0.964514	0.482257	+	0.876030i	$0.339817\pi$
$264$	0	0
$265$	2.36150e12	0.111003
$266$	0	0
$267$	−6.22440e12	−0.280728
$268$	0	0
$269$	−1.16174e13	−0.502889	−0.251444	−	0.967872i	$-0.580906\pi$
$269$	−1.16174e13	−0.502889	−0.251444	+	0.967872i	$0.580906\pi$
$270$	0	0
$271$	−7.46788e12	−0.310360	−0.155180	−	0.987886i	$-0.549596\pi$
$271$	−7.46788e12	−0.310360	−0.155180	+	0.987886i	$0.549596\pi$
$272$	0	0
$273$	−2.40810e13	−0.961125
$274$	0	0
$275$	−6.37197e12	−0.244311
$276$	0	0
$277$	1.59564e13	0.587892	0.293946	−	0.955822i	$-0.405031\pi$
$277$	1.59564e13	0.587892	0.293946	+	0.955822i	$0.405031\pi$
$278$	0	0
$279$	3.40432e12	0.120561
$280$	0	0
$281$	3.96115e13	1.34877	0.674383	−	0.738382i	$-0.264410\pi$
$281$	3.96115e13	1.34877	0.674383	+	0.738382i	$0.264410\pi$
$282$	0	0
$283$	−1.50001e13	−0.491211	−0.245605	−	0.969370i	$-0.578987\pi$
$283$	−1.50001e13	−0.491211	−0.245605	+	0.969370i	$0.578987\pi$
$284$	0	0
$285$	−2.12498e13	−0.669435
$286$	0	0
$287$	−6.20521e13	−1.88107
$288$	0	0
$289$	−4.86390e12	−0.141921
$290$	0	0
$291$	−1.50508e13	−0.422814
$292$	0	0
$293$	3.88255e13	1.05038	0.525188	−	0.850986i	$-0.323995\pi$
$293$	3.88255e13	1.05038	0.525188	+	0.850986i	$0.323995\pi$
$294$	0	0
$295$	1.66226e13	0.433190
$296$	0	0
$297$	5.86796e12	0.147342
$298$	0	0
$299$	7.13784e13	1.72733
$300$	0	0
$301$	−9.02314e13	−2.10495
$302$	0	0
$303$	−2.30763e13	−0.519077
$304$	0	0
$305$	3.30494e13	0.716992
$306$	0	0
$307$	−7.58194e13	−1.58679	−0.793394	−	0.608708i	$-0.791688\pi$
$307$	−7.58194e13	−1.58679	−0.793394	+	0.608708i	$0.791688\pi$
$308$	0	0
$309$	−1.32124e13	−0.266814
$310$	0	0
$311$	7.99907e13	1.55904	0.779520	−	0.626378i	$-0.215463\pi$
$311$	7.99907e13	1.55904	0.779520	+	0.626378i	$0.215463\pi$
$312$	0	0
$313$	−6.85614e13	−1.28999	−0.644994	−	0.764187i	$-0.723140\pi$
$313$	−6.85614e13	−1.28999	−0.644994	+	0.764187i	$0.723140\pi$
$314$	0	0
$315$	−2.46723e13	−0.448231
$316$	0	0
$317$	6.37891e12	0.111923	0.0559617	−	0.998433i	$-0.482178\pi$
$317$	6.37891e12	0.111923	0.0559617	+	0.998433i	$0.482178\pi$
$318$	0	0
$319$	4.84935e13	0.821930
$320$	0	0
$321$	−1.38838e12	−0.0227368
$322$	0	0
$323$	−8.22445e13	−1.30165
$324$	0	0
$325$	−2.13084e13	−0.325981
$326$	0	0
$327$	−1.20704e13	−0.178529
$328$	0	0
$329$	−9.46932e13	−1.35438
$330$	0	0
$331$	−8.90210e13	−1.23151	−0.615756	−	0.787937i	$-0.711149\pi$
$331$	−8.90210e13	−1.23151	−0.615756	+	0.787937i	$0.711149\pi$
$332$	0	0
$333$	−2.22015e13	−0.297126
$334$	0	0
$335$	−2.24486e13	−0.290699
$336$	0	0
$337$	1.42601e14	1.78714	0.893568	−	0.448928i	$-0.148194\pi$
$337$	1.42601e14	1.78714	0.893568	+	0.448928i	$0.148194\pi$
$338$	0	0
$339$	4.32060e13	0.524138
$340$	0	0
$341$	2.35768e13	0.276909
$342$	0	0
$343$	−9.39407e13	−1.06841
$344$	0	0
$345$	7.31311e13	0.805558
$346$	0	0
$347$	1.33726e14	1.42693	0.713467	−	0.700689i	$-0.247124\pi$
$347$	1.33726e14	1.42693	0.713467	+	0.700689i	$0.247124\pi$
$348$	0	0
$349$	−6.30474e13	−0.651820	−0.325910	−	0.945401i	$-0.605671\pi$
$349$	−6.30474e13	−0.651820	−0.325910	+	0.945401i	$0.605671\pi$
$350$	0	0
$351$	1.96230e13	0.196596
$352$	0	0
$353$	−2.89749e13	−0.281359	−0.140680	−	0.990055i	$-0.544929\pi$
$353$	−2.89749e13	−0.281359	−0.140680	+	0.990055i	$0.544929\pi$
$354$	0	0
$355$	5.23316e13	0.492615
$356$	0	0
$357$	−9.54907e13	−0.871538
$358$	0	0
$359$	−4.33108e13	−0.383334	−0.191667	−	0.981460i	$-0.561389\pi$
$359$	−4.33108e13	−0.383334	−0.191667	+	0.981460i	$0.561389\pi$
$360$	0	0
$361$	1.13520e14	0.974505
$362$	0	0
$363$	−2.86918e13	−0.238930
$364$	0	0
$365$	−8.97871e13	−0.725443
$366$	0	0
$367$	1.12471e14	0.881814	0.440907	−	0.897553i	$-0.354657\pi$
$367$	1.12471e14	0.881814	0.440907	+	0.897553i	$0.354657\pi$
$368$	0	0
$369$	5.05646e13	0.384770
$370$	0	0
$371$	−2.96781e13	−0.219220
$372$	0	0
$373$	1.29948e14	0.931900	0.465950	−	0.884811i	$-0.345713\pi$
$373$	1.29948e14	0.931900	0.465950	+	0.884811i	$0.345713\pi$
$374$	0	0
$375$	−9.02466e13	−0.628433
$376$	0	0
$377$	1.62167e14	1.09669
$378$	0	0
$379$	9.45382e13	0.621000	0.310500	−	0.950573i	$-0.399504\pi$
$379$	9.45382e13	0.621000	0.310500	+	0.950573i	$0.399504\pi$
$380$	0	0
$381$	1.15111e14	0.734563
$382$	0	0
$383$	−5.45117e13	−0.337985	−0.168992	−	0.985617i	$-0.554051\pi$
$383$	−5.45117e13	−0.337985	−0.168992	+	0.985617i	$0.554051\pi$
$384$	0	0
$385$	−1.70870e14	−1.02951
$386$	0	0
$387$	7.35272e13	0.430564
$388$	0	0
$389$	−5.11379e13	−0.291085	−0.145543	−	0.989352i	$-0.546493\pi$
$389$	−5.11379e13	−0.291085	−0.145543	+	0.989352i	$0.546493\pi$
$390$	0	0
$391$	2.83044e14	1.56632
$392$	0	0
$393$	−4.09634e13	−0.220413
$394$	0	0
$395$	1.74115e14	0.911068
$396$	0	0
$397$	−4.87234e13	−0.247965	−0.123982	−	0.992284i	$-0.539567\pi$
$397$	−4.87234e13	−0.247965	−0.123982	+	0.992284i	$0.539567\pi$
$398$	0	0
$399$	2.67056e14	1.32206
$400$	0	0
$401$	3.76407e14	1.81286	0.906430	−	0.422356i	$-0.138797\pi$
$401$	3.76407e14	1.81286	0.906430	+	0.422356i	$0.138797\pi$
$402$	0	0
$403$	7.88430e13	0.369475
$404$	0	0
$405$	2.01048e13	0.0916847
$406$	0	0
$407$	−1.53758e14	−0.682448
$408$	0	0
$409$	3.12467e14	1.34998	0.674988	−	0.737828i	$-0.264149\pi$
$409$	3.12467e14	1.34998	0.674988	+	0.737828i	$0.264149\pi$
$410$	0	0
$411$	−1.94759e14	−0.819160
$412$	0	0
$413$	−2.08904e14	−0.855503
$414$	0	0
$415$	−1.33398e14	−0.531967
$416$	0	0
$417$	−1.27536e14	−0.495318
$418$	0	0
$419$	−2.81729e14	−1.06575	−0.532874	−	0.846195i	$-0.678888\pi$
$419$	−2.81729e14	−1.06575	−0.532874	+	0.846195i	$0.678888\pi$
$420$	0	0
$421$	−3.01064e14	−1.10945	−0.554724	−	0.832034i	$-0.687176\pi$
$421$	−3.01064e14	−1.10945	−0.554724	+	0.832034i	$0.687176\pi$
$422$	0	0
$423$	7.71630e13	0.277037
$424$	0	0
$425$	−8.44964e13	−0.295596
$426$	0	0
$427$	−4.15347e14	−1.41598
$428$	0	0
$429$	1.35900e14	0.451549
$430$	0	0
$431$	−3.46694e14	−1.12285	−0.561425	−	0.827528i	$-0.689747\pi$
$431$	−3.46694e14	−1.12285	−0.561425	+	0.827528i	$0.689747\pi$
$432$	0	0
$433$	4.82345e14	1.52291	0.761456	−	0.648217i	$-0.224485\pi$
$433$	4.82345e14	1.52291	0.761456	+	0.648217i	$0.224485\pi$
$434$	0	0
$435$	1.66149e14	0.511453
$436$	0	0
$437$	−7.91581e14	−2.37600
$438$	0	0
$439$	−1.18934e14	−0.348138	−0.174069	−	0.984733i	$-0.555692\pi$
$439$	−1.18934e14	−0.348138	−0.174069	+	0.984733i	$0.555692\pi$
$440$	0	0
$441$	1.93309e14	0.551874
$442$	0	0
$443$	1.99914e14	0.556703	0.278351	−	0.960479i	$-0.410212\pi$
$443$	1.99914e14	0.556703	0.278351	+	0.960479i	$0.410212\pi$
$444$	0	0
$445$	−1.47695e14	−0.401223
$446$	0	0
$447$	3.05456e14	0.809575
$448$	0	0
$449$	−5.06702e14	−1.31038	−0.655191	−	0.755464i	$-0.727412\pi$
$449$	−5.06702e14	−1.31038	−0.655191	+	0.755464i	$0.727412\pi$
$450$	0	0
$451$	3.50189e14	0.883754
$452$	0	0
$453$	1.19585e14	0.294534
$454$	0	0
$455$	−5.71403e14	−1.37366
$456$	0	0
$457$	−3.62768e14	−0.851315	−0.425658	−	0.904884i	$-0.639957\pi$
$457$	−3.62768e14	−0.851315	−0.425658	+	0.904884i	$0.639957\pi$
$458$	0	0
$459$	7.78129e13	0.178271
$460$	0	0
$461$	6.56466e13	0.146844	0.0734221	−	0.997301i	$-0.476608\pi$
$461$	6.56466e13	0.146844	0.0734221	+	0.997301i	$0.476608\pi$
$462$	0	0
$463$	2.06748e14	0.451592	0.225796	−	0.974175i	$-0.427502\pi$
$463$	2.06748e14	0.451592	0.225796	+	0.974175i	$0.427502\pi$
$464$	0	0
$465$	8.07790e13	0.172309
$466$	0	0
$467$	3.79401e14	0.790416	0.395208	−	0.918592i	$-0.370673\pi$
$467$	3.79401e14	0.790416	0.395208	+	0.918592i	$0.370673\pi$
$468$	0	0
$469$	2.82122e14	0.574099
$470$	0	0
$471$	−2.42293e14	−0.481644
$472$	0	0
$473$	5.09218e14	0.988934
$474$	0	0
$475$	2.36309e14	0.448399
$476$	0	0
$477$	2.41839e13	0.0448410
$478$	0	0
$479$	−3.67939e14	−0.666701	−0.333350	−	0.942803i	$-0.608179\pi$
$479$	−3.67939e14	−0.666701	−0.333350	+	0.942803i	$0.608179\pi$
$480$	0	0
$481$	−5.14182e14	−0.910581
$482$	0	0
$483$	−9.19072e14	−1.59089
$484$	0	0
$485$	−3.57132e14	−0.604295
$486$	0	0
$487$	−6.31645e14	−1.04487	−0.522437	−	0.852678i	$-0.674977\pi$
$487$	−6.31645e14	−1.04487	−0.522437	+	0.852678i	$0.674977\pi$
$488$	0	0
$489$	1.17647e14	0.190274
$490$	0	0
$491$	−2.78183e14	−0.439929	−0.219965	−	0.975508i	$-0.570594\pi$
$491$	−2.78183e14	−0.439929	−0.219965	+	0.975508i	$0.570594\pi$
$492$	0	0
$493$	6.43055e14	0.994467
$494$	0	0
$495$	1.39237e14	0.210585
$496$	0	0
$497$	−6.57675e14	−0.972862
$498$	0	0
$499$	−4.59050e14	−0.664213	−0.332106	−	0.943242i	$-0.607759\pi$
$499$	−4.59050e14	−0.664213	−0.332106	+	0.943242i	$0.607759\pi$
$500$	0	0
$501$	−5.62636e14	−0.796379
$502$	0	0
$503$	9.15703e14	1.26803	0.634017	−	0.773319i	$-0.281405\pi$
$503$	9.15703e14	1.26803	0.634017	+	0.773319i	$0.281405\pi$
$504$	0	0
$505$	−5.47563e14	−0.741877
$506$	0	0
$507$	1.89672e13	0.0251455
$508$	0	0
$509$	1.08562e15	1.40842	0.704208	−	0.709993i	$-0.251302\pi$
$509$	1.08562e15	1.40842	0.704208	+	0.709993i	$0.251302\pi$
$510$	0	0
$511$	1.12840e15	1.43267
$512$	0	0
$513$	−2.17617e14	−0.270425
$514$	0	0
$515$	−3.13508e14	−0.381337
$516$	0	0
$517$	5.34398e14	0.636307
$518$	0	0
$519$	8.19291e14	0.955031
$520$	0	0
$521$	−1.63925e14	−0.187084	−0.0935422	−	0.995615i	$-0.529819\pi$
$521$	−1.63925e14	−0.187084	−0.0935422	+	0.995615i	$0.529819\pi$
$522$	0	0
$523$	−4.69127e13	−0.0524241	−0.0262121	−	0.999656i	$-0.508345\pi$
$523$	−4.69127e13	−0.0524241	−0.0262121	+	0.999656i	$0.508345\pi$
$524$	0	0
$525$	2.74368e14	0.300233
$526$	0	0
$527$	3.12644e14	0.335036
$528$	0	0
$529$	1.77141e15	1.85914
$530$	0	0
$531$	1.70230e14	0.174992
$532$	0	0
$533$	1.17106e15	1.17918
$534$	0	0
$535$	−3.29439e13	−0.0324960
$536$	0	0
$537$	−3.99972e14	−0.386520
$538$	0	0
$539$	1.33877e15	1.26756
$540$	0	0
$541$	−1.28196e15	−1.18929	−0.594647	−	0.803987i	$-0.702708\pi$
$541$	−1.28196e15	−1.18929	−0.594647	+	0.803987i	$0.702708\pi$
$542$	0	0
$543$	6.66348e14	0.605762
$544$	0	0
$545$	−2.86411e14	−0.255158
$546$	0	0
$547$	−1.10564e15	−0.965347	−0.482673	−	0.875801i	$-0.660334\pi$
$547$	−1.10564e15	−0.965347	−0.482673	+	0.875801i	$0.660334\pi$
$548$	0	0
$549$	3.38455e14	0.289636
$550$	0	0
$551$	−1.79841e15	−1.50854
$552$	0	0
$553$	−2.18818e15	−1.79926
$554$	0	0
$555$	−5.26807e14	−0.424659
$556$	0	0
$557$	2.12161e15	1.67672	0.838362	−	0.545113i	$-0.183513\pi$
$557$	2.12161e15	1.67672	0.838362	+	0.545113i	$0.183513\pi$
$558$	0	0
$559$	1.70287e15	1.31952
$560$	0	0
$561$	5.38899e14	0.409460
$562$	0	0
$563$	2.44888e15	1.82462	0.912309	−	0.409503i	$-0.134298\pi$
$563$	2.44888e15	1.82462	0.912309	+	0.409503i	$0.134298\pi$
$564$	0	0
$565$	1.02521e15	0.749111
$566$	0	0
$567$	−2.52666e14	−0.181068
$568$	0	0
$569$	−7.38095e14	−0.518794	−0.259397	−	0.965771i	$-0.583524\pi$
$569$	−7.38095e14	−0.518794	−0.259397	+	0.965771i	$0.583524\pi$
$570$	0	0
$571$	2.10103e13	0.0144855	0.00724274	−	0.999974i	$-0.497695\pi$
$571$	2.10103e13	0.0144855	0.00724274	+	0.999974i	$0.497695\pi$
$572$	0	0
$573$	−9.92434e14	−0.671199
$574$	0	0
$575$	−8.13255e14	−0.539577
$576$	0	0
$577$	1.13249e15	0.737168	0.368584	−	0.929594i	$-0.379843\pi$
$577$	1.13249e15	0.737168	0.368584	+	0.929594i	$0.379843\pi$
$578$	0	0
$579$	−1.08679e15	−0.694087
$580$	0	0
$581$	1.67647e15	1.05058
$582$	0	0
$583$	1.67487e14	0.102992
$584$	0	0
$585$	4.65621e14	0.280980
$586$	0	0
$587$	2.91255e15	1.72490	0.862448	−	0.506145i	$-0.168930\pi$
$587$	2.91255e15	1.72490	0.862448	+	0.506145i	$0.168930\pi$
$588$	0	0
$589$	−8.74362e14	−0.508226
$590$	0	0
$591$	−1.79097e15	−1.02178
$592$	0	0
$593$	1.52901e15	0.856270	0.428135	−	0.903715i	$-0.359171\pi$
$593$	1.52901e15	0.856270	0.428135	+	0.903715i	$0.359171\pi$
$594$	0	0
$595$	−2.26584e15	−1.24562
$596$	0	0
$597$	−6.54420e14	−0.353181
$598$	0	0
$599$	−2.71517e15	−1.43863	−0.719315	−	0.694684i	$-0.755544\pi$
$599$	−2.71517e15	−1.43863	−0.719315	+	0.694684i	$0.755544\pi$
$600$	0	0
$601$	−2.59104e15	−1.34792	−0.673960	−	0.738768i	$-0.735408\pi$
$601$	−2.59104e15	−1.34792	−0.673960	+	0.738768i	$0.735408\pi$
$602$	0	0
$603$	−2.29894e14	−0.117431
$604$	0	0
$605$	−6.80810e14	−0.341485
$606$	0	0
$607$	−1.68761e15	−0.831255	−0.415628	−	0.909535i	$-0.636438\pi$
$607$	−1.68761e15	−0.831255	−0.415628	+	0.909535i	$0.636438\pi$
$608$	0	0
$609$	−2.08807e15	−1.01006
$610$	0	0
$611$	1.78707e15	0.849016
$612$	0	0
$613$	−5.35764e14	−0.250000	−0.125000	−	0.992157i	$-0.539893\pi$
$613$	−5.35764e14	−0.250000	−0.125000	+	0.992157i	$0.539893\pi$
$614$	0	0
$615$	1.19982e15	0.549923
$616$	0	0
$617$	−2.84367e15	−1.28030	−0.640148	−	0.768251i	$-0.721127\pi$
$617$	−2.84367e15	−1.28030	−0.640148	+	0.768251i	$0.721127\pi$
$618$	0	0
$619$	−8.14487e14	−0.360235	−0.180117	−	0.983645i	$-0.557648\pi$
$619$	−8.14487e14	−0.360235	−0.180117	+	0.983645i	$0.557648\pi$
$620$	0	0
$621$	7.48928e14	0.325414
$622$	0	0
$623$	1.85615e15	0.792372
$624$	0	0
$625$	−1.38060e15	−0.579065
$626$	0	0
$627$	−1.50712e15	−0.621122
$628$	0	0
$629$	−2.03894e15	−0.825705
$630$	0	0
$631$	1.45868e15	0.580497	0.290248	−	0.956951i	$-0.406262\pi$
$631$	1.45868e15	0.580497	0.290248	+	0.956951i	$0.406262\pi$
$632$	0	0
$633$	−1.61220e15	−0.630520
$634$	0	0
$635$	2.73140e15	1.04986
$636$	0	0
$637$	4.47698e15	1.69129
$638$	0	0
$639$	5.35922e14	0.198997
$640$	0	0
$641$	−1.16297e15	−0.424473	−0.212236	−	0.977218i	$-0.568075\pi$
$641$	−1.16297e15	−0.424473	−0.212236	+	0.977218i	$0.568075\pi$
$642$	0	0
$643$	−4.73461e15	−1.69873	−0.849364	−	0.527807i	$-0.823015\pi$
$643$	−4.73461e15	−1.69873	−0.849364	+	0.527807i	$0.823015\pi$
$644$	0	0
$645$	1.74468e15	0.615372
$646$	0	0
$647$	−5.47527e15	−1.89859	−0.949297	−	0.314380i	$-0.898203\pi$
$647$	−5.47527e15	−1.89859	−0.949297	+	0.314380i	$0.898203\pi$
$648$	0	0
$649$	1.17894e15	0.401927
$650$	0	0
$651$	−1.01519e15	−0.340291
$652$	0	0
$653$	−8.87971e14	−0.292669	−0.146334	−	0.989235i	$-0.546748\pi$
$653$	−8.87971e14	−0.292669	−0.146334	+	0.989235i	$0.546748\pi$
$654$	0	0
$655$	−9.71997e14	−0.315020
$656$	0	0
$657$	−9.19501e14	−0.293051
$658$	0	0
$659$	−3.14454e15	−0.985571	−0.492785	−	0.870151i	$-0.664021\pi$
$659$	−3.14454e15	−0.985571	−0.492785	+	0.870151i	$0.664021\pi$
$660$	0	0
$661$	1.32982e15	0.409907	0.204954	−	0.978772i	$-0.434296\pi$
$661$	1.32982e15	0.409907	0.204954	+	0.978772i	$0.434296\pi$
$662$	0	0
$663$	1.80212e15	0.546337
$664$	0	0
$665$	6.33681e15	1.88952
$666$	0	0
$667$	6.18923e15	1.81528
$668$	0	0
$669$	1.45693e15	0.420334
$670$	0	0
$671$	2.34399e15	0.665247
$672$	0	0
$673$	−6.84438e15	−1.91096	−0.955479	−	0.295058i	$-0.904661\pi$
$673$	−6.84438e15	−1.91096	−0.955479	+	0.295058i	$0.904661\pi$
$674$	0	0
$675$	−2.23576e14	−0.0614121
$676$	0	0
$677$	8.88820e14	0.240202	0.120101	−	0.992762i	$-0.461678\pi$
$677$	8.88820e14	0.240202	0.120101	+	0.992762i	$0.461678\pi$
$678$	0	0
$679$	4.48824e15	1.19342
$680$	0	0
$681$	1.88308e15	0.492674
$682$	0	0
$683$	−2.22402e15	−0.572565	−0.286282	−	0.958145i	$-0.592420\pi$
$683$	−2.22402e15	−0.572565	−0.286282	+	0.958145i	$0.592420\pi$
$684$	0	0
$685$	−4.62133e15	−1.17076
$686$	0	0
$687$	9.13204e14	0.227670
$688$	0	0
$689$	5.60092e14	0.137421
$690$	0	0
$691$	−4.59721e15	−1.11011	−0.555053	−	0.831815i	$-0.687302\pi$
$691$	−4.59721e15	−1.11011	−0.555053	+	0.831815i	$0.687302\pi$
$692$	0	0
$693$	−1.74986e15	−0.415882
$694$	0	0
$695$	−3.02622e15	−0.707921
$696$	0	0
$697$	4.64373e15	1.06927
$698$	0	0
$699$	2.19350e15	0.497180
$700$	0	0
$701$	−2.58701e15	−0.577231	−0.288616	−	0.957445i	$-0.593195\pi$
$701$	−2.58701e15	−0.577231	−0.288616	+	0.957445i	$0.593195\pi$
$702$	0	0
$703$	5.70223e15	1.25254
$704$	0	0
$705$	1.83095e15	0.395947
$706$	0	0
$707$	6.88148e15	1.46513
$708$	0	0
$709$	2.52081e15	0.528427	0.264214	−	0.964464i	$-0.414888\pi$
$709$	2.52081e15	0.528427	0.264214	+	0.964464i	$0.414888\pi$
$710$	0	0
$711$	1.78309e15	0.368036
$712$	0	0
$713$	3.00911e15	0.611570
$714$	0	0
$715$	3.22469e15	0.645365
$716$	0	0
$717$	9.77514e14	0.192649
$718$	0	0
$719$	−3.03716e15	−0.589467	−0.294733	−	0.955580i	$-0.595231\pi$
$719$	−3.03716e15	−0.589467	−0.294733	+	0.955580i	$0.595231\pi$
$720$	0	0
$721$	3.94001e15	0.753100
$722$	0	0
$723$	−3.64492e15	−0.686163
$724$	0	0
$725$	−1.84766e15	−0.342580
$726$	0	0
$727$	−6.52086e15	−1.19087	−0.595436	−	0.803402i	$-0.703021\pi$
$727$	−6.52086e15	−1.19087	−0.595436	+	0.803402i	$0.703021\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	6.75255e15	1.19653
$732$	0	0
$733$	−1.48208e15	−0.258703	−0.129351	−	0.991599i	$-0.541290\pi$
$733$	−1.48208e15	−0.258703	−0.129351	+	0.991599i	$0.541290\pi$
$734$	0	0
$735$	4.58691e15	0.788751
$736$	0	0
$737$	−1.59215e15	−0.269719
$738$	0	0
$739$	−5.39343e15	−0.900161	−0.450081	−	0.892988i	$-0.648605\pi$
$739$	−5.39343e15	−0.900161	−0.450081	+	0.892988i	$0.648605\pi$
$740$	0	0
$741$	−5.03995e15	−0.828754
$742$	0	0
$743$	−8.96192e15	−1.45199	−0.725993	−	0.687702i	$-0.758620\pi$
$743$	−8.96192e15	−1.45199	−0.725993	+	0.687702i	$0.758620\pi$
$744$	0	0
$745$	7.24798e15	1.15706
$746$	0	0
$747$	−1.36611e15	−0.214894
$748$	0	0
$749$	4.14021e14	0.0641760
$750$	0	0
$751$	7.78720e15	1.18949	0.594746	−	0.803913i	$-0.297253\pi$
$751$	7.78720e15	1.18949	0.594746	+	0.803913i	$0.297253\pi$
$752$	0	0
$753$	3.56765e15	0.537044
$754$	0	0
$755$	2.83755e15	0.420955
$756$	0	0
$757$	−2.41922e15	−0.353711	−0.176856	−	0.984237i	$-0.556593\pi$
$757$	−2.41922e15	−0.353711	−0.176856	+	0.984237i	$0.556593\pi$
$758$	0	0
$759$	5.18675e15	0.747422
$760$	0	0
$761$	−5.02552e15	−0.713782	−0.356891	−	0.934146i	$-0.616163\pi$
$761$	−5.02552e15	−0.713782	−0.356891	+	0.934146i	$0.616163\pi$
$762$	0	0
$763$	3.59946e15	0.503909
$764$	0	0
$765$	1.84637e15	0.254790
$766$	0	0
$767$	3.94249e15	0.536285
$768$	0	0
$769$	1.21056e16	1.62327	0.811634	−	0.584166i	$-0.198578\pi$
$769$	1.21056e16	1.62327	0.811634	+	0.584166i	$0.198578\pi$
$770$	0	0
$771$	−3.99678e15	−0.528337
$772$	0	0
$773$	−7.45630e15	−0.971709	−0.485854	−	0.874040i	$-0.661492\pi$
$773$	−7.45630e15	−0.971709	−0.485854	+	0.874040i	$0.661492\pi$
$774$	0	0
$775$	−8.98303e14	−0.115415
$776$	0	0
$777$	6.62063e15	0.838656
$778$	0	0
$779$	−1.29870e16	−1.62200
$780$	0	0
$781$	3.71157e15	0.457064
$782$	0	0
$783$	1.70151e15	0.206607
$784$	0	0
$785$	−5.74923e15	−0.688377
$786$	0	0
$787$	1.12580e16	1.32923	0.664615	−	0.747186i	$-0.268596\pi$
$787$	1.12580e16	1.32923	0.664615	+	0.747186i	$0.268596\pi$
$788$	0	0
$789$	4.78268e15	0.556863
$790$	0	0
$791$	−1.28843e16	−1.47941
$792$	0	0
$793$	7.83852e15	0.887630
$794$	0	0
$795$	5.73845e14	0.0640878
$796$	0	0
$797$	−1.04669e16	−1.15291	−0.576457	−	0.817127i	$-0.695565\pi$
$797$	−1.04669e16	−1.15291	−0.576457	+	0.817127i	$0.695565\pi$
$798$	0	0
$799$	7.08646e15	0.769878
$800$	0	0
$801$	−1.51253e15	−0.162078
$802$	0	0
$803$	−6.36807e15	−0.673088
$804$	0	0
$805$	−2.18081e16	−2.27374
$806$	0	0
$807$	−2.82303e15	−0.290343
$808$	0	0
$809$	−1.28180e16	−1.30048	−0.650238	−	0.759730i	$-0.725331\pi$
$809$	−1.28180e16	−1.30048	−0.650238	+	0.759730i	$0.725331\pi$
$810$	0	0
$811$	−9.79295e15	−0.980164	−0.490082	−	0.871676i	$-0.663033\pi$
$811$	−9.79295e15	−0.980164	−0.490082	+	0.871676i	$0.663033\pi$
$812$	0	0
$813$	−1.81469e15	−0.179187
$814$	0	0
$815$	2.79156e15	0.271945
$816$	0	0
$817$	−1.88847e16	−1.81505
$818$	0	0
$819$	−5.85168e15	−0.554906
$820$	0	0
$821$	7.08410e15	0.662823	0.331411	−	0.943486i	$-0.392475\pi$
$821$	7.08410e15	0.662823	0.331411	+	0.943486i	$0.392475\pi$
$822$	0	0
$823$	−1.98569e16	−1.83321	−0.916606	−	0.399792i	$-0.869082\pi$
$823$	−1.98569e16	−1.83321	−0.916606	+	0.399792i	$0.869082\pi$
$824$	0	0
$825$	−1.54839e15	−0.141053
$826$	0	0
$827$	2.05537e16	1.84760	0.923802	−	0.382871i	$-0.125065\pi$
$827$	2.05537e16	1.84760	0.923802	+	0.382871i	$0.125065\pi$
$828$	0	0
$829$	−1.45920e16	−1.29439	−0.647197	−	0.762323i	$-0.724059\pi$
$829$	−1.45920e16	−1.29439	−0.647197	+	0.762323i	$0.724059\pi$
$830$	0	0
$831$	3.87742e15	0.339419
$832$	0	0
$833$	1.77530e16	1.53364
$834$	0	0
$835$	−1.33504e16	−1.13820
$836$	0	0
$837$	8.27249e14	0.0696059
$838$	0	0
$839$	−6.54963e15	−0.543909	−0.271954	−	0.962310i	$-0.587670\pi$
$839$	−6.54963e15	−0.543909	−0.271954	+	0.962310i	$0.587670\pi$
$840$	0	0
$841$	1.86098e15	0.152533
$842$	0	0
$843$	9.62559e15	0.778710
$844$	0	0
$845$	4.50062e14	0.0359386
$846$	0	0
$847$	8.55606e15	0.674396
$848$	0	0
$849$	−3.64502e15	−0.283601
$850$	0	0
$851$	−1.96242e16	−1.50723
$852$	0	0
$853$	1.99244e16	1.51066	0.755329	−	0.655346i	$-0.227477\pi$
$853$	1.99244e16	1.51066	0.755329	+	0.655346i	$0.227477\pi$
$854$	0	0
$855$	−5.16370e15	−0.386498
$856$	0	0
$857$	1.59149e16	1.17600	0.588002	−	0.808859i	$-0.299915\pi$
$857$	1.59149e16	1.17600	0.588002	+	0.808859i	$0.299915\pi$
$858$	0	0
$859$	1.19020e15	0.0868272	0.0434136	−	0.999057i	$-0.486177\pi$
$859$	1.19020e15	0.0868272	0.0434136	+	0.999057i	$0.486177\pi$
$860$	0	0
$861$	−1.50787e16	−1.08604
$862$	0	0
$863$	1.73343e15	0.123267	0.0616336	−	0.998099i	$-0.480369\pi$
$863$	1.73343e15	0.123267	0.0616336	+	0.998099i	$0.480369\pi$
$864$	0	0
$865$	1.94405e16	1.36495
$866$	0	0
$867$	−1.18193e15	−0.0819381
$868$	0	0
$869$	1.23489e16	0.845317
$870$	0	0
$871$	−5.32428e15	−0.359882
$872$	0	0
$873$	−3.65735e15	−0.244112
$874$	0	0
$875$	2.69121e16	1.77379
$876$	0	0
$877$	1.16602e16	0.758943	0.379471	−	0.925203i	$-0.376106\pi$
$877$	1.16602e16	0.758943	0.379471	+	0.925203i	$0.376106\pi$
$878$	0	0
$879$	9.43459e15	0.606435
$880$	0	0
$881$	−1.66091e16	−1.05434	−0.527168	−	0.849761i	$-0.676746\pi$
$881$	−1.66091e16	−1.05434	−0.527168	+	0.849761i	$0.676746\pi$
$882$	0	0
$883$	1.28205e16	0.803750	0.401875	−	0.915695i	$-0.368359\pi$
$883$	1.28205e16	0.803750	0.401875	+	0.915695i	$0.368359\pi$
$884$	0	0
$885$	4.03929e15	0.250102
$886$	0	0
$887$	2.21358e14	0.0135368	0.00676840	−	0.999977i	$-0.497846\pi$
$887$	2.21358e14	0.0135368	0.00676840	+	0.999977i	$0.497846\pi$
$888$	0	0
$889$	−3.43267e16	−2.07335
$890$	0	0
$891$	1.42591e15	0.0850679
$892$	0	0
$893$	−1.98185e16	−1.16785
$894$	0	0
$895$	−9.49070e15	−0.552423
$896$	0	0
$897$	1.73450e16	0.997275
$898$	0	0
$899$	6.83649e15	0.388289
$900$	0	0
$901$	2.22099e15	0.124612
$902$	0	0
$903$	−2.19262e16	−1.21529
$904$	0	0
$905$	1.58114e16	0.865770
$906$	0	0
$907$	2.48361e16	1.34352	0.671758	−	0.740770i	$-0.265539\pi$
$907$	2.48361e16	1.34352	0.671758	+	0.740770i	$0.265539\pi$
$908$	0	0
$909$	−5.60754e15	−0.299689
$910$	0	0
$911$	−8.10295e15	−0.427851	−0.213925	−	0.976850i	$-0.568625\pi$
$911$	−8.10295e15	−0.427851	−0.213925	+	0.976850i	$0.568625\pi$
$912$	0	0
$913$	−9.46112e15	−0.493575
$914$	0	0
$915$	8.03100e15	0.413955
$916$	0	0
$917$	1.22155e16	0.622130
$918$	0	0
$919$	3.33814e16	1.67984	0.839922	−	0.542707i	$-0.182601\pi$
$919$	3.33814e16	1.67984	0.839922	+	0.542707i	$0.182601\pi$
$920$	0	0
$921$	−1.84241e16	−0.916133
$922$	0	0
$923$	1.24118e16	0.609854
$924$	0	0
$925$	5.85836e15	0.284444
$926$	0	0
$927$	−3.21061e15	−0.154045
$928$	0	0
$929$	3.56713e16	1.69134	0.845672	−	0.533703i	$-0.179200\pi$
$929$	3.56713e16	1.69134	0.845672	+	0.533703i	$0.179200\pi$
$930$	0	0
$931$	−4.96493e16	−2.32643
$932$	0	0
$933$	1.94377e16	0.900112
$934$	0	0
$935$	1.27872e16	0.585210
$936$	0	0
$937$	3.96248e16	1.79225	0.896126	−	0.443800i	$-0.146370\pi$
$937$	3.96248e16	1.79225	0.896126	+	0.443800i	$0.146370\pi$
$938$	0	0
$939$	−1.66604e16	−0.744775
$940$	0	0
$941$	5.37193e15	0.237349	0.118675	−	0.992933i	$-0.462135\pi$
$941$	5.37193e15	0.237349	0.118675	+	0.992933i	$0.462135\pi$
$942$	0	0
$943$	4.46946e16	1.95183
$944$	0	0
$945$	−5.99537e15	−0.258786
$946$	0	0
$947$	−2.22241e16	−0.948199	−0.474099	−	0.880471i	$-0.657226\pi$
$947$	−2.22241e16	−0.948199	−0.474099	+	0.880471i	$0.657226\pi$
$948$	0	0
$949$	−2.12954e16	−0.898092
$950$	0	0
$951$	1.55008e15	0.0646190
$952$	0	0
$953$	7.66867e15	0.316016	0.158008	−	0.987438i	$-0.449493\pi$
$953$	7.66867e15	0.316016	0.158008	+	0.987438i	$0.449493\pi$
$954$	0	0
$955$	−2.35489e16	−0.959293
$956$	0	0
$957$	1.17839e16	0.474542
$958$	0	0
$959$	5.80784e16	2.31213
$960$	0	0
$961$	−2.20847e16	−0.869185
$962$	0	0
$963$	−3.37375e14	−0.0131271
$964$	0	0
$965$	−2.57878e16	−0.992006
$966$	0	0
$967$	−2.80474e16	−1.06671	−0.533355	−	0.845891i	$-0.679069\pi$
$967$	−2.80474e16	−1.06671	−0.533355	+	0.845891i	$0.679069\pi$
$968$	0	0
$969$	−1.99854e16	−0.751505
$970$	0	0
$971$	2.43427e16	0.905031	0.452516	−	0.891757i	$-0.350527\pi$
$971$	2.43427e16	0.905031	0.452516	+	0.891757i	$0.350527\pi$
$972$	0	0
$973$	3.80320e16	1.39807
$974$	0	0
$975$	−5.17795e15	−0.188205
$976$	0	0
$977$	−2.19144e16	−0.787608	−0.393804	−	0.919194i	$-0.628841\pi$
$977$	−2.19144e16	−0.787608	−0.393804	+	0.919194i	$0.628841\pi$
$978$	0	0
$979$	−1.04751e16	−0.372267
$980$	0	0
$981$	−2.93310e15	−0.103074
$982$	0	0
$983$	7.04586e15	0.244844	0.122422	−	0.992478i	$-0.460934\pi$
$983$	7.04586e15	0.244844	0.122422	+	0.992478i	$0.460934\pi$
$984$	0	0
$985$	−4.24969e16	−1.46035
$986$	0	0
$987$	−2.30105e16	−0.781954
$988$	0	0
$989$	6.49915e16	2.18412
$990$	0	0
$991$	−4.04873e16	−1.34559	−0.672796	−	0.739828i	$-0.734907\pi$
$991$	−4.04873e16	−1.34559	−0.672796	+	0.739828i	$0.734907\pi$
$992$	0	0
$993$	−2.16321e16	−0.711013
$994$	0	0
$995$	−1.55283e16	−0.504775
$996$	0	0
$997$	1.79607e16	0.577432	0.288716	−	0.957415i	$-0.406772\pi$
$997$	1.79607e16	0.577432	0.288716	+	0.957415i	$0.406772\pi$
$998$	0	0
$999$	−5.39498e15	−0.171546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.12.a.h.1.1		1
3.2	odd	2		144.12.a.b.1.1		1
4.3	odd	2		6.12.a.a.1.1	✓	1
8.3	odd	2		192.12.a.l.1.1		1
8.5	even	2		192.12.a.b.1.1		1
12.11	even	2		18.12.a.c.1.1		1
20.3	even	4		150.12.c.f.49.2		2
20.7	even	4		150.12.c.f.49.1		2
20.19	odd	2		150.12.a.g.1.1		1
36.7	odd	6		162.12.c.g.109.1		2
36.11	even	6		162.12.c.d.109.1		2
36.23	even	6		162.12.c.d.55.1		2
36.31	odd	6		162.12.c.g.55.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.12.a.a.1.1	✓	1	4.3	odd	2
18.12.a.c.1.1		1	12.11	even	2
48.12.a.h.1.1		1	1.1	even	1	trivial
144.12.a.b.1.1		1	3.2	odd	2
150.12.a.g.1.1		1	20.19	odd	2
150.12.c.f.49.1		2	20.7	even	4
150.12.c.f.49.2		2	20.3	even	4
162.12.c.d.55.1		2	36.23	even	6
162.12.c.d.109.1		2	36.11	even	6
162.12.c.g.55.1		2	36.31	odd	6
162.12.c.g.109.1		2	36.7	odd	6
192.12.a.b.1.1		1	8.5	even	2
192.12.a.l.1.1		1	8.3	odd	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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