LMFDB - Embedded newform 378.2.e.c.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(37,378)]

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([2, 2]))

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

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

chi := DirichletCharacter("378.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.e (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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.3
Root			$0.500000 + 0.224437i$ of defining polynomial
Character	$\chi$	$=$	378.37
Dual form			378.2.e.c.235.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(1.84981 + 3.20397i) q^{5} +(2.64400 - 0.0963576i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +(1.84981 + 3.20397i) q^{5} +(2.64400 - 0.0963576i) q^{7} -1.00000 q^{8} +(-1.84981 - 3.20397i) q^{10} +(-0.738550 + 1.27921i) q^{11} +(-1.34981 + 2.33795i) q^{13} +(-2.64400 + 0.0963576i) q^{14} +1.00000 q^{16} +(-3.28799 - 5.69497i) q^{17} +(-0.444368 + 0.769668i) q^{19} +(1.84981 + 3.20397i) q^{20} +(0.738550 - 1.27921i) q^{22} +(3.14400 + 5.44556i) q^{23} +(-4.34362 + 7.52338i) q^{25} +(1.34981 - 2.33795i) q^{26} +(2.64400 - 0.0963576i) q^{28} +(-1.25526 - 2.17417i) q^{29} +6.81089 q^{31} -1.00000 q^{32} +(3.28799 + 5.69497i) q^{34} +(5.19963 + 8.29305i) q^{35} +(-1.38874 + 2.40536i) q^{37} +(0.444368 - 0.769668i) q^{38} +(-1.84981 - 3.20397i) q^{40} +(2.05563 - 3.56046i) q^{41} +(0.00618986 + 0.0107211i) q^{43} +(-0.738550 + 1.27921i) q^{44} +(-3.14400 - 5.44556i) q^{46} +6.98762 q^{47} +(6.98143 - 0.509538i) q^{49} +(4.34362 - 7.52338i) q^{50} +(-1.34981 + 2.33795i) q^{52} +(1.60507 + 2.78007i) q^{53} -5.46472 q^{55} +(-2.64400 + 0.0963576i) q^{56} +(1.25526 + 2.17417i) q^{58} -6.90978 q^{59} -5.73305 q^{61} -6.81089 q^{62} +1.00000 q^{64} -9.98762 q^{65} -9.46472 q^{67} +(-3.28799 - 5.69497i) q^{68} +(-5.19963 - 8.29305i) q^{70} +5.46472 q^{71} +(-6.03273 - 10.4490i) q^{73} +(1.38874 - 2.40536i) q^{74} +(-0.444368 + 0.769668i) q^{76} +(-1.82946 + 3.45338i) q^{77} +11.4523 q^{79} +(1.84981 + 3.20397i) q^{80} +(-2.05563 + 3.56046i) q^{82} +(-2.23855 - 3.87728i) q^{83} +(12.1643 - 21.0693i) q^{85} +(-0.00618986 - 0.0107211i) q^{86} +(0.738550 - 1.27921i) q^{88} +(4.43818 - 7.68715i) q^{89} +(-3.34362 + 6.31159i) q^{91} +(3.14400 + 5.44556i) q^{92} -6.98762 q^{94} -3.28799 q^{95} +(-6.58836 - 11.4114i) q^{97} +(-6.98143 + 0.509538i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} + 5 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 + 5 * q^5 + 4 * q^7 - 6 * q^8	$ 6 q - 6 q^{2} + 6 q^{4} + 5 q^{5} + 4 q^{7} - 6 q^{8} - 5 q^{10} + q^{11} - 2 q^{13} - 4 q^{14} + 6 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} - q^{22} + 7 q^{23} - 2 q^{25} + 2 q^{26} + 4 q^{28} + 5 q^{29} + 28 q^{31} - 6 q^{32} - 4 q^{34} + 19 q^{35} - 9 q^{37} + 3 q^{38} - 5 q^{40} + 12 q^{41} + 18 q^{43} + q^{44} - 7 q^{46} + 6 q^{47} - 12 q^{49} + 2 q^{50} - 2 q^{52} - 9 q^{53} + 14 q^{55} - 4 q^{56} - 5 q^{58} + 8 q^{59} - 8 q^{61} - 28 q^{62} + 6 q^{64} - 24 q^{65} - 10 q^{67} + 4 q^{68} - 19 q^{70} - 14 q^{71} - 25 q^{73} + 9 q^{74} - 3 q^{76} - 52 q^{77} - 14 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} + 14 q^{85} - 18 q^{86} - q^{88} + 9 q^{89} + 4 q^{91} + 7 q^{92} - 6 q^{94} + 4 q^{95} - 28 q^{97} + 12 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 + 5 * q^5 + 4 * q^7 - 6 * q^8 - 5 * q^10 + q^11 - 2 * q^13 - 4 * q^14 + 6 * q^16 + 4 * q^17 - 3 * q^19 + 5 * q^20 - q^22 + 7 * q^23 - 2 * q^25 + 2 * q^26 + 4 * q^28 + 5 * q^29 + 28 * q^31 - 6 * q^32 - 4 * q^34 + 19 * q^35 - 9 * q^37 + 3 * q^38 - 5 * q^40 + 12 * q^41 + 18 * q^43 + q^44 - 7 * q^46 + 6 * q^47 - 12 * q^49 + 2 * q^50 - 2 * q^52 - 9 * q^53 + 14 * q^55 - 4 * q^56 - 5 * q^58 + 8 * q^59 - 8 * q^61 - 28 * q^62 + 6 * q^64 - 24 * q^65 - 10 * q^67 + 4 * q^68 - 19 * q^70 - 14 * q^71 - 25 * q^73 + 9 * q^74 - 3 * q^76 - 52 * q^77 - 14 * q^79 + 5 * q^80 - 12 * q^82 - 8 * q^83 + 14 * q^85 - 18 * q^86 - q^88 + 9 * q^89 + 4 * q^91 + 7 * q^92 - 6 * q^94 + 4 * q^95 - 28 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	1.84981	+	3.20397i	0.827262	+	1.43286i	0.900178	+	0.435522i	$0.143436\pi$
$5$	1.84981	+	3.20397i	0.827262	+	1.43286i	−0.0729162	+	0.997338i	$0.523231\pi$
$6$		0			0
$7$	2.64400	−	0.0963576i	0.999337	−	0.0364197i
$7$	2.64400	−	0.0963576i	0.999337	−	0.0364197i
$8$	−1.00000			−0.353553
$9$		0			0
$10$	−1.84981	−	3.20397i	−0.584963	−	1.01318i
$11$	−0.738550	+	1.27921i	−0.222681	+	0.385695i	−0.955621	−	0.294598i	$-0.904814\pi$
$11$	−0.738550	+	1.27921i	−0.222681	+	0.385695i	0.732940	+	0.680293i	$0.238148\pi$
$12$		0			0
$13$	−1.34981	+	2.33795i	−0.374371	+	0.648430i	−0.990233	−	0.139425i	$-0.955475\pi$
$13$	−1.34981	+	2.33795i	−0.374371	+	0.648430i	0.615862	+	0.787854i	$0.288808\pi$
$14$	−2.64400	+	0.0963576i	−0.706638	+	0.0257526i
$15$		0			0
$16$	1.00000			0.250000
$17$	−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
$23$	3.14400	+	5.44556i	0.655568	+	1.13548i	0.981751	+	0.190171i	$0.0609043\pi$
$23$	3.14400	+	5.44556i	0.655568	+	1.13548i	−0.326182	+	0.945307i	$0.605762\pi$
$24$		0			0
$25$	−4.34362	+	7.52338i	−0.868725	+	1.50468i
$26$	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	0.725622	−	0.688094i	$-0.241552\pi$
$29$	−1.25526	−	2.17417i	−0.233096	−	0.403734i	−0.958718	+	0.284360i	$0.908219\pi$
$30$		0			0
$31$	6.81089			1.22327			0.611636	−	0.791139i	$-0.290512\pi$
$31$	6.81089			1.22327			0.611636	+	0.791139i	$0.290512\pi$
$32$	−1.00000			−0.176777
$33$		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	$0.239986\pi$
$38$	0.444368	−	0.769668i	0.0720860	−	0.124857i
$39$		0			0
$40$	−1.84981	−	3.20397i	−0.292481	−	0.506592i
$41$	2.05563	−	3.56046i	0.321036	−	0.556050i	−0.659666	−	0.751559i	$-0.729302\pi$
$41$	2.05563	−	3.56046i	0.321036	−	0.556050i	0.980702	+	0.195508i	$0.0626357\pi$
$42$		0			0
$43$	0.00618986	+	0.0107211i	0.000943944	+	0.00163496i	0.866497	−	0.499182i	$-0.166366\pi$
$43$	0.00618986	+	0.0107211i	0.000943944	+	0.00163496i	−0.865553	+	0.500817i	$0.833033\pi$
$44$	−0.738550	+	1.27921i	−0.111341	+	0.192848i
$45$		0			0
$46$	−3.14400	−	5.44556i	−0.463557	−	0.802904i
$47$	6.98762			1.01925			0.509625	−	0.860397i	$-0.329784\pi$
$47$	6.98762			1.01925			0.509625	+	0.860397i	$0.329784\pi$
$48$		0			0
$49$	6.98143	−	0.509538i	0.997347	−	0.0727912i
$50$	4.34362	−	7.52338i	0.614281	−	1.06397i
$51$		0			0
$52$	−1.34981	+	2.33795i	−0.187186	+	0.324215i
$53$	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
$56$	−2.64400	+	0.0963576i	−0.353319	+	0.0128763i
$57$		0			0
$58$	1.25526	+	2.17417i	0.164824	+	0.285483i
$59$	−6.90978			−0.899576			−0.449788	−	0.893135i	$-0.648501\pi$
$59$	−6.90978			−0.899576			−0.449788	+	0.893135i	$0.648501\pi$
$60$		0			0
$61$	−5.73305			−0.734042			−0.367021	−	0.930213i	$-0.619622\pi$
$61$	−5.73305			−0.734042			−0.367021	+	0.930213i	$0.619622\pi$
$62$	−6.81089			−0.864984
$63$		0			0
$64$	1.00000			0.125000
$65$	−9.98762			−1.23881
$66$		0			0
$67$	−9.46472			−1.15630			−0.578150	−	0.815931i	$-0.696225\pi$
$67$	−9.46472			−1.15630			−0.578150	+	0.815931i	$0.696225\pi$
$68$	−3.28799	−	5.69497i	−0.398728	−	0.690616i
$69$		0			0
$70$	−5.19963	−	8.29305i	−0.621474	−	0.991209i
$71$	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	$-0.416204\pi$
$74$	1.38874	−	2.40536i	0.161437	−	0.279618i
$75$		0			0
$76$	−0.444368	+	0.769668i	−0.0509725	+	0.0882870i
$77$	−1.82946	+	3.45338i	−0.208487	+	0.393549i
$78$		0			0
$79$	11.4523			1.28849			0.644244	−	0.764820i	$-0.277172\pi$
$79$	11.4523			1.28849			0.644244	+	0.764820i	$0.277172\pi$
$80$	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	0.716619	−	0.697465i	$-0.245689\pi$
$83$	−2.23855	−	3.87728i	−0.245713	−	0.425587i	−0.962332	+	0.271878i	$0.912355\pi$
$84$		0			0
$85$	12.1643	−	21.0693i	1.31941	−	2.28528i
$86$	−0.00618986	−	0.0107211i	−0.000667469	−	0.00115609i
$87$		0			0
$88$	0.738550	−	1.27921i	0.0787297	−	0.136364i
$89$	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
$94$	−6.98762			−0.720718
$95$	−3.28799			−0.337341
$96$		0			0
$97$	−6.58836	−	11.4114i	−0.668947	−	1.15865i	−0.978199	−	0.207670i	$-0.933412\pi$
$97$	−6.58836	−	11.4114i	−0.668947	−	1.15865i	0.309252	−	0.950980i	$-0.399921\pi$
$98$	−6.98143	+	0.509538i	−0.705231	+	0.0514711i
$99$		0			0
$100$	−4.34362	+	7.52338i	−0.434362	+	0.752338i
$101$	2.62729	−	4.55059i	0.261425	−	0.452801i	−0.705196	−	0.709012i	$-0.749141\pi$
$101$	2.62729	−	4.55059i	0.261425	−	0.452801i	0.966621	+	0.256212i	$0.0824744\pi$
$102$		0			0
$103$	−0.833104	−	1.44298i	−0.0820882	−	0.142181i	0.822059	−	0.569403i	$-0.192826\pi$
$103$	−0.833104	−	1.44298i	−0.0820882	−	0.142181i	−0.904147	+	0.427222i	$0.859492\pi$
$104$	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	$0.836216\pi$
$110$	5.46472			0.521041
$111$		0			0
$112$	2.64400	−	0.0963576i	0.249834	−	0.00910494i
$113$	6.78180	−	11.7464i	0.637978	−	1.10501i	−0.347897	−	0.937533i	$-0.613104\pi$
$113$	6.78180	−	11.7464i	0.637978	−	1.10501i	0.985876	−	0.167478i	$-0.0535624\pi$
$114$		0			0
$115$	−11.6316	+	20.1466i	−1.08465	+	1.87868i
$116$	−1.25526	−	2.17417i	−0.116548	−	0.201867i
$117$		0			0
$118$	6.90978			0.636097
$119$	−9.24219	−	14.7407i	−0.847230	−	1.35127i
$120$		0			0
$121$	4.40909	+	7.63676i	0.400826	+	0.694251i
$122$	5.73305			0.519046
$123$		0			0
$124$	6.81089			0.611636
$125$	−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	$0.540453\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	9.98762			0.875972
$131$	0.0778435	+	0.134829i	0.00680122	+	0.0117801i	0.869406	−	0.494098i	$-0.164502\pi$
$131$	0.0778435	+	0.134829i	0.00680122	+	0.0117801i	−0.862605	+	0.505878i	$0.831168\pi$
$132$		0			0
$133$	−1.10074	+	2.07782i	−0.0954466	+	0.180170i
$134$	9.46472			0.817627
$135$		0			0
$136$	3.28799	+	5.69497i	0.281943	+	0.488340i
$137$	−1.70582	+	2.95456i	−0.145738	+	0.252425i	−0.929648	−	0.368449i	$-0.879889\pi$
$137$	−1.70582	+	2.95456i	−0.145738	+	0.252425i	0.783910	+	0.620874i	$0.213222\pi$
$138$		0			0
$139$	−6.75526	+	11.7005i	−0.572974	+	0.992420i	0.423285	+	0.905997i	$0.360877\pi$
$139$	−6.75526	+	11.7005i	−0.572974	+	0.992420i	−0.996259	+	0.0864229i	$0.972456\pi$
$140$	5.19963	+	8.29305i	0.439449	+	0.700890i
$141$		0			0
$142$	−5.46472			−0.458589
$143$	−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
$149$	0.166896	+	0.289073i	0.0136727	+	0.0236818i	0.872781	−	0.488112i	$-0.162314\pi$
$149$	0.166896	+	0.289073i	0.0136727	+	0.0236818i	−0.859108	+	0.511794i	$0.828981\pi$
$150$		0			0
$151$	9.95489	−	17.2424i	0.810117	−	1.40316i	−0.102664	−	0.994716i	$-0.532737\pi$
$151$	9.95489	−	17.2424i	0.810117	−	1.40316i	0.912781	−	0.408448i	$-0.133930\pi$
$152$	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
$157$	−6.96286			−0.555697			−0.277848	−	0.960625i	$-0.589621\pi$
$157$	−6.96286			−0.555697			−0.277848	+	0.960625i	$0.589621\pi$
$158$	−11.4523			−0.911099
$159$		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	$0.0642572\pi$
$164$	2.05563	−	3.56046i	0.160518	−	0.278025i
$165$		0			0
$166$	2.23855	+	3.87728i	0.173745	+	0.300935i
$167$	−9.74288	+	16.8752i	−0.753927	+	1.30584i	0.191979	+	0.981399i	$0.438509\pi$
$167$	−9.74288	+	16.8752i	−0.753927	+	1.30584i	−0.945906	+	0.324440i	$0.894824\pi$
$168$		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
$173$	−22.5636			−1.71548			−0.857740	−	0.514085i	$-0.828132\pi$
$173$	−22.5636			−1.71548			−0.857740	+	0.514085i	$0.828132\pi$
$174$		0			0
$175$	−10.7596	+	20.3103i	−0.813349	+	1.53532i
$176$	−0.738550	+	1.27921i	−0.0556703	+	0.0964238i
$177$		0			0
$178$	−4.43818	+	7.68715i	−0.332656	+	0.576176i
$179$	−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	$0.168085\pi$
$182$	3.34362	−	6.31159i	0.247846	−	0.467846i
$183$		0			0
$184$	−3.14400	−	5.44556i	−0.231778	−	0.401452i
$185$	−10.2756			−0.755478
$186$		0			0
$187$	9.71339			0.710313
$188$	6.98762			0.509625
$189$		0			0
$190$	3.28799			0.238536
$191$	16.3214			1.18098			0.590488	−	0.807046i	$-0.298935\pi$
$191$	16.3214			1.18098			0.590488	+	0.807046i	$0.298935\pi$
$192$		0			0
$193$	−14.3214			−1.03088			−0.515439	−	0.856926i	$-0.672371\pi$
$193$	−14.3214			−1.03088			−0.515439	+	0.856926i	$0.672371\pi$
$194$	6.58836	+	11.4114i	0.473017	+	0.819289i
$195$		0			0
$196$	6.98143	−	0.509538i	0.498674	−	0.0363956i
$197$	−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	$0.902833\pi$
$200$	4.34362	−	7.52338i	0.307141	−	0.531983i
$201$		0			0
$202$	−2.62729	+	4.55059i	−0.184855	+	0.320179i
$203$	−3.52840	−	5.62755i	−0.247645	−	0.394977i
$204$		0			0
$205$	15.2101			1.06232
$206$	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	−0.599012	−	0.800740i	$-0.704440\pi$
$211$	5.72253	−	9.91171i	0.393955	−	0.682350i	0.992967	+	0.118390i	$0.0377732\pi$
$212$	1.60507	+	2.78007i	0.110237	+	0.190936i
$213$		0			0
$214$	−5.38255	+	9.32284i	−0.367943	+	0.637296i
$215$	−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
$220$	−5.46472			−0.368431
$221$	17.7527			1.19418
$222$		0			0
$223$	−3.61126	−	6.25489i	−0.241828	−	0.418859i	0.719407	−	0.694589i	$-0.244414\pi$
$223$	−3.61126	−	6.25489i	−0.241828	−	0.418859i	−0.961235	+	0.275730i	$0.911080\pi$
$224$	−2.64400	+	0.0963576i	−0.176659	+	0.00643816i
$225$		0			0
$226$	−6.78180	+	11.7464i	−0.451119	+	0.781361i
$227$	6.82760	−	11.8258i	0.453164	−	0.784903i	−0.545417	−	0.838165i	$-0.683629\pi$
$227$	6.82760	−	11.8258i	0.453164	−	0.784903i	0.998581	+	0.0532622i	$0.0169619\pi$
$228$		0			0
$229$	−8.68725	−	15.0468i	−0.574070	−	0.994318i	−0.996142	−	0.0877555i	$-0.972031\pi$
$229$	−8.68725	−	15.0468i	−0.574070	−	0.994318i	0.422073	−	0.906562i	$-0.361303\pi$
$230$	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
$236$	−6.90978			−0.449788
$237$		0			0
$238$	9.24219	+	14.7407i	0.599082	+	0.955495i
$239$	−9.47524	+	16.4116i	−0.612902	+	1.06158i	0.377846	+	0.925868i	$0.376665\pi$
$239$	−9.47524	+	16.4116i	−0.612902	+	1.06158i	−0.990749	+	0.135710i	$0.956669\pi$
$240$		0			0
$241$	12.2527	−	21.2223i	0.789267	−	1.36705i	−0.137150	−	0.990550i	$-0.543794\pi$
$241$	12.2527	−	21.2223i	0.789267	−	1.36705i	0.926417	−	0.376500i	$-0.122872\pi$
$242$	−4.40909	−	7.63676i	−0.283427	−	0.490910i
$243$		0			0
$244$	−5.73305			−0.367021
$245$	14.5469	+	21.4258i	0.929367	+	1.36884i
$246$		0			0
$247$	−1.19963	−	2.07782i	−0.0763305	−	0.132208i
$248$	−6.81089			−0.432492
$249$		0			0
$250$	13.6414			0.862761
$251$	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
$254$	2.85669			0.179245
$255$		0			0
$256$	1.00000			0.0625000
$257$	−4.10439	−	7.10900i	−0.256025	−	0.443448i	0.709149	−	0.705059i	$-0.249079\pi$
$257$	−4.10439	−	7.10900i	−0.256025	−	0.443448i	−0.965173	+	0.261611i	$0.915746\pi$
$258$		0			0
$259$	−3.44004	+	6.49358i	−0.213754	+	0.403491i
$260$	−9.98762			−0.619406
$261$		0			0
$262$	−0.0778435	−	0.134829i	−0.00480919	−	0.00832976i
$263$	−2.67309	+	4.62992i	−0.164830	+	0.285493i	−0.936595	−	0.350414i	$-0.886041\pi$
$263$	−2.67309	+	4.62992i	−0.164830	+	0.285493i	0.771765	+	0.635908i	$0.219374\pi$
$264$		0			0
$265$	−5.93818	+	10.2852i	−0.364779	+	0.631816i
$266$	1.10074	−	2.07782i	0.0674909	−	0.127399i
$267$		0			0
$268$	−9.46472			−0.578150
$269$	−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
$275$	−6.41597	−	11.1128i	−0.386897	−	0.670126i
$276$		0			0
$277$	4.54944	−	7.87987i	0.273349	−	0.473455i	−0.696368	−	0.717685i	$-0.745202\pi$
$277$	4.54944	−	7.87987i	0.273349	−	0.473455i	0.969717	+	0.244230i	$0.0785351\pi$
$278$	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	0.629470	−	0.777025i	$-0.283272\pi$
$281$	−6.00433	−	10.3998i	−0.358188	−	0.620400i	−0.987658	+	0.156624i	$0.949939\pi$
$282$		0			0
$283$	9.84294			0.585102			0.292551	−	0.956250i	$-0.405496\pi$
$283$	9.84294			0.585102			0.292551	+	0.956250i	$0.405496\pi$
$284$	5.46472			0.324271
$285$		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
$290$	−4.64400	+	8.04364i	−0.272705	+	0.472339i
$291$		0			0
$292$	−6.03273	−	10.4490i	−0.353039	−	0.611481i
$293$	−10.7101	+	18.5505i	−0.625694	+	1.08373i	0.362713	+	0.931901i	$0.381851\pi$
$293$	−10.7101	+	18.5505i	−0.625694	+	1.08373i	−0.988406	+	0.151832i	$0.951483\pi$
$294$		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
$299$	−16.9752			−0.981704
$300$		0			0
$301$	0.0173990	+	0.0277502i	0.00100286	+	0.00159950i
$302$	−9.95489	+	17.2424i	−0.572839	+	0.992187i
$303$		0			0
$304$	−0.444368	+	0.769668i	−0.0254862	+	0.0441435i
$305$	−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	$0.551889\pi$
$308$	−1.82946	+	3.45338i	−0.104243	+	0.196775i
$309$		0			0
$310$	−12.5989	−	21.8219i	−0.715569	−	1.23940i
$311$	11.7207			0.664618			0.332309	−	0.943171i	$-0.392172\pi$
$311$	11.7207			0.664618			0.332309	+	0.943171i	$0.392172\pi$
$312$		0			0
$313$	−26.7738			−1.51334			−0.756671	−	0.653796i	$-0.773176\pi$
$313$	−26.7738			−1.51334			−0.756671	+	0.653796i	$0.773176\pi$
$314$	6.96286			0.392937
$315$		0			0
$316$	11.4523			0.644244
$317$	−1.90249			−0.106855			−0.0534273	−	0.998572i	$-0.517015\pi$
$317$	−1.90249			−0.106855			−0.0534273	+	0.998572i	$0.517015\pi$
$318$		0			0
$319$	3.70829			0.207624
$320$	1.84981	+	3.20397i	0.103408	+	0.179107i
$321$		0			0
$322$	−8.83743	−	14.0951i	−0.492491	−	0.785489i
$323$	5.84431			0.325186
$324$		0			0
$325$	−11.7262	−	20.3103i	−0.650451	−	1.12661i
$326$	−4.03706	+	6.99240i	−0.223592	+	0.387273i
$327$		0			0
$328$	−2.05563	+	3.56046i	−0.113503	+	0.196593i
$329$	18.4752	−	0.673310i	1.01857	−	0.0371208i
$330$		0			0
$331$	5.56732			0.306008			0.153004	−	0.988226i	$-0.451105\pi$
$331$	5.56732			0.306008			0.153004	+	0.988226i	$0.451105\pi$
$332$	−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	−0.120302	+	0.992737i	$0.538386\pi$
$337$	−16.8869	+	29.2489i	−0.919887	+	1.59329i	−0.799585	+	0.600553i	$0.794947\pi$
$338$	−2.85600	−	4.94674i	−0.155346	−	0.269067i
$339$		0			0
$340$	12.1643	−	21.0693i	0.659704	−	1.14264i
$341$	−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
$346$	22.5636			1.21303
$347$	30.4065			1.63231			0.816154	−	0.577834i	$-0.196102\pi$
$347$	30.4065			1.63231			0.816154	+	0.577834i	$0.196102\pi$
$348$		0			0
$349$	−6.29782	−	10.9082i	−0.337115	−	0.583900i	0.646774	−	0.762682i	$-0.276118\pi$
$349$	−6.29782	−	10.9082i	−0.337115	−	0.583900i	−0.983889	+	0.178782i	$0.942784\pi$
$350$	10.7596	−	20.3103i	0.575124	−	1.08563i
$351$		0			0
$352$	0.738550	−	1.27921i	0.0393648	−	0.0681819i
$353$	−3.76578	+	6.52252i	−0.200432	+	0.347159i	−0.948668	−	0.316274i	$-0.897568\pi$
$353$	−3.76578	+	6.52252i	−0.200432	+	0.347159i	0.748235	+	0.663433i	$0.230901\pi$
$354$		0			0
$355$	10.1087	+	17.5088i	0.536515	+	0.929271i
$356$	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
$362$	−23.2422			−1.22158
$363$		0			0
$364$	−3.34362	+	6.31159i	−0.175254	+	0.330817i
$365$	22.3189	−	38.6574i	1.16822	−	2.02342i
$366$		0			0
$367$	−11.5618	+	20.0257i	−0.603522	+	1.04533i	0.388761	+	0.921339i	$0.372903\pi$
$367$	−11.5618	+	20.0257i	−0.603522	+	1.04533i	−0.992283	+	0.123992i	$0.960430\pi$
$368$	3.14400	+	5.44556i	0.163892	+	0.283869i
$369$		0			0
$370$	10.2756			0.534204
$371$	4.51169	+	7.19583i	0.234235	+	0.373589i
$372$		0			0
$373$	−14.5822	−	25.2571i	−0.755036	−	1.30776i	−0.945356	−	0.326039i	$-0.894286\pi$
$373$	−14.5822	−	25.2571i	−0.755036	−	1.30776i	0.190320	−	0.981722i	$-0.439047\pi$
$374$	−9.71339			−0.502267
$375$		0			0
$376$	−6.98762			−0.360359
$377$	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	$0.613249\pi$
$380$	−3.28799			−0.168670
$381$		0			0
$382$	−16.3214			−0.835076
$383$	−1.41783	−	2.45575i	−0.0724475	−	0.125483i	0.827526	−	0.561428i	$-0.189748\pi$
$383$	−1.41783	−	2.45575i	−0.0724475	−	0.125483i	−0.899973	+	0.435945i	$0.856414\pi$
$384$		0			0
$385$	−14.4487	+	0.526567i	−0.736374	+	0.0268364i
$386$	14.3214			0.728941
$387$		0			0
$388$	−6.58836	−	11.4114i	−0.334474	−	0.579325i
$389$	−9.30401	+	16.1150i	−0.471732	+	0.817064i	−0.999477	−	0.0323388i	$-0.989704\pi$
$389$	−9.30401	+	16.1150i	−0.471732	+	0.817064i	0.527745	+	0.849403i	$0.323038\pi$
$390$		0			0
$391$	20.6749	−	35.8099i	1.04557	−	1.81099i
$392$	−6.98143	+	0.509538i	−0.352615	+	0.0257356i
$393$		0			0
$394$	2.42402			0.122120
$395$	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
$401$	−3.37704	−	5.84921i	−0.168642	−	0.292096i	0.769301	−	0.638887i	$-0.220605\pi$
$401$	−3.37704	−	5.84921i	−0.168642	−	0.292096i	−0.937942	+	0.346791i	$0.887271\pi$
$402$		0			0
$403$	−9.19344	+	15.9235i	−0.457958	+	0.793206i
$404$	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
$409$	15.3214			0.757595			0.378798	−	0.925480i	$-0.376338\pi$
$409$	15.3214			0.757595			0.378798	+	0.925480i	$0.376338\pi$
$410$	−15.2101			−0.751176
$411$		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
$416$	1.34981	−	2.33795i	0.0661801	−	0.114627i
$417$		0			0
$418$	0.656376	+	1.13688i	0.0321044	+	0.0556064i
$419$	4.32141	−	7.48491i	0.211115	−	0.365662i	−0.740949	−	0.671561i	$-0.765624\pi$
$419$	4.32141	−	7.48491i	0.211115	−	0.365662i	0.952064	+	0.305900i	$0.0989573\pi$
$420$		0			0
$421$	18.5636	+	32.1531i	0.904735	+	1.56705i	0.821273	+	0.570536i	$0.193264\pi$
$421$	18.5636	+	32.1531i	0.904735	+	1.56705i	0.0834618	+	0.996511i	$0.473402\pi$
$422$	−5.72253	+	9.91171i	−0.278568	+	0.482494i
$423$		0			0
$424$	−1.60507	−	2.78007i	−0.0779493	−	0.135012i
$425$	57.1272			2.77108
$426$		0			0
$427$	−15.1582	+	0.552423i	−0.733555	+	0.0267336i
$428$	5.38255	−	9.32284i	0.260175	−	0.450637i
$429$		0			0
$430$	0.0229002	−	0.0396643i	0.00110434	−	0.00191278i
$431$	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	$0.501597\pi$
$434$	−18.0080	+	0.656281i	−0.864410	+	0.0315025i
$435$		0			0
$436$	−0.0945538	−	0.163772i	−0.00452831	−	0.00784326i
$437$	−5.58836			−0.267328
$438$		0			0
$439$	−9.96796			−0.475745			−0.237872	−	0.971296i	$-0.576450\pi$
$439$	−9.96796			−0.475745			−0.237872	+	0.971296i	$0.576450\pi$
$440$	5.46472			0.260520
$441$		0			0
$442$	−17.7527			−0.844410
$443$	15.6996			0.745912			0.372956	−	0.927849i	$-0.378344\pi$
$443$	15.6996			0.745912			0.372956	+	0.927849i	$0.378344\pi$
$444$		0			0
$445$	32.8392			1.55673
$446$	3.61126	+	6.25489i	0.170998	+	0.296178i
$447$		0			0
$448$	2.64400	−	0.0963576i	0.124917	−	0.00455247i
$449$	−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
$452$	6.78180	−	11.7464i	0.318989	−	0.552505i
$453$		0			0
$454$	−6.82760	+	11.8258i	−0.320435	+	0.555010i
$455$	−26.4072	+	0.962383i	−1.23799	+	0.0451172i
$456$		0			0
$457$	32.7083			1.53003			0.765015	−	0.644013i	$-0.222732\pi$
$457$	32.7083			1.53003			0.765015	+	0.644013i	$0.222732\pi$
$458$	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	0.910228	−	0.414107i	$-0.135906\pi$
$461$	2.07165	+	3.58821i	0.0964865	+	0.167120i	−0.813742	+	0.581227i	$0.802573\pi$
$462$		0			0
$463$	−8.34176	+	14.4484i	−0.387675	+	0.671472i	−0.992136	−	0.125162i	$-0.960055\pi$
$463$	−8.34176	+	14.4484i	−0.387675	+	0.671472i	0.604462	+	0.796634i	$0.293388\pi$
$464$	−1.25526	−	2.17417i	−0.0582740	−	0.100934i
$465$		0			0
$466$	7.62110	−	13.2001i	0.353040	−	0.611484i
$467$	−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
$472$	6.90978			0.318048
$473$	−0.0182861			−0.000840794
$474$		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
$479$	−1.47965	+	2.56283i	−0.0676068	+	0.117098i	−0.897847	−	0.440307i	$-0.854870\pi$
$479$	−1.47965	+	2.56283i	−0.0676068	+	0.117098i	0.830241	+	0.557405i	$0.188203\pi$
$480$		0			0
$481$	−3.74907	−	6.49358i	−0.170943	−	0.296082i
$482$	−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	$-0.385996\pi$
$488$	5.73305			0.259523
$489$		0			0
$490$	−14.5469	−	21.4258i	−0.657162	−	0.967917i
$491$	−17.0734	+	29.5721i	−0.770513	+	1.33457i	0.166769	+	0.985996i	$0.446667\pi$
$491$	−17.0734	+	29.5721i	−0.770513	+	1.33457i	−0.937282	+	0.348572i	$0.886667\pi$
$492$		0			0
$493$	−8.25457	+	14.2973i	−0.371767	+	0.643920i
$494$	1.19963	+	2.07782i	0.0539738	+	0.0934854i
$495$		0			0
$496$	6.81089			0.305818
$497$	14.4487	−	0.526567i	0.648113	−	0.0236198i
$498$		0			0
$499$	1.14035	+	1.97515i	0.0510493	+	0.0884199i	0.890421	−	0.455138i	$-0.150410\pi$
$499$	1.14035	+	1.97515i	0.0510493	+	0.0884199i	−0.839372	+	0.543558i	$0.817077\pi$
$500$	−13.6414			−0.610064
$501$		0			0
$502$	−12.1236			−0.541105
$503$	−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
$506$	9.28799			0.412902
$507$		0			0
$508$	−2.85669			−0.126745
$509$	12.8090	+	22.1859i	0.567750	+	0.983373i	0.996788	+	0.0800859i	$0.0255195\pi$
$509$	12.8090	+	22.1859i	0.567750	+	0.983373i	−0.429038	+	0.903287i	$0.641147\pi$
$510$		0			0
$511$	−16.9574	−	27.0458i	−0.750149	−	1.19644i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	4.10439	+	7.10900i	0.181037	+	0.313565i
$515$	3.08217	−	5.33848i	0.135817	−	0.235242i
$516$		0			0
$517$	−5.16071	+	8.93861i	−0.226968	+	0.393119i
$518$	3.44004	−	6.49358i	0.151147	−	0.285312i
$519$		0			0
$520$	9.98762			0.437986
$521$	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
$527$	−22.3942	−	38.7878i	−0.975505	−	1.68962i
$528$		0			0
$529$	−8.26942	+	14.3231i	−0.359540	+	0.622742i
$530$	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
$535$	39.8268			1.72186
$536$	9.46472			0.408814
$537$		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	$0.805031\pi$
$542$	3.67742	−	6.36947i	0.157959	−	0.273592i
$543$		0			0
$544$	3.28799	+	5.69497i	0.140971	+	0.244170i
$545$	0.349814	−	0.605896i	0.0149844	−	0.0259537i
$546$		0			0
$547$	20.3356	+	35.2222i	0.869486	+	1.50599i	0.862522	+	0.506019i	$0.168883\pi$
$547$	20.3356	+	35.2222i	0.869486	+	1.50599i	0.00696400	+	0.999976i	$0.497783\pi$
$548$	−1.70582	+	2.95456i	−0.0728689	+	0.126213i
$549$		0			0
$550$	6.41597	+	11.1128i	0.273578	+	0.473851i
$551$	2.23119			0.0950519
$552$		0			0
$553$	30.2799	−	1.10352i	1.28763	−	0.0469264i
$554$	−4.54944	+	7.87987i	−0.193287	+	0.334783i
$555$		0			0
$556$	−6.75526	+	11.7005i	−0.286487	+	0.496210i
$557$	−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
$560$	5.19963	+	8.29305i	0.219724	+	0.350445i
$561$		0			0
$562$	6.00433	+	10.3998i	0.253277	+	0.438689i
$563$	32.7614			1.38073			0.690364	−	0.723463i	$-0.257451\pi$
$563$	32.7614			1.38073			0.690364	+	0.723463i	$0.257451\pi$
$564$		0			0
$565$	50.1803			2.11110
$566$	−9.84294			−0.413729
$567$		0			0
$568$	−5.46472			−0.229295
$569$	16.7280			0.701272			0.350636	−	0.936512i	$-0.385965\pi$
$569$	16.7280			0.701272			0.350636	+	0.936512i	$0.385965\pi$
$570$		0			0
$571$	−27.4734			−1.14973			−0.574863	−	0.818250i	$-0.694945\pi$
$571$	−27.4734			−1.14973			−0.574863	+	0.818250i	$0.694945\pi$
$572$	−1.99381	−	3.45338i	−0.0833654	−	0.144393i
$573$		0			0
$574$	−5.09201	+	9.61192i	−0.212536	+	0.401194i
$575$	−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	$0.814543\pi$
$578$	13.1218	−	22.7276i	0.545794	−	0.945343i
$579$		0			0
$580$	4.64400	−	8.04364i	0.192831	−	0.333994i
$581$	−6.29232	−	10.0358i	−0.261050	−	0.416356i
$582$		0			0
$583$	−4.74171			−0.196382
$584$	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	0.910404	−	0.413720i	$-0.135771\pi$
$587$	2.34795	+	4.06678i	0.0969105	+	0.167854i	−0.813494	+	0.581573i	$0.802437\pi$
$588$		0			0
$589$	−3.02654	+	5.24212i	−0.124706	+	0.215998i
$590$	12.7818	+	22.1387i	0.526218	+	0.911437i
$591$		0			0
$592$	−1.38874	+	2.40536i	−0.0570767	+	0.0988597i
$593$	−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
$598$	16.9752			0.694169
$599$	−43.8516			−1.79173			−0.895864	−	0.444329i	$-0.853442\pi$
$599$	−43.8516			−1.79173			−0.895864	+	0.444329i	$0.853442\pi$
$600$		0			0
$601$	−6.71634	−	11.6330i	−0.273965	−	0.474522i	0.695908	−	0.718131i	$-0.255002\pi$
$601$	−6.71634	−	11.6330i	−0.273965	−	0.474522i	−0.969874	+	0.243609i	$0.921669\pi$
$602$	−0.0173990	−	0.0277502i	−0.000709131	−	0.00113101i
$603$		0			0
$604$	9.95489	−	17.2424i	0.405059	−	0.701582i
$605$	−16.3120	+	28.2532i	−0.663177	+	1.14866i
$606$		0			0
$607$	2.29232	+	3.97042i	0.0930425	+	0.161154i	0.908790	−	0.417254i	$-0.137007\pi$
$607$	2.29232	+	3.97042i	0.0930425	+	0.161154i	−0.815747	+	0.578408i	$0.803674\pi$
$608$	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	$0.980648\pi$
$614$	5.68725			0.229519
$615$		0			0
$616$	1.82946	−	3.45338i	0.0737111	−	0.139141i
$617$	−6.00433	+	10.3998i	−0.241725	+	0.418680i	−0.961206	−	0.275832i	$-0.911047\pi$
$617$	−6.00433	+	10.3998i	−0.241725	+	0.418680i	0.719481	+	0.694513i	$0.244380\pi$
$618$		0			0
$619$	8.78180	−	15.2105i	0.352970	−	0.611363i	−0.633798	−	0.773499i	$-0.718505\pi$
$619$	8.78180	−	15.2105i	0.352970	−	0.611363i	0.986768	+	0.162136i	$0.0518383\pi$
$620$	12.5989	+	21.8219i	0.505983	+	0.876389i
$621$		0			0
$622$	−11.7207			−0.469956
$623$	10.9938	−	20.7524i	0.440458	−	0.831429i
$624$		0			0
$625$	−3.51602	−	6.08993i	−0.140641	−	0.243597i
$626$	26.7738			1.07009
$627$		0			0
$628$	−6.96286			−0.277848
$629$	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	$0.852453\pi$
$632$	−11.4523			−0.455550
$633$		0			0
$634$	1.90249			0.0755576
$635$	−5.28435	−	9.15276i	−0.209703	−	0.363216i
$636$		0			0
$637$	−8.23236	+	17.0100i	−0.326178	+	0.673960i
$638$	−3.70829			−0.146813
$639$		0			0
$640$	−1.84981	−	3.20397i	−0.0731203	−	0.126648i
$641$	−14.4920	+	25.1008i	−0.572398	+	0.991422i	0.423921	+	0.905699i	$0.360653\pi$
$641$	−14.4920	+	25.1008i	−0.572398	+	0.991422i	−0.996319	+	0.0857228i	$0.972680\pi$
$642$		0			0
$643$	6.03087	−	10.4458i	0.237834	−	0.411941i	−0.722258	−	0.691623i	$-0.756896\pi$
$643$	6.03087	−	10.4458i	0.237834	−	0.411941i	0.960093	+	0.279682i	$0.0902291\pi$
$644$	8.83743	+	14.0951i	0.348244	+	0.555425i
$645$		0			0
$646$	−5.84431			−0.229941
$647$	−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
$653$	18.7040	+	32.3962i	0.731942	+	1.26776i	0.956052	+	0.293198i	$0.0947194\pi$
$653$	18.7040	+	32.3962i	0.731942	+	1.26776i	−0.224109	+	0.974564i	$0.571947\pi$
$654$		0			0
$655$	−0.287992	+	0.498817i	−0.0112528	+	0.0194904i
$656$	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	−0.995265	−	0.0971993i	$-0.969012\pi$
$659$	−14.9356	−	25.8693i	−0.581810	−	1.00772i	0.413455	−	0.910524i	$-0.364322\pi$
$660$		0			0
$661$	5.60803			0.218127			0.109063	−	0.994035i	$-0.465215\pi$
$661$	5.60803			0.218127			0.109063	+	0.994035i	$0.465215\pi$
$662$	−5.56732			−0.216380
$663$		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
$668$	−9.74288	+	16.8752i	−0.376963	+	0.652920i
$669$		0			0
$670$	17.5080	+	30.3247i	0.676392	+	1.17155i
$671$	4.23414	−	7.33375i	0.163457	−	0.283116i
$672$		0			0
$673$	−4.72253	−	8.17966i	−0.182040	−	0.315303i	0.760535	−	0.649297i	$-0.224937\pi$
$673$	−4.72253	−	8.17966i	−0.182040	−	0.315303i	−0.942575	+	0.333994i	$0.891603\pi$
$674$	16.8869	−	29.2489i	0.650458	−	1.12663i
$675$		0			0
$676$	2.85600	+	4.94674i	0.109846	+	0.190259i
$677$	−11.0617			−0.425137			−0.212569	−	0.977146i	$-0.568183\pi$
$677$	−11.0617			−0.425137			−0.212569	+	0.977146i	$0.568183\pi$
$678$		0			0
$679$	−18.5192	−	29.5368i	−0.710701	−	1.13352i
$680$	−12.1643	+	21.0693i	−0.466481	+	0.807970i
$681$		0			0
$682$	5.03018	−	8.71253i	0.192616	−	0.333620i
$683$	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
$686$	−18.4098	+	2.01993i	−0.702889	+	0.0771213i
$687$		0			0
$688$	0.00618986	+	0.0107211i	0.000235986	+	0.000408740i
$689$	−8.66621			−0.330156
$690$		0			0
$691$	25.0617			0.953394			0.476697	−	0.879068i	$-0.341834\pi$
$691$	25.0617			0.953394			0.476697	+	0.879068i	$0.341834\pi$
$692$	−22.5636			−0.857740
$693$		0			0
$694$	−30.4065			−1.15422
$695$	−49.9839			−1.89600
$696$		0			0
$697$	−27.0356			−1.02405
$698$	6.29782	+	10.9082i	0.238376	+	0.412880i
$699$		0			0
$700$	−10.7596	+	20.3103i	−0.406674	+	0.767658i
$701$	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
$704$	−0.738550	+	1.27921i	−0.0278351	+	0.0482119i
$705$		0			0
$706$	3.76578	−	6.52252i	0.141727	−	0.245478i
$707$	6.50805	−	12.2849i	0.244760	−	0.462021i
$708$		0			0
$709$	−22.7403			−0.854031			−0.427016	−	0.904244i	$-0.640435\pi$
$709$	−22.7403			−0.854031			−0.427016	+	0.904244i	$0.640435\pi$
$710$	−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
$716$	−0.166896	−	0.289073i	−0.00623721	−	0.0108032i
$717$		0			0
$718$	−3.44801	+	5.97213i	−0.128679	+	0.222878i
$719$	−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
$724$	23.2422			0.863789
$725$	21.8095			0.809985
$726$		0			0
$727$	23.0908	+	39.9945i	0.856392	+	1.48331i	0.875348	+	0.483494i	$0.160632\pi$
$727$	23.0908	+	39.9945i	0.856392	+	1.48331i	−0.0189562	+	0.999820i	$0.506034\pi$
$728$	3.34362	−	6.31159i	0.123923	−	0.233923i
$729$		0			0
$730$	−22.3189	+	38.6574i	−0.826058	+	1.43077i
$731$	0.0407044	−	0.0705021i	0.00150551	−	0.00260761i
$732$		0			0
$733$	18.0149	+	31.2026i	0.665394	+	1.15250i	0.979178	+	0.203002i	$0.0650696\pi$
$733$	18.0149	+	31.2026i	0.665394	+	1.15250i	−0.313785	+	0.949494i	$0.601597\pi$
$734$	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	$0.507586\pi$
$740$	−10.2756			−0.377739
$741$		0			0
$742$	−4.51169	−	7.19583i	−0.165629	−	0.264167i
$743$	−0.598884	+	1.03730i	−0.0219709	+	0.0380548i	−0.876802	−	0.480852i	$-0.840327\pi$
$743$	−0.598884	+	1.03730i	−0.0219709	+	0.0380548i	0.854831	+	0.518907i	$0.173661\pi$
$744$		0			0
$745$	−0.617454	+	1.06946i	−0.0226218	+	0.0391820i
$746$	14.5822	+	25.2571i	0.533891	+	0.924727i
$747$		0			0
$748$	9.71339			0.355157
$749$	13.3331	−	25.1682i	0.487181	−	0.919626i
$750$		0			0
$751$	−24.0600	−	41.6731i	−0.877961	−	1.52067i	−0.853575	−	0.520970i	$-0.825570\pi$
$751$	−24.0600	−	41.6731i	−0.877961	−	1.52067i	−0.0243853	−	0.999703i	$-0.507763\pi$
$752$	6.98762			0.254812
$753$		0			0
$754$	−6.77747			−0.246821
$755$	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	$0.142554\pi$
$758$	13.5622			0.492602
$759$		0			0
$760$	3.28799			0.119268
$761$	−18.7701	−	32.5108i	−0.680416	−	1.17852i	−0.974854	−	0.222845i	$-0.928466\pi$
$761$	−18.7701	−	32.5108i	−0.680416	−	1.17852i	0.294438	−	0.955671i	$-0.404868\pi$
$762$		0			0
$763$	−0.265781	−	0.423902i	−0.00962191	−	0.0153463i
$764$	16.3214			0.590488
$765$		0			0
$766$	1.41783	+	2.45575i	0.0512281	+	0.0887297i
$767$	9.32691	−	16.1547i	0.336775	−	0.583312i
$768$		0			0
$769$	−13.4592	+	23.3121i	−0.485352	+	0.840654i	−0.999858	−	0.0168324i	$-0.994642\pi$
$769$	−13.4592	+	23.3121i	−0.485352	+	0.840654i	0.514506	+	0.857486i	$0.327975\pi$
$770$	14.4487	−	0.526567i	0.520695	−	0.0189762i
$771$		0			0
$772$	−14.3214			−0.515439
$773$	−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
$779$	1.82691	+	3.16431i	0.0654560	+	0.113373i
$780$		0			0
$781$	−4.03597	+	6.99050i	−0.144418	+	0.250140i
$782$	−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.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(1.84981 + 3.20397i) q^{5} +(2.64400 - 0.0963576i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(1.84981 + 3.20397i) q^{5} +(2.64400 - 0.0963576i) q^{7} -1.00000 q^{8} +(-1.84981 - 3.20397i) q^{10} +(-0.738550 + 1.27921i) q^{11} +(-1.34981 + 2.33795i) q^{13} +(-2.64400 + 0.0963576i) q^{14} +1.00000 q^{16} +(-3.28799 - 5.69497i) q^{17} +(-0.444368 + 0.769668i) q^{19} +(1.84981 + 3.20397i) q^{20} +(0.738550 - 1.27921i) q^{22} +(3.14400 + 5.44556i) q^{23} +(-4.34362 + 7.52338i) q^{25} +(1.34981 - 2.33795i) q^{26} +(2.64400 - 0.0963576i) q^{28} +(-1.25526 - 2.17417i) q^{29} +6.81089 q^{31} -1.00000 q^{32} +(3.28799 + 5.69497i) q^{34} +(5.19963 + 8.29305i) q^{35} +(-1.38874 + 2.40536i) q^{37} +(0.444368 - 0.769668i) q^{38} +(-1.84981 - 3.20397i) q^{40} +(2.05563 - 3.56046i) q^{41} +(0.00618986 + 0.0107211i) q^{43} +(-0.738550 + 1.27921i) q^{44} +(-3.14400 - 5.44556i) q^{46} +6.98762 q^{47} +(6.98143 - 0.509538i) q^{49} +(4.34362 - 7.52338i) q^{50} +(-1.34981 + 2.33795i) q^{52} +(1.60507 + 2.78007i) q^{53} -5.46472 q^{55} +(-2.64400 + 0.0963576i) q^{56} +(1.25526 + 2.17417i) q^{58} -6.90978 q^{59} -5.73305 q^{61} -6.81089 q^{62} +1.00000 q^{64} -9.98762 q^{65} -9.46472 q^{67} +(-3.28799 - 5.69497i) q^{68} +(-5.19963 - 8.29305i) q^{70} +5.46472 q^{71} +(-6.03273 - 10.4490i) q^{73} +(1.38874 - 2.40536i) q^{74} +(-0.444368 + 0.769668i) q^{76} +(-1.82946 + 3.45338i) q^{77} +11.4523 q^{79} +(1.84981 + 3.20397i) q^{80} +(-2.05563 + 3.56046i) q^{82} +(-2.23855 - 3.87728i) q^{83} +(12.1643 - 21.0693i) q^{85} +(-0.00618986 - 0.0107211i) q^{86} +(0.738550 - 1.27921i) q^{88} +(4.43818 - 7.68715i) q^{89} +(-3.34362 + 6.31159i) q^{91} +(3.14400 + 5.44556i) q^{92} -6.98762 q^{94} -3.28799 q^{95} +(-6.58836 - 11.4114i) q^{97} +(-6.98143 + 0.509538i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} + 5 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 + 5 * q^5 + 4 * q^7 - 6 * q^8	\( 6 q - 6 q^{2} + 6 q^{4} + 5 q^{5} + 4 q^{7} - 6 q^{8} - 5 q^{10} + q^{11} - 2 q^{13} - 4 q^{14} + 6 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} - q^{22} + 7 q^{23} - 2 q^{25} + 2 q^{26} + 4 q^{28} + 5 q^{29} + 28 q^{31} - 6 q^{32} - 4 q^{34} + 19 q^{35} - 9 q^{37} + 3 q^{38} - 5 q^{40} + 12 q^{41} + 18 q^{43} + q^{44} - 7 q^{46} + 6 q^{47} - 12 q^{49} + 2 q^{50} - 2 q^{52} - 9 q^{53} + 14 q^{55} - 4 q^{56} - 5 q^{58} + 8 q^{59} - 8 q^{61} - 28 q^{62} + 6 q^{64} - 24 q^{65} - 10 q^{67} + 4 q^{68} - 19 q^{70} - 14 q^{71} - 25 q^{73} + 9 q^{74} - 3 q^{76} - 52 q^{77} - 14 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} + 14 q^{85} - 18 q^{86} - q^{88} + 9 q^{89} + 4 q^{91} + 7 q^{92} - 6 q^{94} + 4 q^{95} - 28 q^{97} + 12 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 + 5 * q^5 + 4 * q^7 - 6 * q^8 - 5 * q^10 + q^11 - 2 * q^13 - 4 * q^14 + 6 * q^16 + 4 * q^17 - 3 * q^19 + 5 * q^20 - q^22 + 7 * q^23 - 2 * q^25 + 2 * q^26 + 4 * q^28 + 5 * q^29 + 28 * q^31 - 6 * q^32 - 4 * q^34 + 19 * q^35 - 9 * q^37 + 3 * q^38 - 5 * q^40 + 12 * q^41 + 18 * q^43 + q^44 - 7 * q^46 + 6 * q^47 - 12 * q^49 + 2 * q^50 - 2 * q^52 - 9 * q^53 + 14 * q^55 - 4 * q^56 - 5 * q^58 + 8 * q^59 - 8 * q^61 - 28 * q^62 + 6 * q^64 - 24 * q^65 - 10 * q^67 + 4 * q^68 - 19 * q^70 - 14 * q^71 - 25 * q^73 + 9 * q^74 - 3 * q^76 - 52 * q^77 - 14 * q^79 + 5 * q^80 - 12 * q^82 - 8 * q^83 + 14 * q^85 - 18 * q^86 - q^88 + 9 * q^89 + 4 * q^91 + 7 * q^92 - 6 * q^94 + 4 * q^95 - 28 * q^97 + 12 * q^98

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