LMFDB - Embedded newform 320.9.b.d.191.6

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			191.6
Root			$2.32463 - 7.96873i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.9.b.d.191.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-39.9624i q^{3} -279.508 q^{5} +2633.20i q^{7} +4964.01 q^{9} +2484.31i q^{11} -41319.8 q^{13} +11169.8i q^{15} -48625.0 q^{17} -241516. i q^{19} +105229. q^{21} +57735.4i q^{23} +78125.0 q^{25} -460567. i q^{27} +729286. q^{29} +134865. i q^{31} +99279.1 q^{33} -736001. i q^{35} -1.68171e6 q^{37} +1.65124e6i q^{39} -761790. q^{41} +2.54364e6i q^{43} -1.38748e6 q^{45} -5.65990e6i q^{47} -1.16893e6 q^{49} +1.94317e6i q^{51} -1.14302e7 q^{53} -694387. i q^{55} -9.65155e6 q^{57} +2.12902e7i q^{59} +2.63701e7 q^{61} +1.30712e7i q^{63} +1.15492e7 q^{65} +1.53982e6i q^{67} +2.30724e6 q^{69} +2.06597e7i q^{71} +2.51296e7 q^{73} -3.12206e6i q^{75} -6.54169e6 q^{77} +1.95344e7i q^{79} +1.41635e7 q^{81} -2.37076e7i q^{83} +1.35911e7 q^{85} -2.91440e7i q^{87} +3.86749e7 q^{89} -1.08803e8i q^{91} +5.38951e6 q^{93} +6.75057e7i q^{95} -9.46443e7 q^{97} +1.23322e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 39.9624i	− 0.493363i	−0.969097	−	0.246681i	$-0.920660\pi$
$3$	− 39.9624i	− 0.493363i	0.969097	−	0.246681i	$-0.0793401\pi$
$4$	0	0
$5$	−279.508	−0.447214
$5$	−279.508	−0.447214
$6$	0	0
$7$	2633.20i	1.09671i	0.836246	+	0.548354i	$0.184745\pi$
$7$	2633.20i	1.09671i	−0.836246	+	0.548354i	$0.815255\pi$
$8$	0	0
$9$	4964.01	0.756593
$10$	0	0
$11$	2484.31i	0.169682i	0.996395	+	0.0848410i	$0.0270382\pi$
$11$	2484.31i	0.169682i	−0.996395	+	0.0848410i	$0.972962\pi$
$12$	0	0
$13$	−41319.8	−1.44672	−0.723360	−	0.690471i	$-0.757403\pi$
$13$	−41319.8	−1.44672	−0.723360	+	0.690471i	$0.757403\pi$
$14$	0	0
$15$	11169.8i	0.220639i
$16$	0	0
$17$	−48625.0	−0.582189	−0.291095	−	0.956694i	$-0.594019\pi$
$17$	−48625.0	−0.582189	−0.291095	+	0.956694i	$0.594019\pi$
$18$	0	0
$19$	− 241516.i	− 1.85324i	−0.376002	−	0.926619i	$-0.622701\pi$
$19$	− 241516.i	− 1.85324i	0.376002	−	0.926619i	$-0.377299\pi$
$20$	0	0
$21$	105229.	0.541075
$22$	0	0
$23$	57735.4i	0.206315i	0.994665	+	0.103158i	$0.0328946\pi$
$23$	57735.4i	0.206315i	−0.994665	+	0.103158i	$0.967105\pi$
$24$	0	0
$25$	78125.0	0.200000
$26$	0	0
$27$	− 460567.i	− 0.866638i
$28$	0	0
$29$	729286.	1.03111	0.515556	−	0.856856i	$-0.327585\pi$
$29$	729286.	1.03111	0.515556	+	0.856856i	$0.327585\pi$
$30$	0	0
$31$	134865.i	0.146033i	0.997331	+	0.0730165i	$0.0232626\pi$
$31$	134865.i	0.146033i	−0.997331	+	0.0730165i	$0.976737\pi$
$32$	0	0
$33$	99279.1	0.0837147
$34$	0	0
$35$	− 736001.i	− 0.490463i
$36$	0	0
$37$	−1.68171e6	−0.897313	−0.448656	−	0.893704i	$-0.648097\pi$
$37$	−1.68171e6	−0.897313	−0.448656	+	0.893704i	$0.648097\pi$
$38$	0	0
$39$	1.65124e6i	0.713758i
$40$	0	0
$41$	−761790.	−0.269588	−0.134794	−	0.990874i	$-0.543037\pi$
$41$	−761790.	−0.269588	−0.134794	+	0.990874i	$0.543037\pi$
$42$	0	0
$43$	2.54364e6i	0.744016i	0.928230	+	0.372008i	$0.121331\pi$
$43$	2.54364e6i	0.744016i	−0.928230	+	0.372008i	$0.878669\pi$
$44$	0	0
$45$	−1.38748e6	−0.338359
$46$	0	0
$47$	− 5.65990e6i	− 1.15989i	−0.814655	−	0.579946i	$-0.803074\pi$
$47$	− 5.65990e6i	− 1.15989i	0.814655	−	0.579946i	$-0.196926\pi$
$48$	0	0
$49$	−1.16893e6	−0.202770
$50$	0	0
$51$	1.94317e6i	0.287230i
$52$	0	0
$53$	−1.14302e7	−1.44861	−0.724305	−	0.689480i	$-0.757839\pi$
$53$	−1.14302e7	−1.44861	−0.724305	+	0.689480i	$0.757839\pi$
$54$	0	0
$55$	− 694387.i	− 0.0758841i
$56$	0	0
$57$	−9.65155e6	−0.914318
$58$	0	0
$59$	2.12902e7i	1.75700i	0.477740	+	0.878501i	$0.341456\pi$
$59$	2.12902e7i	1.75700i	−0.477740	+	0.878501i	$0.658544\pi$
$60$	0	0
$61$	2.63701e7	1.90455	0.952276	−	0.305238i	$-0.0987361\pi$
$61$	2.63701e7	1.90455	0.952276	+	0.305238i	$0.0987361\pi$
$62$	0	0
$63$	1.30712e7i	0.829762i
$64$	0	0
$65$	1.15492e7	0.646993
$66$	0	0
$67$	1.53982e6i	0.0764137i	0.999270	+	0.0382068i	$0.0121646\pi$
$67$	1.53982e6i	0.0764137i	−0.999270	+	0.0382068i	$0.987835\pi$
$68$	0	0
$69$	2.30724e6	0.101788
$70$	0	0
$71$	2.06597e7i	0.813002i	0.913650	+	0.406501i	$0.133251\pi$
$71$	2.06597e7i	0.813002i	−0.913650	+	0.406501i	$0.866749\pi$
$72$	0	0
$73$	2.51296e7	0.884899	0.442449	−	0.896793i	$-0.354110\pi$
$73$	2.51296e7	0.884899	0.442449	+	0.896793i	$0.354110\pi$
$74$	0	0
$75$	− 3.12206e6i	− 0.0986725i
$76$	0	0
$77$	−6.54169e6	−0.186092
$78$	0	0
$79$	1.95344e7i	0.501525i	0.968049	+	0.250762i	$0.0806812\pi$
$79$	1.95344e7i	0.501525i	−0.968049	+	0.250762i	$0.919319\pi$
$80$	0	0
$81$	1.41635e7	0.329027
$82$	0	0
$83$	− 2.37076e7i	− 0.499547i	−0.968304	−	0.249773i	$-0.919644\pi$
$83$	− 2.37076e7i	− 0.499547i	0.968304	−	0.249773i	$-0.0803561\pi$
$84$	0	0
$85$	1.35911e7	0.260363
$86$	0	0
$87$	− 2.91440e7i	− 0.508712i
$88$	0	0
$89$	3.86749e7	0.616410	0.308205	−	0.951320i	$-0.400272\pi$
$89$	3.86749e7	0.616410	0.308205	+	0.951320i	$0.400272\pi$
$90$	0	0
$91$	− 1.08803e8i	− 1.58663i
$92$	0	0
$93$	5.38951e6	0.0720473
$94$	0	0
$95$	6.75057e7i	0.828793i
$96$	0	0
$97$	−9.46443e7	−1.06907	−0.534537	−	0.845145i	$-0.679514\pi$
$97$	−9.46443e7	−1.06907	−0.534537	+	0.845145i	$0.679514\pi$
$98$	0	0
$99$	1.23322e7i	0.128380i
$100$	0	0
$101$	6.52649e7	0.627183	0.313591	−	0.949558i	$-0.398468\pi$
$101$	6.52649e7	0.627183	0.313591	+	0.949558i	$0.398468\pi$
$102$	0	0
$103$	− 1.47000e8i	− 1.30607i	−0.757326	−	0.653037i	$-0.773495\pi$
$103$	− 1.47000e8i	− 1.30607i	0.757326	−	0.653037i	$-0.226505\pi$
$104$	0	0
$105$	−2.94124e7	−0.241976
$106$	0	0
$107$	2.42429e8i	1.84948i	0.380603	+	0.924739i	$0.375716\pi$
$107$	2.42429e8i	1.84948i	−0.380603	+	0.924739i	$0.624284\pi$
$108$	0	0
$109$	5.93486e7	0.420440	0.210220	−	0.977654i	$-0.432582\pi$
$109$	5.93486e7	0.420440	0.210220	+	0.977654i	$0.432582\pi$
$110$	0	0
$111$	6.72051e7i	0.442701i
$112$	0	0
$113$	5.46199e7	0.334994	0.167497	−	0.985873i	$-0.446432\pi$
$113$	5.46199e7	0.334994	0.167497	+	0.985873i	$0.446432\pi$
$114$	0	0
$115$	− 1.61375e7i	− 0.0922669i
$116$	0	0
$117$	−2.05112e8	−1.09458
$118$	0	0
$119$	− 1.28039e8i	− 0.638492i
$120$	0	0
$121$	2.08187e8	0.971208
$122$	0	0
$123$	3.04430e7i	0.133005i
$124$	0	0
$125$	−2.18366e7	−0.0894427
$126$	0	0
$127$	− 2.01099e8i	− 0.773026i	−0.922284	−	0.386513i	$-0.873679\pi$
$127$	− 2.01099e8i	− 0.773026i	0.922284	−	0.386513i	$-0.126321\pi$
$128$	0	0
$129$	1.01650e8	0.367070
$130$	0	0
$131$	− 4.36365e8i	− 1.48172i	−0.671662	−	0.740858i	$-0.734419\pi$
$131$	− 4.36365e8i	− 1.48172i	0.671662	−	0.740858i	$-0.265581\pi$
$132$	0	0
$133$	6.35959e8	2.03246
$134$	0	0
$135$	1.28732e8i	0.387572i
$136$	0	0
$137$	4.48301e8	1.27259	0.636293	−	0.771447i	$-0.280467\pi$
$137$	4.48301e8	1.27259	0.636293	+	0.771447i	$0.280467\pi$
$138$	0	0
$139$	4.73879e8i	1.26943i	0.772747	+	0.634714i	$0.218882\pi$
$139$	4.73879e8i	1.26943i	−0.772747	+	0.634714i	$0.781118\pi$
$140$	0	0
$141$	−2.26183e8	−0.572248
$142$	0	0
$143$	− 1.02651e8i	− 0.245482i
$144$	0	0
$145$	−2.03842e8	−0.461128
$146$	0	0
$147$	4.67132e7i	0.100039i
$148$	0	0
$149$	2.98504e8	0.605627	0.302814	−	0.953050i	$-0.402074\pi$
$149$	2.98504e8	0.605627	0.302814	+	0.953050i	$0.402074\pi$
$150$	0	0
$151$	7.25032e8i	1.39460i	0.716780	+	0.697299i	$0.245615\pi$
$151$	7.25032e8i	1.39460i	−0.716780	+	0.697299i	$0.754385\pi$
$152$	0	0
$153$	−2.41375e8	−0.440480
$154$	0	0
$155$	− 3.76958e7i	− 0.0653080i
$156$	0	0
$157$	6.70288e8	1.10322	0.551611	−	0.834101i	$-0.314013\pi$
$157$	6.70288e8	1.10322	0.551611	+	0.834101i	$0.314013\pi$
$158$	0	0
$159$	4.56779e8i	0.714690i
$160$	0	0
$161$	−1.52029e8	−0.226268
$162$	0	0
$163$	− 9.70735e8i	− 1.37515i	−0.726113	−	0.687575i	$-0.758675\pi$
$163$	− 9.70735e8i	− 1.37515i	0.726113	−	0.687575i	$-0.241325\pi$
$164$	0	0
$165$	−2.77493e7	−0.0374384
$166$	0	0
$167$	− 7.98683e8i	− 1.02685i	−0.858133	−	0.513427i	$-0.828376\pi$
$167$	− 7.98683e8i	− 1.02685i	0.858133	−	0.513427i	$-0.171624\pi$
$168$	0	0
$169$	8.91594e8	1.09300
$170$	0	0
$171$	− 1.19889e9i	− 1.40215i
$172$	0	0
$173$	−6.00425e8	−0.670308	−0.335154	−	0.942163i	$-0.608788\pi$
$173$	−6.00425e8	−0.670308	−0.335154	+	0.942163i	$0.608788\pi$
$174$	0	0
$175$	2.05719e8i	0.219342i
$176$	0	0
$177$	8.50809e8	0.866840
$178$	0	0
$179$	1.15845e9i	1.12840i	0.825637	+	0.564202i	$0.190816\pi$
$179$	1.15845e9i	1.12840i	−0.825637	+	0.564202i	$0.809184\pi$
$180$	0	0
$181$	7.96493e8	0.742109	0.371055	−	0.928611i	$-0.378996\pi$
$181$	7.96493e8	0.742109	0.371055	+	0.928611i	$0.378996\pi$
$182$	0	0
$183$	− 1.05381e9i	− 0.939635i
$184$	0	0
$185$	4.70052e8	0.401290
$186$	0	0
$187$	− 1.20800e8i	− 0.0987870i
$188$	0	0
$189$	1.21276e9	0.950449
$190$	0	0
$191$	− 1.84245e9i	− 1.38440i	−0.721706	−	0.692200i	$-0.756642\pi$
$191$	− 1.84245e9i	− 1.38440i	0.721706	−	0.692200i	$-0.243358\pi$
$192$	0	0
$193$	2.34201e9	1.68795	0.843975	−	0.536382i	$-0.180209\pi$
$193$	2.34201e9	1.68795	0.843975	+	0.536382i	$0.180209\pi$
$194$	0	0
$195$	− 4.61535e8i	− 0.319202i
$196$	0	0
$197$	1.86381e9	1.23747	0.618737	−	0.785598i	$-0.287644\pi$
$197$	1.86381e9	1.23747	0.618737	+	0.785598i	$0.287644\pi$
$198$	0	0
$199$	1.82005e9i	1.16057i	0.814413	+	0.580286i	$0.197059\pi$
$199$	1.82005e9i	1.16057i	−0.814413	+	0.580286i	$0.802941\pi$
$200$	0	0
$201$	6.15349e7	0.0376997
$202$	0	0
$203$	1.92035e9i	1.13083i
$204$	0	0
$205$	2.12927e8	0.120563
$206$	0	0
$207$	2.86599e8i	0.156097i
$208$	0	0
$209$	6.00001e8	0.314461
$210$	0	0
$211$	2.89129e9i	1.45869i	0.684148	+	0.729344i	$0.260174\pi$
$211$	2.89129e9i	1.45869i	−0.684148	+	0.729344i	$0.739826\pi$
$212$	0	0
$213$	8.25613e8	0.401105
$214$	0	0
$215$	− 7.10970e8i	− 0.332734i
$216$	0	0
$217$	−3.55125e8	−0.160156
$218$	0	0
$219$	− 1.00424e9i	− 0.436576i
$220$	0	0
$221$	2.00917e9	0.842265
$222$	0	0
$223$	2.26023e9i	0.913973i	0.889474	+	0.456986i	$0.151071\pi$
$223$	2.26023e9i	0.913973i	−0.889474	+	0.456986i	$0.848929\pi$
$224$	0	0
$225$	3.87813e8	0.151319
$226$	0	0
$227$	1.08021e9i	0.406823i	0.979093	+	0.203412i	$0.0652029\pi$
$227$	1.08021e9i	0.406823i	−0.979093	+	0.203412i	$0.934797\pi$
$228$	0	0
$229$	−6.30352e8	−0.229214	−0.114607	−	0.993411i	$-0.536561\pi$
$229$	−6.30352e8	−0.229214	−0.114607	+	0.993411i	$0.536561\pi$
$230$	0	0
$231$	2.61421e8i	0.0918107i
$232$	0	0
$233$	1.17430e9	0.398435	0.199217	−	0.979955i	$-0.436160\pi$
$233$	1.17430e9	0.398435	0.199217	+	0.979955i	$0.436160\pi$
$234$	0	0
$235$	1.58199e9i	0.518720i
$236$	0	0
$237$	7.80642e8	0.247434
$238$	0	0
$239$	− 3.05182e9i	− 0.935334i	−0.883905	−	0.467667i	$-0.845095\pi$
$239$	− 3.05182e9i	− 0.935334i	0.883905	−	0.467667i	$-0.154905\pi$
$240$	0	0
$241$	2.81871e9	0.835570	0.417785	−	0.908546i	$-0.362807\pi$
$241$	2.81871e9	0.835570	0.417785	+	0.908546i	$0.362807\pi$
$242$	0	0
$243$	− 3.58779e9i	− 1.02897i
$244$	0	0
$245$	3.26726e8	0.0906815
$246$	0	0
$247$	9.97938e9i	2.68112i
$248$	0	0
$249$	−9.47414e8	−0.246458
$250$	0	0
$251$	− 2.92571e8i	− 0.0737116i	−0.999321	−	0.0368558i	$-0.988266\pi$
$251$	− 2.92571e8i	− 0.0737116i	0.999321	−	0.0368558i	$-0.0117342\pi$
$252$	0	0
$253$	−1.43433e8	−0.0350079
$254$	0	0
$255$	− 5.43133e8i	− 0.128453i
$256$	0	0
$257$	3.03746e9	0.696270	0.348135	−	0.937444i	$-0.386815\pi$
$257$	3.03746e9	0.696270	0.348135	+	0.937444i	$0.386815\pi$
$258$	0	0
$259$	− 4.42827e9i	− 0.984091i
$260$	0	0
$261$	3.62018e9	0.780133
$262$	0	0
$263$	1.69827e9i	0.354964i	0.984124	+	0.177482i	$0.0567951\pi$
$263$	1.69827e9i	0.354964i	−0.984124	+	0.177482i	$0.943205\pi$
$264$	0	0
$265$	3.19485e9	0.647838
$266$	0	0
$267$	− 1.54554e9i	− 0.304114i
$268$	0	0
$269$	8.70547e8	0.166258	0.0831291	−	0.996539i	$-0.473509\pi$
$269$	8.70547e8	0.166258	0.0831291	+	0.996539i	$0.473509\pi$
$270$	0	0
$271$	− 7.36841e9i	− 1.36614i	−0.730351	−	0.683072i	$-0.760643\pi$
$271$	− 7.36841e9i	− 1.36614i	0.730351	−	0.683072i	$-0.239357\pi$
$272$	0	0
$273$	−4.34803e9	−0.782785
$274$	0	0
$275$	1.94087e8i	0.0339364i
$276$	0	0
$277$	−2.37339e9	−0.403135	−0.201568	−	0.979475i	$-0.564604\pi$
$277$	−2.37339e9	−0.403135	−0.201568	+	0.979475i	$0.564604\pi$
$278$	0	0
$279$	6.69469e8i	0.110488i
$280$	0	0
$281$	5.50733e9	0.883315	0.441657	−	0.897184i	$-0.354391\pi$
$281$	5.50733e9	0.883315	0.441657	+	0.897184i	$0.354391\pi$
$282$	0	0
$283$	4.55904e9i	0.710768i	0.934720	+	0.355384i	$0.115650\pi$
$283$	4.55904e9i	0.710768i	−0.934720	+	0.355384i	$0.884350\pi$
$284$	0	0
$285$	2.69769e9	0.408896
$286$	0	0
$287$	− 2.00594e9i	− 0.295659i
$288$	0	0
$289$	−4.61137e9	−0.661056
$290$	0	0
$291$	3.78221e9i	0.527441i
$292$	0	0
$293$	8.27611e9	1.12294	0.561469	−	0.827498i	$-0.310237\pi$
$293$	8.27611e9	1.12294	0.561469	+	0.827498i	$0.310237\pi$
$294$	0	0
$295$	− 5.95080e9i	− 0.785756i
$296$	0	0
$297$	1.14419e9	0.147053
$298$	0	0
$299$	− 2.38562e9i	− 0.298480i
$300$	0	0
$301$	−6.69791e9	−0.815969
$302$	0	0
$303$	− 2.60814e9i	− 0.309429i
$304$	0	0
$305$	−7.37067e9	−0.851742
$306$	0	0
$307$	− 7.73582e9i	− 0.870869i	−0.900221	−	0.435434i	$-0.856595\pi$
$307$	− 7.73582e9i	− 0.870869i	0.900221	−	0.435434i	$-0.143405\pi$
$308$	0	0
$309$	−5.87446e9	−0.644368
$310$	0	0
$311$	− 1.08068e10i	− 1.15520i	−0.816321	−	0.577599i	$-0.803990\pi$
$311$	− 1.08068e10i	− 1.15520i	0.816321	−	0.577599i	$-0.196010\pi$
$312$	0	0
$313$	−4.73025e9	−0.492841	−0.246421	−	0.969163i	$-0.579254\pi$
$313$	−4.73025e9	−0.492841	−0.246421	+	0.969163i	$0.579254\pi$
$314$	0	0
$315$	− 3.65352e9i	− 0.371081i
$316$	0	0
$317$	5.44758e9	0.539469	0.269735	−	0.962935i	$-0.413064\pi$
$317$	5.44758e9	0.539469	0.269735	+	0.962935i	$0.413064\pi$
$318$	0	0
$319$	1.81178e9i	0.174961i
$320$	0	0
$321$	9.68803e9	0.912463
$322$	0	0
$323$	1.17437e10i	1.07893i
$324$	0	0
$325$	−3.22811e9	−0.289344
$326$	0	0
$327$	− 2.37171e9i	− 0.207429i
$328$	0	0
$329$	1.49036e10	1.27206
$330$	0	0
$331$	8.65354e8i	0.0720911i	0.999350	+	0.0360456i	$0.0114761\pi$
$331$	8.65354e8i	0.0720911i	−0.999350	+	0.0360456i	$0.988524\pi$
$332$	0	0
$333$	−8.34801e9	−0.678901
$334$	0	0
$335$	− 4.30393e8i	− 0.0341732i
$336$	0	0
$337$	8.64475e8	0.0670244	0.0335122	−	0.999438i	$-0.489331\pi$
$337$	8.64475e8	0.0670244	0.0335122	+	0.999438i	$0.489331\pi$
$338$	0	0
$339$	− 2.18274e9i	− 0.165274i
$340$	0	0
$341$	−3.35046e8	−0.0247792
$342$	0	0
$343$	1.21018e10i	0.874329i
$344$	0	0
$345$	−6.44895e8	−0.0455211
$346$	0	0
$347$	1.10077e10i	0.759239i	0.925143	+	0.379620i	$0.123945\pi$
$347$	1.10077e10i	0.759239i	−0.925143	+	0.379620i	$0.876055\pi$
$348$	0	0
$349$	1.99631e10	1.34563	0.672816	−	0.739810i	$-0.265085\pi$
$349$	1.99631e10	1.34563	0.672816	+	0.739810i	$0.265085\pi$
$350$	0	0
$351$	1.90305e10i	1.25378i
$352$	0	0
$353$	1.32468e10	0.853126	0.426563	−	0.904458i	$-0.359724\pi$
$353$	1.32468e10	0.853126	0.426563	+	0.904458i	$0.359724\pi$
$354$	0	0
$355$	− 5.77457e9i	− 0.363586i
$356$	0	0
$357$	−5.11675e9	−0.315008
$358$	0	0
$359$	2.59685e9i	0.156340i	0.996940	+	0.0781699i	$0.0249076\pi$
$359$	2.59685e9i	0.156340i	−0.996940	+	0.0781699i	$0.975092\pi$
$360$	0	0
$361$	−4.13463e10	−2.43449
$362$	0	0
$363$	− 8.31965e9i	− 0.479158i
$364$	0	0
$365$	−7.02393e9	−0.395739
$366$	0	0
$367$	3.62330e9i	0.199728i	0.995001	+	0.0998642i	$0.0318408\pi$
$367$	3.62330e9i	0.199728i	−0.995001	+	0.0998642i	$0.968159\pi$
$368$	0	0
$369$	−3.78153e9	−0.203968
$370$	0	0
$371$	− 3.00981e10i	− 1.58870i
$372$	0	0
$373$	−4.15162e9	−0.214478	−0.107239	−	0.994233i	$-0.534201\pi$
$373$	−4.15162e9	−0.214478	−0.107239	+	0.994233i	$0.534201\pi$
$374$	0	0
$375$	8.72643e8i	0.0441277i
$376$	0	0
$377$	−3.01339e10	−1.49173
$378$	0	0
$379$	1.89551e10i	0.918692i	0.888257	+	0.459346i	$0.151916\pi$
$379$	1.89551e10i	0.918692i	−0.888257	+	0.459346i	$0.848084\pi$
$380$	0	0
$381$	−8.03638e9	−0.381382
$382$	0	0
$383$	8.87673e9i	0.412532i	0.978496	+	0.206266i	$0.0661313\pi$
$383$	8.87673e9i	0.412532i	−0.978496	+	0.206266i	$0.933869\pi$
$384$	0	0
$385$	1.82846e9	0.0832227
$386$	0	0
$387$	1.26267e10i	0.562918i
$388$	0	0
$389$	−2.05595e10	−0.897872	−0.448936	−	0.893564i	$-0.648197\pi$
$389$	−2.05595e10	−0.897872	−0.448936	+	0.893564i	$0.648197\pi$
$390$	0	0
$391$	− 2.80739e9i	− 0.120114i
$392$	0	0
$393$	−1.74382e10	−0.731023
$394$	0	0
$395$	− 5.46004e9i	− 0.224289i
$396$	0	0
$397$	−2.04868e10	−0.824733	−0.412366	−	0.911018i	$-0.635298\pi$
$397$	−2.04868e10	−0.824733	−0.412366	+	0.911018i	$0.635298\pi$
$398$	0	0
$399$	− 2.54144e10i	− 1.00274i
$400$	0	0
$401$	−7.30576e9	−0.282545	−0.141273	−	0.989971i	$-0.545119\pi$
$401$	−7.30576e9	−0.282545	−0.141273	+	0.989971i	$0.545119\pi$
$402$	0	0
$403$	− 5.57258e9i	− 0.211269i
$404$	0	0
$405$	−3.95882e9	−0.147145
$406$	0	0
$407$	− 4.17789e9i	− 0.152258i
$408$	0	0
$409$	−1.05889e10	−0.378406	−0.189203	−	0.981938i	$-0.560590\pi$
$409$	−1.05889e10	−0.378406	−0.189203	+	0.981938i	$0.560590\pi$
$410$	0	0
$411$	− 1.79152e10i	− 0.627847i
$412$	0	0
$413$	−5.60614e10	−1.92692
$414$	0	0
$415$	6.62649e9i	0.223404i
$416$	0	0
$417$	1.89373e10	0.626289
$418$	0	0
$419$	2.61037e9i	0.0846928i	0.999103	+	0.0423464i	$0.0134833\pi$
$419$	2.61037e9i	0.0846928i	−0.999103	+	0.0423464i	$0.986517\pi$
$420$	0	0
$421$	−9.29505e9	−0.295885	−0.147943	−	0.988996i	$-0.547265\pi$
$421$	−9.29505e9	−0.295885	−0.147943	+	0.988996i	$0.547265\pi$
$422$	0	0
$423$	− 2.80958e10i	− 0.877567i
$424$	0	0
$425$	−3.79883e9	−0.116438
$426$	0	0
$427$	6.94377e10i	2.08874i
$428$	0	0
$429$	−4.10219e9	−0.121112
$430$	0	0
$431$	1.86560e10i	0.540642i	0.962770	+	0.270321i	$0.0871298\pi$
$431$	1.86560e10i	0.540642i	−0.962770	+	0.270321i	$0.912870\pi$
$432$	0	0
$433$	−1.66970e10	−0.474994	−0.237497	−	0.971388i	$-0.576327\pi$
$433$	−1.66970e10	−0.474994	−0.237497	+	0.971388i	$0.576327\pi$
$434$	0	0
$435$	8.14600e9i	0.227503i
$436$	0	0
$437$	1.39440e10	0.382351
$438$	0	0
$439$	5.48360e9i	0.147641i	0.997272	+	0.0738206i	$0.0235192\pi$
$439$	5.48360e9i	0.147641i	−0.997272	+	0.0738206i	$0.976481\pi$
$440$	0	0
$441$	−5.80257e9	−0.153414
$442$	0	0
$443$	− 5.97812e9i	− 0.155221i	−0.996984	−	0.0776104i	$-0.975271\pi$
$443$	− 5.97812e9i	− 0.155221i	0.996984	−	0.0776104i	$-0.0247290\pi$
$444$	0	0
$445$	−1.08100e10	−0.275667
$446$	0	0
$447$	− 1.19289e10i	− 0.298794i
$448$	0	0
$449$	−3.11696e10	−0.766913	−0.383456	−	0.923559i	$-0.625266\pi$
$449$	−3.11696e10	−0.766913	−0.383456	+	0.923559i	$0.625266\pi$
$450$	0	0
$451$	− 1.89253e9i	− 0.0457442i
$452$	0	0
$453$	2.89740e10	0.688043
$454$	0	0
$455$	3.04114e10i	0.709563i
$456$	0	0
$457$	−4.33966e9	−0.0994927	−0.0497463	−	0.998762i	$-0.515841\pi$
$457$	−4.33966e9	−0.0994927	−0.0497463	+	0.998762i	$0.515841\pi$
$458$	0	0
$459$	2.23951e10i	0.504547i
$460$	0	0
$461$	5.16164e10	1.14284	0.571418	−	0.820659i	$-0.306393\pi$
$461$	5.16164e10	1.14284	0.571418	+	0.820659i	$0.306393\pi$
$462$	0	0
$463$	− 6.86820e10i	− 1.49458i	−0.664499	−	0.747289i	$-0.731355\pi$
$463$	− 6.86820e10i	− 1.49458i	0.664499	−	0.747289i	$-0.268645\pi$
$464$	0	0
$465$	−1.50641e9	−0.0322205
$466$	0	0
$467$	− 2.14596e10i	− 0.451185i	−0.974222	−	0.225592i	$-0.927568\pi$
$467$	− 2.14596e10i	− 0.451185i	0.974222	−	0.225592i	$-0.0724317\pi$
$468$	0	0
$469$	−4.05465e9	−0.0838036
$470$	0	0
$471$	− 2.67863e10i	− 0.544289i
$472$	0	0
$473$	−6.31921e9	−0.126246
$474$	0	0
$475$	− 1.88684e10i	− 0.370648i
$476$	0	0
$477$	−5.67397e10	−1.09601
$478$	0	0
$479$	8.24617e10i	1.56643i	0.621753	+	0.783213i	$0.286421\pi$
$479$	8.24617e10i	1.56643i	−0.621753	+	0.783213i	$0.713579\pi$
$480$	0	0
$481$	6.94878e10	1.29816
$482$	0	0
$483$	6.07543e9i	0.111632i
$484$	0	0
$485$	2.64539e10	0.478104
$486$	0	0
$487$	2.94674e9i	0.0523874i	0.999657	+	0.0261937i	$0.00833866\pi$
$487$	2.94674e9i	0.0523874i	−0.999657	+	0.0261937i	$0.991661\pi$
$488$	0	0
$489$	−3.87929e10	−0.678448
$490$	0	0
$491$	− 8.17812e10i	− 1.40711i	−0.710642	−	0.703554i	$-0.751595\pi$
$491$	− 8.17812e10i	− 1.40711i	0.710642	−	0.703554i	$-0.248405\pi$
$492$	0	0
$493$	−3.54616e10	−0.600302
$494$	0	0
$495$	− 3.44694e9i	− 0.0574134i
$496$	0	0
$497$	−5.44012e10	−0.891626
$498$	0	0
$499$	4.69562e10i	0.757340i	0.925532	+	0.378670i	$0.123618\pi$
$499$	4.69562e10i	0.757340i	−0.925532	+	0.378670i	$0.876382\pi$
$500$	0	0
$501$	−3.19173e10	−0.506612
$502$	0	0
$503$	8.89245e10i	1.38915i	0.719420	+	0.694575i	$0.244408\pi$
$503$	8.89245e10i	1.38915i	−0.719420	+	0.694575i	$0.755592\pi$
$504$	0	0
$505$	−1.82421e10	−0.280485
$506$	0	0
$507$	− 3.56302e10i	− 0.539245i
$508$	0	0
$509$	−1.30873e11	−1.94975	−0.974877	−	0.222745i	$-0.928498\pi$
$509$	−1.30873e11	−1.94975	−0.974877	+	0.222745i	$0.928498\pi$
$510$	0	0
$511$	6.61711e10i	0.970476i
$512$	0	0
$513$	−1.11234e11	−1.60609
$514$	0	0
$515$	4.10877e10i	0.584094i
$516$	0	0
$517$	1.40610e10	0.196813
$518$	0	0
$519$	2.39944e10i	0.330705i
$520$	0	0
$521$	7.03085e10	0.954238	0.477119	−	0.878839i	$-0.341681\pi$
$521$	7.03085e10	0.954238	0.477119	+	0.878839i	$0.341681\pi$
$522$	0	0
$523$	− 7.40744e10i	− 0.990060i	−0.868876	−	0.495030i	$-0.835157\pi$
$523$	− 7.40744e10i	− 0.990060i	0.868876	−	0.495030i	$-0.164843\pi$
$524$	0	0
$525$	8.22100e9	0.108215
$526$	0	0
$527$	− 6.55779e9i	− 0.0850189i
$528$	0	0
$529$	7.49776e10	0.957434
$530$	0	0
$531$	1.05685e11i	1.32934i
$532$	0	0
$533$	3.14770e10	0.390018
$534$	0	0
$535$	− 6.77609e10i	− 0.827111i
$536$	0	0
$537$	4.62943e10	0.556712
$538$	0	0
$539$	− 2.90399e9i	− 0.0344064i
$540$	0	0
$541$	−3.35458e9	−0.0391605	−0.0195803	−	0.999808i	$-0.506233\pi$
$541$	−3.35458e9	−0.0391605	−0.0195803	+	0.999808i	$0.506233\pi$
$542$	0	0
$543$	− 3.18298e10i	− 0.366129i
$544$	0	0
$545$	−1.65884e10	−0.188027
$546$	0	0
$547$	− 1.13812e11i	− 1.27128i	−0.771987	−	0.635638i	$-0.780737\pi$
$547$	− 1.13812e11i	− 1.27128i	0.771987	−	0.635638i	$-0.219263\pi$
$548$	0	0
$549$	1.30902e11	1.44097
$550$	0	0
$551$	− 1.76134e11i	− 1.91090i
$552$	0	0
$553$	−5.14380e10	−0.550026
$554$	0	0
$555$	− 1.87844e10i	− 0.197982i
$556$	0	0
$557$	1.05403e11	1.09505	0.547524	−	0.836790i	$-0.315570\pi$
$557$	1.05403e11	1.09505	0.547524	+	0.836790i	$0.315570\pi$
$558$	0	0
$559$	− 1.05103e11i	− 1.07638i
$560$	0	0
$561$	−4.82745e9	−0.0487378
$562$	0	0
$563$	7.97945e10i	0.794217i	0.917772	+	0.397108i	$0.129986\pi$
$563$	7.97945e10i	0.794217i	−0.917772	+	0.397108i	$0.870014\pi$
$564$	0	0
$565$	−1.52667e10	−0.149814
$566$	0	0
$567$	3.72953e10i	0.360846i
$568$	0	0
$569$	−4.57243e10	−0.436212	−0.218106	−	0.975925i	$-0.569988\pi$
$569$	−4.57243e10	−0.436212	−0.218106	+	0.975925i	$0.569988\pi$
$570$	0	0
$571$	2.16780e10i	0.203927i	0.994788	+	0.101964i	$0.0325126\pi$
$571$	2.16780e10i	0.203927i	−0.994788	+	0.101964i	$0.967487\pi$
$572$	0	0
$573$	−7.36286e10	−0.683011
$574$	0	0
$575$	4.51058e9i	0.0412630i
$576$	0	0
$577$	1.69273e11	1.52716	0.763579	−	0.645715i	$-0.223440\pi$
$577$	1.69273e11	1.52716	0.763579	+	0.645715i	$0.223440\pi$
$578$	0	0
$579$	− 9.35923e10i	− 0.832772i
$580$	0	0
$581$	6.24269e10	0.547857
$582$	0	0
$583$	− 2.83963e10i	− 0.245803i
$584$	0	0
$585$	5.73305e10	0.489511
$586$	0	0
$587$	− 3.75401e10i	− 0.316187i	−0.987424	−	0.158093i	$-0.949465\pi$
$587$	− 3.75401e10i	− 0.316187i	0.987424	−	0.158093i	$-0.0505347\pi$
$588$	0	0
$589$	3.25719e10	0.270634
$590$	0	0
$591$	− 7.44822e10i	− 0.610524i
$592$	0	0
$593$	1.04118e11	0.841994	0.420997	−	0.907062i	$-0.361680\pi$
$593$	1.04118e11	0.841994	0.420997	+	0.907062i	$0.361680\pi$
$594$	0	0
$595$	3.57881e10i	0.285542i
$596$	0	0
$597$	7.27337e10	0.572583
$598$	0	0
$599$	4.81442e10i	0.373970i	0.982363	+	0.186985i	$0.0598716\pi$
$599$	4.81442e10i	0.373970i	−0.982363	+	0.186985i	$0.940128\pi$
$600$	0	0
$601$	−1.48443e11	−1.13779	−0.568896	−	0.822410i	$-0.692629\pi$
$601$	−1.48443e11	−1.13779	−0.568896	+	0.822410i	$0.692629\pi$
$602$	0	0
$603$	7.64369e9i	0.0578141i
$604$	0	0
$605$	−5.81901e10	−0.434337
$606$	0	0
$607$	− 5.44460e10i	− 0.401062i	−0.979687	−	0.200531i	$-0.935733\pi$
$607$	− 5.44460e10i	− 0.401062i	0.979687	−	0.200531i	$-0.0642667\pi$
$608$	0	0
$609$	7.67419e10	0.557909
$610$	0	0
$611$	2.33866e11i	1.67804i
$612$	0	0
$613$	−1.16278e11	−0.823484	−0.411742	−	0.911301i	$-0.635079\pi$
$613$	−1.16278e11	−0.823484	−0.411742	+	0.911301i	$0.635079\pi$
$614$	0	0
$615$	− 8.50906e9i	− 0.0594814i
$616$	0	0
$617$	−4.87411e10	−0.336321	−0.168161	−	0.985760i	$-0.553783\pi$
$617$	−4.87411e10	−0.336321	−0.168161	+	0.985760i	$0.553783\pi$
$618$	0	0
$619$	− 4.92280e10i	− 0.335312i	−0.985846	−	0.167656i	$-0.946380\pi$
$619$	− 4.92280e10i	− 0.335312i	0.985846	−	0.167656i	$-0.0536198\pi$
$620$	0	0
$621$	2.65910e10	0.178800
$622$	0	0
$623$	1.01839e11i	0.676022i
$624$	0	0
$625$	6.10352e9	0.0400000
$626$	0	0
$627$	− 2.39775e10i	− 0.155143i
$628$	0	0
$629$	8.17731e10	0.522406
$630$	0	0
$631$	1.10376e11i	0.696240i	0.937450	+	0.348120i	$0.113180\pi$
$631$	1.10376e11i	0.696240i	−0.937450	+	0.348120i	$0.886820\pi$
$632$	0	0
$633$	1.15543e11	0.719662
$634$	0	0
$635$	5.62088e10i	0.345708i
$636$	0	0
$637$	4.82999e10	0.293352
$638$	0	0
$639$	1.02555e11i	0.615112i
$640$	0	0
$641$	2.15342e11	1.27554	0.637772	−	0.770225i	$-0.279856\pi$
$641$	2.15342e11	1.27554	0.637772	+	0.770225i	$0.279856\pi$
$642$	0	0
$643$	− 1.22051e11i	− 0.713996i	−0.934105	−	0.356998i	$-0.883800\pi$
$643$	− 1.22051e11i	− 0.713996i	0.934105	−	0.356998i	$-0.116200\pi$
$644$	0	0
$645$	−2.84120e10	−0.164159
$646$	0	0
$647$	2.92452e11i	1.66893i	0.551064	+	0.834463i	$0.314222\pi$
$647$	2.92452e11i	1.66893i	−0.551064	+	0.834463i	$0.685778\pi$
$648$	0	0
$649$	−5.28916e10	−0.298132
$650$	0	0
$651$	1.41916e10i	0.0790149i
$652$	0	0
$653$	2.80990e10	0.154539	0.0772696	−	0.997010i	$-0.475380\pi$
$653$	2.80990e10	0.154539	0.0772696	+	0.997010i	$0.475380\pi$
$654$	0	0
$655$	1.21968e11i	0.662644i
$656$	0	0
$657$	1.24743e11	0.669509
$658$	0	0
$659$	− 2.51394e11i	− 1.33295i	−0.745529	−	0.666473i	$-0.767803\pi$
$659$	− 2.51394e11i	− 1.33295i	0.745529	−	0.666473i	$-0.232197\pi$
$660$	0	0
$661$	2.53415e11	1.32748	0.663738	−	0.747965i	$-0.268969\pi$
$661$	2.53415e11	1.32748	0.663738	+	0.747965i	$0.268969\pi$
$662$	0	0
$663$	− 8.02914e10i	− 0.415542i
$664$	0	0
$665$	−1.77756e11	−0.908945
$666$	0	0
$667$	4.21056e10i	0.212734i
$668$	0	0
$669$	9.03242e10	0.450920
$670$	0	0
$671$	6.55116e10i	0.323168i
$672$	0	0
$673$	−1.77857e11	−0.866981	−0.433491	−	0.901158i	$-0.642718\pi$
$673$	−1.77857e11	−0.866981	−0.433491	+	0.901158i	$0.642718\pi$
$674$	0	0
$675$	− 3.59818e10i	− 0.173328i
$676$	0	0
$677$	2.43910e11	1.16111	0.580556	−	0.814220i	$-0.302835\pi$
$677$	2.43910e11	1.16111	0.580556	+	0.814220i	$0.302835\pi$
$678$	0	0
$679$	− 2.49217e11i	− 1.17246i
$680$	0	0
$681$	4.31678e10	0.200711
$682$	0	0
$683$	2.74209e11i	1.26008i	0.776562	+	0.630041i	$0.216962\pi$
$683$	2.74209e11i	1.26008i	−0.776562	+	0.630041i	$0.783038\pi$
$684$	0	0
$685$	−1.25304e11	−0.569118
$686$	0	0
$687$	2.51904e10i	0.113086i
$688$	0	0
$689$	4.72295e11	2.09573
$690$	0	0
$691$	1.07912e11i	0.473323i	0.971592	+	0.236661i	$0.0760532\pi$
$691$	1.07912e11i	0.473323i	−0.971592	+	0.236661i	$0.923947\pi$
$692$	0	0
$693$	−3.24730e10	−0.140796
$694$	0	0
$695$	− 1.32453e11i	− 0.567706i
$696$	0	0
$697$	3.70421e10	0.156951
$698$	0	0
$699$	− 4.69280e10i	− 0.196573i
$700$	0	0
$701$	1.00364e11	0.415630	0.207815	−	0.978168i	$-0.433365\pi$
$701$	1.00364e11	0.415630	0.207815	+	0.978168i	$0.433365\pi$
$702$	0	0
$703$	4.06159e11i	1.66293i
$704$	0	0
$705$	6.32201e10	0.255917
$706$	0	0
$707$	1.71855e11i	0.687837i
$708$	0	0
$709$	4.36108e11	1.72587	0.862937	−	0.505311i	$-0.168622\pi$
$709$	4.36108e11	1.72587	0.862937	+	0.505311i	$0.168622\pi$
$710$	0	0
$711$	9.69690e10i	0.379450i
$712$	0	0
$713$	−7.78647e9	−0.0301288
$714$	0	0
$715$	2.86919e10i	0.109783i
$716$	0	0
$717$	−1.21958e11	−0.461459
$718$	0	0
$719$	− 3.29910e11i	− 1.23447i	−0.786780	−	0.617234i	$-0.788253\pi$
$719$	− 3.29910e11i	− 1.23447i	0.786780	−	0.617234i	$-0.211747\pi$
$720$	0	0
$721$	3.87079e11	1.43238
$722$	0	0
$723$	− 1.12642e11i	− 0.412239i
$724$	0	0
$725$	5.69755e10	0.206222
$726$	0	0
$727$	− 2.87638e11i	− 1.02969i	−0.857282	−	0.514847i	$-0.827849\pi$
$727$	− 2.87638e11i	− 1.02969i	0.857282	−	0.514847i	$-0.172151\pi$
$728$	0	0
$729$	−5.04496e10	−0.178627
$730$	0	0
$731$	− 1.23685e11i	− 0.433158i
$732$	0	0
$733$	−7.34639e10	−0.254483	−0.127241	−	0.991872i	$-0.540612\pi$
$733$	−7.34639e10	−0.254483	−0.127241	+	0.991872i	$0.540612\pi$
$734$	0	0
$735$	− 1.30567e10i	− 0.0447389i
$736$	0	0
$737$	−3.82540e9	−0.0129660
$738$	0	0
$739$	3.53166e11i	1.18413i	0.805889	+	0.592067i	$0.201688\pi$
$739$	3.53166e11i	1.18413i	−0.805889	+	0.592067i	$0.798312\pi$
$740$	0	0
$741$	3.98800e11	1.32276
$742$	0	0
$743$	− 2.51184e11i	− 0.824208i	−0.911137	−	0.412104i	$-0.864794\pi$
$743$	− 2.51184e11i	− 0.824208i	0.911137	−	0.412104i	$-0.135206\pi$
$744$	0	0
$745$	−8.34345e10	−0.270845
$746$	0	0
$747$	− 1.17685e11i	− 0.377954i
$748$	0	0
$749$	−6.38363e11	−2.02834
$750$	0	0
$751$	5.06685e11i	1.59286i	0.604729	+	0.796431i	$0.293281\pi$
$751$	5.06685e11i	1.59286i	−0.604729	+	0.796431i	$0.706719\pi$
$752$	0	0
$753$	−1.16918e10	−0.0363666
$754$	0	0
$755$	− 2.02653e11i	− 0.623684i
$756$	0	0
$757$	−9.66712e10	−0.294384	−0.147192	−	0.989108i	$-0.547023\pi$
$757$	−9.66712e10	−0.294384	−0.147192	+	0.989108i	$0.547023\pi$
$758$	0	0
$759$	5.73192e9i	0.0172716i
$760$	0	0
$761$	5.18752e11	1.54675	0.773376	−	0.633947i	$-0.218566\pi$
$761$	5.18752e11	1.54675	0.773376	+	0.633947i	$0.218566\pi$
$762$	0	0
$763$	1.56276e11i	0.461100i
$764$	0	0
$765$	6.74664e10	0.196989
$766$	0	0
$767$	− 8.79708e11i	− 2.54189i
$768$	0	0
$769$	1.06291e11	0.303943	0.151972	−	0.988385i	$-0.451438\pi$
$769$	1.06291e11	0.303943	0.151972	+	0.988385i	$0.451438\pi$
$770$	0	0
$771$	− 1.21384e11i	− 0.343514i
$772$	0	0
$773$	9.89630e10	0.277176	0.138588	−	0.990350i	$-0.455744\pi$
$773$	9.89630e10	0.277176	0.138588	+	0.990350i	$0.455744\pi$
$774$	0	0
$775$	1.05363e10i	0.0292066i
$776$	0	0
$777$	−1.76964e11	−0.485514
$778$	0	0
$779$	1.83984e11i	0.499610i
$780$	0	0
$781$	−5.13253e10	−0.137952
$782$	0	0
$783$	− 3.35885e11i	− 0.893601i
$784$	0	0
$785$	−1.87351e11	−0.493376
$786$	0	0
$787$	− 3.29755e11i	− 0.859592i	−0.902926	−	0.429796i	$-0.858585\pi$
$787$	− 3.29755e11i	− 0.859592i	0.902926	−	0.429796i	$-0.141415\pi$
$788$	0	0
$789$	6.78670e10	0.175126
$790$	0	0
$791$	1.43825e11i	0.367391i
$792$	0	0
$793$	−1.08961e12	−2.75535
$794$	0	0
$795$	− 1.27674e11i	− 0.319619i
$796$	0	0
$797$	−6.85369e10	−0.169860	−0.0849300	−	0.996387i	$-0.527067\pi$
$797$	−6.85369e10	−0.169860	−0.0849300	+	0.996387i	$0.527067\pi$
$798$	0	0
$799$	2.75213e11i	0.675277i
$800$	0	0
$801$	1.91983e11	0.466371
$802$	0	0
$803$	6.24297e10i	0.150151i
$804$	0	0
$805$	4.24933e10	0.101190
$806$	0	0
$807$	− 3.47891e10i	− 0.0820256i
$808$	0	0
$809$	−7.62508e11	−1.78012	−0.890062	−	0.455840i	$-0.849339\pi$
$809$	−7.62508e11	−1.78012	−0.890062	+	0.455840i	$0.849339\pi$
$810$	0	0
$811$	− 4.40805e11i	− 1.01897i	−0.860478	−	0.509487i	$-0.829835\pi$
$811$	− 4.40805e11i	− 1.01897i	0.860478	−	0.509487i	$-0.170165\pi$
$812$	0	0
$813$	−2.94459e11	−0.674005
$814$	0	0
$815$	2.71329e11i	0.614986i
$816$	0	0
$817$	6.14330e11	1.37884
$818$	0	0
$819$	− 5.40100e11i	− 1.20043i
$820$	0	0
$821$	2.67657e11	0.589123	0.294562	−	0.955632i	$-0.404826\pi$
$821$	2.67657e11	0.589123	0.294562	+	0.955632i	$0.404826\pi$
$822$	0	0
$823$	3.84869e11i	0.838906i	0.907777	+	0.419453i	$0.137778\pi$
$823$	3.84869e11i	0.838906i	−0.907777	+	0.419453i	$0.862222\pi$
$824$	0	0
$825$	7.75618e9	0.0167429
$826$	0	0
$827$	4.88174e11i	1.04364i	0.853054	+	0.521822i	$0.174747\pi$
$827$	4.88174e11i	1.04364i	−0.853054	+	0.521822i	$0.825253\pi$
$828$	0	0
$829$	1.47134e11	0.311526	0.155763	−	0.987794i	$-0.450216\pi$
$829$	1.47134e11	0.311526	0.155763	+	0.987794i	$0.450216\pi$
$830$	0	0
$831$	9.48464e10i	0.198892i
$832$	0	0
$833$	5.68392e10	0.118050
$834$	0	0
$835$	2.23239e11i	0.459223i
$836$	0	0
$837$	6.21142e10	0.126558
$838$	0	0
$839$	4.39919e11i	0.887821i	0.896071	+	0.443910i	$0.146409\pi$
$839$	4.39919e11i	0.887821i	−0.896071	+	0.443910i	$0.853591\pi$
$840$	0	0
$841$	3.16120e10	0.0631929
$842$	0	0
$843$	− 2.20086e11i	− 0.435794i
$844$	0	0
$845$	−2.49208e11	−0.488804
$846$	0	0
$847$	5.48198e11i	1.06513i
$848$	0	0
$849$	1.82190e11	0.350666
$850$	0	0
$851$	− 9.70941e10i	− 0.185129i
$852$	0	0
$853$	−6.98819e11	−1.31998	−0.659992	−	0.751273i	$-0.729440\pi$
$853$	−6.98819e11	−1.31998	−0.659992	+	0.751273i	$0.729440\pi$
$854$	0	0
$855$	3.35099e11i	0.627059i
$856$	0	0
$857$	3.28226e10	0.0608485	0.0304242	−	0.999537i	$-0.490314\pi$
$857$	3.28226e10	0.0608485	0.0304242	+	0.999537i	$0.490314\pi$
$858$	0	0
$859$	6.28697e11i	1.15470i	0.816497	+	0.577349i	$0.195913\pi$
$859$	6.28697e11i	1.15470i	−0.816497	+	0.577349i	$0.804087\pi$
$860$	0	0
$861$	−8.01623e10	−0.145867
$862$	0	0
$863$	− 7.90308e11i	− 1.42480i	−0.701775	−	0.712399i	$-0.747609\pi$
$863$	− 7.90308e11i	− 1.42480i	0.701775	−	0.712399i	$-0.252391\pi$
$864$	0	0
$865$	1.67824e11	0.299771
$866$	0	0
$867$	1.84281e11i	0.326140i
$868$	0	0
$869$	−4.85296e10	−0.0850997
$870$	0	0
$871$	− 6.36251e10i	− 0.110549i
$872$	0	0
$873$	−4.69815e11	−0.808854
$874$	0	0
$875$	− 5.75001e10i	− 0.0980926i
$876$	0	0
$877$	−7.21024e10	−0.121885	−0.0609427	−	0.998141i	$-0.519411\pi$
$877$	−7.21024e10	−0.121885	−0.0609427	+	0.998141i	$0.519411\pi$
$878$	0	0
$879$	− 3.30733e11i	− 0.554015i
$880$	0	0
$881$	−5.33265e11	−0.885196	−0.442598	−	0.896720i	$-0.645943\pi$
$881$	−5.33265e11	−0.885196	−0.442598	+	0.896720i	$0.645943\pi$
$882$	0	0
$883$	− 3.49301e11i	− 0.574589i	−0.957842	−	0.287295i	$-0.907244\pi$
$883$	− 3.49301e11i	− 0.574589i	0.957842	−	0.287295i	$-0.0927559\pi$
$884$	0	0
$885$	−2.37808e11	−0.387662
$886$	0	0
$887$	2.17981e11i	0.352148i	0.984377	+	0.176074i	$0.0563398\pi$
$887$	2.17981e11i	0.352148i	−0.984377	+	0.176074i	$0.943660\pi$
$888$	0	0
$889$	5.29532e11	0.847784
$890$	0	0
$891$	3.51866e10i	0.0558299i
$892$	0	0
$893$	−1.36696e12	−2.14956
$894$	0	0
$895$	− 3.23796e11i	− 0.504637i
$896$	0	0
$897$	−9.53348e10	−0.147259
$898$	0	0
$899$	9.83549e10i	0.150577i
$900$	0	0
$901$	5.55795e11	0.843365
$902$	0	0
$903$	2.67665e11i	0.402569i
$904$	0	0
$905$	−2.22627e11	−0.331881
$906$	0	0
$907$	− 6.26742e9i	− 0.00926104i	−0.999989	−	0.00463052i	$-0.998526\pi$
$907$	− 6.26742e9i	− 0.00926104i	0.999989	−	0.00463052i	$-0.00147395\pi$
$908$	0	0
$909$	3.23975e11	0.474522
$910$	0	0
$911$	− 6.22008e11i	− 0.903073i	−0.892253	−	0.451536i	$-0.850876\pi$
$911$	− 6.22008e11i	− 0.903073i	0.892253	−	0.451536i	$-0.149124\pi$
$912$	0	0
$913$	5.88972e10	0.0847640
$914$	0	0
$915$	2.94550e11i	0.420217i
$916$	0	0
$917$	1.14904e12	1.62501
$918$	0	0
$919$	− 8.27760e11i	− 1.16049i	−0.814441	−	0.580246i	$-0.802956\pi$
$919$	− 8.27760e11i	− 1.16049i	0.814441	−	0.580246i	$-0.197044\pi$
$920$	0	0
$921$	−3.09142e11	−0.429654
$922$	0	0
$923$	− 8.53656e11i	− 1.17619i
$924$	0	0
$925$	−1.31383e11	−0.179463
$926$	0	0
$927$	− 7.29708e11i	− 0.988167i
$928$	0	0
$929$	−6.50031e11	−0.872713	−0.436356	−	0.899774i	$-0.643731\pi$
$929$	−6.50031e11	−0.872713	−0.436356	+	0.899774i	$0.643731\pi$
$930$	0	0
$931$	2.82315e11i	0.375781i
$932$	0	0
$933$	−4.31866e11	−0.569932
$934$	0	0
$935$	3.37646e10i	0.0441789i
$936$	0	0
$937$	−6.60322e11	−0.856639	−0.428319	−	0.903627i	$-0.640894\pi$
$937$	−6.60322e11	−0.856639	−0.428319	+	0.903627i	$0.640894\pi$
$938$	0	0
$939$	1.89032e11i	0.243149i
$940$	0	0
$941$	2.13315e11	0.272059	0.136029	−	0.990705i	$-0.456566\pi$
$941$	2.13315e11	0.272059	0.136029	+	0.990705i	$0.456566\pi$
$942$	0	0
$943$	− 4.39823e10i	− 0.0556200i
$944$	0	0
$945$	−3.38978e11	−0.425054
$946$	0	0
$947$	1.23955e10i	0.0154121i	0.999970	+	0.00770607i	$0.00245294\pi$
$947$	1.23955e10i	0.0154121i	−0.999970	+	0.00770607i	$0.997547\pi$
$948$	0	0
$949$	−1.03835e12	−1.28020
$950$	0	0
$951$	− 2.17698e11i	− 0.266154i
$952$	0	0
$953$	−1.59801e12	−1.93734	−0.968672	−	0.248344i	$-0.920114\pi$
$953$	−1.59801e12	−1.93734	−0.968672	+	0.248344i	$0.920114\pi$
$954$	0	0
$955$	5.14980e11i	0.619122i
$956$	0	0
$957$	7.24029e10	0.0863193
$958$	0	0
$959$	1.18046e12i	1.39566i
$960$	0	0
$961$	8.34703e11	0.978674
$962$	0	0
$963$	1.20342e12i	1.39930i
$964$	0	0
$965$	−6.54612e11	−0.754875
$966$	0	0
$967$	4.54304e11i	0.519566i	0.965667	+	0.259783i	$0.0836510\pi$
$967$	4.54304e11i	0.519566i	−0.965667	+	0.259783i	$0.916349\pi$
$968$	0	0
$969$	4.69307e11	0.532306
$970$	0	0
$971$	3.98175e11i	0.447917i	0.974599	+	0.223958i	$0.0718979\pi$
$971$	3.98175e11i	0.447917i	−0.974599	+	0.223958i	$0.928102\pi$
$972$	0	0
$973$	−1.24782e12	−1.39219
$974$	0	0
$975$	1.29003e11i	0.142752i
$976$	0	0
$977$	1.61846e12	1.77633	0.888165	−	0.459525i	$-0.151980\pi$
$977$	1.61846e12	1.77633	0.888165	+	0.459525i	$0.151980\pi$
$978$	0	0
$979$	9.60806e10i	0.104594i
$980$	0	0
$981$	2.94607e11	0.318102
$982$	0	0
$983$	7.83368e10i	0.0838980i	0.999120	+	0.0419490i	$0.0133567\pi$
$983$	7.83368e10i	0.0838980i	−0.999120	+	0.0419490i	$0.986643\pi$
$984$	0	0
$985$	−5.20950e11	−0.553416
$986$	0	0
$987$	− 5.95585e11i	− 0.627589i
$988$	0	0
$989$	−1.46858e11	−0.153502
$990$	0	0
$991$	− 1.11925e12i	− 1.16046i	−0.814451	−	0.580232i	$-0.802962\pi$
$991$	− 1.11925e12i	− 1.16046i	0.814451	−	0.580232i	$-0.197038\pi$
$992$	0	0
$993$	3.45816e10	0.0355671
$994$	0	0
$995$	− 5.08721e11i	− 0.519023i
$996$	0	0
$997$	8.38483e11	0.848621	0.424311	−	0.905517i	$-0.360517\pi$
$997$	8.38483e11	0.848621	0.424311	+	0.905517i	$0.360517\pi$
$998$	0	0
$999$	7.74539e11i	0.777645i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.b.d.191.6		16
4.3	odd	2	inner	320.9.b.d.191.11		16
8.3	odd	2		20.9.b.a.11.8	yes	16
8.5	even	2		20.9.b.a.11.7	✓	16
24.5	odd	2		180.9.c.a.91.10		16
24.11	even	2		180.9.c.a.91.9		16
40.3	even	4		100.9.d.c.99.32		32
40.13	odd	4		100.9.d.c.99.2		32
40.19	odd	2		100.9.b.d.51.9		16
40.27	even	4		100.9.d.c.99.1		32
40.29	even	2		100.9.b.d.51.10		16
40.37	odd	4		100.9.d.c.99.31		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.b.a.11.7	✓	16	8.5	even	2
20.9.b.a.11.8	yes	16	8.3	odd	2
100.9.b.d.51.9		16	40.19	odd	2
100.9.b.d.51.10		16	40.29	even	2
100.9.d.c.99.1		32	40.27	even	4
100.9.d.c.99.2		32	40.13	odd	4
100.9.d.c.99.31		32	40.37	odd	4
100.9.d.c.99.32		32	40.3	even	4
180.9.c.a.91.9		16	24.11	even	2
180.9.c.a.91.10		16	24.5	odd	2
320.9.b.d.191.6		16	1.1	even	1	trivial
320.9.b.d.191.11		16	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-39.9624i q^{3} -279.508 q^{5} +2633.20i q^{7} +4964.01 q^{9} +2484.31i q^{11} -41319.8 q^{13} +11169.8i q^{15} -48625.0 q^{17} -241516. i q^{19} +105229. q^{21} +57735.4i q^{23} +78125.0 q^{25} -460567. i q^{27} +729286. q^{29} +134865. i q^{31} +99279.1 q^{33} -736001. i q^{35} -1.68171e6 q^{37} +1.65124e6i q^{39} -761790. q^{41} +2.54364e6i q^{43} -1.38748e6 q^{45} -5.65990e6i q^{47} -1.16893e6 q^{49} +1.94317e6i q^{51} -1.14302e7 q^{53} -694387. i q^{55} -9.65155e6 q^{57} +2.12902e7i q^{59} +2.63701e7 q^{61} +1.30712e7i q^{63} +1.15492e7 q^{65} +1.53982e6i q^{67} +2.30724e6 q^{69} +2.06597e7i q^{71} +2.51296e7 q^{73} -3.12206e6i q^{75} -6.54169e6 q^{77} +1.95344e7i q^{79} +1.41635e7 q^{81} -2.37076e7i q^{83} +1.35911e7 q^{85} -2.91440e7i q^{87} +3.86749e7 q^{89} -1.08803e8i q^{91} +5.38951e6 q^{93} +6.75057e7i q^{95} -9.46443e7 q^{97} +1.23322e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) \) 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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