LMFDB - Embedded newform 320.9.b.d.191.15

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			191.15
Root			$-6.29294 + 5.63875i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.9.b.d.191.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+137.297i q^{3} +279.508 q^{5} -3940.57i q^{7} -12289.4 q^{9} +17097.0i q^{11} -10938.7 q^{13} +38375.6i q^{15} +101666. q^{17} -93432.1i q^{19} +541029. q^{21} +147346. i q^{23} +78125.0 q^{25} -786497. i q^{27} -41637.4 q^{29} -138577. i q^{31} -2.34736e6 q^{33} -1.10142e6i q^{35} -1.14978e6 q^{37} -1.50185e6i q^{39} +3.83906e6 q^{41} +3.18959e6i q^{43} -3.43500e6 q^{45} +3.51218e6i q^{47} -9.76332e6 q^{49} +1.39584e7i q^{51} -5.66612e6 q^{53} +4.77874e6i q^{55} +1.28279e7 q^{57} +1.69068e7i q^{59} +5.16010e6 q^{61} +4.84274e7i q^{63} -3.05745e6 q^{65} +1.05358e7i q^{67} -2.02302e7 q^{69} -1.85971e7i q^{71} +2.38535e6 q^{73} +1.07263e7i q^{75} +6.73718e7 q^{77} +4.42560e7i q^{79} +2.73525e7 q^{81} +1.51824e7i q^{83} +2.84164e7 q^{85} -5.71668e6i q^{87} -5.42739e7 q^{89} +4.31047e7i q^{91} +1.90261e7 q^{93} -2.61151e7i q^{95} -1.24798e8 q^{97} -2.10112e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	137.297i	1.69502i	0.530777	+	0.847512i	$0.321900\pi$
$3$	137.297i	1.69502i	−0.530777	+	0.847512i	$0.678100\pi$
$4$	0	0
$5$	279.508	0.447214
$5$	279.508	0.447214
$6$	0	0
$7$	− 3940.57i	− 1.64122i	−0.571487	−	0.820611i	$-0.693633\pi$
$7$	− 3940.57i	− 1.64122i	0.571487	−	0.820611i	$-0.306367\pi$
$8$	0	0
$9$	−12289.4	−1.87310
$10$	0	0
$11$	17097.0i	1.16775i	0.811845	+	0.583873i	$0.198463\pi$
$11$	17097.0i	1.16775i	−0.811845	+	0.583873i	$0.801537\pi$
$12$	0	0
$13$	−10938.7	−0.382994	−0.191497	−	0.981493i	$-0.561334\pi$
$13$	−10938.7	−0.382994	−0.191497	+	0.981493i	$0.561334\pi$
$14$	0	0
$15$	38375.6i	0.758037i
$16$	0	0
$17$	101666.	1.21725	0.608624	−	0.793459i	$-0.291722\pi$
$17$	101666.	1.21725	0.608624	+	0.793459i	$0.291722\pi$
$18$	0	0
$19$	− 93432.1i	− 0.716938i	−0.933542	−	0.358469i	$-0.883299\pi$
$19$	− 93432.1i	− 0.716938i	0.933542	−	0.358469i	$-0.116701\pi$
$20$	0	0
$21$	541029.	2.78191
$22$	0	0
$23$	147346.i	0.526536i	0.964723	+	0.263268i	$0.0848003\pi$
$23$	147346.i	0.526536i	−0.964723	+	0.263268i	$0.915200\pi$
$24$	0	0
$25$	78125.0	0.200000
$26$	0	0
$27$	− 786497.i	− 1.47993i
$28$	0	0
$29$	−41637.4	−0.0588696	−0.0294348	−	0.999567i	$-0.509371\pi$
$29$	−41637.4	−0.0588696	−0.0294348	+	0.999567i	$0.509371\pi$
$30$	0	0
$31$	− 138577.i	− 0.150052i	−0.997182	−	0.0750262i	$-0.976096\pi$
$31$	− 138577.i	− 0.150052i	0.997182	−	0.0750262i	$-0.0239040\pi$
$32$	0	0
$33$	−2.34736e6	−1.97936
$34$	0	0
$35$	− 1.10142e6i	− 0.733977i
$36$	0	0
$37$	−1.14978e6	−0.613489	−0.306745	−	0.951792i	$-0.599240\pi$
$37$	−1.14978e6	−0.613489	−0.306745	+	0.951792i	$0.599240\pi$
$38$	0	0
$39$	− 1.50185e6i	− 0.649183i
$40$	0	0
$41$	3.83906e6	1.35859	0.679297	−	0.733864i	$-0.262285\pi$
$41$	3.83906e6	1.35859	0.679297	+	0.733864i	$0.262285\pi$
$42$	0	0
$43$	3.18959e6i	0.932956i	0.884533	+	0.466478i	$0.154477\pi$
$43$	3.18959e6i	0.932956i	−0.884533	+	0.466478i	$0.845523\pi$
$44$	0	0
$45$	−3.43500e6	−0.837678
$46$	0	0
$47$	3.51218e6i	0.719756i	0.932999	+	0.359878i	$0.117182\pi$
$47$	3.51218e6i	0.719756i	−0.932999	+	0.359878i	$0.882818\pi$
$48$	0	0
$49$	−9.76332e6	−1.69361
$50$	0	0
$51$	1.39584e7i	2.06326i
$52$	0	0
$53$	−5.66612e6	−0.718096	−0.359048	−	0.933319i	$-0.616899\pi$
$53$	−5.66612e6	−0.718096	−0.359048	+	0.933319i	$0.616899\pi$
$54$	0	0
$55$	4.77874e6i	0.522231i
$56$	0	0
$57$	1.28279e7	1.21523
$58$	0	0
$59$	1.69068e7i	1.39525i	0.716461	+	0.697627i	$0.245761\pi$
$59$	1.69068e7i	1.39525i	−0.716461	+	0.697627i	$0.754239\pi$
$60$	0	0
$61$	5.16010e6	0.372682	0.186341	−	0.982485i	$-0.440337\pi$
$61$	5.16010e6	0.372682	0.186341	+	0.982485i	$0.440337\pi$
$62$	0	0
$63$	4.84274e7i	3.07418i
$64$	0	0
$65$	−3.05745e6	−0.171280
$66$	0	0
$67$	1.05358e7i	0.522838i	0.965225	+	0.261419i	$0.0841904\pi$
$67$	1.05358e7i	0.522838i	−0.965225	+	0.261419i	$0.915810\pi$
$68$	0	0
$69$	−2.02302e7	−0.892490
$70$	0	0
$71$	− 1.85971e7i	− 0.731833i	−0.930648	−	0.365916i	$-0.880756\pi$
$71$	− 1.85971e7i	− 0.731833i	0.930648	−	0.365916i	$-0.119244\pi$
$72$	0	0
$73$	2.38535e6	0.0839964	0.0419982	−	0.999118i	$-0.486628\pi$
$73$	2.38535e6	0.0839964	0.0419982	+	0.999118i	$0.486628\pi$
$74$	0	0
$75$	1.07263e7i	0.339005i
$76$	0	0
$77$	6.73718e7	1.91653
$78$	0	0
$79$	4.42560e7i	1.13622i	0.822951	+	0.568112i	$0.192326\pi$
$79$	4.42560e7i	1.13622i	−0.822951	+	0.568112i	$0.807674\pi$
$80$	0	0
$81$	2.73525e7	0.635415
$82$	0	0
$83$	1.51824e7i	0.319910i	0.987124	+	0.159955i	$0.0511349\pi$
$83$	1.51824e7i	0.319910i	−0.987124	+	0.159955i	$0.948865\pi$
$84$	0	0
$85$	2.84164e7	0.544369
$86$	0	0
$87$	− 5.71668e6i	− 0.0997854i
$88$	0	0
$89$	−5.42739e7	−0.865030	−0.432515	−	0.901627i	$-0.642374\pi$
$89$	−5.42739e7	−0.865030	−0.432515	+	0.901627i	$0.642374\pi$
$90$	0	0
$91$	4.31047e7i	0.628577i
$92$	0	0
$93$	1.90261e7	0.254342
$94$	0	0
$95$	− 2.61151e7i	− 0.320624i
$96$	0	0
$97$	−1.24798e8	−1.40968	−0.704841	−	0.709365i	$-0.748982\pi$
$97$	−1.24798e8	−1.40968	−0.704841	+	0.709365i	$0.748982\pi$
$98$	0	0
$99$	− 2.10112e8i	− 2.18731i
$100$	0	0
$101$	−1.96354e8	−1.88693	−0.943463	−	0.331478i	$-0.892453\pi$
$101$	−1.96354e8	−1.88693	−0.943463	+	0.331478i	$0.892453\pi$
$102$	0	0
$103$	1.08404e8i	0.963152i	0.876404	+	0.481576i	$0.159935\pi$
$103$	1.08404e8i	0.963152i	−0.876404	+	0.481576i	$0.840065\pi$
$104$	0	0
$105$	1.51222e8	1.24411
$106$	0	0
$107$	− 5.19293e7i	− 0.396166i	−0.980185	−	0.198083i	$-0.936528\pi$
$107$	− 5.19293e7i	− 0.396166i	0.980185	−	0.198083i	$-0.0634715\pi$
$108$	0	0
$109$	−2.01131e8	−1.42486	−0.712432	−	0.701741i	$-0.752406\pi$
$109$	−2.01131e8	−1.42486	−0.712432	+	0.701741i	$0.752406\pi$
$110$	0	0
$111$	− 1.57861e8i	− 1.03988i
$112$	0	0
$113$	−6.82663e7	−0.418690	−0.209345	−	0.977842i	$-0.567133\pi$
$113$	−6.82663e7	−0.418690	−0.209345	+	0.977842i	$0.567133\pi$
$114$	0	0
$115$	4.11845e7i	0.235474i
$116$	0	0
$117$	1.34430e8	0.717387
$118$	0	0
$119$	− 4.00621e8i	− 1.99777i
$120$	0	0
$121$	−7.79470e7	−0.363629
$122$	0	0
$123$	5.27091e8i	2.30285i
$124$	0	0
$125$	2.18366e7	0.0894427
$126$	0	0
$127$	3.48984e8i	1.34150i	0.741683	+	0.670751i	$0.234028\pi$
$127$	3.48984e8i	1.34150i	−0.741683	+	0.670751i	$0.765972\pi$
$128$	0	0
$129$	−4.37921e8	−1.58138
$130$	0	0
$131$	1.49192e8i	0.506596i	0.967388	+	0.253298i	$0.0815153\pi$
$131$	1.49192e8i	0.506596i	−0.967388	+	0.253298i	$0.918485\pi$
$132$	0	0
$133$	−3.68176e8	−1.17665
$134$	0	0
$135$	− 2.19832e8i	− 0.661846i
$136$	0	0
$137$	6.05661e8	1.71928	0.859642	−	0.510897i	$-0.170687\pi$
$137$	6.05661e8	1.71928	0.859642	+	0.510897i	$0.170687\pi$
$138$	0	0
$139$	− 5.19290e8i	− 1.39107i	−0.718490	−	0.695537i	$-0.755166\pi$
$139$	− 5.19290e8i	− 1.39107i	0.718490	−	0.695537i	$-0.244834\pi$
$140$	0	0
$141$	−4.82211e8	−1.22000
$142$	0	0
$143$	− 1.87018e8i	− 0.447239i
$144$	0	0
$145$	−1.16380e7	−0.0263273
$146$	0	0
$147$	− 1.34047e9i	− 2.87071i
$148$	0	0
$149$	6.13964e8	1.24565	0.622827	−	0.782359i	$-0.285984\pi$
$149$	6.13964e8	1.24565	0.622827	+	0.782359i	$0.285984\pi$
$150$	0	0
$151$	5.41162e8i	1.04093i	0.853884	+	0.520463i	$0.174241\pi$
$151$	5.41162e8i	1.04093i	−0.853884	+	0.520463i	$0.825759\pi$
$152$	0	0
$153$	−1.24941e9	−2.28003
$154$	0	0
$155$	− 3.87333e7i	− 0.0671055i
$156$	0	0
$157$	6.43982e8	1.05993	0.529963	−	0.848021i	$-0.322206\pi$
$157$	6.43982e8	1.05993	0.529963	+	0.848021i	$0.322206\pi$
$158$	0	0
$159$	− 7.77941e8i	− 1.21719i
$160$	0	0
$161$	5.80629e8	0.864162
$162$	0	0
$163$	− 6.74607e8i	− 0.955653i	−0.878454	−	0.477826i	$-0.841425\pi$
$163$	− 6.74607e8i	− 0.955653i	0.878454	−	0.477826i	$-0.158575\pi$
$164$	0	0
$165$	−6.56107e8	−0.885195
$166$	0	0
$167$	− 4.55554e8i	− 0.585699i	−0.956159	−	0.292849i	$-0.905397\pi$
$167$	− 4.55554e8i	− 0.585699i	0.956159	−	0.292849i	$-0.0946035\pi$
$168$	0	0
$169$	−6.96076e8	−0.853316
$170$	0	0
$171$	1.14823e9i	1.34290i
$172$	0	0
$173$	−9.64344e8	−1.07658	−0.538292	−	0.842759i	$-0.680930\pi$
$173$	−9.64344e8	−1.07658	−0.538292	+	0.842759i	$0.680930\pi$
$174$	0	0
$175$	− 3.07857e8i	− 0.328244i
$176$	0	0
$177$	−2.32125e9	−2.36499
$178$	0	0
$179$	− 1.01365e9i	− 0.987363i	−0.869643	−	0.493682i	$-0.835651\pi$
$179$	− 1.01365e9i	− 0.987363i	0.869643	−	0.493682i	$-0.164349\pi$
$180$	0	0
$181$	−1.51507e8	−0.141162	−0.0705812	−	0.997506i	$-0.522485\pi$
$181$	−1.51507e8	−0.141162	−0.0705812	+	0.997506i	$0.522485\pi$
$182$	0	0
$183$	7.08465e8i	0.631705i
$184$	0	0
$185$	−3.21373e8	−0.274361
$186$	0	0
$187$	1.73817e9i	1.42143i
$188$	0	0
$189$	−3.09925e9	−2.42890
$190$	0	0
$191$	− 5.46186e8i	− 0.410399i	−0.978720	−	0.205200i	$-0.934216\pi$
$191$	− 5.46186e8i	− 0.410399i	0.978720	−	0.205200i	$-0.0657844\pi$
$192$	0	0
$193$	1.52005e9	1.09554	0.547771	−	0.836628i	$-0.315477\pi$
$193$	1.52005e9	1.09554	0.547771	+	0.836628i	$0.315477\pi$
$194$	0	0
$195$	− 4.19779e8i	− 0.290323i
$196$	0	0
$197$	−1.98770e9	−1.31974	−0.659868	−	0.751382i	$-0.729388\pi$
$197$	−1.98770e9	−1.31974	−0.659868	+	0.751382i	$0.729388\pi$
$198$	0	0
$199$	1.68735e9i	1.07595i	0.842960	+	0.537976i	$0.180811\pi$
$199$	1.68735e9i	1.07595i	−0.842960	+	0.537976i	$0.819189\pi$
$200$	0	0
$201$	−1.44653e9	−0.886222
$202$	0	0
$203$	1.64075e8i	0.0966182i
$204$	0	0
$205$	1.07305e9	0.607582
$206$	0	0
$207$	− 1.81080e9i	− 0.986256i
$208$	0	0
$209$	1.59740e9	0.837201
$210$	0	0
$211$	3.55239e9i	1.79222i	0.443834	+	0.896109i	$0.353618\pi$
$211$	3.55239e9i	1.79222i	−0.443834	+	0.896109i	$0.646382\pi$
$212$	0	0
$213$	2.55332e9	1.24047
$214$	0	0
$215$	8.91518e8i	0.417231i
$216$	0	0
$217$	−5.46071e8	−0.246269
$218$	0	0
$219$	3.27501e8i	0.142376i
$220$	0	0
$221$	−1.11209e9	−0.466198
$222$	0	0
$223$	1.41187e9i	0.570920i	0.958391	+	0.285460i	$0.0921465\pi$
$223$	1.41187e9i	0.570920i	−0.958391	+	0.285460i	$0.907854\pi$
$224$	0	0
$225$	−9.60112e8	−0.374621
$226$	0	0
$227$	1.08981e9i	0.410438i	0.978716	+	0.205219i	$0.0657906\pi$
$227$	1.08981e9i	0.410438i	−0.978716	+	0.205219i	$0.934209\pi$
$228$	0	0
$229$	2.25807e9	0.821097	0.410549	−	0.911839i	$-0.365337\pi$
$229$	2.25807e9	0.821097	0.410549	+	0.911839i	$0.365337\pi$
$230$	0	0
$231$	9.24994e9i	3.24856i
$232$	0	0
$233$	6.52483e8	0.221384	0.110692	−	0.993855i	$-0.464693\pi$
$233$	6.52483e8	0.221384	0.110692	+	0.993855i	$0.464693\pi$
$234$	0	0
$235$	9.81684e8i	0.321885i
$236$	0	0
$237$	−6.07622e9	−1.92593
$238$	0	0
$239$	− 4.79300e9i	− 1.46898i	−0.678619	−	0.734490i	$-0.737421\pi$
$239$	− 4.79300e9i	− 1.46898i	0.678619	−	0.734490i	$-0.262579\pi$
$240$	0	0
$241$	−5.18381e9	−1.53667	−0.768335	−	0.640048i	$-0.778915\pi$
$241$	−5.18381e9	−1.53667	−0.768335	+	0.640048i	$0.778915\pi$
$242$	0	0
$243$	− 1.40479e9i	− 0.402889i
$244$	0	0
$245$	−2.72893e9	−0.757405
$246$	0	0
$247$	1.02202e9i	0.274583i
$248$	0	0
$249$	−2.08449e9	−0.542255
$250$	0	0
$251$	2.41878e9i	0.609398i	0.952449	+	0.304699i	$0.0985558\pi$
$251$	2.41878e9i	0.609398i	−0.952449	+	0.304699i	$0.901444\pi$
$252$	0	0
$253$	−2.51917e9	−0.614859
$254$	0	0
$255$	3.90149e9i	0.922719i
$256$	0	0
$257$	−5.25554e9	−1.20472	−0.602358	−	0.798226i	$-0.705772\pi$
$257$	−5.25554e9	−1.20472	−0.602358	+	0.798226i	$0.705772\pi$
$258$	0	0
$259$	4.53078e9i	1.00687i
$260$	0	0
$261$	5.11700e8	0.110269
$262$	0	0
$263$	− 1.24951e9i	− 0.261165i	−0.991437	−	0.130583i	$-0.958315\pi$
$263$	− 1.24951e9i	− 0.261165i	0.991437	−	0.130583i	$-0.0416848\pi$
$264$	0	0
$265$	−1.58373e9	−0.321142
$266$	0	0
$267$	− 7.45164e9i	− 1.46625i
$268$	0	0
$269$	5.36357e9	1.02434	0.512171	−	0.858884i	$-0.328841\pi$
$269$	5.36357e9	1.02434	0.512171	+	0.858884i	$0.328841\pi$
$270$	0	0
$271$	6.72329e8i	0.124654i	0.998056	+	0.0623268i	$0.0198521\pi$
$271$	6.72329e8i	0.124654i	−0.998056	+	0.0623268i	$0.980148\pi$
$272$	0	0
$273$	−5.91814e9	−1.06545
$274$	0	0
$275$	1.33570e9i	0.233549i
$276$	0	0
$277$	−8.55117e9	−1.45247	−0.726234	−	0.687448i	$-0.758731\pi$
$277$	−8.55117e9	−1.45247	−0.726234	+	0.687448i	$0.758731\pi$
$278$	0	0
$279$	1.70303e9i	0.281064i
$280$	0	0
$281$	−8.47677e8	−0.135958	−0.0679791	−	0.997687i	$-0.521655\pi$
$281$	−8.47677e8	−0.135958	−0.0679791	+	0.997687i	$0.521655\pi$
$282$	0	0
$283$	2.59848e9i	0.405111i	0.979271	+	0.202555i	$0.0649246\pi$
$283$	2.59848e9i	0.405111i	−0.979271	+	0.202555i	$0.935075\pi$
$284$	0	0
$285$	3.58552e9	0.543466
$286$	0	0
$287$	− 1.51281e10i	− 2.22975i
$288$	0	0
$289$	3.36016e9	0.481691
$290$	0	0
$291$	− 1.71344e10i	− 2.38944i
$292$	0	0
$293$	−1.18893e10	−1.61319	−0.806594	−	0.591107i	$-0.798691\pi$
$293$	−1.18893e10	−1.61319	−0.806594	+	0.591107i	$0.798691\pi$
$294$	0	0
$295$	4.72559e9i	0.623976i
$296$	0	0
$297$	1.34467e10	1.72818
$298$	0	0
$299$	− 1.61177e9i	− 0.201660i
$300$	0	0
$301$	1.25688e10	1.53119
$302$	0	0
$303$	− 2.69588e10i	− 3.19838i
$304$	0	0
$305$	1.44229e9	0.166669
$306$	0	0
$307$	− 1.74109e10i	− 1.96005i	−0.198868	−	0.980026i	$-0.563727\pi$
$307$	− 1.74109e10i	− 1.96005i	0.198868	−	0.980026i	$-0.436273\pi$
$308$	0	0
$309$	−1.48835e10	−1.63257
$310$	0	0
$311$	8.61717e9i	0.921135i	0.887625	+	0.460567i	$0.152354\pi$
$311$	8.61717e9i	0.921135i	−0.887625	+	0.460567i	$0.847646\pi$
$312$	0	0
$313$	−9.76978e9	−1.01791	−0.508953	−	0.860794i	$-0.669967\pi$
$313$	−9.76978e9	−1.01791	−0.508953	+	0.860794i	$0.669967\pi$
$314$	0	0
$315$	1.35359e10i	1.37482i
$316$	0	0
$317$	6.57561e9	0.651177	0.325589	−	0.945512i	$-0.394438\pi$
$317$	6.57561e9	0.651177	0.325589	+	0.945512i	$0.394438\pi$
$318$	0	0
$319$	− 7.11872e8i	− 0.0687447i
$320$	0	0
$321$	7.12973e9	0.671510
$322$	0	0
$323$	− 9.49884e9i	− 0.872691i
$324$	0	0
$325$	−8.54584e8	−0.0765987
$326$	0	0
$327$	− 2.76147e10i	− 2.41518i
$328$	0	0
$329$	1.38400e10	1.18128
$330$	0	0
$331$	2.32794e10i	1.93936i	0.244370	+	0.969682i	$0.421419\pi$
$331$	2.32794e10i	1.93936i	−0.244370	+	0.969682i	$0.578581\pi$
$332$	0	0
$333$	1.41301e10	1.14913
$334$	0	0
$335$	2.94484e9i	0.233820i
$336$	0	0
$337$	−1.99301e9	−0.154522	−0.0772610	−	0.997011i	$-0.524617\pi$
$337$	−1.99301e9	−0.154522	−0.0772610	+	0.997011i	$0.524617\pi$
$338$	0	0
$339$	− 9.37275e9i	− 0.709690i
$340$	0	0
$341$	2.36924e9	0.175223
$342$	0	0
$343$	1.57565e10i	1.13837i
$344$	0	0
$345$	−5.65451e9	−0.399134
$346$	0	0
$347$	6.31855e9i	0.435813i	0.975970	+	0.217906i	$0.0699227\pi$
$347$	6.31855e9i	0.435813i	−0.975970	+	0.217906i	$0.930077\pi$
$348$	0	0
$349$	1.87010e9	0.126056	0.0630279	−	0.998012i	$-0.479924\pi$
$349$	1.87010e9	0.126056	0.0630279	+	0.998012i	$0.479924\pi$
$350$	0	0
$351$	8.60323e9i	0.566804i
$352$	0	0
$353$	3.02132e10	1.94580	0.972900	−	0.231225i	$-0.0742734\pi$
$353$	3.02132e10	1.94580	0.972900	+	0.231225i	$0.0742734\pi$
$354$	0	0
$355$	− 5.19805e9i	− 0.327286i
$356$	0	0
$357$	5.50041e10	3.38627
$358$	0	0
$359$	1.82078e10i	1.09618i	0.836421	+	0.548088i	$0.184644\pi$
$359$	1.82078e10i	1.09618i	−0.836421	+	0.548088i	$0.815356\pi$
$360$	0	0
$361$	8.25401e9	0.486000
$362$	0	0
$363$	− 1.07019e10i	− 0.616359i
$364$	0	0
$365$	6.66725e8	0.0375643
$366$	0	0
$367$	1.46648e10i	0.808372i	0.914677	+	0.404186i	$0.132445\pi$
$367$	1.46648e10i	0.808372i	−0.914677	+	0.404186i	$0.867555\pi$
$368$	0	0
$369$	−4.71799e10	−2.54479
$370$	0	0
$371$	2.23278e10i	1.17855i
$372$	0	0
$373$	−1.41277e10	−0.729857	−0.364928	−	0.931036i	$-0.618906\pi$
$373$	−1.41277e10	−0.729857	−0.364928	+	0.931036i	$0.618906\pi$
$374$	0	0
$375$	2.99810e9i	0.151607i
$376$	0	0
$377$	4.55458e8	0.0225467
$378$	0	0
$379$	− 3.94069e9i	− 0.190992i	−0.995430	−	0.0954960i	$-0.969556\pi$
$379$	− 3.94069e9i	− 0.190992i	0.995430	−	0.0954960i	$-0.0304437\pi$
$380$	0	0
$381$	−4.79145e10	−2.27388
$382$	0	0
$383$	3.43063e10i	1.59433i	0.603760	+	0.797166i	$0.293668\pi$
$383$	3.43063e10i	1.59433i	−0.603760	+	0.797166i	$0.706332\pi$
$384$	0	0
$385$	1.88310e10	0.857098
$386$	0	0
$387$	− 3.91983e10i	− 1.74752i
$388$	0	0
$389$	−4.14309e8	−0.0180936	−0.00904681	−	0.999959i	$-0.502880\pi$
$389$	−4.14309e8	−0.0180936	−0.00904681	+	0.999959i	$0.502880\pi$
$390$	0	0
$391$	1.49801e10i	0.640924i
$392$	0	0
$393$	−2.04837e10	−0.858692
$394$	0	0
$395$	1.23699e10i	0.508135i
$396$	0	0
$397$	−2.25453e10	−0.907598	−0.453799	−	0.891104i	$-0.649932\pi$
$397$	−2.25453e10	−0.907598	−0.453799	+	0.891104i	$0.649932\pi$
$398$	0	0
$399$	− 5.05494e10i	− 1.99446i
$400$	0	0
$401$	−2.53666e10	−0.981037	−0.490519	−	0.871431i	$-0.663193\pi$
$401$	−2.53666e10	−0.981037	−0.490519	+	0.871431i	$0.663193\pi$
$402$	0	0
$403$	1.51584e9i	0.0574691i
$404$	0	0
$405$	7.64527e9	0.284166
$406$	0	0
$407$	− 1.96577e10i	− 0.716399i
$408$	0	0
$409$	1.47057e10	0.525523	0.262761	−	0.964861i	$-0.415367\pi$
$409$	1.47057e10	0.525523	0.262761	+	0.964861i	$0.415367\pi$
$410$	0	0
$411$	8.31554e10i	2.91423i
$412$	0	0
$413$	6.66225e10	2.28992
$414$	0	0
$415$	4.24361e9i	0.143068i
$416$	0	0
$417$	7.12968e10	2.35790
$418$	0	0
$419$	7.30915e9i	0.237143i	0.992946	+	0.118572i	$0.0378315\pi$
$419$	7.30915e9i	0.237143i	−0.992946	+	0.118572i	$0.962168\pi$
$420$	0	0
$421$	4.09140e10	1.30240	0.651199	−	0.758907i	$-0.274266\pi$
$421$	4.09140e10	1.30240	0.651199	+	0.758907i	$0.274266\pi$
$422$	0	0
$423$	− 4.31627e10i	− 1.34818i
$424$	0	0
$425$	7.94263e9	0.243449
$426$	0	0
$427$	− 2.03337e10i	− 0.611654i
$428$	0	0
$429$	2.56770e10	0.758080
$430$	0	0
$431$	3.95128e9i	0.114506i	0.998360	+	0.0572531i	$0.0182342\pi$
$431$	3.95128e9i	0.114506i	−0.998360	+	0.0572531i	$0.981766\pi$
$432$	0	0
$433$	2.88725e10	0.821358	0.410679	−	0.911780i	$-0.365292\pi$
$433$	2.88725e10	0.821358	0.410679	+	0.911780i	$0.365292\pi$
$434$	0	0
$435$	− 1.59786e9i	− 0.0446254i
$436$	0	0
$437$	1.37669e10	0.377493
$438$	0	0
$439$	− 1.66340e10i	− 0.447857i	−0.974606	−	0.223929i	$-0.928112\pi$
$439$	− 1.66340e10i	− 0.447857i	0.974606	−	0.223929i	$-0.0718883\pi$
$440$	0	0
$441$	1.19986e11	3.17231
$442$	0	0
$443$	− 1.61256e10i	− 0.418699i	−0.977841	−	0.209350i	$-0.932865\pi$
$443$	− 1.61256e10i	− 0.418699i	0.977841	−	0.209350i	$-0.0671346\pi$
$444$	0	0
$445$	−1.51700e10	−0.386853
$446$	0	0
$447$	8.42953e10i	2.11141i
$448$	0	0
$449$	1.31471e10	0.323478	0.161739	−	0.986834i	$-0.448290\pi$
$449$	1.31471e10	0.323478	0.161739	+	0.986834i	$0.448290\pi$
$450$	0	0
$451$	6.56363e10i	1.58649i
$452$	0	0
$453$	−7.42999e10	−1.76439
$454$	0	0
$455$	1.20481e10i	0.281108i
$456$	0	0
$457$	−6.79985e10	−1.55896	−0.779480	−	0.626427i	$-0.784516\pi$
$457$	−6.79985e10	−1.55896	−0.779480	+	0.626427i	$0.784516\pi$
$458$	0	0
$459$	− 7.99597e10i	− 1.80144i
$460$	0	0
$461$	2.86783e10	0.634966	0.317483	−	0.948264i	$-0.397162\pi$
$461$	2.86783e10	0.634966	0.317483	+	0.948264i	$0.397162\pi$
$462$	0	0
$463$	4.22729e9i	0.0919894i	0.998942	+	0.0459947i	$0.0146457\pi$
$463$	4.22729e9i	0.0919894i	−0.998942	+	0.0459947i	$0.985354\pi$
$464$	0	0
$465$	5.31796e9	0.113745
$466$	0	0
$467$	4.27269e10i	0.898325i	0.893450	+	0.449163i	$0.148278\pi$
$467$	4.27269e10i	0.898325i	−0.893450	+	0.449163i	$0.851722\pi$
$468$	0	0
$469$	4.15170e10	0.858093
$470$	0	0
$471$	8.84167e10i	1.79660i
$472$	0	0
$473$	−5.45323e10	−1.08946
$474$	0	0
$475$	− 7.29938e9i	− 0.143388i
$476$	0	0
$477$	6.96334e10	1.34507
$478$	0	0
$479$	6.73017e10i	1.27845i	0.769020	+	0.639225i	$0.220745\pi$
$479$	6.73017e10i	1.27845i	−0.769020	+	0.639225i	$0.779255\pi$
$480$	0	0
$481$	1.25770e10	0.234962
$482$	0	0
$483$	7.97185e10i	1.46477i
$484$	0	0
$485$	−3.48821e10	−0.630429
$486$	0	0
$487$	− 9.69928e10i	− 1.72434i	−0.506616	−	0.862172i	$-0.669104\pi$
$487$	− 9.69928e10i	− 1.72434i	0.506616	−	0.862172i	$-0.330896\pi$
$488$	0	0
$489$	9.26214e10	1.61985
$490$	0	0
$491$	− 1.73575e10i	− 0.298648i	−0.988788	−	0.149324i	$-0.952290\pi$
$491$	− 1.73575e10i	− 0.298648i	0.988788	−	0.149324i	$-0.0477098\pi$
$492$	0	0
$493$	−4.23309e9	−0.0716589
$494$	0	0
$495$	− 5.87281e10i	− 0.978194i
$496$	0	0
$497$	−7.32833e10	−1.20110
$498$	0	0
$499$	1.42959e10i	0.230573i	0.993332	+	0.115287i	$0.0367787\pi$
$499$	1.42959e10i	0.230573i	−0.993332	+	0.115287i	$0.963221\pi$
$500$	0	0
$501$	6.25462e10	0.992773
$502$	0	0
$503$	6.90584e10i	1.07881i	0.842047	+	0.539404i	$0.181351\pi$
$503$	6.90584e10i	1.07881i	−0.842047	+	0.539404i	$0.818649\pi$
$504$	0	0
$505$	−5.48827e10	−0.843859
$506$	0	0
$507$	− 9.55691e10i	− 1.44639i
$508$	0	0
$509$	5.98939e10	0.892302	0.446151	−	0.894958i	$-0.352794\pi$
$509$	5.98939e10	0.892302	0.446151	+	0.894958i	$0.352794\pi$
$510$	0	0
$511$	− 9.39964e9i	− 0.137857i
$512$	0	0
$513$	−7.34840e10	−1.06102
$514$	0	0
$515$	3.02997e10i	0.430735i
$516$	0	0
$517$	−6.00476e10	−0.840491
$518$	0	0
$519$	− 1.32401e11i	− 1.82483i
$520$	0	0
$521$	2.97388e10	0.403620	0.201810	−	0.979425i	$-0.435318\pi$
$521$	2.97388e10	0.403620	0.201810	+	0.979425i	$0.435318\pi$
$522$	0	0
$523$	3.20935e10i	0.428954i	0.976729	+	0.214477i	$0.0688047\pi$
$523$	3.20935e10i	0.428954i	−0.976729	+	0.214477i	$0.931195\pi$
$524$	0	0
$525$	4.22679e10	0.556382
$526$	0	0
$527$	− 1.40885e10i	− 0.182651i
$528$	0	0
$529$	5.66001e10	0.722760
$530$	0	0
$531$	− 2.07775e11i	− 2.61345i
$532$	0	0
$533$	−4.19943e10	−0.520333
$534$	0	0
$535$	− 1.45147e10i	− 0.177171i
$536$	0	0
$537$	1.39171e11	1.67360
$538$	0	0
$539$	− 1.66923e11i	− 1.97770i
$540$	0	0
$541$	−3.92268e10	−0.457924	−0.228962	−	0.973435i	$-0.573533\pi$
$541$	−3.92268e10	−0.457924	−0.228962	+	0.973435i	$0.573533\pi$
$542$	0	0
$543$	− 2.08015e10i	− 0.239274i
$544$	0	0
$545$	−5.62179e10	−0.637219
$546$	0	0
$547$	9.39638e10i	1.04957i	0.851235	+	0.524785i	$0.175854\pi$
$547$	9.39638e10i	1.04957i	−0.851235	+	0.524785i	$0.824146\pi$
$548$	0	0
$549$	−6.34147e10	−0.698073
$550$	0	0
$551$	3.89027e9i	0.0422059i
$552$	0	0
$553$	1.74394e11	1.86480
$554$	0	0
$555$	− 4.41235e10i	− 0.465048i
$556$	0	0
$557$	−8.98122e10	−0.933071	−0.466535	−	0.884503i	$-0.654498\pi$
$557$	−8.98122e10	−0.933071	−0.466535	+	0.884503i	$0.654498\pi$
$558$	0	0
$559$	− 3.48899e10i	− 0.357316i
$560$	0	0
$561$	−2.38646e11	−2.40936
$562$	0	0
$563$	1.80533e10i	0.179689i	0.995956	+	0.0898446i	$0.0286371\pi$
$563$	1.80533e10i	0.179689i	−0.995956	+	0.0898446i	$0.971363\pi$
$564$	0	0
$565$	−1.90810e10	−0.187244
$566$	0	0
$567$	− 1.07785e11i	− 1.04286i
$568$	0	0
$569$	−6.90947e10	−0.659167	−0.329584	−	0.944126i	$-0.606908\pi$
$569$	−6.90947e10	−0.659167	−0.329584	+	0.944126i	$0.606908\pi$
$570$	0	0
$571$	− 7.86073e10i	− 0.739467i	−0.929138	−	0.369733i	$-0.879449\pi$
$571$	− 7.86073e10i	− 0.739467i	0.929138	−	0.369733i	$-0.120551\pi$
$572$	0	0
$573$	7.49896e10	0.695637
$574$	0	0
$575$	1.15114e10i	0.105307i
$576$	0	0
$577$	−1.80277e11	−1.62643	−0.813216	−	0.581962i	$-0.802285\pi$
$577$	−1.80277e11	−1.62643	−0.813216	+	0.581962i	$0.802285\pi$
$578$	0	0
$579$	2.08698e11i	1.85697i
$580$	0	0
$581$	5.98273e10	0.525043
$582$	0	0
$583$	− 9.68734e10i	− 0.838553i
$584$	0	0
$585$	3.75744e10	0.320825
$586$	0	0
$587$	3.62809e9i	0.0305581i	0.999883	+	0.0152790i	$0.00486366\pi$
$587$	3.62809e9i	0.0305581i	−0.999883	+	0.0152790i	$0.995136\pi$
$588$	0	0
$589$	−1.29475e10	−0.107578
$590$	0	0
$591$	− 2.72906e11i	− 2.23698i
$592$	0	0
$593$	8.72511e10	0.705589	0.352795	−	0.935701i	$-0.385231\pi$
$593$	8.72511e10	0.705589	0.352795	+	0.935701i	$0.385231\pi$
$594$	0	0
$595$	− 1.11977e11i	− 0.893431i
$596$	0	0
$597$	−2.31668e11	−1.82376
$598$	0	0
$599$	− 1.90496e10i	− 0.147972i	−0.997259	−	0.0739858i	$-0.976428\pi$
$599$	− 1.90496e10i	− 0.147972i	0.997259	−	0.0739858i	$-0.0235720\pi$
$600$	0	0
$601$	1.75085e11	1.34200	0.670998	−	0.741459i	$-0.265866\pi$
$601$	1.75085e11	1.34200	0.670998	+	0.741459i	$0.265866\pi$
$602$	0	0
$603$	− 1.29479e11i	− 0.979329i
$604$	0	0
$605$	−2.17869e10	−0.162620
$606$	0	0
$607$	1.75308e11i	1.29136i	0.763608	+	0.645680i	$0.223426\pi$
$607$	1.75308e11i	1.29136i	−0.763608	+	0.645680i	$0.776574\pi$
$608$	0	0
$609$	−2.25270e10	−0.163770
$610$	0	0
$611$	− 3.84186e10i	− 0.275662i
$612$	0	0
$613$	−1.26999e10	−0.0899412	−0.0449706	−	0.998988i	$-0.514319\pi$
$613$	−1.26999e10	−0.0899412	−0.0449706	+	0.998988i	$0.514319\pi$
$614$	0	0
$615$	1.47326e11i	1.02987i
$616$	0	0
$617$	2.29769e11	1.58544	0.792720	−	0.609586i	$-0.208664\pi$
$617$	2.29769e11	1.58544	0.792720	+	0.609586i	$0.208664\pi$
$618$	0	0
$619$	1.67400e11i	1.14023i	0.821564	+	0.570116i	$0.193102\pi$
$619$	1.67400e11i	1.14023i	−0.821564	+	0.570116i	$0.806898\pi$
$620$	0	0
$621$	1.15887e11	0.779237
$622$	0	0
$623$	2.13870e11i	1.41971i
$624$	0	0
$625$	6.10352e9	0.0400000
$626$	0	0
$627$	2.19319e11i	1.41907i
$628$	0	0
$629$	−1.16893e11	−0.746768
$630$	0	0
$631$	2.92637e9i	0.0184591i	0.999957	+	0.00922957i	$0.00293791\pi$
$631$	2.92637e9i	0.0184591i	−0.999957	+	0.00922957i	$0.997062\pi$
$632$	0	0
$633$	−4.87732e11	−3.03785
$634$	0	0
$635$	9.75441e10i	0.599938i
$636$	0	0
$637$	1.06798e11	0.648642
$638$	0	0
$639$	2.28548e11i	1.37080i
$640$	0	0
$641$	−1.15318e11	−0.683071	−0.341535	−	0.939869i	$-0.610947\pi$
$641$	−1.15318e11	−0.683071	−0.341535	+	0.939869i	$0.610947\pi$
$642$	0	0
$643$	4.99010e10i	0.291921i	0.989290	+	0.145961i	$0.0466273\pi$
$643$	4.99010e10i	0.291921i	−0.989290	+	0.145961i	$0.953373\pi$
$644$	0	0
$645$	−1.22403e11	−0.707216
$646$	0	0
$647$	5.43112e10i	0.309936i	0.987919	+	0.154968i	$0.0495275\pi$
$647$	5.43112e10i	0.309936i	−0.987919	+	0.154968i	$0.950473\pi$
$648$	0	0
$649$	−2.89055e11	−1.62930
$650$	0	0
$651$	− 7.49739e10i	− 0.417432i
$652$	0	0
$653$	−1.40265e11	−0.771433	−0.385716	−	0.922617i	$-0.626046\pi$
$653$	−1.40265e11	−0.771433	−0.385716	+	0.922617i	$0.626046\pi$
$654$	0	0
$655$	4.17006e10i	0.226557i
$656$	0	0
$657$	−2.93146e10	−0.157334
$658$	0	0
$659$	2.51334e11i	1.33263i	0.745670	+	0.666315i	$0.232129\pi$
$659$	2.51334e11i	1.33263i	−0.745670	+	0.666315i	$0.767871\pi$
$660$	0	0
$661$	−1.18006e10	−0.0618155	−0.0309077	−	0.999522i	$-0.509840\pi$
$661$	−1.18006e10	−0.0618155	−0.0309077	+	0.999522i	$0.509840\pi$
$662$	0	0
$663$	− 1.52686e11i	− 0.790216i
$664$	0	0
$665$	−1.02908e11	−0.526216
$666$	0	0
$667$	− 6.13511e9i	− 0.0309970i
$668$	0	0
$669$	−1.93846e11	−0.967724
$670$	0	0
$671$	8.82220e10i	0.435198i
$672$	0	0
$673$	−3.32886e11	−1.62269	−0.811345	−	0.584567i	$-0.801264\pi$
$673$	−3.32886e11	−1.62269	−0.811345	+	0.584567i	$0.801264\pi$
$674$	0	0
$675$	− 6.14450e10i	− 0.295986i
$676$	0	0
$677$	1.55980e11	0.742530	0.371265	−	0.928527i	$-0.378924\pi$
$677$	1.55980e11	0.742530	0.371265	+	0.928527i	$0.378924\pi$
$678$	0	0
$679$	4.91776e11i	2.31360i
$680$	0	0
$681$	−1.49627e11	−0.695701
$682$	0	0
$683$	2.38756e10i	0.109716i	0.998494	+	0.0548581i	$0.0174707\pi$
$683$	2.38756e10i	0.109716i	−0.998494	+	0.0548581i	$0.982529\pi$
$684$	0	0
$685$	1.69287e11	0.768887
$686$	0	0
$687$	3.10025e11i	1.39178i
$688$	0	0
$689$	6.19799e10	0.275026
$690$	0	0
$691$	− 3.49556e11i	− 1.53322i	−0.642113	−	0.766610i	$-0.721942\pi$
$691$	− 3.49556e11i	− 1.53322i	0.642113	−	0.766610i	$-0.278058\pi$
$692$	0	0
$693$	−8.27962e11	−3.58986
$694$	0	0
$695$	− 1.45146e11i	− 0.622107i
$696$	0	0
$697$	3.90301e11	1.65374
$698$	0	0
$699$	8.95839e10i	0.375250i
$700$	0	0
$701$	2.57887e10	0.106796	0.0533982	−	0.998573i	$-0.482995\pi$
$701$	2.57887e10	0.106796	0.0533982	+	0.998573i	$0.482995\pi$
$702$	0	0
$703$	1.07426e11i	0.439834i
$704$	0	0
$705$	−1.34782e11	−0.545602
$706$	0	0
$707$	7.73749e11i	3.09686i
$708$	0	0
$709$	1.18322e11	0.468251	0.234126	−	0.972206i	$-0.424777\pi$
$709$	1.18322e11	0.468251	0.234126	+	0.972206i	$0.424777\pi$
$710$	0	0
$711$	− 5.43882e11i	− 2.12827i
$712$	0	0
$713$	2.04187e10	0.0790079
$714$	0	0
$715$	− 5.22731e10i	− 0.200011i
$716$	0	0
$717$	6.58064e11	2.48996
$718$	0	0
$719$	1.28179e11i	0.479624i	0.970819	+	0.239812i	$0.0770858\pi$
$719$	1.28179e11i	0.479624i	−0.970819	+	0.239812i	$0.922914\pi$
$720$	0	0
$721$	4.27173e11	1.58075
$722$	0	0
$723$	− 7.11721e11i	− 2.60469i
$724$	0	0
$725$	−3.25292e9	−0.0117739
$726$	0	0
$727$	2.11824e11i	0.758295i	0.925336	+	0.379148i	$0.123783\pi$
$727$	2.11824e11i	0.758295i	−0.925336	+	0.379148i	$0.876217\pi$
$728$	0	0
$729$	3.72333e11	1.31832
$730$	0	0
$731$	3.24272e11i	1.13564i
$732$	0	0
$733$	−2.18401e11	−0.756552	−0.378276	−	0.925693i	$-0.623483\pi$
$733$	−2.18401e11	−0.756552	−0.378276	+	0.925693i	$0.623483\pi$
$734$	0	0
$735$	− 3.74674e11i	− 1.28382i
$736$	0	0
$737$	−1.80129e11	−0.610541
$738$	0	0
$739$	1.87281e11i	0.627936i	0.949433	+	0.313968i	$0.101659\pi$
$739$	1.87281e11i	0.627936i	−0.949433	+	0.313968i	$0.898341\pi$
$740$	0	0
$741$	−1.40321e11	−0.465424
$742$	0	0
$743$	− 4.43366e11i	− 1.45481i	−0.686207	−	0.727406i	$-0.740726\pi$
$743$	− 4.43366e11i	− 1.45481i	0.686207	−	0.727406i	$-0.259274\pi$
$744$	0	0
$745$	1.71608e11	0.557074
$746$	0	0
$747$	− 1.86583e11i	− 0.599224i
$748$	0	0
$749$	−2.04631e11	−0.650196
$750$	0	0
$751$	− 1.63129e11i	− 0.512827i	−0.966567	−	0.256414i	$-0.917459\pi$
$751$	− 1.63129e11i	− 0.512827i	0.966567	−	0.256414i	$-0.0825409\pi$
$752$	0	0
$753$	−3.32090e11	−1.03294
$754$	0	0
$755$	1.51260e11i	0.465516i
$756$	0	0
$757$	−8.66867e10	−0.263979	−0.131989	−	0.991251i	$-0.542136\pi$
$757$	−8.66867e10	−0.263979	−0.131989	+	0.991251i	$0.542136\pi$
$758$	0	0
$759$	− 3.45875e11i	− 1.04220i
$760$	0	0
$761$	3.59912e11	1.07314	0.536571	−	0.843855i	$-0.319719\pi$
$761$	3.59912e11	1.07314	0.536571	+	0.843855i	$0.319719\pi$
$762$	0	0
$763$	7.92573e11i	2.33852i
$764$	0	0
$765$	−3.49222e11	−1.01966
$766$	0	0
$767$	− 1.84938e11i	− 0.534373i
$768$	0	0
$769$	−6.61724e11	−1.89222	−0.946110	−	0.323845i	$-0.895024\pi$
$769$	−6.61724e11	−1.89222	−0.946110	+	0.323845i	$0.895024\pi$
$770$	0	0
$771$	− 7.21569e11i	− 2.04202i
$772$	0	0
$773$	2.24564e11	0.628958	0.314479	−	0.949264i	$-0.398170\pi$
$773$	2.24564e11	0.628958	0.314479	+	0.949264i	$0.398170\pi$
$774$	0	0
$775$	− 1.08263e10i	− 0.0300105i
$776$	0	0
$777$	−6.22063e11	−1.70667
$778$	0	0
$779$	− 3.58691e11i	− 0.974027i
$780$	0	0
$781$	3.17954e11	0.854594
$782$	0	0
$783$	3.27477e10i	0.0871231i
$784$	0	0
$785$	1.79998e11	0.474013
$786$	0	0
$787$	− 6.78642e11i	− 1.76906i	−0.466486	−	0.884528i	$-0.654480\pi$
$787$	− 6.78642e11i	− 1.76906i	0.466486	−	0.884528i	$-0.345520\pi$
$788$	0	0
$789$	1.71553e11	0.442682
$790$	0	0
$791$	2.69008e11i	0.687163i
$792$	0	0
$793$	−5.64447e10	−0.142735
$794$	0	0
$795$	− 2.17441e11i	− 0.544343i
$796$	0	0
$797$	−2.55700e11	−0.633720	−0.316860	−	0.948472i	$-0.602628\pi$
$797$	−2.55700e11	−0.633720	−0.316860	+	0.948472i	$0.602628\pi$
$798$	0	0
$799$	3.57068e11i	0.876121i
$800$	0	0
$801$	6.66996e11	1.62029
$802$	0	0
$803$	4.07822e10i	0.0980863i
$804$	0	0
$805$	1.62291e11	0.386465
$806$	0	0
$807$	7.36401e11i	1.73628i
$808$	0	0
$809$	4.12911e10	0.0963968	0.0481984	−	0.998838i	$-0.484652\pi$
$809$	4.12911e10	0.0963968	0.0481984	+	0.998838i	$0.484652\pi$
$810$	0	0
$811$	2.06089e11i	0.476399i	0.971216	+	0.238200i	$0.0765573\pi$
$811$	2.06089e11i	0.476399i	−0.971216	+	0.238200i	$0.923443\pi$
$812$	0	0
$813$	−9.23087e10	−0.211291
$814$	0	0
$815$	− 1.88558e11i	− 0.427381i
$816$	0	0
$817$	2.98010e11	0.668872
$818$	0	0
$819$	− 5.29732e11i	− 1.17739i
$820$	0	0
$821$	−8.40243e11	−1.84941	−0.924703	−	0.380688i	$-0.875687\pi$
$821$	−8.40243e11	−1.84941	−0.924703	+	0.380688i	$0.875687\pi$
$822$	0	0
$823$	− 3.94949e11i	− 0.860878i	−0.902620	−	0.430439i	$-0.858359\pi$
$823$	− 3.94949e11i	− 0.860878i	0.902620	−	0.430439i	$-0.141641\pi$
$824$	0	0
$825$	−1.83387e11	−0.395871
$826$	0	0
$827$	− 3.63473e11i	− 0.777051i	−0.921438	−	0.388526i	$-0.872984\pi$
$827$	− 3.63473e11i	− 0.777051i	0.921438	−	0.388526i	$-0.127016\pi$
$828$	0	0
$829$	4.62310e11	0.978847	0.489424	−	0.872046i	$-0.337207\pi$
$829$	4.62310e11	0.978847	0.489424	+	0.872046i	$0.337207\pi$
$830$	0	0
$831$	− 1.17405e12i	− 2.46197i
$832$	0	0
$833$	−9.92595e11	−2.06154
$834$	0	0
$835$	− 1.27331e11i	− 0.261933i
$836$	0	0
$837$	−1.08990e11	−0.222067
$838$	0	0
$839$	1.39033e11i	0.280588i	0.990110	+	0.140294i	$0.0448049\pi$
$839$	1.39033e11i	0.280588i	−0.990110	+	0.140294i	$0.955195\pi$
$840$	0	0
$841$	−4.98513e11	−0.996534
$842$	0	0
$843$	− 1.16383e11i	− 0.230452i
$844$	0	0
$845$	−1.94559e11	−0.381614
$846$	0	0
$847$	3.07156e11i	0.596795i
$848$	0	0
$849$	−3.56763e11	−0.686672
$850$	0	0
$851$	− 1.69415e11i	− 0.323024i
$852$	0	0
$853$	2.95120e11	0.557446	0.278723	−	0.960372i	$-0.410089\pi$
$853$	2.95120e11	0.557446	0.278723	+	0.960372i	$0.410089\pi$
$854$	0	0
$855$	3.20939e11i	0.600563i
$856$	0	0
$857$	2.93094e11	0.543355	0.271678	−	0.962388i	$-0.412422\pi$
$857$	2.93094e11	0.543355	0.271678	+	0.962388i	$0.412422\pi$
$858$	0	0
$859$	5.58133e11i	1.02510i	0.858658	+	0.512549i	$0.171299\pi$
$859$	5.58133e11i	1.02510i	−0.858658	+	0.512549i	$0.828701\pi$
$860$	0	0
$861$	2.07704e12	3.77949
$862$	0	0
$863$	3.25698e11i	0.587181i	0.955931	+	0.293590i	$0.0948502\pi$
$863$	3.25698e11i	0.587181i	−0.955931	+	0.293590i	$0.905150\pi$
$864$	0	0
$865$	−2.69542e11	−0.481463
$866$	0	0
$867$	4.61339e11i	0.816477i
$868$	0	0
$869$	−7.56643e11	−1.32682
$870$	0	0
$871$	− 1.15247e11i	− 0.200243i
$872$	0	0
$873$	1.53370e12	2.64048
$874$	0	0
$875$	− 8.60487e10i	− 0.146795i
$876$	0	0
$877$	2.02891e11	0.342976	0.171488	−	0.985186i	$-0.445143\pi$
$877$	2.02891e11	0.342976	0.171488	+	0.985186i	$0.445143\pi$
$878$	0	0
$879$	− 1.63236e12i	− 2.73439i
$880$	0	0
$881$	−6.22197e10	−0.103282	−0.0516409	−	0.998666i	$-0.516445\pi$
$881$	−6.22197e10	−0.103282	−0.0516409	+	0.998666i	$0.516445\pi$
$882$	0	0
$883$	− 7.69290e10i	− 0.126546i	−0.997996	−	0.0632729i	$-0.979846\pi$
$883$	− 7.69290e10i	− 0.126546i	0.997996	−	0.0632729i	$-0.0201538\pi$
$884$	0	0
$885$	−6.48809e11	−1.05765
$886$	0	0
$887$	− 7.30650e10i	− 0.118036i	−0.998257	−	0.0590180i	$-0.981203\pi$
$887$	− 7.30650e10i	− 0.118036i	0.998257	−	0.0590180i	$-0.0187969\pi$
$888$	0	0
$889$	1.37520e12	2.20170
$890$	0	0
$891$	4.67645e11i	0.742003i
$892$	0	0
$893$	3.28150e11	0.516020
$894$	0	0
$895$	− 2.83324e11i	− 0.441562i
$896$	0	0
$897$	2.21291e11	0.341818
$898$	0	0
$899$	5.76996e9i	0.00883353i
$900$	0	0
$901$	−5.76050e11	−0.874100
$902$	0	0
$903$	1.72566e12i	2.59540i
$904$	0	0
$905$	−4.23475e10	−0.0631297
$906$	0	0
$907$	7.63733e11i	1.12853i	0.825594	+	0.564264i	$0.190840\pi$
$907$	7.63733e11i	1.12853i	−0.825594	+	0.564264i	$0.809160\pi$
$908$	0	0
$909$	2.41308e12	3.53441
$910$	0	0
$911$	6.04985e11i	0.878357i	0.898400	+	0.439178i	$0.144730\pi$
$911$	6.04985e11i	0.878357i	−0.898400	+	0.439178i	$0.855270\pi$
$912$	0	0
$913$	−2.59573e11	−0.373573
$914$	0	0
$915$	1.98022e11i	0.282507i
$916$	0	0
$917$	5.87904e11	0.831436
$918$	0	0
$919$	1.09773e12i	1.53898i	0.638657	+	0.769492i	$0.279490\pi$
$919$	1.09773e12i	1.53898i	−0.638657	+	0.769492i	$0.720510\pi$
$920$	0	0
$921$	2.39046e12	3.32233
$922$	0	0
$923$	2.03428e11i	0.280287i
$924$	0	0
$925$	−8.98264e10	−0.122698
$926$	0	0
$927$	− 1.33222e12i	− 1.80408i
$928$	0	0
$929$	5.68272e11	0.762945	0.381473	−	0.924380i	$-0.375417\pi$
$929$	5.68272e11	0.762945	0.381473	+	0.924380i	$0.375417\pi$
$930$	0	0
$931$	9.12208e11i	1.21421i
$932$	0	0
$933$	−1.18311e12	−1.56135
$934$	0	0
$935$	4.85834e11i	0.635685i
$936$	0	0
$937$	1.49600e11	0.194076	0.0970382	−	0.995281i	$-0.469063\pi$
$937$	1.49600e11	0.194076	0.0970382	+	0.995281i	$0.469063\pi$
$938$	0	0
$939$	− 1.34136e12i	− 1.72537i
$940$	0	0
$941$	5.28765e11	0.674379	0.337190	−	0.941437i	$-0.390524\pi$
$941$	5.28765e11	0.674379	0.337190	+	0.941437i	$0.390524\pi$
$942$	0	0
$943$	5.65671e11i	0.715348i
$944$	0	0
$945$	−8.66266e11	−1.08624
$946$	0	0
$947$	− 9.04370e11i	− 1.12447i	−0.826979	−	0.562233i	$-0.809943\pi$
$947$	− 9.04370e11i	− 1.12447i	0.826979	−	0.562233i	$-0.190057\pi$
$948$	0	0
$949$	−2.60926e10	−0.0321701
$950$	0	0
$951$	9.02811e11i	1.10376i
$952$	0	0
$953$	−5.71895e11	−0.693337	−0.346669	−	0.937988i	$-0.612687\pi$
$953$	−5.71895e11	−0.693337	−0.346669	+	0.937988i	$0.612687\pi$
$954$	0	0
$955$	− 1.52664e11i	− 0.183536i
$956$	0	0
$957$	9.77379e10	0.116524
$958$	0	0
$959$	− 2.38665e12i	− 2.82173i
$960$	0	0
$961$	8.33688e11	0.977484
$962$	0	0
$963$	6.38181e11i	0.742060i
$964$	0	0
$965$	4.24867e11	0.489941
$966$	0	0
$967$	− 9.92056e11i	− 1.13457i	−0.823522	−	0.567284i	$-0.807994\pi$
$967$	− 9.92056e11i	− 1.13457i	0.823522	−	0.567284i	$-0.192006\pi$
$968$	0	0
$969$	1.30416e12	1.47923
$970$	0	0
$971$	− 4.62553e11i	− 0.520336i	−0.965563	−	0.260168i	$-0.916222\pi$
$971$	− 4.62553e11i	− 0.520336i	0.965563	−	0.260168i	$-0.0837780\pi$
$972$	0	0
$973$	−2.04630e12	−2.28306
$974$	0	0
$975$	− 1.17332e11i	− 0.129837i
$976$	0	0
$977$	1.35557e11	0.148780	0.0743899	−	0.997229i	$-0.476299\pi$
$977$	1.35557e11	0.148780	0.0743899	+	0.997229i	$0.476299\pi$
$978$	0	0
$979$	− 9.27919e11i	− 1.01013i
$980$	0	0
$981$	2.47179e12	2.66892
$982$	0	0
$983$	− 1.95656e10i	− 0.0209546i	−0.999945	−	0.0104773i	$-0.996665\pi$
$983$	− 1.95656e10i	− 0.0209546i	0.999945	−	0.0104773i	$-0.00333509\pi$
$984$	0	0
$985$	−5.55580e11	−0.590204
$986$	0	0
$987$	1.90019e12i	2.00230i
$988$	0	0
$989$	−4.69975e11	−0.491235
$990$	0	0
$991$	4.63152e11i	0.480208i	0.970747	+	0.240104i	$0.0771815\pi$
$991$	4.63152e11i	0.480208i	−0.970747	+	0.240104i	$0.922818\pi$
$992$	0	0
$993$	−3.19619e12	−3.28727
$994$	0	0
$995$	4.71628e11i	0.481180i
$996$	0	0
$997$	5.69988e11	0.576879	0.288439	−	0.957498i	$-0.406864\pi$
$997$	5.69988e11	0.576879	0.288439	+	0.957498i	$0.406864\pi$
$998$	0	0
$999$	9.04296e11i	0.907922i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.b.d.191.15		16
4.3	odd	2	inner	320.9.b.d.191.2		16
8.3	odd	2		20.9.b.a.11.11	✓	16
8.5	even	2		20.9.b.a.11.12	yes	16
24.5	odd	2		180.9.c.a.91.5		16
24.11	even	2		180.9.c.a.91.6		16
40.3	even	4		100.9.d.c.99.7		32
40.13	odd	4		100.9.d.c.99.25		32
40.19	odd	2		100.9.b.d.51.6		16
40.27	even	4		100.9.d.c.99.26		32
40.29	even	2		100.9.b.d.51.5		16
40.37	odd	4		100.9.d.c.99.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.b.a.11.11	✓	16	8.3	odd	2
20.9.b.a.11.12	yes	16	8.5	even	2
100.9.b.d.51.5		16	40.29	even	2
100.9.b.d.51.6		16	40.19	odd	2
100.9.d.c.99.7		32	40.3	even	4
100.9.d.c.99.8		32	40.37	odd	4
100.9.d.c.99.25		32	40.13	odd	4
100.9.d.c.99.26		32	40.27	even	4
180.9.c.a.91.5		16	24.5	odd	2
180.9.c.a.91.6		16	24.11	even	2
320.9.b.d.191.2		16	4.3	odd	2	inner
320.9.b.d.191.15		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)

\(f(q)\)	\(=\)	\(q+137.297i q^{3} +279.508 q^{5} -3940.57i q^{7} -12289.4 q^{9} +17097.0i q^{11} -10938.7 q^{13} +38375.6i q^{15} +101666. q^{17} -93432.1i q^{19} +541029. q^{21} +147346. i q^{23} +78125.0 q^{25} -786497. i q^{27} -41637.4 q^{29} -138577. i q^{31} -2.34736e6 q^{33} -1.10142e6i q^{35} -1.14978e6 q^{37} -1.50185e6i q^{39} +3.83906e6 q^{41} +3.18959e6i q^{43} -3.43500e6 q^{45} +3.51218e6i q^{47} -9.76332e6 q^{49} +1.39584e7i q^{51} -5.66612e6 q^{53} +4.77874e6i q^{55} +1.28279e7 q^{57} +1.69068e7i q^{59} +5.16010e6 q^{61} +4.84274e7i q^{63} -3.05745e6 q^{65} +1.05358e7i q^{67} -2.02302e7 q^{69} -1.85971e7i q^{71} +2.38535e6 q^{73} +1.07263e7i q^{75} +6.73718e7 q^{77} +4.42560e7i q^{79} +2.73525e7 q^{81} +1.51824e7i q^{83} +2.84164e7 q^{85} -5.71668e6i q^{87} -5.42739e7 q^{89} +4.31047e7i q^{91} +1.90261e7 q^{93} -2.61151e7i q^{95} -1.24798e8 q^{97} -2.10112e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) \) 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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