LMFDB - Embedded newform 378.2.h.d.361.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(289,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 2]))

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

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

chi := DirichletCharacter("378.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.h (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.3
Root			$0.500000 + 2.05195i$ of defining polynomial
Character	$\chi$	$=$	378.361
Dual form			378.2.h.d.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +0.460505 q^{5} +(2.25729 + 1.38008i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +0.460505 q^{5} +(2.25729 + 1.38008i) q^{7} -1.00000 q^{8} +(0.230252 - 0.398809i) q^{10} +3.64766 q^{11} +(0.730252 - 1.26483i) q^{13} +(2.32383 - 1.26483i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.86693 - 3.23361i) q^{17} +(-2.02704 - 3.51094i) q^{19} +(-0.230252 - 0.398809i) q^{20} +(1.82383 - 3.15897i) q^{22} -1.13307 q^{23} -4.78794 q^{25} +(-0.730252 - 1.26483i) q^{26} +(0.0665372 - 2.64491i) q^{28} +(4.48755 + 7.77266i) q^{29} +(0.257295 + 0.445647i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.86693 - 3.23361i) q^{34} +(1.03950 + 0.635534i) q^{35} +(-4.55408 - 7.88791i) q^{37} -4.05408 q^{38} -0.460505 q^{40} +(0.472958 - 0.819187i) q^{41} +(4.66372 + 8.07779i) q^{43} +(-1.82383 - 3.15897i) q^{44} +(-0.566537 + 0.981271i) q^{46} +(1.16372 - 2.01561i) q^{47} +(3.19076 + 6.23049i) q^{49} +(-2.39397 + 4.14647i) q^{50} -1.46050 q^{52} +(-6.21780 + 10.7695i) q^{53} +1.67977 q^{55} +(-2.25729 - 1.38008i) q^{56} +8.97509 q^{58} +(-6.44805 - 11.1684i) q^{59} +(-6.04163 + 10.4644i) q^{61} +0.514589 q^{62} +1.00000 q^{64} +(0.336285 - 0.582462i) q^{65} +(1.16012 + 2.00938i) q^{67} -3.73385 q^{68} +(1.07014 - 0.582462i) q^{70} -1.67977 q^{71} +(-6.62062 + 11.4673i) q^{73} -9.10817 q^{74} +(-2.02704 + 3.51094i) q^{76} +(8.23385 + 5.03407i) q^{77} +(2.50360 - 4.33636i) q^{79} +(-0.230252 + 0.398809i) q^{80} +(-0.472958 - 0.819187i) q^{82} +(-3.32383 - 5.75705i) q^{83} +(0.859728 - 1.48909i) q^{85} +9.32743 q^{86} -3.64766 q^{88} +(1.36333 + 2.36135i) q^{89} +(3.39397 - 1.84730i) q^{91} +(0.566537 + 0.981271i) q^{92} +(-1.16372 - 2.01561i) q^{94} +(-0.933463 - 1.61680i) q^{95} +(-5.59358 - 9.68836i) q^{97} +(6.99115 + 0.351971i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8	$ 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8} - 5 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} - q^{22} - 14 q^{23} + 4 q^{25} + 2 q^{26} + 4 q^{28} + 5 q^{29} - 14 q^{31} + 3 q^{32} - 4 q^{34} + 19 q^{35} - 9 q^{37} - 6 q^{38} + 10 q^{40} + 12 q^{41} + 18 q^{43} + q^{44} - 7 q^{46} - 3 q^{47} + 2 q^{50} + 4 q^{52} - 9 q^{53} + 14 q^{55} + 2 q^{56} + 10 q^{58} - 4 q^{59} + 4 q^{61} - 28 q^{62} + 6 q^{64} + 12 q^{65} + 5 q^{67} - 8 q^{68} + 2 q^{70} - 14 q^{71} - 25 q^{73} - 18 q^{74} - 3 q^{76} + 35 q^{77} + 7 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} + 14 q^{85} + 36 q^{86} + 2 q^{88} + 9 q^{89} + 4 q^{91} + 7 q^{92} + 3 q^{94} - 2 q^{95} - 28 q^{97} + 12 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8 - 5 * q^10 - 2 * q^11 - 2 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^19 + 5 * q^20 - q^22 - 14 * q^23 + 4 * q^25 + 2 * q^26 + 4 * q^28 + 5 * q^29 - 14 * q^31 + 3 * q^32 - 4 * q^34 + 19 * q^35 - 9 * q^37 - 6 * q^38 + 10 * q^40 + 12 * q^41 + 18 * q^43 + q^44 - 7 * q^46 - 3 * q^47 + 2 * q^50 + 4 * q^52 - 9 * q^53 + 14 * q^55 + 2 * q^56 + 10 * q^58 - 4 * q^59 + 4 * q^61 - 28 * q^62 + 6 * q^64 + 12 * q^65 + 5 * q^67 - 8 * q^68 + 2 * q^70 - 14 * q^71 - 25 * q^73 - 18 * q^74 - 3 * q^76 + 35 * q^77 + 7 * q^79 + 5 * q^80 - 12 * q^82 - 8 * q^83 + 14 * q^85 + 36 * q^86 + 2 * q^88 + 9 * q^89 + 4 * q^91 + 7 * q^92 + 3 * q^94 - 2 * q^95 - 28 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	0.460505			0.205944			0.102972	−	0.994684i	$-0.467165\pi$
$5$	0.460505			0.205944			0.102972	+	0.994684i	$0.467165\pi$
$6$		0			0
$7$	2.25729	+	1.38008i	0.853177	+	0.521621i
$7$	2.25729	+	1.38008i	0.853177	+	0.521621i
$8$	−1.00000			−0.353553
$9$		0			0
$10$	0.230252	−	0.398809i	0.0728122	−	0.126114i
$11$	3.64766			1.09981			0.549906	−	0.835227i	$-0.314664\pi$
$11$	3.64766			1.09981			0.549906	+	0.835227i	$0.314664\pi$
$12$		0			0
$13$	0.730252	−	1.26483i	0.202536	−	0.350802i	−0.746809	−	0.665038i	$-0.768415\pi$
$13$	0.730252	−	1.26483i	0.202536	−	0.350802i	0.949345	+	0.314236i	$0.101748\pi$
$14$	2.32383	−	1.26483i	0.621070	−	0.338041i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	1.86693	−	3.23361i	0.452796	−	0.784266i	−0.545763	−	0.837940i	$-0.683760\pi$
$17$	1.86693	−	3.23361i	0.452796	−	0.784266i	0.998558	+	0.0536743i	$0.0170933\pi$
$18$		0			0
$19$	−2.02704	−	3.51094i	−0.465035	−	0.805465i	0.534168	−	0.845378i	$-0.320625\pi$
$19$	−2.02704	−	3.51094i	−0.465035	−	0.805465i	−0.999203	+	0.0399136i	$0.987292\pi$
$20$	−0.230252	−	0.398809i	−0.0514860	−	0.0891764i
$21$		0			0
$22$	1.82383	−	3.15897i	0.388842	−	0.673495i
$23$	−1.13307			−0.236262			−0.118131	−	0.992998i	$-0.537690\pi$
$23$	−1.13307			−0.236262			−0.118131	+	0.992998i	$0.537690\pi$
$24$		0			0
$25$	−4.78794			−0.957587
$26$	−0.730252	−	1.26483i	−0.143214	−	0.248054i
$27$		0			0
$28$	0.0665372	−	2.64491i	0.0125744	−	0.499842i
$29$	4.48755	+	7.77266i	0.833317	+	1.44335i	0.895394	+	0.445275i	$0.146894\pi$
$29$	4.48755	+	7.77266i	0.833317	+	1.44335i	−0.0620772	+	0.998071i	$0.519772\pi$
$30$		0			0
$31$	0.257295	+	0.445647i	0.0462115	+	0.0800406i	0.888206	−	0.459446i	$-0.151952\pi$
$31$	0.257295	+	0.445647i	0.0462115	+	0.0800406i	−0.841994	+	0.539486i	$0.818619\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	−1.86693	−	3.23361i	−0.320175	−	0.554560i
$35$	1.03950	+	0.635534i	0.175707	+	0.107425i
$36$		0			0
$37$	−4.55408	−	7.88791i	−0.748687	−	1.29676i	−0.948452	−	0.316920i	$-0.897351\pi$
$37$	−4.55408	−	7.88791i	−0.748687	−	1.29676i	0.199765	−	0.979844i	$-0.435982\pi$
$38$	−4.05408			−0.657659
$39$		0			0
$40$	−0.460505			−0.0728122
$41$	0.472958	−	0.819187i	0.0738636	−	0.127936i	−0.826728	−	0.562602i	$-0.809800\pi$
$41$	0.472958	−	0.819187i	0.0738636	−	0.127936i	0.900592	+	0.434666i	$0.143134\pi$
$42$		0			0
$43$	4.66372	+	8.07779i	0.711210	+	1.23185i	0.964403	+	0.264436i	$0.0851858\pi$
$43$	4.66372	+	8.07779i	0.711210	+	1.23185i	−0.253193	+	0.967416i	$0.581481\pi$
$44$	−1.82383	−	3.15897i	−0.274953	−	0.476233i
$45$		0			0
$46$	−0.566537	+	0.981271i	−0.0835314	+	0.144681i
$47$	1.16372	−	2.01561i	0.169745	−	0.294007i	−0.768585	−	0.639748i	$-0.779039\pi$
$47$	1.16372	−	2.01561i	0.169745	−	0.294007i	0.938330	+	0.345740i	$0.112372\pi$
$48$		0			0
$49$	3.19076	+	6.23049i	0.455822	+	0.890071i
$50$	−2.39397	+	4.14647i	−0.338558	+	0.586400i
$51$		0			0
$52$	−1.46050			−0.202536
$53$	−6.21780	+	10.7695i	−0.854080	+	1.47931i	0.0234151	+	0.999726i	$0.492546\pi$
$53$	−6.21780	+	10.7695i	−0.854080	+	1.47931i	−0.877495	+	0.479585i	$0.840787\pi$
$54$		0			0
$55$	1.67977			0.226500
$56$	−2.25729	−	1.38008i	−0.301644	−	0.184421i
$57$		0			0
$58$	8.97509			1.17849
$59$	−6.44805	−	11.1684i	−0.839465	−	1.45400i	−0.890343	−	0.455291i	$-0.849535\pi$
$59$	−6.44805	−	11.1684i	−0.839465	−	1.45400i	0.0508779	−	0.998705i	$-0.483798\pi$
$60$		0			0
$61$	−6.04163	+	10.4644i	−0.773552	+	1.33983i	0.162053	+	0.986782i	$0.448188\pi$
$61$	−6.04163	+	10.4644i	−0.773552	+	1.33983i	−0.935605	+	0.353049i	$0.885145\pi$
$62$	0.514589			0.0653529
$63$		0			0
$64$	1.00000			0.125000
$65$	0.336285	−	0.582462i	0.0417110	−	0.0722456i
$66$		0			0
$67$	1.16012	+	2.00938i	0.141731	+	0.245485i	0.928148	−	0.372210i	$-0.121400\pi$
$67$	1.16012	+	2.00938i	0.141731	+	0.245485i	−0.786418	+	0.617695i	$0.788067\pi$
$68$	−3.73385			−0.452796
$69$		0			0
$70$	1.07014	−	0.582462i	0.127906	−	0.0696176i
$71$	−1.67977			−0.199352			−0.0996758	−	0.995020i	$-0.531781\pi$
$71$	−1.67977			−0.199352			−0.0996758	+	0.995020i	$0.531781\pi$
$72$		0			0
$73$	−6.62062	+	11.4673i	−0.774885	+	1.34214i	0.159974	+	0.987121i	$0.448859\pi$
$73$	−6.62062	+	11.4673i	−0.774885	+	1.34214i	−0.934859	+	0.355019i	$0.884474\pi$
$74$	−9.10817			−1.05880
$75$		0			0
$76$	−2.02704	+	3.51094i	−0.232518	+	0.402732i
$77$	8.23385	+	5.03407i	0.938334	+	0.573685i
$78$		0			0
$79$	2.50360	−	4.33636i	0.281677	−	0.487879i	−0.690121	−	0.723694i	$-0.742443\pi$
$79$	2.50360	−	4.33636i	0.281677	−	0.487879i	0.971798	+	0.235815i	$0.0757761\pi$
$80$	−0.230252	+	0.398809i	−0.0257430	+	0.0445882i
$81$		0			0
$82$	−0.472958	−	0.819187i	−0.0522295	−	0.0904641i
$83$	−3.32383	−	5.75705i	−0.364838	−	0.631918i	0.623912	−	0.781494i	$-0.285542\pi$
$83$	−3.32383	−	5.75705i	−0.364838	−	0.631918i	−0.988750	+	0.149577i	$0.952209\pi$
$84$		0			0
$85$	0.859728	−	1.48909i	0.0932506	−	0.161515i
$86$	9.32743			1.00580
$87$		0			0
$88$	−3.64766			−0.388842
$89$	1.36333	+	2.36135i	0.144512	+	0.250303i	0.929191	−	0.369600i	$-0.120505\pi$
$89$	1.36333	+	2.36135i	0.144512	+	0.250303i	−0.784679	+	0.619903i	$0.787172\pi$
$90$		0			0
$91$	3.39397	−	1.84730i	0.355784	−	0.193649i
$92$	0.566537	+	0.981271i	0.0590656	+	0.102305i
$93$		0			0
$94$	−1.16372	−	2.01561i	−0.120028	−	0.207895i
$95$	−0.933463	−	1.61680i	−0.0957713	−	0.165881i
$96$		0			0
$97$	−5.59358	−	9.68836i	−0.567942	−	0.983704i	−0.996769	−	0.0803178i	$-0.974406\pi$
$97$	−5.59358	−	9.68836i	−0.567942	−	0.983704i	0.428827	−	0.903386i	$-0.358927\pi$
$98$	6.99115	+	0.351971i	0.706212	+	0.0355544i
$99$		0			0
$100$	2.39397	+	4.14647i	0.239397	+	0.414647i
$101$	−13.7558			−1.36876			−0.684378	−	0.729127i	$-0.739926\pi$
$101$	−13.7558			−1.36876			−0.684378	+	0.729127i	$0.739926\pi$
$102$		0			0
$103$	11.1623			1.09985			0.549925	−	0.835214i	$-0.314656\pi$
$103$	11.1623			1.09985			0.549925	+	0.835214i	$0.314656\pi$
$104$	−0.730252	+	1.26483i	−0.0716071	+	0.124027i
$105$		0			0
$106$	6.21780	+	10.7695i	0.603926	+	1.04603i
$107$	3.89037	+	6.73832i	0.376096	+	0.651418i	0.990490	−	0.137581i	$-0.0439329\pi$
$107$	3.89037	+	6.73832i	0.376096	+	0.651418i	−0.614394	+	0.788999i	$0.710600\pi$
$108$		0			0
$109$	−3.75729	+	6.50783i	−0.359884	+	0.623337i	−0.987941	−	0.154830i	$-0.950517\pi$
$109$	−3.75729	+	6.50783i	−0.359884	+	0.623337i	0.628058	+	0.778167i	$0.283850\pi$
$110$	0.839883	−	1.45472i	0.0800797	−	0.138702i
$111$		0			0
$112$	−2.32383	+	1.26483i	−0.219581	+	0.119516i
$113$	−3.03064	+	5.24922i	−0.285099	+	0.493805i	−0.972633	−	0.232346i	$-0.925360\pi$
$113$	−3.03064	+	5.24922i	−0.285099	+	0.493805i	0.687534	+	0.726152i	$0.258693\pi$
$114$		0			0
$115$	−0.521786			−0.0486568
$116$	4.48755	−	7.77266i	0.416658	−	0.721673i
$117$		0			0
$118$	−12.8961			−1.18718
$119$	8.67684	−	4.72270i	0.795405	−	0.432929i
$120$		0			0
$121$	2.30545			0.209586
$122$	6.04163	+	10.4644i	0.546984	+	0.947403i
$123$		0			0
$124$	0.257295	−	0.445647i	0.0231057	−	0.0400203i
$125$	−4.50739			−0.403153
$126$		0			0
$127$	8.80992			0.781754			0.390877	−	0.920443i	$-0.372172\pi$
$127$	8.80992			0.781754			0.390877	+	0.920443i	$0.372172\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$	−0.336285	−	0.582462i	−0.0294941	−	0.0510853i
$131$	−21.1373			−1.84678			−0.923389	−	0.383865i	$-0.874593\pi$
$131$	−21.1373			−1.84678			−0.923389	+	0.383865i	$0.874593\pi$
$132$		0			0
$133$	0.269748	−	10.7227i	0.0233901	−	0.929777i
$134$	2.32023			0.200438
$135$		0			0
$136$	−1.86693	+	3.23361i	−0.160088	+	0.277280i
$137$	4.40642			0.376466			0.188233	−	0.982124i	$-0.439724\pi$
$137$	4.40642			0.376466			0.188233	+	0.982124i	$0.439724\pi$
$138$		0			0
$139$	−1.01245	+	1.75362i	−0.0858751	+	0.148740i	−0.905764	−	0.423783i	$-0.860702\pi$
$139$	−1.01245	+	1.75362i	−0.0858751	+	0.148740i	0.819889	+	0.572523i	$0.194035\pi$
$140$	0.0306407	−	1.21800i	0.00258961	−	0.102939i
$141$		0			0
$142$	−0.839883	+	1.45472i	−0.0704815	+	0.122077i
$143$	2.66372	−	4.61369i	0.222751	−	0.385816i
$144$		0			0
$145$	2.06654	+	3.57935i	0.171617	+	0.297249i
$146$	6.62062	+	11.4673i	0.547927	+	0.949037i
$147$		0			0
$148$	−4.55408	+	7.88791i	−0.374343	+	0.648382i
$149$	9.16225			0.750601			0.375300	−	0.926903i	$-0.377540\pi$
$149$	9.16225			0.750601			0.375300	+	0.926903i	$0.377540\pi$
$150$		0			0
$151$	−0.103896			−0.00845496			−0.00422748	−	0.999991i	$-0.501346\pi$
$151$	−0.103896			−0.00845496			−0.00422748	+	0.999991i	$0.501346\pi$
$152$	2.02704	+	3.51094i	0.164415	+	0.284775i
$153$		0			0
$154$	8.47656	−	4.61369i	0.683060	−	0.371782i
$155$	0.118485	+	0.205223i	0.00951698	+	0.0164839i
$156$		0			0
$157$	−10.4911	−	18.1712i	−0.837285	−	1.45022i	−0.892157	−	0.451726i	$-0.850808\pi$
$157$	−10.4911	−	18.1712i	−0.837285	−	1.45022i	0.0548721	−	0.998493i	$-0.482525\pi$
$158$	−2.50360	−	4.33636i	−0.199176	−	0.344982i
$159$		0			0
$160$	0.230252	+	0.398809i	0.0182031	+	0.0315286i
$161$	−2.55768	−	1.56373i	−0.201574	−	0.123239i
$162$		0			0
$163$	−11.5182	−	19.9501i	−0.902174	−	1.56261i	−0.824666	−	0.565620i	$-0.808637\pi$
$163$	−11.5182	−	19.9501i	−0.902174	−	1.56261i	−0.0775078	−	0.996992i	$-0.524696\pi$
$164$	−0.945916			−0.0738636
$165$		0			0
$166$	−6.64766			−0.515959
$167$	5.31498	−	9.20581i	0.411285	−	0.712367i	−0.583745	−	0.811937i	$-0.698413\pi$
$167$	5.31498	−	9.20581i	0.411285	−	0.712367i	0.995031	+	0.0995698i	$0.0317467\pi$
$168$		0			0
$169$	5.43346	+	9.41103i	0.417959	+	0.723926i
$170$	−0.859728	−	1.48909i	−0.0659382	−	0.114208i
$171$		0			0
$172$	4.66372	−	8.07779i	0.355605	−	0.615926i
$173$	1.46936	−	2.54500i	0.111713	−	0.193493i	−0.804748	−	0.593617i	$-0.797699\pi$
$173$	1.46936	−	2.54500i	0.111713	−	0.193493i	0.916461	+	0.400124i	$0.131033\pi$
$174$		0			0
$175$	−10.8078	−	6.60773i	−0.816991	−	0.499498i
$176$	−1.82383	+	3.15897i	−0.137476	+	0.238116i
$177$		0			0
$178$	2.72665			0.204371
$179$	4.58113	−	7.93474i	0.342409	−	0.593071i	−0.642470	−	0.766311i	$-0.722090\pi$
$179$	4.58113	−	7.93474i	0.342409	−	0.593071i	0.984880	+	0.173240i	$0.0554237\pi$
$180$		0			0
$181$	22.4284			1.66709			0.833545	−	0.552452i	$-0.186308\pi$
$181$	22.4284			1.66709			0.833545	+	0.552452i	$0.186308\pi$
$182$	0.0971780	−	3.86291i	0.00720331	−	0.286338i
$183$		0			0
$184$	1.13307			0.0835314
$185$	−2.09718	−	3.63242i	−0.154188	−	0.267061i
$186$		0			0
$187$	6.80992	−	11.7951i	0.497990	−	0.862545i
$188$	−2.32743			−0.169745
$189$		0			0
$190$	−1.86693			−0.135441
$191$	1.24484	−	2.15613i	0.0900736	−	0.156012i	−0.817468	−	0.575974i	$-0.804623\pi$
$191$	1.24484	−	2.15613i	0.0900736	−	0.156012i	0.907542	+	0.419962i	$0.137956\pi$
$192$		0			0
$193$	−2.24484	−	3.88818i	−0.161587	−	0.279877i	0.773851	−	0.633368i	$-0.218328\pi$
$193$	−2.24484	−	3.88818i	−0.161587	−	0.279877i	−0.935438	+	0.353491i	$0.884995\pi$
$194$	−11.1872			−0.803191
$195$		0			0
$196$	3.80039	−	5.87852i	0.271456	−	0.419895i
$197$	−12.7339			−0.907249			−0.453625	−	0.891193i	$-0.649869\pi$
$197$	−12.7339			−0.907249			−0.453625	+	0.891193i	$0.649869\pi$
$198$		0			0
$199$	−1.47296	+	2.55124i	−0.104415	+	0.180852i	−0.913499	−	0.406841i	$-0.866630\pi$
$199$	−1.47296	+	2.55124i	−0.104415	+	0.180852i	0.809084	+	0.587693i	$0.199964\pi$
$200$	4.78794			0.338558
$201$		0			0
$202$	−6.87792	+	11.9129i	−0.483928	+	0.838189i
$203$	−0.597178	+	23.7384i	−0.0419137	+	1.66611i
$204$		0			0
$205$	0.217799	−	0.377240i	0.0152118	−	0.0263476i
$206$	5.58113	−	9.66679i	0.388855	−	0.673517i
$207$		0			0
$208$	0.730252	+	1.26483i	0.0506339	+	0.0877005i
$209$	−7.39397	−	12.8067i	−0.511451	−	0.885860i
$210$		0			0
$211$	−0.608168	+	1.05338i	−0.0418680	+	0.0725176i	−0.886200	−	0.463303i	$-0.846664\pi$
$211$	−0.608168	+	1.05338i	−0.0418680	+	0.0725176i	0.844332	+	0.535820i	$0.179998\pi$
$212$	12.4356			0.854080
$213$		0			0
$214$	7.78074			0.531880
$215$	2.14766	+	3.71986i	0.146469	+	0.253693i
$216$		0			0
$217$	−0.0342393	+	1.36104i	−0.00232432	+	0.0923937i
$218$	3.75729	+	6.50783i	0.254476	+	0.440766i
$219$		0			0
$220$	−0.839883	−	1.45472i	−0.0566249	−	0.0980773i
$221$	−2.72665	−	4.72270i	−0.183415	−	0.317683i
$222$		0			0
$223$	−0.445916	−	0.772349i	−0.0298607	−	0.0517203i	0.850709	−	0.525637i	$-0.176173\pi$
$223$	−0.445916	−	0.772349i	−0.0298607	−	0.0517203i	−0.880570	+	0.473917i	$0.842840\pi$
$224$	−0.0665372	+	2.64491i	−0.00444571	+	0.176721i
$225$		0			0
$226$	3.03064	+	5.24922i	0.201595	+	0.349173i
$227$	14.6519			0.972483			0.486242	−	0.873824i	$-0.338368\pi$
$227$	14.6519			0.972483			0.486242	+	0.873824i	$0.338368\pi$
$228$		0			0
$229$	−9.57587			−0.632791			−0.316396	−	0.948627i	$-0.602473\pi$
$229$	−9.57587			−0.632791			−0.316396	+	0.948627i	$0.602473\pi$
$230$	−0.260893	+	0.451880i	−0.0172028	+	0.0297961i
$231$		0			0
$232$	−4.48755	−	7.77266i	−0.294622	−	0.510300i
$233$	−7.21420	−	12.4954i	−0.472618	−	0.818598i	0.526891	−	0.849933i	$-0.323358\pi$
$233$	−7.21420	−	12.4954i	−0.472618	−	0.818598i	−0.999509	+	0.0313345i	$0.990024\pi$
$234$		0			0
$235$	0.535897	−	0.928200i	0.0349580	−	0.0605491i
$236$	−6.44805	+	11.1684i	−0.419732	+	0.726998i
$237$		0			0
$238$	0.248440	−	9.87572i	0.0161040	−	0.640148i
$239$	9.15486	−	15.8567i	0.592179	−	1.02568i	−0.401760	−	0.915745i	$-0.631601\pi$
$239$	9.15486	−	15.8567i	0.592179	−	1.02568i	0.993938	−	0.109938i	$-0.0350654\pi$
$240$		0			0
$241$	0.0933847			0.00601544			0.00300772	−	0.999995i	$-0.499043\pi$
$241$	0.0933847			0.00601544			0.00300772	+	0.999995i	$0.499043\pi$
$242$	1.15272	−	1.99658i	0.0741000	−	0.128345i
$243$		0			0
$244$	12.0833			0.773552
$245$	1.46936	+	2.86917i	0.0938739	+	0.183305i
$246$		0			0
$247$	−5.92101			−0.376745
$248$	−0.257295	−	0.445647i	−0.0163382	−	0.0282986i
$249$		0			0
$250$	−2.25370	+	3.90352i	−0.142536	+	0.246880i
$251$	18.2733			1.15340			0.576702	−	0.816955i	$-0.304339\pi$
$251$	18.2733			1.15340			0.576702	+	0.816955i	$0.304339\pi$
$252$		0			0
$253$	−4.13307			−0.259844
$254$	4.40496	−	7.62961i	0.276392	−	0.478724i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	21.0512			1.31314			0.656568	−	0.754267i	$-0.272008\pi$
$257$	21.0512			1.31314			0.656568	+	0.754267i	$0.272008\pi$
$258$		0			0
$259$	0.606032	−	24.0903i	0.0376570	−	1.49690i
$260$	−0.672570			−0.0417110
$261$		0			0
$262$	−10.5687	+	18.3055i	−0.652935	+	1.13092i
$263$	5.16518			0.318499			0.159249	−	0.987238i	$-0.449093\pi$
$263$	5.16518			0.318499			0.159249	+	0.987238i	$0.449093\pi$
$264$		0			0
$265$	−2.86333	+	4.95943i	−0.175893	+	0.304655i
$266$	−9.15126	−	5.59496i	−0.561100	−	0.343049i
$267$		0			0
$268$	1.16012	−	2.00938i	0.0708654	−	0.122742i
$269$	−8.42840	+	14.5984i	−0.513889	+	0.890081i	0.485981	+	0.873969i	$0.338462\pi$
$269$	−8.42840	+	14.5984i	−0.513889	+	0.890081i	−0.999870	+	0.0161123i	$0.994871\pi$
$270$		0			0
$271$	12.5562	+	21.7480i	0.762736	+	1.32110i	0.941435	+	0.337194i	$0.109478\pi$
$271$	12.5562	+	21.7480i	0.762736	+	1.32110i	−0.178699	+	0.983904i	$0.557189\pi$
$272$	1.86693	+	3.23361i	0.113199	+	0.196066i
$273$		0			0
$274$	2.20321	−	3.81607i	0.133101	−	0.230537i
$275$	−17.4648			−1.05317
$276$		0			0
$277$	3.38151			0.203176			0.101588	−	0.994827i	$-0.467608\pi$
$277$	3.38151			0.203176			0.101588	+	0.994827i	$0.467608\pi$
$278$	1.01245	+	1.75362i	0.0607229	+	0.105175i
$279$		0			0
$280$	−1.03950	−	0.635534i	−0.0621217	−	0.0379804i
$281$	10.1388	+	17.5609i	0.604831	+	1.04760i	0.992078	+	0.125622i	$0.0400925\pi$
$281$	10.1388	+	17.5609i	0.604831	+	1.04760i	−0.387248	+	0.921976i	$0.626574\pi$
$282$		0			0
$283$	−8.67471	−	15.0250i	−0.515658	−	0.893145i	−0.999835	−	0.0181754i	$-0.994214\pi$
$283$	−8.67471	−	15.0250i	−0.515658	−	0.893145i	0.484177	−	0.874970i	$-0.339119\pi$
$284$	0.839883	+	1.45472i	0.0498379	+	0.0863218i
$285$		0			0
$286$	−2.66372	−	4.61369i	−0.157509	−	0.272813i
$287$	2.19815	−	1.19643i	0.129753	−	0.0706228i
$288$		0			0
$289$	1.52918	+	2.64861i	0.0899517	+	0.155801i
$290$	4.13307			0.242702
$291$		0			0
$292$	13.2412			0.774885
$293$	4.93560	−	8.54871i	0.288341	−	0.499421i	−0.685073	−	0.728474i	$-0.740230\pi$
$293$	4.93560	−	8.54871i	0.288341	−	0.499421i	0.973414	+	0.229054i	$0.0735631\pi$
$294$		0			0
$295$	−2.96936	−	5.14308i	−0.172883	−	0.299442i
$296$	4.55408	+	7.88791i	0.264701	+	0.458475i
$297$		0			0
$298$	4.58113	−	7.93474i	0.265378	−	0.459647i
$299$	−0.827430	+	1.43315i	−0.0478515	+	0.0828813i
$300$		0			0
$301$	−0.620621	+	24.6703i	−0.0357720	+	1.42197i
$302$	−0.0519482	+	0.0899768i	−0.00298928	+	0.00517759i
$303$		0			0
$304$	4.05408			0.232518
$305$	−2.78220	+	4.81891i	−0.159308	+	0.275930i
$306$		0			0
$307$	7.78794			0.444481			0.222240	−	0.974992i	$-0.428663\pi$
$307$	7.78794			0.444481			0.222240	+	0.974992i	$0.428663\pi$
$308$	0.242705	−	9.64776i	0.0138294	−	0.549732i
$309$		0			0
$310$	0.236971			0.0134590
$311$	7.70535	+	13.3461i	0.436930	+	0.756785i	0.997451	−	0.0713552i	$-0.0227324\pi$
$311$	7.70535	+	13.3461i	0.436930	+	0.756785i	−0.560521	+	0.828140i	$0.689399\pi$
$312$		0			0
$313$	−4.24844	+	7.35851i	−0.240136	+	0.415928i	−0.960753	−	0.277406i	$-0.910525\pi$
$313$	−4.24844	+	7.35851i	−0.240136	+	0.415928i	0.720617	+	0.693334i	$0.243859\pi$
$314$	−20.9823			−1.18410
$315$		0			0
$316$	−5.00720			−0.281677
$317$	−7.05262	+	12.2155i	−0.396115	+	0.686091i	−0.993243	−	0.116055i	$-0.962975\pi$
$317$	−7.05262	+	12.2155i	−0.396115	+	0.686091i	0.597128	+	0.802146i	$0.296308\pi$
$318$		0			0
$319$	16.3691	+	28.3520i	0.916491	+	1.58741i
$320$	0.460505			0.0257430
$321$		0			0
$322$	−2.63307	+	1.43315i	−0.146736	+	0.0798664i
$323$	−15.1373			−0.842264
$324$		0			0
$325$	−3.49640	+	6.05594i	−0.193945	+	0.335923i
$326$	−23.0364			−1.27587
$327$		0			0
$328$	−0.472958	+	0.819187i	−0.0261147	+	0.0452320i
$329$	5.40856	−	2.94381i	0.298183	−	0.162298i
$330$		0			0
$331$	−13.7719	+	23.8536i	−0.756971	+	1.31111i	0.187417	+	0.982280i	$0.439988\pi$
$331$	−13.7719	+	23.8536i	−0.756971	+	1.31111i	−0.944388	+	0.328832i	$0.893345\pi$
$332$	−3.32383	+	5.75705i	−0.182419	+	0.315959i
$333$		0			0
$334$	−5.31498	−	9.20581i	−0.290823	−	0.503720i
$335$	0.534239	+	0.925330i	0.0291886	+	0.0505562i
$336$		0			0
$337$	0.748440	−	1.29634i	0.0407701	−	0.0706159i	−0.844920	−	0.534892i	$-0.820352\pi$
$337$	0.748440	−	1.29634i	0.0407701	−	0.0706159i	0.885690	+	0.464276i	$0.153686\pi$
$338$	10.8669			0.591083
$339$		0			0
$340$	−1.71946			−0.0932506
$341$	0.938524	+	1.62557i	0.0508239	+	0.0880296i
$342$		0			0
$343$	−1.39610	+	18.4676i	−0.0753825	+	0.997155i
$344$	−4.66372	−	8.07779i	−0.251451	−	0.435525i
$345$		0			0
$346$	−1.46936	−	2.54500i	−0.0789932	−	0.136820i
$347$	−9.14406	−	15.8380i	−0.490879	−	0.850228i	0.509066	−	0.860728i	$-0.329991\pi$
$347$	−9.14406	−	15.8380i	−0.490879	−	0.850228i	−0.999945	+	0.0105001i	$0.996658\pi$
$348$		0			0
$349$	−3.90136	−	6.75735i	−0.208835	−	0.361713i	0.742513	−	0.669832i	$-0.233634\pi$
$349$	−3.90136	−	6.75735i	−0.208835	−	0.361713i	−0.951348	+	0.308119i	$0.900300\pi$
$350$	−11.1264	+	6.05594i	−0.594729	+	0.323704i
$351$		0			0
$352$	1.82383	+	3.15897i	0.0972106	+	0.168374i
$353$	−26.9253			−1.43309			−0.716544	−	0.697542i	$-0.754277\pi$
$353$	−26.9253			−1.43309			−0.716544	+	0.697542i	$0.754277\pi$
$354$		0			0
$355$	−0.773541			−0.0410553
$356$	1.36333	−	2.36135i	0.0722562	−	0.125151i
$357$		0			0
$358$	−4.58113	−	7.93474i	−0.242120	−	0.419364i
$359$	3.13161	+	5.42411i	0.165280	+	0.286274i	0.936755	−	0.349987i	$-0.113814\pi$
$359$	3.13161	+	5.42411i	0.165280	+	0.286274i	−0.771475	+	0.636260i	$0.780481\pi$
$360$		0			0
$361$	1.28220	−	2.22084i	0.0674842	−	0.116886i
$362$	11.2142	−	19.4236i	0.589405	−	1.02088i
$363$		0			0
$364$	−3.29679	−	2.01561i	−0.172799	−	0.105647i
$365$	−3.04883	+	5.28073i	−0.159583	+	0.276406i
$366$		0			0
$367$	29.2733			1.52806			0.764028	−	0.645183i	$-0.223219\pi$
$367$	29.2733			1.52806			0.764028	+	0.645183i	$0.223219\pi$
$368$	0.566537	−	0.981271i	0.0295328	−	0.0511523i
$369$		0			0
$370$	−4.19436			−0.218054
$371$	−28.8982	+	15.7290i	−1.50032	+	0.816608i
$372$		0			0
$373$	17.8597			0.924742			0.462371	−	0.886687i	$-0.346999\pi$
$373$	17.8597			0.924742			0.462371	+	0.886687i	$0.346999\pi$
$374$	−6.80992	−	11.7951i	−0.352132	−	0.609911i
$375$		0			0
$376$	−1.16372	+	2.01561i	−0.0600140	+	0.103947i
$377$	13.1082			0.675105
$378$		0			0
$379$	−22.4255			−1.15192			−0.575960	−	0.817478i	$-0.695371\pi$
$379$	−22.4255			−1.15192			−0.575960	+	0.817478i	$0.695371\pi$
$380$	−0.933463	+	1.61680i	−0.0478856	+	0.0829403i
$381$		0			0
$382$	−1.24484	−	2.15613i	−0.0636916	−	0.110317i
$383$	14.1403			0.722534			0.361267	−	0.932462i	$-0.382344\pi$
$383$	14.1403			0.722534			0.361267	+	0.932462i	$0.382344\pi$
$384$		0			0
$385$	3.79173	+	2.31821i	0.193244	+	0.118147i
$386$	−4.48968			−0.228519
$387$		0			0
$388$	−5.59358	+	9.68836i	−0.283971	+	0.491852i
$389$	23.1301			1.17275			0.586373	−	0.810041i	$-0.300555\pi$
$389$	23.1301			1.17275			0.586373	+	0.810041i	$0.300555\pi$
$390$		0			0
$391$	−2.11537	+	3.66392i	−0.106979	+	0.185292i
$392$	−3.19076	−	6.23049i	−0.161158	−	0.314688i
$393$		0			0
$394$	−6.36693	+	11.0278i	−0.320761	+	0.555574i
$395$	1.15292	−	1.99691i	0.0580097	−	0.100476i
$396$		0			0
$397$	−5.13307	−	8.89075i	−0.257622	−	0.446214i	0.707983	−	0.706230i	$-0.249605\pi$
$397$	−5.13307	−	8.89075i	−0.257622	−	0.446214i	−0.965604	+	0.260016i	$0.916272\pi$
$398$	1.47296	+	2.55124i	0.0738327	+	0.127882i
$399$		0			0
$400$	2.39397	−	4.14647i	0.119698	−	0.207324i
$401$	−34.0335			−1.69955			−0.849775	−	0.527146i	$-0.823262\pi$
$401$	−34.0335			−1.69955			−0.849775	+	0.527146i	$0.823262\pi$
$402$		0			0
$403$	0.751560			0.0374379
$404$	6.87792	+	11.9129i	0.342189	+	0.592689i
$405$		0			0
$406$	20.2594	+	12.3863i	1.00546	+	0.614724i
$407$	−16.6118	−	28.7724i	−0.823415	−	1.42620i
$408$		0			0
$409$	1.74484	+	3.02215i	0.0862769	+	0.149436i	0.905935	−	0.423418i	$-0.139170\pi$
$409$	1.74484	+	3.02215i	0.0862769	+	0.149436i	−0.819658	+	0.572854i	$0.805836\pi$
$410$	−0.217799	−	0.377240i	−0.0107563	−	0.0186305i
$411$		0			0
$412$	−5.58113	−	9.66679i	−0.274962	−	0.476249i
$413$	0.858071	−	34.1091i	0.0422229	−	1.67840i
$414$		0			0
$415$	−1.53064	−	2.65115i	−0.0751362	−	0.130140i
$416$	1.46050			0.0716071
$417$		0			0
$418$	−14.7879			−0.723302
$419$	−14.4897	+	25.0969i	−0.707867	+	1.22606i	0.257779	+	0.966204i	$0.417009\pi$
$419$	−14.4897	+	25.0969i	−0.707867	+	1.22606i	−0.965647	+	0.259858i	$0.916324\pi$
$420$		0			0
$421$	−1.06128	−	1.83819i	−0.0517237	−	0.0895881i	0.839004	−	0.544125i	$-0.183138\pi$
$421$	−1.06128	−	1.83819i	−0.0517237	−	0.0895881i	−0.890728	+	0.454537i	$0.849805\pi$
$422$	0.608168	+	1.05338i	0.0296052	+	0.0512777i
$423$		0			0
$424$	6.21780	−	10.7695i	0.301963	−	0.523015i
$425$	−8.93872	+	15.4823i	−0.433592	+	0.751003i
$426$		0			0
$427$	−28.0795	+	15.2833i	−1.35886	+	0.739612i
$428$	3.89037	−	6.73832i	0.188048	−	0.325709i
$429$		0			0
$430$	4.29533			0.207139
$431$	−10.9356	+	18.9410i	−0.526749	+	0.912356i	0.472765	+	0.881189i	$0.343256\pi$
$431$	−10.9356	+	18.9410i	−0.526749	+	0.912356i	−0.999514	+	0.0311679i	$0.990077\pi$
$432$		0			0
$433$	−13.0512			−0.627199			−0.313599	−	0.949555i	$-0.601535\pi$
$433$	−13.0512			−0.627199			−0.313599	+	0.949555i	$0.601535\pi$
$434$	1.16158	+	0.710174i	0.0557576	+	0.0340895i
$435$		0			0
$436$	7.51459			0.359884
$437$	2.29679	+	3.97816i	0.109870	+	0.190301i
$438$		0			0
$439$	−2.43200	+	4.21235i	−0.116073	+	0.201044i	−0.918208	−	0.396098i	$-0.870364\pi$
$439$	−2.43200	+	4.21235i	−0.116073	+	0.201044i	0.802135	+	0.597143i	$0.203697\pi$
$440$	−1.67977			−0.0800797
$441$		0			0
$442$	−5.45331			−0.259387
$443$	−5.76975	+	9.99350i	−0.274129	+	0.474805i	−0.969915	−	0.243444i	$-0.921723\pi$
$443$	−5.76975	+	9.99350i	−0.274129	+	0.474805i	0.695786	+	0.718249i	$0.255056\pi$
$444$		0			0
$445$	0.627819	+	1.08741i	0.0297615	+	0.0515484i
$446$	−0.891832			−0.0422294
$447$		0			0
$448$	2.25729	+	1.38008i	0.106647	+	0.0652027i
$449$	26.4251			1.24708			0.623538	−	0.781793i	$-0.285694\pi$
$449$	26.4251			1.24708			0.623538	+	0.781793i	$0.285694\pi$
$450$		0			0
$451$	1.72519	−	2.98812i	0.0812361	−	0.140705i
$452$	6.06128			0.285099
$453$		0			0
$454$	7.32597	−	12.6889i	0.343825	−	0.595522i
$455$	1.56294	−	0.850689i	0.0732717	−	0.0398809i
$456$		0			0
$457$	1.86906	−	3.23731i	0.0874310	−	0.151435i	−0.818994	−	0.573803i	$-0.805468\pi$
$457$	1.86906	−	3.23731i	0.0874310	−	0.151435i	0.906425	+	0.422368i	$0.138801\pi$
$458$	−4.78794	+	8.29295i	−0.223726	+	0.387504i
$459$		0			0
$460$	0.260893	+	0.451880i	0.0121642	+	0.0210690i
$461$	7.90496	+	13.6918i	0.368171	+	0.637690i	0.989280	−	0.146034i	$-0.0466509\pi$
$461$	7.90496	+	13.6918i	0.368171	+	0.637690i	−0.621109	+	0.783724i	$0.713318\pi$
$462$		0			0
$463$	19.1965	−	33.2493i	0.892137	−	1.54523i	0.0548278	−	0.998496i	$-0.482539\pi$
$463$	19.1965	−	33.2493i	0.892137	−	1.54523i	0.837309	−	0.546730i	$-0.184128\pi$
$464$	−8.97509			−0.416658
$465$		0			0
$466$	−14.4284			−0.668383
$467$	−3.15652	−	5.46725i	−0.146066	−	0.252994i	0.783704	−	0.621134i	$-0.213328\pi$
$467$	−3.15652	−	5.46725i	−0.146066	−	0.252994i	−0.929770	+	0.368140i	$0.879995\pi$
$468$		0			0
$469$	−0.154382	+	6.13682i	−0.00712869	+	0.283372i
$470$	−0.535897	−	0.928200i	−0.0247191	−	0.0428147i
$471$		0			0
$472$	6.44805	+	11.1684i	0.296796	+	0.514065i
$473$	17.0117	+	29.4651i	0.782197	+	1.35481i
$474$		0			0
$475$	9.70535	+	16.8102i	0.445312	+	0.771303i
$476$	−8.42840	−	5.15301i	−0.386315	−	0.236188i
$477$		0			0
$478$	−9.15486	−	15.8567i	−0.418734	−	0.725268i
$479$	20.4136			0.932722			0.466361	−	0.884594i	$-0.345565\pi$
$479$	20.4136			0.932722			0.466361	+	0.884594i	$0.345565\pi$
$480$		0			0
$481$	−13.3025			−0.606543
$482$	0.0466924	−	0.0808735i	0.00212678	−	0.00368369i
$483$		0			0
$484$	−1.15272	−	1.99658i	−0.0523966	−	0.0907535i
$485$	−2.57587	−	4.46154i	−0.116964	−	0.202588i
$486$		0			0
$487$	6.18190	−	10.7074i	0.280129	−	0.485197i	−0.691287	−	0.722580i	$-0.742956\pi$
$487$	6.18190	−	10.7074i	0.280129	−	0.485197i	0.971416	+	0.237383i	$0.0762895\pi$
$488$	6.04163	−	10.4644i	0.273492	−	0.473702i
$489$		0			0
$490$	3.21946	+	0.162084i	0.145440	+	0.00732222i
$491$	−0.207004	+	0.358541i	−0.00934194	+	0.0161807i	−0.870659	−	0.491888i	$-0.836307\pi$
$491$	−0.207004	+	0.358541i	−0.00934194	+	0.0161807i	0.861317	+	0.508069i	$0.169640\pi$
$492$		0			0
$493$	33.5117			1.50929
$494$	−2.96050	+	5.12774i	−0.133199	+	0.230708i
$495$		0			0
$496$	−0.514589			−0.0231057
$497$	−3.79173	−	2.31821i	−0.170082	−	0.103986i
$498$		0			0
$499$	−0.923935			−0.0413610			−0.0206805	−	0.999786i	$-0.506583\pi$
$499$	−0.923935			−0.0413610			−0.0206805	+	0.999786i	$0.506583\pi$
$500$	2.25370	+	3.90352i	0.100788	+	0.174571i
$501$		0			0
$502$	9.13667	−	15.8252i	0.407790	−	0.706312i
$503$	23.8142			1.06182			0.530911	−	0.847428i	$-0.321850\pi$
$503$	23.8142			1.06182			0.530911	+	0.847428i	$0.321850\pi$
$504$		0			0
$505$	−6.33463			−0.281887
$506$	−2.06654	+	3.57935i	−0.0918688	+	0.159121i
$507$		0			0
$508$	−4.40496	−	7.62961i	−0.195438	−	0.338509i
$509$	30.6342			1.35784			0.678919	−	0.734213i	$-0.262449\pi$
$509$	30.6342			1.35784			0.678919	+	0.734213i	$0.262449\pi$
$510$		0			0
$511$	−30.7704	+	16.7480i	−1.36120	+	0.740887i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	10.5256	−	18.2308i	0.464263	−	0.804128i
$515$	5.14027			0.226507
$516$		0			0
$517$	4.24484	−	7.35228i	0.186688	−	0.323353i
$518$	−20.5598	−	12.5700i	−0.903347	−	0.552294i
$519$		0			0
$520$	−0.336285	+	0.582462i	−0.0147471	+	0.0255427i
$521$	13.4518	−	23.2993i	0.589336	−	1.02076i	−0.404984	−	0.914324i	$-0.632723\pi$
$521$	13.4518	−	23.2993i	0.589336	−	1.02076i	0.994320	−	0.106436i	$-0.0339439\pi$
$522$		0			0
$523$	−7.85301	−	13.6018i	−0.343388	−	0.594766i	0.641671	−	0.766980i	$-0.278241\pi$
$523$	−7.85301	−	13.6018i	−0.343388	−	0.594766i	−0.985060	+	0.172214i	$0.944908\pi$
$524$	10.5687	+	18.3055i	0.461695	+	0.799679i
$525$		0			0
$526$	2.58259	−	4.47318i	0.112606	−	0.195040i
$527$	1.92140			0.0836975
$528$		0			0
$529$	−21.7161			−0.944180
$530$	2.86333	+	4.95943i	0.124375	+	0.215424i
$531$		0			0
$532$	−9.42101	+	5.12774i	−0.408453	+	0.222316i
$533$	−0.690757	−	1.19643i	−0.0299200	−	0.0518230i
$534$		0			0
$535$	1.79153	+	3.10303i	0.0774548	+	0.134156i
$536$	−1.16012	−	2.00938i	−0.0501094	−	0.0867920i
$537$		0			0
$538$	8.42840	+	14.5984i	0.363374	+	0.629383i
$539$	11.6388	+	22.7267i	0.501319	+	0.978910i
$540$		0			0
$541$	−2.05934	−	3.56688i	−0.0885379	−	0.153352i	0.818355	−	0.574713i	$-0.194886\pi$
$541$	−2.05934	−	3.56688i	−0.0885379	−	0.153352i	−0.906893	+	0.421360i	$0.861553\pi$
$542$	25.1124			1.07867
$543$		0			0
$544$	3.73385			0.160088
$545$	−1.73025	+	2.99689i	−0.0741159	+	0.128372i
$546$		0			0
$547$	−11.8602	−	20.5425i	−0.507106	−	0.878333i	−0.999966	−	0.00822465i	$-0.997382\pi$
$547$	−11.8602	−	20.5425i	−0.507106	−	0.878333i	0.492860	−	0.870108i	$-0.335951\pi$
$548$	−2.20321	−	3.81607i	−0.0941165	−	0.163015i
$549$		0			0
$550$	−8.73239	+	15.1249i	−0.372350	+	0.644930i
$551$	18.1929	−	31.5110i	0.775043	−	1.34241i
$552$		0			0
$553$	11.6359	−	6.33327i	0.494808	−	0.269318i
$554$	1.69076	−	2.92848i	0.0718334	−	0.124419i
$555$		0			0
$556$	2.02491			0.0858751
$557$	21.0313	−	36.4273i	0.891125	−	1.54347i	0.0525975	−	0.998616i	$-0.483250\pi$
$557$	21.0313	−	36.4273i	0.891125	−	1.54347i	0.838528	−	0.544859i	$-0.183417\pi$
$558$		0			0
$559$	13.6228			0.576181
$560$	−1.07014	+	0.582462i	−0.0452215	+	0.0246135i
$561$		0			0
$562$	20.2776			0.855360
$563$	5.91216	+	10.2402i	0.249168	+	0.431571i	0.963295	−	0.268445i	$-0.0865097\pi$
$563$	5.91216	+	10.2402i	0.249168	+	0.431571i	−0.714127	+	0.700016i	$0.753176\pi$
$564$		0			0
$565$	−1.39562	+	2.41729i	−0.0587144	+	0.101696i
$566$	−17.3494			−0.729250
$567$		0			0
$568$	1.67977			0.0704815
$569$	7.10078	−	12.2989i	0.297680	−	0.515597i	−0.677925	−	0.735131i	$-0.737120\pi$
$569$	7.10078	−	12.2989i	0.297680	−	0.515597i	0.975605	+	0.219534i	$0.0704538\pi$
$570$		0			0
$571$	−5.97869	−	10.3554i	−0.250200	−	0.433360i	0.713380	−	0.700777i	$-0.247163\pi$
$571$	−5.97869	−	10.3554i	−0.250200	−	0.433360i	−0.963581	+	0.267417i	$0.913830\pi$
$572$	−5.32743			−0.222751
$573$		0			0
$574$	0.0629386	−	2.50187i	0.00262701	−	0.104426i
$575$	5.42509			0.226242
$576$		0			0
$577$	21.3135	−	36.9161i	0.887293	−	1.53684i	0.0442307	−	0.999021i	$-0.485916\pi$
$577$	21.3135	−	36.9161i	0.887293	−	1.53684i	0.843062	−	0.537816i	$-0.180750\pi$
$578$	3.05836			0.127211
$579$		0			0
$580$	2.06654	−	3.57935i	0.0858083	−	0.148624i
$581$	0.442317	−	17.5825i	0.0183504	−	0.729445i
$582$		0			0
$583$	−22.6804	+	39.2837i	−0.939328	+	1.62696i
$584$	6.62062	−	11.4673i	0.273963	−	0.474518i
$585$		0			0
$586$	−4.93560	−	8.54871i	−0.203888	−	0.353144i
$587$	−20.5328	−	35.5638i	−0.847478	−	1.46788i	−0.883451	−	0.468523i	$-0.844786\pi$
$587$	−20.5328	−	35.5638i	−0.847478	−	1.46788i	0.0359730	−	0.999353i	$-0.488547\pi$
$588$		0			0
$589$	1.04309	−	1.80669i	0.0429799	−	0.0744434i
$590$	−5.93872			−0.244493
$591$		0			0
$592$	9.10817			0.374343
$593$	−16.1008	−	27.8874i	−0.661180	−	1.14520i	−0.980306	−	0.197485i	$-0.936723\pi$
$593$	−16.1008	−	27.8874i	−0.661180	−	1.14520i	0.319126	−	0.947712i	$-0.396611\pi$
$594$		0			0
$595$	3.99573	−	2.17483i	0.163809	−	0.0891592i
$596$	−4.58113	−	7.93474i	−0.187650	−	0.325020i
$597$		0			0
$598$	0.827430	+	1.43315i	0.0338361	+	0.0586059i
$599$	9.53590	+	16.5167i	0.389626	+	0.674852i	0.992399	−	0.123060i	$-0.0392709\pi$
$599$	9.53590	+	16.5167i	0.389626	+	0.674852i	−0.602773	+	0.797913i	$0.705938\pi$
$600$		0			0
$601$	4.27188	+	7.39912i	0.174254	+	0.301816i	0.939903	−	0.341442i	$-0.110915\pi$
$601$	4.27188	+	7.39912i	0.174254	+	0.301816i	−0.765649	+	0.643259i	$0.777582\pi$
$602$	21.0548	+	12.8726i	0.858128	+	0.524648i
$603$		0			0
$604$	0.0519482	+	0.0899768i	0.00211374	+	0.00366111i
$605$	1.06167			0.0431631
$606$		0			0
$607$	38.0115			1.54284			0.771419	−	0.636328i	$-0.219547\pi$
$607$	38.0115			1.54284			0.771419	+	0.636328i	$0.219547\pi$
$608$	2.02704	−	3.51094i	0.0822074	−	0.142387i
$609$		0			0
$610$	2.78220	+	4.81891i	0.112648	+	0.195112i
$611$	−1.69961	−	2.94381i	−0.0687589	−	0.119094i
$612$		0			0
$613$	11.3296	−	19.6234i	0.457597	−	0.792581i	−0.541237	−	0.840870i	$-0.682044\pi$
$613$	11.3296	−	19.6234i	0.457597	−	0.792581i	0.998833	+	0.0482894i	$0.0153770\pi$
$614$	3.89397	−	6.74455i	0.157148	−	0.272188i
$615$		0			0
$616$	−8.23385	−	5.03407i	−0.331751	−	0.202828i
$617$	10.1388	−	17.5609i	0.408173	−	0.706977i	−0.586512	−	0.809941i	$-0.699499\pi$
$617$	10.1388	−	17.5609i	0.408173	−	0.706977i	0.994685	+	0.102964i	$0.0328327\pi$
$618$		0			0
$619$	2.06128			0.0828499			0.0414249	−	0.999142i	$-0.486810\pi$
$619$	2.06128			0.0828499			0.0414249	+	0.999142i	$0.486810\pi$
$620$	0.118485	−	0.205223i	0.00475849	−	0.00824194i
$621$		0			0
$622$	15.4107			0.617912
$623$	−0.181424	+	7.21177i	−0.00726860	+	0.288933i
$624$		0			0
$625$	21.8640			0.874560
$626$	4.24844	+	7.35851i	0.169802	+	0.294105i
$627$		0			0
$628$	−10.4911	+	18.1712i	−0.418642	+	0.725110i
$629$	−34.0085			−1.35601
$630$		0			0
$631$	1.63715			0.0651740			0.0325870	−	0.999469i	$-0.489625\pi$
$631$	1.63715			0.0651740			0.0325870	+	0.999469i	$0.489625\pi$
$632$	−2.50360	+	4.33636i	−0.0995878	+	0.172491i
$633$		0			0
$634$	7.05262	+	12.2155i	0.280095	+	0.485139i
$635$	4.05701			0.160998
$636$		0			0
$637$	10.2106	+	0.514055i	0.404559	+	0.0203676i
$638$	32.7381			1.29611
$639$		0			0
$640$	0.230252	−	0.398809i	0.00910153	−	0.0157643i
$641$	−21.9325			−0.866281			−0.433140	−	0.901326i	$-0.642595\pi$
$641$	−21.9325			−0.866281			−0.433140	+	0.901326i	$0.642595\pi$
$642$		0			0
$643$	−14.1819	+	24.5638i	−0.559280	+	0.968701i	0.438277	+	0.898840i	$0.355589\pi$
$643$	−14.1819	+	24.5638i	−0.559280	+	0.968701i	−0.997557	+	0.0698609i	$0.977744\pi$
$644$	−0.0753916	+	2.99689i	−0.00297085	+	0.118094i
$645$		0			0
$646$	−7.56867	+	13.1093i	−0.297785	+	0.515780i
$647$	−17.3904	+	30.1210i	−0.683686	+	1.18418i	0.290162	+	0.956978i	$0.406291\pi$
$647$	−17.3904	+	30.1210i	−0.683686	+	1.18418i	−0.973848	+	0.227201i	$0.927042\pi$
$648$		0			0
$649$	−23.5203	−	40.7384i	−0.923253	−	1.59912i
$650$	3.49640	+	6.05594i	0.137140	+	0.237534i
$651$		0			0
$652$	−11.5182	+	19.9501i	−0.451087	+	0.781306i
$653$	3.19863			0.125172			0.0625860	−	0.998040i	$-0.480065\pi$
$653$	3.19863			0.125172			0.0625860	+	0.998040i	$0.480065\pi$
$654$		0			0
$655$	−9.73385			−0.380333
$656$	0.472958	+	0.819187i	0.0184659	+	0.0319839i
$657$		0			0
$658$	0.154861	−	6.15585i	0.00603710	−	0.239980i
$659$	−5.30418	−	9.18711i	−0.206622	−	0.357879i	0.744027	−	0.668150i	$-0.232914\pi$
$659$	−5.30418	−	9.18711i	−0.206622	−	0.357879i	−0.950648	+	0.310271i	$0.899580\pi$
$660$		0			0
$661$	−5.06507	−	8.77297i	−0.197009	−	0.341229i	0.750549	−	0.660815i	$-0.229789\pi$
$661$	−5.06507	−	8.77297i	−0.197009	−	0.341229i	−0.947557	+	0.319586i	$0.896456\pi$
$662$	13.7719	+	23.8536i	0.535259	+	0.927097i
$663$		0			0
$664$	3.32383	+	5.75705i	0.128990	+	0.223417i
$665$	0.124220	−	4.93786i	0.00481705	−	0.191482i
$666$		0			0
$667$	−5.08472	−	8.80700i	−0.196881	−	0.341008i
$668$	−10.6300			−0.411285
$669$		0			0
$670$	1.06848			0.0412789
$671$	−22.0378	+	38.1707i	−0.850761	+	1.47356i
$672$		0			0
$673$	1.60817	+	2.78543i	0.0619903	+	0.107370i	0.895355	−	0.445353i	$-0.146922\pi$
$673$	1.60817	+	2.78543i	0.0619903	+	0.107370i	−0.833365	+	0.552724i	$0.813589\pi$
$674$	−0.748440	−	1.29634i	−0.0288288	−	0.0499330i
$675$		0			0
$676$	5.43346	−	9.41103i	0.208979	−	0.361963i
$677$	−14.6819	+	25.4298i	−0.564271	+	0.977347i	0.432846	+	0.901468i	$0.357509\pi$
$677$	−14.6819	+	25.4298i	−0.564271	+	0.977347i	−0.997117	+	0.0758786i	$0.975824\pi$
$678$		0			0
$679$	0.744363	−	29.5891i	0.0285660	−	1.13552i
$680$	−0.859728	+	1.48909i	−0.0329691	+	0.0571041i
$681$		0			0
$682$	1.87705			0.0718759
$683$	−12.6278	+	21.8720i	−0.483190	+	0.836910i	−0.999814	−	0.0193029i	$-0.993855\pi$
$683$	−12.6278	+	21.8720i	−0.483190	+	0.836910i	0.516624	+	0.856213i	$0.327189\pi$
$684$		0			0
$685$	2.02918			0.0775309
$686$	15.2953	+	10.4428i	0.583978	+	0.398710i
$687$		0			0
$688$	−9.32743			−0.355605
$689$	9.08113	+	15.7290i	0.345963	+	0.599226i
$690$		0			0
$691$	7.68190	−	13.3054i	0.292233	−	0.506163i	−0.682104	−	0.731255i	$-0.738935\pi$
$691$	7.68190	−	13.3054i	0.292233	−	0.506163i	0.974338	+	0.225092i	$0.0722683\pi$
$692$	−2.93872			−0.111713
$693$		0			0
$694$	−18.2881			−0.694208
$695$	−0.466240	+	0.807551i	−0.0176855	+	0.0306321i
$696$		0			0
$697$	−1.76595	−	3.05872i	−0.0668903	−	0.115857i
$698$	−7.80272			−0.295337
$699$		0			0
$700$	−0.318576	+	12.6637i	−0.0120410	+	0.478642i
$701$	13.3700			0.504980			0.252490	−	0.967600i	$-0.418751\pi$
$701$	13.3700			0.504980			0.252490	+	0.967600i	$0.418751\pi$
$702$		0			0
$703$	−18.4626	+	31.9782i	−0.696332	+	1.20608i
$704$	3.64766			0.137476
$705$		0			0
$706$	−13.4626	+	23.3180i	−0.506673	+	0.877584i
$707$	−31.0510	−	18.9842i	−1.16779	−	0.713972i
$708$		0			0
$709$	0.562939	−	0.975038i	0.0211416	−	0.0366183i	−0.855261	−	0.518197i	$-0.826603\pi$
$709$	0.562939	−	0.975038i	0.0211416	−	0.0366183i	0.876403	+	0.481579i	$0.159937\pi$
$710$	−0.386770	+	0.669906i	−0.0145152	+	0.0251411i
$711$		0			0
$712$	−1.36333	−	2.36135i	−0.0510928	−	0.0884954i
$713$	−0.291534	−	0.504951i	−0.0109180	−	0.0189106i
$714$		0			0
$715$	1.22665	−	2.12463i	0.0458743	−	0.0794565i
$716$	−9.16225			−0.342409
$717$		0			0
$718$	6.26322			0.233741
$719$	−9.13667	−	15.8252i	−0.340740	−	0.590180i	0.643830	−	0.765169i	$-0.277344\pi$
$719$	−9.13667	−	15.8252i	−0.340740	−	0.590180i	−0.984570	+	0.174989i	$0.944011\pi$
$720$		0			0
$721$	25.1965	+	15.4048i	0.938366	+	0.573705i
$722$	−1.28220	−	2.22084i	−0.0477186	−	0.0826510i
$723$		0			0
$724$	−11.2142	−	19.4236i	−0.416772	−	0.721871i
$725$	−21.4861	−	37.2150i	−0.797973	−	1.38213i
$726$		0			0
$727$	−14.8478	−	25.7171i	−0.550673	−	0.953793i	−0.998226	−	0.0595359i	$-0.981038\pi$
$727$	−14.8478	−	25.7171i	−0.550673	−	0.953793i	0.447553	−	0.894257i	$-0.352295\pi$
$728$	−3.39397	+	1.84730i	−0.125789	+	0.0684654i
$729$		0			0
$730$	3.04883	+	5.28073i	0.112842	+	0.195448i
$731$	34.8272			1.28813
$732$		0			0
$733$	19.2278			0.710195			0.355098	−	0.934829i	$-0.384448\pi$
$733$	19.2278			0.710195			0.355098	+	0.934829i	$0.384448\pi$
$734$	14.6367	−	25.3515i	0.540249	−	0.935740i
$735$		0			0
$736$	−0.566537	−	0.981271i	−0.0208828	−	0.0361701i
$737$	4.23171	+	7.32955i	0.155877	+	0.269987i
$738$		0			0
$739$	−15.1336	+	26.2121i	−0.556697	+	0.964227i	0.441073	+	0.897471i	$0.354598\pi$
$739$	−15.1336	+	26.2121i	−0.556697	+	0.964227i	−0.997769	+	0.0667556i	$0.978735\pi$
$740$	−2.09718	+	3.63242i	−0.0770938	+	0.133530i
$741$		0			0
$742$	−0.827430	+	32.8911i	−0.0303759	+	1.20747i
$743$	11.8815	−	20.5794i	0.435890	−	0.754984i	−0.561477	−	0.827492i	$-0.689767\pi$
$743$	11.8815	−	20.5794i	0.435890	−	0.754984i	0.997368	+	0.0725076i	$0.0231002\pi$
$744$		0			0
$745$	4.21926			0.154582
$746$	8.92986	−	15.4670i	0.326946	−	0.566286i
$747$		0			0
$748$	−13.6198			−0.497990
$749$	−0.517709	+	20.5794i	−0.0189167	+	0.751954i
$750$		0			0
$751$	12.6683			0.462273			0.231136	−	0.972921i	$-0.425756\pi$
$751$	12.6683			0.462273			0.231136	+	0.972921i	$0.425756\pi$
$752$	1.16372	+	2.01561i	0.0424363	+	0.0735019i
$753$		0			0
$754$	6.55408	−	11.3520i	0.238686	−	0.413416i
$755$	−0.0478448			−0.00174125
$756$		0			0
$757$	−29.0799			−1.05693			−0.528464	−	0.848955i	$-0.677232\pi$
$757$	−29.0799			−1.05693			−0.528464	+	0.848955i	$0.677232\pi$
$758$	−11.2127	+	19.4210i	−0.407265	+	0.705404i
$759$		0			0
$760$	0.933463	+	1.61680i	0.0338603	+	0.0586477i
$761$	−29.2029			−1.05860			−0.529302	−	0.848433i	$-0.677546\pi$
$761$	−29.2029			−1.05860			−0.529302	+	0.848433i	$0.677546\pi$
$762$		0			0
$763$	−17.4626	+	9.50471i	−0.632190	+	0.344094i
$764$	−2.48968			−0.0900736
$765$		0			0
$766$	7.07014	−	12.2458i	0.255454	−	0.442460i
$767$	−18.8348			−0.680086
$768$		0			0
$769$	12.5869	−	21.8011i	0.453894	−	0.786167i	−0.544730	−	0.838611i	$-0.683368\pi$
$769$	12.5869	−	21.8011i	0.453894	−	0.786167i	0.998624	+	0.0524443i	$0.0167012\pi$
$770$	3.90350	−	2.12463i	0.140672	−	0.0765662i
$771$		0			0
$772$	−2.24484	+	3.88818i	−0.0807936	+	0.139939i
$773$	0.752039	−	1.30257i	0.0270490	−	0.0468502i	−0.852184	−	0.523242i	$-0.824722\pi$
$773$	0.752039	−	1.30257i	0.0270490	−	0.0468502i	0.879233	+	0.476392i	$0.158056\pi$
$774$		0			0
$775$	−1.23191	−	2.13373i	−0.0442515	−	0.0766458i
$776$	5.59358	+	9.68836i	0.200798	+	0.347792i
$777$		0			0
$778$	11.5651	−	20.0313i	0.414628	−	0.718157i
$779$	−3.83482			−0.137397
$780$		0			0
$781$	−6.12722			−0.219249
$782$	2.11537	+	3.66392i	0.0756453	+	0.131022i
$783$		0			0
$784$	−6.99115

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.h (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +0.460505 q^{5} +(2.25729 + 1.38008i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +0.460505 q^{5} +(2.25729 + 1.38008i) q^{7} -1.00000 q^{8} +(0.230252 - 0.398809i) q^{10} +3.64766 q^{11} +(0.730252 - 1.26483i) q^{13} +(2.32383 - 1.26483i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.86693 - 3.23361i) q^{17} +(-2.02704 - 3.51094i) q^{19} +(-0.230252 - 0.398809i) q^{20} +(1.82383 - 3.15897i) q^{22} -1.13307 q^{23} -4.78794 q^{25} +(-0.730252 - 1.26483i) q^{26} +(0.0665372 - 2.64491i) q^{28} +(4.48755 + 7.77266i) q^{29} +(0.257295 + 0.445647i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.86693 - 3.23361i) q^{34} +(1.03950 + 0.635534i) q^{35} +(-4.55408 - 7.88791i) q^{37} -4.05408 q^{38} -0.460505 q^{40} +(0.472958 - 0.819187i) q^{41} +(4.66372 + 8.07779i) q^{43} +(-1.82383 - 3.15897i) q^{44} +(-0.566537 + 0.981271i) q^{46} +(1.16372 - 2.01561i) q^{47} +(3.19076 + 6.23049i) q^{49} +(-2.39397 + 4.14647i) q^{50} -1.46050 q^{52} +(-6.21780 + 10.7695i) q^{53} +1.67977 q^{55} +(-2.25729 - 1.38008i) q^{56} +8.97509 q^{58} +(-6.44805 - 11.1684i) q^{59} +(-6.04163 + 10.4644i) q^{61} +0.514589 q^{62} +1.00000 q^{64} +(0.336285 - 0.582462i) q^{65} +(1.16012 + 2.00938i) q^{67} -3.73385 q^{68} +(1.07014 - 0.582462i) q^{70} -1.67977 q^{71} +(-6.62062 + 11.4673i) q^{73} -9.10817 q^{74} +(-2.02704 + 3.51094i) q^{76} +(8.23385 + 5.03407i) q^{77} +(2.50360 - 4.33636i) q^{79} +(-0.230252 + 0.398809i) q^{80} +(-0.472958 - 0.819187i) q^{82} +(-3.32383 - 5.75705i) q^{83} +(0.859728 - 1.48909i) q^{85} +9.32743 q^{86} -3.64766 q^{88} +(1.36333 + 2.36135i) q^{89} +(3.39397 - 1.84730i) q^{91} +(0.566537 + 0.981271i) q^{92} +(-1.16372 - 2.01561i) q^{94} +(-0.933463 - 1.61680i) q^{95} +(-5.59358 - 9.68836i) q^{97} +(6.99115 + 0.351971i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8	\( 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8} - 5 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} - q^{22} - 14 q^{23} + 4 q^{25} + 2 q^{26} + 4 q^{28} + 5 q^{29} - 14 q^{31} + 3 q^{32} - 4 q^{34} + 19 q^{35} - 9 q^{37} - 6 q^{38} + 10 q^{40} + 12 q^{41} + 18 q^{43} + q^{44} - 7 q^{46} - 3 q^{47} + 2 q^{50} + 4 q^{52} - 9 q^{53} + 14 q^{55} + 2 q^{56} + 10 q^{58} - 4 q^{59} + 4 q^{61} - 28 q^{62} + 6 q^{64} + 12 q^{65} + 5 q^{67} - 8 q^{68} + 2 q^{70} - 14 q^{71} - 25 q^{73} - 18 q^{74} - 3 q^{76} + 35 q^{77} + 7 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} + 14 q^{85} + 36 q^{86} + 2 q^{88} + 9 q^{89} + 4 q^{91} + 7 q^{92} + 3 q^{94} - 2 q^{95} - 28 q^{97} + 12 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8 - 5 * q^10 - 2 * q^11 - 2 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^19 + 5 * q^20 - q^22 - 14 * q^23 + 4 * q^25 + 2 * q^26 + 4 * q^28 + 5 * q^29 - 14 * q^31 + 3 * q^32 - 4 * q^34 + 19 * q^35 - 9 * q^37 - 6 * q^38 + 10 * q^40 + 12 * q^41 + 18 * q^43 + q^44 - 7 * q^46 - 3 * q^47 + 2 * q^50 + 4 * q^52 - 9 * q^53 + 14 * q^55 + 2 * q^56 + 10 * q^58 - 4 * q^59 + 4 * q^61 - 28 * q^62 + 6 * q^64 + 12 * q^65 + 5 * q^67 - 8 * q^68 + 2 * q^70 - 14 * q^71 - 25 * q^73 - 18 * q^74 - 3 * q^76 + 35 * q^77 + 7 * q^79 + 5 * q^80 - 12 * q^82 - 8 * q^83 + 14 * q^85 + 36 * q^86 + 2 * q^88 + 9 * q^89 + 4 * q^91 + 7 * q^92 + 3 * q^94 - 2 * q^95 - 28 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more