LMFDB - Embedded newform 378.2.h.d.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("378.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.224437i$ of defining polynomial
Character	$\chi$	$=$	378.289
Dual form			378.2.h.d.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -3.69963 q^{5} +(-1.40545 - 2.24159i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -3.69963 q^{5} +(-1.40545 - 2.24159i) q^{7} -1.00000 q^{8} +(-1.84981 - 3.20397i) q^{10} +1.47710 q^{11} +(-1.34981 - 2.33795i) q^{13} +(1.23855 - 2.33795i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.28799 - 5.69497i) q^{17} +(-0.444368 + 0.769668i) q^{19} +(1.84981 - 3.20397i) q^{20} +(0.738550 + 1.27921i) q^{22} -6.28799 q^{23} +8.68725 q^{25} +(1.34981 - 2.33795i) q^{26} +(2.64400 - 0.0963576i) q^{28} +(-1.25526 + 2.17417i) q^{29} +(-3.40545 + 5.89841i) q^{31} +(0.500000 - 0.866025i) q^{32} +(3.28799 - 5.69497i) q^{34} +(5.19963 + 8.29305i) q^{35} +(-1.38874 + 2.40536i) q^{37} -0.888736 q^{38} +3.69963 q^{40} +(2.05563 + 3.56046i) q^{41} +(0.00618986 - 0.0107211i) q^{43} +(-0.738550 + 1.27921i) q^{44} +(-3.14400 - 5.44556i) q^{46} +(-3.49381 - 6.05146i) q^{47} +(-3.04944 + 6.30087i) q^{49} +(4.34362 + 7.52338i) q^{50} +2.69963 q^{52} +(1.60507 + 2.78007i) q^{53} -5.46472 q^{55} +(1.40545 + 2.24159i) q^{56} -2.51052 q^{58} +(3.45489 - 5.98404i) q^{59} +(2.86652 + 4.96497i) q^{61} -6.81089 q^{62} +1.00000 q^{64} +(4.99381 + 8.64953i) q^{65} +(4.73236 - 8.19669i) q^{67} +6.57598 q^{68} +(-4.58217 + 8.64953i) q^{70} +5.46472 q^{71} +(-6.03273 - 10.4490i) q^{73} -2.77747 q^{74} +(-0.444368 - 0.769668i) q^{76} +(-2.07598 - 3.31105i) q^{77} +(-5.72617 - 9.91802i) q^{79} +(1.84981 + 3.20397i) q^{80} +(-2.05563 + 3.56046i) q^{82} +(-2.23855 + 3.87728i) q^{83} +(12.1643 + 21.0693i) q^{85} +0.0123797 q^{86} -1.47710 q^{88} +(4.43818 - 7.68715i) q^{89} +(-3.34362 + 6.31159i) q^{91} +(3.14400 - 5.44556i) q^{92} +(3.49381 - 6.05146i) q^{94} +(1.64400 - 2.84748i) q^{95} +(-6.58836 + 11.4114i) q^{97} +(-6.98143 + 0.509538i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8	$ 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8} - 5 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} - q^{22} - 14 q^{23} + 4 q^{25} + 2 q^{26} + 4 q^{28} + 5 q^{29} - 14 q^{31} + 3 q^{32} - 4 q^{34} + 19 q^{35} - 9 q^{37} - 6 q^{38} + 10 q^{40} + 12 q^{41} + 18 q^{43} + q^{44} - 7 q^{46} - 3 q^{47} + 2 q^{50} + 4 q^{52} - 9 q^{53} + 14 q^{55} + 2 q^{56} + 10 q^{58} - 4 q^{59} + 4 q^{61} - 28 q^{62} + 6 q^{64} + 12 q^{65} + 5 q^{67} - 8 q^{68} + 2 q^{70} - 14 q^{71} - 25 q^{73} - 18 q^{74} - 3 q^{76} + 35 q^{77} + 7 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} + 14 q^{85} + 36 q^{86} + 2 q^{88} + 9 q^{89} + 4 q^{91} + 7 q^{92} + 3 q^{94} - 2 q^{95} - 28 q^{97} + 12 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8 - 5 * q^10 - 2 * q^11 - 2 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^19 + 5 * q^20 - q^22 - 14 * q^23 + 4 * q^25 + 2 * q^26 + 4 * q^28 + 5 * q^29 - 14 * q^31 + 3 * q^32 - 4 * q^34 + 19 * q^35 - 9 * q^37 - 6 * q^38 + 10 * q^40 + 12 * q^41 + 18 * q^43 + q^44 - 7 * q^46 - 3 * q^47 + 2 * q^50 + 4 * q^52 - 9 * q^53 + 14 * q^55 + 2 * q^56 + 10 * q^58 - 4 * q^59 + 4 * q^61 - 28 * q^62 + 6 * q^64 + 12 * q^65 + 5 * q^67 - 8 * q^68 + 2 * q^70 - 14 * q^71 - 25 * q^73 - 18 * q^74 - 3 * q^76 + 35 * q^77 + 7 * q^79 + 5 * q^80 - 12 * q^82 - 8 * q^83 + 14 * q^85 + 36 * q^86 + 2 * q^88 + 9 * q^89 + 4 * q^91 + 7 * q^92 + 3 * q^94 - 2 * q^95 - 28 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	0.500000	+	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
$5$	−3.69963			−1.65452			−0.827262	−	0.561816i	$-0.810103\pi$
$5$	−3.69963			−1.65452			−0.827262	+	0.561816i	$0.810103\pi$
$6$		0			0
$7$	−1.40545	−	2.24159i	−0.531209	−	0.847241i
$7$	−1.40545	−	2.24159i	−0.531209	−	0.847241i
$8$	−1.00000			−0.353553
$9$		0			0
$10$	−1.84981	−	3.20397i	−0.584963	−	1.01318i
$11$	1.47710			0.445362			0.222681	−	0.974891i	$-0.428519\pi$
$11$	1.47710			0.445362			0.222681	+	0.974891i	$0.428519\pi$
$12$		0			0
$13$	−1.34981	−	2.33795i	−0.374371	−	0.648430i	0.615862	−	0.787854i	$-0.288808\pi$
$13$	−1.34981	−	2.33795i	−0.374371	−	0.648430i	−0.990233	+	0.139425i	$0.955475\pi$
$14$	1.23855	−	2.33795i	0.331016	−	0.624843i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−3.28799	−	5.69497i	−0.797455	−	1.38123i	−0.921268	−	0.388927i	$-0.872846\pi$
$17$	−3.28799	−	5.69497i	−0.797455	−	1.38123i	0.123813	−	0.992306i	$-0.460488\pi$
$18$		0			0
$19$	−0.444368	+	0.769668i	−0.101945	+	0.176574i	−0.912486	−	0.409108i	$-0.865840\pi$
$19$	−0.444368	+	0.769668i	−0.101945	+	0.176574i	0.810541	+	0.585682i	$0.199173\pi$
$20$	1.84981	−	3.20397i	0.413631	−	0.716430i
$21$		0			0
$22$	0.738550	+	1.27921i	0.157459	+	0.272728i
$23$	−6.28799			−1.31114			−0.655568	−	0.755136i	$-0.727571\pi$
$23$	−6.28799			−1.31114			−0.655568	+	0.755136i	$0.727571\pi$
$24$		0			0
$25$	8.68725			1.73745
$26$	1.34981	−	2.33795i	0.264720	−	0.458509i
$27$		0			0
$28$	2.64400	−	0.0963576i	0.499668	−	0.0182099i
$29$	−1.25526	+	2.17417i	−0.233096	+	0.403734i	−0.958718	−	0.284360i	$-0.908219\pi$
$29$	−1.25526	+	2.17417i	−0.233096	+	0.403734i	0.725622	+	0.688094i	$0.241552\pi$
$30$		0			0
$31$	−3.40545	+	5.89841i	−0.611636	+	1.05938i	0.379329	+	0.925262i	$0.376155\pi$
$31$	−3.40545	+	5.89841i	−0.611636	+	1.05938i	−0.990965	+	0.134123i	$0.957178\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$	3.28799	−	5.69497i	0.563886	−	0.976679i
$35$	5.19963	+	8.29305i	0.878898	+	1.40178i
$36$		0			0
$37$	−1.38874	+	2.40536i	−0.228307	+	0.395439i	−0.957306	−	0.289075i	$-0.906652\pi$
$37$	−1.38874	+	2.40536i	−0.228307	+	0.395439i	0.729000	+	0.684514i	$0.239986\pi$
$38$	−0.888736			−0.144172
$39$		0			0
$40$	3.69963			0.584963
$41$	2.05563	+	3.56046i	0.321036	+	0.556050i	0.980702	−	0.195508i	$-0.0626357\pi$
$41$	2.05563	+	3.56046i	0.321036	+	0.556050i	−0.659666	+	0.751559i	$0.729302\pi$
$42$		0			0
$43$	0.00618986	−	0.0107211i	0.000943944	−	0.00163496i	−0.865553	−	0.500817i	$-0.833033\pi$
$43$	0.00618986	−	0.0107211i	0.000943944	−	0.00163496i	0.866497	+	0.499182i	$0.166366\pi$
$44$	−0.738550	+	1.27921i	−0.111341	+	0.192848i
$45$		0			0
$46$	−3.14400	−	5.44556i	−0.463557	−	0.802904i
$47$	−3.49381	−	6.05146i	−0.509625	−	0.882696i	−0.999938	−	0.0111494i	$-0.996451\pi$
$47$	−3.49381	−	6.05146i	−0.509625	−	0.882696i	0.490313	−	0.871546i	$-0.336882\pi$
$48$		0			0
$49$	−3.04944	+	6.30087i	−0.435635	+	0.900124i
$50$	4.34362	+	7.52338i	0.614281	+	1.06397i
$51$		0			0
$52$	2.69963			0.374371
$53$	1.60507	+	2.78007i	0.220474	+	0.381872i	0.954952	−	0.296760i	$-0.0959063\pi$
$53$	1.60507	+	2.78007i	0.220474	+	0.381872i	−0.734478	+	0.678632i	$0.762573\pi$
$54$		0			0
$55$	−5.46472			−0.736863
$56$	1.40545	+	2.24159i	0.187811	+	0.299545i
$57$		0			0
$58$	−2.51052			−0.329647
$59$	3.45489	−	5.98404i	0.449788	−	0.779056i	−0.548584	−	0.836096i	$-0.684833\pi$
$59$	3.45489	−	5.98404i	0.449788	−	0.779056i	0.998372	+	0.0570397i	$0.0181661\pi$
$60$		0			0
$61$	2.86652	+	4.96497i	0.367021	+	0.635699i	0.989098	−	0.147257i	$-0.0470444\pi$
$61$	2.86652	+	4.96497i	0.367021	+	0.635699i	−0.622077	+	0.782956i	$0.713711\pi$
$62$	−6.81089			−0.864984
$63$		0			0
$64$	1.00000			0.125000
$65$	4.99381	+	8.64953i	0.619406	+	1.07284i
$66$		0			0
$67$	4.73236	−	8.19669i	0.578150	−	1.00138i	−0.417542	−	0.908658i	$-0.637108\pi$
$67$	4.73236	−	8.19669i	0.578150	−	1.00138i	0.995692	−	0.0927271i	$-0.0295584\pi$
$68$	6.57598			0.797455
$69$		0			0
$70$	−4.58217	+	8.64953i	−0.547675	+	1.03382i
$71$	5.46472			0.648543			0.324271	−	0.945964i	$-0.394881\pi$
$71$	5.46472			0.648543			0.324271	+	0.945964i	$0.394881\pi$
$72$		0			0
$73$	−6.03273	−	10.4490i	−0.706078	−	1.22296i	−0.966301	−	0.257414i	$-0.917130\pi$
$73$	−6.03273	−	10.4490i	−0.706078	−	1.22296i	0.260223	−	0.965548i	$-0.416204\pi$
$74$	−2.77747			−0.322875
$75$		0			0
$76$	−0.444368	−	0.769668i	−0.0509725	−	0.0882870i
$77$	−2.07598	−	3.31105i	−0.236580	−	0.377329i
$78$		0			0
$79$	−5.72617	−	9.91802i	−0.644244	−	1.11586i	−0.984475	−	0.175522i	$-0.943839\pi$
$79$	−5.72617	−	9.91802i	−0.644244	−	1.11586i	0.340231	−	0.940342i	$-0.389495\pi$
$80$	1.84981	+	3.20397i	0.206816	+	0.358215i
$81$		0			0
$82$	−2.05563	+	3.56046i	−0.227007	+	0.393187i
$83$	−2.23855	+	3.87728i	−0.245713	+	0.425587i	−0.962332	−	0.271878i	$-0.912355\pi$
$83$	−2.23855	+	3.87728i	−0.245713	+	0.425587i	0.716619	+	0.697465i	$0.245689\pi$
$84$		0			0
$85$	12.1643	+	21.0693i	1.31941	+	2.28528i
$86$	0.0123797			0.00133494
$87$		0			0
$88$	−1.47710			−0.157459
$89$	4.43818	−	7.68715i	0.470446	−	0.814836i	−0.528983	−	0.848633i	$-0.677426\pi$
$89$	4.43818	−	7.68715i	0.470446	−	0.814836i	0.999429	+	0.0337963i	$0.0107597\pi$
$90$		0			0
$91$	−3.34362	+	6.31159i	−0.350507	+	0.661634i
$92$	3.14400	−	5.44556i	0.327784	−	0.567739i
$93$		0			0
$94$	3.49381	−	6.05146i	0.360359	−	0.624160i
$95$	1.64400	−	2.84748i	0.168670	−	0.292146i
$96$		0			0
$97$	−6.58836	+	11.4114i	−0.668947	+	1.15865i	0.309252	+	0.950980i	$0.399921\pi$
$97$	−6.58836	+	11.4114i	−0.668947	+	1.15865i	−0.978199	+	0.207670i	$0.933412\pi$
$98$	−6.98143	+	0.509538i	−0.705231	+	0.0514711i
$99$		0			0
$100$	−4.34362	+	7.52338i	−0.434362	+	0.752338i
$101$	−5.25457			−0.522849			−0.261425	−	0.965224i	$-0.584192\pi$
$101$	−5.25457			−0.522849			−0.261425	+	0.965224i	$0.584192\pi$
$102$		0			0
$103$	1.66621			0.164176			0.0820882	−	0.996625i	$-0.473841\pi$
$103$	1.66621			0.164176			0.0820882	+	0.996625i	$0.473841\pi$
$104$	1.34981	+	2.33795i	0.132360	+	0.229255i
$105$		0			0
$106$	−1.60507	+	2.78007i	−0.155899	+	0.270024i
$107$	5.38255	−	9.32284i	0.520350	−	0.901273i	−0.479370	−	0.877613i	$-0.659135\pi$
$107$	5.38255	−	9.32284i	0.520350	−	0.901273i	0.999720	−	0.0236602i	$-0.00753198\pi$
$108$		0			0
$109$	−0.0945538	−	0.163772i	−0.00905662	−	0.0156865i	0.861462	−	0.507823i	$-0.169550\pi$
$109$	−0.0945538	−	0.163772i	−0.00905662	−	0.0156865i	−0.870518	+	0.492136i	$0.836216\pi$
$110$	−2.73236	−	4.73259i	−0.260520	−	0.451234i
$111$		0			0
$112$	−1.23855	+	2.33795i	−0.117032	+	0.220915i
$113$	6.78180	+	11.7464i	0.637978	+	1.10501i	0.985876	+	0.167478i	$0.0535624\pi$
$113$	6.78180	+	11.7464i	0.637978	+	1.10501i	−0.347897	+	0.937533i	$0.613104\pi$
$114$		0			0
$115$	23.2632			2.16931
$116$	−1.25526	−	2.17417i	−0.116548	−	0.201867i
$117$		0			0
$118$	6.90978			0.636097
$119$	−8.14468	+	15.3743i	−0.746622	+	1.40936i
$120$		0			0
$121$	−8.81818			−0.801652
$122$	−2.86652	+	4.96497i	−0.259523	+	0.449507i
$123$		0			0
$124$	−3.40545	−	5.89841i	−0.305818	−	0.529692i
$125$	−13.6414			−1.22013
$126$		0			0
$127$	−2.85669			−0.253490			−0.126745	−	0.991935i	$-0.540453\pi$
$127$	−2.85669			−0.253490			−0.126745	+	0.991935i	$0.540453\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$	−4.99381	+	8.64953i	−0.437986	+	0.758614i
$131$	−0.155687			−0.0136024			−0.00680122	−	0.999977i	$-0.502165\pi$
$131$	−0.155687			−0.0136024			−0.00680122	+	0.999977i	$0.502165\pi$
$132$		0			0
$133$	2.34981	−	0.0856364i	0.203755	−	0.00742562i
$134$	9.46472			0.817627
$135$		0			0
$136$	3.28799	+	5.69497i	0.281943	+	0.488340i
$137$	3.41164			0.291476			0.145738	−	0.989323i	$-0.453444\pi$
$137$	3.41164			0.291476			0.145738	+	0.989323i	$0.453444\pi$
$138$		0			0
$139$	−6.75526	−	11.7005i	−0.572974	−	0.992420i	−0.996259	−	0.0864229i	$-0.972456\pi$
$139$	−6.75526	−	11.7005i	−0.572974	−	0.992420i	0.423285	−	0.905997i	$-0.360877\pi$
$140$	−9.78180	+	0.356487i	−0.826713	+	0.0301287i
$141$		0			0
$142$	2.73236	+	4.73259i	0.229295	+	0.397150i
$143$	−1.99381	−	3.45338i	−0.166731	−	0.288786i
$144$		0			0
$145$	4.64400	−	8.04364i	0.385663	−	0.667988i
$146$	6.03273	−	10.4490i	0.499272	−	0.864765i
$147$		0			0
$148$	−1.38874	−	2.40536i	−0.114153	−	0.197719i
$149$	−0.333792			−0.0273453			−0.0136727	−	0.999907i	$-0.504352\pi$
$149$	−0.333792			−0.0273453			−0.0136727	+	0.999907i	$0.504352\pi$
$150$		0			0
$151$	−19.9098			−1.62023			−0.810117	−	0.586268i	$-0.800597\pi$
$151$	−19.9098			−1.62023			−0.810117	+	0.586268i	$0.800597\pi$
$152$	0.444368	−	0.769668i	0.0360430	−	0.0624283i
$153$		0			0
$154$	1.82946	−	3.45338i	0.147422	−	0.278281i
$155$	12.5989	−	21.8219i	1.01197	−	1.75278i
$156$		0			0
$157$	3.48143	−	6.03001i	0.277848	−	0.481248i	−0.693001	−	0.720936i	$-0.743712\pi$
$157$	3.48143	−	6.03001i	0.277848	−	0.481248i	0.970850	+	0.239689i	$0.0770454\pi$
$158$	5.72617	−	9.91802i	0.455550	−	0.789035i
$159$		0			0
$160$	−1.84981	+	3.20397i	−0.146241	+	0.253296i
$161$	8.83743	+	14.0951i	0.696487	+	1.11085i
$162$		0			0
$163$	4.03706	−	6.99240i	0.316207	−	0.547687i	−0.663486	−	0.748189i	$-0.730924\pi$
$163$	4.03706	−	6.99240i	0.316207	−	0.547687i	0.979693	+	0.200502i	$0.0642572\pi$
$164$	−4.11126			−0.321036
$165$		0			0
$166$	−4.47710			−0.347490
$167$	−9.74288	−	16.8752i	−0.753927	−	1.30584i	−0.945906	−	0.324440i	$-0.894824\pi$
$167$	−9.74288	−	16.8752i	−0.753927	−	1.30584i	0.191979	−	0.981399i	$-0.438509\pi$
$168$		0			0
$169$	2.85600	−	4.94674i	0.219693	−	0.380519i
$170$	−12.1643	+	21.0693i	−0.932963	+	1.61594i
$171$		0			0
$172$	0.00618986	+	0.0107211i	0.000471972	+	0.000817480i
$173$	11.2818	+	19.5407i	0.857740	+	1.48565i	0.874080	+	0.485782i	$0.161465\pi$
$173$	11.2818	+	19.5407i	0.857740	+	1.48565i	−0.0163405	+	0.999866i	$0.505202\pi$
$174$		0			0
$175$	−12.2095	−	19.4732i	−0.922948	−	1.47204i
$176$	−0.738550	−	1.27921i	−0.0556703	−	0.0964238i
$177$		0			0
$178$	8.87636			0.665311
$179$	−0.166896	−	0.289073i	−0.0124744	−	0.0216063i	0.859721	−	0.510764i	$-0.170637\pi$
$179$	−0.166896	−	0.289073i	−0.0124744	−	0.0216063i	−0.872195	+	0.489158i	$0.837304\pi$
$180$		0			0
$181$	23.2422			1.72758			0.863789	−	0.503853i	$-0.168085\pi$
$181$	23.2422			1.72758			0.863789	+	0.503853i	$0.168085\pi$
$182$	−7.13781	+	0.260130i	−0.529089	+	0.0192821i
$183$		0			0
$184$	6.28799			0.463557
$185$	5.13781	−	8.89894i	0.377739	−	0.654263i
$186$		0			0
$187$	−4.85669	−	8.41204i	−0.355157	−	0.615149i
$188$	6.98762			0.509625
$189$		0			0
$190$	3.28799			0.238536
$191$	−8.16071	−	14.1348i	−0.590488	−	1.02276i	−0.994167	−	0.107854i	$-0.965602\pi$
$191$	−8.16071	−	14.1348i	−0.590488	−	1.02276i	0.403679	−	0.914901i	$-0.367731\pi$
$192$		0			0
$193$	7.16071	−	12.4027i	0.515439	−	0.892766i	−0.484400	−	0.874846i	$-0.660962\pi$
$193$	7.16071	−	12.4027i	0.515439	−	0.892766i	0.999839	−	0.0179200i	$-0.00570443\pi$
$194$	−13.1767			−0.946034
$195$		0			0
$196$	−3.93199	−	5.79133i	−0.280856	−	0.413666i
$197$	−2.42402			−0.172704			−0.0863520	−	0.996265i	$-0.527521\pi$
$197$	−2.42402			−0.172704			−0.0863520	+	0.996265i	$0.527521\pi$
$198$		0			0
$199$	−3.05563	−	5.29251i	−0.216608	−	0.375176i	0.737161	−	0.675717i	$-0.236166\pi$
$199$	−3.05563	−	5.29251i	−0.216608	−	0.375176i	−0.953769	+	0.300541i	$0.902833\pi$
$200$	−8.68725			−0.614281
$201$		0			0
$202$	−2.62729	−	4.55059i	−0.184855	−	0.320179i
$203$	6.63781	−	0.241908i	0.465883	−	0.0169786i
$204$		0			0
$205$	−7.60507	−	13.1724i	−0.531161	−	0.919999i
$206$	0.833104	+	1.44298i	0.0580451	+	0.100537i
$207$		0			0
$208$	−1.34981	+	2.33795i	−0.0935928	+	0.162107i
$209$	−0.656376	+	1.13688i	−0.0454025	+	0.0786394i
$210$		0			0
$211$	5.72253	+	9.91171i	0.393955	+	0.682350i	0.992967	−	0.118390i	$-0.0377732\pi$
$211$	5.72253	+	9.91171i	0.393955	+	0.682350i	−0.599012	+	0.800740i	$0.704440\pi$
$212$	−3.21015			−0.220474
$213$		0			0
$214$	10.7651			0.735887
$215$	−0.0229002	+	0.0396643i	−0.00156178	+	0.00270508i
$216$		0			0
$217$	18.0080	−	0.656281i	1.22246	−	0.0445513i
$218$	0.0945538	−	0.163772i	0.00640399	−	0.0110920i
$219$		0			0
$220$	2.73236	−	4.73259i	0.184216	−	0.319071i
$221$	−8.87636	+	15.3743i	−0.597088	+	1.03419i
$222$		0			0
$223$	−3.61126	+	6.25489i	−0.241828	+	0.418859i	−0.961235	−	0.275730i	$-0.911080\pi$
$223$	−3.61126	+	6.25489i	−0.241828	+	0.418859i	0.719407	+	0.694589i	$0.244414\pi$
$224$	−2.64400	+	0.0963576i	−0.176659	+	0.00643816i
$225$		0			0
$226$	−6.78180	+	11.7464i	−0.451119	+	0.781361i
$227$	−13.6552			−0.906328			−0.453164	−	0.891427i	$-0.649705\pi$
$227$	−13.6552			−0.906328			−0.453164	+	0.891427i	$0.649705\pi$
$228$		0			0
$229$	17.3745			1.14814			0.574070	−	0.818807i	$-0.305364\pi$
$229$	17.3745			1.14814			0.574070	+	0.818807i	$0.305364\pi$
$230$	11.6316	+	20.1466i	0.766966	+	1.32842i
$231$		0			0
$232$	1.25526	−	2.17417i	0.0824119	−	0.142742i
$233$	−7.62110	+	13.2001i	−0.499275	+	0.864769i	−1.00000		0.000837426i	$-0.999733\pi$
$233$	−7.62110	+	13.2001i	−0.499275	+	0.864769i	0.500725	+	0.865606i	$0.333067\pi$
$234$		0			0
$235$	12.9258	+	22.3881i	0.843186	+	1.46044i
$236$	3.45489	+	5.98404i	0.224894	+	0.389528i
$237$		0			0
$238$	−17.3869	+	0.633646i	−1.12702	+	0.0410732i
$239$	−9.47524	−	16.4116i	−0.612902	−	1.06158i	−0.990749	−	0.135710i	$-0.956669\pi$
$239$	−9.47524	−	16.4116i	−0.612902	−	1.06158i	0.377846	−	0.925868i	$-0.376665\pi$
$240$		0			0
$241$	−24.5054			−1.57853			−0.789267	−	0.614051i	$-0.789539\pi$
$241$	−24.5054			−1.57853			−0.789267	+	0.614051i	$0.789539\pi$
$242$	−4.40909	−	7.63676i	−0.283427	−	0.490910i
$243$		0			0
$244$	−5.73305			−0.367021
$245$	11.2818	−	23.3109i	0.720768	−	1.48928i
$246$		0			0
$247$	2.39926			0.152661
$248$	3.40545	−	5.89841i	0.216246	−	0.374549i
$249$		0			0
$250$	−6.82072	−	11.8138i	−0.431380	−	0.747173i
$251$	12.1236			0.765238			0.382619	−	0.923906i	$-0.375022\pi$
$251$	12.1236			0.765238			0.382619	+	0.923906i	$0.375022\pi$
$252$		0			0
$253$	−9.28799			−0.583931
$254$	−1.42835	−	2.47397i	−0.0896224	−	0.155231i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	8.20877			0.512049			0.256025	−	0.966670i	$-0.417587\pi$
$257$	8.20877			0.512049			0.256025	+	0.966670i	$0.417587\pi$
$258$		0			0
$259$	7.34362	−	0.267630i	0.456311	−	0.0166297i
$260$	−9.98762			−0.619406
$261$		0			0
$262$	−0.0778435	−	0.134829i	−0.00480919	−	0.00832976i
$263$	5.34617			0.329659			0.164830	−	0.986322i	$-0.447293\pi$
$263$	5.34617			0.329659			0.164830	+	0.986322i	$0.447293\pi$
$264$		0			0
$265$	−5.93818	−	10.2852i	−0.364779	−	0.631816i
$266$	1.24907	+	1.99218i	0.0765854	+	0.122148i
$267$		0			0
$268$	4.73236	+	8.19669i	0.289075	+	0.500692i
$269$	−9.24219	−	16.0079i	−0.563506	−	0.976022i	−0.997187	−	0.0749550i	$-0.976119\pi$
$269$	−9.24219	−	16.0079i	−0.563506	−	0.976022i	0.433681	−	0.901067i	$-0.357215\pi$
$270$		0			0
$271$	−3.67742	+	6.36947i	−0.223387	+	0.386918i	−0.955834	−	0.293906i	$-0.905045\pi$
$271$	−3.67742	+	6.36947i	−0.223387	+	0.386918i	0.732447	+	0.680824i	$0.238378\pi$
$272$	−3.28799	+	5.69497i	−0.199364	+	0.345308i
$273$		0			0
$274$	1.70582	+	2.95456i	0.103052	+	0.178492i
$275$	12.8319			0.773795
$276$		0			0
$277$	−9.09888			−0.546699			−0.273349	−	0.961915i	$-0.588132\pi$
$277$	−9.09888			−0.546699			−0.273349	+	0.961915i	$0.588132\pi$
$278$	6.75526	−	11.7005i	0.405154	−	0.701747i
$279$		0			0
$280$	−5.19963	−	8.29305i	−0.310737	−	0.495604i
$281$	−6.00433	+	10.3998i	−0.358188	+	0.620400i	−0.987658	−	0.156624i	$-0.949939\pi$
$281$	−6.00433	+	10.3998i	−0.358188	+	0.620400i	0.629470	+	0.777025i	$0.283272\pi$
$282$		0			0
$283$	−4.92147	+	8.52423i	−0.292551	+	0.506713i	−0.974412	−	0.224768i	$-0.927837\pi$
$283$	−4.92147	+	8.52423i	−0.292551	+	0.506713i	0.681861	+	0.731481i	$0.261171\pi$
$284$	−2.73236	+	4.73259i	−0.162136	+	0.280827i
$285$		0			0
$286$	1.99381	−	3.45338i	0.117896	−	0.204203i
$287$	5.09201	−	9.61192i	0.300572	−	0.567373i
$288$		0			0
$289$	−13.1218	+	22.7276i	−0.771870	+	1.33692i
$290$	9.28799			0.545410
$291$		0			0
$292$	12.0655			0.706078
$293$	−10.7101	−	18.5505i	−0.625694	−	1.08373i	−0.988406	−	0.151832i	$-0.951483\pi$
$293$	−10.7101	−	18.5505i	−0.625694	−	1.08373i	0.362713	−	0.931901i	$-0.381851\pi$
$294$		0			0
$295$	−12.7818	+	22.1387i	−0.744185	+	1.28897i
$296$	1.38874	−	2.40536i	0.0807186	−	0.139809i
$297$		0			0
$298$	−0.166896	−	0.289073i	−0.00966804	−	0.0167455i
$299$	8.48762	+	14.7010i	0.490852	+	0.850180i
$300$		0			0
$301$	−0.0327319	+	0.00119288i	−0.00188664	+	6.87564e-5i
$302$	−9.95489	−	17.2424i	−0.572839	−	0.992187i
$303$		0			0
$304$	0.888736			0.0509725
$305$	−10.6051	−	18.3685i	−0.607245	−	1.05178i
$306$		0			0
$307$	−5.68725			−0.324588			−0.162294	−	0.986742i	$-0.551889\pi$
$307$	−5.68725			−0.324588			−0.162294	+	0.986742i	$0.551889\pi$
$308$	3.90545	−	0.142330i	0.222533	−	0.00810999i
$309$		0			0
$310$	25.1978			1.43114
$311$	−5.86033	+	10.1504i	−0.332309	+	0.575576i	−0.982964	−	0.183797i	$-0.941161\pi$
$311$	−5.86033	+	10.1504i	−0.332309	+	0.575576i	0.650655	+	0.759373i	$0.274494\pi$
$312$		0			0
$313$	13.3869	+	23.1868i	0.756671	+	1.31059i	0.944539	+	0.328398i	$0.106509\pi$
$313$	13.3869	+	23.1868i	0.756671	+	1.31059i	−0.187868	+	0.982194i	$0.560158\pi$
$314$	6.96286			0.392937
$315$		0			0
$316$	11.4523			0.644244
$317$	0.951246	+	1.64761i	0.0534273	+	0.0925388i	0.891502	−	0.453016i	$-0.149652\pi$
$317$	0.951246	+	1.64761i	0.0534273	+	0.0925388i	−0.838075	+	0.545555i	$0.816319\pi$
$318$		0			0
$319$	−1.85414	+	3.21147i	−0.103812	+	0.179808i
$320$	−3.69963			−0.206816
$321$		0			0
$322$	−7.78799	+	14.7010i	−0.434008	+	0.819254i
$323$	5.84431			0.325186
$324$		0			0
$325$	−11.7262	−	20.3103i	−0.650451	−	1.12661i
$326$	8.07413			0.447184
$327$		0			0
$328$	−2.05563	−	3.56046i	−0.113503	−	0.196593i
$329$	−8.65452	+	16.3367i	−0.477139	+	0.900670i
$330$		0			0
$331$	−2.78366	−	4.82144i	−0.153004	−	0.265010i	0.779327	−	0.626618i	$-0.215561\pi$
$331$	−2.78366	−	4.82144i	−0.153004	−	0.265010i	−0.932330	+	0.361608i	$0.882228\pi$
$332$	−2.23855	−	3.87728i	−0.122856	−	0.212794i
$333$		0			0
$334$	9.74288	−	16.8752i	0.533107	−	0.923368i
$335$	−17.5080	+	30.3247i	−0.956563	+	1.65682i
$336$		0			0
$337$	−16.8869	−	29.2489i	−0.919887	−	1.59329i	−0.799585	−	0.600553i	$-0.794947\pi$
$337$	−16.8869	−	29.2489i	−0.919887	−	1.59329i	−0.120302	−	0.992737i	$-0.538386\pi$
$338$	5.71201			0.310692
$339$		0			0
$340$	−24.3287			−1.31941
$341$	−5.03018	+	8.71253i	−0.272400	+	0.471810i
$342$		0			0
$343$	18.4098	−	2.01993i	0.994035	−	0.109066i
$344$	−0.00618986	+	0.0107211i	−0.000333735	+	0.000578045i
$345$		0			0
$346$	−11.2818	+	19.5407i	−0.606513	+	1.05051i
$347$	−15.2033	+	26.3328i	−0.816154	+	1.41362i	0.0923418	+	0.995727i	$0.470565\pi$
$347$	−15.2033	+	26.3328i	−0.816154	+	1.41362i	−0.908496	+	0.417893i	$0.862769\pi$
$348$		0			0
$349$	−6.29782	+	10.9082i	−0.337115	+	0.583900i	−0.983889	−	0.178782i	$-0.942784\pi$
$349$	−6.29782	+	10.9082i	−0.337115	+	0.583900i	0.646774	+	0.762682i	$0.276118\pi$
$350$	10.7596	−	20.3103i	0.575124	−	1.08563i
$351$		0			0
$352$	0.738550	−	1.27921i	0.0393648	−	0.0681819i
$353$	7.53156			0.400865			0.200432	−	0.979708i	$-0.435765\pi$
$353$	7.53156			0.400865			0.200432	+	0.979708i	$0.435765\pi$
$354$		0			0
$355$	−20.2174			−1.07303
$356$	4.43818	+	7.68715i	0.235223	+	0.407418i
$357$		0			0
$358$	0.166896	−	0.289073i	0.00882074	−	0.0152780i
$359$	3.44801	−	5.97213i	0.181979	−	0.315197i	−0.760575	−	0.649250i	$-0.775083\pi$
$359$	3.44801	−	5.97213i	0.181979	−	0.315197i	0.942554	+	0.334053i	$0.108416\pi$
$360$		0			0
$361$	9.10507	+	15.7705i	0.479214	+	0.830024i
$362$	11.6211	+	20.1283i	0.610791	+	1.05792i
$363$		0			0
$364$	−3.79418	−	6.05146i	−0.198869	−	0.317183i
$365$	22.3189	+	38.6574i	1.16822	+	2.02342i
$366$		0			0
$367$	23.1236			1.20704			0.603522	−	0.797346i	$-0.293763\pi$
$367$	23.1236			1.20704			0.603522	+	0.797346i	$0.293763\pi$
$368$	3.14400	+	5.44556i	0.163892	+	0.283869i
$369$		0			0
$370$	10.2756			0.534204
$371$	3.97593	−	7.50516i	0.206420	−	0.389648i
$372$		0			0
$373$	29.1643			1.51007			0.755036	−	0.655683i	$-0.227619\pi$
$373$	29.1643			1.51007			0.755036	+	0.655683i	$0.227619\pi$
$374$	4.85669	−	8.41204i	0.251134	−	0.434976i
$375$		0			0
$376$	3.49381	+	6.05146i	0.180180	+	0.312080i
$377$	6.77747			0.349058
$378$		0			0
$379$	−13.5622			−0.696645			−0.348322	−	0.937375i	$-0.613249\pi$
$379$	−13.5622			−0.696645			−0.348322	+	0.937375i	$0.613249\pi$
$380$	1.64400	+	2.84748i	0.0843352	+	0.146073i
$381$		0			0
$382$	8.16071	−	14.1348i	0.417538	−	0.723197i
$383$	2.83565			0.144895			0.0724475	−	0.997372i	$-0.476919\pi$
$383$	2.83565			0.144895			0.0724475	+	0.997372i	$0.476919\pi$
$384$		0			0
$385$	7.68037	+	12.2497i	0.391428	+	0.624300i
$386$	14.3214			0.728941
$387$		0			0
$388$	−6.58836	−	11.4114i	−0.334474	−	0.579325i
$389$	18.6080			0.943464			0.471732	−	0.881742i	$-0.343629\pi$
$389$	18.6080			0.943464			0.471732	+	0.881742i	$0.343629\pi$
$390$		0			0
$391$	20.6749	+	35.8099i	1.04557	+	1.81099i
$392$	3.04944	−	6.30087i	0.154020	−	0.318242i
$393$		0			0
$394$	−1.21201	−	2.09926i	−0.0610601	−	0.105759i
$395$	21.1847	+	36.6930i	1.06592	+	1.84622i
$396$		0			0
$397$	−10.2880	+	17.8193i	−0.516340	+	0.894326i	0.483481	+	0.875355i	$0.339372\pi$
$397$	−10.2880	+	17.8193i	−0.516340	+	0.894326i	−0.999820	+	0.0189712i	$0.993961\pi$
$398$	3.05563	−	5.29251i	0.153165	−	0.265290i
$399$		0			0
$400$	−4.34362	−	7.52338i	−0.217181	−	0.376169i
$401$	6.75409			0.337283			0.168642	−	0.985677i	$-0.446062\pi$
$401$	6.75409			0.337283			0.168642	+	0.985677i	$0.446062\pi$
$402$		0			0
$403$	18.3869			0.915916
$404$	2.62729	−	4.55059i	0.130712	−	0.226400i
$405$		0			0
$406$	3.52840	+	5.62755i	0.175112	+	0.279291i
$407$	−2.05130	+	3.55296i	−0.101679	+	0.176114i
$408$		0			0
$409$	−7.66071	+	13.2687i	−0.378798	+	0.656097i	−0.990888	−	0.134691i	$-0.956996\pi$
$409$	−7.66071	+	13.2687i	−0.378798	+	0.656097i	0.612090	+	0.790788i	$0.290329\pi$
$410$	7.60507	−	13.1724i	0.375588	−	0.650537i
$411$		0			0
$412$	−0.833104	+	1.44298i	−0.0410441	+	0.0710904i
$413$	−18.2694	+	0.665809i	−0.898980	+	0.0327623i
$414$		0			0
$415$	8.28180	−	14.3445i	0.406538	−	0.704144i
$416$	−2.69963			−0.132360
$417$		0			0
$418$	−1.31275			−0.0642088
$419$	4.32141	+	7.48491i	0.211115	+	0.365662i	0.952064	−	0.305900i	$-0.0989573\pi$
$419$	4.32141	+	7.48491i	0.211115	+	0.365662i	−0.740949	+	0.671561i	$0.765624\pi$
$420$		0			0
$421$	18.5636	−	32.1531i	0.904735	−	1.56705i	0.0834618	−	0.996511i	$-0.473402\pi$
$421$	18.5636	−	32.1531i	0.904735	−	1.56705i	0.821273	−	0.570536i	$-0.193264\pi$
$422$	−5.72253	+	9.91171i	−0.278568	+	0.482494i
$423$		0			0
$424$	−1.60507	−	2.78007i	−0.0779493	−	0.135012i
$425$	−28.5636	−	49.4736i	−1.38554	−	2.39982i
$426$		0			0
$427$	7.10067	−	13.4036i	0.343625	−	0.648644i
$428$	5.38255	+	9.32284i	0.260175	+	0.450637i
$429$		0			0
$430$	−0.0458003			−0.00220869
$431$	4.71015	+	8.15822i	0.226880	+	0.392967i	0.956882	−	0.290478i	$-0.0938142\pi$
$431$	4.71015	+	8.15822i	0.226880	+	0.392967i	−0.730002	+	0.683445i	$0.760481\pi$
$432$		0			0
$433$	−0.208771			−0.0100329			−0.00501645	−	0.999987i	$-0.501597\pi$
$433$	−0.208771			−0.0100329			−0.00501645	+	0.999987i	$0.501597\pi$
$434$	9.57234	+	15.2672i	0.459487	+	0.732850i
$435$		0			0
$436$	0.189108			0.00905662
$437$	2.79418	−	4.83967i	0.133664	−	0.231513i
$438$		0			0
$439$	4.98398	+	8.63250i	0.237872	+	0.412007i	0.960104	−	0.279645i	$-0.0902167\pi$
$439$	4.98398	+	8.63250i	0.237872	+	0.412007i	−0.722231	+	0.691652i	$0.756883\pi$
$440$	5.46472			0.260520
$441$		0			0
$442$	−17.7527			−0.844410
$443$	−7.84981	−	13.5963i	−0.372956	−	0.645979i	0.617063	−	0.786914i	$-0.288322\pi$
$443$	−7.84981	−	13.5963i	−0.372956	−	0.645979i	−0.990019	+	0.140935i	$0.954989\pi$
$444$		0			0
$445$	−16.4196	+	28.4396i	−0.778364	+	1.34817i
$446$	−7.22253			−0.341997
$447$		0			0
$448$	−1.40545	−	2.24159i	−0.0664011	−	0.105905i
$449$	−33.6253			−1.58688			−0.793439	−	0.608650i	$-0.791712\pi$
$449$	−33.6253			−1.58688			−0.793439	+	0.608650i	$0.791712\pi$
$450$		0			0
$451$	3.03637	+	5.25915i	0.142977	+	0.247644i
$452$	−13.5636			−0.637978
$453$		0			0
$454$	−6.82760	−	11.8258i	−0.320435	−	0.555010i
$455$	12.3702	−	23.3505i	0.579922	−	1.09469i
$456$		0			0
$457$	−16.3541	−	28.3262i	−0.765015	−	1.32504i	−0.940239	−	0.340516i	$-0.889398\pi$
$457$	−16.3541	−	28.3262i	−0.765015	−	1.32504i	0.175224	−	0.984529i	$-0.443935\pi$
$458$	8.68725	+	15.0468i	0.405928	+	0.703089i
$459$		0			0
$460$	−11.6316	+	20.1466i	−0.542327	+	0.939338i
$461$	2.07165	−	3.58821i	0.0964865	−	0.167120i	−0.813742	−	0.581227i	$-0.802573\pi$
$461$	2.07165	−	3.58821i	0.0964865	−	0.167120i	0.910228	+	0.414107i	$0.135906\pi$
$462$		0			0
$463$	−8.34176	−	14.4484i	−0.387675	−	0.671472i	0.604462	−	0.796634i	$-0.293388\pi$
$463$	−8.34176	−	14.4484i	−0.387675	−	0.671472i	−0.992136	+	0.125162i	$0.960055\pi$
$464$	2.51052			0.116548
$465$		0			0
$466$	−15.2422			−0.706081
$467$	−14.9585	+	25.9089i	−0.692198	+	1.19892i	0.278918	+	0.960315i	$0.410024\pi$
$467$	−14.9585	+	25.9089i	−0.692198	+	1.19892i	−0.971116	+	0.238608i	$0.923309\pi$
$468$		0			0
$469$	−25.0247	+	0.911998i	−1.15553	+	0.0421121i
$470$	−12.9258	+	22.3881i	−0.596223	+	1.03269i
$471$		0			0
$472$	−3.45489	+	5.98404i	−0.159024	+	0.275438i
$473$	0.00914304	−	0.0158362i	0.000420397	−	0.000728149i
$474$		0			0
$475$	−3.86033	+	6.68630i	−0.177124	+	0.306788i
$476$	−9.24219	−	14.7407i	−0.423615	−	0.675637i
$477$		0			0
$478$	9.47524	−	16.4116i	0.433387	−	0.750649i
$479$	2.95930			0.135214			0.0676068	−	0.997712i	$-0.478464\pi$
$479$	2.95930			0.135214			0.0676068	+	0.997712i	$0.478464\pi$
$480$		0			0
$481$	7.49814			0.341886
$482$	−12.2527	−	21.2223i	−0.558096	−	0.966650i
$483$		0			0
$484$	4.40909	−	7.63676i	0.200413	−	0.347126i
$485$	24.3745	−	42.2179i	1.10679	−	1.91701i
$486$		0			0
$487$	−14.0309	−	24.3022i	−0.635800	−	1.10124i	−0.986345	−	0.164691i	$-0.947337\pi$
$487$	−14.0309	−	24.3022i	−0.635800	−	1.10124i	0.350546	−	0.936546i	$-0.385996\pi$
$488$	−2.86652	−	4.96497i	−0.129761	−	0.224753i
$489$		0			0
$490$	25.8287	−	1.88510i	1.16682	−	0.0851602i
$491$	−17.0734	−	29.5721i	−0.770513	−	1.33457i	−0.937282	−	0.348572i	$-0.886667\pi$
$491$	−17.0734	−	29.5721i	−0.770513	−	1.33457i	0.166769	−	0.985996i	$-0.446667\pi$
$492$		0			0
$493$	16.5091			0.743534
$494$	1.19963	+	2.07782i	0.0539738	+	0.0934854i
$495$		0			0
$496$	6.81089			0.305818
$497$	−7.68037	−	12.2497i	−0.344512	−	0.549472i
$498$		0			0
$499$	−2.28071			−0.102099			−0.0510493	−	0.998696i	$-0.516257\pi$
$499$	−2.28071			−0.102099			−0.0510493	+	0.998696i	$0.516257\pi$
$500$	6.82072	−	11.8138i	0.305032	−	0.528331i
$501$		0			0
$502$	6.06182	+	10.4994i	0.270552	+	0.468610i
$503$	−13.9890			−0.623739			−0.311869	−	0.950125i	$-0.600955\pi$
$503$	−13.9890			−0.623739			−0.311869	+	0.950125i	$0.600955\pi$
$504$		0			0
$505$	19.4400			0.865067
$506$	−4.64400	−	8.04364i	−0.206451	−	0.357583i
$507$		0			0
$508$	1.42835	−	2.47397i	0.0633726	−	0.109765i
$509$	−25.6181			−1.13550			−0.567750	−	0.823201i	$-0.692186\pi$
$509$	−25.6181			−1.13550			−0.567750	+	0.823201i	$0.692186\pi$
$510$		0			0
$511$	−14.9437	+	28.2084i	−0.661069	+	1.24787i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	4.10439	+	7.10900i	0.181037	+	0.313565i
$515$	−6.16435			−0.271634
$516$		0			0
$517$	−5.16071	−	8.93861i	−0.226968	−	0.393119i
$518$	3.90359	+	6.22595i	0.171514	+	0.273553i
$519$		0			0
$520$	−4.99381	−	8.64953i	−0.218993	−	0.379307i
$521$	20.9127	+	36.2219i	0.916203	+	1.58691i	0.805130	+	0.593099i	$0.202096\pi$
$521$	20.9127	+	36.2219i	0.916203	+	1.58691i	0.111073	+	0.993812i	$0.464571\pi$
$522$		0			0
$523$	7.88323	−	13.6542i	0.344710	−	0.597055i	−0.640591	−	0.767882i	$-0.721311\pi$
$523$	7.88323	−	13.6542i	0.344710	−	0.597055i	0.985301	+	0.170827i	$0.0546440\pi$
$524$	0.0778435	−	0.134829i	0.00340061	−	0.00589003i
$525$		0			0
$526$	2.67309	+	4.62992i	0.116552	+	0.201874i
$527$	44.7883			1.95101
$528$		0			0
$529$	16.5388			0.719080
$530$	5.93818	−	10.2852i	0.257938	−	0.446762i
$531$		0			0
$532$	−1.10074	+	2.07782i	−0.0477233	+	0.0900848i
$533$	5.54944	−	9.61192i	0.240373	−	0.416338i
$534$		0			0
$535$	−19.9134	+	34.4911i	−0.860932	+	1.49118i
$536$	−4.73236	+	8.19669i	−0.204407	+	0.354043i
$537$		0			0
$538$	9.24219	−	16.0079i	0.398459	−	0.690152i
$539$	−4.50433	+	9.30701i	−0.194015	+	0.400881i
$540$		0			0
$541$	−21.0963	+	36.5399i	−0.907002	+	1.57097i	−0.0887957	+	0.996050i	$0.528302\pi$
$541$	−21.0963	+	36.5399i	−0.907002	+	1.57097i	−0.818207	+	0.574924i	$0.805031\pi$
$542$	−7.35483			−0.315917
$543$		0			0
$544$	−6.57598			−0.281943
$545$	0.349814	+	0.605896i	0.0149844	+	0.0259537i
$546$		0			0
$547$	20.3356	−	35.2222i	0.869486	−	1.50599i	0.00696400	−	0.999976i	$-0.497783\pi$
$547$	20.3356	−	35.2222i	0.869486	−	1.50599i	0.862522	−	0.506019i	$-0.168883\pi$
$548$	−1.70582	+	2.95456i	−0.0728689	+	0.126213i
$549$		0			0
$550$	6.41597	+	11.1128i	0.273578	+	0.473851i
$551$	−1.11559	−	1.93227i	−0.0475259	−	0.0823173i
$552$		0			0
$553$	−14.1843	+	26.7750i	−0.603178	+	1.13859i
$554$	−4.54944	−	7.87987i	−0.193287	−	0.334783i
$555$		0			0
$556$	13.5105			0.572974
$557$	−6.68794	−	11.5838i	−0.283377	−	0.490823i	0.688837	−	0.724916i	$-0.258121\pi$
$557$	−6.68794	−	11.5838i	−0.283377	−	0.490823i	−0.972214	+	0.234093i	$0.924788\pi$
$558$		0			0
$559$	−0.0334206			−0.00141354
$560$	4.58217	−	8.64953i	0.193632	−	0.365509i
$561$		0			0
$562$	−12.0087			−0.506555
$563$	−16.3807	+	28.3722i	−0.690364	+	1.19574i	0.281355	+	0.959604i	$0.409216\pi$
$563$	−16.3807	+	28.3722i	−0.690364	+	1.19574i	−0.971719	+	0.236141i	$0.924117\pi$
$564$		0			0
$565$	−25.0901	−	43.4574i	−1.05555	−	1.82827i
$566$	−9.84294			−0.413729
$567$		0			0
$568$	−5.46472			−0.229295
$569$	−8.36398	−	14.4868i	−0.350636	−	0.607320i	0.635725	−	0.771916i	$-0.280701\pi$
$569$	−8.36398	−	14.4868i	−0.350636	−	0.607320i	−0.986361	+	0.164596i	$0.947368\pi$
$570$		0			0
$571$	13.7367	−	23.7926i	0.574863	−	0.995691i	−0.421194	−	0.906971i	$-0.638389\pi$
$571$	13.7367	−	23.7926i	0.574863	−	0.995691i	0.996057	−	0.0887207i	$-0.0282778\pi$
$572$	3.98762			0.166731
$573$		0			0
$574$	10.8702	−	0.396151i	0.453712	−	0.0165350i
$575$	−54.6253			−2.27803
$576$		0			0
$577$	1.41714	+	2.45455i	0.0589962	+	0.102184i	0.894015	−	0.448037i	$-0.147877\pi$
$577$	1.41714	+	2.45455i	0.0589962	+	0.102184i	−0.835019	+	0.550221i	$0.814543\pi$
$578$	−26.2436			−1.09159
$579$		0			0
$580$	4.64400	+	8.04364i	0.192831	+	0.333994i
$581$	11.8374	−	0.431403i	0.491100	−	0.0178976i
$582$		0			0
$583$	2.37085	+	4.10644i	0.0981908	+	0.170071i
$584$	6.03273	+	10.4490i	0.249636	+	0.432383i
$585$		0			0
$586$	10.7101	−	18.5505i	0.442432	−	0.766315i
$587$	2.34795	−	4.06678i	0.0969105	−	0.167854i	−0.813494	−	0.581573i	$-0.802437\pi$
$587$	2.34795	−	4.06678i	0.0969105	−	0.167854i	0.910404	+	0.413720i	$0.135771\pi$
$588$		0			0
$589$	−3.02654	−	5.24212i	−0.124706	−	0.215998i
$590$	−25.5636			−1.05244
$591$		0			0
$592$	2.77747			0.114153
$593$	−0.636024	+	1.10163i	−0.0261184	+	0.0452383i	−0.878789	−	0.477210i	$-0.841648\pi$
$593$	−0.636024	+	1.10163i	−0.0261184	+	0.0452383i	0.852671	+	0.522449i	$0.174981\pi$
$594$		0			0
$595$	30.1323	−	56.8792i	1.23530	−	2.33182i
$596$	0.166896	−	0.289073i	0.00683634	−	0.0118409i
$597$		0			0
$598$	−8.48762	+	14.7010i	−0.347085	+	0.601168i
$599$	21.9258	−	37.9766i	0.895864	−	1.55168i	0.0631320	−	0.998005i	$-0.479891\pi$
$599$	21.9258	−	37.9766i	0.895864	−	1.55168i	0.832732	−	0.553676i	$-0.186776\pi$
$600$		0			0
$601$	−6.71634	+	11.6330i	−0.273965	+	0.474522i	−0.969874	−	0.243609i	$-0.921669\pi$
$601$	−6.71634	+	11.6330i	−0.273965	+	0.474522i	0.695908	+	0.718131i	$0.255002\pi$
$602$	−0.0173990	−	0.0277502i	−0.000709131	−	0.00113101i
$603$		0			0
$604$	9.95489	−	17.2424i	0.405059	−	0.701582i
$605$	32.6240			1.32635
$606$		0			0
$607$	−4.58465			−0.186085			−0.0930425	−	0.995662i	$-0.529659\pi$
$607$	−4.58465			−0.186085			−0.0930425	+	0.995662i	$0.529659\pi$
$608$	0.444368	+	0.769668i	0.0180215	+	0.0312142i
$609$		0			0
$610$	10.6051	−	18.3685i	0.429387	−	0.743720i
$611$	−9.43199	+	16.3367i	−0.381577	+	0.660911i
$612$		0			0
$613$	−11.0538	−	19.1457i	−0.446458	−	0.773287i	0.551695	−	0.834046i	$-0.313981\pi$
$613$	−11.0538	−	19.1457i	−0.446458	−	0.773287i	−0.998152	+	0.0607587i	$0.980648\pi$
$614$	−2.84362	−	4.92530i	−0.114759	−	0.198769i
$615$		0			0
$616$	2.07598	+	3.31105i	0.0836438	+	0.133406i
$617$	−6.00433	−	10.3998i	−0.241725	−	0.418680i	0.719481	−	0.694513i	$-0.244380\pi$
$617$	−6.00433	−	10.3998i	−0.241725	−	0.418680i	−0.961206	+	0.275832i	$0.911047\pi$
$618$		0			0
$619$	−17.5636			−0.705941			−0.352970	−	0.935634i	$-0.614828\pi$
$619$	−17.5636			−0.705941			−0.352970	+	0.935634i	$0.614828\pi$
$620$	12.5989	+	21.8219i	0.505983	+	0.876389i
$621$		0			0
$622$	−11.7207			−0.469956
$623$	−23.4691	+	0.855304i	−0.940268	+	0.0342670i
$624$		0			0
$625$	7.03204			0.281282
$626$	−13.3869	+	23.1868i	−0.535047	+	0.926729i
$627$		0			0
$628$	3.48143	+	6.03001i	0.138924	+	0.240624i
$629$	18.2646			0.728258
$630$		0			0
$631$	−44.9381			−1.78896			−0.894479	−	0.447110i	$-0.852453\pi$
$631$	−44.9381			−1.78896			−0.894479	+	0.447110i	$0.852453\pi$
$632$	5.72617	+	9.91802i	0.227775	+	0.394518i
$633$		0			0
$634$	−0.951246	+	1.64761i	−0.0377788	+	0.0654348i
$635$	10.5687			0.419406
$636$		0			0
$637$	18.8473	−	1.37556i	0.746756	−	0.0545018i
$638$	−3.70829			−0.146813
$639$		0			0
$640$	−1.84981	−	3.20397i	−0.0731203	−	0.126648i
$641$	28.9839			1.14480			0.572398	−	0.819976i	$-0.306013\pi$
$641$	28.9839			1.14480			0.572398	+	0.819976i	$0.306013\pi$
$642$		0			0
$643$	6.03087	+	10.4458i	0.237834	+	0.411941i	0.960093	−	0.279682i	$-0.0902291\pi$
$643$	6.03087	+	10.4458i	0.237834	+	0.411941i	−0.722258	+	0.691623i	$0.756896\pi$
$644$	−16.6254	+	0.605896i	−0.655134	+	0.0238756i
$645$		0			0
$646$	2.92216	+	5.06132i	0.114971	+	0.199135i
$647$	−18.8825	−	32.7055i	−0.742349	−	1.28579i	−0.951423	−	0.307887i	$-0.900378\pi$
$647$	−18.8825	−	32.7055i	−0.742349	−	1.28579i	0.209073	−	0.977900i	$-0.432955\pi$
$648$		0			0
$649$	5.10322	−	8.83903i	0.200319	−	0.346962i
$650$	11.7262	−	20.3103i	0.459938	−	0.796636i
$651$		0			0
$652$	4.03706	+	6.99240i	0.158104	+	0.273843i
$653$	−37.4079			−1.46388			−0.731942	−	0.681366i	$-0.761386\pi$
$653$	−37.4079			−1.46388			−0.731942	+	0.681366i	$0.761386\pi$
$654$		0			0
$655$	0.575984			0.0225056
$656$	2.05563	−	3.56046i	0.0802589	−	0.139013i
$657$		0			0
$658$	−18.4752	+	0.673310i	−0.720240	+	0.0262484i
$659$	−14.9356	+	25.8693i	−0.581810	+	1.00772i	0.413455	+	0.910524i	$0.364322\pi$
$659$	−14.9356	+	25.8693i	−0.581810	+	1.00772i	−0.995265	+	0.0971993i	$0.969012\pi$
$660$		0			0
$661$	−2.80401	+	4.85669i	−0.109063	+	0.188904i	−0.915391	−	0.402566i	$-0.868119\pi$
$661$	−2.80401	+	4.85669i	−0.109063	+	0.188904i	0.806328	+	0.591469i	$0.201452\pi$
$662$	2.78366	−	4.82144i	0.108190	−	0.187391i
$663$		0			0
$664$	2.23855	−	3.87728i	0.0868726	−	0.150468i
$665$	−8.69344	+	0.316823i	−0.337117	+	0.0122859i
$666$		0			0
$667$	7.89307	−	13.6712i	0.305621	−	0.529351i
$668$	19.4858			0.753927
$669$		0			0
$670$	−35.0159			−1.35278
$671$	4.23414	+	7.33375i	0.163457	+	0.283116i
$672$		0			0
$673$	−4.72253	+	8.17966i	−0.182040	+	0.315303i	−0.942575	−	0.333994i	$-0.891603\pi$
$673$	−4.72253	+	8.17966i	−0.182040	+	0.315303i	0.760535	+	0.649297i	$0.224937\pi$
$674$	16.8869	−	29.2489i	0.650458	−	1.12663i
$675$		0			0
$676$	2.85600	+	4.94674i	0.109846	+	0.190259i
$677$	5.53087	+	9.57975i	0.212569	+	0.368180i	0.952518	−	0.304483i	$-0.0984837\pi$
$677$	5.53087	+	9.57975i	0.212569	+	0.368180i	−0.739949	+	0.672663i	$0.765150\pi$
$678$		0			0
$679$	34.8392	−	1.26968i	1.33701	−	0.0487258i
$680$	−12.1643	−	21.0693i	−0.466481	−	0.807970i
$681$		0			0
$682$	−10.0604			−0.385231
$683$	4.41961	+	7.65499i	0.169112	+	0.292910i	0.938108	−	0.346343i	$-0.112577\pi$
$683$	4.41961	+	7.65499i	0.169112	+	0.292910i	−0.768996	+	0.639253i	$0.779243\pi$
$684$		0			0
$685$	−12.6218			−0.482254
$686$	10.9542	+	14.9334i	0.418233	+	0.570159i
$687$		0			0
$688$	−0.0123797			−0.000471972
$689$	4.33310	−	7.50516i	0.165078	−	0.285924i
$690$		0			0
$691$	−12.5309	−	21.7041i	−0.476697	−	0.825663i	0.522947	−	0.852365i	$-0.324833\pi$
$691$	−12.5309	−	21.7041i	−0.476697	−	0.825663i	−0.999643	+	0.0267023i	$0.991499\pi$
$692$	−22.5636			−0.857740
$693$		0			0
$694$	−30.4065			−1.15422
$695$	24.9920	+	43.2873i	0.947999	+	1.64198i
$696$		0			0
$697$	13.5178	−	23.4135i	0.512023	−	0.886850i
$698$	−12.5956			−0.476752
$699$		0			0
$700$	22.9691	−	0.837082i	0.868149	−	0.0316387i
$701$	43.4858			1.64243			0.821217	−	0.570616i	$-0.193295\pi$
$701$	43.4858			1.64243			0.821217	+	0.570616i	$0.193295\pi$
$702$		0			0
$703$	−1.23422	−	2.13773i	−0.0465495	−	0.0806260i
$704$	1.47710			0.0556703
$705$		0			0
$706$	3.76578	+	6.52252i	0.141727	+	0.245478i
$707$	7.38502	+	11.7786i	0.277742	+	0.442979i
$708$		0			0
$709$	11.3702	+	19.6937i	0.427016	+	0.739613i	0.996606	−	0.0823158i	$-0.0262316\pi$
$709$	11.3702	+	19.6937i	0.427016	+	0.739613i	−0.569591	+	0.821928i	$0.692898\pi$
$710$	−10.1087	−	17.5088i	−0.379373	−	0.657094i
$711$		0			0
$712$	−4.43818	+	7.68715i	−0.166328	+	0.288088i
$713$	21.4134	−	37.0891i	0.801939	−	1.38900i
$714$		0			0
$715$	7.37636	+	12.7762i	0.275860	+	0.477804i
$716$	0.333792			0.0124744
$717$		0			0
$718$	6.89602			0.257357
$719$	−6.06182	+	10.4994i	−0.226068	+	0.391561i	−0.956639	−	0.291275i	$-0.905920\pi$
$719$	−6.06182	+	10.4994i	−0.226068	+	0.391561i	0.730571	+	0.682836i	$0.239254\pi$
$720$		0			0
$721$	−2.34176	−	3.73495i	−0.0872119	−	0.139097i
$722$	−9.10507	+	15.7705i	−0.338856	+	0.586915i
$723$		0			0
$724$	−11.6211	+	20.1283i	−0.431895	+	0.748063i
$725$	−10.9048	+	18.8876i	−0.404993	+	0.701468i
$726$		0			0
$727$	23.0908	−	39.9945i	0.856392	−	1.48331i	−0.0189562	−	0.999820i	$-0.506034\pi$
$727$	23.0908	−	39.9945i	0.856392	−	1.48331i	0.875348	−	0.483494i	$-0.160632\pi$
$728$	3.34362	−	6.31159i	0.123923	−	0.233923i
$729$		0			0
$730$	−22.3189	+	38.6574i	−0.826058	+	1.43077i
$731$	−0.0814088			−0.00301101
$732$		0			0
$733$	−36.0297			−1.33079			−0.665394	−	0.746493i	$-0.731736\pi$
$733$	−36.0297			−1.33079			−0.665394	+	0.746493i	$0.731736\pi$
$734$	11.5618	+	20.0257i	0.426755	+	0.739161i
$735$		0			0
$736$	−3.14400	+	5.44556i	−0.115889	+	0.200726i
$737$	6.99017	−	12.1073i	0.257486	−	0.445979i
$738$		0			0
$739$	23.2119	+	40.2042i	0.853865	+	1.47894i	0.877694	+	0.479221i	$0.159081\pi$
$739$	23.2119	+	40.2042i	0.853865	+	1.47894i	−0.0238296	+	0.999716i	$0.507586\pi$
$740$	5.13781	+	8.89894i	0.188870	+	0.327132i
$741$		0			0
$742$	8.48762	−	0.309322i	0.311590	−	0.0113556i
$743$	−0.598884	−	1.03730i	−0.0219709	−	0.0380548i	0.854831	−	0.518907i	$-0.173661\pi$
$743$	−0.598884	−	1.03730i	−0.0219709	−	0.0380548i	−0.876802	+	0.480852i	$0.840327\pi$
$744$		0			0
$745$	1.23491			0.0452435
$746$	14.5822	+	25.2571i	0.533891	+	0.924727i
$747$		0			0
$748$	9.71339			0.355157
$749$	−28.4629	+	1.03730i	−1.04001	+	0.0379021i
$750$		0			0
$751$	48.1199			1.75592			0.877961	−	0.478733i	$-0.158904\pi$
$751$	48.1199			1.75592			0.877961	+	0.478733i	$0.158904\pi$
$752$	−3.49381	+	6.05146i	−0.127406	+	0.220674i
$753$		0			0
$754$	3.38874	+	5.86946i	0.123410	+	0.213753i
$755$	73.6588			2.68072
$756$		0			0
$757$	49.6006			1.80276			0.901382	−	0.433025i	$-0.142554\pi$
$757$	49.6006			1.80276			0.901382	+	0.433025i	$0.142554\pi$
$758$	−6.78111	−	11.7452i	−0.246301	−	0.426606i
$759$		0			0
$760$	−1.64400	+	2.84748i	−0.0596340	+	0.103289i
$761$	37.5402			1.36083			0.680416	−	0.732826i	$-0.261799\pi$
$761$	37.5402			1.36083			0.680416	+	0.732826i	$0.261799\pi$
$762$		0			0
$763$	−0.234219	+	0.442124i	−0.00847931	+	0.0160060i
$764$	16.3214			0.590488
$765$		0			0
$766$	1.41783	+	2.45575i	0.0512281	+	0.0887297i
$767$	−18.6538			−0.673551
$768$		0			0
$769$	−13.4592	−	23.3121i	−0.485352	−	0.840654i	0.514506	−	0.857486i	$-0.327975\pi$
$769$	−13.4592	−	23.3121i	−0.485352	−	0.840654i	−0.999858	+	0.0168324i	$0.994642\pi$
$770$	−6.76833	+	12.7762i	−0.243914	+	0.460423i
$771$		0			0
$772$	7.16071	+	12.4027i	0.257719	+	0.446383i
$773$	−25.1130	−	43.4971i	−0.903254	−	1.56448i	−0.823245	−	0.567687i	$-0.807838\pi$
$773$	−25.1130	−	43.4971i	−0.903254	−	1.56448i	−0.0800089	−	0.996794i	$-0.525495\pi$
$774$		0			0
$775$	−29.5840	+	51.2409i	−1.06269	+	1.84063i
$776$	6.58836	−	11.4114i	0.236508	−	0.409645i
$777$		0			0
$778$	9.30401	+	16.1150i	0.333565	+	0.577752i
$779$	−3.65383			−0.130912
$780$		0			0
$781$	8.07194			0.288837
$782$	−20.6749	+	35.8099i	−0.739332	+	1.28056i

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -3.69963 q^{5} +(-1.40545 - 2.24159i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -3.69963 q^{5} +(-1.40545 - 2.24159i) q^{7} -1.00000 q^{8} +(-1.84981 - 3.20397i) q^{10} +1.47710 q^{11} +(-1.34981 - 2.33795i) q^{13} +(1.23855 - 2.33795i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.28799 - 5.69497i) q^{17} +(-0.444368 + 0.769668i) q^{19} +(1.84981 - 3.20397i) q^{20} +(0.738550 + 1.27921i) q^{22} -6.28799 q^{23} +8.68725 q^{25} +(1.34981 - 2.33795i) q^{26} +(2.64400 - 0.0963576i) q^{28} +(-1.25526 + 2.17417i) q^{29} +(-3.40545 + 5.89841i) q^{31} +(0.500000 - 0.866025i) q^{32} +(3.28799 - 5.69497i) q^{34} +(5.19963 + 8.29305i) q^{35} +(-1.38874 + 2.40536i) q^{37} -0.888736 q^{38} +3.69963 q^{40} +(2.05563 + 3.56046i) q^{41} +(0.00618986 - 0.0107211i) q^{43} +(-0.738550 + 1.27921i) q^{44} +(-3.14400 - 5.44556i) q^{46} +(-3.49381 - 6.05146i) q^{47} +(-3.04944 + 6.30087i) q^{49} +(4.34362 + 7.52338i) q^{50} +2.69963 q^{52} +(1.60507 + 2.78007i) q^{53} -5.46472 q^{55} +(1.40545 + 2.24159i) q^{56} -2.51052 q^{58} +(3.45489 - 5.98404i) q^{59} +(2.86652 + 4.96497i) q^{61} -6.81089 q^{62} +1.00000 q^{64} +(4.99381 + 8.64953i) q^{65} +(4.73236 - 8.19669i) q^{67} +6.57598 q^{68} +(-4.58217 + 8.64953i) q^{70} +5.46472 q^{71} +(-6.03273 - 10.4490i) q^{73} -2.77747 q^{74} +(-0.444368 - 0.769668i) q^{76} +(-2.07598 - 3.31105i) q^{77} +(-5.72617 - 9.91802i) q^{79} +(1.84981 + 3.20397i) q^{80} +(-2.05563 + 3.56046i) q^{82} +(-2.23855 + 3.87728i) q^{83} +(12.1643 + 21.0693i) q^{85} +0.0123797 q^{86} -1.47710 q^{88} +(4.43818 - 7.68715i) q^{89} +(-3.34362 + 6.31159i) q^{91} +(3.14400 - 5.44556i) q^{92} +(3.49381 - 6.05146i) q^{94} +(1.64400 - 2.84748i) q^{95} +(-6.58836 + 11.4114i) q^{97} +(-6.98143 + 0.509538i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8	\( 6 q + 3 q^{2} - 3 q^{4} - 10 q^{5} - 2 q^{7} - 6 q^{8} - 5 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} - q^{22} - 14 q^{23} + 4 q^{25} + 2 q^{26} + 4 q^{28} + 5 q^{29} - 14 q^{31} + 3 q^{32} - 4 q^{34} + 19 q^{35} - 9 q^{37} - 6 q^{38} + 10 q^{40} + 12 q^{41} + 18 q^{43} + q^{44} - 7 q^{46} - 3 q^{47} + 2 q^{50} + 4 q^{52} - 9 q^{53} + 14 q^{55} + 2 q^{56} + 10 q^{58} - 4 q^{59} + 4 q^{61} - 28 q^{62} + 6 q^{64} + 12 q^{65} + 5 q^{67} - 8 q^{68} + 2 q^{70} - 14 q^{71} - 25 q^{73} - 18 q^{74} - 3 q^{76} + 35 q^{77} + 7 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} + 14 q^{85} + 36 q^{86} + 2 q^{88} + 9 q^{89} + 4 q^{91} + 7 q^{92} + 3 q^{94} - 2 q^{95} - 28 q^{97} + 12 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 - 3 * q^4 - 10 * q^5 - 2 * q^7 - 6 * q^8 - 5 * q^10 - 2 * q^11 - 2 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^19 + 5 * q^20 - q^22 - 14 * q^23 + 4 * q^25 + 2 * q^26 + 4 * q^28 + 5 * q^29 - 14 * q^31 + 3 * q^32 - 4 * q^34 + 19 * q^35 - 9 * q^37 - 6 * q^38 + 10 * q^40 + 12 * q^41 + 18 * q^43 + q^44 - 7 * q^46 - 3 * q^47 + 2 * q^50 + 4 * q^52 - 9 * q^53 + 14 * q^55 + 2 * q^56 + 10 * q^58 - 4 * q^59 + 4 * q^61 - 28 * q^62 + 6 * q^64 + 12 * q^65 + 5 * q^67 - 8 * q^68 + 2 * q^70 - 14 * q^71 - 25 * q^73 - 18 * q^74 - 3 * q^76 + 35 * q^77 + 7 * q^79 + 5 * q^80 - 12 * q^82 - 8 * q^83 + 14 * q^85 + 36 * q^86 + 2 * q^88 + 9 * q^89 + 4 * q^91 + 7 * q^92 + 3 * q^94 - 2 * q^95 - 28 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more