LMFDB - Embedded newform 320.9.b.d.191.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,9,Mod(191,320)] 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, 0])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.191"); S:= CuspForms(chi, 9); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-38800] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$130.361155220$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 $ x^16 - 5x^15 + 26x^14 - 834x^13 + 4390x^12 - 61783x^11 + 466168x^10 - 1105435x^9 + 46850799x^8 - 116275535x^7 + 626274432x^6 - 8558999923x^5 + 34408048994x^4 - 448299413930x^3 + 1712647133330x^2 + 15986651928135*x + 206161212459445
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{120}\cdot 5^{16} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.14
Root			$3.05707 + 7.10588i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.9.b.d.191.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+110.171i q^{3} +279.508 q^{5} +3540.70i q^{7} -5576.56 q^{9} -15969.8i q^{11} +24176.2 q^{13} +30793.6i q^{15} +43810.0 q^{17} -50919.1i q^{19} -390081. q^{21} +270875. i q^{23} +78125.0 q^{25} +108456. i q^{27} +1.32588e6 q^{29} +1.18623e6i q^{31} +1.75941e6 q^{33} +989655. i q^{35} -2.97108e6 q^{37} +2.66351e6i q^{39} -4.92072e6 q^{41} +2.86229e6i q^{43} -1.55870e6 q^{45} -12696.5i q^{47} -6.77174e6 q^{49} +4.82658e6i q^{51} +5.50364e6 q^{53} -4.46371e6i q^{55} +5.60979e6 q^{57} +6.68863e6i q^{59} +2.50550e6 q^{61} -1.97449e7i q^{63} +6.75746e6 q^{65} -3.86718e6i q^{67} -2.98424e7 q^{69} +3.21701e7i q^{71} +1.91362e7 q^{73} +8.60708e6i q^{75} +5.65444e7 q^{77} +5.33471e7i q^{79} -4.85365e7 q^{81} -6.22776e6i q^{83} +1.22453e7 q^{85} +1.46072e8i q^{87} +3.83322e7 q^{89} +8.56008e7i q^{91} -1.30687e8 q^{93} -1.42323e7i q^{95} +7.87237e7 q^{97} +8.90568e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) $ 16 * q - 38800 * q^9 - 51392 * q^13 + 27552 * q^17 - 414496 * q^21 + 1250000 * q^25 - 2764896 * q^29 - 5521600 * q^33 - 9009472 * q^37 - 8576448 * q^41 - 1580000 * q^45 - 32803600 * q^49 - 2452032 * q^53 + 11957760 * q^57 - 8371712 * q^61 + 9060000 * q^65 - 7527264 * q^69 + 61907232 * q^73 + 156997440 * q^77 + 140586672 * q^81 + 106960000 * q^85 + 106647456 * q^89 - 105563840 * q^93 + 171851232 * q^97

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
$3$	110.171i	1.36013i	0.733151	+	0.680065i	$0.238049\pi$
$3$	110.171i	1.36013i	−0.733151	+	0.680065i	$0.761951\pi$
$4$	0	0
$5$	279.508	0.447214
$5$	279.508	0.447214
$6$	0	0
$7$	3540.70i	1.47468i	0.675524	+	0.737338i	$0.263918\pi$
$7$	3540.70i	1.47468i	−0.675524	+	0.737338i	$0.736082\pi$
$8$	0	0
$9$	−5576.56	−0.849956
$10$	0	0
$11$	− 15969.8i	− 1.09076i	−0.838188	−	0.545381i	$-0.816385\pi$
$11$	− 15969.8i	− 1.09076i	0.838188	−	0.545381i	$-0.183615\pi$
$12$	0	0
$13$	24176.2	0.846477	0.423239	−	0.906018i	$-0.360893\pi$
$13$	24176.2	0.846477	0.423239	+	0.906018i	$0.360893\pi$
$14$	0	0
$15$	30793.6i	0.608269i
$16$	0	0
$17$	43810.0	0.524539	0.262269	−	0.964995i	$-0.415529\pi$
$17$	43810.0	0.524539	0.262269	+	0.964995i	$0.415529\pi$
$18$	0	0
$19$	− 50919.1i	− 0.390721i	−0.980732	−	0.195360i	$-0.937412\pi$
$19$	− 50919.1i	− 0.390721i	0.980732	−	0.195360i	$-0.0625876\pi$
$20$	0	0
$21$	−390081.	−2.00575
$22$	0	0
$23$	270875.i	0.967959i	0.875079	+	0.483980i	$0.160809\pi$
$23$	270875.i	0.967959i	−0.875079	+	0.483980i	$0.839191\pi$
$24$	0	0
$25$	78125.0	0.200000
$26$	0	0
$27$	108456.i	0.204079i
$28$	0	0
$29$	1.32588e6	1.87461	0.937305	−	0.348511i	$-0.113313\pi$
$29$	1.32588e6	1.87461	0.937305	+	0.348511i	$0.113313\pi$
$30$	0	0
$31$	1.18623e6i	1.28446i	0.766512	+	0.642230i	$0.221991\pi$
$31$	1.18623e6i	1.28446i	−0.766512	+	0.642230i	$0.778009\pi$
$32$	0	0
$33$	1.75941e6	1.48358
$34$	0	0
$35$	989655.i	0.659495i
$36$	0	0
$37$	−2.97108e6	−1.58529	−0.792643	−	0.609686i	$-0.791295\pi$
$37$	−2.97108e6	−1.58529	−0.792643	+	0.609686i	$0.791295\pi$
$38$	0	0
$39$	2.66351e6i	1.15132i
$40$	0	0
$41$	−4.92072e6	−1.74138	−0.870689	−	0.491834i	$-0.836327\pi$
$41$	−4.92072e6	−1.74138	−0.870689	+	0.491834i	$0.836327\pi$
$42$	0	0
$43$	2.86229e6i	0.837221i	0.908166	+	0.418611i	$0.137483\pi$
$43$	2.86229e6i	0.837221i	−0.908166	+	0.418611i	$0.862517\pi$
$44$	0	0
$45$	−1.55870e6	−0.380112
$46$	0	0
$47$	− 12696.5i	− 0.00260192i	−0.999999	−	0.00130096i	$-0.999586\pi$
$47$	− 12696.5i	− 0.00260192i	0.999999	−	0.00130096i	$-0.000414108\pi$
$48$	0	0
$49$	−6.77174e6	−1.17467
$50$	0	0
$51$	4.82658e6i	0.713442i
$52$	0	0
$53$	5.50364e6	0.697504	0.348752	−	0.937215i	$-0.386606\pi$
$53$	5.50364e6	0.697504	0.348752	+	0.937215i	$0.386606\pi$
$54$	0	0
$55$	− 4.46371e6i	− 0.487803i
$56$	0	0
$57$	5.60979e6	0.531431
$58$	0	0
$59$	6.68863e6i	0.551987i	0.961159	+	0.275994i	$0.0890068\pi$
$59$	6.68863e6i	0.551987i	−0.961159	+	0.275994i	$0.910993\pi$
$60$	0	0
$61$	2.50550e6	0.180957	0.0904784	−	0.995898i	$-0.471160\pi$
$61$	2.50550e6	0.180957	0.0904784	+	0.995898i	$0.471160\pi$
$62$	0	0
$63$	− 1.97449e7i	− 1.25341i
$64$	0	0
$65$	6.75746e6	0.378556
$66$	0	0
$67$	− 3.86718e6i	− 0.191909i	−0.995386	−	0.0959546i	$-0.969410\pi$
$67$	− 3.86718e6i	− 0.191909i	0.995386	−	0.0959546i	$-0.0305904\pi$
$68$	0	0
$69$	−2.98424e7	−1.31655
$70$	0	0
$71$	3.21701e7i	1.26596i	0.774170	+	0.632978i	$0.218168\pi$
$71$	3.21701e7i	1.26596i	−0.774170	+	0.632978i	$0.781832\pi$
$72$	0	0
$73$	1.91362e7	0.673853	0.336927	−	0.941531i	$-0.390613\pi$
$73$	1.91362e7	0.673853	0.336927	+	0.941531i	$0.390613\pi$
$74$	0	0
$75$	8.60708e6i	0.272026i
$76$	0	0
$77$	5.65444e7	1.60852
$78$	0	0
$79$	5.33471e7i	1.36963i	0.728718	+	0.684814i	$0.240116\pi$
$79$	5.33471e7i	1.36963i	−0.728718	+	0.684814i	$0.759884\pi$
$80$	0	0
$81$	−4.85365e7	−1.12753
$82$	0	0
$83$	− 6.22776e6i	− 0.131226i	−0.997845	−	0.0656129i	$-0.979100\pi$
$83$	− 6.22776e6i	− 0.131226i	0.997845	−	0.0656129i	$-0.0209003\pi$
$84$	0	0
$85$	1.22453e7	0.234581
$86$	0	0
$87$	1.46072e8i	2.54971i
$88$	0	0
$89$	3.83322e7	0.610947	0.305473	−	0.952201i	$-0.401185\pi$
$89$	3.83322e7	0.610947	0.305473	+	0.952201i	$0.401185\pi$
$90$	0	0
$91$	8.56008e7i	1.24828i
$92$	0	0
$93$	−1.30687e8	−1.74703
$94$	0	0
$95$	− 1.42323e7i	− 0.174736i
$96$	0	0
$97$	7.87237e7	0.889239	0.444620	−	0.895720i	$-0.353339\pi$
$97$	7.87237e7	0.889239	0.444620	+	0.895720i	$0.353339\pi$
$98$	0	0
$99$	8.90568e7i	0.927100i
$100$	0	0
$101$	1.16450e7	0.111906	0.0559530	−	0.998433i	$-0.482180\pi$
$101$	1.16450e7	0.111906	0.0559530	+	0.998433i	$0.482180\pi$
$102$	0	0
$103$	− 6.88569e7i	− 0.611785i	−0.952066	−	0.305892i	$-0.901045\pi$
$103$	− 6.88569e7i	− 0.611785i	0.952066	−	0.305892i	$-0.0989547\pi$
$104$	0	0
$105$	−1.09031e8	−0.897000
$106$	0	0
$107$	− 7.69690e7i	− 0.587193i	−0.955929	−	0.293596i	$-0.905148\pi$
$107$	− 7.69690e7i	− 0.587193i	0.955929	−	0.293596i	$-0.0948521\pi$
$108$	0	0
$109$	4.29484e6	0.0304257	0.0152129	−	0.999884i	$-0.495157\pi$
$109$	4.29484e6	0.0304257	0.0152129	+	0.999884i	$0.495157\pi$
$110$	0	0
$111$	− 3.27326e8i	− 2.15620i
$112$	0	0
$113$	−2.34309e8	−1.43706	−0.718530	−	0.695496i	$-0.755185\pi$
$113$	−2.34309e8	−1.43706	−0.718530	+	0.695496i	$0.755185\pi$
$114$	0	0
$115$	7.57118e7i	0.432884i
$116$	0	0
$117$	−1.34820e8	−0.719469
$118$	0	0
$119$	1.55118e8i	0.773525i
$120$	0	0
$121$	−4.06770e7	−0.189761
$122$	0	0
$123$	− 5.42118e8i	− 2.36850i
$124$	0	0
$125$	2.18366e7	0.0894427
$126$	0	0
$127$	− 2.94344e8i	− 1.13146i	−0.824589	−	0.565732i	$-0.808594\pi$
$127$	− 2.94344e8i	− 1.13146i	0.824589	−	0.565732i	$-0.191406\pi$
$128$	0	0
$129$	−3.15341e8	−1.13873
$130$	0	0
$131$	3.43317e8i	1.16576i	0.812558	+	0.582881i	$0.198075\pi$
$131$	3.43317e8i	1.16576i	−0.812558	+	0.582881i	$0.801925\pi$
$132$	0	0
$133$	1.80289e8	0.576186
$134$	0	0
$135$	3.03144e7i	0.0912670i
$136$	0	0
$137$	2.03539e8	0.577784	0.288892	−	0.957362i	$-0.406713\pi$
$137$	2.03539e8	0.577784	0.288892	+	0.957362i	$0.406713\pi$
$138$	0	0
$139$	− 2.51305e8i	− 0.673197i	−0.941648	−	0.336599i	$-0.890723\pi$
$139$	− 2.51305e8i	− 0.673197i	0.941648	−	0.336599i	$-0.109277\pi$
$140$	0	0
$141$	1.39878e6	0.00353895
$142$	0	0
$143$	− 3.86091e8i	− 0.923305i
$144$	0	0
$145$	3.70593e8	0.838351
$146$	0	0
$147$	− 7.46047e8i	− 1.59771i
$148$	0	0
$149$	−7.65026e8	−1.55214	−0.776070	−	0.630647i	$-0.782790\pi$
$149$	−7.65026e8	−1.55214	−0.776070	+	0.630647i	$0.782790\pi$
$150$	0	0
$151$	− 5.12585e8i	− 0.985958i	−0.870041	−	0.492979i	$-0.835908\pi$
$151$	− 5.12585e8i	− 0.985958i	0.870041	−	0.492979i	$-0.164092\pi$
$152$	0	0
$153$	−2.44309e8	−0.445835
$154$	0	0
$155$	3.31560e8i	0.574428i
$156$	0	0
$157$	−1.16971e9	−1.92522	−0.962610	−	0.270890i	$-0.912682\pi$
$157$	−1.16971e9	−1.92522	−0.962610	+	0.270890i	$0.912682\pi$
$158$	0	0
$159$	6.06339e8i	0.948696i
$160$	0	0
$161$	−9.59085e8	−1.42743
$162$	0	0
$163$	− 8.79268e8i	− 1.24558i	−0.782390	−	0.622789i	$-0.786000\pi$
$163$	− 8.79268e8i	− 1.24558i	0.782390	−	0.622789i	$-0.214000\pi$
$164$	0	0
$165$	4.91769e8	0.663477
$166$	0	0
$167$	4.63319e8i	0.595682i	0.954616	+	0.297841i	$0.0962665\pi$
$167$	4.63319e8i	0.595682i	−0.954616	+	0.297841i	$0.903734\pi$
$168$	0	0
$169$	−2.31240e8	−0.283476
$170$	0	0
$171$	2.83954e8i	0.332095i
$172$	0	0
$173$	−4.71680e8	−0.526579	−0.263289	−	0.964717i	$-0.584807\pi$
$173$	−4.71680e8	−0.526579	−0.263289	+	0.964717i	$0.584807\pi$
$174$	0	0
$175$	2.76617e8i	0.294935i
$176$	0	0
$177$	−7.36890e8	−0.750775
$178$	0	0
$179$	8.30768e8i	0.809222i	0.914489	+	0.404611i	$0.132593\pi$
$179$	8.30768e8i	0.809222i	−0.914489	+	0.404611i	$0.867407\pi$
$180$	0	0
$181$	−6.19754e8	−0.577437	−0.288719	−	0.957414i	$-0.593229\pi$
$181$	−6.19754e8	−0.577437	−0.288719	+	0.957414i	$0.593229\pi$
$182$	0	0
$183$	2.76032e8i	0.246125i
$184$	0	0
$185$	−8.30442e8	−0.708961
$186$	0	0
$187$	− 6.99639e8i	− 0.572147i
$188$	0	0
$189$	−3.84010e8	−0.300951
$190$	0	0
$191$	− 4.24020e7i	− 0.0318605i	−0.999873	−	0.0159303i	$-0.994929\pi$
$191$	− 4.24020e7i	− 0.0318605i	0.999873	−	0.0159303i	$-0.00507097\pi$
$192$	0	0
$193$	4.21151e8	0.303535	0.151767	−	0.988416i	$-0.451504\pi$
$193$	4.21151e8	0.303535	0.151767	+	0.988416i	$0.451504\pi$
$194$	0	0
$195$	7.44474e8i	0.514886i
$196$	0	0
$197$	1.93944e9	1.28769	0.643846	−	0.765155i	$-0.277338\pi$
$197$	1.93944e9	1.28769	0.643846	+	0.765155i	$0.277338\pi$
$198$	0	0
$199$	2.11255e8i	0.134708i	0.997729	+	0.0673542i	$0.0214557\pi$
$199$	2.11255e8i	0.134708i	−0.997729	+	0.0673542i	$0.978544\pi$
$200$	0	0
$201$	4.26050e8	0.261022
$202$	0	0
$203$	4.69452e9i	2.76444i
$204$	0	0
$205$	−1.37538e9	−0.778768
$206$	0	0
$207$	− 1.51055e9i	− 0.822723i
$208$	0	0
$209$	−8.13170e8	−0.426183
$210$	0	0
$211$	− 2.01406e9i	− 1.01611i	−0.861324	−	0.508056i	$-0.830364\pi$
$211$	− 2.01406e9i	− 1.01611i	0.861324	−	0.508056i	$-0.169636\pi$
$212$	0	0
$213$	−3.54420e9	−1.72187
$214$	0	0
$215$	8.00035e8i	0.374417i
$216$	0	0
$217$	−4.20007e9	−1.89416
$218$	0	0
$219$	2.10825e9i	0.916528i
$220$	0	0
$221$	1.05916e9	0.444010
$222$	0	0
$223$	− 9.06863e8i	− 0.366710i	−0.983047	−	0.183355i	$-0.941304\pi$
$223$	− 9.06863e8i	− 0.366710i	0.983047	−	0.183355i	$-0.0586957\pi$
$224$	0	0
$225$	−4.35669e8	−0.169991
$226$	0	0
$227$	− 9.31268e7i	− 0.0350729i	−0.999846	−	0.0175364i	$-0.994418\pi$
$227$	− 9.31268e7i	− 0.0350729i	0.999846	−	0.0175364i	$-0.00558230\pi$
$228$	0	0
$229$	5.17840e8	0.188301	0.0941507	−	0.995558i	$-0.469986\pi$
$229$	5.17840e8	0.188301	0.0941507	+	0.995558i	$0.469986\pi$
$230$	0	0
$231$	6.22953e9i	2.18780i
$232$	0	0
$233$	−1.20417e9	−0.408569	−0.204284	−	0.978912i	$-0.565487\pi$
$233$	−1.20417e9	−0.408569	−0.204284	+	0.978912i	$0.565487\pi$
$234$	0	0
$235$	− 3.54879e6i	− 0.00116361i
$236$	0	0
$237$	−5.87728e9	−1.86287
$238$	0	0
$239$	− 2.12088e9i	− 0.650018i	−0.945711	−	0.325009i	$-0.894633\pi$
$239$	− 2.12088e9i	− 0.650018i	0.945711	−	0.325009i	$-0.105367\pi$
$240$	0	0
$241$	−6.85448e8	−0.203192	−0.101596	−	0.994826i	$-0.532395\pi$
$241$	−6.85448e8	−0.203192	−0.101596	+	0.994826i	$0.532395\pi$
$242$	0	0
$243$	− 4.63572e9i	− 1.32951i
$244$	0	0
$245$	−1.89276e9	−0.525329
$246$	0	0
$247$	− 1.23103e9i	− 0.330736i
$248$	0	0
$249$	6.86116e8	0.178484
$250$	0	0
$251$	4.44219e9i	1.11919i	0.828768	+	0.559593i	$0.189042\pi$
$251$	4.44219e9i	1.11919i	−0.828768	+	0.559593i	$0.810958\pi$
$252$	0	0
$253$	4.32582e9	1.05581
$254$	0	0
$255$	1.34907e9i	0.319061i
$256$	0	0
$257$	1.71181e9	0.392395	0.196198	−	0.980564i	$-0.437141\pi$
$257$	1.71181e9	0.392395	0.196198	+	0.980564i	$0.437141\pi$
$258$	0	0
$259$	− 1.05197e10i	− 2.33778i
$260$	0	0
$261$	−7.39383e9	−1.59334
$262$	0	0
$263$	2.50597e9i	0.523785i	0.965097	+	0.261892i	$0.0843465\pi$
$263$	2.50597e9i	0.523785i	−0.965097	+	0.261892i	$0.915653\pi$
$264$	0	0
$265$	1.53831e9	0.311933
$266$	0	0
$267$	4.22308e9i	0.830967i
$268$	0	0
$269$	−3.62847e8	−0.0692970	−0.0346485	−	0.999400i	$-0.511031\pi$
$269$	−3.62847e8	−0.0692970	−0.0346485	+	0.999400i	$0.511031\pi$
$270$	0	0
$271$	1.05662e10i	1.95903i	0.201368	+	0.979516i	$0.435461\pi$
$271$	1.05662e10i	1.95903i	−0.201368	+	0.979516i	$0.564539\pi$
$272$	0	0
$273$	−9.43069e9	−1.69782
$274$	0	0
$275$	− 1.24764e9i	− 0.218152i
$276$	0	0
$277$	1.36227e9	0.231390	0.115695	−	0.993285i	$-0.463091\pi$
$277$	1.36227e9	0.231390	0.115695	+	0.993285i	$0.463091\pi$
$278$	0	0
$279$	− 6.61506e9i	− 1.09173i
$280$	0	0
$281$	−3.01499e9	−0.483572	−0.241786	−	0.970330i	$-0.577733\pi$
$281$	−3.01499e9	−0.483572	−0.241786	+	0.970330i	$0.577733\pi$
$282$	0	0
$283$	1.38220e9i	0.215490i	0.994179	+	0.107745i	$0.0343629\pi$
$283$	1.38220e9i	0.215490i	−0.994179	+	0.107745i	$0.965637\pi$
$284$	0	0
$285$	1.56798e9	0.237663
$286$	0	0
$287$	− 1.74228e10i	− 2.56797i
$288$	0	0
$289$	−5.05644e9	−0.724859
$290$	0	0
$291$	8.67304e9i	1.20948i
$292$	0	0
$293$	−5.86750e9	−0.796127	−0.398064	−	0.917358i	$-0.630318\pi$
$293$	−5.86750e9	−0.796127	−0.398064	+	0.917358i	$0.630318\pi$
$294$	0	0
$295$	1.86953e9i	0.246856i
$296$	0	0
$297$	1.73203e9	0.222602
$298$	0	0
$299$	6.54873e9i	0.819355i
$300$	0	0
$301$	−1.01345e10	−1.23463
$302$	0	0
$303$	1.28293e9i	0.152207i
$304$	0	0
$305$	7.00308e8	0.0809263
$306$	0	0
$307$	− 5.79386e9i	− 0.652250i	−0.945327	−	0.326125i	$-0.894257\pi$
$307$	− 5.79386e9i	− 0.652250i	0.945327	−	0.326125i	$-0.105743\pi$
$308$	0	0
$309$	7.58601e9	0.832107
$310$	0	0
$311$	8.37260e9i	0.894991i	0.894286	+	0.447496i	$0.147684\pi$
$311$	8.37260e9i	0.894991i	−0.894286	+	0.447496i	$0.852316\pi$
$312$	0	0
$313$	1.22771e10	1.27914	0.639572	−	0.768731i	$-0.279111\pi$
$313$	1.22771e10	1.27914	0.639572	+	0.768731i	$0.279111\pi$
$314$	0	0
$315$	− 5.51887e9i	− 0.560542i
$316$	0	0
$317$	1.38443e10	1.37099	0.685495	−	0.728077i	$-0.259586\pi$
$317$	1.38443e10	1.37099	0.685495	+	0.728077i	$0.259586\pi$
$318$	0	0
$319$	− 2.11740e10i	− 2.04475i
$320$	0	0
$321$	8.47972e9	0.798659
$322$	0	0
$323$	− 2.23077e9i	− 0.204948i
$324$	0	0
$325$	1.88877e9	0.169295
$326$	0	0
$327$	4.73165e8i	0.0413830i
$328$	0	0
$329$	4.49546e7	0.00383699
$330$	0	0
$331$	7.71740e9i	0.642923i	0.946923	+	0.321462i	$0.104174\pi$
$331$	7.71740e9i	0.642923i	−0.946923	+	0.321462i	$0.895826\pi$
$332$	0	0
$333$	1.65684e10	1.34742
$334$	0	0
$335$	− 1.08091e9i	− 0.0858244i
$336$	0	0
$337$	−1.47787e10	−1.14582	−0.572910	−	0.819618i	$-0.694186\pi$
$337$	−1.47787e10	−1.14582	−0.572910	+	0.819618i	$0.694186\pi$
$338$	0	0
$339$	− 2.58139e10i	− 1.95459i
$340$	0	0
$341$	1.89438e10	1.40104
$342$	0	0
$343$	− 3.56527e9i	− 0.257582i
$344$	0	0
$345$	−8.34121e9	−0.588780
$346$	0	0
$347$	− 2.69899e10i	− 1.86158i	−0.365549	−	0.930792i	$-0.619119\pi$
$347$	− 2.69899e10i	− 1.86158i	0.365549	−	0.930792i	$-0.380881\pi$
$348$	0	0
$349$	−2.02677e10	−1.36617	−0.683083	−	0.730340i	$-0.739361\pi$
$349$	−2.02677e10	−1.36617	−0.683083	+	0.730340i	$0.739361\pi$
$350$	0	0
$351$	2.62206e9i	0.172748i
$352$	0	0
$353$	9.41747e9	0.606506	0.303253	−	0.952910i	$-0.401927\pi$
$353$	9.41747e9	0.606506	0.303253	+	0.952910i	$0.401927\pi$
$354$	0	0
$355$	8.99181e9i	0.566153i
$356$	0	0
$357$	−1.70894e10	−1.05210
$358$	0	0
$359$	3.30369e9i	0.198894i	0.995043	+	0.0994469i	$0.0317073\pi$
$359$	3.30369e9i	0.198894i	−0.995043	+	0.0994469i	$0.968293\pi$
$360$	0	0
$361$	1.43908e10	0.847337
$362$	0	0
$363$	− 4.48141e9i	− 0.258100i
$364$	0	0
$365$	5.34874e9	0.301356
$366$	0	0
$367$	− 1.14531e10i	− 0.631335i	−0.948870	−	0.315667i	$-0.897772\pi$
$367$	− 1.14531e10i	− 0.631335i	0.948870	−	0.315667i	$-0.102228\pi$
$368$	0	0
$369$	2.74407e10	1.48009
$370$	0	0
$371$	1.94867e10i	1.02859i
$372$	0	0
$373$	−2.26449e10	−1.16986	−0.584932	−	0.811083i	$-0.698879\pi$
$373$	−2.26449e10	−1.16986	−0.584932	+	0.811083i	$0.698879\pi$
$374$	0	0
$375$	2.40575e9i	0.121654i
$376$	0	0
$377$	3.20547e10	1.58681
$378$	0	0
$379$	− 5.61760e9i	− 0.272266i	−0.990691	−	0.136133i	$-0.956533\pi$
$379$	− 5.61760e9i	− 0.272266i	0.990691	−	0.136133i	$-0.0434675\pi$
$380$	0	0
$381$	3.24281e10	1.53894
$382$	0	0
$383$	− 4.10535e9i	− 0.190790i	−0.995440	−	0.0953950i	$-0.969589\pi$
$383$	− 4.10535e9i	− 0.190790i	0.995440	−	0.0953950i	$-0.0304114\pi$
$384$	0	0
$385$	1.58046e10	0.719352
$386$	0	0
$387$	− 1.59618e10i	− 0.711602i
$388$	0	0
$389$	1.37663e9	0.0601199	0.0300599	−	0.999548i	$-0.490430\pi$
$389$	1.37663e9	0.0601199	0.0300599	+	0.999548i	$0.490430\pi$
$390$	0	0
$391$	1.18670e10i	0.507732i
$392$	0	0
$393$	−3.78234e10	−1.58559
$394$	0	0
$395$	1.49110e10i	0.612516i
$396$	0	0
$397$	3.04208e10	1.22464	0.612321	−	0.790610i	$-0.290236\pi$
$397$	3.04208e10	1.22464	0.612321	+	0.790610i	$0.290236\pi$
$398$	0	0
$399$	1.98626e10i	0.783689i
$400$	0	0
$401$	−1.95473e10	−0.755977	−0.377989	−	0.925810i	$-0.623384\pi$
$401$	−1.95473e10	−0.755977	−0.377989	+	0.925810i	$0.623384\pi$
$402$	0	0
$403$	2.86785e10i	1.08727i
$404$	0	0
$405$	−1.35664e10	−0.504247
$406$	0	0
$407$	4.74477e10i	1.72917i
$408$	0	0
$409$	−4.37475e10	−1.56336	−0.781681	−	0.623678i	$-0.785638\pi$
$409$	−4.37475e10	−1.56336	−0.781681	+	0.623678i	$0.785638\pi$
$410$	0	0
$411$	2.24240e10i	0.785862i
$412$	0	0
$413$	−2.36824e10	−0.814002
$414$	0	0
$415$	− 1.74071e9i	− 0.0586860i
$416$	0	0
$417$	2.76865e10	0.915637
$418$	0	0
$419$	1.93115e10i	0.626555i	0.949662	+	0.313277i	$0.101427\pi$
$419$	1.93115e10i	0.626555i	−0.949662	+	0.313277i	$0.898573\pi$
$420$	0	0
$421$	3.65545e10	1.16362	0.581812	−	0.813323i	$-0.302344\pi$
$421$	3.65545e10	1.16362	0.581812	+	0.813323i	$0.302344\pi$
$422$	0	0
$423$	7.08030e7i	0.00221152i
$424$	0	0
$425$	3.42266e9	0.104908
$426$	0	0
$427$	8.87121e9i	0.266853i
$428$	0	0
$429$	4.25359e10	1.25582
$430$	0	0
$431$	3.98669e10i	1.15532i	0.816276	+	0.577662i	$0.196035\pi$
$431$	3.98669e10i	1.15532i	−0.816276	+	0.577662i	$0.803965\pi$
$432$	0	0
$433$	3.21564e10	0.914779	0.457389	−	0.889266i	$-0.348785\pi$
$433$	3.21564e10	0.914779	0.457389	+	0.889266i	$0.348785\pi$
$434$	0	0
$435$	4.08285e10i	1.14027i
$436$	0	0
$437$	1.37927e10	0.378201
$438$	0	0
$439$	2.72016e10i	0.732379i	0.930540	+	0.366189i	$0.119338\pi$
$439$	2.72016e10i	0.732379i	−0.930540	+	0.366189i	$0.880662\pi$
$440$	0	0
$441$	3.77630e10	0.998418
$442$	0	0
$443$	1.32228e10i	0.343327i	0.985156	+	0.171663i	$0.0549142\pi$
$443$	1.32228e10i	0.343327i	−0.985156	+	0.171663i	$0.945086\pi$
$444$	0	0
$445$	1.07142e10	0.273224
$446$	0	0
$447$	− 8.42834e10i	− 2.11111i
$448$	0	0
$449$	2.82958e10	0.696203	0.348102	−	0.937457i	$-0.386826\pi$
$449$	2.82958e10	0.696203	0.348102	+	0.937457i	$0.386826\pi$
$450$	0	0
$451$	7.85831e10i	1.89943i
$452$	0	0
$453$	5.64718e10	1.34103
$454$	0	0
$455$	2.39261e10i	0.558248i
$456$	0	0
$457$	7.89688e10	1.81047	0.905234	−	0.424914i	$-0.139696\pi$
$457$	7.89688e10	1.81047	0.905234	+	0.424914i	$0.139696\pi$
$458$	0	0
$459$	4.75146e9i	0.107047i
$460$	0	0
$461$	−2.63713e10	−0.583885	−0.291943	−	0.956436i	$-0.594302\pi$
$461$	−2.63713e10	−0.583885	−0.291943	+	0.956436i	$0.594302\pi$
$462$	0	0
$463$	2.74281e10i	0.596858i	0.954432	+	0.298429i	$0.0964627\pi$
$463$	2.74281e10i	0.596858i	−0.954432	+	0.298429i	$0.903537\pi$
$464$	0	0
$465$	−3.65282e10	−0.781297
$466$	0	0
$467$	− 8.24023e10i	− 1.73249i	−0.499616	−	0.866247i	$-0.666525\pi$
$467$	− 8.24023e10i	− 1.73249i	0.499616	−	0.866247i	$-0.333475\pi$
$468$	0	0
$469$	1.36925e10	0.283004
$470$	0	0
$471$	− 1.28868e11i	− 2.61855i
$472$	0	0
$473$	4.57104e10	0.913209
$474$	0	0
$475$	− 3.97805e9i	− 0.0781441i
$476$	0	0
$477$	−3.06914e10	−0.592847
$478$	0	0
$479$	1.28142e10i	0.243415i	0.992566	+	0.121708i	$0.0388370\pi$
$479$	1.28142e10i	0.243415i	−0.992566	+	0.121708i	$0.961163\pi$
$480$	0	0
$481$	−7.18296e10	−1.34191
$482$	0	0
$483$	− 1.05663e11i	− 1.94149i
$484$	0	0
$485$	2.20039e10	0.397680
$486$	0	0
$487$	2.27188e10i	0.403895i	0.979396	+	0.201948i	$0.0647271\pi$
$487$	2.27188e10i	0.403895i	−0.979396	+	0.201948i	$0.935273\pi$
$488$	0	0
$489$	9.68695e10	1.69415
$490$	0	0
$491$	− 8.70411e10i	− 1.49761i	−0.662791	−	0.748805i	$-0.730628\pi$
$491$	− 8.70411e10i	− 1.49761i	0.662791	−	0.748805i	$-0.269372\pi$
$492$	0	0
$493$	5.80866e10	0.983305
$494$	0	0
$495$	2.48921e10i	0.414612i
$496$	0	0
$497$	−1.13904e11	−1.86688
$498$	0	0
$499$	− 1.34972e10i	− 0.217691i	−0.994059	−	0.108846i	$-0.965285\pi$
$499$	− 1.34972e10i	− 0.217691i	0.994059	−	0.108846i	$-0.0347154\pi$
$500$	0	0
$501$	−5.10441e10	−0.810205
$502$	0	0
$503$	− 4.13249e10i	− 0.645565i	−0.946473	−	0.322782i	$-0.895382\pi$
$503$	− 4.13249e10i	− 0.645565i	0.946473	−	0.322782i	$-0.104618\pi$
$504$	0	0
$505$	3.25487e9	0.0500459
$506$	0	0
$507$	− 2.54759e10i	− 0.385564i
$508$	0	0
$509$	3.37390e9	0.0502645	0.0251322	−	0.999684i	$-0.491999\pi$
$509$	3.37390e9	0.0502645	0.0251322	+	0.999684i	$0.491999\pi$
$510$	0	0
$511$	6.77557e10i	0.993715i
$512$	0	0
$513$	5.52248e9	0.0797379
$514$	0	0
$515$	− 1.92461e10i	− 0.273598i
$516$	0	0
$517$	−2.02761e8	−0.00283807
$518$	0	0
$519$	− 5.19653e10i	− 0.716216i
$520$	0	0
$521$	1.24497e11	1.68969	0.844847	−	0.535007i	$-0.179691\pi$
$521$	1.24497e11	1.68969	0.844847	+	0.535007i	$0.179691\pi$
$522$	0	0
$523$	− 4.47960e10i	− 0.598733i	−0.954138	−	0.299366i	$-0.903225\pi$
$523$	− 4.47960e10i	− 0.598733i	0.954138	−	0.299366i	$-0.0967753\pi$
$524$	0	0
$525$	−3.04751e10	−0.401151
$526$	0	0
$527$	5.19686e10i	0.673749i
$528$	0	0
$529$	4.93792e9	0.0630553
$530$	0	0
$531$	− 3.72995e10i	− 0.469165i
$532$	0	0
$533$	−1.18964e11	−1.47404
$534$	0	0
$535$	− 2.15135e10i	− 0.262601i
$536$	0	0
$537$	−9.15263e10	−1.10065
$538$	0	0
$539$	1.08144e11i	1.28129i
$540$	0	0
$541$	8.68526e10	1.01390	0.506948	−	0.861977i	$-0.330774\pi$
$541$	8.68526e10	1.01390	0.506948	+	0.861977i	$0.330774\pi$
$542$	0	0
$543$	− 6.82787e10i	− 0.785391i
$544$	0	0
$545$	1.20044e9	0.0136068
$546$	0	0
$547$	1.65768e11i	1.85162i	0.377993	+	0.925808i	$0.376614\pi$
$547$	1.65768e11i	1.85162i	−0.377993	+	0.925808i	$0.623386\pi$
$548$	0	0
$549$	−1.39721e10	−0.153805
$550$	0	0
$551$	− 6.75124e10i	− 0.732448i
$552$	0	0
$553$	−1.88886e11	−2.01976
$554$	0	0
$555$	− 9.14903e10i	− 0.964280i
$556$	0	0
$557$	9.73769e10	1.01166	0.505831	−	0.862633i	$-0.331186\pi$
$557$	9.73769e10	1.01166	0.505831	+	0.862633i	$0.331186\pi$
$558$	0	0
$559$	6.91995e10i	0.708689i
$560$	0	0
$561$	7.70796e10	0.778195
$562$	0	0
$563$	1.96004e9i	0.0195088i	0.999952	+	0.00975440i	$0.00310497\pi$
$563$	1.96004e9i	0.0195088i	−0.999952	+	0.00975440i	$0.996895\pi$
$564$	0	0
$565$	−6.54913e10	−0.642673
$566$	0	0
$567$	− 1.71853e11i	− 1.66274i
$568$	0	0
$569$	7.88598e10	0.752327	0.376164	−	0.926553i	$-0.377243\pi$
$569$	7.88598e10	0.752327	0.376164	+	0.926553i	$0.377243\pi$
$570$	0	0
$571$	− 5.71331e10i	− 0.537457i	−0.963216	−	0.268728i	$-0.913397\pi$
$571$	− 5.71331e10i	− 0.537457i	0.963216	−	0.268728i	$-0.0866034\pi$
$572$	0	0
$573$	4.67145e9	0.0433345
$574$	0	0
$575$	2.11621e10i	0.193592i
$576$	0	0
$577$	1.51459e11	1.36644	0.683221	−	0.730212i	$-0.260579\pi$
$577$	1.51459e11	1.36644	0.683221	+	0.730212i	$0.260579\pi$
$578$	0	0
$579$	4.63984e10i	0.412847i
$580$	0	0
$581$	2.20506e10	0.193516
$582$	0	0
$583$	− 8.78922e10i	− 0.760810i
$584$	0	0
$585$	−3.76834e10	−0.321756
$586$	0	0
$587$	1.33645e11i	1.12564i	0.826580	+	0.562819i	$0.190283\pi$
$587$	1.33645e11i	1.12564i	−0.826580	+	0.562819i	$0.809717\pi$
$588$	0	0
$589$	6.04015e10	0.501865
$590$	0	0
$591$	2.13670e11i	1.75143i
$592$	0	0
$593$	−1.58419e11	−1.28112	−0.640558	−	0.767910i	$-0.721297\pi$
$593$	−1.58419e11	−1.28112	−0.640558	+	0.767910i	$0.721297\pi$
$594$	0	0
$595$	4.33568e10i	0.345931i
$596$	0	0
$597$	−2.32741e10	−0.183221
$598$	0	0
$599$	− 4.31245e10i	− 0.334978i	−0.985874	−	0.167489i	$-0.946434\pi$
$599$	− 4.31245e10i	− 0.334978i	0.985874	−	0.167489i	$-0.0535659\pi$
$600$	0	0
$601$	4.15092e10	0.318160	0.159080	−	0.987266i	$-0.449147\pi$
$601$	4.15092e10	0.318160	0.159080	+	0.987266i	$0.449147\pi$
$602$	0	0
$603$	2.15656e10i	0.163114i
$604$	0	0
$605$	−1.13696e10	−0.0848637
$606$	0	0
$607$	2.83174e10i	0.208592i	0.994546	+	0.104296i	$0.0332590\pi$
$607$	2.83174e10i	0.208592i	−0.994546	+	0.104296i	$0.966741\pi$
$608$	0	0
$609$	−5.17199e11	−3.76000
$610$	0	0
$611$	− 3.06954e8i	− 0.00220246i
$612$	0	0
$613$	−2.40024e11	−1.69986	−0.849928	−	0.526899i	$-0.823355\pi$
$613$	−2.40024e11	−1.69986	−0.849928	+	0.526899i	$0.823355\pi$
$614$	0	0
$615$	− 1.51527e11i	− 1.05923i
$616$	0	0
$617$	−1.16795e10	−0.0805904	−0.0402952	−	0.999188i	$-0.512830\pi$
$617$	−1.16795e10	−0.0805904	−0.0402952	+	0.999188i	$0.512830\pi$
$618$	0	0
$619$	− 2.19401e11i	− 1.49443i	−0.664583	−	0.747215i	$-0.731391\pi$
$619$	− 2.19401e11i	− 1.49443i	0.664583	−	0.747215i	$-0.268609\pi$
$620$	0	0
$621$	−2.93780e10	−0.197540
$622$	0	0
$623$	1.35723e11i	0.900948i
$624$	0	0
$625$	6.10352e9	0.0400000
$626$	0	0
$627$	− 8.95874e10i	− 0.579665i
$628$	0	0
$629$	−1.30163e11	−0.831544
$630$	0	0
$631$	2.35335e10i	0.148446i	0.997242	+	0.0742230i	$0.0236477\pi$
$631$	2.35335e10i	0.148446i	−0.997242	+	0.0742230i	$0.976352\pi$
$632$	0	0
$633$	2.21890e11	1.38205
$634$	0	0
$635$	− 8.22718e10i	− 0.506006i
$636$	0	0
$637$	−1.63715e11	−0.994332
$638$	0	0
$639$	− 1.79398e11i	− 1.07601i
$640$	0	0
$641$	−8.47231e10	−0.501845	−0.250923	−	0.968007i	$-0.580734\pi$
$641$	−8.47231e10	−0.501845	−0.250923	+	0.968007i	$0.580734\pi$
$642$	0	0
$643$	2.37102e11i	1.38705i	0.720435	+	0.693523i	$0.243942\pi$
$643$	2.37102e11i	1.38705i	−0.720435	+	0.693523i	$0.756058\pi$
$644$	0	0
$645$	−8.81404e10	−0.509256
$646$	0	0
$647$	2.88712e11i	1.64758i	0.566892	+	0.823792i	$0.308146\pi$
$647$	2.88712e11i	1.64758i	−0.566892	+	0.823792i	$0.691854\pi$
$648$	0	0
$649$	1.06816e11	0.602086
$650$	0	0
$651$	− 4.62724e11i	− 2.57631i
$652$	0	0
$653$	−7.01205e10	−0.385649	−0.192824	−	0.981233i	$-0.561765\pi$
$653$	−7.01205e10	−0.385649	−0.192824	+	0.981233i	$0.561765\pi$
$654$	0	0
$655$	9.59600e10i	0.521345i
$656$	0	0
$657$	−1.06714e11	−0.572746
$658$	0	0
$659$	1.30280e11i	0.690774i	0.938460	+	0.345387i	$0.112252\pi$
$659$	1.30280e11i	0.690774i	−0.938460	+	0.345387i	$0.887748\pi$
$660$	0	0
$661$	9.13098e10	0.478312	0.239156	−	0.970981i	$-0.423129\pi$
$661$	9.13098e10	0.478312	0.239156	+	0.970981i	$0.423129\pi$
$662$	0	0
$663$	1.16688e11i	0.603912i
$664$	0	0
$665$	5.03923e10	0.257678
$666$	0	0
$667$	3.59146e11i	1.81454i
$668$	0	0
$669$	9.99097e10	0.498773
$670$	0	0
$671$	− 4.00124e10i	− 0.197381i
$672$	0	0
$673$	−2.19105e11	−1.06805	−0.534026	−	0.845468i	$-0.679321\pi$
$673$	−2.19105e11	−1.06805	−0.534026	+	0.845468i	$0.679321\pi$
$674$	0	0
$675$	8.47313e9i	0.0408158i
$676$	0	0
$677$	8.12354e10	0.386715	0.193357	−	0.981128i	$-0.438062\pi$
$677$	8.12354e10	0.386715	0.193357	+	0.981128i	$0.438062\pi$
$678$	0	0
$679$	2.78737e11i	1.31134i
$680$	0	0
$681$	1.02598e10	0.0477037
$682$	0	0
$683$	2.91761e11i	1.34074i	0.742028	+	0.670369i	$0.233864\pi$
$683$	2.91761e11i	1.34074i	−0.742028	+	0.670369i	$0.766136\pi$
$684$	0	0
$685$	5.68909e10	0.258393
$686$	0	0
$687$	5.70507e10i	0.256115i
$688$	0	0
$689$	1.33057e11	0.590421
$690$	0	0
$691$	1.35594e11i	0.594743i	0.954762	+	0.297371i	$0.0961100\pi$
$691$	1.35594e11i	0.594743i	−0.954762	+	0.297371i	$0.903890\pi$
$692$	0	0
$693$	−3.15323e11	−1.36717
$694$	0	0
$695$	− 7.02420e10i	− 0.301063i
$696$	0	0
$697$	−2.15577e11	−0.913420
$698$	0	0
$699$	− 1.32664e11i	− 0.555707i
$700$	0	0
$701$	−1.48441e11	−0.614725	−0.307363	−	0.951592i	$-0.599446\pi$
$701$	−1.48441e11	−0.614725	−0.307363	+	0.951592i	$0.599446\pi$
$702$	0	0
$703$	1.51285e11i	0.619404i
$704$	0	0
$705$	3.90972e8	0.00158267
$706$	0	0
$707$	4.12314e10i	0.165025i
$708$	0	0
$709$	−4.79726e11	−1.89849	−0.949245	−	0.314536i	$-0.898151\pi$
$709$	−4.79726e11	−1.89849	−0.949245	+	0.314536i	$0.898151\pi$
$710$	0	0
$711$	− 2.97493e11i	− 1.16412i
$712$	0	0
$713$	−3.21318e11	−1.24330
$714$	0	0
$715$	− 1.07916e11i	− 0.412915i
$716$	0	0
$717$	2.33659e11	0.884110
$718$	0	0
$719$	− 4.02314e11i	− 1.50539i	−0.658368	−	0.752696i	$-0.728753\pi$
$719$	− 4.02314e11i	− 1.50539i	0.658368	−	0.752696i	$-0.271247\pi$
$720$	0	0
$721$	2.43801e11	0.902184
$722$	0	0
$723$	− 7.55163e10i	− 0.276368i
$724$	0	0
$725$	1.03584e11	0.374922
$726$	0	0
$727$	3.49293e11i	1.25041i	0.780461	+	0.625204i	$0.214984\pi$
$727$	3.49293e11i	1.25041i	−0.780461	+	0.625204i	$0.785016\pi$
$728$	0	0
$729$	1.92272e11	0.680777
$730$	0	0
$731$	1.25397e11i	0.439155i
$732$	0	0
$733$	1.75651e11	0.608464	0.304232	−	0.952598i	$-0.401600\pi$
$733$	1.75651e11	0.608464	0.304232	+	0.952598i	$0.401600\pi$
$734$	0	0
$735$	− 2.08526e11i	− 0.714516i
$736$	0	0
$737$	−6.17583e10	−0.209327
$738$	0	0
$739$	1.85701e11i	0.622640i	0.950305	+	0.311320i	$0.100771\pi$
$739$	1.85701e11i	0.622640i	−0.950305	+	0.311320i	$0.899229\pi$
$740$	0	0
$741$	1.35624e11	0.449844
$742$	0	0
$743$	2.85901e11i	0.938126i	0.883165	+	0.469063i	$0.155408\pi$
$743$	2.85901e11i	0.938126i	−0.883165	+	0.469063i	$0.844592\pi$
$744$	0	0
$745$	−2.13831e11	−0.694138
$746$	0	0
$747$	3.47295e10i	0.111536i
$748$	0	0
$749$	2.72524e11	0.865919
$750$	0	0
$751$	− 6.18679e11i	− 1.94494i	−0.233031	−	0.972469i	$-0.574864\pi$
$751$	− 6.18679e11i	− 1.94494i	0.233031	−	0.972469i	$-0.425136\pi$
$752$	0	0
$753$	−4.89399e11	−1.52224
$754$	0	0
$755$	− 1.43272e11i	− 0.440934i
$756$	0	0
$757$	−9.34034e10	−0.284433	−0.142216	−	0.989836i	$-0.545423\pi$
$757$	−9.34034e10	−0.284433	−0.142216	+	0.989836i	$0.545423\pi$
$758$	0	0
$759$	4.76579e11i	1.43604i
$760$	0	0
$761$	−3.21388e11	−0.958276	−0.479138	−	0.877740i	$-0.659051\pi$
$761$	−3.21388e11	−0.958276	−0.479138	+	0.877740i	$0.659051\pi$
$762$	0	0
$763$	1.52067e10i	0.0448681i
$764$	0	0
$765$	−6.82865e10	−0.199384
$766$	0	0
$767$	1.61706e11i	0.467245i
$768$	0	0
$769$	2.44957e10	0.0700461	0.0350231	−	0.999387i	$-0.488850\pi$
$769$	2.44957e10	0.0700461	0.0350231	+	0.999387i	$0.488850\pi$
$770$	0	0
$771$	1.88591e11i	0.533709i
$772$	0	0
$773$	−1.61194e11	−0.451472	−0.225736	−	0.974189i	$-0.572479\pi$
$773$	−1.61194e11	−0.451472	−0.225736	+	0.974189i	$0.572479\pi$
$774$	0	0
$775$	9.26739e10i	0.256892i
$776$	0	0
$777$	1.15896e12	3.17969
$778$	0	0
$779$	2.50558e11i	0.680392i
$780$	0	0
$781$	5.13751e11	1.38086
$782$	0	0
$783$	1.43799e11i	0.382569i
$784$	0	0
$785$	−3.26945e11	−0.860985
$786$	0	0
$787$	− 4.76190e11i	− 1.24131i	−0.784083	−	0.620657i	$-0.786866\pi$
$787$	− 4.76190e11i	− 1.24131i	0.784083	−	0.620657i	$-0.213134\pi$
$788$	0	0
$789$	−2.76084e11	−0.712416
$790$	0	0
$791$	− 8.29617e11i	− 2.11920i
$792$	0	0
$793$	6.05735e10	0.153176
$794$	0	0
$795$	1.69477e11i	0.424270i
$796$	0	0
$797$	1.59422e11	0.395107	0.197553	−	0.980292i	$-0.436700\pi$
$797$	1.59422e11	0.395107	0.197553	+	0.980292i	$0.436700\pi$
$798$	0	0
$799$	− 5.56235e8i	− 0.00136481i
$800$	0	0
$801$	−2.13762e11	−0.519278
$802$	0	0
$803$	− 3.05603e11i	− 0.735013i
$804$	0	0
$805$	−2.68072e11	−0.638364
$806$	0	0
$807$	− 3.99751e10i	− 0.0942530i
$808$	0	0
$809$	−4.86253e11	−1.13519	−0.567594	−	0.823308i	$-0.692126\pi$
$809$	−4.86253e11	−1.13519	−0.567594	+	0.823308i	$0.692126\pi$
$810$	0	0
$811$	− 3.39760e11i	− 0.785396i	−0.919667	−	0.392698i	$-0.871542\pi$
$811$	− 3.39760e11i	− 0.785396i	0.919667	−	0.392698i	$-0.128458\pi$
$812$	0	0
$813$	−1.16408e12	−2.66454
$814$	0	0
$815$	− 2.45763e11i	− 0.557039i
$816$	0	0
$817$	1.45745e11	0.327120
$818$	0	0
$819$	− 4.77358e11i	− 1.06098i
$820$	0	0
$821$	−6.74268e11	−1.48409	−0.742044	−	0.670351i	$-0.766144\pi$
$821$	−6.74268e11	−1.48409	−0.742044	+	0.670351i	$0.766144\pi$
$822$	0	0
$823$	− 3.11660e11i	− 0.679332i	−0.940546	−	0.339666i	$-0.889686\pi$
$823$	− 3.11660e11i	− 0.679332i	0.940546	−	0.339666i	$-0.110314\pi$
$824$	0	0
$825$	1.37454e11	0.296716
$826$	0	0
$827$	3.10210e11i	0.663183i	0.943423	+	0.331592i	$0.107586\pi$
$827$	3.10210e11i	0.663183i	−0.943423	+	0.331592i	$0.892414\pi$
$828$	0	0
$829$	3.09076e11	0.654406	0.327203	−	0.944954i	$-0.393894\pi$
$829$	3.09076e11	0.654406	0.327203	+	0.944954i	$0.393894\pi$
$830$	0	0
$831$	1.50082e11i	0.314720i
$832$	0	0
$833$	−2.96670e11	−0.616160
$834$	0	0
$835$	1.29502e11i	0.266397i
$836$	0	0
$837$	−1.28653e11	−0.262132
$838$	0	0
$839$	5.22492e11i	1.05446i	0.849722	+	0.527232i	$0.176770\pi$
$839$	5.22492e11i	1.05446i	−0.849722	+	0.527232i	$0.823230\pi$
$840$	0	0
$841$	1.25770e12	2.51416
$842$	0	0
$843$	− 3.32163e11i	− 0.657721i
$844$	0	0
$845$	−6.46336e10	−0.126774
$846$	0	0
$847$	− 1.44025e11i	− 0.279836i
$848$	0	0
$849$	−1.52278e11	−0.293094
$850$	0	0
$851$	− 8.04790e11i	− 1.53449i
$852$	0	0
$853$	5.25270e11	0.992171	0.496086	−	0.868274i	$-0.334770\pi$
$853$	5.25270e11	0.992171	0.496086	+	0.868274i	$0.334770\pi$
$854$	0	0
$855$	7.93674e10i	0.148518i
$856$	0	0
$857$	3.18270e11	0.590028	0.295014	−	0.955493i	$-0.404676\pi$
$857$	3.18270e11	0.590028	0.295014	+	0.955493i	$0.404676\pi$
$858$	0	0
$859$	− 8.61467e11i	− 1.58222i	−0.611676	−	0.791108i	$-0.709504\pi$
$859$	− 8.61467e11i	− 1.58222i	0.611676	−	0.791108i	$-0.290496\pi$
$860$	0	0
$861$	1.91948e12	3.49277
$862$	0	0
$863$	− 1.40931e11i	− 0.254075i	−0.991898	−	0.127038i	$-0.959453\pi$
$863$	− 1.40931e11i	− 0.254075i	0.991898	−	0.127038i	$-0.0405469\pi$
$864$	0	0
$865$	−1.31839e11	−0.235493
$866$	0	0
$867$	− 5.57071e11i	− 0.985903i
$868$	0	0
$869$	8.51945e11	1.49394
$870$	0	0
$871$	− 9.34940e10i	− 0.162447i
$872$	0	0
$873$	−4.39008e11	−0.755814
$874$	0	0
$875$	7.73168e10i	0.131899i
$876$	0	0
$877$	4.00362e11	0.676790	0.338395	−	0.941004i	$-0.390116\pi$
$877$	4.00362e11	0.676790	0.338395	+	0.941004i	$0.390116\pi$
$878$	0	0
$879$	− 6.46426e11i	− 1.08284i
$880$	0	0
$881$	4.94890e11	0.821496	0.410748	−	0.911749i	$-0.365268\pi$
$881$	4.94890e11	0.821496	0.410748	+	0.911749i	$0.365268\pi$
$882$	0	0
$883$	− 7.88039e11i	− 1.29630i	−0.761513	−	0.648149i	$-0.775543\pi$
$883$	− 7.88039e11i	− 1.29630i	0.761513	−	0.648149i	$-0.224457\pi$
$884$	0	0
$885$	−2.05967e11	−0.335757
$886$	0	0
$887$	− 2.78785e10i	− 0.0450375i	−0.999746	−	0.0225188i	$-0.992831\pi$
$887$	− 2.78785e10i	− 0.0450375i	0.999746	−	0.0225188i	$-0.00716855\pi$
$888$	0	0
$889$	1.04218e12	1.66854
$890$	0	0
$891$	7.75120e11i	1.22987i
$892$	0	0
$893$	−6.46496e8	−0.00101662
$894$	0	0
$895$	2.32207e11i	0.361895i
$896$	0	0
$897$	−7.21478e11	−1.11443
$898$	0	0
$899$	1.57279e12i	2.40786i
$900$	0	0
$901$	2.41114e11	0.365868
$902$	0	0
$903$	− 1.11653e12i	− 1.67926i
$904$	0	0
$905$	−1.73226e11	−0.258238
$906$	0	0
$907$	5.94476e11i	0.878426i	0.898383	+	0.439213i	$0.144743\pi$
$907$	5.94476e11i	0.878426i	−0.898383	+	0.439213i	$0.855257\pi$
$908$	0	0
$909$	−6.49390e10	−0.0951152
$910$	0	0
$911$	5.11180e11i	0.742165i	0.928600	+	0.371082i	$0.121013\pi$
$911$	5.11180e11i	0.742165i	−0.928600	+	0.371082i	$0.878987\pi$
$912$	0	0
$913$	−9.94564e10	−0.143136
$914$	0	0
$915$	7.71534e10i	0.110070i
$916$	0	0
$917$	−1.21558e12	−1.71912
$918$	0	0
$919$	7.89745e11i	1.10720i	0.832784	+	0.553598i	$0.186746\pi$
$919$	7.89745e11i	1.10720i	−0.832784	+	0.553598i	$0.813254\pi$
$920$	0	0
$921$	6.38313e11	0.887146
$922$	0	0
$923$	7.77751e11i	1.07160i
$924$	0	0
$925$	−2.32116e11	−0.317057
$926$	0	0
$927$	3.83985e11i	0.519990i
$928$	0	0
$929$	1.28241e12	1.72172	0.860862	−	0.508839i	$-0.169925\pi$
$929$	1.28241e12	1.72172	0.860862	+	0.508839i	$0.169925\pi$
$930$	0	0
$931$	3.44811e11i	0.458968i
$932$	0	0
$933$	−9.22414e11	−1.21730
$934$	0	0
$935$	− 1.95555e11i	− 0.255872i
$936$	0	0
$937$	−4.16365e11	−0.540152	−0.270076	−	0.962839i	$-0.587049\pi$
$937$	−4.16365e11	−0.540152	−0.270076	+	0.962839i	$0.587049\pi$
$938$	0	0
$939$	1.35258e12i	1.73980i
$940$	0	0
$941$	1.33006e12	1.69634	0.848171	−	0.529723i	$-0.177704\pi$
$941$	1.33006e12	1.69634	0.848171	+	0.529723i	$0.177704\pi$
$942$	0	0
$943$	− 1.33290e12i	− 1.68558i
$944$	0	0
$945$	−1.07334e11	−0.134589
$946$	0	0
$947$	− 1.36285e11i	− 0.169453i	−0.996404	−	0.0847265i	$-0.972998\pi$
$947$	− 1.36285e11i	− 0.169453i	0.996404	−	0.0847265i	$-0.0270017\pi$
$948$	0	0
$949$	4.62642e11	0.570401
$950$	0	0
$951$	1.52524e12i	1.86473i
$952$	0	0
$953$	1.22326e12	1.48302	0.741509	−	0.670943i	$-0.234111\pi$
$953$	1.22326e12	1.48302	0.741509	+	0.670943i	$0.234111\pi$
$954$	0	0
$955$	− 1.18517e10i	− 0.0142485i
$956$	0	0
$957$	2.33275e12	2.78113
$958$	0	0
$959$	7.20671e11i	0.852045i
$960$	0	0
$961$	−5.54240e11	−0.649837
$962$	0	0
$963$	4.29222e11i	0.499088i
$964$	0	0
$965$	1.17715e11	0.135745
$966$	0	0
$967$	− 7.99217e10i	− 0.0914027i	−0.998955	−	0.0457014i	$-0.985448\pi$
$967$	− 7.99217e10i	− 0.0914027i	0.998955	−	0.0457014i	$-0.0145523\pi$
$968$	0	0
$969$	2.45765e11	0.278756
$970$	0	0
$971$	3.43166e11i	0.386036i	0.981195	+	0.193018i	$0.0618276\pi$
$971$	3.43166e11i	0.386036i	−0.981195	+	0.193018i	$0.938172\pi$
$972$	0	0
$973$	8.89796e11	0.992748
$974$	0	0
$975$	2.08087e11i	0.230264i
$976$	0	0
$977$	−2.96831e11	−0.325785	−0.162893	−	0.986644i	$-0.552082\pi$
$977$	−2.96831e11	−0.325785	−0.162893	+	0.986644i	$0.552082\pi$
$978$	0	0
$979$	− 6.12158e11i	− 0.666397i
$980$	0	0
$981$	−2.39505e10	−0.0258605
$982$	0	0
$983$	9.12247e11i	0.977009i	0.872561	+	0.488504i	$0.162457\pi$
$983$	9.12247e11i	0.977009i	−0.872561	+	0.488504i	$0.837543\pi$
$984$	0	0
$985$	5.42091e11	0.575873
$986$	0	0
$987$	4.95267e9i	0.00521880i
$988$	0	0
$989$	−7.75323e11	−0.810396
$990$	0	0
$991$	9.63646e11i	0.999132i	0.866276	+	0.499566i	$0.166507\pi$
$991$	9.63646e11i	0.999132i	−0.866276	+	0.499566i	$0.833493\pi$
$992$	0	0
$993$	−8.50231e11	−0.874460
$994$	0	0
$995$	5.90475e10i	0.0602434i
$996$	0	0
$997$	1.13678e12	1.15052	0.575261	−	0.817970i	$-0.304900\pi$
$997$	1.13678e12	1.15052	0.575261	+	0.817970i	$0.304900\pi$
$998$	0	0
$999$	− 3.22232e11i	− 0.323524i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.b.d.191.14		16
4.3	odd	2	inner	320.9.b.d.191.3		16
8.3	odd	2		20.9.b.a.11.5	✓	16
8.5	even	2		20.9.b.a.11.6	yes	16
24.5	odd	2		180.9.c.a.91.11		16
24.11	even	2		180.9.c.a.91.12		16
40.3	even	4		100.9.d.c.99.6		32
40.13	odd	4		100.9.d.c.99.28		32
40.19	odd	2		100.9.b.d.51.12		16
40.27	even	4		100.9.d.c.99.27		32
40.29	even	2		100.9.b.d.51.11		16
40.37	odd	4		100.9.d.c.99.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.b.a.11.5	✓	16	8.3	odd	2
20.9.b.a.11.6	yes	16	8.5	even	2
100.9.b.d.51.11		16	40.29	even	2
100.9.b.d.51.12		16	40.19	odd	2
100.9.d.c.99.5		32	40.37	odd	4
100.9.d.c.99.6		32	40.3	even	4
100.9.d.c.99.27		32	40.27	even	4
100.9.d.c.99.28		32	40.13	odd	4
180.9.c.a.91.11		16	24.5	odd	2
180.9.c.a.91.12		16	24.11	even	2
320.9.b.d.191.3		16	4.3	odd	2	inner
320.9.b.d.191.14		16	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+110.171i q^{3} +279.508 q^{5} +3540.70i q^{7} -5576.56 q^{9} -15969.8i q^{11} +24176.2 q^{13} +30793.6i q^{15} +43810.0 q^{17} -50919.1i q^{19} -390081. q^{21} +270875. i q^{23} +78125.0 q^{25} +108456. i q^{27} +1.32588e6 q^{29} +1.18623e6i q^{31} +1.75941e6 q^{33} +989655. i q^{35} -2.97108e6 q^{37} +2.66351e6i q^{39} -4.92072e6 q^{41} +2.86229e6i q^{43} -1.55870e6 q^{45} -12696.5i q^{47} -6.77174e6 q^{49} +4.82658e6i q^{51} +5.50364e6 q^{53} -4.46371e6i q^{55} +5.60979e6 q^{57} +6.68863e6i q^{59} +2.50550e6 q^{61} -1.97449e7i q^{63} +6.75746e6 q^{65} -3.86718e6i q^{67} -2.98424e7 q^{69} +3.21701e7i q^{71} +1.91362e7 q^{73} +8.60708e6i q^{75} +5.65444e7 q^{77} +5.33471e7i q^{79} -4.85365e7 q^{81} -6.22776e6i q^{83} +1.22453e7 q^{85} +1.46072e8i q^{87} +3.83322e7 q^{89} +8.56008e7i q^{91} -1.30687e8 q^{93} -1.42323e7i q^{95} +7.87237e7 q^{97} +8.90568e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) \) 16 * q - 38800 * q^9 - 51392 * q^13 + 27552 * q^17 - 414496 * q^21 + 1250000 * q^25 - 2764896 * q^29 - 5521600 * q^33 - 9009472 * q^37 - 8576448 * q^41 - 1580000 * q^45 - 32803600 * q^49 - 2452032 * q^53 + 11957760 * q^57 - 8371712 * q^61 + 9060000 * q^65 - 7527264 * q^69 + 61907232 * q^73 + 156997440 * q^77 + 140586672 * q^81 + 106960000 * q^85 + 106647456 * q^89 - 105563840 * q^93 + 171851232 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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