LMFDB - Embedded newform 320.9.h.g.319.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,9,Mod(319,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.319"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	320.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,1420] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$130.361155220$
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} + 94 x^{18} + 5343 x^{16} + 172772 x^{14} + 36131456 x^{12} + 3044563968 x^{10} + \cdots + 11\!\cdots\!76 $ x^20 + 94x^18 + 5343x^16 + 172772x^14 + 36131456x^12 + 3044563968x^10 + 147994443776x^8 + 2898633162752x^6 + 367168164200448x^4 + 26458647810801664*x^2 + 1152921504606846976
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{140}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.10
Root			$7.15078 - 3.58697i$ of defining polynomial
Character	$\chi$	$=$	320.319
Dual form			320.9.h.g.319.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-42.6663 q^{3} +(560.298 + 276.932i) q^{5} +869.685 q^{7} -4740.59 q^{9} -24735.0i q^{11} -41128.5i q^{13} +(-23905.8 - 11815.7i) q^{15} -40900.7i q^{17} -79502.2i q^{19} -37106.2 q^{21} -428560. q^{23} +(237242. + 310329. i) q^{25} +482197. q^{27} -184740. q^{29} +154752. i q^{31} +1.05535e6i q^{33} +(487283. + 240844. i) q^{35} +1.72203e6i q^{37} +1.75480e6i q^{39} -2.00720e6 q^{41} -1.43073e6 q^{43} +(-2.65614e6 - 1.31282e6i) q^{45} +4.98494e6 q^{47} -5.00845e6 q^{49} +1.74508e6i q^{51} +6.97640e6i q^{53} +(6.84992e6 - 1.38590e7i) q^{55} +3.39206e6i q^{57} -5.65200e6i q^{59} -6.73713e6 q^{61} -4.12282e6 q^{63} +(1.13898e7 - 2.30442e7i) q^{65} +4.99529e6 q^{67} +1.82850e7 q^{69} -2.07710e7i q^{71} +3.31772e7i q^{73} +(-1.01222e7 - 1.32406e7i) q^{75} -2.15117e7i q^{77} +7.49445e7i q^{79} +1.05295e7 q^{81} +8.75346e7 q^{83} +(1.13267e7 - 2.29165e7i) q^{85} +7.88215e6 q^{87} -1.14653e7 q^{89} -3.57688e7i q^{91} -6.60268e6i q^{93} +(2.20167e7 - 4.45449e7i) q^{95} -8.74640e7i q^{97} +1.17258e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 1420 q^{5} + 2556 q^{9} - 410256 q^{21} - 657260 q^{25} - 660136 q^{29} + 7068520 q^{41} + 18729060 q^{45} + 11719036 q^{49} + 17660440 q^{61} - 44202240 q^{65} - 111747216 q^{69} - 154212444 q^{81}+ \cdots + 105006376 q^{89}+O(q^{100}) $ 20 * q + 1420 * q^5 + 2556 * q^9 - 410256 * q^21 - 657260 * q^25 - 660136 * q^29 + 7068520 * q^41 + 18729060 * q^45 + 11719036 * q^49 + 17660440 * q^61 - 44202240 * q^65 - 111747216 * q^69 - 154212444 * q^81 - 68800000 * q^85 + 105006376 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−42.6663			−0.526744			−0.263372	−	0.964694i	$-0.584835\pi$
$3$	−42.6663			−0.526744			−0.263372	+	0.964694i	$0.584835\pi$
$4$		0			0
$5$	560.298	+	276.932i	0.896476	+	0.443091i
$5$	560.298	+	276.932i	0.896476	+	0.443091i
$6$		0			0
$7$	869.685			0.362218			0.181109	−	0.983463i	$-0.442031\pi$
$7$	869.685			0.362218			0.181109	+	0.983463i	$0.442031\pi$
$8$		0			0
$9$	−4740.59			−0.722541
$10$		0			0
$11$		−	24735.0i		−	1.68943i	−0.535214	−	0.844717i	$-0.679769\pi$
$11$		−	24735.0i		−	1.68943i	0.535214	−	0.844717i	$-0.320231\pi$
$12$		0			0
$13$		−	41128.5i		−	1.44002i	−0.693962	−	0.720011i	$-0.744137\pi$
$13$		−	41128.5i		−	1.44002i	0.693962	−	0.720011i	$-0.255863\pi$
$14$		0			0
$15$	−23905.8	−	11815.7i	−0.472214	−	0.233396i
$16$		0			0
$17$		−	40900.7i		−	0.489705i	−0.969560	−	0.244853i	$-0.921260\pi$
$17$		−	40900.7i		−	0.489705i	0.969560	−	0.244853i	$-0.0787395\pi$
$18$		0			0
$19$		−	79502.2i		−	0.610049i	−0.952345	−	0.305024i	$-0.901335\pi$
$19$		−	79502.2i		−	0.610049i	0.952345	−	0.305024i	$-0.0986646\pi$
$20$		0			0
$21$	−37106.2			−0.190796
$22$		0			0
$23$	−428560.			−1.53144			−0.765720	−	0.643174i	$-0.777617\pi$
$23$	−428560.			−1.53144			−0.765720	+	0.643174i	$0.777617\pi$
$24$		0			0
$25$	237242.	+	310329.i	0.607340	+	0.794442i
$26$		0			0
$27$	482197.			0.907338
$28$		0			0
$29$	−184740.			−0.261197			−0.130598	−	0.991435i	$-0.541690\pi$
$29$	−184740.			−0.261197			−0.130598	+	0.991435i	$0.541690\pi$
$30$		0			0
$31$			154752.i			0.167567i	0.996484	+	0.0837835i	$0.0267004\pi$
$31$			154752.i			0.167567i	−0.996484	+	0.0837835i	$0.973300\pi$
$32$		0			0
$33$			1.05535e6i			0.889899i
$34$		0			0
$35$	487283.	+	240844.i	0.324720	+	0.160496i
$36$		0			0
$37$			1.72203e6i			0.918825i	0.888223	+	0.459413i	$0.151940\pi$
$37$			1.72203e6i			0.918825i	−0.888223	+	0.459413i	$0.848060\pi$
$38$		0			0
$39$			1.75480e6i			0.758523i
$40$		0			0
$41$	−2.00720e6			−0.710323			−0.355162	−	0.934805i	$-0.615574\pi$
$41$	−2.00720e6			−0.710323			−0.355162	+	0.934805i	$0.615574\pi$
$42$		0			0
$43$	−1.43073e6			−0.418488			−0.209244	−	0.977863i	$-0.567100\pi$
$43$	−1.43073e6			−0.418488			−0.209244	+	0.977863i	$0.567100\pi$
$44$		0			0
$45$	−2.65614e6	−	1.31282e6i	−0.647741	−	0.320152i
$46$		0			0
$47$	4.98494e6			1.02157			0.510786	−	0.859708i	$-0.329354\pi$
$47$	4.98494e6			1.02157			0.510786	+	0.859708i	$0.329354\pi$
$48$		0			0
$49$	−5.00845e6			−0.868798
$50$		0			0
$51$			1.74508e6i			0.257949i
$52$		0			0
$53$			6.97640e6i			0.884154i	0.896977	+	0.442077i	$0.145758\pi$
$53$			6.97640e6i			0.884154i	−0.896977	+	0.442077i	$0.854242\pi$
$54$		0			0
$55$	6.84992e6	−	1.38590e7i	0.748574	−	1.51454i
$56$		0			0
$57$			3.39206e6i			0.321340i
$58$		0			0
$59$		−	5.65200e6i		−	0.466438i	−0.972424	−	0.233219i	$-0.925074\pi$
$59$		−	5.65200e6i		−	0.466438i	0.972424	−	0.233219i	$-0.0749259\pi$
$60$		0			0
$61$	−6.73713e6			−0.486582			−0.243291	−	0.969953i	$-0.578227\pi$
$61$	−6.73713e6			−0.486582			−0.243291	+	0.969953i	$0.578227\pi$
$62$		0			0
$63$	−4.12282e6			−0.261717
$64$		0			0
$65$	1.13898e7	−	2.30442e7i	0.638062	−	1.29095i
$66$		0			0
$67$	4.99529e6			0.247891			0.123946	−	0.992289i	$-0.460445\pi$
$67$	4.99529e6			0.247891			0.123946	+	0.992289i	$0.460445\pi$
$68$		0			0
$69$	1.82850e7			0.806677
$70$		0			0
$71$		−	2.07710e7i		−	0.817378i	−0.912674	−	0.408689i	$-0.865986\pi$
$71$		−	2.07710e7i		−	0.817378i	0.912674	−	0.408689i	$-0.134014\pi$
$72$		0			0
$73$			3.31772e7i			1.16828i	0.811652	+	0.584141i	$0.198569\pi$
$73$			3.31772e7i			1.16828i	−0.811652	+	0.584141i	$0.801431\pi$
$74$		0			0
$75$	−1.01222e7	−	1.32406e7i	−0.319913	−	0.418468i
$76$		0			0
$77$		−	2.15117e7i		−	0.611943i
$78$		0			0
$79$			7.49445e7i			1.92412i	0.272844	+	0.962058i	$0.412036\pi$
$79$			7.49445e7i			1.92412i	−0.272844	+	0.962058i	$0.587964\pi$
$80$		0			0
$81$	1.05295e7			0.244606
$82$		0			0
$83$	8.75346e7			1.84445			0.922226	−	0.386651i	$-0.126368\pi$
$83$	8.75346e7			1.84445			0.922226	+	0.386651i	$0.126368\pi$
$84$		0			0
$85$	1.13267e7	−	2.29165e7i	0.216984	−	0.439009i
$86$		0			0
$87$	7.88215e6			0.137584
$88$		0			0
$89$	−1.14653e7			−0.182737			−0.0913685	−	0.995817i	$-0.529124\pi$
$89$	−1.14653e7			−0.182737			−0.0913685	+	0.995817i	$0.529124\pi$
$90$		0			0
$91$		−	3.57688e7i		−	0.521602i
$92$		0			0
$93$		−	6.60268e6i		−	0.0882649i
$94$		0			0
$95$	2.20167e7	−	4.45449e7i	0.270307	−	0.546894i
$96$		0			0
$97$		−	8.74640e7i		−	0.987967i	−0.869471	−	0.493983i	$-0.835540\pi$
$97$		−	8.74640e7i		−	0.987967i	0.869471	−	0.493983i	$-0.164460\pi$
$98$		0			0
$99$			1.17258e8i			1.22068i
$100$		0			0
$101$	−1.07960e8			−1.03747			−0.518736	−	0.854934i	$-0.673597\pi$
$101$	−1.07960e8			−1.03747			−0.518736	+	0.854934i	$0.673597\pi$
$102$		0			0
$103$	−9.62475e7			−0.855147			−0.427573	−	0.903981i	$-0.640631\pi$
$103$	−9.62475e7			−0.855147			−0.427573	+	0.903981i	$0.640631\pi$
$104$		0			0
$105$	−2.07905e7	−	1.02759e7i	−0.171044	−	0.0845401i
$106$		0			0
$107$	−8.83513e7			−0.674028			−0.337014	−	0.941500i	$-0.609417\pi$
$107$	−8.83513e7			−0.674028			−0.337014	+	0.941500i	$0.609417\pi$
$108$		0			0
$109$	−4.82155e7			−0.341571			−0.170785	−	0.985308i	$-0.554630\pi$
$109$	−4.82155e7			−0.341571			−0.170785	+	0.985308i	$0.554630\pi$
$110$		0			0
$111$		−	7.34724e7i		−	0.483986i
$112$		0			0
$113$			1.83862e8i			1.12766i	0.825891	+	0.563830i	$0.190672\pi$
$113$			1.83862e8i			1.12766i	−0.825891	+	0.563830i	$0.809328\pi$
$114$		0			0
$115$	−2.40121e8	−	1.18682e8i	−1.37290	−	0.678568i
$116$		0			0
$117$			1.94973e8i			1.04048i
$118$		0			0
$119$		−	3.55707e7i		−	0.177380i
$120$		0			0
$121$	−3.97461e8			−1.85419
$122$		0			0
$123$	8.56399e7			0.374158
$124$		0			0
$125$	4.69862e7	+	2.39577e8i	0.192455	+	0.981306i
$126$		0			0
$127$	3.52941e8			1.35671			0.678355	−	0.734734i	$-0.262693\pi$
$127$	3.52941e8			1.35671			0.678355	+	0.734734i	$0.262693\pi$
$128$		0			0
$129$	6.10437e7			0.220436
$130$		0			0
$131$		−	1.20638e8i		−	0.409638i	−0.978800	−	0.204819i	$-0.934339\pi$
$131$		−	1.20638e8i		−	0.409638i	0.978800	−	0.204819i	$-0.0656606\pi$
$132$		0			0
$133$		−	6.91419e7i		−	0.220971i
$134$		0			0
$135$	2.70174e8	+	1.33536e8i	0.813407	+	0.402034i
$136$		0			0
$137$			2.98009e8i			0.845955i	0.906140	+	0.422978i	$0.139015\pi$
$137$			2.98009e8i			0.845955i	−0.906140	+	0.422978i	$0.860985\pi$
$138$		0			0
$139$			4.00098e8i			1.07178i	0.844287	+	0.535891i	$0.180024\pi$
$139$			4.00098e8i			1.07178i	−0.844287	+	0.535891i	$0.819976\pi$
$140$		0			0
$141$	−2.12689e8			−0.538107
$142$		0			0
$143$	−1.01731e9			−2.43282
$144$		0			0
$145$	−1.03509e8	−	5.11603e7i	−0.234157	−	0.115734i
$146$		0			0
$147$	2.13692e8			0.457634
$148$		0			0
$149$	−7.70799e8			−1.56385			−0.781927	−	0.623370i	$-0.785763\pi$
$149$	−7.70799e8			−1.56385			−0.781927	+	0.623370i	$0.785763\pi$
$150$		0			0
$151$			5.11492e8i			0.983856i	0.870636	+	0.491928i	$0.163708\pi$
$151$			5.11492e8i			0.983856i	−0.870636	+	0.491928i	$0.836292\pi$
$152$		0			0
$153$			1.93893e8i			0.353832i
$154$		0			0
$155$	−4.28557e7	+	8.67070e7i	−0.0742475	+	0.150220i
$156$		0			0
$157$		−	8.51365e7i		−	0.140126i	−0.997543	−	0.0700628i	$-0.977680\pi$
$157$		−	8.51365e7i		−	0.140126i	0.997543	−	0.0700628i	$-0.0223200\pi$
$158$		0			0
$159$		−	2.97657e8i		−	0.465723i
$160$		0			0
$161$	−3.72712e8			−0.554715
$162$		0			0
$163$	2.79284e8			0.395636			0.197818	−	0.980239i	$-0.436614\pi$
$163$	2.79284e8			0.395636			0.197818	+	0.980239i	$0.436614\pi$
$164$		0			0
$165$	−2.92260e8	+	5.91310e8i	−0.394307	+	0.797773i
$166$		0			0
$167$	−4.30699e8			−0.553743			−0.276871	−	0.960907i	$-0.589298\pi$
$167$	−4.30699e8			−0.553743			−0.276871	+	0.960907i	$0.589298\pi$
$168$		0			0
$169$	−8.75822e8			−1.07367
$170$		0			0
$171$			3.76887e8i			0.440785i
$172$		0			0
$173$		−	6.81243e8i		−	0.760533i	−0.924877	−	0.380266i	$-0.875832\pi$
$173$		−	6.81243e8i		−	0.760533i	0.924877	−	0.380266i	$-0.124168\pi$
$174$		0			0
$175$	2.06326e8	+	2.69888e8i	0.219989	+	0.287761i
$176$		0			0
$177$			2.41150e8i			0.245693i
$178$		0			0
$179$		−	1.86210e9i		−	1.81381i	−0.421336	−	0.906905i	$-0.638439\pi$
$179$		−	1.86210e9i		−	1.81381i	0.421336	−	0.906905i	$-0.361561\pi$
$180$		0			0
$181$	−9.45023e8			−0.880498			−0.440249	−	0.897876i	$-0.645110\pi$
$181$	−9.45023e8			−0.880498			−0.440249	+	0.897876i	$0.645110\pi$
$182$		0			0
$183$	2.87448e8			0.256304
$184$		0			0
$185$	−4.76885e8	+	9.64848e8i	−0.407124	+	0.823705i
$186$		0			0
$187$	−1.01168e9			−0.827324
$188$		0			0
$189$	4.19359e8			0.328654
$190$		0			0
$191$		−	6.13867e8i		−	0.461255i	−0.973042	−	0.230627i	$-0.925922\pi$
$191$		−	6.13867e8i		−	0.461255i	0.973042	−	0.230627i	$-0.0740778\pi$
$192$		0			0
$193$		−	7.45004e8i		−	0.536945i	−0.963287	−	0.268472i	$-0.913481\pi$
$193$		−	7.45004e8i		−	0.536945i	0.963287	−	0.268472i	$-0.0865188\pi$
$194$		0			0
$195$	−4.85960e8	+	9.83210e8i	−0.336095	+	0.679998i
$196$		0			0
$197$			2.47398e9i			1.64260i	0.570498	+	0.821299i	$0.306750\pi$
$197$			2.47398e9i			1.64260i	−0.570498	+	0.821299i	$0.693250\pi$
$198$		0			0
$199$		−	8.62436e8i		−	0.549939i	−0.961453	−	0.274969i	$-0.911332\pi$
$199$		−	8.62436e8i		−	0.549939i	0.961453	−	0.274969i	$-0.0886677\pi$
$200$		0			0
$201$	−2.13130e8			−0.130575
$202$		0			0
$203$	−1.60665e8			−0.0946102
$204$		0			0
$205$	−1.12463e9	−	5.55859e8i	−0.636788	−	0.314738i
$206$		0			0
$207$	2.03163e9			1.10653
$208$		0			0
$209$	−1.96649e9			−1.03064
$210$		0			0
$211$			2.06407e9i			1.04135i	0.853756	+	0.520673i	$0.174319\pi$
$211$			2.06407e9i			1.04135i	−0.853756	+	0.520673i	$0.825681\pi$
$212$		0			0
$213$			8.86219e8i			0.430549i
$214$		0			0
$215$	−8.01633e8	−	3.96214e8i	−0.375164	−	0.185428i
$216$		0			0
$217$			1.34585e8i			0.0606958i
$218$		0			0
$219$		−	1.41555e9i		−	0.615386i
$220$		0			0
$221$	−1.68218e9			−0.705186
$222$		0			0
$223$	2.49560e9			1.00915			0.504574	−	0.863368i	$-0.331649\pi$
$223$	2.49560e9			1.00915			0.504574	+	0.863368i	$0.331649\pi$
$224$		0			0
$225$	−1.12467e9	−	1.47114e9i	−0.438828	−	0.574017i
$226$		0			0
$227$	4.05348e9			1.52660			0.763300	−	0.646045i	$-0.223578\pi$
$227$	4.05348e9			1.52660			0.763300	+	0.646045i	$0.223578\pi$
$228$		0			0
$229$	−2.28775e9			−0.831890			−0.415945	−	0.909390i	$-0.636549\pi$
$229$	−2.28775e9			−0.831890			−0.415945	+	0.909390i	$0.636549\pi$
$230$		0			0
$231$			9.17822e8i			0.322337i
$232$		0			0
$233$			3.13148e9i			1.06249i	0.847217	+	0.531247i	$0.178276\pi$
$233$			3.13148e9i			1.06249i	−0.847217	+	0.531247i	$0.821724\pi$
$234$		0			0
$235$	2.79305e9	+	1.38049e9i	0.915815	+	0.452650i
$236$		0			0
$237$		−	3.19760e9i		−	1.01352i
$238$		0			0
$239$			1.66159e9i			0.509251i	0.967040	+	0.254626i	$0.0819523\pi$
$239$			1.66159e9i			0.509251i	−0.967040	+	0.254626i	$0.918048\pi$
$240$		0			0
$241$	−1.60357e9			−0.475358			−0.237679	−	0.971344i	$-0.576387\pi$
$241$	−1.60357e9			−0.475358			−0.237679	+	0.971344i	$0.576387\pi$
$242$		0			0
$243$	−3.61295e9			−1.03618
$244$		0			0
$245$	−2.80622e9	−	1.38700e9i	−0.778857	−	0.384957i
$246$		0			0
$247$	−3.26980e9			−0.878484
$248$		0			0
$249$	−3.73477e9			−0.971554
$250$		0			0
$251$		−	6.18770e9i		−	1.55896i	−0.626429	−	0.779478i	$-0.715484\pi$
$251$		−	6.18770e9i		−	1.55896i	0.626429	−	0.779478i	$-0.284516\pi$
$252$		0			0
$253$			1.06004e10i			2.58727i
$254$		0			0
$255$	−4.83268e8	+	9.77763e8i	−0.114295	+	0.231245i
$256$		0			0
$257$			6.20387e9i			1.42210i	0.703142	+	0.711050i	$0.251780\pi$
$257$			6.20387e9i			1.42210i	−0.703142	+	0.711050i	$0.748220\pi$
$258$		0			0
$259$			1.49762e9i			0.332815i
$260$		0			0
$261$	8.75775e8			0.188725
$262$		0			0
$263$	−1.96327e9			−0.410353			−0.205177	−	0.978725i	$-0.565777\pi$
$263$	−1.96327e9			−0.410353			−0.205177	+	0.978725i	$0.565777\pi$
$264$		0			0
$265$	−1.93199e9	+	3.90886e9i	−0.391761	+	0.792623i
$266$		0			0
$267$	4.89183e8			0.0962557
$268$		0			0
$269$	−6.31601e9			−1.20624			−0.603120	−	0.797651i	$-0.706076\pi$
$269$	−6.31601e9			−1.20624			−0.603120	+	0.797651i	$0.706076\pi$
$270$		0			0
$271$		−	4.71258e9i		−	0.873739i	−0.899525	−	0.436869i	$-0.856087\pi$
$271$		−	4.71258e9i		−	0.873739i	0.899525	−	0.436869i	$-0.143913\pi$
$272$		0			0
$273$			1.52612e9i			0.274751i
$274$		0			0
$275$	7.67599e9	−	5.86818e9i	1.34216	−	1.02606i
$276$		0			0
$277$			1.00765e8i			0.0171156i	0.999963	+	0.00855780i	$0.00272406\pi$
$277$			1.00765e8i			0.0171156i	−0.999963	+	0.00855780i	$0.997276\pi$
$278$		0			0
$279$		−	7.33614e8i		−	0.121074i
$280$		0			0
$281$	−4.26355e9			−0.683827			−0.341914	−	0.939731i	$-0.611075\pi$
$281$	−4.26355e9			−0.683827			−0.341914	+	0.939731i	$0.611075\pi$
$282$		0			0
$283$	−1.16260e10			−1.81253			−0.906265	−	0.422710i	$-0.861079\pi$
$283$	−1.16260e10			−1.81253			−0.906265	+	0.422710i	$0.861079\pi$
$284$		0			0
$285$	−9.39371e8	+	1.90056e9i	−0.142383	+	0.288073i
$286$		0			0
$287$	−1.74563e9			−0.257292
$288$		0			0
$289$	5.30289e9			0.760189
$290$		0			0
$291$			3.73176e9i			0.520405i
$292$		0			0
$293$		−	3.75705e9i		−	0.509772i	−0.966971	−	0.254886i	$-0.917962\pi$
$293$		−	3.75705e9i		−	0.509772i	0.966971	−	0.254886i	$-0.0820380\pi$
$294$		0			0
$295$	1.56522e9	−	3.16680e9i	0.206675	−	0.418151i
$296$		0			0
$297$		−	1.19271e10i		−	1.53289i
$298$		0			0
$299$			1.76260e10i			2.20531i
$300$		0			0
$301$	−1.24428e9			−0.151584
$302$		0			0
$303$	4.60624e9			0.546483
$304$		0			0
$305$	−3.77480e9	−	1.86573e9i	−0.436209	−	0.215600i
$306$		0			0
$307$	−5.80347e9			−0.653333			−0.326666	−	0.945140i	$-0.605925\pi$
$307$	−5.80347e9			−0.653333			−0.326666	+	0.945140i	$0.605925\pi$
$308$		0			0
$309$	4.10652e9			0.450443
$310$		0			0
$311$			1.22144e10i			1.30566i	0.757503	+	0.652832i	$0.226419\pi$
$311$			1.22144e10i			1.30566i	−0.757503	+	0.652832i	$0.773581\pi$
$312$		0			0
$313$		−	8.99814e9i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$313$		−	8.99814e9i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$314$		0			0
$315$	−2.31001e9	−	1.14174e9i	−0.234623	−	0.115965i
$316$		0			0
$317$			2.25206e9i			0.223019i	0.993763	+	0.111510i	$0.0355686\pi$
$317$			2.25206e9i			0.223019i	−0.993763	+	0.111510i	$0.964431\pi$
$318$		0			0
$319$			4.56953e9i			0.441275i
$320$		0			0
$321$	3.76962e9			0.355040
$322$		0			0
$323$	−3.25169e9			−0.298744
$324$		0			0
$325$	1.27634e10	−	9.75741e9i	1.14401	−	0.874583i
$326$		0			0
$327$	2.05718e9			0.179920
$328$		0			0
$329$	4.33533e9			0.370032
$330$		0			0
$331$		−	1.68572e9i		−	0.140434i	−0.997532	−	0.0702172i	$-0.977631\pi$
$331$		−	1.68572e9i		−	0.140434i	0.997532	−	0.0702172i	$-0.0223692\pi$
$332$		0			0
$333$		−	8.16342e9i		−	0.663889i
$334$		0			0
$335$	2.79885e9	+	1.38336e9i	0.222229	+	0.109839i
$336$		0			0
$337$			7.73250e9i			0.599515i	0.954015	+	0.299758i	$0.0969058\pi$
$337$			7.73250e9i			0.599515i	−0.954015	+	0.299758i	$0.903094\pi$
$338$		0			0
$339$		−	7.84470e9i		−	0.593988i
$340$		0			0
$341$	3.82778e9			0.283093
$342$		0			0
$343$	−9.36933e9			−0.676912
$344$		0			0
$345$	1.02451e10	+	5.06372e9i	0.723167	+	0.357432i
$346$		0			0
$347$	3.07433e9			0.212047			0.106023	−	0.994364i	$-0.466188\pi$
$347$	3.07433e9			0.212047			0.106023	+	0.994364i	$0.466188\pi$
$348$		0			0
$349$	1.30882e10			0.882224			0.441112	−	0.897452i	$-0.354584\pi$
$349$	1.30882e10			0.882224			0.441112	+	0.897452i	$0.354584\pi$
$350$		0			0
$351$		−	1.98320e10i		−	1.30659i
$352$		0			0
$353$		−	6.61462e9i		−	0.425996i	−0.977053	−	0.212998i	$-0.931677\pi$
$353$		−	6.61462e9i		−	0.425996i	0.977053	−	0.212998i	$-0.0683229\pi$
$354$		0			0
$355$	5.75215e9	−	1.16379e10i	0.362173	−	0.732760i
$356$		0			0
$357$			1.51767e9i			0.0934338i
$358$		0			0
$359$			3.34086e9i			0.201132i	0.994930	+	0.100566i	$0.0320653\pi$
$359$			3.34086e9i			0.201132i	−0.994930	+	0.100566i	$0.967935\pi$
$360$		0			0
$361$	1.06630e10			0.627840
$362$		0			0
$363$	1.69582e10			0.976681
$364$		0			0
$365$	−9.18783e9	+	1.85891e10i	−0.517656	+	1.04734i
$366$		0			0
$367$	−9.60705e9			−0.529573			−0.264787	−	0.964307i	$-0.585302\pi$
$367$	−9.60705e9			−0.529573			−0.264787	+	0.964307i	$0.585302\pi$
$368$		0			0
$369$	9.51533e9			0.513237
$370$		0			0
$371$			6.06727e9i			0.320256i
$372$		0			0
$373$		−	1.82578e10i		−	0.943222i	−0.881807	−	0.471611i	$-0.843673\pi$
$373$		−	1.82578e10i		−	0.943222i	0.881807	−	0.471611i	$-0.156327\pi$
$374$		0			0
$375$	−2.00472e9	−	1.02218e10i	−0.101375	−	0.516897i
$376$		0			0
$377$			7.59806e9i			0.376129i
$378$		0			0
$379$		−	2.08206e10i		−	1.00910i	−0.863382	−	0.504551i	$-0.831658\pi$
$379$		−	2.08206e10i		−	1.00910i	0.863382	−	0.504551i	$-0.168342\pi$
$380$		0			0
$381$	−1.50587e10			−0.714639
$382$		0			0
$383$	−2.29859e10			−1.06824			−0.534118	−	0.845410i	$-0.679356\pi$
$383$	−2.29859e10			−1.06824			−0.534118	+	0.845410i	$0.679356\pi$
$384$		0			0
$385$	5.95727e9	−	1.20529e10i	0.271147	−	0.548592i
$386$		0			0
$387$	6.78249e9			0.302374
$388$		0			0
$389$	1.60076e10			0.699083			0.349541	−	0.936921i	$-0.386337\pi$
$389$	1.60076e10			0.699083			0.349541	+	0.936921i	$0.386337\pi$
$390$		0			0
$391$			1.75284e10i			0.749954i
$392$		0			0
$393$			5.14719e9i			0.215774i
$394$		0			0
$395$	−2.07545e10	+	4.19912e10i	−0.852560	+	1.72492i
$396$		0			0
$397$		−	1.72716e10i		−	0.695299i	−0.937625	−	0.347650i	$-0.886980\pi$
$397$		−	1.72716e10i		−	0.695299i	0.937625	−	0.347650i	$-0.113020\pi$
$398$		0			0
$399$			2.95002e9i			0.116395i
$400$		0			0
$401$	−1.56829e10			−0.606524			−0.303262	−	0.952907i	$-0.598076\pi$
$401$	−1.56829e10			−0.606524			−0.303262	+	0.952907i	$0.598076\pi$
$402$		0			0
$403$	6.36470e9			0.241300
$404$		0			0
$405$	5.89965e9	+	2.91595e9i	0.219284	+	0.108383i
$406$		0			0
$407$	4.25943e10			1.55229
$408$		0			0
$409$	−1.05519e8			−0.00377082			−0.00188541	−	0.999998i	$-0.500600\pi$
$409$	−1.05519e8			−0.00377082			−0.00188541	+	0.999998i	$0.500600\pi$
$410$		0			0
$411$		−	1.27149e10i		−	0.445602i
$412$		0			0
$413$		−	4.91546e9i		−	0.168952i
$414$		0			0
$415$	4.90454e10	+	2.42412e10i	1.65351	+	0.817261i
$416$		0			0
$417$		−	1.70707e10i		−	0.564555i
$418$		0			0
$419$		−	1.28448e10i		−	0.416746i	−0.978049	−	0.208373i	$-0.933183\pi$
$419$		−	1.28448e10i		−	0.416746i	0.978049	−	0.208373i	$-0.0668168\pi$
$420$		0			0
$421$	3.08274e10			0.981315			0.490658	−	0.871353i	$-0.336757\pi$
$421$	3.08274e10			0.981315			0.490658	+	0.871353i	$0.336757\pi$
$422$		0			0
$423$	−2.36316e10			−0.738127
$424$		0			0
$425$	1.26927e10	−	9.70336e9i	0.389042	−	0.297417i
$426$		0			0
$427$	−5.85918e9			−0.176249
$428$		0			0
$429$	4.34049e10			1.28147
$430$		0			0
$431$		−	9.59665e9i		−	0.278106i	−0.990285	−	0.139053i	$-0.955594\pi$
$431$		−	9.59665e9i		−	0.278106i	0.990285	−	0.139053i	$-0.0444058\pi$
$432$		0			0
$433$			6.69168e8i			0.0190363i	0.999955	+	0.00951817i	$0.00302977\pi$
$433$			6.69168e8i			0.0190363i	−0.999955	+	0.00951817i	$0.996970\pi$
$434$		0			0
$435$	4.41635e9	+	2.18282e9i	0.123341	+	0.0609622i
$436$		0			0
$437$			3.40714e10i			0.934254i
$438$		0			0
$439$		−	3.75455e10i		−	1.01088i	−0.862862	−	0.505440i	$-0.831330\pi$
$439$		−	3.75455e10i		−	1.01088i	0.862862	−	0.505440i	$-0.168670\pi$
$440$		0			0
$441$	2.37430e10			0.627742
$442$		0			0
$443$	−1.19986e10			−0.311542			−0.155771	−	0.987793i	$-0.549786\pi$
$443$	−1.19986e10			−0.311542			−0.155771	+	0.987793i	$0.549786\pi$
$444$		0			0
$445$	−6.42400e9	−	3.17512e9i	−0.163819	−	0.0809693i
$446$		0			0
$447$	3.28871e10			0.823751
$448$		0			0
$449$	6.86153e10			1.68825			0.844123	−	0.536150i	$-0.180122\pi$
$449$	6.86153e10			1.68825			0.844123	+	0.536150i	$0.180122\pi$
$450$		0			0
$451$			4.96482e10i			1.20004i
$452$		0			0
$453$		−	2.18235e10i		−	0.518240i
$454$		0			0
$455$	9.90554e9	−	2.00412e10i	0.231117	−	0.467604i
$456$		0			0
$457$		−	5.08092e10i		−	1.16487i	−0.812877	−	0.582435i	$-0.802100\pi$
$457$		−	5.08092e10i		−	1.16487i	0.812877	−	0.582435i	$-0.197900\pi$
$458$		0			0
$459$		−	1.97222e10i		−	0.444328i
$460$		0			0
$461$	3.46473e10			0.767124			0.383562	−	0.923515i	$-0.374697\pi$
$461$	3.46473e10			0.767124			0.383562	+	0.923515i	$0.374697\pi$
$462$		0			0
$463$	−4.92524e10			−1.07178			−0.535888	−	0.844289i	$-0.680023\pi$
$463$	−4.92524e10			−1.07178			−0.535888	+	0.844289i	$0.680023\pi$
$464$		0			0
$465$	1.82849e9	−	3.69946e9i	0.0391094	−	0.0791274i
$466$		0			0
$467$	−7.83687e9			−0.164769			−0.0823844	−	0.996601i	$-0.526254\pi$
$467$	−7.83687e9			−0.164769			−0.0823844	+	0.996601i	$0.526254\pi$
$468$		0			0
$469$	4.34433e9			0.0897907
$470$		0			0
$471$			3.63246e9i			0.0738103i
$472$		0			0
$473$			3.53890e10i			0.707007i
$474$		0			0
$475$	2.46718e10	−	1.88613e10i	0.484649	−	0.370507i
$476$		0			0
$477$		−	3.30723e10i		−	0.638838i
$478$		0			0
$479$		−	2.94112e10i		−	0.558689i	−0.960191	−	0.279345i	$-0.909883\pi$
$479$		−	2.94112e10i		−	0.558689i	0.960191	−	0.279345i	$-0.0901172\pi$
$480$		0			0
$481$	7.08244e10			1.32313
$482$		0			0
$483$	1.59022e10			0.292193
$484$		0			0
$485$	2.42216e10	−	4.90059e10i	0.437760	−	0.885689i
$486$		0			0
$487$	−5.85444e10			−1.04080			−0.520402	−	0.853921i	$-0.674218\pi$
$487$	−5.85444e10			−1.04080			−0.520402	+	0.853921i	$0.674218\pi$
$488$		0			0
$489$	−1.19160e10			−0.208399
$490$		0			0
$491$		−	8.05466e10i		−	1.38587i	−0.721002	−	0.692933i	$-0.756318\pi$
$491$		−	8.05466e10i		−	1.38587i	0.721002	−	0.692933i	$-0.243682\pi$
$492$		0			0
$493$			7.55597e9i			0.127909i
$494$		0			0
$495$	−3.24726e10	+	6.56997e10i	−0.540875	+	1.09431i
$496$		0			0
$497$		−	1.80642e10i		−	0.296069i
$498$		0			0
$499$			2.25270e10i			0.363330i	0.983360	+	0.181665i	$0.0581487\pi$
$499$			2.25270e10i			0.363330i	−0.983360	+	0.181665i	$0.941851\pi$
$500$		0			0
$501$	1.83763e10			0.291681
$502$		0			0
$503$	4.17996e10			0.652981			0.326491	−	0.945200i	$-0.394134\pi$
$503$	4.17996e10			0.652981			0.326491	+	0.945200i	$0.394134\pi$
$504$		0			0
$505$	−6.04897e10	−	2.98976e10i	−0.930070	−	0.459695i
$506$		0			0
$507$	3.73680e10			0.565547
$508$		0			0
$509$	−1.06239e10			−0.158276			−0.0791378	−	0.996864i	$-0.525217\pi$
$509$	−1.06239e10			−0.158276			−0.0791378	+	0.996864i	$0.525217\pi$
$510$		0			0
$511$			2.88537e10i			0.423173i
$512$		0			0
$513$		−	3.83357e10i		−	0.553521i
$514$		0			0
$515$	−5.39273e10	−	2.66540e10i	−0.766619	−	0.378908i
$516$		0			0
$517$		−	1.23303e11i		−	1.72588i
$518$		0			0
$519$			2.90661e10i			0.400606i
$520$		0			0
$521$	−6.29819e10			−0.854800			−0.427400	−	0.904063i	$-0.640570\pi$
$521$	−6.29819e10			−0.854800			−0.427400	+	0.904063i	$0.640570\pi$
$522$		0			0
$523$	2.56642e10			0.343021			0.171511	−	0.985182i	$-0.445135\pi$
$523$	2.56642e10			0.343021			0.171511	+	0.985182i	$0.445135\pi$
$524$		0			0
$525$	−8.80316e9	−	1.15151e10i	−0.115878	−	0.151576i
$526$		0			0
$527$	6.32945e9			0.0820584
$528$		0			0
$529$	1.05353e11			1.34531
$530$		0			0
$531$			2.67938e10i			0.337021i
$532$		0			0
$533$			8.25532e10i			1.02288i
$534$		0			0
$535$	−4.95031e10	−	2.44673e10i	−0.604250	−	0.298656i
$536$		0			0
$537$			7.94490e10i			0.955413i
$538$		0			0
$539$			1.23884e11i			1.46778i
$540$		0			0
$541$	−7.25189e10			−0.846569			−0.423285	−	0.905997i	$-0.639123\pi$
$541$	−7.25189e10			−0.846569			−0.423285	+	0.905997i	$0.639123\pi$
$542$		0			0
$543$	4.03206e10			0.463797
$544$		0			0
$545$	−2.70150e10	−	1.33524e10i	−0.306210	−	0.151347i
$546$		0			0
$547$	2.29110e10			0.255914			0.127957	−	0.991780i	$-0.459158\pi$
$547$	2.29110e10			0.255914			0.127957	+	0.991780i	$0.459158\pi$
$548$		0			0
$549$	3.19380e10			0.351575
$550$		0			0
$551$			1.46872e10i			0.159343i
$552$		0			0
$553$			6.51781e10i			0.696949i
$554$		0			0
$555$	2.03469e10	−	4.11664e10i	0.214450	−	0.433882i
$556$		0			0
$557$			1.28630e11i			1.33636i	0.744001	+	0.668178i	$0.232926\pi$
$557$			1.28630e11i			1.33636i	−0.744001	+	0.668178i	$0.767074\pi$
$558$		0			0
$559$			5.88436e10i			0.602632i
$560$		0			0
$561$	4.31645e10			0.435788
$562$		0			0
$563$	−1.71738e11			−1.70936			−0.854678	−	0.519159i	$-0.826245\pi$
$563$	−1.71738e11			−1.70936			−0.854678	+	0.519159i	$0.826245\pi$
$564$		0			0
$565$	−5.09173e10	+	1.03017e11i	−0.499656	+	1.01092i
$566$		0			0
$567$	9.15734e9			0.0886007
$568$		0			0
$569$	−8.20809e10			−0.783056			−0.391528	−	0.920166i	$-0.628053\pi$
$569$	−8.20809e10			−0.783056			−0.391528	+	0.920166i	$0.628053\pi$
$570$		0			0
$571$			1.07462e11i			1.01090i	0.862855	+	0.505452i	$0.168674\pi$
$571$			1.07462e11i			1.01090i	−0.862855	+	0.505452i	$0.831326\pi$
$572$		0			0
$573$			2.61914e10i			0.242963i
$574$		0			0
$575$	−1.01672e11	−	1.32995e11i	−0.930105	−	1.21664i
$576$		0			0
$577$		−	3.37727e10i		−	0.304693i	−0.988327	−	0.152346i	$-0.951317\pi$
$577$		−	3.37727e10i		−	0.304693i	0.988327	−	0.152346i	$-0.0486830\pi$
$578$		0			0
$579$			3.17866e10i			0.282832i
$580$		0			0
$581$	7.61276e10			0.668094
$582$		0			0
$583$	1.72561e11			1.49372
$584$		0			0
$585$	−5.39944e10	+	1.09243e11i	−0.461026	+	0.932761i
$586$		0			0
$587$	−1.01250e11			−0.852793			−0.426396	−	0.904536i	$-0.640217\pi$
$587$	−1.01250e11			−0.852793			−0.426396	+	0.904536i	$0.640217\pi$
$588$		0			0
$589$	1.23031e10			0.102224
$590$		0			0
$591$		−	1.05555e11i		−	0.865229i
$592$		0			0
$593$		−	1.86134e11i		−	1.50525i	−0.658451	−	0.752624i	$-0.728788\pi$
$593$		−	1.86134e11i		−	1.50525i	0.658451	−	0.752624i	$-0.271212\pi$
$594$		0			0
$595$	9.85067e9	−	1.99302e10i	0.0785955	−	0.159017i
$596$		0			0
$597$			3.67969e10i			0.289677i
$598$		0			0
$599$			7.77748e10i			0.604131i	0.953287	+	0.302066i	$0.0976762\pi$
$599$			7.77748e10i			0.604131i	−0.953287	+	0.302066i	$0.902324\pi$
$600$		0			0
$601$	−1.76175e11			−1.35035			−0.675176	−	0.737657i	$-0.735932\pi$
$601$	−1.76175e11			−1.35035			−0.675176	+	0.737657i	$0.735932\pi$
$602$		0			0
$603$	−2.36806e10			−0.179112
$604$		0			0
$605$	−2.22697e11	−	1.10070e11i	−1.66223	−	0.821574i
$606$		0			0
$607$	−1.04822e11			−0.772143			−0.386071	−	0.922469i	$-0.626168\pi$
$607$	−1.04822e11			−0.772143			−0.386071	+	0.922469i	$0.626168\pi$
$608$		0			0
$609$	6.85499e9			0.0498353
$610$		0			0
$611$		−	2.05023e11i		−	1.47109i
$612$		0			0
$613$		−	1.64383e10i		−	0.116417i	−0.998304	−	0.0582085i	$-0.981461\pi$
$613$		−	1.64383e10i		−	0.116417i	0.998304	−	0.0582085i	$-0.0185388\pi$
$614$		0			0
$615$	4.79838e10	+	2.37164e10i	0.335424	+	0.165786i
$616$		0			0
$617$			2.58887e11i			1.78636i	0.449698	+	0.893181i	$0.351532\pi$
$617$			2.58887e11i			1.78636i	−0.449698	+	0.893181i	$0.648468\pi$
$618$		0			0
$619$		−	1.40280e11i		−	0.955508i	−0.878494	−	0.477754i	$-0.841451\pi$
$619$		−	1.40280e11i		−	0.955508i	0.878494	−	0.477754i	$-0.158549\pi$
$620$		0			0
$621$	−2.06650e11			−1.38953
$622$		0			0
$623$	−9.97123e9			−0.0661906
$624$		0			0
$625$	−4.00202e10	+	1.47246e11i	−0.262277	+	0.964993i
$626$		0			0
$627$	8.39026e10			0.542882
$628$		0			0
$629$	7.04320e10			0.449953
$630$		0			0
$631$		−	3.75788e10i		−	0.237042i	−0.992952	−	0.118521i	$-0.962185\pi$
$631$		−	3.75788e10i		−	0.237042i	0.992952	−	0.118521i	$-0.0378153\pi$
$632$		0			0
$633$		−	8.80662e10i		−	0.548523i
$634$		0			0
$635$	1.97752e11	+	9.77406e10i	1.21626	+	0.601146i
$636$		0			0
$637$			2.05990e11i			1.25109i
$638$		0			0
$639$			9.84666e10i			0.590589i
$640$		0			0
$641$	−9.09734e10			−0.538868			−0.269434	−	0.963019i	$-0.586837\pi$
$641$	−9.09734e10			−0.538868			−0.269434	+	0.963019i	$0.586837\pi$
$642$		0			0
$643$	−3.82679e10			−0.223868			−0.111934	−	0.993716i	$-0.535704\pi$
$643$	−3.82679e10			−0.223868			−0.111934	+	0.993716i	$0.535704\pi$
$644$		0			0
$645$	3.42027e10	+	1.69050e10i	0.197616	+	0.0976733i
$646$		0			0
$647$	−1.00170e11			−0.571635			−0.285818	−	0.958284i	$-0.592265\pi$
$647$	−1.00170e11			−0.571635			−0.285818	+	0.958284i	$0.592265\pi$
$648$		0			0
$649$	−1.39802e11			−0.788016
$650$		0			0
$651$		−	5.74225e9i		−	0.0319711i
$652$		0			0
$653$		−	1.26162e11i		−	0.693869i	−0.937889	−	0.346934i	$-0.887223\pi$
$653$		−	1.26162e11i		−	0.693869i	0.937889	−	0.346934i	$-0.112777\pi$
$654$		0			0
$655$	3.34087e10	−	6.75934e10i	0.181507	−	0.367231i
$656$		0			0
$657$		−	1.57279e11i		−	0.844132i
$658$		0			0
$659$			1.89374e9i			0.0100410i	0.999987	+	0.00502051i	$0.00159809\pi$
$659$			1.89374e9i			0.0100410i	−0.999987	+	0.00502051i	$0.998402\pi$
$660$		0			0
$661$	−2.06198e11			−1.08014			−0.540068	−	0.841622i	$-0.681601\pi$
$661$	−2.06198e11			−1.08014			−0.540068	+	0.841622i	$0.681601\pi$
$662$		0			0
$663$	7.17724e10			0.371453
$664$		0			0
$665$	1.91476e10	−	3.87400e10i	0.0979102	−	0.198095i
$666$		0			0
$667$	7.91720e10			0.400008
$668$		0			0
$669$	−1.06478e11			−0.531563
$670$		0			0
$671$			1.66643e11i			0.822047i
$672$		0			0
$673$			2.75305e11i			1.34200i	0.741455	+	0.671002i	$0.234136\pi$
$673$			2.75305e11i			1.34200i	−0.741455	+	0.671002i	$0.765864\pi$
$674$		0			0
$675$	1.14397e11	+	1.49640e11i	0.551063	+	0.720827i
$676$		0			0
$677$		−	2.05068e11i		−	0.976208i	−0.872785	−	0.488104i	$-0.837689\pi$
$677$		−	2.05068e11i		−	0.976208i	0.872785	−	0.488104i	$-0.162311\pi$
$678$		0			0
$679$		−	7.60661e10i		−	0.357859i
$680$		0			0
$681$	−1.72947e11			−0.804127
$682$		0			0
$683$	3.92855e11			1.80530			0.902649	−	0.430377i	$-0.141620\pi$
$683$	3.92855e11			1.80530			0.902649	+	0.430377i	$0.141620\pi$
$684$		0			0
$685$	−8.25284e10	+	1.66974e11i	−0.374836	+	0.758379i
$686$		0			0
$687$	9.76096e10			0.438193
$688$		0			0
$689$	2.86929e11			1.27320
$690$		0			0
$691$			1.92324e11i			0.843570i	0.906696	+	0.421785i	$0.138596\pi$
$691$			1.92324e11i			0.843570i	−0.906696	+	0.421785i	$0.861404\pi$
$692$		0			0
$693$			1.01978e11i			0.442154i
$694$		0			0
$695$	−1.10800e11	+	2.24174e11i	−0.474898	+	0.960828i
$696$		0			0
$697$			8.20959e10i			0.347849i
$698$		0			0
$699$		−	1.33609e11i		−	0.559662i
$700$		0			0
$701$	−2.51795e11			−1.04274			−0.521369	−	0.853331i	$-0.674579\pi$
$701$	−2.51795e11			−1.04274			−0.521369	+	0.853331i	$0.674579\pi$
$702$		0			0
$703$	1.36905e11			0.560528
$704$		0			0
$705$	−1.19169e11	−	5.89004e10i	−0.482400	−	0.238431i
$706$		0			0
$707$	−9.38911e10			−0.375791
$708$		0			0
$709$	4.10364e11			1.62400			0.811998	−	0.583661i	$-0.198380\pi$
$709$	4.10364e11			1.62400			0.811998	+	0.583661i	$0.198380\pi$
$710$		0			0
$711$		−	3.55281e11i		−	1.39025i
$712$		0			0
$713$		−	6.63204e10i		−	0.256619i
$714$		0			0
$715$	−5.69998e11	−	2.81727e11i	−2.18097	−	1.07796i
$716$		0			0
$717$		−	7.08938e10i		−	0.268245i
$718$		0			0
$719$		−	2.36375e11i		−	0.884474i	−0.896898	−	0.442237i	$-0.854185\pi$
$719$		−	2.36375e11i		−	0.884474i	0.896898	−	0.442237i	$-0.145815\pi$
$720$		0			0
$721$	−8.37050e10			−0.309749
$722$		0			0
$723$	6.84185e10			0.250392
$724$		0			0
$725$	−4.38280e10	−	5.73300e10i	−0.158635	−	0.207506i
$726$		0			0
$727$	1.16942e11			0.418631			0.209316	−	0.977848i	$-0.432876\pi$
$727$	1.16942e11			0.418631			0.209316	+	0.977848i	$0.432876\pi$
$728$		0			0
$729$	8.50669e10			0.301197
$730$		0			0
$731$			5.85176e10i			0.204936i
$732$		0			0
$733$			4.81267e11i			1.66713i	0.552420	+	0.833566i	$0.313704\pi$
$733$			4.81267e11i			1.66713i	−0.552420	+	0.833566i	$0.686296\pi$
$734$		0			0
$735$	1.19731e11	+	5.91781e10i	0.410258	+	0.202774i
$736$		0			0
$737$		−	1.23558e11i		−	0.418796i
$738$		0			0
$739$		−	9.92996e10i		−	0.332943i	−0.986046	−	0.166471i	$-0.946763\pi$
$739$		−	9.92996e10i		−	0.332943i	0.986046	−	0.166471i	$-0.0532374\pi$
$740$		0			0
$741$	1.39510e11			0.462736
$742$		0			0
$743$	−6.32997e10			−0.207705			−0.103852	−	0.994593i	$-0.533117\pi$
$743$	−6.32997e10			−0.207705			−0.103852	+	0.994593i	$0.533117\pi$
$744$		0			0
$745$	−4.31877e11	−	2.13459e11i	−1.40196	−	0.692931i
$746$		0			0
$747$	−4.14966e11			−1.33269
$748$		0			0
$749$	−7.68378e10			−0.244145
$750$		0			0
$751$			4.98905e11i			1.56840i	0.620505	+	0.784202i	$0.286928\pi$
$751$			4.98905e11i			1.56840i	−0.620505	+	0.784202i	$0.713072\pi$
$752$		0			0
$753$			2.64006e11i			0.821171i
$754$		0			0
$755$	−1.41649e11	+	2.86588e11i	−0.435938	+	0.882003i
$756$		0			0
$757$		−	1.79684e11i		−	0.547174i	−0.961847	−	0.273587i	$-0.911790\pi$
$757$		−	1.79684e11i		−	0.547174i	0.961847	−	0.273587i	$-0.0882101\pi$
$758$		0			0
$759$		−	4.52281e11i		−	1.36283i
$760$		0			0
$761$	−3.04576e11			−0.908148			−0.454074	−	0.890964i	$-0.650030\pi$
$761$	−3.04576e11			−0.908148			−0.454074	+	0.890964i	$0.650030\pi$
$762$		0			0
$763$	−4.19323e10			−0.123723
$764$		0			0
$765$	−5.36953e10	+	1.08638e11i	−0.156780	+	0.317202i
$766$		0			0
$767$	−2.32458e11			−0.671682
$768$		0			0
$769$	2.22920e11			0.637446			0.318723	−	0.947848i	$-0.396746\pi$
$769$	2.22920e11			0.637446			0.318723	+	0.947848i	$0.396746\pi$
$770$		0			0
$771$		−	2.64696e11i		−	0.749082i
$772$		0			0
$773$		−	5.59182e11i		−	1.56616i	−0.621924	−	0.783078i	$-0.713649\pi$
$773$		−	5.59182e11i		−	1.56616i	0.621924	−	0.783078i	$-0.286351\pi$
$774$		0			0
$775$	−4.80239e10	+	3.67136e10i	−0.133122	+	0.101770i
$776$		0			0
$777$		−	6.38979e10i		−	0.175308i
$778$		0			0
$779$			1.59577e11i			0.433332i
$780$		0			0
$781$	−5.13770e11			−1.38091
$782$		0			0
$783$	−8.90808e10			−0.236994
$784$		0			0
$785$	2.35770e10	−	4.77018e10i	0.0620884	−	0.125619i
$786$		0			0
$787$	1.13289e11			0.295318			0.147659	−	0.989038i	$-0.452826\pi$
$787$	1.13289e11			0.295318			0.147659	+	0.989038i	$0.452826\pi$
$788$		0			0
$789$	8.37655e10			0.216151
$790$		0			0
$791$			1.59902e11i			0.408458i
$792$		0			0
$793$			2.77088e11i			0.700689i
$794$		0			0
$795$	8.24308e10	−	1.66777e11i	0.206358	−	0.417510i
$796$		0			0
$797$			7.63543e11i			1.89234i	0.323663	+	0.946172i	$0.395086\pi$
$797$			7.63543e11i			1.89234i	−0.323663	+	0.946172i	$0.604914\pi$
$798$		0			0
$799$		−	2.03888e11i		−	0.500269i
$800$		0			0
$801$	5.43525e10			0.132035
$802$		0			0
$803$	8.20637e11			1.97374
$804$		0			0
$805$	−2.08830e11	−	1.03216e11i	−0.497289	−	0.245790i
$806$		0			0
$807$	2.69480e11			0.635379
$808$		0			0
$809$	4.67881e11			1.09230			0.546149	−	0.837688i	$-0.316093\pi$
$809$	4.67881e11			1.09230			0.546149	+	0.837688i	$0.316093\pi$
$810$		0			0
$811$			6.66951e11i			1.54174i	0.636994	+	0.770869i	$0.280178\pi$
$811$			6.66951e11i			1.54174i	−0.636994	+	0.770869i	$0.719822\pi$
$812$		0			0
$813$			2.01068e11i			0.460237i
$814$		0			0
$815$	1.56482e11	+	7.73428e10i	0.354678	+	0.175303i
$816$		0			0
$817$			1.13746e11i			0.255298i
$818$		0			0
$819$			1.69565e11i			0.376879i
$820$		0			0
$821$	4.34647e11			0.956674			0.478337	−	0.878176i	$-0.341240\pi$
$821$	4.34647e11			0.956674			0.478337	+	0.878176i	$0.341240\pi$
$822$		0			0
$823$	−6.53323e11			−1.42406			−0.712030	−	0.702149i	$-0.752224\pi$
$823$	−6.53323e11			−1.42406			−0.712030	+	0.702149i	$0.752224\pi$
$824$		0			0
$825$	−3.27506e11	+	2.50373e11i	−0.706973	+	0.540471i
$826$		0			0
$827$	−8.42913e10			−0.180203			−0.0901013	−	0.995933i	$-0.528719\pi$
$827$	−8.42913e10			−0.180203			−0.0901013	+	0.995933i	$0.528719\pi$
$828$		0			0
$829$	−1.67984e11			−0.355672			−0.177836	−	0.984060i	$-0.556910\pi$
$829$	−1.67984e11			−0.355672			−0.177836	+	0.984060i	$0.556910\pi$
$830$		0			0
$831$		−	4.29928e9i		−	0.00901553i
$832$		0			0
$833$			2.04849e11i			0.425455i
$834$		0			0
$835$	−2.41320e11	−	1.19274e11i	−0.496417	−	0.245359i
$836$		0			0
$837$			7.46207e10i			0.152040i
$838$		0			0
$839$			8.94664e11i			1.80556i	0.430102	+	0.902780i	$0.358477\pi$
$839$			8.94664e11i			1.80556i	−0.430102	+	0.902780i	$0.641523\pi$
$840$		0			0
$841$	−4.66118e11			−0.931776
$842$		0			0
$843$	1.81910e11			0.360202
$844$		0			0
$845$	−4.90721e11	−	2.42543e11i	−0.962516	−	0.475732i
$846$		0			0
$847$	−3.45666e11			−0.671619
$848$		0			0
$849$	4.96039e11			0.954739
$850$		0			0
$851$		−	7.37991e11i		−	1.40713i
$852$		0			0
$853$		−	1.57864e11i		−	0.298186i	−0.988823	−	0.149093i	$-0.952365\pi$
$853$		−	1.57864e11i		−	0.298186i	0.988823	−	0.149093i	$-0.0476353\pi$
$854$		0			0
$855$	−1.04372e11	+	2.11169e11i	−0.195308	+	0.395154i
$856$		0			0
$857$		−	1.15753e10i		−	0.0214590i	−0.999942	−	0.0107295i	$-0.996585\pi$
$857$		−	1.15753e10i		−	0.0214590i	0.999942	−	0.0107295i	$-0.00341538\pi$
$858$		0			0
$859$		−	5.96689e10i		−	0.109591i	−0.998498	−	0.0547956i	$-0.982549\pi$
$859$		−	5.96689e10i		−	0.109591i	0.998498	−	0.0547956i	$-0.0174507\pi$
$860$		0			0
$861$	7.44797e10			0.135527
$862$		0			0
$863$	−1.49504e11			−0.269532			−0.134766	−	0.990877i	$-0.543028\pi$
$863$	−1.49504e11			−0.269532			−0.134766	+	0.990877i	$0.543028\pi$
$864$		0			0
$865$	1.88658e11	−	3.81699e11i	0.336986	−	0.681800i
$866$		0			0
$867$	−2.26255e11			−0.400425
$868$		0			0
$869$	1.85375e12			3.25067
$870$		0			0
$871$		−	2.05449e11i		−	0.356969i
$872$		0			0
$873$			4.14631e11i			0.713846i
$874$		0			0
$875$	4.08632e10	+	2.08356e11i	0.0697108	+	0.355446i
$876$		0			0
$877$		−	8.95446e11i		−	1.51370i	−0.653587	−	0.756852i	$-0.726736\pi$
$877$		−	8.95446e11i		−	1.51370i	0.653587	−	0.756852i	$-0.273264\pi$
$878$		0			0
$879$			1.60299e11i			0.268519i
$880$		0			0
$881$	9.41351e11			1.56260			0.781300	−	0.624155i	$-0.214557\pi$
$881$	9.41351e11			1.56260			0.781300	+	0.624155i	$0.214557\pi$
$882$		0			0
$883$	−8.12182e11			−1.33601			−0.668006	−	0.744156i	$-0.732852\pi$
$883$	−8.12182e11			−1.33601			−0.668006	+	0.744156i	$0.732852\pi$
$884$		0			0
$885$	−6.67821e10	+	1.35116e11i	−0.108865	+	0.220258i
$886$		0			0
$887$	4.35414e11			0.703409			0.351704	−	0.936111i	$-0.385602\pi$
$887$	4.35414e11			0.703409			0.351704	+	0.936111i	$0.385602\pi$
$888$		0			0
$889$	3.06947e11			0.491424
$890$		0			0
$891$		−	2.60447e11i		−	0.413246i
$892$		0			0
$893$		−	3.96314e11i		−	0.623209i
$894$		0			0
$895$	5.15676e11	−	1.04333e12i	0.803683	−	1.62604i
$896$		0			0
$897$		−	7.52036e11i		−	1.16163i
$898$		0			0
$899$		−	2.85888e10i		−	0.0437680i
$900$		0			0
$901$	2.85339e11			0.432975
$902$		0			0
$903$	5.30888e10			0.0798458
$904$		0			0
$905$	−5.29494e11	−	2.61707e11i	−0.789345	−	0.390141i
$906$		0			0
$907$	8.59055e11			1.26938			0.634690	−	0.772766i	$-0.281128\pi$
$907$	8.59055e11			1.26938			0.634690	+	0.772766i	$0.281128\pi$
$908$		0			0
$909$	5.11793e11			0.749617
$910$		0			0
$911$		−	9.78674e11i		−	1.42090i	−0.703746	−	0.710452i	$-0.748491\pi$
$911$		−	9.78674e11i		−	1.42090i	0.703746	−	0.710452i	$-0.251509\pi$
$912$		0			0
$913$		−	2.16517e12i		−	3.11608i
$914$		0			0
$915$	1.61057e11	+	7.96037e10i	0.229770	+	0.113566i
$916$		0			0
$917$		−	1.04917e11i		−	0.148378i
$918$		0			0
$919$			3.76307e11i			0.527571i	0.964581	+	0.263785i	$0.0849711\pi$
$919$			3.76307e11i			0.527571i	−0.964581	+	0.263785i	$0.915029\pi$
$920$		0			0
$921$	2.47613e11			0.344139
$922$		0			0
$923$	−8.54278e11			−1.17704
$924$		0			0
$925$	−5.34395e11	+	4.08537e11i	−0.729954	+	0.558039i
$926$		0			0
$927$	4.56270e11			0.617878
$928$		0			0
$929$	1.22156e12			1.64003			0.820013	−	0.572345i	$-0.193966\pi$
$929$	1.22156e12			1.64003			0.820013	+	0.572345i	$0.193966\pi$
$930$		0			0
$931$			3.98183e11i			0.530009i
$932$		0			0
$933$		−	5.21144e11i		−	0.687751i
$934$		0			0
$935$	−5.66841e11	−	2.80166e11i	−0.741677	−	0.366580i
$936$		0			0
$937$			5.06116e11i			0.656586i	0.944576	+	0.328293i	$0.106473\pi$
$937$			5.06116e11i			0.656586i	−0.944576	+	0.328293i	$0.893527\pi$
$938$		0			0
$939$			3.83917e11i			0.493827i
$940$		0			0
$941$	6.00360e9			0.00765690			0.00382845	−	0.999993i	$-0.498781\pi$
$941$	6.00360e9			0.00765690			0.00382845	+	0.999993i	$0.498781\pi$
$942$		0			0
$943$	8.60207e11			1.08782
$944$		0			0
$945$	2.34966e11	+	1.16134e11i	0.294631	+	0.145624i
$946$		0			0
$947$	−3.08522e11			−0.383606			−0.191803	−	0.981433i	$-0.561433\pi$
$947$	−3.08522e11			−0.383606			−0.191803	+	0.981433i	$0.561433\pi$
$948$		0			0
$949$	1.36453e12			1.68235
$950$		0			0
$951$		−	9.60868e10i		−	0.117474i
$952$		0			0
$953$		−	6.14491e11i		−	0.744979i	−0.928036	−	0.372489i	$-0.878504\pi$
$953$		−	6.14491e11i		−	0.744979i	0.928036	−	0.372489i	$-0.121496\pi$
$954$		0			0
$955$	1.69999e11	−	3.43948e11i	0.204378	−	0.413504i
$956$		0			0
$957$		−	1.94965e11i		−	0.232439i
$958$		0			0
$959$			2.59174e11i			0.306420i
$960$		0			0
$961$	8.28943e11			0.971921
$962$		0			0
$963$	4.18838e11			0.487013
$964$		0			0
$965$	2.06316e11	−	4.17424e11i	0.237916	−	0.481358i
$966$		0			0
$967$	−7.29374e11			−0.834150			−0.417075	−	0.908872i	$-0.636945\pi$
$967$	−7.29374e11			−0.834150			−0.417075	+	0.908872i	$0.636945\pi$
$968$		0			0
$969$	1.38738e11			0.157362
$970$		0			0
$971$			4.05501e11i			0.456158i	0.973643	+	0.228079i	$0.0732444\pi$
$971$			4.05501e11i			0.456158i	−0.973643	+	0.228079i	$0.926756\pi$
$972$		0			0
$973$			3.47959e11i			0.388219i
$974$		0			0
$975$	−5.44565e11	+	4.16312e11i	−0.602603	+	0.460681i
$976$		0			0
$977$			5.72274e11i			0.628096i	0.949407	+	0.314048i	$0.101685\pi$
$977$			5.72274e11i			0.628096i	−0.949407	+	0.314048i	$0.898315\pi$
$978$		0			0
$979$			2.83595e11i			0.308722i
$980$		0			0
$981$	2.28570e11			0.246799
$982$		0			0
$983$	8.31746e11			0.890792			0.445396	−	0.895334i	$-0.353063\pi$
$983$	8.31746e11			0.890792			0.445396	+	0.895334i	$0.353063\pi$
$984$		0			0
$985$	−6.85125e11	+	1.38617e12i	−0.727821	+	1.47255i
$986$		0			0
$987$	−1.84972e11			−0.194912
$988$		0			0
$989$	6.13152e11			0.640889
$990$		0			0
$991$			1.13988e12i			1.18186i	0.806724	+	0.590928i	$0.201238\pi$
$991$			1.13988e12i			1.18186i	−0.806724	+	0.590928i	$0.798762\pi$
$992$		0			0
$993$			7.19234e10i			0.0739729i
$994$		0			0
$995$	2.38836e11	−	4.83221e11i	0.243673	−	0.493007i
$996$		0			0
$997$			1.49273e11i			0.151078i	0.997143	+	0.0755390i	$0.0240677\pi$
$997$			1.49273e11i			0.151078i	−0.997143	+	0.0755390i	$0.975932\pi$
$998$		0			0
$999$			8.30355e11i			0.833685i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.h.g.319.10		20
4.3	odd	2	inner	320.9.h.g.319.12		20
5.4	even	2	inner	320.9.h.g.319.11		20
8.3	odd	2		20.9.d.c.19.17	yes	20
8.5	even	2		20.9.d.c.19.3	✓	20
20.19	odd	2	inner	320.9.h.g.319.9		20
40.3	even	4		100.9.b.g.51.7		20
40.13	odd	4		100.9.b.g.51.8		20
40.19	odd	2		20.9.d.c.19.4	yes	20
40.27	even	4		100.9.b.g.51.14		20
40.29	even	2		20.9.d.c.19.18	yes	20
40.37	odd	4		100.9.b.g.51.13		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.d.c.19.3	✓	20	8.5	even	2
20.9.d.c.19.4	yes	20	40.19	odd	2
20.9.d.c.19.17	yes	20	8.3	odd	2
20.9.d.c.19.18	yes	20	40.29	even	2
100.9.b.g.51.7		20	40.3	even	4
100.9.b.g.51.8		20	40.13	odd	4
100.9.b.g.51.13		20	40.37	odd	4
100.9.b.g.51.14		20	40.27	even	4
320.9.h.g.319.9		20	20.19	odd	2	inner
320.9.h.g.319.10		20	1.1	even	1	trivial
320.9.h.g.319.11		20	5.4	even	2	inner
320.9.h.g.319.12		20	4.3	odd	2	inner

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	320.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-42.6663 q^{3} +(560.298 + 276.932i) q^{5} +869.685 q^{7} -4740.59 q^{9} -24735.0i q^{11} -41128.5i q^{13} +(-23905.8 - 11815.7i) q^{15} -40900.7i q^{17} -79502.2i q^{19} -37106.2 q^{21} -428560. q^{23} +(237242. + 310329. i) q^{25} +482197. q^{27} -184740. q^{29} +154752. i q^{31} +1.05535e6i q^{33} +(487283. + 240844. i) q^{35} +1.72203e6i q^{37} +1.75480e6i q^{39} -2.00720e6 q^{41} -1.43073e6 q^{43} +(-2.65614e6 - 1.31282e6i) q^{45} +4.98494e6 q^{47} -5.00845e6 q^{49} +1.74508e6i q^{51} +6.97640e6i q^{53} +(6.84992e6 - 1.38590e7i) q^{55} +3.39206e6i q^{57} -5.65200e6i q^{59} -6.73713e6 q^{61} -4.12282e6 q^{63} +(1.13898e7 - 2.30442e7i) q^{65} +4.99529e6 q^{67} +1.82850e7 q^{69} -2.07710e7i q^{71} +3.31772e7i q^{73} +(-1.01222e7 - 1.32406e7i) q^{75} -2.15117e7i q^{77} +7.49445e7i q^{79} +1.05295e7 q^{81} +8.75346e7 q^{83} +(1.13267e7 - 2.29165e7i) q^{85} +7.88215e6 q^{87} -1.14653e7 q^{89} -3.57688e7i q^{91} -6.60268e6i q^{93} +(2.20167e7 - 4.45449e7i) q^{95} -8.74640e7i q^{97} +1.17258e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 1420 q^{5} + 2556 q^{9} - 410256 q^{21} - 657260 q^{25} - 660136 q^{29} + 7068520 q^{41} + 18729060 q^{45} + 11719036 q^{49} + 17660440 q^{61} - 44202240 q^{65} - 111747216 q^{69} - 154212444 q^{81}+ \cdots + 105006376 q^{89}+O(q^{100}) \) 20 * q + 1420 * q^5 + 2556 * q^9 - 410256 * q^21 - 657260 * q^25 - 660136 * q^29 + 7068520 * q^41 + 18729060 * q^45 + 11719036 * q^49 + 17660440 * q^61 - 44202240 * q^65 - 111747216 * q^69 - 154212444 * q^81 - 68800000 * q^85 + 105006376 * q^89

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