LMFDB - Embedded newform 3240.2.q.l.2161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$, degree $2$, not 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:	$25.8715302549$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2161.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.2161
Dual form			3240.2.q.l.1081.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-2.00000 - 3.46410i) q^{13} -1.00000 q^{17} -5.00000 q^{19} +(2.50000 + 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(4.00000 - 6.92820i) q^{29} +(-3.50000 - 6.06218i) q^{31} -4.00000 q^{35} -6.00000 q^{37} +(3.00000 + 5.19615i) q^{41} +(1.00000 - 1.73205i) q^{43} +(4.00000 - 6.92820i) q^{47} +(-4.50000 - 7.79423i) q^{49} -9.00000 q^{53} -2.00000 q^{55} +(2.00000 + 3.46410i) q^{59} +(-6.50000 + 11.2583i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(5.00000 + 8.66025i) q^{67} +6.00000 q^{71} -6.00000 q^{73} +(-4.00000 - 6.92820i) q^{77} +(-4.50000 + 7.79423i) q^{79} +(-8.50000 + 14.7224i) q^{83} +(0.500000 + 0.866025i) q^{85} +6.00000 q^{89} -16.0000 q^{91} +(2.50000 + 4.33013i) q^{95} +(4.00000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - q^5 + 4 * q^7	$ 2 q - q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{17} - 10 q^{19} + 5 q^{23} - q^{25} + 8 q^{29} - 7 q^{31} - 8 q^{35} - 12 q^{37} + 6 q^{41} + 2 q^{43} + 8 q^{47} - 9 q^{49} - 18 q^{53} - 4 q^{55} + 4 q^{59} - 13 q^{61} - 4 q^{65} + 10 q^{67} + 12 q^{71} - 12 q^{73} - 8 q^{77} - 9 q^{79} - 17 q^{83} + q^{85} + 12 q^{89} - 32 q^{91} + 5 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q - q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^17 - 10 * q^19 + 5 * q^23 - q^25 + 8 * q^29 - 7 * q^31 - 8 * q^35 - 12 * q^37 + 6 * q^41 + 2 * q^43 + 8 * q^47 - 9 * q^49 - 18 * q^53 - 4 * q^55 + 4 * q^59 - 13 * q^61 - 4 * q^65 + 10 * q^67 + 12 * q^71 - 12 * q^73 - 8 * q^77 - 9 * q^79 - 17 * q^83 + q^85 + 12 * q^89 - 32 * q^91 + 5 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	2.00000	−	3.46410i	0.755929	−	1.30931i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	2.00000	−	3.46410i	0.755929	−	1.30931i	0.944911	−	0.327327i	$-0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	0.443227	−	0.896410i	$-0.353834\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000			−0.242536			−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000			−0.242536			−0.121268	+	0.992620i	$0.538696\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.50000	+	4.33013i	0.521286	+	0.902894i	0.999694	+	0.0247559i	$0.00788087\pi$
$23$	2.50000	+	4.33013i	0.521286	+	0.902894i	−0.478407	+	0.878138i	$0.658786\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.00000	−	6.92820i	0.742781	−	1.28654i	−0.208443	−	0.978035i	$-0.566840\pi$
$29$	4.00000	−	6.92820i	0.742781	−	1.28654i	0.951224	−	0.308500i	$-0.0998271\pi$
$30$		0			0
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	0.359211	−	0.933257i	$-0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	−0.779647	−	0.626219i	$-0.784601\pi$
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	0.932145	+	0.362084i	$0.117935\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$	−6.50000	+	11.2583i	−0.832240	+	1.44148i	0.0640184	+	0.997949i	$0.479608\pi$
$61$	−6.50000	+	11.2583i	−0.832240	+	1.44148i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	0.991098	+	0.133135i	$0.0425044\pi$
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	−0.380251	+	0.924883i	$0.624162\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000	−	6.92820i	−0.455842	−	0.789542i
$78$		0			0
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	0.493684	+	0.869641i	$0.335650\pi$
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	−0.999974	+	0.00727784i	$0.997683\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.50000	+	14.7224i	−0.932996	+	1.61600i	−0.154828	+	0.987942i	$0.549482\pi$
$83$	−8.50000	+	14.7224i	−0.932996	+	1.61600i	−0.778169	+	0.628055i	$0.783851\pi$
$84$		0			0
$85$	0.500000	+	0.866025i	0.0542326	+	0.0939336i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	−16.0000			−1.67726
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.50000	+	4.33013i	0.256495	+	0.444262i
$96$		0			0
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	−0.588315	−	0.808632i	$-0.700208\pi$
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	0.994453	+	0.105180i	$0.0335417\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	−0.396236	−	0.918149i	$-0.629684\pi$
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	0.993258	−	0.115924i	$-0.0369830\pi$
$102$		0			0
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	0.750510	−	0.660859i	$-0.229808\pi$
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	−0.947576	+	0.319531i	$0.896475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	17.0000			1.62830			0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000			1.62830			0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$	2.50000	−	4.33013i	0.233126	−	0.403786i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.00000	+	3.46410i	−0.183340	+	0.317554i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−18.0000			−1.59724			−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000			−1.59724			−0.798621	+	0.601834i	$0.794437\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	0.966803	−	0.255524i	$-0.0822479\pi$
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	−0.704692	+	0.709514i	$0.748915\pi$
$132$		0			0
$133$	−10.0000	+	17.3205i	−0.867110	+	1.50188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.50000	−	9.52628i	0.469897	−	0.813885i	−0.529511	−	0.848303i	$-0.677624\pi$
$137$	5.50000	−	9.52628i	0.469897	−	0.813885i	0.999408	+	0.0344182i	$0.0109578\pi$
$138$		0			0
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	−0.999947	−	0.0103230i	$-0.996714\pi$
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	0.491033	−	0.871141i	$-0.336619\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.00000			−0.668994
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	−0.996207	−	0.0870170i	$-0.972267\pi$
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	0.422744	−	0.906249i	$-0.361067\pi$
$150$		0			0
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	−0.331842	−	0.943335i	$-0.607670\pi$
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	0.982873	−	0.184284i	$-0.0589965\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.50000	+	6.06218i	−0.281127	+	0.486926i
$156$		0			0
$157$	−11.0000	−	19.0526i	−0.877896	−	1.52056i	−0.853646	−	0.520854i	$-0.825614\pi$
$157$	−11.0000	−	19.0526i	−0.877896	−	1.52056i	−0.0242497	−	0.999706i	$-0.507720\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	20.0000			1.57622
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.50000	−	12.9904i	−0.580367	−	1.00523i	−0.995436	−	0.0954356i	$-0.969576\pi$
$167$	−7.50000	−	12.9904i	−0.580367	−	1.00523i	0.415068	−	0.909790i	$-0.363758\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.50000	−	6.06218i	0.266100	−	0.460899i	−0.701751	−	0.712422i	$-0.747598\pi$
$173$	3.50000	−	6.06218i	0.266100	−	0.460899i	0.967851	+	0.251523i	$0.0809315\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	+	5.19615i	0.220564	+	0.382029i
$186$		0			0
$187$	−1.00000	+	1.73205i	−0.0731272	+	0.126660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.0000	+	22.5167i	−0.940647	+	1.62925i	−0.176406	+	0.984317i	$0.556447\pi$
$191$	−13.0000	+	22.5167i	−0.940647	+	1.62925i	−0.764241	+	0.644931i	$0.776886\pi$
$192$		0			0
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	−0.995948	−	0.0899262i	$-0.971337\pi$
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	0.420096	−	0.907480i	$-0.361996\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	25.0000			1.78118			0.890588	−	0.454811i	$-0.150293\pi$
$197$	25.0000			1.78118			0.890588	+	0.454811i	$0.150293\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16.0000	−	27.7128i	−1.12298	−	1.94506i
$204$		0			0
$205$	3.00000	−	5.19615i	0.209529	−	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.00000	+	8.66025i	−0.345857	+	0.599042i
$210$		0			0
$211$	−8.50000	−	14.7224i	−0.585164	−	1.01353i	−0.994855	−	0.101310i	$-0.967697\pi$
$211$	−8.50000	−	14.7224i	−0.585164	−	1.01353i	0.409691	−	0.912224i	$-0.365637\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.00000			−0.136399
$216$		0			0
$217$	−28.0000			−1.90076
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.00000	+	3.46410i	0.134535	+	0.233021i
$222$		0			0
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	−0.897564	−	0.440884i	$-0.854665\pi$
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	0.830599	+	0.556871i	$0.187998\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.50000	+	7.79423i	−0.298675	+	0.517321i	−0.975833	−	0.218517i	$-0.929878\pi$
$227$	−4.50000	+	7.79423i	−0.298675	+	0.517321i	0.677158	+	0.735838i	$0.263211\pi$
$228$		0			0
$229$	−9.50000	−	16.4545i	−0.627778	−	1.08734i	−0.987997	−	0.154475i	$-0.950631\pi$
$229$	−9.50000	−	16.4545i	−0.627778	−	1.08734i	0.360219	−	0.932868i	$-0.382702\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	−0.849472	−	0.527633i	$-0.823079\pi$
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	0.881680	+	0.471848i	$0.156413\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50000	+	7.79423i	−0.287494	+	0.497955i
$246$		0			0
$247$	10.0000	+	17.3205i	0.636285	+	1.10208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	10.0000			0.628695
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	0.909010	−	0.416775i	$-0.136840\pi$
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	−0.815442	+	0.578838i	$0.803506\pi$
$258$		0			0
$259$	−12.0000	+	20.7846i	−0.745644	+	1.29149i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	4.50000	+	7.79423i	0.276433	+	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	26.0000			1.58525			0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000			1.58525			0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$	15.0000			0.911185			0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000			0.911185			0.455593	+	0.890188i	$0.349427\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	+	1.73205i	0.0603023	+	0.104447i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.00000	+	13.8564i	−0.477240	+	0.826604i	−0.999660	−	0.0260845i	$-0.991696\pi$
$281$	−8.00000	+	13.8564i	−0.477240	+	0.826604i	0.522420	+	0.852688i	$0.325029\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	24.0000			1.41668
$288$		0			0
$289$	−16.0000			−0.941176
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.500000	−	0.866025i	−0.0292103	−	0.0505937i	0.851051	−	0.525084i	$-0.175966\pi$
$293$	−0.500000	−	0.866025i	−0.0292103	−	0.0505937i	−0.880261	+	0.474490i	$0.842633\pi$
$294$		0			0
$295$	2.00000	−	3.46410i	0.116445	−	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.0000	−	17.3205i	0.578315	−	1.00167i
$300$		0			0
$301$	−4.00000	−	6.92820i	−0.230556	−	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	13.0000			0.744378
$306$		0			0
$307$	10.0000			0.570730			0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000			0.570730			0.285365	+	0.958419i	$0.407885\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.00000	+	13.8564i	0.453638	+	0.785725i	0.998609	−	0.0527306i	$-0.0167924\pi$
$311$	8.00000	+	13.8564i	0.453638	+	0.785725i	−0.544970	+	0.838455i	$0.683459\pi$
$312$		0			0
$313$	14.0000	−	24.2487i	0.791327	−	1.37062i	−0.133819	−	0.991006i	$-0.542724\pi$
$313$	14.0000	−	24.2487i	0.791327	−	1.37062i	0.925146	−	0.379612i	$-0.123943\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.5000	−	28.5788i	0.926732	−	1.60515i	0.137981	−	0.990435i	$-0.455939\pi$
$317$	16.5000	−	28.5788i	0.926732	−	1.60515i	0.788751	−	0.614713i	$-0.210728\pi$
$318$		0			0
$319$	−8.00000	−	13.8564i	−0.447914	−	0.775810i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.00000			0.278207
$324$		0			0
$325$	4.00000			0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−16.0000	−	27.7128i	−0.882109	−	1.52786i
$330$		0			0
$331$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$331$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.00000	−	8.66025i	0.273179	−	0.473160i
$336$		0			0
$337$	10.0000	+	17.3205i	0.544735	+	0.943508i	0.998624	+	0.0524499i	$0.0167030\pi$
$337$	10.0000	+	17.3205i	0.544735	+	0.943508i	−0.453889	+	0.891058i	$0.649964\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.0000			−0.758143
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.0000	−	17.3205i	−0.536828	−	0.929814i	−0.999072	−	0.0430610i	$-0.986289\pi$
$347$	−10.0000	−	17.3205i	−0.536828	−	0.929814i	0.462244	−	0.886753i	$-0.347044\pi$
$348$		0			0
$349$	−4.50000	+	7.79423i	−0.240879	+	0.417215i	−0.960965	−	0.276670i	$-0.910769\pi$
$349$	−4.50000	+	7.79423i	−0.240879	+	0.417215i	0.720086	+	0.693885i	$0.244103\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.0000	−	29.4449i	0.904819	−	1.56719i	0.0836583	−	0.996495i	$-0.473340\pi$
$353$	17.0000	−	29.4449i	0.904819	−	1.56719i	0.821160	−	0.570697i	$-0.193327\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.00000	+	5.19615i	0.157027	+	0.271979i
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000	+	31.1769i	−0.934513	+	1.61862i
$372$		0			0
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	0.999787	+	0.0206542i	$0.00657489\pi$
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	−0.482006	+	0.876168i	$0.660092\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−32.0000			−1.64808
$378$		0			0
$379$	13.0000			0.667765			0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000			0.667765			0.333883	+	0.942615i	$0.391641\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.5000	+	18.1865i	0.536525	+	0.929288i	0.999088	+	0.0427020i	$0.0135966\pi$
$383$	10.5000	+	18.1865i	0.536525	+	0.929288i	−0.462563	+	0.886586i	$0.653070\pi$
$384$		0			0
$385$	−4.00000	+	6.92820i	−0.203859	+	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	−0.932002	−	0.362454i	$-0.881939\pi$
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	0.779895	+	0.625910i	$0.215272\pi$
$390$		0			0
$391$	−2.50000	−	4.33013i	−0.126430	−	0.218984i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.00000			0.452839
$396$		0			0
$397$	−32.0000			−1.60603			−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000			−1.60603			−0.803017	+	0.595956i	$0.796773\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$		0			0
$403$	−14.0000	+	24.2487i	−0.697390	+	1.20791i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.00000	+	10.3923i	−0.297409	+	0.515127i
$408$		0			0
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	0.996701	+	0.0811615i	$0.0258630\pi$
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	−0.428063	+	0.903749i	$0.640804\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16.0000			0.787309
$414$		0			0
$415$	17.0000			0.834497
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.00000	−	5.19615i	−0.146560	−	0.253849i	0.783394	−	0.621525i	$-0.213487\pi$
$419$	−3.00000	−	5.19615i	−0.146560	−	0.253849i	−0.929954	+	0.367677i	$0.880153\pi$
$420$		0			0
$421$	−7.50000	+	12.9904i	−0.365528	+	0.633112i	−0.988861	−	0.148844i	$-0.952445\pi$
$421$	−7.50000	+	12.9904i	−0.365528	+	0.633112i	0.623333	+	0.781956i	$0.285778\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.500000	−	0.866025i	0.0242536	−	0.0420084i
$426$		0			0
$427$	26.0000	+	45.0333i	1.25823	+	2.17932i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.0000			0.963366			0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000			0.963366			0.481683	+	0.876346i	$0.340026\pi$
$432$		0			0
$433$	20.0000			0.961139			0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000			0.961139			0.480569	+	0.876957i	$0.340430\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.5000	−	21.6506i	−0.597956	−	1.03569i
$438$		0			0
$439$	2.50000	−	4.33013i	0.119318	−	0.206666i	−0.800179	−	0.599761i	$-0.795262\pi$
$439$	2.50000	−	4.33013i	0.119318	−	0.206666i	0.919498	+	0.393095i	$0.128596\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.50000	−	2.59808i	0.0712672	−	0.123438i	−0.828190	−	0.560448i	$-0.810629\pi$
$443$	1.50000	−	2.59808i	0.0712672	−	0.123438i	0.899457	+	0.437009i	$0.143962\pi$
$444$		0			0
$445$	−3.00000	−	5.19615i	−0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.00000	+	13.8564i	0.375046	+	0.649598i
$456$		0			0
$457$	4.00000	−	6.92820i	0.187112	−	0.324088i	−0.757174	−	0.653213i	$-0.773421\pi$
$457$	4.00000	−	6.92820i	0.187112	−	0.324088i	0.944286	+	0.329125i	$0.106754\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	−0.927392	−	0.374091i	$-0.877955\pi$
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	0.787668	+	0.616100i	$0.211288\pi$
$462$		0			0
$463$	−3.00000	−	5.19615i	−0.139422	−	0.241486i	0.787856	−	0.615859i	$-0.211191\pi$
$463$	−3.00000	−	5.19615i	−0.139422	−	0.241486i	−0.927278	+	0.374374i	$0.877858\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−31.0000			−1.43451			−0.717254	−	0.696811i	$-0.754601\pi$
$467$	−31.0000			−1.43451			−0.717254	+	0.696811i	$0.754601\pi$
$468$		0			0
$469$	40.0000			1.84703
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.00000	−	3.46410i	−0.0919601	−	0.159280i
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.0000	−	22.5167i	0.593985	−	1.02881i	−0.399704	−	0.916644i	$-0.630887\pi$
$479$	13.0000	−	22.5167i	0.593985	−	1.02881i	0.993689	−	0.112168i	$-0.0357796\pi$
$480$		0			0
$481$	12.0000	+	20.7846i	0.547153	+	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.00000			−0.363261
$486$		0			0
$487$	34.0000			1.54069			0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000			1.54069			0.770344	+	0.637629i	$0.220085\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	0.975898	+	0.218229i	$0.0700279\pi$
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	−0.298957	+	0.954267i	$0.596639\pi$
$492$		0			0
$493$	−4.00000	+	6.92820i	−0.180151	+	0.312031i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.0000	−	20.7846i	0.538274	−	0.932317i
$498$		0			0
$499$	0.500000	+	0.866025i	0.0223831	+	0.0387686i	0.877000	−	0.480490i	$-0.159541\pi$
$499$	0.500000	+	0.866025i	0.0223831	+	0.0387686i	−0.854617	+	0.519259i	$0.826208\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.0000			1.20387			0.601935	−	0.798545i	$-0.294397\pi$
$503$	27.0000			1.20387			0.601935	+	0.798545i	$0.294397\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.0000	−	24.2487i	−0.620539	−	1.07481i	−0.989385	−	0.145315i	$-0.953580\pi$
$509$	−14.0000	−	24.2487i	−0.620539	−	1.07481i	0.368846	−	0.929490i	$-0.379753\pi$
$510$		0			0
$511$	−12.0000	+	20.7846i	−0.530849	+	0.919457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.00000	+	3.46410i	−0.0881305	+	0.152647i
$516$		0			0
$517$	−8.00000	−	13.8564i	−0.351840	−	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14.0000			−0.613351			−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000			−0.613351			−0.306676	+	0.951814i	$0.599217\pi$
$522$		0			0
$523$	−8.00000			−0.349816			−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000			−0.349816			−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.50000	+	6.06218i	0.152462	+	0.264073i
$528$		0			0
$529$	−1.00000	+	1.73205i	−0.0434783	+	0.0753066i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.0000	−	20.7846i	0.519778	−	0.900281i
$534$		0			0
$535$	6.00000	+	10.3923i	0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.0000			−0.775315
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.50000	−	14.7224i	−0.364100	−	0.630640i
$546$		0			0
$547$	5.00000	−	8.66025i	0.213785	−	0.370286i	−0.739111	−	0.673583i	$-0.764754\pi$
$547$	5.00000	−	8.66025i	0.213785	−	0.370286i	0.952896	+	0.303298i	$0.0980876\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−20.0000	+	34.6410i	−0.852029	+	1.47576i
$552$		0			0
$553$	18.0000	+	31.1769i	0.765438	+	1.32578i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.0000			−0.932170			−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000			−0.932170			−0.466085	+	0.884740i	$0.654336\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	0.820798	−	0.571218i	$-0.193529\pi$
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	−0.905088	+	0.425223i	$0.860196\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14.0000	−	24.2487i	0.586911	−	1.01656i	−0.407724	−	0.913105i	$-0.633677\pi$
$569$	14.0000	−	24.2487i	0.586911	−	1.01656i	0.994634	−	0.103454i	$-0.0329893\pi$
$570$		0			0
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	0.913584	−	0.406651i	$-0.133303\pi$
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	−0.808962	+	0.587861i	$0.799970\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.00000			−0.208514
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	34.0000	+	58.8897i	1.41056	+	2.44316i
$582$		0			0
$583$	−9.00000	+	15.5885i	−0.372742	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.5000	−	28.5788i	0.681028	−	1.17957i	−0.293640	−	0.955916i	$-0.594867\pi$
$587$	16.5000	−	28.5788i	0.681028	−	1.17957i	0.974668	−	0.223659i	$-0.0718001\pi$
$588$		0			0
$589$	17.5000	+	30.3109i	0.721075	+	1.24894i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.0000			−0.615976			−0.307988	−	0.951390i	$-0.599656\pi$
$593$	−15.0000			−0.615976			−0.307988	+	0.951390i	$0.599656\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	0.954521	+	0.298143i	$0.0963673\pi$
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	−0.219061	+	0.975711i	$0.570299\pi$
$600$		0			0
$601$	23.5000	−	40.7032i	0.958585	−	1.66032i	0.232643	−	0.972562i	$-0.425263\pi$
$601$	23.5000	−	40.7032i	0.958585	−	1.66032i	0.725942	−	0.687756i	$-0.241404\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.50000	−	6.06218i	0.142295	−	0.246463i
$606$		0			0
$607$	−15.0000	−	25.9808i	−0.608831	−	1.05453i	−0.991433	−	0.130613i	$-0.958305\pi$
$607$	−15.0000	−	25.9808i	−0.608831	−	1.05453i	0.382602	−	0.923913i	$-0.375028\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−32.0000			−1.29458
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.5000	−	23.3827i	−0.543490	−	0.941351i	−0.998700	−	0.0509678i	$-0.983769\pi$
$617$	−13.5000	−	23.3827i	−0.543490	−	0.941351i	0.455211	−	0.890384i	$-0.349564\pi$
$618$		0			0
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	−0.434551	−	0.900647i	$-0.643093\pi$
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	0.997259	−	0.0739910i	$-0.0235736\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0000	−	20.7846i	0.480770	−	0.832718i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.00000			0.239236
$630$		0			0
$631$	1.00000			0.0398094			0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000			0.0398094			0.0199047	+	0.999802i	$0.493664\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.00000	+	15.5885i	0.357154	+	0.618609i
$636$		0			0
$637$	−18.0000	+	31.1769i	−0.713186	+	1.23527i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−24.0000	+	41.5692i	−0.947943	+	1.64189i	−0.198194	+	0.980163i	$0.563508\pi$
$641$	−24.0000	+	41.5692i	−0.947943	+	1.64189i	−0.749749	+	0.661723i	$0.769826\pi$
$642$		0			0
$643$	−6.00000	−	10.3923i	−0.236617	−	0.409832i	0.723124	−	0.690718i	$-0.242705\pi$
$643$	−6.00000	−	10.3923i	−0.236617	−	0.409832i	−0.959741	+	0.280885i	$0.909372\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−33.0000			−1.29736			−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000			−1.29736			−0.648682	+	0.761060i	$0.724679\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.5000	+	21.6506i	0.489163	+	0.847255i	0.999922	−	0.0124688i	$-0.00396906\pi$
$653$	12.5000	+	21.6506i	0.489163	+	0.847255i	−0.510759	+	0.859724i	$0.670636\pi$
$654$		0			0
$655$	3.00000	−	5.19615i	0.117220	−	0.203030i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	9.00000	+	15.5885i	0.350059	+	0.606321i	0.986260	−	0.165203i	$-0.0528281\pi$
$661$	9.00000	+	15.5885i	0.350059	+	0.606321i	−0.636200	+	0.771524i	$0.719495\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.0000			0.775567
$666$		0			0
$667$	40.0000			1.54881
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.0000	+	22.5167i	0.501859	+	0.869246i
$672$		0			0
$673$	18.0000	−	31.1769i	0.693849	−	1.20178i	−0.276718	−	0.960951i	$-0.589247\pi$
$673$	18.0000	−	31.1769i	0.693849	−	1.20178i	0.970567	−	0.240831i	$-0.0774198\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.0000	+	29.4449i	−0.653363	+	1.13166i	0.328938	+	0.944351i	$0.393309\pi$
$677$	−17.0000	+	29.4449i	−0.653363	+	1.13166i	−0.982301	+	0.187307i	$0.940024\pi$
$678$		0			0
$679$	−16.0000	−	27.7128i	−0.614024	−	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.00000			0.344375			0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000			0.344375			0.172188	+	0.985064i	$0.444916\pi$
$684$		0			0
$685$	−11.0000			−0.420288
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18.0000	+	31.1769i	0.685745	+	1.18775i
$690$		0			0
$691$	−8.50000	+	14.7224i	−0.323355	+	0.560068i	−0.981178	−	0.193105i	$-0.938144\pi$
$691$	−8.50000	+	14.7224i	−0.323355	+	0.560068i	0.657823	+	0.753173i	$0.271478\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.00000	+	10.3923i	−0.227593	+	0.394203i
$696$		0			0
$697$	−3.00000	−	5.19615i	−0.113633	−	0.196818i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$	30.0000			1.13147
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.0000	−	41.5692i	−0.902613	−	1.56337i
$708$		0			0
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	−0.00446726	−	0.999990i	$-0.501422\pi$
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	0.868250	−	0.496126i	$-0.165245\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.5000	−	30.3109i	0.655380	−	1.13515i
$714$		0			0
$715$	4.00000	+	6.92820i	0.149592	+	0.259100i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	34.0000			1.26799			0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000			1.26799			0.633993	+	0.773339i	$0.281415\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.00000	+	6.92820i	0.148556	+	0.257307i
$726$		0			0
$727$	2.00000	−	3.46410i	0.0741759	−	0.128476i	−0.826552	−	0.562861i	$-0.809701\pi$
$727$	2.00000	−	3.46410i	0.0741759	−	0.128476i	0.900728	+	0.434384i	$0.143034\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.00000	+	1.73205i	−0.0369863	+	0.0640622i
$732$		0			0
$733$	−12.0000	−	20.7846i	−0.443230	−	0.767697i	0.554697	−	0.832052i	$-0.312834\pi$
$733$	−12.0000	−	20.7846i	−0.443230	−	0.767697i	−0.997927	+	0.0643554i	$0.979501\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	20.0000			0.736709
$738$		0			0
$739$	9.00000			0.331070			0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000			0.331070			0.165535	+	0.986204i	$0.447065\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.0000	−	41.5692i	−0.880475	−	1.52503i	−0.850814	−	0.525467i	$-0.823891\pi$
$743$	−24.0000	−	41.5692i	−0.880475	−	1.52503i	−0.0296605	−	0.999560i	$-0.509443\pi$
$744$		0			0
$745$	−7.00000	+	12.1244i	−0.256460	+	0.444202i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−24.0000	+	41.5692i	−0.876941	+	1.51891i
$750$		0			0
$751$	19.5000	+	33.7750i	0.711565	+	1.23247i	0.964269	+	0.264923i	$0.0853467\pi$
$751$	19.5000	+	33.7750i	0.711565	+	1.23247i	−0.252704	+	0.967544i	$0.581320\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.0000	−	20.7846i	−0.435000	−	0.753442i	0.562296	−	0.826936i	$-0.309918\pi$
$761$	−12.0000	−	20.7846i	−0.435000	−	0.753442i	−0.997296	+	0.0734946i	$0.976585\pi$
$762$		0			0
$763$	34.0000	−	58.8897i	1.23088	−	2.13195i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.00000	−	13.8564i	0.288863	−	0.500326i
$768$		0			0
$769$	17.5000	+	30.3109i	0.631066	+	1.09304i	0.987334	+	0.158655i	$0.0507157\pi$
$769$	17.5000	+	30.3109i	0.631066	+	1.09304i	−0.356268	+	0.934384i	$0.615951\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−33.0000			−1.18693			−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000			−1.18693			−0.593464	+	0.804861i	$0.702240\pi$
$774$		0			0
$775$	7.00000			0.251447
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.0000	−	25.9808i	−0.537431	−	0.930857i
$780$		0			0
$781$	6.00000	−	10.3923i	0.214697	−	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.0000	+	19.0526i	−0.392607	+	0.680015i
$786$		0			0
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	−0.975798	−	0.218675i	$-0.929827\pi$
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	0.298521	−	0.954403i	$-0.403507\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$	52.0000			1.84657
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.5000	−	18.1865i	−0.371929	−	0.644200i	0.617933	−	0.786231i	$-0.287970\pi$
$797$	−10.5000	−	18.1865i	−0.371929	−	0.644200i	−0.989862	+	0.142031i	$0.954637\pi$
$798$		0			0
$799$	−4.00000	+	6.92820i	−0.141510	+	0.245102i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.00000	+	10.3923i	−0.211735	+	0.366736i
$804$		0			0
$805$	−10.0000	−	17.3205i	−0.352454	−	0.610468i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.00000	+	1.73205i	0.0350285	+	0.0606711i
$816$		0			0
$817$	−5.00000	+	8.66025i	−0.174928	+	0.302984i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000	−	1.73205i	0.0349002	−	0.0604490i	−0.848048	−	0.529920i	$-0.822222\pi$
$821$	1.00000	−	1.73205i	0.0349002	−	0.0604490i	0.882948	+	0.469471i	$0.155555\pi$
$822$		0			0
$823$	25.0000	+	43.3013i	0.871445	+	1.50939i	0.860502	+	0.509447i	$0.170150\pi$
$823$	25.0000	+	43.3013i	0.871445	+	1.50939i	0.0109433	+	0.999940i	$0.496517\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.00000			0.104320			0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000			0.104320			0.0521601	+	0.998639i	$0.483389\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.50000	+	7.79423i	0.155916	+	0.270054i
$834$		0			0
$835$	−7.50000	+	12.9904i	−0.259548	+	0.449551i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	14.0000	−	24.2487i	0.483334	−	0.837158i	−0.516483	−	0.856297i	$-0.672759\pi$
$839$	14.0000	−	24.2487i	0.483334	−	0.837158i	0.999817	+	0.0191389i	$0.00609246\pi$
$840$		0			0
$841$	−17.5000	−	30.3109i	−0.603448	−	1.04520i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.00000			0.103203
$846$		0			0
$847$	28.0000			0.962091
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−15.0000	−	25.9808i	−0.514193	−	0.890609i
$852$		0			0
$853$	−16.0000	+	27.7128i	−0.547830	+	0.948869i	0.450593	+	0.892729i	$0.351212\pi$
$853$	−16.0000	+	27.7128i	−0.547830	+	0.948869i	−0.998423	+	0.0561393i	$0.982121\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.5000	−	38.9711i	0.768585	−	1.33123i	−0.169745	−	0.985488i	$-0.554294\pi$
$857$	22.5000	−	38.9711i	0.768585	−	1.33123i	0.938330	−	0.345741i	$-0.112372\pi$
$858$		0			0
$859$	15.5000	+	26.8468i	0.528853	+	0.916001i	0.999434	+	0.0336436i	$0.0107111\pi$
$859$	15.5000	+	26.8468i	0.528853	+	0.916001i	−0.470581	+	0.882357i	$0.655956\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	39.0000			1.32758			0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000			1.32758			0.663788	+	0.747921i	$0.268948\pi$
$864$		0			0
$865$	−7.00000			−0.238007
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.00000	+	15.5885i	0.305304	+	0.528802i
$870$		0			0
$871$	20.0000	−	34.6410i	0.677674	−	1.17377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.00000	−	3.46410i	0.0676123	−	0.117108i
$876$		0			0
$877$	7.00000	+	12.1244i	0.236373	+	0.409410i	0.959671	−	0.281126i	$-0.0907079\pi$
$877$	7.00000	+	12.1244i	0.236373	+	0.409410i	−0.723298	+	0.690536i	$0.757375\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.0000			0.808581			0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000			0.808581			0.404290	+	0.914631i	$0.367519\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.50000	+	14.7224i	0.285402	+	0.494331i	0.972707	−	0.232038i	$-0.0745395\pi$
$887$	8.50000	+	14.7224i	0.285402	+	0.494331i	−0.687305	+	0.726369i	$0.741206\pi$
$888$		0			0
$889$	−36.0000	+	62.3538i	−1.20740	+	2.09128i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−20.0000	+	34.6410i	−0.669274	+	1.15922i
$894$		0			0
$895$	8.00000	+	13.8564i	0.267411	+	0.463169i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−56.0000			−1.86770
$900$		0			0
$901$	9.00000			0.299833
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.50000	+	6.06218i	0.116344	+	0.201514i
$906$		0			0
$907$	6.00000	−	10.3923i	0.199227	−	0.345071i	−0.749051	−	0.662512i	$-0.769490\pi$
$907$	6.00000	−	10.3923i	0.199227	−	0.345071i	0.948278	+	0.317441i	$0.102824\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.0000	−	27.7128i	0.530104	−	0.918166i	−0.469280	−	0.883050i	$-0.655486\pi$
$911$	16.0000	−	27.7128i	0.530104	−	0.918166i	0.999383	−	0.0351168i	$-0.0111803\pi$
$912$		0			0
$913$	17.0000	+	29.4449i	0.562618	+	0.974483i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.0000			0.792550
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12.0000	−	20.7846i	−0.394985	−	0.684134i
$924$		0			0
$925$	3.00000	−	5.19615i	0.0986394	−	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.0000	−	32.9090i	0.623370	−	1.07971i	−0.365484	−	0.930818i	$-0.619096\pi$
$929$	19.0000	−	32.9090i	0.623370	−	1.07971i	0.988854	−	0.148890i	$-0.0475702\pi$
$930$		0			0
$931$	22.5000	+	38.9711i	0.737408	+	1.27723i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.00000			0.0654070
$936$		0			0
$937$	28.0000			0.914720			0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000			0.914720			0.457360	+	0.889282i	$0.348795\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.0000	+	32.9090i	0.619382	+	1.07280i	0.989599	+	0.143856i	$0.0459502\pi$
$941$	19.0000	+	32.9090i	0.619382	+	1.07280i	−0.370216	+	0.928946i	$0.620716\pi$
$942$		0			0
$943$	−15.0000	+	25.9808i	−0.488467	+	0.846050i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.50000	−	9.52628i	0.178726	−	0.309562i	−0.762718	−	0.646731i	$-0.776136\pi$
$947$	5.50000	−	9.52628i	0.178726	−	0.309562i	0.941444	+	0.337168i	$0.109469\pi$
$948$		0			0
$949$	12.0000	+	20.7846i	0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	26.0000			0.841340
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−22.0000	−	38.1051i	−0.710417	−	1.23048i
$960$		0			0
$961$	−9.00000	+	15.5885i	−0.290323	+	0.502853i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.00000	+	13.8564i	−0.257529	+	0.446054i
$966$		0			0
$967$	21.0000	+	36.3731i	0.675314	+	1.16968i	0.976377	+	0.216075i	$0.0693254\pi$
$967$	21.0000	+	36.3731i	0.675314	+	1.16968i	−0.301062	+	0.953604i	$0.597341\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	26.0000			0.834380			0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000			0.834380			0.417190	+	0.908819i	$0.363015\pi$
$972$		0			0
$973$	−48.0000			−1.53881
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.0000	−	32.9090i	−0.607864	−	1.05285i	−0.991592	−	0.129405i	$-0.958693\pi$
$977$	−19.0000	−	32.9090i	−0.607864	−	1.05285i	0.383728	−	0.923446i	$-0.374640\pi$
$978$		0			0
$979$	6.00000	−	10.3923i	0.191761	−	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.5000	+	23.3827i	−0.430583	+	0.745792i	−0.996924	−	0.0783795i	$-0.975025\pi$
$983$	−13.5000	+	23.3827i	−0.430583	+	0.745792i	0.566340	+	0.824171i	$0.308359\pi$
$984$		0			0
$985$	−12.5000	−	21.6506i	−0.398283	−	0.689847i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.0000			0.317982
$990$		0			0
$991$	5.00000			0.158830			0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000			0.158830			0.0794151	+	0.996842i	$0.474695\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.00000	+	3.46410i	0.0634043	+	0.109819i
$996$		0			0
$997$	−21.0000	+	36.3731i	−0.665077	+	1.15195i	0.314188	+	0.949361i	$0.398268\pi$
$997$	−21.0000	+	36.3731i	−0.665077	+	1.15195i	−0.979265	+	0.202586i	$0.935066\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.l.2161.1		2
3.2	odd	2		3240.2.q.w.2161.1		2
9.2	odd	6		1080.2.a.a.1.1	✓	1
9.4	even	3	inner	3240.2.q.l.1081.1		2
9.5	odd	6		3240.2.q.w.1081.1		2
9.7	even	3		1080.2.a.g.1.1	yes	1
36.7	odd	6		2160.2.a.w.1.1		1
36.11	even	6		2160.2.a.l.1.1		1
45.2	even	12		5400.2.f.s.649.1		2
45.7	odd	12		5400.2.f.l.649.1		2
45.29	odd	6		5400.2.a.bu.1.1		1
45.34	even	6		5400.2.a.br.1.1		1
45.38	even	12		5400.2.f.s.649.2		2
45.43	odd	12		5400.2.f.l.649.2		2
72.11	even	6		8640.2.a.cg.1.1		1
72.29	odd	6		8640.2.a.bf.1.1		1
72.43	odd	6		8640.2.a.bd.1.1		1
72.61	even	6		8640.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.a.1.1	✓	1	9.2	odd	6
1080.2.a.g.1.1	yes	1	9.7	even	3
2160.2.a.l.1.1		1	36.11	even	6
2160.2.a.w.1.1		1	36.7	odd	6
3240.2.q.l.1081.1		2	9.4	even	3	inner
3240.2.q.l.2161.1		2	1.1	even	1	trivial
3240.2.q.w.1081.1		2	9.5	odd	6
3240.2.q.w.2161.1		2	3.2	odd	2
5400.2.a.br.1.1		1	45.34	even	6
5400.2.a.bu.1.1		1	45.29	odd	6
5400.2.f.l.649.1		2	45.7	odd	12
5400.2.f.l.649.2		2	45.43	odd	12
5400.2.f.s.649.1		2	45.2	even	12
5400.2.f.s.649.2		2	45.38	even	12
8640.2.a.a.1.1		1	72.61	even	6
8640.2.a.bd.1.1		1	72.43	odd	6
8640.2.a.bf.1.1		1	72.29	odd	6
8640.2.a.cg.1.1		1	72.11	even	6

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-2.00000 - 3.46410i) q^{13} -1.00000 q^{17} -5.00000 q^{19} +(2.50000 + 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(4.00000 - 6.92820i) q^{29} +(-3.50000 - 6.06218i) q^{31} -4.00000 q^{35} -6.00000 q^{37} +(3.00000 + 5.19615i) q^{41} +(1.00000 - 1.73205i) q^{43} +(4.00000 - 6.92820i) q^{47} +(-4.50000 - 7.79423i) q^{49} -9.00000 q^{53} -2.00000 q^{55} +(2.00000 + 3.46410i) q^{59} +(-6.50000 + 11.2583i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(5.00000 + 8.66025i) q^{67} +6.00000 q^{71} -6.00000 q^{73} +(-4.00000 - 6.92820i) q^{77} +(-4.50000 + 7.79423i) q^{79} +(-8.50000 + 14.7224i) q^{83} +(0.500000 + 0.866025i) q^{85} +6.00000 q^{89} -16.0000 q^{91} +(2.50000 + 4.33013i) q^{95} +(4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - q^5 + 4 * q^7	\( 2 q - q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{17} - 10 q^{19} + 5 q^{23} - q^{25} + 8 q^{29} - 7 q^{31} - 8 q^{35} - 12 q^{37} + 6 q^{41} + 2 q^{43} + 8 q^{47} - 9 q^{49} - 18 q^{53} - 4 q^{55} + 4 q^{59} - 13 q^{61} - 4 q^{65} + 10 q^{67} + 12 q^{71} - 12 q^{73} - 8 q^{77} - 9 q^{79} - 17 q^{83} + q^{85} + 12 q^{89} - 32 q^{91} + 5 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q - q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^17 - 10 * q^19 + 5 * q^23 - q^25 + 8 * q^29 - 7 * q^31 - 8 * q^35 - 12 * q^37 + 6 * q^41 + 2 * q^43 + 8 * q^47 - 9 * q^49 - 18 * q^53 - 4 * q^55 + 4 * q^59 - 13 * q^61 - 4 * q^65 + 10 * q^67 + 12 * q^71 - 12 * q^73 - 8 * q^77 - 9 * q^79 - 17 * q^83 + q^85 + 12 * q^89 - 32 * q^91 + 5 * q^95 + 8 * q^97

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