LMFDB - Embedded newform 48.15.g.a.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,15,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, 15, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 15 $
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:	$59.6779047129$
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} + 16897x^{2} + 16896x + 285474816 $ x^4 - x^3 + 16897x^2 + 16896x + 285474816
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.2
Root			$65.2428 + 113.004i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.15.g.a.31.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1262.67i q^{3} +48265.2 q^{5} -988060. i q^{7} -1.59432e6 q^{9} +O(q^{10})$	$q-1262.67i q^{3} +48265.2 q^{5} -988060. i q^{7} -1.59432e6 q^{9} +2.32961e7i q^{11} -1.22817e8 q^{13} -6.09427e7i q^{15} +5.67127e8 q^{17} +1.44150e9i q^{19} -1.24759e9 q^{21} -4.82619e9i q^{23} -3.77399e9 q^{25} +2.01310e9i q^{27} -2.25157e10 q^{29} +9.74889e9i q^{31} +2.94152e10 q^{33} -4.76889e10i q^{35} -4.31253e10 q^{37} +1.55077e11i q^{39} +2.29915e11 q^{41} +1.08622e11i q^{43} -7.69502e10 q^{45} +1.42608e11i q^{47} -2.98039e11 q^{49} -7.16092e11i q^{51} +5.07891e11 q^{53} +1.12439e12i q^{55} +1.82013e12 q^{57} +4.16206e12i q^{59} -1.20121e12 q^{61} +1.57529e12i q^{63} -5.92778e12 q^{65} +3.78597e12i q^{67} -6.09386e12 q^{69} +1.50156e13i q^{71} +1.11912e12 q^{73} +4.76529e12i q^{75} +2.30179e13 q^{77} +2.64523e13i q^{79} +2.54187e12 q^{81} -5.32326e12i q^{83} +2.73725e13 q^{85} +2.84297e13i q^{87} -6.78257e13 q^{89} +1.21350e14i q^{91} +1.23096e13 q^{93} +6.95741e13i q^{95} +3.20889e13 q^{97} -3.71415e13i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18360 q^{5} - 6377292 q^{9}+O(q^{10}) $ 4 * q + 18360 * q^5 - 6377292 * q^9	$ 4 q + 18360 q^{5} - 6377292 q^{9} - 158388152 q^{13} + 262496808 q^{17} - 2206700496 q^{21} - 16699713940 q^{25} - 59940048168 q^{29} + 16466640336 q^{33} + 14727873320 q^{37} + 446330350440 q^{41} - 29271770280 q^{45} + 734268733828 q^{49} + 837011171736 q^{53} + 1309414251888 q^{57} - 3828732927064 q^{61} - 15265566735120 q^{65} - 6884591879232 q^{69} - 8960085606968 q^{73} + 49868530746048 q^{77} + 10167463313316 q^{81} + 88817754861360 q^{85} - 79296711493176 q^{89} + 42094011836880 q^{93} - 81693961168376 q^{97}+O(q^{100}) $ 4 * q + 18360 * q^5 - 6377292 * q^9 - 158388152 * q^13 + 262496808 * q^17 - 2206700496 * q^21 - 16699713940 * q^25 - 59940048168 * q^29 + 16466640336 * q^33 + 14727873320 * q^37 + 446330350440 * q^41 - 29271770280 * q^45 + 734268733828 * q^49 + 837011171736 * q^53 + 1309414251888 * q^57 - 3828732927064 * q^61 - 15265566735120 * q^65 - 6884591879232 * q^69 - 8960085606968 * q^73 + 49868530746048 * q^77 + 10167463313316 * q^81 + 88817754861360 * q^85 - 79296711493176 * q^89 + 42094011836880 * q^93 - 81693961168376 * 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
$3$	− 1262.67i	− 0.577350i
$3$	− 1262.67i	− 0.577350i
$4$	0	0
$5$	48265.2	0.617794	0.308897	−	0.951096i	$-0.400040\pi$
$5$	48265.2	0.617794	0.308897	+	0.951096i	$0.400040\pi$
$6$	0	0
$7$	− 988060.i	− 1.19977i	−0.800087	−	0.599884i	$-0.795214\pi$
$7$	− 988060.i	− 1.19977i	0.800087	−	0.599884i	$-0.204786\pi$
$8$	0	0
$9$	−1.59432e6	−0.333333
$10$	0	0
$11$	2.32961e7i	1.19546i	0.801698	+	0.597729i	$0.203930\pi$
$11$	2.32961e7i	1.19546i	−0.801698	+	0.597729i	$0.796070\pi$
$12$	0	0
$13$	−1.22817e8	−1.95729	−0.978644	−	0.205562i	$-0.934098\pi$
$13$	−1.22817e8	−1.95729	−0.978644	+	0.205562i	$0.934098\pi$
$14$	0	0
$15$	− 6.09427e7i	− 0.356684i
$16$	0	0
$17$	5.67127e8	1.38210	0.691048	−	0.722809i	$-0.257149\pi$
$17$	5.67127e8	1.38210	0.691048	+	0.722809i	$0.257149\pi$
$18$	0	0
$19$	1.44150e9i	1.61265i	0.591476	+	0.806323i	$0.298545\pi$
$19$	1.44150e9i	1.61265i	−0.591476	+	0.806323i	$0.701455\pi$
$20$	0	0
$21$	−1.24759e9	−0.692686
$22$	0	0
$23$	− 4.82619e9i	− 1.41746i	−0.705482	−	0.708728i	$-0.749269\pi$
$23$	− 4.82619e9i	− 1.41746i	0.705482	−	0.708728i	$-0.250731\pi$
$24$	0	0
$25$	−3.77399e9	−0.618331
$26$	0	0
$27$	2.01310e9i	0.192450i
$28$	0	0
$29$	−2.25157e10	−1.30526	−0.652632	−	0.757675i	$-0.726335\pi$
$29$	−2.25157e10	−1.30526	−0.652632	+	0.757675i	$0.726335\pi$
$30$	0	0
$31$	9.74889e9i	0.354343i	0.984180	+	0.177171i	$0.0566947\pi$
$31$	9.74889e9i	0.354343i	−0.984180	+	0.177171i	$0.943305\pi$
$32$	0	0
$33$	2.94152e10	0.690198
$34$	0	0
$35$	− 4.76889e10i	− 0.741209i
$36$	0	0
$37$	−4.31253e10	−0.454277	−0.227138	−	0.973862i	$-0.572937\pi$
$37$	−4.31253e10	−0.454277	−0.227138	+	0.973862i	$0.572937\pi$
$38$	0	0
$39$	1.55077e11i	1.13004i
$40$	0	0
$41$	2.29915e11	1.18054	0.590270	−	0.807206i	$-0.299021\pi$
$41$	2.29915e11	1.18054	0.590270	+	0.807206i	$0.299021\pi$
$42$	0	0
$43$	1.08622e11i	0.399611i	0.979836	+	0.199805i	$0.0640310\pi$
$43$	1.08622e11i	0.399611i	−0.979836	+	0.199805i	$0.935969\pi$
$44$	0	0
$45$	−7.69502e10	−0.205931
$46$	0	0
$47$	1.42608e11i	0.281488i	0.990046	+	0.140744i	$0.0449494\pi$
$47$	1.42608e11i	0.281488i	−0.990046	+	0.140744i	$0.955051\pi$
$48$	0	0
$49$	−2.98039e11	−0.439442
$50$	0	0
$51$	− 7.16092e11i	− 0.797953i
$52$	0	0
$53$	5.07891e11	0.432354	0.216177	−	0.976354i	$-0.430641\pi$
$53$	5.07891e11	0.432354	0.216177	+	0.976354i	$0.430641\pi$
$54$	0	0
$55$	1.12439e12i	0.738547i
$56$	0	0
$57$	1.82013e12	0.931061
$58$	0	0
$59$	4.16206e12i	1.67241i	0.548413	+	0.836207i	$0.315232\pi$
$59$	4.16206e12i	1.67241i	−0.548413	+	0.836207i	$0.684768\pi$
$60$	0	0
$61$	−1.20121e12	−0.382218	−0.191109	−	0.981569i	$-0.561208\pi$
$61$	−1.20121e12	−0.382218	−0.191109	+	0.981569i	$0.561208\pi$
$62$	0	0
$63$	1.57529e12i	0.399922i
$64$	0	0
$65$	−5.92778e12	−1.20920
$66$	0	0
$67$	3.78597e12i	0.624674i	0.949971	+	0.312337i	$0.101112\pi$
$67$	3.78597e12i	0.624674i	−0.949971	+	0.312337i	$0.898888\pi$
$68$	0	0
$69$	−6.09386e12	−0.818368
$70$	0	0
$71$	1.50156e13i	1.65095i	0.564435	+	0.825477i	$0.309094\pi$
$71$	1.50156e13i	1.65095i	−0.564435	+	0.825477i	$0.690906\pi$
$72$	0	0
$73$	1.11912e12	0.101302	0.0506510	−	0.998716i	$-0.483870\pi$
$73$	1.11912e12	0.101302	0.0506510	+	0.998716i	$0.483870\pi$
$74$	0	0
$75$	4.76529e12i	0.356993i
$76$	0	0
$77$	2.30179e13	1.43427
$78$	0	0
$79$	2.64523e13i	1.37744i	0.725025	+	0.688722i	$0.241828\pi$
$79$	2.64523e13i	1.37744i	−0.725025	+	0.688722i	$0.758172\pi$
$80$	0	0
$81$	2.54187e12	0.111111
$82$	0	0
$83$	− 5.32326e12i	− 0.196169i	−0.995178	−	0.0980846i	$-0.968728\pi$
$83$	− 5.32326e12i	− 0.196169i	0.995178	−	0.0980846i	$-0.0312716\pi$
$84$	0	0
$85$	2.73725e13	0.853850
$86$	0	0
$87$	2.84297e13i	0.753595i
$88$	0	0
$89$	−6.78257e13	−1.53343	−0.766715	−	0.641988i	$-0.778110\pi$
$89$	−6.78257e13	−1.53343	−0.766715	+	0.641988i	$0.778110\pi$
$90$	0	0
$91$	1.21350e14i	2.34829i
$92$	0	0
$93$	1.23096e13	0.204580
$94$	0	0
$95$	6.95741e13i	0.996283i
$96$	0	0
$97$	3.20889e13	0.397149	0.198574	−	0.980086i	$-0.436369\pi$
$97$	3.20889e13	0.397149	0.198574	+	0.980086i	$0.436369\pi$
$98$	0	0
$99$	− 3.71415e13i	− 0.398486i
$100$	0	0
$101$	4.40282e13	0.410659	0.205329	−	0.978693i	$-0.434173\pi$
$101$	4.40282e13	0.410659	0.205329	+	0.978693i	$0.434173\pi$
$102$	0	0
$103$	− 1.27057e14i	− 1.03309i	−0.856260	−	0.516545i	$-0.827218\pi$
$103$	− 1.27057e14i	− 1.03309i	0.856260	−	0.516545i	$-0.172782\pi$
$104$	0	0
$105$	−6.02151e13	−0.427937
$106$	0	0
$107$	2.57222e14i	1.60185i	0.598767	+	0.800924i	$0.295658\pi$
$107$	2.57222e14i	1.60185i	−0.598767	+	0.800924i	$0.704342\pi$
$108$	0	0
$109$	−2.19172e14	−1.19895	−0.599473	−	0.800395i	$-0.704623\pi$
$109$	−2.19172e14	−1.19895	−0.599473	+	0.800395i	$0.704623\pi$
$110$	0	0
$111$	5.44529e13i	0.262277i
$112$	0	0
$113$	1.23218e14	0.523751	0.261876	−	0.965102i	$-0.415659\pi$
$113$	1.23218e14	0.523751	0.261876	+	0.965102i	$0.415659\pi$
$114$	0	0
$115$	− 2.32937e14i	− 0.875696i
$116$	0	0
$117$	1.95810e14	0.652429
$118$	0	0
$119$	− 5.60356e14i	− 1.65819i
$120$	0	0
$121$	−1.62958e14	−0.429120
$122$	0	0
$123$	− 2.90306e14i	− 0.681585i
$124$	0	0
$125$	−4.76739e14	−0.999795
$126$	0	0
$127$	3.78796e14i	0.710851i	0.934705	+	0.355426i	$0.115664\pi$
$127$	3.78796e14i	0.710851i	−0.934705	+	0.355426i	$0.884336\pi$
$128$	0	0
$129$	1.37153e14	0.230715
$130$	0	0
$131$	− 1.99337e14i	− 0.301085i	−0.988604	−	0.150542i	$-0.951898\pi$
$131$	− 1.99337e14i	− 0.301085i	0.988604	−	0.150542i	$-0.0481020\pi$
$132$	0	0
$133$	1.42429e15	1.93480
$134$	0	0
$135$	9.71624e13i	0.118895i
$136$	0	0
$137$	7.95247e13	0.0877926	0.0438963	−	0.999036i	$-0.486023\pi$
$137$	7.95247e13	0.0877926	0.0438963	+	0.999036i	$0.486023\pi$
$138$	0	0
$139$	− 1.68115e15i	− 1.67688i	−0.544992	−	0.838441i	$-0.683467\pi$
$139$	− 1.68115e15i	− 1.67688i	0.544992	−	0.838441i	$-0.316533\pi$
$140$	0	0
$141$	1.80066e14	0.162517
$142$	0	0
$143$	− 2.86116e15i	− 2.33986i
$144$	0	0
$145$	−1.08672e15	−0.806385
$146$	0	0
$147$	3.76324e14i	0.253712i
$148$	0	0
$149$	3.69734e13	0.0226770	0.0113385	−	0.999936i	$-0.496391\pi$
$149$	3.69734e13	0.0226770	0.0113385	+	0.999936i	$0.496391\pi$
$150$	0	0
$151$	− 6.80534e14i	− 0.380199i	−0.981765	−	0.190100i	$-0.939119\pi$
$151$	− 6.80534e14i	− 0.380199i	0.981765	−	0.190100i	$-0.0608811\pi$
$152$	0	0
$153$	−9.04184e14	−0.460699
$154$	0	0
$155$	4.70532e14i	0.218911i
$156$	0	0
$157$	−9.15795e14	−0.389494	−0.194747	−	0.980854i	$-0.562389\pi$
$157$	−9.15795e14	−0.389494	−0.194747	+	0.980854i	$0.562389\pi$
$158$	0	0
$159$	− 6.41296e14i	− 0.249619i
$160$	0	0
$161$	−4.76856e15	−1.70062
$162$	0	0
$163$	− 1.59110e15i	− 0.520456i	−0.965547	−	0.260228i	$-0.916202\pi$
$163$	− 1.59110e15i	− 0.520456i	0.965547	−	0.260228i	$-0.0837977\pi$
$164$	0	0
$165$	1.41973e15	0.426400
$166$	0	0
$167$	5.05801e15i	1.39625i	0.715974	+	0.698127i	$0.245983\pi$
$167$	5.05801e15i	1.39625i	−0.715974	+	0.698127i	$0.754017\pi$
$168$	0	0
$169$	1.11466e16	2.83098
$170$	0	0
$171$	− 2.29821e15i	− 0.537548i
$172$	0	0
$173$	−3.84738e15	−0.829549	−0.414775	−	0.909924i	$-0.636140\pi$
$173$	−3.84738e15	−0.829549	−0.414775	+	0.909924i	$0.636140\pi$
$174$	0	0
$175$	3.72893e15i	0.741853i
$176$	0	0
$177$	5.25528e15	0.965569
$178$	0	0
$179$	3.62594e15i	0.615814i	0.951416	+	0.307907i	$0.0996286\pi$
$179$	3.62594e15i	0.615814i	−0.951416	+	0.307907i	$0.900371\pi$
$180$	0	0
$181$	−1.03797e16	−1.63093	−0.815465	−	0.578807i	$-0.803519\pi$
$181$	−1.03797e16	−1.63093	−0.815465	+	0.578807i	$0.803519\pi$
$182$	0	0
$183$	1.51673e15i	0.220674i
$184$	0	0
$185$	−2.08145e15	−0.280649
$186$	0	0
$187$	1.32119e16i	1.65224i
$188$	0	0
$189$	1.98906e15	0.230895
$190$	0	0
$191$	− 6.34062e15i	− 0.683751i	−0.939745	−	0.341875i	$-0.888938\pi$
$191$	− 6.34062e15i	− 0.683751i	0.939745	−	0.341875i	$-0.111062\pi$
$192$	0	0
$193$	4.42606e15	0.443727	0.221863	−	0.975078i	$-0.428786\pi$
$193$	4.42606e15	0.443727	0.221863	+	0.975078i	$0.428786\pi$
$194$	0	0
$195$	7.48480e15i	0.698132i
$196$	0	0
$197$	4.06093e15	0.352665	0.176333	−	0.984331i	$-0.443577\pi$
$197$	4.06093e15	0.352665	0.176333	+	0.984331i	$0.443577\pi$
$198$	0	0
$199$	− 5.32846e15i	− 0.431152i	−0.976487	−	0.215576i	$-0.930837\pi$
$199$	− 5.32846e15i	− 0.431152i	0.976487	−	0.215576i	$-0.0691629\pi$
$200$	0	0
$201$	4.78041e15	0.360655
$202$	0	0
$203$	2.22468e16i	1.56601i
$204$	0	0
$205$	1.10969e16	0.729331
$206$	0	0
$207$	7.69450e15i	0.472485i
$208$	0	0
$209$	−3.35813e16	−1.92785
$210$	0	0
$211$	1.74049e16i	0.934749i	0.884060	+	0.467374i	$0.154800\pi$
$211$	1.74049e16i	0.934749i	−0.884060	+	0.467374i	$0.845200\pi$
$212$	0	0
$213$	1.89597e16	0.953179
$214$	0	0
$215$	5.24264e15i	0.246877i
$216$	0	0
$217$	9.63249e15	0.425129
$218$	0	0
$219$	− 1.41308e15i	− 0.0584867i
$220$	0	0
$221$	−6.96528e16	−2.70516
$222$	0	0
$223$	3.68823e15i	0.134488i	0.997737	+	0.0672441i	$0.0214206\pi$
$223$	3.68823e15i	0.134488i	−0.997737	+	0.0672441i	$0.978579\pi$
$224$	0	0
$225$	6.01696e15	0.206110
$226$	0	0
$227$	− 4.23682e16i	− 1.36414i	−0.731286	−	0.682071i	$-0.761080\pi$
$227$	− 4.23682e16i	− 1.36414i	0.731286	−	0.682071i	$-0.238920\pi$
$228$	0	0
$229$	2.81851e16	0.853436	0.426718	−	0.904385i	$-0.359670\pi$
$229$	2.81851e16	0.853436	0.426718	+	0.904385i	$0.359670\pi$
$230$	0	0
$231$	− 2.90640e16i	− 0.828077i
$232$	0	0
$233$	−5.09104e15	−0.136557	−0.0682787	−	0.997666i	$-0.521751\pi$
$233$	−5.09104e15	−0.136557	−0.0682787	+	0.997666i	$0.521751\pi$
$234$	0	0
$235$	6.88300e15i	0.173901i
$236$	0	0
$237$	3.34004e16	0.795268
$238$	0	0
$239$	2.69545e16i	0.605126i	0.953129	+	0.302563i	$0.0978423\pi$
$239$	2.69545e16i	0.605126i	−0.953129	+	0.302563i	$0.902158\pi$
$240$	0	0
$241$	2.01381e16	0.426480	0.213240	−	0.977000i	$-0.431598\pi$
$241$	2.01381e16	0.426480	0.213240	+	0.977000i	$0.431598\pi$
$242$	0	0
$243$	− 3.20953e15i	− 0.0641500i
$244$	0	0
$245$	−1.43849e16	−0.271484
$246$	0	0
$247$	− 1.77040e17i	− 3.15641i
$248$	0	0
$249$	−6.72149e15	−0.113258
$250$	0	0
$251$	− 4.33496e15i	− 0.0690667i	−0.999404	−	0.0345334i	$-0.989006\pi$
$251$	− 4.33496e15i	− 0.0690667i	0.999404	−	0.0345334i	$-0.0109945\pi$
$252$	0	0
$253$	1.12431e17	1.69451
$254$	0	0
$255$	− 3.45623e16i	− 0.492971i
$256$	0	0
$257$	−9.59521e16	−1.29575	−0.647877	−	0.761745i	$-0.724343\pi$
$257$	−9.59521e16	−1.29575	−0.647877	+	0.761745i	$0.724343\pi$
$258$	0	0
$259$	4.26104e16i	0.545026i
$260$	0	0
$261$	3.58972e16	0.435088
$262$	0	0
$263$	− 1.11253e17i	− 1.27826i	−0.769097	−	0.639132i	$-0.779294\pi$
$263$	− 1.11253e17i	− 1.27826i	0.769097	−	0.639132i	$-0.220706\pi$
$264$	0	0
$265$	2.45134e16	0.267105
$266$	0	0
$267$	8.56411e16i	0.885326i
$268$	0	0
$269$	−1.61075e17	−1.58039	−0.790193	−	0.612858i	$-0.790020\pi$
$269$	−1.61075e17	−1.58039	−0.790193	+	0.612858i	$0.790020\pi$
$270$	0	0
$271$	4.56194e16i	0.424976i	0.977164	+	0.212488i	$0.0681566\pi$
$271$	4.56194e16i	0.424976i	−0.977164	+	0.212488i	$0.931843\pi$
$272$	0	0
$273$	1.53225e17	1.35579
$274$	0	0
$275$	− 8.79193e16i	− 0.739188i
$276$	0	0
$277$	−1.63997e17	−1.31062	−0.655311	−	0.755359i	$-0.727462\pi$
$277$	−1.63997e17	−1.31062	−0.655311	+	0.755359i	$0.727462\pi$
$278$	0	0
$279$	− 1.55429e16i	− 0.118114i
$280$	0	0
$281$	−2.21492e17	−1.60108	−0.800541	−	0.599278i	$-0.795455\pi$
$281$	−2.21492e17	−1.60108	−0.800541	+	0.599278i	$0.795455\pi$
$282$	0	0
$283$	− 1.46578e17i	− 1.00824i	−0.863634	−	0.504120i	$-0.831817\pi$
$283$	− 1.46578e17i	− 1.00824i	0.863634	−	0.504120i	$-0.168183\pi$
$284$	0	0
$285$	8.78488e16	0.575204
$286$	0	0
$287$	− 2.27170e17i	− 1.41637i
$288$	0	0
$289$	1.53256e17	0.910188
$290$	0	0
$291$	− 4.05176e16i	− 0.229294i
$292$	0	0
$293$	2.00771e17	1.08300	0.541498	−	0.840702i	$-0.317857\pi$
$293$	2.00771e17	1.08300	0.541498	+	0.840702i	$0.317857\pi$
$294$	0	0
$295$	2.00882e17i	1.03321i
$296$	0	0
$297$	−4.68973e16	−0.230066
$298$	0	0
$299$	5.92738e17i	2.77437i
$300$	0	0
$301$	1.07325e17	0.479440
$302$	0	0
$303$	− 5.55928e16i	− 0.237094i
$304$	0	0
$305$	−5.79768e16	−0.236132
$306$	0	0
$307$	3.87213e17i	1.50654i	0.657709	+	0.753272i	$0.271526\pi$
$307$	3.87213e17i	1.50654i	−0.657709	+	0.753272i	$0.728474\pi$
$308$	0	0
$309$	−1.60431e17	−0.596455
$310$	0	0
$311$	− 1.39856e17i	− 0.497002i	−0.968632	−	0.248501i	$-0.920062\pi$
$311$	− 1.39856e17i	− 0.497002i	0.968632	−	0.248501i	$-0.0799380\pi$
$312$	0	0
$313$	−1.80451e17	−0.613126	−0.306563	−	0.951850i	$-0.599179\pi$
$313$	−1.80451e17	−0.613126	−0.306563	+	0.951850i	$0.599179\pi$
$314$	0	0
$315$	7.60315e16i	0.247070i
$316$	0	0
$317$	−3.01450e17	−0.937130	−0.468565	−	0.883429i	$-0.655229\pi$
$317$	−3.01450e17	−0.937130	−0.468565	+	0.883429i	$0.655229\pi$
$318$	0	0
$319$	− 5.24527e17i	− 1.56039i
$320$	0	0
$321$	3.24785e17	0.924827
$322$	0	0
$323$	8.17513e17i	2.22883i
$324$	0	0
$325$	4.63510e17	1.21025
$326$	0	0
$327$	2.76741e17i	0.692211i
$328$	0	0
$329$	1.40905e17	0.337720
$330$	0	0
$331$	7.22226e17i	1.65912i	0.558420	+	0.829558i	$0.311408\pi$
$331$	7.22226e17i	1.65912i	−0.558420	+	0.829558i	$0.688592\pi$
$332$	0	0
$333$	6.87557e16	0.151426
$334$	0	0
$335$	1.82730e17i	0.385920i
$336$	0	0
$337$	1.55312e17	0.314626	0.157313	−	0.987549i	$-0.449717\pi$
$337$	1.55312e17	0.314626	0.157313	+	0.987549i	$0.449717\pi$
$338$	0	0
$339$	− 1.55583e17i	− 0.302388i
$340$	0	0
$341$	−2.27111e17	−0.423602
$342$	0	0
$343$	− 3.75644e17i	− 0.672540i
$344$	0	0
$345$	−2.94121e17	−0.505583
$346$	0	0
$347$	− 2.64463e17i	− 0.436575i	−0.975885	−	0.218287i	$-0.929953\pi$
$347$	− 2.64463e17i	− 0.436575i	0.975885	−	0.218287i	$-0.0700470\pi$
$348$	0	0
$349$	−2.66338e16	−0.0422333	−0.0211167	−	0.999777i	$-0.506722\pi$
$349$	−2.66338e16	−0.0422333	−0.0211167	+	0.999777i	$0.506722\pi$
$350$	0	0
$351$	− 2.47242e17i	− 0.376680i
$352$	0	0
$353$	3.90609e17	0.571900	0.285950	−	0.958245i	$-0.407691\pi$
$353$	3.90609e17	0.571900	0.285950	+	0.958245i	$0.407691\pi$
$354$	0	0
$355$	7.24732e17i	1.01995i
$356$	0	0
$357$	−7.07542e17	−0.957358
$358$	0	0
$359$	− 1.03460e18i	− 1.34621i	−0.739547	−	0.673105i	$-0.764960\pi$
$359$	− 1.03460e18i	− 1.34621i	0.739547	−	0.673105i	$-0.235040\pi$
$360$	0	0
$361$	−1.27891e18	−1.60063
$362$	0	0
$363$	2.05762e17i	0.247753i
$364$	0	0
$365$	5.40146e16	0.0625837
$366$	0	0
$367$	4.77333e17i	0.532303i	0.963931	+	0.266152i	$0.0857521\pi$
$367$	4.77333e17i	0.532303i	−0.963931	+	0.266152i	$0.914248\pi$
$368$	0	0
$369$	−3.66559e17	−0.393513
$370$	0	0
$371$	− 5.01826e17i	− 0.518724i
$372$	0	0
$373$	−7.73235e17	−0.769750	−0.384875	−	0.922969i	$-0.625755\pi$
$373$	−7.73235e17	−0.769750	−0.384875	+	0.922969i	$0.625755\pi$
$374$	0	0
$375$	6.01962e17i	0.577232i
$376$	0	0
$377$	2.76530e18	2.55478
$378$	0	0
$379$	− 1.12169e18i	− 0.998617i	−0.866424	−	0.499308i	$-0.833587\pi$
$379$	− 1.12169e18i	− 0.998617i	0.866424	−	0.499308i	$-0.166413\pi$
$380$	0	0
$381$	4.78292e17	0.410410
$382$	0	0
$383$	− 7.84075e17i	− 0.648584i	−0.945957	−	0.324292i	$-0.894874\pi$
$383$	− 7.84075e17i	− 0.648584i	0.945957	−	0.324292i	$-0.105126\pi$
$384$	0	0
$385$	1.11096e18	0.886084
$386$	0	0
$387$	− 1.73178e17i	− 0.133204i
$388$	0	0
$389$	6.84400e17	0.507765	0.253883	−	0.967235i	$-0.418292\pi$
$389$	6.84400e17	0.507765	0.253883	+	0.967235i	$0.418292\pi$
$390$	0	0
$391$	− 2.73706e18i	− 1.95906i
$392$	0	0
$393$	−2.51696e17	−0.173832
$394$	0	0
$395$	1.27673e18i	0.850977i
$396$	0	0
$397$	−1.94763e18	−1.25306	−0.626532	−	0.779396i	$-0.715526\pi$
$397$	−1.94763e18	−1.25306	−0.626532	+	0.779396i	$0.715526\pi$
$398$	0	0
$399$	− 1.79840e18i	− 1.11706i
$400$	0	0
$401$	−5.07795e17	−0.304563	−0.152282	−	0.988337i	$-0.548662\pi$
$401$	−5.07795e17	−0.304563	−0.152282	+	0.988337i	$0.548662\pi$
$402$	0	0
$403$	− 1.19733e18i	− 0.693551i
$404$	0	0
$405$	1.22684e17	0.0686438
$406$	0	0
$407$	− 1.00465e18i	− 0.543069i
$408$	0	0
$409$	−1.02120e18	−0.533392	−0.266696	−	0.963781i	$-0.585932\pi$
$409$	−1.02120e18	−0.533392	−0.266696	+	0.963781i	$0.585932\pi$
$410$	0	0
$411$	− 1.00413e17i	− 0.0506871i
$412$	0	0
$413$	4.11236e18	2.00651
$414$	0	0
$415$	− 2.56928e17i	− 0.121192i
$416$	0	0
$417$	−2.12273e18	−0.968148
$418$	0	0
$419$	− 5.76380e17i	− 0.254220i	−0.991889	−	0.127110i	$-0.959430\pi$
$419$	− 5.76380e17i	− 0.254220i	0.991889	−	0.127110i	$-0.0405702\pi$
$420$	0	0
$421$	1.73770e18	0.741312	0.370656	−	0.928770i	$-0.379133\pi$
$421$	1.73770e18	0.741312	0.370656	+	0.928770i	$0.379133\pi$
$422$	0	0
$423$	− 2.27363e17i	− 0.0938292i
$424$	0	0
$425$	−2.14033e18	−0.854592
$426$	0	0
$427$	1.18687e18i	0.458573i
$428$	0	0
$429$	−3.61268e18	−1.35092
$430$	0	0
$431$	− 5.66339e17i	− 0.204991i	−0.994733	−	0.102496i	$-0.967317\pi$
$431$	− 5.66339e17i	− 0.204991i	0.994733	−	0.102496i	$-0.0326828\pi$
$432$	0	0
$433$	−3.30265e18	−1.15730	−0.578651	−	0.815575i	$-0.696421\pi$
$433$	−3.30265e18	−1.15730	−0.578651	+	0.815575i	$0.696421\pi$
$434$	0	0
$435$	1.37217e18i	0.465566i
$436$	0	0
$437$	6.95694e18	2.28585
$438$	0	0
$439$	1.99529e18i	0.634972i	0.948263	+	0.317486i	$0.102839\pi$
$439$	1.99529e18i	0.634972i	−0.948263	+	0.317486i	$0.897161\pi$
$440$	0	0
$441$	4.75171e17	0.146481
$442$	0	0
$443$	− 3.68211e17i	− 0.109969i	−0.998487	−	0.0549846i	$-0.982489\pi$
$443$	− 3.68211e17i	− 0.109969i	0.998487	−	0.0549846i	$-0.0175110\pi$
$444$	0	0
$445$	−3.27362e18	−0.947344
$446$	0	0
$447$	− 4.66850e16i	− 0.0130926i
$448$	0	0
$449$	2.79373e18	0.759382	0.379691	−	0.925113i	$-0.376030\pi$
$449$	2.79373e18	0.759382	0.379691	+	0.925113i	$0.376030\pi$
$450$	0	0
$451$	5.35613e18i	1.41129i
$452$	0	0
$453$	−8.59286e17	−0.219508
$454$	0	0
$455$	5.85700e18i	1.45076i
$456$	0	0
$457$	1.74659e18	0.419544	0.209772	−	0.977750i	$-0.432728\pi$
$457$	1.74659e18	0.419544	0.209772	+	0.977750i	$0.432728\pi$
$458$	0	0
$459$	1.14168e18i	0.265984i
$460$	0	0
$461$	−4.93335e18	−1.11490	−0.557449	−	0.830211i	$-0.688220\pi$
$461$	−4.93335e18	−1.11490	−0.557449	+	0.830211i	$0.688220\pi$
$462$	0	0
$463$	2.53867e17i	0.0556595i	0.999613	+	0.0278298i	$0.00885963\pi$
$463$	2.53867e17i	0.0556595i	−0.999613	+	0.0278298i	$0.991140\pi$
$464$	0	0
$465$	5.94124e17	0.126388
$466$	0	0
$467$	− 4.03491e18i	− 0.832944i	−0.909148	−	0.416472i	$-0.863266\pi$
$467$	− 4.03491e18i	− 0.832944i	0.909148	−	0.416472i	$-0.136734\pi$
$468$	0	0
$469$	3.74076e18	0.749463
$470$	0	0
$471$	1.15634e18i	0.224875i
$472$	0	0
$473$	−2.53046e18	−0.477718
$474$	0	0
$475$	− 5.44020e18i	− 0.997148i
$476$	0	0
$477$	−8.09742e17	−0.144118
$478$	0	0
$479$	5.06724e18i	0.875837i	0.899015	+	0.437918i	$0.144284\pi$
$479$	5.06724e18i	0.875837i	−0.899015	+	0.437918i	$0.855716\pi$
$480$	0	0
$481$	5.29652e18	0.889150
$482$	0	0
$483$	6.02110e18i	0.981852i
$484$	0	0
$485$	1.54878e18	0.245356
$486$	0	0
$487$	6.73830e18i	1.03716i	0.855028	+	0.518582i	$0.173540\pi$
$487$	6.73830e18i	1.03716i	−0.855028	+	0.518582i	$0.826460\pi$
$488$	0	0
$489$	−2.00902e18	−0.300485
$490$	0	0
$491$	6.14735e18i	0.893547i	0.894647	+	0.446773i	$0.147427\pi$
$491$	6.14735e18i	0.893547i	−0.894647	+	0.446773i	$0.852573\pi$
$492$	0	0
$493$	−1.27692e19	−1.80400
$494$	0	0
$495$	− 1.79264e18i	− 0.246182i
$496$	0	0
$497$	1.48363e19	1.98076
$498$	0	0
$499$	9.70589e18i	1.25989i	0.776641	+	0.629943i	$0.216922\pi$
$499$	9.70589e18i	1.25989i	−0.776641	+	0.629943i	$0.783078\pi$
$500$	0	0
$501$	6.38658e18	0.806128
$502$	0	0
$503$	4.85404e18i	0.595838i	0.954591	+	0.297919i	$0.0962925\pi$
$503$	4.85404e18i	0.595838i	−0.954591	+	0.297919i	$0.903708\pi$
$504$	0	0
$505$	2.12503e18	0.253702
$506$	0	0
$507$	− 1.40744e19i	− 1.63447i
$508$	0	0
$509$	4.60469e18	0.520207	0.260103	−	0.965581i	$-0.416243\pi$
$509$	4.60469e18	0.520207	0.260103	+	0.965581i	$0.416243\pi$
$510$	0	0
$511$	− 1.10576e18i	− 0.121539i
$512$	0	0
$513$	−2.90187e18	−0.310354
$514$	0	0
$515$	− 6.13243e18i	− 0.638237i
$516$	0	0
$517$	−3.32221e18	−0.336507
$518$	0	0
$519$	4.85795e18i	0.478940i
$520$	0	0
$521$	−1.66774e19	−1.60054	−0.800268	−	0.599643i	$-0.795309\pi$
$521$	−1.66774e19	−1.60054	−0.800268	+	0.599643i	$0.795309\pi$
$522$	0	0
$523$	− 1.97930e19i	− 1.84927i	−0.380859	−	0.924633i	$-0.624372\pi$
$523$	− 1.97930e19i	− 1.84927i	0.380859	−	0.924633i	$-0.375628\pi$
$524$	0	0
$525$	4.70839e18	0.428309
$526$	0	0
$527$	5.52886e18i	0.489735i
$528$	0	0
$529$	−1.16993e19	−1.00918
$530$	0	0
$531$	− 6.63566e18i	− 0.557472i
$532$	0	0
$533$	−2.82375e19	−2.31066
$534$	0	0
$535$	1.24148e19i	0.989612i
$536$	0	0
$537$	4.57835e18	0.355540
$538$	0	0
$539$	− 6.94316e18i	− 0.525334i
$540$	0	0
$541$	2.35383e19	1.73538	0.867689	−	0.497107i	$-0.165604\pi$
$541$	2.35383e19	1.73538	0.867689	+	0.497107i	$0.165604\pi$
$542$	0	0
$543$	1.31061e19i	0.941618i
$544$	0	0
$545$	−1.05784e19	−0.740701
$546$	0	0
$547$	− 2.88009e19i	− 1.96560i	−0.184676	−	0.982800i	$-0.559123\pi$
$547$	− 2.88009e19i	− 1.96560i	0.184676	−	0.982800i	$-0.440877\pi$
$548$	0	0
$549$	1.91512e18	0.127406
$550$	0	0
$551$	− 3.24563e19i	− 2.10493i
$552$	0	0
$553$	2.61365e19	1.65261
$554$	0	0
$555$	2.62818e18i	0.162033i
$556$	0	0
$557$	8.51606e18	0.511979	0.255990	−	0.966680i	$-0.417599\pi$
$557$	8.51606e18	0.511979	0.255990	+	0.966680i	$0.417599\pi$
$558$	0	0
$559$	− 1.33406e19i	− 0.782153i
$560$	0	0
$561$	1.66821e19	0.953920
$562$	0	0
$563$	− 2.92502e19i	− 1.63144i	−0.578449	−	0.815718i	$-0.696342\pi$
$563$	− 2.92502e19i	− 1.63144i	0.578449	−	0.815718i	$-0.303658\pi$
$564$	0	0
$565$	5.94714e18	0.323570
$566$	0	0
$567$	− 2.51152e18i	− 0.133307i
$568$	0	0
$569$	−1.81906e19	−0.942022	−0.471011	−	0.882127i	$-0.656111\pi$
$569$	−1.81906e19	−0.942022	−0.471011	+	0.882127i	$0.656111\pi$
$570$	0	0
$571$	1.74800e19i	0.883258i	0.897198	+	0.441629i	$0.145599\pi$
$571$	1.74800e19i	0.883258i	−0.897198	+	0.441629i	$0.854401\pi$
$572$	0	0
$573$	−8.00608e18	−0.394764
$574$	0	0
$575$	1.82140e19i	0.876456i
$576$	0	0
$577$	2.14406e18	0.100695	0.0503473	−	0.998732i	$-0.483967\pi$
$577$	2.14406e18	0.100695	0.0503473	+	0.998732i	$0.483967\pi$
$578$	0	0
$579$	− 5.58863e18i	− 0.256186i
$580$	0	0
$581$	−5.25970e18	−0.235357
$582$	0	0
$583$	1.18319e19i	0.516861i
$584$	0	0
$585$	9.45079e18	0.403067
$586$	0	0
$587$	2.27567e19i	0.947637i	0.880623	+	0.473818i	$0.157125\pi$
$587$	2.27567e19i	0.947637i	−0.880623	+	0.473818i	$0.842875\pi$
$588$	0	0
$589$	−1.40530e19	−0.571429
$590$	0	0
$591$	− 5.12760e18i	− 0.203611i
$592$	0	0
$593$	−1.61609e19	−0.626734	−0.313367	−	0.949632i	$-0.601457\pi$
$593$	−1.61609e19	−0.626734	−0.313367	+	0.949632i	$0.601457\pi$
$594$	0	0
$595$	− 2.70457e19i	− 1.02442i
$596$	0	0
$597$	−6.72806e18	−0.248926
$598$	0	0
$599$	− 1.29804e19i	− 0.469139i	−0.972099	−	0.234569i	$-0.924632\pi$
$599$	− 1.29804e19i	− 0.469139i	0.972099	−	0.234569i	$-0.0753680\pi$
$600$	0	0
$601$	2.14780e19	0.758353	0.379177	−	0.925324i	$-0.376207\pi$
$601$	2.14780e19	0.758353	0.379177	+	0.925324i	$0.376207\pi$
$602$	0	0
$603$	− 6.03605e18i	− 0.208225i
$604$	0	0
$605$	−7.86521e18	−0.265108
$606$	0	0
$607$	1.67305e19i	0.551048i	0.961294	+	0.275524i	$0.0888513\pi$
$607$	1.67305e19i	0.551048i	−0.961294	+	0.275524i	$0.911149\pi$
$608$	0	0
$609$	2.80903e19	0.904138
$610$	0	0
$611$	− 1.75147e19i	− 0.550952i
$612$	0	0
$613$	2.11132e19	0.649129	0.324565	−	0.945864i	$-0.394782\pi$
$613$	2.11132e19	0.649129	0.324565	+	0.945864i	$0.394782\pi$
$614$	0	0
$615$	− 1.40117e19i	− 0.421079i
$616$	0	0
$617$	−5.84940e19	−1.71836	−0.859181	−	0.511671i	$-0.829027\pi$
$617$	−5.84940e19	−1.71836	−0.859181	+	0.511671i	$0.829027\pi$
$618$	0	0
$619$	− 3.86175e18i	− 0.110905i	−0.998461	−	0.0554523i	$-0.982340\pi$
$619$	− 3.86175e18i	− 0.110905i	0.998461	−	0.0554523i	$-0.0176601\pi$
$620$	0	0
$621$	9.71558e18	0.272789
$622$	0	0
$623$	6.70158e19i	1.83976i
$624$	0	0
$625$	2.47123e16	0.000663366 0
$626$	0	0
$627$	4.24019e19i	1.11304i
$628$	0	0
$629$	−2.44576e19	−0.627854
$630$	0	0
$631$	5.88197e19i	1.47678i	0.674371	+	0.738392i	$0.264415\pi$
$631$	5.88197e19i	1.47678i	−0.674371	+	0.738392i	$0.735585\pi$
$632$	0	0
$633$	2.19766e19	0.539677
$634$	0	0
$635$	1.82826e19i	0.439160i
$636$	0	0
$637$	3.66043e19	0.860114
$638$	0	0
$639$	− 2.39398e19i	− 0.550318i
$640$	0	0
$641$	−2.21396e19	−0.497925	−0.248963	−	0.968513i	$-0.580090\pi$
$641$	−2.21396e19	−0.497925	−0.248963	+	0.968513i	$0.580090\pi$
$642$	0	0
$643$	− 2.96363e19i	− 0.652149i	−0.945344	−	0.326075i	$-0.894274\pi$
$643$	− 2.96363e19i	− 0.652149i	0.945344	−	0.326075i	$-0.105726\pi$
$644$	0	0
$645$	6.61969e18	0.142534
$646$	0	0
$647$	4.28835e19i	0.903566i	0.892128	+	0.451783i	$0.149212\pi$
$647$	4.28835e19i	0.903566i	−0.892128	+	0.451783i	$0.850788\pi$
$648$	0	0
$649$	−9.69597e19	−1.99930
$650$	0	0
$651$	− 1.21626e19i	− 0.245448i
$652$	0	0
$653$	2.57770e19	0.509142	0.254571	−	0.967054i	$-0.418066\pi$
$653$	2.57770e19	0.509142	0.254571	+	0.967054i	$0.418066\pi$
$654$	0	0
$655$	− 9.62104e18i	− 0.186008i
$656$	0	0
$657$	−1.78424e18	−0.0337673
$658$	0	0
$659$	6.04554e19i	1.12005i	0.828476	+	0.560025i	$0.189209\pi$
$659$	6.04554e19i	1.12005i	−0.828476	+	0.560025i	$0.810791\pi$
$660$	0	0
$661$	−3.86845e19	−0.701661	−0.350830	−	0.936439i	$-0.614101\pi$
$661$	−3.86845e19	−0.701661	−0.350830	+	0.936439i	$0.614101\pi$
$662$	0	0
$663$	8.79482e19i	1.56182i
$664$	0	0
$665$	6.87434e19	1.19531
$666$	0	0
$667$	1.08665e20i	1.85015i
$668$	0	0
$669$	4.65700e18	0.0776468
$670$	0	0
$671$	− 2.79836e19i	− 0.456926i
$672$	0	0
$673$	6.62671e19	1.05972	0.529862	−	0.848084i	$-0.322244\pi$
$673$	6.62671e19	1.05972	0.529862	+	0.848084i	$0.322244\pi$
$674$	0	0
$675$	− 7.59741e18i	− 0.118998i
$676$	0	0
$677$	9.37507e19	1.43831	0.719157	−	0.694847i	$-0.244528\pi$
$677$	9.37507e19	1.43831	0.719157	+	0.694847i	$0.244528\pi$
$678$	0	0
$679$	− 3.17058e19i	− 0.476486i
$680$	0	0
$681$	−5.34969e19	−0.787587
$682$	0	0
$683$	− 1.11901e20i	− 1.61394i	−0.590591	−	0.806971i	$-0.701105\pi$
$683$	− 1.11901e20i	− 1.61394i	0.590591	−	0.806971i	$-0.298895\pi$
$684$	0	0
$685$	3.83827e18	0.0542378
$686$	0	0
$687$	− 3.55883e19i	− 0.492732i
$688$	0	0
$689$	−6.23776e19	−0.846241
$690$	0	0
$691$	1.28716e20i	1.71114i	0.517687	+	0.855570i	$0.326793\pi$
$691$	1.28716e20i	1.71114i	−0.517687	+	0.855570i	$0.673207\pi$
$692$	0	0
$693$	−3.66980e19	−0.478091
$694$	0	0
$695$	− 8.11409e19i	− 1.03597i
$696$	0	0
$697$	1.30391e20	1.63162
$698$	0	0
$699$	6.42828e18i	0.0788414i
$700$	0	0
$701$	−7.39963e19	−0.889577	−0.444789	−	0.895636i	$-0.646721\pi$
$701$	−7.39963e19	−0.889577	−0.444789	+	0.895636i	$0.646721\pi$
$702$	0	0
$703$	− 6.21651e19i	− 0.732587i
$704$	0	0
$705$	8.69093e18	0.100402
$706$	0	0
$707$	− 4.35025e19i	− 0.492695i
$708$	0	0
$709$	−1.08505e19	−0.120483	−0.0602416	−	0.998184i	$-0.519187\pi$
$709$	−1.08505e19	−0.120483	−0.0602416	+	0.998184i	$0.519187\pi$
$710$	0	0
$711$	− 4.21736e19i	− 0.459148i
$712$	0	0
$713$	4.70500e19	0.502265
$714$	0	0
$715$	− 1.38094e20i	− 1.44555i
$716$	0	0
$717$	3.40345e19	0.349370
$718$	0	0
$719$	1.30501e20i	1.31375i	0.753999	+	0.656875i	$0.228122\pi$
$719$	1.30501e20i	1.31375i	−0.753999	+	0.656875i	$0.771878\pi$
$720$	0	0
$721$	−1.25540e20	−1.23947
$722$	0	0
$723$	− 2.54276e19i	− 0.246228i
$724$	0	0
$725$	8.49739e19	0.807085
$726$	0	0
$727$	− 1.19103e20i	− 1.10964i	−0.831971	−	0.554820i	$-0.812787\pi$
$727$	− 1.19103e20i	− 1.10964i	0.831971	−	0.554820i	$-0.187213\pi$
$728$	0	0
$729$	−4.05256e18	−0.0370370
$730$	0	0
$731$	6.16023e19i	0.552300i
$732$	0	0
$733$	−2.04057e20	−1.79483	−0.897415	−	0.441187i	$-0.854558\pi$
$733$	−2.04057e20	−1.79483	−0.897415	+	0.441187i	$0.854558\pi$
$734$	0	0
$735$	1.81633e19i	0.156742i
$736$	0	0
$737$	−8.81982e19	−0.746771
$738$	0	0
$739$	− 9.96867e19i	− 0.828183i	−0.910235	−	0.414091i	$-0.864099\pi$
$739$	− 9.96867e19i	− 0.828183i	0.910235	−	0.414091i	$-0.135901\pi$
$740$	0	0
$741$	−2.23543e20	−1.82236
$742$	0	0
$743$	− 4.48171e18i	− 0.0358527i	−0.999839	−	0.0179263i	$-0.994294\pi$
$743$	− 4.48171e18i	− 0.0358527i	0.999839	−	0.0179263i	$-0.00570644\pi$
$744$	0	0
$745$	1.78452e18	0.0140097
$746$	0	0
$747$	8.48699e18i	0.0653897i
$748$	0	0
$749$	2.54150e20	1.92184
$750$	0	0
$751$	− 1.75149e20i	− 1.29996i	−0.759953	−	0.649978i	$-0.774778\pi$
$751$	− 1.75149e20i	− 1.29996i	0.759953	−	0.649978i	$-0.225222\pi$
$752$	0	0
$753$	−5.47360e18	−0.0398757
$754$	0	0
$755$	− 3.28461e19i	− 0.234885i
$756$	0	0
$757$	6.03204e19	0.423441	0.211721	−	0.977330i	$-0.432093\pi$
$757$	6.03204e19	0.423441	0.211721	+	0.977330i	$0.432093\pi$
$758$	0	0
$759$	− 1.41963e20i	− 0.978325i
$760$	0	0
$761$	2.90495e20	1.96538	0.982688	−	0.185266i	$-0.0593148\pi$
$761$	2.90495e20	1.96538	0.982688	+	0.185266i	$0.0593148\pi$
$762$	0	0
$763$	2.16555e20i	1.43846i
$764$	0	0
$765$	−4.36406e19	−0.284617
$766$	0	0
$767$	− 5.11171e20i	− 3.27340i
$768$	0	0
$769$	8.31873e19	0.523086	0.261543	−	0.965192i	$-0.415769\pi$
$769$	8.31873e19	0.523086	0.261543	+	0.965192i	$0.415769\pi$
$770$	0	0
$771$	1.21155e20i	0.748104i
$772$	0	0
$773$	1.51800e20	0.920481	0.460241	−	0.887794i	$-0.347763\pi$
$773$	1.51800e20	0.920481	0.460241	+	0.887794i	$0.347763\pi$
$774$	0	0
$775$	− 3.67922e19i	− 0.219101i
$776$	0	0
$777$	5.38027e19	0.314671
$778$	0	0
$779$	3.31422e20i	1.90379i
$780$	0	0
$781$	−3.49806e20	−1.97365
$782$	0	0
$783$	− 4.53262e19i	− 0.251198i
$784$	0	0
$785$	−4.42010e19	−0.240627
$786$	0	0
$787$	5.16371e19i	0.276146i	0.990422	+	0.138073i	$0.0440908\pi$
$787$	5.16371e19i	0.276146i	−0.990422	+	0.138073i	$0.955909\pi$
$788$	0	0
$789$	−1.40475e20	−0.738007
$790$	0	0
$791$	− 1.21747e20i	− 0.628380i
$792$	0	0
$793$	1.47529e20	0.748111
$794$	0	0
$795$	− 3.09522e19i	− 0.154213i
$796$	0	0
$797$	2.17303e20	1.06380	0.531898	−	0.846809i	$-0.321479\pi$
$797$	2.17303e20	1.06380	0.531898	+	0.846809i	$0.321479\pi$
$798$	0	0
$799$	8.08769e19i	0.389043i
$800$	0	0
$801$	1.08136e20	0.511143
$802$	0	0
$803$	2.60712e19i	0.121102i
$804$	0	0
$805$	−2.30155e20	−1.05063
$806$	0	0
$807$	2.03384e20i	0.912436i
$808$	0	0
$809$	−3.96253e20	−1.74716	−0.873581	−	0.486679i	$-0.838208\pi$
$809$	−3.96253e20	−1.74716	−0.873581	+	0.486679i	$0.838208\pi$
$810$	0	0
$811$	− 5.62010e19i	− 0.243556i	−0.992557	−	0.121778i	$-0.961140\pi$
$811$	− 5.62010e19i	− 0.243556i	0.992557	−	0.121778i	$-0.0388596\pi$
$812$	0	0
$813$	5.76020e19	0.245360
$814$	0	0
$815$	− 7.67946e19i	− 0.321534i
$816$	0	0
$817$	−1.56578e20	−0.644430
$818$	0	0
$819$	− 1.93472e20i	− 0.782763i
$820$	0	0
$821$	4.46915e20	1.77755	0.888776	−	0.458341i	$-0.151556\pi$
$821$	4.46915e20	1.77755	0.888776	+	0.458341i	$0.151556\pi$
$822$	0	0
$823$	3.52023e20i	1.37649i	0.725479	+	0.688245i	$0.241618\pi$
$823$	3.52023e20i	1.37649i	−0.725479	+	0.688245i	$0.758382\pi$
$824$	0	0
$825$	−1.11013e20	−0.426771
$826$	0	0
$827$	2.57157e20i	0.971987i	0.873962	+	0.485993i	$0.161542\pi$
$827$	2.57157e20i	0.971987i	−0.873962	+	0.485993i	$0.838458\pi$
$828$	0	0
$829$	2.04550e20	0.760182	0.380091	−	0.924949i	$-0.375893\pi$
$829$	2.04550e20	0.760182	0.380091	+	0.924949i	$0.375893\pi$
$830$	0	0
$831$	2.07073e20i	0.756688i
$832$	0	0
$833$	−1.69026e20	−0.607350
$834$	0	0
$835$	2.44126e20i	0.862598i
$836$	0	0
$837$	−1.96255e19	−0.0681933
$838$	0	0
$839$	4.46857e20i	1.52699i	0.645816	+	0.763493i	$0.276517\pi$
$839$	4.46857e20i	1.52699i	−0.645816	+	0.763493i	$0.723483\pi$
$840$	0	0
$841$	2.09396e20	0.703715
$842$	0	0
$843$	2.79670e20i	0.924386i
$844$	0	0
$845$	5.37993e20	1.74896
$846$	0	0
$847$	1.61013e20i	0.514844i
$848$	0	0
$849$	−1.85079e20	−0.582108
$850$	0	0
$851$	2.08131e20i	0.643917i
$852$	0	0
$853$	−2.43984e20	−0.742538	−0.371269	−	0.928525i	$-0.621077\pi$
$853$	−2.43984e20	−0.742538	−0.371269	+	0.928525i	$0.621077\pi$
$854$	0	0
$855$	− 1.10924e20i	− 0.332094i
$856$	0	0
$857$	−3.30060e20	−0.972137	−0.486068	−	0.873921i	$-0.661569\pi$
$857$	−3.30060e20	−0.972137	−0.486068	+	0.873921i	$0.661569\pi$
$858$	0	0
$859$	− 3.07504e20i	− 0.891043i	−0.895271	−	0.445522i	$-0.853018\pi$
$859$	− 3.07504e20i	− 0.891043i	0.895271	−	0.445522i	$-0.146982\pi$
$860$	0	0
$861$	−2.86840e20	−0.817744
$862$	0	0
$863$	− 2.92528e20i	− 0.820525i	−0.911967	−	0.410262i	$-0.865437\pi$
$863$	− 2.92528e20i	− 0.820525i	0.911967	−	0.410262i	$-0.134563\pi$
$864$	0	0
$865$	−1.85694e20	−0.512490
$866$	0	0
$867$	− 1.93510e20i	− 0.525497i
$868$	0	0
$869$	−6.16236e20	−1.64668
$870$	0	0
$871$	− 4.64981e20i	− 1.22267i
$872$	0	0
$873$	−5.11601e19	−0.132383
$874$	0	0
$875$	4.71047e20i	1.19952i
$876$	0	0
$877$	2.73744e20	0.686038	0.343019	−	0.939329i	$-0.388551\pi$
$877$	2.73744e20	0.686038	0.343019	+	0.939329i	$0.388551\pi$
$878$	0	0
$879$	− 2.53506e20i	− 0.625268i
$880$	0	0
$881$	3.72617e20	0.904547	0.452274	−	0.891879i	$-0.350613\pi$
$881$	3.72617e20	0.904547	0.452274	+	0.891879i	$0.350613\pi$
$882$	0	0
$883$	3.43070e20i	0.819705i	0.912152	+	0.409853i	$0.134420\pi$
$883$	3.43070e20i	0.819705i	−0.912152	+	0.409853i	$0.865580\pi$
$884$	0	0
$885$	2.53647e20	0.596523
$886$	0	0
$887$	4.98820e20i	1.15473i	0.816488	+	0.577363i	$0.195918\pi$
$887$	4.98820e20i	1.15473i	−0.816488	+	0.577363i	$0.804082\pi$
$888$	0	0
$889$	3.74273e20	0.852856
$890$	0	0
$891$	5.92156e19i	0.132829i
$892$	0	0
$893$	−2.05569e20	−0.453940
$894$	0	0
$895$	1.75007e20i	0.380446i
$896$	0	0
$897$	7.48429e20	1.60178
$898$	0	0
$899$	− 2.19503e20i	− 0.462511i
$900$	0	0
$901$	2.88039e20	0.597554
$902$	0	0
$903$	− 1.35515e20i	− 0.276805i
$904$	0	0
$905$	−5.00978e20	−1.00758
$906$	0	0
$907$	4.49105e20i	0.889399i	0.895680	+	0.444700i	$0.146690\pi$
$907$	4.49105e20i	0.889399i	−0.895680	+	0.444700i	$0.853310\pi$
$908$	0	0
$909$	−7.01951e19	−0.136886
$910$	0	0
$911$	− 4.87755e20i	− 0.936641i	−0.883559	−	0.468320i	$-0.844859\pi$
$911$	− 4.87755e20i	− 0.936641i	0.883559	−	0.468320i	$-0.155141\pi$
$912$	0	0
$913$	1.24011e20	0.234512
$914$	0	0
$915$	7.32052e19i	0.136331i
$916$	0	0
$917$	−1.96957e20	−0.361232
$918$	0	0
$919$	− 4.16251e19i	− 0.0751877i	−0.999293	−	0.0375938i	$-0.988031\pi$
$919$	− 4.16251e19i	− 0.0751877i	0.999293	−	0.0375938i	$-0.0119693\pi$
$920$	0	0
$921$	4.88921e20	0.869804
$922$	0	0
$923$	− 1.84417e21i	− 3.23139i
$924$	0	0
$925$	1.62755e20	0.280893
$926$	0	0
$927$	2.02570e20i	0.344363i
$928$	0	0
$929$	−3.12297e20	−0.522947	−0.261473	−	0.965211i	$-0.584208\pi$
$929$	−3.12297e20	−0.522947	−0.261473	+	0.965211i	$0.584208\pi$
$930$	0	0
$931$	− 4.29623e20i	− 0.708663i
$932$	0	0
$933$	−1.76591e20	−0.286945
$934$	0	0
$935$	6.37672e20i	1.02074i
$936$	0	0
$937$	−1.84262e20	−0.290575	−0.145288	−	0.989389i	$-0.546411\pi$
$937$	−1.84262e20	−0.290575	−0.145288	+	0.989389i	$0.546411\pi$
$938$	0	0
$939$	2.27850e20i	0.353988i
$940$	0	0
$941$	3.97291e20	0.608109	0.304054	−	0.952655i	$-0.401660\pi$
$941$	3.97291e20	0.608109	0.304054	+	0.952655i	$0.401660\pi$
$942$	0	0
$943$	− 1.10961e21i	− 1.67336i
$944$	0	0
$945$	9.60023e19	0.142646
$946$	0	0
$947$	− 1.96271e20i	− 0.287347i	−0.989625	−	0.143674i	$-0.954108\pi$
$947$	− 1.96271e20i	− 0.287347i	0.989625	−	0.143674i	$-0.0458915\pi$
$948$	0	0
$949$	−1.37447e20	−0.198277
$950$	0	0
$951$	3.80630e20i	0.541052i
$952$	0	0
$953$	−1.41143e21	−1.97701	−0.988507	−	0.151177i	$-0.951694\pi$
$953$	−1.41143e21	−1.97701	−0.988507	+	0.151177i	$0.951694\pi$
$954$	0	0
$955$	− 3.06031e20i	− 0.422417i
$956$	0	0
$957$	−6.62302e20	−0.900891
$958$	0	0
$959$	− 7.85752e19i	− 0.105331i
$960$	0	0
$961$	6.61903e20	0.874441
$962$	0	0
$963$	− 4.10094e20i	− 0.533949i
$964$	0	0
$965$	2.13624e20	0.274132
$966$	0	0
$967$	− 4.30311e19i	− 0.0544249i	−0.999630	−	0.0272124i	$-0.991337\pi$
$967$	− 4.30311e19i	− 0.0544249i	0.999630	−	0.0272124i	$-0.00866306\pi$
$968$	0	0
$969$	1.03224e21	1.28682
$970$	0	0
$971$	− 5.43559e20i	− 0.667901i	−0.942591	−	0.333951i	$-0.891618\pi$
$971$	− 5.43559e20i	− 0.667901i	0.942591	−	0.333951i	$-0.108382\pi$
$972$	0	0
$973$	−1.66108e21	−2.01187
$974$	0	0
$975$	− 5.85258e20i	− 0.698739i
$976$	0	0
$977$	−6.68715e20	−0.787008	−0.393504	−	0.919323i	$-0.628737\pi$
$977$	−6.68715e20	−0.787008	−0.393504	+	0.919323i	$0.628737\pi$
$978$	0	0
$979$	− 1.58007e21i	− 1.83315i
$980$	0	0
$981$	3.49431e20	0.399648
$982$	0	0
$983$	− 1.25760e20i	− 0.141798i	−0.997484	−	0.0708988i	$-0.977413\pi$
$983$	− 1.25760e20i	− 0.141798i	0.997484	−	0.0708988i	$-0.0225868\pi$
$984$	0	0
$985$	1.96002e20	0.217874
$986$	0	0
$987$	− 1.77916e20i	− 0.194982i
$988$	0	0
$989$	5.24228e20	0.566430
$990$	0	0
$991$	5.59246e20i	0.595782i	0.954600	+	0.297891i	$0.0962832\pi$
$991$	5.59246e20i	0.595782i	−0.954600	+	0.297891i	$0.903717\pi$
$992$	0	0
$993$	9.11929e20	0.957892
$994$	0	0
$995$	− 2.57179e20i	− 0.266363i
$996$	0	0
$997$	1.53050e21	1.56303	0.781514	−	0.623888i	$-0.214448\pi$
$997$	1.53050e21	1.56303	0.781514	+	0.623888i	$0.214448\pi$
$998$	0	0
$999$	− 8.68155e19i	− 0.0874256i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.15.g.a.31.2	✓	4
3.2	odd	2		144.15.g.e.127.1		4
4.3	odd	2	inner	48.15.g.a.31.4	yes	4
12.11	even	2		144.15.g.e.127.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.15.g.a.31.2	✓	4	1.1	even	1	trivial
48.15.g.a.31.4	yes	4	4.3	odd	2	inner
144.15.g.e.127.1		4	3.2	odd	2
144.15.g.e.127.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1262.67i q^{3} +48265.2 q^{5} -988060. i q^{7} -1.59432e6 q^{9} +O(q^{10})\)	\(q-1262.67i q^{3} +48265.2 q^{5} -988060. i q^{7} -1.59432e6 q^{9} +2.32961e7i q^{11} -1.22817e8 q^{13} -6.09427e7i q^{15} +5.67127e8 q^{17} +1.44150e9i q^{19} -1.24759e9 q^{21} -4.82619e9i q^{23} -3.77399e9 q^{25} +2.01310e9i q^{27} -2.25157e10 q^{29} +9.74889e9i q^{31} +2.94152e10 q^{33} -4.76889e10i q^{35} -4.31253e10 q^{37} +1.55077e11i q^{39} +2.29915e11 q^{41} +1.08622e11i q^{43} -7.69502e10 q^{45} +1.42608e11i q^{47} -2.98039e11 q^{49} -7.16092e11i q^{51} +5.07891e11 q^{53} +1.12439e12i q^{55} +1.82013e12 q^{57} +4.16206e12i q^{59} -1.20121e12 q^{61} +1.57529e12i q^{63} -5.92778e12 q^{65} +3.78597e12i q^{67} -6.09386e12 q^{69} +1.50156e13i q^{71} +1.11912e12 q^{73} +4.76529e12i q^{75} +2.30179e13 q^{77} +2.64523e13i q^{79} +2.54187e12 q^{81} -5.32326e12i q^{83} +2.73725e13 q^{85} +2.84297e13i q^{87} -6.78257e13 q^{89} +1.21350e14i q^{91} +1.23096e13 q^{93} +6.95741e13i q^{95} +3.20889e13 q^{97} -3.71415e13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18360 q^{5} - 6377292 q^{9}+O(q^{10}) \) 4 * q + 18360 * q^5 - 6377292 * q^9	\( 4 q + 18360 q^{5} - 6377292 q^{9} - 158388152 q^{13} + 262496808 q^{17} - 2206700496 q^{21} - 16699713940 q^{25} - 59940048168 q^{29} + 16466640336 q^{33} + 14727873320 q^{37} + 446330350440 q^{41} - 29271770280 q^{45} + 734268733828 q^{49} + 837011171736 q^{53} + 1309414251888 q^{57} - 3828732927064 q^{61} - 15265566735120 q^{65} - 6884591879232 q^{69} - 8960085606968 q^{73} + 49868530746048 q^{77} + 10167463313316 q^{81} + 88817754861360 q^{85} - 79296711493176 q^{89} + 42094011836880 q^{93} - 81693961168376 q^{97}+O(q^{100}) \) 4 * q + 18360 * q^5 - 6377292 * q^9 - 158388152 * q^13 + 262496808 * q^17 - 2206700496 * q^21 - 16699713940 * q^25 - 59940048168 * q^29 + 16466640336 * q^33 + 14727873320 * q^37 + 446330350440 * q^41 - 29271770280 * q^45 + 734268733828 * q^49 + 837011171736 * q^53 + 1309414251888 * q^57 - 3828732927064 * q^61 - 15265566735120 * q^65 - 6884591879232 * q^69 - 8960085606968 * q^73 + 49868530746048 * q^77 + 10167463313316 * q^81 + 88817754861360 * q^85 - 79296711493176 * q^89 + 42094011836880 * q^93 - 81693961168376 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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