LMFDB - Embedded newform 855.2.k.h.406.2

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			406.2
Root			$-1.02359 - 1.77290i$ of defining polynomial
Character	$\chi$	$=$	855.406
Dual form			855.2.k.h.676.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.595455 - 1.03136i) q^{2} +(0.290867 - 0.503797i) q^{4} +(0.500000 + 0.866025i) q^{5} -0.609175 q^{7} -3.07461 q^{8} +O(q^{10})$	\(q+(-0.595455 - 1.03136i) q^{2} +(0.290867 - 0.503797i) q^{4} +(0.500000 + 0.866025i) q^{5} -0.609175 q^{7} -3.07461 q^{8} +(0.595455 - 1.03136i) q^{10} -4.48517 q^{11} +(-2.21900 + 3.84342i) q^{13} +(0.362736 + 0.628278i) q^{14} +(1.24906 + 2.16343i) q^{16} +(1.45172 + 2.51445i) q^{17} +(3.60532 + 2.44983i) q^{19} +0.581734 q^{20} +(2.67071 + 4.62581i) q^{22} +(-1.42363 + 2.46580i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.28525 q^{26} +(-0.177189 + 0.306901i) q^{28} +(0.558149 - 0.966742i) q^{29} -6.22908 q^{31} +(-1.58710 + 2.74893i) q^{32} +(1.72886 - 2.99448i) q^{34} +(-0.304588 - 0.527561i) q^{35} -3.77264 q^{37} +(0.379847 - 5.17714i) q^{38} +(-1.53731 - 2.66269i) q^{40} +(-4.15184 - 7.19120i) q^{41} +(4.99438 + 8.65053i) q^{43} +(-1.30459 + 2.25961i) q^{44} +3.39082 q^{46} +(-2.94250 + 5.09656i) q^{47} -6.62891 q^{49} +1.19091 q^{50} +(1.29087 + 2.23585i) q^{52} +(4.22436 - 7.31681i) q^{53} +(-2.24258 - 3.88427i) q^{55} +1.87298 q^{56} -1.32941 q^{58} +(5.11793 + 8.86451i) q^{59} +(2.49099 - 4.31453i) q^{61} +(3.70913 + 6.42441i) q^{62} +8.77641 q^{64} -4.43800 q^{65} +(-4.23808 + 7.34057i) q^{67} +1.68903 q^{68} +(-0.362736 + 0.628278i) q^{70} +(5.80995 + 10.0631i) q^{71} +(-1.86162 - 3.22443i) q^{73} +(2.24644 + 3.89095i) q^{74} +(2.28289 - 1.10377i) q^{76} +2.73225 q^{77} +(-4.51908 - 7.82728i) q^{79} +(-1.24906 + 2.16343i) q^{80} +(-4.94447 + 8.56407i) q^{82} +2.12178 q^{83} +(-1.45172 + 2.51445i) q^{85} +(5.94786 - 10.3020i) q^{86} +13.7901 q^{88} +(3.96608 - 6.86946i) q^{89} +(1.35176 - 2.34131i) q^{91} +(0.828173 + 1.43444i) q^{92} +7.00850 q^{94} +(-0.318955 + 4.34721i) q^{95} +(4.83628 + 8.37668i) q^{97} +(3.94721 + 6.83677i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.595455	−	1.03136i	−0.421050	−	0.729280i	0.574992	−	0.818159i	$-0.305005\pi$
$2$	−0.595455	−	1.03136i	−0.421050	−	0.729280i	−0.996042	+	0.0888786i	$0.971672\pi$
$3$		0			0
$3$		0			0
$4$	0.290867	−	0.503797i	0.145434	−	0.251898i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	−0.609175			−0.230247			−0.115123	−	0.993351i	$-0.536726\pi$
$7$	−0.609175			−0.230247			−0.115123	+	0.993351i	$0.536726\pi$
$8$	−3.07461			−1.08704
$9$		0			0
$10$	0.595455	−	1.03136i	0.188299	−	0.326144i
$11$	−4.48517			−1.35233			−0.676164	−	0.736751i	$-0.736359\pi$
$11$	−4.48517			−1.35233			−0.676164	+	0.736751i	$0.736359\pi$
$12$		0			0
$13$	−2.21900	+	3.84342i	−0.615439	+	1.06597i	0.374868	+	0.927078i	$0.377688\pi$
$13$	−2.21900	+	3.84342i	−0.615439	+	1.06597i	−0.990307	+	0.138894i	$0.955645\pi$
$14$	0.362736	+	0.628278i	0.0969454	+	0.167914i
$15$		0			0
$16$	1.24906	+	2.16343i	0.312265	+	0.540858i
$17$	1.45172	+	2.51445i	0.352093	+	0.609843i	0.986616	−	0.163061i	$-0.0521368\pi$
$17$	1.45172	+	2.51445i	0.352093	+	0.609843i	−0.634523	+	0.772904i	$0.718803\pi$
$18$		0			0
$19$	3.60532	+	2.44983i	0.827117	+	0.562030i
$19$	3.60532	+	2.44983i	0.827117	+	0.562030i
$20$	0.581734			0.130080
$21$		0			0
$22$	2.67071	+	4.62581i	0.569398	+	0.986227i
$23$	−1.42363	+	2.46580i	−0.296847	+	0.514154i	−0.975413	−	0.220386i	$-0.929268\pi$
$23$	−1.42363	+	2.46580i	−0.296847	+	0.514154i	0.678566	+	0.734540i	$0.262602\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	5.28525			1.03652
$27$		0			0
$28$	−0.177189	+	0.306901i	−0.0334856	+	0.0579987i
$29$	0.558149	−	0.966742i	0.103646	−	0.179519i	−0.809538	−	0.587067i	$-0.800283\pi$
$29$	0.558149	−	0.966742i	0.103646	−	0.179519i	0.913184	+	0.407547i	$0.133616\pi$
$30$		0			0
$31$	−6.22908			−1.11877			−0.559387	−	0.828906i	$-0.688964\pi$
$31$	−6.22908			−1.11877			−0.559387	+	0.828906i	$0.688964\pi$
$32$	−1.58710	+	2.74893i	−0.280562	+	0.485947i
$33$		0			0
$34$	1.72886	−	2.99448i	0.296498	−	0.513549i
$35$	−0.304588	−	0.527561i	−0.0514847	−	0.0891741i
$36$		0			0
$37$	−3.77264			−0.620219			−0.310109	−	0.950701i	$-0.600366\pi$
$37$	−3.77264			−0.620219			−0.310109	+	0.950701i	$0.600366\pi$
$38$	0.379847	−	5.17714i	0.0616193	−	0.839843i
$39$		0			0
$40$	−1.53731	−	2.66269i	−0.243069	−	0.421009i
$41$	−4.15184	−	7.19120i	−0.648409	−	1.12308i	−0.983503	−	0.180893i	$-0.942101\pi$
$41$	−4.15184	−	7.19120i	−0.648409	−	1.12308i	0.335094	−	0.942185i	$-0.391232\pi$
$42$		0			0
$43$	4.99438	+	8.65053i	0.761637	+	1.31919i	0.942007	+	0.335594i	$0.108937\pi$
$43$	4.99438	+	8.65053i	0.761637	+	1.31919i	−0.180370	+	0.983599i	$0.557730\pi$
$44$	−1.30459	+	2.25961i	−0.196674	+	0.340649i
$45$		0			0
$46$	3.39082			0.499950
$47$	−2.94250	+	5.09656i	−0.429208	+	0.743409i	−0.996803	−	0.0798983i	$-0.974540\pi$
$47$	−2.94250	+	5.09656i	−0.429208	+	0.743409i	0.567595	+	0.823308i	$0.307874\pi$
$48$		0			0
$49$	−6.62891			−0.946986
$50$	1.19091			0.168420
$51$		0			0
$52$	1.29087	+	2.23585i	0.179011	+	0.310056i
$53$	4.22436	−	7.31681i	0.580261	−	1.00504i	−0.415188	−	0.909736i	$-0.636284\pi$
$53$	4.22436	−	7.31681i	0.580261	−	1.00504i	0.995448	−	0.0953049i	$-0.0303826\pi$
$54$		0			0
$55$	−2.24258	−	3.88427i	−0.302390	−	0.523755i
$56$	1.87298			0.250287
$57$		0			0
$58$	−1.32941			−0.174560
$59$	5.11793	+	8.86451i	0.666297	+	1.15406i	0.978932	+	0.204187i	$0.0654552\pi$
$59$	5.11793	+	8.86451i	0.666297	+	1.15406i	−0.312634	+	0.949874i	$0.601211\pi$
$60$		0			0
$61$	2.49099	−	4.31453i	0.318939	−	0.552419i	−0.661328	−	0.750097i	$-0.730007\pi$
$61$	2.49099	−	4.31453i	0.318939	−	0.552419i	0.980267	+	0.197678i	$0.0633401\pi$
$62$	3.70913	+	6.42441i	0.471060	+	0.815900i
$63$		0			0
$64$	8.77641			1.09705
$65$	−4.43800			−0.550466
$66$		0			0
$67$	−4.23808	+	7.34057i	−0.517764	+	0.896794i	0.482023	+	0.876159i	$0.339902\pi$
$67$	−4.23808	+	7.34057i	−0.517764	+	0.896794i	−0.999787	+	0.0206350i	$0.993431\pi$
$68$	1.68903			0.204825
$69$		0			0
$70$	−0.362736	+	0.628278i	−0.0433553	+	0.0750936i
$71$	5.80995	+	10.0631i	0.689514	+	1.19427i	0.971995	+	0.235001i	$0.0755093\pi$
$71$	5.80995	+	10.0631i	0.689514	+	1.19427i	−0.282481	+	0.959273i	$0.591157\pi$
$72$		0			0
$73$	−1.86162	−	3.22443i	−0.217887	−	0.377391i	0.736275	−	0.676682i	$-0.236583\pi$
$73$	−1.86162	−	3.22443i	−0.217887	−	0.377391i	−0.954162	+	0.299292i	$0.903250\pi$
$74$	2.24644	+	3.89095i	0.261143	+	0.452313i
$75$		0			0
$76$	2.28289	−	1.10377i	0.261865	−	0.126611i
$77$	2.73225			0.311369
$78$		0			0
$79$	−4.51908	−	7.82728i	−0.508437	−	0.880638i	−0.999952	−	0.00976923i	$-0.996890\pi$
$79$	−4.51908	−	7.82728i	−0.508437	−	0.880638i	0.491516	−	0.870869i	$-0.336443\pi$
$80$	−1.24906	+	2.16343i	−0.139649	+	0.241879i
$81$		0			0
$82$	−4.94447	+	8.56407i	−0.546025	+	0.945744i
$83$	2.12178			0.232896			0.116448	−	0.993197i	$-0.462849\pi$
$83$	2.12178			0.232896			0.116448	+	0.993197i	$0.462849\pi$
$84$		0			0
$85$	−1.45172	+	2.51445i	−0.157461	+	0.272730i
$86$	5.94786	−	10.3020i	0.641374	−	1.11089i
$87$		0			0
$88$	13.7901			1.47003
$89$	3.96608	−	6.86946i	0.420404	−	0.728161i	−0.575575	−	0.817749i	$-0.695222\pi$
$89$	3.96608	−	6.86946i	0.420404	−	0.728161i	0.995979	+	0.0895879i	$0.0285550\pi$
$90$		0			0
$91$	1.35176	−	2.34131i	0.141703	−	0.245436i
$92$	0.828173	+	1.43444i	0.0863430	+	0.149551i
$93$		0			0
$94$	7.00850			0.722872
$95$	−0.318955	+	4.34721i	−0.0327241	+	0.446015i
$96$		0			0
$97$	4.83628	+	8.37668i	0.491050	+	0.850523i	0.999947	−	0.0103043i	$-0.00328001\pi$
$97$	4.83628	+	8.37668i	0.491050	+	0.850523i	−0.508897	+	0.860827i	$0.669947\pi$
$98$	3.94721	+	6.83677i	0.398729	+	0.690619i
$99$		0			0
$100$	0.290867	+	0.503797i	0.0290867	+	0.0503797i
$101$	−0.485632	+	0.841140i	−0.0483222	+	0.0836965i	−0.889175	−	0.457568i	$-0.848721\pi$
$101$	−0.485632	+	0.841140i	−0.0483222	+	0.0836965i	0.840853	+	0.541264i	$0.182054\pi$
$102$		0			0
$103$	−3.34143			−0.329241			−0.164620	−	0.986357i	$-0.552640\pi$
$103$	−3.34143			−0.329241			−0.164620	+	0.986357i	$0.552640\pi$
$104$	6.82256	−	11.8170i	0.669007	−	1.15875i
$105$		0			0
$106$	−10.0617			−0.977275
$107$	−9.51655			−0.920000			−0.460000	−	0.887919i	$-0.652151\pi$
$107$	−9.51655			−0.920000			−0.460000	+	0.887919i	$0.652151\pi$
$108$		0			0
$109$	−2.77178	−	4.80087i	−0.265489	−	0.459840i	0.702203	−	0.711977i	$-0.252200\pi$
$109$	−2.77178	−	4.80087i	−0.265489	−	0.459840i	−0.967692	+	0.252137i	$0.918867\pi$
$110$	−2.67071	+	4.62581i	−0.254643	+	0.441054i
$111$		0			0
$112$	−0.760896	−	1.31791i	−0.0718979	−	0.124531i
$113$	−1.54134			−0.144997			−0.0724987	−	0.997369i	$-0.523097\pi$
$113$	−1.54134			−0.144997			−0.0724987	+	0.997369i	$0.523097\pi$
$114$		0			0
$115$	−2.84726			−0.265508
$116$	−0.324694	−	0.562387i	−0.0301471	−	0.0522163i
$117$		0			0
$118$	6.09499	−	10.5568i	0.561089	−	0.971835i
$119$	−0.884350	−	1.53174i	−0.0810682	−	0.140414i
$120$		0			0
$121$	9.11672			0.828793
$122$	−5.93310			−0.537158
$123$		0			0
$124$	−1.81183	+	3.13819i	−0.162707	+	0.281818i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−1.15274	+	1.99661i	−0.102289	+	0.177171i	−0.912628	−	0.408792i	$-0.865950\pi$
$127$	−1.15274	+	1.99661i	−0.102289	+	0.177171i	0.810338	+	0.585963i	$0.199283\pi$
$128$	−2.05176	−	3.55376i	−0.181352	−	0.314111i
$129$		0			0
$130$	2.64263	+	4.57716i	0.231774	+	0.401444i
$131$	−6.45905	−	11.1874i	−0.564330	−	0.977448i	−0.997112	−	0.0759493i	$-0.975801\pi$
$131$	−6.45905	−	11.1874i	−0.564330	−	0.977448i	0.432782	−	0.901499i	$-0.357532\pi$
$132$		0			0
$133$	−2.19627	−	1.49238i	−0.190441	−	0.129405i
$134$	10.0943			0.872018
$135$		0			0
$136$	−4.46346	−	7.73095i	−0.382739	−	0.662923i
$137$	−6.36677	+	11.0276i	−0.543950	+	0.942149i	0.454722	+	0.890633i	$0.349739\pi$
$137$	−6.36677	+	11.0276i	−0.543950	+	0.942149i	−0.998672	+	0.0515159i	$0.983595\pi$
$138$		0			0
$139$	−5.30433	+	9.18738i	−0.449908	+	0.779263i	−0.998380	−	0.0569059i	$-0.981876\pi$
$139$	−5.30433	+	9.18738i	−0.449908	+	0.779263i	0.548472	+	0.836169i	$0.315210\pi$
$140$	−0.354378			−0.0299504
$141$		0			0
$142$	6.91913	−	11.9843i	0.580640	−	1.00570i
$143$	9.95258	−	17.2384i	0.832276	−	1.44154i
$144$		0			0
$145$	1.11630			0.0927035
$146$	−2.21703	+	3.84000i	−0.183482	+	0.317801i
$147$		0			0
$148$	−1.09734	+	1.90065i	−0.0902006	+	0.156232i
$149$	1.88653	+	3.26757i	0.154551	+	0.267690i	0.932895	−	0.360147i	$-0.117274\pi$
$149$	1.88653	+	3.26757i	0.154551	+	0.267690i	−0.778344	+	0.627837i	$0.783940\pi$
$150$		0			0
$151$	−9.51562			−0.774370			−0.387185	−	0.922002i	$-0.626553\pi$
$151$	−9.51562			−0.774370			−0.387185	+	0.922002i	$0.626553\pi$
$152$	−11.0850	−	7.53228i	−0.899109	−	0.610948i
$153$		0			0
$154$	−1.62693	−	2.81793i	−0.131102	−	0.227075i
$155$	−3.11454	−	5.39454i	−0.250166	−	0.433300i
$156$		0			0
$157$	1.72822	+	2.99336i	0.137927	+	0.238896i	0.926712	−	0.375773i	$-0.122623\pi$
$157$	1.72822	+	2.99336i	0.137927	+	0.238896i	−0.788785	+	0.614669i	$0.789290\pi$
$158$	−5.38182	+	9.32158i	−0.428155	+	0.741585i
$159$		0			0
$160$	−3.17419			−0.250942
$161$	0.867239	−	1.50210i	0.0683480	−	0.118382i
$162$		0			0
$163$	6.65283			0.521090			0.260545	−	0.965462i	$-0.416098\pi$
$163$	6.65283			0.521090			0.260545	+	0.965462i	$0.416098\pi$
$164$	−4.83054			−0.377202
$165$		0			0
$166$	−1.26343	−	2.18832i	−0.0980609	−	0.169846i
$167$	−8.22775	+	14.2509i	−0.636682	+	1.10277i	0.349474	+	0.936946i	$0.386360\pi$
$167$	−8.22775	+	14.2509i	−0.636682	+	1.10277i	−0.986156	+	0.165820i	$0.946973\pi$
$168$		0			0
$169$	−3.34790	−	5.79874i	−0.257531	−	0.446057i
$170$	3.45773			0.265195
$171$		0			0
$172$	5.81081			0.443070
$173$	−11.3912	−	19.7302i	−0.866058	−	1.50006i	−0.865992	−	0.500057i	$-0.833312\pi$
$173$	−11.3912	−	19.7302i	−0.866058	−	1.50006i	−6.58713e−5	−	1.00000i	$-0.500021\pi$
$174$		0			0
$175$	0.304588	−	0.527561i	0.0230247	−	0.0398799i
$176$	−5.60224	−	9.70336i	−0.422284	−	0.731418i
$177$		0			0
$178$	−9.44650			−0.708045
$179$	2.32916			0.174090			0.0870449	−	0.996204i	$-0.472258\pi$
$179$	2.32916			0.174090			0.0870449	+	0.996204i	$0.472258\pi$
$180$		0			0
$181$	11.1696	−	19.3463i	0.830230	−	1.43800i	−0.0676258	−	0.997711i	$-0.521542\pi$
$181$	11.1696	−	19.3463i	0.830230	−	1.43800i	0.897856	−	0.440290i	$-0.145124\pi$
$182$	−3.21965			−0.238656
$183$		0			0
$184$	4.37710	−	7.58137i	0.322684	−	0.558906i
$185$	−1.88632	−	3.26721i	−0.138685	−	0.240210i
$186$		0			0
$187$	−6.51119	−	11.2777i	−0.476145	−	0.824708i
$188$	1.71175	+	2.96484i	0.124842	+	0.216233i
$189$		0			0
$190$	4.67346	−	2.25961i	0.339048	−	0.163929i
$191$	−2.23766			−0.161911			−0.0809556	−	0.996718i	$-0.525797\pi$
$191$	−2.23766			−0.161911			−0.0809556	+	0.996718i	$0.525797\pi$
$192$		0			0
$193$	2.27153	+	3.93441i	0.163508	+	0.283205i	0.936125	−	0.351669i	$-0.114386\pi$
$193$	2.27153	+	3.93441i	0.163508	+	0.283205i	−0.772616	+	0.634873i	$0.781052\pi$
$194$	5.75957	−	9.97587i	0.413513	−	0.716226i
$195$		0			0
$196$	−1.92813	+	3.33962i	−0.137724	+	0.238544i
$197$	19.2236			1.36962			0.684812	−	0.728720i	$-0.259884\pi$
$197$	19.2236			1.36962			0.684812	+	0.728720i	$0.259884\pi$
$198$		0			0
$199$	3.07547	−	5.32687i	0.218014	−	0.377612i	−0.736186	−	0.676779i	$-0.763375\pi$
$199$	3.07547	−	5.32687i	0.218014	−	0.377612i	0.954201	+	0.299167i	$0.0967087\pi$
$200$	1.53731	−	2.66269i	0.108704	−	0.188281i
$201$		0			0
$202$	1.15669			0.0813843
$203$	−0.340010	+	0.588915i	−0.0238641	+	0.0413338i
$204$		0			0
$205$	4.15184	−	7.19120i	0.289977	−	0.502255i
$206$	1.98967	+	3.44621i	0.138627	+	0.240109i
$207$		0			0
$208$	−11.0866			−0.768720
$209$	−16.1705	−	10.9879i	−1.11853	−	0.760049i
$210$		0			0
$211$	−6.34661	−	10.9926i	−0.436919	−	0.756765i	0.560531	−	0.828133i	$-0.310597\pi$
$211$	−6.34661	−	10.9926i	−0.436919	−	0.756765i	−0.997450	+	0.0713679i	$0.977264\pi$
$212$	−2.45746	−	4.25644i	−0.168779	−	0.292333i
$213$		0			0
$214$	5.66668	+	9.81497i	0.387366	+	0.670938i
$215$	−4.99438	+	8.65053i	−0.340614	+	0.589961i
$216$		0			0
$217$	3.79460			0.257594
$218$	−3.30094	+	5.71740i	−0.223568	+	0.387231i
$219$		0			0
$220$	−2.60918			−0.175911
$221$	−12.8854			−0.866767
$222$		0			0
$223$	−11.2688	−	19.5181i	−0.754614	−	1.30703i	−0.945566	−	0.325430i	$-0.894491\pi$
$223$	−11.2688	−	19.5181i	−0.754614	−	1.30703i	0.190952	−	0.981599i	$-0.438842\pi$
$224$	0.966820	−	1.67458i	0.0645984	−	0.111888i
$225$		0			0
$226$	0.917800	+	1.58968i	0.0610512	+	0.105744i
$227$	−18.1124			−1.20216			−0.601080	−	0.799189i	$-0.705263\pi$
$227$	−18.1124			−1.20216			−0.601080	+	0.799189i	$0.705263\pi$
$228$		0			0
$229$	−9.41604			−0.622229			−0.311115	−	0.950372i	$-0.600702\pi$
$229$	−9.41604			−0.622229			−0.311115	+	0.950372i	$0.600702\pi$
$230$	1.69541	+	2.93654i	0.111792	+	0.193630i
$231$		0			0
$232$	−1.71609	+	2.97236i	−0.112667	+	0.195145i
$233$	7.85000	+	13.5966i	0.514271	+	0.890743i	0.999863	+	0.0165573i	$0.00527061\pi$
$233$	7.85000	+	13.5966i	0.514271	+	0.890743i	−0.485592	+	0.874185i	$0.661396\pi$
$234$		0			0
$235$	−5.88500			−0.383895
$236$	5.95455			0.387608
$237$		0			0
$238$	−1.05318	+	1.82416i	−0.0682676	+	0.118243i
$239$	23.4610			1.51757			0.758783	−	0.651344i	$-0.225795\pi$
$239$	23.4610			1.51757			0.758783	+	0.651344i	$0.225795\pi$
$240$		0			0
$241$	−6.58469	+	11.4050i	−0.424157	+	0.734662i	−0.996341	−	0.0854634i	$-0.972763\pi$
$241$	−6.58469	+	11.4050i	−0.424157	+	0.734662i	0.572184	+	0.820125i	$0.306096\pi$
$242$	−5.42860	−	9.40260i	−0.348963	−	0.604422i
$243$		0			0
$244$	−1.44910	−	2.50991i	−0.0927689	−	0.160680i
$245$	−3.31445	−	5.74080i	−0.211753	−	0.366766i
$246$		0			0
$247$	−17.4159	+	8.42058i	−1.10815	+	0.535789i
$248$	19.1520			1.21615
$249$		0			0
$250$	0.595455	+	1.03136i	0.0376599	+	0.0652288i
$251$	−8.66257	+	15.0040i	−0.546776	+	0.947045i	0.451716	+	0.892162i	$0.350812\pi$
$251$	−8.66257	+	15.0040i	−0.546776	+	0.947045i	−0.998493	+	0.0548830i	$0.982521\pi$
$252$		0			0
$253$	6.38521	−	11.0595i	0.401435	−	0.695305i
$254$	2.74563			0.172276
$255$		0			0
$256$	6.33295	−	10.9690i	0.395809	−	0.685562i
$257$	−2.83980	+	4.91867i	−0.177142	+	0.306818i	−0.940900	−	0.338683i	$-0.890018\pi$
$257$	−2.83980	+	4.91867i	−0.177142	+	0.306818i	0.763759	+	0.645502i	$0.223352\pi$
$258$		0			0
$259$	2.29820			0.142803
$260$	−1.29087	+	2.23585i	−0.0800562	+	0.138661i
$261$		0			0
$262$	−7.69215	+	13.3232i	−0.475222	+	0.823109i
$263$	2.82882	+	4.89966i	0.174433	+	0.302126i	0.939965	−	0.341272i	$-0.110858\pi$
$263$	2.82882	+	4.89966i	0.174433	+	0.302126i	−0.765532	+	0.643398i	$0.777524\pi$
$264$		0			0
$265$	8.44872			0.519001
$266$	−0.231393	+	3.15379i	−0.0141876	+	0.193371i
$267$		0			0
$268$	2.46544	+	4.27026i	0.150601	+	0.260848i
$269$	−11.9959	−	20.7775i	−0.731402	−	1.26683i	−0.956284	−	0.292440i	$-0.905533\pi$
$269$	−11.9959	−	20.7775i	−0.731402	−	1.26683i	0.224881	−	0.974386i	$-0.427801\pi$
$270$		0			0
$271$	10.6497	+	18.4459i	0.646926	+	1.12051i	0.983853	+	0.178978i	$0.0572791\pi$
$271$	10.6497	+	18.4459i	0.646926	+	1.12051i	−0.336927	+	0.941531i	$0.609388\pi$
$272$	−3.62656	+	6.28138i	−0.219892	+	0.380865i
$273$		0			0
$274$	15.1645			0.916121
$275$	2.24258	−	3.88427i	0.135233	−	0.234230i
$276$		0			0
$277$	−0.821109			−0.0493357			−0.0246678	−	0.999696i	$-0.507853\pi$
$277$	−0.821109			−0.0493357			−0.0246678	+	0.999696i	$0.507853\pi$
$278$	12.6340			0.757735
$279$		0			0
$280$	0.936489	+	1.62205i	0.0559659	+	0.0969358i
$281$	−0.293739	+	0.508772i	−0.0175230	+	0.0303508i	−0.874654	−	0.484748i	$-0.838911\pi$
$281$	−0.293739	+	0.508772i	−0.0175230	+	0.0303508i	0.857131	+	0.515099i	$0.172245\pi$
$282$		0			0
$283$	−15.4712	−	26.7969i	−0.919667	−	1.59291i	−0.799921	−	0.600105i	$-0.795125\pi$
$283$	−15.4712	−	26.7969i	−0.919667	−	1.59291i	−0.119746	−	0.992805i	$-0.538208\pi$
$284$	6.75969			0.401114
$285$		0			0
$286$	−23.7052			−1.40172
$287$	2.52920	+	4.38070i	0.149294	+	0.258585i
$288$		0			0
$289$	4.28504	−	7.42191i	0.252061	−	0.436583i
$290$	−0.664705	−	1.15130i	−0.0390328	−	0.0676068i
$291$		0			0
$292$	−2.16594			−0.126752
$293$	−3.76271			−0.219820			−0.109910	−	0.993942i	$-0.535056\pi$
$293$	−3.76271			−0.219820			−0.109910	+	0.993942i	$0.535056\pi$
$294$		0			0
$295$	−5.11793	+	8.86451i	−0.297977	+	0.516112i
$296$	11.5994			0.674202
$297$		0			0
$298$	2.24669	−	3.89138i	0.130147	−	0.225422i
$299$	−6.31806	−	10.9432i	−0.365383	−	0.632861i
$300$		0			0
$301$	−3.04246	−	5.26969i	−0.175364	−	0.303740i
$302$	5.66612	+	9.81401i	0.326049	+	0.564733i
$303$		0			0
$304$	−0.796788	+	10.8598i	−0.0456989	+	0.622855i
$305$	4.98199			0.285268
$306$		0			0
$307$	−10.1709	−	17.6166i	−0.580485	−	1.00543i	−0.995422	−	0.0955798i	$-0.969529\pi$
$307$	−10.1709	−	17.6166i	−0.580485	−	1.00543i	0.414936	−	0.909850i	$-0.363804\pi$
$308$	0.794723	−	1.37650i	0.0452835	−	0.0784334i
$309$		0			0
$310$	−3.70913	+	6.42441i	−0.210665	+	0.364882i
$311$	7.67830			0.435397			0.217698	−	0.976016i	$-0.430145\pi$
$311$	7.67830			0.435397			0.217698	+	0.976016i	$0.430145\pi$
$312$		0			0
$313$	−11.9964	+	20.7783i	−0.678074	+	1.17446i	0.297486	+	0.954726i	$0.403852\pi$
$313$	−11.9964	+	20.7783i	−0.678074	+	1.17446i	−0.975560	+	0.219733i	$0.929481\pi$
$314$	2.05815	−	3.56482i	0.116148	−	0.201174i
$315$		0			0
$316$	−5.25781			−0.295775
$317$	0.519518	−	0.899831i	0.0291790	−	0.0505395i	−0.851067	−	0.525057i	$-0.824044\pi$
$317$	0.519518	−	0.899831i	0.0291790	−	0.0505395i	0.880246	+	0.474517i	$0.157377\pi$
$318$		0			0
$319$	−2.50339	+	4.33600i	−0.140163	+	0.242769i
$320$	4.38821	+	7.60059i	0.245308	+	0.424886i
$321$		0			0
$322$	−2.06561			−0.115112
$323$	−0.926066	+	12.6218i	−0.0515277	+	0.702298i
$324$		0			0
$325$	−2.21900	−	3.84342i	−0.123088	−	0.213194i
$326$	−3.96146	−	6.86145i	−0.219405	−	0.380020i
$327$		0			0
$328$	12.7653	+	22.1102i	0.704846	+	1.22083i
$329$	1.79250	−	3.10470i	0.0988236	−	0.171167i
$330$		0			0
$331$	30.8316			1.69466			0.847328	−	0.531069i	$-0.178210\pi$
$331$	30.8316			1.69466			0.847328	+	0.531069i	$0.178210\pi$
$332$	0.617157	−	1.06895i	0.0338709	−	0.0586661i
$333$		0			0
$334$	19.5970			1.07230
$335$	−8.47616			−0.463102
$336$		0			0
$337$	−10.3576	−	17.9400i	−0.564217	−	0.977252i	−0.997122	−	0.0758124i	$-0.975845\pi$
$337$	−10.3576	−	17.9400i	−0.564217	−	0.977252i	0.432906	−	0.901439i	$-0.357488\pi$
$338$	−3.98705	+	6.90577i	−0.216867	+	0.375625i
$339$		0			0
$340$	0.844513	+	1.46274i	0.0458002	+	0.0793282i
$341$	27.9384			1.51295
$342$		0			0
$343$	8.30239			0.448287
$344$	−15.3558	−	26.5970i	−0.827929	−	1.43402i
$345$		0			0
$346$	−13.5659	+	23.4968i	−0.729308	+	1.26320i
$347$	4.11068	+	7.11991i	0.220673	+	0.382217i	0.955013	−	0.296566i	$-0.0958413\pi$
$347$	4.11068	+	7.11991i	0.220673	+	0.382217i	−0.734340	+	0.678782i	$0.762508\pi$
$348$		0			0
$349$	11.9216			0.638150			0.319075	−	0.947730i	$-0.396628\pi$
$349$	11.9216			0.638150			0.319075	+	0.947730i	$0.396628\pi$
$350$	−0.725473			−0.0387782
$351$		0			0
$352$	7.11839	−	12.3294i	0.379412	−	0.657160i
$353$	11.7983			0.627959			0.313980	−	0.949430i	$-0.398338\pi$
$353$	11.7983			0.627959			0.313980	+	0.949430i	$0.398338\pi$
$354$		0			0
$355$	−5.80995	+	10.0631i	−0.308360	+	0.534095i
$356$	−2.30721	−	3.99620i	−0.122282	−	0.211798i
$357$		0			0
$358$	−1.38691	−	2.40220i	−0.0733005	−	0.126960i
$359$	0.0554058	+	0.0959656i	0.00292420	+	0.00506487i	0.867484	−	0.497465i	$-0.165736\pi$
$359$	0.0554058	+	0.0959656i	0.00292420	+	0.00506487i	−0.864560	+	0.502530i	$0.832403\pi$
$360$		0			0
$361$	6.99666	+	17.6648i	0.368245	+	0.929729i
$362$	−26.6040			−1.39827
$363$		0			0
$364$	−0.786364	−	1.36202i	−0.0412167	−	0.0713894i
$365$	1.86162	−	3.22443i	0.0974418	−	0.168774i
$366$		0			0
$367$	−5.86986	+	10.1669i	−0.306404	+	0.530708i	−0.977573	−	0.210597i	$-0.932459\pi$
$367$	−5.86986	+	10.1669i	−0.306404	+	0.530708i	0.671169	+	0.741305i	$0.265793\pi$
$368$	−7.11278			−0.370779
$369$		0			0
$370$	−2.24644	+	3.89095i	−0.116787	+	0.202281i
$371$	−2.57338	+	4.45722i	−0.133603	+	0.231407i
$372$		0			0
$373$	14.5190			0.751763			0.375882	−	0.926668i	$-0.377340\pi$
$373$	14.5190			0.751763			0.375882	+	0.926668i	$0.377340\pi$
$374$	−7.75424	+	13.4307i	−0.400962	+	0.694487i
$375$		0			0
$376$	9.04704	−	15.6699i	0.466566	−	0.808115i
$377$	2.47706	+	4.29040i	0.127575	+	0.220967i
$378$		0			0
$379$	−6.59023			−0.338518			−0.169259	−	0.985572i	$-0.554137\pi$
$379$	−6.59023			−0.338518			−0.169259	+	0.985572i	$0.554137\pi$
$380$	2.09734	+	1.42515i	0.107591	+	0.0731087i
$381$		0			0
$382$	1.33242	+	2.30782i	0.0681727	+	0.118079i
$383$	−1.43461	−	2.48481i	−0.0733049	−	0.126968i	0.827043	−	0.562139i	$-0.190021\pi$
$383$	−1.43461	−	2.48481i	−0.0733049	−	0.126968i	−0.900348	+	0.435171i	$0.856688\pi$
$384$		0			0
$385$	1.36613	+	2.36620i	0.0696243	+	0.120593i
$386$	2.70519	−	4.68552i	0.137690	−	0.238487i
$387$		0			0
$388$	5.62686			0.285660
$389$	−3.16575	+	5.48323i	−0.160510	+	0.278011i	−0.935052	−	0.354512i	$-0.884647\pi$
$389$	−3.16575	+	5.48323i	−0.160510	+	0.278011i	0.774542	+	0.632523i	$0.217980\pi$
$390$		0			0
$391$	−8.26682			−0.418071
$392$	20.3813			1.02941
$393$		0			0
$394$	−11.4468	−	19.8264i	−0.576680	−	0.998840i
$395$	4.51908	−	7.82728i	0.227380	−	0.393833i
$396$		0			0
$397$	15.2749	+	26.4569i	0.766626	+	1.32784i	0.939382	+	0.342871i	$0.111399\pi$
$397$	15.2749	+	26.4569i	0.766626	+	1.32784i	−0.172756	+	0.984965i	$0.555267\pi$
$398$	−7.32522			−0.367180
$399$		0			0
$400$	−2.49812			−0.124906
$401$	15.1711	+	26.2771i	0.757609	+	1.31222i	0.944067	+	0.329754i	$0.106966\pi$
$401$	15.1711	+	26.2771i	0.757609	+	1.31222i	−0.186458	+	0.982463i	$0.559701\pi$
$402$		0			0
$403$	13.8223	−	23.9409i	0.688538	−	1.19258i
$404$	0.282509	+	0.489320i	0.0140553	+	0.0243446i
$405$		0			0
$406$	0.809843			0.0401919
$407$	16.9209			0.838740
$408$		0			0
$409$	7.48628	−	12.9666i	0.370173	−	0.641158i	−0.619419	−	0.785060i	$-0.712632\pi$
$409$	7.48628	−	12.9666i	0.370173	−	0.641158i	0.989592	+	0.143903i	$0.0459652\pi$
$410$	−9.88894			−0.488380
$411$		0			0
$412$	−0.971912	+	1.68340i	−0.0478827	+	0.0829352i
$413$	−3.11772	−	5.40004i	−0.153413	−	0.265719i
$414$		0			0
$415$	1.06089	+	1.83752i	0.0520771	+	0.0902002i
$416$	−7.04353	−	12.1997i	−0.345337	−	0.598142i
$417$		0			0
$418$	−1.70368	+	23.2203i	−0.0833296	+	1.13574i
$419$	6.17419			0.301629			0.150815	−	0.988562i	$-0.451810\pi$
$419$	6.17419			0.301629			0.150815	+	0.988562i	$0.451810\pi$
$420$		0			0
$421$	13.7714	+	23.8528i	0.671177	+	1.16251i	0.977571	+	0.210608i	$0.0675443\pi$
$421$	13.7714	+	23.8528i	0.671177	+	1.16251i	−0.306394	+	0.951905i	$0.599122\pi$
$422$	−7.55824	+	13.0913i	−0.367929	+	0.637272i
$423$		0			0
$424$	−12.9883	+	22.4963i	−0.630766	+	1.09252i
$425$	−2.90343			−0.140837
$426$		0			0
$427$	−1.51745	+	2.62830i	−0.0734347	+	0.127193i
$428$	−2.76805	+	4.79441i	−0.133799	+	0.231746i
$429$		0			0
$430$	11.8957			0.573663
$431$	−7.52941	+	13.0413i	−0.362679	+	0.628179i	−0.988401	−	0.151868i	$-0.951471\pi$
$431$	−7.52941	+	13.0413i	−0.362679	+	0.628179i	0.625722	+	0.780046i	$0.284805\pi$
$432$		0			0
$433$	0.485420	−	0.840772i	0.0233278	−	0.0404049i	−0.854126	−	0.520066i	$-0.825907\pi$
$433$	0.485420	−	0.840772i	0.0233278	−	0.0404049i	0.877454	+	0.479661i	$0.159241\pi$
$434$	−2.25951	−	3.91359i	−0.108460	−	0.187858i
$435$		0			0
$436$	−3.22488			−0.154444
$437$	−11.1734	+	5.40234i	−0.534497	+	0.258429i
$438$		0			0
$439$	13.7187	+	23.7616i	0.654760	+	1.13408i	0.981954	+	0.189120i	$0.0605636\pi$
$439$	13.7187	+	23.7616i	0.654760	+	1.13408i	−0.327194	+	0.944957i	$0.606103\pi$
$440$	6.89507	+	11.9426i	0.328710	+	0.569342i
$441$		0			0
$442$	7.67269	+	13.2895i	0.364952	+	0.632116i
$443$	−4.38272	+	7.59109i	−0.208229	+	0.360664i	−0.951157	−	0.308708i	$-0.900103\pi$
$443$	−4.38272	+	7.59109i	−0.208229	+	0.360664i	0.742928	+	0.669372i	$0.233437\pi$
$444$		0			0
$445$	7.93217			0.376021
$446$	−13.4201	+	23.2443i	−0.635461	+	1.10065i
$447$		0			0
$448$	−5.34637			−0.252592
$449$	9.63397			0.454655			0.227327	−	0.973818i	$-0.427001\pi$
$449$	9.63397			0.454655			0.227327	+	0.973818i	$0.427001\pi$
$450$		0			0
$451$	18.6217	+	32.2538i	0.876862	+	1.51877i
$452$	−0.448326	+	0.776524i	−0.0210875	+	0.0365246i
$453$		0			0
$454$	10.7851	+	18.6803i	0.506169	+	0.876711i
$455$	2.70352			0.126743
$456$		0			0
$457$	−10.6708			−0.499161			−0.249580	−	0.968354i	$-0.580293\pi$
$457$	−10.6708			−0.499161			−0.249580	+	0.968354i	$0.580293\pi$
$458$	5.60683	+	9.71131i	0.261990	+	0.453780i
$459$		0			0
$460$	−0.828173	+	1.43444i	−0.0386138	+	0.0668810i
$461$	2.84340	+	4.92491i	0.132430	+	0.229376i	0.924613	−	0.380908i	$-0.124389\pi$
$461$	2.84340	+	4.92491i	0.132430	+	0.229376i	−0.792183	+	0.610284i	$0.791055\pi$
$462$		0			0
$463$	35.3550			1.64309			0.821543	−	0.570147i	$-0.193114\pi$
$463$	35.3550			1.64309			0.821543	+	0.570147i	$0.193114\pi$
$464$	2.78864			0.129459
$465$		0			0
$466$	9.34864	−	16.1923i	0.433067	−	0.750095i
$467$	32.9071			1.52276			0.761380	−	0.648306i	$-0.224522\pi$
$467$	32.9071			1.52276			0.761380	+	0.648306i	$0.224522\pi$
$468$		0			0
$469$	2.58173	−	4.47169i	0.119213	−	0.206484i
$470$	3.50425	+	6.06954i	0.161639	+	0.279967i
$471$		0			0
$472$	−15.7356	−	27.2549i	−0.724292	−	1.25451i
$473$	−22.4006	−	38.7991i	−1.02998	−	1.78398i
$474$		0			0
$475$	−3.92428	+	1.89738i	−0.180058	+	0.0870579i
$476$	−1.02891			−0.0471602
$477$		0			0
$478$	−13.9700	−	24.1967i	−0.638971	−	1.10673i
$479$	−4.52861	+	7.84378i	−0.206917	+	0.358391i	−0.950742	−	0.309984i	$-0.899676\pi$
$479$	−4.52861	+	7.84378i	−0.206917	+	0.358391i	0.743825	+	0.668375i	$0.233010\pi$
$480$		0			0
$481$	8.37149	−	14.4998i	0.381707	−	0.661136i
$482$	15.6835			0.714366
$483$		0			0
$484$	2.65175	−	4.59297i	0.120534	−	0.208772i
$485$	−4.83628	+	8.37668i	−0.219604	+	0.380365i
$486$		0			0
$487$	−16.5206			−0.748620			−0.374310	−	0.927304i	$-0.622120\pi$
$487$	−16.5206			−0.748620			−0.374310	+	0.927304i	$0.622120\pi$
$488$	−7.65884	+	13.2655i	−0.346700	+	0.600501i
$489$		0			0
$490$	−3.94721	+	6.83677i	−0.178317	+	0.308854i
$491$	−0.695625	−	1.20486i	−0.0313931	−	0.0543745i	0.849902	−	0.526941i	$-0.176661\pi$
$491$	−0.695625	−	1.20486i	−0.0313931	−	0.0543745i	−0.881295	+	0.472566i	$0.843328\pi$
$492$		0			0
$493$	3.24109			0.145972
$494$	19.0550	+	12.9480i	0.857326	+	0.582557i
$495$		0			0
$496$	−7.78048	−	13.4762i	−0.349354	−	0.605099i
$497$	−3.53928	−	6.13021i	−0.158758	−	0.274977i
$498$		0			0
$499$	8.33255	+	14.4324i	0.373016	+	0.646083i	0.990028	−	0.140871i	$-0.0449902\pi$
$499$	8.33255	+	14.4324i	0.373016	+	0.646083i	−0.617012	+	0.786954i	$0.711657\pi$
$500$	−0.290867	+	0.503797i	−0.0130080	+	0.0225305i
$501$		0			0
$502$	20.6327			0.920881
$503$	7.81956	−	13.5439i	0.348657	−	0.603892i	−0.637354	−	0.770571i	$-0.719971\pi$
$503$	7.81956	−	13.5439i	0.348657	−	0.603892i	0.986011	+	0.166679i	$0.0533045\pi$
$504$		0			0
$505$	−0.971265			−0.0432207
$506$	−15.2084			−0.676096
$507$		0			0
$508$	0.670591	+	1.16150i	0.0297526	+	0.0515331i
$509$	−9.57702	+	16.5879i	−0.424494	+	0.735245i	−0.996373	−	0.0850929i	$-0.972881\pi$
$509$	−9.57702	+	16.5879i	−0.424494	+	0.735245i	0.571879	+	0.820338i	$0.306215\pi$
$510$		0			0
$511$	1.13406	+	1.96424i	0.0501677	+	0.0868929i
$512$	−23.2910			−1.02933
$513$		0			0
$514$	6.76389			0.298342
$515$	−1.67071	−	2.89376i	−0.0736205	−	0.127514i
$516$		0			0
$517$	13.1976	−	22.8589i	0.580430	−	1.00533i
$518$	−1.36848	−	2.37027i	−0.0601273	−	0.104144i
$519$		0			0
$520$	13.6451			0.598378
$521$	19.2394			0.842892			0.421446	−	0.906853i	$-0.361523\pi$
$521$	19.2394			0.842892			0.421446	+	0.906853i	$0.361523\pi$
$522$		0			0
$523$	−3.31973	+	5.74993i	−0.145161	+	0.251427i	−0.929433	−	0.368990i	$-0.879704\pi$
$523$	−3.31973	+	5.74993i	−0.145161	+	0.251427i	0.784272	+	0.620418i	$0.213037\pi$
$524$	−7.51490			−0.328290
$525$		0			0
$526$	3.36887	−	5.83505i	0.146890	−	0.254420i
$527$	−9.04285	−	15.6627i	−0.393913	−	0.682277i
$528$		0			0
$529$	7.44657	+	12.8978i	0.323764	+	0.560775i
$530$	−5.03083	−	8.71366i	−0.218525	−	0.378497i
$531$		0			0
$532$	−1.39068	+	0.672391i	−0.0602935	+	0.0291519i
$533$	36.8517			1.59623
$534$		0			0
$535$	−4.75828	−	8.24158i	−0.205718	−	0.356314i
$536$	13.0305	−	22.5694i	0.562830	−	0.974850i
$537$		0			0
$538$	−14.2860	+	24.7441i	−0.615914	+	1.06679i
$539$	29.7317			1.28064
$540$		0			0
$541$	−20.8756	+	36.1575i	−0.897510	+	1.55453i	−0.0668435	+	0.997763i	$0.521293\pi$
$541$	−20.8756	+	36.1575i	−0.897510	+	1.55453i	−0.830667	+	0.556770i	$0.812040\pi$
$542$	12.6829	−	21.9674i	0.544777	−	0.943581i
$543$		0			0
$544$	−9.21606			−0.395135
$545$	2.77178	−	4.80087i	0.118730	−	0.205647i
$546$		0			0
$547$	−6.10258	+	10.5700i	−0.260927	+	0.451939i	−0.966489	−	0.256710i	$-0.917362\pi$
$547$	−6.10258	+	10.5700i	−0.260927	+	0.451939i	0.705561	+	0.708649i	$0.250695\pi$
$548$	3.70377	+	6.41512i	0.158217	+	0.274040i
$549$		0			0
$550$	−5.34143			−0.227759
$551$	4.38066	−	2.11804i	0.186622	−	0.0902317i
$552$		0			0
$553$	2.75291	+	4.76819i	0.117066	+	0.202764i
$554$	0.488934	+	0.846858i	0.0207728	+	0.0359795i
$555$		0			0
$556$	3.08571	+	5.34461i	0.130863	+	0.226662i
$557$	17.5774	−	30.4450i	0.744779	−	1.28999i	−0.205519	−	0.978653i	$-0.565888\pi$
$557$	17.5774	−	30.4450i	0.744779	−	1.28999i	0.950298	−	0.311342i	$-0.100778\pi$
$558$		0			0
$559$	−44.3301			−1.87496
$560$	0.760896	−	1.31791i	0.0321537	−	0.0556919i
$561$		0			0
$562$	0.699634			0.0295123
$563$	−17.8406			−0.751891			−0.375945	−	0.926642i	$-0.622682\pi$
$563$	−17.8406			−0.751891			−0.375945	+	0.926642i	$0.622682\pi$
$564$		0			0
$565$	−0.770672	−	1.33484i	−0.0324224	−	0.0561572i
$566$	−18.4248	+	31.9127i	−0.774452	+	1.34139i
$567$		0			0
$568$	−17.8633	−	30.9402i	−0.749529	−	1.29822i
$569$	−31.6042			−1.32492			−0.662459	−	0.749098i	$-0.730487\pi$
$569$	−31.6042			−1.32492			−0.662459	+	0.749098i	$0.730487\pi$
$570$		0			0
$571$	−4.73053			−0.197967			−0.0989833	−	0.995089i	$-0.531559\pi$
$571$	−4.73053			−0.197967			−0.0989833	+	0.995089i	$0.531559\pi$
$572$	−5.78975	−	10.0281i	−0.242082	−	0.419298i
$573$		0			0
$574$	3.01205	−	5.21702i	0.125721	−	0.217754i
$575$	−1.42363	−	2.46580i	−0.0593694	−	0.102831i
$576$		0			0
$577$	24.4074			1.01609			0.508047	−	0.861330i	$-0.330368\pi$
$577$	24.4074			1.01609			0.508047	+	0.861330i	$0.330368\pi$
$578$	−10.2062			−0.424522
$579$		0			0
$580$	0.324694	−	0.562387i	0.0134822	−	0.0233518i
$581$	−1.29254			−0.0536235
$582$		0			0
$583$	−18.9470	+	32.8171i	−0.784703	+	1.35915i
$584$	5.72377	+	9.91386i	0.236851	+	0.410239i
$585$		0			0
$586$	2.24052	+	3.88070i	0.0925551	+	0.160310i
$587$	−14.3077	−	24.7817i	−0.590543	−	1.02285i	−0.994159	−	0.107922i	$-0.965580\pi$
$587$	−14.3077	−	24.7817i	−0.590543	−	1.02285i	0.403616	−	0.914928i	$-0.367753\pi$
$588$		0			0
$589$	−22.4578	−	15.2602i	−0.925358	−	0.628785i
$590$	12.1900			0.501853
$591$		0			0
$592$	−4.71225	−	8.16186i	−0.193672	−	0.335450i
$593$	1.85756	−	3.21738i	0.0762807	−	0.132122i	−0.825362	−	0.564604i	$-0.809029\pi$
$593$	1.85756	−	3.21738i	0.0762807	−	0.132122i	0.901642	+	0.432482i	$0.142362\pi$
$594$		0			0
$595$	0.884350	−	1.53174i	0.0362548	−	0.0627952i
$596$	2.19492			0.0899076
$597$		0			0
$598$	−7.52423	+	13.0324i	−0.307689	+	0.532933i
$599$	3.54970	−	6.14826i	0.145037	−	0.251211i	−0.784350	−	0.620319i	$-0.787003\pi$
$599$	3.54970	−	6.14826i	0.145037	−	0.251211i	0.929387	+	0.369108i	$0.120337\pi$
$600$		0			0
$601$	−11.0596			−0.451131			−0.225566	−	0.974228i	$-0.572423\pi$
$601$	−11.0596			−0.451131			−0.225566	+	0.974228i	$0.572423\pi$
$602$	−3.62329	+	6.27572i	−0.147674	+	0.255779i
$603$		0			0
$604$	−2.76778	+	4.79394i	−0.112619	+	0.195063i
$605$	4.55836	+	7.89531i	0.185324	+	0.320990i
$606$		0			0
$607$	27.1193			1.10074			0.550369	−	0.834921i	$-0.314487\pi$
$607$	27.1193			1.10074			0.550369	+	0.834921i	$0.314487\pi$
$608$	−12.4564	+	6.02266i	−0.505174	+	0.244251i
$609$		0			0
$610$	−2.96655	−	5.13821i	−0.120112	−	0.208040i
$611$	−13.0588	−	22.6185i	−0.528302	−	0.915046i
$612$		0			0
$613$	−20.9156	−	36.2268i	−0.844772	−	1.46319i	−0.885819	−	0.464031i	$-0.846403\pi$
$613$	−20.9156	−	36.2268i	−0.844772	−	1.46319i	0.0410468	−	0.999157i	$-0.486931\pi$
$614$	−12.1127	+	20.9797i	−0.488827	+	0.846673i
$615$		0			0
$616$	−8.40062			−0.338471
$617$	−8.85262	+	15.3332i	−0.356393	+	0.617291i	−0.987355	−	0.158523i	$-0.949327\pi$
$617$	−8.85262	+	15.3332i	−0.356393	+	0.617291i	0.630962	+	0.775813i	$0.282660\pi$
$618$		0			0
$619$	−39.8064			−1.59995			−0.799976	−	0.600032i	$-0.795155\pi$
$619$	−39.8064			−1.59995			−0.799976	+	0.600032i	$0.795155\pi$
$620$	−3.62367			−0.145530
$621$		0			0
$622$	−4.57208	−	7.91908i	−0.183324	−	0.317526i
$623$	−2.41604	+	4.18471i	−0.0967966	+	0.167657i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	28.5732			1.14201
$627$		0			0
$628$	2.01072			0.0802366
$629$	−5.47681	−	9.48611i	−0.218375	−	0.378236i
$630$		0			0
$631$	−23.0990	+	40.0087i	−0.919557	+	1.59272i	−0.119469	+	0.992838i	$0.538119\pi$
$631$	−23.0990	+	40.0087i	−0.919557	+	1.59272i	−0.800088	+	0.599882i	$0.795214\pi$
$632$	13.8944	+	24.0659i	0.552691	+	0.957288i
$633$		0			0
$634$	−1.23740			−0.0491433
$635$	−2.30549			−0.0914905
$636$		0			0
$637$	14.7095	−	25.4776i	0.582813	−	1.00946i
$638$	5.96262			0.236062
$639$		0			0
$640$	2.05176	−	3.55376i	0.0811030	−	0.140475i
$641$	3.30674	+	5.72744i	0.130608	+	0.226220i	0.923911	−	0.382607i	$-0.124974\pi$
$641$	3.30674	+	5.72744i	0.130608	+	0.226220i	−0.793303	+	0.608827i	$0.791640\pi$
$642$		0			0
$643$	15.3076	+	26.5135i	0.603673	+	1.04559i	0.992260	+	0.124180i	$0.0396300\pi$
$643$	15.3076	+	26.5135i	0.603673	+	1.04559i	−0.388587	+	0.921412i	$0.627037\pi$
$644$	−0.504503	−	0.873824i	−0.0198802	−	0.0344335i
$645$		0			0
$646$	13.5691	−	6.56063i	0.533868	−	0.258125i
$647$	11.8979			0.467753			0.233877	−	0.972266i	$-0.424859\pi$
$647$	11.8979			0.467753			0.233877	+	0.972266i	$0.424859\pi$
$648$		0			0
$649$	−22.9548	−	39.7588i	−0.901053	−	1.56067i
$650$	−2.64263	+	4.57716i	−0.103652	+	0.179531i
$651$		0			0
$652$	1.93509	−	3.35167i	0.0757839	−	0.131262i
$653$	1.42899			0.0559207			0.0279604	−	0.999609i	$-0.491099\pi$
$653$	1.42899			0.0559207			0.0279604	+	0.999609i	$0.491099\pi$
$654$		0			0
$655$	6.45905	−	11.1874i	0.252376	−	0.437128i
$656$	10.3718	−	17.9645i	0.404950	−	0.701395i
$657$		0			0
$658$	−4.26941			−0.166439
$659$	−12.2485	+	21.2150i	−0.477134	+	0.826420i	−0.999657	−	0.0262051i	$-0.991658\pi$
$659$	−12.2485	+	21.2150i	−0.477134	+	0.826420i	0.522523	+	0.852625i	$0.324991\pi$
$660$		0			0
$661$	1.61303	−	2.79385i	0.0627396	−	0.108668i	−0.832949	−	0.553349i	$-0.813350\pi$
$661$	1.61303	−	2.79385i	0.0627396	−	0.108668i	0.895689	+	0.444681i	$0.146683\pi$
$662$	−18.3588	−	31.7984i	−0.713535	−	1.23588i
$663$		0			0
$664$	−6.52366			−0.253167
$665$	0.194300	−	2.64822i	0.00753462	−	0.102693i
$666$		0			0
$667$	1.58919	+	2.75256i	0.0615338	+	0.106580i
$668$	4.78636	+	8.29023i	0.185190	+	0.320758i
$669$		0			0
$670$	5.04717	+	8.74196i	0.194989	+	0.337731i
$671$	−11.1725	+	19.3514i	−0.431311	+	0.747052i
$672$		0			0
$673$	37.1424			1.43173			0.715866	−	0.698237i	$-0.246032\pi$
$673$	37.1424			1.43173			0.715866	+	0.698237i	$0.246032\pi$
$674$	−12.3350	+	21.3649i	−0.475127	+	0.822944i
$675$		0			0
$676$	−3.89518			−0.149815
$677$	24.7550			0.951412			0.475706	−	0.879604i	$-0.342193\pi$
$677$	24.7550			0.951412			0.475706	+	0.879604i	$0.342193\pi$
$678$		0			0
$679$	−2.94614	−	5.10287i	−0.113063	−	0.195830i
$680$	4.46346	−	7.73095i	0.171166	−	0.296468i
$681$		0			0
$682$	−16.6361	−	28.8145i	−0.637028	−	1.10337i
$683$	−40.1153			−1.53497			−0.767484	−	0.641068i	$-0.778492\pi$
$683$	−40.1153			−1.53497			−0.767484	+	0.641068i	$0.778492\pi$
$684$		0			0
$685$	−12.7335			−0.486524
$686$	−4.94370	−	8.56274i	−0.188751	−	0.326927i
$687$		0			0
$688$	−12.4766	+	21.6100i	−0.475664	+	0.823875i
$689$	18.7477	+	32.4720i	0.714230	+	1.23708i
$690$		0			0
$691$	39.4963			1.50251			0.751254	−	0.660013i	$-0.229449\pi$
$691$	39.4963			1.50251			0.751254	+	0.660013i	$0.229449\pi$
$692$	−13.2533			−0.503816
$693$		0			0
$694$	4.89545	−	8.47917i	0.185829	−	0.321865i
$695$	−10.6087			−0.402410
$696$		0			0
$697$	12.0546	−	20.8792i	0.456600	−	0.790855i
$698$	−7.09879	−	12.2955i	−0.268693	−	0.465390i
$699$		0			0
$700$	−0.177189	−	0.306901i	−0.00669712	−	0.0115997i
$701$	0.0109776	+	0.0190137i	0.000414618	+	0.000718139i	0.866233	−	0.499641i	$-0.166535\pi$
$701$	0.0109776	+	0.0190137i	0.000414618	+	0.000718139i	−0.865818	+	0.500359i	$0.833201\pi$
$702$		0			0
$703$	−13.6016	−	9.24234i	−0.512994	−	0.348581i
$704$	−39.3637			−1.48357
$705$		0			0
$706$	−7.02535	−	12.1683i	−0.264402	−	0.457958i
$707$	0.295835	−	0.512402i	0.0111260	−	0.0192708i
$708$		0			0
$709$	8.90087	−	15.4168i	0.334279	−	0.578989i	−0.649067	−	0.760731i	$-0.724840\pi$
$709$	8.90087	−	15.4168i	0.334279	−	0.578989i	0.983346	+	0.181743i	$0.0581738\pi$
$710$	13.8383			0.519340
$711$		0			0
$712$	−12.1942	+	21.1209i	−0.456996	+	0.791540i
$713$	8.86789	−	15.3596i	0.332105	−	0.575223i
$714$		0			0
$715$	19.9052			0.744410
$716$	0.677477	−	1.17342i	0.0253185	−	0.0438529i
$717$		0			0
$718$	0.0659833	−	0.114286i	0.00246247	−	0.00426513i
$719$	9.40515	+	16.2902i	0.350753	+	0.607522i	0.986382	−	0.164473i	$-0.0525923\pi$
$719$	9.40515	+	16.2902i	0.350753	+	0.607522i	−0.635629	+	0.771995i	$0.719259\pi$
$720$		0			0
$721$	2.03552			0.0758066
$722$	14.0526	−	17.7347i	0.522983	−	0.660016i
$723$		0			0
$724$	−6.49774	−	11.2544i	−0.241487	−	0.418267i
$725$	0.558149	+	0.966742i	0.0207291	+	0.0359039i
$726$		0			0
$727$	−2.50151	−	4.33274i	−0.0927758	−	0.160692i	0.815902	−	0.578190i	$-0.196241\pi$
$727$	−2.50151	−	4.33274i	−0.0927758	−	0.160692i	−0.908678	+	0.417497i	$0.862907\pi$
$728$	−4.15613	+	7.19864i	−0.154037	+	0.266799i
$729$		0			0
$730$	−4.43405			−0.164112
$731$	−14.5009	+	25.1162i	−0.536334	+	0.928957i
$732$		0			0
$733$	−23.5259			−0.868950			−0.434475	−	0.900684i	$-0.643066\pi$
$733$	−23.5259			−0.868950			−0.434475	+	0.900684i	$0.643066\pi$
$734$	13.9809			0.516046
$735$		0			0
$736$	−4.51887	−	7.82691i	−0.166568	−	0.288504i
$737$	19.0085	−	32.9237i	0.700187	−	1.21276i
$738$		0			0
$739$	18.6918	+	32.3752i	0.687590	+	1.19094i	0.972615	+	0.232421i	$0.0746646\pi$
$739$	18.6918	+	32.3752i	0.687590	+	1.19094i	−0.285026	+	0.958520i	$0.592002\pi$
$740$	−2.19468			−0.0806779
$741$		0			0
$742$	6.12932			0.225014
$743$	5.19430	+	8.99679i	0.190560	+	0.330060i	0.945436	−	0.325808i	$-0.105636\pi$
$743$	5.19430	+	8.99679i	0.190560	+	0.330060i	−0.754876	+	0.655868i	$0.772303\pi$
$744$		0			0
$745$	−1.88653	+	3.26757i	−0.0691173	+	0.119715i
$746$	−8.64539	−	14.9742i	−0.316530	−	0.548246i
$747$		0			0
$748$	−7.57556			−0.276990
$749$	5.79725			0.211827
$750$		0			0
$751$	15.6413	−	27.0915i	0.570758	−	0.988581i	−0.425731	−	0.904850i	$-0.639983\pi$
$751$	15.6413	−	27.0915i	0.570758	−	0.988581i	0.996488	−	0.0837314i	$-0.0266838\pi$
$752$	−14.7014			−0.536105
$753$		0			0
$754$	2.94996	−	5.10947i	0.107431	−	0.186076i
$755$	−4.75781	−	8.24077i	−0.173154	−	0.299912i
$756$		0			0
$757$	6.31205	+	10.9328i	0.229415	+	0.397359i	0.957635	−	0.287985i	$-0.0929853\pi$
$757$	6.31205	+	10.9328i	0.229415	+	0.397359i	−0.728220	+	0.685344i	$0.759652\pi$
$758$	3.92419	+	6.79689i	0.142533	+	0.246874i
$759$		0			0
$760$	0.980664	−	13.3660i	0.0355724	−	0.484836i
$761$	−11.5495			−0.418668			−0.209334	−	0.977844i	$-0.567130\pi$
$761$	−11.5495			−0.418668			−0.209334	+	0.977844i	$0.567130\pi$
$762$		0			0
$763$	1.68850	+	2.92457i	0.0611279	+	0.105877i
$764$	−0.650861	+	1.12732i	−0.0235473	+	0.0407851i
$765$		0			0
$766$	−1.70849	+	2.95918i	−0.0617301	+	0.106920i
$767$	−45.4267			−1.64026
$768$		0			0
$769$	13.4603	−	23.3140i	0.485392	−	0.840724i	−0.514467	−	0.857510i	$-0.672010\pi$
$769$	13.4603	−	23.3140i	0.485392	−	0.840724i	0.999859	+	0.0167864i	$0.00534353\pi$
$770$	1.62693	−	2.81793i	0.0586306	−	0.101551i
$771$		0			0
$772$	2.64286			0.0951184
$773$	10.6666	−	18.4750i	0.383649	−	0.664500i	−0.607932	−	0.793989i	$-0.708001\pi$
$773$	10.6666	−	18.4750i	0.383649	−	0.664500i	0.991581	+	0.129489i	$0.0413338\pi$
$774$		0			0
$775$	3.11454	−	5.39454i	0.111877	−	0.193778i
$776$	−14.8697	−	25.7550i	−0.533790	−	0.924552i
$777$		0			0
$778$	7.54024			0.270331
$779$	2.64851	−	36.0979i	0.0948926	−	1.29334i
$780$		0			0
$781$	−26.0586	−	45.1348i	−0.932450	−	1.61505i
$782$	4.92252	+	8.52605i	0.176029	+	0.304891i
$783$		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)

\(f(q)\)	\(=\)	\(q+(-0.595455 - 1.03136i) q^{2} +(0.290867 - 0.503797i) q^{4} +(0.500000 + 0.866025i) q^{5} -0.609175 q^{7} -3.07461 q^{8} +O(q^{10})\)	\(q+(-0.595455 - 1.03136i) q^{2} +(0.290867 - 0.503797i) q^{4} +(0.500000 + 0.866025i) q^{5} -0.609175 q^{7} -3.07461 q^{8} +(0.595455 - 1.03136i) q^{10} -4.48517 q^{11} +(-2.21900 + 3.84342i) q^{13} +(0.362736 + 0.628278i) q^{14} +(1.24906 + 2.16343i) q^{16} +(1.45172 + 2.51445i) q^{17} +(3.60532 + 2.44983i) q^{19} +0.581734 q^{20} +(2.67071 + 4.62581i) q^{22} +(-1.42363 + 2.46580i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.28525 q^{26} +(-0.177189 + 0.306901i) q^{28} +(0.558149 - 0.966742i) q^{29} -6.22908 q^{31} +(-1.58710 + 2.74893i) q^{32} +(1.72886 - 2.99448i) q^{34} +(-0.304588 - 0.527561i) q^{35} -3.77264 q^{37} +(0.379847 - 5.17714i) q^{38} +(-1.53731 - 2.66269i) q^{40} +(-4.15184 - 7.19120i) q^{41} +(4.99438 + 8.65053i) q^{43} +(-1.30459 + 2.25961i) q^{44} +3.39082 q^{46} +(-2.94250 + 5.09656i) q^{47} -6.62891 q^{49} +1.19091 q^{50} +(1.29087 + 2.23585i) q^{52} +(4.22436 - 7.31681i) q^{53} +(-2.24258 - 3.88427i) q^{55} +1.87298 q^{56} -1.32941 q^{58} +(5.11793 + 8.86451i) q^{59} +(2.49099 - 4.31453i) q^{61} +(3.70913 + 6.42441i) q^{62} +8.77641 q^{64} -4.43800 q^{65} +(-4.23808 + 7.34057i) q^{67} +1.68903 q^{68} +(-0.362736 + 0.628278i) q^{70} +(5.80995 + 10.0631i) q^{71} +(-1.86162 - 3.22443i) q^{73} +(2.24644 + 3.89095i) q^{74} +(2.28289 - 1.10377i) q^{76} +2.73225 q^{77} +(-4.51908 - 7.82728i) q^{79} +(-1.24906 + 2.16343i) q^{80} +(-4.94447 + 8.56407i) q^{82} +2.12178 q^{83} +(-1.45172 + 2.51445i) q^{85} +(5.94786 - 10.3020i) q^{86} +13.7901 q^{88} +(3.96608 - 6.86946i) q^{89} +(1.35176 - 2.34131i) q^{91} +(0.828173 + 1.43444i) q^{92} +7.00850 q^{94} +(-0.318955 + 4.34721i) q^{95} +(4.83628 + 8.37668i) q^{97} +(3.94721 + 6.83677i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8}+O(q^{10}) \) 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