LMFDB - Embedded newform 1008.2.cs.f.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(271,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))

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

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

chi := DirichletCharacter("1008.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1008.cs (of order $6$, 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:	$8.04892052375$
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 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1008.271
Dual form			1008.2.cs.f.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(-1.50000 - 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +(1.50000 - 0.866025i) q^{11} +(4.50000 - 2.59808i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-7.50000 - 4.33013i) q^{23} +(-1.00000 - 1.73205i) q^{25} +6.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-4.50000 + 0.866025i) q^{35} +(2.50000 - 4.33013i) q^{37} -6.92820i q^{41} -3.46410i q^{43} +(1.50000 - 2.59808i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-4.50000 - 7.79423i) q^{53} -3.00000 q^{55} +(4.50000 + 7.79423i) q^{59} +(7.50000 + 4.33013i) q^{61} +(4.50000 - 2.59808i) q^{67} -3.46410i q^{71} +(1.50000 - 0.866025i) q^{73} +(1.50000 - 4.33013i) q^{77} +(-4.50000 - 2.59808i) q^{79} -12.0000 q^{83} -9.00000 q^{85} +(10.5000 + 6.06218i) q^{89} +(10.5000 - 6.06218i) q^{95} +6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 3 * q^5 + 4 * q^7	$ 2 q - 3 q^{5} + 4 q^{7} + 3 q^{11} + 9 q^{17} - 7 q^{19} - 15 q^{23} - 2 q^{25} + 12 q^{29} - 5 q^{31} - 9 q^{35} + 5 q^{37} + 3 q^{47} + 2 q^{49} - 9 q^{53} - 6 q^{55} + 9 q^{59} + 15 q^{61} + 9 q^{67} + 3 q^{73} + 3 q^{77} - 9 q^{79} - 24 q^{83} - 18 q^{85} + 21 q^{89} + 21 q^{95}+O(q^{100}) $ 2 * q - 3 * q^5 + 4 * q^7 + 3 * q^11 + 9 * q^17 - 7 * q^19 - 15 * q^23 - 2 * q^25 + 12 * q^29 - 5 * q^31 - 9 * q^35 + 5 * q^37 + 3 * q^47 + 2 * q^49 - 9 * q^53 - 6 * q^55 + 9 * q^59 + 15 * q^61 + 9 * q^67 + 3 * q^73 + 3 * q^77 - 9 * q^79 - 24 * q^83 - 18 * q^85 + 21 * q^89 + 21 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	0.125567	−	0.992085i	$-0.459925\pi$
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	−0.796387	+	0.604787i	$0.793258\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	−	0.866025i	0.452267	−	0.261116i	−0.256520	−	0.966539i	$-0.582576\pi$
$11$	1.50000	−	0.866025i	0.452267	−	0.261116i	0.708787	+	0.705422i	$0.249243\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.50000	−	2.59808i	1.09141	−	0.630126i	0.157459	−	0.987526i	$-0.449670\pi$
$17$	4.50000	−	2.59808i	1.09141	−	0.630126i	0.933952	+	0.357400i	$0.116337\pi$
$18$		0			0
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	$0.463407\pi$
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.50000	−	4.33013i	−1.56386	−	0.902894i	−0.996861	−	0.0791743i	$-0.974772\pi$
$23$	−7.50000	−	4.33013i	−1.56386	−	0.902894i	−0.566997	−	0.823720i	$-0.691895\pi$
$24$		0			0
$25$	−1.00000	−	1.73205i	−0.200000	−	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	$-0.314891\pi$
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.50000	+	0.866025i	−0.760639	+	0.146385i
$36$		0			0
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	−0.584002	−	0.811752i	$-0.698514\pi$
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	0.994999	+	0.0998840i	$0.0318472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	−0.735643	−	0.677369i	$-0.763120\pi$
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	0.954441	+	0.298401i	$0.0964533\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	−0.989828	−	0.142269i	$-0.954560\pi$
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	0.371706	−	0.928351i	$-0.378773\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	0.994769	+	0.102151i	$0.0325726\pi$
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	−0.408919	+	0.912571i	$0.634094\pi$
$60$		0			0
$61$	7.50000	+	4.33013i	0.960277	+	0.554416i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	7.50000	+	4.33013i	0.960277	+	0.554416i	0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.50000	−	2.59808i	0.549762	−	0.317406i	−0.199264	−	0.979946i	$-0.563855\pi$
$67$	4.50000	−	2.59808i	0.549762	−	0.317406i	0.749026	+	0.662540i	$0.230522\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.46410i		−	0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$		−	3.46410i		−	0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$		0			0
$73$	1.50000	−	0.866025i	0.175562	−	0.101361i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	1.50000	−	0.866025i	0.175562	−	0.101361i	0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.50000	−	4.33013i	0.170941	−	0.493464i
$78$		0			0
$79$	−4.50000	−	2.59808i	−0.506290	−	0.292306i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	−4.50000	−	2.59808i	−0.506290	−	0.292306i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	−9.00000			−0.976187
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.5000	+	6.06218i	1.11300	+	0.642590i	0.939604	−	0.342263i	$-0.111193\pi$
$89$	10.5000	+	6.06218i	1.11300	+	0.642590i	0.173394	+	0.984853i	$0.444527\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.5000	−	6.06218i	1.07728	−	0.621966i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.50000	−	2.59808i	0.447767	−	0.258518i	−0.259120	−	0.965845i	$-0.583432\pi$
$101$	4.50000	−	2.59808i	0.447767	−	0.258518i	0.706887	+	0.707327i	$0.250099\pi$
$102$		0			0
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	−0.840341	−	0.542059i	$-0.817645\pi$
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	0.889607	+	0.456727i	$0.150978\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.50000	+	2.59808i	0.435031	+	0.251166i	0.701488	−	0.712681i	$-0.252519\pi$
$107$	4.50000	+	2.59808i	0.435031	+	0.251166i	−0.266456	+	0.963847i	$0.585853\pi$
$108$		0			0
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	0.472708	−	0.881219i	$-0.343277\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	7.50000	+	12.9904i	0.699379	+	1.21136i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.50000	−	12.9904i	0.412514	−	1.19083i
$120$		0			0
$121$	−4.00000	+	6.92820i	−0.363636	+	0.629837i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.1244i			1.08444i
$126$		0			0
$127$			3.46410i			0.307389i	0.988118	+	0.153695i	$0.0491172\pi$
$127$			3.46410i			0.307389i	−0.988118	+	0.153695i	$0.950883\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.5000	+	18.1865i	−0.917389	+	1.58896i	−0.114024	+	0.993478i	$0.536374\pi$
$131$	−10.5000	+	18.1865i	−0.917389	+	1.58896i	−0.803365	+	0.595487i	$0.796959\pi$
$132$		0			0
$133$	3.50000	+	18.1865i	0.303488	+	1.57697i
$134$		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
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−9.00000	−	5.19615i	−0.747409	−	0.431517i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	10.5000	−	6.06218i	0.854478	−	0.493333i	−0.00768132	−	0.999970i	$-0.502445\pi$
$151$	10.5000	−	6.06218i	0.854478	−	0.493333i	0.862159	+	0.506637i	$0.169112\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			8.66025i			0.695608i
$156$		0			0
$157$	13.5000	−	7.79423i	1.07742	−	0.622047i	0.147219	−	0.989104i	$-0.452968\pi$
$157$	13.5000	−	7.79423i	1.07742	−	0.622047i	0.930199	+	0.367057i	$0.119635\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−22.5000	+	4.33013i	−1.77325	+	0.341262i
$162$		0			0
$163$	−10.5000	−	6.06218i	−0.822423	−	0.474826i	0.0288280	−	0.999584i	$-0.490822\pi$
$163$	−10.5000	−	6.06218i	−0.822423	−	0.474826i	−0.851251	+	0.524758i	$0.824156\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.50000	−	4.33013i	−0.570214	−	0.329213i	0.187021	−	0.982356i	$-0.440117\pi$
$173$	−7.50000	−	4.33013i	−0.570214	−	0.329213i	−0.757235	+	0.653143i	$0.773450\pi$
$174$		0			0
$175$	−5.00000	−	1.73205i	−0.377964	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.5000	−	7.79423i	1.00904	−	0.582568i	0.0981277	−	0.995174i	$-0.468715\pi$
$179$	13.5000	−	7.79423i	1.00904	−	0.582568i	0.910910	+	0.412606i	$0.135381\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.50000	+	4.33013i	−0.551411	+	0.318357i
$186$		0			0
$187$	4.50000	−	7.79423i	0.329073	−	0.569970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.5000	+	6.06218i	0.759753	+	0.438644i	0.829207	−	0.558941i	$-0.188792\pi$
$191$	10.5000	+	6.06218i	0.759753	+	0.438644i	−0.0694538	+	0.997585i	$0.522126\pi$
$192$		0			0
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	0.941865	−	0.335993i	$-0.109072\pi$
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	−0.761911	+	0.647682i	$0.775738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	−	10.3923i	0.842235	−	0.729397i
$204$		0			0
$205$	−6.00000	+	10.3923i	−0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			12.1244i			0.838659i
$210$		0			0
$211$			24.2487i			1.66935i	0.550743	+	0.834675i	$0.314345\pi$
$211$			24.2487i			1.66935i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.00000	+	5.19615i	−0.204598	+	0.354375i
$216$		0			0
$217$	−12.5000	−	4.33013i	−0.848555	−	0.293948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.5000	+	18.1865i	0.696909	+	1.20708i	0.969533	+	0.244962i	$0.0787754\pi$
$227$	10.5000	+	18.1865i	0.696909	+	1.20708i	−0.272623	+	0.962121i	$0.587891\pi$
$228$		0			0
$229$	−10.5000	−	6.06218i	−0.693860	−	0.400600i	0.111197	−	0.993798i	$-0.464532\pi$
$229$	−10.5000	−	6.06218i	−0.693860	−	0.400600i	−0.805056	+	0.593198i	$0.797865\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.50000	−	2.59808i	0.0982683	−	0.170206i	−0.812700	−	0.582683i	$-0.802003\pi$
$233$	1.50000	−	2.59808i	0.0982683	−	0.170206i	0.910968	+	0.412477i	$0.135336\pi$
$234$		0			0
$235$	−4.50000	+	2.59808i	−0.293548	+	0.169480i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$	−10.5000	+	6.06218i	−0.676364	+	0.390499i	−0.798484	−	0.602016i	$-0.794364\pi$
$241$	−10.5000	+	6.06218i	−0.676364	+	0.390499i	0.122119	+	0.992515i	$0.461031\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.50000	+	9.52628i	−0.479157	+	0.608612i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−15.0000			−0.943042
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.5000	+	9.52628i	1.02924	+	0.594233i	0.916767	−	0.399422i	$-0.130789\pi$
$257$	16.5000	+	9.52628i	1.02924	+	0.594233i	0.112474	+	0.993655i	$0.464122\pi$
$258$		0			0
$259$	−2.50000	−	12.9904i	−0.155342	−	0.807183i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.50000	−	4.33013i	0.462470	−	0.267007i	−0.250612	−	0.968088i	$-0.580632\pi$
$263$	7.50000	−	4.33013i	0.462470	−	0.267007i	0.713082	+	0.701080i	$0.247299\pi$
$264$		0			0
$265$			15.5885i			0.957591i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.50000	+	0.866025i	−0.0914566	+	0.0528025i	−0.545031	−	0.838416i	$-0.683482\pi$
$269$	−1.50000	+	0.866025i	−0.0914566	+	0.0528025i	0.453574	+	0.891219i	$0.350149\pi$
$270$		0			0
$271$	0.500000	−	0.866025i	0.0303728	−	0.0526073i	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	0.500000	−	0.866025i	0.0303728	−	0.0526073i	0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	−	1.73205i	−0.180907	−	0.104447i
$276$		0			0
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	0.999923	+	0.0124177i	$0.00395278\pi$
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	−0.489207	+	0.872167i	$0.662714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	5.50000	+	9.52628i	0.326941	+	0.566279i	0.981903	−	0.189383i	$-0.0606488\pi$
$283$	5.50000	+	9.52628i	0.326941	+	0.566279i	−0.654962	+	0.755662i	$0.727315\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0000	−	13.8564i	−0.708338	−	0.817918i
$288$		0			0
$289$	5.00000	−	8.66025i	0.294118	−	0.509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			20.7846i			1.21425i	0.794606	+	0.607125i	$0.207677\pi$
$293$			20.7846i			1.21425i	−0.794606	+	0.607125i	$0.792323\pi$
$294$		0			0
$295$		−	15.5885i		−	0.907595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−6.00000	−	6.92820i	−0.345834	−	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.50000	−	12.9904i	−0.429449	−	0.743827i
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.5000	+	28.5788i	0.935629	+	1.62056i	0.773508	+	0.633786i	$0.218500\pi$
$311$	16.5000	+	28.5788i	0.935629	+	1.62056i	0.162121	+	0.986771i	$0.448167\pi$
$312$		0			0
$313$	25.5000	+	14.7224i	1.44135	+	0.832161i	0.997940	−	0.0641600i	$-0.0204368\pi$
$313$	25.5000	+	14.7224i	1.44135	+	0.832161i	0.443406	+	0.896321i	$0.353770\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.5000	+	18.1865i	−0.589739	+	1.02146i	0.404528	+	0.914526i	$0.367436\pi$
$317$	−10.5000	+	18.1865i	−0.589739	+	1.02146i	−0.994266	+	0.106932i	$0.965897\pi$
$318$		0			0
$319$	9.00000	−	5.19615i	0.503903	−	0.290929i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			36.3731i			2.02385i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.50000	−	7.79423i	−0.0826977	−	0.429710i
$330$		0			0
$331$	25.5000	+	14.7224i	1.40161	+	0.809218i	0.994558	−	0.104188i	$-0.0332244\pi$
$331$	25.5000	+	14.7224i	1.40161	+	0.809218i	0.407049	+	0.913406i	$0.366558\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.00000			−0.491723
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.50000	−	4.33013i	−0.406148	−	0.234490i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.50000	−	4.33013i	0.402621	−	0.232453i	−0.284993	−	0.958530i	$-0.591991\pi$
$347$	7.50000	−	4.33013i	0.402621	−	0.232453i	0.687614	+	0.726076i	$0.258658\pi$
$348$		0			0
$349$			27.7128i			1.48343i	0.670714	+	0.741716i	$0.265988\pi$
$349$			27.7128i			1.48343i	−0.670714	+	0.741716i	$0.734012\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.5000	−	9.52628i	0.878206	−	0.507033i	0.00813978	−	0.999967i	$-0.497409\pi$
$353$	16.5000	−	9.52628i	0.878206	−	0.507033i	0.870067	+	0.492934i	$0.164076\pi$
$354$		0			0
$355$	−3.00000	+	5.19615i	−0.159223	+	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.5000	−	7.79423i	−0.712503	−	0.411364i	0.0994843	−	0.995039i	$-0.468281\pi$
$359$	−13.5000	−	7.79423i	−0.712503	−	0.411364i	−0.811987	+	0.583675i	$0.801614\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.00000			−0.157027
$366$		0			0
$367$	−0.500000	−	0.866025i	−0.0260998	−	0.0452062i	0.852680	−	0.522433i	$-0.174975\pi$
$367$	−0.500000	−	0.866025i	−0.0260998	−	0.0452062i	−0.878780	+	0.477227i	$0.841642\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.5000	−	7.79423i	−1.16814	−	0.404656i
$372$		0			0
$373$	6.50000	−	11.2583i	0.336557	−	0.582934i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	6.50000	−	11.2583i	0.336557	−	0.582934i	0.983783	+	0.179364i	$0.0574041\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	10.3923i		−	0.533817i	−0.963722	−	0.266908i	$-0.913998\pi$
$379$		−	10.3923i		−	0.533817i	0.963722	−	0.266908i	$-0.0860021\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.5000	−	23.3827i	0.689818	−	1.19480i	−0.282079	−	0.959391i	$-0.591024\pi$
$383$	13.5000	−	23.3827i	0.689818	−	1.19480i	0.971897	−	0.235408i	$-0.0756427\pi$
$384$		0			0
$385$	−6.00000	+	5.19615i	−0.305788	+	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.5000	−	18.1865i	−0.532371	−	0.922094i	−0.999286	−	0.0377914i	$-0.987968\pi$
$389$	−10.5000	−	18.1865i	−0.532371	−	0.922094i	0.466915	−	0.884302i	$-0.345366\pi$
$390$		0			0
$391$	−45.0000			−2.27575
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.50000	+	7.79423i	0.226420	+	0.392170i
$396$		0			0
$397$	−10.5000	−	6.06218i	−0.526980	−	0.304252i	0.212806	−	0.977095i	$-0.431740\pi$
$397$	−10.5000	−	6.06218i	−0.526980	−	0.304252i	−0.739786	+	0.672843i	$0.765073\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	−0.826139	−	0.563466i	$-0.809468\pi$
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	0.901046	+	0.433724i	$0.142801\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.66025i		−	0.429273i
$408$		0			0
$409$	−4.50000	+	2.59808i	−0.222511	+	0.128467i	−0.607112	−	0.794616i	$-0.707672\pi$
$409$	−4.50000	+	2.59808i	−0.222511	+	0.128467i	0.384602	+	0.923083i	$0.374339\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	22.5000	+	7.79423i	1.10715	+	0.383529i
$414$		0			0
$415$	18.0000	+	10.3923i	0.883585	+	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.00000	−	5.19615i	−0.436564	−	0.252050i
$426$		0			0
$427$	22.5000	−	4.33013i	1.08885	−	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	25.5000	−	14.7224i	1.22829	−	0.709155i	0.261619	−	0.965171i	$-0.415743\pi$
$431$	25.5000	−	14.7224i	1.22829	−	0.709155i	0.966672	+	0.256017i	$0.0824102\pi$
$432$		0			0
$433$		−	34.6410i		−	1.66474i	−0.554220	−	0.832370i	$-0.686983\pi$
$433$		−	34.6410i		−	1.66474i	0.554220	−	0.832370i	$-0.313017\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	52.5000	−	30.3109i	2.51142	−	1.44997i
$438$		0			0
$439$	−9.50000	+	16.4545i	−0.453410	+	0.785330i	−0.998595	−	0.0529862i	$-0.983126\pi$
$439$	−9.50000	+	16.4545i	−0.453410	+	0.785330i	0.545185	+	0.838316i	$0.316459\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.5000	−	11.2583i	−0.926473	−	0.534899i	−0.0407786	−	0.999168i	$-0.512984\pi$
$443$	−19.5000	−	11.2583i	−0.926473	−	0.534899i	−0.885694	+	0.464269i	$0.846317\pi$
$444$		0			0
$445$	−10.5000	−	18.1865i	−0.497748	−	0.862124i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$	−6.00000	−	10.3923i	−0.282529	−	0.489355i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	−0.854094	−	0.520119i	$-0.825888\pi$
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	0.877483	+	0.479608i	$0.159221\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	27.7128i		−	1.29071i	−0.763881	−	0.645357i	$-0.776709\pi$
$461$		−	27.7128i		−	1.29071i	0.763881	−	0.645357i	$-0.223291\pi$
$462$		0			0
$463$			3.46410i			0.160990i	0.996755	+	0.0804952i	$0.0256502\pi$
$463$			3.46410i			0.160990i	−0.996755	+	0.0804952i	$0.974350\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	−0.829231	−	0.558906i	$-0.811221\pi$
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	0.898642	+	0.438682i	$0.144554\pi$
$468$		0			0
$469$	4.50000	−	12.9904i	0.207791	−	0.599840i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.00000	−	5.19615i	−0.137940	−	0.238919i
$474$		0			0
$475$	14.0000			0.642364
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.50000	+	7.79423i	0.205610	+	0.356127i	0.950327	−	0.311253i	$-0.100749\pi$
$479$	4.50000	+	7.79423i	0.205610	+	0.356127i	−0.744717	+	0.667381i	$0.767415\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.00000	−	10.3923i	0.272446	−	0.471890i
$486$		0			0
$487$	−7.50000	+	4.33013i	−0.339857	+	0.196217i	−0.660209	−	0.751082i	$-0.729532\pi$
$487$	−7.50000	+	4.33013i	−0.339857	+	0.196217i	0.320352	+	0.947299i	$0.396199\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.3205i		−	0.781664i	−0.920462	−	0.390832i	$-0.872187\pi$
$491$		−	17.3205i		−	0.781664i	0.920462	−	0.390832i	$-0.127813\pi$
$492$		0			0
$493$	27.0000	−	15.5885i	1.21602	−	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000	−	6.92820i	−0.269137	−	0.310772i
$498$		0			0
$499$	−10.5000	−	6.06218i	−0.470045	−	0.271380i	0.246214	−	0.969216i	$-0.420813\pi$
$499$	−10.5000	−	6.06218i	−0.470045	−	0.271380i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	−9.00000			−0.400495
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.5000	−	7.79423i	−0.598377	−	0.345473i	0.170026	−	0.985440i	$-0.445615\pi$
$509$	−13.5000	−	7.79423i	−0.598377	−	0.345473i	−0.768403	+	0.639966i	$0.778948\pi$
$510$		0			0
$511$	1.50000	−	4.33013i	0.0663561	−	0.191554i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.50000	+	0.866025i	−0.0660979	+	0.0381616i
$516$		0			0
$517$		−	5.19615i		−	0.228527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−25.5000	+	14.7224i	−1.11718	+	0.645001i	−0.940678	−	0.339300i	$-0.889810\pi$
$521$	−25.5000	+	14.7224i	−1.11718	+	0.645001i	−0.176497	+	0.984301i	$0.556477\pi$
$522$		0			0
$523$	−11.5000	+	19.9186i	−0.502860	+	0.870979i	0.497135	+	0.867673i	$0.334385\pi$
$523$	−11.5000	+	19.9186i	−0.502860	+	0.870979i	−0.999995	+	0.00330547i	$0.998948\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22.5000	−	12.9904i	−0.980115	−	0.565870i
$528$		0			0
$529$	26.0000	+	45.0333i	1.13043	+	1.95797i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−4.50000	−	7.79423i	−0.194552	−	0.336974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.50000	−	11.2583i	−0.193829	−	0.484931i
$540$		0			0
$541$	−3.50000	+	6.06218i	−0.150477	+	0.260633i	−0.931403	−	0.363990i	$-0.881414\pi$
$541$	−3.50000	+	6.06218i	−0.150477	+	0.260633i	0.780926	+	0.624623i	$0.214748\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			19.0526i			0.816122i
$546$		0			0
$547$		−	24.2487i		−	1.03680i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$		−	24.2487i		−	1.03680i	0.855138	−	0.518400i	$-0.173472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21.0000	+	36.3731i	−0.894630	+	1.54954i
$552$		0			0
$553$	−13.5000	+	2.59808i	−0.574078	+	0.110481i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.50000	+	2.59808i	0.0635570	+	0.110084i	0.896053	−	0.443947i	$-0.146422\pi$
$557$	1.50000	+	2.59808i	0.0635570	+	0.110084i	−0.832496	+	0.554031i	$0.813089\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.50000	−	12.9904i	−0.316087	−	0.547479i	0.663581	−	0.748105i	$-0.269036\pi$
$563$	−7.50000	−	12.9904i	−0.316087	−	0.547479i	−0.979668	+	0.200625i	$0.935703\pi$
$564$		0			0
$565$	9.00000	+	5.19615i	0.378633	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.5000	−	33.7750i	0.817483	−	1.41592i	−0.0900490	−	0.995937i	$-0.528702\pi$
$569$	19.5000	−	33.7750i	0.817483	−	1.41592i	0.907532	−	0.419984i	$-0.137964\pi$
$570$		0			0
$571$	10.5000	−	6.06218i	0.439411	−	0.253694i	−0.263937	−	0.964540i	$-0.585021\pi$
$571$	10.5000	−	6.06218i	0.439411	−	0.253694i	0.703348	+	0.710846i	$0.251688\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			17.3205i			0.722315i
$576$		0			0
$577$	13.5000	−	7.79423i	0.562012	−	0.324478i	−0.191940	−	0.981407i	$-0.561478\pi$
$577$	13.5000	−	7.79423i	0.562012	−	0.324478i	0.753953	+	0.656929i	$0.228145\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.0000	+	20.7846i	−0.995688	+	0.862291i
$582$		0			0
$583$	−13.5000	−	7.79423i	−0.559113	−	0.322804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	35.0000			1.44215
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16.5000	+	9.52628i	0.677574	+	0.391197i	0.798940	−	0.601410i	$-0.205394\pi$
$593$	16.5000	+	9.52628i	0.677574	+	0.391197i	−0.121367	+	0.992608i	$0.538728\pi$
$594$		0			0
$595$	−18.0000	+	15.5885i	−0.737928	+	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.5000	+	16.4545i	−1.16448	+	0.672312i	−0.952373	−	0.304935i	$-0.901365\pi$
$599$	−28.5000	+	16.4545i	−1.16448	+	0.672312i	−0.212105	+	0.977247i	$0.568032\pi$
$600$		0			0
$601$		−	6.92820i		−	0.282607i	−0.989966	−	0.141304i	$-0.954871\pi$
$601$		−	6.92820i		−	0.282607i	0.989966	−	0.141304i	$-0.0451294\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.0000	−	6.92820i	0.487869	−	0.281672i
$606$		0			0
$607$	14.5000	−	25.1147i	0.588537	−	1.01938i	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	14.5000	−	25.1147i	0.588537	−	1.01938i	0.994424	−	0.105453i	$-0.0336291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−5.50000	−	9.52628i	−0.222143	−	0.384763i	0.733316	−	0.679888i	$-0.237972\pi$
$613$	−5.50000	−	9.52628i	−0.222143	−	0.384763i	−0.955458	+	0.295126i	$0.904638\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		0			0
$619$	3.50000	+	6.06218i	0.140677	+	0.243659i	0.927752	−	0.373198i	$-0.121739\pi$
$619$	3.50000	+	6.06218i	0.140677	+	0.243659i	−0.787075	+	0.616858i	$0.788405\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	31.5000	−	6.06218i	1.26202	−	0.242876i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	25.9808i		−	1.03592i
$630$		0			0
$631$			45.0333i			1.79275i	0.443298	+	0.896374i	$0.353808\pi$
$631$			45.0333i			1.79275i	−0.443298	+	0.896374i	$0.646192\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.00000	−	5.19615i	0.119051	−	0.206203i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.50000	−	7.79423i	−0.177739	−	0.307854i	0.763367	−	0.645966i	$-0.223545\pi$
$641$	−4.50000	−	7.79423i	−0.177739	−	0.307854i	−0.941106	+	0.338112i	$0.890212\pi$
$642$		0			0
$643$	4.00000			0.157745			0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000			0.157745			0.0788723	+	0.996885i	$0.474868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	0.680096	−	0.733123i	$-0.261938\pi$
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	−0.974951	+	0.222419i	$0.928605\pi$
$648$		0			0
$649$	13.5000	+	7.79423i	0.529921	+	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.50000	−	2.59808i	0.0586995	−	0.101671i	−0.835182	−	0.549973i	$-0.814638\pi$
$653$	1.50000	−	2.59808i	0.0586995	−	0.101671i	0.893882	+	0.448303i	$0.147971\pi$
$654$		0			0
$655$	31.5000	−	18.1865i	1.23081	−	0.710607i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	24.2487i		−	0.944596i	−0.881439	−	0.472298i	$-0.843425\pi$
$659$		−	24.2487i		−	0.944596i	0.881439	−	0.472298i	$-0.156575\pi$
$660$		0			0
$661$	−34.5000	+	19.9186i	−1.34189	+	0.774743i	−0.987085	−	0.160196i	$-0.948788\pi$
$661$	−34.5000	+	19.9186i	−1.34189	+	0.774743i	−0.354809	+	0.934939i	$0.615454\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.5000	−	30.3109i	0.407173	−	1.17541i
$666$		0			0
$667$	−45.0000	−	25.9808i	−1.74241	−	1.00598i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.0000			0.579069
$672$		0			0
$673$	−50.0000			−1.92736			−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000			−1.92736			−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	34.5000	+	19.9186i	1.32594	+	0.765533i	0.984669	−	0.174431i	$-0.0558085\pi$
$677$	34.5000	+	19.9186i	1.32594	+	0.765533i	0.341273	+	0.939964i	$0.389142\pi$
$678$		0			0
$679$	12.0000	+	13.8564i	0.460518	+	0.531760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	37.5000	−	21.6506i	1.43490	−	0.828439i	0.437409	−	0.899263i	$-0.355896\pi$
$683$	37.5000	−	21.6506i	1.43490	−	0.828439i	0.997489	+	0.0708242i	$0.0225629\pi$
$684$		0			0
$685$		−	5.19615i		−	0.198535i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	6.50000	−	11.2583i	0.247272	−	0.428287i	−0.715496	−	0.698617i	$-0.753799\pi$
$691$	6.50000	−	11.2583i	0.247272	−	0.428287i	0.962768	+	0.270330i	$0.0871327\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	+	3.46410i	0.227593	+	0.131401i
$696$		0			0
$697$	−18.0000	−	31.1769i	−0.681799	−	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	17.5000	+	30.3109i	0.660025	+	1.14320i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.50000	−	12.9904i	0.169240	−	0.488554i
$708$		0			0
$709$	−17.5000	+	30.3109i	−0.657226	+	1.13835i	0.324104	+	0.946021i	$0.394937\pi$
$709$	−17.5000	+	30.3109i	−0.657226	+	1.13835i	−0.981331	+	0.192328i	$0.938396\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			43.3013i			1.62165i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.5000	+	28.5788i	−0.615346	+	1.06581i	0.374978	+	0.927034i	$0.377650\pi$
$719$	−16.5000	+	28.5788i	−0.615346	+	1.06581i	−0.990324	+	0.138777i	$0.955683\pi$
$720$		0			0
$721$	−0.500000	−	2.59808i	−0.0186210	−	0.0967574i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.00000	−	10.3923i	−0.222834	−	0.385961i
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.00000	−	15.5885i	−0.332877	−	0.576560i
$732$		0			0
$733$	−16.5000	−	9.52628i	−0.609441	−	0.351861i	0.163305	−	0.986576i	$-0.447784\pi$
$733$	−16.5000	−	9.52628i	−0.609441	−	0.351861i	−0.772747	+	0.634714i	$0.781118\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.50000	−	7.79423i	0.165760	−	0.287104i
$738$		0			0
$739$	−1.50000	+	0.866025i	−0.0551784	+	0.0318573i	−0.527335	−	0.849657i	$-0.676809\pi$
$739$	−1.50000	+	0.866025i	−0.0551784	+	0.0318573i	0.472157	+	0.881514i	$0.343476\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			38.1051i			1.39794i	0.715150	+	0.698971i	$0.246358\pi$
$743$			38.1051i			1.39794i	−0.715150	+	0.698971i	$0.753642\pi$
$744$		0			0
$745$	13.5000	−	7.79423i	0.494602	−	0.285558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.5000	−	2.59808i	0.493279	−	0.0949316i
$750$		0			0
$751$	−34.5000	−	19.9186i	−1.25892	−	0.726839i	−0.286058	−	0.958212i	$-0.592345\pi$
$751$	−34.5000	−	19.9186i	−1.25892	−	0.726839i	−0.972865	+	0.231373i	$0.925678\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.0000			−0.764268
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.50000	+	2.59808i	0.163125	+	0.0941802i	0.579340	−	0.815086i	$-0.303310\pi$
$761$	4.50000	+	2.59808i	0.163125	+	0.0941802i	−0.416215	+	0.909266i	$0.636644\pi$
$762$		0			0
$763$	−27.5000	−	9.52628i	−0.995567	−	0.344874i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	6.92820i		−	0.249837i	−0.992167	−	0.124919i	$-0.960133\pi$
$769$		−	6.92820i		−	0.249837i	0.992167	−	0.124919i	$-0.0398670\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.50000	+	4.33013i	−0.269756	+	0.155744i	−0.628777	−	0.777586i	$-0.716444\pi$
$773$	−7.50000	+	4.33013i	−0.269756	+	0.155744i	0.359021	+	0.933330i	$0.383111\pi$
$774$		0			0
$775$	−5.00000	+	8.66025i	−0.179605	+	0.311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	42.0000	+	24.2487i	1.50481	+	0.868800i
$780$		0			0
$781$	−3.00000	−	5.19615i	−0.107348	−	0.185933i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−27.0000			−0.963671
$786$		0			0
$787$	3.50000	+	6.06218i	0.124762	+	0.216093i	0.921640	−	0.388047i	$-0.126850\pi$
$787$	3.50000	+	6.06218i	0.124762	+	0.216093i	−0.796878	+	0.604140i	$0.793517\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000	+	10.3923i	−0.426671	+	0.369508i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.8564i		−	0.490819i	−0.969419	−	0.245410i	$-0.921078\pi$
$797$		−	13.8564i		−	0.490819i	0.969419	−	0.245410i	$-0.0789224\pi$
$798$		0			0
$799$		−	15.5885i		−	0.551480i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.50000	−	2.59808i	0.0529339	−	0.0916841i
$804$		0			0
$805$	37.5000	+	12.9904i	1.32170	+	0.457851i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.50000	+	12.9904i	0.263686	+	0.456717i	0.967219	−	0.253946i	$-0.0817284\pi$
$809$	7.50000	+	12.9904i	0.263686	+	0.456717i	−0.703533	+	0.710663i	$0.748395\pi$
$810$		0			0
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.5000	+	18.1865i	0.367799	+	0.637046i
$816$		0			0
$817$	21.0000	+	12.1244i	0.734697	+	0.424178i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.5000	+	49.3634i	−0.994657	+	1.72280i	−0.407923	+	0.913016i	$0.633747\pi$
$821$	−28.5000	+	49.3634i	−0.994657	+	1.72280i	−0.586734	+	0.809780i	$0.699586\pi$
$822$		0			0
$823$	−19.5000	+	11.2583i	−0.679727	+	0.392441i	−0.799752	−	0.600330i	$-0.795036\pi$
$823$	−19.5000	+	11.2583i	−0.679727	+	0.392441i	0.120025	+	0.992771i	$0.461703\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	38.1051i		−	1.32504i	−0.749042	−	0.662522i	$-0.769486\pi$
$827$		−	38.1051i		−	1.32504i	0.749042	−	0.662522i	$-0.230514\pi$
$828$		0			0
$829$	7.50000	−	4.33013i	0.260486	−	0.150392i	−0.364070	−	0.931371i	$-0.618613\pi$
$829$	7.50000	−	4.33013i	0.260486	−	0.150392i	0.624556	+	0.780980i	$0.285280\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.5000	−	33.7750i	−0.467747	−	1.17023i
$834$		0			0
$835$	18.0000	+	10.3923i	0.622916	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.5000	−	11.2583i	−0.670820	−	0.387298i
$846$		0			0
$847$	4.00000	+	20.7846i	0.137442	+	0.714168i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−37.5000	+	21.6506i	−1.28548	+	0.742174i
$852$		0			0
$853$		−	13.8564i		−	0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$		−	13.8564i		−	0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.5000	+	21.6506i	−1.28098	+	0.739572i	−0.977027	−	0.213117i	$-0.931639\pi$
$857$	−37.5000	+	21.6506i	−1.28098	+	0.739572i	−0.303949	+	0.952688i	$0.598305\pi$
$858$		0			0
$859$	20.5000	−	35.5070i	0.699451	−	1.21148i	−0.269206	−	0.963083i	$-0.586761\pi$
$859$	20.5000	−	35.5070i	0.699451	−	1.21148i	0.968657	−	0.248402i	$-0.0799054\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	40.5000	+	23.3827i	1.37864	+	0.795956i	0.991995	−	0.126275i	$-0.0403020\pi$
$863$	40.5000	+	23.3827i	1.37864	+	0.795956i	0.386641	+	0.922230i	$0.373635\pi$
$864$		0			0
$865$	7.50000	+	12.9904i	0.255008	+	0.441686i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.00000			−0.305304
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	21.0000	+	24.2487i	0.709930	+	0.819756i
$876$		0			0
$877$	−23.5000	+	40.7032i	−0.793539	+	1.37445i	0.130224	+	0.991485i	$0.458430\pi$
$877$	−23.5000	+	40.7032i	−0.793539	+	1.37445i	−0.923763	+	0.382965i	$0.874903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			13.8564i			0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$			13.8564i			0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$		0			0
$883$		−	45.0333i		−	1.51549i	−0.652550	−	0.757746i	$-0.726301\pi$
$883$		−	45.0333i		−	1.51549i	0.652550	−	0.757746i	$-0.273699\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	−0.839744	−	0.542983i	$-0.817295\pi$
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	0.890109	+	0.455748i	$0.150628\pi$
$888$		0			0
$889$	6.00000	+	6.92820i	0.201234	+	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.5000	+	18.1865i	0.351369	+	0.608589i
$894$		0			0
$895$	−27.0000			−0.902510
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15.0000	−	25.9808i	−0.500278	−	0.866507i
$900$		0			0
$901$	−40.5000	−	23.3827i	−1.34925	−	0.778990i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.00000	+	10.3923i	−0.199447	+	0.345452i
$906$		0			0
$907$	4.50000	−	2.59808i	0.149420	−	0.0862677i	−0.423426	−	0.905931i	$-0.639173\pi$
$907$	4.50000	−	2.59808i	0.149420	−	0.0862677i	0.572846	+	0.819663i	$0.305839\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			10.3923i			0.344312i	0.985070	+	0.172156i	$0.0550734\pi$
$911$			10.3923i			0.344312i	−0.985070	+	0.172156i	$0.944927\pi$
$912$		0			0
$913$	−18.0000	+	10.3923i	−0.595713	+	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.5000	+	54.5596i	0.346741	+	1.80172i
$918$		0			0
$919$	37.5000	+	21.6506i	1.23701	+	0.714189i	0.968482	−	0.249083i	$-0.0801292\pi$
$919$	37.5000	+	21.6506i	1.23701	+	0.714189i	0.268529	+	0.963272i	$0.413463\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−10.0000			−0.328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22.5000	+	12.9904i	0.738201	+	0.426201i	0.821415	−	0.570331i	$-0.193185\pi$
$929$	22.5000	+	12.9904i	0.738201	+	0.426201i	−0.0832138	+	0.996532i	$0.526518\pi$
$930$		0			0
$931$	38.5000	+	30.3109i	1.26179	+	0.993399i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−13.5000	+	7.79423i	−0.441497	+	0.254899i
$936$		0			0
$937$			41.5692i			1.35801i	0.734135	+	0.679004i	$0.237588\pi$
$937$			41.5692i			1.35801i	−0.734135	+	0.679004i	$0.762412\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25.5000	+	14.7224i	−0.831276	+	0.479938i	−0.854289	−	0.519798i	$-0.826007\pi$
$941$	−25.5000	+	14.7224i	−0.831276	+	0.479938i	0.0230132	+	0.999735i	$0.492674\pi$
$942$		0			0
$943$	−30.0000	+	51.9615i	−0.976934	+	1.69210i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.5000	−	14.7224i	−0.828639	−	0.478415i	0.0247477	−	0.999694i	$-0.492122\pi$
$947$	−25.5000	−	14.7224i	−0.828639	−	0.478415i	−0.853386	+	0.521279i	$0.825455\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.0000			0.583077			0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000			0.583077			0.291539	+	0.956559i	$0.405833\pi$
$954$		0			0
$955$	−10.5000	−	18.1865i	−0.339772	−	0.588502i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.50000	+	2.59808i	0.242188	+	0.0838963i
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	8.66025i		−	0.278783i
$966$		0			0
$967$			31.1769i			1.00258i	0.865279	+	0.501291i	$0.167141\pi$
$967$			31.1769i			1.00258i	−0.865279	+	0.501291i	$0.832859\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.50000	−	2.59808i	0.0481373	−	0.0833762i	−0.840953	−	0.541108i	$-0.818005\pi$
$971$	1.50000	−	2.59808i	0.0481373	−	0.0833762i	0.889090	+	0.457732i	$0.151338\pi$
$972$		0			0
$973$	−8.00000	+	6.92820i	−0.256468	+	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	13.5000	+	23.3827i	0.431903	+	0.748078i	0.997037	−	0.0769208i	$-0.0245089\pi$
$977$	13.5000	+	23.3827i	0.431903	+	0.748078i	−0.565134	+	0.824999i	$0.691176\pi$
$978$		0			0
$979$	21.0000			0.671163
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.50000	−	2.59808i	−0.0478426	−	0.0828658i	0.841112	−	0.540860i	$-0.181901\pi$
$983$	−1.50000	−	2.59808i	−0.0478426	−	0.0828658i	−0.888955	+	0.457995i	$0.848568\pi$
$984$		0			0
$985$	−9.00000	−	5.19615i	−0.286764	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15.0000	+	25.9808i	−0.476972	+	0.826140i
$990$		0			0
$991$	4.50000	−	2.59808i	0.142947	−	0.0825306i	−0.426821	−	0.904336i	$-0.640366\pi$
$991$	4.50000	−	2.59808i	0.142947	−	0.0825306i	0.569768	+	0.821806i	$0.307033\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	19.0526i		−	0.604007i
$996$		0			0
$997$	1.50000	−	0.866025i	0.0475055	−	0.0274273i	−0.476059	−	0.879413i	$-0.657935\pi$
$997$	1.50000	−	0.866025i	0.0475055	−	0.0274273i	0.523565	+	0.851986i	$0.324602\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.cs.f.271.1		2
3.2	odd	2		112.2.p.a.47.1	yes	2
4.3	odd	2		1008.2.cs.c.271.1		2
7.2	even	3		7056.2.b.m.1567.2		2
7.3	odd	6		1008.2.cs.c.703.1		2
7.5	odd	6		7056.2.b.b.1567.1		2
12.11	even	2		112.2.p.b.47.1	yes	2
21.2	odd	6		784.2.f.b.783.1		2
21.5	even	6		784.2.f.a.783.2		2
21.11	odd	6		784.2.p.c.31.1		2
21.17	even	6		112.2.p.b.31.1	yes	2
21.20	even	2		784.2.p.d.607.1		2
24.5	odd	2		448.2.p.b.383.1		2
24.11	even	2		448.2.p.a.383.1		2
28.3	even	6	inner	1008.2.cs.f.703.1		2
28.19	even	6		7056.2.b.m.1567.1		2
28.23	odd	6		7056.2.b.b.1567.2		2
84.11	even	6		784.2.p.d.31.1		2
84.23	even	6		784.2.f.a.783.1		2
84.47	odd	6		784.2.f.b.783.2		2
84.59	odd	6		112.2.p.a.31.1	✓	2
84.83	odd	2		784.2.p.c.607.1		2
168.5	even	6		3136.2.f.b.3135.1		2
168.59	odd	6		448.2.p.b.255.1		2
168.101	even	6		448.2.p.a.255.1		2
168.107	even	6		3136.2.f.b.3135.2		2
168.131	odd	6		3136.2.f.a.3135.1		2
168.149	odd	6		3136.2.f.a.3135.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.p.a.31.1	✓	2	84.59	odd	6
112.2.p.a.47.1	yes	2	3.2	odd	2
112.2.p.b.31.1	yes	2	21.17	even	6
112.2.p.b.47.1	yes	2	12.11	even	2
448.2.p.a.255.1		2	168.101	even	6
448.2.p.a.383.1		2	24.11	even	2
448.2.p.b.255.1		2	168.59	odd	6
448.2.p.b.383.1		2	24.5	odd	2
784.2.f.a.783.1		2	84.23	even	6
784.2.f.a.783.2		2	21.5	even	6
784.2.f.b.783.1		2	21.2	odd	6
784.2.f.b.783.2		2	84.47	odd	6
784.2.p.c.31.1		2	21.11	odd	6
784.2.p.c.607.1		2	84.83	odd	2
784.2.p.d.31.1		2	84.11	even	6
784.2.p.d.607.1		2	21.20	even	2
1008.2.cs.c.271.1		2	4.3	odd	2
1008.2.cs.c.703.1		2	7.3	odd	6
1008.2.cs.f.271.1		2	1.1	even	1	trivial
1008.2.cs.f.703.1		2	28.3	even	6	inner
3136.2.f.a.3135.1		2	168.131	odd	6
3136.2.f.a.3135.2		2	168.149	odd	6
3136.2.f.b.3135.1		2	168.5	even	6
3136.2.f.b.3135.2		2	168.107	even	6
7056.2.b.b.1567.1		2	7.5	odd	6
7056.2.b.b.1567.2		2	28.23	odd	6
7056.2.b.m.1567.1		2	28.19	even	6
7056.2.b.m.1567.2		2	7.2	even	3

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cs (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +(1.50000 - 0.866025i) q^{11} +(4.50000 - 2.59808i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-7.50000 - 4.33013i) q^{23} +(-1.00000 - 1.73205i) q^{25} +6.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-4.50000 + 0.866025i) q^{35} +(2.50000 - 4.33013i) q^{37} -6.92820i q^{41} -3.46410i q^{43} +(1.50000 - 2.59808i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-4.50000 - 7.79423i) q^{53} -3.00000 q^{55} +(4.50000 + 7.79423i) q^{59} +(7.50000 + 4.33013i) q^{61} +(4.50000 - 2.59808i) q^{67} -3.46410i q^{71} +(1.50000 - 0.866025i) q^{73} +(1.50000 - 4.33013i) q^{77} +(-4.50000 - 2.59808i) q^{79} -12.0000 q^{83} -9.00000 q^{85} +(10.5000 + 6.06218i) q^{89} +(10.5000 - 6.06218i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 3 * q^5 + 4 * q^7	\( 2 q - 3 q^{5} + 4 q^{7} + 3 q^{11} + 9 q^{17} - 7 q^{19} - 15 q^{23} - 2 q^{25} + 12 q^{29} - 5 q^{31} - 9 q^{35} + 5 q^{37} + 3 q^{47} + 2 q^{49} - 9 q^{53} - 6 q^{55} + 9 q^{59} + 15 q^{61} + 9 q^{67} + 3 q^{73} + 3 q^{77} - 9 q^{79} - 24 q^{83} - 18 q^{85} + 21 q^{89} + 21 q^{95}+O(q^{100}) \) 2 * q - 3 * q^5 + 4 * q^7 + 3 * q^11 + 9 * q^17 - 7 * q^19 - 15 * q^23 - 2 * q^25 + 12 * q^29 - 5 * q^31 - 9 * q^35 + 5 * q^37 + 3 * q^47 + 2 * q^49 - 9 * q^53 - 6 * q^55 + 9 * q^59 + 15 * q^61 + 9 * q^67 + 3 * q^73 + 3 * q^77 - 9 * q^79 - 24 * q^83 - 18 * q^85 + 21 * q^89 + 21 * q^95

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