LMFDB - Embedded newform 48.8.a.d.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 12)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} +270.000 q^{5} -1112.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} +270.000 q^{5} -1112.00 q^{7} +729.000 q^{9} +5724.00 q^{11} -4570.00 q^{13} -7290.00 q^{15} -36558.0 q^{17} -51740.0 q^{19} +30024.0 q^{21} -22248.0 q^{23} -5225.00 q^{25} -19683.0 q^{27} -157194. q^{29} +103936. q^{31} -154548. q^{33} -300240. q^{35} -94834.0 q^{37} +123390. q^{39} +659610. q^{41} +75772.0 q^{43} +196830. q^{45} -405648. q^{47} +413001. q^{49} +987066. q^{51} -1.34627e6 q^{53} +1.54548e6 q^{55} +1.39698e6 q^{57} +1.30388e6 q^{59} +1.83378e6 q^{61} -810648. q^{63} -1.23390e6 q^{65} -1.36939e6 q^{67} +600696. q^{69} -2.71404e6 q^{71} +2.86879e6 q^{73} +141075. q^{75} -6.36509e6 q^{77} +1.12965e6 q^{79} +531441. q^{81} -5.91203e6 q^{83} -9.87066e6 q^{85} +4.24424e6 q^{87} -897750. q^{89} +5.08184e6 q^{91} -2.80627e6 q^{93} -1.39698e7 q^{95} +1.37191e7 q^{97} +4.17280e6 q^{99} +O(q^{100})$

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
$5$	270.000	0.965981	0.482991	−	0.875625i	$-0.339550\pi$
$5$	270.000	0.965981	0.482991	+	0.875625i	$0.339550\pi$
$6$	0	0
$7$	−1112.00	−1.22535	−0.612677	−	0.790333i	$-0.709907\pi$
$7$	−1112.00	−1.22535	−0.612677	+	0.790333i	$0.709907\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	5724.00	1.29666	0.648329	−	0.761361i	$-0.275468\pi$
$11$	5724.00	1.29666	0.648329	+	0.761361i	$0.275468\pi$
$12$	0	0
$13$	−4570.00	−0.576919	−0.288459	−	0.957492i	$-0.593143\pi$
$13$	−4570.00	−0.576919	−0.288459	+	0.957492i	$0.593143\pi$
$14$	0	0
$15$	−7290.00	−0.557710
$16$	0	0
$17$	−36558.0	−1.80473	−0.902363	−	0.430977i	$-0.858169\pi$
$17$	−36558.0	−1.80473	−0.902363	+	0.430977i	$0.858169\pi$
$18$	0	0
$19$	−51740.0	−1.73057	−0.865284	−	0.501281i	$-0.832862\pi$
$19$	−51740.0	−1.73057	−0.865284	+	0.501281i	$0.832862\pi$
$20$	0	0
$21$	30024.0	0.707459
$22$	0	0
$23$	−22248.0	−0.381280	−0.190640	−	0.981660i	$-0.561056\pi$
$23$	−22248.0	−0.381280	−0.190640	+	0.981660i	$0.561056\pi$
$24$	0	0
$25$	−5225.00	−0.0668800
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−157194.	−1.19686	−0.598429	−	0.801175i	$-0.704208\pi$
$29$	−157194.	−1.19686	−0.598429	+	0.801175i	$0.704208\pi$
$30$	0	0
$31$	103936.	0.626614	0.313307	−	0.949652i	$-0.398563\pi$
$31$	103936.	0.626614	0.313307	+	0.949652i	$0.398563\pi$
$32$	0	0
$33$	−154548.	−0.748625
$34$	0	0
$35$	−300240.	−1.18367
$36$	0	0
$37$	−94834.0	−0.307793	−0.153896	−	0.988087i	$-0.549182\pi$
$37$	−94834.0	−0.307793	−0.153896	+	0.988087i	$0.549182\pi$
$38$	0	0
$39$	123390.	0.333084
$40$	0	0
$41$	659610.	1.49466	0.747332	−	0.664451i	$-0.231334\pi$
$41$	659610.	1.49466	0.747332	+	0.664451i	$0.231334\pi$
$42$	0	0
$43$	75772.0	0.145335	0.0726673	−	0.997356i	$-0.476849\pi$
$43$	75772.0	0.145335	0.0726673	+	0.997356i	$0.476849\pi$
$44$	0	0
$45$	196830.	0.321994
$46$	0	0
$47$	−405648.	−0.569911	−0.284955	−	0.958541i	$-0.591979\pi$
$47$	−405648.	−0.569911	−0.284955	+	0.958541i	$0.591979\pi$
$48$	0	0
$49$	413001.	0.501493
$50$	0	0
$51$	987066.	1.04196
$52$	0	0
$53$	−1.34627e6	−1.24213	−0.621066	−	0.783758i	$-0.713300\pi$
$53$	−1.34627e6	−1.24213	−0.621066	+	0.783758i	$0.713300\pi$
$54$	0	0
$55$	1.54548e6	1.25255
$56$	0	0
$57$	1.39698e6	0.999144
$58$	0	0
$59$	1.30388e6	0.826527	0.413263	−	0.910612i	$-0.364389\pi$
$59$	1.30388e6	0.826527	0.413263	+	0.910612i	$0.364389\pi$
$60$	0	0
$61$	1.83378e6	1.03441	0.517206	−	0.855861i	$-0.326972\pi$
$61$	1.83378e6	1.03441	0.517206	+	0.855861i	$0.326972\pi$
$62$	0	0
$63$	−810648.	−0.408451
$64$	0	0
$65$	−1.23390e6	−0.557293
$66$	0	0
$67$	−1.36939e6	−0.556243	−0.278122	−	0.960546i	$-0.589712\pi$
$67$	−1.36939e6	−0.556243	−0.278122	+	0.960546i	$0.589712\pi$
$68$	0	0
$69$	600696.	0.220132
$70$	0	0
$71$	−2.71404e6	−0.899937	−0.449968	−	0.893044i	$-0.648565\pi$
$71$	−2.71404e6	−0.899937	−0.449968	+	0.893044i	$0.648565\pi$
$72$	0	0
$73$	2.86879e6	0.863116	0.431558	−	0.902085i	$-0.357964\pi$
$73$	2.86879e6	0.863116	0.431558	+	0.902085i	$0.357964\pi$
$74$	0	0
$75$	141075.	0.0386132
$76$	0	0
$77$	−6.36509e6	−1.58886
$78$	0	0
$79$	1.12965e6	0.257779	0.128890	−	0.991659i	$-0.458859\pi$
$79$	1.12965e6	0.257779	0.128890	+	0.991659i	$0.458859\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−5.91203e6	−1.13491	−0.567457	−	0.823403i	$-0.692073\pi$
$83$	−5.91203e6	−1.13491	−0.567457	+	0.823403i	$0.692073\pi$
$84$	0	0
$85$	−9.87066e6	−1.74333
$86$	0	0
$87$	4.24424e6	0.691007
$88$	0	0
$89$	−897750.	−0.134987	−0.0674933	−	0.997720i	$-0.521500\pi$
$89$	−897750.	−0.134987	−0.0674933	+	0.997720i	$0.521500\pi$
$90$	0	0
$91$	5.08184e6	0.706930
$92$	0	0
$93$	−2.80627e6	−0.361776
$94$	0	0
$95$	−1.39698e7	−1.67170
$96$	0	0
$97$	1.37191e7	1.52624	0.763122	−	0.646255i	$-0.223666\pi$
$97$	1.37191e7	1.52624	0.763122	+	0.646255i	$0.223666\pi$
$98$	0	0
$99$	4.17280e6	0.432219
$100$	0	0
$101$	1.93304e6	0.186688	0.0933438	−	0.995634i	$-0.470244\pi$
$101$	1.93304e6	0.186688	0.0933438	+	0.995634i	$0.470244\pi$
$102$	0	0
$103$	−4.76127e6	−0.429331	−0.214666	−	0.976688i	$-0.568866\pi$
$103$	−4.76127e6	−0.429331	−0.214666	+	0.976688i	$0.568866\pi$
$104$	0	0
$105$	8.10648e6	0.683392
$106$	0	0
$107$	−2.21044e6	−0.174435	−0.0872177	−	0.996189i	$-0.527798\pi$
$107$	−2.21044e6	−0.174435	−0.0872177	+	0.996189i	$0.527798\pi$
$108$	0	0
$109$	−2.00929e7	−1.48610	−0.743052	−	0.669234i	$-0.766622\pi$
$109$	−2.00929e7	−1.48610	−0.743052	+	0.669234i	$0.766622\pi$
$110$	0	0
$111$	2.56052e6	0.177704
$112$	0	0
$113$	7.94075e6	0.517711	0.258855	−	0.965916i	$-0.416655\pi$
$113$	7.94075e6	0.517711	0.258855	+	0.965916i	$0.416655\pi$
$114$	0	0
$115$	−6.00696e6	−0.368309
$116$	0	0
$117$	−3.33153e6	−0.192306
$118$	0	0
$119$	4.06525e7	2.21143
$120$	0	0
$121$	1.32770e7	0.681320
$122$	0	0
$123$	−1.78095e7	−0.862945
$124$	0	0
$125$	−2.25045e7	−1.03059
$126$	0	0
$127$	2.27395e7	0.985071	0.492536	−	0.870292i	$-0.336070\pi$
$127$	2.27395e7	0.985071	0.492536	+	0.870292i	$0.336070\pi$
$128$	0	0
$129$	−2.04584e6	−0.0839090
$130$	0	0
$131$	2.27784e7	0.885265	0.442633	−	0.896703i	$-0.354045\pi$
$131$	2.27784e7	0.885265	0.442633	+	0.896703i	$0.354045\pi$
$132$	0	0
$133$	5.75349e7	2.12056
$134$	0	0
$135$	−5.31441e6	−0.185903
$136$	0	0
$137$	4.16987e7	1.38548	0.692741	−	0.721186i	$-0.256403\pi$
$137$	4.16987e7	1.38548	0.692741	+	0.721186i	$0.256403\pi$
$138$	0	0
$139$	6.19472e6	0.195645	0.0978227	−	0.995204i	$-0.468812\pi$
$139$	6.19472e6	0.195645	0.0978227	+	0.995204i	$0.468812\pi$
$140$	0	0
$141$	1.09525e7	0.329038
$142$	0	0
$143$	−2.61587e7	−0.748066
$144$	0	0
$145$	−4.24424e7	−1.15614
$146$	0	0
$147$	−1.11510e7	−0.289537
$148$	0	0
$149$	−5.41658e6	−0.134145	−0.0670723	−	0.997748i	$-0.521366\pi$
$149$	−5.41658e6	−0.134145	−0.0670723	+	0.997748i	$0.521366\pi$
$150$	0	0
$151$	−7.08444e7	−1.67450	−0.837252	−	0.546818i	$-0.815839\pi$
$151$	−7.08444e7	−1.67450	−0.837252	+	0.546818i	$0.815839\pi$
$152$	0	0
$153$	−2.66508e7	−0.601575
$154$	0	0
$155$	2.80627e7	0.605297
$156$	0	0
$157$	−7.90801e7	−1.63087	−0.815433	−	0.578851i	$-0.803501\pi$
$157$	−7.90801e7	−1.63087	−0.815433	+	0.578851i	$0.803501\pi$
$158$	0	0
$159$	3.63494e7	0.717146
$160$	0	0
$161$	2.47398e7	0.467203
$162$	0	0
$163$	2.44491e7	0.442188	0.221094	−	0.975253i	$-0.429037\pi$
$163$	2.44491e7	0.442188	0.221094	+	0.975253i	$0.429037\pi$
$164$	0	0
$165$	−4.17280e7	−0.723158
$166$	0	0
$167$	−3.55538e7	−0.590716	−0.295358	−	0.955387i	$-0.595439\pi$
$167$	−3.55538e7	−0.590716	−0.295358	+	0.955387i	$0.595439\pi$
$168$	0	0
$169$	−4.18636e7	−0.667165
$170$	0	0
$171$	−3.77185e7	−0.576856
$172$	0	0
$173$	1.02794e8	1.50941	0.754704	−	0.656066i	$-0.227781\pi$
$173$	1.02794e8	1.50941	0.754704	+	0.656066i	$0.227781\pi$
$174$	0	0
$175$	5.81020e6	0.0819517
$176$	0	0
$177$	−3.52049e7	−0.477195
$178$	0	0
$179$	4.15587e7	0.541597	0.270799	−	0.962636i	$-0.412712\pi$
$179$	4.15587e7	0.541597	0.270799	+	0.962636i	$0.412712\pi$
$180$	0	0
$181$	−3.35013e7	−0.419939	−0.209969	−	0.977708i	$-0.567336\pi$
$181$	−3.35013e7	−0.419939	−0.209969	+	0.977708i	$0.567336\pi$
$182$	0	0
$183$	−4.95121e7	−0.597218
$184$	0	0
$185$	−2.56052e7	−0.297322
$186$	0	0
$187$	−2.09258e8	−2.34011
$188$	0	0
$189$	2.18875e7	0.235820
$190$	0	0
$191$	−4.90804e7	−0.509672	−0.254836	−	0.966984i	$-0.582022\pi$
$191$	−4.90804e7	−0.509672	−0.254836	+	0.966984i	$0.582022\pi$
$192$	0	0
$193$	6.04483e7	0.605248	0.302624	−	0.953110i	$-0.402137\pi$
$193$	6.04483e7	0.605248	0.302624	+	0.953110i	$0.402137\pi$
$194$	0	0
$195$	3.33153e7	0.321753
$196$	0	0
$197$	5.00456e6	0.0466374	0.0233187	−	0.999728i	$-0.492577\pi$
$197$	5.00456e6	0.0466374	0.0233187	+	0.999728i	$0.492577\pi$
$198$	0	0
$199$	2.21135e8	1.98917	0.994584	−	0.103935i	$-0.0331434\pi$
$199$	2.21135e8	1.98917	0.994584	+	0.103935i	$0.0331434\pi$
$200$	0	0
$201$	3.69735e7	0.321147
$202$	0	0
$203$	1.74800e8	1.46658
$204$	0	0
$205$	1.78095e8	1.44382
$206$	0	0
$207$	−1.62188e7	−0.127093
$208$	0	0
$209$	−2.96160e8	−2.24395
$210$	0	0
$211$	7.16641e7	0.525186	0.262593	−	0.964907i	$-0.415422\pi$
$211$	7.16641e7	0.525186	0.262593	+	0.964907i	$0.415422\pi$
$212$	0	0
$213$	7.32791e7	0.519579
$214$	0	0
$215$	2.04584e7	0.140391
$216$	0	0
$217$	−1.15577e8	−0.767824
$218$	0	0
$219$	−7.74574e7	−0.498320
$220$	0	0
$221$	1.67070e8	1.04118
$222$	0	0
$223$	−2.61440e7	−0.157872	−0.0789360	−	0.996880i	$-0.525152\pi$
$223$	−2.61440e7	−0.157872	−0.0789360	+	0.996880i	$0.525152\pi$
$224$	0	0
$225$	−3.80902e6	−0.0222933
$226$	0	0
$227$	3.40994e7	0.193489	0.0967444	−	0.995309i	$-0.469157\pi$
$227$	3.40994e7	0.193489	0.0967444	+	0.995309i	$0.469157\pi$
$228$	0	0
$229$	2.27428e8	1.25147	0.625734	−	0.780037i	$-0.284800\pi$
$229$	2.27428e8	1.25147	0.625734	+	0.780037i	$0.284800\pi$
$230$	0	0
$231$	1.71857e8	0.917331
$232$	0	0
$233$	5.69060e7	0.294722	0.147361	−	0.989083i	$-0.452922\pi$
$233$	5.69060e7	0.294722	0.147361	+	0.989083i	$0.452922\pi$
$234$	0	0
$235$	−1.09525e8	−0.550523
$236$	0	0
$237$	−3.05005e7	−0.148829
$238$	0	0
$239$	−3.02716e8	−1.43431	−0.717154	−	0.696915i	$-0.754556\pi$
$239$	−3.02716e8	−1.43431	−0.717154	+	0.696915i	$0.754556\pi$
$240$	0	0
$241$	−2.53696e7	−0.116749	−0.0583746	−	0.998295i	$-0.518592\pi$
$241$	−2.53696e7	−0.116749	−0.0583746	+	0.998295i	$0.518592\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	1.11510e8	0.484433
$246$	0	0
$247$	2.36452e8	0.998397
$248$	0	0
$249$	1.59625e8	0.655243
$250$	0	0
$251$	1.46544e7	0.0584939	0.0292469	−	0.999572i	$-0.490689\pi$
$251$	1.46544e7	0.0584939	0.0292469	+	0.999572i	$0.490689\pi$
$252$	0	0
$253$	−1.27348e8	−0.494389
$254$	0	0
$255$	2.66508e8	1.00651
$256$	0	0
$257$	−2.34652e7	−0.0862300	−0.0431150	−	0.999070i	$-0.513728\pi$
$257$	−2.34652e7	−0.0862300	−0.0431150	+	0.999070i	$0.513728\pi$
$258$	0	0
$259$	1.05455e8	0.377155
$260$	0	0
$261$	−1.14594e8	−0.398953
$262$	0	0
$263$	−4.25583e8	−1.44258	−0.721289	−	0.692634i	$-0.756450\pi$
$263$	−4.25583e8	−1.44258	−0.721289	+	0.692634i	$0.756450\pi$
$264$	0	0
$265$	−3.63494e8	−1.19988
$266$	0	0
$267$	2.42393e7	0.0779345
$268$	0	0
$269$	−3.93062e8	−1.23120	−0.615599	−	0.788060i	$-0.711086\pi$
$269$	−3.93062e8	−1.23120	−0.615599	+	0.788060i	$0.711086\pi$
$270$	0	0
$271$	−5.89560e7	−0.179943	−0.0899716	−	0.995944i	$-0.528678\pi$
$271$	−5.89560e7	−0.179943	−0.0899716	+	0.995944i	$0.528678\pi$
$272$	0	0
$273$	−1.37210e8	−0.408146
$274$	0	0
$275$	−2.99079e7	−0.0867204
$276$	0	0
$277$	−2.15973e7	−0.0610550	−0.0305275	−	0.999534i	$-0.509719\pi$
$277$	−2.15973e7	−0.0610550	−0.0305275	+	0.999534i	$0.509719\pi$
$278$	0	0
$279$	7.57693e7	0.208871
$280$	0	0
$281$	−3.34718e8	−0.899926	−0.449963	−	0.893047i	$-0.648563\pi$
$281$	−3.34718e8	−0.899926	−0.449963	+	0.893047i	$0.648563\pi$
$282$	0	0
$283$	−3.26806e8	−0.857113	−0.428557	−	0.903515i	$-0.640978\pi$
$283$	−3.26806e8	−0.857113	−0.428557	+	0.903515i	$0.640978\pi$
$284$	0	0
$285$	3.77185e8	0.965155
$286$	0	0
$287$	−7.33486e8	−1.83149
$288$	0	0
$289$	9.26149e8	2.25703
$290$	0	0
$291$	−3.70415e8	−0.881177
$292$	0	0
$293$	−7.28964e8	−1.69305	−0.846525	−	0.532350i	$-0.821309\pi$
$293$	−7.28964e8	−1.69305	−0.846525	+	0.532350i	$0.821309\pi$
$294$	0	0
$295$	3.52049e8	0.798409
$296$	0	0
$297$	−1.12665e8	−0.249542
$298$	0	0
$299$	1.01673e8	0.219967
$300$	0	0
$301$	−8.42585e7	−0.178086
$302$	0	0
$303$	−5.21920e7	−0.107784
$304$	0	0
$305$	4.95121e8	0.999222
$306$	0	0
$307$	1.26039e8	0.248612	0.124306	−	0.992244i	$-0.460330\pi$
$307$	1.26039e8	0.248612	0.124306	+	0.992244i	$0.460330\pi$
$308$	0	0
$309$	1.28554e8	0.247875
$310$	0	0
$311$	7.15768e8	1.34931	0.674654	−	0.738134i	$-0.264293\pi$
$311$	7.15768e8	1.34931	0.674654	+	0.738134i	$0.264293\pi$
$312$	0	0
$313$	−2.91196e8	−0.536760	−0.268380	−	0.963313i	$-0.586488\pi$
$313$	−2.91196e8	−0.536760	−0.268380	+	0.963313i	$0.586488\pi$
$314$	0	0
$315$	−2.18875e8	−0.394556
$316$	0	0
$317$	4.24285e7	0.0748084	0.0374042	−	0.999300i	$-0.488091\pi$
$317$	4.24285e7	0.0748084	0.0374042	+	0.999300i	$0.488091\pi$
$318$	0	0
$319$	−8.99778e8	−1.55192
$320$	0	0
$321$	5.96818e7	0.100710
$322$	0	0
$323$	1.89151e9	3.12320
$324$	0	0
$325$	2.38782e7	0.0385843
$326$	0	0
$327$	5.42507e8	0.858002
$328$	0	0
$329$	4.51081e8	0.698343
$330$	0	0
$331$	5.47257e8	0.829455	0.414728	−	0.909946i	$-0.363877\pi$
$331$	5.47257e8	0.829455	0.414728	+	0.909946i	$0.363877\pi$
$332$	0	0
$333$	−6.91340e7	−0.102598
$334$	0	0
$335$	−3.69735e8	−0.537321
$336$	0	0
$337$	−1.85332e8	−0.263783	−0.131891	−	0.991264i	$-0.542105\pi$
$337$	−1.85332e8	−0.263783	−0.131891	+	0.991264i	$0.542105\pi$
$338$	0	0
$339$	−2.14400e8	−0.298900
$340$	0	0
$341$	5.94930e8	0.812504
$342$	0	0
$343$	4.56523e8	0.610848
$344$	0	0
$345$	1.62188e8	0.212643
$346$	0	0
$347$	−1.29711e9	−1.66657	−0.833286	−	0.552842i	$-0.813543\pi$
$347$	−1.29711e9	−1.66657	−0.833286	+	0.552842i	$0.813543\pi$
$348$	0	0
$349$	−6.22136e8	−0.783423	−0.391711	−	0.920088i	$-0.628117\pi$
$349$	−6.22136e8	−0.783423	−0.391711	+	0.920088i	$0.628117\pi$
$350$	0	0
$351$	8.99513e7	0.111028
$352$	0	0
$353$	6.20687e8	0.751037	0.375519	−	0.926815i	$-0.377465\pi$
$353$	6.20687e8	0.751037	0.375519	+	0.926815i	$0.377465\pi$
$354$	0	0
$355$	−7.32791e8	−0.869322
$356$	0	0
$357$	−1.09762e9	−1.27677
$358$	0	0
$359$	−6.06882e8	−0.692268	−0.346134	−	0.938185i	$-0.612506\pi$
$359$	−6.06882e8	−0.692268	−0.346134	+	0.938185i	$0.612506\pi$
$360$	0	0
$361$	1.78316e9	1.99487
$362$	0	0
$363$	−3.58479e8	−0.393360
$364$	0	0
$365$	7.74574e8	0.833754
$366$	0	0
$367$	1.40538e9	1.48410	0.742051	−	0.670343i	$-0.233853\pi$
$367$	1.40538e9	1.48410	0.742051	+	0.670343i	$0.233853\pi$
$368$	0	0
$369$	4.80856e8	0.498222
$370$	0	0
$371$	1.49706e9	1.52205
$372$	0	0
$373$	−1.12925e9	−1.12670	−0.563350	−	0.826218i	$-0.690488\pi$
$373$	−1.12925e9	−1.12670	−0.563350	+	0.826218i	$0.690488\pi$
$374$	0	0
$375$	6.07622e8	0.595009
$376$	0	0
$377$	7.18377e8	0.690490
$378$	0	0
$379$	−1.63515e9	−1.54284	−0.771419	−	0.636328i	$-0.780453\pi$
$379$	−1.63515e9	−1.54284	−0.771419	+	0.636328i	$0.780453\pi$
$380$	0	0
$381$	−6.13966e8	−0.568731
$382$	0	0
$383$	2.27685e8	0.207080	0.103540	−	0.994625i	$-0.466983\pi$
$383$	2.27685e8	0.207080	0.103540	+	0.994625i	$0.466983\pi$
$384$	0	0
$385$	−1.71857e9	−1.53481
$386$	0	0
$387$	5.52378e7	0.0484449
$388$	0	0
$389$	−6.23120e8	−0.536720	−0.268360	−	0.963319i	$-0.586482\pi$
$389$	−6.23120e8	−0.536720	−0.268360	+	0.963319i	$0.586482\pi$
$390$	0	0
$391$	8.13342e8	0.688105
$392$	0	0
$393$	−6.15016e8	−0.511108
$394$	0	0
$395$	3.05005e8	0.249010
$396$	0	0
$397$	−1.49583e6	−0.00119982	−0.000599911	−	1.00000i	$-0.500191\pi$
$397$	−1.49583e6	−0.00119982	−0.000599911		1.00000i	$0.500191\pi$
$398$	0	0
$399$	−1.55344e9	−1.22431
$400$	0	0
$401$	−3.06501e8	−0.237370	−0.118685	−	0.992932i	$-0.537868\pi$
$401$	−3.06501e8	−0.237370	−0.118685	+	0.992932i	$0.537868\pi$
$402$	0	0
$403$	−4.74988e8	−0.361505
$404$	0	0
$405$	1.43489e8	0.107331
$406$	0	0
$407$	−5.42830e8	−0.399101
$408$	0	0
$409$	3.56868e8	0.257915	0.128957	−	0.991650i	$-0.458837\pi$
$409$	3.56868e8	0.257915	0.128957	+	0.991650i	$0.458837\pi$
$410$	0	0
$411$	−1.12587e9	−0.799909
$412$	0	0
$413$	−1.44992e9	−1.01279
$414$	0	0
$415$	−1.59625e9	−1.09631
$416$	0	0
$417$	−1.67257e8	−0.112956
$418$	0	0
$419$	2.28428e9	1.51705	0.758526	−	0.651642i	$-0.225920\pi$
$419$	2.28428e9	1.51705	0.758526	+	0.651642i	$0.225920\pi$
$420$	0	0
$421$	2.63135e8	0.171866	0.0859332	−	0.996301i	$-0.472613\pi$
$421$	2.63135e8	0.171866	0.0859332	+	0.996301i	$0.472613\pi$
$422$	0	0
$423$	−2.95717e8	−0.189970
$424$	0	0
$425$	1.91016e8	0.120700
$426$	0	0
$427$	−2.03917e9	−1.26752
$428$	0	0
$429$	7.06284e8	0.431896
$430$	0	0
$431$	−1.49053e9	−0.896747	−0.448374	−	0.893846i	$-0.647997\pi$
$431$	−1.49053e9	−0.896747	−0.448374	+	0.893846i	$0.647997\pi$
$432$	0	0
$433$	1.10539e9	0.654347	0.327174	−	0.944964i	$-0.393904\pi$
$433$	1.10539e9	0.654347	0.327174	+	0.944964i	$0.393904\pi$
$434$	0	0
$435$	1.14594e9	0.667500
$436$	0	0
$437$	1.15111e9	0.659830
$438$	0	0
$439$	−2.96353e8	−0.167180	−0.0835898	−	0.996500i	$-0.526639\pi$
$439$	−2.96353e8	−0.167180	−0.0835898	+	0.996500i	$0.526639\pi$
$440$	0	0
$441$	3.01078e8	0.167164
$442$	0	0
$443$	−3.58944e8	−0.196162	−0.0980808	−	0.995178i	$-0.531270\pi$
$443$	−3.58944e8	−0.196162	−0.0980808	+	0.995178i	$0.531270\pi$
$444$	0	0
$445$	−2.42393e8	−0.130394
$446$	0	0
$447$	1.46248e8	0.0774484
$448$	0	0
$449$	−3.41948e9	−1.78278	−0.891391	−	0.453235i	$-0.850270\pi$
$449$	−3.41948e9	−1.78278	−0.891391	+	0.453235i	$0.850270\pi$
$450$	0	0
$451$	3.77561e9	1.93807
$452$	0	0
$453$	1.91280e9	0.966775
$454$	0	0
$455$	1.37210e9	0.682881
$456$	0	0
$457$	−2.85828e9	−1.40087	−0.700434	−	0.713717i	$-0.747010\pi$
$457$	−2.85828e9	−1.40087	−0.700434	+	0.713717i	$0.747010\pi$
$458$	0	0
$459$	7.19571e8	0.347320
$460$	0	0
$461$	9.62700e8	0.457654	0.228827	−	0.973467i	$-0.426511\pi$
$461$	9.62700e8	0.457654	0.228827	+	0.973467i	$0.426511\pi$
$462$	0	0
$463$	−6.01893e8	−0.281829	−0.140914	−	0.990022i	$-0.545004\pi$
$463$	−6.01893e8	−0.281829	−0.140914	+	0.990022i	$0.545004\pi$
$464$	0	0
$465$	−7.57693e8	−0.349469
$466$	0	0
$467$	−1.72060e9	−0.781755	−0.390878	−	0.920443i	$-0.627828\pi$
$467$	−1.72060e9	−0.781755	−0.390878	+	0.920443i	$0.627828\pi$
$468$	0	0
$469$	1.52276e9	0.681595
$470$	0	0
$471$	2.13516e9	0.941581
$472$	0	0
$473$	4.33719e8	0.188449
$474$	0	0
$475$	2.70342e8	0.115740
$476$	0	0
$477$	−9.81434e8	−0.414044
$478$	0	0
$479$	5.37262e8	0.223363	0.111682	−	0.993744i	$-0.464376\pi$
$479$	5.37262e8	0.223363	0.111682	+	0.993744i	$0.464376\pi$
$480$	0	0
$481$	4.33391e8	0.177571
$482$	0	0
$483$	−6.67974e8	−0.269740
$484$	0	0
$485$	3.70415e9	1.47432
$486$	0	0
$487$	4.49621e9	1.76399	0.881993	−	0.471262i	$-0.156201\pi$
$487$	4.49621e9	1.76399	0.881993	+	0.471262i	$0.156201\pi$
$488$	0	0
$489$	−6.60127e8	−0.255297
$490$	0	0
$491$	−3.32917e9	−1.26926	−0.634631	−	0.772815i	$-0.718848\pi$
$491$	−3.32917e9	−1.26926	−0.634631	+	0.772815i	$0.718848\pi$
$492$	0	0
$493$	5.74670e9	2.16000
$494$	0	0
$495$	1.12665e9	0.417516
$496$	0	0
$497$	3.01801e9	1.10274
$498$	0	0
$499$	2.04986e9	0.738536	0.369268	−	0.929323i	$-0.379608\pi$
$499$	2.04986e9	0.738536	0.369268	+	0.929323i	$0.379608\pi$
$500$	0	0
$501$	9.59953e8	0.341050
$502$	0	0
$503$	−7.76491e8	−0.272050	−0.136025	−	0.990705i	$-0.543433\pi$
$503$	−7.76491e8	−0.272050	−0.136025	+	0.990705i	$0.543433\pi$
$504$	0	0
$505$	5.21920e8	0.180337
$506$	0	0
$507$	1.13032e9	0.385188
$508$	0	0
$509$	−4.56679e9	−1.53497	−0.767483	−	0.641069i	$-0.778491\pi$
$509$	−4.56679e9	−1.53497	−0.767483	+	0.641069i	$0.778491\pi$
$510$	0	0
$511$	−3.19010e9	−1.05762
$512$	0	0
$513$	1.01840e9	0.333048
$514$	0	0
$515$	−1.28554e9	−0.414726
$516$	0	0
$517$	−2.32193e9	−0.738979
$518$	0	0
$519$	−2.77544e9	−0.871457
$520$	0	0
$521$	−2.21159e9	−0.685130	−0.342565	−	0.939494i	$-0.611296\pi$
$521$	−2.21159e9	−0.685130	−0.342565	+	0.939494i	$0.611296\pi$
$522$	0	0
$523$	2.34154e9	0.715724	0.357862	−	0.933774i	$-0.383506\pi$
$523$	2.34154e9	0.715724	0.357862	+	0.933774i	$0.383506\pi$
$524$	0	0
$525$	−1.56875e8	−0.0473148
$526$	0	0
$527$	−3.79969e9	−1.13087
$528$	0	0
$529$	−2.90985e9	−0.854626
$530$	0	0
$531$	9.50531e8	0.275509
$532$	0	0
$533$	−3.01442e9	−0.862300
$534$	0	0
$535$	−5.96818e8	−0.168501
$536$	0	0
$537$	−1.12209e9	−0.312691
$538$	0	0
$539$	2.36402e9	0.650265
$540$	0	0
$541$	2.35497e9	0.639433	0.319717	−	0.947513i	$-0.396412\pi$
$541$	2.35497e9	0.639433	0.319717	+	0.947513i	$0.396412\pi$
$542$	0	0
$543$	9.04534e8	0.242452
$544$	0	0
$545$	−5.42507e9	−1.43555
$546$	0	0
$547$	−3.41493e9	−0.892127	−0.446063	−	0.895001i	$-0.647174\pi$
$547$	−3.41493e9	−0.892127	−0.446063	+	0.895001i	$0.647174\pi$
$548$	0	0
$549$	1.33683e9	0.344804
$550$	0	0
$551$	8.13322e9	2.07125
$552$	0	0
$553$	−1.25617e9	−0.315871
$554$	0	0
$555$	6.91340e8	0.171659
$556$	0	0
$557$	−4.79293e8	−0.117519	−0.0587595	−	0.998272i	$-0.518715\pi$
$557$	−4.79293e8	−0.117519	−0.0587595	+	0.998272i	$0.518715\pi$
$558$	0	0
$559$	−3.46278e8	−0.0838462
$560$	0	0
$561$	5.64997e9	1.35106
$562$	0	0
$563$	−6.50302e9	−1.53580	−0.767902	−	0.640567i	$-0.778699\pi$
$563$	−6.50302e9	−1.53580	−0.767902	+	0.640567i	$0.778699\pi$
$564$	0	0
$565$	2.14400e9	0.500099
$566$	0	0
$567$	−5.90962e8	−0.136150
$568$	0	0
$569$	−4.51625e9	−1.02774	−0.513872	−	0.857867i	$-0.671789\pi$
$569$	−4.51625e9	−1.02774	−0.513872	+	0.857867i	$0.671789\pi$
$570$	0	0
$571$	−2.47375e9	−0.556071	−0.278035	−	0.960571i	$-0.589683\pi$
$571$	−2.47375e9	−0.556071	−0.278035	+	0.960571i	$0.589683\pi$
$572$	0	0
$573$	1.32517e9	0.294260
$574$	0	0
$575$	1.16246e8	0.0255000
$576$	0	0
$577$	−5.31721e8	−0.115231	−0.0576154	−	0.998339i	$-0.518350\pi$
$577$	−5.31721e8	−0.115231	−0.0576154	+	0.998339i	$0.518350\pi$
$578$	0	0
$579$	−1.63210e9	−0.349440
$580$	0	0
$581$	6.57418e9	1.39067
$582$	0	0
$583$	−7.70607e9	−1.61062
$584$	0	0
$585$	−8.99513e8	−0.185764
$586$	0	0
$587$	−8.17110e9	−1.66743	−0.833715	−	0.552195i	$-0.813790\pi$
$587$	−8.17110e9	−1.66743	−0.833715	+	0.552195i	$0.813790\pi$
$588$	0	0
$589$	−5.37765e9	−1.08440
$590$	0	0
$591$	−1.35123e8	−0.0269261
$592$	0	0
$593$	−1.59923e9	−0.314934	−0.157467	−	0.987524i	$-0.550333\pi$
$593$	−1.59923e9	−0.314934	−0.157467	+	0.987524i	$0.550333\pi$
$594$	0	0
$595$	1.09762e10	2.13620
$596$	0	0
$597$	−5.97064e9	−1.14845
$598$	0	0
$599$	8.80330e9	1.67360	0.836800	−	0.547509i	$-0.184424\pi$
$599$	8.80330e9	1.67360	0.836800	+	0.547509i	$0.184424\pi$
$600$	0	0
$601$	8.21235e9	1.54314	0.771572	−	0.636142i	$-0.219471\pi$
$601$	8.21235e9	1.54314	0.771572	+	0.636142i	$0.219471\pi$
$602$	0	0
$603$	−9.98284e8	−0.185414
$604$	0	0
$605$	3.58479e9	0.658143
$606$	0	0
$607$	−5.09192e9	−0.924105	−0.462052	−	0.886853i	$-0.652887\pi$
$607$	−5.09192e9	−0.924105	−0.462052	+	0.886853i	$0.652887\pi$
$608$	0	0
$609$	−4.71959e9	−0.846728
$610$	0	0
$611$	1.85381e9	0.328792
$612$	0	0
$613$	−7.01282e9	−1.22965	−0.614824	−	0.788664i	$-0.710773\pi$
$613$	−7.01282e9	−1.22965	−0.614824	+	0.788664i	$0.710773\pi$
$614$	0	0
$615$	−4.80856e9	−0.833589
$616$	0	0
$617$	4.63714e9	0.794789	0.397394	−	0.917648i	$-0.369914\pi$
$617$	4.63714e9	0.794789	0.397394	+	0.917648i	$0.369914\pi$
$618$	0	0
$619$	−1.33917e9	−0.226943	−0.113472	−	0.993541i	$-0.536197\pi$
$619$	−1.33917e9	−0.226943	−0.113472	+	0.993541i	$0.536197\pi$
$620$	0	0
$621$	4.37907e8	0.0733773
$622$	0	0
$623$	9.98298e8	0.165406
$624$	0	0
$625$	−5.66801e9	−0.928647
$626$	0	0
$627$	7.99631e9	1.29555
$628$	0	0
$629$	3.46694e9	0.555481
$630$	0	0
$631$	1.12354e10	1.78027	0.890133	−	0.455700i	$-0.150611\pi$
$631$	1.12354e10	1.78027	0.890133	+	0.455700i	$0.150611\pi$
$632$	0	0
$633$	−1.93493e9	−0.303216
$634$	0	0
$635$	6.13966e9	0.951561
$636$	0	0
$637$	−1.88741e9	−0.289321
$638$	0	0
$639$	−1.97854e9	−0.299979
$640$	0	0
$641$	9.82380e9	1.47325	0.736625	−	0.676301i	$-0.236418\pi$
$641$	9.82380e9	1.47325	0.736625	+	0.676301i	$0.236418\pi$
$642$	0	0
$643$	−3.56240e9	−0.528450	−0.264225	−	0.964461i	$-0.585116\pi$
$643$	−3.56240e9	−0.528450	−0.264225	+	0.964461i	$0.585116\pi$
$644$	0	0
$645$	−5.52378e8	−0.0810545
$646$	0	0
$647$	−5.91119e9	−0.858045	−0.429022	−	0.903294i	$-0.641142\pi$
$647$	−5.91119e9	−0.858045	−0.429022	+	0.903294i	$0.641142\pi$
$648$	0	0
$649$	7.46343e9	1.07172
$650$	0	0
$651$	3.12057e9	0.443303
$652$	0	0
$653$	−7.70538e8	−0.108292	−0.0541462	−	0.998533i	$-0.517244\pi$
$653$	−7.70538e8	−0.108292	−0.0541462	+	0.998533i	$0.517244\pi$
$654$	0	0
$655$	6.15016e9	0.855150
$656$	0	0
$657$	2.09135e9	0.287705
$658$	0	0
$659$	−4.95271e9	−0.674130	−0.337065	−	0.941481i	$-0.609434\pi$
$659$	−4.95271e9	−0.674130	−0.337065	+	0.941481i	$0.609434\pi$
$660$	0	0
$661$	−1.20319e9	−0.162042	−0.0810210	−	0.996712i	$-0.525818\pi$
$661$	−1.20319e9	−0.162042	−0.0810210	+	0.996712i	$0.525818\pi$
$662$	0	0
$663$	−4.51089e9	−0.601125
$664$	0	0
$665$	1.55344e10	2.04842
$666$	0	0
$667$	3.49725e9	0.456338
$668$	0	0
$669$	7.05888e8	0.0911474
$670$	0	0
$671$	1.04966e10	1.34128
$672$	0	0
$673$	1.30823e10	1.65436	0.827181	−	0.561936i	$-0.189943\pi$
$673$	1.30823e10	1.65436	0.827181	+	0.561936i	$0.189943\pi$
$674$	0	0
$675$	1.02844e8	0.0128711
$676$	0	0
$677$	−1.49380e10	−1.85025	−0.925126	−	0.379660i	$-0.876041\pi$
$677$	−1.49380e10	−1.85025	−0.925126	+	0.379660i	$0.876041\pi$
$678$	0	0
$679$	−1.52556e10	−1.87019
$680$	0	0
$681$	−9.20683e8	−0.111711
$682$	0	0
$683$	9.50415e9	1.14141	0.570704	−	0.821156i	$-0.306670\pi$
$683$	9.50415e9	1.14141	0.570704	+	0.821156i	$0.306670\pi$
$684$	0	0
$685$	1.12587e10	1.33835
$686$	0	0
$687$	−6.14055e9	−0.722535
$688$	0	0
$689$	6.15247e9	0.716609
$690$	0	0
$691$	−1.87893e9	−0.216639	−0.108320	−	0.994116i	$-0.534547\pi$
$691$	−1.87893e9	−0.216639	−0.108320	+	0.994116i	$0.534547\pi$
$692$	0	0
$693$	−4.64015e9	−0.529622
$694$	0	0
$695$	1.67257e9	0.188990
$696$	0	0
$697$	−2.41140e10	−2.69746
$698$	0	0
$699$	−1.53646e9	−0.170158
$700$	0	0
$701$	−1.06591e10	−1.16872	−0.584359	−	0.811496i	$-0.698654\pi$
$701$	−1.06591e10	−1.16872	−0.584359	+	0.811496i	$0.698654\pi$
$702$	0	0
$703$	4.90671e9	0.532656
$704$	0	0
$705$	2.95717e9	0.317845
$706$	0	0
$707$	−2.14954e9	−0.228758
$708$	0	0
$709$	−1.58464e10	−1.66982	−0.834911	−	0.550385i	$-0.814481\pi$
$709$	−1.58464e10	−1.66982	−0.834911	+	0.550385i	$0.814481\pi$
$710$	0	0
$711$	8.23513e8	0.0859265
$712$	0	0
$713$	−2.31237e9	−0.238915
$714$	0	0
$715$	−7.06284e9	−0.722617
$716$	0	0
$717$	8.17333e9	0.828098
$718$	0	0
$719$	1.78482e10	1.79079	0.895394	−	0.445274i	$-0.146894\pi$
$719$	1.78482e10	1.79079	0.895394	+	0.445274i	$0.146894\pi$
$720$	0	0
$721$	5.29453e9	0.526083
$722$	0	0
$723$	6.84979e8	0.0674051
$724$	0	0
$725$	8.21339e8	0.0800459
$726$	0	0
$727$	−1.41123e10	−1.36216	−0.681081	−	0.732208i	$-0.738490\pi$
$727$	−1.41123e10	−1.36216	−0.681081	+	0.732208i	$0.738490\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−2.77007e9	−0.262289
$732$	0	0
$733$	−5.55324e9	−0.520814	−0.260407	−	0.965499i	$-0.583857\pi$
$733$	−5.55324e9	−0.520814	−0.260407	+	0.965499i	$0.583857\pi$
$734$	0	0
$735$	−3.01078e9	−0.279687
$736$	0	0
$737$	−7.83838e9	−0.721257
$738$	0	0
$739$	−9.72614e9	−0.886513	−0.443256	−	0.896395i	$-0.646177\pi$
$739$	−9.72614e9	−0.886513	−0.443256	+	0.896395i	$0.646177\pi$
$740$	0	0
$741$	−6.38420e9	−0.576425
$742$	0	0
$743$	−8.37462e8	−0.0749038	−0.0374519	−	0.999298i	$-0.511924\pi$
$743$	−8.37462e8	−0.0749038	−0.0374519	+	0.999298i	$0.511924\pi$
$744$	0	0
$745$	−1.46248e9	−0.129581
$746$	0	0
$747$	−4.30987e9	−0.378305
$748$	0	0
$749$	2.45800e9	0.213745
$750$	0	0
$751$	3.05422e9	0.263124	0.131562	−	0.991308i	$-0.458001\pi$
$751$	3.05422e9	0.263124	0.131562	+	0.991308i	$0.458001\pi$
$752$	0	0
$753$	−3.95669e8	−0.0337715
$754$	0	0
$755$	−1.91280e10	−1.61754
$756$	0	0
$757$	1.77250e10	1.48508	0.742541	−	0.669800i	$-0.233620\pi$
$757$	1.77250e10	1.48508	0.742541	+	0.669800i	$0.233620\pi$
$758$	0	0
$759$	3.43838e9	0.285436
$760$	0	0
$761$	1.28268e9	0.105505	0.0527524	−	0.998608i	$-0.483201\pi$
$761$	1.28268e9	0.105505	0.0527524	+	0.998608i	$0.483201\pi$
$762$	0	0
$763$	2.23433e10	1.82100
$764$	0	0
$765$	−7.19571e9	−0.581110
$766$	0	0
$767$	−5.95875e9	−0.476839
$768$	0	0
$769$	1.50778e10	1.19562	0.597812	−	0.801636i	$-0.296037\pi$
$769$	1.50778e10	1.19562	0.597812	+	0.801636i	$0.296037\pi$
$770$	0	0
$771$	6.33561e8	0.0497849
$772$	0	0
$773$	1.26737e10	0.986902	0.493451	−	0.869774i	$-0.335735\pi$
$773$	1.26737e10	0.986902	0.493451	+	0.869774i	$0.335735\pi$
$774$	0	0
$775$	−5.43066e8	−0.0419079
$776$	0	0
$777$	−2.84730e9	−0.217750
$778$	0	0
$779$	−3.41282e10	−2.58662
$780$	0	0
$781$	−1.55352e10	−1.16691
$782$	0	0
$783$	3.09405e9	0.230336
$784$	0	0
$785$	−2.13516e10	−1.57539
$786$	0	0
$787$	1.46684e9	0.107268	0.0536342	−	0.998561i	$-0.482919\pi$
$787$	1.46684e9	0.107268	0.0536342	+	0.998561i	$0.482919\pi$
$788$	0	0
$789$	1.14907e10	0.832873
$790$	0	0
$791$	−8.83012e9	−0.634379
$792$	0	0
$793$	−8.38038e9	−0.596771
$794$	0	0
$795$	9.81434e9	0.692749
$796$	0	0
$797$	−1.49629e10	−1.04692	−0.523459	−	0.852051i	$-0.675359\pi$
$797$	−1.49629e10	−1.04692	−0.523459	+	0.852051i	$0.675359\pi$
$798$	0	0
$799$	1.48297e10	1.02853
$800$	0	0
$801$	−6.54460e8	−0.0449955
$802$	0	0
$803$	1.64210e10	1.11917
$804$	0	0
$805$	6.67974e9	0.451309
$806$	0	0
$807$	1.06127e10	0.710832
$808$	0	0
$809$	1.06263e10	0.705606	0.352803	−	0.935698i	$-0.385229\pi$
$809$	1.06263e10	0.705606	0.352803	+	0.935698i	$0.385229\pi$
$810$	0	0
$811$	2.44306e10	1.60828	0.804139	−	0.594442i	$-0.202627\pi$
$811$	2.44306e10	1.60828	0.804139	+	0.594442i	$0.202627\pi$
$812$	0	0
$813$	1.59181e9	0.103890
$814$	0	0
$815$	6.60127e9	0.427145
$816$	0	0
$817$	−3.92044e9	−0.251512
$818$	0	0
$819$	3.70466e9	0.235643
$820$	0	0
$821$	2.68568e10	1.69376	0.846882	−	0.531781i	$-0.178477\pi$
$821$	2.68568e10	1.69376	0.846882	+	0.531781i	$0.178477\pi$
$822$	0	0
$823$	−5.82914e9	−0.364506	−0.182253	−	0.983252i	$-0.558339\pi$
$823$	−5.82914e9	−0.364506	−0.182253	+	0.983252i	$0.558339\pi$
$824$	0	0
$825$	8.07513e8	0.0500681
$826$	0	0
$827$	1.31761e10	0.810062	0.405031	−	0.914303i	$-0.367261\pi$
$827$	1.31761e10	0.810062	0.405031	+	0.914303i	$0.367261\pi$
$828$	0	0
$829$	1.99634e10	1.21701	0.608504	−	0.793551i	$-0.291770\pi$
$829$	1.99634e10	1.21701	0.608504	+	0.793551i	$0.291770\pi$
$830$	0	0
$831$	5.83128e8	0.0352501
$832$	0	0
$833$	−1.50985e10	−0.905057
$834$	0	0
$835$	−9.59953e9	−0.570620
$836$	0	0
$837$	−2.04577e9	−0.120592
$838$	0	0
$839$	−2.19433e10	−1.28273	−0.641365	−	0.767236i	$-0.721632\pi$
$839$	−2.19433e10	−1.28273	−0.641365	+	0.767236i	$0.721632\pi$
$840$	0	0
$841$	7.46008e9	0.432471
$842$	0	0
$843$	9.03739e9	0.519573
$844$	0	0
$845$	−1.13032e10	−0.644469
$846$	0	0
$847$	−1.47640e10	−0.834859
$848$	0	0
$849$	8.82378e9	0.494854
$850$	0	0
$851$	2.10987e9	0.117355
$852$	0	0
$853$	−2.13102e10	−1.17562	−0.587808	−	0.809000i	$-0.700009\pi$
$853$	−2.13102e10	−1.17562	−0.587808	+	0.809000i	$0.700009\pi$
$854$	0	0
$855$	−1.01840e10	−0.557232
$856$	0	0
$857$	4.65996e9	0.252900	0.126450	−	0.991973i	$-0.459642\pi$
$857$	4.65996e9	0.252900	0.126450	+	0.991973i	$0.459642\pi$
$858$	0	0
$859$	1.68420e9	0.0906606	0.0453303	−	0.998972i	$-0.485566\pi$
$859$	1.68420e9	0.0906606	0.0453303	+	0.998972i	$0.485566\pi$
$860$	0	0
$861$	1.98041e10	1.05741
$862$	0	0
$863$	−1.08632e9	−0.0575336	−0.0287668	−	0.999586i	$-0.509158\pi$
$863$	−1.08632e9	−0.0575336	−0.0287668	+	0.999586i	$0.509158\pi$
$864$	0	0
$865$	2.77544e10	1.45806
$866$	0	0
$867$	−2.50060e10	−1.30310
$868$	0	0
$869$	6.46611e9	0.334252
$870$	0	0
$871$	6.25810e9	0.320907
$872$	0	0
$873$	1.00012e10	0.508748
$874$	0	0
$875$	2.50250e10	1.26283
$876$	0	0
$877$	1.11443e10	0.557897	0.278948	−	0.960306i	$-0.410014\pi$
$877$	1.11443e10	0.557897	0.278948	+	0.960306i	$0.410014\pi$
$878$	0	0
$879$	1.96820e10	0.977482
$880$	0	0
$881$	−3.78845e10	−1.86658	−0.933288	−	0.359130i	$-0.883074\pi$
$881$	−3.78845e10	−1.86658	−0.933288	+	0.359130i	$0.883074\pi$
$882$	0	0
$883$	−1.58131e10	−0.772956	−0.386478	−	0.922299i	$-0.626308\pi$
$883$	−1.58131e10	−0.772956	−0.386478	+	0.922299i	$0.626308\pi$
$884$	0	0
$885$	−9.50531e9	−0.460962
$886$	0	0
$887$	1.52171e10	0.732148	0.366074	−	0.930586i	$-0.380702\pi$
$887$	1.52171e10	0.732148	0.366074	+	0.930586i	$0.380702\pi$
$888$	0	0
$889$	−2.52863e10	−1.20706
$890$	0	0
$891$	3.04197e9	0.144073
$892$	0	0
$893$	2.09882e10	0.986270
$894$	0	0
$895$	1.12209e10	0.523173
$896$	0	0
$897$	−2.74518e9	−0.126998
$898$	0	0
$899$	−1.63381e10	−0.749969
$900$	0	0
$901$	4.92171e10	2.24171
$902$	0	0
$903$	2.27498e9	0.102818
$904$	0	0
$905$	−9.04534e9	−0.405653
$906$	0	0
$907$	1.52395e10	0.678181	0.339091	−	0.940754i	$-0.389881\pi$
$907$	1.52395e10	0.678181	0.339091	+	0.940754i	$0.389881\pi$
$908$	0	0
$909$	1.40918e9	0.0622292
$910$	0	0
$911$	1.49447e10	0.654895	0.327448	−	0.944869i	$-0.393811\pi$
$911$	1.49447e10	0.654895	0.327448	+	0.944869i	$0.393811\pi$
$912$	0	0
$913$	−3.38404e10	−1.47160
$914$	0	0
$915$	−1.33683e10	−0.576901
$916$	0	0
$917$	−2.53296e10	−1.08476
$918$	0	0
$919$	1.65291e10	0.702500	0.351250	−	0.936282i	$-0.385757\pi$
$919$	1.65291e10	0.702500	0.351250	+	0.936282i	$0.385757\pi$
$920$	0	0
$921$	−3.40306e9	−0.143536
$922$	0	0
$923$	1.24032e10	0.519190
$924$	0	0
$925$	4.95508e8	0.0205852
$926$	0	0
$927$	−3.47097e9	−0.143110
$928$	0	0
$929$	−2.58170e10	−1.05646	−0.528228	−	0.849102i	$-0.677143\pi$
$929$	−2.58170e10	−1.05646	−0.528228	+	0.849102i	$0.677143\pi$
$930$	0	0
$931$	−2.13687e10	−0.867868
$932$	0	0
$933$	−1.93257e10	−0.779023
$934$	0	0
$935$	−5.64997e10	−2.26050
$936$	0	0
$937$	−1.83428e10	−0.728411	−0.364206	−	0.931319i	$-0.618659\pi$
$937$	−1.83428e10	−0.728411	−0.364206	+	0.931319i	$0.618659\pi$
$938$	0	0
$939$	7.86229e9	0.309899
$940$	0	0
$941$	−6.33346e9	−0.247786	−0.123893	−	0.992296i	$-0.539538\pi$
$941$	−6.33346e9	−0.247786	−0.123893	+	0.992296i	$0.539538\pi$
$942$	0	0
$943$	−1.46750e10	−0.569885
$944$	0	0
$945$	5.90962e9	0.227797
$946$	0	0
$947$	−2.01958e10	−0.772744	−0.386372	−	0.922343i	$-0.626272\pi$
$947$	−2.01958e10	−0.772744	−0.386372	+	0.922343i	$0.626272\pi$
$948$	0	0
$949$	−1.31104e10	−0.497948
$950$	0	0
$951$	−1.14557e9	−0.0431906
$952$	0	0
$953$	4.86172e10	1.81955	0.909777	−	0.415096i	$-0.136252\pi$
$953$	4.86172e10	1.81955	0.909777	+	0.415096i	$0.136252\pi$
$954$	0	0
$955$	−1.32517e10	−0.492334
$956$	0	0
$957$	2.42940e10	0.895999
$958$	0	0
$959$	−4.63690e10	−1.69771
$960$	0	0
$961$	−1.67099e10	−0.607355
$962$	0	0
$963$	−1.61141e9	−0.0581451
$964$	0	0
$965$	1.63210e10	0.584659
$966$	0	0
$967$	−1.23778e10	−0.440201	−0.220100	−	0.975477i	$-0.570638\pi$
$967$	−1.23778e10	−0.440201	−0.220100	+	0.975477i	$0.570638\pi$
$968$	0	0
$969$	−5.10708e10	−1.80318
$970$	0	0
$971$	1.70597e10	0.598003	0.299001	−	0.954253i	$-0.403346\pi$
$971$	1.70597e10	0.598003	0.299001	+	0.954253i	$0.403346\pi$
$972$	0	0
$973$	−6.88852e9	−0.239735
$974$	0	0
$975$	−6.44713e8	−0.0222767
$976$	0	0
$977$	3.96945e10	1.36176	0.680878	−	0.732397i	$-0.261598\pi$
$977$	3.96945e10	1.36176	0.680878	+	0.732397i	$0.261598\pi$
$978$	0	0
$979$	−5.13872e9	−0.175031
$980$	0	0
$981$	−1.46477e10	−0.495368
$982$	0	0
$983$	7.55397e7	0.00253652	0.00126826	−	0.999999i	$-0.499596\pi$
$983$	7.55397e7	0.00253652	0.00126826	+	0.999999i	$0.499596\pi$
$984$	0	0
$985$	1.35123e9	0.0450508
$986$	0	0
$987$	−1.21792e10	−0.403188
$988$	0	0
$989$	−1.68578e9	−0.0554131
$990$	0	0
$991$	2.99026e10	0.976002	0.488001	−	0.872843i	$-0.337726\pi$
$991$	2.99026e10	0.976002	0.488001	+	0.872843i	$0.337726\pi$
$992$	0	0
$993$	−1.47759e10	−0.478886
$994$	0	0
$995$	5.97064e10	1.92150
$996$	0	0
$997$	1.24254e10	0.397081	0.198540	−	0.980093i	$-0.436380\pi$
$997$	1.24254e10	0.397081	0.198540	+	0.980093i	$0.436380\pi$
$998$	0	0
$999$	1.86662e9	0.0592347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.8.a.d.1.1		1
3.2	odd	2		144.8.a.c.1.1		1
4.3	odd	2		12.8.a.b.1.1	✓	1
8.3	odd	2		192.8.a.b.1.1		1
8.5	even	2		192.8.a.j.1.1		1
12.11	even	2		36.8.a.a.1.1		1
20.3	even	4		300.8.d.a.49.2		2
20.7	even	4		300.8.d.a.49.1		2
20.19	odd	2		300.8.a.a.1.1		1
24.5	odd	2		576.8.a.u.1.1		1
24.11	even	2		576.8.a.v.1.1		1
28.3	even	6		588.8.i.g.373.1		2
28.11	odd	6		588.8.i.b.373.1		2
28.19	even	6		588.8.i.g.361.1		2
28.23	odd	6		588.8.i.b.361.1		2
28.27	even	2		588.8.a.a.1.1		1
36.7	odd	6		324.8.e.b.109.1		2
36.11	even	6		324.8.e.e.109.1		2
36.23	even	6		324.8.e.e.217.1		2
36.31	odd	6		324.8.e.b.217.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.8.a.b.1.1	✓	1	4.3	odd	2
36.8.a.a.1.1		1	12.11	even	2
48.8.a.d.1.1		1	1.1	even	1	trivial
144.8.a.c.1.1		1	3.2	odd	2
192.8.a.b.1.1		1	8.3	odd	2
192.8.a.j.1.1		1	8.5	even	2
300.8.a.a.1.1		1	20.19	odd	2
300.8.d.a.49.1		2	20.7	even	4
300.8.d.a.49.2		2	20.3	even	4
324.8.e.b.109.1		2	36.7	odd	6
324.8.e.b.217.1		2	36.31	odd	6
324.8.e.e.109.1		2	36.11	even	6
324.8.e.e.217.1		2	36.23	even	6
576.8.a.u.1.1		1	24.5	odd	2
576.8.a.v.1.1		1	24.11	even	2
588.8.a.a.1.1		1	28.27	even	2
588.8.i.b.361.1		2	28.23	odd	6
588.8.i.b.373.1		2	28.11	odd	6
588.8.i.g.361.1		2	28.19	even	6
588.8.i.g.373.1		2	28.3	even	6

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

Related objects

Downloads

Learn more