LMFDB - Embedded newform 48.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,8,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, 8, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
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:	$14.9944812232$
Analytic rank:	$1$
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)

$f(q)$

$=$

$q-27.0000 q^{3} -114.000 q^{5} +1576.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} -114.000 q^{5} +1576.00 q^{7} +729.000 q^{9} -7332.00 q^{11} -3802.00 q^{13} +3078.00 q^{15} -6606.00 q^{17} -24860.0 q^{19} -42552.0 q^{21} -41448.0 q^{23} -65129.0 q^{25} -19683.0 q^{27} -41610.0 q^{29} -33152.0 q^{31} +197964. q^{33} -179664. q^{35} -36466.0 q^{37} +102654. q^{39} -639078. q^{41} +156412. q^{43} -83106.0 q^{45} +433776. q^{47} +1.66023e6 q^{49} +178362. q^{51} +786078. q^{53} +835848. q^{55} +671220. q^{57} -745140. q^{59} -1.66062e6 q^{61} +1.14890e6 q^{63} +433428. q^{65} +3.29084e6 q^{67} +1.11910e6 q^{69} -5.71615e6 q^{71} +2.65990e6 q^{73} +1.75848e6 q^{75} -1.15552e7 q^{77} -3.80744e6 q^{79} +531441. q^{81} -2.22947e6 q^{83} +753084. q^{85} +1.12347e6 q^{87} +5.99121e6 q^{89} -5.99195e6 q^{91} +895104. q^{93} +2.83404e6 q^{95} -4.06013e6 q^{97} -5.34503e6 q^{99} +O(q^{100})$

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−114.000	−0.407859	−0.203929	−	0.978986i	$-0.565371\pi$
$5$	−114.000	−0.407859	−0.203929	+	0.978986i	$0.565371\pi$
$6$	0	0
$7$	1576.00	1.73665	0.868327	−	0.495993i	$-0.165196\pi$
$7$	1576.00	1.73665	0.868327	+	0.495993i	$0.165196\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−7332.00	−1.66092	−0.830459	−	0.557080i	$-0.811922\pi$
$11$	−7332.00	−1.66092	−0.830459	+	0.557080i	$0.811922\pi$
$12$	0	0
$13$	−3802.00	−0.479966	−0.239983	−	0.970777i	$-0.577142\pi$
$13$	−3802.00	−0.479966	−0.239983	+	0.970777i	$0.577142\pi$
$14$	0	0
$15$	3078.00	0.235477
$16$	0	0
$17$	−6606.00	−0.326112	−0.163056	−	0.986617i	$-0.552135\pi$
$17$	−6606.00	−0.326112	−0.163056	+	0.986617i	$0.552135\pi$
$18$	0	0
$19$	−24860.0	−0.831502	−0.415751	−	0.909478i	$-0.636481\pi$
$19$	−24860.0	−0.831502	−0.415751	+	0.909478i	$0.636481\pi$
$20$	0	0
$21$	−42552.0	−1.00266
$22$	0	0
$23$	−41448.0	−0.710323	−0.355162	−	0.934805i	$-0.615574\pi$
$23$	−41448.0	−0.710323	−0.355162	+	0.934805i	$0.615574\pi$
$24$	0	0
$25$	−65129.0	−0.833651
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−41610.0	−0.316814	−0.158407	−	0.987374i	$-0.550636\pi$
$29$	−41610.0	−0.316814	−0.158407	+	0.987374i	$0.550636\pi$
$30$	0	0
$31$	−33152.0	−0.199868	−0.0999341	−	0.994994i	$-0.531863\pi$
$31$	−33152.0	−0.199868	−0.0999341	+	0.994994i	$0.531863\pi$
$32$	0	0
$33$	197964.	0.958931
$34$	0	0
$35$	−179664.	−0.708309
$36$	0	0
$37$	−36466.0	−0.118354	−0.0591769	−	0.998248i	$-0.518848\pi$
$37$	−36466.0	−0.118354	−0.0591769	+	0.998248i	$0.518848\pi$
$38$	0	0
$39$	102654.	0.277108
$40$	0	0
$41$	−639078.	−1.44814	−0.724070	−	0.689727i	$-0.757731\pi$
$41$	−639078.	−1.44814	−0.724070	+	0.689727i	$0.757731\pi$
$42$	0	0
$43$	156412.	0.300006	0.150003	−	0.988686i	$-0.452072\pi$
$43$	156412.	0.300006	0.150003	+	0.988686i	$0.452072\pi$
$44$	0	0
$45$	−83106.0	−0.135953
$46$	0	0
$47$	433776.	0.609429	0.304714	−	0.952444i	$-0.401439\pi$
$47$	433776.	0.609429	0.304714	+	0.952444i	$0.401439\pi$
$48$	0	0
$49$	1.66023e6	2.01596
$50$	0	0
$51$	178362.	0.188281
$52$	0	0
$53$	786078.	0.725271	0.362635	−	0.931931i	$-0.381877\pi$
$53$	786078.	0.725271	0.362635	+	0.931931i	$0.381877\pi$
$54$	0	0
$55$	835848.	0.677420
$56$	0	0
$57$	671220.	0.480068
$58$	0	0
$59$	−745140.	−0.472341	−0.236171	−	0.971712i	$-0.575892\pi$
$59$	−745140.	−0.472341	−0.236171	+	0.971712i	$0.575892\pi$
$60$	0	0
$61$	−1.66062e6	−0.936732	−0.468366	−	0.883535i	$-0.655157\pi$
$61$	−1.66062e6	−0.936732	−0.468366	+	0.883535i	$0.655157\pi$
$62$	0	0
$63$	1.14890e6	0.578884
$64$	0	0
$65$	433428.	0.195758
$66$	0	0
$67$	3.29084e6	1.33673	0.668366	−	0.743832i	$-0.266994\pi$
$67$	3.29084e6	1.33673	0.668366	+	0.743832i	$0.266994\pi$
$68$	0	0
$69$	1.11910e6	0.410105
$70$	0	0
$71$	−5.71615e6	−1.89539	−0.947697	−	0.319171i	$-0.896596\pi$
$71$	−5.71615e6	−1.89539	−0.947697	+	0.319171i	$0.896596\pi$
$72$	0	0
$73$	2.65990e6	0.800267	0.400134	−	0.916457i	$-0.368964\pi$
$73$	2.65990e6	0.800267	0.400134	+	0.916457i	$0.368964\pi$
$74$	0	0
$75$	1.75848e6	0.481309
$76$	0	0
$77$	−1.15552e7	−2.88444
$78$	0	0
$79$	−3.80744e6	−0.868837	−0.434418	−	0.900711i	$-0.643046\pi$
$79$	−3.80744e6	−0.868837	−0.434418	+	0.900711i	$0.643046\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−2.22947e6	−0.427984	−0.213992	−	0.976835i	$-0.568647\pi$
$83$	−2.22947e6	−0.427984	−0.213992	+	0.976835i	$0.568647\pi$
$84$	0	0
$85$	753084.	0.133008
$86$	0	0
$87$	1.12347e6	0.182913
$88$	0	0
$89$	5.99121e6	0.900844	0.450422	−	0.892816i	$-0.351274\pi$
$89$	5.99121e6	0.900844	0.450422	+	0.892816i	$0.351274\pi$
$90$	0	0
$91$	−5.99195e6	−0.833534
$92$	0	0
$93$	895104.	0.115394
$94$	0	0
$95$	2.83404e6	0.339136
$96$	0	0
$97$	−4.06013e6	−0.451688	−0.225844	−	0.974163i	$-0.572514\pi$
$97$	−4.06013e6	−0.451688	−0.225844	+	0.974163i	$0.572514\pi$
$98$	0	0
$99$	−5.34503e6	−0.553639
$100$	0	0
$101$	−1.72819e7	−1.66904	−0.834522	−	0.550975i	$-0.814256\pi$
$101$	−1.72819e7	−1.66904	−0.834522	+	0.550975i	$0.814256\pi$
$102$	0	0
$103$	1.43623e7	1.29507	0.647536	−	0.762035i	$-0.275799\pi$
$103$	1.43623e7	1.29507	0.647536	+	0.762035i	$0.275799\pi$
$104$	0	0
$105$	4.85093e6	0.408943
$106$	0	0
$107$	−6.45440e6	−0.509346	−0.254673	−	0.967027i	$-0.581968\pi$
$107$	−6.45440e6	−0.509346	−0.254673	+	0.967027i	$0.581968\pi$
$108$	0	0
$109$	−884410.	−0.0654125	−0.0327063	−	0.999465i	$-0.510413\pi$
$109$	−884410.	−0.0654125	−0.0327063	+	0.999465i	$0.510413\pi$
$110$	0	0
$111$	984582.	0.0683316
$112$	0	0
$113$	1.21325e7	0.790999	0.395499	−	0.918466i	$-0.370572\pi$
$113$	1.21325e7	0.790999	0.395499	+	0.918466i	$0.370572\pi$
$114$	0	0
$115$	4.72507e6	0.289712
$116$	0	0
$117$	−2.77166e6	−0.159989
$118$	0	0
$119$	−1.04111e7	−0.566344
$120$	0	0
$121$	3.42711e7	1.75865
$122$	0	0
$123$	1.72551e7	0.836084
$124$	0	0
$125$	1.63310e7	0.747871
$126$	0	0
$127$	−6.86806e6	−0.297524	−0.148762	−	0.988873i	$-0.547529\pi$
$127$	−6.86806e6	−0.297524	−0.148762	+	0.988873i	$0.547529\pi$
$128$	0	0
$129$	−4.22312e6	−0.173209
$130$	0	0
$131$	3.95208e7	1.53595	0.767973	−	0.640482i	$-0.221265\pi$
$131$	3.95208e7	1.53595	0.767973	+	0.640482i	$0.221265\pi$
$132$	0	0
$133$	−3.91794e7	−1.44403
$134$	0	0
$135$	2.24386e6	0.0784925
$136$	0	0
$137$	1.91741e7	0.637078	0.318539	−	0.947910i	$-0.396808\pi$
$137$	1.91741e7	0.637078	0.318539	+	0.947910i	$0.396808\pi$
$138$	0	0
$139$	−1.32449e7	−0.418309	−0.209154	−	0.977883i	$-0.567071\pi$
$139$	−1.32449e7	−0.418309	−0.209154	+	0.977883i	$0.567071\pi$
$140$	0	0
$141$	−1.17120e7	−0.351854
$142$	0	0
$143$	2.78763e7	0.797184
$144$	0	0
$145$	4.74354e6	0.129215
$146$	0	0
$147$	−4.48263e7	−1.16392
$148$	0	0
$149$	5.73624e7	1.42061	0.710306	−	0.703893i	$-0.248556\pi$
$149$	5.73624e7	1.42061	0.710306	+	0.703893i	$0.248556\pi$
$150$	0	0
$151$	3.10873e7	0.734790	0.367395	−	0.930065i	$-0.380250\pi$
$151$	3.10873e7	0.734790	0.367395	+	0.930065i	$0.380250\pi$
$152$	0	0
$153$	−4.81577e6	−0.108704
$154$	0	0
$155$	3.77933e6	0.0815180
$156$	0	0
$157$	−3.37835e7	−0.696715	−0.348358	−	0.937362i	$-0.613261\pi$
$157$	−3.37835e7	−0.696715	−0.348358	+	0.937362i	$0.613261\pi$
$158$	0	0
$159$	−2.12241e7	−0.418735
$160$	0	0
$161$	−6.53220e7	−1.23359
$162$	0	0
$163$	−6.26659e7	−1.13338	−0.566689	−	0.823932i	$-0.691776\pi$
$163$	−6.26659e7	−1.13338	−0.566689	+	0.823932i	$0.691776\pi$
$164$	0	0
$165$	−2.25679e7	−0.391109
$166$	0	0
$167$	−6.27072e7	−1.04186	−0.520931	−	0.853599i	$-0.674415\pi$
$167$	−6.27072e7	−1.04186	−0.520931	+	0.853599i	$0.674415\pi$
$168$	0	0
$169$	−4.82933e7	−0.769633
$170$	0	0
$171$	−1.81229e7	−0.277167
$172$	0	0
$173$	−2.70521e7	−0.397228	−0.198614	−	0.980078i	$-0.563644\pi$
$173$	−2.70521e7	−0.397228	−0.198614	+	0.980078i	$0.563644\pi$
$174$	0	0
$175$	−1.02643e8	−1.44776
$176$	0	0
$177$	2.01188e7	0.272706
$178$	0	0
$179$	1.34281e8	1.74996	0.874981	−	0.484157i	$-0.160874\pi$
$179$	1.34281e8	1.74996	0.874981	+	0.484157i	$0.160874\pi$
$180$	0	0
$181$	1.14661e8	1.43727	0.718636	−	0.695386i	$-0.244767\pi$
$181$	1.14661e8	1.43727	0.718636	+	0.695386i	$0.244767\pi$
$182$	0	0
$183$	4.48367e7	0.540822
$184$	0	0
$185$	4.15712e6	0.0482716
$186$	0	0
$187$	4.84352e7	0.541646
$188$	0	0
$189$	−3.10204e7	−0.334219
$190$	0	0
$191$	−1.63605e7	−0.169895	−0.0849474	−	0.996385i	$-0.527072\pi$
$191$	−1.63605e7	−0.169895	−0.0849474	+	0.996385i	$0.527072\pi$
$192$	0	0
$193$	−1.54198e8	−1.54394	−0.771968	−	0.635661i	$-0.780728\pi$
$193$	−1.54198e8	−1.54394	−0.771968	+	0.635661i	$0.780728\pi$
$194$	0	0
$195$	−1.17026e7	−0.113021
$196$	0	0
$197$	8.32288e7	0.775607	0.387804	−	0.921742i	$-0.373234\pi$
$197$	8.32288e7	0.775607	0.387804	+	0.921742i	$0.373234\pi$
$198$	0	0
$199$	7.61722e7	0.685190	0.342595	−	0.939483i	$-0.388694\pi$
$199$	7.61722e7	0.685190	0.342595	+	0.939483i	$0.388694\pi$
$200$	0	0
$201$	−8.88526e7	−0.771763
$202$	0	0
$203$	−6.55774e7	−0.550196
$204$	0	0
$205$	7.28549e7	0.590636
$206$	0	0
$207$	−3.02156e7	−0.236774
$208$	0	0
$209$	1.82274e8	1.38106
$210$	0	0
$211$	−3.52446e7	−0.258288	−0.129144	−	0.991626i	$-0.541223\pi$
$211$	−3.52446e7	−0.258288	−0.129144	+	0.991626i	$0.541223\pi$
$212$	0	0
$213$	1.54336e8	1.09431
$214$	0	0
$215$	−1.78310e7	−0.122360
$216$	0	0
$217$	−5.22476e7	−0.347102
$218$	0	0
$219$	−7.18172e7	−0.462034
$220$	0	0
$221$	2.51160e7	0.156523
$222$	0	0
$223$	1.89131e8	1.14208	0.571040	−	0.820922i	$-0.306540\pi$
$223$	1.89131e8	1.14208	0.571040	+	0.820922i	$0.306540\pi$
$224$	0	0
$225$	−4.74790e7	−0.277884
$226$	0	0
$227$	1.76100e8	0.999239	0.499620	−	0.866245i	$-0.333473\pi$
$227$	1.76100e8	0.999239	0.499620	+	0.866245i	$0.333473\pi$
$228$	0	0
$229$	6.50396e7	0.357894	0.178947	−	0.983859i	$-0.442731\pi$
$229$	6.50396e7	0.357894	0.178947	+	0.983859i	$0.442731\pi$
$230$	0	0
$231$	3.11991e8	1.66533
$232$	0	0
$233$	−2.51319e8	−1.30160	−0.650802	−	0.759248i	$-0.725567\pi$
$233$	−2.51319e8	−1.30160	−0.650802	+	0.759248i	$0.725567\pi$
$234$	0	0
$235$	−4.94505e7	−0.248561
$236$	0	0
$237$	1.02801e8	0.501623
$238$	0	0
$239$	−2.13079e8	−1.00960	−0.504799	−	0.863237i	$-0.668434\pi$
$239$	−2.13079e8	−1.00960	−0.504799	+	0.863237i	$0.668434\pi$
$240$	0	0
$241$	2.57284e8	1.18400	0.592001	−	0.805937i	$-0.298338\pi$
$241$	2.57284e8	1.18400	0.592001	+	0.805937i	$0.298338\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−1.89267e8	−0.822229
$246$	0	0
$247$	9.45177e7	0.399093
$248$	0	0
$249$	6.01956e7	0.247097
$250$	0	0
$251$	−1.23058e8	−0.491193	−0.245596	−	0.969372i	$-0.578984\pi$
$251$	−1.23058e8	−0.491193	−0.245596	+	0.969372i	$0.578984\pi$
$252$	0	0
$253$	3.03897e8	1.17979
$254$	0	0
$255$	−2.03333e7	−0.0767921
$256$	0	0
$257$	−4.43334e8	−1.62916	−0.814582	−	0.580048i	$-0.803034\pi$
$257$	−4.43334e8	−1.62916	−0.814582	+	0.580048i	$0.803034\pi$
$258$	0	0
$259$	−5.74704e7	−0.205539
$260$	0	0
$261$	−3.03337e7	−0.105605
$262$	0	0
$263$	−2.98925e8	−1.01325	−0.506625	−	0.862166i	$-0.669107\pi$
$263$	−2.98925e8	−1.01325	−0.506625	+	0.862166i	$0.669107\pi$
$264$	0	0
$265$	−8.96129e7	−0.295808
$266$	0	0
$267$	−1.61763e8	−0.520102
$268$	0	0
$269$	2.08908e8	0.654368	0.327184	−	0.944961i	$-0.393900\pi$
$269$	2.08908e8	0.654368	0.327184	+	0.944961i	$0.393900\pi$
$270$	0	0
$271$	1.12749e7	0.0344129	0.0172064	−	0.999852i	$-0.494523\pi$
$271$	1.12749e7	0.0344129	0.0172064	+	0.999852i	$0.494523\pi$
$272$	0	0
$273$	1.61783e8	0.481241
$274$	0	0
$275$	4.77526e8	1.38463
$276$	0	0
$277$	−6.58964e8	−1.86287	−0.931435	−	0.363907i	$-0.881443\pi$
$277$	−6.58964e8	−1.86287	−0.931435	+	0.363907i	$0.881443\pi$
$278$	0	0
$279$	−2.41678e7	−0.0666227
$280$	0	0
$281$	−1.05123e8	−0.282634	−0.141317	−	0.989964i	$-0.545134\pi$
$281$	−1.05123e8	−0.282634	−0.141317	+	0.989964i	$0.545134\pi$
$282$	0	0
$283$	−3.30161e8	−0.865911	−0.432956	−	0.901415i	$-0.642529\pi$
$283$	−3.30161e8	−0.865911	−0.432956	+	0.901415i	$0.642529\pi$
$284$	0	0
$285$	−7.65191e7	−0.195800
$286$	0	0
$287$	−1.00719e9	−2.51492
$288$	0	0
$289$	−3.66699e8	−0.893651
$290$	0	0
$291$	1.09623e8	0.260782
$292$	0	0
$293$	−8.71002e7	−0.202294	−0.101147	−	0.994871i	$-0.532251\pi$
$293$	−8.71002e7	−0.202294	−0.101147	+	0.994871i	$0.532251\pi$
$294$	0	0
$295$	8.49460e7	0.192649
$296$	0	0
$297$	1.44316e8	0.319644
$298$	0	0
$299$	1.57585e8	0.340931
$300$	0	0
$301$	2.46505e8	0.521007
$302$	0	0
$303$	4.66612e8	0.963623
$304$	0	0
$305$	1.89310e8	0.382054
$306$	0	0
$307$	3.91709e8	0.772644	0.386322	−	0.922364i	$-0.373745\pi$
$307$	3.91709e8	0.772644	0.386322	+	0.922364i	$0.373745\pi$
$308$	0	0
$309$	−3.87782e8	−0.747710
$310$	0	0
$311$	2.04936e8	0.386328	0.193164	−	0.981166i	$-0.438125\pi$
$311$	2.04936e8	0.386328	0.193164	+	0.981166i	$0.438125\pi$
$312$	0	0
$313$	8.77202e8	1.61694	0.808471	−	0.588536i	$-0.200295\pi$
$313$	8.77202e8	1.61694	0.808471	+	0.588536i	$0.200295\pi$
$314$	0	0
$315$	−1.30975e8	−0.236103
$316$	0	0
$317$	−4.40831e8	−0.777256	−0.388628	−	0.921395i	$-0.627051\pi$
$317$	−4.40831e8	−0.777256	−0.388628	+	0.921395i	$0.627051\pi$
$318$	0	0
$319$	3.05085e8	0.526202
$320$	0	0
$321$	1.74269e8	0.294071
$322$	0	0
$323$	1.64225e8	0.271163
$324$	0	0
$325$	2.47620e8	0.400124
$326$	0	0
$327$	2.38791e7	0.0377659
$328$	0	0
$329$	6.83631e8	1.05837
$330$	0	0
$331$	−1.11223e9	−1.68576	−0.842882	−	0.538099i	$-0.819143\pi$
$331$	−1.11223e9	−1.68576	−0.842882	+	0.538099i	$0.819143\pi$
$332$	0	0
$333$	−2.65837e7	−0.0394513
$334$	0	0
$335$	−3.75155e8	−0.545198
$336$	0	0
$337$	2.88198e8	0.410191	0.205096	−	0.978742i	$-0.434249\pi$
$337$	2.88198e8	0.410191	0.205096	+	0.978742i	$0.434249\pi$
$338$	0	0
$339$	−3.27577e8	−0.456683
$340$	0	0
$341$	2.43070e8	0.331965
$342$	0	0
$343$	1.31862e9	1.76438
$344$	0	0
$345$	−1.27577e8	−0.167265
$346$	0	0
$347$	1.10601e9	1.42103	0.710517	−	0.703680i	$-0.248461\pi$
$347$	1.10601e9	1.42103	0.710517	+	0.703680i	$0.248461\pi$
$348$	0	0
$349$	−1.32184e9	−1.66453	−0.832264	−	0.554379i	$-0.812956\pi$
$349$	−1.32184e9	−1.66453	−0.832264	+	0.554379i	$0.812956\pi$
$350$	0	0
$351$	7.48348e7	0.0923695
$352$	0	0
$353$	1.20395e9	1.45679	0.728396	−	0.685157i	$-0.240266\pi$
$353$	1.20395e9	1.45679	0.728396	+	0.685157i	$0.240266\pi$
$354$	0	0
$355$	6.51641e8	0.773053
$356$	0	0
$357$	2.81099e8	0.326979
$358$	0	0
$359$	1.32057e9	1.50637	0.753185	−	0.657809i	$-0.228516\pi$
$359$	1.32057e9	1.50637	0.753185	+	0.657809i	$0.228516\pi$
$360$	0	0
$361$	−2.75852e8	−0.308604
$362$	0	0
$363$	−9.25318e8	−1.01536
$364$	0	0
$365$	−3.03228e8	−0.326396
$366$	0	0
$367$	−1.75107e9	−1.84915	−0.924575	−	0.381000i	$-0.875580\pi$
$367$	−1.75107e9	−1.84915	−0.924575	+	0.381000i	$0.875580\pi$
$368$	0	0
$369$	−4.65888e8	−0.482713
$370$	0	0
$371$	1.23886e9	1.25954
$372$	0	0
$373$	−4.87945e8	−0.486844	−0.243422	−	0.969920i	$-0.578270\pi$
$373$	−4.87945e8	−0.486844	−0.243422	+	0.969920i	$0.578270\pi$
$374$	0	0
$375$	−4.40936e8	−0.431783
$376$	0	0
$377$	1.58201e8	0.152060
$378$	0	0
$379$	−1.11007e9	−1.04740	−0.523700	−	0.851903i	$-0.675449\pi$
$379$	−1.11007e9	−1.04740	−0.523700	+	0.851903i	$0.675449\pi$
$380$	0	0
$381$	1.85438e8	0.171775
$382$	0	0
$383$	−1.86912e9	−1.69997	−0.849983	−	0.526810i	$-0.823388\pi$
$383$	−1.86912e9	−1.69997	−0.849983	+	0.526810i	$0.823388\pi$
$384$	0	0
$385$	1.31730e9	1.17644
$386$	0	0
$387$	1.14024e8	0.100002
$388$	0	0
$389$	−2.73895e8	−0.235918	−0.117959	−	0.993018i	$-0.537635\pi$
$389$	−2.73895e8	−0.235918	−0.117959	+	0.993018i	$0.537635\pi$
$390$	0	0
$391$	2.73805e8	0.231645
$392$	0	0
$393$	−1.06706e9	−0.886779
$394$	0	0
$395$	4.34048e8	0.354363
$396$	0	0
$397$	6.24552e8	0.500958	0.250479	−	0.968122i	$-0.419412\pi$
$397$	6.24552e8	0.500958	0.250479	+	0.968122i	$0.419412\pi$
$398$	0	0
$399$	1.05784e9	0.833712
$400$	0	0
$401$	5.55500e8	0.430208	0.215104	−	0.976591i	$-0.430991\pi$
$401$	5.55500e8	0.430208	0.215104	+	0.976591i	$0.430991\pi$
$402$	0	0
$403$	1.26044e8	0.0959299
$404$	0	0
$405$	−6.05843e7	−0.0453176
$406$	0	0
$407$	2.67369e8	0.196576
$408$	0	0
$409$	−2.15770e9	−1.55941	−0.779704	−	0.626149i	$-0.784630\pi$
$409$	−2.15770e9	−1.55941	−0.779704	+	0.626149i	$0.784630\pi$
$410$	0	0
$411$	−5.17700e8	−0.367817
$412$	0	0
$413$	−1.17434e9	−0.820293
$414$	0	0
$415$	2.54159e8	0.174557
$416$	0	0
$417$	3.57612e8	0.241511
$418$	0	0
$419$	−1.67797e9	−1.11438	−0.557191	−	0.830384i	$-0.688121\pi$
$419$	−1.67797e9	−1.11438	−0.557191	+	0.830384i	$0.688121\pi$
$420$	0	0
$421$	−5.25233e8	−0.343056	−0.171528	−	0.985179i	$-0.554870\pi$
$421$	−5.25233e8	−0.343056	−0.171528	+	0.985179i	$0.554870\pi$
$422$	0	0
$423$	3.16223e8	0.203143
$424$	0	0
$425$	4.30242e8	0.271864
$426$	0	0
$427$	−2.61713e9	−1.62678
$428$	0	0
$429$	−7.52659e8	−0.460254
$430$	0	0
$431$	−1.70593e8	−0.102634	−0.0513169	−	0.998682i	$-0.516342\pi$
$431$	−1.70593e8	−0.102634	−0.0513169	+	0.998682i	$0.516342\pi$
$432$	0	0
$433$	−1.68797e9	−0.999210	−0.499605	−	0.866253i	$-0.666522\pi$
$433$	−1.68797e9	−0.999210	−0.499605	+	0.866253i	$0.666522\pi$
$434$	0	0
$435$	−1.28076e8	−0.0746026
$436$	0	0
$437$	1.03040e9	0.590636
$438$	0	0
$439$	1.17850e9	0.664817	0.332409	−	0.943135i	$-0.392139\pi$
$439$	1.17850e9	0.664817	0.332409	+	0.943135i	$0.392139\pi$
$440$	0	0
$441$	1.21031e9	0.671988
$442$	0	0
$443$	−7.15755e8	−0.391157	−0.195579	−	0.980688i	$-0.562658\pi$
$443$	−7.15755e8	−0.391157	−0.195579	+	0.980688i	$0.562658\pi$
$444$	0	0
$445$	−6.82998e8	−0.367417
$446$	0	0
$447$	−1.54879e9	−0.820191
$448$	0	0
$449$	−1.37358e9	−0.716132	−0.358066	−	0.933696i	$-0.616564\pi$
$449$	−1.37358e9	−0.716132	−0.358066	+	0.933696i	$0.616564\pi$
$450$	0	0
$451$	4.68572e9	2.40524
$452$	0	0
$453$	−8.39357e8	−0.424231
$454$	0	0
$455$	6.83083e8	0.339964
$456$	0	0
$457$	1.84752e9	0.905488	0.452744	−	0.891641i	$-0.350445\pi$
$457$	1.84752e9	0.905488	0.452744	+	0.891641i	$0.350445\pi$
$458$	0	0
$459$	1.30026e8	0.0627604
$460$	0	0
$461$	3.09414e9	1.47091	0.735455	−	0.677573i	$-0.236968\pi$
$461$	3.09414e9	1.47091	0.735455	+	0.677573i	$0.236968\pi$
$462$	0	0
$463$	−3.00451e9	−1.40682	−0.703412	−	0.710782i	$-0.748341\pi$
$463$	−3.00451e9	−1.40682	−0.703412	+	0.710782i	$0.748341\pi$
$464$	0	0
$465$	−1.02042e8	−0.0470645
$466$	0	0
$467$	2.99252e9	1.35965	0.679825	−	0.733374i	$-0.262056\pi$
$467$	2.99252e9	1.35965	0.679825	+	0.733374i	$0.262056\pi$
$468$	0	0
$469$	5.18636e9	2.32144
$470$	0	0
$471$	9.12154e8	0.402249
$472$	0	0
$473$	−1.14681e9	−0.498286
$474$	0	0
$475$	1.61911e9	0.693183
$476$	0	0
$477$	5.73051e8	0.241757
$478$	0	0
$479$	−1.84041e9	−0.765141	−0.382570	−	0.923926i	$-0.624961\pi$
$479$	−1.84041e9	−0.765141	−0.382570	+	0.923926i	$0.624961\pi$
$480$	0	0
$481$	1.38644e8	0.0568058
$482$	0	0
$483$	1.76370e9	0.712211
$484$	0	0
$485$	4.62854e8	0.184225
$486$	0	0
$487$	4.26676e8	0.167397	0.0836983	−	0.996491i	$-0.473327\pi$
$487$	4.26676e8	0.167397	0.0836983	+	0.996491i	$0.473327\pi$
$488$	0	0
$489$	1.69198e9	0.654356
$490$	0	0
$491$	−6.07547e7	−0.0231630	−0.0115815	−	0.999933i	$-0.503687\pi$
$491$	−6.07547e7	−0.0231630	−0.0115815	+	0.999933i	$0.503687\pi$
$492$	0	0
$493$	2.74876e8	0.103317
$494$	0	0
$495$	6.09333e8	0.225807
$496$	0	0
$497$	−9.00866e9	−3.29164
$498$	0	0
$499$	−3.24588e9	−1.16945	−0.584723	−	0.811233i	$-0.698797\pi$
$499$	−3.24588e9	−1.16945	−0.584723	+	0.811233i	$0.698797\pi$
$500$	0	0
$501$	1.69310e9	0.601519
$502$	0	0
$503$	7.44381e8	0.260800	0.130400	−	0.991461i	$-0.458374\pi$
$503$	7.44381e8	0.260800	0.130400	+	0.991461i	$0.458374\pi$
$504$	0	0
$505$	1.97014e9	0.680734
$506$	0	0
$507$	1.30392e9	0.444348
$508$	0	0
$509$	−4.44155e8	−0.149287	−0.0746436	−	0.997210i	$-0.523782\pi$
$509$	−4.44155e8	−0.149287	−0.0746436	+	0.997210i	$0.523782\pi$
$510$	0	0
$511$	4.19200e9	1.38979
$512$	0	0
$513$	4.89319e8	0.160023
$514$	0	0
$515$	−1.63730e9	−0.528207
$516$	0	0
$517$	−3.18045e9	−1.01221
$518$	0	0
$519$	7.30407e8	0.229340
$520$	0	0
$521$	3.04963e9	0.944745	0.472372	−	0.881399i	$-0.343398\pi$
$521$	3.04963e9	0.944745	0.472372	+	0.881399i	$0.343398\pi$
$522$	0	0
$523$	1.40306e9	0.428866	0.214433	−	0.976739i	$-0.431210\pi$
$523$	1.40306e9	0.428866	0.214433	+	0.976739i	$0.431210\pi$
$524$	0	0
$525$	2.77137e9	0.835866
$526$	0	0
$527$	2.19002e8	0.0651795
$528$	0	0
$529$	−1.68689e9	−0.495441
$530$	0	0
$531$	−5.43207e8	−0.157447
$532$	0	0
$533$	2.42977e9	0.695058
$534$	0	0
$535$	7.35802e8	0.207741
$536$	0	0
$537$	−3.62558e9	−1.01034
$538$	0	0
$539$	−1.21728e10	−3.34835
$540$	0	0
$541$	4.21106e9	1.14341	0.571704	−	0.820460i	$-0.306283\pi$
$541$	4.21106e9	1.14341	0.571704	+	0.820460i	$0.306283\pi$
$542$	0	0
$543$	−3.09583e9	−0.829809
$544$	0	0
$545$	1.00823e8	0.0266791
$546$	0	0
$547$	−1.99956e9	−0.522371	−0.261185	−	0.965289i	$-0.584113\pi$
$547$	−1.99956e9	−0.522371	−0.261185	+	0.965289i	$0.584113\pi$
$548$	0	0
$549$	−1.21059e9	−0.312244
$550$	0	0
$551$	1.03442e9	0.263432
$552$	0	0
$553$	−6.00053e9	−1.50887
$554$	0	0
$555$	−1.12242e8	−0.0278696
$556$	0	0
$557$	−3.37403e9	−0.827287	−0.413643	−	0.910439i	$-0.635744\pi$
$557$	−3.37403e9	−0.827287	−0.413643	+	0.910439i	$0.635744\pi$
$558$	0	0
$559$	−5.94678e8	−0.143993
$560$	0	0
$561$	−1.30775e9	−0.312719
$562$	0	0
$563$	5.58021e9	1.31787	0.658933	−	0.752201i	$-0.271008\pi$
$563$	5.58021e9	1.31787	0.658933	+	0.752201i	$0.271008\pi$
$564$	0	0
$565$	−1.38310e9	−0.322616
$566$	0	0
$567$	8.37551e8	0.192961
$568$	0	0
$569$	−8.88310e8	−0.202149	−0.101074	−	0.994879i	$-0.532228\pi$
$569$	−8.88310e8	−0.202149	−0.101074	+	0.994879i	$0.532228\pi$
$570$	0	0
$571$	1.79171e9	0.402755	0.201377	−	0.979514i	$-0.435458\pi$
$571$	1.79171e9	0.402755	0.201377	+	0.979514i	$0.435458\pi$
$572$	0	0
$573$	4.41734e8	0.0980888
$574$	0	0
$575$	2.69947e9	0.592162
$576$	0	0
$577$	3.82103e9	0.828066	0.414033	−	0.910262i	$-0.364120\pi$
$577$	3.82103e9	0.828066	0.414033	+	0.910262i	$0.364120\pi$
$578$	0	0
$579$	4.16336e9	0.891392
$580$	0	0
$581$	−3.51364e9	−0.743260
$582$	0	0
$583$	−5.76352e9	−1.20461
$584$	0	0
$585$	3.15969e8	0.0652528
$586$	0	0
$587$	−4.36219e9	−0.890166	−0.445083	−	0.895489i	$-0.646826\pi$
$587$	−4.36219e9	−0.890166	−0.445083	+	0.895489i	$0.646826\pi$
$588$	0	0
$589$	8.24159e8	0.166191
$590$	0	0
$591$	−2.24718e9	−0.447797
$592$	0	0
$593$	6.38531e9	1.25745	0.628724	−	0.777628i	$-0.283577\pi$
$593$	6.38531e9	1.25745	0.628724	+	0.777628i	$0.283577\pi$
$594$	0	0
$595$	1.18686e9	0.230988
$596$	0	0
$597$	−2.05665e9	−0.395594
$598$	0	0
$599$	−8.04297e8	−0.152905	−0.0764527	−	0.997073i	$-0.524359\pi$
$599$	−8.04297e8	−0.152905	−0.0764527	+	0.997073i	$0.524359\pi$
$600$	0	0
$601$	−4.87162e9	−0.915403	−0.457702	−	0.889106i	$-0.651327\pi$
$601$	−4.87162e9	−0.915403	−0.457702	+	0.889106i	$0.651327\pi$
$602$	0	0
$603$	2.39902e9	0.445577
$604$	0	0
$605$	−3.90690e9	−0.717280
$606$	0	0
$607$	−7.17517e9	−1.30218	−0.651091	−	0.759000i	$-0.725688\pi$
$607$	−7.17517e9	−1.30218	−0.651091	+	0.759000i	$0.725688\pi$
$608$	0	0
$609$	1.77059e9	0.317656
$610$	0	0
$611$	−1.64922e9	−0.292505
$612$	0	0
$613$	3.47891e9	0.610002	0.305001	−	0.952352i	$-0.401343\pi$
$613$	3.47891e9	0.610002	0.305001	+	0.952352i	$0.401343\pi$
$614$	0	0
$615$	−1.96708e9	−0.341004
$616$	0	0
$617$	−2.39378e8	−0.0410286	−0.0205143	−	0.999790i	$-0.506530\pi$
$617$	−2.39378e8	−0.0410286	−0.0205143	+	0.999790i	$0.506530\pi$
$618$	0	0
$619$	5.52959e9	0.937078	0.468539	−	0.883443i	$-0.344781\pi$
$619$	5.52959e9	0.937078	0.468539	+	0.883443i	$0.344781\pi$
$620$	0	0
$621$	8.15821e8	0.136702
$622$	0	0
$623$	9.44215e9	1.56445
$624$	0	0
$625$	3.22647e9	0.528626
$626$	0	0
$627$	−4.92139e9	−0.797354
$628$	0	0
$629$	2.40894e8	0.0385966
$630$	0	0
$631$	6.13683e9	0.972392	0.486196	−	0.873850i	$-0.338384\pi$
$631$	6.13683e9	0.972392	0.486196	+	0.873850i	$0.338384\pi$
$632$	0	0
$633$	9.51603e8	0.149122
$634$	0	0
$635$	7.82959e8	0.121348
$636$	0	0
$637$	−6.31221e9	−0.967594
$638$	0	0
$639$	−4.16707e9	−0.631798
$640$	0	0
$641$	1.07038e10	1.60522	0.802611	−	0.596503i	$-0.203444\pi$
$641$	1.07038e10	1.60522	0.802611	+	0.596503i	$0.203444\pi$
$642$	0	0
$643$	1.39803e9	0.207385	0.103692	−	0.994609i	$-0.466934\pi$
$643$	1.39803e9	0.207385	0.103692	+	0.994609i	$0.466934\pi$
$644$	0	0
$645$	4.81436e8	0.0706447
$646$	0	0
$647$	5.31605e9	0.771656	0.385828	−	0.922571i	$-0.373916\pi$
$647$	5.31605e9	0.771656	0.385828	+	0.922571i	$0.373916\pi$
$648$	0	0
$649$	5.46337e9	0.784520
$650$	0	0
$651$	1.41068e9	0.200399
$652$	0	0
$653$	3.24403e9	0.455921	0.227960	−	0.973670i	$-0.426794\pi$
$653$	3.24403e9	0.455921	0.227960	+	0.973670i	$0.426794\pi$
$654$	0	0
$655$	−4.50537e9	−0.626449
$656$	0	0
$657$	1.93907e9	0.266756
$658$	0	0
$659$	5.16506e9	0.703034	0.351517	−	0.936181i	$-0.385666\pi$
$659$	5.16506e9	0.703034	0.351517	+	0.936181i	$0.385666\pi$
$660$	0	0
$661$	−3.22515e9	−0.434356	−0.217178	−	0.976132i	$-0.569685\pi$
$661$	−3.22515e9	−0.434356	−0.217178	+	0.976132i	$0.569685\pi$
$662$	0	0
$663$	−6.78132e8	−0.0903685
$664$	0	0
$665$	4.46645e9	0.588961
$666$	0	0
$667$	1.72465e9	0.225041
$668$	0	0
$669$	−5.10655e9	−0.659380
$670$	0	0
$671$	1.21757e10	1.55583
$672$	0	0
$673$	−2.00633e9	−0.253718	−0.126859	−	0.991921i	$-0.540490\pi$
$673$	−2.00633e9	−0.253718	−0.126859	+	0.991921i	$0.540490\pi$
$674$	0	0
$675$	1.28193e9	0.160436
$676$	0	0
$677$	1.00211e10	1.24124	0.620619	−	0.784112i	$-0.286881\pi$
$677$	1.00211e10	1.24124	0.620619	+	0.784112i	$0.286881\pi$
$678$	0	0
$679$	−6.39876e9	−0.784425
$680$	0	0
$681$	−4.75471e9	−0.576911
$682$	0	0
$683$	5.84861e9	0.702393	0.351196	−	0.936302i	$-0.385775\pi$
$683$	5.84861e9	0.702393	0.351196	+	0.936302i	$0.385775\pi$
$684$	0	0
$685$	−2.18584e9	−0.259838
$686$	0	0
$687$	−1.75607e9	−0.206630
$688$	0	0
$689$	−2.98867e9	−0.348105
$690$	0	0
$691$	−2.58686e9	−0.298263	−0.149131	−	0.988817i	$-0.547648\pi$
$691$	−2.58686e9	−0.298263	−0.149131	+	0.988817i	$0.547648\pi$
$692$	0	0
$693$	−8.42376e9	−0.961479
$694$	0	0
$695$	1.50992e9	0.170611
$696$	0	0
$697$	4.22175e9	0.472256
$698$	0	0
$699$	6.78560e9	0.751481
$700$	0	0
$701$	−1.74460e9	−0.191286	−0.0956429	−	0.995416i	$-0.530491\pi$
$701$	−1.74460e9	−0.191286	−0.0956429	+	0.995416i	$0.530491\pi$
$702$	0	0
$703$	9.06545e8	0.0984114
$704$	0	0
$705$	1.33516e9	0.143507
$706$	0	0
$707$	−2.72363e10	−2.89855
$708$	0	0
$709$	−1.12051e10	−1.18074	−0.590368	−	0.807134i	$-0.701017\pi$
$709$	−1.12051e10	−1.18074	−0.590368	+	0.807134i	$0.701017\pi$
$710$	0	0
$711$	−2.77562e9	−0.289612
$712$	0	0
$713$	1.37408e9	0.141971
$714$	0	0
$715$	−3.17789e9	−0.325138
$716$	0	0
$717$	5.75314e9	0.582892
$718$	0	0
$719$	9.36568e8	0.0939698	0.0469849	−	0.998896i	$-0.485039\pi$
$719$	9.36568e8	0.0939698	0.0469849	+	0.998896i	$0.485039\pi$
$720$	0	0
$721$	2.26350e10	2.24909
$722$	0	0
$723$	−6.94666e9	−0.683584
$724$	0	0
$725$	2.71002e9	0.264113
$726$	0	0
$727$	−4.20445e9	−0.405825	−0.202913	−	0.979197i	$-0.565041\pi$
$727$	−4.20445e9	−0.405825	−0.202913	+	0.979197i	$0.565041\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.03326e9	−0.0978358
$732$	0	0
$733$	−1.15491e10	−1.08314	−0.541571	−	0.840655i	$-0.682170\pi$
$733$	−1.15491e10	−1.08314	−0.541571	+	0.840655i	$0.682170\pi$
$734$	0	0
$735$	5.11020e9	0.474714
$736$	0	0
$737$	−2.41284e10	−2.22020
$738$	0	0
$739$	1.39655e10	1.27292	0.636460	−	0.771310i	$-0.280398\pi$
$739$	1.39655e10	1.27292	0.636460	+	0.771310i	$0.280398\pi$
$740$	0	0
$741$	−2.55198e9	−0.230416
$742$	0	0
$743$	−1.43832e10	−1.28646	−0.643230	−	0.765673i	$-0.722406\pi$
$743$	−1.43832e10	−1.28646	−0.643230	+	0.765673i	$0.722406\pi$
$744$	0	0
$745$	−6.53932e9	−0.579409
$746$	0	0
$747$	−1.62528e9	−0.142661
$748$	0	0
$749$	−1.01721e10	−0.884557
$750$	0	0
$751$	6.70841e8	0.0577936	0.0288968	−	0.999582i	$-0.490801\pi$
$751$	6.70841e8	0.0577936	0.0288968	+	0.999582i	$0.490801\pi$
$752$	0	0
$753$	3.32257e9	0.283590
$754$	0	0
$755$	−3.54395e9	−0.299691
$756$	0	0
$757$	−1.91569e10	−1.60506	−0.802529	−	0.596613i	$-0.796513\pi$
$757$	−1.91569e10	−1.60506	−0.802529	+	0.596613i	$0.796513\pi$
$758$	0	0
$759$	−8.20521e9	−0.681151
$760$	0	0
$761$	1.79120e10	1.47332	0.736660	−	0.676263i	$-0.236402\pi$
$761$	1.79120e10	1.47332	0.736660	+	0.676263i	$0.236402\pi$
$762$	0	0
$763$	−1.39383e9	−0.113599
$764$	0	0
$765$	5.48998e8	0.0443359
$766$	0	0
$767$	2.83302e9	0.226708
$768$	0	0
$769$	−2.14072e10	−1.69753	−0.848765	−	0.528771i	$-0.822653\pi$
$769$	−2.14072e10	−1.69753	−0.848765	+	0.528771i	$0.822653\pi$
$770$	0	0
$771$	1.19700e10	0.940599
$772$	0	0
$773$	−7.55163e8	−0.0588047	−0.0294024	−	0.999568i	$-0.509360\pi$
$773$	−7.55163e8	−0.0588047	−0.0294024	+	0.999568i	$0.509360\pi$
$774$	0	0
$775$	2.15916e9	0.166620
$776$	0	0
$777$	1.55170e9	0.118668
$778$	0	0
$779$	1.58875e10	1.20413
$780$	0	0
$781$	4.19108e10	3.14809
$782$	0	0
$783$	8.19010e8	0.0609709
$784$	0	0
$785$	3.85132e9	0.284162
$786$	0	0
$787$	−2.04665e10	−1.49669	−0.748347	−	0.663308i	$-0.769152\pi$
$787$	−2.04665e10	−1.49669	−0.748347	+	0.663308i	$0.769152\pi$
$788$	0	0
$789$	8.07097e9	0.585000
$790$	0	0
$791$	1.91208e10	1.37369
$792$	0	0
$793$	6.31367e9	0.449599
$794$	0	0
$795$	2.41955e9	0.170785
$796$	0	0
$797$	−1.03098e10	−0.721348	−0.360674	−	0.932692i	$-0.617453\pi$
$797$	−1.03098e10	−0.721348	−0.360674	+	0.932692i	$0.617453\pi$
$798$	0	0
$799$	−2.86552e9	−0.198742
$800$	0	0
$801$	4.36759e9	0.300281
$802$	0	0
$803$	−1.95024e10	−1.32918
$804$	0	0
$805$	7.44671e9	0.503129
$806$	0	0
$807$	−5.64051e9	−0.377799
$808$	0	0
$809$	−4.16428e8	−0.0276516	−0.0138258	−	0.999904i	$-0.504401\pi$
$809$	−4.16428e8	−0.0276516	−0.0138258	+	0.999904i	$0.504401\pi$
$810$	0	0
$811$	5.82687e9	0.383586	0.191793	−	0.981435i	$-0.438570\pi$
$811$	5.82687e9	0.383586	0.191793	+	0.981435i	$0.438570\pi$
$812$	0	0
$813$	−3.04423e8	−0.0198683
$814$	0	0
$815$	7.14391e9	0.462258
$816$	0	0
$817$	−3.88840e9	−0.249456
$818$	0	0
$819$	−4.36813e9	−0.277845
$820$	0	0
$821$	−2.08333e10	−1.31388	−0.656941	−	0.753942i	$-0.728150\pi$
$821$	−2.08333e10	−1.31388	−0.656941	+	0.753942i	$0.728150\pi$
$822$	0	0
$823$	−4.23403e9	−0.264761	−0.132381	−	0.991199i	$-0.542262\pi$
$823$	−4.23403e9	−0.264761	−0.132381	+	0.991199i	$0.542262\pi$
$824$	0	0
$825$	−1.28932e10	−0.799414
$826$	0	0
$827$	−5.70597e9	−0.350800	−0.175400	−	0.984497i	$-0.556122\pi$
$827$	−5.70597e9	−0.350800	−0.175400	+	0.984497i	$0.556122\pi$
$828$	0	0
$829$	2.51612e10	1.53388	0.766940	−	0.641719i	$-0.221779\pi$
$829$	2.51612e10	1.53388	0.766940	+	0.641719i	$0.221779\pi$
$830$	0	0
$831$	1.77920e10	1.07553
$832$	0	0
$833$	−1.09675e10	−0.657431
$834$	0	0
$835$	7.14862e9	0.424932
$836$	0	0
$837$	6.52531e8	0.0384647
$838$	0	0
$839$	−2.27048e10	−1.32724	−0.663622	−	0.748068i	$-0.730982\pi$
$839$	−2.27048e10	−1.32724	−0.663622	+	0.748068i	$0.730982\pi$
$840$	0	0
$841$	−1.55185e10	−0.899629
$842$	0	0
$843$	2.83832e9	0.163179
$844$	0	0
$845$	5.50544e9	0.313901
$846$	0	0
$847$	5.40112e10	3.05416
$848$	0	0
$849$	8.91435e9	0.499934
$850$	0	0
$851$	1.51144e9	0.0840695
$852$	0	0
$853$	2.86872e10	1.58258	0.791292	−	0.611439i	$-0.209409\pi$
$853$	2.86872e10	1.58258	0.791292	+	0.611439i	$0.209409\pi$
$854$	0	0
$855$	2.06602e9	0.113045
$856$	0	0
$857$	5.34950e9	0.290322	0.145161	−	0.989408i	$-0.453630\pi$
$857$	5.34950e9	0.290322	0.145161	+	0.989408i	$0.453630\pi$
$858$	0	0
$859$	1.40330e10	0.755394	0.377697	−	0.925929i	$-0.376716\pi$
$859$	1.40330e10	0.755394	0.377697	+	0.925929i	$0.376716\pi$
$860$	0	0
$861$	2.71940e10	1.45199
$862$	0	0
$863$	−3.24994e10	−1.72122	−0.860612	−	0.509261i	$-0.829919\pi$
$863$	−3.24994e10	−1.72122	−0.860612	+	0.509261i	$0.829919\pi$
$864$	0	0
$865$	3.08394e9	0.162013
$866$	0	0
$867$	9.90088e9	0.515949
$868$	0	0
$869$	2.79162e10	1.44307
$870$	0	0
$871$	−1.25118e10	−0.641586
$872$	0	0
$873$	−2.95983e9	−0.150563
$874$	0	0
$875$	2.57376e10	1.29879
$876$	0	0
$877$	−3.38694e10	−1.69554	−0.847772	−	0.530361i	$-0.822056\pi$
$877$	−3.38694e10	−1.69554	−0.847772	+	0.530361i	$0.822056\pi$
$878$	0	0
$879$	2.35171e9	0.116794
$880$	0	0
$881$	−1.52708e10	−0.752397	−0.376198	−	0.926539i	$-0.622769\pi$
$881$	−1.52708e10	−0.752397	−0.376198	+	0.926539i	$0.622769\pi$
$882$	0	0
$883$	1.12045e10	0.547685	0.273842	−	0.961775i	$-0.411705\pi$
$883$	1.12045e10	0.547685	0.273842	+	0.961775i	$0.411705\pi$
$884$	0	0
$885$	−2.29354e9	−0.111226
$886$	0	0
$887$	−6.97232e9	−0.335463	−0.167732	−	0.985833i	$-0.553644\pi$
$887$	−6.97232e9	−0.335463	−0.167732	+	0.985833i	$0.553644\pi$
$888$	0	0
$889$	−1.08241e10	−0.516695
$890$	0	0
$891$	−3.89653e9	−0.184546
$892$	0	0
$893$	−1.07837e10	−0.506742
$894$	0	0
$895$	−1.53080e10	−0.713737
$896$	0	0
$897$	−4.25480e9	−0.196837
$898$	0	0
$899$	1.37945e9	0.0633211
$900$	0	0
$901$	−5.19283e9	−0.236520
$902$	0	0
$903$	−6.65564e9	−0.300804
$904$	0	0
$905$	−1.30713e10	−0.586204
$906$	0	0
$907$	−1.18095e9	−0.0525539	−0.0262770	−	0.999655i	$-0.508365\pi$
$907$	−1.18095e9	−0.0525539	−0.0262770	+	0.999655i	$0.508365\pi$
$908$	0	0
$909$	−1.25985e10	−0.556348
$910$	0	0
$911$	−1.27915e10	−0.560541	−0.280271	−	0.959921i	$-0.590424\pi$
$911$	−1.27915e10	−0.560541	−0.280271	+	0.959921i	$0.590424\pi$
$912$	0	0
$913$	1.63465e10	0.710847
$914$	0	0
$915$	−5.11138e9	−0.220579
$916$	0	0
$917$	6.22848e10	2.66741
$918$	0	0
$919$	−3.52353e10	−1.49752	−0.748761	−	0.662840i	$-0.769351\pi$
$919$	−3.52353e10	−1.49752	−0.748761	+	0.662840i	$0.769351\pi$
$920$	0	0
$921$	−1.05761e10	−0.446086
$922$	0	0
$923$	2.17328e10	0.909725
$924$	0	0
$925$	2.37499e9	0.0986658
$926$	0	0
$927$	1.04701e10	0.431691
$928$	0	0
$929$	−2.17764e10	−0.891111	−0.445556	−	0.895254i	$-0.646994\pi$
$929$	−2.17764e10	−0.891111	−0.445556	+	0.895254i	$0.646994\pi$
$930$	0	0
$931$	−4.12734e10	−1.67628
$932$	0	0
$933$	−5.53327e9	−0.223047
$934$	0	0
$935$	−5.52161e9	−0.220915
$936$	0	0
$937$	1.15795e10	0.459833	0.229916	−	0.973210i	$-0.426155\pi$
$937$	1.15795e10	0.459833	0.229916	+	0.973210i	$0.426155\pi$
$938$	0	0
$939$	−2.36845e10	−0.933542
$940$	0	0
$941$	−3.83930e10	−1.50207	−0.751033	−	0.660265i	$-0.770444\pi$
$941$	−3.83930e10	−1.50207	−0.751033	+	0.660265i	$0.770444\pi$
$942$	0	0
$943$	2.64885e10	1.02865
$944$	0	0
$945$	3.53633e9	0.136314
$946$	0	0
$947$	−1.11841e10	−0.427933	−0.213966	−	0.976841i	$-0.568638\pi$
$947$	−1.11841e10	−0.427933	−0.213966	+	0.976841i	$0.568638\pi$
$948$	0	0
$949$	−1.01129e10	−0.384101
$950$	0	0
$951$	1.19024e10	0.448749
$952$	0	0
$953$	−5.19835e9	−0.194554	−0.0972770	−	0.995257i	$-0.531013\pi$
$953$	−5.19835e9	−0.194554	−0.0972770	+	0.995257i	$0.531013\pi$
$954$	0	0
$955$	1.86510e9	0.0692931
$956$	0	0
$957$	−8.23728e9	−0.303803
$958$	0	0
$959$	3.02183e10	1.10638
$960$	0	0
$961$	−2.64136e10	−0.960053
$962$	0	0
$963$	−4.70526e9	−0.169782
$964$	0	0
$965$	1.75786e10	0.629708
$966$	0	0
$967$	2.69243e8	0.00957529	0.00478765	−	0.999989i	$-0.498476\pi$
$967$	2.69243e8	0.00957529	0.00478765	+	0.999989i	$0.498476\pi$
$968$	0	0
$969$	−4.43408e9	−0.156556
$970$	0	0
$971$	−4.37283e9	−0.153284	−0.0766418	−	0.997059i	$-0.524420\pi$
$971$	−4.37283e9	−0.153284	−0.0766418	+	0.997059i	$0.524420\pi$
$972$	0	0
$973$	−2.08740e10	−0.726457
$974$	0	0
$975$	−6.68575e9	−0.231012
$976$	0	0
$977$	−3.74991e10	−1.28644	−0.643220	−	0.765681i	$-0.722402\pi$
$977$	−3.74991e10	−1.28644	−0.643220	+	0.765681i	$0.722402\pi$
$978$	0	0
$979$	−4.39276e10	−1.49623
$980$	0	0
$981$	−6.44735e8	−0.0218042
$982$	0	0
$983$	3.06190e10	1.02814	0.514071	−	0.857748i	$-0.328137\pi$
$983$	3.06190e10	1.02814	0.514071	+	0.857748i	$0.328137\pi$
$984$	0	0
$985$	−9.48808e9	−0.316338
$986$	0	0
$987$	−1.84580e10	−0.611048
$988$	0	0
$989$	−6.48296e9	−0.213102
$990$	0	0
$991$	1.72703e10	0.563693	0.281847	−	0.959459i	$-0.409053\pi$
$991$	1.72703e10	0.563693	0.281847	+	0.959459i	$0.409053\pi$
$992$	0	0
$993$	3.00302e10	0.973276
$994$	0	0
$995$	−8.68363e9	−0.279461
$996$	0	0
$997$	−3.75077e9	−0.119864	−0.0599319	−	0.998202i	$-0.519088\pi$
$997$	−3.75077e9	−0.119864	−0.0599319	+	0.998202i	$0.519088\pi$
$998$	0	0
$999$	7.17760e8	0.0227772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.8.a.b.1.1		1
3.2	odd	2		144.8.a.h.1.1		1
4.3	odd	2		6.8.a.a.1.1	✓	1
8.3	odd	2		192.8.a.f.1.1		1
8.5	even	2		192.8.a.n.1.1		1
12.11	even	2		18.8.a.a.1.1		1
20.3	even	4		150.8.c.k.49.1		2
20.7	even	4		150.8.c.k.49.2		2
20.19	odd	2		150.8.a.e.1.1		1
24.5	odd	2		576.8.a.i.1.1		1
24.11	even	2		576.8.a.h.1.1		1
28.3	even	6		294.8.e.d.79.1		2
28.11	odd	6		294.8.e.c.79.1		2
28.19	even	6		294.8.e.d.67.1		2
28.23	odd	6		294.8.e.c.67.1		2
28.27	even	2		294.8.a.l.1.1		1
36.7	odd	6		162.8.c.d.109.1		2
36.11	even	6		162.8.c.i.109.1		2
36.23	even	6		162.8.c.i.55.1		2
36.31	odd	6		162.8.c.d.55.1		2
60.23	odd	4		450.8.c.a.199.2		2
60.47	odd	4		450.8.c.a.199.1		2
60.59	even	2		450.8.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.8.a.a.1.1	✓	1	4.3	odd	2
18.8.a.a.1.1		1	12.11	even	2
48.8.a.b.1.1		1	1.1	even	1	trivial
144.8.a.h.1.1		1	3.2	odd	2
150.8.a.e.1.1		1	20.19	odd	2
150.8.c.k.49.1		2	20.3	even	4
150.8.c.k.49.2		2	20.7	even	4
162.8.c.d.55.1		2	36.31	odd	6
162.8.c.d.109.1		2	36.7	odd	6
162.8.c.i.55.1		2	36.23	even	6
162.8.c.i.109.1		2	36.11	even	6
192.8.a.f.1.1		1	8.3	odd	2
192.8.a.n.1.1		1	8.5	even	2
294.8.a.l.1.1		1	28.27	even	2
294.8.e.c.67.1		2	28.23	odd	6
294.8.e.c.79.1		2	28.11	odd	6
294.8.e.d.67.1		2	28.19	even	6
294.8.e.d.79.1		2	28.3	even	6
450.8.a.ba.1.1		1	60.59	even	2
450.8.c.a.199.1		2	60.47	odd	4
450.8.c.a.199.2		2	60.23	odd	4
576.8.a.h.1.1		1	24.11	even	2
576.8.a.i.1.1		1	24.5	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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