LMFDB - Embedded newform 832.2.ba.g.673.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(225,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("832.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.ba (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:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.592240896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 $ x^8 - 7x^6 + 40x^4 - 63*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			673.4
Root			$-1.12824 - 0.651388i$ of defining polynomial
Character	$\chi$	$=$	832.673
Dual form			832.2.ba.g.225.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.866025 - 0.500000i) q^{3} +(3.12250 + 1.80278i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(-0.866025 - 1.50000i) q^{11} +3.60555 q^{13} +(1.50000 - 2.59808i) q^{17} +(-0.866025 + 1.50000i) q^{19} +3.60555 q^{21} +(3.12250 + 5.40833i) q^{23} -5.00000 q^{25} +5.00000i q^{27} +(5.40833 - 3.12250i) q^{29} +7.21110i q^{31} +(-1.50000 - 0.866025i) q^{33} +(-1.80278 - 3.12250i) q^{37} +(3.12250 - 1.80278i) q^{39} +(4.50000 - 2.59808i) q^{41} +(-0.866025 - 0.500000i) q^{43} +(3.00000 + 5.19615i) q^{49} -3.00000i q^{51} -12.4900i q^{53} +1.73205i q^{57} +(6.06218 - 10.5000i) q^{59} +(5.40833 + 3.12250i) q^{61} +(-6.24500 + 3.60555i) q^{63} +(0.866025 + 1.50000i) q^{67} +(5.40833 + 3.12250i) q^{69} +(9.36750 + 5.40833i) q^{71} +3.46410i q^{73} +(-4.33013 + 2.50000i) q^{75} -6.24500i q^{77} -12.4900 q^{79} +(-0.500000 - 0.866025i) q^{81} -6.92820 q^{83} +(3.12250 - 5.40833i) q^{87} +(7.50000 - 4.33013i) q^{89} +(11.2583 + 6.50000i) q^{91} +(3.60555 + 6.24500i) q^{93} +(-10.5000 - 6.06218i) q^{97} +3.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} + 12 q^{17} - 40 q^{25} - 12 q^{33} + 36 q^{41} + 24 q^{49} - 4 q^{81} + 60 q^{89} - 84 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 + 12 * q^17 - 40 * q^25 - 12 * q^33 + 36 * q^41 + 24 * q^49 - 4 * q^81 + 60 * q^89 - 84 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i	−0.228714	−	0.973494i	$-0.573452\pi$
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i	0.728714	+	0.684819i	$0.240119\pi$
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	3.12250	+	1.80278i	1.18019	+	0.681385i	0.956059	−	0.293173i	$-0.0947112\pi$
$7$	3.12250	+	1.80278i	1.18019	+	0.681385i	0.224134	+	0.974558i	$0.428045\pi$
$8$		0			0
$9$	−1.00000	+	1.73205i	−0.333333	+	0.577350i
$10$		0			0
$11$	−0.866025	−	1.50000i	−0.261116	−	0.452267i	0.705422	−	0.708787i	$-0.250757\pi$
$11$	−0.866025	−	1.50000i	−0.261116	−	0.452267i	−0.966539	+	0.256520i	$0.917424\pi$
$12$		0			0
$13$	3.60555			1.00000
$13$	3.60555			1.00000
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$		0			0
$19$	−0.866025	+	1.50000i	−0.198680	+	0.344124i	−0.948101	−	0.317970i	$-0.896999\pi$
$19$	−0.866025	+	1.50000i	−0.198680	+	0.344124i	0.749421	+	0.662094i	$0.230332\pi$
$20$		0			0
$21$	3.60555			0.786796
$22$		0			0
$23$	3.12250	+	5.40833i	0.651086	+	1.12771i	0.982860	+	0.184355i	$0.0590196\pi$
$23$	3.12250	+	5.40833i	0.651086	+	1.12771i	−0.331774	+	0.943359i	$0.607647\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	5.40833	−	3.12250i	1.00430	−	0.579834i	0.0947831	−	0.995498i	$-0.469784\pi$
$29$	5.40833	−	3.12250i	1.00430	−	0.579834i	0.909518	+	0.415664i	$0.136451\pi$
$30$		0			0
$31$			7.21110i			1.29515i	0.762001	+	0.647576i	$0.224217\pi$
$31$			7.21110i			1.29515i	−0.762001	+	0.647576i	$0.775783\pi$
$32$		0			0
$33$	−1.50000	−	0.866025i	−0.261116	−	0.150756i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.80278	−	3.12250i	−0.296374	−	0.513336i	0.678929	−	0.734204i	$-0.262444\pi$
$37$	−1.80278	−	3.12250i	−0.296374	−	0.513336i	−0.975304	+	0.220868i	$0.929111\pi$
$38$		0			0
$39$	3.12250	−	1.80278i	0.500000	−	0.288675i
$40$		0			0
$41$	4.50000	−	2.59808i	0.702782	−	0.405751i	−0.105601	−	0.994409i	$-0.533677\pi$
$41$	4.50000	−	2.59808i	0.702782	−	0.405751i	0.808383	+	0.588657i	$0.200343\pi$
$42$		0			0
$43$	−0.866025	−	0.500000i	−0.132068	−	0.0762493i	0.432511	−	0.901629i	$-0.357628\pi$
$43$	−0.866025	−	0.500000i	−0.132068	−	0.0762493i	−0.564578	+	0.825380i	$0.690961\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$		0			0
$51$		−	3.00000i		−	0.420084i
$52$		0			0
$53$		−	12.4900i		−	1.71563i	−0.513956	−	0.857816i	$-0.671821\pi$
$53$		−	12.4900i		−	1.71563i	0.513956	−	0.857816i	$-0.328179\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			1.73205i			0.229416i
$58$		0			0
$59$	6.06218	−	10.5000i	0.789228	−	1.36698i	−0.137212	−	0.990542i	$-0.543814\pi$
$59$	6.06218	−	10.5000i	0.789228	−	1.36698i	0.926440	−	0.376442i	$-0.122853\pi$
$60$		0			0
$61$	5.40833	+	3.12250i	0.692465	+	0.399795i	0.804535	−	0.593905i	$-0.202415\pi$
$61$	5.40833	+	3.12250i	0.692465	+	0.399795i	−0.112070	+	0.993700i	$0.535748\pi$
$62$		0			0
$63$	−6.24500	+	3.60555i	−0.786796	+	0.454257i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.866025	+	1.50000i	0.105802	+	0.183254i	0.914066	−	0.405567i	$-0.132926\pi$
$67$	0.866025	+	1.50000i	0.105802	+	0.183254i	−0.808264	+	0.588821i	$0.799592\pi$
$68$		0			0
$69$	5.40833	+	3.12250i	0.651086	+	0.375905i
$70$		0			0
$71$	9.36750	+	5.40833i	1.11172	+	0.641850i	0.939274	−	0.343169i	$-0.111500\pi$
$71$	9.36750	+	5.40833i	1.11172	+	0.641850i	0.172444	+	0.985019i	$0.444834\pi$
$72$		0			0
$73$			3.46410i			0.405442i	0.979236	+	0.202721i	$0.0649785\pi$
$73$			3.46410i			0.405442i	−0.979236	+	0.202721i	$0.935021\pi$
$74$		0			0
$75$	−4.33013	+	2.50000i	−0.500000	+	0.288675i
$76$		0			0
$77$		−	6.24500i		−	0.711684i
$78$		0			0
$79$	−12.4900			−1.40523			−0.702617	−	0.711568i	$-0.747985\pi$
$79$	−12.4900			−1.40523			−0.702617	+	0.711568i	$0.747985\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−6.92820			−0.760469			−0.380235	−	0.924890i	$-0.624157\pi$
$83$	−6.92820			−0.760469			−0.380235	+	0.924890i	$0.624157\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.12250	−	5.40833i	0.334767	−	0.579834i
$88$		0			0
$89$	7.50000	−	4.33013i	0.794998	−	0.458993i	−0.0467209	−	0.998908i	$-0.514877\pi$
$89$	7.50000	−	4.33013i	0.794998	−	0.458993i	0.841719	+	0.539915i	$0.181544\pi$
$90$		0			0
$91$	11.2583	+	6.50000i	1.18019	+	0.681385i
$92$		0			0
$93$	3.60555	+	6.24500i	0.373878	+	0.647576i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−10.5000	−	6.06218i	−1.06611	−	0.615521i	−0.138996	−	0.990293i	$-0.544388\pi$
$97$	−10.5000	−	6.06218i	−1.06611	−	0.615521i	−0.927117	+	0.374772i	$0.877721\pi$
$98$		0			0
$99$	3.46410			0.348155
$100$		0			0
$101$	−5.40833	+	3.12250i	−0.538149	+	0.310700i	−0.744328	−	0.667814i	$-0.767230\pi$
$101$	−5.40833	+	3.12250i	−0.538149	+	0.310700i	0.206180	+	0.978514i	$0.433897\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.9904	+	7.50000i	−1.25583	+	0.725052i	−0.972261	−	0.233900i	$-0.924851\pi$
$107$	−12.9904	+	7.50000i	−1.25583	+	0.725052i	−0.283567	+	0.958952i	$0.591518\pi$
$108$		0			0
$109$	−14.4222			−1.38140			−0.690698	−	0.723143i	$-0.742697\pi$
$109$	−14.4222			−1.38140			−0.690698	+	0.723143i	$0.742697\pi$
$110$		0			0
$111$	−3.12250	−	1.80278i	−0.296374	−	0.171112i
$112$		0			0
$113$	−4.50000	+	7.79423i	−0.423324	+	0.733219i	−0.996262	−	0.0863794i	$-0.972470\pi$
$113$	−4.50000	+	7.79423i	−0.423324	+	0.733219i	0.572938	+	0.819599i	$0.305804\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.60555	+	6.24500i	−0.333333	+	0.577350i
$118$		0			0
$119$	9.36750	−	5.40833i	0.858717	−	0.495781i
$120$		0			0
$121$	4.00000	−	6.92820i	0.363636	−	0.629837i
$122$		0			0
$123$	2.59808	−	4.50000i	0.234261	−	0.405751i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.12250	−	5.40833i	−0.277077	−	0.479911i	0.693580	−	0.720380i	$-0.256032\pi$
$127$	−3.12250	−	5.40833i	−0.277077	−	0.479911i	−0.970657	+	0.240468i	$0.922699\pi$
$128$		0			0
$129$	−1.00000			−0.0880451
$130$		0			0
$131$			6.00000i			0.524222i	0.965038	+	0.262111i	$0.0844187\pi$
$131$			6.00000i			0.524222i	−0.965038	+	0.262111i	$0.915581\pi$
$132$		0			0
$133$	−5.40833	+	3.12250i	−0.468961	+	0.270755i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.5000	−	6.06218i	−0.897076	−	0.517927i	−0.0208253	−	0.999783i	$-0.506629\pi$
$137$	−10.5000	−	6.06218i	−0.897076	−	0.517927i	−0.876250	+	0.481856i	$0.839963\pi$
$138$		0			0
$139$	−9.52628	−	5.50000i	−0.808008	−	0.466504i	0.0382553	−	0.999268i	$-0.487820\pi$
$139$	−9.52628	−	5.50000i	−0.808008	−	0.466504i	−0.846264	+	0.532764i	$0.821153\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.12250	−	5.40833i	−0.261116	−	0.452267i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.19615	+	3.00000i	0.428571	+	0.247436i
$148$		0			0
$149$	−5.40833	+	9.36750i	−0.443067	+	0.767415i	−0.997915	−	0.0645365i	$-0.979443\pi$
$149$	−5.40833	+	9.36750i	−0.443067	+	0.767415i	0.554848	+	0.831952i	$0.312776\pi$
$150$		0			0
$151$		−	7.21110i		−	0.586831i	−0.955985	−	0.293416i	$-0.905208\pi$
$151$		−	7.21110i		−	0.586831i	0.955985	−	0.293416i	$-0.0947920\pi$
$152$		0			0
$153$	3.00000	+	5.19615i	0.242536	+	0.420084i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			12.4900i			0.996810i	0.866944	+	0.498405i	$0.166081\pi$
$157$			12.4900i			0.996810i	−0.866944	+	0.498405i	$0.833919\pi$
$158$		0			0
$159$	−6.24500	−	10.8167i	−0.495261	−	0.857816i
$160$		0			0
$161$			22.5167i			1.77456i
$162$		0			0
$163$	7.79423	−	13.5000i	0.610491	−	1.05740i	−0.380667	−	0.924712i	$-0.624305\pi$
$163$	7.79423	−	13.5000i	0.610491	−	1.05740i	0.991158	−	0.132689i	$-0.0423612\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.36750	−	5.40833i	0.724879	−	0.418509i	−0.0916670	−	0.995790i	$-0.529220\pi$
$167$	9.36750	−	5.40833i	0.724879	−	0.418509i	0.816546	+	0.577281i	$0.195886\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	−1.73205	−	3.00000i	−0.132453	−	0.229416i
$172$		0			0
$173$	−16.2250	−	9.36750i	−1.23356	−	0.712198i	−0.265792	−	0.964030i	$-0.585633\pi$
$173$	−16.2250	−	9.36750i	−1.23356	−	0.712198i	−0.967771	+	0.251833i	$0.918967\pi$
$174$		0			0
$175$	−15.6125	−	9.01388i	−1.18019	−	0.681385i
$176$		0			0
$177$		−	12.1244i		−	0.911322i
$178$		0			0
$179$	7.79423	−	4.50000i	0.582568	−	0.336346i	−0.179585	−	0.983742i	$-0.557476\pi$
$179$	7.79423	−	4.50000i	0.582568	−	0.336346i	0.762153	+	0.647397i	$0.224142\pi$
$180$		0			0
$181$			12.4900i			0.928374i	0.885737	+	0.464187i	$0.153653\pi$
$181$			12.4900i			0.928374i	−0.885737	+	0.464187i	$0.846347\pi$
$182$		0			0
$183$	6.24500			0.461644
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−5.19615			−0.379980
$188$		0			0
$189$	−9.01388	+	15.6125i	−0.655663	+	1.13564i
$190$		0			0
$191$	−3.12250	+	5.40833i	−0.225936	+	0.391333i	−0.956600	−	0.291405i	$-0.905877\pi$
$191$	−3.12250	+	5.40833i	−0.225936	+	0.391333i	0.730664	+	0.682737i	$0.239211\pi$
$192$		0			0
$193$	22.5000	−	12.9904i	1.61959	−	0.935068i	0.632559	−	0.774512i	$-0.282005\pi$
$193$	22.5000	−	12.9904i	1.61959	−	0.935068i	0.987027	−	0.160556i	$-0.0513286\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.40833	−	9.36750i	−0.385327	−	0.667407i	0.606487	−	0.795093i	$-0.292578\pi$
$197$	−5.40833	−	9.36750i	−0.385327	−	0.667407i	−0.991815	+	0.127687i	$0.959245\pi$
$198$		0			0
$199$	3.12250	−	5.40833i	0.221348	−	0.383386i	−0.733869	−	0.679290i	$-0.762288\pi$
$199$	3.12250	−	5.40833i	0.221348	−	0.383386i	0.955218	+	0.295904i	$0.0956210\pi$
$200$		0			0
$201$	1.50000	+	0.866025i	0.105802	+	0.0610847i
$202$		0			0
$203$	22.5167			1.58036
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−12.4900			−0.868115
$208$		0			0
$209$	3.00000			0.207514
$210$		0			0
$211$	−16.4545	+	9.50000i	−1.13277	+	0.654007i	−0.944631	−	0.328135i	$-0.893580\pi$
$211$	−16.4545	+	9.50000i	−1.13277	+	0.654007i	−0.188142	+	0.982142i	$0.560247\pi$
$212$		0			0
$213$	10.8167			0.741145
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.0000	+	22.5167i	−0.882498	+	1.52853i
$218$		0			0
$219$	1.73205	+	3.00000i	0.117041	+	0.202721i
$220$		0			0
$221$	5.40833	−	9.36750i	0.363803	−	0.630126i
$222$		0			0
$223$	−15.6125	+	9.01388i	−1.04549	+	0.603614i	−0.921383	−	0.388655i	$-0.872940\pi$
$223$	−15.6125	+	9.01388i	−1.04549	+	0.603614i	−0.124107	+	0.992269i	$0.539606\pi$
$224$		0			0
$225$	5.00000	−	8.66025i	0.333333	−	0.577350i
$226$		0			0
$227$	−7.79423	+	13.5000i	−0.517321	+	0.896026i	0.482476	+	0.875909i	$0.339737\pi$
$227$	−7.79423	+	13.5000i	−0.517321	+	0.896026i	−0.999798	+	0.0201176i	$0.993596\pi$
$228$		0			0
$229$	−14.4222			−0.953046			−0.476523	−	0.879162i	$-0.658103\pi$
$229$	−14.4222			−0.953046			−0.476523	+	0.879162i	$0.658103\pi$
$230$		0			0
$231$	−3.12250	−	5.40833i	−0.205445	−	0.355842i
$232$		0			0
$233$	−24.0000			−1.57229			−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000			−1.57229			−0.786146	+	0.618041i	$0.787927\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−10.8167	+	6.24500i	−0.702617	+	0.405656i
$238$		0			0
$239$		−	21.6333i		−	1.39934i	−0.714465	−	0.699671i	$-0.753330\pi$
$239$		−	21.6333i		−	1.39934i	0.714465	−	0.699671i	$-0.246670\pi$
$240$		0			0
$241$	−19.5000	−	11.2583i	−1.25611	−	0.725213i	−0.283790	−	0.958886i	$-0.591592\pi$
$241$	−19.5000	−	11.2583i	−1.25611	−	0.725213i	−0.972315	+	0.233674i	$0.924925\pi$
$242$		0			0
$243$	−13.8564	−	8.00000i	−0.888889	−	0.513200i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.12250	+	5.40833i	−0.198680	+	0.344124i
$248$		0			0
$249$	−6.00000	+	3.46410i	−0.380235	+	0.219529i
$250$		0			0
$251$	2.59808	+	1.50000i	0.163989	+	0.0946792i	0.579748	−	0.814795i	$-0.303151\pi$
$251$	2.59808	+	1.50000i	0.163989	+	0.0946792i	−0.415759	+	0.909475i	$0.636484\pi$
$252$		0			0
$253$	5.40833	−	9.36750i	0.340019	−	0.588929i
$254$		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
$259$		−	13.0000i		−	0.807781i
$260$		0			0
$261$			12.4900i			0.773111i
$262$		0			0
$263$	3.12250	+	5.40833i	0.192542	+	0.333492i	0.946092	−	0.323899i	$-0.104994\pi$
$263$	3.12250	+	5.40833i	0.192542	+	0.333492i	−0.753550	+	0.657390i	$0.771660\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.33013	−	7.50000i	0.264999	−	0.458993i
$268$		0			0
$269$	−27.0416	−	15.6125i	−1.64876	−	0.951911i	−0.977567	−	0.210626i	$-0.932450\pi$
$269$	−27.0416	−	15.6125i	−1.64876	−	0.951911i	−0.671191	−	0.741285i	$-0.734217\pi$
$270$		0			0
$271$	−3.12250	+	1.80278i	−0.189678	+	0.109511i	−0.591832	−	0.806061i	$-0.701595\pi$
$271$	−3.12250	+	1.80278i	−0.189678	+	0.109511i	0.402154	+	0.915572i	$0.368262\pi$
$272$		0			0
$273$	13.0000			0.786796
$274$		0			0
$275$	4.33013	+	7.50000i	0.261116	+	0.452267i
$276$		0			0
$277$	16.2250	+	9.36750i	0.974865	+	0.562838i	0.900716	−	0.434409i	$-0.143043\pi$
$277$	16.2250	+	9.36750i	0.974865	+	0.562838i	0.0741488	+	0.997247i	$0.476376\pi$
$278$		0			0
$279$	−12.4900	−	7.21110i	−0.747757	−	0.431717i
$280$		0			0
$281$		−	17.3205i		−	1.03325i	−0.856210	−	0.516627i	$-0.827187\pi$
$281$		−	17.3205i		−	1.03325i	0.856210	−	0.516627i	$-0.172813\pi$
$282$		0			0
$283$	−19.9186	+	11.5000i	−1.18404	+	0.683604i	−0.956945	−	0.290269i	$-0.906255\pi$
$283$	−19.9186	+	11.5000i	−1.18404	+	0.683604i	−0.227092	+	0.973873i	$0.572922\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	18.7350			1.10589
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$	−12.1244			−0.710742
$292$		0			0
$293$	16.2250	−	28.1025i	0.947873	−	1.64177i	0.197980	−	0.980206i	$-0.436562\pi$
$293$	16.2250	−	28.1025i	0.947873	−	1.64177i	0.749893	−	0.661559i	$-0.230105\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	7.50000	−	4.33013i	0.435194	−	0.251259i
$298$		0			0
$299$	11.2583	+	19.5000i	0.651086	+	1.12771i
$300$		0			0
$301$	−1.80278	−	3.12250i	−0.103910	−	0.179978i
$302$		0			0
$303$	−3.12250	+	5.40833i	−0.179383	+	0.310700i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	27.7128			1.58165			0.790827	−	0.612040i	$-0.209651\pi$
$307$	27.7128			1.58165			0.790827	+	0.612040i	$0.209651\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.9800			1.41649			0.708243	−	0.705969i	$-0.249488\pi$
$311$	24.9800			1.41649			0.708243	+	0.705969i	$0.249488\pi$
$312$		0			0
$313$	8.00000			0.452187			0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000			0.452187			0.226093	+	0.974106i	$0.427405\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	21.6333			1.21505			0.607524	−	0.794301i	$-0.292163\pi$
$317$	21.6333			1.21505			0.607524	+	0.794301i	$0.292163\pi$
$318$		0			0
$319$	−9.36750	−	5.40833i	−0.524479	−	0.302808i
$320$		0			0
$321$	−7.50000	+	12.9904i	−0.418609	+	0.725052i
$322$		0			0
$323$	2.59808	+	4.50000i	0.144561	+	0.250387i
$324$		0			0
$325$	−18.0278			−1.00000
$326$		0			0
$327$	−12.4900	+	7.21110i	−0.690698	+	0.398775i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	11.2583	−	19.5000i	0.618814	−	1.07182i	−0.370889	−	0.928677i	$-0.620947\pi$
$331$	11.2583	−	19.5000i	0.618814	−	1.07182i	0.989703	−	0.143140i	$-0.0457198\pi$
$332$		0			0
$333$	7.21110			0.395166
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.00000			−0.217894			−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000			−0.217894			−0.108947	+	0.994048i	$0.534748\pi$
$338$		0			0
$339$			9.00000i			0.488813i
$340$		0			0
$341$	10.8167	−	6.24500i	0.585755	−	0.338186i
$342$		0			0
$343$		−	3.60555i		−	0.194681i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−28.5788	−	16.5000i	−1.53419	−	0.885766i	−0.999162	−	0.0409337i	$-0.986967\pi$
$347$	−28.5788	−	16.5000i	−1.53419	−	0.885766i	−0.535031	−	0.844833i	$-0.679700\pi$
$348$		0			0
$349$	12.6194	+	21.8575i	0.675503	+	1.17000i	0.976322	+	0.216324i	$0.0694066\pi$
$349$	12.6194	+	21.8575i	0.675503	+	1.17000i	−0.300819	+	0.953681i	$0.597260\pi$
$350$		0			0
$351$			18.0278i			0.962250i
$352$		0			0
$353$	7.50000	−	4.33013i	0.399185	−	0.230469i	−0.286947	−	0.957946i	$-0.592641\pi$
$353$	7.50000	−	4.33013i	0.399185	−	0.230469i	0.686132	+	0.727477i	$0.259307\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	5.40833	−	9.36750i	0.286239	−	0.495781i
$358$		0			0
$359$			21.6333i			1.14176i	0.821033	+	0.570881i	$0.193398\pi$
$359$			21.6333i			1.14176i	−0.821033	+	0.570881i	$0.806602\pi$
$360$		0			0
$361$	8.00000	+	13.8564i	0.421053	+	0.729285i
$362$		0			0
$363$		−	8.00000i		−	0.419891i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	15.6125	+	27.0416i	0.814966	+	1.41156i	0.909353	+	0.416026i	$0.136578\pi$
$367$	15.6125	+	27.0416i	0.814966	+	1.41156i	−0.0943870	+	0.995536i	$0.530089\pi$
$368$		0			0
$369$			10.3923i			0.541002i
$370$		0			0
$371$	22.5167	−	39.0000i	1.16901	−	2.02478i
$372$		0			0
$373$	16.2250	+	9.36750i	0.840098	+	0.485031i	0.857297	−	0.514821i	$-0.172142\pi$
$373$	16.2250	+	9.36750i	0.840098	+	0.485031i	−0.0171998	+	0.999852i	$0.505475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.5000	−	11.2583i	1.00430	−	0.579834i
$378$		0			0
$379$	−0.866025	−	1.50000i	−0.0444847	−	0.0770498i	0.842926	−	0.538030i	$-0.180831\pi$
$379$	−0.866025	−	1.50000i	−0.0444847	−	0.0770498i	−0.887410	+	0.460980i	$0.847498\pi$
$380$		0			0
$381$	−5.40833	−	3.12250i	−0.277077	−	0.159970i
$382$		0			0
$383$	28.1025	+	16.2250i	1.43597	+	0.829058i	0.997567	−	0.0697204i	$-0.0222107\pi$
$383$	28.1025	+	16.2250i	1.43597	+	0.829058i	0.438404	+	0.898778i	$0.355544\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.73205	−	1.00000i	0.0880451	−	0.0508329i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	18.7350			0.947469
$392$		0			0
$393$	3.00000	+	5.19615i	0.151330	+	0.262111i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.80278	+	3.12250i	−0.0904787	+	0.156714i	−0.907713	−	0.419592i	$-0.862173\pi$
$397$	−1.80278	+	3.12250i	−0.0904787	+	0.156714i	0.817234	+	0.576306i	$0.195506\pi$
$398$		0			0
$399$	−3.12250	+	5.40833i	−0.156320	+	0.270755i
$400$		0			0
$401$	−1.50000	+	0.866025i	−0.0749064	+	0.0432472i	−0.536985	−	0.843592i	$-0.680437\pi$
$401$	−1.50000	+	0.866025i	−0.0749064	+	0.0432472i	0.462079	+	0.886839i	$0.347104\pi$
$402$		0			0
$403$			26.0000i			1.29515i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.12250	+	5.40833i	−0.154777	+	0.268081i
$408$		0			0
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	0.127609	−	0.991825i	$-0.459270\pi$
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	−0.795141	+	0.606425i	$0.792603\pi$
$410$		0			0
$411$	−12.1244			−0.598050
$412$		0			0
$413$	37.8583	−	21.8575i	1.86288	−	1.07554i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−11.0000			−0.538672
$418$		0			0
$419$	−18.1865	+	10.5000i	−0.888470	+	0.512959i	−0.873442	−	0.486928i	$-0.838117\pi$
$419$	−18.1865	+	10.5000i	−0.888470	+	0.512959i	−0.0150285	+	0.999887i	$0.504784\pi$
$420$		0			0
$421$	14.4222			0.702895			0.351448	−	0.936208i	$-0.385690\pi$
$421$	14.4222			0.702895			0.351448	+	0.936208i	$0.385690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.50000	+	12.9904i	−0.363803	+	0.630126i
$426$		0			0
$427$	11.2583	+	19.5000i	0.544829	+	0.943671i
$428$		0			0
$429$	−5.40833	−	3.12250i	−0.261116	−	0.150756i
$430$		0			0
$431$	28.1025	−	16.2250i	1.35365	−	0.781530i	0.364891	−	0.931050i	$-0.381106\pi$
$431$	28.1025	−	16.2250i	1.35365	−	0.781530i	0.988759	+	0.149521i	$0.0477730\pi$
$432$		0			0
$433$	3.50000	−	6.06218i	0.168199	−	0.291330i	−0.769588	−	0.638541i	$-0.779538\pi$
$433$	3.50000	−	6.06218i	0.168199	−	0.291330i	0.937787	+	0.347212i	$0.112871\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.8167			−0.517431
$438$		0			0
$439$	−3.12250	−	5.40833i	−0.149029	−	0.258125i	0.781840	−	0.623479i	$-0.214281\pi$
$439$	−3.12250	−	5.40833i	−0.149029	−	0.258125i	−0.930869	+	0.365354i	$0.880948\pi$
$440$		0			0
$441$	−12.0000			−0.571429
$442$		0			0
$443$			18.0000i			0.855206i	0.903967	+	0.427603i	$0.140642\pi$
$443$			18.0000i			0.855206i	−0.903967	+	0.427603i	$0.859358\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			10.8167i			0.511610i
$448$		0			0
$449$	−31.5000	−	18.1865i	−1.48658	−	0.858276i	−0.486694	−	0.873573i	$-0.661797\pi$
$449$	−31.5000	−	18.1865i	−1.48658	−	0.858276i	−0.999883	+	0.0152970i	$0.995131\pi$
$450$		0			0
$451$	−7.79423	−	4.50000i	−0.367016	−	0.211897i
$452$		0			0
$453$	−3.60555	−	6.24500i	−0.169404	−	0.293416i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.5000	−	11.2583i	0.912172	−	0.526642i	0.0310423	−	0.999518i	$-0.490117\pi$
$457$	19.5000	−	11.2583i	0.912172	−	0.526642i	0.881129	+	0.472876i	$0.156784\pi$
$458$		0			0
$459$	12.9904	+	7.50000i	0.606339	+	0.350070i
$460$		0			0
$461$	5.40833	−	9.36750i	0.251891	−	0.436288i	−0.712156	−	0.702022i	$-0.752281\pi$
$461$	5.40833	−	9.36750i	0.251891	−	0.436288i	0.964046	+	0.265734i	$0.0856143\pi$
$462$		0			0
$463$			7.21110i			0.335128i	0.985861	+	0.167564i	$0.0535901\pi$
$463$			7.21110i			0.335128i	−0.985861	+	0.167564i	$0.946410\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$		0			0
$469$			6.24500i			0.288367i
$470$		0			0
$471$	6.24500	+	10.8167i	0.287754	+	0.498405i
$472$		0			0
$473$			1.73205i			0.0796398i
$474$		0			0
$475$	4.33013	−	7.50000i	0.198680	−	0.344124i
$476$		0			0
$477$	21.6333	+	12.4900i	0.990521	+	0.571878i
$478$		0			0
$479$	28.1025	−	16.2250i	1.28404	−	0.741338i	0.306452	−	0.951886i	$-0.400858\pi$
$479$	28.1025	−	16.2250i	1.28404	−	0.741338i	0.977584	+	0.210548i	$0.0675248\pi$
$480$		0			0
$481$	−6.50000	−	11.2583i	−0.296374	−	0.513336i
$482$		0			0
$483$	11.2583	+	19.5000i	0.512272	+	0.887281i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.12250	−	1.80278i	−0.141494	−	0.0816916i	0.427582	−	0.903977i	$-0.359366\pi$
$487$	−3.12250	−	1.80278i	−0.141494	−	0.0816916i	−0.569076	+	0.822285i	$0.692699\pi$
$488$		0			0
$489$		−	15.5885i		−	0.704934i
$490$		0			0
$491$	−18.1865	+	10.5000i	−0.820747	+	0.473858i	−0.850674	−	0.525694i	$-0.823806\pi$
$491$	−18.1865	+	10.5000i	−0.820747	+	0.473858i	0.0299272	+	0.999552i	$0.490472\pi$
$492$		0			0
$493$		−	18.7350i		−	0.843782i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	19.5000	+	33.7750i	0.874695	+	1.51502i
$498$		0			0
$499$	−41.5692			−1.86089			−0.930447	−	0.366427i	$-0.880581\pi$
$499$	−41.5692			−1.86089			−0.930447	+	0.366427i	$0.880581\pi$
$500$		0			0
$501$	5.40833	−	9.36750i	0.241626	−	0.418509i
$502$		0			0
$503$	−15.6125	+	27.0416i	−0.696127	+	1.20573i	0.273673	+	0.961823i	$0.411762\pi$
$503$	−15.6125	+	27.0416i	−0.696127	+	1.20573i	−0.969799	+	0.243904i	$0.921572\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	11.2583	−	6.50000i	0.500000	−	0.288675i
$508$		0			0
$509$	5.40833	+	9.36750i	0.239720	+	0.415207i	0.960634	−	0.277818i	$-0.0896111\pi$
$509$	5.40833	+	9.36750i	0.239720	+	0.415207i	−0.720914	+	0.693025i	$0.756278\pi$
$510$		0			0
$511$	−6.24500	+	10.8167i	−0.276262	+	0.478501i
$512$		0			0
$513$	−7.50000	−	4.33013i	−0.331133	−	0.191180i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−18.7350			−0.822375
$520$		0			0
$521$	36.0000			1.57719			0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000			1.57719			0.788594	+	0.614914i	$0.210809\pi$
$522$		0			0
$523$	16.4545	−	9.50000i	0.719504	−	0.415406i	−0.0950659	−	0.995471i	$-0.530306\pi$
$523$	16.4545	−	9.50000i	0.719504	−	0.415406i	0.814570	+	0.580065i	$0.196973\pi$
$524$		0			0
$525$	−18.0278			−0.786796
$526$		0			0
$527$	18.7350	+	10.8167i	0.816109	+	0.471181i
$528$		0			0
$529$	−8.00000	+	13.8564i	−0.347826	+	0.602452i
$530$		0			0
$531$	12.1244	+	21.0000i	0.526152	+	0.911322i
$532$		0			0
$533$	16.2250	−	9.36750i	0.702782	−	0.405751i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.50000	−	7.79423i	0.194189	−	0.336346i
$538$		0			0
$539$	5.19615	−	9.00000i	0.223814	−	0.387657i
$540$		0			0
$541$	36.0555			1.55015			0.775074	−	0.631871i	$-0.217713\pi$
$541$	36.0555			1.55015			0.775074	+	0.631871i	$0.217713\pi$
$542$		0			0
$543$	6.24500	+	10.8167i	0.267999	+	0.464187i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	10.0000i		−	0.427569i	−0.976881	−	0.213785i	$-0.931421\pi$
$547$		−	10.0000i		−	0.427569i	0.976881	−	0.213785i	$-0.0685791\pi$
$548$		0			0
$549$	−10.8167	+	6.24500i	−0.461644	+	0.266530i
$550$		0			0
$551$			10.8167i			0.460805i
$552$		0			0
$553$	−39.0000	−	22.5167i	−1.65845	−	0.957506i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.40833	−	9.36750i	−0.229158	−	0.396914i	0.728401	−	0.685151i	$-0.240264\pi$
$557$	−5.40833	−	9.36750i	−0.229158	−	0.396914i	−0.957559	+	0.288238i	$0.906931\pi$
$558$		0			0
$559$	−3.12250	−	1.80278i	−0.132068	−	0.0762493i
$560$		0			0
$561$	−4.50000	+	2.59808i	−0.189990	+	0.109691i
$562$		0			0
$563$	7.79423	+	4.50000i	0.328488	+	0.189652i	0.655169	−	0.755482i	$-0.272597\pi$
$563$	7.79423	+	4.50000i	0.328488	+	0.189652i	−0.326682	+	0.945134i	$0.605931\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	3.60555i		−	0.151419i
$568$		0			0
$569$	10.5000	+	18.1865i	0.440183	+	0.762419i	0.997703	−	0.0677445i	$-0.0215803\pi$
$569$	10.5000	+	18.1865i	0.440183	+	0.762419i	−0.557520	+	0.830164i	$0.688247\pi$
$570$		0			0
$571$		−	2.00000i		−	0.0836974i	−0.999124	−	0.0418487i	$-0.986675\pi$
$571$		−	2.00000i		−	0.0836974i	0.999124	−	0.0418487i	$-0.0133247\pi$
$572$		0			0
$573$			6.24500i			0.260889i
$574$		0			0
$575$	−15.6125	−	27.0416i	−0.651086	−	1.12771i
$576$		0			0
$577$			38.1051i			1.58634i	0.609002	+	0.793168i	$0.291570\pi$
$577$			38.1051i			1.58634i	−0.609002	+	0.793168i	$0.708430\pi$
$578$		0			0
$579$	12.9904	−	22.5000i	0.539862	−	0.935068i
$580$		0			0
$581$	−21.6333	−	12.4900i	−0.897501	−	0.518172i
$582$		0			0
$583$	−18.7350	+	10.8167i	−0.775924	+	0.447980i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.9186	−	34.5000i	−0.822128	−	1.42397i	−0.904095	−	0.427332i	$-0.859453\pi$
$587$	−19.9186	−	34.5000i	−0.822128	−	1.42397i	0.0819667	−	0.996635i	$-0.473880\pi$
$588$		0			0
$589$	−10.8167	−	6.24500i	−0.445692	−	0.257321i
$590$		0			0
$591$	−9.36750	−	5.40833i	−0.385327	−	0.222469i
$592$		0			0
$593$		−	17.3205i		−	0.711268i	−0.934625	−	0.355634i	$-0.884265\pi$
$593$		−	17.3205i		−	0.711268i	0.934625	−	0.355634i	$-0.115735\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	6.24500i		−	0.255591i
$598$		0			0
$599$	−37.4700			−1.53098			−0.765491	−	0.643446i	$-0.777504\pi$
$599$	−37.4700			−1.53098			−0.765491	+	0.643446i	$0.777504\pi$
$600$		0			0
$601$	5.50000	+	9.52628i	0.224350	+	0.388585i	0.956124	−	0.292962i	$-0.0946409\pi$
$601$	5.50000	+	9.52628i	0.224350	+	0.388585i	−0.731774	+	0.681547i	$0.761308\pi$
$602$		0			0
$603$	−3.46410			−0.141069
$604$		0			0
$605$		0			0
$606$		0			0
$607$	15.6125	−	27.0416i	0.633692	−	1.09759i	−0.353099	−	0.935586i	$-0.614872\pi$
$607$	15.6125	−	27.0416i	0.633692	−	1.09759i	0.986791	−	0.162000i	$-0.0517945\pi$
$608$		0			0
$609$	19.5000	−	11.2583i	0.790180	−	0.456211i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.80278	−	3.12250i	−0.0728134	−	0.126117i	0.827320	−	0.561731i	$-0.189864\pi$
$613$	−1.80278	−	3.12250i	−0.0728134	−	0.126117i	−0.900133	+	0.435615i	$0.856531\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.5000	+	7.79423i	0.543490	+	0.313784i	0.746492	−	0.665394i	$-0.231737\pi$
$617$	13.5000	+	7.79423i	0.543490	+	0.313784i	−0.203002	+	0.979178i	$0.565070\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−27.0416	+	15.6125i	−1.08514	+	0.626508i
$622$		0			0
$623$	31.2250			1.25100
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	2.59808	−	1.50000i	0.103757	−	0.0599042i
$628$		0			0
$629$	−10.8167			−0.431288
$630$		0			0
$631$	−3.12250	−	1.80278i	−0.124305	−	0.0717674i	0.436558	−	0.899676i	$-0.356197\pi$
$631$	−3.12250	−	1.80278i	−0.124305	−	0.0717674i	−0.560863	+	0.827909i	$0.689531\pi$
$632$		0			0
$633$	−9.50000	+	16.4545i	−0.377591	+	0.654007i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	10.8167	+	18.7350i	0.428571	+	0.742307i
$638$		0			0
$639$	−18.7350	+	10.8167i	−0.741145	+	0.427900i
$640$		0			0
$641$	13.5000	−	23.3827i	0.533218	−	0.923561i	−0.466029	−	0.884769i	$-0.654316\pi$
$641$	13.5000	−	23.3827i	0.533218	−	0.923561i	0.999247	−	0.0387913i	$-0.0123508\pi$
$642$		0			0
$643$	0.866025	−	1.50000i	0.0341527	−	0.0591542i	−0.848444	−	0.529285i	$-0.822460\pi$
$643$	0.866025	−	1.50000i	0.0341527	−	0.0591542i	0.882597	+	0.470131i	$0.155793\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.12250	−	5.40833i	−0.122758	−	0.212623i	0.798096	−	0.602530i	$-0.205841\pi$
$647$	−3.12250	−	5.40833i	−0.122758	−	0.212623i	−0.920854	+	0.389907i	$0.872507\pi$
$648$		0			0
$649$	−21.0000			−0.824322
$650$		0			0
$651$			26.0000i			1.01902i
$652$		0			0
$653$	16.2250	−	9.36750i	0.634933	−	0.366578i	−0.147727	−	0.989028i	$-0.547196\pi$
$653$	16.2250	−	9.36750i	0.634933	−	0.366578i	0.782660	+	0.622450i	$0.213862\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6.00000	−	3.46410i	−0.234082	−	0.135147i
$658$		0			0
$659$	−7.79423	−	4.50000i	−0.303620	−	0.175295i	0.340448	−	0.940263i	$-0.389421\pi$
$659$	−7.79423	−	4.50000i	−0.303620	−	0.175295i	−0.644068	+	0.764968i	$0.722755\pi$
$660$		0			0
$661$	−9.01388	−	15.6125i	−0.350599	−	0.607256i	0.635755	−	0.771891i	$-0.280689\pi$
$661$	−9.01388	−	15.6125i	−0.350599	−	0.607256i	−0.986355	+	0.164635i	$0.947355\pi$
$662$		0			0
$663$		−	10.8167i		−	0.420084i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	33.7750	+	19.5000i	1.30777	+	0.755043i
$668$		0			0
$669$	−9.01388	+	15.6125i	−0.348497	+	0.603614i
$670$		0			0
$671$		−	10.8167i		−	0.417572i
$672$		0			0
$673$	2.50000	+	4.33013i	0.0963679	+	0.166914i	0.910179	−	0.414216i	$-0.135944\pi$
$673$	2.50000	+	4.33013i	0.0963679	+	0.166914i	−0.813811	+	0.581130i	$0.802611\pi$
$674$		0			0
$675$		−	25.0000i		−	0.962250i
$676$		0			0
$677$		−	37.4700i		−	1.44009i	−0.693928	−	0.720044i	$-0.744121\pi$
$677$		−	37.4700i		−	1.44009i	0.693928	−	0.720044i	$-0.255879\pi$
$678$		0			0
$679$	−21.8575	−	37.8583i	−0.838814	−	1.45287i
$680$		0			0
$681$			15.5885i			0.597351i
$682$		0			0
$683$	−4.33013	+	7.50000i	−0.165688	+	0.286980i	−0.936899	−	0.349599i	$-0.886318\pi$
$683$	−4.33013	+	7.50000i	−0.165688	+	0.286980i	0.771212	+	0.636579i	$0.219651\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−12.4900	+	7.21110i	−0.476523	+	0.275121i
$688$		0			0
$689$		−	45.0333i		−	1.71563i
$690$		0			0
$691$	19.9186	+	34.5000i	0.757739	+	1.31244i	0.944001	+	0.329942i	$0.107029\pi$
$691$	19.9186	+	34.5000i	0.757739	+	1.31244i	−0.186263	+	0.982500i	$0.559638\pi$
$692$		0			0
$693$	10.8167	+	6.24500i	0.410891	+	0.237228i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	15.5885i		−	0.590455i
$698$		0			0
$699$	−20.7846	+	12.0000i	−0.786146	+	0.453882i
$700$		0			0
$701$			37.4700i			1.41522i	0.706602	+	0.707611i	$0.250227\pi$
$701$			37.4700i			1.41522i	−0.706602	+	0.707611i	$0.749773\pi$
$702$		0			0
$703$	6.24500			0.235535
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.5167			−0.846826
$708$		0			0
$709$	−23.4361	+	40.5925i	−0.880161	+	1.52448i	−0.0289988	+	0.999579i	$0.509232\pi$
$709$	−23.4361	+	40.5925i	−0.880161	+	1.52448i	−0.851162	+	0.524903i	$0.824101\pi$
$710$		0			0
$711$	12.4900	−	21.6333i	0.468411	−	0.811312i
$712$		0			0
$713$	−39.0000	+	22.5167i	−1.46056	+	0.843256i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−10.8167	−	18.7350i	−0.403955	−	0.699671i
$718$		0			0
$719$	21.8575	−	37.8583i	0.815147	−	1.41188i	−0.0940754	−	0.995565i	$-0.529989\pi$
$719$	21.8575	−	37.8583i	0.815147	−	1.41188i	0.909222	−	0.416311i	$-0.136677\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−22.5167			−0.837404
$724$		0			0
$725$	−27.0416	+	15.6125i	−1.00430	+	0.579834i
$726$		0			0
$727$	−49.9600			−1.85291			−0.926457	−	0.376402i	$-0.877161\pi$
$727$	−49.9600			−1.85291			−0.926457	+	0.376402i	$0.877161\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	−2.59808	+	1.50000i	−0.0960933	+	0.0554795i
$732$		0			0
$733$	−7.21110			−0.266348			−0.133174	−	0.991093i	$-0.542517\pi$
$733$	−7.21110			−0.266348			−0.133174	+	0.991093i	$0.542517\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.50000	−	2.59808i	0.0552532	−	0.0957014i
$738$		0			0
$739$	18.1865	+	31.5000i	0.669002	+	1.15875i	0.978183	+	0.207743i	$0.0666118\pi$
$739$	18.1865	+	31.5000i	0.669002	+	1.15875i	−0.309181	+	0.951003i	$0.600055\pi$
$740$		0			0
$741$			6.24500i			0.229416i
$742$		0			0
$743$	−9.36750	+	5.40833i	−0.343660	+	0.198412i	−0.661889	−	0.749602i	$-0.730245\pi$
$743$	−9.36750	+	5.40833i	−0.343660	+	0.198412i	0.318229	+	0.948014i	$0.396912\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	6.92820	−	12.0000i	0.253490	−	0.439057i
$748$		0			0
$749$	−54.0833			−1.97616
$750$		0			0
$751$	15.6125	+	27.0416i	0.569708	+	0.986763i	0.996595	+	0.0824580i	$0.0262770\pi$
$751$	15.6125	+	27.0416i	0.569708	+	0.986763i	−0.426887	+	0.904305i	$0.640390\pi$
$752$		0			0
$753$	3.00000			0.109326
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−5.40833	+	3.12250i	−0.196569	+	0.113489i	−0.595054	−	0.803686i	$-0.702869\pi$
$757$	−5.40833	+	3.12250i	−0.196569	+	0.113489i	0.398485	+	0.917175i	$0.369536\pi$
$758$		0			0
$759$		−	10.8167i		−	0.392620i
$760$		0			0
$761$	−25.5000	−	14.7224i	−0.924374	−	0.533688i	−0.0393463	−	0.999226i	$-0.512528\pi$
$761$	−25.5000	−	14.7224i	−0.924374	−	0.533688i	−0.885028	+	0.465538i	$0.845861\pi$
$762$		0			0
$763$	−45.0333	−	26.0000i	−1.63032	−	0.941263i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	21.8575	−	37.8583i	0.789228	−	1.36698i
$768$		0			0
$769$	−7.50000	+	4.33013i	−0.270457	+	0.156148i	−0.629095	−	0.777328i	$-0.716574\pi$
$769$	−7.50000	+	4.33013i	−0.270457	+	0.156148i	0.358638	+	0.933477i	$0.383241\pi$
$770$		0			0
$771$	2.59808	+	1.50000i	0.0935674	+	0.0540212i
$772$		0			0
$773$	−16.2250	+	28.1025i	−0.583572	+	1.01078i	0.411480	+	0.911419i	$0.365012\pi$
$773$	−16.2250	+	28.1025i	−0.583572	+	1.01078i	−0.995052	+	0.0993575i	$0.968321\pi$
$774$		0			0
$775$		−	36.0555i		−	1.29515i
$776$		0			0
$777$	−6.50000	−	11.2583i	−0.233186	−	0.403890i
$778$		0			0
$779$			9.00000i			0.322458i
$780$		0			0
$781$		−	18.7350i		−	0.670391i
$782$		0			0
$783$	15.6125	+	27.0416i	0.557945	+	0.966389i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	14.7224	−	25.5000i	0.524798	−	0.908977i	−0.474785	−	0.880102i	$-0.657474\pi$
$787$	14.7224	−	25.5000i	0.524798	−	0.908977i	0.999583	−	0.0288750i	$-0.00919248\pi$
$788$		0			0
$789$	5.40833	+	3.12250i	0.192542	+	0.111164i
$790$		0			0
$791$	−28.1025	+	16.2250i	−0.999210	+	0.576894i
$792$		0			0
$793$	19.5000	+	11.2583i	0.692465	+	0.399795i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	27.0416	+	15.6125i	0.957864	+	0.553023i	0.895515	−	0.445031i	$-0.146808\pi$
$797$	27.0416	+	15.6125i	0.957864	+	0.553023i	0.0623489	+	0.998054i	$0.480141\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			17.3205i			0.611990i
$802$		0			0
$803$	5.19615	−	3.00000i	0.183368	−	0.105868i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−31.2250			−1.09917
$808$		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
$811$	−3.46410			−0.121641			−0.0608205	−	0.998149i	$-0.519372\pi$
$811$	−3.46410			−0.121641			−0.0608205	+	0.998149i	$0.519372\pi$
$812$		0			0
$813$	−1.80278	+	3.12250i	−0.0632261	+	0.109511i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.50000	−	0.866025i	0.0524784	−	0.0302984i
$818$		0			0
$819$	−22.5167	+	13.0000i	−0.786796	+	0.454257i
$820$		0			0
$821$	16.2250	+	28.1025i	0.566256	+	0.980784i	0.996932	+	0.0782770i	$0.0249419\pi$
$821$	16.2250	+	28.1025i	0.566256	+	0.980784i	−0.430676	+	0.902507i	$0.641725\pi$
$822$		0			0
$823$	3.12250	−	5.40833i	0.108843	−	0.188522i	−0.806459	−	0.591291i	$-0.798619\pi$
$823$	3.12250	−	5.40833i	0.108843	−	0.188522i	0.915302	+	0.402768i	$0.131952\pi$
$824$		0			0
$825$	7.50000	+	4.33013i	0.261116	+	0.150756i
$826$		0			0
$827$	48.4974			1.68642			0.843210	−	0.537584i	$-0.180663\pi$
$827$	48.4974			1.68642			0.843210	+	0.537584i	$0.180663\pi$
$828$		0			0
$829$	−37.8583	+	21.8575i	−1.31487	+	0.759142i	−0.982899	−	0.184146i	$-0.941048\pi$
$829$	−37.8583	+	21.8575i	−1.31487	+	0.759142i	−0.331974	+	0.943289i	$0.607715\pi$
$830$		0			0
$831$	18.7350			0.649910
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−36.0555			−1.24626
$838$		0			0
$839$	−28.1025	−	16.2250i	−0.970206	−	0.560149i	−0.0709068	−	0.997483i	$-0.522589\pi$
$839$	−28.1025	−	16.2250i	−0.970206	−	0.560149i	−0.899299	+	0.437334i	$0.855923\pi$
$840$		0			0
$841$	5.00000	−	8.66025i	0.172414	−	0.298629i
$842$		0			0
$843$	−8.66025	−	15.0000i	−0.298275	−	0.516627i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	24.9800	−	14.4222i	0.858323	−	0.495553i
$848$		0			0
$849$	−11.5000	+	19.9186i	−0.394679	+	0.683604i
$850$		0			0
$851$	11.2583	−	19.5000i	0.385931	−	0.668451i
$852$		0			0
$853$	28.8444			0.987614			0.493807	−	0.869572i	$-0.335605\pi$
$853$	28.8444			0.987614			0.493807	+	0.869572i	$0.335605\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.0000			0.409912			0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000			0.409912			0.204956	+	0.978771i	$0.434295\pi$
$858$		0			0
$859$		−	10.0000i		−	0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$859$		−	10.0000i		−	0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$860$		0			0
$861$	16.2250	−	9.36750i	0.552946	−	0.319243i
$862$		0			0
$863$			21.6333i			0.736406i	0.929745	+	0.368203i	$0.120027\pi$
$863$			21.6333i			0.736406i	−0.929745	+	0.368203i	$0.879973\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	6.92820	+	4.00000i	0.235294	+	0.135847i
$868$		0			0
$869$	10.8167	+	18.7350i	0.366930	+	0.635541i
$870$		0			0
$871$	3.12250	+	5.40833i	0.105802	+	0.183254i
$872$		0			0
$873$	21.0000	−	12.1244i	0.710742	−	0.410347i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−12.6194	+	21.8575i	−0.426128	+	0.738075i	−0.996525	−	0.0832939i	$-0.973456\pi$
$877$	−12.6194	+	21.8575i	−0.426128	+	0.738075i	0.570397	+	0.821369i	$0.306789\pi$
$878$		0			0
$879$		−	32.4500i		−	1.09451i
$880$		0			0
$881$	−4.50000	−	7.79423i	−0.151609	−	0.262594i	0.780210	−	0.625517i	$-0.215112\pi$
$881$	−4.50000	−	7.79423i	−0.151609	−	0.262594i	−0.931819	+	0.362923i	$0.881779\pi$
$882$		0			0
$883$		−	38.0000i		−	1.27880i	−0.768874	−	0.639401i	$-0.779182\pi$
$883$		−	38.0000i		−	1.27880i	0.768874	−	0.639401i	$-0.220818\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.12250	−	5.40833i	−0.104843	−	0.181594i	0.808831	−	0.588041i	$-0.200101\pi$
$887$	−3.12250	−	5.40833i	−0.104843	−	0.181594i	−0.913674	+	0.406447i	$0.866767\pi$
$888$		0			0
$889$		−	22.5167i		−	0.755185i
$890$		0			0
$891$	−0.866025	+	1.50000i	−0.0290129	+	0.0502519i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	19.5000	+	11.2583i	0.651086	+	0.375905i
$898$		0			0
$899$	22.5167	+	39.0000i	0.750973	+	1.30072i
$900$		0			0
$901$	−32.4500	−	18.7350i	−1.08106	−	0.624153i
$902$		0			0
$903$	−3.12250	−	1.80278i	−0.103910	−	0.0599926i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−26.8468	+	15.5000i	−0.891433	+	0.514669i	−0.874411	−	0.485186i	$-0.838752\pi$
$907$	−26.8468	+	15.5000i	−0.891433	+	0.514669i	−0.0170220	+	0.999855i	$0.505419\pi$
$908$		0			0
$909$		−	12.4900i		−	0.414267i
$910$		0			0
$911$	−24.9800			−0.827624			−0.413812	−	0.910362i	$-0.635803\pi$
$911$	−24.9800			−0.827624			−0.413812	+	0.910362i	$0.635803\pi$
$912$		0			0
$913$	6.00000	+	10.3923i	0.198571	+	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.8167	+	18.7350i	−0.357197	+	0.618684i
$918$		0			0
$919$	9.36750	−	16.2250i	0.309005	−	0.535213i	−0.669140	−	0.743137i	$-0.733337\pi$
$919$	9.36750	−	16.2250i	0.309005	−	0.535213i	0.978145	+	0.207924i	$0.0666706\pi$
$920$		0			0
$921$	24.0000	−	13.8564i	0.790827	−	0.456584i
$922$		0			0
$923$	33.7750	+	19.5000i	1.11172	+	0.641850i
$924$		0			0
$925$	9.01388	+	15.6125i	0.296374	+	0.513336i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.50000	−	0.866025i	−0.0492134	−	0.0284134i	0.475191	−	0.879882i	$-0.342379\pi$
$929$	−1.50000	−	0.866025i	−0.0492134	−	0.0284134i	−0.524405	+	0.851469i	$0.675712\pi$
$930$		0			0
$931$	−10.3923			−0.340594
$932$		0			0
$933$	21.6333	−	12.4900i	0.708243	−	0.408904i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	20.0000			0.653372			0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000			0.653372			0.326686	+	0.945133i	$0.394068\pi$
$938$		0			0
$939$	6.92820	−	4.00000i	0.226093	−	0.130535i
$940$		0			0
$941$	−21.6333			−0.705226			−0.352613	−	0.935769i	$-0.614707\pi$
$941$	−21.6333			−0.705226			−0.352613	+	0.935769i	$0.614707\pi$
$942$		0			0
$943$	28.1025	+	16.2250i	0.915143	+	0.528358i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.52628	+	16.5000i	0.309562	+	0.536178i	0.978267	−	0.207351i	$-0.0664841\pi$
$947$	9.52628	+	16.5000i	0.309562	+	0.536178i	−0.668704	+	0.743529i	$0.733151\pi$
$948$		0			0
$949$			12.4900i			0.405442i
$950$		0			0
$951$	18.7350	−	10.8167i	0.607524	−	0.350754i
$952$		0			0
$953$	−25.5000	+	44.1673i	−0.826026	+	1.43072i	0.0751066	+	0.997176i	$0.476070\pi$
$953$	−25.5000	+	44.1673i	−0.826026	+	1.43072i	−0.901133	+	0.433544i	$0.857263\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−10.8167			−0.349653
$958$		0			0
$959$	−21.8575	−	37.8583i	−0.705815	−	1.22251i
$960$		0			0
$961$	−21.0000			−0.677419
$962$		0			0
$963$		−	30.0000i		−	0.966736i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			7.21110i			0.231893i	0.993255	+	0.115947i	$0.0369902\pi$
$967$			7.21110i			0.231893i	−0.993255	+	0.115947i	$0.963010\pi$
$968$		0			0
$969$	4.50000	+	2.59808i	0.144561	+	0.0834622i
$970$		0			0
$971$	−18.1865	−	10.5000i	−0.583634	−	0.336961i	0.178942	−	0.983860i	$-0.442732\pi$
$971$	−18.1865	−	10.5000i	−0.583634	−	0.336961i	−0.762576	+	0.646899i	$0.776066\pi$
$972$		0			0
$973$	−19.8305	−	34.3475i	−0.635738	−	1.10113i
$974$		0			0
$975$	−15.6125	+	9.01388i	−0.500000	+	0.288675i
$976$		0			0
$977$	−34.5000	+	19.9186i	−1.10375	+	0.637252i	−0.937204	−	0.348781i	$-0.886596\pi$
$977$	−34.5000	+	19.9186i	−1.10375	+	0.637252i	−0.166549	+	0.986033i	$0.553262\pi$
$978$		0			0
$979$	−12.9904	−	7.50000i	−0.415174	−	0.239701i
$980$		0			0
$981$	14.4222	−	24.9800i	0.460466	−	0.797550i
$982$		0			0
$983$			21.6333i			0.689995i	0.938604	+	0.344998i	$0.112120\pi$
$983$			21.6333i			0.689995i	−0.938604	+	0.344998i	$0.887880\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	6.24500i		−	0.198579i
$990$		0			0
$991$	−15.6125	−	27.0416i	−0.495947	−	0.859006i	0.504042	−	0.863679i	$-0.331846\pi$
$991$	−15.6125	−	27.0416i	−0.495947	−	0.859006i	−0.999989	+	0.00467341i	$0.998512\pi$
$992$		0			0
$993$		−	22.5167i		−	0.714545i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.40833	+	3.12250i	0.171283	+	0.0988905i	0.583191	−	0.812335i	$-0.301804\pi$
$997$	5.40833	+	3.12250i	0.171283	+	0.0988905i	−0.411907	+	0.911226i	$0.635137\pi$
$998$		0			0
$999$	15.6125	−	9.01388i	0.493957	−	0.285186i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.ba.g.673.4	yes	8
4.3	odd	2	inner	832.2.ba.g.673.1	yes	8
8.3	odd	2	inner	832.2.ba.g.673.3	yes	8
8.5	even	2	inner	832.2.ba.g.673.2	yes	8
13.4	even	6	inner	832.2.ba.g.225.2	yes	8
52.43	odd	6	inner	832.2.ba.g.225.3	yes	8
104.43	odd	6	inner	832.2.ba.g.225.1	✓	8
104.69	even	6	inner	832.2.ba.g.225.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.ba.g.225.1	✓	8	104.43	odd	6	inner
832.2.ba.g.225.2	yes	8	13.4	even	6	inner
832.2.ba.g.225.3	yes	8	52.43	odd	6	inner
832.2.ba.g.225.4	yes	8	104.69	even	6	inner
832.2.ba.g.673.1	yes	8	4.3	odd	2	inner
832.2.ba.g.673.2	yes	8	8.5	even	2	inner
832.2.ba.g.673.3	yes	8	8.3	odd	2	inner
832.2.ba.g.673.4	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.ba (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{3} +(3.12250 + 1.80278i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(-0.866025 - 1.50000i) q^{11} +3.60555 q^{13} +(1.50000 - 2.59808i) q^{17} +(-0.866025 + 1.50000i) q^{19} +3.60555 q^{21} +(3.12250 + 5.40833i) q^{23} -5.00000 q^{25} +5.00000i q^{27} +(5.40833 - 3.12250i) q^{29} +7.21110i q^{31} +(-1.50000 - 0.866025i) q^{33} +(-1.80278 - 3.12250i) q^{37} +(3.12250 - 1.80278i) q^{39} +(4.50000 - 2.59808i) q^{41} +(-0.866025 - 0.500000i) q^{43} +(3.00000 + 5.19615i) q^{49} -3.00000i q^{51} -12.4900i q^{53} +1.73205i q^{57} +(6.06218 - 10.5000i) q^{59} +(5.40833 + 3.12250i) q^{61} +(-6.24500 + 3.60555i) q^{63} +(0.866025 + 1.50000i) q^{67} +(5.40833 + 3.12250i) q^{69} +(9.36750 + 5.40833i) q^{71} +3.46410i q^{73} +(-4.33013 + 2.50000i) q^{75} -6.24500i q^{77} -12.4900 q^{79} +(-0.500000 - 0.866025i) q^{81} -6.92820 q^{83} +(3.12250 - 5.40833i) q^{87} +(7.50000 - 4.33013i) q^{89} +(11.2583 + 6.50000i) q^{91} +(3.60555 + 6.24500i) q^{93} +(-10.5000 - 6.06218i) q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} + 12 q^{17} - 40 q^{25} - 12 q^{33} + 36 q^{41} + 24 q^{49} - 4 q^{81} + 60 q^{89} - 84 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 + 12 * q^17 - 40 * q^25 - 12 * q^33 + 36 * q^41 + 24 * q^49 - 4 * q^81 + 60 * q^89 - 84 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more