LMFDB - Embedded newform 384.10.d.f.193.18

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			193.18
Root			$5.69422 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.f.193.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+81.0000i q^{3} +1655.59i q^{5} -11370.0 q^{7} -6561.00 q^{9} +O(q^{10})$	$q+81.0000i q^{3} +1655.59i q^{5} -11370.0 q^{7} -6561.00 q^{9} -94507.4i q^{11} -140234. i q^{13} -134103. q^{15} -531000. q^{17} -387485. i q^{19} -920969. i q^{21} -2.36100e6 q^{23} -787863. q^{25} -531441. i q^{27} +2.70715e6i q^{29} -5.18531e6 q^{31} +7.65510e6 q^{33} -1.88241e7i q^{35} -142841. i q^{37} +1.13589e7 q^{39} +1.92825e6 q^{41} -1.65243e7i q^{43} -1.08623e7i q^{45} -1.78851e7 q^{47} +8.89229e7 q^{49} -4.30110e7i q^{51} -2.01018e6i q^{53} +1.56466e8 q^{55} +3.13863e7 q^{57} -9.71099e7i q^{59} +7.23555e6i q^{61} +7.45985e7 q^{63} +2.32170e8 q^{65} +2.34701e8i q^{67} -1.91241e8i q^{69} -2.62515e7 q^{71} -3.59965e8 q^{73} -6.38169e7i q^{75} +1.07455e9i q^{77} -3.01072e8 q^{79} +4.30467e7 q^{81} +3.48363e8i q^{83} -8.79119e8i q^{85} -2.19279e8 q^{87} +9.32242e8 q^{89} +1.59446e9i q^{91} -4.20010e8i q^{93} +6.41517e8 q^{95} -9.09752e8 q^{97} +6.20063e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1655.59i	1.18465i	0.805701	+	0.592323i	$0.201789\pi$
$5$	1655.59i	1.18465i	−0.805701	+	0.592323i	$0.798211\pi$
$6$	0	0
$7$	−11370.0	−1.78986	−0.894929	−	0.446208i	$-0.852774\pi$
$7$	−11370.0	−1.78986	−0.894929	+	0.446208i	$0.852774\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	− 94507.4i	− 1.94625i	−0.230276	−	0.973125i	$-0.573963\pi$
$11$	− 94507.4i	− 1.94625i	0.230276	−	0.973125i	$-0.426037\pi$
$12$	0	0
$13$	− 140234.i	− 1.36178i	−0.732385	−	0.680891i	$-0.761593\pi$
$13$	− 140234.i	− 1.36178i	0.732385	−	0.680891i	$-0.238407\pi$
$14$	0	0
$15$	−134103.	−0.683956
$16$	0	0
$17$	−531000.	−1.54196	−0.770982	−	0.636857i	$-0.780234\pi$
$17$	−531000.	−1.54196	−0.770982	+	0.636857i	$0.780234\pi$
$18$	0	0
$19$	− 387485.i	− 0.682124i	−0.940041	−	0.341062i	$-0.889213\pi$
$19$	− 387485.i	− 0.682124i	0.940041	−	0.341062i	$-0.110787\pi$
$20$	0	0
$21$	− 920969.i	− 1.03338i
$22$	0	0
$23$	−2.36100e6	−1.75922	−0.879610	−	0.475695i	$-0.842197\pi$
$23$	−2.36100e6	−1.75922	−0.879610	+	0.475695i	$0.842197\pi$
$24$	0	0
$25$	−787863.	−0.403386
$26$	0	0
$27$	− 531441.i	− 0.192450i
$28$	0	0
$29$	2.70715e6i	0.710756i	0.934723	+	0.355378i	$0.115648\pi$
$29$	2.70715e6i	0.710756i	−0.934723	+	0.355378i	$0.884352\pi$
$30$	0	0
$31$	−5.18531e6	−1.00843	−0.504216	−	0.863577i	$-0.668219\pi$
$31$	−5.18531e6	−1.00843	−0.504216	+	0.863577i	$0.668219\pi$
$32$	0	0
$33$	7.65510e6	1.12367
$34$	0	0
$35$	− 1.88241e7i	− 2.12035i
$36$	0	0
$37$	− 142841.i	− 0.0125298i	−0.999980	−	0.00626492i	$-0.998006\pi$
$37$	− 142841.i	− 0.0125298i	0.999980	−	0.00626492i	$-0.00199420\pi$
$38$	0	0
$39$	1.13589e7	0.786225
$40$	0	0
$41$	1.92825e6	0.106570	0.0532851	−	0.998579i	$-0.483031\pi$
$41$	1.92825e6	0.106570	0.0532851	+	0.998579i	$0.483031\pi$
$42$	0	0
$43$	− 1.65243e7i	− 0.737080i	−0.929612	−	0.368540i	$-0.879858\pi$
$43$	− 1.65243e7i	− 0.737080i	0.929612	−	0.368540i	$-0.120142\pi$
$44$	0	0
$45$	− 1.08623e7i	− 0.394882i
$46$	0	0
$47$	−1.78851e7	−0.534627	−0.267314	−	0.963610i	$-0.586136\pi$
$47$	−1.78851e7	−0.534627	−0.267314	+	0.963610i	$0.586136\pi$
$48$	0	0
$49$	8.89229e7	2.20359
$50$	0	0
$51$	− 4.30110e7i	− 0.890253i
$52$	0	0
$53$	− 2.01018e6i	− 0.0349940i	−0.999847	−	0.0174970i	$-0.994430\pi$
$53$	− 2.01018e6i	− 0.0349940i	0.999847	−	0.0174970i	$-0.00556975\pi$
$54$	0	0
$55$	1.56466e8	2.30562
$56$	0	0
$57$	3.13863e7	0.393825
$58$	0	0
$59$	− 9.71099e7i	− 1.04335i	−0.853145	−	0.521674i	$-0.825308\pi$
$59$	− 9.71099e7i	− 1.04335i	0.853145	−	0.521674i	$-0.174692\pi$
$60$	0	0
$61$	7.23555e6i	0.0669094i	0.999440	+	0.0334547i	$0.0106509\pi$
$61$	7.23555e6i	0.0669094i	−0.999440	+	0.0334547i	$0.989349\pi$
$62$	0	0
$63$	7.45985e7	0.596619
$64$	0	0
$65$	2.32170e8	1.61323
$66$	0	0
$67$	2.34701e8i	1.42291i	0.702730	+	0.711457i	$0.251964\pi$
$67$	2.34701e8i	1.42291i	−0.702730	+	0.711457i	$0.748036\pi$
$68$	0	0
$69$	− 1.91241e8i	− 1.01569i
$70$	0	0
$71$	−2.62515e7	−0.122600	−0.0613002	−	0.998119i	$-0.519525\pi$
$71$	−2.62515e7	−0.122600	−0.0613002	+	0.998119i	$0.519525\pi$
$72$	0	0
$73$	−3.59965e8	−1.48357	−0.741785	−	0.670638i	$-0.766020\pi$
$73$	−3.59965e8	−1.48357	−0.741785	+	0.670638i	$0.766020\pi$
$74$	0	0
$75$	− 6.38169e7i	− 0.232895i
$76$	0	0
$77$	1.07455e9i	3.48351i
$78$	0	0
$79$	−3.01072e8	−0.869658	−0.434829	−	0.900513i	$-0.643191\pi$
$79$	−3.01072e8	−0.869658	−0.434829	+	0.900513i	$0.643191\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	3.48363e8i	0.805714i	0.915263	+	0.402857i	$0.131983\pi$
$83$	3.48363e8i	0.805714i	−0.915263	+	0.402857i	$0.868017\pi$
$84$	0	0
$85$	− 8.79119e8i	− 1.82668i
$86$	0	0
$87$	−2.19279e8	−0.410355
$88$	0	0
$89$	9.32242e8	1.57497	0.787487	−	0.616331i	$-0.211382\pi$
$89$	9.32242e8	1.57497	0.787487	+	0.616331i	$0.211382\pi$
$90$	0	0
$91$	1.59446e9i	2.43740i
$92$	0	0
$93$	− 4.20010e8i	− 0.582219i
$94$	0	0
$95$	6.41517e8	0.808075
$96$	0	0
$97$	−9.09752e8	−1.04340	−0.521699	−	0.853129i	$-0.674702\pi$
$97$	−9.09752e8	−1.04340	−0.521699	+	0.853129i	$0.674702\pi$
$98$	0	0
$99$	6.20063e8i	0.648750i
$100$	0	0
$101$	− 3.56743e8i	− 0.341121i	−0.985347	−	0.170561i	$-0.945442\pi$
$101$	− 3.56743e8i	− 0.341121i	0.985347	−	0.170561i	$-0.0545579\pi$
$102$	0	0
$103$	1.27419e9	1.11550	0.557748	−	0.830010i	$-0.311666\pi$
$103$	1.27419e9	1.11550	0.557748	+	0.830010i	$0.311666\pi$
$104$	0	0
$105$	1.52475e9	1.22418
$106$	0	0
$107$	− 1.90069e9i	− 1.40179i	−0.713264	−	0.700896i	$-0.752784\pi$
$107$	− 1.90069e9i	− 1.40179i	0.713264	−	0.700896i	$-0.247216\pi$
$108$	0	0
$109$	1.92409e8i	0.130559i	0.997867	+	0.0652793i	$0.0207938\pi$
$109$	1.92409e8i	0.130559i	−0.997867	+	0.0652793i	$0.979206\pi$
$110$	0	0
$111$	1.15701e7	0.00723411
$112$	0	0
$113$	−2.29060e9	−1.32159	−0.660793	−	0.750568i	$-0.729780\pi$
$113$	−2.29060e9	−1.32159	−0.660793	+	0.750568i	$0.729780\pi$
$114$	0	0
$115$	− 3.90885e9i	− 2.08405i
$116$	0	0
$117$	9.20074e8i	0.453927i
$118$	0	0
$119$	6.03746e9	2.75990
$120$	0	0
$121$	−6.57370e9	−2.78789
$122$	0	0
$123$	1.56188e8i	0.0615283i
$124$	0	0
$125$	1.92920e9i	0.706776i
$126$	0	0
$127$	−9.60539e7	−0.0327641	−0.0163821	−	0.999866i	$-0.505215\pi$
$127$	−9.60539e7	−0.0327641	−0.0163821	+	0.999866i	$0.505215\pi$
$128$	0	0
$129$	1.33847e9	0.425554
$130$	0	0
$131$	− 2.10259e9i	− 0.623783i	−0.950118	−	0.311892i	$-0.899037\pi$
$131$	− 2.10259e9i	− 0.623783i	0.950118	−	0.311892i	$-0.100963\pi$
$132$	0	0
$133$	4.40569e9i	1.22091i
$134$	0	0
$135$	8.79850e8	0.227985
$136$	0	0
$137$	−8.18137e8	−0.198419	−0.0992095	−	0.995067i	$-0.531631\pi$
$137$	−8.18137e8	−0.198419	−0.0992095	+	0.995067i	$0.531631\pi$
$138$	0	0
$139$	− 7.00619e8i	− 0.159190i	−0.996827	−	0.0795950i	$-0.974637\pi$
$139$	− 7.00619e8i	− 0.159190i	0.996827	−	0.0795950i	$-0.0253627\pi$
$140$	0	0
$141$	− 1.44869e9i	− 0.308667i
$142$	0	0
$143$	−1.32531e10	−2.65037
$144$	0	0
$145$	−4.48193e9	−0.841994
$146$	0	0
$147$	7.20276e9i	1.27224i
$148$	0	0
$149$	− 4.80548e9i	− 0.798728i	−0.916792	−	0.399364i	$-0.869231\pi$
$149$	− 4.80548e9i	− 0.798728i	0.916792	−	0.399364i	$-0.130769\pi$
$150$	0	0
$151$	−1.05075e9	−0.164477	−0.0822383	−	0.996613i	$-0.526207\pi$
$151$	−1.05075e9	−0.164477	−0.0822383	+	0.996613i	$0.526207\pi$
$152$	0	0
$153$	3.48389e9	0.513988
$154$	0	0
$155$	− 8.58476e9i	− 1.19464i
$156$	0	0
$157$	2.84280e9i	0.373421i	0.982415	+	0.186710i	$0.0597826\pi$
$157$	2.84280e9i	0.373421i	−0.982415	+	0.186710i	$0.940217\pi$
$158$	0	0
$159$	1.62825e8	0.0202038
$160$	0	0
$161$	2.68445e10	3.14876
$162$	0	0
$163$	1.36183e9i	0.151105i	0.997142	+	0.0755527i	$0.0240721\pi$
$163$	1.36183e9i	0.151105i	−0.997142	+	0.0755527i	$0.975928\pi$
$164$	0	0
$165$	1.26737e10i	1.33115i
$166$	0	0
$167$	−3.81425e9	−0.379477	−0.189738	−	0.981835i	$-0.560764\pi$
$167$	−3.81425e9	−0.379477	−0.189738	+	0.981835i	$0.560764\pi$
$168$	0	0
$169$	−9.06101e9	−0.854450
$170$	0	0
$171$	2.54229e9i	0.227375i
$172$	0	0
$173$	− 1.35181e10i	− 1.14738i	−0.819071	−	0.573692i	$-0.805511\pi$
$173$	− 1.35181e10i	− 1.14738i	0.819071	−	0.573692i	$-0.194489\pi$
$174$	0	0
$175$	8.95799e9	0.722004
$176$	0	0
$177$	7.86590e9	0.602378
$178$	0	0
$179$	− 2.34665e9i	− 0.170848i	−0.996345	−	0.0854240i	$-0.972776\pi$
$179$	− 2.34665e9i	− 0.170848i	0.996345	−	0.0854240i	$-0.0272245\pi$
$180$	0	0
$181$	2.69581e10i	1.86696i	0.358627	+	0.933481i	$0.383245\pi$
$181$	2.69581e10i	1.86696i	−0.358627	+	0.933481i	$0.616755\pi$
$182$	0	0
$183$	−5.86079e8	−0.0386302
$184$	0	0
$185$	2.36487e8	0.0148434
$186$	0	0
$187$	5.01834e10i	3.00105i
$188$	0	0
$189$	6.04248e9i	0.344458i
$190$	0	0
$191$	3.44999e9	0.187572	0.0937859	−	0.995592i	$-0.470103\pi$
$191$	3.44999e9	0.187572	0.0937859	+	0.995592i	$0.470103\pi$
$192$	0	0
$193$	1.93617e10	1.00447	0.502234	−	0.864732i	$-0.332512\pi$
$193$	1.93617e10	1.00447	0.502234	+	0.864732i	$0.332512\pi$
$194$	0	0
$195$	1.88058e10i	0.931398i
$196$	0	0
$197$	2.11248e10i	0.999298i	0.866228	+	0.499649i	$0.166538\pi$
$197$	2.11248e10i	0.999298i	−0.866228	+	0.499649i	$0.833462\pi$
$198$	0	0
$199$	−3.06345e10	−1.38475	−0.692376	−	0.721536i	$-0.743436\pi$
$199$	−3.06345e10	−1.38475	−0.692376	+	0.721536i	$0.743436\pi$
$200$	0	0
$201$	−1.90108e10	−0.821519
$202$	0	0
$203$	− 3.07802e10i	− 1.27215i
$204$	0	0
$205$	3.19239e9i	0.126248i
$206$	0	0
$207$	1.54905e10	0.586407
$208$	0	0
$209$	−3.66202e10	−1.32758
$210$	0	0
$211$	3.75426e9i	0.130393i	0.997872	+	0.0651964i	$0.0207674\pi$
$211$	3.75426e9i	0.130393i	−0.997872	+	0.0651964i	$0.979233\pi$
$212$	0	0
$213$	− 2.12637e9i	− 0.0707834i
$214$	0	0
$215$	2.73575e10	0.873179
$216$	0	0
$217$	5.89569e10	1.80495
$218$	0	0
$219$	− 2.91572e10i	− 0.856539i
$220$	0	0
$221$	7.44641e10i	2.09982i
$222$	0	0
$223$	5.80461e9	0.157181	0.0785907	−	0.996907i	$-0.474958\pi$
$223$	5.80461e9	0.157181	0.0785907	+	0.996907i	$0.474958\pi$
$224$	0	0
$225$	5.16917e9	0.134462
$226$	0	0
$227$	− 4.37575e10i	− 1.09379i	−0.837200	−	0.546897i	$-0.815809\pi$
$227$	− 4.37575e10i	− 1.09379i	0.837200	−	0.546897i	$-0.184191\pi$
$228$	0	0
$229$	− 5.00629e10i	− 1.20298i	−0.798882	−	0.601488i	$-0.794575\pi$
$229$	− 5.00629e10i	− 1.20298i	0.798882	−	0.601488i	$-0.205425\pi$
$230$	0	0
$231$	−8.70384e10	−2.01121
$232$	0	0
$233$	−5.78995e10	−1.28698	−0.643492	−	0.765453i	$-0.722515\pi$
$233$	−5.78995e10	−1.28698	−0.643492	+	0.765453i	$0.722515\pi$
$234$	0	0
$235$	− 2.96104e10i	− 0.633344i
$236$	0	0
$237$	− 2.43868e10i	− 0.502097i
$238$	0	0
$239$	−6.59252e10	−1.30696	−0.653478	−	0.756945i	$-0.726691\pi$
$239$	−6.59252e10	−1.30696	−0.653478	+	0.756945i	$0.726691\pi$
$240$	0	0
$241$	−1.03616e10	−0.197857	−0.0989284	−	0.995095i	$-0.531541\pi$
$241$	−1.03616e10	−0.197857	−0.0989284	+	0.995095i	$0.531541\pi$
$242$	0	0
$243$	3.48678e9i	0.0641500i
$244$	0	0
$245$	1.47220e11i	2.61048i
$246$	0	0
$247$	−5.43384e10	−0.928904
$248$	0	0
$249$	−2.82174e10	−0.465179
$250$	0	0
$251$	− 7.88299e10i	− 1.25360i	−0.779180	−	0.626800i	$-0.784364\pi$
$251$	− 7.88299e10i	− 1.25360i	0.779180	−	0.626800i	$-0.215636\pi$
$252$	0	0
$253$	2.23132e11i	3.42388i
$254$	0	0
$255$	7.12087e10	1.05463
$256$	0	0
$257$	4.14179e9	0.0592228	0.0296114	−	0.999561i	$-0.490573\pi$
$257$	4.14179e9	0.0592228	0.0296114	+	0.999561i	$0.490573\pi$
$258$	0	0
$259$	1.62410e9i	0.0224267i
$260$	0	0
$261$	− 1.77616e10i	− 0.236919i
$262$	0	0
$263$	6.51316e10	0.839442	0.419721	−	0.907653i	$-0.362128\pi$
$263$	6.51316e10	0.839442	0.419721	+	0.907653i	$0.362128\pi$
$264$	0	0
$265$	3.32804e9	0.0414555
$266$	0	0
$267$	7.55116e10i	0.909312i
$268$	0	0
$269$	4.43073e10i	0.515930i	0.966154	+	0.257965i	$0.0830519\pi$
$269$	4.43073e10i	0.515930i	−0.966154	+	0.257965i	$0.916948\pi$
$270$	0	0
$271$	9.19147e10	1.03520	0.517598	−	0.855624i	$-0.326826\pi$
$271$	9.19147e10	1.03520	0.517598	+	0.855624i	$0.326826\pi$
$272$	0	0
$273$	−1.29151e11	−1.40723
$274$	0	0
$275$	7.44589e10i	0.785090i
$276$	0	0
$277$	− 3.23505e10i	− 0.330158i	−0.986280	−	0.165079i	$-0.947212\pi$
$277$	− 3.23505e10i	− 0.330158i	0.986280	−	0.165079i	$-0.0527880\pi$
$278$	0	0
$279$	3.40208e10	0.336144
$280$	0	0
$281$	−5.98617e10	−0.572757	−0.286379	−	0.958117i	$-0.592451\pi$
$281$	−5.98617e10	−0.572757	−0.286379	+	0.958117i	$0.592451\pi$
$282$	0	0
$283$	− 1.24438e11i	− 1.15323i	−0.817017	−	0.576614i	$-0.804374\pi$
$283$	− 1.24438e11i	− 1.15323i	0.817017	−	0.576614i	$-0.195626\pi$
$284$	0	0
$285$	5.19629e10i	0.466543i
$286$	0	0
$287$	−2.19241e10	−0.190745
$288$	0	0
$289$	1.63373e11	1.37765
$290$	0	0
$291$	− 7.36899e10i	− 0.602407i
$292$	0	0
$293$	9.22003e10i	0.730850i	0.930841	+	0.365425i	$0.119076\pi$
$293$	9.22003e10i	0.730850i	−0.930841	+	0.365425i	$0.880924\pi$
$294$	0	0
$295$	1.60774e11	1.23600
$296$	0	0
$297$	−5.02251e10	−0.374556
$298$	0	0
$299$	3.31092e11i	2.39567i
$300$	0	0
$301$	1.87881e11i	1.31927i
$302$	0	0
$303$	2.88961e10	0.196946
$304$	0	0
$305$	−1.19791e10	−0.0792639
$306$	0	0
$307$	2.08578e11i	1.34013i	0.742303	+	0.670064i	$0.233733\pi$
$307$	2.08578e11i	1.34013i	−0.742303	+	0.670064i	$0.766267\pi$
$308$	0	0
$309$	1.03210e11i	0.644032i
$310$	0	0
$311$	1.15604e11	0.700728	0.350364	−	0.936614i	$-0.386058\pi$
$311$	1.15604e11	0.700728	0.350364	+	0.936614i	$0.386058\pi$
$312$	0	0
$313$	−1.38691e11	−0.816769	−0.408384	−	0.912810i	$-0.633908\pi$
$313$	−1.38691e11	−0.816769	−0.408384	+	0.912810i	$0.633908\pi$
$314$	0	0
$315$	1.23505e11i	0.706783i
$316$	0	0
$317$	5.75006e10i	0.319820i	0.987132	+	0.159910i	$0.0511204\pi$
$317$	5.75006e10i	0.319820i	−0.987132	+	0.159910i	$0.948880\pi$
$318$	0	0
$319$	2.55845e11	1.38331
$320$	0	0
$321$	1.53956e11	0.809325
$322$	0	0
$323$	2.05754e11i	1.05181i
$324$	0	0
$325$	1.10485e11i	0.549324i
$326$	0	0
$327$	−1.55851e10	−0.0753780
$328$	0	0
$329$	2.03353e11	0.956907
$330$	0	0
$331$	− 3.43458e11i	− 1.57271i	−0.617776	−	0.786354i	$-0.711966\pi$
$331$	− 3.43458e11i	− 1.57271i	0.617776	−	0.786354i	$-0.288034\pi$
$332$	0	0
$333$	9.37181e8i	0.00417662i
$334$	0	0
$335$	−3.88569e11	−1.68565
$336$	0	0
$337$	3.16926e11	1.33851	0.669257	−	0.743031i	$-0.266612\pi$
$337$	3.16926e11	1.33851	0.669257	+	0.743031i	$0.266612\pi$
$338$	0	0
$339$	− 1.85538e11i	− 0.763019i
$340$	0	0
$341$	4.90050e11i	1.96266i
$342$	0	0
$343$	−5.52232e11	−2.15426
$344$	0	0
$345$	3.16617e11	1.20323
$346$	0	0
$347$	− 6.34668e10i	− 0.234998i	−0.993073	−	0.117499i	$-0.962512\pi$
$347$	− 6.34668e10i	− 0.234998i	0.993073	−	0.117499i	$-0.0374877\pi$
$348$	0	0
$349$	4.80264e11i	1.73287i	0.499291	+	0.866434i	$0.333594\pi$
$349$	4.80264e11i	1.73287i	−0.499291	+	0.866434i	$0.666406\pi$
$350$	0	0
$351$	−7.45260e10	−0.262075
$352$	0	0
$353$	5.25576e10	0.180156	0.0900781	−	0.995935i	$-0.471288\pi$
$353$	5.25576e10	0.180156	0.0900781	+	0.995935i	$0.471288\pi$
$354$	0	0
$355$	− 4.34618e10i	− 0.145238i
$356$	0	0
$357$	4.89034e11i	1.59343i
$358$	0	0
$359$	3.68799e11	1.17183	0.585916	−	0.810372i	$-0.300735\pi$
$359$	3.68799e11	1.17183	0.585916	+	0.810372i	$0.300735\pi$
$360$	0	0
$361$	1.72543e11	0.534707
$362$	0	0
$363$	− 5.32470e11i	− 1.60959i
$364$	0	0
$365$	− 5.95956e11i	− 1.75750i
$366$	0	0
$367$	−2.50239e11	−0.720043	−0.360022	−	0.932944i	$-0.617231\pi$
$367$	−2.50239e11	−0.720043	−0.360022	+	0.932944i	$0.617231\pi$
$368$	0	0
$369$	−1.26512e10	−0.0355234
$370$	0	0
$371$	2.28557e10i	0.0626343i
$372$	0	0
$373$	6.47052e11i	1.73081i	0.501073	+	0.865405i	$0.332939\pi$
$373$	6.47052e11i	1.73081i	−0.501073	+	0.865405i	$0.667061\pi$
$374$	0	0
$375$	−1.56265e11	−0.408058
$376$	0	0
$377$	3.79633e11	0.967895
$378$	0	0
$379$	− 3.44133e11i	− 0.856742i	−0.903603	−	0.428371i	$-0.859088\pi$
$379$	− 3.44133e11i	− 0.856742i	0.903603	−	0.428371i	$-0.140912\pi$
$380$	0	0
$381$	− 7.78037e9i	− 0.0189164i
$382$	0	0
$383$	2.36072e11	0.560595	0.280298	−	0.959913i	$-0.409567\pi$
$383$	2.36072e11	0.560595	0.280298	+	0.959913i	$0.409567\pi$
$384$	0	0
$385$	−1.77901e12	−4.12673
$386$	0	0
$387$	1.08416e11i	0.245693i
$388$	0	0
$389$	− 2.69565e11i	− 0.596885i	−0.954428	−	0.298442i	$-0.903533\pi$
$389$	− 2.69565e11i	− 0.596885i	0.954428	−	0.298442i	$-0.0964671\pi$
$390$	0	0
$391$	1.25369e12	2.71265
$392$	0	0
$393$	1.70310e11	0.360141
$394$	0	0
$395$	− 4.98453e11i	− 1.03024i
$396$	0	0
$397$	− 2.24972e11i	− 0.454540i	−0.973832	−	0.227270i	$-0.927020\pi$
$397$	− 2.24972e11i	− 0.454540i	0.973832	−	0.227270i	$-0.0729800\pi$
$398$	0	0
$399$	−3.56861e11	−0.704890
$400$	0	0
$401$	−2.94935e11	−0.569609	−0.284805	−	0.958586i	$-0.591929\pi$
$401$	−2.94935e11	−0.569609	−0.284805	+	0.958586i	$0.591929\pi$
$402$	0	0
$403$	7.27155e11i	1.37327i
$404$	0	0
$405$	7.12678e10i	0.131627i
$406$	0	0
$407$	−1.34996e10	−0.0243862
$408$	0	0
$409$	−7.05914e11	−1.24737	−0.623687	−	0.781674i	$-0.714366\pi$
$409$	−7.05914e11	−1.24737	−0.623687	+	0.781674i	$0.714366\pi$
$410$	0	0
$411$	− 6.62691e10i	− 0.114557i
$412$	0	0
$413$	1.10414e12i	1.86745i
$414$	0	0
$415$	−5.76748e11	−0.954486
$416$	0	0
$417$	5.67502e10	0.0919083
$418$	0	0
$419$	6.73207e11i	1.06705i	0.845784	+	0.533526i	$0.179133\pi$
$419$	6.73207e11i	1.06705i	−0.845784	+	0.533526i	$0.820867\pi$
$420$	0	0
$421$	− 7.27739e11i	− 1.12903i	−0.825422	−	0.564516i	$-0.809063\pi$
$421$	− 7.27739e11i	− 1.12903i	0.825422	−	0.564516i	$-0.190937\pi$
$422$	0	0
$423$	1.17344e11	0.178209
$424$	0	0
$425$	4.18355e11	0.622006
$426$	0	0
$427$	− 8.22680e10i	− 0.119758i
$428$	0	0
$429$	− 1.07350e12i	− 1.53019i
$430$	0	0
$431$	−5.32994e11	−0.744003	−0.372001	−	0.928232i	$-0.621328\pi$
$431$	−5.32994e11	−0.744003	−0.372001	+	0.928232i	$0.621328\pi$
$432$	0	0
$433$	2.23588e11	0.305670	0.152835	−	0.988252i	$-0.451160\pi$
$433$	2.23588e11	0.305670	0.152835	+	0.988252i	$0.451160\pi$
$434$	0	0
$435$	− 3.63036e11i	− 0.486126i
$436$	0	0
$437$	9.14851e11i	1.20001i
$438$	0	0
$439$	−9.15099e10	−0.117592	−0.0587960	−	0.998270i	$-0.518726\pi$
$439$	−9.15099e10	−0.117592	−0.0587960	+	0.998270i	$0.518726\pi$
$440$	0	0
$441$	−5.83423e11	−0.734531
$442$	0	0
$443$	6.99027e11i	0.862337i	0.902271	+	0.431169i	$0.141899\pi$
$443$	6.99027e11i	0.862337i	−0.902271	+	0.431169i	$0.858101\pi$
$444$	0	0
$445$	1.54341e12i	1.86579i
$446$	0	0
$447$	3.89244e11	0.461146
$448$	0	0
$449$	2.69460e11	0.312886	0.156443	−	0.987687i	$-0.449997\pi$
$449$	2.69460e11	0.312886	0.156443	+	0.987687i	$0.449997\pi$
$450$	0	0
$451$	− 1.82234e11i	− 0.207412i
$452$	0	0
$453$	− 8.51109e10i	− 0.0949606i
$454$	0	0
$455$	−2.63977e12	−2.88745
$456$	0	0
$457$	7.28944e11	0.781756	0.390878	−	0.920443i	$-0.372171\pi$
$457$	7.28944e11	0.781756	0.390878	+	0.920443i	$0.372171\pi$
$458$	0	0
$459$	2.82195e11i	0.296751i
$460$	0	0
$461$	1.09764e12i	1.13190i	0.824440	+	0.565949i	$0.191490\pi$
$461$	1.09764e12i	1.13190i	−0.824440	+	0.565949i	$0.808510\pi$
$462$	0	0
$463$	−2.32754e11	−0.235387	−0.117693	−	0.993050i	$-0.537550\pi$
$463$	−2.32754e11	−0.235387	−0.117693	+	0.993050i	$0.537550\pi$
$464$	0	0
$465$	6.95365e11	0.689723
$466$	0	0
$467$	6.68871e11i	0.650754i	0.945584	+	0.325377i	$0.105491\pi$
$467$	6.68871e11i	0.650754i	−0.945584	+	0.325377i	$0.894509\pi$
$468$	0	0
$469$	− 2.66855e12i	− 2.54681i
$470$	0	0
$471$	−2.30267e11	−0.215595
$472$	0	0
$473$	−1.56167e12	−1.43454
$474$	0	0
$475$	3.05285e11i	0.275159i
$476$	0	0
$477$	1.31888e10i	0.0116647i
$478$	0	0
$479$	1.61992e12	1.40599	0.702996	−	0.711194i	$-0.251845\pi$
$479$	1.61992e12	1.40599	0.702996	+	0.711194i	$0.251845\pi$
$480$	0	0
$481$	−2.00312e10	−0.0170629
$482$	0	0
$483$	2.17441e12i	1.81794i
$484$	0	0
$485$	− 1.50618e12i	− 1.23606i
$486$	0	0
$487$	−2.49759e11	−0.201206	−0.100603	−	0.994927i	$-0.532077\pi$
$487$	−2.49759e11	−0.201206	−0.100603	+	0.994927i	$0.532077\pi$
$488$	0	0
$489$	−1.10309e11	−0.0872407
$490$	0	0
$491$	− 1.89367e12i	− 1.47040i	−0.677848	−	0.735202i	$-0.737087\pi$
$491$	− 1.89367e12i	− 1.47040i	0.677848	−	0.735202i	$-0.262913\pi$
$492$	0	0
$493$	− 1.43749e12i	− 1.09596i
$494$	0	0
$495$	−1.02657e12	−0.768539
$496$	0	0
$497$	2.98479e11	0.219437
$498$	0	0
$499$	4.18338e11i	0.302047i	0.988530	+	0.151024i	$0.0482569\pi$
$499$	4.18338e11i	0.302047i	−0.988530	+	0.151024i	$0.951743\pi$
$500$	0	0
$501$	− 3.08954e11i	− 0.219091i
$502$	0	0
$503$	−2.09661e12	−1.46036	−0.730182	−	0.683253i	$-0.760565\pi$
$503$	−2.09661e12	−1.46036	−0.730182	+	0.683253i	$0.760565\pi$
$504$	0	0
$505$	5.90620e11	0.404108
$506$	0	0
$507$	− 7.33942e11i	− 0.493317i
$508$	0	0
$509$	− 1.81715e12i	− 1.19994i	−0.800022	−	0.599970i	$-0.795179\pi$
$509$	− 1.81715e12i	− 1.19994i	0.800022	−	0.599970i	$-0.204821\pi$
$510$	0	0
$511$	4.09280e12	2.65538
$512$	0	0
$513$	−2.05925e11	−0.131275
$514$	0	0
$515$	2.10955e12i	1.32147i
$516$	0	0
$517$	1.69027e12i	1.04052i
$518$	0	0
$519$	1.09497e12	0.662443
$520$	0	0
$521$	−8.17238e11	−0.485936	−0.242968	−	0.970034i	$-0.578121\pi$
$521$	−8.17238e11	−0.485936	−0.242968	+	0.970034i	$0.578121\pi$
$522$	0	0
$523$	− 1.36193e12i	− 0.795969i	−0.917392	−	0.397984i	$-0.869710\pi$
$523$	− 1.36193e12i	− 0.795969i	0.917392	−	0.397984i	$-0.130290\pi$
$524$	0	0
$525$	7.25597e11i	0.416849i
$526$	0	0
$527$	2.75340e12	1.55497
$528$	0	0
$529$	3.77316e12	2.09486
$530$	0	0
$531$	6.37138e11i	0.347783i
$532$	0	0
$533$	− 2.70406e11i	− 0.145125i
$534$	0	0
$535$	3.14676e12	1.66063
$536$	0	0
$537$	1.90079e11	0.0986392
$538$	0	0
$539$	− 8.40387e12i	− 4.28874i
$540$	0	0
$541$	− 2.08586e12i	− 1.04688i	−0.852063	−	0.523440i	$-0.824649\pi$
$541$	− 2.08586e12i	− 1.04688i	0.852063	−	0.523440i	$-0.175351\pi$
$542$	0	0
$543$	−2.18360e12	−1.07789
$544$	0	0
$545$	−3.18550e11	−0.154666
$546$	0	0
$547$	− 2.54074e12i	− 1.21344i	−0.794917	−	0.606719i	$-0.792485\pi$
$547$	− 2.54074e12i	− 1.21344i	0.794917	−	0.606719i	$-0.207515\pi$
$548$	0	0
$549$	− 4.74724e10i	− 0.0223031i
$550$	0	0
$551$	1.04898e12	0.484824
$552$	0	0
$553$	3.42319e12	1.55657
$554$	0	0
$555$	1.91554e10i	0.00856986i
$556$	0	0
$557$	− 2.62379e12i	− 1.15500i	−0.816391	−	0.577499i	$-0.804029\pi$
$557$	− 2.62379e12i	− 1.15500i	0.816391	−	0.577499i	$-0.195971\pi$
$558$	0	0
$559$	−2.31726e12	−1.00374
$560$	0	0
$561$	−4.06485e12	−1.73266
$562$	0	0
$563$	− 1.67317e12i	− 0.701863i	−0.936401	−	0.350932i	$-0.885865\pi$
$563$	− 1.67317e12i	− 0.701863i	0.936401	−	0.350932i	$-0.114135\pi$
$564$	0	0
$565$	− 3.79230e12i	− 1.56561i
$566$	0	0
$567$	−4.89441e11	−0.198873
$568$	0	0
$569$	3.63787e12	1.45493	0.727466	−	0.686144i	$-0.240698\pi$
$569$	3.63787e12	1.45493	0.727466	+	0.686144i	$0.240698\pi$
$570$	0	0
$571$	2.12669e12i	0.837223i	0.908165	+	0.418612i	$0.137483\pi$
$571$	2.12669e12i	0.837223i	−0.908165	+	0.418612i	$0.862517\pi$
$572$	0	0
$573$	2.79449e11i	0.108295i
$574$	0	0
$575$	1.86014e12	0.709645
$576$	0	0
$577$	2.52863e12	0.949716	0.474858	−	0.880062i	$-0.342499\pi$
$577$	2.52863e12	0.949716	0.474858	+	0.880062i	$0.342499\pi$
$578$	0	0
$579$	1.56830e12i	0.579930i
$580$	0	0
$581$	− 3.96088e12i	− 1.44211i
$582$	0	0
$583$	−1.89977e11	−0.0681071
$584$	0	0
$585$	−1.52327e12	−0.537743
$586$	0	0
$587$	1.94162e12i	0.674981i	0.941329	+	0.337491i	$0.109578\pi$
$587$	1.94162e12i	0.674981i	−0.941329	+	0.337491i	$0.890422\pi$
$588$	0	0
$589$	2.00923e12i	0.687876i
$590$	0	0
$591$	−1.71111e12	−0.576945
$592$	0	0
$593$	−2.39070e12	−0.793926	−0.396963	−	0.917835i	$-0.629936\pi$
$593$	−2.39070e12	−0.793926	−0.396963	+	0.917835i	$0.629936\pi$
$594$	0	0
$595$	9.99557e12i	3.26950i
$596$	0	0
$597$	− 2.48140e12i	− 0.799488i
$598$	0	0
$599$	−1.02268e11	−0.0324576	−0.0162288	−	0.999868i	$-0.505166\pi$
$599$	−1.02268e11	−0.0324576	−0.0162288	+	0.999868i	$0.505166\pi$
$600$	0	0
$601$	−3.63671e12	−1.13704	−0.568518	−	0.822671i	$-0.692483\pi$
$601$	−3.63671e12	−1.13704	−0.568518	+	0.822671i	$0.692483\pi$
$602$	0	0
$603$	− 1.53987e12i	− 0.474304i
$604$	0	0
$605$	− 1.08834e13i	− 3.30266i
$606$	0	0
$607$	3.46914e12	1.03723	0.518613	−	0.855009i	$-0.326449\pi$
$607$	3.46914e12	1.03723	0.518613	+	0.855009i	$0.326449\pi$
$608$	0	0
$609$	2.49320e12	0.734478
$610$	0	0
$611$	2.50810e12i	0.728046i
$612$	0	0
$613$	3.39770e12i	0.971881i	0.873992	+	0.485941i	$0.161523\pi$
$613$	3.39770e12i	0.971881i	−0.873992	+	0.485941i	$0.838477\pi$
$614$	0	0
$615$	−2.58584e11	−0.0728893
$616$	0	0
$617$	−2.17197e12	−0.603351	−0.301676	−	0.953411i	$-0.597546\pi$
$617$	−2.17197e12	−0.603351	−0.301676	+	0.953411i	$0.597546\pi$
$618$	0	0
$619$	− 4.30312e12i	− 1.17808i	−0.808103	−	0.589041i	$-0.799506\pi$
$619$	− 4.30312e12i	− 1.17808i	0.808103	−	0.589041i	$-0.200494\pi$
$620$	0	0
$621$	1.25473e12i	0.338562i
$622$	0	0
$623$	−1.05996e13	−2.81898
$624$	0	0
$625$	−4.73276e12	−1.24067
$626$	0	0
$627$	− 2.96623e12i	− 0.766481i
$628$	0	0
$629$	7.58486e10i	0.0193206i
$630$	0	0
$631$	4.20938e12	1.05703	0.528513	−	0.848925i	$-0.322750\pi$
$631$	4.20938e12	1.05703	0.528513	+	0.848925i	$0.322750\pi$
$632$	0	0
$633$	−3.04095e11	−0.0752823
$634$	0	0
$635$	− 1.59026e11i	− 0.0388139i
$636$	0	0
$637$	− 1.24700e13i	− 3.00081i
$638$	0	0
$639$	1.72236e11	0.0408668
$640$	0	0
$641$	−7.60479e11	−0.177920	−0.0889602	−	0.996035i	$-0.528354\pi$
$641$	−7.60479e11	−0.177920	−0.0889602	+	0.996035i	$0.528354\pi$
$642$	0	0
$643$	7.42913e12i	1.71391i	0.515389	+	0.856957i	$0.327648\pi$
$643$	7.42913e12i	1.71391i	−0.515389	+	0.856957i	$0.672352\pi$
$644$	0	0
$645$	2.21596e12i	0.504130i
$646$	0	0
$647$	−2.60122e12	−0.583590	−0.291795	−	0.956481i	$-0.594252\pi$
$647$	−2.60122e12	−0.583590	−0.291795	+	0.956481i	$0.594252\pi$
$648$	0	0
$649$	−9.17760e12	−2.03062
$650$	0	0
$651$	4.77551e12i	1.04209i
$652$	0	0
$653$	3.06819e12i	0.660348i	0.943920	+	0.330174i	$0.107107\pi$
$653$	3.06819e12i	0.660348i	−0.943920	+	0.330174i	$0.892893\pi$
$654$	0	0
$655$	3.48103e12	0.738962
$656$	0	0
$657$	2.36173e12	0.494523
$658$	0	0
$659$	2.46719e12i	0.509586i	0.966996	+	0.254793i	$0.0820073\pi$
$659$	2.46719e12i	0.509586i	−0.966996	+	0.254793i	$0.917993\pi$
$660$	0	0
$661$	4.01622e12i	0.818297i	0.912468	+	0.409148i	$0.134174\pi$
$661$	4.01622e12i	0.818297i	−0.912468	+	0.409148i	$0.865826\pi$
$662$	0	0
$663$	−6.03159e12	−1.21233
$664$	0	0
$665$	−7.29404e12	−1.44634
$666$	0	0
$667$	− 6.39157e12i	− 1.25038i
$668$	0	0
$669$	4.70173e11i	0.0907487i
$670$	0	0
$671$	6.83813e11	0.130222
$672$	0	0
$673$	7.90823e12	1.48598	0.742988	−	0.669305i	$-0.233408\pi$
$673$	7.90823e12	1.48598	0.742988	+	0.669305i	$0.233408\pi$
$674$	0	0
$675$	4.18703e11i	0.0776316i
$676$	0	0
$677$	2.08763e11i	0.0381949i	0.999818	+	0.0190974i	$0.00607927\pi$
$677$	2.08763e11i	0.0381949i	−0.999818	+	0.0190974i	$0.993921\pi$
$678$	0	0
$679$	1.03439e13	1.86754
$680$	0	0
$681$	3.54435e12	0.631503
$682$	0	0
$683$	− 3.71560e12i	− 0.653335i	−0.945139	−	0.326667i	$-0.894074\pi$
$683$	− 3.71560e12i	− 0.653335i	0.945139	−	0.326667i	$-0.105926\pi$
$684$	0	0
$685$	− 1.35450e12i	− 0.235056i
$686$	0	0
$687$	4.05510e12	0.694538
$688$	0	0
$689$	−2.81895e11	−0.0476542
$690$	0	0
$691$	6.47722e12i	1.08078i	0.841415	+	0.540390i	$0.181723\pi$
$691$	6.47722e12i	1.08078i	−0.841415	+	0.540390i	$0.818277\pi$
$692$	0	0
$693$	− 7.05011e12i	− 1.16117i
$694$	0	0
$695$	1.15994e12	0.188584
$696$	0	0
$697$	−1.02390e12	−0.164327
$698$	0	0
$699$	− 4.68986e12i	− 0.743040i
$700$	0	0
$701$	7.68628e12i	1.20222i	0.799165	+	0.601112i	$0.205275\pi$
$701$	7.68628e12i	1.20222i	−0.799165	+	0.601112i	$0.794725\pi$
$702$	0	0
$703$	−5.53488e10	−0.00854691
$704$	0	0
$705$	2.39845e12	0.365661
$706$	0	0
$707$	4.05616e12i	0.610559i
$708$	0	0
$709$	− 1.27745e13i	− 1.89861i	−0.314358	−	0.949304i	$-0.601789\pi$
$709$	− 1.27745e13i	− 1.89861i	0.314358	−	0.949304i	$-0.398211\pi$
$710$	0	0
$711$	1.97533e12	0.289886
$712$	0	0
$713$	1.22425e13	1.77406
$714$	0	0
$715$	− 2.19418e13i	− 3.13975i
$716$	0	0
$717$	− 5.33994e12i	− 0.754571i
$718$	0	0
$719$	−9.64154e12	−1.34545	−0.672723	−	0.739894i	$-0.734876\pi$
$719$	−9.64154e12	−1.34545	−0.672723	+	0.739894i	$0.734876\pi$
$720$	0	0
$721$	−1.44876e13	−1.99658
$722$	0	0
$723$	− 8.39292e11i	− 0.114233i
$724$	0	0
$725$	− 2.13286e12i	− 0.286709i
$726$	0	0
$727$	1.93455e12	0.256848	0.128424	−	0.991719i	$-0.459008\pi$
$727$	1.93455e12	0.256848	0.128424	+	0.991719i	$0.459008\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	8.77439e12i	1.13655i
$732$	0	0
$733$	− 4.64764e12i	− 0.594654i	−0.954776	−	0.297327i	$-0.903905\pi$
$733$	− 4.64764e12i	− 0.594654i	0.954776	−	0.297327i	$-0.0960951\pi$
$734$	0	0
$735$	−1.19248e13	−1.50716
$736$	0	0
$737$	2.21810e13	2.76935
$738$	0	0
$739$	− 1.50955e13i	− 1.86186i	−0.365194	−	0.930932i	$-0.618997\pi$
$739$	− 1.50955e13i	− 1.86186i	0.365194	−	0.930932i	$-0.381003\pi$
$740$	0	0
$741$	− 4.40141e12i	− 0.536303i
$742$	0	0
$743$	−1.13339e13	−1.36437	−0.682183	−	0.731182i	$-0.738969\pi$
$743$	−1.13339e13	−1.36437	−0.682183	+	0.731182i	$0.738969\pi$
$744$	0	0
$745$	7.95593e12	0.946210
$746$	0	0
$747$	− 2.28561e12i	− 0.268571i
$748$	0	0
$749$	2.16108e13i	2.50901i
$750$	0	0
$751$	6.43395e12	0.738071	0.369035	−	0.929415i	$-0.379688\pi$
$751$	6.43395e12	0.738071	0.369035	+	0.929415i	$0.379688\pi$
$752$	0	0
$753$	6.38522e12	0.723766
$754$	0	0
$755$	− 1.73962e12i	− 0.194847i
$756$	0	0
$757$	1.59289e11i	0.0176301i	0.999961	+	0.00881507i	$0.00280596\pi$
$757$	1.59289e11i	0.0176301i	−0.999961	+	0.00881507i	$0.997194\pi$
$758$	0	0
$759$	−1.80737e13	−1.97678
$760$	0	0
$761$	−1.38855e12	−0.150083	−0.0750413	−	0.997180i	$-0.523909\pi$
$761$	−1.38855e12	−0.150083	−0.0750413	+	0.997180i	$0.523909\pi$
$762$	0	0
$763$	− 2.18768e12i	− 0.233681i
$764$	0	0
$765$	5.76790e12i	0.608894i
$766$	0	0
$767$	−1.36181e13	−1.42081
$768$	0	0
$769$	−7.69105e12	−0.793080	−0.396540	−	0.918017i	$-0.629789\pi$
$769$	−7.69105e12	−0.793080	−0.396540	+	0.918017i	$0.629789\pi$
$770$	0	0
$771$	3.35485e11i	0.0341923i
$772$	0	0
$773$	− 1.48855e13i	− 1.49954i	−0.661701	−	0.749768i	$-0.730165\pi$
$773$	− 1.48855e13i	− 1.49954i	0.661701	−	0.749768i	$-0.269835\pi$
$774$	0	0
$775$	4.08531e12	0.406787
$776$	0	0
$777$	−1.31552e11	−0.0129480
$778$	0	0
$779$	− 7.47167e11i	− 0.0726941i
$780$	0	0
$781$	2.48096e12i	0.238611i
$782$	0	0
$783$	1.43869e12	0.136785
$784$	0	0
$785$	−4.70653e12	−0.442372
$786$	0	0
$787$	8.96061e12i	0.832629i	0.909221	+	0.416314i	$0.136678\pi$
$787$	8.96061e12i	0.832629i	−0.909221	+	0.416314i	$0.863322\pi$
$788$	0	0
$789$	5.27566e12i	0.484652i
$790$	0	0
$791$	2.60441e13	2.36545
$792$	0	0
$793$	1.01467e12	0.0911160
$794$	0	0
$795$	2.69571e11i	0.0239343i
$796$	0	0
$797$	6.13567e12i	0.538641i	0.963051	+	0.269320i	$0.0867991\pi$
$797$	6.13567e12i	0.538641i	−0.963051	+	0.269320i	$0.913201\pi$
$798$	0	0
$799$	9.49698e12	0.824375
$800$	0	0
$801$	−6.11644e12	−0.524992
$802$	0	0
$803$	3.40194e13i	2.88740i
$804$	0	0
$805$	4.44436e13i	3.73016i
$806$	0	0
$807$	−3.58890e12	−0.297872
$808$	0	0
$809$	1.68903e13	1.38634	0.693170	−	0.720774i	$-0.256214\pi$
$809$	1.68903e13	1.38634	0.693170	+	0.720774i	$0.256214\pi$
$810$	0	0
$811$	1.31863e13i	1.07036i	0.844739	+	0.535179i	$0.179756\pi$
$811$	1.31863e13i	1.07036i	−0.844739	+	0.535179i	$0.820244\pi$
$812$	0	0
$813$	7.44509e12i	0.597671i
$814$	0	0
$815$	−2.25464e12	−0.179006
$816$	0	0
$817$	−6.40291e12	−0.502780
$818$	0	0
$819$	− 1.04612e13i	− 0.812466i
$820$	0	0
$821$	− 6.47323e12i	− 0.497252i	−0.968600	−	0.248626i	$-0.920021\pi$
$821$	− 6.47323e12i	− 0.497252i	0.968600	−	0.248626i	$-0.0799790\pi$
$822$	0	0
$823$	−1.72880e13	−1.31355	−0.656773	−	0.754088i	$-0.728079\pi$
$823$	−1.72880e13	−1.31355	−0.656773	+	0.754088i	$0.728079\pi$
$824$	0	0
$825$	−6.03117e12	−0.453272
$826$	0	0
$827$	− 1.84398e12i	− 0.137082i	−0.997648	−	0.0685410i	$-0.978166\pi$
$827$	− 1.84398e12i	− 0.137082i	0.997648	−	0.0685410i	$-0.0218344\pi$
$828$	0	0
$829$	1.85743e12i	0.136589i	0.997665	+	0.0682947i	$0.0217558\pi$
$829$	1.85743e12i	0.136589i	−0.997665	+	0.0682947i	$0.978244\pi$
$830$	0	0
$831$	2.62039e12	0.190617
$832$	0	0
$833$	−4.72180e13	−3.39786
$834$	0	0
$835$	− 6.31485e12i	− 0.449546i
$836$	0	0
$837$	2.75568e12i	0.194073i
$838$	0	0
$839$	1.04502e12	0.0728106	0.0364053	−	0.999337i	$-0.488409\pi$
$839$	1.04502e12	0.0728106	0.0364053	+	0.999337i	$0.488409\pi$
$840$	0	0
$841$	7.17851e12	0.494826
$842$	0	0
$843$	− 4.84880e12i	− 0.330681i
$844$	0	0
$845$	− 1.50013e13i	− 1.01222i
$846$	0	0
$847$	7.47429e13	4.98993
$848$	0	0
$849$	1.00795e13	0.665816
$850$	0	0
$851$	3.37248e11i	0.0220428i
$852$	0	0
$853$	− 4.28041e12i	− 0.276831i	−0.990374	−	0.138415i	$-0.955799\pi$
$853$	− 4.28041e12i	− 0.276831i	0.990374	−	0.138415i	$-0.0442009\pi$
$854$	0	0
$855$	−4.20899e12	−0.269358
$856$	0	0
$857$	9.25941e12	0.586367	0.293184	−	0.956056i	$-0.405285\pi$
$857$	9.25941e12	0.586367	0.293184	+	0.956056i	$0.405285\pi$
$858$	0	0
$859$	9.57537e12i	0.600049i	0.953932	+	0.300024i	$0.0969948\pi$
$859$	9.57537e12i	0.600049i	−0.953932	+	0.300024i	$0.903005\pi$
$860$	0	0
$861$	− 1.77586e12i	− 0.110127i
$862$	0	0
$863$	−1.56057e13	−0.957713	−0.478856	−	0.877893i	$-0.658949\pi$
$863$	−1.56057e13	−0.957713	−0.478856	+	0.877893i	$0.658949\pi$
$864$	0	0
$865$	2.23805e13	1.35924
$866$	0	0
$867$	1.32332e13i	0.795387i
$868$	0	0
$869$	2.84535e13i	1.69257i
$870$	0	0
$871$	3.29130e13	1.93770
$872$	0	0
$873$	5.96888e12	0.347800
$874$	0	0
$875$	− 2.19350e13i	− 1.26503i
$876$	0	0
$877$	2.51400e13i	1.43505i	0.696532	+	0.717526i	$0.254726\pi$
$877$	2.51400e13i	1.43505i	−0.696532	+	0.717526i	$0.745274\pi$
$878$	0	0
$879$	−7.46822e12	−0.421956
$880$	0	0
$881$	1.06427e12	0.0595198	0.0297599	−	0.999557i	$-0.490526\pi$
$881$	1.06427e12	0.0595198	0.0297599	+	0.999557i	$0.490526\pi$
$882$	0	0
$883$	3.43027e13i	1.89891i	0.313902	+	0.949455i	$0.398364\pi$
$883$	3.43027e13i	1.89891i	−0.313902	+	0.949455i	$0.601636\pi$
$884$	0	0
$885$	1.30227e13i	0.713604i
$886$	0	0
$887$	−1.38759e13	−0.752670	−0.376335	−	0.926484i	$-0.622816\pi$
$887$	−1.38759e13	−0.752670	−0.376335	+	0.926484i	$0.622816\pi$
$888$	0	0
$889$	1.09213e12	0.0586431
$890$	0	0
$891$	− 4.06823e12i	− 0.216250i
$892$	0	0
$893$	6.93020e12i	0.364682i
$894$	0	0
$895$	3.88510e12	0.202394
$896$	0	0
$897$	−2.68184e13	−1.38314
$898$	0	0
$899$	− 1.40374e13i	− 0.716750i
$900$	0	0
$901$	1.06740e12i	0.0539595i
$902$	0	0
$903$	−1.52184e13	−0.761681
$904$	0	0
$905$	−4.46316e13	−2.21169
$906$	0	0
$907$	7.19852e12i	0.353192i	0.984283	+	0.176596i	$0.0565086\pi$
$907$	7.19852e12i	0.353192i	−0.984283	+	0.176596i	$0.943491\pi$
$908$	0	0
$909$	2.34059e12i	0.113707i
$910$	0	0
$911$	−2.48977e12	−0.119764	−0.0598820	−	0.998205i	$-0.519072\pi$
$911$	−2.48977e12	−0.119764	−0.0598820	+	0.998205i	$0.519072\pi$
$912$	0	0
$913$	3.29229e13	1.56812
$914$	0	0
$915$	− 9.70309e11i	− 0.0457631i
$916$	0	0
$917$	2.39064e13i	1.11648i
$918$	0	0
$919$	1.73700e13	0.803303	0.401652	−	0.915793i	$-0.368436\pi$
$919$	1.73700e13	0.803303	0.401652	+	0.915793i	$0.368436\pi$
$920$	0	0
$921$	−1.68948e13	−0.773723
$922$	0	0
$923$	3.68135e12i	0.166955i
$924$	0	0
$925$	1.12539e11i	0.00505436i
$926$	0	0
$927$	−8.35999e12	−0.371832
$928$	0	0
$929$	−4.98860e12	−0.219739	−0.109870	−	0.993946i	$-0.535043\pi$
$929$	−4.98860e12	−0.219739	−0.109870	+	0.993946i	$0.535043\pi$
$930$	0	0
$931$	− 3.44563e13i	− 1.50312i
$932$	0	0
$933$	9.36389e12i	0.404565i
$934$	0	0
$935$	−8.30833e13	−3.55518
$936$	0	0
$937$	2.01453e12	0.0853781	0.0426890	−	0.999088i	$-0.486408\pi$
$937$	2.01453e12	0.0853781	0.0426890	+	0.999088i	$0.486408\pi$
$938$	0	0
$939$	− 1.12340e13i	− 0.471562i
$940$	0	0
$941$	− 2.75127e13i	− 1.14388i	−0.820296	−	0.571939i	$-0.806191\pi$
$941$	− 2.75127e13i	− 1.14388i	0.820296	−	0.571939i	$-0.193809\pi$
$942$	0	0
$943$	−4.55259e12	−0.187480
$944$	0	0
$945$	−1.00039e13	−0.408061
$946$	0	0
$947$	− 2.57827e13i	− 1.04173i	−0.853640	−	0.520863i	$-0.825610\pi$
$947$	− 2.57827e13i	− 1.04173i	0.853640	−	0.520863i	$-0.174390\pi$
$948$	0	0
$949$	5.04793e13i	2.02030i
$950$	0	0
$951$	−4.65755e12	−0.184648
$952$	0	0
$953$	−2.06001e13	−0.809004	−0.404502	−	0.914537i	$-0.632555\pi$
$953$	−2.06001e13	−0.809004	−0.404502	+	0.914537i	$0.632555\pi$
$954$	0	0
$955$	5.71178e12i	0.222206i
$956$	0	0
$957$	2.07235e13i	0.798654i
$958$	0	0
$959$	9.30220e12	0.355142
$960$	0	0
$961$	4.47786e11	0.0169362
$962$	0	0
$963$	1.24704e13i	0.467264i
$964$	0	0
$965$	3.20551e13i	1.18994i
$966$	0	0
$967$	3.94563e13	1.45110	0.725550	−	0.688170i	$-0.241586\pi$
$967$	3.94563e13	1.45110	0.725550	+	0.688170i	$0.241586\pi$
$968$	0	0
$969$	−1.66661e13	−0.607263
$970$	0	0
$971$	− 8.55423e12i	− 0.308812i	−0.988007	−	0.154406i	$-0.950654\pi$
$971$	− 8.55423e12i	− 0.308812i	0.988007	−	0.154406i	$-0.0493464\pi$
$972$	0	0
$973$	7.96603e12i	0.284927i
$974$	0	0
$975$	−8.94929e12	−0.317152
$976$	0	0
$977$	−1.68374e12	−0.0591221	−0.0295611	−	0.999563i	$-0.509411\pi$
$977$	−1.68374e12	−0.0591221	−0.0295611	+	0.999563i	$0.509411\pi$
$978$	0	0
$979$	− 8.81038e13i	− 3.06530i
$980$	0	0
$981$	− 1.26239e12i	− 0.0435195i
$982$	0	0
$983$	5.66001e13	1.93342	0.966711	−	0.255872i	$-0.0823626\pi$
$983$	5.66001e13	1.93342	0.966711	+	0.255872i	$0.0823626\pi$
$984$	0	0
$985$	−3.49741e13	−1.18381
$986$	0	0
$987$	1.64716e13i	0.552470i
$988$	0	0
$989$	3.90138e13i	1.29669i
$990$	0	0
$991$	−7.49565e12	−0.246875	−0.123438	−	0.992352i	$-0.539392\pi$
$991$	−7.49565e12	−0.246875	−0.123438	+	0.992352i	$0.539392\pi$
$992$	0	0
$993$	2.78201e13	0.908003
$994$	0	0
$995$	− 5.07183e13i	− 1.64044i
$996$	0	0
$997$	− 5.52784e13i	− 1.77185i	−0.463828	−	0.885925i	$-0.653524\pi$
$997$	− 5.52784e13i	− 1.77185i	0.463828	−	0.885925i	$-0.346476\pi$
$998$	0	0
$999$	−7.59117e10	−0.00241137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.f.193.18	yes	20
4.3	odd	2	inner	384.10.d.f.193.8	yes	20
8.3	odd	2	inner	384.10.d.f.193.13	yes	20
8.5	even	2	inner	384.10.d.f.193.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.f.193.3	✓	20	8.5	even	2	inner
384.10.d.f.193.8	yes	20	4.3	odd	2	inner
384.10.d.f.193.13	yes	20	8.3	odd	2	inner
384.10.d.f.193.18	yes	20	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+81.0000i q^{3} +1655.59i q^{5} -11370.0 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q+81.0000i q^{3} +1655.59i q^{5} -11370.0 q^{7} -6561.00 q^{9} -94507.4i q^{11} -140234. i q^{13} -134103. q^{15} -531000. q^{17} -387485. i q^{19} -920969. i q^{21} -2.36100e6 q^{23} -787863. q^{25} -531441. i q^{27} +2.70715e6i q^{29} -5.18531e6 q^{31} +7.65510e6 q^{33} -1.88241e7i q^{35} -142841. i q^{37} +1.13589e7 q^{39} +1.92825e6 q^{41} -1.65243e7i q^{43} -1.08623e7i q^{45} -1.78851e7 q^{47} +8.89229e7 q^{49} -4.30110e7i q^{51} -2.01018e6i q^{53} +1.56466e8 q^{55} +3.13863e7 q^{57} -9.71099e7i q^{59} +7.23555e6i q^{61} +7.45985e7 q^{63} +2.32170e8 q^{65} +2.34701e8i q^{67} -1.91241e8i q^{69} -2.62515e7 q^{71} -3.59965e8 q^{73} -6.38169e7i q^{75} +1.07455e9i q^{77} -3.01072e8 q^{79} +4.30467e7 q^{81} +3.48363e8i q^{83} -8.79119e8i q^{85} -2.19279e8 q^{87} +9.32242e8 q^{89} +1.59446e9i q^{91} -4.20010e8i q^{93} +6.41517e8 q^{95} -9.09752e8 q^{97} +6.20063e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 131220 q^{9}+O(q^{10}) \) 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

Related objects

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