LMFDB - Embedded newform 48.17.g.a.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,17,Mod(31,48)]

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([1, 0, 0]))

N = Newforms(chi, 17, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 17);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 17 $
Character orbit:	$[\chi]$	$=$	48.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$77.9157810512$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 379501x^{2} + 379500x + 144020250000 $ x^4 - x^3 + 379501x^2 + 379500x + 144020250000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.1
Root			$308.268 + 533.936i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.17.g.a.31.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3788.00i q^{3} -360373. q^{5} -5.22136e6i q^{7} -1.43489e7 q^{9} +O(q^{10})$	$q-3788.00i q^{3} -360373. q^{5} -5.22136e6i q^{7} -1.43489e7 q^{9} +8.78416e7i q^{11} -6.19491e8 q^{13} +1.36509e9i q^{15} -6.62095e9 q^{17} -3.53050e9i q^{19} -1.97785e10 q^{21} -5.95950e10i q^{23} -2.27192e10 q^{25} +5.43536e10i q^{27} -2.15437e11 q^{29} +1.08591e12i q^{31} +3.32744e11 q^{33} +1.88164e12i q^{35} +2.55550e12 q^{37} +2.34663e12i q^{39} -1.08366e13 q^{41} -8.66774e12i q^{43} +5.17096e12 q^{45} -1.07301e13i q^{47} +5.97034e12 q^{49} +2.50801e13i q^{51} -2.35418e11 q^{53} -3.16557e13i q^{55} -1.33735e13 q^{57} +2.62648e13i q^{59} +3.38835e14 q^{61} +7.49208e13i q^{63} +2.23248e14 q^{65} -4.91860e14i q^{67} -2.25746e14 q^{69} +1.07798e15i q^{71} -2.37507e14 q^{73} +8.60603e13i q^{75} +4.58653e14 q^{77} +1.62935e15i q^{79} +2.05891e14 q^{81} +3.26625e14i q^{83} +2.38601e15 q^{85} +8.16075e14i q^{87} +7.39996e15 q^{89} +3.23459e15i q^{91} +4.11341e15 q^{93} +1.27229e15i q^{95} +1.26360e16 q^{97} -1.26043e15i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 140424 q^{5} - 57395628 q^{9}+O(q^{10}) $ 4 * q - 140424 * q^5 - 57395628 * q^9	$ 4 q - 140424 q^{5} - 57395628 q^{9} - 586685528 q^{13} - 571005336 q^{17} - 8495392752 q^{21} - 182227446772 q^{25} - 585621416424 q^{29} - 176071520880 q^{33} - 2770470511096 q^{37} - 9744697113624 q^{41} + 2014930916568 q^{45} + 44786525712964 q^{49} + 93950973394200 q^{53} + 40139973246384 q^{57} + 374162346312392 q^{61} + 635766937476528 q^{65} + 165370244765184 q^{69} + 13\!\cdots\!20 q^{73}+ \cdots + 45\!\cdots\!40 q^{97}+O(q^{100}) $ 4 * q - 140424 * q^5 - 57395628 * q^9 - 586685528 * q^13 - 571005336 * q^17 - 8495392752 * q^21 - 182227446772 * q^25 - 585621416424 * q^29 - 176071520880 * q^33 - 2770470511096 * q^37 - 9744697113624 * q^41 + 2014930916568 * q^45 + 44786525712964 * q^49 + 93950973394200 * q^53 + 40139973246384 * q^57 + 374162346312392 * q^61 + 635766937476528 * q^65 + 165370244765184 * q^69 + 1379949716544520 * q^73 + 1828180800500544 * q^77 + 823564528378596 * q^81 + 8448616033483824 * q^85 + 20666558628633864 * q^89 + 7887770592692400 * q^93 + 45712018082632840 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 3788.00i	− 0.577350i
$3$	− 3788.00i	− 0.577350i
$4$	0	0
$5$	−360373.	−0.922555	−0.461277	−	0.887256i	$-0.652609\pi$
$5$	−360373.	−0.922555	−0.461277	+	0.887256i	$0.652609\pi$
$6$	0	0
$7$	− 5.22136e6i	− 0.905731i	−0.891579	−	0.452865i	$-0.850402\pi$
$7$	− 5.22136e6i	− 0.905731i	0.891579	−	0.452865i	$-0.149598\pi$
$8$	0	0
$9$	−1.43489e7	−0.333333
$10$	0	0
$11$	8.78416e7i	0.409788i	0.978784	+	0.204894i	$0.0656849\pi$
$11$	8.78416e7i	0.409788i	−0.978784	+	0.204894i	$0.934315\pi$
$12$	0	0
$13$	−6.19491e8	−0.759431	−0.379716	−	0.925103i	$-0.623978\pi$
$13$	−6.19491e8	−0.759431	−0.379716	+	0.925103i	$0.623978\pi$
$14$	0	0
$15$	1.36509e9i	0.532637i
$16$	0	0
$17$	−6.62095e9	−0.949136	−0.474568	−	0.880219i	$-0.657396\pi$
$17$	−6.62095e9	−0.949136	−0.474568	+	0.880219i	$0.657396\pi$
$18$	0	0
$19$	− 3.53050e9i	− 0.207877i	−0.994584	−	0.103939i	$-0.966855\pi$
$19$	− 3.53050e9i	− 0.207877i	0.994584	−	0.103939i	$-0.0331445\pi$
$20$	0	0
$21$	−1.97785e10	−0.522924
$22$	0	0
$23$	− 5.95950e10i	− 0.761005i	−0.924780	−	0.380502i	$-0.875751\pi$
$23$	− 5.95950e10i	− 0.761005i	0.924780	−	0.380502i	$-0.124249\pi$
$24$	0	0
$25$	−2.27192e10	−0.148893
$26$	0	0
$27$	5.43536e10i	0.192450i
$28$	0	0
$29$	−2.15437e11	−0.430662	−0.215331	−	0.976541i	$-0.569083\pi$
$29$	−2.15437e11	−0.430662	−0.215331	+	0.976541i	$0.569083\pi$
$30$	0	0
$31$	1.08591e12i	1.27321i	0.771191	+	0.636603i	$0.219661\pi$
$31$	1.08591e12i	1.27321i	−0.771191	+	0.636603i	$0.780339\pi$
$32$	0	0
$33$	3.32744e11	0.236591
$34$	0	0
$35$	1.88164e12i	0.835586i
$36$	0	0
$37$	2.55550e12	0.727549	0.363774	−	0.931487i	$-0.381488\pi$
$37$	2.55550e12	0.727549	0.363774	+	0.931487i	$0.381488\pi$
$38$	0	0
$39$	2.34663e12i	0.438458i
$40$	0	0
$41$	−1.08366e13	−1.35713	−0.678566	−	0.734540i	$-0.737398\pi$
$41$	−1.08366e13	−1.35713	−0.678566	+	0.734540i	$0.737398\pi$
$42$	0	0
$43$	− 8.66774e12i	− 0.741580i	−0.928717	−	0.370790i	$-0.879087\pi$
$43$	− 8.66774e12i	− 0.741580i	0.928717	−	0.370790i	$-0.120913\pi$
$44$	0	0
$45$	5.17096e12	0.307518
$46$	0	0
$47$	− 1.07301e13i	− 0.450631i	−0.974286	−	0.225316i	$-0.927659\pi$
$47$	− 1.07301e13i	− 0.450631i	0.974286	−	0.225316i	$-0.0723414\pi$
$48$	0	0
$49$	5.97034e12	0.179651
$50$	0	0
$51$	2.50801e13i	0.547984i
$52$	0	0
$53$	−2.35418e11	−0.00378122	−0.00189061	−	0.999998i	$-0.500602\pi$
$53$	−2.35418e11	−0.00378122	−0.00189061	+	0.999998i	$0.500602\pi$
$54$	0	0
$55$	− 3.16557e13i	− 0.378052i
$56$	0	0
$57$	−1.33735e13	−0.120018
$58$	0	0
$59$	2.62648e13i	0.178878i	0.995992	+	0.0894392i	$0.0285075\pi$
$59$	2.62648e13i	0.178878i	−0.995992	+	0.0894392i	$0.971493\pi$
$60$	0	0
$61$	3.38835e14	1.76746	0.883731	−	0.467995i	$-0.155024\pi$
$61$	3.38835e14	1.76746	0.883731	+	0.467995i	$0.155024\pi$
$62$	0	0
$63$	7.49208e13i	0.301910i
$64$	0	0
$65$	2.23248e14	0.700617
$66$	0	0
$67$	− 4.91860e14i	− 1.21128i	−0.795740	−	0.605638i	$-0.792918\pi$
$67$	− 4.91860e14i	− 1.21128i	0.795740	−	0.605638i	$-0.207082\pi$
$68$	0	0
$69$	−2.25746e14	−0.439366
$70$	0	0
$71$	1.07798e15i	1.66934i	0.550753	+	0.834668i	$0.314341\pi$
$71$	1.07798e15i	1.66934i	−0.550753	+	0.834668i	$0.685659\pi$
$72$	0	0
$73$	−2.37507e14	−0.294505	−0.147253	−	0.989099i	$-0.547043\pi$
$73$	−2.37507e14	−0.294505	−0.147253	+	0.989099i	$0.547043\pi$
$74$	0	0
$75$	8.60603e13i	0.0859632i
$76$	0	0
$77$	4.58653e14	0.371157
$78$	0	0
$79$	1.62935e15i	1.07398i	0.843588	+	0.536991i	$0.180439\pi$
$79$	1.62935e15i	1.07398i	−0.843588	+	0.536991i	$0.819561\pi$
$80$	0	0
$81$	2.05891e14	0.111111
$82$	0	0
$83$	3.26625e14i	0.145019i	0.997368	+	0.0725094i	$0.0231007\pi$
$83$	3.26625e14i	0.145019i	−0.997368	+	0.0725094i	$0.976899\pi$
$84$	0	0
$85$	2.38601e15	0.875630
$86$	0	0
$87$	8.16075e14i	0.248643i
$88$	0	0
$89$	7.39996e15	1.87979	0.939895	−	0.341464i	$-0.110923\pi$
$89$	7.39996e15	1.87979	0.939895	+	0.341464i	$0.110923\pi$
$90$	0	0
$91$	3.23459e15i	0.687840i
$92$	0	0
$93$	4.11341e15	0.735086
$94$	0	0
$95$	1.27229e15i	0.191778i
$96$	0	0
$97$	1.26360e16	1.61226	0.806132	−	0.591735i	$-0.201557\pi$
$97$	1.26360e16	1.61226	0.806132	+	0.591735i	$0.201557\pi$
$98$	0	0
$99$	− 1.26043e15i	− 0.136596i
$100$	0	0
$101$	−7.85837e14	−0.0725707	−0.0362853	−	0.999341i	$-0.511553\pi$
$101$	−7.85837e14	−0.0725707	−0.0362853	+	0.999341i	$0.511553\pi$
$102$	0	0
$103$	1.58920e16i	1.25453i	0.778807	+	0.627264i	$0.215825\pi$
$103$	1.58920e16i	1.25453i	−0.778807	+	0.627264i	$0.784175\pi$
$104$	0	0
$105$	7.12763e15	0.482426
$106$	0	0
$107$	− 2.80800e15i	− 0.163428i	−0.996656	−	0.0817141i	$-0.973961\pi$
$107$	− 2.80800e15i	− 0.163428i	0.996656	−	0.0817141i	$-0.0260394\pi$
$108$	0	0
$109$	5.39528e15	0.270771	0.135385	−	0.990793i	$-0.456773\pi$
$109$	5.39528e15	0.270771	0.135385	+	0.990793i	$0.456773\pi$
$110$	0	0
$111$	− 9.68022e15i	− 0.420050i
$112$	0	0
$113$	3.50564e16	1.31868	0.659340	−	0.751845i	$-0.270836\pi$
$113$	3.50564e16	1.31868	0.659340	+	0.751845i	$0.270836\pi$
$114$	0	0
$115$	2.14764e16i	0.702068i
$116$	0	0
$117$	8.88902e15	0.253144
$118$	0	0
$119$	3.45703e16i	0.859662i
$120$	0	0
$121$	3.82336e16	0.832074
$122$	0	0
$123$	4.10490e16i	0.783540i
$124$	0	0
$125$	6.31759e16	1.05992
$126$	0	0
$127$	− 1.12415e16i	− 0.166110i	−0.996545	−	0.0830550i	$-0.973532\pi$
$127$	− 1.12415e16i	− 0.166110i	0.996545	−	0.0830550i	$-0.0264677\pi$
$128$	0	0
$129$	−3.28334e16	−0.428152
$130$	0	0
$131$	9.97967e16i	1.15066i	0.817923	+	0.575328i	$0.195126\pi$
$131$	9.97967e16i	1.15066i	−0.817923	+	0.575328i	$0.804874\pi$
$132$	0	0
$133$	−1.84340e16	−0.188281
$134$	0	0
$135$	− 1.95876e16i	− 0.177546i
$136$	0	0
$137$	−4.85606e16	−0.391308	−0.195654	−	0.980673i	$-0.562683\pi$
$137$	−4.85606e16	−0.391308	−0.195654	+	0.980673i	$0.562683\pi$
$138$	0	0
$139$	− 7.44929e16i	− 0.534560i	−0.963619	−	0.267280i	$-0.913875\pi$
$139$	− 7.44929e16i	− 0.534560i	0.963619	−	0.267280i	$-0.0861249\pi$
$140$	0	0
$141$	−4.06456e16	−0.260172
$142$	0	0
$143$	− 5.44171e16i	− 0.311205i
$144$	0	0
$145$	7.76377e16	0.397309
$146$	0	0
$147$	− 2.26156e16i	− 0.103722i
$148$	0	0
$149$	−2.86718e17	−1.18023	−0.590113	−	0.807320i	$-0.700917\pi$
$149$	−2.86718e17	−1.18023	−0.590113	+	0.807320i	$0.700917\pi$
$150$	0	0
$151$	− 3.85192e17i	− 1.42515i	−0.701594	−	0.712577i	$-0.747528\pi$
$151$	− 3.85192e17i	− 1.42515i	0.701594	−	0.712577i	$-0.252472\pi$
$152$	0	0
$153$	9.50033e16	0.316379
$154$	0	0
$155$	− 3.91331e17i	− 1.17460i
$156$	0	0
$157$	−2.55142e17	−0.691169	−0.345585	−	0.938388i	$-0.612319\pi$
$157$	−2.55142e17	−0.691169	−0.345585	+	0.938388i	$0.612319\pi$
$158$	0	0
$159$	8.91761e14i	0.00218309i
$160$	0	0
$161$	−3.11167e17	−0.689265
$162$	0	0
$163$	2.33321e17i	0.468223i	0.972210	+	0.234111i	$0.0752180\pi$
$163$	2.33321e17i	0.468223i	−0.972210	+	0.234111i	$0.924782\pi$
$164$	0	0
$165$	−1.19912e17	−0.218268
$166$	0	0
$167$	3.51040e17i	0.580262i	0.956987	+	0.290131i	$0.0936989\pi$
$167$	3.51040e17i	0.580262i	−0.956987	+	0.290131i	$0.906301\pi$
$168$	0	0
$169$	−2.81647e17	−0.423264
$170$	0	0
$171$	5.06587e16i	0.0692924i
$172$	0	0
$173$	−6.54566e17	−0.815801	−0.407901	−	0.913026i	$-0.633739\pi$
$173$	−6.54566e17	−0.815801	−0.407901	+	0.913026i	$0.633739\pi$
$174$	0	0
$175$	1.18625e17i	0.134857i
$176$	0	0
$177$	9.94909e16	0.103276
$178$	0	0
$179$	− 1.27709e18i	− 1.21170i	−0.795578	−	0.605851i	$-0.792833\pi$
$179$	− 1.27709e18i	− 1.21170i	0.795578	−	0.605851i	$-0.207167\pi$
$180$	0	0
$181$	−2.13681e18	−1.85497	−0.927486	−	0.373858i	$-0.878035\pi$
$181$	−2.13681e18	−1.85497	−0.927486	+	0.373858i	$0.878035\pi$
$182$	0	0
$183$	− 1.28351e18i	− 1.02044i
$184$	0	0
$185$	−9.20933e17	−0.671203
$186$	0	0
$187$	− 5.81595e17i	− 0.388944i
$188$	0	0
$189$	2.83800e17	0.174308
$190$	0	0
$191$	1.99687e18i	1.12741i	0.825976	+	0.563706i	$0.190625\pi$
$191$	1.99687e18i	1.12741i	−0.825976	+	0.563706i	$0.809375\pi$
$192$	0	0
$193$	−1.72651e17	−0.0896832	−0.0448416	−	0.998994i	$-0.514278\pi$
$193$	−1.72651e17	−0.0896832	−0.0448416	+	0.998994i	$0.514278\pi$
$194$	0	0
$195$	− 8.45662e17i	− 0.404501i
$196$	0	0
$197$	9.90654e17	0.436709	0.218354	−	0.975870i	$-0.429931\pi$
$197$	9.90654e17	0.436709	0.218354	+	0.975870i	$0.429931\pi$
$198$	0	0
$199$	− 2.53456e17i	− 0.103057i	−0.998672	−	0.0515286i	$-0.983591\pi$
$199$	− 2.53456e17i	− 0.103057i	0.998672	−	0.0515286i	$-0.0164093\pi$
$200$	0	0
$201$	−1.86316e18	−0.699331
$202$	0	0
$203$	1.12487e18i	0.390064i
$204$	0	0
$205$	3.90522e18	1.25203
$206$	0	0
$207$	8.55123e17i	0.253668i
$208$	0	0
$209$	3.10124e17	0.0851855
$210$	0	0
$211$	− 1.56670e18i	− 0.398773i	−0.979921	−	0.199386i	$-0.936105\pi$
$211$	− 1.56670e18i	− 0.398773i	0.979921	−	0.199386i	$-0.0638948\pi$
$212$	0	0
$213$	4.08338e18	0.963792
$214$	0	0
$215$	3.12362e18i	0.684148i
$216$	0	0
$217$	5.66991e18	1.15318
$218$	0	0
$219$	8.99674e17i	0.170033i
$220$	0	0
$221$	4.10162e18	0.720804
$222$	0	0
$223$	− 3.77973e18i	− 0.618047i	−0.951054	−	0.309023i	$-0.899998\pi$
$223$	− 3.77973e18i	− 0.618047i	0.951054	−	0.309023i	$-0.100002\pi$
$224$	0	0
$225$	3.25996e17	0.0496309
$226$	0	0
$227$	5.04768e18i	0.715953i	0.933731	+	0.357977i	$0.116533\pi$
$227$	5.04768e18i	0.715953i	−0.933731	+	0.357977i	$0.883467\pi$
$228$	0	0
$229$	1.53931e18	0.203536	0.101768	−	0.994808i	$-0.467550\pi$
$229$	1.53931e18	0.203536	0.101768	+	0.994808i	$0.467550\pi$
$230$	0	0
$231$	− 1.73737e18i	− 0.214288i
$232$	0	0
$233$	−1.07934e19	−1.24254	−0.621269	−	0.783597i	$-0.713383\pi$
$233$	−1.07934e19	−1.24254	−0.621269	+	0.783597i	$0.713383\pi$
$234$	0	0
$235$	3.86684e18i	0.415732i
$236$	0	0
$237$	6.17196e18	0.620064
$238$	0	0
$239$	1.56042e19i	1.46575i	0.680365	+	0.732874i	$0.261821\pi$
$239$	1.56042e19i	1.46575i	−0.680365	+	0.732874i	$0.738179\pi$
$240$	0	0
$241$	1.65950e19	1.45828	0.729141	−	0.684363i	$-0.239920\pi$
$241$	1.65950e19	1.45828	0.729141	+	0.684363i	$0.239920\pi$
$242$	0	0
$243$	− 7.79915e17i	− 0.0641500i
$244$	0	0
$245$	−2.15155e18	−0.165738
$246$	0	0
$247$	2.18711e18i	0.157868i
$248$	0	0
$249$	1.23725e18	0.0837266
$250$	0	0
$251$	2.87255e19i	1.82338i	0.410878	+	0.911690i	$0.365222\pi$
$251$	2.87255e19i	1.82338i	−0.410878	+	0.911690i	$0.634778\pi$
$252$	0	0
$253$	5.23492e18	0.311850
$254$	0	0
$255$	− 9.03819e18i	− 0.505545i
$256$	0	0
$257$	1.31331e19	0.690085	0.345042	−	0.938587i	$-0.387865\pi$
$257$	1.31331e19	0.690085	0.345042	+	0.938587i	$0.387865\pi$
$258$	0	0
$259$	− 1.33432e19i	− 0.658963i
$260$	0	0
$261$	3.09129e18	0.143554
$262$	0	0
$263$	− 4.65385e18i	− 0.203314i	−0.994820	−	0.101657i	$-0.967586\pi$
$263$	− 4.65385e18i	− 0.203314i	0.994820	−	0.101657i	$-0.0324144\pi$
$264$	0	0
$265$	8.48381e16	0.00348838
$266$	0	0
$267$	− 2.80310e19i	− 1.08530i
$268$	0	0
$269$	6.70644e18	0.244610	0.122305	−	0.992493i	$-0.460971\pi$
$269$	6.70644e18	0.244610	0.122305	+	0.992493i	$0.460971\pi$
$270$	0	0
$271$	1.37433e19i	0.472428i	0.971701	+	0.236214i	$0.0759067\pi$
$271$	1.37433e19i	0.472428i	−0.971701	+	0.236214i	$0.924093\pi$
$272$	0	0
$273$	1.22526e19	0.397125
$274$	0	0
$275$	− 1.99569e18i	− 0.0610144i
$276$	0	0
$277$	−3.77201e19	−1.08827	−0.544133	−	0.838999i	$-0.683141\pi$
$277$	−3.77201e19	−1.08827	−0.544133	+	0.838999i	$0.683141\pi$
$278$	0	0
$279$	− 1.55816e19i	− 0.424402i
$280$	0	0
$281$	−5.98631e19	−1.53996	−0.769979	−	0.638069i	$-0.779733\pi$
$281$	−5.98631e19	−1.53996	−0.769979	+	0.638069i	$0.779733\pi$
$282$	0	0
$283$	− 6.75587e19i	− 1.64206i	−0.570883	−	0.821031i	$-0.693399\pi$
$283$	− 6.75587e19i	− 1.64206i	0.570883	−	0.821031i	$-0.306601\pi$
$284$	0	0
$285$	4.81945e18	0.110723
$286$	0	0
$287$	5.65818e19i	1.22920i
$288$	0	0
$289$	−4.82427e18	−0.0991400
$290$	0	0
$291$	− 4.78652e19i	− 0.930841i
$292$	0	0
$293$	−7.68743e19	−1.41527	−0.707637	−	0.706576i	$-0.750239\pi$
$293$	−7.68743e19	−1.41527	−0.707637	+	0.706576i	$0.750239\pi$
$294$	0	0
$295$	− 9.46513e18i	− 0.165025i
$296$	0	0
$297$	−4.77451e18	−0.0788637
$298$	0	0
$299$	3.69186e19i	0.577931i
$300$	0	0
$301$	−4.52574e19	−0.671672
$302$	0	0
$303$	2.97674e18i	0.0418987i
$304$	0	0
$305$	−1.22107e20	−1.63058
$306$	0	0
$307$	− 1.35153e20i	− 1.71284i	−0.516276	−	0.856422i	$-0.672682\pi$
$307$	− 1.35153e20i	− 1.71284i	0.516276	−	0.856422i	$-0.327318\pi$
$308$	0	0
$309$	6.01988e19	0.724302
$310$	0	0
$311$	1.07981e20i	1.23386i	0.787020	+	0.616928i	$0.211623\pi$
$311$	1.07981e20i	1.23386i	−0.787020	+	0.616928i	$0.788377\pi$
$312$	0	0
$313$	1.37030e20	1.48752	0.743759	−	0.668448i	$-0.233041\pi$
$313$	1.37030e20	1.48752	0.743759	+	0.668448i	$0.233041\pi$
$314$	0	0
$315$	− 2.69994e19i	− 0.278529i
$316$	0	0
$317$	−1.08994e20	−1.06888	−0.534439	−	0.845207i	$-0.679477\pi$
$317$	−1.08994e20	−1.06888	−0.534439	+	0.845207i	$0.679477\pi$
$318$	0	0
$319$	− 1.89243e19i	− 0.176480i
$320$	0	0
$321$	−1.06367e19	−0.0943553
$322$	0	0
$323$	2.33752e19i	0.197304i
$324$	0	0
$325$	1.40744e19	0.113074
$326$	0	0
$327$	− 2.04373e19i	− 0.156330i
$328$	0	0
$329$	−5.60258e19	−0.408151
$330$	0	0
$331$	− 3.08309e18i	− 0.0213975i	−0.999943	−	0.0106987i	$-0.996594\pi$
$331$	− 3.08309e18i	− 0.0213975i	0.999943	−	0.0106987i	$-0.00340558\pi$
$332$	0	0
$333$	−3.66686e19	−0.242516
$334$	0	0
$335$	1.77253e20i	1.11747i
$336$	0	0
$337$	−1.44317e20	−0.867520	−0.433760	−	0.901029i	$-0.642813\pi$
$337$	−1.44317e20	−0.867520	−0.433760	+	0.901029i	$0.642813\pi$
$338$	0	0
$339$	− 1.32793e20i	− 0.761340i
$340$	0	0
$341$	−9.53878e19	−0.521744
$342$	0	0
$343$	− 2.04694e20i	− 1.06845i
$344$	0	0
$345$	8.13526e19	0.405339
$346$	0	0
$347$	1.64935e20i	0.784654i	0.919826	+	0.392327i	$0.128330\pi$
$347$	1.64935e20i	0.784654i	−0.919826	+	0.392327i	$0.871670\pi$
$348$	0	0
$349$	2.35205e20	1.06867	0.534333	−	0.845274i	$-0.320563\pi$
$349$	2.35205e20	1.06867	0.534333	+	0.845274i	$0.320563\pi$
$350$	0	0
$351$	− 3.36716e19i	− 0.146153i
$352$	0	0
$353$	3.33026e20	1.38127	0.690637	−	0.723201i	$-0.257330\pi$
$353$	3.33026e20	1.38127	0.690637	+	0.723201i	$0.257330\pi$
$354$	0	0
$355$	− 3.88475e20i	− 1.54005i
$356$	0	0
$357$	1.30952e20	0.496326
$358$	0	0
$359$	4.21282e20i	1.52692i	0.645853	+	0.763462i	$0.276502\pi$
$359$	4.21282e20i	1.52692i	−0.645853	+	0.763462i	$0.723498\pi$
$360$	0	0
$361$	2.75977e20	0.956787
$362$	0	0
$363$	− 1.44829e20i	− 0.480398i
$364$	0	0
$365$	8.55910e19	0.271697
$366$	0	0
$367$	6.13373e20i	1.86379i	0.362732	+	0.931894i	$0.381844\pi$
$367$	6.13373e20i	1.86379i	−0.362732	+	0.931894i	$0.618156\pi$
$368$	0	0
$369$	1.55493e20	0.452377
$370$	0	0
$371$	1.22920e18i	0.00342477i
$372$	0	0
$373$	2.17396e20	0.580203	0.290102	−	0.956996i	$-0.406311\pi$
$373$	2.17396e20	0.580203	0.290102	+	0.956996i	$0.406311\pi$
$374$	0	0
$375$	− 2.39310e20i	− 0.611943i
$376$	0	0
$377$	1.33461e20	0.327058
$378$	0	0
$379$	4.69384e20i	1.10259i	0.834310	+	0.551296i	$0.185866\pi$
$379$	4.69384e20i	1.10259i	−0.834310	+	0.551296i	$0.814134\pi$
$380$	0	0
$381$	−4.25829e19	−0.0959037
$382$	0	0
$383$	6.41933e20i	1.38644i	0.720728	+	0.693218i	$0.243807\pi$
$383$	6.41933e20i	1.38644i	−0.720728	+	0.693218i	$0.756193\pi$
$384$	0	0
$385$	−1.65286e20	−0.342413
$386$	0	0
$387$	1.24373e20i	0.247193i
$388$	0	0
$389$	3.55635e19	0.0678278	0.0339139	−	0.999425i	$-0.489203\pi$
$389$	3.55635e19	0.0678278	0.0339139	+	0.999425i	$0.489203\pi$
$390$	0	0
$391$	3.94575e20i	0.722297i
$392$	0	0
$393$	3.78029e20	0.664332
$394$	0	0
$395$	− 5.87173e20i	− 0.990808i
$396$	0	0
$397$	5.57406e20	0.903333	0.451666	−	0.892187i	$-0.350830\pi$
$397$	5.57406e20	0.903333	0.451666	+	0.892187i	$0.350830\pi$
$398$	0	0
$399$	6.98278e19i	0.108704i
$400$	0	0
$401$	−7.48750e20	−1.11991	−0.559953	−	0.828524i	$-0.689181\pi$
$401$	−7.48750e20	−1.11991	−0.559953	+	0.828524i	$0.689181\pi$
$402$	0	0
$403$	− 6.72710e20i	− 0.966913i
$404$	0	0
$405$	−7.41976e19	−0.102506
$406$	0	0
$407$	2.24479e20i	0.298140i
$408$	0	0
$409$	−7.57698e20	−0.967631	−0.483816	−	0.875170i	$-0.660749\pi$
$409$	−7.57698e20	−0.967631	−0.483816	+	0.875170i	$0.660749\pi$
$410$	0	0
$411$	1.83947e20i	0.225922i
$412$	0	0
$413$	1.37138e20	0.162016
$414$	0	0
$415$	− 1.17707e20i	− 0.133788i
$416$	0	0
$417$	−2.82179e20	−0.308628
$418$	0	0
$419$	− 9.76317e20i	− 1.02773i	−0.857871	−	0.513864i	$-0.828213\pi$
$419$	− 9.76317e20i	− 1.02773i	0.857871	−	0.513864i	$-0.171787\pi$
$420$	0	0
$421$	1.18650e21	1.20230	0.601148	−	0.799138i	$-0.294710\pi$
$421$	1.18650e21	1.20230	0.601148	+	0.799138i	$0.294710\pi$
$422$	0	0
$423$	1.53965e20i	0.150210i
$424$	0	0
$425$	1.50423e20	0.141319
$426$	0	0
$427$	− 1.76918e21i	− 1.60084i
$428$	0	0
$429$	−2.06132e20	−0.179675
$430$	0	0
$431$	1.90027e21i	1.59587i	0.602743	+	0.797935i	$0.294074\pi$
$431$	1.90027e21i	1.59587i	−0.602743	+	0.797935i	$0.705926\pi$
$432$	0	0
$433$	−6.21560e20	−0.503014	−0.251507	−	0.967856i	$-0.580926\pi$
$433$	−6.21560e20	−0.503014	−0.251507	+	0.967856i	$0.580926\pi$
$434$	0	0
$435$	− 2.94091e20i	− 0.229387i
$436$	0	0
$437$	−2.10400e20	−0.158195
$438$	0	0
$439$	1.04264e21i	0.755819i	0.925842	+	0.377910i	$0.123357\pi$
$439$	1.04264e21i	0.755819i	−0.925842	+	0.377910i	$0.876643\pi$
$440$	0	0
$441$	−8.56679e19	−0.0598838
$442$	0	0
$443$	− 2.89989e21i	− 1.95502i	−0.210889	−	0.977510i	$-0.567636\pi$
$443$	− 2.89989e21i	− 1.95502i	0.210889	−	0.977510i	$-0.432364\pi$
$444$	0	0
$445$	−2.66675e21	−1.73421
$446$	0	0
$447$	1.08609e21i	0.681404i
$448$	0	0
$449$	−2.46046e20	−0.148952	−0.0744760	−	0.997223i	$-0.523728\pi$
$449$	−2.46046e20	−0.148952	−0.0744760	+	0.997223i	$0.523728\pi$
$450$	0	0
$451$	− 9.51904e20i	− 0.556136i
$452$	0	0
$453$	−1.45911e21	−0.822813
$454$	0	0
$455$	− 1.16566e21i	− 0.634570i
$456$	0	0
$457$	1.72123e19	0.00904711	0.00452355	−	0.999990i	$-0.498560\pi$
$457$	1.72123e19	0.00904711	0.00452355	+	0.999990i	$0.498560\pi$
$458$	0	0
$459$	− 3.59872e20i	− 0.182661i
$460$	0	0
$461$	5.03857e20	0.247002	0.123501	−	0.992344i	$-0.460588\pi$
$461$	5.03857e20	0.247002	0.123501	+	0.992344i	$0.460588\pi$
$462$	0	0
$463$	3.86309e20i	0.182931i	0.995808	+	0.0914655i	$0.0291551\pi$
$463$	3.86309e20i	0.182931i	−0.995808	+	0.0914655i	$0.970845\pi$
$464$	0	0
$465$	−1.48236e21	−0.678157
$466$	0	0
$467$	− 7.69940e20i	− 0.340347i	−0.985414	−	0.170173i	$-0.945567\pi$
$467$	− 7.69940e20i	− 0.340347i	0.985414	−	0.170173i	$-0.0544328\pi$
$468$	0	0
$469$	−2.56818e21	−1.09709
$470$	0	0
$471$	9.66476e20i	0.399047i
$472$	0	0
$473$	7.61388e20	0.303890
$474$	0	0
$475$	8.02101e19i	0.0309514i
$476$	0	0
$477$	3.37798e18	0.00126041
$478$	0	0
$479$	− 4.90335e21i	− 1.76933i	−0.466228	−	0.884664i	$-0.654388\pi$
$479$	− 4.90335e21i	− 1.76933i	0.466228	−	0.884664i	$-0.345612\pi$
$480$	0	0
$481$	−1.58311e21	−0.552523
$482$	0	0
$483$	1.17870e21i	0.397948i
$484$	0	0
$485$	−4.55368e21	−1.48740
$486$	0	0
$487$	5.33875e21i	1.68736i	0.536845	+	0.843681i	$0.319616\pi$
$487$	5.33875e21i	1.68736i	−0.536845	+	0.843681i	$0.680384\pi$
$488$	0	0
$489$	8.83817e20	0.270328
$490$	0	0
$491$	− 2.05807e21i	− 0.609268i	−0.952469	−	0.304634i	$-0.901466\pi$
$491$	− 2.05807e21i	− 0.609268i	0.952469	−	0.304634i	$-0.0985342\pi$
$492$	0	0
$493$	1.42640e21	0.408757
$494$	0	0
$495$	4.54225e20i	0.126017i
$496$	0	0
$497$	5.62852e21	1.51197
$498$	0	0
$499$	3.99610e21i	1.03952i	0.854313	+	0.519759i	$0.173978\pi$
$499$	3.99610e21i	1.03952i	−0.854313	+	0.519759i	$0.826022\pi$
$500$	0	0
$501$	1.32974e21	0.335015
$502$	0	0
$503$	1.54278e21i	0.376497i	0.982121	+	0.188248i	$0.0602810\pi$
$503$	1.54278e21i	0.376497i	−0.982121	+	0.188248i	$0.939719\pi$
$504$	0	0
$505$	2.83194e20	0.0669504
$506$	0	0
$507$	1.06688e21i	0.244372i
$508$	0	0
$509$	6.53591e21	1.45065	0.725326	−	0.688405i	$-0.241689\pi$
$509$	6.53591e21	1.45065	0.725326	+	0.688405i	$0.241689\pi$
$510$	0	0
$511$	1.24011e21i	0.266742i
$512$	0	0
$513$	1.91895e20	0.0400060
$514$	0	0
$515$	− 5.72704e21i	− 1.15737i
$516$	0	0
$517$	9.42551e20	0.184663
$518$	0	0
$519$	2.47949e21i	0.471003i
$520$	0	0
$521$	3.84224e21	0.707755	0.353878	−	0.935292i	$-0.384863\pi$
$521$	3.84224e21	0.707755	0.353878	+	0.935292i	$0.384863\pi$
$522$	0	0
$523$	1.05856e21i	0.189104i	0.995520	+	0.0945522i	$0.0301419\pi$
$523$	1.05856e21i	0.189104i	−0.995520	+	0.0945522i	$0.969858\pi$
$524$	0	0
$525$	4.49352e20	0.0778595
$526$	0	0
$527$	− 7.18973e21i	− 1.20845i
$528$	0	0
$529$	2.58104e21	0.420872
$530$	0	0
$531$	− 3.76871e20i	− 0.0596262i
$532$	0	0
$533$	6.71318e21	1.03065
$534$	0	0
$535$	1.01193e21i	0.150771i
$536$	0	0
$537$	−4.83759e21	−0.699576
$538$	0	0
$539$	5.24445e20i	0.0736189i
$540$	0	0
$541$	2.76090e21	0.376247	0.188124	−	0.982145i	$-0.439759\pi$
$541$	2.76090e21	0.376247	0.188124	+	0.982145i	$0.439759\pi$
$542$	0	0
$543$	8.09423e21i	1.07097i
$544$	0	0
$545$	−1.94431e21	−0.249801
$546$	0	0
$547$	− 4.11011e20i	− 0.0512808i	−0.999671	−	0.0256404i	$-0.991838\pi$
$547$	− 4.11011e20i	− 0.0512808i	0.999671	−	0.0256404i	$-0.00816248\pi$
$548$	0	0
$549$	−4.86192e21	−0.589154
$550$	0	0
$551$	7.60600e20i	0.0895248i
$552$	0	0
$553$	8.50741e21	0.972739
$554$	0	0
$555$	3.48849e21i	0.387519i
$556$	0	0
$557$	9.57546e21	1.03352	0.516759	−	0.856131i	$-0.327138\pi$
$557$	9.57546e21	1.03352	0.516759	+	0.856131i	$0.327138\pi$
$558$	0	0
$559$	5.36959e21i	0.563179i
$560$	0	0
$561$	−2.20308e21	−0.224557
$562$	0	0
$563$	1.65058e22i	1.63519i	0.575791	+	0.817597i	$0.304694\pi$
$563$	1.65058e22i	1.63519i	−0.575791	+	0.817597i	$0.695306\pi$
$564$	0	0
$565$	−1.26334e22	−1.21655
$566$	0	0
$567$	− 1.07503e21i	− 0.100637i
$568$	0	0
$569$	7.25823e21	0.660591	0.330295	−	0.943878i	$-0.392852\pi$
$569$	7.25823e21	0.660591	0.330295	+	0.943878i	$0.392852\pi$
$570$	0	0
$571$	2.38406e21i	0.210974i	0.994421	+	0.105487i	$0.0336401\pi$
$571$	2.38406e21i	0.210974i	−0.994421	+	0.105487i	$0.966360\pi$
$572$	0	0
$573$	7.56413e21	0.650912
$574$	0	0
$575$	1.35395e21i	0.113308i
$576$	0	0
$577$	4.77664e21	0.388791	0.194395	−	0.980923i	$-0.437726\pi$
$577$	4.77664e21	0.388791	0.194395	+	0.980923i	$0.437726\pi$
$578$	0	0
$579$	6.54002e20i	0.0517786i
$580$	0	0
$581$	1.70542e21	0.131348
$582$	0	0
$583$	− 2.06795e19i	− 0.00154950i
$584$	0	0
$585$	−3.20336e21	−0.233539
$586$	0	0
$587$	− 2.61793e22i	− 1.85717i	−0.371115	−	0.928587i	$-0.621024\pi$
$587$	− 2.61793e22i	− 1.85717i	0.371115	−	0.928587i	$-0.378976\pi$
$588$	0	0
$589$	3.83379e21	0.264671
$590$	0	0
$591$	− 3.75259e21i	− 0.252134i
$592$	0	0
$593$	−2.14609e22	−1.40349	−0.701747	−	0.712427i	$-0.747596\pi$
$593$	−2.14609e22	−1.40349	−0.701747	+	0.712427i	$0.747596\pi$
$594$	0	0
$595$	− 1.24582e22i	− 0.793086i
$596$	0	0
$597$	−9.60091e20	−0.0595001
$598$	0	0
$599$	2.55608e21i	0.154227i	0.997022	+	0.0771135i	$0.0245704\pi$
$599$	2.55608e21i	0.154227i	−0.997022	+	0.0771135i	$0.975430\pi$
$600$	0	0
$601$	−6.44355e21	−0.378555	−0.189278	−	0.981924i	$-0.560615\pi$
$601$	−6.44355e21	−0.378555	−0.189278	+	0.981924i	$0.560615\pi$
$602$	0	0
$603$	7.05766e21i	0.403759i
$604$	0	0
$605$	−1.37783e22	−0.767634
$606$	0	0
$607$	− 6.83436e21i	− 0.370842i	−0.982659	−	0.185421i	$-0.940635\pi$
$607$	− 6.83436e21i	− 0.370842i	0.982659	−	0.185421i	$-0.0593648\pi$
$608$	0	0
$609$	4.26102e21	0.225203
$610$	0	0
$611$	6.64721e21i	0.342224i
$612$	0	0
$613$	−1.83424e22	−0.919968	−0.459984	−	0.887927i	$-0.652145\pi$
$613$	−1.83424e22	−0.919968	−0.459984	+	0.887927i	$0.652145\pi$
$614$	0	0
$615$	− 1.47929e22i	− 0.722859i
$616$	0	0
$617$	−1.23087e22	−0.586043	−0.293022	−	0.956106i	$-0.594661\pi$
$617$	−1.23087e22	−0.586043	−0.293022	+	0.956106i	$0.594661\pi$
$618$	0	0
$619$	1.45935e22i	0.677073i	0.940953	+	0.338537i	$0.109932\pi$
$619$	1.45935e22i	0.677073i	−0.940953	+	0.338537i	$0.890068\pi$
$620$	0	0
$621$	3.23920e21	0.146455
$622$	0	0
$623$	− 3.86378e22i	− 1.70258i
$624$	0	0
$625$	−1.93002e22	−0.828938
$626$	0	0
$627$	− 1.17475e21i	− 0.0491819i
$628$	0	0
$629$	−1.69198e22	−0.690543
$630$	0	0
$631$	− 3.34572e22i	− 1.33124i	−0.746292	−	0.665618i	$-0.768168\pi$
$631$	− 3.34572e22i	− 1.33124i	0.746292	−	0.665618i	$-0.231832\pi$
$632$	0	0
$633$	−5.93464e21	−0.230231
$634$	0	0
$635$	4.05115e21i	0.153246i
$636$	0	0
$637$	−3.69858e21	−0.136433
$638$	0	0
$639$	− 1.54678e22i	− 0.556445i
$640$	0	0
$641$	−1.92372e21	−0.0674958	−0.0337479	−	0.999430i	$-0.510744\pi$
$641$	−1.92372e21	−0.0674958	−0.0337479	+	0.999430i	$0.510744\pi$
$642$	0	0
$643$	3.52972e22i	1.20796i	0.797000	+	0.603979i	$0.206419\pi$
$643$	3.52972e22i	1.20796i	−0.797000	+	0.603979i	$0.793581\pi$
$644$	0	0
$645$	1.18323e22	0.394993
$646$	0	0
$647$	1.78800e22i	0.582281i	0.956680	+	0.291141i	$0.0940347\pi$
$647$	1.78800e22i	0.582281i	−0.956680	+	0.291141i	$0.905965\pi$
$648$	0	0
$649$	−2.30714e21	−0.0733022
$650$	0	0
$651$	− 2.14776e22i	− 0.665790i
$652$	0	0
$653$	−1.59480e22	−0.482393	−0.241196	−	0.970476i	$-0.577540\pi$
$653$	−1.59480e22	−0.482393	−0.241196	+	0.970476i	$0.577540\pi$
$654$	0	0
$655$	− 3.59640e22i	− 1.06154i
$656$	0	0
$657$	3.40796e21	0.0981684
$658$	0	0
$659$	5.67794e22i	1.59627i	0.602476	+	0.798137i	$0.294181\pi$
$659$	5.67794e22i	1.59627i	−0.602476	+	0.798137i	$0.705819\pi$
$660$	0	0
$661$	5.15514e22	1.41458	0.707292	−	0.706921i	$-0.249916\pi$
$661$	5.15514e22	1.41458	0.707292	+	0.706921i	$0.249916\pi$
$662$	0	0
$663$	− 1.55369e22i	− 0.416156i
$664$	0	0
$665$	6.64311e21	0.173699
$666$	0	0
$667$	1.28390e22i	0.327736i
$668$	0	0
$669$	−1.43176e22	−0.356830
$670$	0	0
$671$	2.97638e22i	0.724284i
$672$	0	0
$673$	−6.12492e22	−1.45539	−0.727696	−	0.685900i	$-0.759409\pi$
$673$	−6.12492e22	−1.45539	−0.727696	+	0.685900i	$0.759409\pi$
$674$	0	0
$675$	− 1.23487e21i	− 0.0286544i
$676$	0	0
$677$	−1.46668e22	−0.332374	−0.166187	−	0.986094i	$-0.553146\pi$
$677$	−1.46668e22	−0.332374	−0.166187	+	0.986094i	$0.553146\pi$
$678$	0	0
$679$	− 6.59772e22i	− 1.46028i
$680$	0	0
$681$	1.91206e22	0.413356
$682$	0	0
$683$	8.91229e22i	1.88202i	0.338383	+	0.941009i	$0.390120\pi$
$683$	8.91229e22i	1.88202i	−0.338383	+	0.941009i	$0.609880\pi$
$684$	0	0
$685$	1.74999e22	0.361004
$686$	0	0
$687$	− 5.83088e21i	− 0.117511i
$688$	0	0
$689$	1.45839e20	0.00287158
$690$	0	0
$691$	− 2.26063e22i	− 0.434916i	−0.976070	−	0.217458i	$-0.930224\pi$
$691$	− 2.26063e22i	− 0.434916i	0.976070	−	0.217458i	$-0.0697764\pi$
$692$	0	0
$693$	−6.58116e21	−0.123719
$694$	0	0
$695$	2.68452e22i	0.493161i
$696$	0	0
$697$	7.17485e22	1.28810
$698$	0	0
$699$	4.08852e22i	0.717380i
$700$	0	0
$701$	−6.45959e22	−1.10780	−0.553899	−	0.832584i	$-0.686861\pi$
$701$	−6.45959e22	−1.10780	−0.553899	+	0.832584i	$0.686861\pi$
$702$	0	0
$703$	− 9.02218e21i	− 0.151241i
$704$	0	0
$705$	1.46476e22	0.240023
$706$	0	0
$707$	4.10313e21i	0.0657295i
$708$	0	0
$709$	4.64290e22	0.727142	0.363571	−	0.931567i	$-0.381557\pi$
$709$	4.64290e22	0.727142	0.363571	+	0.931567i	$0.381557\pi$
$710$	0	0
$711$	− 2.33794e22i	− 0.357994i
$712$	0	0
$713$	6.47146e22	0.968916
$714$	0	0
$715$	1.96105e22i	0.287104i
$716$	0	0
$717$	5.91087e22	0.846250
$718$	0	0
$719$	2.18704e22i	0.306215i	0.988210	+	0.153108i	$0.0489281\pi$
$719$	2.18704e22i	0.306215i	−0.988210	+	0.153108i	$0.951072\pi$
$720$	0	0
$721$	8.29777e22	1.13626
$722$	0	0
$723$	− 6.28619e22i	− 0.841940i
$724$	0	0
$725$	4.89456e21	0.0641224
$726$	0	0
$727$	− 4.68917e22i	− 0.600925i	−0.953794	−	0.300463i	$-0.902859\pi$
$727$	− 4.68917e22i	− 0.600925i	0.953794	−	0.300463i	$-0.0971411\pi$
$728$	0	0
$729$	−2.95431e21	−0.0370370
$730$	0	0
$731$	5.73886e22i	0.703861i
$732$	0	0
$733$	6.19965e22	0.743936	0.371968	−	0.928246i	$-0.378683\pi$
$733$	6.19965e22	0.743936	0.371968	+	0.928246i	$0.378683\pi$
$734$	0	0
$735$	8.15006e21i	0.0956890i
$736$	0	0
$737$	4.32058e22	0.496366
$738$	0	0
$739$	2.15220e22i	0.241950i	0.992656	+	0.120975i	$0.0386022\pi$
$739$	2.15220e22i	0.241950i	−0.992656	+	0.120975i	$0.961398\pi$
$740$	0	0
$741$	8.28477e21	0.0911454
$742$	0	0
$743$	9.74537e21i	0.104927i	0.998623	+	0.0524636i	$0.0167073\pi$
$743$	9.74537e21i	0.104927i	−0.998623	+	0.0524636i	$0.983293\pi$
$744$	0	0
$745$	1.03326e23	1.08882
$746$	0	0
$747$	− 4.68671e21i	− 0.0483396i
$748$	0	0
$749$	−1.46616e22	−0.148022
$750$	0	0
$751$	1.20764e23i	1.19349i	0.802432	+	0.596744i	$0.203539\pi$
$751$	1.20764e23i	1.19349i	−0.802432	+	0.596744i	$0.796461\pi$
$752$	0	0
$753$	1.08812e23	1.05273
$754$	0	0
$755$	1.38813e23i	1.31478i
$756$	0	0
$757$	−1.13124e22	−0.104903	−0.0524515	−	0.998623i	$-0.516704\pi$
$757$	−1.13124e22	−0.104903	−0.0524515	+	0.998623i	$0.516704\pi$
$758$	0	0
$759$	− 1.98299e22i	− 0.180047i
$760$	0	0
$761$	−1.10144e23	−0.979228	−0.489614	−	0.871939i	$-0.662862\pi$
$761$	−1.10144e23	−0.979228	−0.489614	+	0.871939i	$0.662862\pi$
$762$	0	0
$763$	− 2.81707e22i	− 0.245246i
$764$	0	0
$765$	−3.42366e22	−0.291877
$766$	0	0
$767$	− 1.62708e22i	− 0.135846i
$768$	0	0
$769$	−1.93201e23	−1.57979	−0.789893	−	0.613245i	$-0.789864\pi$
$769$	−1.93201e23	−1.57979	−0.789893	+	0.613245i	$0.789864\pi$
$770$	0	0
$771$	− 4.97481e22i	− 0.398421i
$772$	0	0
$773$	−6.85136e22	−0.537453	−0.268727	−	0.963216i	$-0.586603\pi$
$773$	−6.85136e22	−0.537453	−0.268727	+	0.963216i	$0.586603\pi$
$774$	0	0
$775$	− 2.46710e22i	− 0.189571i
$776$	0	0
$777$	−5.05439e22	−0.380453
$778$	0	0
$779$	3.82585e22i	0.282117i
$780$	0	0
$781$	−9.46915e22	−0.684073
$782$	0	0
$783$	− 1.17098e22i	− 0.0828809i
$784$	0	0
$785$	9.19462e22	0.637642
$786$	0	0
$787$	− 9.28055e22i	− 0.630632i	−0.948987	−	0.315316i	$-0.897890\pi$
$787$	− 9.28055e22i	− 0.630632i	0.948987	−	0.315316i	$-0.102110\pi$
$788$	0	0
$789$	−1.76288e22	−0.117383
$790$	0	0
$791$	− 1.83042e23i	− 1.19437i
$792$	0	0
$793$	−2.09906e23	−1.34227
$794$	0	0
$795$	− 3.21366e20i	− 0.00201402i
$796$	0	0
$797$	9.87144e22	0.606337	0.303168	−	0.952937i	$-0.401956\pi$
$797$	9.87144e22	0.606337	0.303168	+	0.952937i	$0.401956\pi$
$798$	0	0
$799$	7.10435e22i	0.427711i
$800$	0	0
$801$	−1.06181e23	−0.626597
$802$	0	0
$803$	− 2.08630e22i	− 0.120685i
$804$	0	0
$805$	1.12136e23	0.635885
$806$	0	0
$807$	− 2.54040e22i	− 0.141226i
$808$	0	0
$809$	3.73633e22	0.203637	0.101819	−	0.994803i	$-0.467534\pi$
$809$	3.73633e22	0.203637	0.101819	+	0.994803i	$0.467534\pi$
$810$	0	0
$811$	1.25266e23i	0.669369i	0.942330	+	0.334685i	$0.108630\pi$
$811$	1.25266e23i	0.669369i	−0.942330	+	0.334685i	$0.891370\pi$
$812$	0	0
$813$	5.20594e22	0.272757
$814$	0	0
$815$	− 8.40824e22i	− 0.431961i
$816$	0	0
$817$	−3.06014e22	−0.154158
$818$	0	0
$819$	− 4.64128e22i	− 0.229280i
$820$	0	0
$821$	−2.55103e23	−1.23586	−0.617931	−	0.786232i	$-0.712029\pi$
$821$	−2.55103e23	−1.23586	−0.617931	+	0.786232i	$0.712029\pi$
$822$	0	0
$823$	− 1.17112e23i	− 0.556421i	−0.960520	−	0.278210i	$-0.910259\pi$
$823$	− 1.17112e23i	− 0.556421i	0.960520	−	0.278210i	$-0.0897413\pi$
$824$	0	0
$825$	−7.55968e21	−0.0352267
$826$	0	0
$827$	− 3.30491e23i	− 1.51048i	−0.655446	−	0.755242i	$-0.727519\pi$
$827$	− 3.30491e23i	− 1.51048i	0.655446	−	0.755242i	$-0.272481\pi$
$828$	0	0
$829$	−1.31366e23	−0.588907	−0.294454	−	0.955666i	$-0.595138\pi$
$829$	−1.31366e23	−0.588907	−0.294454	+	0.955666i	$0.595138\pi$
$830$	0	0
$831$	1.42884e23i	0.628311i
$832$	0	0
$833$	−3.95293e22	−0.170514
$834$	0	0
$835$	− 1.26505e23i	− 0.535324i
$836$	0	0
$837$	−5.90229e22	−0.245029
$838$	0	0
$839$	2.42312e23i	0.986914i	0.869770	+	0.493457i	$0.164267\pi$
$839$	2.42312e23i	0.986914i	−0.869770	+	0.493457i	$0.835733\pi$
$840$	0	0
$841$	−2.03833e23	−0.814530
$842$	0	0
$843$	2.26761e23i	0.889095i
$844$	0	0
$845$	1.01498e23	0.390485
$846$	0	0
$847$	− 1.99631e23i	− 0.753635i
$848$	0	0
$849$	−2.55912e23	−0.948045
$850$	0	0
$851$	− 1.52295e23i	− 0.553668i
$852$	0	0
$853$	3.34753e23	1.19435	0.597176	−	0.802110i	$-0.296289\pi$
$853$	3.34753e23	1.19435	0.597176	+	0.802110i	$0.296289\pi$
$854$	0	0
$855$	− 1.82560e22i	− 0.0639260i
$856$	0	0
$857$	−3.53238e23	−1.21400	−0.607002	−	0.794700i	$-0.707628\pi$
$857$	−3.53238e23	−1.21400	−0.607002	+	0.794700i	$0.707628\pi$
$858$	0	0
$859$	− 2.25702e23i	− 0.761361i	−0.924707	−	0.380680i	$-0.875690\pi$
$859$	− 2.25702e23i	− 0.761361i	0.924707	−	0.380680i	$-0.124310\pi$
$860$	0	0
$861$	2.14331e23	0.709677
$862$	0	0
$863$	− 4.65349e23i	− 1.51249i	−0.654288	−	0.756245i	$-0.727032\pi$
$863$	− 4.65349e23i	− 1.51249i	0.654288	−	0.756245i	$-0.272968\pi$
$864$	0	0
$865$	2.35888e23	0.752621
$866$	0	0
$867$	1.82743e22i	0.0572385i
$868$	0	0
$869$	−1.43125e23	−0.440105
$870$	0	0
$871$	3.04703e23i	0.919881i
$872$	0	0
$873$	−1.81313e23	−0.537422
$874$	0	0
$875$	− 3.29864e23i	− 0.959999i
$876$	0	0
$877$	−5.03888e23	−1.43992	−0.719958	−	0.694018i	$-0.755839\pi$
$877$	−5.03888e23	−1.43992	−0.719958	+	0.694018i	$0.755839\pi$
$878$	0	0
$879$	2.91200e23i	0.817109i
$880$	0	0
$881$	5.94146e23	1.63714	0.818570	−	0.574407i	$-0.194767\pi$
$881$	5.94146e23	1.63714	0.818570	+	0.574407i	$0.194767\pi$
$882$	0	0
$883$	2.33388e23i	0.631528i	0.948838	+	0.315764i	$0.102261\pi$
$883$	2.33388e23i	0.631528i	−0.948838	+	0.315764i	$0.897739\pi$
$884$	0	0
$885$	−3.58538e22	−0.0952773
$886$	0	0
$887$	4.85675e22i	0.126753i	0.997990	+	0.0633763i	$0.0201868\pi$
$887$	4.85675e22i	0.126753i	−0.997990	+	0.0633763i	$0.979813\pi$
$888$	0	0
$889$	−5.86961e22	−0.150451
$890$	0	0
$891$	1.80858e22i	0.0455320i
$892$	0	0
$893$	−3.78826e22	−0.0936760
$894$	0	0
$895$	4.60227e23i	1.11786i
$896$	0	0
$897$	1.39847e23	0.333668
$898$	0	0
$899$	− 2.33945e23i	− 0.548322i
$900$	0	0
$901$	1.55869e21	0.00358889
$902$	0	0
$903$	1.71435e23i	0.387790i
$904$	0	0
$905$	7.70049e23	1.71131
$906$	0	0
$907$	2.95630e23i	0.645492i	0.946486	+	0.322746i	$0.104606\pi$
$907$	2.95630e23i	0.645492i	−0.946486	+	0.322746i	$0.895394\pi$
$908$	0	0
$909$	1.12759e22	0.0241902
$910$	0	0
$911$	5.80586e23i	1.22383i	0.790925	+	0.611913i	$0.209600\pi$
$911$	5.80586e23i	1.22383i	−0.790925	+	0.611913i	$0.790400\pi$
$912$	0	0
$913$	−2.86912e22	−0.0594269
$914$	0	0
$915$	4.62541e23i	0.941416i
$916$	0	0
$917$	5.21074e23	1.04219
$918$	0	0
$919$	6.73665e23i	1.32410i	0.749461	+	0.662049i	$0.230313\pi$
$919$	6.73665e23i	1.32410i	−0.749461	+	0.662049i	$0.769687\pi$
$920$	0	0
$921$	−5.11958e23	−0.988911
$922$	0	0
$923$	− 6.67799e23i	− 1.26775i
$924$	0	0
$925$	−5.80590e22	−0.108327
$926$	0	0
$927$	− 2.28033e23i	− 0.418176i
$928$	0	0
$929$	8.03203e23	1.44777	0.723885	−	0.689921i	$-0.242355\pi$
$929$	8.03203e23	1.44777	0.723885	+	0.689921i	$0.242355\pi$
$930$	0	0
$931$	− 2.10783e22i	− 0.0373454i
$932$	0	0
$933$	4.09032e23	0.712367
$934$	0	0
$935$	2.09591e23i	0.358823i
$936$	0	0
$937$	9.63691e23	1.62189	0.810945	−	0.585123i	$-0.198954\pi$
$937$	9.63691e23	1.62189	0.810945	+	0.585123i	$0.198954\pi$
$938$	0	0
$939$	− 5.19070e23i	− 0.858819i
$940$	0	0
$941$	3.57623e23	0.581712	0.290856	−	0.956767i	$-0.406060\pi$
$941$	3.57623e23	0.581712	0.290856	+	0.956767i	$0.406060\pi$
$942$	0	0
$943$	6.45807e23i	1.03278i
$944$	0	0
$945$	−1.02274e23	−0.160809
$946$	0	0
$947$	− 5.56883e23i	− 0.860923i	−0.902609	−	0.430461i	$-0.858351\pi$
$947$	− 5.56883e23i	− 0.860923i	0.902609	−	0.430461i	$-0.141649\pi$
$948$	0	0
$949$	1.47133e23	0.223656
$950$	0	0
$951$	4.12869e23i	0.617117i
$952$	0	0
$953$	−3.29464e23	−0.484245	−0.242122	−	0.970246i	$-0.577844\pi$
$953$	−3.29464e23	−0.484245	−0.242122	+	0.970246i	$0.577844\pi$
$954$	0	0
$955$	− 7.19618e23i	− 1.04010i
$956$	0	0
$957$	−7.16853e22	−0.101891
$958$	0	0
$959$	2.53552e23i	0.354420i
$960$	0	0
$961$	−4.51770e23	−0.621056
$962$	0	0
$963$	4.02917e22i	0.0544761i
$964$	0	0
$965$	6.22188e22	0.0827377
$966$	0	0
$967$	− 9.49275e22i	− 0.124160i	−0.998071	−	0.0620798i	$-0.980227\pi$
$967$	− 9.49275e22i	− 0.124160i	0.998071	−	0.0620798i	$-0.0197733\pi$
$968$	0	0
$969$	8.85452e22	0.113913
$970$	0	0
$971$	− 1.21561e24i	− 1.53829i	−0.639073	−	0.769146i	$-0.720682\pi$
$971$	− 1.21561e24i	− 1.53829i	0.639073	−	0.769146i	$-0.279318\pi$
$972$	0	0
$973$	−3.88954e23	−0.484168
$974$	0	0
$975$	− 5.33136e22i	− 0.0652831i
$976$	0	0
$977$	6.98630e23	0.841571	0.420785	−	0.907160i	$-0.361755\pi$
$977$	6.98630e23	0.841571	0.420785	+	0.907160i	$0.361755\pi$
$978$	0	0
$979$	6.50025e23i	0.770315i
$980$	0	0
$981$	−7.74164e22	−0.0902570
$982$	0	0
$983$	2.91314e23i	0.334144i	0.985945	+	0.167072i	$0.0534312\pi$
$983$	2.91314e23i	0.334144i	−0.985945	+	0.167072i	$0.946569\pi$
$984$	0	0
$985$	−3.57005e23	−0.402888
$986$	0	0
$987$	2.12225e23i	0.235646i
$988$	0	0
$989$	−5.16554e23	−0.564346
$990$	0	0
$991$	1.29800e23i	0.139535i	0.997563	+	0.0697677i	$0.0222258\pi$
$991$	1.29800e23i	0.139535i	−0.997563	+	0.0697677i	$0.977774\pi$
$992$	0	0
$993$	−1.16787e22	−0.0123538
$994$	0	0
$995$	9.13388e22i	0.0950759i
$996$	0	0
$997$	1.19324e24	1.22227	0.611133	−	0.791528i	$-0.290714\pi$
$997$	1.19324e24	1.22227	0.611133	+	0.791528i	$0.290714\pi$
$998$	0	0
$999$	1.38901e23i	0.140017i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.17.g.a.31.1	✓	4
4.3	odd	2	inner	48.17.g.a.31.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.17.g.a.31.1	✓	4	1.1	even	1	trivial
48.17.g.a.31.3	yes	4	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3788.00i q^{3} -360373. q^{5} -5.22136e6i q^{7} -1.43489e7 q^{9} +O(q^{10})\)	\(q-3788.00i q^{3} -360373. q^{5} -5.22136e6i q^{7} -1.43489e7 q^{9} +8.78416e7i q^{11} -6.19491e8 q^{13} +1.36509e9i q^{15} -6.62095e9 q^{17} -3.53050e9i q^{19} -1.97785e10 q^{21} -5.95950e10i q^{23} -2.27192e10 q^{25} +5.43536e10i q^{27} -2.15437e11 q^{29} +1.08591e12i q^{31} +3.32744e11 q^{33} +1.88164e12i q^{35} +2.55550e12 q^{37} +2.34663e12i q^{39} -1.08366e13 q^{41} -8.66774e12i q^{43} +5.17096e12 q^{45} -1.07301e13i q^{47} +5.97034e12 q^{49} +2.50801e13i q^{51} -2.35418e11 q^{53} -3.16557e13i q^{55} -1.33735e13 q^{57} +2.62648e13i q^{59} +3.38835e14 q^{61} +7.49208e13i q^{63} +2.23248e14 q^{65} -4.91860e14i q^{67} -2.25746e14 q^{69} +1.07798e15i q^{71} -2.37507e14 q^{73} +8.60603e13i q^{75} +4.58653e14 q^{77} +1.62935e15i q^{79} +2.05891e14 q^{81} +3.26625e14i q^{83} +2.38601e15 q^{85} +8.16075e14i q^{87} +7.39996e15 q^{89} +3.23459e15i q^{91} +4.11341e15 q^{93} +1.27229e15i q^{95} +1.26360e16 q^{97} -1.26043e15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 140424 q^{5} - 57395628 q^{9}+O(q^{10}) \) 4 * q - 140424 * q^5 - 57395628 * q^9	\( 4 q - 140424 q^{5} - 57395628 q^{9} - 586685528 q^{13} - 571005336 q^{17} - 8495392752 q^{21} - 182227446772 q^{25} - 585621416424 q^{29} - 176071520880 q^{33} - 2770470511096 q^{37} - 9744697113624 q^{41} + 2014930916568 q^{45} + 44786525712964 q^{49} + 93950973394200 q^{53} + 40139973246384 q^{57} + 374162346312392 q^{61} + 635766937476528 q^{65} + 165370244765184 q^{69} + 13\!\cdots\!20 q^{73}+ \cdots + 45\!\cdots\!40 q^{97}+O(q^{100}) \) 4 * q - 140424 * q^5 - 57395628 * q^9 - 586685528 * q^13 - 571005336 * q^17 - 8495392752 * q^21 - 182227446772 * q^25 - 585621416424 * q^29 - 176071520880 * q^33 - 2770470511096 * q^37 - 9744697113624 * q^41 + 2014930916568 * q^45 + 44786525712964 * q^49 + 93950973394200 * q^53 + 40139973246384 * q^57 + 374162346312392 * q^61 + 635766937476528 * q^65 + 165370244765184 * q^69 + 1379949716544520 * q^73 + 1828180800500544 * q^77 + 823564528378596 * q^81 + 8448616033483824 * q^85 + 20666558628633864 * q^89 + 7887770592692400 * q^93 + 45712018082632840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more