LMFDB - Embedded newform 855.2.k.h.676.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(406,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("855.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.k (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.4601315889.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 $ x^8 - x^7 + 6x^6 - 3x^5 + 26x^4 - 14x^3 + 31x^2 + 12x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			676.1
Root			$1.07988 - 1.87040i$ of defining polynomial
Character	$\chi$	$=$	855.676
Dual form			855.2.k.h.406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.832272 + 1.44154i) q^{2} +(-0.385355 - 0.667454i) q^{4} +(0.500000 - 0.866025i) q^{5} -2.43525 q^{7} -2.04621 q^{8} +O(q^{10})$	\(q+(-0.832272 + 1.44154i) q^{2} +(-0.385355 - 0.667454i) q^{4} +(0.500000 - 0.866025i) q^{5} -2.43525 q^{7} -2.04621 q^{8} +(0.832272 + 1.44154i) q^{10} +5.75477 q^{11} +(0.797505 + 1.38132i) q^{13} +(2.02680 - 3.51051i) q^{14} +(2.47371 - 4.28460i) q^{16} +(-2.99203 + 5.18234i) q^{17} +(0.149412 + 4.35634i) q^{19} -0.770710 q^{20} +(-4.78953 + 8.29572i) q^{22} +(-0.470022 - 0.814102i) q^{23} +(-0.500000 - 0.866025i) q^{25} -2.65497 q^{26} +(0.938437 + 1.62542i) q^{28} +(1.30917 + 2.26755i) q^{29} -5.26913 q^{31} +(2.07140 + 3.58777i) q^{32} +(-4.98037 - 8.62625i) q^{34} +(-1.21763 + 2.10899i) q^{35} -2.89384 q^{37} +(-6.40418 - 3.41028i) q^{38} +(-1.02310 + 1.77207i) q^{40} +(-3.15767 + 5.46925i) q^{41} +(-2.26961 + 3.93108i) q^{43} +(-2.21763 - 3.84104i) q^{44} +1.56475 q^{46} +(4.47718 + 7.75471i) q^{47} -1.06953 q^{49} +1.66454 q^{50} +(0.614645 - 1.06460i) q^{52} +(-1.09819 - 1.90213i) q^{53} +(2.87738 - 4.98377i) q^{55} +4.98304 q^{56} -4.35834 q^{58} +(-5.39939 + 9.35202i) q^{59} +(5.26434 + 9.11811i) q^{61} +(4.38535 - 7.59566i) q^{62} +2.99898 q^{64} +1.59501 q^{65} +(-0.504789 - 0.874320i) q^{67} +4.61197 q^{68} +(-2.02680 - 3.51051i) q^{70} +(4.41694 - 7.65036i) q^{71} +(5.12499 - 8.87674i) q^{73} +(2.40846 - 4.17157i) q^{74} +(2.85008 - 1.77846i) q^{76} -14.0143 q^{77} +(-3.80229 + 6.58577i) q^{79} +(-2.47371 - 4.28460i) q^{80} +(-5.25609 - 9.10381i) q^{82} -3.11355 q^{83} +(2.99203 + 5.18234i) q^{85} +(-3.77787 - 6.54346i) q^{86} -11.7755 q^{88} +(-5.55706 - 9.62511i) q^{89} +(-1.94213 - 3.36387i) q^{91} +(-0.362251 + 0.627436i) q^{92} -14.9049 q^{94} +(3.84741 + 2.04877i) q^{95} +(-2.02888 + 3.51412i) q^{97} +(0.890144 - 1.54177i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8}+O(q^{10}) $ 8 * q + q^2 - 5 * q^4 + 4 * q^5 - 8 * q^7 - 24 * q^8	$ 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8} - q^{10} + 4 q^{11} - 7 q^{13} - q^{14} - 7 q^{16} - q^{17} + 5 q^{19} - 10 q^{20} - 2 q^{22} + 2 q^{23} - 4 q^{25} - 6 q^{26} + 19 q^{28} - q^{29} + 30 q^{32} - 15 q^{34} - 4 q^{35} - 4 q^{37} - 13 q^{38} - 12 q^{40} - 8 q^{41} - q^{43} - 12 q^{44} + 24 q^{46} - 12 q^{47} - 20 q^{49} - 2 q^{50} + 3 q^{52} - 5 q^{53} + 2 q^{55} + 82 q^{56} - 54 q^{58} - 5 q^{59} + 37 q^{62} + 112 q^{64} - 14 q^{65} - 4 q^{67} - 32 q^{68} + q^{70} + 20 q^{71} + 20 q^{73} + 25 q^{74} + 63 q^{76} - 28 q^{77} - 17 q^{79} + 7 q^{80} - 21 q^{82} - 2 q^{83} + q^{85} + 8 q^{86} - 14 q^{88} + 11 q^{89} - 6 q^{91} - q^{92} - 62 q^{94} + 4 q^{95} - q^{97} + 9 q^{98}+O(q^{100}) $ 8 * q + q^2 - 5 * q^4 + 4 * q^5 - 8 * q^7 - 24 * q^8 - q^10 + 4 * q^11 - 7 * q^13 - q^14 - 7 * q^16 - q^17 + 5 * q^19 - 10 * q^20 - 2 * q^22 + 2 * q^23 - 4 * q^25 - 6 * q^26 + 19 * q^28 - q^29 + 30 * q^32 - 15 * q^34 - 4 * q^35 - 4 * q^37 - 13 * q^38 - 12 * q^40 - 8 * q^41 - q^43 - 12 * q^44 + 24 * q^46 - 12 * q^47 - 20 * q^49 - 2 * q^50 + 3 * q^52 - 5 * q^53 + 2 * q^55 + 82 * q^56 - 54 * q^58 - 5 * q^59 + 37 * q^62 + 112 * q^64 - 14 * q^65 - 4 * q^67 - 32 * q^68 + q^70 + 20 * q^71 + 20 * q^73 + 25 * q^74 + 63 * q^76 - 28 * q^77 - 17 * q^79 + 7 * q^80 - 21 * q^82 - 2 * q^83 + q^85 + 8 * q^86 - 14 * q^88 + 11 * q^89 - 6 * q^91 - q^92 - 62 * q^94 + 4 * q^95 - q^97 + 9 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	−0.832272	+	1.44154i	−0.588506	+	1.01932i	0.405923	+	0.913907i	$0.366950\pi$
$2$	−0.832272	+	1.44154i	−0.588506	+	1.01932i	−0.994428	+	0.105414i	$0.966383\pi$
$3$		0			0
$3$		0			0
$4$	−0.385355	−	0.667454i	−0.192677	−	0.333727i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−2.43525			−0.920440			−0.460220	−	0.887805i	$-0.652229\pi$
$7$	−2.43525			−0.920440			−0.460220	+	0.887805i	$0.652229\pi$
$8$	−2.04621			−0.723444
$9$		0			0
$10$	0.832272	+	1.44154i	0.263188	+	0.455854i
$11$	5.75477			1.73513			0.867564	−	0.497326i	$-0.165685\pi$
$11$	5.75477			1.73513			0.867564	+	0.497326i	$0.165685\pi$
$12$		0			0
$13$	0.797505	+	1.38132i	0.221188	+	0.383109i	0.955169	−	0.296061i	$-0.0956732\pi$
$13$	0.797505	+	1.38132i	0.221188	+	0.383109i	−0.733981	+	0.679170i	$0.762340\pi$
$14$	2.02680	−	3.51051i	0.541684	−	0.938224i
$15$		0			0
$16$	2.47371	−	4.28460i	0.618428	−	1.07115i
$17$	−2.99203	+	5.18234i	−0.725673	+	1.25690i	0.233023	+	0.972471i	$0.425138\pi$
$17$	−2.99203	+	5.18234i	−0.725673	+	1.25690i	−0.958696	+	0.284432i	$0.908195\pi$
$18$		0			0
$19$	0.149412	+	4.35634i	0.0342775	+	0.999412i
$19$	0.149412	+	4.35634i	0.0342775	+	0.999412i
$20$	−0.770710			−0.172336
$21$		0			0
$22$	−4.78953	+	8.29572i	−1.02113	+	1.76865i
$23$	−0.470022	−	0.814102i	−0.0980064	−	0.169752i	0.812853	−	0.582469i	$-0.197913\pi$
$23$	−0.470022	−	0.814102i	−0.0980064	−	0.169752i	−0.910859	+	0.412717i	$0.864580\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	−2.65497			−0.520682
$27$		0			0
$28$	0.938437	+	1.62542i	0.177348	+	0.307176i
$29$	1.30917	+	2.26755i	0.243106	+	0.421073i	0.961598	−	0.274463i	$-0.0885002\pi$
$29$	1.30917	+	2.26755i	0.243106	+	0.421073i	−0.718491	+	0.695536i	$0.755167\pi$
$30$		0			0
$31$	−5.26913			−0.946364			−0.473182	−	0.880965i	$-0.656895\pi$
$31$	−5.26913			−0.946364			−0.473182	+	0.880965i	$0.656895\pi$
$32$	2.07140	+	3.58777i	0.366175	+	0.634233i
$33$		0			0
$34$	−4.98037	−	8.62625i	−0.854126	−	1.47939i
$35$	−1.21763	+	2.10899i	−0.205817	+	0.356485i
$36$		0			0
$37$	−2.89384			−0.475744			−0.237872	−	0.971297i	$-0.576450\pi$
$37$	−2.89384			−0.475744			−0.237872	+	0.971297i	$0.576450\pi$
$38$	−6.40418	−	3.41028i	−1.03889	−	0.553220i
$39$		0			0
$40$	−1.02310	+	1.77207i	−0.161767	+	0.280189i
$41$	−3.15767	+	5.46925i	−0.493145	+	0.854153i	−0.999969	−	0.00789701i	$-0.997486\pi$
$41$	−3.15767	+	5.46925i	−0.493145	+	0.854153i	0.506823	+	0.862050i	$0.330820\pi$
$42$		0			0
$43$	−2.26961	+	3.93108i	−0.346113	+	0.599485i	−0.985555	−	0.169354i	$-0.945832\pi$
$43$	−2.26961	+	3.93108i	−0.346113	+	0.599485i	0.639443	+	0.768839i	$0.279165\pi$
$44$	−2.21763	−	3.84104i	−0.334320	−	0.579059i
$45$		0			0
$46$	1.56475			0.230709
$47$	4.47718	+	7.75471i	0.653064	+	1.13114i	0.982375	+	0.186919i	$0.0598502\pi$
$47$	4.47718	+	7.75471i	0.653064	+	1.13114i	−0.329311	+	0.944221i	$0.606816\pi$
$48$		0			0
$49$	−1.06953			−0.152791
$50$	1.66454			0.235402
$51$		0			0
$52$	0.614645	−	1.06460i	0.0852359	−	0.147633i
$53$	−1.09819	−	1.90213i	−0.150848	−	0.261277i	0.780691	−	0.624917i	$-0.214867\pi$
$53$	−1.09819	−	1.90213i	−0.150848	−	0.261277i	−0.931540	+	0.363640i	$0.881534\pi$
$54$		0			0
$55$	2.87738	−	4.98377i	0.387986	−	0.672012i
$56$	4.98304			0.665887
$57$		0			0
$58$	−4.35834			−0.572278
$59$	−5.39939	+	9.35202i	−0.702941	+	1.21753i	0.264489	+	0.964389i	$0.414797\pi$
$59$	−5.39939	+	9.35202i	−0.702941	+	1.21753i	−0.967430	+	0.253140i	$0.918537\pi$
$60$		0			0
$61$	5.26434	+	9.11811i	0.674030	+	1.16745i	0.976751	+	0.214375i	$0.0687716\pi$
$61$	5.26434	+	9.11811i	0.674030	+	1.16745i	−0.302721	+	0.953079i	$0.597895\pi$
$62$	4.38535	−	7.59566i	0.556941	−	0.964649i
$63$		0			0
$64$	2.99898			0.374873
$65$	1.59501			0.197837
$66$		0			0
$67$	−0.504789	−	0.874320i	−0.0616698	−	0.106815i	0.833542	−	0.552456i	$-0.186309\pi$
$67$	−0.504789	−	0.874320i	−0.0616698	−	0.106815i	−0.895212	+	0.445641i	$0.852976\pi$
$68$	4.61197			0.559284
$69$		0			0
$70$	−2.02680	−	3.51051i	−0.242248	−	0.419587i
$71$	4.41694	−	7.65036i	0.524194	−	0.907931i	−0.475409	−	0.879765i	$-0.657700\pi$
$71$	4.41694	−	7.65036i	0.524194	−	0.907931i	0.999603	−	0.0281662i	$-0.00896677\pi$
$72$		0			0
$73$	5.12499	−	8.87674i	0.599835	−	1.03894i	−0.393011	−	0.919534i	$-0.628566\pi$
$73$	5.12499	−	8.87674i	0.599835	−	1.03894i	0.992845	−	0.119410i	$-0.0381003\pi$
$74$	2.40846	−	4.17157i	0.279978	−	0.484936i
$75$		0			0
$76$	2.85008	−	1.77846i	0.326927	−	0.204004i
$77$	−14.0143			−1.59708
$78$		0			0
$79$	−3.80229	+	6.58577i	−0.427792	+	0.740957i	−0.996677	−	0.0814604i	$-0.974042\pi$
$79$	−3.80229	+	6.58577i	−0.427792	+	0.740957i	0.568885	+	0.822417i	$0.307375\pi$
$80$	−2.47371	−	4.28460i	−0.276570	−	0.479032i
$81$		0			0
$82$	−5.25609	−	9.10381i	−0.580438	−	1.00535i
$83$	−3.11355			−0.341756			−0.170878	−	0.985292i	$-0.554660\pi$
$83$	−3.11355			−0.341756			−0.170878	+	0.985292i	$0.554660\pi$
$84$		0			0
$85$	2.99203	+	5.18234i	0.324531	+	0.562104i
$86$	−3.77787	−	6.54346i	−0.407378	−	0.705600i
$87$		0			0
$88$	−11.7755			−1.25527
$89$	−5.55706	−	9.62511i	−0.589047	−	1.02026i	−0.994358	−	0.106081i	$-0.966170\pi$
$89$	−5.55706	−	9.62511i	−0.589047	−	1.02026i	0.405310	−	0.914179i	$-0.367163\pi$
$90$		0			0
$91$	−1.94213	−	3.36387i	−0.203590	−	0.352629i
$92$	−0.362251	+	0.627436i	−0.0377672	+	0.0654148i
$93$		0			0
$94$	−14.9049			−1.53733
$95$	3.84741	+	2.04877i	0.394735	+	0.210200i
$96$		0			0
$97$	−2.02888	+	3.51412i	−0.206002	+	0.356805i	−0.950451	−	0.310873i	$-0.899379\pi$
$97$	−2.02888	+	3.51412i	−0.206002	+	0.356805i	0.744450	+	0.667678i	$0.232712\pi$
$98$	0.890144	−	1.54177i	0.0899181	−	0.155743i
$99$		0			0
$100$	−0.385355	+	0.667454i	−0.0385355	+	0.0667454i
$101$	−5.56503	−	9.63892i	−0.553741	−	0.959108i	−0.998000	−	0.0632098i	$-0.979866\pi$
$101$	−5.56503	−	9.63892i	−0.553741	−	0.959108i	0.444259	−	0.895898i	$-0.353467\pi$
$102$		0			0
$103$	11.5791			1.14092			0.570460	−	0.821326i	$-0.306765\pi$
$103$	11.5791			1.14092			0.570460	+	0.821326i	$0.306765\pi$
$104$	−1.63186	−	2.82647i	−0.160017	−	0.277158i
$105$		0			0
$106$	3.65598			0.355101
$107$	−17.9177			−1.73217			−0.866086	−	0.499894i	$-0.833372\pi$
$107$	−17.9177			−1.73217			−0.866086	+	0.499894i	$0.833372\pi$
$108$		0			0
$109$	−2.81235	+	4.87113i	−0.269374	+	0.466570i	−0.968700	−	0.248233i	$-0.920150\pi$
$109$	−2.81235	+	4.87113i	−0.269374	+	0.466570i	0.699326	+	0.714803i	$0.253484\pi$
$110$	4.78953	+	8.29572i	0.456664	+	0.790965i
$111$		0			0
$112$	−6.02412	+	10.4341i	−0.569226	+	0.985928i
$113$	15.6789			1.47494			0.737472	−	0.675378i	$-0.236019\pi$
$113$	15.6789			1.47494			0.737472	+	0.675378i	$0.236019\pi$
$114$		0			0
$115$	−0.940044			−0.0876595
$116$	1.00899	−	1.74762i	0.0936822	−	0.162262i
$117$		0			0
$118$	−8.98753	−	15.5669i	−0.827369	−	1.43304i
$119$	7.28635	−	12.6203i	0.667939	−	1.15690i
$120$		0			0
$121$	22.1173			2.01067
$122$	−17.5255			−1.58668
$123$		0			0
$124$	2.03049	+	3.51691i	0.182343	+	0.315827i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−3.05996	−	5.30000i	−0.271527	−	0.470299i	0.697726	−	0.716365i	$-0.254195\pi$
$127$	−3.05996	−	5.30000i	−0.271527	−	0.470299i	−0.969253	+	0.246066i	$0.920862\pi$
$128$	−6.63877	+	11.4987i	−0.586790	+	1.01635i
$129$		0			0
$130$	−1.32748	+	2.29927i	−0.116428	+	0.201659i
$131$	−7.44055	+	12.8874i	−0.650084	+	1.12598i	0.333018	+	0.942920i	$0.391933\pi$
$131$	−7.44055	+	12.8874i	−0.650084	+	1.12598i	−0.983102	+	0.183058i	$0.941400\pi$
$132$		0			0
$133$	−0.363857	−	10.6088i	−0.0315504	−	0.919899i
$134$	1.68049			0.145172
$135$		0			0
$136$	6.12231	−	10.6042i	0.524984	−	0.909299i
$137$	8.67518	+	15.0258i	0.741170	+	1.28374i	0.951963	+	0.306214i	$0.0990622\pi$
$137$	8.67518	+	15.0258i	0.741170	+	1.28374i	−0.210793	+	0.977531i	$0.567604\pi$
$138$		0			0
$139$	3.35267	+	5.80700i	0.284370	+	0.492543i	0.972456	−	0.233086i	$-0.0748823\pi$
$139$	3.35267	+	5.80700i	0.284370	+	0.492543i	−0.688086	+	0.725629i	$0.741549\pi$
$140$	1.87687			0.158625
$141$		0			0
$142$	7.35219	+	12.7344i	0.616982	+	1.06864i
$143$	4.58946	+	7.94917i	0.383790	+	0.664743i
$144$		0			0
$145$	2.61834			0.217441
$146$	8.53077	+	14.7757i	0.706012	+	1.22285i
$147$		0			0
$148$	1.11515	+	1.93150i	0.0916651	+	0.158769i
$149$	7.19642	−	12.4646i	0.589553	−	1.02114i	−0.404737	−	0.914433i	$-0.632637\pi$
$149$	7.19642	−	12.4646i	0.589553	−	1.02114i	0.994291	−	0.106704i	$-0.0340296\pi$
$150$		0			0
$151$	12.7219			1.03529			0.517645	−	0.855595i	$-0.326809\pi$
$151$	12.7219			1.03529			0.517645	+	0.855595i	$0.326809\pi$
$152$	−0.305729	−	8.91398i	−0.0247979	−	0.723019i
$153$		0			0
$154$	11.6637	−	20.2022i	0.939890	−	1.62794i
$155$	−2.63457	+	4.56320i	−0.211614	+	0.366525i
$156$		0			0
$157$	1.68765	−	2.92309i	0.134689	−	0.233288i	−0.790790	−	0.612088i	$-0.790330\pi$
$157$	1.68765	−	2.92309i	0.134689	−	0.233288i	0.925479	+	0.378800i	$0.123663\pi$
$158$	−6.32909	−	10.9623i	−0.503515	−	0.872114i
$159$		0			0
$160$	4.14280			0.327517
$161$	1.14462	+	1.98255i	0.0902089	+	0.156246i
$162$		0			0
$163$	0.307960			0.0241213			0.0120607	−	0.999927i	$-0.496161\pi$
$163$	0.307960			0.0241213			0.0120607	+	0.999927i	$0.496161\pi$
$164$	4.86730			0.380072
$165$		0			0
$166$	2.59132	−	4.48830i	0.201125	−	0.348359i
$167$	7.13215	+	12.3532i	0.551902	+	0.955923i	0.998137	+	0.0610070i	$0.0194312\pi$
$167$	7.13215	+	12.3532i	0.551902	+	0.955923i	−0.446235	+	0.894916i	$0.647235\pi$
$168$		0			0
$169$	5.22797	−	9.05511i	0.402152	−	0.696547i
$170$	−9.96073			−0.763953
$171$		0			0
$172$	3.49842			0.266752
$173$	6.67357	−	11.5590i	0.507382	−	0.878811i	−0.492581	−	0.870266i	$-0.663947\pi$
$173$	6.67357	−	11.5590i	0.507382	−	0.878811i	0.999963	−	0.00854514i	$-0.00272003\pi$
$174$		0			0
$175$	1.21763	+	2.10899i	0.0920440	+	0.159425i
$176$	14.2356	−	24.6569i	1.07305	−	1.85858i
$177$		0			0
$178$	18.5000			1.38663
$179$	14.2207			1.06291			0.531454	−	0.847087i	$-0.321646\pi$
$179$	14.2207			1.06291			0.531454	+	0.847087i	$0.321646\pi$
$180$		0			0
$181$	−4.94132	−	8.55861i	−0.367285	−	0.636157i	0.621855	−	0.783133i	$-0.286379\pi$
$181$	−4.94132	−	8.55861i	−0.367285	−	0.636157i	−0.989140	+	0.146976i	$0.953046\pi$
$182$	6.46552			0.479256
$183$		0			0
$184$	0.961763	+	1.66582i	0.0709021	+	0.122806i
$185$	−1.44692	+	2.50613i	−0.106379	+	0.184255i
$186$		0			0
$187$	−17.2184	+	29.8232i	−1.25914	+	2.18089i
$188$	3.45061	−	5.97663i	0.251661	−	0.435891i
$189$		0			0
$190$	−6.15548	+	3.84104i	−0.446565	+	0.278659i
$191$	12.9942			0.940228			0.470114	−	0.882606i	$-0.344213\pi$
$191$	12.9942			0.940228			0.470114	+	0.882606i	$0.344213\pi$
$192$		0			0
$193$	−7.25795	+	12.5711i	−0.522439	+	0.904890i	0.477221	+	0.878784i	$0.341644\pi$
$193$	−7.25795	+	12.5711i	−0.522439	+	0.904890i	−0.999659	+	0.0261066i	$0.991689\pi$
$194$	−3.37716	−	5.84942i	−0.242466	−	0.419964i
$195$		0			0
$196$	0.412150	+	0.713865i	0.0294393	+	0.0509904i
$197$	25.0010			1.78125			0.890624	−	0.454740i	$-0.150268\pi$
$197$	25.0010			1.78125			0.890624	+	0.454740i	$0.150268\pi$
$198$		0			0
$199$	1.12769	+	1.95322i	0.0799401	+	0.138460i	0.903224	−	0.429170i	$-0.141194\pi$
$199$	1.12769	+	1.95322i	0.0799401	+	0.138460i	−0.823284	+	0.567630i	$0.807860\pi$
$200$	1.02310	+	1.77207i	0.0723444	+	0.125304i
$201$		0			0
$202$	18.5265			1.30352
$203$	−3.18816	−	5.52205i	−0.223765	−	0.387572i
$204$		0			0
$205$	3.15767	+	5.46925i	0.220541	+	0.381989i
$206$	−9.63694	+	16.6917i	−0.671437	+	1.16296i
$207$		0			0
$208$	7.89120			0.547156
$209$	0.859833	+	25.0697i	0.0594759	+	1.73411i
$210$		0			0
$211$	−11.1081	+	19.2397i	−0.764710	+	1.32452i	0.175689	+	0.984446i	$0.443785\pi$
$211$	−11.1081	+	19.2397i	−0.764710	+	1.32452i	−0.940400	+	0.340071i	$0.889549\pi$
$212$	−0.846388	+	1.46599i	−0.0581302	+	0.100684i
$213$		0			0
$214$	14.9124	−	25.8291i	1.01939	−	1.76564i
$215$	2.26961	+	3.93108i	0.154786	+	0.268098i
$216$		0			0
$217$	12.8317			0.871071
$218$	−4.68128	−	8.10822i	−0.317057	−	0.549158i
$219$		0			0
$220$	−4.43525			−0.299025
$221$	−9.54463			−0.642041
$222$		0			0
$223$	−5.10799	+	8.84730i	−0.342056	+	0.592459i	−0.984814	−	0.173610i	$-0.944457\pi$
$223$	−5.10799	+	8.84730i	−0.342056	+	0.592459i	0.642758	+	0.766069i	$0.277790\pi$
$224$	−5.04438	−	8.73712i	−0.337042	−	0.583774i
$225$		0			0
$226$	−13.0491	+	22.6017i	−0.868012	+	1.50344i
$227$	−4.15180			−0.275565			−0.137782	−	0.990463i	$-0.543997\pi$
$227$	−4.15180			−0.275565			−0.137782	+	0.990463i	$0.543997\pi$
$228$		0			0
$229$	6.53286			0.431703			0.215852	−	0.976426i	$-0.430747\pi$
$229$	6.53286			0.431703			0.215852	+	0.976426i	$0.430747\pi$
$230$	0.782373	−	1.35511i	0.0515881	−	0.0893533i
$231$		0			0
$232$	−2.67883	−	4.63987i	−0.175874	−	0.304623i
$233$	2.57410	−	4.45848i	0.168635	−	0.292084i	−0.769305	−	0.638882i	$-0.779397\pi$
$233$	2.57410	−	4.45848i	0.168635	−	0.292084i	0.937940	+	0.346797i	$0.112731\pi$
$234$		0			0
$235$	8.95437			0.584118
$236$	8.32272			0.541763
$237$		0			0
$238$	12.1285	+	21.0071i	0.786171	+	1.36169i
$239$	−13.9962			−0.905338			−0.452669	−	0.891679i	$-0.649528\pi$
$239$	−13.9962			−0.905338			−0.452669	+	0.891679i	$0.649528\pi$
$240$		0			0
$241$	−7.61285	−	13.1858i	−0.490387	−	0.849375i	0.509552	−	0.860440i	$-0.329811\pi$
$241$	−7.61285	−	13.1858i	−0.490387	−	0.849375i	−0.999939	+	0.0110652i	$0.996478\pi$
$242$	−18.4076	+	31.8830i	−1.18329	+	2.04952i
$243$		0			0
$244$	4.05728	−	7.02742i	0.259741	−	0.449884i
$245$	−0.534767	+	0.926244i	−0.0341650	+	0.0591756i
$246$		0			0
$247$	−5.89834	+	3.68059i	−0.375302	+	0.234190i
$248$	10.7817			0.684642
$249$		0			0
$250$	0.832272	−	1.44154i	0.0526375	−	0.0911709i
$251$	−3.05630	−	5.29366i	−0.192912	−	0.334133i	0.753302	−	0.657674i	$-0.228460\pi$
$251$	−3.05630	−	5.29366i	−0.192912	−	0.334133i	−0.946214	+	0.323542i	$0.895126\pi$
$252$		0			0
$253$	−2.70487	−	4.68497i	−0.170053	−	0.294541i
$254$	10.1869			0.639181
$255$		0			0
$256$	−8.05154	−	13.9457i	−0.503221	−	0.871605i
$257$	0.0613414	+	0.106246i	0.00382637	+	0.00662747i	0.867932	−	0.496683i	$-0.165449\pi$
$257$	0.0613414	+	0.106246i	0.00382637	+	0.00662747i	−0.864106	+	0.503310i	$0.832115\pi$
$258$		0			0
$259$	7.04723			0.437893
$260$	−0.614645	−	1.06460i	−0.0381187	−	0.0660235i
$261$		0			0
$262$	−12.3851	−	21.4517i	−0.765156	−	1.32529i
$263$	−5.03027	+	8.71267i	−0.310179	+	0.537247i	−0.978401	−	0.206716i	$-0.933722\pi$
$263$	−5.03027	+	8.71267i	−0.310179	+	0.537247i	0.668222	+	0.743962i	$0.267056\pi$
$264$		0			0
$265$	−2.19639			−0.134923
$266$	15.5958	+	8.30489i	0.956240	+	0.509206i
$267$		0			0
$268$	−0.389046	+	0.673847i	−0.0237648	+	0.0411618i
$269$	2.85614	−	4.94698i	0.174142	−	0.301623i	−0.765722	−	0.643172i	$-0.777618\pi$
$269$	2.85614	−	4.94698i	0.174142	−	0.301623i	0.939864	+	0.341549i	$0.110951\pi$
$270$		0			0
$271$	6.35560	−	11.0082i	0.386075	−	0.668702i	−0.605843	−	0.795585i	$-0.707164\pi$
$271$	6.35560	−	11.0082i	0.386075	−	0.668702i	0.991918	+	0.126883i	$0.0404972\pi$
$272$	14.8028	+	25.6393i	0.897554	+	1.55461i
$273$		0			0
$274$	−28.8804			−1.74473
$275$	−2.87738	−	4.98377i	−0.173513	−	0.300533i
$276$		0			0
$277$	17.6019			1.05760			0.528799	−	0.848747i	$-0.322642\pi$
$277$	17.6019			1.05760			0.528799	+	0.848747i	$0.322642\pi$
$278$	−11.1613			−0.669413
$279$		0			0
$280$	2.49152	−	4.31544i	0.148897	−	0.257897i
$281$	−10.2502	−	17.7539i	−0.611476	−	1.05911i	−0.990992	−	0.133922i	$-0.957243\pi$
$281$	−10.2502	−	17.7539i	−0.611476	−	1.05911i	0.379516	−	0.925185i	$-0.376090\pi$
$282$		0			0
$283$	5.92805	−	10.2677i	0.352386	−	0.610350i	−0.634281	−	0.773103i	$-0.718704\pi$
$283$	5.92805	−	10.2677i	0.352386	−	0.610350i	0.986667	+	0.162752i	$0.0520371\pi$
$284$	−6.80836			−0.404002
$285$		0			0
$286$	−15.2787			−0.903449
$287$	7.68973	−	13.3190i	0.453911	−	0.786196i
$288$		0			0
$289$	−9.40447	−	16.2890i	−0.553204	−	0.958177i
$290$	−2.17917	+	3.77443i	−0.127965	+	0.221642i
$291$		0			0
$292$	−7.89976			−0.462298
$293$	24.9814			1.45943			0.729715	−	0.683751i	$-0.239653\pi$
$293$	24.9814			1.45943			0.729715	+	0.683751i	$0.239653\pi$
$294$		0			0
$295$	5.39939	+	9.35202i	0.314365	+	0.544495i
$296$	5.92139			0.344174
$297$		0			0
$298$	11.9788	+	20.7478i	0.693911	+	1.20189i
$299$	0.749690	−	1.29850i	0.0433557	−	0.0750943i
$300$		0			0
$301$	5.52708	−	9.57319i	0.318576	−	0.551789i
$302$	−10.5881	+	18.3391i	−0.609274	+	1.05529i
$303$		0			0
$304$	19.0348	+	10.1362i	1.09172	+	0.581348i
$305$	10.5287			0.602871
$306$		0			0
$307$	−8.45997	+	14.6531i	−0.482836	+	0.836296i	−0.999806	−	0.0197074i	$-0.993727\pi$
$307$	−8.45997	+	14.6531i	−0.482836	+	0.836296i	0.516970	+	0.856003i	$0.327060\pi$
$308$	5.40049	+	9.35392i	0.307721	+	0.532989i
$309$		0			0
$310$	−4.38535	−	7.59566i	−0.249071	−	0.431404i
$311$	15.2133			0.862670			0.431335	−	0.902192i	$-0.358043\pi$
$311$	15.2133			0.862670			0.431335	+	0.902192i	$0.358043\pi$
$312$		0			0
$313$	−12.4637	−	21.5877i	−0.704488	−	1.22021i	−0.966876	−	0.255246i	$-0.917844\pi$
$313$	−12.4637	−	21.5877i	−0.704488	−	1.22021i	0.262389	−	0.964962i	$-0.415490\pi$
$314$	2.80917	+	4.86562i	0.158531	+	0.274583i
$315$		0			0
$316$	5.86093			0.329703
$317$	−12.6152	−	21.8502i	−0.708541	−	1.22723i	−0.965398	−	0.260780i	$-0.916020\pi$
$317$	−12.6152	−	21.8502i	−0.708541	−	1.22723i	0.256857	−	0.966449i	$-0.417313\pi$
$318$		0			0
$319$	7.53396	+	13.0492i	0.421821	+	0.730615i
$320$	1.49949	−	2.59720i	0.0838241	−	0.145188i
$321$		0			0
$322$	−3.81055			−0.212354
$323$	−23.0231	−	12.2600i	−1.28104	−	0.682163i
$324$		0			0
$325$	0.797505	−	1.38132i	0.0442376	−	0.0766218i
$326$	−0.256307	+	0.443937i	−0.0141955	+	0.0245874i
$327$		0			0
$328$	6.46125	−	11.1912i	0.356763	−	0.617932i
$329$	−10.9031	−	18.8847i	−0.601106	−	1.04115i
$330$		0			0
$331$	−20.2063			−1.11064			−0.555320	−	0.831637i	$-0.687404\pi$
$331$	−20.2063			−1.11064			−0.555320	+	0.831637i	$0.687404\pi$
$332$	1.19982	+	2.07815i	0.0658487	+	0.114053i
$333$		0			0
$334$	−23.7436			−1.29919
$335$	−1.00958			−0.0551592
$336$		0			0
$337$	15.9123	−	27.5610i	0.866800	−	1.50134i	0.00155051	−	0.999999i	$-0.499506\pi$
$337$	15.9123	−	27.5610i	0.866800	−	1.50134i	0.865249	−	0.501342i	$-0.167160\pi$
$338$	8.70219	+	15.0726i	0.473337	+	0.819843i
$339$		0			0
$340$	2.30599	−	3.99408i	0.125060	−	0.216610i
$341$	−30.3226			−1.64206
$342$		0			0
$343$	19.6514			1.06107
$344$	4.64410	−	8.04382i	0.250393	−	0.433693i
$345$		0			0
$346$	11.1085	+	19.2404i	0.597194	+	1.03437i
$347$	−1.65128	+	2.86009i	−0.0886451	+	0.153538i	−0.906939	−	0.421263i	$-0.861587\pi$
$347$	−1.65128	+	2.86009i	−0.0886451	+	0.153538i	0.818294	+	0.574801i	$0.194920\pi$
$348$		0			0
$349$	17.8486			0.955416			0.477708	−	0.878519i	$-0.341468\pi$
$349$	17.8486			0.955416			0.477708	+	0.878519i	$0.341468\pi$
$350$	−4.05359			−0.216674
$351$		0			0
$352$	11.9204	+	20.6468i	0.635360	+	1.10048i
$353$	8.29523			0.441511			0.220755	−	0.975329i	$-0.429148\pi$
$353$	8.29523			0.441511			0.220755	+	0.975329i	$0.429148\pi$
$354$		0			0
$355$	−4.41694	−	7.65036i	−0.234427	−	0.406039i
$356$	−4.28288	+	7.41817i	−0.226992	+	0.393162i
$357$		0			0
$358$	−11.8355	+	20.4997i	−0.625527	+	1.08344i
$359$	4.17511	−	7.23150i	0.220354	−	0.381664i	−0.734562	−	0.678542i	$-0.762612\pi$
$359$	4.17511	−	7.23150i	0.220354	−	0.381664i	0.954915	+	0.296878i	$0.0959455\pi$
$360$		0			0
$361$	−18.9554	+	1.30178i	−0.997650	+	0.0685148i
$362$	16.4501			0.864598
$363$		0			0
$364$	−1.49682	+	2.59256i	−0.0784545	+	0.135887i
$365$	−5.12499	−	8.87674i	−0.268254	−	0.464630i
$366$		0			0
$367$	−7.20988	−	12.4879i	−0.376353	−	0.651862i	0.614176	−	0.789169i	$-0.289489\pi$
$367$	−7.20988	−	12.4879i	−0.376353	−	0.651862i	−0.990528	+	0.137307i	$0.956155\pi$
$368$	−4.65080			−0.242440
$369$		0			0
$370$	−2.40846	−	4.17157i	−0.125210	−	0.216870i
$371$	2.67438	+	4.63216i	0.138847	+	0.240490i
$372$		0			0
$373$	−24.1157			−1.24866			−0.624332	−	0.781159i	$-0.714629\pi$
$373$	−24.1157			−1.24866			−0.624332	+	0.781159i	$0.714629\pi$
$374$	−28.6608	−	49.6420i	−1.48202	−	2.56693i
$375$		0			0
$376$	−9.16125	−	15.8678i	−0.472455	−	0.818317i
$377$	−2.08814	+	3.61676i	−0.107545	+	0.186273i
$378$		0			0
$379$	16.6757			0.856571			0.428285	−	0.903644i	$-0.359118\pi$
$379$	16.6757			0.856571			0.428285	+	0.903644i	$0.359118\pi$
$380$	−0.115154	−	3.35747i	−0.00590725	−	0.172235i
$381$		0			0
$382$	−10.8147	+	18.7316i	−0.553329	+	0.958395i
$383$	−5.43895	+	9.42053i	−0.277917	+	0.481367i	−0.970867	−	0.239619i	$-0.922977\pi$
$383$	−5.43895	+	9.42053i	−0.277917	+	0.481367i	0.692950	+	0.720986i	$0.256311\pi$
$384$		0			0
$385$	−7.00716	+	12.1368i	−0.357118	+	0.618546i
$386$	−12.0812	−	20.9252i	−0.614916	−	1.06507i
$387$		0			0
$388$	3.12736			0.158767
$389$	18.2272	+	31.5704i	0.924154	+	1.60068i	0.792917	+	0.609330i	$0.208562\pi$
$389$	18.2272	+	31.5704i	0.924154	+	1.60068i	0.131237	+	0.991351i	$0.458105\pi$
$390$		0			0
$391$	5.62528			0.284482
$392$	2.18849			0.110535
$393$		0			0
$394$	−20.8077	+	36.0399i	−1.04827	+	1.81566i
$395$	3.80229	+	6.58577i	0.191314	+	0.331366i
$396$		0			0
$397$	−4.29191	+	7.43380i	−0.215405	+	0.373092i	−0.953398	−	0.301717i	$-0.902440\pi$
$397$	−4.29191	+	7.43380i	−0.215405	+	0.373092i	0.737993	+	0.674808i	$0.235774\pi$
$398$	−3.75419			−0.188181
$399$		0			0
$400$	−4.94743			−0.247371
$401$	−8.52785	+	14.7707i	−0.425860	+	0.737612i	−0.996500	−	0.0835885i	$-0.973362\pi$
$401$	−8.52785	+	14.7707i	−0.425860	+	0.737612i	0.570640	+	0.821200i	$0.306695\pi$
$402$		0			0
$403$	−4.20216	−	7.27836i	−0.209325	−	0.362561i
$404$	−4.28903	+	7.42881i	−0.213387	+	0.369597i
$405$		0			0
$406$	10.6137			0.526747
$407$	−16.6533			−0.825476
$408$		0			0
$409$	5.89702	+	10.2139i	0.291589	+	0.505047i	0.974186	−	0.225749i	$-0.0724828\pi$
$409$	5.89702	+	10.2139i	0.291589	+	0.505047i	−0.682597	+	0.730795i	$0.739149\pi$
$410$	−10.5122			−0.519159
$411$		0			0
$412$	−4.46205	−	7.72850i	−0.219829	−	0.380756i
$413$	13.1489	−	22.7745i	0.647015	−	1.12066i
$414$		0			0
$415$	−1.55677	+	2.69641i	−0.0764190	+	0.132362i
$416$	−3.30390	+	5.72252i	−0.161987	+	0.280570i
$417$		0			0
$418$	−36.8546	−	19.6253i	−1.80261	−	0.959907i
$419$	−1.14280			−0.0558292			−0.0279146	−	0.999610i	$-0.508887\pi$
$419$	−1.14280			−0.0558292			−0.0279146	+	0.999610i	$0.508887\pi$
$420$		0			0
$421$	−9.75944	+	16.9039i	−0.475646	+	0.823843i	−0.999611	−	0.0278967i	$-0.991119\pi$
$421$	−9.75944	+	16.9039i	−0.475646	+	0.823843i	0.523965	+	0.851740i	$0.324452\pi$
$422$	−18.4899	−	32.0254i	−0.900072	−	1.55897i
$423$		0			0
$424$	2.24713	+	3.89215i	0.109130	+	0.189019i
$425$	5.98406			0.290269
$426$		0			0
$427$	−12.8200	−	22.2049i	−0.620404	−	1.07457i
$428$	6.90469	+	11.9593i	0.333751	+	0.578073i
$429$		0			0
$430$	−7.55574			−0.364370
$431$	−18.4392	−	31.9377i	−0.888187	−	1.53838i	−0.842017	−	0.539451i	$-0.818632\pi$
$431$	−18.4392	−	31.9377i	−0.888187	−	1.53838i	−0.0461694	−	0.998934i	$-0.514701\pi$
$432$		0			0
$433$	−0.184467	−	0.319506i	−0.00886490	−	0.0153545i	0.861559	−	0.507658i	$-0.169488\pi$
$433$	−0.184467	−	0.319506i	−0.00886490	−	0.0153545i	−0.870424	+	0.492303i	$0.836155\pi$
$434$	−10.6795	+	18.4974i	−0.512630	+	0.887902i
$435$		0			0
$436$	4.33501			0.207609
$437$	3.47628	−	2.16921i	0.166293	−	0.103767i
$438$		0			0
$439$	1.13220	−	1.96102i	0.0540367	−	0.0935944i	−0.837742	−	0.546067i	$-0.816125\pi$
$439$	1.13220	−	1.96102i	0.0540367	−	0.0935944i	0.891778	+	0.452472i	$0.149458\pi$
$440$	−5.88773	+	10.1978i	−0.280686	+	0.486163i
$441$		0			0
$442$	7.94373	−	13.7590i	0.377845	−	0.654447i
$443$	−8.23137	−	14.2572i	−0.391084	−	0.677378i	0.601508	−	0.798866i	$-0.294567\pi$
$443$	−8.23137	−	14.2572i	−0.391084	−	0.677378i	−0.992593	+	0.121488i	$0.961233\pi$
$444$		0			0
$445$	−11.1141			−0.526860
$446$	−8.50248	−	14.7267i	−0.402604	−	0.697331i
$447$		0			0
$448$	−7.30329			−0.345048
$449$	−14.1613			−0.668315			−0.334158	−	0.942517i	$-0.608452\pi$
$449$	−14.1613			−0.668315			−0.334158	+	0.942517i	$0.608452\pi$
$450$		0			0
$451$	−18.1717	+	31.4742i	−0.855670	+	1.48206i
$452$	−6.04193	−	10.4649i	−0.284188	−	0.492229i
$453$		0			0
$454$	3.45543	−	5.98498i	0.162171	−	0.280889i
$455$	−3.88426			−0.182097
$456$		0			0
$457$	1.22073			0.0571033			0.0285516	−	0.999592i	$-0.490910\pi$
$457$	1.22073			0.0571033			0.0285516	+	0.999592i	$0.490910\pi$
$458$	−5.43712	+	9.41736i	−0.254060	+	0.440045i
$459$		0			0
$460$	0.362251	+	0.627436i	0.0168900	+	0.0292544i
$461$	−4.34580	+	7.52714i	−0.202404	+	0.350574i	−0.949303	−	0.314364i	$-0.898209\pi$
$461$	−4.34580	+	7.52714i	−0.202404	+	0.350574i	0.746898	+	0.664938i	$0.231542\pi$
$462$		0			0
$463$	−19.7149			−0.916229			−0.458114	−	0.888893i	$-0.651475\pi$
$463$	−19.7149			−0.916229			−0.458114	+	0.888893i	$0.651475\pi$
$464$	12.9540			0.601375
$465$		0			0
$466$	4.28471	+	7.42133i	0.198485	+	0.343787i
$467$	11.4795			0.531207			0.265604	−	0.964082i	$-0.414429\pi$
$467$	11.4795			0.531207			0.265604	+	0.964082i	$0.414429\pi$
$468$		0			0
$469$	1.22929	+	2.12919i	0.0567633	+	0.0983170i
$470$	−7.45247	+	12.9081i	−0.343757	+	0.595404i
$471$		0			0
$472$	11.0483	−	19.1362i	0.508538	−	0.880814i
$473$	−13.0611	+	22.6225i	−0.600549	+	1.04018i
$474$		0			0
$475$	3.69799	−	2.30756i	0.169676	−	0.105878i
$476$	−11.2313			−0.514787
$477$		0			0
$478$	11.6486	−	20.1760i	0.532796	−	0.922830i
$479$	19.6316	+	34.0029i	0.896989	+	1.55363i	0.831324	+	0.555789i	$0.187584\pi$
$479$	19.6316	+	34.0029i	0.896989	+	1.55363i	0.0656652	+	0.997842i	$0.479083\pi$
$480$		0			0
$481$	−2.30785	−	3.99731i	−0.105229	−	0.182262i
$482$	25.3439			1.15438
$483$		0			0
$484$	−8.52302	−	14.7623i	−0.387410	−	0.671014i
$485$	2.02888	+	3.51412i	0.0921267	+	0.159568i
$486$		0			0
$487$	31.5943			1.43168			0.715838	−	0.698266i	$-0.246045\pi$
$487$	31.5943			1.43168			0.715838	+	0.698266i	$0.246045\pi$
$488$	−10.7719	−	18.6576i	−0.487623	−	0.844588i
$489$		0			0
$490$	−0.890144	−	1.54177i	−0.0402126	−	0.0696503i
$491$	−5.53187	+	9.58148i	−0.249650	+	0.432406i	−0.963429	−	0.267965i	$-0.913649\pi$
$491$	−5.53187	+	9.58148i	−0.249650	+	0.432406i	0.713779	+	0.700371i	$0.246982\pi$
$492$		0			0
$493$	−15.6683			−0.705663
$494$	−0.396685	−	11.5659i	−0.0178477	−	0.520376i
$495$		0			0
$496$	−13.0343	+	22.5761i	−0.585258	+	1.01370i
$497$	−10.7564	+	18.6306i	−0.482489	+	0.835696i
$498$		0			0
$499$	10.1868	−	17.6440i	0.456023	−	0.789854i	−0.542724	−	0.839911i	$-0.682607\pi$
$499$	10.1868	−	17.6440i	0.456023	−	0.789854i	0.998746	+	0.0500570i	$0.0159403\pi$
$500$	0.385355	+	0.667454i	0.0172336	+	0.0298495i
$501$		0			0
$502$	10.1747			0.454118
$503$	−6.83622	−	11.8407i	−0.304812	−	0.527950i	0.672407	−	0.740181i	$-0.265260\pi$
$503$	−6.83622	−	11.8407i	−0.304812	−	0.527950i	−0.977219	+	0.212231i	$0.931927\pi$
$504$		0			0
$505$	−11.1301			−0.495281
$506$	9.00474			0.400310
$507$		0			0
$508$	−2.35834	+	4.08476i	−0.104634	+	0.181232i
$509$	−3.86196	−	6.68912i	−0.171179	−	0.296490i	0.767654	−	0.640865i	$-0.221424\pi$
$509$	−3.86196	−	6.68912i	−0.171179	−	0.296490i	−0.938832	+	0.344375i	$0.888091\pi$
$510$		0			0
$511$	−12.4807	+	21.6171i	−0.552112	+	0.956285i
$512$	0.249240			0.0110150
$513$		0			0
$514$	−0.204211			−0.00900736
$515$	5.78953	−	10.0278i	0.255117	−	0.441876i
$516$		0			0
$517$	25.7651	+	44.6265i	1.13315	+	1.96267i
$518$	−5.86521	+	10.1588i	−0.257703	+	0.446354i
$519$		0			0
$520$	−3.26372			−0.143124
$521$	2.16876			0.0950151			0.0475075	−	0.998871i	$-0.484872\pi$
$521$	2.16876			0.0950151			0.0475075	+	0.998871i	$0.484872\pi$
$522$		0			0
$523$	11.9466	+	20.6921i	0.522389	+	0.904804i	0.999661	+	0.0260485i	$0.00829243\pi$
$523$	11.9466	+	20.6921i	0.522389	+	0.904804i	−0.477272	+	0.878756i	$0.658374\pi$
$524$	11.4690			0.501026
$525$		0			0
$526$	−8.37310	−	14.5026i	−0.365085	−	0.632345i
$527$	15.7654	−	27.3065i	0.686751	−	1.18949i
$528$		0			0
$529$	11.0582	−	19.1533i	0.480790	−	0.832752i
$530$	1.82799	−	3.16617i	0.0794029	−	0.137530i
$531$		0			0
$532$	−6.94067	+	4.33101i	−0.300916	+	0.187773i
$533$	−10.0730			−0.436312
$534$		0			0
$535$	−8.95887	+	15.5172i	−0.387326	+	0.670868i
$536$	1.03290	+	1.78904i	0.0446147	+	0.0772749i
$537$		0			0
$538$	4.75418	+	8.23448i	0.204967	+	0.355013i
$539$	−6.15492			−0.265111
$540$		0			0
$541$	−21.2275	−	36.7671i	−0.912641	−	1.58074i	−0.810319	−	0.585990i	$-0.800706\pi$
$541$	−21.2275	−	36.7671i	−0.912641	−	1.58074i	−0.102323	−	0.994751i	$-0.532627\pi$
$542$	10.5792	+	18.3237i	0.454415	+	0.787069i
$543$		0			0
$544$	−24.7907			−1.06289
$545$	2.81235	+	4.87113i	0.120468	+	0.208656i
$546$		0			0
$547$	−6.01535	−	10.4189i	−0.257198	−	0.445480i	0.708292	−	0.705919i	$-0.249466\pi$
$547$	−6.01535	−	10.4189i	−0.257198	−	0.445480i	−0.965490	+	0.260439i	$0.916133\pi$
$548$	6.68604	−	11.5806i	0.285614	−	0.494697i
$549$		0			0
$550$	9.57907			0.408453
$551$	−9.68259	+	6.04198i	−0.412492	+	0.257397i
$552$		0			0
$553$	9.25956	−	16.0380i	0.393756	−	0.682006i
$554$	−14.6496	+	25.3739i	−0.622403	+	1.07803i
$555$		0			0
$556$	2.58394	−	4.47551i	0.109583	−	0.189804i
$557$	−4.37635	−	7.58006i	−0.185432	−	0.321178i	0.758290	−	0.651917i	$-0.226035\pi$
$557$	−4.37635	−	7.58006i	−0.185432	−	0.321178i	−0.943722	+	0.330740i	$0.892702\pi$
$558$		0			0
$559$	−7.24011			−0.306224
$560$	6.02412	+	10.4341i	0.254566	+	0.440921i
$561$		0			0
$562$	34.1238			1.43943
$563$	35.9707			1.51598			0.757991	−	0.652265i	$-0.226181\pi$
$563$	35.9707			1.51598			0.757991	+	0.652265i	$0.226181\pi$
$564$		0			0
$565$	7.83943	−	13.5783i	0.329807	−	0.571243i
$566$	9.86750	+	17.0910i	0.414762	+	0.718389i
$567$		0			0
$568$	−9.03798	+	15.6542i	−0.379225	+	0.656837i
$569$	20.3125			0.851543			0.425772	−	0.904831i	$-0.360003\pi$
$569$	20.3125			0.851543			0.425772	+	0.904831i	$0.360003\pi$
$570$		0			0
$571$	10.1773			0.425906			0.212953	−	0.977062i	$-0.431692\pi$
$571$	10.1773			0.425906			0.212953	+	0.977062i	$0.431692\pi$
$572$	3.53714	−	6.12650i	0.147895	−	0.256162i
$573$		0			0
$574$	12.7999	+	22.1701i	0.534258	+	0.925362i
$575$	−0.470022	+	0.814102i	−0.0196013	+	0.0339504i
$576$		0			0
$577$	−32.7441			−1.36316			−0.681578	−	0.731745i	$-0.738706\pi$
$577$	−32.7441			−1.36316			−0.681578	+	0.731745i	$0.738706\pi$
$578$	31.3083			1.30225
$579$		0			0
$580$	−1.00899	−	1.74762i	−0.0418960	−	0.0725660i
$581$	7.58228			0.314566
$582$		0			0
$583$	−6.31984	−	10.9463i	−0.261741	−	0.453349i
$584$	−10.4868	+	18.1637i	−0.433947	+	0.751618i
$585$		0			0
$586$	−20.7913	+	36.0117i	−0.858883	+	1.48763i
$587$	4.38663	−	7.59786i	0.181056	−	0.313597i	−0.761185	−	0.648535i	$-0.775382\pi$
$587$	4.38663	−	7.59786i	0.181056	−	0.313597i	0.942240	+	0.334938i	$0.108715\pi$
$588$		0			0
$589$	−0.787274	−	22.9541i	−0.0324390	−	0.945808i
$590$	−17.9751			−0.740021
$591$		0			0
$592$	−7.15852	+	12.3989i	−0.294213	+	0.509592i
$593$	−16.1603	−	27.9905i	−0.663625	−	1.14943i	−0.979656	−	0.200684i	$-0.935684\pi$
$593$	−16.1603	−	27.9905i	−0.663625	−	1.14943i	0.316031	−	0.948749i	$-0.397650\pi$
$594$		0			0
$595$	−7.28635	−	12.6203i	−0.298711	−	0.517383i
$596$	−11.0927			−0.454375
$597$		0			0
$598$	1.24789	+	2.16141i	0.0510301	+	0.0883868i
$599$	−9.77520	−	16.9311i	−0.399404	−	0.691787i	0.594249	−	0.804281i	$-0.297449\pi$
$599$	−9.77520	−	16.9311i	−0.399404	−	0.691787i	−0.993652	+	0.112494i	$0.964116\pi$
$600$		0			0
$601$	−0.401837			−0.0163913			−0.00819564	−	0.999966i	$-0.502609\pi$
$601$	−0.401837			−0.0163913			−0.00819564	+	0.999966i	$0.502609\pi$
$602$	9.20008	+	15.9350i	0.374967	+	0.649462i
$603$		0			0
$604$	−4.90243	−	8.49126i	−0.199477	−	0.345505i
$605$	11.0587	−	19.1542i	0.449599	−	0.778728i
$606$		0			0
$607$	13.4453			0.545727			0.272863	−	0.962053i	$-0.412029\pi$
$607$	13.4453			0.545727			0.272863	+	0.962053i	$0.412029\pi$
$608$	−15.3200	+	9.55976i	−0.621309	+	0.387700i
$609$		0			0
$610$	−8.76274	+	15.1775i	−0.354793	+	0.614519i
$611$	−7.14115	+	12.3688i	−0.288900	+	0.500390i
$612$		0			0
$613$	10.3527	−	17.9313i	0.418140	−	0.724239i	−0.577613	−	0.816311i	$-0.696016\pi$
$613$	10.3527	−	17.9313i	0.418140	−	0.724239i	0.995752	+	0.0920716i	$0.0293489\pi$
$614$	−14.0820	−	24.3907i	−0.568303	−	0.984330i
$615$		0			0
$616$	28.6762			1.15540
$617$	−4.63936	−	8.03560i	−0.186773	−	0.323501i	0.757399	−	0.652952i	$-0.226470\pi$
$617$	−4.63936	−	8.03560i	−0.186773	−	0.323501i	−0.944173	+	0.329451i	$0.893136\pi$
$618$		0			0
$619$	−2.89129			−0.116211			−0.0581053	−	0.998310i	$-0.518506\pi$
$619$	−2.89129			−0.116211			−0.0581053	+	0.998310i	$0.518506\pi$
$620$	4.06097			0.163093
$621$		0			0
$622$	−12.6616	+	21.9306i	−0.507686	+	0.879338i
$623$	13.5329	+	23.4396i	0.542183	+	0.939088i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	41.4926			1.65838
$627$		0			0
$628$	−2.60138			−0.103806
$629$	8.65844	−	14.9969i	0.345234	−	0.597964i
$630$		0			0
$631$	−15.2270	−	26.3740i	−0.606178	−	1.04993i	−0.991864	−	0.127301i	$-0.959369\pi$
$631$	−15.2270	−	26.3740i	−0.606178	−	1.04993i	0.385686	−	0.922630i	$-0.373965\pi$
$632$	7.78029	−	13.4759i	0.309483	−	0.536041i
$633$		0			0
$634$	41.9972			1.66792
$635$	−6.11991			−0.242861
$636$		0			0
$637$	−0.852959	−	1.47737i	−0.0337955	−	0.0585355i
$638$	−25.0812			−0.992975
$639$		0			0
$640$	6.63877	+	11.4987i	0.262420	+	0.454525i
$641$	−10.0369	+	17.3845i	−0.396434	+	0.686645i	−0.993283	−	0.115709i	$-0.963086\pi$
$641$	−10.0369	+	17.3845i	−0.396434	+	0.686645i	0.596849	+	0.802354i	$0.296419\pi$
$642$		0			0
$643$	1.04457	−	1.80924i	0.0411937	−	0.0713496i	−0.844693	−	0.535250i	$-0.820217\pi$
$643$	1.04457	−	1.80924i	0.0411937	−	0.0713496i	0.885887	+	0.463901i	$0.153551\pi$
$644$	0.882172	−	1.52797i	0.0347625	−	0.0602103i
$645$		0			0
$646$	36.8347	−	22.9850i	1.44924	−	0.904334i
$647$	2.10623			0.0828043			0.0414021	−	0.999143i	$-0.486818\pi$
$647$	2.10623			0.0828043			0.0414021	+	0.999143i	$0.486818\pi$
$648$		0			0
$649$	−31.0722	+	53.8187i	−1.21969	+	2.11257i
$650$	1.32748	+	2.29927i	0.0520682	+	0.0901847i
$651$		0			0
$652$	−0.118674	−	0.205549i	−0.00464763	−	0.00804994i
$653$	−1.83067			−0.0716395			−0.0358197	−	0.999358i	$-0.511404\pi$
$653$	−1.83067			−0.0716395			−0.0358197	+	0.999358i	$0.511404\pi$
$654$		0			0
$655$	7.44055	+	12.8874i	0.290726	+	0.503553i
$656$	15.6223	+	27.0587i	0.609950	+	1.05646i
$657$		0			0
$658$	36.2973			1.41502
$659$	12.0268	+	20.8310i	0.468497	+	0.811460i	0.999352	−	0.0360024i	$-0.0114624\pi$
$659$	12.0268	+	20.8310i	0.468497	+	0.811460i	−0.530855	+	0.847463i	$0.678129\pi$
$660$		0			0
$661$	8.72110	+	15.1054i	0.339211	+	0.587531i	0.984285	−	0.176589i	$-0.0565065\pi$
$661$	8.72110	+	15.1054i	0.339211	+	0.587531i	−0.645073	+	0.764121i	$0.723173\pi$
$662$	16.8172	−	29.1282i	0.653617	−	1.13210i
$663$		0			0
$664$	6.37097			0.247241
$665$	−9.36941	−	4.98929i	−0.363330	−	0.193476i
$666$		0			0
$667$	1.23068	−	2.13159i	0.0476519	−	0.0825356i
$668$	5.49682	−	9.52077i	0.212678	−	0.368370i
$669$		0			0
$670$	0.840244	−	1.45535i	0.0324615	−	0.0562249i
$671$	30.2951	+	52.4726i	1.16953	+	2.02568i
$672$		0			0
$673$	47.5187			1.83171			0.915856	−	0.401506i	$-0.131513\pi$
$673$	47.5187			1.83171			0.915856	+	0.401506i	$0.131513\pi$
$674$	26.4868	+	45.8765i	1.02023	+	1.76709i
$675$		0			0
$676$	−8.05850			−0.309942
$677$	−14.5531			−0.559321			−0.279661	−	0.960099i	$-0.590222\pi$
$677$	−14.5531			−0.559321			−0.279661	+	0.960099i	$0.590222\pi$
$678$		0			0
$679$	4.94084	−	8.55778i	0.189612	−	0.328418i
$680$	−6.12231	−	10.6042i	−0.234780	−	0.406651i
$681$		0			0
$682$	25.2367	−	43.7112i	0.966363	−	1.67379i
$683$	−3.33714			−0.127692			−0.0638460	−	0.997960i	$-0.520337\pi$
$683$	−3.33714			−0.127692			−0.0638460	+	0.997960i	$0.520337\pi$
$684$		0			0
$685$	17.3504			0.662923
$686$	−16.3553	+	28.3282i	−0.624448	+	1.08158i
$687$		0			0
$688$	11.2287	+	19.4487i	0.428092	+	0.741476i
$689$	1.75163	−	3.03391i	0.0667318	−	0.115583i
$690$		0			0
$691$	−19.3318			−0.735415			−0.367708	−	0.929941i	$-0.619857\pi$
$691$	−19.3318			−0.735415			−0.367708	+	0.929941i	$0.619857\pi$
$692$	−10.2868			−0.391044
$693$		0			0
$694$	−2.74862	−	4.76075i	−0.104336	−	0.180716i
$695$	6.70534			0.254348
$696$		0			0
$697$	−18.8957	−	32.7283i	−0.715725	−	1.23967i
$698$	−14.8549	+	25.7295i	−0.562268	+	0.973876i
$699$		0			0
$700$	0.938437	−	1.62542i	0.0354696	−	0.0614351i
$701$	4.96892	−	8.60643i	0.187674	−	0.325060i	−0.756801	−	0.653646i	$-0.773239\pi$
$701$	4.96892	−	8.60643i	0.187674	−	0.325060i	0.944474	+	0.328586i	$0.106572\pi$
$702$		0			0
$703$	−0.432375	−	12.6065i	−0.0163073	−	0.475464i
$704$	17.2584			0.650452
$705$		0			0
$706$	−6.90389	+	11.9579i	−0.259831	+	0.450041i
$707$	13.5523	+	23.4732i	0.509686	+	0.882801i
$708$		0			0
$709$	−18.6059	−	32.2264i	−0.698760	−	1.21029i	−0.968897	−	0.247466i	$-0.920402\pi$
$709$	−18.6059	−	32.2264i	−0.698760	−	1.21029i	0.270136	−	0.962822i	$-0.412931\pi$
$710$	14.7044			0.551846
$711$		0			0
$712$	11.3709	+	19.6950i	0.426143	+	0.738101i
$713$	2.47661	+	4.28961i	0.0927497	+	0.160647i
$714$		0			0
$715$	9.17891			0.343272
$716$	−5.48003	−	9.49169i	−0.204798	−	0.354721i
$717$		0			0
$718$	6.94966	+	12.0372i	0.259359	+	0.449223i
$719$	−1.32109	+	2.28819i	−0.0492683	+	0.0853351i	−0.889608	−	0.456725i	$-0.849022\pi$
$719$	−1.32109	+	2.28819i	−0.0492683	+	0.0853351i	0.840340	+	0.542060i	$0.182356\pi$
$720$		0			0
$721$	−28.1980			−1.05015
$722$	13.8995	−	28.4083i	0.517284	−	1.05725i
$723$		0			0
$724$	−3.80832	+	6.59621i	−0.141535	+	0.245146i
$725$	1.30917	−	2.26755i	0.0486213	−	0.0842145i
$726$		0			0
$727$	5.08653	−	8.81013i	0.188649	−	0.326750i	−0.756151	−	0.654397i	$-0.772923\pi$
$727$	5.08653	−	8.81013i	0.188649	−	0.326750i	0.944800	+	0.327647i	$0.106256\pi$
$728$	3.97400	+	6.88317i	0.147286	+	0.255107i
$729$		0			0
$730$	17.0615			0.631476
$731$	−13.5815	−	23.5238i	−0.502329	−	0.870060i
$732$		0			0
$733$	14.8222			0.547472			0.273736	−	0.961805i	$-0.411741\pi$
$733$	14.8222			0.547472			0.273736	+	0.961805i	$0.411741\pi$
$734$	24.0023			0.885942
$735$		0			0
$736$	1.94720	−	3.37266i	0.0717749	−	0.124318i
$737$	−2.90494	−	5.03151i	−0.107005	−	0.185338i
$738$		0			0
$739$	−17.7433	+	30.7323i	−0.652697	+	1.13050i	0.329769	+	0.944062i	$0.393029\pi$
$739$	−17.7433	+	30.7323i	−0.652697	+	1.13050i	−0.982466	+	0.186443i	$0.940304\pi$
$740$	2.23031			0.0819877
$741$		0			0
$742$	−8.90325			−0.326849
$743$	−4.36941	+	7.56804i	−0.160298	+	0.277645i	−0.934976	−	0.354712i	$-0.884579\pi$
$743$	−4.36941	+	7.56804i	−0.160298	+	0.277645i	0.774677	+	0.632357i	$0.217912\pi$
$744$		0			0
$745$	−7.19642	−	12.4646i	−0.263656	−	0.456666i
$746$	20.0708	−	34.7637i	0.734846	−	1.27279i
$747$		0			0
$748$	26.5408			0.970428
$749$	43.6342			1.59436
$750$		0			0
$751$	−6.54957	−	11.3442i	−0.238997	−	0.413955i	0.721430	−	0.692488i	$-0.243485\pi$
$751$	−6.54957	−	11.3442i	−0.238997	−	0.413955i	−0.960427	+	0.278533i	$0.910152\pi$
$752$	44.3011			1.61549
$753$		0			0
$754$	−3.47580	−	6.02026i	−0.126581	−	0.219245i
$755$	6.36093	−	11.0175i	0.231498	−	0.400966i
$756$		0			0
$757$	8.21901	−	14.2357i	0.298725	−	0.517407i	−0.677119	−	0.735873i	$-0.736772\pi$
$757$	8.21901	−	14.2357i	0.298725	−	0.517407i	0.975845	+	0.218466i	$0.0701053\pi$
$758$	−13.8787	+	24.0386i	−0.504097	+	0.873121i
$759$		0			0
$760$	−7.87259	−	4.19222i	−0.285569	−	0.152068i
$761$	−16.3918			−0.594203			−0.297101	−	0.954846i	$-0.596020\pi$
$761$	−16.3918			−0.594203			−0.297101	+	0.954846i	$0.596020\pi$
$762$		0			0
$763$	6.84879	−	11.8625i	0.247943	−	0.429450i
$764$	−5.00738	−	8.67304i	−0.181161	−	0.313780i
$765$		0			0
$766$	−9.05337	−	15.6809i	−0.327112	−	0.566574i
$767$	−17.2242			−0.621929
$768$		0			0
$769$	25.0210	+	43.3377i	0.902282	+	1.56280i	0.824525	+	0.565825i	$0.191442\pi$
$769$	25.0210	+	43.3377i	0.902282	+	1.56280i	0.0777564	+	0.996972i	$0.475224\pi$
$770$	−11.6637	−	20.2022i	−0.420332	−	0.728036i
$771$		0			0
$772$	11.1875			0.402649
$773$	24.3436	+	42.1644i	0.875580	+	1.51655i	0.856144	+	0.516737i	$0.172854\pi$
$773$	24.3436	+	42.1644i	0.875580	+	1.51655i	0.0194356	+	0.999811i	$0.493813\pi$
$774$		0			0
$775$	2.63457	+	4.56320i	0.0946364	+	0.163915i
$776$	4.15151	−	7.19063i	0.149031	−	0.258129i
$777$		0			0
$778$	−60.6799			−2.17548
$779$	−24.2977	−	12.9387i	−0.870555	−	0.463577i
$780$		0			0
$781$	25.4185	−	44.0261i	0.909544	−	1.57538i
$782$	−4.68176	+	8.10905i	−0.167419	+	0.289979i
$783$		0			0

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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.832272 + 1.44154i) q^{2} +(-0.385355 - 0.667454i) q^{4} +(0.500000 - 0.866025i) q^{5} -2.43525 q^{7} -2.04621 q^{8} +O(q^{10})\)	\(q+(-0.832272 + 1.44154i) q^{2} +(-0.385355 - 0.667454i) q^{4} +(0.500000 - 0.866025i) q^{5} -2.43525 q^{7} -2.04621 q^{8} +(0.832272 + 1.44154i) q^{10} +5.75477 q^{11} +(0.797505 + 1.38132i) q^{13} +(2.02680 - 3.51051i) q^{14} +(2.47371 - 4.28460i) q^{16} +(-2.99203 + 5.18234i) q^{17} +(0.149412 + 4.35634i) q^{19} -0.770710 q^{20} +(-4.78953 + 8.29572i) q^{22} +(-0.470022 - 0.814102i) q^{23} +(-0.500000 - 0.866025i) q^{25} -2.65497 q^{26} +(0.938437 + 1.62542i) q^{28} +(1.30917 + 2.26755i) q^{29} -5.26913 q^{31} +(2.07140 + 3.58777i) q^{32} +(-4.98037 - 8.62625i) q^{34} +(-1.21763 + 2.10899i) q^{35} -2.89384 q^{37} +(-6.40418 - 3.41028i) q^{38} +(-1.02310 + 1.77207i) q^{40} +(-3.15767 + 5.46925i) q^{41} +(-2.26961 + 3.93108i) q^{43} +(-2.21763 - 3.84104i) q^{44} +1.56475 q^{46} +(4.47718 + 7.75471i) q^{47} -1.06953 q^{49} +1.66454 q^{50} +(0.614645 - 1.06460i) q^{52} +(-1.09819 - 1.90213i) q^{53} +(2.87738 - 4.98377i) q^{55} +4.98304 q^{56} -4.35834 q^{58} +(-5.39939 + 9.35202i) q^{59} +(5.26434 + 9.11811i) q^{61} +(4.38535 - 7.59566i) q^{62} +2.99898 q^{64} +1.59501 q^{65} +(-0.504789 - 0.874320i) q^{67} +4.61197 q^{68} +(-2.02680 - 3.51051i) q^{70} +(4.41694 - 7.65036i) q^{71} +(5.12499 - 8.87674i) q^{73} +(2.40846 - 4.17157i) q^{74} +(2.85008 - 1.77846i) q^{76} -14.0143 q^{77} +(-3.80229 + 6.58577i) q^{79} +(-2.47371 - 4.28460i) q^{80} +(-5.25609 - 9.10381i) q^{82} -3.11355 q^{83} +(2.99203 + 5.18234i) q^{85} +(-3.77787 - 6.54346i) q^{86} -11.7755 q^{88} +(-5.55706 - 9.62511i) q^{89} +(-1.94213 - 3.36387i) q^{91} +(-0.362251 + 0.627436i) q^{92} -14.9049 q^{94} +(3.84741 + 2.04877i) q^{95} +(-2.02888 + 3.51412i) q^{97} +(0.890144 - 1.54177i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8}+O(q^{10}) \) 8 * q + q^2 - 5 * q^4 + 4 * q^5 - 8 * q^7 - 24 * q^8	\( 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8} - q^{10} + 4 q^{11} - 7 q^{13} - q^{14} - 7 q^{16} - q^{17} + 5 q^{19} - 10 q^{20} - 2 q^{22} + 2 q^{23} - 4 q^{25} - 6 q^{26} + 19 q^{28} - q^{29} + 30 q^{32} - 15 q^{34} - 4 q^{35} - 4 q^{37} - 13 q^{38} - 12 q^{40} - 8 q^{41} - q^{43} - 12 q^{44} + 24 q^{46} - 12 q^{47} - 20 q^{49} - 2 q^{50} + 3 q^{52} - 5 q^{53} + 2 q^{55} + 82 q^{56} - 54 q^{58} - 5 q^{59} + 37 q^{62} + 112 q^{64} - 14 q^{65} - 4 q^{67} - 32 q^{68} + q^{70} + 20 q^{71} + 20 q^{73} + 25 q^{74} + 63 q^{76} - 28 q^{77} - 17 q^{79} + 7 q^{80} - 21 q^{82} - 2 q^{83} + q^{85} + 8 q^{86} - 14 q^{88} + 11 q^{89} - 6 q^{91} - q^{92} - 62 q^{94} + 4 q^{95} - q^{97} + 9 q^{98}+O(q^{100}) \) 8 * q + q^2 - 5 * q^4 + 4 * q^5 - 8 * q^7 - 24 * q^8 - q^10 + 4 * q^11 - 7 * q^13 - q^14 - 7 * q^16 - q^17 + 5 * q^19 - 10 * q^20 - 2 * q^22 + 2 * q^23 - 4 * q^25 - 6 * q^26 + 19 * q^28 - q^29 + 30 * q^32 - 15 * q^34 - 4 * q^35 - 4 * q^37 - 13 * q^38 - 12 * q^40 - 8 * q^41 - q^43 - 12 * q^44 + 24 * q^46 - 12 * q^47 - 20 * q^49 - 2 * q^50 + 3 * q^52 - 5 * q^53 + 2 * q^55 + 82 * q^56 - 54 * q^58 - 5 * q^59 + 37 * q^62 + 112 * q^64 - 14 * q^65 - 4 * q^67 - 32 * q^68 + q^70 + 20 * q^71 + 20 * q^73 + 25 * q^74 + 63 * q^76 - 28 * q^77 - 17 * q^79 + 7 * q^80 - 21 * q^82 - 2 * q^83 + q^85 + 8 * q^86 - 14 * q^88 + 11 * q^89 - 6 * q^91 - q^92 - 62 * q^94 + 4 * q^95 - q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more