LMFDB - Embedded newform 384.10.d.f.193.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(193,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$197.773761087$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 $ x^20 - 8x^19 - 300700x^18 + 6140664x^17 + 35387063979x^16 - 1130222504088x^15 - 2027709461599244x^14 + 84850591004671480x^13 + 57031067836759165823x^12 - 2761881829653049004184x^11 - 715774112122919877685260x^10 + 33804686703728196561023832x^9 + 4440127592352179578492608037x^8 - 186062650255909170075184345736x^7 - 13450754766452318411315895909412x^6 + 457952106473696272472169073091112x^5 + 15298236539262291502286172763910964x^4 - 382483105264444074951018570546548352x^3 + 3006361776222337276289989101322653304x^2 - 10187939951443559461324435287012809264*x + 12842550816412602186535854197192532452
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{175}\cdot 3^{32} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.11
Root			$257.371 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.f.193.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+81.0000i q^{3} -2358.11i q^{5} +1310.82 q^{7} -6561.00 q^{9} +O(q^{10})$	$q+81.0000i q^{3} -2358.11i q^{5} +1310.82 q^{7} -6561.00 q^{9} +12659.0i q^{11} -149414. i q^{13} +191007. q^{15} -242843. q^{17} +713999. i q^{19} +106177. i q^{21} +634045. q^{23} -3.60758e6 q^{25} -531441. i q^{27} +919977. i q^{29} -5.06417e6 q^{31} -1.02538e6 q^{33} -3.09107e6i q^{35} -1.14011e7i q^{37} +1.21025e7 q^{39} -8.44272e6 q^{41} +1.27628e7i q^{43} +1.54716e7i q^{45} +1.08809e6 q^{47} -3.86353e7 q^{49} -1.96703e7i q^{51} +9.16666e6i q^{53} +2.98514e7 q^{55} -5.78339e7 q^{57} -2.45165e6i q^{59} -4.46880e7i q^{61} -8.60032e6 q^{63} -3.52335e8 q^{65} +2.55851e8i q^{67} +5.13577e7i q^{69} -1.45406e8 q^{71} -3.89644e8 q^{73} -2.92214e8i q^{75} +1.65937e7i q^{77} +6.65328e8 q^{79} +4.30467e7 q^{81} -3.04812e8i q^{83} +5.72653e8i q^{85} -7.45182e7 q^{87} +3.38397e8 q^{89} -1.95855e8i q^{91} -4.10198e8i q^{93} +1.68369e9 q^{95} +1.28207e9 q^{97} -8.30558e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 131220 q^{9}+O(q^{10}) $ 20 * q - 131220 * q^9	$ 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) $ 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	81.0000i	0.577350i
$3$	81.0000i	0.577350i
$4$	0	0
$5$	− 2358.11i	− 1.68733i	−0.536870	−	0.843665i	$-0.680394\pi$
$5$	− 2358.11i	− 1.68733i	0.536870	−	0.843665i	$-0.319606\pi$
$6$	0	0
$7$	1310.82	0.206349	0.103175	−	0.994663i	$-0.467100\pi$
$7$	1310.82	0.206349	0.103175	+	0.994663i	$0.467100\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	12659.0i	0.260695i	0.991468	+	0.130348i	$0.0416093\pi$
$11$	12659.0i	0.260695i	−0.991468	+	0.130348i	$0.958391\pi$
$12$	0	0
$13$	− 149414.i	− 1.45093i	−0.688261	−	0.725463i	$-0.741626\pi$
$13$	− 149414.i	− 1.45093i	0.688261	−	0.725463i	$-0.258374\pi$
$14$	0	0
$15$	191007.	0.974180
$16$	0	0
$17$	−242843.	−0.705190	−0.352595	−	0.935776i	$-0.614701\pi$
$17$	−242843.	−0.705190	−0.352595	+	0.935776i	$0.614701\pi$
$18$	0	0
$19$	713999.i	1.25692i	0.777844	+	0.628458i	$0.216314\pi$
$19$	713999.i	1.25692i	−0.777844	+	0.628458i	$0.783686\pi$
$20$	0	0
$21$	106177.i	0.119136i
$22$	0	0
$23$	634045.	0.472438	0.236219	−	0.971700i	$-0.424092\pi$
$23$	634045.	0.472438	0.236219	+	0.971700i	$0.424092\pi$
$24$	0	0
$25$	−3.60758e6	−1.84708
$26$	0	0
$27$	− 531441.i	− 0.192450i
$28$	0	0
$29$	919977.i	0.241538i	0.992681	+	0.120769i	$0.0385361\pi$
$29$	919977.i	0.241538i	−0.992681	+	0.120769i	$0.961464\pi$
$30$	0	0
$31$	−5.06417e6	−0.984874	−0.492437	−	0.870348i	$-0.663894\pi$
$31$	−5.06417e6	−0.984874	−0.492437	+	0.870348i	$0.663894\pi$
$32$	0	0
$33$	−1.02538e6	−0.150512
$34$	0	0
$35$	− 3.09107e6i	− 0.348179i
$36$	0	0
$37$	− 1.14011e7i	− 1.00009i	−0.866000	−	0.500045i	$-0.833317\pi$
$37$	− 1.14011e7i	− 1.00009i	0.866000	−	0.500045i	$-0.166683\pi$
$38$	0	0
$39$	1.21025e7	0.837693
$40$	0	0
$41$	−8.44272e6	−0.466611	−0.233306	−	0.972403i	$-0.574954\pi$
$41$	−8.44272e6	−0.466611	−0.233306	+	0.972403i	$0.574954\pi$
$42$	0	0
$43$	1.27628e7i	0.569297i	0.958632	+	0.284649i	$0.0918769\pi$
$43$	1.27628e7i	0.569297i	−0.958632	+	0.284649i	$0.908123\pi$
$44$	0	0
$45$	1.54716e7i	0.562443i
$46$	0	0
$47$	1.08809e6	0.0325255	0.0162628	−	0.999868i	$-0.494823\pi$
$47$	1.08809e6	0.0325255	0.0162628	+	0.999868i	$0.494823\pi$
$48$	0	0
$49$	−3.86353e7	−0.957420
$50$	0	0
$51$	− 1.96703e7i	− 0.407142i
$52$	0	0
$53$	9.16666e6i	0.159577i	0.996812	+	0.0797884i	$0.0254245\pi$
$53$	9.16666e6i	0.159577i	−0.996812	+	0.0797884i	$0.974576\pi$
$54$	0	0
$55$	2.98514e7	0.439879
$56$	0	0
$57$	−5.78339e7	−0.725681
$58$	0	0
$59$	− 2.45165e6i	− 0.0263406i	−0.999913	−	0.0131703i	$-0.995808\pi$
$59$	− 2.45165e6i	− 0.0263406i	0.999913	−	0.0131703i	$-0.00419235\pi$
$60$	0	0
$61$	− 4.46880e7i	− 0.413244i	−0.978421	−	0.206622i	$-0.933753\pi$
$61$	− 4.46880e7i	− 0.413244i	0.978421	−	0.206622i	$-0.0662471\pi$
$62$	0	0
$63$	−8.60032e6	−0.0687831
$64$	0	0
$65$	−3.52335e8	−2.44819
$66$	0	0
$67$	2.55851e8i	1.55114i	0.631264	+	0.775568i	$0.282537\pi$
$67$	2.55851e8i	1.55114i	−0.631264	+	0.775568i	$0.717463\pi$
$68$	0	0
$69$	5.13577e7i	0.272762i
$70$	0	0
$71$	−1.45406e8	−0.679079	−0.339539	−	0.940592i	$-0.610271\pi$
$71$	−1.45406e8	−0.679079	−0.339539	+	0.940592i	$0.610271\pi$
$72$	0	0
$73$	−3.89644e8	−1.60589	−0.802944	−	0.596054i	$-0.796734\pi$
$73$	−3.89644e8	−1.60589	−0.802944	+	0.596054i	$0.796734\pi$
$74$	0	0
$75$	− 2.92214e8i	− 1.06641i
$76$	0	0
$77$	1.65937e7i	0.0537943i
$78$	0	0
$79$	6.65328e8	1.92183	0.960913	−	0.276850i	$-0.0892904\pi$
$79$	6.65328e8	1.92183	0.960913	+	0.276850i	$0.0892904\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	− 3.04812e8i	− 0.704986i	−0.935814	−	0.352493i	$-0.885334\pi$
$83$	− 3.04812e8i	− 0.704986i	0.935814	−	0.352493i	$-0.114666\pi$
$84$	0	0
$85$	5.72653e8i	1.18989i
$86$	0	0
$87$	−7.45182e7	−0.139452
$88$	0	0
$89$	3.38397e8	0.571705	0.285852	−	0.958274i	$-0.407723\pi$
$89$	3.38397e8	0.571705	0.285852	+	0.958274i	$0.407723\pi$
$90$	0	0
$91$	− 1.95855e8i	− 0.299398i
$92$	0	0
$93$	− 4.10198e8i	− 0.568617i
$94$	0	0
$95$	1.68369e9	2.12083
$96$	0	0
$97$	1.28207e9	1.47041	0.735203	−	0.677847i	$-0.237087\pi$
$97$	1.28207e9	1.47041	0.735203	+	0.677847i	$0.237087\pi$
$98$	0	0
$99$	− 8.30558e7i	− 0.0868984i
$100$	0	0
$101$	− 1.21088e9i	− 1.15786i	−0.815379	−	0.578928i	$-0.803471\pi$
$101$	− 1.21088e9i	− 1.15786i	0.815379	−	0.578928i	$-0.196529\pi$
$102$	0	0
$103$	−1.71434e9	−1.50082	−0.750412	−	0.660970i	$-0.770145\pi$
$103$	−1.71434e9	−1.50082	−0.750412	+	0.660970i	$0.770145\pi$
$104$	0	0
$105$	2.50377e8	0.201021
$106$	0	0
$107$	2.28518e9i	1.68536i	0.538412	+	0.842682i	$0.319024\pi$
$107$	2.28518e9i	1.68536i	−0.538412	+	0.842682i	$0.680976\pi$
$108$	0	0
$109$	− 1.16653e8i	− 0.0791549i	−0.999217	−	0.0395774i	$-0.987399\pi$
$109$	− 1.16653e8i	− 0.0791549i	0.999217	−	0.0395774i	$-0.0126012\pi$
$110$	0	0
$111$	9.23489e8	0.577402
$112$	0	0
$113$	2.33117e9	1.34500	0.672499	−	0.740098i	$-0.265221\pi$
$113$	2.33117e9	1.34500	0.672499	+	0.740098i	$0.265221\pi$
$114$	0	0
$115$	− 1.49515e9i	− 0.797159i
$116$	0	0
$117$	9.80303e8i	0.483642i
$118$	0	0
$119$	−3.18325e8	−0.145516
$120$	0	0
$121$	2.19770e9	0.932038
$122$	0	0
$123$	− 6.83860e8i	− 0.269398i
$124$	0	0
$125$	3.90140e9i	1.42931i
$126$	0	0
$127$	3.43019e9	1.17004	0.585021	−	0.811018i	$-0.301086\pi$
$127$	3.43019e9	1.17004	0.585021	+	0.811018i	$0.301086\pi$
$128$	0	0
$129$	−1.03379e9	−0.328684
$130$	0	0
$131$	5.02477e9i	1.49072i	0.666664	+	0.745358i	$0.267721\pi$
$131$	5.02477e9i	1.49072i	−0.666664	+	0.745358i	$0.732279\pi$
$132$	0	0
$133$	9.35926e8i	0.259364i
$134$	0	0
$135$	−1.25320e9	−0.324727
$136$	0	0
$137$	−1.35961e9	−0.329741	−0.164870	−	0.986315i	$-0.552721\pi$
$137$	−1.35961e9	−0.329741	−0.164870	+	0.986315i	$0.552721\pi$
$138$	0	0
$139$	− 7.29504e9i	− 1.65753i	−0.559597	−	0.828765i	$-0.689044\pi$
$139$	− 7.29504e9i	− 1.65753i	0.559597	−	0.828765i	$-0.310956\pi$
$140$	0	0
$141$	8.81353e7i	0.0187786i
$142$	0	0
$143$	1.89143e9	0.378249
$144$	0	0
$145$	2.16941e9	0.407555
$146$	0	0
$147$	− 3.12946e9i	− 0.552767i
$148$	0	0
$149$	6.34307e8i	0.105429i	0.998610	+	0.0527146i	$0.0167874\pi$
$149$	6.34307e8i	0.105429i	−0.998610	+	0.0527146i	$0.983213\pi$
$150$	0	0
$151$	2.80634e9	0.439283	0.219642	−	0.975581i	$-0.429511\pi$
$151$	2.80634e9	0.439283	0.219642	+	0.975581i	$0.429511\pi$
$152$	0	0
$153$	1.59330e9	0.235063
$154$	0	0
$155$	1.19419e10i	1.66181i
$156$	0	0
$157$	− 4.70003e9i	− 0.617380i	−0.951163	−	0.308690i	$-0.900110\pi$
$157$	− 4.70003e9i	− 0.617380i	0.951163	−	0.308690i	$-0.0998905\pi$
$158$	0	0
$159$	−7.42500e8	−0.0921317
$160$	0	0
$161$	8.31122e8	0.0974874
$162$	0	0
$163$	− 9.00023e9i	− 0.998641i	−0.866417	−	0.499320i	$-0.833583\pi$
$163$	− 9.00023e9i	− 0.998641i	0.866417	−	0.499320i	$-0.166417\pi$
$164$	0	0
$165$	2.41796e9i	0.253964i
$166$	0	0
$167$	−1.51957e10	−1.51181	−0.755904	−	0.654683i	$-0.772802\pi$
$167$	−1.51957e10	−1.51181	−0.755904	+	0.654683i	$0.772802\pi$
$168$	0	0
$169$	−1.17200e10	−1.10519
$170$	0	0
$171$	− 4.68454e9i	− 0.418972i
$172$	0	0
$173$	1.48179e10i	1.25771i	0.777524	+	0.628853i	$0.216475\pi$
$173$	1.48179e10i	1.25771i	−0.777524	+	0.628853i	$0.783525\pi$
$174$	0	0
$175$	−4.72890e9	−0.381144
$176$	0	0
$177$	1.98584e8	0.0152077
$178$	0	0
$179$	2.46574e10i	1.79518i	0.440832	+	0.897590i	$0.354683\pi$
$179$	2.46574e10i	1.79518i	−0.440832	+	0.897590i	$0.645317\pi$
$180$	0	0
$181$	2.24515e10i	1.55486i	0.628969	+	0.777430i	$0.283477\pi$
$181$	2.24515e10i	1.55486i	−0.628969	+	0.777430i	$0.716523\pi$
$182$	0	0
$183$	3.61973e9	0.238587
$184$	0	0
$185$	−2.68851e10	−1.68748
$186$	0	0
$187$	− 3.07416e9i	− 0.183840i
$188$	0	0
$189$	− 6.96626e8i	− 0.0397120i
$190$	0	0
$191$	6.05951e8	0.0329448	0.0164724	−	0.999864i	$-0.494756\pi$
$191$	6.05951e8	0.0329448	0.0164724	+	0.999864i	$0.494756\pi$
$192$	0	0
$193$	−1.24435e10	−0.645559	−0.322780	−	0.946474i	$-0.604617\pi$
$193$	−1.24435e10	−0.645559	−0.322780	+	0.946474i	$0.604617\pi$
$194$	0	0
$195$	− 2.85391e10i	− 1.41346i
$196$	0	0
$197$	− 6.81666e9i	− 0.322458i	−0.986917	−	0.161229i	$-0.948454\pi$
$197$	− 6.81666e9i	− 0.322458i	0.986917	−	0.161229i	$-0.0515458\pi$
$198$	0	0
$199$	−1.91661e10	−0.866351	−0.433175	−	0.901310i	$-0.642607\pi$
$199$	−1.91661e10	−0.866351	−0.433175	+	0.901310i	$0.642607\pi$
$200$	0	0
$201$	−2.07239e10	−0.895549
$202$	0	0
$203$	1.20593e9i	0.0498413i
$204$	0	0
$205$	1.99089e10i	0.787327i
$206$	0	0
$207$	−4.15997e9	−0.157479
$208$	0	0
$209$	−9.03852e9	−0.327672
$210$	0	0
$211$	9.48425e9i	0.329406i	0.986343	+	0.164703i	$0.0526666\pi$
$211$	9.48425e9i	0.329406i	−0.986343	+	0.164703i	$0.947333\pi$
$212$	0	0
$213$	− 1.17779e10i	− 0.392066i
$214$	0	0
$215$	3.00962e10	0.960592
$216$	0	0
$217$	−6.63824e9	−0.203228
$218$	0	0
$219$	− 3.15612e10i	− 0.927160i
$220$	0	0
$221$	3.62841e10i	1.02318i
$222$	0	0
$223$	−4.82364e10	−1.30618	−0.653089	−	0.757281i	$-0.726527\pi$
$223$	−4.82364e10	−1.30618	−0.653089	+	0.757281i	$0.726527\pi$
$224$	0	0
$225$	2.36693e10	0.615694
$226$	0	0
$227$	3.93639e10i	0.983970i	0.870604	+	0.491985i	$0.163729\pi$
$227$	3.93639e10i	0.983970i	−0.870604	+	0.491985i	$0.836271\pi$
$228$	0	0
$229$	6.21477e10i	1.49336i	0.665182	+	0.746681i	$0.268354\pi$
$229$	6.21477e10i	1.49336i	−0.665182	+	0.746681i	$0.731646\pi$
$230$	0	0
$231$	−1.34409e9	−0.0310581
$232$	0	0
$233$	7.09062e10	1.57610	0.788048	−	0.615614i	$-0.211092\pi$
$233$	7.09062e10	1.57610	0.788048	+	0.615614i	$0.211092\pi$
$234$	0	0
$235$	− 2.56584e9i	− 0.0548813i
$236$	0	0
$237$	5.38916e10i	1.10957i
$238$	0	0
$239$	−7.01178e10	−1.39007	−0.695036	−	0.718975i	$-0.744612\pi$
$239$	−7.01178e10	−1.39007	−0.695036	+	0.718975i	$0.744612\pi$
$240$	0	0
$241$	7.33658e10	1.40093	0.700466	−	0.713686i	$-0.252975\pi$
$241$	7.33658e10	1.40093	0.700466	+	0.713686i	$0.252975\pi$
$242$	0	0
$243$	3.48678e9i	0.0641500i
$244$	0	0
$245$	9.11066e10i	1.61548i
$246$	0	0
$247$	1.06681e11	1.82369
$248$	0	0
$249$	2.46898e10	0.407024
$250$	0	0
$251$	5.19577e10i	0.826263i	0.910671	+	0.413132i	$0.135565\pi$
$251$	5.19577e10i	0.826263i	−0.910671	+	0.413132i	$0.864435\pi$
$252$	0	0
$253$	8.02639e9i	0.123162i
$254$	0	0
$255$	−4.63849e10	−0.686982
$256$	0	0
$257$	1.06918e11	1.52881	0.764405	−	0.644736i	$-0.223033\pi$
$257$	1.06918e11	1.52881	0.764405	+	0.644736i	$0.223033\pi$
$258$	0	0
$259$	− 1.49448e10i	− 0.206368i
$260$	0	0
$261$	− 6.03597e9i	− 0.0805128i
$262$	0	0
$263$	7.44592e10	0.959660	0.479830	−	0.877362i	$-0.340698\pi$
$263$	7.44592e10	0.959660	0.479830	+	0.877362i	$0.340698\pi$
$264$	0	0
$265$	2.16160e10	0.269259
$266$	0	0
$267$	2.74102e10i	0.330074i
$268$	0	0
$269$	4.17901e10i	0.486618i	0.969949	+	0.243309i	$0.0782329\pi$
$269$	4.17901e10i	0.486618i	−0.969949	+	0.243309i	$0.921767\pi$
$270$	0	0
$271$	4.20098e10	0.473139	0.236570	−	0.971615i	$-0.423977\pi$
$271$	4.20098e10	0.473139	0.236570	+	0.971615i	$0.423977\pi$
$272$	0	0
$273$	1.58643e10	0.172857
$274$	0	0
$275$	− 4.56684e10i	− 0.481525i
$276$	0	0
$277$	1.22601e11i	1.25122i	0.780136	+	0.625610i	$0.215150\pi$
$277$	1.22601e11i	1.25122i	−0.780136	+	0.625610i	$0.784850\pi$
$278$	0	0
$279$	3.32260e10	0.328291
$280$	0	0
$281$	−1.14643e11	−1.09690	−0.548452	−	0.836182i	$-0.684783\pi$
$281$	−1.14643e11	−1.09690	−0.548452	+	0.836182i	$0.684783\pi$
$282$	0	0
$283$	− 1.15293e11i	− 1.06848i	−0.845334	−	0.534239i	$-0.820598\pi$
$283$	− 1.15293e11i	− 1.06848i	0.845334	−	0.534239i	$-0.179402\pi$
$284$	0	0
$285$	1.36379e11i	1.22446i
$286$	0	0
$287$	−1.10669e10	−0.0962849
$288$	0	0
$289$	−5.96149e10	−0.502707
$290$	0	0
$291$	1.03847e11i	0.848939i
$292$	0	0
$293$	1.62732e11i	1.28994i	0.764207	+	0.644971i	$0.223130\pi$
$293$	1.62732e11i	1.28994i	−0.764207	+	0.644971i	$0.776870\pi$
$294$	0	0
$295$	−5.78128e9	−0.0444452
$296$	0	0
$297$	6.72752e9	0.0501708
$298$	0	0
$299$	− 9.47351e10i	− 0.685473i
$300$	0	0
$301$	1.67298e10i	0.117474i
$302$	0	0
$303$	9.80812e10	0.668488
$304$	0	0
$305$	−1.05380e11	−0.697280
$306$	0	0
$307$	4.86945e10i	0.312865i	0.987689	+	0.156433i	$0.0499994\pi$
$307$	4.86945e10i	0.312865i	−0.987689	+	0.156433i	$0.950001\pi$
$308$	0	0
$309$	− 1.38862e11i	− 0.866501i
$310$	0	0
$311$	−1.73861e11	−1.05386	−0.526928	−	0.849910i	$-0.676656\pi$
$311$	−1.73861e11	−1.05386	−0.526928	+	0.849910i	$0.676656\pi$
$312$	0	0
$313$	−3.89931e10	−0.229635	−0.114818	−	0.993387i	$-0.536628\pi$
$313$	−3.89931e10	−0.229635	−0.114818	+	0.993387i	$0.536628\pi$
$314$	0	0
$315$	2.02805e10i	0.116060i
$316$	0	0
$317$	2.07976e11i	1.15677i	0.815764	+	0.578385i	$0.196317\pi$
$317$	2.07976e11i	1.15677i	−0.815764	+	0.578385i	$0.803683\pi$
$318$	0	0
$319$	−1.16460e10	−0.0629679
$320$	0	0
$321$	−1.85100e11	−0.973045
$322$	0	0
$323$	− 1.73390e11i	− 0.886365i
$324$	0	0
$325$	5.39022e11i	2.67998i
$326$	0	0
$327$	9.44892e9	0.0457001
$328$	0	0
$329$	1.42629e9	0.00671163
$330$	0	0
$331$	1.26118e11i	0.577498i	0.957405	+	0.288749i	$0.0932393\pi$
$331$	1.26118e11i	0.577498i	−0.957405	+	0.288749i	$0.906761\pi$
$332$	0	0
$333$	7.48026e10i	0.333363i
$334$	0	0
$335$	6.03325e11	2.61728
$336$	0	0
$337$	−1.49166e11	−0.629991	−0.314996	−	0.949093i	$-0.602003\pi$
$337$	−1.49166e11	−0.629991	−0.314996	+	0.949093i	$0.602003\pi$
$338$	0	0
$339$	1.88825e11i	0.776535i
$340$	0	0
$341$	− 6.41074e10i	− 0.256752i
$342$	0	0
$343$	−1.03541e11	−0.403912
$344$	0	0
$345$	1.21107e11	0.460240
$346$	0	0
$347$	− 3.56632e11i	− 1.32050i	−0.751046	−	0.660250i	$-0.770450\pi$
$347$	− 3.56632e11i	− 1.32050i	0.751046	−	0.660250i	$-0.229550\pi$
$348$	0	0
$349$	5.37695e10i	0.194009i	0.995284	+	0.0970044i	$0.0309261\pi$
$349$	5.37695e10i	0.194009i	−0.995284	+	0.0970044i	$0.969074\pi$
$350$	0	0
$351$	−7.94046e10	−0.279231
$352$	0	0
$353$	−1.75541e11	−0.601718	−0.300859	−	0.953669i	$-0.597273\pi$
$353$	−1.75541e11	−0.601718	−0.300859	+	0.953669i	$0.597273\pi$
$354$	0	0
$355$	3.42884e11i	1.14583i
$356$	0	0
$357$	− 2.57843e10i	− 0.0840134i
$358$	0	0
$359$	−2.70333e11	−0.858963	−0.429482	−	0.903076i	$-0.641304\pi$
$359$	−2.70333e11	−0.858963	−0.429482	+	0.903076i	$0.641304\pi$
$360$	0	0
$361$	−1.87106e11	−0.579837
$362$	0	0
$363$	1.78013e11i	0.538112i
$364$	0	0
$365$	9.18826e11i	2.70966i
$366$	0	0
$367$	6.72371e11	1.93469	0.967345	−	0.253463i	$-0.0815695\pi$
$367$	6.72371e11	1.93469	0.967345	+	0.253463i	$0.0815695\pi$
$368$	0	0
$369$	5.53927e10	0.155537
$370$	0	0
$371$	1.20159e10i	0.0329286i
$372$	0	0
$373$	5.33707e11i	1.42762i	0.700339	+	0.713811i	$0.253032\pi$
$373$	5.33707e11i	1.42762i	−0.700339	+	0.713811i	$0.746968\pi$
$374$	0	0
$375$	−3.16013e11	−0.825210
$376$	0	0
$377$	1.37457e11	0.350454
$378$	0	0
$379$	2.28214e11i	0.568155i	0.958801	+	0.284077i	$0.0916872\pi$
$379$	2.28214e11i	0.568155i	−0.958801	+	0.284077i	$0.908313\pi$
$380$	0	0
$381$	2.77845e11i	0.675524i
$382$	0	0
$383$	−7.21713e11	−1.71384	−0.856919	−	0.515451i	$-0.827624\pi$
$383$	−7.21713e11	−1.71384	−0.856919	+	0.515451i	$0.827624\pi$
$384$	0	0
$385$	3.91299e10	0.0907687
$386$	0	0
$387$	− 8.37370e10i	− 0.189766i
$388$	0	0
$389$	− 3.78958e11i	− 0.839107i	−0.907731	−	0.419554i	$-0.862187\pi$
$389$	− 3.78958e11i	− 0.839107i	0.907731	−	0.419554i	$-0.137813\pi$
$390$	0	0
$391$	−1.53974e11	−0.333159
$392$	0	0
$393$	−4.07006e11	−0.860665
$394$	0	0
$395$	− 1.56892e12i	− 3.24275i
$396$	0	0
$397$	− 7.33971e11i	− 1.48293i	−0.670989	−	0.741467i	$-0.734130\pi$
$397$	− 7.33971e11i	− 1.48293i	0.670989	−	0.741467i	$-0.265870\pi$
$398$	0	0
$399$	−7.58100e10	−0.149744
$400$	0	0
$401$	−2.66124e11	−0.513967	−0.256983	−	0.966416i	$-0.582729\pi$
$401$	−2.66124e11	−0.513967	−0.256983	+	0.966416i	$0.582729\pi$
$402$	0	0
$403$	7.56657e11i	1.42898i
$404$	0	0
$405$	− 1.01509e11i	− 0.187481i
$406$	0	0
$407$	1.44327e11	0.260718
$408$	0	0
$409$	6.86135e11	1.21242	0.606212	−	0.795303i	$-0.292688\pi$
$409$	6.86135e11	1.21242	0.606212	+	0.795303i	$0.292688\pi$
$410$	0	0
$411$	− 1.10129e11i	− 0.190376i
$412$	0	0
$413$	− 3.21369e9i	− 0.00543536i
$414$	0	0
$415$	−7.18781e11	−1.18954
$416$	0	0
$417$	5.90898e11	0.956975
$418$	0	0
$419$	3.83589e11i	0.608000i	0.952672	+	0.304000i	$0.0983223\pi$
$419$	3.83589e11i	0.608000i	−0.952672	+	0.304000i	$0.901678\pi$
$420$	0	0
$421$	6.77263e11i	1.05072i	0.850879	+	0.525361i	$0.176070\pi$
$421$	6.77263e11i	1.05072i	−0.850879	+	0.525361i	$0.823930\pi$
$422$	0	0
$423$	−7.13896e9	−0.0108418
$424$	0	0
$425$	8.76077e11	1.30254
$426$	0	0
$427$	− 5.85782e10i	− 0.0852727i
$428$	0	0
$429$	1.53206e11i	0.218382i
$430$	0	0
$431$	3.46217e11	0.483282	0.241641	−	0.970366i	$-0.422314\pi$
$431$	3.46217e11	0.483282	0.241641	+	0.970366i	$0.422314\pi$
$432$	0	0
$433$	7.60273e11	1.03938	0.519689	−	0.854355i	$-0.326048\pi$
$433$	7.60273e11	1.03938	0.519689	+	0.854355i	$0.326048\pi$
$434$	0	0
$435$	1.75722e11i	0.235302i
$436$	0	0
$437$	4.52708e11i	0.593815i
$438$	0	0
$439$	1.35206e11	0.173742	0.0868710	−	0.996220i	$-0.472313\pi$
$439$	1.35206e11	0.173742	0.0868710	+	0.996220i	$0.472313\pi$
$440$	0	0
$441$	2.53487e11	0.319140
$442$	0	0
$443$	9.04003e11i	1.11520i	0.830110	+	0.557600i	$0.188278\pi$
$443$	9.04003e11i	1.11520i	−0.830110	+	0.557600i	$0.811722\pi$
$444$	0	0
$445$	− 7.97980e11i	− 0.964655i
$446$	0	0
$447$	−5.13789e10	−0.0608696
$448$	0	0
$449$	5.50987e11	0.639784	0.319892	−	0.947454i	$-0.396353\pi$
$449$	5.50987e11	0.639784	0.319892	+	0.947454i	$0.396353\pi$
$450$	0	0
$451$	− 1.06877e11i	− 0.121643i
$452$	0	0
$453$	2.27314e11i	0.253620i
$454$	0	0
$455$	−4.61849e11	−0.505183
$456$	0	0
$457$	1.15885e12	1.24281	0.621405	−	0.783489i	$-0.286562\pi$
$457$	1.15885e12	1.24281	0.621405	+	0.783489i	$0.286562\pi$
$458$	0	0
$459$	1.29057e11i	0.135714i
$460$	0	0
$461$	6.46065e11i	0.666227i	0.942887	+	0.333113i	$0.108099\pi$
$461$	6.46065e11i	0.666227i	−0.942887	+	0.333113i	$0.891901\pi$
$462$	0	0
$463$	8.66388e11	0.876189	0.438094	−	0.898929i	$-0.355654\pi$
$463$	8.66388e11	0.876189	0.438094	+	0.898929i	$0.355654\pi$
$464$	0	0
$465$	−9.67294e11	−0.959445
$466$	0	0
$467$	2.75888e11i	0.268415i	0.990953	+	0.134207i	$0.0428488\pi$
$467$	2.75888e11i	0.268415i	−0.990953	+	0.134207i	$0.957151\pi$
$468$	0	0
$469$	3.35375e11i	0.320076i
$470$	0	0
$471$	3.80702e11	0.356444
$472$	0	0
$473$	−1.61565e11	−0.148413
$474$	0	0
$475$	− 2.57581e12i	− 2.32163i
$476$	0	0
$477$	− 6.01425e10i	− 0.0531923i
$478$	0	0
$479$	−2.20650e12	−1.91511	−0.957555	−	0.288249i	$-0.906927\pi$
$479$	−2.20650e12	−1.91511	−0.957555	+	0.288249i	$0.906927\pi$
$480$	0	0
$481$	−1.70348e12	−1.45106
$482$	0	0
$483$	6.73209e10i	0.0562844i
$484$	0	0
$485$	− 3.02326e12i	− 2.48106i
$486$	0	0
$487$	3.21730e11	0.259186	0.129593	−	0.991567i	$-0.458633\pi$
$487$	3.21730e11	0.259186	0.129593	+	0.991567i	$0.458633\pi$
$488$	0	0
$489$	7.29018e11	0.576566
$490$	0	0
$491$	3.80381e11i	0.295360i	0.989035	+	0.147680i	$0.0471806\pi$
$491$	3.80381e11i	0.295360i	−0.989035	+	0.147680i	$0.952819\pi$
$492$	0	0
$493$	− 2.23410e11i	− 0.170330i
$494$	0	0
$495$	−1.95855e11	−0.146626
$496$	0	0
$497$	−1.90602e11	−0.140127
$498$	0	0
$499$	− 3.33496e11i	− 0.240790i	−0.992726	−	0.120395i	$-0.961584\pi$
$499$	− 3.33496e11i	− 0.240790i	0.992726	−	0.120395i	$-0.0384161\pi$
$500$	0	0
$501$	− 1.23085e12i	− 0.872843i
$502$	0	0
$503$	1.16862e12	0.813990	0.406995	−	0.913430i	$-0.366577\pi$
$503$	1.16862e12	0.813990	0.406995	+	0.913430i	$0.366577\pi$
$504$	0	0
$505$	−2.85539e12	−1.95368
$506$	0	0
$507$	− 9.49317e11i	− 0.638080i
$508$	0	0
$509$	− 1.20618e12i	− 0.796494i	−0.917278	−	0.398247i	$-0.869619\pi$
$509$	− 1.20618e12i	− 0.796494i	0.917278	−	0.398247i	$-0.130381\pi$
$510$	0	0
$511$	−5.10755e11	−0.331374
$512$	0	0
$513$	3.79448e11	0.241894
$514$	0	0
$515$	4.04261e12i	2.53238i
$516$	0	0
$517$	1.37742e10i	0.00847925i
$518$	0	0
$519$	−1.20025e12	−0.726137
$520$	0	0
$521$	−3.06301e12	−1.82129	−0.910645	−	0.413189i	$-0.864415\pi$
$521$	−3.06301e12	−1.82129	−0.910645	+	0.413189i	$0.864415\pi$
$522$	0	0
$523$	− 2.23351e12i	− 1.30536i	−0.757633	−	0.652681i	$-0.773644\pi$
$523$	− 2.23351e12i	− 1.30536i	0.757633	−	0.652681i	$-0.226356\pi$
$524$	0	0
$525$	− 3.83041e11i	− 0.220054i
$526$	0	0
$527$	1.22980e12	0.694524
$528$	0	0
$529$	−1.39914e12	−0.776802
$530$	0	0
$531$	1.60853e10i	0.00878019i
$532$	0	0
$533$	1.26146e12i	0.677018i
$534$	0	0
$535$	5.38872e12	2.84376
$536$	0	0
$537$	−1.99725e12	−1.03645
$538$	0	0
$539$	− 4.89085e11i	− 0.249595i
$540$	0	0
$541$	− 5.90938e11i	− 0.296588i	−0.988943	−	0.148294i	$-0.952622\pi$
$541$	− 5.90938e11i	− 0.296588i	0.988943	−	0.148294i	$-0.0473782\pi$
$542$	0	0
$543$	−1.81857e12	−0.897699
$544$	0	0
$545$	−2.75082e11	−0.133560
$546$	0	0
$547$	3.09173e12i	1.47659i	0.674479	+	0.738294i	$0.264368\pi$
$547$	3.09173e12i	1.47659i	−0.674479	+	0.738294i	$0.735632\pi$
$548$	0	0
$549$	2.93198e11i	0.137748i
$550$	0	0
$551$	−6.56862e11	−0.303593
$552$	0	0
$553$	8.72128e11	0.396568
$554$	0	0
$555$	− 2.17769e12i	− 0.974267i
$556$	0	0
$557$	− 2.94524e12i	− 1.29650i	−0.761428	−	0.648249i	$-0.775501\pi$
$557$	− 2.94524e12i	− 1.29650i	0.761428	−	0.648249i	$-0.224499\pi$
$558$	0	0
$559$	1.90694e12	0.826009
$560$	0	0
$561$	2.49007e11	0.106140
$562$	0	0
$563$	2.62039e12i	1.09920i	0.835427	+	0.549602i	$0.185221\pi$
$563$	2.62039e12i	1.09920i	−0.835427	+	0.549602i	$0.814779\pi$
$564$	0	0
$565$	− 5.49718e12i	− 2.26946i
$566$	0	0
$567$	5.64267e10	0.0229277
$568$	0	0
$569$	9.57955e11	0.383124	0.191562	−	0.981480i	$-0.438645\pi$
$569$	9.57955e11	0.383124	0.191562	+	0.981480i	$0.438645\pi$
$570$	0	0
$571$	− 1.78103e12i	− 0.701148i	−0.936535	−	0.350574i	$-0.885986\pi$
$571$	− 1.78103e12i	− 0.701148i	0.936535	−	0.350574i	$-0.114014\pi$
$572$	0	0
$573$	4.90820e10i	0.0190207i
$574$	0	0
$575$	−2.28737e12	−0.872632
$576$	0	0
$577$	4.57175e12	1.71708	0.858542	−	0.512743i	$-0.171371\pi$
$577$	4.57175e12	1.71708	0.858542	+	0.512743i	$0.171371\pi$
$578$	0	0
$579$	− 1.00793e12i	− 0.372714i
$580$	0	0
$581$	− 3.99555e11i	− 0.145473i
$582$	0	0
$583$	−1.16041e11	−0.0416009
$584$	0	0
$585$	2.31167e12	0.816064
$586$	0	0
$587$	2.32819e12i	0.809368i	0.914456	+	0.404684i	$0.132619\pi$
$587$	2.32819e12i	0.809368i	−0.914456	+	0.404684i	$0.867381\pi$
$588$	0	0
$589$	− 3.61581e12i	− 1.23790i
$590$	0	0
$591$	5.52149e11	0.186171
$592$	0	0
$593$	7.75980e11	0.257694	0.128847	−	0.991664i	$-0.458872\pi$
$593$	7.75980e11	0.257694	0.128847	+	0.991664i	$0.458872\pi$
$594$	0	0
$595$	7.50647e11i	0.245533i
$596$	0	0
$597$	− 1.55245e12i	− 0.500188i
$598$	0	0
$599$	3.58904e12	1.13909	0.569544	−	0.821961i	$-0.307120\pi$
$599$	3.58904e12	1.13909	0.569544	+	0.821961i	$0.307120\pi$
$600$	0	0
$601$	−2.43528e12	−0.761400	−0.380700	−	0.924699i	$-0.624317\pi$
$601$	−2.43528e12	−0.761400	−0.380700	+	0.924699i	$0.624317\pi$
$602$	0	0
$603$	− 1.67864e12i	− 0.517045i
$604$	0	0
$605$	− 5.18242e12i	− 1.57266i
$606$	0	0
$607$	2.16109e11	0.0646136	0.0323068	−	0.999478i	$-0.489715\pi$
$607$	2.16109e11	0.0646136	0.0323068	+	0.999478i	$0.489715\pi$
$608$	0	0
$609$	−9.76802e10	−0.0287759
$610$	0	0
$611$	− 1.62576e11i	− 0.0471922i
$612$	0	0
$613$	− 5.34985e12i	− 1.53028i	−0.643866	−	0.765138i	$-0.722671\pi$
$613$	− 5.34985e12i	− 1.53028i	0.643866	−	0.765138i	$-0.277329\pi$
$614$	0	0
$615$	−1.61262e12	−0.454563
$616$	0	0
$617$	4.02003e12	1.11673	0.558363	−	0.829597i	$-0.311430\pi$
$617$	4.02003e12	1.11673	0.558363	+	0.829597i	$0.311430\pi$
$618$	0	0
$619$	− 2.33115e12i	− 0.638209i	−0.947720	−	0.319104i	$-0.896618\pi$
$619$	− 2.33115e12i	− 0.638209i	0.947720	−	0.319104i	$-0.103382\pi$
$620$	0	0
$621$	− 3.36958e11i	− 0.0909208i
$622$	0	0
$623$	4.43579e11	0.117971
$624$	0	0
$625$	2.15389e12	0.564629
$626$	0	0
$627$	− 7.32120e11i	− 0.189181i
$628$	0	0
$629$	2.76868e12i	0.705253i
$630$	0	0
$631$	−1.60959e12	−0.404188	−0.202094	−	0.979366i	$-0.564775\pi$
$631$	−1.60959e12	−0.404188	−0.202094	+	0.979366i	$0.564775\pi$
$632$	0	0
$633$	−7.68225e11	−0.190183
$634$	0	0
$635$	− 8.08878e12i	− 1.97425i
$636$	0	0
$637$	5.77265e12i	1.38915i
$638$	0	0
$639$	9.54010e11	0.226360
$640$	0	0
$641$	9.07113e10	0.0212227	0.0106113	−	0.999944i	$-0.496622\pi$
$641$	9.07113e10	0.0212227	0.0106113	+	0.999944i	$0.496622\pi$
$642$	0	0
$643$	3.99830e12i	0.922415i	0.887292	+	0.461207i	$0.152584\pi$
$643$	3.99830e12i	0.922415i	−0.887292	+	0.461207i	$0.847416\pi$
$644$	0	0
$645$	2.43780e12i	0.554598i
$646$	0	0
$647$	−7.74626e12	−1.73789	−0.868946	−	0.494907i	$-0.835202\pi$
$647$	−7.74626e12	−1.73789	−0.868946	+	0.494907i	$0.835202\pi$
$648$	0	0
$649$	3.10355e10	0.00686685
$650$	0	0
$651$	− 5.37697e11i	− 0.117334i
$652$	0	0
$653$	− 3.69315e12i	− 0.794854i	−0.917634	−	0.397427i	$-0.869903\pi$
$653$	− 3.69315e12i	− 0.794854i	0.917634	−	0.397427i	$-0.130097\pi$
$654$	0	0
$655$	1.18490e13	2.51533
$656$	0	0
$657$	2.55646e12	0.535296
$658$	0	0
$659$	9.56399e12i	1.97540i	0.156363	+	0.987700i	$0.450023\pi$
$659$	9.56399e12i	1.97540i	−0.156363	+	0.987700i	$0.549977\pi$
$660$	0	0
$661$	− 5.90211e12i	− 1.20254i	−0.799044	−	0.601272i	$-0.794661\pi$
$661$	− 5.90211e12i	− 1.20254i	0.799044	−	0.601272i	$-0.205339\pi$
$662$	0	0
$663$	−2.93902e12	−0.590733
$664$	0	0
$665$	2.20702e12	0.437632
$666$	0	0
$667$	5.83307e11i	0.114112i
$668$	0	0
$669$	− 3.90714e12i	− 0.754123i
$670$	0	0
$671$	5.65707e11	0.107731
$672$	0	0
$673$	2.04043e12	0.383402	0.191701	−	0.981453i	$-0.438600\pi$
$673$	2.04043e12	0.383402	0.191701	+	0.981453i	$0.438600\pi$
$674$	0	0
$675$	1.91722e12i	0.355471i
$676$	0	0
$677$	− 5.46220e12i	− 0.999352i	−0.866212	−	0.499676i	$-0.833452\pi$
$677$	− 5.46220e12i	− 0.999352i	0.866212	−	0.499676i	$-0.166548\pi$
$678$	0	0
$679$	1.68056e12	0.303417
$680$	0	0
$681$	−3.18848e12	−0.568096
$682$	0	0
$683$	− 1.00775e13i	− 1.77198i	−0.463703	−	0.885991i	$-0.653480\pi$
$683$	− 1.00775e13i	− 1.77198i	0.463703	−	0.885991i	$-0.346520\pi$
$684$	0	0
$685$	3.20612e12i	0.556381i
$686$	0	0
$687$	−5.03396e12	−0.862193
$688$	0	0
$689$	1.36962e12	0.231534
$690$	0	0
$691$	4.61451e12i	0.769970i	0.922923	+	0.384985i	$0.125793\pi$
$691$	4.61451e12i	0.769970i	−0.922923	+	0.384985i	$0.874207\pi$
$692$	0	0
$693$	− 1.08872e11i	− 0.0179314i
$694$	0	0
$695$	−1.72025e13	−2.79680
$696$	0	0
$697$	2.05026e12	0.329050
$698$	0	0
$699$	5.74340e12i	0.909959i
$700$	0	0
$701$	− 1.75605e12i	− 0.274666i	−0.990525	−	0.137333i	$-0.956147\pi$
$701$	− 1.75605e12i	− 0.274666i	0.990525	−	0.137333i	$-0.0438531\pi$
$702$	0	0
$703$	8.14037e12	1.25703
$704$	0	0
$705$	2.07833e11	0.0316857
$706$	0	0
$707$	− 1.58725e12i	− 0.238923i
$708$	0	0
$709$	− 1.66308e12i	− 0.247175i	−0.992334	−	0.123587i	$-0.960560\pi$
$709$	− 1.66308e12i	− 0.247175i	0.992334	−	0.123587i	$-0.0394399\pi$
$710$	0	0
$711$	−4.36522e12	−0.640609
$712$	0	0
$713$	−3.21091e12	−0.465292
$714$	0	0
$715$	− 4.46021e12i	− 0.638231i
$716$	0	0
$717$	− 5.67954e12i	− 0.802559i
$718$	0	0
$719$	−2.86871e12	−0.400319	−0.200160	−	0.979763i	$-0.564146\pi$
$719$	−2.86871e12	−0.400319	−0.200160	+	0.979763i	$0.564146\pi$
$720$	0	0
$721$	−2.24720e12	−0.309694
$722$	0	0
$723$	5.94263e12i	0.808829i
$724$	0	0
$725$	− 3.31889e12i	− 0.446141i
$726$	0	0
$727$	3.23515e12	0.429526	0.214763	−	0.976666i	$-0.431102\pi$
$727$	3.23515e12	0.429526	0.214763	+	0.976666i	$0.431102\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	− 3.09937e12i	− 0.401463i
$732$	0	0
$733$	3.30921e12i	0.423405i	0.977334	+	0.211703i	$0.0679008\pi$
$733$	3.30921e12i	0.423405i	−0.977334	+	0.211703i	$0.932099\pi$
$734$	0	0
$735$	−7.37963e12	−0.932700
$736$	0	0
$737$	−3.23882e12	−0.404374
$738$	0	0
$739$	− 3.46011e12i	− 0.426766i	−0.976969	−	0.213383i	$-0.931552\pi$
$739$	− 3.46011e12i	− 0.426766i	0.976969	−	0.213383i	$-0.0684482\pi$
$740$	0	0
$741$	8.64118e12i	1.05291i
$742$	0	0
$743$	4.38595e12	0.527976	0.263988	−	0.964526i	$-0.414962\pi$
$743$	4.38595e12	0.527976	0.263988	+	0.964526i	$0.414962\pi$
$744$	0	0
$745$	1.49577e12	0.177894
$746$	0	0
$747$	1.99987e12i	0.234995i
$748$	0	0
$749$	2.99547e12i	0.347774i
$750$	0	0
$751$	−1.34891e12	−0.154741	−0.0773703	−	0.997002i	$-0.524652\pi$
$751$	−1.34891e12	−0.154741	−0.0773703	+	0.997002i	$0.524652\pi$
$752$	0	0
$753$	−4.20858e12	−0.477043
$754$	0	0
$755$	− 6.61768e12i	− 0.741216i
$756$	0	0
$757$	− 8.21137e10i	− 0.00908833i	−0.999990	−	0.00454417i	$-0.998554\pi$
$757$	− 8.21137e10i	− 0.00908833i	0.999990	−	0.00454417i	$-0.00144646\pi$
$758$	0	0
$759$	−6.50138e11	−0.0711078
$760$	0	0
$761$	−1.70402e13	−1.84180	−0.920901	−	0.389796i	$-0.872545\pi$
$761$	−1.70402e13	−1.84180	−0.920901	+	0.389796i	$0.872545\pi$
$762$	0	0
$763$	− 1.52912e11i	− 0.0163336i
$764$	0	0
$765$	− 3.75717e12i	− 0.396629i
$766$	0	0
$767$	−3.66311e11	−0.0382182
$768$	0	0
$769$	−1.31929e13	−1.36041	−0.680206	−	0.733021i	$-0.738109\pi$
$769$	−1.31929e13	−1.36041	−0.680206	+	0.733021i	$0.738109\pi$
$770$	0	0
$771$	8.66040e12i	0.882659i
$772$	0	0
$773$	9.36151e12i	0.943057i	0.881851	+	0.471529i	$0.156298\pi$
$773$	9.36151e12i	0.943057i	−0.881851	+	0.471529i	$0.843702\pi$
$774$	0	0
$775$	1.82694e13	1.81914
$776$	0	0
$777$	1.21053e12	0.119147
$778$	0	0
$779$	− 6.02809e12i	− 0.586491i
$780$	0	0
$781$	− 1.84070e12i	− 0.177033i
$782$	0	0
$783$	4.88914e11	0.0464841
$784$	0	0
$785$	−1.10832e13	−1.04172
$786$	0	0
$787$	4.69737e12i	0.436484i	0.975895	+	0.218242i	$0.0700323\pi$
$787$	4.69737e12i	0.436484i	−0.975895	+	0.218242i	$0.929968\pi$
$788$	0	0
$789$	6.03119e12i	0.554060i
$790$	0	0
$791$	3.05576e12	0.277540
$792$	0	0
$793$	−6.67701e12	−0.599587
$794$	0	0
$795$	1.75090e12i	0.155457i
$796$	0	0
$797$	− 1.57832e12i	− 0.138558i	−0.997597	−	0.0692791i	$-0.977930\pi$
$797$	− 1.57832e12i	− 0.138558i	0.997597	−	0.0692791i	$-0.0220699\pi$
$798$	0	0
$799$	−2.64236e11	−0.0229367
$800$	0	0
$801$	−2.22022e12	−0.190568
$802$	0	0
$803$	− 4.93251e12i	− 0.418647i
$804$	0	0
$805$	− 1.95988e12i	− 0.164493i
$806$	0	0
$807$	−3.38500e12	−0.280949
$808$	0	0
$809$	1.60915e13	1.32077	0.660387	−	0.750925i	$-0.270392\pi$
$809$	1.60915e13	1.32077	0.660387	+	0.750925i	$0.270392\pi$
$810$	0	0
$811$	− 2.29953e12i	− 0.186657i	−0.995635	−	0.0933287i	$-0.970249\pi$
$811$	− 2.29953e12i	− 0.186657i	0.995635	−	0.0933287i	$-0.0297507\pi$
$812$	0	0
$813$	3.40280e12i	0.273167i
$814$	0	0
$815$	−2.12236e13	−1.68504
$816$	0	0
$817$	−9.11265e12	−0.715559
$818$	0	0
$819$	1.28501e12i	0.0997993i
$820$	0	0
$821$	1.01265e13i	0.777887i	0.921262	+	0.388944i	$0.127160\pi$
$821$	1.01265e13i	0.777887i	−0.921262	+	0.388944i	$0.872840\pi$
$822$	0	0
$823$	1.20614e13	0.916426	0.458213	−	0.888842i	$-0.348490\pi$
$823$	1.20614e13	0.916426	0.458213	+	0.888842i	$0.348490\pi$
$824$	0	0
$825$	3.69914e12	0.278009
$826$	0	0
$827$	− 1.44216e13i	− 1.07211i	−0.844183	−	0.536054i	$-0.819914\pi$
$827$	− 1.44216e13i	− 1.07211i	0.844183	−	0.536054i	$-0.180086\pi$
$828$	0	0
$829$	2.01515e13i	1.48187i	0.671574	+	0.740937i	$0.265618\pi$
$829$	2.01515e13i	1.48187i	−0.671574	+	0.740937i	$0.734382\pi$
$830$	0	0
$831$	−9.93065e12	−0.722392
$832$	0	0
$833$	9.38234e12	0.675163
$834$	0	0
$835$	3.58332e13i	2.55092i
$836$	0	0
$837$	2.69131e12i	0.189539i
$838$	0	0
$839$	1.41876e13	0.988507	0.494253	−	0.869318i	$-0.335442\pi$
$839$	1.41876e13	0.988507	0.494253	+	0.869318i	$0.335442\pi$
$840$	0	0
$841$	1.36608e13	0.941659
$842$	0	0
$843$	− 9.28607e12i	− 0.633298i
$844$	0	0
$845$	2.76370e13i	1.86482i
$846$	0	0
$847$	2.88079e12	0.192325
$848$	0	0
$849$	9.33877e12	0.616886
$850$	0	0
$851$	− 7.22881e12i	− 0.472481i
$852$	0	0
$853$	1.03561e13i	0.669768i	0.942259	+	0.334884i	$0.108697\pi$
$853$	1.03561e13i	0.669768i	−0.942259	+	0.334884i	$0.891303\pi$
$854$	0	0
$855$	−1.10467e13	−0.706944
$856$	0	0
$857$	1.15824e13	0.733476	0.366738	−	0.930324i	$-0.380475\pi$
$857$	1.15824e13	0.733476	0.366738	+	0.930324i	$0.380475\pi$
$858$	0	0
$859$	4.06490e12i	0.254730i	0.991856	+	0.127365i	$0.0406520\pi$
$859$	4.06490e12i	0.254730i	−0.991856	+	0.127365i	$0.959348\pi$
$860$	0	0
$861$	− 8.96421e11i	− 0.0555901i
$862$	0	0
$863$	−3.08032e12	−0.189037	−0.0945186	−	0.995523i	$-0.530131\pi$
$863$	−3.08032e12	−0.189037	−0.0945186	+	0.995523i	$0.530131\pi$
$864$	0	0
$865$	3.49423e13	2.12217
$866$	0	0
$867$	− 4.82881e12i	− 0.290238i
$868$	0	0
$869$	8.42240e12i	0.501011i
$870$	0	0
$871$	3.82276e13	2.25058
$872$	0	0
$873$	−8.41163e12	−0.490135
$874$	0	0
$875$	5.11405e12i	0.294936i
$876$	0	0
$877$	− 2.13243e11i	− 0.0121724i	−0.999981	−	0.00608620i	$-0.998063\pi$
$877$	− 2.13243e11i	− 0.0121724i	0.999981	−	0.00608620i	$-0.00193731\pi$
$878$	0	0
$879$	−1.31813e13	−0.744748
$880$	0	0
$881$	−7.30543e10	−0.00408558	−0.00204279	−	0.999998i	$-0.500650\pi$
$881$	−7.30543e10	−0.00408558	−0.00204279	+	0.999998i	$0.500650\pi$
$882$	0	0
$883$	− 1.02997e13i	− 0.570164i	−0.958503	−	0.285082i	$-0.907979\pi$
$883$	− 1.02997e13i	− 0.570164i	0.958503	−	0.285082i	$-0.0920209\pi$
$884$	0	0
$885$	− 4.68284e11i	− 0.0256605i
$886$	0	0
$887$	−2.79649e13	−1.51690	−0.758450	−	0.651732i	$-0.774043\pi$
$887$	−2.79649e13	−1.51690	−0.758450	+	0.651732i	$0.774043\pi$
$888$	0	0
$889$	4.49637e12	0.241437
$890$	0	0
$891$	5.44929e11i	0.0289661i
$892$	0	0
$893$	7.76895e11i	0.0408819i
$894$	0	0
$895$	5.81449e13	3.02906
$896$	0	0
$897$	7.67354e12	0.395758
$898$	0	0
$899$	− 4.65892e12i	− 0.237885i
$900$	0	0
$901$	− 2.22606e12i	− 0.112532i
$902$	0	0
$903$	−1.35512e12	−0.0678237
$904$	0	0
$905$	5.29432e13	2.62356
$906$	0	0
$907$	− 2.31646e12i	− 0.113656i	−0.998384	−	0.0568279i	$-0.981901\pi$
$907$	− 2.31646e12i	− 0.113656i	0.998384	−	0.0568279i	$-0.0180986\pi$
$908$	0	0
$909$	7.94457e12i	0.385952i
$910$	0	0
$911$	−1.61506e13	−0.776883	−0.388442	−	0.921473i	$-0.626986\pi$
$911$	−1.61506e13	−0.776883	−0.388442	+	0.921473i	$0.626986\pi$
$912$	0	0
$913$	3.85862e12	0.183786
$914$	0	0
$915$	− 8.53574e12i	− 0.402575i
$916$	0	0
$917$	6.58658e12i	0.307608i
$918$	0	0
$919$	3.40206e13	1.57334	0.786670	−	0.617374i	$-0.211803\pi$
$919$	3.40206e13	1.57334	0.786670	+	0.617374i	$0.211803\pi$
$920$	0	0
$921$	−3.94425e12	−0.180633
$922$	0	0
$923$	2.17257e13i	0.985293i
$924$	0	0
$925$	4.11304e13i	1.84725i
$926$	0	0
$927$	1.12478e13	0.500275
$928$	0	0
$929$	−2.99915e13	−1.32108	−0.660538	−	0.750793i	$-0.729672\pi$
$929$	−2.99915e13	−1.32108	−0.660538	+	0.750793i	$0.729672\pi$
$930$	0	0
$931$	− 2.75856e13i	− 1.20340i
$932$	0	0
$933$	− 1.40828e13i	− 0.608444i
$934$	0	0
$935$	−7.24922e12	−0.310198
$936$	0	0
$937$	−2.38419e13	−1.01044	−0.505222	−	0.862990i	$-0.668589\pi$
$937$	−2.38419e13	−1.01044	−0.505222	+	0.862990i	$0.668589\pi$
$938$	0	0
$939$	− 3.15844e12i	− 0.132580i
$940$	0	0
$941$	3.93491e12i	0.163599i	0.996649	+	0.0817996i	$0.0260667\pi$
$941$	3.93491e12i	0.163599i	−0.996649	+	0.0817996i	$0.973933\pi$
$942$	0	0
$943$	−5.35307e12	−0.220445
$944$	0	0
$945$	−1.64272e12	−0.0670072
$946$	0	0
$947$	2.29195e13i	0.926042i	0.886347	+	0.463021i	$0.153235\pi$
$947$	2.29195e13i	0.926042i	−0.886347	+	0.463021i	$0.846765\pi$
$948$	0	0
$949$	5.82182e13i	2.33003i
$950$	0	0
$951$	−1.68461e13	−0.667862
$952$	0	0
$953$	−2.69689e13	−1.05912	−0.529561	−	0.848272i	$-0.677643\pi$
$953$	−2.69689e13	−1.05912	−0.529561	+	0.848272i	$0.677643\pi$
$954$	0	0
$955$	− 1.42890e12i	− 0.0555888i
$956$	0	0
$957$	− 9.43326e11i	− 0.0363545i
$958$	0	0
$959$	−1.78221e12	−0.0680418
$960$	0	0
$961$	−7.93797e11	−0.0300230
$962$	0	0
$963$	− 1.49931e13i	− 0.561788i
$964$	0	0
$965$	2.93433e13i	1.08927i
$966$	0	0
$967$	−1.79377e13	−0.659703	−0.329852	−	0.944033i	$-0.606999\pi$
$967$	−1.79377e13	−0.659703	−0.329852	+	0.944033i	$0.606999\pi$
$968$	0	0
$969$	1.40446e13	0.511743
$970$	0	0
$971$	3.36215e13i	1.21375i	0.794796	+	0.606876i	$0.207578\pi$
$971$	3.36215e13i	1.21375i	−0.794796	+	0.606876i	$0.792422\pi$
$972$	0	0
$973$	− 9.56252e12i	− 0.342030i
$974$	0	0
$975$	−4.36608e13	−1.54729
$976$	0	0
$977$	−3.54989e13	−1.24649	−0.623245	−	0.782026i	$-0.714186\pi$
$977$	−3.54989e13	−1.24649	−0.623245	+	0.782026i	$0.714186\pi$
$978$	0	0
$979$	4.28378e12i	0.149041i
$980$	0	0
$981$	7.65362e11i	0.0263850i
$982$	0	0
$983$	7.84871e11	0.0268107	0.0134053	−	0.999910i	$-0.495733\pi$
$983$	7.84871e11	0.0268107	0.0134053	+	0.999910i	$0.495733\pi$
$984$	0	0
$985$	−1.60745e13	−0.544093
$986$	0	0
$987$	1.15530e11i	0.00387496i
$988$	0	0
$989$	8.09222e12i	0.268958i
$990$	0	0
$991$	4.31644e13	1.42165	0.710827	−	0.703367i	$-0.248321\pi$
$991$	4.31644e13	1.42165	0.710827	+	0.703367i	$0.248321\pi$
$992$	0	0
$993$	−1.02155e13	−0.333419
$994$	0	0
$995$	4.51958e13i	1.46182i
$996$	0	0
$997$	− 3.72331e12i	− 0.119344i	−0.998218	−	0.0596720i	$-0.980995\pi$
$997$	− 3.72331e12i	− 0.119344i	0.998218	−	0.0596720i	$-0.0190055\pi$
$998$	0	0
$999$	−6.05901e12	−0.192467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.f.193.11	yes	20
4.3	odd	2	inner	384.10.d.f.193.1	✓	20
8.3	odd	2	inner	384.10.d.f.193.20	yes	20
8.5	even	2	inner	384.10.d.f.193.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.f.193.1	✓	20	4.3	odd	2	inner
384.10.d.f.193.10	yes	20	8.5	even	2	inner
384.10.d.f.193.11	yes	20	1.1	even	1	trivial
384.10.d.f.193.20	yes	20	8.3	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+81.0000i q^{3} -2358.11i q^{5} +1310.82 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q+81.0000i q^{3} -2358.11i q^{5} +1310.82 q^{7} -6561.00 q^{9} +12659.0i q^{11} -149414. i q^{13} +191007. q^{15} -242843. q^{17} +713999. i q^{19} +106177. i q^{21} +634045. q^{23} -3.60758e6 q^{25} -531441. i q^{27} +919977. i q^{29} -5.06417e6 q^{31} -1.02538e6 q^{33} -3.09107e6i q^{35} -1.14011e7i q^{37} +1.21025e7 q^{39} -8.44272e6 q^{41} +1.27628e7i q^{43} +1.54716e7i q^{45} +1.08809e6 q^{47} -3.86353e7 q^{49} -1.96703e7i q^{51} +9.16666e6i q^{53} +2.98514e7 q^{55} -5.78339e7 q^{57} -2.45165e6i q^{59} -4.46880e7i q^{61} -8.60032e6 q^{63} -3.52335e8 q^{65} +2.55851e8i q^{67} +5.13577e7i q^{69} -1.45406e8 q^{71} -3.89644e8 q^{73} -2.92214e8i q^{75} +1.65937e7i q^{77} +6.65328e8 q^{79} +4.30467e7 q^{81} -3.04812e8i q^{83} +5.72653e8i q^{85} -7.45182e7 q^{87} +3.38397e8 q^{89} -1.95855e8i q^{91} -4.10198e8i q^{93} +1.68369e9 q^{95} +1.28207e9 q^{97} -8.30558e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 131220 q^{9}+O(q^{10}) \) 20 * q - 131220 * q^9	\( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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