LMFDB - Embedded newform 836.2.i.c.353.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [836,2,Mod(45,836)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(836, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("836.45");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 836 = 2^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	836.i (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.67549360898$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			353.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	836.353
Dual form			836.2.i.c.45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 + 2.59808i) q^{5} -3.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q+(-1.50000 + 2.59808i) q^{5} -3.00000 q^{7} +(1.50000 + 2.59808i) q^{9} -1.00000 q^{11} +(-1.00000 - 1.73205i) q^{13} +(1.00000 - 1.73205i) q^{17} +(0.500000 - 4.33013i) q^{19} +(-4.00000 - 6.92820i) q^{23} +(-2.00000 - 3.46410i) q^{25} -10.0000 q^{31} +(4.50000 - 7.79423i) q^{35} -3.00000 q^{37} +(2.00000 - 3.46410i) q^{41} +(-2.00000 + 3.46410i) q^{43} -9.00000 q^{45} +(3.00000 + 5.19615i) q^{47} +2.00000 q^{49} +(-2.50000 - 4.33013i) q^{53} +(1.50000 - 2.59808i) q^{55} +(-2.00000 + 3.46410i) q^{59} +(6.00000 + 10.3923i) q^{61} +(-4.50000 - 7.79423i) q^{63} +6.00000 q^{65} +(-6.00000 - 10.3923i) q^{67} +(-1.00000 + 1.73205i) q^{71} +(-3.00000 + 5.19615i) q^{73} +3.00000 q^{77} +(-2.50000 + 4.33013i) q^{79} +(-4.50000 + 7.79423i) q^{81} +9.00000 q^{83} +(3.00000 + 5.19615i) q^{85} +(-3.00000 - 5.19615i) q^{89} +(3.00000 + 5.19615i) q^{91} +(10.5000 + 7.79423i) q^{95} +(-2.50000 + 4.33013i) q^{97} +(-1.50000 - 2.59808i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 3 * q^5 - 6 * q^7 + 3 * q^9	$ 2 q - 3 q^{5} - 6 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} + q^{19} - 8 q^{23} - 4 q^{25} - 20 q^{31} + 9 q^{35} - 6 q^{37} + 4 q^{41} - 4 q^{43} - 18 q^{45} + 6 q^{47} + 4 q^{49} - 5 q^{53} + 3 q^{55} - 4 q^{59} + 12 q^{61} - 9 q^{63} + 12 q^{65} - 12 q^{67} - 2 q^{71} - 6 q^{73} + 6 q^{77} - 5 q^{79} - 9 q^{81} + 18 q^{83} + 6 q^{85} - 6 q^{89} + 6 q^{91} + 21 q^{95} - 5 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - 3 * q^5 - 6 * q^7 + 3 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^17 + q^19 - 8 * q^23 - 4 * q^25 - 20 * q^31 + 9 * q^35 - 6 * q^37 + 4 * q^41 - 4 * q^43 - 18 * q^45 + 6 * q^47 + 4 * q^49 - 5 * q^53 + 3 * q^55 - 4 * q^59 + 12 * q^61 - 9 * q^63 + 12 * q^65 - 12 * q^67 - 2 * q^71 - 6 * q^73 + 6 * q^77 - 5 * q^79 - 9 * q^81 + 18 * q^83 + 6 * q^85 - 6 * q^89 + 6 * q^91 + 21 * q^95 - 5 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/836\mathbb{Z}\right)^\times$.

$n$	$419$	$705$	$761$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$		0			0
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	0.306851	+	0.951757i	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	−3.00000			−1.13389			−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000			−1.13389			−0.566947	+	0.823754i	$0.691875\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	0.0607377	−	0.998154i	$-0.480655\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−10.0000			−1.79605			−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000			−1.79605			−0.898027	+	0.439941i	$0.854999\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.50000	−	7.79423i	0.760639	−	1.31747i
$36$		0			0
$37$	−3.00000			−0.493197			−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000			−0.493197			−0.246598	+	0.969118i	$0.579313\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	−0.666523	−	0.745485i	$-0.732218\pi$
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	0.978870	+	0.204483i	$0.0655513\pi$
$42$		0			0
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	−0.977261	−	0.212041i	$-0.931989\pi$
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	0.672264	+	0.740312i	$0.265322\pi$
$44$		0			0
$45$	−9.00000			−1.34164
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.50000	−	4.33013i	−0.343401	−	0.594789i	0.641661	−	0.766989i	$-0.278246\pi$
$53$	−2.50000	−	4.33013i	−0.343401	−	0.594789i	−0.985062	+	0.172200i	$0.944912\pi$
$54$		0			0
$55$	1.50000	−	2.59808i	0.202260	−	0.350325i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	6.00000	+	10.3923i	0.768221	+	1.33060i	0.938527	+	0.345207i	$0.112191\pi$
$61$	6.00000	+	10.3923i	0.768221	+	1.33060i	−0.170305	+	0.985391i	$0.554475\pi$
$62$		0			0
$63$	−4.50000	−	7.79423i	−0.566947	−	0.981981i
$64$		0			0
$65$	6.00000			0.744208
$66$		0			0
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	−0.955588	−	0.294706i	$-0.904778\pi$
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	0.222571	−	0.974916i	$-0.428555\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	−0.919244	−	0.393688i	$-0.871199\pi$
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	0.800566	+	0.599245i	$0.204532\pi$
$72$		0			0
$73$	−3.00000	+	5.19615i	−0.351123	+	0.608164i	−0.986447	−	0.164083i	$-0.947534\pi$
$73$	−3.00000	+	5.19615i	−0.351123	+	0.608164i	0.635323	+	0.772246i	$0.280867\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.00000			0.341882
$78$		0			0
$79$	−2.50000	+	4.33013i	−0.281272	+	0.487177i	−0.971698	−	0.236225i	$-0.924090\pi$
$79$	−2.50000	+	4.33013i	−0.281272	+	0.487177i	0.690426	+	0.723403i	$0.257423\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$	9.00000			0.987878			0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000			0.987878			0.493939	+	0.869496i	$0.335557\pi$
$84$		0			0
$85$	3.00000	+	5.19615i	0.325396	+	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	$-0.269678\pi$
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	−0.980071	+	0.198650i	$0.936344\pi$
$90$		0			0
$91$	3.00000	+	5.19615i	0.314485	+	0.544705i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.5000	+	7.79423i	1.07728	+	0.799671i
$96$		0			0
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	0.710742	+	0.703452i	$0.248359\pi$
$98$		0			0
$99$	−1.50000	−	2.59808i	−0.150756	−	0.261116i
$100$		0			0
$101$	−4.00000	−	6.92820i	−0.398015	−	0.689382i	0.595466	−	0.803380i	$-0.296967\pi$
$101$	−4.00000	−	6.92820i	−0.398015	−	0.689382i	−0.993481	+	0.113998i	$0.963634\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000			0.290021			0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000			0.290021			0.145010	+	0.989430i	$0.453678\pi$
$108$		0			0
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	−0.754204	−	0.656640i	$-0.771977\pi$
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	0.945769	+	0.324840i	$0.105310\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	24.0000			2.23801
$116$		0			0
$117$	3.00000	−	5.19615i	0.277350	−	0.480384i
$118$		0			0
$119$	−3.00000	+	5.19615i	−0.275010	+	0.476331i
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	−8.00000	−	13.8564i	−0.709885	−	1.22956i	−0.964899	−	0.262620i	$-0.915413\pi$
$127$	−8.00000	−	13.8564i	−0.709885	−	1.22956i	0.255014	−	0.966937i	$-0.417920\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−1.50000	+	12.9904i	−0.130066	+	1.12641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	0.922961	−	0.384893i	$-0.125762\pi$
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	−0.794808	+	0.606861i	$0.792428\pi$
$138$		0			0
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.00000	+	1.73205i	0.0836242	+	0.144841i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	0.326411	+	0.945228i	$0.394160\pi$
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	−0.981797	+	0.189933i	$0.939173\pi$
$150$		0			0
$151$	−9.00000			−0.732410			−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000			−0.732410			−0.366205	+	0.930534i	$0.619343\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$	15.0000	−	25.9808i	1.20483	−	2.08683i
$156$		0			0
$157$	−1.50000	+	2.59808i	−0.119713	+	0.207349i	−0.919654	−	0.392730i	$-0.871531\pi$
$157$	−1.50000	+	2.59808i	−0.119713	+	0.207349i	0.799941	+	0.600079i	$0.204864\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.0000	+	20.7846i	0.945732	+	1.63806i
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.5000	−	21.6506i	−0.967279	−	1.67538i	−0.703363	−	0.710831i	$-0.748319\pi$
$167$	−12.5000	−	21.6506i	−0.967279	−	1.67538i	−0.263916	−	0.964546i	$-0.585014\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$	12.0000	−	5.19615i	0.917663	−	0.397360i
$172$		0			0
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	−0.542583	−	0.840002i	$-0.682554\pi$
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	0.998755	+	0.0498898i	$0.0158870\pi$
$174$		0			0
$175$	6.00000	+	10.3923i	0.453557	+	0.785584i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$	11.5000	+	19.9186i	0.854788	+	1.48054i	0.876841	+	0.480780i	$0.159646\pi$
$181$	11.5000	+	19.9186i	0.854788	+	1.48054i	−0.0220530	+	0.999757i	$0.507020\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.50000	−	7.79423i	0.330847	−	0.573043i
$186$		0			0
$187$	−1.00000	+	1.73205i	−0.0731272	+	0.126660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$	−13.0000	+	22.5167i	−0.935760	+	1.62078i	−0.162488	+	0.986710i	$0.551952\pi$
$193$	−13.0000	+	22.5167i	−0.935760	+	1.62078i	−0.773272	+	0.634074i	$0.781381\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	0.965272	+	0.261245i	$0.0841331\pi$
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	−0.256391	+	0.966573i	$0.582534\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$		0			0
$207$	12.0000	−	20.7846i	0.834058	−	1.44463i
$208$		0			0
$209$	−0.500000	+	4.33013i	−0.0345857	+	0.299521i
$210$		0			0
$211$	−11.5000	+	19.9186i	−0.791693	+	1.37125i	0.133226	+	0.991086i	$0.457467\pi$
$211$	−11.5000	+	19.9186i	−0.791693	+	1.37125i	−0.924918	+	0.380166i	$0.875867\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.00000	−	10.3923i	−0.409197	−	0.708749i
$216$		0			0
$217$	30.0000			2.03653
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$	1.00000	−	1.73205i	0.0669650	−	0.115987i	−0.830599	−	0.556871i	$-0.812002\pi$
$223$	1.00000	−	1.73205i	0.0669650	−	0.115987i	0.897564	+	0.440884i	$0.145335\pi$
$224$		0			0
$225$	6.00000	−	10.3923i	0.400000	−	0.692820i
$226$		0			0
$227$	11.0000			0.730096			0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000			0.730096			0.365048	+	0.930989i	$0.381053\pi$
$228$		0			0
$229$	−23.0000			−1.51988			−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−23.0000			−1.51988			−0.759941	+	0.649992i	$0.774772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	+	8.66025i	−0.327561	+	0.567352i	−0.982027	−	0.188739i	$-0.939560\pi$
$233$	−5.00000	+	8.66025i	−0.327561	+	0.567352i	0.654466	+	0.756091i	$0.272893\pi$
$234$		0			0
$235$	−18.0000			−1.17419
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.0000			−1.35838			−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000			−1.35838			−0.679189	+	0.733964i	$0.737668\pi$
$240$		0			0
$241$	4.00000	+	6.92820i	0.257663	+	0.446285i	0.965615	−	0.259975i	$-0.0837143\pi$
$241$	4.00000	+	6.92820i	0.257663	+	0.446285i	−0.707953	+	0.706260i	$0.750381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	+	5.19615i	−0.191663	+	0.331970i
$246$		0			0
$247$	−8.00000	+	3.46410i	−0.509028	+	0.220416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	0.612159	−	0.790735i	$-0.290301\pi$
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	−0.990876	+	0.134778i	$0.956968\pi$
$252$		0			0
$253$	4.00000	+	6.92820i	0.251478	+	0.435572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.50000	+	11.2583i	0.405459	+	0.702275i	0.994375	−	0.105919i	$-0.0337784\pi$
$257$	6.50000	+	11.2583i	0.405459	+	0.702275i	−0.588916	+	0.808194i	$0.700445\pi$
$258$		0			0
$259$	9.00000			0.559233
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.500000	+	0.866025i	−0.0308313	+	0.0534014i	−0.881029	−	0.473062i	$-0.843149\pi$
$263$	−0.500000	+	0.866025i	−0.0308313	+	0.0534014i	0.850198	+	0.526463i	$0.176482\pi$
$264$		0			0
$265$	15.0000			0.921443
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	0.345192	+	0.938532i	$0.387814\pi$
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	−0.985389	+	0.170321i	$0.945520\pi$
$270$		0			0
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	−0.952531	−	0.304443i	$-0.901530\pi$
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	0.739921	+	0.672694i	$0.234863\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$	−15.0000	−	25.9808i	−0.898027	−	1.55543i
$280$		0			0
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	−5.50000	+	9.52628i	−0.326941	+	0.566279i	−0.981903	−	0.189383i	$-0.939351\pi$
$283$	−5.50000	+	9.52628i	−0.326941	+	0.566279i	0.654962	+	0.755662i	$0.272685\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	+	10.3923i	−0.354169	+	0.613438i
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	24.0000			1.40209			0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.00000	+	13.8564i	−0.462652	+	0.801337i
$300$		0			0
$301$	6.00000	−	10.3923i	0.345834	−	0.599002i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−36.0000			−2.06135
$306$		0			0
$307$	−5.50000	+	9.52628i	−0.313902	+	0.543693i	−0.979203	−	0.202881i	$-0.934970\pi$
$307$	−5.50000	+	9.52628i	−0.313902	+	0.543693i	0.665302	+	0.746575i	$0.268303\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	−9.50000	−	16.4545i	−0.536972	−	0.930062i	−0.999065	−	0.0432311i	$-0.986235\pi$
$313$	−9.50000	−	16.4545i	−0.536972	−	0.930062i	0.462093	−	0.886831i	$-0.347098\pi$
$314$		0			0
$315$	27.0000			1.52128
$316$		0			0
$317$	−5.00000	−	8.66025i	−0.280828	−	0.486408i	0.690761	−	0.723083i	$-0.257276\pi$
$317$	−5.00000	−	8.66025i	−0.280828	−	0.486408i	−0.971589	+	0.236675i	$0.923942\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.00000	−	5.19615i	−0.389490	−	0.289122i
$324$		0			0
$325$	−4.00000	+	6.92820i	−0.221880	+	0.384308i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.00000	−	15.5885i	−0.496186	−	0.859419i
$330$		0			0
$331$	−14.0000			−0.769510			−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000			−0.769510			−0.384755	+	0.923019i	$0.625714\pi$
$332$		0			0
$333$	−4.50000	−	7.79423i	−0.246598	−	0.427121i
$334$		0			0
$335$	36.0000			1.96689
$336$		0			0
$337$	13.0000	−	22.5167i	0.708155	−	1.22656i	−0.257386	−	0.966309i	$-0.582861\pi$
$337$	13.0000	−	22.5167i	0.708155	−	1.22656i	0.965541	−	0.260252i	$-0.0838056\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.0000			0.541530
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.5000	−	25.1147i	0.778401	−	1.34823i	−0.154462	−	0.987999i	$-0.549365\pi$
$347$	14.5000	−	25.1147i	0.778401	−	1.34823i	0.932863	−	0.360231i	$-0.117302\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000			−0.479022			−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000			−0.479022			−0.239511	+	0.970894i	$0.576987\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.50000	−	4.33013i	0.131945	−	0.228535i	−0.792481	−	0.609896i	$-0.791211\pi$
$359$	2.50000	−	4.33013i	0.131945	−	0.228535i	0.924426	+	0.381361i	$0.124544\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.00000	−	15.5885i	−0.471082	−	0.815937i
$366$		0			0
$367$	−8.00000	−	13.8564i	−0.417597	−	0.723299i	0.578101	−	0.815966i	$-0.303794\pi$
$367$	−8.00000	−	13.8564i	−0.417597	−	0.723299i	−0.995697	+	0.0926670i	$0.970461\pi$
$368$		0			0
$369$	12.0000			0.624695
$370$		0			0
$371$	7.50000	+	12.9904i	0.389381	+	0.674427i
$372$		0			0
$373$	34.0000			1.76045			0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000			1.76045			0.880227	+	0.474554i	$0.157390\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.0000	+	31.1769i	−0.919757	+	1.59307i	−0.119974	+	0.992777i	$0.538281\pi$
$383$	−18.0000	+	31.1769i	−0.919757	+	1.59307i	−0.799783	+	0.600289i	$0.795052\pi$
$384$		0			0
$385$	−4.50000	+	7.79423i	−0.229341	+	0.397231i
$386$		0			0
$387$	−12.0000			−0.609994
$388$		0			0
$389$	−15.5000	−	26.8468i	−0.785881	−	1.36119i	−0.928471	−	0.371404i	$-0.878876\pi$
$389$	−15.5000	−	26.8468i	−0.785881	−	1.36119i	0.142590	−	0.989782i	$-0.454457\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.50000	−	12.9904i	−0.377366	−	0.653617i
$396$		0			0
$397$	−7.50000	+	12.9904i	−0.376414	+	0.651969i	−0.990538	−	0.137241i	$-0.956176\pi$
$397$	−7.50000	+	12.9904i	−0.376414	+	0.651969i	0.614123	+	0.789210i	$0.289510\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	10.0000	+	17.3205i	0.498135	+	0.862796i
$404$		0			0
$405$	−13.5000	−	23.3827i	−0.670820	−	1.16190i
$406$		0			0
$407$	3.00000			0.148704
$408$		0			0
$409$	−11.0000	−	19.0526i	−0.543915	−	0.942088i	−0.998674	−	0.0514740i	$-0.983608\pi$
$409$	−11.0000	−	19.0526i	−0.543915	−	0.942088i	0.454759	−	0.890614i	$-0.349725\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.00000	−	10.3923i	0.295241	−	0.511372i
$414$		0			0
$415$	−13.5000	+	23.3827i	−0.662689	+	1.14781i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.0000			0.683945			0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000			0.683945			0.341972	+	0.939710i	$0.388905\pi$
$420$		0			0
$421$	18.5000	−	32.0429i	0.901635	−	1.56168i	0.0762630	−	0.997088i	$-0.475701\pi$
$421$	18.5000	−	32.0429i	0.901635	−	1.56168i	0.825372	−	0.564590i	$-0.190966\pi$
$422$		0			0
$423$	−9.00000	+	15.5885i	−0.437595	+	0.757937i
$424$		0			0
$425$	−8.00000			−0.388057
$426$		0			0
$427$	−18.0000	−	31.1769i	−0.871081	−	1.50876i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.5000	−	28.5788i	−0.794777	−	1.37659i	−0.922981	−	0.384846i	$-0.874254\pi$
$431$	−16.5000	−	28.5788i	−0.794777	−	1.37659i	0.128204	−	0.991748i	$-0.459079\pi$
$432$		0			0
$433$	−1.00000	−	1.73205i	−0.0480569	−	0.0832370i	0.840996	−	0.541041i	$-0.181970\pi$
$433$	−1.00000	−	1.73205i	−0.0480569	−	0.0832370i	−0.889053	+	0.457804i	$0.848636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−32.0000	+	13.8564i	−1.53077	+	0.662842i
$438$		0			0
$439$	−18.5000	+	32.0429i	−0.882957	+	1.52933i	−0.0349192	+	0.999390i	$0.511117\pi$
$439$	−18.5000	+	32.0429i	−0.882957	+	1.52933i	−0.848038	+	0.529936i	$0.822216\pi$
$440$		0			0
$441$	3.00000	+	5.19615i	0.142857	+	0.247436i
$442$		0			0
$443$	−8.00000	−	13.8564i	−0.380091	−	0.658338i	0.610984	−	0.791643i	$-0.290774\pi$
$443$	−8.00000	−	13.8564i	−0.380091	−	0.658338i	−0.991075	+	0.133306i	$0.957441\pi$
$444$		0			0
$445$	18.0000			0.853282
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.0000			−0.613508			−0.306754	−	0.951789i	$-0.599243\pi$
$449$	−13.0000			−0.613508			−0.306754	+	0.951789i	$0.599243\pi$
$450$		0			0
$451$	−2.00000	+	3.46410i	−0.0941763	+	0.163118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−18.0000			−0.843853
$456$		0			0
$457$	−32.0000			−1.49690			−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000			−1.49690			−0.748448	+	0.663193i	$0.769201\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.0000	−	31.1769i	0.838344	−	1.45205i	−0.0529352	−	0.998598i	$-0.516858\pi$
$461$	18.0000	−	31.1769i	0.838344	−	1.45205i	0.891279	−	0.453456i	$-0.149809\pi$
$462$		0			0
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.00000			0.185098			0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000			0.185098			0.0925490	+	0.995708i	$0.470499\pi$
$468$		0			0
$469$	18.0000	+	31.1769i	0.831163	+	1.43962i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.00000	−	3.46410i	0.0919601	−	0.159280i
$474$		0			0
$475$	−16.0000	+	6.92820i	−0.734130	+	0.317888i
$476$		0			0
$477$	7.50000	−	12.9904i	0.343401	−	0.594789i
$478$		0			0
$479$	16.0000	+	27.7128i	0.731059	+	1.26623i	0.956431	+	0.291958i	$0.0943068\pi$
$479$	16.0000	+	27.7128i	0.731059	+	1.26623i	−0.225372	+	0.974273i	$0.572360\pi$
$480$		0			0
$481$	3.00000	+	5.19615i	0.136788	+	0.236924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.50000	−	12.9904i	−0.340557	−	0.589863i
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.500000	−	0.866025i	0.0225647	−	0.0390832i	−0.854523	−	0.519414i	$-0.826150\pi$
$491$	0.500000	−	0.866025i	0.0225647	−	0.0390832i	0.877087	+	0.480331i	$0.159483\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	9.00000			0.404520
$496$		0			0
$497$	3.00000	−	5.19615i	0.134568	−	0.233079i
$498$		0			0
$499$	−1.00000	+	1.73205i	−0.0447661	+	0.0775372i	−0.887540	−	0.460730i	$-0.847588\pi$
$499$	−1.00000	+	1.73205i	−0.0447661	+	0.0775372i	0.842774	+	0.538267i	$0.180921\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.00000	+	10.3923i	0.267527	+	0.463370i	0.968223	−	0.250090i	$-0.0804603\pi$
$503$	6.00000	+	10.3923i	0.267527	+	0.463370i	−0.700696	+	0.713460i	$0.747127\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.500000	−	0.866025i	−0.0221621	−	0.0383859i	0.854732	−	0.519070i	$-0.173722\pi$
$509$	−0.500000	−	0.866025i	−0.0221621	−	0.0383859i	−0.876894	+	0.480684i	$0.840388\pi$
$510$		0			0
$511$	9.00000	−	15.5885i	0.398137	−	0.689593i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	−	10.3923i	0.264392	−	0.457940i
$516$		0			0
$517$	−3.00000	−	5.19615i	−0.131940	−	0.228527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$	−1.50000	−	2.59808i	−0.0655904	−	0.113606i	0.831365	−	0.555726i	$-0.187560\pi$
$523$	−1.50000	−	2.59808i	−0.0655904	−	0.113606i	−0.896956	+	0.442120i	$0.854226\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	+	17.3205i	−0.435607	+	0.754493i
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$	−8.00000			−0.346518
$534$		0			0
$535$	−4.50000	+	7.79423i	−0.194552	+	0.336974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	0.886721	−	0.462304i	$-0.152977\pi$
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	−0.843728	+	0.536771i	$0.819644\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.00000	+	10.3923i	0.257012	+	0.445157i
$546$		0			0
$547$	−8.50000	−	14.7224i	−0.363434	−	0.629486i	0.625090	−	0.780553i	$-0.285062\pi$
$547$	−8.50000	−	14.7224i	−0.363434	−	0.629486i	−0.988524	+	0.151067i	$0.951729\pi$
$548$		0			0
$549$	−18.0000	+	31.1769i	−0.768221	+	1.33060i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	7.50000	−	12.9904i	0.318932	−	0.552407i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.0000	−	25.9808i	−0.635570	−	1.10084i	−0.986394	−	0.164399i	$-0.947432\pi$
$557$	−15.0000	−	25.9808i	−0.635570	−	1.10084i	0.350824	−	0.936442i	$-0.385902\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.0000			1.22220			0.611102	−	0.791552i	$-0.290726\pi$
$563$	29.0000			1.22220			0.611102	+	0.791552i	$0.290726\pi$
$564$		0			0
$565$	21.0000	−	36.3731i	0.883477	−	1.53023i
$566$		0			0
$567$	13.5000	−	23.3827i	0.566947	−	0.981981i
$568$		0			0
$569$	34.0000			1.42535			0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000			1.42535			0.712677	+	0.701492i	$0.247483\pi$
$570$		0			0
$571$	−11.0000			−0.460336			−0.230168	−	0.973151i	$-0.573928\pi$
$571$	−11.0000			−0.460336			−0.230168	+	0.973151i	$0.573928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.0000	+	27.7128i	−0.667246	+	1.15570i
$576$		0			0
$577$	17.0000			0.707719			0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000			0.707719			0.353860	+	0.935299i	$0.384869\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−27.0000			−1.12015
$582$		0			0
$583$	2.50000	+	4.33013i	0.103539	+	0.179336i
$584$		0			0
$585$	9.00000	+	15.5885i	0.372104	+	0.644503i
$586$		0			0
$587$	1.00000	−	1.73205i	0.0412744	−	0.0714894i	−0.844650	−	0.535319i	$-0.820192\pi$
$587$	1.00000	−	1.73205i	0.0412744	−	0.0714894i	0.885925	+	0.463829i	$0.153525\pi$
$588$		0			0
$589$	−5.00000	+	43.3013i	−0.206021	+	1.78420i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.0000	+	29.4449i	0.698106	+	1.20916i	0.969122	+	0.246581i	$0.0793071\pi$
$593$	17.0000	+	29.4449i	0.698106	+	1.20916i	−0.271016	+	0.962575i	$0.587360\pi$
$594$		0			0
$595$	−9.00000	−	15.5885i	−0.368964	−	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	0.990752	+	0.135686i	$0.0433238\pi$
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	−0.377869	+	0.925859i	$0.623343\pi$
$600$		0			0
$601$	−24.0000			−0.978980			−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000			−0.978980			−0.489490	+	0.872009i	$0.662817\pi$
$602$		0			0
$603$	18.0000	−	31.1769i	0.733017	−	1.26962i
$604$		0			0
$605$	−1.50000	+	2.59808i	−0.0609837	+	0.105627i
$606$		0			0
$607$	40.0000			1.62355			0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000			1.62355			0.811775	+	0.583970i	$0.198502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.00000	−	10.3923i	0.242734	−	0.420428i
$612$		0			0
$613$	4.00000	−	6.92820i	0.161558	−	0.279827i	−0.773869	−	0.633345i	$-0.781681\pi$
$613$	4.00000	−	6.92820i	0.161558	−	0.279827i	0.935428	+	0.353518i	$0.115015\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.5000	−	28.5788i	−0.664265	−	1.15054i	−0.979484	−	0.201522i	$-0.935411\pi$
$617$	−16.5000	−	28.5788i	−0.664265	−	1.15054i	0.315219	−	0.949019i	$-0.397922\pi$
$618$		0			0
$619$	32.0000			1.28619			0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000			1.28619			0.643094	+	0.765787i	$0.277650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.00000	+	15.5885i	0.360577	+	0.624538i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.00000	+	5.19615i	−0.119618	+	0.207184i
$630$		0			0
$631$	17.0000	+	29.4449i	0.676759	+	1.17218i	0.975951	+	0.217989i	$0.0699496\pi$
$631$	17.0000	+	29.4449i	0.676759	+	1.17218i	−0.299192	+	0.954193i	$0.596717\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	48.0000			1.90482
$636$		0			0
$637$	−2.00000	−	3.46410i	−0.0792429	−	0.137253i
$638$		0			0
$639$	−6.00000			−0.237356
$640$		0			0
$641$	24.5000	−	42.4352i	0.967692	−	1.67609i	0.265490	−	0.964114i	$-0.414466\pi$
$641$	24.5000	−	42.4352i	0.967692	−	1.67609i	0.702202	−	0.711978i	$-0.252200\pi$
$642$		0			0
$643$	−17.0000	+	29.4449i	−0.670415	+	1.16119i	0.307372	+	0.951589i	$0.400550\pi$
$643$	−17.0000	+	29.4449i	−0.670415	+	1.16119i	−0.977787	+	0.209603i	$0.932783\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.00000			−0.235884			−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000			−0.235884			−0.117942	+	0.993020i	$0.537630\pi$
$648$		0			0
$649$	2.00000	−	3.46410i	0.0785069	−	0.135978i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.0000			−0.391330			−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000			−0.391330			−0.195665	+	0.980671i	$0.562687\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−18.0000			−0.702247
$658$		0			0
$659$	−17.5000	−	30.3109i	−0.681703	−	1.18074i	−0.974461	−	0.224558i	$-0.927906\pi$
$659$	−17.5000	−	30.3109i	−0.681703	−	1.18074i	0.292758	−	0.956187i	$-0.405427\pi$
$660$		0			0
$661$	−21.5000	−	37.2391i	−0.836253	−	1.44843i	−0.893006	−	0.450045i	$-0.851408\pi$
$661$	−21.5000	−	37.2391i	−0.836253	−	1.44843i	0.0567530	−	0.998388i	$-0.481925\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−31.5000	−	23.3827i	−1.22152	−	0.906742i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.00000	−	10.3923i	−0.231627	−	0.401190i
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	50.0000			1.92166			0.960828	−	0.277145i	$-0.0893883\pi$
$677$	50.0000			1.92166			0.960828	+	0.277145i	$0.0893883\pi$
$678$		0			0
$679$	7.50000	−	12.9904i	0.287824	−	0.498525i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.00000			0.229584			0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000			0.229584			0.114792	+	0.993390i	$0.463380\pi$
$684$		0			0
$685$	−9.00000			−0.343872
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.00000	+	8.66025i	−0.190485	+	0.329929i
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$		0			0
$693$	4.50000	+	7.79423i	0.170941	+	0.296078i
$694$		0			0
$695$	−21.0000			−0.796575
$696$		0			0
$697$	−4.00000	−	6.92820i	−0.151511	−	0.262424i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.00000	−	5.19615i	0.113308	−	0.196256i	−0.803794	−	0.594908i	$-0.797189\pi$
$701$	3.00000	−	5.19615i	0.113308	−	0.196256i	0.917102	+	0.398652i	$0.130522\pi$
$702$		0			0
$703$	−1.50000	+	12.9904i	−0.0565736	+	0.489942i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.0000	+	20.7846i	0.451306	+	0.781686i
$708$		0			0
$709$	17.5000	+	30.3109i	0.657226	+	1.13835i	0.981331	+	0.192328i	$0.0616038\pi$
$709$	17.5000	+	30.3109i	0.657226	+	1.13835i	−0.324104	+	0.946021i	$0.605063\pi$
$710$		0			0
$711$	−15.0000			−0.562544
$712$		0			0
$713$	40.0000	+	69.2820i	1.49801	+	2.59463i
$714$		0			0
$715$	−6.00000			−0.224387
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.0000	+	29.4449i	−0.633993	+	1.09811i	0.352735	+	0.935723i	$0.385252\pi$
$719$	−17.0000	+	29.4449i	−0.633993	+	1.09811i	−0.986728	+	0.162385i	$0.948081\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	7.00000	−	12.1244i	0.259616	−	0.449667i	−0.706523	−	0.707690i	$-0.749737\pi$
$727$	7.00000	−	12.1244i	0.259616	−	0.449667i	0.966139	+	0.258022i	$0.0830708\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	4.00000	+	6.92820i	0.147945	+	0.256249i
$732$		0			0
$733$	30.0000			1.10808			0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000			1.10808			0.554038	+	0.832492i	$0.313086\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.00000	+	10.3923i	0.221013	+	0.382805i
$738$		0			0
$739$	4.00000	−	6.92820i	0.147142	−	0.254858i	−0.783028	−	0.621987i	$-0.786326\pi$
$739$	4.00000	−	6.92820i	0.147142	−	0.254858i	0.930170	+	0.367129i	$0.119659\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.50000	−	9.52628i	0.201775	−	0.349485i	−0.747325	−	0.664459i	$-0.768662\pi$
$743$	5.50000	−	9.52628i	0.201775	−	0.349485i	0.949101	+	0.314973i	$0.101996\pi$
$744$		0			0
$745$	−24.0000	−	41.5692i	−0.879292	−	1.52298i
$746$		0			0
$747$	13.5000	+	23.3827i	0.493939	+	0.855528i
$748$		0			0
$749$	−9.00000			−0.328853
$750$		0			0
$751$	11.0000	+	19.0526i	0.401396	+	0.695238i	0.993895	−	0.110333i	$-0.0351919\pi$
$751$	11.0000	+	19.0526i	0.401396	+	0.695238i	−0.592499	+	0.805571i	$0.701859\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13.5000	−	23.3827i	0.491315	−	0.850983i
$756$		0			0
$757$	−13.0000	+	22.5167i	−0.472493	+	0.818382i	−0.999505	−	0.0314762i	$-0.989979\pi$
$757$	−13.0000	+	22.5167i	−0.472493	+	0.818382i	0.527011	+	0.849858i	$0.323312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	28.0000			1.01500			0.507500	−	0.861652i	$-0.330570\pi$
$761$	28.0000			1.01500			0.507500	+	0.861652i	$0.330570\pi$
$762$		0			0
$763$	−6.00000	+	10.3923i	−0.217215	+	0.376227i
$764$		0			0
$765$	−9.00000	+	15.5885i	−0.325396	+	0.563602i
$766$		0			0
$767$	8.00000			0.288863
$768$		0			0
$769$	−13.0000	−	22.5167i	−0.468792	−	0.811972i	0.530572	−	0.847640i	$-0.321977\pi$
$769$	−13.0000	−	22.5167i	−0.468792	−	0.811972i	−0.999364	+	0.0356685i	$0.988644\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.00000	+	5.19615i	0.107903	+	0.186893i	0.914920	−	0.403634i	$-0.132253\pi$
$773$	3.00000	+	5.19615i	0.107903	+	0.186893i	−0.807018	+	0.590527i	$0.798920\pi$
$774$		0			0
$775$	20.0000	+	34.6410i	0.718421	+	1.24434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.0000	−	10.3923i	−0.501602	−	0.372343i
$780$		0			0
$781$	1.00000	−	1.73205i	0.0357828	−	0.0619777i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.50000	−	7.79423i	−0.160612	−	0.278188i
$786$		0			0
$787$	−11.0000			−0.392108			−0.196054	−	0.980593i	$-0.562813\pi$
$787$	−11.0000			−0.392108			−0.196054	+	0.980593i	$0.562813\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	42.0000			1.49335
$792$		0			0
$793$	12.0000	−	20.7846i	0.426132	−	0.738083i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	43.0000			1.52314			0.761569	−	0.648084i	$-0.224429\pi$
$797$	43.0000			1.52314			0.761569	+	0.648084i	$0.224429\pi$
$798$		0			0
$799$	12.0000			0.424529
$800$		0			0
$801$	9.00000	−	15.5885i	0.317999	−	0.550791i
$802$		0			0
$803$	3.00000	−	5.19615i	0.105868	−	0.183368i
$804$		0			0
$805$	−72.0000			−2.53767
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24.0000			0.843795			0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000			0.843795			0.421898	+	0.906644i	$0.361364\pi$
$810$		0			0
$811$	17.5000	+	30.3109i	0.614508	+	1.06436i	0.990471	+	0.137724i	$0.0439788\pi$
$811$	17.5000	+	30.3109i	0.614508	+	1.06436i	−0.375962	+	0.926635i	$0.622688\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.00000	+	10.3923i	−0.210171	+	0.364027i
$816$		0			0
$817$	14.0000	+	10.3923i	0.489798	+	0.363581i
$818$		0			0
$819$	−9.00000	+	15.5885i	−0.314485	+	0.544705i
$820$		0			0
$821$	27.0000	+	46.7654i	0.942306	+	1.63212i	0.761056	+	0.648686i	$0.224681\pi$
$821$	27.0000	+	46.7654i	0.942306	+	1.63212i	0.181250	+	0.983437i	$0.441986\pi$
$822$		0			0
$823$	−10.0000	−	17.3205i	−0.348578	−	0.603755i	0.637419	−	0.770517i	$-0.280002\pi$
$823$	−10.0000	−	17.3205i	−0.348578	−	0.603755i	−0.985997	+	0.166762i	$0.946669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.0000	−	31.1769i	−0.625921	−	1.08413i	−0.988362	−	0.152121i	$-0.951390\pi$
$827$	−18.0000	−	31.1769i	−0.625921	−	1.08413i	0.362441	−	0.932007i	$-0.381944\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.00000	−	3.46410i	0.0692959	−	0.120024i
$834$		0			0
$835$	75.0000			2.59548
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−27.0000	+	46.7654i	−0.932144	+	1.61452i	−0.152493	+	0.988304i	$0.548730\pi$
$839$	−27.0000	+	46.7654i	−0.932144	+	1.61452i	−0.779650	+	0.626215i	$0.784603\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.5000	+	23.3827i	0.464414	+	0.804389i
$846$		0			0
$847$	−3.00000			−0.103081
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.0000	+	20.7846i	0.411355	+	0.712487i
$852$		0			0
$853$	−21.0000	+	36.3731i	−0.719026	+	1.24539i	0.242360	+	0.970186i	$0.422079\pi$
$853$	−21.0000	+	36.3731i	−0.719026	+	1.24539i	−0.961386	+	0.275204i	$0.911255\pi$
$854$		0			0
$855$	−4.50000	+	38.9711i	−0.153897	+	1.33278i
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	6.00000	+	10.3923i	0.204717	+	0.354581i	0.950043	−	0.312120i	$-0.101039\pi$
$859$	6.00000	+	10.3923i	0.204717	+	0.354581i	−0.745325	+	0.666701i	$0.767706\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	56.0000			1.90626			0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000			1.90626			0.953131	+	0.302558i	$0.0978405\pi$
$864$		0			0
$865$	18.0000	+	31.1769i	0.612018	+	1.06005i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.50000	−	4.33013i	0.0848067	−	0.146889i
$870$		0			0
$871$	−12.0000	+	20.7846i	−0.406604	+	0.704260i
$872$		0			0
$873$	−15.0000			−0.507673
$874$		0			0
$875$	9.00000			0.304256
$876$		0			0
$877$	27.0000	−	46.7654i	0.911725	−	1.57915i	0.100099	−	0.994977i	$-0.468084\pi$
$877$	27.0000	−	46.7654i	0.911725	−	1.57915i	0.811626	−	0.584177i	$-0.198583\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.0000			0.875962			0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000			0.875962			0.437981	+	0.898984i	$0.355694\pi$
$882$		0			0
$883$	−1.00000	−	1.73205i	−0.0336527	−	0.0582882i	0.848709	−	0.528861i	$-0.177381\pi$
$883$	−1.00000	−	1.73205i	−0.0336527	−	0.0582882i	−0.882361	+	0.470573i	$0.844047\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.5000	−	18.1865i	−0.352555	−	0.610644i	0.634141	−	0.773217i	$-0.281354\pi$
$887$	−10.5000	−	18.1865i	−0.352555	−	0.610644i	−0.986696	+	0.162573i	$0.948021\pi$
$888$		0			0
$889$	24.0000	+	41.5692i	0.804934	+	1.39419i
$890$		0			0
$891$	4.50000	−	7.79423i	0.150756	−	0.261116i
$892$		0			0
$893$	24.0000	−	10.3923i	0.803129	−	0.347765i
$894$		0			0
$895$	−27.0000	+	46.7654i	−0.902510	+	1.56319i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−10.0000			−0.333148
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−69.0000			−2.29364
$906$		0			0
$907$	−2.00000	+	3.46410i	−0.0664089	+	0.115024i	−0.897318	−	0.441384i	$-0.854488\pi$
$907$	−2.00000	+	3.46410i	−0.0664089	+	0.115024i	0.830909	+	0.556408i	$0.187821\pi$
$908$		0			0
$909$	12.0000	−	20.7846i	0.398015	−	0.689382i
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$	−9.00000			−0.297857
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−3.00000			−0.0989609			−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000			−0.0989609			−0.0494804	+	0.998775i	$0.515757\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.00000			0.131662
$924$		0			0
$925$	6.00000	+	10.3923i	0.197279	+	0.341697i
$926$		0			0
$927$	−6.00000	−	10.3923i	−0.197066	−	0.341328i
$928$		0			0
$929$	23.0000	−	39.8372i	0.754606	−	1.30702i	−0.190965	−	0.981597i	$-0.561162\pi$
$929$	23.0000	−	39.8372i	0.754606	−	1.30702i	0.945570	−	0.325418i	$-0.105505\pi$
$930$		0			0
$931$	1.00000	−	8.66025i	0.0327737	−	0.283828i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.00000	−	5.19615i	−0.0981105	−	0.169932i
$936$		0			0
$937$	−4.00000	−	6.92820i	−0.130674	−	0.226335i	0.793262	−	0.608880i	$-0.208381\pi$
$937$	−4.00000	−	6.92820i	−0.130674	−	0.226335i	−0.923937	+	0.382545i	$0.875048\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.0000	−	20.7846i	−0.391189	−	0.677559i	0.601418	−	0.798935i	$-0.294603\pi$
$941$	−12.0000	−	20.7846i	−0.391189	−	0.677559i	−0.992607	+	0.121376i	$0.961269\pi$
$942$		0			0
$943$	−32.0000			−1.04206
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.0000	−	27.7128i	0.519930	−	0.900545i	−0.479801	−	0.877377i	$-0.659291\pi$
$947$	16.0000	−	27.7128i	0.519930	−	0.900545i	0.999732	−	0.0231683i	$-0.00737536\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.00000	−	6.92820i	0.129573	−	0.224427i	−0.793938	−	0.607998i	$-0.791973\pi$
$953$	4.00000	−	6.92820i	0.129573	−	0.224427i	0.923511	+	0.383572i	$0.125306\pi$
$954$		0			0
$955$	−18.0000	+	31.1769i	−0.582466	+	1.00886i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.50000	−	7.79423i	−0.145313	−	0.251689i
$960$		0			0
$961$	69.0000			2.22581
$962$		0			0
$963$	4.50000	+	7.79423i	0.145010	+	0.251166i
$964$		0			0
$965$	−39.0000	−	67.5500i	−1.25545	−	2.17451i
$966$		0			0
$967$	−17.5000	+	30.3109i	−0.562762	+	0.974732i	0.434492	+	0.900676i	$0.356928\pi$
$967$	−17.5000	+	30.3109i	−0.562762	+	0.974732i	−0.997254	+	0.0740568i	$0.976405\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	0.167258	+	0.985913i	$0.446509\pi$
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	−0.937455	+	0.348107i	$0.886825\pi$
$972$		0			0
$973$	−10.5000	−	18.1865i	−0.336615	−	0.583033i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.00000			0.287936			0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000			0.287936			0.143968	+	0.989582i	$0.454014\pi$
$978$		0			0
$979$	3.00000	+	5.19615i	0.0958804	+	0.166070i
$980$		0			0
$981$	12.0000			0.383131
$982$		0			0
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	−0.945705	−	0.325027i	$-0.894626\pi$
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	0.754334	+	0.656490i	$0.227960\pi$
$984$		0			0
$985$	−3.00000	+	5.19615i	−0.0955879	+	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0000			1.01754
$990$		0			0
$991$	29.0000	−	50.2295i	0.921215	−	1.59559i	0.123678	−	0.992322i	$-0.460531\pi$
$991$	29.0000	−	50.2295i	0.921215	−	1.59559i	0.797537	−	0.603269i	$-0.206136\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−60.0000			−1.90213
$996$		0			0
$997$	10.0000	+	17.3205i	0.316703	+	0.548546i	0.979798	−	0.199989i	$-0.0640908\pi$
$997$	10.0000	+	17.3205i	0.316703	+	0.548546i	−0.663095	+	0.748535i	$0.730757\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	836.2.i.c.353.1	yes	2
19.7	even	3	inner	836.2.i.c.45.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.i.c.45.1	✓	2	19.7	even	3	inner
836.2.i.c.353.1	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 836 = 2^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	836.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 2.59808i) q^{5} -3.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 + 2.59808i) q^{5} -3.00000 q^{7} +(1.50000 + 2.59808i) q^{9} -1.00000 q^{11} +(-1.00000 - 1.73205i) q^{13} +(1.00000 - 1.73205i) q^{17} +(0.500000 - 4.33013i) q^{19} +(-4.00000 - 6.92820i) q^{23} +(-2.00000 - 3.46410i) q^{25} -10.0000 q^{31} +(4.50000 - 7.79423i) q^{35} -3.00000 q^{37} +(2.00000 - 3.46410i) q^{41} +(-2.00000 + 3.46410i) q^{43} -9.00000 q^{45} +(3.00000 + 5.19615i) q^{47} +2.00000 q^{49} +(-2.50000 - 4.33013i) q^{53} +(1.50000 - 2.59808i) q^{55} +(-2.00000 + 3.46410i) q^{59} +(6.00000 + 10.3923i) q^{61} +(-4.50000 - 7.79423i) q^{63} +6.00000 q^{65} +(-6.00000 - 10.3923i) q^{67} +(-1.00000 + 1.73205i) q^{71} +(-3.00000 + 5.19615i) q^{73} +3.00000 q^{77} +(-2.50000 + 4.33013i) q^{79} +(-4.50000 + 7.79423i) q^{81} +9.00000 q^{83} +(3.00000 + 5.19615i) q^{85} +(-3.00000 - 5.19615i) q^{89} +(3.00000 + 5.19615i) q^{91} +(10.5000 + 7.79423i) q^{95} +(-2.50000 + 4.33013i) q^{97} +(-1.50000 - 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 3 * q^5 - 6 * q^7 + 3 * q^9	\( 2 q - 3 q^{5} - 6 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} + q^{19} - 8 q^{23} - 4 q^{25} - 20 q^{31} + 9 q^{35} - 6 q^{37} + 4 q^{41} - 4 q^{43} - 18 q^{45} + 6 q^{47} + 4 q^{49} - 5 q^{53} + 3 q^{55} - 4 q^{59} + 12 q^{61} - 9 q^{63} + 12 q^{65} - 12 q^{67} - 2 q^{71} - 6 q^{73} + 6 q^{77} - 5 q^{79} - 9 q^{81} + 18 q^{83} + 6 q^{85} - 6 q^{89} + 6 q^{91} + 21 q^{95} - 5 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - 3 * q^5 - 6 * q^7 + 3 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^17 + q^19 - 8 * q^23 - 4 * q^25 - 20 * q^31 + 9 * q^35 - 6 * q^37 + 4 * q^41 - 4 * q^43 - 18 * q^45 + 6 * q^47 + 4 * q^49 - 5 * q^53 + 3 * q^55 - 4 * q^59 + 12 * q^61 - 9 * q^63 + 12 * q^65 - 12 * q^67 - 2 * q^71 - 6 * q^73 + 6 * q^77 - 5 * q^79 - 9 * q^81 + 18 * q^83 + 6 * q^85 - 6 * q^89 + 6 * q^91 + 21 * q^95 - 5 * q^97 - 3 * q^99

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