LMFDB - Embedded newform 48.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,10,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, 10, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
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:	$24.7217201359$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
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)

$f(q)$

$=$

$q-81.0000 q^{3} -794.000 q^{5} +5880.00 q^{7} +6561.00 q^{9} +O(q^{10})$

$q-81.0000 q^{3} -794.000 q^{5} +5880.00 q^{7} +6561.00 q^{9} +30644.0 q^{11} -15314.0 q^{13} +64314.0 q^{15} -575086. q^{17} +617644. q^{19} -476280. q^{21} -441880. q^{23} -1.32269e6 q^{25} -531441. q^{27} -2.32864e6 q^{29} -9.58851e6 q^{31} -2.48216e6 q^{33} -4.66872e6 q^{35} +9.27668e6 q^{37} +1.24043e6 q^{39} -5.90377e6 q^{41} -3.35935e7 q^{43} -5.20943e6 q^{45} -2.11354e7 q^{47} -5.77921e6 q^{49} +4.65820e7 q^{51} -1.08576e8 q^{53} -2.43313e7 q^{55} -5.00292e7 q^{57} +1.27637e8 q^{59} +1.47189e8 q^{61} +3.85787e7 q^{63} +1.21593e7 q^{65} +3.31578e7 q^{67} +3.57923e7 q^{69} +9.29375e6 q^{71} +3.51080e8 q^{73} +1.07138e8 q^{75} +1.80187e8 q^{77} +1.26193e8 q^{79} +4.30467e7 q^{81} -4.75038e8 q^{83} +4.56618e8 q^{85} +1.88620e8 q^{87} -5.66134e8 q^{89} -9.00463e7 q^{91} +7.76669e8 q^{93} -4.90409e8 q^{95} -1.47468e9 q^{97} +2.01055e8 q^{99} +O(q^{100})$

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
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	0	0
$5$	−794.000	−0.568140	−0.284070	−	0.958804i	$-0.591685\pi$
$5$	−794.000	−0.568140	−0.284070	+	0.958804i	$0.591685\pi$
$6$	0	0
$7$	5880.00	0.925627	0.462814	−	0.886456i	$-0.346840\pi$
$7$	5880.00	0.925627	0.462814	+	0.886456i	$0.346840\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	30644.0	0.631071	0.315536	−	0.948914i	$-0.397816\pi$
$11$	30644.0	0.631071	0.315536	+	0.948914i	$0.397816\pi$
$12$	0	0
$13$	−15314.0	−0.148711	−0.0743556	−	0.997232i	$-0.523690\pi$
$13$	−15314.0	−0.148711	−0.0743556	+	0.997232i	$0.523690\pi$
$14$	0	0
$15$	64314.0	0.328016
$16$	0	0
$17$	−575086.	−1.66999	−0.834993	−	0.550261i	$-0.814528\pi$
$17$	−575086.	−1.66999	−0.834993	+	0.550261i	$0.814528\pi$
$18$	0	0
$19$	617644.	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$19$	617644.	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$20$	0	0
$21$	−476280.	−0.534411
$22$	0	0
$23$	−441880.	−0.329253	−0.164626	−	0.986356i	$-0.552642\pi$
$23$	−441880.	−0.329253	−0.164626	+	0.986356i	$0.552642\pi$
$24$	0	0
$25$	−1.32269e6	−0.677217
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	−2.32864e6	−0.611381	−0.305690	−	0.952131i	$-0.598887\pi$
$29$	−2.32864e6	−0.611381	−0.305690	+	0.952131i	$0.598887\pi$
$30$	0	0
$31$	−9.58851e6	−1.86476	−0.932381	−	0.361476i	$-0.882273\pi$
$31$	−9.58851e6	−1.86476	−0.932381	+	0.361476i	$0.882273\pi$
$32$	0	0
$33$	−2.48216e6	−0.364349
$34$	0	0
$35$	−4.66872e6	−0.525886
$36$	0	0
$37$	9.27668e6	0.813738	0.406869	−	0.913486i	$-0.366621\pi$
$37$	9.27668e6	0.813738	0.406869	+	0.913486i	$0.366621\pi$
$38$	0	0
$39$	1.24043e6	0.0858584
$40$	0	0
$41$	−5.90377e6	−0.326289	−0.163144	−	0.986602i	$-0.552164\pi$
$41$	−5.90377e6	−0.326289	−0.163144	+	0.986602i	$0.552164\pi$
$42$	0	0
$43$	−3.35935e7	−1.49846	−0.749232	−	0.662307i	$-0.769577\pi$
$43$	−3.35935e7	−1.49846	−0.749232	+	0.662307i	$0.769577\pi$
$44$	0	0
$45$	−5.20943e6	−0.189380
$46$	0	0
$47$	−2.11354e7	−0.631786	−0.315893	−	0.948795i	$-0.602304\pi$
$47$	−2.11354e7	−0.631786	−0.315893	+	0.948795i	$0.602304\pi$
$48$	0	0
$49$	−5.77921e6	−0.143214
$50$	0	0
$51$	4.65820e7	0.964166
$52$	0	0
$53$	−1.08576e8	−1.89013	−0.945063	−	0.326888i	$-0.894000\pi$
$53$	−1.08576e8	−1.89013	−0.945063	+	0.326888i	$0.894000\pi$
$54$	0	0
$55$	−2.43313e7	−0.358537
$56$	0	0
$57$	−5.00292e7	−0.627750
$58$	0	0
$59$	1.27637e8	1.37133	0.685665	−	0.727917i	$-0.259511\pi$
$59$	1.27637e8	1.37133	0.685665	+	0.727917i	$0.259511\pi$
$60$	0	0
$61$	1.47189e8	1.36111	0.680553	−	0.732699i	$-0.261740\pi$
$61$	1.47189e8	1.36111	0.680553	+	0.732699i	$0.261740\pi$
$62$	0	0
$63$	3.85787e7	0.308542
$64$	0	0
$65$	1.21593e7	0.0844888
$66$	0	0
$67$	3.31578e7	0.201024	0.100512	−	0.994936i	$-0.467952\pi$
$67$	3.31578e7	0.201024	0.100512	+	0.994936i	$0.467952\pi$
$68$	0	0
$69$	3.57923e7	0.190094
$70$	0	0
$71$	9.29375e6	0.0434039	0.0217019	−	0.999764i	$-0.493092\pi$
$71$	9.29375e6	0.0434039	0.0217019	+	0.999764i	$0.493092\pi$
$72$	0	0
$73$	3.51080e8	1.44695	0.723475	−	0.690351i	$-0.242544\pi$
$73$	3.51080e8	1.44695	0.723475	+	0.690351i	$0.242544\pi$
$74$	0	0
$75$	1.07138e8	0.390991
$76$	0	0
$77$	1.80187e8	0.584137
$78$	0	0
$79$	1.26193e8	0.364514	0.182257	−	0.983251i	$-0.441660\pi$
$79$	1.26193e8	0.364514	0.182257	+	0.983251i	$0.441660\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	−4.75038e8	−1.09869	−0.549347	−	0.835594i	$-0.685123\pi$
$83$	−4.75038e8	−1.09869	−0.549347	+	0.835594i	$0.685123\pi$
$84$	0	0
$85$	4.56618e8	0.948786
$86$	0	0
$87$	1.88620e8	0.352981
$88$	0	0
$89$	−5.66134e8	−0.956454	−0.478227	−	0.878236i	$-0.658720\pi$
$89$	−5.66134e8	−0.956454	−0.478227	+	0.878236i	$0.658720\pi$
$90$	0	0
$91$	−9.00463e7	−0.137651
$92$	0	0
$93$	7.76669e8	1.07662
$94$	0	0
$95$	−4.90409e8	−0.617735
$96$	0	0
$97$	−1.47468e9	−1.69132	−0.845661	−	0.533721i	$-0.820793\pi$
$97$	−1.47468e9	−1.69132	−0.845661	+	0.533721i	$0.820793\pi$
$98$	0	0
$99$	2.01055e8	0.210357
$100$	0	0
$101$	−6.43692e8	−0.615506	−0.307753	−	0.951466i	$-0.599577\pi$
$101$	−6.43692e8	−0.615506	−0.307753	+	0.951466i	$0.599577\pi$
$102$	0	0
$103$	2.15834e9	1.88952	0.944761	−	0.327760i	$-0.106294\pi$
$103$	2.15834e9	1.88952	0.944761	+	0.327760i	$0.106294\pi$
$104$	0	0
$105$	3.78166e8	0.303620
$106$	0	0
$107$	−8.54143e8	−0.629946	−0.314973	−	0.949101i	$-0.601995\pi$
$107$	−8.54143e8	−0.629946	−0.314973	+	0.949101i	$0.601995\pi$
$108$	0	0
$109$	−6.67211e8	−0.452735	−0.226368	−	0.974042i	$-0.572685\pi$
$109$	−6.67211e8	−0.452735	−0.226368	+	0.974042i	$0.572685\pi$
$110$	0	0
$111$	−7.51411e8	−0.469812
$112$	0	0
$113$	−2.48382e9	−1.43307	−0.716534	−	0.697552i	$-0.754272\pi$
$113$	−2.48382e9	−1.43307	−0.716534	+	0.697552i	$0.754272\pi$
$114$	0	0
$115$	3.50853e8	0.187062
$116$	0	0
$117$	−1.00475e8	−0.0495704
$118$	0	0
$119$	−3.38151e9	−1.54578
$120$	0	0
$121$	−1.41889e9	−0.601749
$122$	0	0
$123$	4.78205e8	0.188383
$124$	0	0
$125$	2.60100e9	0.952894
$126$	0	0
$127$	−5.52123e8	−0.188330	−0.0941649	−	0.995557i	$-0.530018\pi$
$127$	−5.52123e8	−0.188330	−0.0941649	+	0.995557i	$0.530018\pi$
$128$	0	0
$129$	2.72107e9	0.865139
$130$	0	0
$131$	−4.45164e9	−1.32068	−0.660342	−	0.750965i	$-0.729589\pi$
$131$	−4.45164e9	−1.32068	−0.660342	+	0.750965i	$0.729589\pi$
$132$	0	0
$133$	3.63175e9	1.00643
$134$	0	0
$135$	4.21964e8	0.109339
$136$	0	0
$137$	−6.54857e9	−1.58819	−0.794097	−	0.607791i	$-0.792056\pi$
$137$	−6.54857e9	−1.58819	−0.794097	+	0.607791i	$0.792056\pi$
$138$	0	0
$139$	2.97897e9	0.676862	0.338431	−	0.940991i	$-0.390104\pi$
$139$	2.97897e9	0.676862	0.338431	+	0.940991i	$0.390104\pi$
$140$	0	0
$141$	1.71197e9	0.364762
$142$	0	0
$143$	−4.69282e8	−0.0938473
$144$	0	0
$145$	1.84894e9	0.347350
$146$	0	0
$147$	4.68116e8	0.0826847
$148$	0	0
$149$	3.49168e9	0.580359	0.290179	−	0.956972i	$-0.406285\pi$
$149$	3.49168e9	0.580359	0.290179	+	0.956972i	$0.406285\pi$
$150$	0	0
$151$	−2.36345e9	−0.369957	−0.184978	−	0.982743i	$-0.559221\pi$
$151$	−2.36345e9	−0.369957	−0.184978	+	0.982743i	$0.559221\pi$
$152$	0	0
$153$	−3.77314e9	−0.556662
$154$	0	0
$155$	7.61328e9	1.05945
$156$	0	0
$157$	1.18383e10	1.55503	0.777517	−	0.628862i	$-0.216479\pi$
$157$	1.18383e10	1.55503	0.777517	+	0.628862i	$0.216479\pi$
$158$	0	0
$159$	8.79462e9	1.09126
$160$	0	0
$161$	−2.59825e9	−0.304765
$162$	0	0
$163$	4.87356e9	0.540757	0.270379	−	0.962754i	$-0.412851\pi$
$163$	4.87356e9	0.540757	0.270379	+	0.962754i	$0.412851\pi$
$164$	0	0
$165$	1.97084e9	0.207001
$166$	0	0
$167$	1.55717e10	1.54921	0.774606	−	0.632444i	$-0.217948\pi$
$167$	1.55717e10	1.54921	0.774606	+	0.632444i	$0.217948\pi$
$168$	0	0
$169$	−1.03700e10	−0.977885
$170$	0	0
$171$	4.05236e9	0.362431
$172$	0	0
$173$	5.64996e9	0.479554	0.239777	−	0.970828i	$-0.422926\pi$
$173$	5.64996e9	0.479554	0.239777	+	0.970828i	$0.422926\pi$
$174$	0	0
$175$	−7.77741e9	−0.626850
$176$	0	0
$177$	−1.03386e10	−0.791738
$178$	0	0
$179$	2.06473e9	0.150323	0.0751613	−	0.997171i	$-0.476053\pi$
$179$	2.06473e9	0.150323	0.0751613	+	0.997171i	$0.476053\pi$
$180$	0	0
$181$	8.33213e9	0.577035	0.288518	−	0.957475i	$-0.406838\pi$
$181$	8.33213e9	0.577035	0.288518	+	0.957475i	$0.406838\pi$
$182$	0	0
$183$	−1.19223e10	−0.785835
$184$	0	0
$185$	−7.36568e9	−0.462317
$186$	0	0
$187$	−1.76229e10	−1.05388
$188$	0	0
$189$	−3.12487e9	−0.178137
$190$	0	0
$191$	−1.06452e10	−0.578767	−0.289383	−	0.957213i	$-0.593450\pi$
$191$	−1.06452e10	−0.578767	−0.289383	+	0.957213i	$0.593450\pi$
$192$	0	0
$193$	1.00851e10	0.523206	0.261603	−	0.965176i	$-0.415749\pi$
$193$	1.00851e10	0.523206	0.261603	+	0.965176i	$0.415749\pi$
$194$	0	0
$195$	−9.84905e8	−0.0487796
$196$	0	0
$197$	2.73516e9	0.129385	0.0646927	−	0.997905i	$-0.479393\pi$
$197$	2.73516e9	0.129385	0.0646927	+	0.997905i	$0.479393\pi$
$198$	0	0
$199$	−1.63515e10	−0.739124	−0.369562	−	0.929206i	$-0.620492\pi$
$199$	−1.63515e10	−0.739124	−0.369562	+	0.929206i	$0.620492\pi$
$200$	0	0
$201$	−2.68578e9	−0.116061
$202$	0	0
$203$	−1.36924e10	−0.565911
$204$	0	0
$205$	4.68759e9	0.185378
$206$	0	0
$207$	−2.89917e9	−0.109751
$208$	0	0
$209$	1.89271e10	0.686160
$210$	0	0
$211$	3.23788e8	0.0112458	0.00562290	−	0.999984i	$-0.498210\pi$
$211$	3.23788e8	0.0112458	0.00562290	+	0.999984i	$0.498210\pi$
$212$	0	0
$213$	−7.52794e8	−0.0250592
$214$	0	0
$215$	2.66732e10	0.851338
$216$	0	0
$217$	−5.63805e10	−1.72608
$218$	0	0
$219$	−2.84375e10	−0.835397
$220$	0	0
$221$	8.80687e9	0.248345
$222$	0	0
$223$	1.77933e10	0.481821	0.240910	−	0.970547i	$-0.422554\pi$
$223$	1.77933e10	0.481821	0.240910	+	0.970547i	$0.422554\pi$
$224$	0	0
$225$	−8.67816e9	−0.225739
$226$	0	0
$227$	−2.16706e10	−0.541696	−0.270848	−	0.962622i	$-0.587304\pi$
$227$	−2.16706e10	−0.541696	−0.270848	+	0.962622i	$0.587304\pi$
$228$	0	0
$229$	−7.36907e10	−1.77073	−0.885366	−	0.464894i	$-0.846092\pi$
$229$	−7.36907e10	−1.77073	−0.885366	+	0.464894i	$0.846092\pi$
$230$	0	0
$231$	−1.45951e10	−0.337251
$232$	0	0
$233$	4.17259e10	0.927480	0.463740	−	0.885971i	$-0.346507\pi$
$233$	4.17259e10	0.927480	0.463740	+	0.885971i	$0.346507\pi$
$234$	0	0
$235$	1.67815e10	0.358943
$236$	0	0
$237$	−1.02217e10	−0.210452
$238$	0	0
$239$	7.03773e10	1.39522	0.697609	−	0.716479i	$-0.254247\pi$
$239$	7.03773e10	1.39522	0.697609	+	0.716479i	$0.254247\pi$
$240$	0	0
$241$	−2.19843e10	−0.419793	−0.209897	−	0.977724i	$-0.567313\pi$
$241$	−2.19843e10	−0.419793	−0.209897	+	0.977724i	$0.567313\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	4.58869e9	0.0813657
$246$	0	0
$247$	−9.45860e9	−0.161693
$248$	0	0
$249$	3.84780e10	0.634331
$250$	0	0
$251$	6.07417e10	0.965951	0.482975	−	0.875634i	$-0.339556\pi$
$251$	6.07417e10	0.965951	0.482975	+	0.875634i	$0.339556\pi$
$252$	0	0
$253$	−1.35410e10	−0.207782
$254$	0	0
$255$	−3.69861e10	−0.547782
$256$	0	0
$257$	2.99210e10	0.427836	0.213918	−	0.976852i	$-0.431377\pi$
$257$	2.99210e10	0.427836	0.213918	+	0.976852i	$0.431377\pi$
$258$	0	0
$259$	5.45469e10	0.753218
$260$	0	0
$261$	−1.52782e10	−0.203794
$262$	0	0
$263$	−5.03592e10	−0.649050	−0.324525	−	0.945877i	$-0.605204\pi$
$263$	−5.03592e10	−0.649050	−0.324525	+	0.945877i	$0.605204\pi$
$264$	0	0
$265$	8.62090e10	1.07386
$266$	0	0
$267$	4.58569e10	0.552209
$268$	0	0
$269$	8.98188e10	1.04588	0.522940	−	0.852369i	$-0.324835\pi$
$269$	8.98188e10	1.04588	0.522940	+	0.852369i	$0.324835\pi$
$270$	0	0
$271$	5.74877e10	0.647460	0.323730	−	0.946150i	$-0.395063\pi$
$271$	5.74877e10	0.647460	0.323730	+	0.946150i	$0.395063\pi$
$272$	0	0
$273$	7.29375e9	0.0794729
$274$	0	0
$275$	−4.05325e10	−0.427372
$276$	0	0
$277$	4.70498e10	0.480174	0.240087	−	0.970751i	$-0.422824\pi$
$277$	4.70498e10	0.480174	0.240087	+	0.970751i	$0.422824\pi$
$278$	0	0
$279$	−6.29102e10	−0.621588
$280$	0	0
$281$	5.79723e10	0.554679	0.277340	−	0.960772i	$-0.410547\pi$
$281$	5.79723e10	0.554679	0.277340	+	0.960772i	$0.410547\pi$
$282$	0	0
$283$	−3.19354e10	−0.295960	−0.147980	−	0.988990i	$-0.547277\pi$
$283$	−3.19354e10	−0.295960	−0.147980	+	0.988990i	$0.547277\pi$
$284$	0	0
$285$	3.97232e10	0.356650
$286$	0	0
$287$	−3.47141e10	−0.302022
$288$	0	0
$289$	2.12136e11	1.78885
$290$	0	0
$291$	1.19449e11	0.976485
$292$	0	0
$293$	−1.47197e11	−1.16680	−0.583398	−	0.812187i	$-0.698277\pi$
$293$	−1.47197e11	−1.16680	−0.583398	+	0.812187i	$0.698277\pi$
$294$	0	0
$295$	−1.01344e11	−0.779108
$296$	0	0
$297$	−1.62855e10	−0.121450
$298$	0	0
$299$	6.76695e9	0.0489635
$300$	0	0
$301$	−1.97529e11	−1.38702
$302$	0	0
$303$	5.21391e10	0.355363
$304$	0	0
$305$	−1.16868e11	−0.773299
$306$	0	0
$307$	−8.30102e10	−0.533345	−0.266673	−	0.963787i	$-0.585924\pi$
$307$	−8.30102e10	−0.533345	−0.266673	+	0.963787i	$0.585924\pi$
$308$	0	0
$309$	−1.74825e11	−1.09092
$310$	0	0
$311$	2.56030e11	1.55192	0.775960	−	0.630782i	$-0.217266\pi$
$311$	2.56030e11	1.55192	0.775960	+	0.630782i	$0.217266\pi$
$312$	0	0
$313$	9.94512e10	0.585680	0.292840	−	0.956161i	$-0.405400\pi$
$313$	9.94512e10	0.585680	0.292840	+	0.956161i	$0.405400\pi$
$314$	0	0
$315$	−3.06315e10	−0.175295
$316$	0	0
$317$	−2.76175e11	−1.53610	−0.768048	−	0.640392i	$-0.778772\pi$
$317$	−2.76175e11	−1.53610	−0.768048	+	0.640392i	$0.778772\pi$
$318$	0	0
$319$	−7.13589e10	−0.385825
$320$	0	0
$321$	6.91856e10	0.363700
$322$	0	0
$323$	−3.55198e11	−1.81577
$324$	0	0
$325$	2.02557e10	0.100710
$326$	0	0
$327$	5.40441e10	0.261387
$328$	0	0
$329$	−1.24276e11	−0.584799
$330$	0	0
$331$	1.56680e11	0.717442	0.358721	−	0.933445i	$-0.383213\pi$
$331$	1.56680e11	0.717442	0.358721	+	0.933445i	$0.383213\pi$
$332$	0	0
$333$	6.08643e10	0.271246
$334$	0	0
$335$	−2.63273e10	−0.114210
$336$	0	0
$337$	−8.42586e10	−0.355861	−0.177930	−	0.984043i	$-0.556940\pi$
$337$	−8.42586e10	−0.355861	−0.177930	+	0.984043i	$0.556940\pi$
$338$	0	0
$339$	2.01189e11	0.827382
$340$	0	0
$341$	−2.93830e11	−1.17680
$342$	0	0
$343$	−2.71261e11	−1.05819
$344$	0	0
$345$	−2.84191e10	−0.108000
$346$	0	0
$347$	−2.04007e11	−0.755374	−0.377687	−	0.925933i	$-0.623280\pi$
$347$	−2.04007e11	−0.755374	−0.377687	+	0.925933i	$0.623280\pi$
$348$	0	0
$349$	5.36485e10	0.193572	0.0967861	−	0.995305i	$-0.469144\pi$
$349$	5.36485e10	0.193572	0.0967861	+	0.995305i	$0.469144\pi$
$350$	0	0
$351$	8.13849e9	0.0286195
$352$	0	0
$353$	3.90043e11	1.33698	0.668492	−	0.743720i	$-0.266940\pi$
$353$	3.90043e11	1.33698	0.668492	+	0.743720i	$0.266940\pi$
$354$	0	0
$355$	−7.37924e9	−0.0246595
$356$	0	0
$357$	2.73902e11	0.892459
$358$	0	0
$359$	−5.66317e10	−0.179943	−0.0899714	−	0.995944i	$-0.528678\pi$
$359$	−5.66317e10	−0.179943	−0.0899714	+	0.995944i	$0.528678\pi$
$360$	0	0
$361$	5.87964e10	0.182208
$362$	0	0
$363$	1.14930e11	0.347420
$364$	0	0
$365$	−2.78758e11	−0.822070
$366$	0	0
$367$	1.20927e11	0.347957	0.173979	−	0.984749i	$-0.444338\pi$
$367$	1.20927e11	0.347957	0.173979	+	0.984749i	$0.444338\pi$
$368$	0	0
$369$	−3.87346e10	−0.108763
$370$	0	0
$371$	−6.38424e11	−1.74955
$372$	0	0
$373$	6.35288e11	1.69934	0.849671	−	0.527313i	$-0.176801\pi$
$373$	6.35288e11	1.69934	0.849671	+	0.527313i	$0.176801\pi$
$374$	0	0
$375$	−2.10681e11	−0.550154
$376$	0	0
$377$	3.56608e10	0.0909191
$378$	0	0
$379$	6.42559e11	1.59969	0.799847	−	0.600204i	$-0.204914\pi$
$379$	6.42559e11	1.59969	0.799847	+	0.600204i	$0.204914\pi$
$380$	0	0
$381$	4.47219e10	0.108732
$382$	0	0
$383$	4.37826e11	1.03970	0.519848	−	0.854259i	$-0.325988\pi$
$383$	4.37826e11	1.03970	0.519848	+	0.854259i	$0.325988\pi$
$384$	0	0
$385$	−1.43068e11	−0.331872
$386$	0	0
$387$	−2.20407e11	−0.499488
$388$	0	0
$389$	7.57236e11	1.67671	0.838356	−	0.545124i	$-0.183517\pi$
$389$	7.57236e11	1.67671	0.838356	+	0.545124i	$0.183517\pi$
$390$	0	0
$391$	2.54119e11	0.549847
$392$	0	0
$393$	3.60583e11	0.762498
$394$	0	0
$395$	−1.00198e11	−0.207095
$396$	0	0
$397$	−5.69444e11	−1.15052	−0.575259	−	0.817971i	$-0.695099\pi$
$397$	−5.69444e11	−1.15052	−0.575259	+	0.817971i	$0.695099\pi$
$398$	0	0
$399$	−2.94171e11	−0.581062
$400$	0	0
$401$	−8.01976e11	−1.54886	−0.774429	−	0.632661i	$-0.781963\pi$
$401$	−8.01976e11	−1.54886	−0.774429	+	0.632661i	$0.781963\pi$
$402$	0	0
$403$	1.46838e11	0.277311
$404$	0	0
$405$	−3.41791e10	−0.0631267
$406$	0	0
$407$	2.84275e11	0.513527
$408$	0	0
$409$	−5.70013e11	−1.00723	−0.503616	−	0.863927i	$-0.667997\pi$
$409$	−5.70013e11	−1.00723	−0.503616	+	0.863927i	$0.667997\pi$
$410$	0	0
$411$	5.30434e11	0.916944
$412$	0	0
$413$	7.50505e11	1.26934
$414$	0	0
$415$	3.77180e11	0.624212
$416$	0	0
$417$	−2.41297e11	−0.390786
$418$	0	0
$419$	4.58353e11	0.726503	0.363251	−	0.931691i	$-0.381667\pi$
$419$	4.58353e11	0.726503	0.363251	+	0.931691i	$0.381667\pi$
$420$	0	0
$421$	−2.52732e11	−0.392095	−0.196047	−	0.980594i	$-0.562811\pi$
$421$	−2.52732e11	−0.392095	−0.196047	+	0.980594i	$0.562811\pi$
$422$	0	0
$423$	−1.38669e11	−0.210595
$424$	0	0
$425$	7.60660e11	1.13094
$426$	0	0
$427$	8.65473e11	1.25988
$428$	0	0
$429$	3.80119e10	0.0541828
$430$	0	0
$431$	1.07248e12	1.49706	0.748531	−	0.663099i	$-0.230759\pi$
$431$	1.07248e12	1.49706	0.748531	+	0.663099i	$0.230759\pi$
$432$	0	0
$433$	2.49809e11	0.341517	0.170759	−	0.985313i	$-0.445378\pi$
$433$	2.49809e11	0.341517	0.170759	+	0.985313i	$0.445378\pi$
$434$	0	0
$435$	−1.49764e11	−0.200543
$436$	0	0
$437$	−2.72925e11	−0.357994
$438$	0	0
$439$	−1.31438e12	−1.68900	−0.844502	−	0.535552i	$-0.820103\pi$
$439$	−1.31438e12	−1.68900	−0.844502	+	0.535552i	$0.820103\pi$
$440$	0	0
$441$	−3.79174e10	−0.0477380
$442$	0	0
$443$	−1.27922e12	−1.57808	−0.789038	−	0.614345i	$-0.789420\pi$
$443$	−1.27922e12	−1.57808	−0.789038	+	0.614345i	$0.789420\pi$
$444$	0	0
$445$	4.49510e11	0.543400
$446$	0	0
$447$	−2.82826e11	−0.335070
$448$	0	0
$449$	−6.95230e11	−0.807273	−0.403636	−	0.914920i	$-0.632254\pi$
$449$	−6.95230e11	−0.807273	−0.403636	+	0.914920i	$0.632254\pi$
$450$	0	0
$451$	−1.80915e11	−0.205911
$452$	0	0
$453$	1.91440e11	0.213595
$454$	0	0
$455$	7.14968e10	0.0782051
$456$	0	0
$457$	−9.55669e11	−1.02491	−0.512454	−	0.858715i	$-0.671263\pi$
$457$	−9.55669e11	−1.02491	−0.512454	+	0.858715i	$0.671263\pi$
$458$	0	0
$459$	3.05624e11	0.321389
$460$	0	0
$461$	9.84273e11	1.01499	0.507494	−	0.861655i	$-0.330572\pi$
$461$	9.84273e11	1.01499	0.507494	+	0.861655i	$0.330572\pi$
$462$	0	0
$463$	−1.75436e12	−1.77420	−0.887102	−	0.461574i	$-0.847285\pi$
$463$	−1.75436e12	−1.77420	−0.887102	+	0.461574i	$0.847285\pi$
$464$	0	0
$465$	−6.16676e11	−0.611672
$466$	0	0
$467$	−1.12622e12	−1.09572	−0.547859	−	0.836571i	$-0.684557\pi$
$467$	−1.12622e12	−1.09572	−0.547859	+	0.836571i	$0.684557\pi$
$468$	0	0
$469$	1.94968e11	0.186074
$470$	0	0
$471$	−9.58900e11	−0.897799
$472$	0	0
$473$	−1.02944e12	−0.945638
$474$	0	0
$475$	−8.16951e11	−0.736334
$476$	0	0
$477$	−7.12364e11	−0.630042
$478$	0	0
$479$	1.36421e12	1.18406	0.592029	−	0.805917i	$-0.298327\pi$
$479$	1.36421e12	1.18406	0.592029	+	0.805917i	$0.298327\pi$
$480$	0	0
$481$	−1.42063e11	−0.121012
$482$	0	0
$483$	2.10459e11	0.175956
$484$	0	0
$485$	1.17090e12	0.960908
$486$	0	0
$487$	−1.05490e12	−0.849825	−0.424913	−	0.905234i	$-0.639695\pi$
$487$	−1.05490e12	−0.849825	−0.424913	+	0.905234i	$0.639695\pi$
$488$	0	0
$489$	−3.94758e11	−0.312206
$490$	0	0
$491$	2.42794e12	1.88526	0.942629	−	0.333843i	$-0.108346\pi$
$491$	2.42794e12	1.88526	0.942629	+	0.333843i	$0.108346\pi$
$492$	0	0
$493$	1.33917e12	1.02100
$494$	0	0
$495$	−1.59638e11	−0.119512
$496$	0	0
$497$	5.46473e10	0.0401758
$498$	0	0
$499$	−1.45453e12	−1.05020	−0.525099	−	0.851041i	$-0.675972\pi$
$499$	−1.45453e12	−1.05020	−0.525099	+	0.851041i	$0.675972\pi$
$500$	0	0
$501$	−1.26131e12	−0.894438
$502$	0	0
$503$	−7.10774e11	−0.495080	−0.247540	−	0.968878i	$-0.579622\pi$
$503$	−7.10774e11	−0.495080	−0.247540	+	0.968878i	$0.579622\pi$
$504$	0	0
$505$	5.11092e11	0.349694
$506$	0	0
$507$	8.39968e11	0.564582
$508$	0	0
$509$	−3.98321e11	−0.263028	−0.131514	−	0.991314i	$-0.541984\pi$
$509$	−3.98321e11	−0.263028	−0.131514	+	0.991314i	$0.541984\pi$
$510$	0	0
$511$	2.06435e12	1.33934
$512$	0	0
$513$	−3.28241e11	−0.209250
$514$	0	0
$515$	−1.71372e12	−1.07351
$516$	0	0
$517$	−6.47673e11	−0.398702
$518$	0	0
$519$	−4.57647e11	−0.276871
$520$	0	0
$521$	−3.44462e10	−0.0204820	−0.0102410	−	0.999948i	$-0.503260\pi$
$521$	−3.44462e10	−0.0204820	−0.0102410	+	0.999948i	$0.503260\pi$
$522$	0	0
$523$	6.91554e11	0.404174	0.202087	−	0.979368i	$-0.435228\pi$
$523$	6.91554e11	0.404174	0.202087	+	0.979368i	$0.435228\pi$
$524$	0	0
$525$	6.29970e11	0.361912
$526$	0	0
$527$	5.51422e12	3.11413
$528$	0	0
$529$	−1.60589e12	−0.891593
$530$	0	0
$531$	8.37425e11	0.457110
$532$	0	0
$533$	9.04103e10	0.0485227
$534$	0	0
$535$	6.78189e11	0.357898
$536$	0	0
$537$	−1.67243e11	−0.0867888
$538$	0	0
$539$	−1.77098e11	−0.0903783
$540$	0	0
$541$	8.87587e11	0.445475	0.222737	−	0.974878i	$-0.428501\pi$
$541$	8.87587e11	0.445475	0.222737	+	0.974878i	$0.428501\pi$
$542$	0	0
$543$	−6.74902e11	−0.333151
$544$	0	0
$545$	5.29766e11	0.257217
$546$	0	0
$547$	−1.30771e11	−0.0624551	−0.0312276	−	0.999512i	$-0.509942\pi$
$547$	−1.30771e11	−0.0624551	−0.0312276	+	0.999512i	$0.509942\pi$
$548$	0	0
$549$	9.65708e11	0.453702
$550$	0	0
$551$	−1.43827e12	−0.664751
$552$	0	0
$553$	7.42017e11	0.337404
$554$	0	0
$555$	5.96620e11	0.266919
$556$	0	0
$557$	−7.24083e11	−0.318742	−0.159371	−	0.987219i	$-0.550947\pi$
$557$	−7.24083e11	−0.318742	−0.159371	+	0.987219i	$0.550947\pi$
$558$	0	0
$559$	5.14450e11	0.222838
$560$	0	0
$561$	1.42746e12	0.608458
$562$	0	0
$563$	−2.25839e12	−0.947351	−0.473675	−	0.880700i	$-0.657073\pi$
$563$	−2.25839e12	−0.947351	−0.473675	+	0.880700i	$0.657073\pi$
$564$	0	0
$565$	1.97215e12	0.814183
$566$	0	0
$567$	2.53115e11	0.102847
$568$	0	0
$569$	2.70124e12	1.08033	0.540167	−	0.841558i	$-0.318361\pi$
$569$	2.70124e12	1.08033	0.540167	+	0.841558i	$0.318361\pi$
$570$	0	0
$571$	2.97612e12	1.17162	0.585812	−	0.810447i	$-0.300776\pi$
$571$	2.97612e12	1.17162	0.585812	+	0.810447i	$0.300776\pi$
$572$	0	0
$573$	8.62261e11	0.334151
$574$	0	0
$575$	5.84470e11	0.222975
$576$	0	0
$577$	2.18579e12	0.820952	0.410476	−	0.911871i	$-0.365362\pi$
$577$	2.18579e12	0.820952	0.410476	+	0.911871i	$0.365362\pi$
$578$	0	0
$579$	−8.16894e11	−0.302073
$580$	0	0
$581$	−2.79322e12	−1.01698
$582$	0	0
$583$	−3.32719e12	−1.19280
$584$	0	0
$585$	7.97773e10	0.0281629
$586$	0	0
$587$	−3.99972e12	−1.39046	−0.695229	−	0.718788i	$-0.744697\pi$
$587$	−3.99972e12	−1.39046	−0.695229	+	0.718788i	$0.744697\pi$
$588$	0	0
$589$	−5.92229e12	−2.02755
$590$	0	0
$591$	−2.21548e11	−0.0747007
$592$	0	0
$593$	5.59604e12	1.85838	0.929190	−	0.369602i	$-0.120506\pi$
$593$	5.59604e12	1.85838	0.929190	+	0.369602i	$0.120506\pi$
$594$	0	0
$595$	2.68492e12	0.878222
$596$	0	0
$597$	1.32447e12	0.426734
$598$	0	0
$599$	−1.47476e12	−0.468058	−0.234029	−	0.972230i	$-0.575191\pi$
$599$	−1.47476e12	−0.468058	−0.234029	+	0.972230i	$0.575191\pi$
$600$	0	0
$601$	6.94109e10	0.0217016	0.0108508	−	0.999941i	$-0.496546\pi$
$601$	6.94109e10	0.0217016	0.0108508	+	0.999941i	$0.496546\pi$
$602$	0	0
$603$	2.17548e11	0.0670081
$604$	0	0
$605$	1.12660e12	0.341878
$606$	0	0
$607$	−4.38348e11	−0.131060	−0.0655300	−	0.997851i	$-0.520874\pi$
$607$	−4.38348e11	−0.131060	−0.0655300	+	0.997851i	$0.520874\pi$
$608$	0	0
$609$	1.10909e12	0.326729
$610$	0	0
$611$	3.23668e11	0.0939537
$612$	0	0
$613$	−2.40882e12	−0.689021	−0.344511	−	0.938782i	$-0.611955\pi$
$613$	−2.40882e12	−0.689021	−0.344511	+	0.938782i	$0.611955\pi$
$614$	0	0
$615$	−3.79695e11	−0.107028
$616$	0	0
$617$	−6.31670e12	−1.75472	−0.877358	−	0.479836i	$-0.840696\pi$
$617$	−6.31670e12	−1.75472	−0.877358	+	0.479836i	$0.840696\pi$
$618$	0	0
$619$	6.92000e12	1.89451	0.947257	−	0.320474i	$-0.103842\pi$
$619$	6.92000e12	1.89451	0.947257	+	0.320474i	$0.103842\pi$
$620$	0	0
$621$	2.34833e11	0.0633647
$622$	0	0
$623$	−3.32887e12	−0.885320
$624$	0	0
$625$	5.18186e11	0.135839
$626$	0	0
$627$	−1.53309e12	−0.396155
$628$	0	0
$629$	−5.33489e12	−1.35893
$630$	0	0
$631$	2.13258e12	0.535518	0.267759	−	0.963486i	$-0.413717\pi$
$631$	2.13258e12	0.535518	0.267759	+	0.963486i	$0.413717\pi$
$632$	0	0
$633$	−2.62268e10	−0.00649276
$634$	0	0
$635$	4.38385e11	0.106998
$636$	0	0
$637$	8.85028e10	0.0212975
$638$	0	0
$639$	6.09763e10	0.0144680
$640$	0	0
$641$	−2.01044e12	−0.470359	−0.235180	−	0.971952i	$-0.575568\pi$
$641$	−2.01044e12	−0.470359	−0.235180	+	0.971952i	$0.575568\pi$
$642$	0	0
$643$	−6.81806e12	−1.57294	−0.786468	−	0.617630i	$-0.788093\pi$
$643$	−6.81806e12	−1.57294	−0.786468	+	0.617630i	$0.788093\pi$
$644$	0	0
$645$	−2.16053e12	−0.491520
$646$	0	0
$647$	−2.45838e12	−0.551543	−0.275771	−	0.961223i	$-0.588933\pi$
$647$	−2.45838e12	−0.551543	−0.275771	+	0.961223i	$0.588933\pi$
$648$	0	0
$649$	3.91130e12	0.865407
$650$	0	0
$651$	4.56682e12	0.996550
$652$	0	0
$653$	−6.69587e12	−1.44111	−0.720556	−	0.693396i	$-0.756114\pi$
$653$	−6.69587e12	−1.44111	−0.720556	+	0.693396i	$0.756114\pi$
$654$	0	0
$655$	3.53460e12	0.750334
$656$	0	0
$657$	2.30344e12	0.482316
$658$	0	0
$659$	−6.82323e12	−1.40931	−0.704654	−	0.709551i	$-0.748898\pi$
$659$	−6.82323e12	−1.40931	−0.704654	+	0.709551i	$0.748898\pi$
$660$	0	0
$661$	−4.68862e12	−0.955298	−0.477649	−	0.878551i	$-0.658511\pi$
$661$	−4.68862e12	−0.955298	−0.477649	+	0.878551i	$0.658511\pi$
$662$	0	0
$663$	−7.13356e11	−0.143382
$664$	0	0
$665$	−2.88361e12	−0.571793
$666$	0	0
$667$	1.02898e12	0.201299
$668$	0	0
$669$	−1.44126e12	−0.278179
$670$	0	0
$671$	4.51047e12	0.858954
$672$	0	0
$673$	−1.71841e12	−0.322892	−0.161446	−	0.986882i	$-0.551616\pi$
$673$	−1.71841e12	−0.322892	−0.161446	+	0.986882i	$0.551616\pi$
$674$	0	0
$675$	7.02931e11	0.130330
$676$	0	0
$677$	1.99328e12	0.364687	0.182343	−	0.983235i	$-0.441632\pi$
$677$	1.99328e12	0.364687	0.182343	+	0.983235i	$0.441632\pi$
$678$	0	0
$679$	−8.67114e12	−1.56553
$680$	0	0
$681$	1.75532e12	0.312748
$682$	0	0
$683$	4.95462e12	0.871198	0.435599	−	0.900141i	$-0.356537\pi$
$683$	4.95462e12	0.871198	0.435599	+	0.900141i	$0.356537\pi$
$684$	0	0
$685$	5.19956e12	0.902317
$686$	0	0
$687$	5.96895e12	1.02233
$688$	0	0
$689$	1.66273e12	0.281083
$690$	0	0
$691$	−6.34842e12	−1.05929	−0.529645	−	0.848220i	$-0.677675\pi$
$691$	−6.34842e12	−1.05929	−0.529645	+	0.848220i	$0.677675\pi$
$692$	0	0
$693$	1.18221e12	0.194712
$694$	0	0
$695$	−2.36530e12	−0.384552
$696$	0	0
$697$	3.39517e12	0.544897
$698$	0	0
$699$	−3.37980e12	−0.535481
$700$	0	0
$701$	6.38577e12	0.998808	0.499404	−	0.866369i	$-0.333552\pi$
$701$	6.38577e12	0.998808	0.499404	+	0.866369i	$0.333552\pi$
$702$	0	0
$703$	5.72968e12	0.884773
$704$	0	0
$705$	−1.35930e12	−0.207236
$706$	0	0
$707$	−3.78491e12	−0.569729
$708$	0	0
$709$	4.51248e12	0.670668	0.335334	−	0.942099i	$-0.391151\pi$
$709$	4.51248e12	0.670668	0.335334	+	0.942099i	$0.391151\pi$
$710$	0	0
$711$	8.27954e11	0.121505
$712$	0	0
$713$	4.23697e12	0.613978
$714$	0	0
$715$	3.72610e11	0.0533184
$716$	0	0
$717$	−5.70056e12	−0.805529
$718$	0	0
$719$	−2.66483e12	−0.371869	−0.185934	−	0.982562i	$-0.559531\pi$
$719$	−2.66483e12	−0.371869	−0.185934	+	0.982562i	$0.559531\pi$
$720$	0	0
$721$	1.26910e13	1.74899
$722$	0	0
$723$	1.78073e12	0.242368
$724$	0	0
$725$	3.08007e12	0.414037
$726$	0	0
$727$	−9.74251e12	−1.29350	−0.646750	−	0.762702i	$-0.723872\pi$
$727$	−9.74251e12	−1.29350	−0.646750	+	0.762702i	$0.723872\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	1.93191e13	2.50241
$732$	0	0
$733$	6.63328e12	0.848712	0.424356	−	0.905495i	$-0.360500\pi$
$733$	6.63328e12	0.848712	0.424356	+	0.905495i	$0.360500\pi$
$734$	0	0
$735$	−3.71684e11	−0.0469765
$736$	0	0
$737$	1.01609e12	0.126861
$738$	0	0
$739$	1.74099e12	0.214732	0.107366	−	0.994220i	$-0.465758\pi$
$739$	1.74099e12	0.214732	0.107366	+	0.994220i	$0.465758\pi$
$740$	0	0
$741$	7.66147e11	0.0933534
$742$	0	0
$743$	1.32134e13	1.59061	0.795307	−	0.606207i	$-0.207310\pi$
$743$	1.32134e13	1.59061	0.795307	+	0.606207i	$0.207310\pi$
$744$	0	0
$745$	−2.77239e12	−0.329725
$746$	0	0
$747$	−3.11672e12	−0.366231
$748$	0	0
$749$	−5.02236e12	−0.583095
$750$	0	0
$751$	5.17129e12	0.593224	0.296612	−	0.954998i	$-0.404143\pi$
$751$	5.17129e12	0.593224	0.296612	+	0.954998i	$0.404143\pi$
$752$	0	0
$753$	−4.92008e12	−0.557692
$754$	0	0
$755$	1.87658e12	0.210187
$756$	0	0
$757$	−1.00161e13	−1.10858	−0.554291	−	0.832323i	$-0.687010\pi$
$757$	−1.00161e13	−1.10858	−0.554291	+	0.832323i	$0.687010\pi$
$758$	0	0
$759$	1.09682e12	0.119963
$760$	0	0
$761$	8.63252e12	0.933054	0.466527	−	0.884507i	$-0.345505\pi$
$761$	8.63252e12	0.933054	0.466527	+	0.884507i	$0.345505\pi$
$762$	0	0
$763$	−3.92320e12	−0.419064
$764$	0	0
$765$	2.99587e12	0.316262
$766$	0	0
$767$	−1.95463e12	−0.203932
$768$	0	0
$769$	−1.58592e13	−1.63536	−0.817679	−	0.575674i	$-0.804740\pi$
$769$	−1.58592e13	−1.63536	−0.817679	+	0.575674i	$0.804740\pi$
$770$	0	0
$771$	−2.42360e12	−0.247011
$772$	0	0
$773$	−2.96693e12	−0.298882	−0.149441	−	0.988771i	$-0.547747\pi$
$773$	−2.96693e12	−0.298882	−0.149441	+	0.988771i	$0.547747\pi$
$774$	0	0
$775$	1.26826e13	1.26285
$776$	0	0
$777$	−4.41830e12	−0.434871
$778$	0	0
$779$	−3.64643e12	−0.354772
$780$	0	0
$781$	2.84798e11	0.0273909
$782$	0	0
$783$	1.23754e12	0.117660
$784$	0	0
$785$	−9.39958e12	−0.883477
$786$	0	0
$787$	−5.91607e12	−0.549727	−0.274863	−	0.961483i	$-0.588633\pi$
$787$	−5.91607e12	−0.549727	−0.274863	+	0.961483i	$0.588633\pi$
$788$	0	0
$789$	4.07910e12	0.374729
$790$	0	0
$791$	−1.46048e13	−1.32649
$792$	0	0
$793$	−2.25406e12	−0.202412
$794$	0	0
$795$	−6.98293e12	−0.619991
$796$	0	0
$797$	1.02288e13	0.897970	0.448985	−	0.893539i	$-0.351786\pi$
$797$	1.02288e13	0.897970	0.448985	+	0.893539i	$0.351786\pi$
$798$	0	0
$799$	1.21547e13	1.05507
$800$	0	0
$801$	−3.71441e12	−0.318818
$802$	0	0
$803$	1.07585e13	0.913128
$804$	0	0
$805$	2.06301e12	0.173149
$806$	0	0
$807$	−7.27532e12	−0.603839
$808$	0	0
$809$	−2.67249e12	−0.219355	−0.109678	−	0.993967i	$-0.534982\pi$
$809$	−2.67249e12	−0.219355	−0.109678	+	0.993967i	$0.534982\pi$
$810$	0	0
$811$	−4.47993e12	−0.363645	−0.181822	−	0.983331i	$-0.558200\pi$
$811$	−4.47993e12	−0.363645	−0.181822	+	0.983331i	$0.558200\pi$
$812$	0	0
$813$	−4.65650e12	−0.373811
$814$	0	0
$815$	−3.86961e12	−0.307226
$816$	0	0
$817$	−2.07488e13	−1.62927
$818$	0	0
$819$	−5.90794e11	−0.0458837
$820$	0	0
$821$	2.23289e13	1.71523	0.857615	−	0.514292i	$-0.171945\pi$
$821$	2.23289e13	1.71523	0.857615	+	0.514292i	$0.171945\pi$
$822$	0	0
$823$	−3.16277e12	−0.240308	−0.120154	−	0.992755i	$-0.538339\pi$
$823$	−3.16277e12	−0.240308	−0.120154	+	0.992755i	$0.538339\pi$
$824$	0	0
$825$	3.28313e12	0.246743
$826$	0	0
$827$	1.39072e13	1.03387	0.516934	−	0.856025i	$-0.327073\pi$
$827$	1.39072e13	1.03387	0.516934	+	0.856025i	$0.327073\pi$
$828$	0	0
$829$	2.46033e12	0.180925	0.0904625	−	0.995900i	$-0.471165\pi$
$829$	2.46033e12	0.180925	0.0904625	+	0.995900i	$0.471165\pi$
$830$	0	0
$831$	−3.81104e12	−0.277229
$832$	0	0
$833$	3.32354e12	0.239166
$834$	0	0
$835$	−1.23639e13	−0.880170
$836$	0	0
$837$	5.09573e12	0.358874
$838$	0	0
$839$	−1.37792e13	−0.960054	−0.480027	−	0.877254i	$-0.659373\pi$
$839$	−1.37792e13	−0.960054	−0.480027	+	0.877254i	$0.659373\pi$
$840$	0	0
$841$	−9.08457e12	−0.626214
$842$	0	0
$843$	−4.69575e12	−0.320244
$844$	0	0
$845$	8.23376e12	0.555576
$846$	0	0
$847$	−8.34309e12	−0.556995
$848$	0	0
$849$	2.58677e12	0.170873
$850$	0	0
$851$	−4.09918e12	−0.267925
$852$	0	0
$853$	−3.65300e12	−0.236254	−0.118127	−	0.992998i	$-0.537689\pi$
$853$	−3.65300e12	−0.236254	−0.118127	+	0.992998i	$0.537689\pi$
$854$	0	0
$855$	−3.21758e12	−0.205912
$856$	0	0
$857$	−8.68822e12	−0.550196	−0.275098	−	0.961416i	$-0.588710\pi$
$857$	−8.68822e12	−0.550196	−0.275098	+	0.961416i	$0.588710\pi$
$858$	0	0
$859$	−9.89500e12	−0.620079	−0.310039	−	0.950724i	$-0.600342\pi$
$859$	−9.89500e12	−0.620079	−0.310039	+	0.950724i	$0.600342\pi$
$860$	0	0
$861$	2.81185e12	0.174372
$862$	0	0
$863$	−8.67018e12	−0.532083	−0.266042	−	0.963962i	$-0.585716\pi$
$863$	−8.67018e12	−0.532083	−0.266042	+	0.963962i	$0.585716\pi$
$864$	0	0
$865$	−4.48607e12	−0.272454
$866$	0	0
$867$	−1.71830e13	−1.03279
$868$	0	0
$869$	3.86707e12	0.230034
$870$	0	0
$871$	−5.07778e11	−0.0298946
$872$	0	0
$873$	−9.67540e12	−0.563774
$874$	0	0
$875$	1.52939e13	0.882025
$876$	0	0
$877$	−1.91428e13	−1.09271	−0.546357	−	0.837552i	$-0.683986\pi$
$877$	−1.91428e13	−1.09271	−0.546357	+	0.837552i	$0.683986\pi$
$878$	0	0
$879$	1.19230e13	0.673650
$880$	0	0
$881$	1.11285e13	0.622363	0.311181	−	0.950351i	$-0.399275\pi$
$881$	1.11285e13	0.622363	0.311181	+	0.950351i	$0.399275\pi$
$882$	0	0
$883$	2.79241e13	1.54581	0.772906	−	0.634521i	$-0.218803\pi$
$883$	2.79241e13	1.54581	0.772906	+	0.634521i	$0.218803\pi$
$884$	0	0
$885$	8.20884e12	0.449818
$886$	0	0
$887$	−3.11172e13	−1.68789	−0.843946	−	0.536429i	$-0.819773\pi$
$887$	−3.11172e13	−1.68789	−0.843946	+	0.536429i	$0.819773\pi$
$888$	0	0
$889$	−3.24648e12	−0.174323
$890$	0	0
$891$	1.31912e12	0.0701190
$892$	0	0
$893$	−1.30542e13	−0.686938
$894$	0	0
$895$	−1.63940e12	−0.0854044
$896$	0	0
$897$	−5.48123e11	−0.0282691
$898$	0	0
$899$	2.23282e13	1.14008
$900$	0	0
$901$	6.24403e13	3.15648
$902$	0	0
$903$	1.59999e13	0.800796
$904$	0	0
$905$	−6.61571e12	−0.327837
$906$	0	0
$907$	−4.59824e11	−0.0225610	−0.0112805	−	0.999936i	$-0.503591\pi$
$907$	−4.59824e11	−0.0225610	−0.0112805	+	0.999936i	$0.503591\pi$
$908$	0	0
$909$	−4.22327e12	−0.205169
$910$	0	0
$911$	2.40747e13	1.15805	0.579026	−	0.815309i	$-0.303433\pi$
$911$	2.40747e13	1.15805	0.579026	+	0.815309i	$0.303433\pi$
$912$	0	0
$913$	−1.45571e13	−0.693354
$914$	0	0
$915$	9.46633e12	0.446464
$916$	0	0
$917$	−2.61756e13	−1.22246
$918$	0	0
$919$	2.66234e13	1.23124	0.615621	−	0.788042i	$-0.288905\pi$
$919$	2.66234e13	1.23124	0.615621	+	0.788042i	$0.288905\pi$
$920$	0	0
$921$	6.72382e12	0.307927
$922$	0	0
$923$	−1.42325e11	−0.00645464
$924$	0	0
$925$	−1.22702e13	−0.551077
$926$	0	0
$927$	1.41609e13	0.629841
$928$	0	0
$929$	−8.26068e12	−0.363869	−0.181935	−	0.983311i	$-0.558236\pi$
$929$	−8.26068e12	−0.363869	−0.181935	+	0.983311i	$0.558236\pi$
$930$	0	0
$931$	−3.56949e12	−0.155716
$932$	0	0
$933$	−2.07384e13	−0.896001
$934$	0	0
$935$	1.39926e13	0.598751
$936$	0	0
$937$	3.89553e13	1.65097	0.825484	−	0.564425i	$-0.190902\pi$
$937$	3.89553e13	1.65097	0.825484	+	0.564425i	$0.190902\pi$
$938$	0	0
$939$	−8.05555e12	−0.338143
$940$	0	0
$941$	2.68466e13	1.11619	0.558093	−	0.829779i	$-0.311533\pi$
$941$	2.68466e13	1.11619	0.558093	+	0.829779i	$0.311533\pi$
$942$	0	0
$943$	2.60876e12	0.107431
$944$	0	0
$945$	2.48115e12	0.101207
$946$	0	0
$947$	−9.07791e12	−0.366784	−0.183392	−	0.983040i	$-0.558708\pi$
$947$	−9.07791e12	−0.366784	−0.183392	+	0.983040i	$0.558708\pi$
$948$	0	0
$949$	−5.37644e12	−0.215178
$950$	0	0
$951$	2.23702e13	0.886866
$952$	0	0
$953$	−4.15532e13	−1.63187	−0.815937	−	0.578141i	$-0.803778\pi$
$953$	−4.15532e13	−1.63187	−0.815937	+	0.578141i	$0.803778\pi$
$954$	0	0
$955$	8.45229e12	0.328821
$956$	0	0
$957$	5.78007e12	0.222756
$958$	0	0
$959$	−3.85056e13	−1.47008
$960$	0	0
$961$	6.54999e13	2.47734
$962$	0	0
$963$	−5.60403e12	−0.209982
$964$	0	0
$965$	−8.00758e12	−0.297254
$966$	0	0
$967$	−1.57423e13	−0.578960	−0.289480	−	0.957184i	$-0.593482\pi$
$967$	−1.57423e13	−0.578960	−0.289480	+	0.957184i	$0.593482\pi$
$968$	0	0
$969$	2.87711e13	1.04833
$970$	0	0
$971$	−3.60515e12	−0.130148	−0.0650740	−	0.997880i	$-0.520728\pi$
$971$	−3.60515e12	−0.130148	−0.0650740	+	0.997880i	$0.520728\pi$
$972$	0	0
$973$	1.75164e13	0.626521
$974$	0	0
$975$	−1.64071e12	−0.0581448
$976$	0	0
$977$	1.16364e13	0.408595	0.204297	−	0.978909i	$-0.434509\pi$
$977$	1.16364e13	0.408595	0.204297	+	0.978909i	$0.434509\pi$
$978$	0	0
$979$	−1.73486e13	−0.603591
$980$	0	0
$981$	−4.37757e12	−0.150912
$982$	0	0
$983$	4.11223e13	1.40471	0.702355	−	0.711826i	$-0.252132\pi$
$983$	4.11223e13	1.40471	0.702355	+	0.711826i	$0.252132\pi$
$984$	0	0
$985$	−2.17172e12	−0.0735090
$986$	0	0
$987$	1.00664e13	0.337634
$988$	0	0
$989$	1.48443e13	0.493373
$990$	0	0
$991$	1.01984e13	0.335891	0.167946	−	0.985796i	$-0.446287\pi$
$991$	1.01984e13	0.335891	0.167946	+	0.985796i	$0.446287\pi$
$992$	0	0
$993$	−1.26910e13	−0.414215
$994$	0	0
$995$	1.29831e13	0.419926
$996$	0	0
$997$	−1.72567e13	−0.553132	−0.276566	−	0.960995i	$-0.589196\pi$
$997$	−1.72567e13	−0.553132	−0.276566	+	0.960995i	$0.589196\pi$
$998$	0	0
$999$	−4.93001e12	−0.156604

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.10.a.b.1.1		1
3.2	odd	2		144.10.a.k.1.1		1
4.3	odd	2		24.10.a.b.1.1	✓	1
8.3	odd	2		192.10.a.e.1.1		1
8.5	even	2		192.10.a.l.1.1		1
12.11	even	2		72.10.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.10.a.b.1.1	✓	1	4.3	odd	2
48.10.a.b.1.1		1	1.1	even	1	trivial
72.10.a.d.1.1		1	12.11	even	2
144.10.a.k.1.1		1	3.2	odd	2
192.10.a.e.1.1		1	8.3	odd	2
192.10.a.l.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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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