LMFDB - Embedded newform 1638.2.j.k.235.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(235,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("1638.235");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.j (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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			235.2
Root			$1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	1638.235
Dual form			1638.2.j.k.1171.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.822876 + 1.42526i) q^{5} +(-1.32288 - 2.29129i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.822876 + 1.42526i) q^{5} +(-1.32288 - 2.29129i) q^{7} +1.00000 q^{8} +(0.822876 - 1.42526i) q^{10} +(-2.32288 + 4.02334i) q^{11} +1.00000 q^{13} +(-1.32288 + 2.29129i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.677124 + 1.17281i) q^{17} +(-2.50000 - 4.33013i) q^{19} -1.64575 q^{20} +4.64575 q^{22} +(-3.82288 - 6.62141i) q^{23} +(1.14575 - 1.98450i) q^{25} +(-0.500000 - 0.866025i) q^{26} +2.64575 q^{28} +6.29150 q^{29} +(3.64575 - 6.31463i) q^{31} +(-0.500000 + 0.866025i) q^{32} +1.35425 q^{34} +(2.17712 - 3.77089i) q^{35} +(-0.177124 - 0.306788i) q^{37} +(-2.50000 + 4.33013i) q^{38} +(0.822876 + 1.42526i) q^{40} +7.64575 q^{41} +5.29150 q^{43} +(-2.32288 - 4.02334i) q^{44} +(-3.82288 + 6.62141i) q^{46} +(1.50000 + 2.59808i) q^{47} +(-3.50000 + 6.06218i) q^{49} -2.29150 q^{50} +(-0.500000 + 0.866025i) q^{52} +(-1.50000 + 2.59808i) q^{53} -7.64575 q^{55} +(-1.32288 - 2.29129i) q^{56} +(-3.14575 - 5.44860i) q^{58} +(3.96863 - 6.87386i) q^{59} +(-1.96863 - 3.40976i) q^{61} -7.29150 q^{62} +1.00000 q^{64} +(0.822876 + 1.42526i) q^{65} +(6.79150 - 11.7632i) q^{67} +(-0.677124 - 1.17281i) q^{68} -4.35425 q^{70} -5.70850 q^{71} +(4.17712 - 7.23499i) q^{73} +(-0.177124 + 0.306788i) q^{74} +5.00000 q^{76} +12.2915 q^{77} +(5.00000 + 8.66025i) q^{79} +(0.822876 - 1.42526i) q^{80} +(-3.82288 - 6.62141i) q^{82} +2.70850 q^{83} -2.22876 q^{85} +(-2.64575 - 4.58258i) q^{86} +(-2.32288 + 4.02334i) q^{88} +(0.531373 + 0.920365i) q^{89} +(-1.32288 - 2.29129i) q^{91} +7.64575 q^{92} +(1.50000 - 2.59808i) q^{94} +(4.11438 - 7.12631i) q^{95} -14.9373 q^{97} +7.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 + 4 * q^8	$ 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{8} - 2 q^{10} - 4 q^{11} + 4 q^{13} - 2 q^{16} - 8 q^{17} - 10 q^{19} + 4 q^{20} + 8 q^{22} - 10 q^{23} - 6 q^{25} - 2 q^{26} + 4 q^{29} + 4 q^{31} - 2 q^{32} + 16 q^{34} + 14 q^{35} - 6 q^{37} - 10 q^{38} - 2 q^{40} + 20 q^{41} - 4 q^{44} - 10 q^{46} + 6 q^{47} - 14 q^{49} + 12 q^{50} - 2 q^{52} - 6 q^{53} - 20 q^{55} - 2 q^{58} + 8 q^{61} - 8 q^{62} + 4 q^{64} - 2 q^{65} + 6 q^{67} - 8 q^{68} - 28 q^{70} - 44 q^{71} + 22 q^{73} - 6 q^{74} + 20 q^{76} + 28 q^{77} + 20 q^{79} - 2 q^{80} - 10 q^{82} + 32 q^{83} + 44 q^{85} - 4 q^{88} + 18 q^{89} + 20 q^{92} + 6 q^{94} - 10 q^{95} - 28 q^{97} + 28 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 + 4 * q^8 - 2 * q^10 - 4 * q^11 + 4 * q^13 - 2 * q^16 - 8 * q^17 - 10 * q^19 + 4 * q^20 + 8 * q^22 - 10 * q^23 - 6 * q^25 - 2 * q^26 + 4 * q^29 + 4 * q^31 - 2 * q^32 + 16 * q^34 + 14 * q^35 - 6 * q^37 - 10 * q^38 - 2 * q^40 + 20 * q^41 - 4 * q^44 - 10 * q^46 + 6 * q^47 - 14 * q^49 + 12 * q^50 - 2 * q^52 - 6 * q^53 - 20 * q^55 - 2 * q^58 + 8 * q^61 - 8 * q^62 + 4 * q^64 - 2 * q^65 + 6 * q^67 - 8 * q^68 - 28 * q^70 - 44 * q^71 + 22 * q^73 - 6 * q^74 + 20 * q^76 + 28 * q^77 + 20 * q^79 - 2 * q^80 - 10 * q^82 + 32 * q^83 + 44 * q^85 - 4 * q^88 + 18 * q^89 + 20 * q^92 + 6 * q^94 - 10 * q^95 - 28 * q^97 + 28 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	0.822876	+	1.42526i	0.368001	+	0.637397i	0.989253	−	0.146214i	$-0.0467089\pi$
$5$	0.822876	+	1.42526i	0.368001	+	0.637397i	−0.621252	+	0.783611i	$0.713376\pi$
$6$		0			0
$7$	−1.32288	−	2.29129i	−0.500000	−	0.866025i
$7$	−1.32288	−	2.29129i	−0.500000	−	0.866025i
$8$	1.00000			0.353553
$9$		0			0
$10$	0.822876	−	1.42526i	0.260216	−	0.450708i
$11$	−2.32288	+	4.02334i	−0.700373	+	1.21308i	0.267962	+	0.963429i	$0.413650\pi$
$11$	−2.32288	+	4.02334i	−0.700373	+	1.21308i	−0.968335	+	0.249653i	$0.919684\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$	−1.32288	+	2.29129i	−0.353553	+	0.612372i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−0.677124	+	1.17281i	−0.164227	+	0.284449i	−0.936380	−	0.350987i	$-0.885846\pi$
$17$	−0.677124	+	1.17281i	−0.164227	+	0.284449i	0.772154	+	0.635436i	$0.219180\pi$
$18$		0			0
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	−0.996199	−	0.0871106i	$-0.972237\pi$
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	0.422659	−	0.906289i	$-0.361097\pi$
$20$	−1.64575			−0.368001
$21$		0			0
$22$	4.64575			0.990478
$23$	−3.82288	−	6.62141i	−0.797125	−	1.38066i	−0.921481	−	0.388423i	$-0.873020\pi$
$23$	−3.82288	−	6.62141i	−0.797125	−	1.38066i	0.124357	−	0.992238i	$-0.460313\pi$
$24$		0			0
$25$	1.14575	−	1.98450i	0.229150	−	0.396900i
$26$	−0.500000	−	0.866025i	−0.0980581	−	0.169842i
$27$		0			0
$28$	2.64575			0.500000
$29$	6.29150			1.16830			0.584151	−	0.811645i	$-0.301427\pi$
$29$	6.29150			1.16830			0.584151	+	0.811645i	$0.301427\pi$
$30$		0			0
$31$	3.64575	−	6.31463i	0.654796	−	1.13414i	−0.327149	−	0.944973i	$-0.606088\pi$
$31$	3.64575	−	6.31463i	0.654796	−	1.13414i	0.981945	−	0.189167i	$-0.0605789\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	1.35425			0.232252
$35$	2.17712	−	3.77089i	0.368001	−	0.637397i
$36$		0			0
$37$	−0.177124	−	0.306788i	−0.0291191	−	0.0504357i	0.851099	−	0.525006i	$-0.175937\pi$
$37$	−0.177124	−	0.306788i	−0.0291191	−	0.0504357i	−0.880218	+	0.474570i	$0.842604\pi$
$38$	−2.50000	+	4.33013i	−0.405554	+	0.702439i
$39$		0			0
$40$	0.822876	+	1.42526i	0.130108	+	0.225354i
$41$	7.64575			1.19407			0.597033	−	0.802217i	$-0.296346\pi$
$41$	7.64575			1.19407			0.597033	+	0.802217i	$0.296346\pi$
$42$		0			0
$43$	5.29150			0.806947			0.403473	−	0.914991i	$-0.367803\pi$
$43$	5.29150			0.806947			0.403473	+	0.914991i	$0.367803\pi$
$44$	−2.32288	−	4.02334i	−0.350187	−	0.606541i
$45$		0			0
$46$	−3.82288	+	6.62141i	−0.563652	+	0.976274i
$47$	1.50000	+	2.59808i	0.218797	+	0.378968i	0.954441	−	0.298401i	$-0.0964533\pi$
$47$	1.50000	+	2.59808i	0.218797	+	0.378968i	−0.735643	+	0.677369i	$0.763120\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$	−2.29150			−0.324067
$51$		0			0
$52$	−0.500000	+	0.866025i	−0.0693375	+	0.120096i
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	−0.950464	−	0.310835i	$-0.899391\pi$
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	0.744423	+	0.667708i	$0.232725\pi$
$54$		0			0
$55$	−7.64575			−1.03095
$56$	−1.32288	−	2.29129i	−0.176777	−	0.306186i
$57$		0			0
$58$	−3.14575	−	5.44860i	−0.413057	−	0.715436i
$59$	3.96863	−	6.87386i	0.516671	−	0.894901i	−0.483141	−	0.875542i	$-0.660504\pi$
$59$	3.96863	−	6.87386i	0.516671	−	0.894901i	0.999813	−	0.0193585i	$-0.00616237\pi$
$60$		0			0
$61$	−1.96863	−	3.40976i	−0.252057	−	0.436575i	0.712035	−	0.702144i	$-0.247774\pi$
$61$	−1.96863	−	3.40976i	−0.252057	−	0.436575i	−0.964092	+	0.265569i	$0.914440\pi$
$62$	−7.29150			−0.926022
$63$		0			0
$64$	1.00000			0.125000
$65$	0.822876	+	1.42526i	0.102065	+	0.176782i
$66$		0			0
$67$	6.79150	−	11.7632i	0.829714	−	1.43711i	−0.0685485	−	0.997648i	$-0.521837\pi$
$67$	6.79150	−	11.7632i	0.829714	−	1.43711i	0.898263	−	0.439459i	$-0.144830\pi$
$68$	−0.677124	−	1.17281i	−0.0821134	−	0.142225i
$69$		0			0
$70$	−4.35425			−0.520432
$71$	−5.70850			−0.677474			−0.338737	−	0.940881i	$-0.610000\pi$
$71$	−5.70850			−0.677474			−0.338737	+	0.940881i	$0.610000\pi$
$72$		0			0
$73$	4.17712	−	7.23499i	0.488895	−	0.846792i	−0.511023	−	0.859567i	$-0.670733\pi$
$73$	4.17712	−	7.23499i	0.488895	−	0.846792i	0.999918	+	0.0127753i	$0.00406663\pi$
$74$	−0.177124	+	0.306788i	−0.0205903	+	0.0356634i
$75$		0			0
$76$	5.00000			0.573539
$77$	12.2915			1.40075
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$	0.822876	−	1.42526i	0.0920003	−	0.159349i
$81$		0			0
$82$	−3.82288	−	6.62141i	−0.422166	−	0.731213i
$83$	2.70850			0.297296			0.148648	−	0.988890i	$-0.452508\pi$
$83$	2.70850			0.297296			0.148648	+	0.988890i	$0.452508\pi$
$84$		0			0
$85$	−2.22876			−0.241743
$86$	−2.64575	−	4.58258i	−0.285299	−	0.494152i
$87$		0			0
$88$	−2.32288	+	4.02334i	−0.247619	+	0.428889i
$89$	0.531373	+	0.920365i	0.0563254	+	0.0975585i	0.892813	−	0.450427i	$-0.148728\pi$
$89$	0.531373	+	0.920365i	0.0563254	+	0.0975585i	−0.836488	+	0.547985i	$0.815395\pi$
$90$		0			0
$91$	−1.32288	−	2.29129i	−0.138675	−	0.240192i
$92$	7.64575			0.797125
$93$		0			0
$94$	1.50000	−	2.59808i	0.154713	−	0.267971i
$95$	4.11438	−	7.12631i	0.422126	−	0.731144i
$96$		0			0
$97$	−14.9373			−1.51665			−0.758324	−	0.651878i	$-0.773982\pi$
$97$	−14.9373			−1.51665			−0.758324	+	0.651878i	$0.773982\pi$
$98$	7.00000			0.707107
$99$		0			0
$100$	1.14575	+	1.98450i	0.114575	+	0.198450i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−2.64575	−	4.58258i	−0.260694	−	0.451535i	0.705733	−	0.708478i	$-0.250618\pi$
$103$	−2.64575	−	4.58258i	−0.260694	−	0.451535i	−0.966426	+	0.256943i	$0.917285\pi$
$104$	1.00000			0.0980581
$105$		0			0
$106$	3.00000			0.291386
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	−0.995474	−	0.0950377i	$-0.969703\pi$
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	0.415432	−	0.909624i	$-0.363630\pi$
$108$		0			0
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	−0.754204	−	0.656640i	$-0.771977\pi$
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	0.945769	+	0.324840i	$0.105310\pi$
$110$	3.82288	+	6.62141i	0.364497	+	0.631327i
$111$		0			0
$112$	−1.32288	+	2.29129i	−0.125000	+	0.216506i
$113$	11.2288			1.05631			0.528156	−	0.849147i	$-0.322884\pi$
$113$	11.2288			1.05631			0.528156	+	0.849147i	$0.322884\pi$
$114$		0			0
$115$	6.29150	−	10.8972i	0.586686	−	1.01617i
$116$	−3.14575	+	5.44860i	−0.292076	+	0.505890i
$117$		0			0
$118$	−7.93725			−0.730683
$119$	3.58301			0.328454
$120$		0			0
$121$	−5.29150	−	9.16515i	−0.481046	−	0.833196i
$122$	−1.96863	+	3.40976i	−0.178231	+	0.308705i
$123$		0			0
$124$	3.64575	+	6.31463i	0.327398	+	0.567070i
$125$	12.0000			1.07331
$126$		0			0
$127$	16.2288			1.44007			0.720035	−	0.693938i	$-0.244126\pi$
$127$	16.2288			1.44007			0.720035	+	0.693938i	$0.244126\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	0.822876	−	1.42526i	0.0721710	−	0.125004i
$131$	−2.46863	−	4.27579i	−0.215685	−	0.373577i	0.737799	−	0.675020i	$-0.235865\pi$
$131$	−2.46863	−	4.27579i	−0.215685	−	0.373577i	−0.953484	+	0.301443i	$0.902532\pi$
$132$		0			0
$133$	−6.61438	+	11.4564i	−0.573539	+	0.993399i
$134$	−13.5830			−1.17339
$135$		0			0
$136$	−0.677124	+	1.17281i	−0.0580629	+	0.100568i
$137$	8.76013	−	15.1730i	0.748428	−	1.29632i	−0.200147	−	0.979766i	$-0.564142\pi$
$137$	8.76013	−	15.1730i	0.748428	−	1.29632i	0.948576	−	0.316550i	$-0.102525\pi$
$138$		0			0
$139$	4.22876			0.358678			0.179339	−	0.983787i	$-0.442604\pi$
$139$	4.22876			0.358678			0.179339	+	0.983787i	$0.442604\pi$
$140$	2.17712	+	3.77089i	0.184001	+	0.318698i
$141$		0			0
$142$	2.85425	+	4.94370i	0.239523	+	0.414866i
$143$	−2.32288	+	4.02334i	−0.194249	+	0.336448i
$144$		0			0
$145$	5.17712	+	8.96704i	0.429937	+	0.744672i
$146$	−8.35425			−0.691403
$147$		0			0
$148$	0.354249			0.0291191
$149$	−11.4686	−	19.8642i	−0.939547	−	1.62734i	−0.766319	−	0.642461i	$-0.777914\pi$
$149$	−11.4686	−	19.8642i	−0.939547	−	1.62734i	−0.173228	−	0.984882i	$-0.555420\pi$
$150$		0			0
$151$	−10.9686	+	18.9982i	−0.892614	+	1.54605i	−0.0558844	+	0.998437i	$0.517798\pi$
$151$	−10.9686	+	18.9982i	−0.892614	+	1.54605i	−0.836730	+	0.547616i	$0.815536\pi$
$152$	−2.50000	−	4.33013i	−0.202777	−	0.351220i
$153$		0			0
$154$	−6.14575	−	10.6448i	−0.495239	−	0.857779i
$155$	12.0000			0.963863
$156$		0			0
$157$	1.32288	−	2.29129i	0.105577	−	0.182865i	−0.808397	−	0.588638i	$-0.799664\pi$
$157$	1.32288	−	2.29129i	0.105577	−	0.182865i	0.913974	+	0.405773i	$0.132998\pi$
$158$	5.00000	−	8.66025i	0.397779	−	0.688973i
$159$		0			0
$160$	−1.64575			−0.130108
$161$	−10.1144	+	17.5186i	−0.797125	+	1.38066i
$162$		0			0
$163$	−11.7915	−	20.4235i	−0.923582	−	1.59969i	−0.793826	−	0.608145i	$-0.791914\pi$
$163$	−11.7915	−	20.4235i	−0.923582	−	1.59969i	−0.129755	−	0.991546i	$-0.541419\pi$
$164$	−3.82288	+	6.62141i	−0.298516	+	0.517046i
$165$		0			0
$166$	−1.35425	−	2.34563i	−0.105110	−	0.182056i
$167$	−24.8745			−1.92485			−0.962424	−	0.271553i	$-0.912463\pi$
$167$	−24.8745			−1.92485			−0.962424	+	0.271553i	$0.912463\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$	1.11438	+	1.93016i	0.0854689	+	0.148036i
$171$		0			0
$172$	−2.64575	+	4.58258i	−0.201737	+	0.349418i
$173$	−5.85425	−	10.1399i	−0.445090	−	0.770919i	0.552968	−	0.833202i	$-0.313495\pi$
$173$	−5.85425	−	10.1399i	−0.445090	−	0.770919i	−0.998059	+	0.0622834i	$0.980162\pi$
$174$		0			0
$175$	−6.06275			−0.458301
$176$	4.64575			0.350187
$177$		0			0
$178$	0.531373	−	0.920365i	0.0398281	−	0.0689843i
$179$	−3.00000	+	5.19615i	−0.224231	+	0.388379i	−0.956088	−	0.293079i	$-0.905320\pi$
$179$	−3.00000	+	5.19615i	−0.224231	+	0.388379i	0.731858	+	0.681457i	$0.238654\pi$
$180$		0			0
$181$	9.35425			0.695296			0.347648	−	0.937625i	$-0.386980\pi$
$181$	9.35425			0.695296			0.347648	+	0.937625i	$0.386980\pi$
$182$	−1.32288	+	2.29129i	−0.0980581	+	0.169842i
$183$		0			0
$184$	−3.82288	−	6.62141i	−0.281826	−	0.488137i
$185$	0.291503	−	0.504897i	0.0214317	−	0.0371208i
$186$		0			0
$187$	−3.14575	−	5.44860i	−0.230040	−	0.398441i
$188$	−3.00000			−0.218797
$189$		0			0
$190$	−8.22876			−0.596977
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−1.53137	+	2.65242i	−0.110231	+	0.190925i	−0.915863	−	0.401490i	$-0.868492\pi$
$193$	−1.53137	+	2.65242i	−0.110231	+	0.190925i	0.805633	+	0.592416i	$0.201826\pi$
$194$	7.46863	+	12.9360i	0.536216	+	0.928754i
$195$		0			0
$196$	−3.50000	−	6.06218i	−0.250000	−	0.433013i
$197$	−20.8118			−1.48278			−0.741388	−	0.671076i	$-0.765832\pi$
$197$	−20.8118			−1.48278			−0.741388	+	0.671076i	$0.765832\pi$
$198$		0			0
$199$	−2.11438	+	3.66221i	−0.149884	+	0.259607i	−0.931185	−	0.364548i	$-0.881223\pi$
$199$	−2.11438	+	3.66221i	−0.149884	+	0.259607i	0.781300	+	0.624155i	$0.214557\pi$
$200$	1.14575	−	1.98450i	0.0810169	−	0.140325i
$201$		0			0
$202$		0			0
$203$	−8.32288	−	14.4156i	−0.584151	−	1.01178i
$204$		0			0
$205$	6.29150	+	10.8972i	0.439418	+	0.761094i
$206$	−2.64575	+	4.58258i	−0.184338	+	0.319283i
$207$		0			0
$208$	−0.500000	−	0.866025i	−0.0346688	−	0.0600481i
$209$	23.2288			1.60677
$210$		0			0
$211$	−11.6458			−0.801727			−0.400863	−	0.916138i	$-0.631290\pi$
$211$	−11.6458			−0.801727			−0.400863	+	0.916138i	$0.631290\pi$
$212$	−1.50000	−	2.59808i	−0.103020	−	0.178437i
$213$		0			0
$214$	−6.00000	+	10.3923i	−0.410152	+	0.710403i
$215$	4.35425	+	7.54178i	0.296957	+	0.514345i
$216$		0			0
$217$	−19.2915			−1.30959
$218$	−4.00000			−0.270914
$219$		0			0
$220$	3.82288	−	6.62141i	0.257738	−	0.446416i
$221$	−0.677124	+	1.17281i	−0.0455483	+	0.0788920i
$222$		0			0
$223$	16.5203			1.10628			0.553139	−	0.833089i	$-0.313430\pi$
$223$	16.5203			1.10628			0.553139	+	0.833089i	$0.313430\pi$
$224$	2.64575			0.176777
$225$		0			0
$226$	−5.61438	−	9.72439i	−0.373463	−	0.646857i
$227$	−7.64575	+	13.2428i	−0.507466	+	0.878957i	0.492496	+	0.870315i	$0.336085\pi$
$227$	−7.64575	+	13.2428i	−0.507466	+	0.878957i	−0.999963	+	0.00864295i	$0.997249\pi$
$228$		0			0
$229$	13.2288	+	22.9129i	0.874181	+	1.51413i	0.857633	+	0.514263i	$0.171934\pi$
$229$	13.2288	+	22.9129i	0.874181	+	1.51413i	0.0165480	+	0.999863i	$0.494732\pi$
$230$	−12.5830			−0.829699
$231$		0			0
$232$	6.29150			0.413057
$233$	−2.32288	−	4.02334i	−0.152177	−	0.263578i	0.779851	−	0.625965i	$-0.215295\pi$
$233$	−2.32288	−	4.02334i	−0.152177	−	0.263578i	−0.932027	+	0.362388i	$0.881962\pi$
$234$		0			0
$235$	−2.46863	+	4.27579i	−0.161035	+	0.278922i
$236$	3.96863	+	6.87386i	0.258336	+	0.447450i
$237$		0			0
$238$	−1.79150	−	3.10297i	−0.116126	−	0.201136i
$239$	−3.00000			−0.194054			−0.0970269	−	0.995282i	$-0.530933\pi$
$239$	−3.00000			−0.194054			−0.0970269	+	0.995282i	$0.530933\pi$
$240$		0			0
$241$	−8.93725	+	15.4798i	−0.575699	+	0.997140i	0.420266	+	0.907401i	$0.361937\pi$
$241$	−8.93725	+	15.4798i	−0.575699	+	0.997140i	−0.995965	+	0.0897393i	$0.971397\pi$
$242$	−5.29150	+	9.16515i	−0.340151	+	0.589158i
$243$		0			0
$244$	3.93725			0.252057
$245$	−11.5203			−0.736002
$246$		0			0
$247$	−2.50000	−	4.33013i	−0.159071	−	0.275519i
$248$	3.64575	−	6.31463i	0.231505	−	0.400979i
$249$		0			0
$250$	−6.00000	−	10.3923i	−0.379473	−	0.657267i
$251$	2.70850			0.170959			0.0854794	−	0.996340i	$-0.472758\pi$
$251$	2.70850			0.170959			0.0854794	+	0.996340i	$0.472758\pi$
$252$		0			0
$253$	35.5203			2.23314
$254$	−8.11438	−	14.0545i	−0.509141	−	0.881859i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	7.93725	+	13.7477i	0.495112	+	0.857560i	0.999984	−	0.00563467i	$-0.00179358\pi$
$257$	7.93725	+	13.7477i	0.495112	+	0.857560i	−0.504872	+	0.863194i	$0.668460\pi$
$258$		0			0
$259$	−0.468627	+	0.811686i	−0.0291191	+	0.0504357i
$260$	−1.64575			−0.102065
$261$		0			0
$262$	−2.46863	+	4.27579i	−0.152512	+	0.264159i
$263$	0.822876	−	1.42526i	0.0507407	−	0.0878854i	−0.839539	−	0.543299i	$-0.817175\pi$
$263$	0.822876	−	1.42526i	0.0507407	−	0.0878854i	0.890280	+	0.455413i	$0.150508\pi$
$264$		0			0
$265$	−4.93725			−0.303293
$266$	13.2288			0.811107
$267$		0			0
$268$	6.79150	+	11.7632i	0.414857	+	0.718553i
$269$	−12.4373	+	21.5420i	−0.758313	+	1.31344i	0.185398	+	0.982664i	$0.440643\pi$
$269$	−12.4373	+	21.5420i	−0.758313	+	1.31344i	−0.943710	+	0.330773i	$0.892691\pi$
$270$		0			0
$271$	8.67712	+	15.0292i	0.527098	+	0.912960i	0.999501	+	0.0315777i	$0.0100532\pi$
$271$	8.67712	+	15.0292i	0.527098	+	0.912960i	−0.472404	+	0.881382i	$0.656613\pi$
$272$	1.35425			0.0821134
$273$		0			0
$274$	−17.5203			−1.05844
$275$	5.32288	+	9.21949i	0.320981	+	0.555956i
$276$		0			0
$277$	9.26013	−	16.0390i	0.556387	−	0.963691i	−0.441407	−	0.897307i	$-0.645520\pi$
$277$	9.26013	−	16.0390i	0.556387	−	0.963691i	0.997794	−	0.0663840i	$-0.0211462\pi$
$278$	−2.11438	−	3.66221i	−0.126812	−	0.219645i
$279$		0			0
$280$	2.17712	−	3.77089i	0.130108	−	0.225354i
$281$	−18.5830			−1.10857			−0.554285	−	0.832327i	$-0.687008\pi$
$281$	−18.5830			−1.10857			−0.554285	+	0.832327i	$0.687008\pi$
$282$		0			0
$283$	13.4686	−	23.3283i	0.800627	−	1.38673i	−0.118577	−	0.992945i	$-0.537833\pi$
$283$	13.4686	−	23.3283i	0.800627	−	1.38673i	0.919204	−	0.393781i	$-0.128833\pi$
$284$	2.85425	−	4.94370i	0.169368	−	0.293355i
$285$		0			0
$286$	4.64575			0.274709
$287$	−10.1144	−	17.5186i	−0.597033	−	1.03409i
$288$		0			0
$289$	7.58301	+	13.1342i	0.446059	+	0.772597i
$290$	5.17712	−	8.96704i	0.304011	−	0.526563i
$291$		0			0
$292$	4.17712	+	7.23499i	0.244448	+	0.423396i
$293$	29.5203			1.72459			0.862296	−	0.506405i	$-0.169026\pi$
$293$	29.5203			1.72459			0.862296	+	0.506405i	$0.169026\pi$
$294$		0			0
$295$	13.0627			0.760542
$296$	−0.177124	−	0.306788i	−0.0102951	−	0.0178317i
$297$		0			0
$298$	−11.4686	+	19.8642i	−0.664360	+	1.15070i
$299$	−3.82288	−	6.62141i	−0.221083	−	0.382926i
$300$		0			0
$301$	−7.00000	−	12.1244i	−0.403473	−	0.698836i
$302$	21.9373			1.26235
$303$		0			0
$304$	−2.50000	+	4.33013i	−0.143385	+	0.248350i
$305$	3.23987	−	5.61162i	0.185514	−	0.321320i
$306$		0			0
$307$	11.5830			0.661077			0.330539	−	0.943793i	$-0.392770\pi$
$307$	11.5830			0.661077			0.330539	+	0.943793i	$0.392770\pi$
$308$	−6.14575	+	10.6448i	−0.350187	+	0.606541i
$309$		0			0
$310$	−6.00000	−	10.3923i	−0.340777	−	0.590243i
$311$	−8.46863	+	14.6681i	−0.480212	+	0.831751i	−0.999742	−	0.0227007i	$-0.992774\pi$
$311$	−8.46863	+	14.6681i	−0.480212	+	0.831751i	0.519531	+	0.854452i	$0.326107\pi$
$312$		0			0
$313$	14.5830	+	25.2585i	0.824280	+	1.42770i	0.902468	+	0.430757i	$0.141753\pi$
$313$	14.5830	+	25.2585i	0.824280	+	1.42770i	−0.0781880	+	0.996939i	$0.524913\pi$
$314$	−2.64575			−0.149308
$315$		0			0
$316$	−10.0000			−0.562544
$317$	−1.35425	−	2.34563i	−0.0760622	−	0.131744i	0.825486	−	0.564423i	$-0.190901\pi$
$317$	−1.35425	−	2.34563i	−0.0760622	−	0.131744i	−0.901548	+	0.432680i	$0.857568\pi$
$318$		0			0
$319$	−14.6144	+	25.3128i	−0.818248	+	1.41725i
$320$	0.822876	+	1.42526i	0.0460001	+	0.0796746i
$321$		0			0
$322$	20.2288			1.12730
$323$	6.77124			0.376762
$324$		0			0
$325$	1.14575	−	1.98450i	0.0635548	−	0.110080i
$326$	−11.7915	+	20.4235i	−0.653071	+	1.13115i
$327$		0			0
$328$	7.64575			0.422166
$329$	3.96863	−	6.87386i	0.218797	−	0.378968i
$330$		0			0
$331$	−16.2915	−	28.2177i	−0.895462	−	1.55099i	−0.833232	−	0.552924i	$-0.813512\pi$
$331$	−16.2915	−	28.2177i	−0.895462	−	1.55099i	−0.0622301	−	0.998062i	$-0.519821\pi$
$332$	−1.35425	+	2.34563i	−0.0743241	+	0.128733i
$333$		0			0
$334$	12.4373	+	21.5420i	0.680536	+	1.17872i
$335$	22.3542			1.22134
$336$		0			0
$337$	5.58301			0.304126			0.152063	−	0.988371i	$-0.451408\pi$
$337$	5.58301			0.304126			0.152063	+	0.988371i	$0.451408\pi$
$338$	−0.500000	−	0.866025i	−0.0271964	−	0.0471056i
$339$		0			0
$340$	1.11438	−	1.93016i	0.0604356	−	0.104678i
$341$	16.9373	+	29.3362i	0.917204	+	1.58864i
$342$		0			0
$343$	18.5203			1.00000
$344$	5.29150			0.285299
$345$		0			0
$346$	−5.85425	+	10.1399i	−0.314726	+	0.545122i
$347$	−13.1144	+	22.7148i	−0.704017	+	1.21939i	0.263029	+	0.964788i	$0.415279\pi$
$347$	−13.1144	+	22.7148i	−0.704017	+	1.21939i	−0.967045	+	0.254605i	$0.918055\pi$
$348$		0			0
$349$	3.06275			0.163945			0.0819725	−	0.996635i	$-0.473878\pi$
$349$	3.06275			0.163945			0.0819725	+	0.996635i	$0.473878\pi$
$350$	3.03137	+	5.25049i	0.162034	+	0.280651i
$351$		0			0
$352$	−2.32288	−	4.02334i	−0.123810	−	0.214445i
$353$	0.291503	−	0.504897i	0.0155151	−	0.0268730i	−0.858164	−	0.513376i	$-0.828395\pi$
$353$	0.291503	−	0.504897i	0.0155151	−	0.0268730i	0.873679	+	0.486503i	$0.161728\pi$
$354$		0			0
$355$	−4.69738	−	8.13611i	−0.249311	−	0.431820i
$356$	−1.06275			−0.0563254
$357$		0			0
$358$	6.00000			0.317110
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	0.775934	−	0.630814i	$-0.217279\pi$
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	−0.934268	+	0.356572i	$0.883946\pi$
$360$		0			0
$361$	−3.00000	+	5.19615i	−0.157895	+	0.273482i
$362$	−4.67712	−	8.10102i	−0.245824	−	0.425780i
$363$		0			0
$364$	2.64575			0.138675
$365$	13.7490			0.719656
$366$		0			0
$367$	−10.2915	+	17.8254i	−0.537212	+	0.930479i	0.461841	+	0.886963i	$0.347189\pi$
$367$	−10.2915	+	17.8254i	−0.537212	+	0.930479i	−0.999053	+	0.0435157i	$0.986144\pi$
$368$	−3.82288	+	6.62141i	−0.199281	+	0.345165i
$369$		0			0
$370$	−0.583005			−0.0303090
$371$	7.93725			0.412082
$372$		0			0
$373$	−4.67712	−	8.10102i	−0.242172	−	0.419455i	0.719160	−	0.694844i	$-0.244527\pi$
$373$	−4.67712	−	8.10102i	−0.242172	−	0.419455i	−0.961333	+	0.275389i	$0.911193\pi$
$374$	−3.14575	+	5.44860i	−0.162663	+	0.281740i
$375$		0			0
$376$	1.50000	+	2.59808i	0.0773566	+	0.133986i
$377$	6.29150			0.324029
$378$		0			0
$379$	33.1660			1.70362			0.851812	−	0.523848i	$-0.175504\pi$
$379$	33.1660			1.70362			0.851812	+	0.523848i	$0.175504\pi$
$380$	4.11438	+	7.12631i	0.211063	+	0.365572i
$381$		0			0
$382$		0			0
$383$	−18.8745	−	32.6916i	−0.964442	−	1.67046i	−0.711106	−	0.703085i	$-0.751805\pi$
$383$	−18.8745	−	32.6916i	−0.964442	−	1.67046i	−0.253336	−	0.967378i	$-0.581528\pi$
$384$		0			0
$385$	10.1144	+	17.5186i	0.515476	+	0.892831i
$386$	3.06275			0.155890
$387$		0			0
$388$	7.46863	−	12.9360i	0.379162	−	0.656728i
$389$	9.14575	−	15.8409i	0.463708	−	0.803166i	−0.535434	−	0.844577i	$-0.679852\pi$
$389$	9.14575	−	15.8409i	0.463708	−	0.803166i	0.999142	+	0.0414111i	$0.0131853\pi$
$390$		0			0
$391$	10.3542			0.523637
$392$	−3.50000	+	6.06218i	−0.176777	+	0.306186i
$393$		0			0
$394$	10.4059	+	18.0235i	0.524241	+	0.908012i
$395$	−8.22876	+	14.2526i	−0.414034	+	0.717127i
$396$		0			0
$397$	−0.468627	−	0.811686i	−0.0235197	−	0.0407373i	0.854026	−	0.520230i	$-0.174154\pi$
$397$	−0.468627	−	0.811686i	−0.0235197	−	0.0407373i	−0.877546	+	0.479493i	$0.840821\pi$
$398$	4.22876			0.211968
$399$		0			0
$400$	−2.29150			−0.114575
$401$	4.11438	+	7.12631i	0.205462	+	0.355871i	0.950280	−	0.311397i	$-0.100797\pi$
$401$	4.11438	+	7.12631i	0.205462	+	0.355871i	−0.744818	+	0.667268i	$0.767464\pi$
$402$		0			0
$403$	3.64575	−	6.31463i	0.181608	−	0.314554i
$404$		0			0
$405$		0			0
$406$	−8.32288	+	14.4156i	−0.413057	+	0.715436i
$407$	1.64575			0.0815769
$408$		0			0
$409$	5.53137	−	9.58062i	0.273509	−	0.473731i	−0.696249	−	0.717800i	$-0.745149\pi$
$409$	5.53137	−	9.58062i	0.273509	−	0.473731i	0.969758	+	0.244069i	$0.0784824\pi$
$410$	6.29150	−	10.8972i	0.310715	−	0.538174i
$411$		0			0
$412$	5.29150			0.260694
$413$	−21.0000			−1.03334
$414$		0			0
$415$	2.22876	+	3.86032i	0.109405	+	0.189496i
$416$	−0.500000	+	0.866025i	−0.0245145	+	0.0424604i
$417$		0			0
$418$	−11.6144	−	20.1167i	−0.568078	−	0.983940i
$419$	22.4575			1.09712			0.548561	−	0.836111i	$-0.315176\pi$
$419$	22.4575			1.09712			0.548561	+	0.836111i	$0.315176\pi$
$420$		0			0
$421$	−29.6458			−1.44485			−0.722423	−	0.691452i	$-0.756972\pi$
$421$	−29.6458			−1.44485			−0.722423	+	0.691452i	$0.756972\pi$
$422$	5.82288	+	10.0855i	0.283453	+	0.490955i
$423$		0			0
$424$	−1.50000	+	2.59808i	−0.0728464	+	0.126174i
$425$	1.55163	+	2.68751i	0.0752652	+	0.130363i
$426$		0			0
$427$	−5.20850	+	9.02138i	−0.252057	+	0.436575i
$428$	12.0000			0.580042
$429$		0			0
$430$	4.35425	−	7.54178i	0.209981	−	0.363697i
$431$	−0.291503	+	0.504897i	−0.0140412	+	0.0243200i	−0.872961	−	0.487791i	$-0.837803\pi$
$431$	−0.291503	+	0.504897i	−0.0140412	+	0.0243200i	0.858919	+	0.512111i	$0.171136\pi$
$432$		0			0
$433$	−12.4170			−0.596723			−0.298361	−	0.954453i	$-0.596440\pi$
$433$	−12.4170			−0.596723			−0.298361	+	0.954453i	$0.596440\pi$
$434$	9.64575	+	16.7069i	0.463011	+	0.801958i
$435$		0			0
$436$	2.00000	+	3.46410i	0.0957826	+	0.165900i
$437$	−19.1144	+	33.1071i	−0.914365	+	1.58373i
$438$		0			0
$439$	−9.17712	−	15.8952i	−0.438000	−	0.758639i	0.559535	−	0.828807i	$-0.310980\pi$
$439$	−9.17712	−	15.8952i	−0.438000	−	0.758639i	−0.997535	+	0.0701681i	$0.977646\pi$
$440$	−7.64575			−0.364497
$441$		0			0
$442$	1.35425			0.0644150
$443$	17.4686	+	30.2565i	0.829960	+	1.43753i	0.898069	+	0.439854i	$0.144970\pi$
$443$	17.4686	+	30.2565i	0.829960	+	1.43753i	−0.0681097	+	0.997678i	$0.521697\pi$
$444$		0			0
$445$	−0.874508	+	1.51469i	−0.0414556	+	0.0718033i
$446$	−8.26013	−	14.3070i	−0.391128	−	0.677454i
$447$		0			0
$448$	−1.32288	−	2.29129i	−0.0625000	−	0.108253i
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$	−17.7601	+	30.7614i	−0.836292	+	1.44850i
$452$	−5.61438	+	9.72439i	−0.264078	+	0.457397i
$453$		0			0
$454$	15.2915			0.717666
$455$	2.17712	−	3.77089i	0.102065	−	0.176782i
$456$		0			0
$457$	−2.88562	−	4.99804i	−0.134984	−	0.233799i	0.790608	−	0.612323i	$-0.209765\pi$
$457$	−2.88562	−	4.99804i	−0.134984	−	0.233799i	−0.925591	+	0.378525i	$0.876432\pi$
$458$	13.2288	−	22.9129i	0.618139	−	1.07065i
$459$		0			0
$460$	6.29150	+	10.8972i	0.293343	+	0.508085i
$461$	−16.4575			−0.766503			−0.383251	−	0.923644i	$-0.625196\pi$
$461$	−16.4575			−0.766503			−0.383251	+	0.923644i	$0.625196\pi$
$462$		0			0
$463$	−17.1660			−0.797772			−0.398886	−	0.917000i	$-0.630603\pi$
$463$	−17.1660			−0.797772			−0.398886	+	0.917000i	$0.630603\pi$
$464$	−3.14575	−	5.44860i	−0.146038	−	0.252945i
$465$		0			0
$466$	−2.32288	+	4.02334i	−0.107605	+	0.186378i
$467$	−0.239870	−	0.415468i	−0.0110999	−	0.0192256i	0.860422	−	0.509582i	$-0.170200\pi$
$467$	−0.239870	−	0.415468i	−0.0110999	−	0.0192256i	−0.871522	+	0.490356i	$0.836867\pi$
$468$		0			0
$469$	−35.9373			−1.65943
$470$	4.93725			0.227739
$471$		0			0
$472$	3.96863	−	6.87386i	0.182671	−	0.316395i
$473$	−12.2915	+	21.2895i	−0.565164	+	0.978893i
$474$		0			0
$475$	−11.4575			−0.525707
$476$	−1.79150	+	3.10297i	−0.0821134	+	0.142225i
$477$		0			0
$478$	1.50000	+	2.59808i	0.0686084	+	0.118833i
$479$	14.3745	−	24.8974i	0.656788	−	1.13759i	−0.324654	−	0.945833i	$-0.605248\pi$
$479$	14.3745	−	24.8974i	0.656788	−	1.13759i	0.981442	−	0.191757i	$-0.0614187\pi$
$480$		0			0
$481$	−0.177124	−	0.306788i	−0.00807617	−	0.0139883i
$482$	17.8745			0.814162
$483$		0			0
$484$	10.5830			0.481046
$485$	−12.2915	−	21.2895i	−0.558128	−	0.966707i
$486$		0			0
$487$	4.03137	−	6.98254i	0.182679	−	0.316409i	−0.760113	−	0.649791i	$-0.774856\pi$
$487$	4.03137	−	6.98254i	0.182679	−	0.316409i	0.942792	+	0.333382i	$0.108190\pi$
$488$	−1.96863	−	3.40976i	−0.0891156	−	0.154353i
$489$		0			0
$490$	5.76013	+	9.97684i	0.260216	+	0.450708i
$491$	31.1660			1.40650			0.703251	−	0.710941i	$-0.251731\pi$
$491$	31.1660			1.40650			0.703251	+	0.710941i	$0.251731\pi$
$492$		0			0
$493$	−4.26013	+	7.37876i	−0.191867	+	0.332323i
$494$	−2.50000	+	4.33013i	−0.112480	+	0.194822i
$495$		0			0
$496$	−7.29150			−0.327398
$497$	7.55163	+	13.0798i	0.338737	+	0.586710i
$498$		0			0
$499$	9.35425	+	16.2020i	0.418754	+	0.725303i	0.995814	−	0.0913986i	$-0.0291337\pi$
$499$	9.35425	+	16.2020i	0.418754	+	0.725303i	−0.577061	+	0.816701i	$0.695800\pi$
$500$	−6.00000	+	10.3923i	−0.268328	+	0.464758i
$501$		0			0
$502$	−1.35425	−	2.34563i	−0.0604431	−	0.104690i
$503$	16.4575			0.733804			0.366902	−	0.930260i	$-0.380418\pi$
$503$	16.4575			0.733804			0.366902	+	0.930260i	$0.380418\pi$
$504$		0			0
$505$		0			0
$506$	−17.7601	−	30.7614i	−0.789534	−	1.36751i
$507$		0			0
$508$	−8.11438	+	14.0545i	−0.360017	+	0.623568i
$509$	12.0000	+	20.7846i	0.531891	+	0.921262i	0.999307	+	0.0372243i	$0.0118516\pi$
$509$	12.0000	+	20.7846i	0.531891	+	0.921262i	−0.467416	+	0.884037i	$0.654815\pi$
$510$		0			0
$511$	−22.1033			−0.977791
$512$	1.00000			0.0441942
$513$		0			0
$514$	7.93725	−	13.7477i	0.350097	−	0.606386i
$515$	4.35425	−	7.54178i	0.191871	−	0.332331i
$516$		0			0
$517$	−13.9373			−0.612960
$518$	0.937254			0.0411806
$519$		0			0
$520$	0.822876	+	1.42526i	0.0360855	+	0.0625019i
$521$	9.00000	−	15.5885i	0.394297	−	0.682943i	−0.598714	−	0.800963i	$-0.704321\pi$
$521$	9.00000	−	15.5885i	0.394297	−	0.682943i	0.993011	+	0.118020i	$0.0376547\pi$
$522$		0			0
$523$	18.6458	+	32.2954i	0.815322	+	1.41218i	0.909097	+	0.416585i	$0.136773\pi$
$523$	18.6458	+	32.2954i	0.815322	+	1.41218i	−0.0937748	+	0.995593i	$0.529893\pi$
$524$	4.93725			0.215685
$525$		0			0
$526$	−1.64575			−0.0717582
$527$	4.93725	+	8.55157i	0.215070	+	0.372512i
$528$		0			0
$529$	−17.7288	+	30.7071i	−0.770816	+	1.33509i
$530$	2.46863	+	4.27579i	0.107230	+	0.185728i
$531$		0			0
$532$	−6.61438	−	11.4564i	−0.286770	−	0.496700i
$533$	7.64575			0.331174
$534$		0			0
$535$	9.87451	−	17.1031i	0.426912	−	0.739434i
$536$	6.79150	−	11.7632i	0.293348	−	0.508094i
$537$		0			0
$538$	24.8745			1.07242
$539$	−16.2601	−	28.1634i	−0.700373	−	1.21308i
$540$		0			0
$541$	−18.1771	−	31.4837i	−0.781496	−	1.35359i	−0.931070	−	0.364840i	$-0.881124\pi$
$541$	−18.1771	−	31.4837i	−0.781496	−	1.35359i	0.149575	−	0.988750i	$-0.452210\pi$
$542$	8.67712	−	15.0292i	0.372714	−	0.645560i
$543$		0			0
$544$	−0.677124	−	1.17281i	−0.0290315	−	0.0502840i
$545$	6.58301			0.281985
$546$		0			0
$547$	−5.06275			−0.216467			−0.108234	−	0.994125i	$-0.534519\pi$
$547$	−5.06275			−0.216467			−0.108234	+	0.994125i	$0.534519\pi$
$548$	8.76013	+	15.1730i	0.374214	+	0.648158i
$549$		0			0
$550$	5.32288	−	9.21949i	0.226968	−	0.393120i
$551$	−15.7288	−	27.2430i	−0.670068	−	1.16059i
$552$		0			0
$553$	13.2288	−	22.9129i	0.562544	−	0.974355i
$554$	−18.5203			−0.786850
$555$		0			0
$556$	−2.11438	+	3.66221i	−0.0896696	+	0.155312i
$557$	−15.5830	+	26.9906i	−0.660273	+	1.14363i	0.320271	+	0.947326i	$0.396226\pi$
$557$	−15.5830	+	26.9906i	−0.660273	+	1.14363i	−0.980544	+	0.196301i	$0.937107\pi$
$558$		0			0
$559$	5.29150			0.223807
$560$	−4.35425			−0.184001
$561$		0			0
$562$	9.29150	+	16.0934i	0.391938	+	0.678857i
$563$	23.5203	−	40.7383i	0.991261	−	1.71691i	0.381387	−	0.924416i	$-0.375447\pi$
$563$	23.5203	−	40.7383i	0.991261	−	1.71691i	0.609874	−	0.792498i	$-0.291220\pi$
$564$		0			0
$565$	9.23987	+	16.0039i	0.388724	+	0.673290i
$566$	−26.9373			−1.13226
$567$		0			0
$568$	−5.70850			−0.239523
$569$	−3.38562	−	5.86407i	−0.141933	−	0.245835i	0.786292	−	0.617855i	$-0.211998\pi$
$569$	−3.38562	−	5.86407i	−0.141933	−	0.245835i	−0.928224	+	0.372021i	$0.878665\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$	−2.32288	−	4.02334i	−0.0971243	−	0.168224i
$573$		0			0
$574$	−10.1144	+	17.5186i	−0.422166	+	0.731213i
$575$	−17.5203			−0.730645
$576$		0			0
$577$	−4.29150	+	7.43310i	−0.178658	+	0.309444i	−0.941421	−	0.337234i	$-0.890509\pi$
$577$	−4.29150	+	7.43310i	−0.178658	+	0.309444i	0.762763	+	0.646678i	$0.223842\pi$
$578$	7.58301	−	13.1342i	0.315411	−	0.546309i
$579$		0			0
$580$	−10.3542			−0.429937
$581$	−3.58301	−	6.20595i	−0.148648	−	0.257466i
$582$		0			0
$583$	−6.96863	−	12.0700i	−0.288611	−	0.499889i
$584$	4.17712	−	7.23499i	0.172851	−	0.299386i
$585$		0			0
$586$	−14.7601	−	25.5653i	−0.609735	−	1.05609i
$587$	−16.0627			−0.662980			−0.331490	−	0.943459i	$-0.607551\pi$
$587$	−16.0627			−0.662980			−0.331490	+	0.943459i	$0.607551\pi$
$588$		0			0
$589$	−36.4575			−1.50221
$590$	−6.53137	−	11.3127i	−0.268892	−	0.465735i
$591$		0			0
$592$	−0.177124	+	0.306788i	−0.00727977	+	0.0126089i
$593$	3.23987	+	5.61162i	0.133046	+	0.230442i	0.924849	−	0.380334i	$-0.124191\pi$
$593$	3.23987	+	5.61162i	0.133046	+	0.230442i	−0.791804	+	0.610776i	$0.790858\pi$
$594$		0			0
$595$	2.94837	+	5.10672i	0.120871	+	0.209355i
$596$	22.9373			0.939547
$597$		0			0
$598$	−3.82288	+	6.62141i	−0.156329	+	0.270770i
$599$	7.40588	−	12.8274i	0.302596	−	0.524112i	−0.674127	−	0.738615i	$-0.735480\pi$
$599$	7.40588	−	12.8274i	0.302596	−	0.524112i	0.976723	+	0.214504i	$0.0688133\pi$
$600$		0			0
$601$	−16.8745			−0.688326			−0.344163	−	0.938910i	$-0.611837\pi$
$601$	−16.8745			−0.688326			−0.344163	+	0.938910i	$0.611837\pi$
$602$	−7.00000	+	12.1244i	−0.285299	+	0.494152i
$603$		0			0
$604$	−10.9686	−	18.9982i	−0.446307	−	0.773027i
$605$	8.70850	−	15.0836i	0.354051	−	0.613234i
$606$		0			0
$607$	15.4059	+	26.6838i	0.625305	+	1.08306i	0.988482	+	0.151340i	$0.0483589\pi$
$607$	15.4059	+	26.6838i	0.625305	+	1.08306i	−0.363176	+	0.931720i	$0.618308\pi$
$608$	5.00000			0.202777
$609$		0			0
$610$	−6.47974			−0.262357
$611$	1.50000	+	2.59808i	0.0606835	+	0.105107i
$612$		0			0
$613$	−5.06275	+	8.76893i	−0.204482	+	0.354174i	−0.949968	−	0.312348i	$-0.898884\pi$
$613$	−5.06275	+	8.76893i	−0.204482	+	0.354174i	0.745485	+	0.666522i	$0.232218\pi$
$614$	−5.79150	−	10.0312i	−0.233726	−	0.404825i
$615$		0			0
$616$	12.2915			0.495239
$617$	21.2915			0.857164			0.428582	−	0.903503i	$-0.359013\pi$
$617$	21.2915			0.857164			0.428582	+	0.903503i	$0.359013\pi$
$618$		0			0
$619$	−10.8745	+	18.8352i	−0.437083	+	0.757051i	−0.997463	−	0.0711852i	$-0.977322\pi$
$619$	−10.8745	+	18.8352i	−0.437083	+	0.757051i	0.560380	+	0.828236i	$0.310655\pi$
$620$	−6.00000	+	10.3923i	−0.240966	+	0.417365i
$621$		0			0
$622$	16.9373			0.679122
$623$	1.40588	−	2.43506i	0.0563254	−	0.0975585i
$624$		0			0
$625$	4.14575	+	7.18065i	0.165830	+	0.287226i
$626$	14.5830	−	25.2585i	0.582854	−	1.00953i
$627$		0			0
$628$	1.32288	+	2.29129i	0.0527885	+	0.0914323i
$629$	0.479741			0.0191285
$630$		0			0
$631$	−38.4575			−1.53097			−0.765485	−	0.643454i	$-0.777501\pi$
$631$	−38.4575			−1.53097			−0.765485	+	0.643454i	$0.777501\pi$
$632$	5.00000	+	8.66025i	0.198889	+	0.344486i
$633$		0			0
$634$	−1.35425	+	2.34563i	−0.0537841	+	0.0931568i
$635$	13.3542	+	23.1302i	0.529947	+	0.917895i
$636$		0			0
$637$	−3.50000	+	6.06218i	−0.138675	+	0.240192i
$638$	29.2288			1.15718
$639$		0			0
$640$	0.822876	−	1.42526i	0.0325270	−	0.0563384i
$641$	−21.2915	+	36.8780i	−0.840964	+	1.45659i	0.0481170	+	0.998842i	$0.484678\pi$
$641$	−21.2915	+	36.8780i	−0.840964	+	1.45659i	−0.889081	+	0.457750i	$0.848655\pi$
$642$		0			0
$643$	20.8745			0.823210			0.411605	−	0.911362i	$-0.364968\pi$
$643$	20.8745			0.823210			0.411605	+	0.911362i	$0.364968\pi$
$644$	−10.1144	−	17.5186i	−0.398562	−	0.690330i
$645$		0			0
$646$	−3.38562	−	5.86407i	−0.133206	−	0.230719i
$647$	−18.8745	+	32.6916i	−0.742033	+	1.28524i	0.209535	+	0.977801i	$0.432805\pi$
$647$	−18.8745	+	32.6916i	−0.742033	+	1.28524i	−0.951568	+	0.307438i	$0.900528\pi$
$648$		0			0
$649$	18.4373	+	31.9343i	0.723726	+	1.25353i
$650$	−2.29150			−0.0898801
$651$		0			0
$652$	23.5830			0.923582
$653$	6.00000	+	10.3923i	0.234798	+	0.406682i	0.959214	−	0.282681i	$-0.0912238\pi$
$653$	6.00000	+	10.3923i	0.234798	+	0.406682i	−0.724416	+	0.689363i	$0.757890\pi$
$654$		0			0
$655$	4.06275	−	7.03688i	0.158745	−	0.274954i
$656$	−3.82288	−	6.62141i	−0.149258	−	0.258523i
$657$		0			0
$658$	−7.93725			−0.309426
$659$	1.06275			0.0413987			0.0206994	−	0.999786i	$-0.493411\pi$
$659$	1.06275			0.0413987			0.0206994	+	0.999786i	$0.493411\pi$
$660$		0			0
$661$	10.7601	−	18.6371i	0.418521	−	0.724899i	−0.577270	−	0.816553i	$-0.695882\pi$
$661$	10.7601	−	18.6371i	0.418521	−	0.724899i	0.995791	+	0.0916543i	$0.0292155\pi$
$662$	−16.2915	+	28.2177i	−0.633187	+	1.09671i
$663$		0			0
$664$	2.70850			0.105110
$665$	−21.7712			−0.844253
$666$		0			0
$667$	−24.0516	−	41.6586i	−0.931283	−	1.61303i
$668$	12.4373	−	21.5420i	0.481212	−	0.833483i
$669$		0			0
$670$	−11.1771	−	19.3593i	−0.431810	−	0.747917i
$671$	18.2915			0.706136
$672$		0			0
$673$	14.5830			0.562134			0.281067	−	0.959688i	$-0.409312\pi$
$673$	14.5830			0.562134			0.281067	+	0.959688i	$0.409312\pi$
$674$	−2.79150	−	4.83502i	−0.107525	−	0.186238i
$675$		0			0
$676$	−0.500000	+	0.866025i	−0.0192308	+	0.0333087i
$677$	−11.0830	−	19.1963i	−0.425954	−	0.737775i	0.570555	−	0.821260i	$-0.306728\pi$
$677$	−11.0830	−	19.1963i	−0.425954	−	0.737775i	−0.996509	+	0.0834849i	$0.973395\pi$
$678$		0			0
$679$	19.7601	+	34.2255i	0.758324	+	1.31346i
$680$	−2.22876			−0.0854689
$681$		0			0
$682$	16.9373	−	29.3362i	0.648561	−	1.12334i
$683$	4.35425	−	7.54178i	0.166611	−	0.288578i	−0.770615	−	0.637300i	$-0.780051\pi$
$683$	4.35425	−	7.54178i	0.166611	−	0.288578i	0.937226	+	0.348722i	$0.113384\pi$
$684$		0			0
$685$	28.8340			1.10169
$686$	−9.26013	−	16.0390i	−0.353553	−	0.612372i
$687$		0			0
$688$	−2.64575	−	4.58258i	−0.100868	−	0.174709i
$689$	−1.50000	+	2.59808i	−0.0571454	+	0.0989788i
$690$		0			0
$691$	−17.0203	−	29.4800i	−0.647481	−	1.12147i	−0.983722	−	0.179694i	$-0.942489\pi$
$691$	−17.0203	−	29.4800i	−0.647481	−	1.12147i	0.336241	−	0.941776i	$-0.390844\pi$
$692$	11.7085			0.445090
$693$		0			0
$694$	26.2288			0.995630
$695$	3.47974	+	6.02709i	0.131994	+	0.228620i
$696$		0			0
$697$	−5.17712	+	8.96704i	−0.196098	+	0.339651i
$698$	−1.53137	−	2.65242i	−0.0579633	−	0.100395i
$699$		0			0
$700$	3.03137	−	5.25049i	0.114575	−	0.198450i
$701$	−12.0000			−0.453234			−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000			−0.453234			−0.226617	+	0.973984i	$0.572767\pi$
$702$		0			0
$703$	−0.885622	+	1.53394i	−0.0334019	+	0.0578537i
$704$	−2.32288	+	4.02334i	−0.0875467	+	0.151635i
$705$		0			0
$706$	−0.583005			−0.0219417
$707$		0			0
$708$		0			0
$709$	−21.7601	−	37.6897i	−0.817219	−	1.41546i	−0.907724	−	0.419569i	$-0.862181\pi$
$709$	−21.7601	−	37.6897i	−0.817219	−	1.41546i	0.0905049	−	0.995896i	$-0.471152\pi$
$710$	−4.69738	+	8.13611i	−0.176290	+	0.305343i
$711$		0			0
$712$	0.531373	+	0.920365i	0.0199140	+	0.0344921i
$713$	−55.7490			−2.08782
$714$		0			0
$715$	−7.64575			−0.285935
$716$	−3.00000	−	5.19615i	−0.112115	−	0.194189i
$717$		0			0
$718$	−3.00000	+	5.19615i	−0.111959	+	0.193919i
$719$	11.7085	+	20.2797i	0.436653	+	0.756306i	0.997429	−	0.0716621i	$-0.0228303\pi$
$719$	11.7085	+	20.2797i	0.436653	+	0.756306i	−0.560776	+	0.827968i	$0.689497\pi$
$720$		0			0
$721$	−7.00000	+	12.1244i	−0.260694	+	0.451535i
$722$	6.00000			0.223297
$723$		0			0
$724$	−4.67712	+	8.10102i	−0.173824	+	0.301072i
$725$	7.20850	−	12.4855i	0.267717	−	0.463699i
$726$		0			0
$727$	36.4575			1.35213			0.676067	−	0.736840i	$-0.263683\pi$
$727$	36.4575			1.35213			0.676067	+	0.736840i	$0.263683\pi$
$728$	−1.32288	−	2.29129i	−0.0490290	−	0.0849208i
$729$		0			0
$730$	−6.87451	−	11.9070i	−0.254437	−	0.440698i
$731$	−3.58301	+	6.20595i	−0.132522	+	0.229535i
$732$		0			0
$733$	6.11438	+	10.5904i	0.225840	+	0.391166i	0.956571	−	0.291499i	$-0.0941541\pi$
$733$	6.11438	+	10.5904i	0.225840	+	0.391166i	−0.730731	+	0.682665i	$0.760821\pi$
$734$	20.5830			0.759733
$735$		0			0
$736$	7.64575			0.281826
$737$	31.5516	+	54.6490i	1.16222	+	2.01302i
$738$		0			0
$739$	3.35425	−	5.80973i	0.123388	−	0.213714i	−0.797714	−	0.603036i	$-0.793957\pi$
$739$	3.35425	−	5.80973i	0.123388	−	0.213714i	0.921102	+	0.389322i	$0.127291\pi$
$740$	0.291503	+	0.504897i	0.0107158	+	0.0185604i
$741$		0			0
$742$	−3.96863	−	6.87386i	−0.145693	−	0.252347i
$743$	−1.45751			−0.0534710			−0.0267355	−	0.999643i	$-0.508511\pi$
$743$	−1.45751			−0.0534710			−0.0267355	+	0.999643i	$0.508511\pi$
$744$		0			0
$745$	18.8745	−	32.6916i	0.691508	−	1.19773i
$746$	−4.67712	+	8.10102i	−0.171242	+	0.296599i
$747$		0			0
$748$	6.29150			0.230040
$749$	−15.8745	+	27.4955i	−0.580042	+	1.00466i
$750$		0			0
$751$	−4.58301	−	7.93800i	−0.167236	−	0.289662i	0.770211	−	0.637789i	$-0.220151\pi$
$751$	−4.58301	−	7.93800i	−0.167236	−	0.289662i	−0.937447	+	0.348128i	$0.886818\pi$
$752$	1.50000	−	2.59808i	0.0546994	−	0.0947421i
$753$		0			0
$754$	−3.14575	−	5.44860i	−0.114562	−	0.198426i
$755$	−36.1033			−1.31393
$756$		0			0
$757$	21.3542			0.776133			0.388067	−	0.921631i	$-0.373143\pi$
$757$	21.3542			0.776133			0.388067	+	0.921631i	$0.373143\pi$
$758$	−16.5830	−	28.7226i	−0.602322	−	1.04325i
$759$		0			0
$760$	4.11438	−	7.12631i	0.149244	−	0.258499i
$761$	−1.64575	−	2.85052i	−0.0596584	−	0.103331i	0.834654	−	0.550775i	$-0.185668\pi$
$761$	−1.64575	−	2.85052i	−0.0596584	−	0.103331i	−0.894312	+	0.447444i	$0.852334\pi$
$762$		0			0
$763$	−10.5830			−0.383131
$764$		0			0
$765$		0			0
$766$	−18.8745	+	32.6916i	−0.681964	+	1.18120i
$767$	3.96863	−	6.87386i	0.143299	−	0.248201i
$768$		0			0
$769$	0.354249			0.0127745			0.00638727	−	0.999980i	$-0.497967\pi$
$769$	0.354249			0.0127745			0.00638727	+	0.999980i	$0.497967\pi$
$770$	10.1144	−	17.5186i	0.364497	−	0.631327i
$771$		0			0
$772$	−1.53137	−	2.65242i	−0.0551153	−	0.0954625i
$773$	−22.1660	+	38.3927i	−0.797256	+	1.38089i	0.124141	+	0.992265i	$0.460383\pi$
$773$	−22.1660	+	38.3927i	−0.797256	+	1.38089i	−0.921397	+	0.388623i	$0.872951\pi$
$774$		0			0
$775$	−8.35425	−	14.4700i	−0.300093	−	0.519777i
$776$	−14.9373			−0.536216
$777$		0			0
$778$	−18.2915			−0.655782
$779$	−19.1144	−	33.1071i	−0.684844	−	1.18618i
$780$		0			0
$781$	13.2601	−	22.9672i	0.474485	−	0.821832i
$782$	−5.17712	−	8.96704i	−0.185134	−	0.320661i
$783$		0			0
$784$	7.00000			0.250000
$785$	4.35425			0.155410
$786$		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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.j (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.822876 + 1.42526i) q^{5} +(-1.32288 - 2.29129i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.822876 + 1.42526i) q^{5} +(-1.32288 - 2.29129i) q^{7} +1.00000 q^{8} +(0.822876 - 1.42526i) q^{10} +(-2.32288 + 4.02334i) q^{11} +1.00000 q^{13} +(-1.32288 + 2.29129i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.677124 + 1.17281i) q^{17} +(-2.50000 - 4.33013i) q^{19} -1.64575 q^{20} +4.64575 q^{22} +(-3.82288 - 6.62141i) q^{23} +(1.14575 - 1.98450i) q^{25} +(-0.500000 - 0.866025i) q^{26} +2.64575 q^{28} +6.29150 q^{29} +(3.64575 - 6.31463i) q^{31} +(-0.500000 + 0.866025i) q^{32} +1.35425 q^{34} +(2.17712 - 3.77089i) q^{35} +(-0.177124 - 0.306788i) q^{37} +(-2.50000 + 4.33013i) q^{38} +(0.822876 + 1.42526i) q^{40} +7.64575 q^{41} +5.29150 q^{43} +(-2.32288 - 4.02334i) q^{44} +(-3.82288 + 6.62141i) q^{46} +(1.50000 + 2.59808i) q^{47} +(-3.50000 + 6.06218i) q^{49} -2.29150 q^{50} +(-0.500000 + 0.866025i) q^{52} +(-1.50000 + 2.59808i) q^{53} -7.64575 q^{55} +(-1.32288 - 2.29129i) q^{56} +(-3.14575 - 5.44860i) q^{58} +(3.96863 - 6.87386i) q^{59} +(-1.96863 - 3.40976i) q^{61} -7.29150 q^{62} +1.00000 q^{64} +(0.822876 + 1.42526i) q^{65} +(6.79150 - 11.7632i) q^{67} +(-0.677124 - 1.17281i) q^{68} -4.35425 q^{70} -5.70850 q^{71} +(4.17712 - 7.23499i) q^{73} +(-0.177124 + 0.306788i) q^{74} +5.00000 q^{76} +12.2915 q^{77} +(5.00000 + 8.66025i) q^{79} +(0.822876 - 1.42526i) q^{80} +(-3.82288 - 6.62141i) q^{82} +2.70850 q^{83} -2.22876 q^{85} +(-2.64575 - 4.58258i) q^{86} +(-2.32288 + 4.02334i) q^{88} +(0.531373 + 0.920365i) q^{89} +(-1.32288 - 2.29129i) q^{91} +7.64575 q^{92} +(1.50000 - 2.59808i) q^{94} +(4.11438 - 7.12631i) q^{95} -14.9373 q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 + 4 * q^8	\( 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{8} - 2 q^{10} - 4 q^{11} + 4 q^{13} - 2 q^{16} - 8 q^{17} - 10 q^{19} + 4 q^{20} + 8 q^{22} - 10 q^{23} - 6 q^{25} - 2 q^{26} + 4 q^{29} + 4 q^{31} - 2 q^{32} + 16 q^{34} + 14 q^{35} - 6 q^{37} - 10 q^{38} - 2 q^{40} + 20 q^{41} - 4 q^{44} - 10 q^{46} + 6 q^{47} - 14 q^{49} + 12 q^{50} - 2 q^{52} - 6 q^{53} - 20 q^{55} - 2 q^{58} + 8 q^{61} - 8 q^{62} + 4 q^{64} - 2 q^{65} + 6 q^{67} - 8 q^{68} - 28 q^{70} - 44 q^{71} + 22 q^{73} - 6 q^{74} + 20 q^{76} + 28 q^{77} + 20 q^{79} - 2 q^{80} - 10 q^{82} + 32 q^{83} + 44 q^{85} - 4 q^{88} + 18 q^{89} + 20 q^{92} + 6 q^{94} - 10 q^{95} - 28 q^{97} + 28 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 - 2 * q^5 + 4 * q^8 - 2 * q^10 - 4 * q^11 + 4 * q^13 - 2 * q^16 - 8 * q^17 - 10 * q^19 + 4 * q^20 + 8 * q^22 - 10 * q^23 - 6 * q^25 - 2 * q^26 + 4 * q^29 + 4 * q^31 - 2 * q^32 + 16 * q^34 + 14 * q^35 - 6 * q^37 - 10 * q^38 - 2 * q^40 + 20 * q^41 - 4 * q^44 - 10 * q^46 + 6 * q^47 - 14 * q^49 + 12 * q^50 - 2 * q^52 - 6 * q^53 - 20 * q^55 - 2 * q^58 + 8 * q^61 - 8 * q^62 + 4 * q^64 - 2 * q^65 + 6 * q^67 - 8 * q^68 - 28 * q^70 - 44 * q^71 + 22 * q^73 - 6 * q^74 + 20 * q^76 + 28 * q^77 + 20 * q^79 - 2 * q^80 - 10 * q^82 + 32 * q^83 + 44 * q^85 - 4 * q^88 + 18 * q^89 + 20 * q^92 + 6 * q^94 - 10 * q^95 - 28 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more