LMFDB - Embedded newform 900.3.c.r.451.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(451,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("900.451");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.c (of order $2$, degree $1$, 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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6080256576.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 7x^{6} - 12x^{5} + 12x^{4} - 48x^{3} + 112x^{2} - 192x + 256 $ x^8 - 3x^7 + 7x^6 - 12x^5 + 12x^4 - 48x^3 + 112x^2 - 192*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			451.2
Root			$-1.51328 + 1.30766i$ of defining polynomial
Character	$\chi$	$=$	900.451
Dual form			900.3.c.r.451.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.88911 + 0.656712i) q^{2} +(3.13746 - 2.48120i) q^{4} -9.55505i q^{7} +(-4.29756 + 6.74766i) q^{8} +O(q^{10})$	\(q+(-1.88911 + 0.656712i) q^{2} +(3.13746 - 2.48120i) q^{4} -9.55505i q^{7} +(-4.29756 + 6.74766i) q^{8} +9.92480i q^{11} +7.55643 q^{13} +(6.27492 + 18.0505i) q^{14} +(3.68729 - 15.5693i) q^{16} -17.1903 q^{17} +26.1762i q^{19} +(-6.51774 - 18.7490i) q^{22} +1.67451i q^{23} +(-14.2749 + 4.96240i) q^{26} +(-23.7080 - 29.9786i) q^{28} -0.350497 q^{29} +46.0258i q^{31} +(3.25887 + 31.8336i) q^{32} +(32.4743 - 11.2890i) q^{34} +22.6693 q^{37} +(-17.1903 - 49.4498i) q^{38} +77.2990 q^{41} -41.7994i q^{43} +(24.6254 + 31.1386i) q^{44} +(-1.09967 - 3.16332i) q^{46} +14.0866i q^{47} -42.2990 q^{49} +(23.7080 - 18.7490i) q^{52} +22.6693 q^{53} +(64.4743 + 41.0634i) q^{56} +(0.662126 - 0.230175i) q^{58} -94.7802i q^{59} +38.0000 q^{61} +(-30.2257 - 86.9478i) q^{62} +(-27.0619 - 57.9970i) q^{64} +29.8477i q^{67} +(-53.9337 + 42.6525i) q^{68} +7.19630i q^{71} -34.3805 q^{73} +(-42.8248 + 14.8872i) q^{74} +(64.9485 + 82.1269i) q^{76} +94.8320 q^{77} +46.0258i q^{79} +(-146.026 + 50.7632i) q^{82} -24.1336i q^{83} +(27.4502 + 78.9636i) q^{86} +(-66.9692 - 42.6525i) q^{88} +100.199 q^{89} -72.2021i q^{91} +(4.15479 + 5.25370i) q^{92} +(-9.25083 - 26.6111i) q^{94} +131.861 q^{97} +(79.9074 - 27.7783i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{4}+O(q^{10}) $ 8 * q + 10 * q^4	$ 8 q + 10 q^{4} + 20 q^{14} - 46 q^{16} - 84 q^{26} - 184 q^{29} - 12 q^{34} + 256 q^{41} + 348 q^{44} + 112 q^{46} + 24 q^{49} + 244 q^{56} + 304 q^{61} + 10 q^{64} - 252 q^{74} - 24 q^{76} + 280 q^{86} + 560 q^{89} - 376 q^{94}+O(q^{100}) $ 8 * q + 10 * q^4 + 20 * q^14 - 46 * q^16 - 84 * q^26 - 184 * q^29 - 12 * q^34 + 256 * q^41 + 348 * q^44 + 112 * q^46 + 24 * q^49 + 244 * q^56 + 304 * q^61 + 10 * q^64 - 252 * q^74 - 24 * q^76 + 280 * q^86 + 560 * q^89 - 376 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	−1.88911	+	0.656712i	−0.944554	+	0.328356i
$2$	−1.88911	+	0.656712i	−0.944554	+	0.328356i
$3$		0			0
$3$		0			0
$4$	3.13746	−	2.48120i	0.784365	−	0.620300i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	9.55505i		−	1.36501i	−0.730882	−	0.682504i	$-0.760891\pi$
$7$		−	9.55505i		−	1.36501i	0.730882	−	0.682504i	$-0.239109\pi$
$8$	−4.29756	+	6.74766i	−0.537196	+	0.843458i
$9$		0			0
$10$		0			0
$11$			9.92480i			0.902255i	0.892460	+	0.451127i	$0.148978\pi$
$11$			9.92480i			0.902255i	−0.892460	+	0.451127i	$0.851022\pi$
$12$		0			0
$13$	7.55643			0.581264			0.290632	−	0.956835i	$-0.406134\pi$
$13$	7.55643			0.581264			0.290632	+	0.956835i	$0.406134\pi$
$14$	6.27492	+	18.0505i	0.448208	+	1.28932i
$15$		0			0
$16$	3.68729	−	15.5693i	0.230456	−	0.973083i
$17$	−17.1903			−1.01119			−0.505596	−	0.862770i	$-0.668727\pi$
$17$	−17.1903			−1.01119			−0.505596	+	0.862770i	$0.668727\pi$
$18$		0			0
$19$			26.1762i			1.37770i	0.724905	+	0.688849i	$0.241884\pi$
$19$			26.1762i			1.37770i	−0.724905	+	0.688849i	$0.758116\pi$
$20$		0			0
$21$		0			0
$22$	−6.51774	−	18.7490i	−0.296261	−	0.852228i
$23$			1.67451i			0.0728047i	0.999337	+	0.0364023i	$0.0115898\pi$
$23$			1.67451i			0.0728047i	−0.999337	+	0.0364023i	$0.988410\pi$
$24$		0			0
$25$		0			0
$26$	−14.2749	+	4.96240i	−0.549035	+	0.190862i
$27$		0			0
$28$	−23.7080	−	29.9786i	−0.846714	−	1.07066i
$29$	−0.350497			−0.0120861			−0.00604305	−	0.999982i	$-0.501924\pi$
$29$	−0.350497			−0.0120861			−0.00604305	+	0.999982i	$0.501924\pi$
$30$		0			0
$31$			46.0258i			1.48470i	0.670010	+	0.742352i	$0.266290\pi$
$31$			46.0258i			1.48470i	−0.670010	+	0.742352i	$0.733710\pi$
$32$	3.25887	+	31.8336i	0.101840	+	0.994801i
$33$		0			0
$34$	32.4743	−	11.2890i	0.955125	−	0.332031i
$35$		0			0
$36$		0			0
$37$	22.6693			0.612684			0.306342	−	0.951922i	$-0.400895\pi$
$37$	22.6693			0.612684			0.306342	+	0.951922i	$0.400895\pi$
$38$	−17.1903	−	49.4498i	−0.452375	−	1.30131i
$39$		0			0
$40$		0			0
$41$	77.2990			1.88534			0.942671	−	0.333724i	$-0.108305\pi$
$41$	77.2990			1.88534			0.942671	+	0.333724i	$0.108305\pi$
$42$		0			0
$43$		−	41.7994i		−	0.972079i	−0.873937	−	0.486039i	$-0.838441\pi$
$43$		−	41.7994i		−	0.972079i	0.873937	−	0.486039i	$-0.161559\pi$
$44$	24.6254	+	31.1386i	0.559668	+	0.707697i
$45$		0			0
$46$	−1.09967	−	3.16332i	−0.0239058	−	0.0687679i
$47$			14.0866i			0.299715i	0.988708	+	0.149857i	$0.0478814\pi$
$47$			14.0866i			0.299715i	−0.988708	+	0.149857i	$0.952119\pi$
$48$		0			0
$49$	−42.2990			−0.863245
$50$		0			0
$51$		0			0
$52$	23.7080	−	18.7490i	0.455923	−	0.360558i
$53$	22.6693			0.427723			0.213861	−	0.976864i	$-0.431396\pi$
$53$	22.6693			0.427723			0.213861	+	0.976864i	$0.431396\pi$
$54$		0			0
$55$		0			0
$56$	64.4743	+	41.0634i	1.15133	+	0.733276i
$57$		0			0
$58$	0.662126	−	0.230175i	0.0114160	−	0.00396854i
$59$		−	94.7802i		−	1.60644i	−0.595680	−	0.803222i	$-0.703117\pi$
$59$		−	94.7802i		−	1.60644i	0.595680	−	0.803222i	$-0.296883\pi$
$60$		0			0
$61$	38.0000			0.622951			0.311475	−	0.950254i	$-0.399177\pi$
$61$	38.0000			0.622951			0.311475	+	0.950254i	$0.399177\pi$
$62$	−30.2257	−	86.9478i	−0.487512	−	1.40238i
$63$		0			0
$64$	−27.0619	−	57.9970i	−0.422842	−	0.906203i
$65$		0			0
$66$		0			0
$67$			29.8477i			0.445488i	0.974877	+	0.222744i	$0.0715013\pi$
$67$			29.8477i			0.445488i	−0.974877	+	0.222744i	$0.928499\pi$
$68$	−53.9337	+	42.6525i	−0.793143	+	0.627242i
$69$		0			0
$70$		0			0
$71$			7.19630i			0.101356i	0.998715	+	0.0506782i	$0.0161383\pi$
$71$			7.19630i			0.101356i	−0.998715	+	0.0506782i	$0.983862\pi$
$72$		0			0
$73$	−34.3805			−0.470966			−0.235483	−	0.971878i	$-0.575667\pi$
$73$	−34.3805			−0.470966			−0.235483	+	0.971878i	$0.575667\pi$
$74$	−42.8248	+	14.8872i	−0.578713	+	0.201178i
$75$		0			0
$76$	64.9485	+	82.1269i	0.854586	+	1.08062i
$77$	94.8320			1.23158
$78$		0			0
$79$			46.0258i			0.582606i	0.956631	+	0.291303i	$0.0940887\pi$
$79$			46.0258i			0.582606i	−0.956631	+	0.291303i	$0.905911\pi$
$80$		0			0
$81$		0			0
$82$	−146.026	+	50.7632i	−1.78081	+	0.619063i
$83$		−	24.1336i		−	0.290767i	−0.989375	−	0.145383i	$-0.953558\pi$
$83$		−	24.1336i		−	0.290767i	0.989375	−	0.145383i	$-0.0464415\pi$
$84$		0			0
$85$		0			0
$86$	27.4502	+	78.9636i	0.319188	+	0.918181i
$87$		0			0
$88$	−66.9692	−	42.6525i	−0.761014	−	0.484687i
$89$	100.199			1.12584			0.562918	−	0.826513i	$-0.309679\pi$
$89$	100.199			1.12584			0.562918	+	0.826513i	$0.309679\pi$
$90$		0			0
$91$		−	72.2021i		−	0.793430i
$92$	4.15479	+	5.25370i	0.0451607	+	0.0571054i
$93$		0			0
$94$	−9.25083	−	26.6111i	−0.0984131	−	0.283097i
$95$		0			0
$96$		0			0
$97$	131.861			1.35939			0.679696	−	0.733494i	$-0.262112\pi$
$97$	131.861			1.35939			0.679696	+	0.733494i	$0.262112\pi$
$98$	79.9074	−	27.7783i	0.815382	−	0.283452i
$99$		0			0
$100$		0			0
$101$	−29.4502			−0.291586			−0.145793	−	0.989315i	$-0.546573\pi$
$101$	−29.4502			−0.291586			−0.145793	+	0.989315i	$0.546573\pi$
$102$		0			0
$103$			143.786i			1.39598i	0.716107	+	0.697991i	$0.245922\pi$
$103$			143.786i			1.39598i	−0.716107	+	0.697991i	$0.754078\pi$
$104$	−32.4743	+	50.9882i	−0.312252	+	0.490272i
$105$		0			0
$106$	−42.8248	+	14.8872i	−0.404007	+	0.140445i
$107$		−	35.1014i		−	0.328050i	−0.986456	−	0.164025i	$-0.947552\pi$
$107$		−	35.1014i		−	0.328050i	0.986456	−	0.164025i	$-0.0524478\pi$
$108$		0			0
$109$	151.498			1.38989			0.694947	−	0.719061i	$-0.255428\pi$
$109$	151.498			1.38989			0.694947	+	0.719061i	$0.255428\pi$
$110$		0			0
$111$		0			0
$112$	−148.766	−	35.2323i	−1.32827	−	0.314574i
$113$	−32.3031			−0.285868			−0.142934	−	0.989732i	$-0.545654\pi$
$113$	−32.3031			−0.285868			−0.142934	+	0.989732i	$0.545654\pi$
$114$		0			0
$115$		0			0
$116$	−1.09967	+	0.869652i	−0.00947990	+	0.00749700i
$117$		0			0
$118$	62.2433	+	179.050i	0.527486	+	1.51737i
$119$			164.254i			1.38028i
$120$		0			0
$121$	22.4983			0.185937
$122$	−71.7861	+	24.9551i	−0.588411	+	0.204550i
$123$		0			0
$124$	114.199	+	144.404i	0.920962	+	1.16455i
$125$		0			0
$126$		0			0
$127$			192.053i			1.51223i	0.654438	+	0.756116i	$0.272905\pi$
$127$			192.053i			1.51223i	−0.654438	+	0.756116i	$0.727095\pi$
$128$	89.2102	+	91.7908i	0.696954	+	0.717115i
$129$		0			0
$130$		0			0
$131$			42.4277i			0.323876i	0.986801	+	0.161938i	$0.0517744\pi$
$131$			42.4277i			0.323876i	−0.986801	+	0.161938i	$0.948226\pi$
$132$		0			0
$133$	250.115			1.88057
$134$	−19.6013	−	56.3855i	−0.146279	−	0.420787i
$135$		0			0
$136$	73.8762	−	115.994i	0.543208	−	0.852897i
$137$	206.854			1.50988			0.754942	−	0.655791i	$-0.227665\pi$
$137$	206.854			1.50988			0.754942	+	0.655791i	$0.227665\pi$
$138$		0			0
$139$		−	46.0258i		−	0.331121i	−0.986200	−	0.165561i	$-0.947057\pi$
$139$		−	46.0258i		−	0.331121i	0.986200	−	0.165561i	$-0.0529433\pi$
$140$		0			0
$141$		0			0
$142$	−4.72590	−	13.5946i	−0.0332810	−	0.0957366i
$143$			74.9961i			0.524448i
$144$		0			0
$145$		0			0
$146$	64.9485	−	22.5781i	0.444853	−	0.154645i
$147$		0			0
$148$	71.1240	−	56.2471i	0.480567	−	0.380048i
$149$	11.6495			0.0781846			0.0390923	−	0.999236i	$-0.487553\pi$
$149$	11.6495			0.0781846			0.0390923	+	0.999236i	$0.487553\pi$
$150$		0			0
$151$			125.424i			0.830624i	0.909679	+	0.415312i	$0.136328\pi$
$151$			125.424i			0.830624i	−0.909679	+	0.415312i	$0.863672\pi$
$152$	−176.628	−	112.494i	−1.16203	−	0.740093i
$153$		0			0
$154$	−179.148	+	62.2773i	−1.16330	+	0.404398i
$155$		0			0
$156$		0			0
$157$	197.220			1.25618			0.628090	−	0.778140i	$-0.283837\pi$
$157$	197.220			1.25618			0.628090	+	0.778140i	$0.283837\pi$
$158$	−30.2257	−	86.9478i	−0.191302	−	0.550303i
$159$		0			0
$160$		0			0
$161$	16.0000			0.0993789
$162$		0			0
$163$		−	18.4196i		−	0.113004i	−0.998402	−	0.0565018i	$-0.982005\pi$
$163$		−	18.4196i		−	0.113004i	0.998402	−	0.0565018i	$-0.0179947\pi$
$164$	242.522	−	191.794i	1.47880	−	1.16948i
$165$		0			0
$166$	15.8488	+	45.5910i	0.0954749	+	0.274645i
$167$			92.8920i			0.556240i	0.960546	+	0.278120i	$0.0897112\pi$
$167$			92.8920i			0.556240i	−0.960546	+	0.278120i	$0.910289\pi$
$168$		0			0
$169$	−111.900			−0.662132
$170$		0			0
$171$		0			0
$172$	−103.713	−	131.144i	−0.602981	−	0.762464i
$173$	117.501			0.679198			0.339599	−	0.940570i	$-0.389709\pi$
$173$	117.501			0.679198			0.339599	+	0.940570i	$0.389709\pi$
$174$		0			0
$175$		0			0
$176$	154.522	+	36.5956i	0.877968	+	0.207930i
$177$		0			0
$178$	−189.287	+	65.8021i	−1.06341	+	0.369675i
$179$			231.988i			1.29602i	0.761631	+	0.648011i	$0.224399\pi$
$179$			231.988i			1.29602i	−0.761631	+	0.648011i	$0.775601\pi$
$180$		0			0
$181$	−218.096			−1.20495			−0.602476	−	0.798137i	$-0.705819\pi$
$181$	−218.096			−1.20495			−0.602476	+	0.798137i	$0.705819\pi$
$182$	47.4160	+	136.398i	0.260527	+	0.749437i
$183$		0			0
$184$	−11.2990	−	7.19630i	−0.0614076	−	0.0391103i
$185$		0			0
$186$		0			0
$187$		−	170.610i		−	0.912352i
$188$	34.9516	+	44.1961i	0.185913	+	0.235085i
$189$		0			0
$190$		0			0
$191$			137.208i			0.718366i	0.933267	+	0.359183i	$0.116945\pi$
$191$			137.208i			0.718366i	−0.933267	+	0.359183i	$0.883055\pi$
$192$		0			0
$193$	−37.0290			−0.191860			−0.0959301	−	0.995388i	$-0.530583\pi$
$193$	−37.0290			−0.191860			−0.0959301	+	0.995388i	$0.530583\pi$
$194$	−249.100	+	86.5947i	−1.28402	+	0.446364i
$195$		0			0
$196$	−132.711	+	104.952i	−0.677099	+	0.535471i
$197$	−194.572			−0.987674			−0.493837	−	0.869554i	$-0.664406\pi$
$197$	−194.572			−0.987674			−0.493837	+	0.869554i	$0.664406\pi$
$198$		0			0
$199$		−	176.037i		−	0.884610i	−0.896865	−	0.442305i	$-0.854161\pi$
$199$		−	176.037i		−	0.884610i	0.896865	−	0.442305i	$-0.145839\pi$
$200$		0			0
$201$		0			0
$202$	55.6345	−	19.3403i	0.275419	−	0.0957440i
$203$			3.34901i			0.0164976i
$204$		0			0
$205$		0			0
$206$	−94.4261	−	271.628i	−0.458379	−	1.31858i
$207$		0			0
$208$	27.8628	−	117.649i	0.133956	−	0.565618i
$209$	−259.794			−1.24303
$210$		0			0
$211$		−	20.7193i		−	0.0981955i	−0.998794	−	0.0490978i	$-0.984365\pi$
$211$		−	20.7193i		−	0.0981955i	0.998794	−	0.0490978i	$-0.0156346\pi$
$212$	71.1240	−	56.2471i	0.335490	−	0.265316i
$213$		0			0
$214$	23.0515	+	66.3103i	0.107717	+	0.309861i
$215$		0			0
$216$		0			0
$217$	439.779			2.02663
$218$	−286.197	+	99.4908i	−1.31283	+	0.456380i
$219$		0			0
$220$		0			0
$221$	−129.897			−0.587769
$222$		0			0
$223$			97.0265i			0.435096i	0.976050	+	0.217548i	$0.0698059\pi$
$223$			97.0265i			0.435096i	−0.976050	+	0.217548i	$0.930194\pi$
$224$	304.172	−	31.1386i	1.35791	−	0.139012i
$225$		0			0
$226$	61.0241	−	21.2138i	0.270018	−	0.0938666i
$227$			407.256i			1.79408i	0.441948	+	0.897040i	$0.354287\pi$
$227$			407.256i			1.79408i	−0.441948	+	0.897040i	$0.645713\pi$
$228$		0			0
$229$	7.89702			0.0344848			0.0172424	−	0.999851i	$-0.494511\pi$
$229$	7.89702			0.0344848			0.0172424	+	0.999851i	$0.494511\pi$
$230$		0			0
$231$		0			0
$232$	1.50628	−	2.36503i	0.00649260	−	0.0101941i
$233$	−28.1483			−0.120808			−0.0604042	−	0.998174i	$-0.519239\pi$
$233$	−28.1483			−0.120808			−0.0604042	+	0.998174i	$0.519239\pi$
$234$		0			0
$235$		0			0
$236$	−235.169	−	297.369i	−0.996477	−	1.26004i
$237$		0			0
$238$	−107.867	−	310.293i	−0.453225	−	1.30375i
$239$		−	296.005i		−	1.23851i	−0.785189	−	0.619257i	$-0.787434\pi$
$239$		−	296.005i		−	1.23851i	0.785189	−	0.619257i	$-0.212566\pi$
$240$		0			0
$241$	465.794			1.93276			0.966378	−	0.257127i	$-0.0827758\pi$
$241$	465.794			1.93276			0.966378	+	0.257127i	$0.0827758\pi$
$242$	−42.5018	+	14.7749i	−0.175627	+	0.0610534i
$243$		0			0
$244$	119.223	−	94.2856i	0.488621	−	0.386416i
$245$		0			0
$246$		0			0
$247$			197.799i			0.800806i
$248$	−310.567	−	197.799i	−1.25229	−	0.797577i
$249$		0			0
$250$		0			0
$251$		−	141.676i		−	0.564445i	−0.959349	−	0.282223i	$-0.908928\pi$
$251$		−	141.676i		−	0.564445i	0.959349	−	0.282223i	$-0.0910716\pi$
$252$		0			0
$253$	−16.6191			−0.0656883
$254$	−126.124	−	362.810i	−0.496550	−	1.42838i
$255$		0			0
$256$	−228.808	−	114.817i	−0.893780	−	0.448505i
$257$	41.7549			0.162470			0.0812352	−	0.996695i	$-0.474113\pi$
$257$	41.7549			0.162470			0.0812352	+	0.996695i	$0.474113\pi$
$258$		0			0
$259$		−	216.606i		−	0.836318i
$260$		0			0
$261$		0			0
$262$	−27.8628	−	80.1505i	−0.106346	−	0.305918i
$263$			203.283i			0.772939i	0.922302	+	0.386469i	$0.126306\pi$
$263$			203.283i			0.772939i	−0.922302	+	0.386469i	$0.873694\pi$
$264$		0			0
$265$		0			0
$266$	−472.495	+	164.254i	−1.77630	+	0.617495i
$267$		0			0
$268$	74.0580	+	93.6458i	0.276336	+	0.349425i
$269$	−244.048			−0.907242			−0.453621	−	0.891195i	$-0.649868\pi$
$269$	−244.048			−0.907242			−0.453621	+	0.891195i	$0.649868\pi$
$270$		0			0
$271$		−	466.585i		−	1.72172i	−0.508845	−	0.860858i	$-0.669927\pi$
$271$		−	466.585i		−	1.72172i	0.508845	−	0.860858i	$-0.330073\pi$
$272$	−63.3855	+	267.641i	−0.233035	+	0.983973i
$273$		0			0
$274$	−390.770	+	135.844i	−1.42617	+	0.495780i
$275$		0			0
$276$		0			0
$277$	−494.181			−1.78405			−0.892023	−	0.451990i	$-0.850714\pi$
$277$	−494.181			−1.78405			−0.892023	+	0.451990i	$0.850714\pi$
$278$	30.2257	+	86.9478i	0.108726	+	0.312762i
$279$		0			0
$280$		0			0
$281$	43.4020			0.154455			0.0772277	−	0.997013i	$-0.475393\pi$
$281$	43.4020			0.154455			0.0772277	+	0.997013i	$0.475393\pi$
$282$		0			0
$283$		−	310.785i		−	1.09818i	−0.835763	−	0.549090i	$-0.814974\pi$
$283$		−	310.785i		−	1.09818i	0.835763	−	0.549090i	$-0.185026\pi$
$284$	17.8555	+	22.5781i	0.0628714	+	0.0795003i
$285$		0			0
$286$	−49.2508	−	141.676i	−0.172206	−	0.495370i
$287$		−	738.596i		−	2.57351i
$288$		0			0
$289$	6.50497			0.0225085
$290$		0			0
$291$		0			0
$292$	−107.867	+	85.3049i	−0.369409	+	0.292140i
$293$	245.207			0.836886			0.418443	−	0.908243i	$-0.362576\pi$
$293$	245.207			0.836886			0.418443	+	0.908243i	$0.362576\pi$
$294$		0			0
$295$		0			0
$296$	−97.4228	+	152.965i	−0.329131	+	0.516773i
$297$		0			0
$298$	−22.0072	+	7.65037i	−0.0738496	+	0.0256724i
$299$			12.6533i			0.0423187i
$300$		0			0
$301$	−399.395			−1.32689
$302$	−82.3676	−	236.940i	−0.272740	−	0.784569i
$303$		0			0
$304$	407.547	+	96.5195i	1.34061	+	0.317498i
$305$		0			0
$306$		0			0
$307$		−	337.514i		−	1.09939i	−0.835364	−	0.549697i	$-0.814743\pi$
$307$		−	337.514i		−	1.09939i	0.835364	−	0.549697i	$-0.185257\pi$
$308$	297.531	−	235.297i	0.966011	−	0.763952i
$309$		0			0
$310$		0			0
$311$		−	427.756i		−	1.37542i	−0.725986	−	0.687710i	$-0.758616\pi$
$311$		−	427.756i		−	1.37542i	0.725986	−	0.687710i	$-0.241384\pi$
$312$		0			0
$313$	83.8739			0.267968			0.133984	−	0.990984i	$-0.457223\pi$
$313$	83.8739			0.267968			0.133984	+	0.990984i	$0.457223\pi$
$314$	−372.571	+	129.517i	−1.18653	+	0.412475i
$315$		0			0
$316$	114.199	+	144.404i	0.361390	+	0.456975i
$317$	112.204			0.353957			0.176978	−	0.984215i	$-0.443368\pi$
$317$	112.204			0.353957			0.176978	+	0.984215i	$0.443368\pi$
$318$		0			0
$319$		−	3.47861i		−	0.0109047i
$320$		0			0
$321$		0			0
$322$	−30.2257	+	10.5074i	−0.0938687	+	0.0326317i
$323$		−	449.976i		−	1.39312i
$324$		0			0
$325$		0			0
$326$	12.0964	+	34.7966i	0.0371054	+	0.106738i
$327$		0			0
$328$	−332.197	+	521.588i	−1.01280	+	1.59021i
$329$	134.598			0.409113
$330$		0			0
$331$			132.621i			0.400666i	0.979728	+	0.200333i	$0.0642025\pi$
$331$			132.621i			0.400666i	−0.979728	+	0.200333i	$0.935798\pi$
$332$	−59.8803	−	75.7182i	−0.180362	−	0.228067i
$333$		0			0
$334$	−61.0033	−	175.483i	−0.182645	−	0.525398i
$335$		0			0
$336$		0			0
$337$	−20.7739			−0.0616437			−0.0308219	−	0.999525i	$-0.509812\pi$
$337$	−20.7739			−0.0616437			−0.0308219	+	0.999525i	$0.509812\pi$
$338$	211.392	−	73.4863i	0.625420	−	0.217415i
$339$		0			0
$340$		0			0
$341$	−456.797			−1.33958
$342$		0			0
$343$		−	64.0283i		−	0.186672i
$344$	282.048	+	179.636i	0.819907	+	0.522196i
$345$		0			0
$346$	−221.973	+	77.1645i	−0.641539	+	0.223019i
$347$			8.89616i			0.0256374i	0.999918	+	0.0128187i	$0.00408042\pi$
$347$			8.89616i			0.0256374i	−0.999918	+	0.0128187i	$0.995920\pi$
$348$		0			0
$349$	−19.4020			−0.0555931			−0.0277965	−	0.999614i	$-0.508849\pi$
$349$	−19.4020			−0.0555931			−0.0277965	+	0.999614i	$0.508849\pi$
$350$		0			0
$351$		0			0
$352$	−315.942	+	32.3436i	−0.897564	+	0.0918853i
$353$	−80.2902			−0.227451			−0.113726	−	0.993512i	$-0.536278\pi$
$353$	−80.2902			−0.227451			−0.113726	+	0.993512i	$0.536278\pi$
$354$		0			0
$355$		0			0
$356$	314.371	−	248.615i	0.883065	−	0.698356i
$357$		0			0
$358$	−152.349	−	438.251i	−0.425557	−	1.22416i
$359$		−	314.115i		−	0.874972i	−0.899225	−	0.437486i	$-0.855869\pi$
$359$		−	314.115i		−	0.874972i	0.899225	−	0.437486i	$-0.144131\pi$
$360$		0			0
$361$	−324.196			−0.898050
$362$	412.008	−	143.227i	1.13814	−	0.395653i
$363$		0			0
$364$	−179.148	−	226.531i	−0.492164	−	0.622338i
$365$		0			0
$366$		0			0
$367$			476.800i			1.29918i	0.760283	+	0.649592i	$0.225060\pi$
$367$			476.800i			1.29918i	−0.760283	+	0.649592i	$0.774940\pi$
$368$	26.0709	+	6.17440i	0.0708450	+	0.0167783i
$369$		0			0
$370$		0			0
$371$		−	216.606i		−	0.583844i
$372$		0			0
$373$	−86.1333			−0.230920			−0.115460	−	0.993312i	$-0.536834\pi$
$373$	−86.1333			−0.230920			−0.115460	+	0.993312i	$0.536834\pi$
$374$	112.042	+	322.300i	0.299576	+	0.861766i
$375$		0			0
$376$	−95.0515	−	60.5380i	−0.252797	−	0.161005i
$377$	−2.64850			−0.00702521
$378$		0			0
$379$			638.035i			1.68347i	0.539891	+	0.841735i	$0.318466\pi$
$379$			638.035i			1.68347i	−0.539891	+	0.841735i	$0.681534\pi$
$380$		0			0
$381$		0			0
$382$	−90.1061	−	259.201i	−0.235880	−	0.678535i
$383$		−	216.742i		−	0.565907i	−0.959134	−	0.282953i	$-0.908686\pi$
$383$		−	216.742i		−	0.565907i	0.959134	−	0.282953i	$-0.0913142\pi$
$384$		0			0
$385$		0			0
$386$	69.9518	−	24.3174i	0.181222	−	0.0629985i
$387$		0			0
$388$	413.708	−	327.174i	1.06626	−	0.843231i
$389$	476.640			1.22529			0.612647	−	0.790356i	$-0.290105\pi$
$389$	476.640			1.22529			0.612647	+	0.790356i	$0.290105\pi$
$390$		0			0
$391$		−	28.7852i		−	0.0736195i
$392$	181.783	−	285.419i	0.463731	−	0.728111i
$393$		0			0
$394$	367.567	−	127.778i	0.932912	−	0.324309i
$395$		0			0
$396$		0			0
$397$	−43.0792			−0.108512			−0.0542559	−	0.998527i	$-0.517279\pi$
$397$	−43.0792			−0.108512			−0.0542559	+	0.998527i	$0.517279\pi$
$398$	115.606	+	332.554i	0.290467	+	0.835562i
$399$		0			0
$400$		0			0
$401$	168.694			0.420684			0.210342	−	0.977628i	$-0.432542\pi$
$401$	168.694			0.420684			0.210342	+	0.977628i	$0.432542\pi$
$402$		0			0
$403$			347.791i			0.863005i
$404$	−92.3987	+	73.0718i	−0.228710	+	0.180871i
$405$		0			0
$406$	−2.19934	−	6.32665i	−0.00541709	−	0.0155829i
$407$			224.988i			0.552797i
$408$		0			0
$409$	−373.890			−0.914157			−0.457079	−	0.889426i	$-0.651104\pi$
$409$	−373.890			−0.914157			−0.457079	+	0.889426i	$0.651104\pi$
$410$		0			0
$411$		0			0
$412$	356.762	+	451.123i	0.865927	+	1.09496i
$413$	−905.630			−2.19281
$414$		0			0
$415$		0			0
$416$	24.6254	+	240.549i	0.0591957	+	0.578242i
$417$		0			0
$418$	490.779	−	170.610i	1.17411	−	0.408158i
$419$		−	87.5839i		−	0.209031i	−0.994523	−	0.104515i	$-0.966671\pi$
$419$		−	87.5839i		−	0.209031i	0.994523	−	0.104515i	$-0.0333291\pi$
$420$		0			0
$421$	70.3023			0.166989			0.0834944	−	0.996508i	$-0.473392\pi$
$421$	70.3023			0.166989			0.0834944	+	0.996508i	$0.473392\pi$
$422$	13.6066	+	39.1409i	0.0322431	+	0.0927510i
$423$		0			0
$424$	−97.4228	+	152.965i	−0.229771	+	0.360766i
$425$		0			0
$426$		0			0
$427$		−	363.092i		−	0.850332i
$428$	−87.0935	−	110.129i	−0.203490	−	0.257311i
$429$		0			0
$430$		0			0
$431$			247.370i			0.573944i	0.957939	+	0.286972i	$0.0926487\pi$
$431$			247.370i			0.573944i	−0.957939	+	0.286972i	$0.907351\pi$
$432$		0			0
$433$	−636.247			−1.46939			−0.734696	−	0.678397i	$-0.762675\pi$
$433$	−636.247			−1.46939			−0.734696	+	0.678397i	$0.762675\pi$
$434$	−830.791	+	288.808i	−1.91426	+	0.665457i
$435$		0			0
$436$	475.320	−	375.898i	1.09018	−	0.862151i
$437$	−43.8323			−0.100303
$438$		0			0
$439$			769.786i			1.75350i	0.480947	+	0.876750i	$0.340293\pi$
$439$			769.786i			1.75350i	−0.480947	+	0.876750i	$0.659707\pi$
$440$		0			0
$441$		0			0
$442$	245.390	−	85.3049i	0.555180	−	0.192998i
$443$			612.214i			1.38197i	0.722868	+	0.690986i	$0.242823\pi$
$443$			612.214i			1.38197i	−0.722868	+	0.690986i	$0.757177\pi$
$444$		0			0
$445$		0			0
$446$	−63.7185	−	183.294i	−0.142867	−	0.410972i
$447$		0			0
$448$	−554.165	+	258.578i	−1.23697	+	0.577182i
$449$	175.897			0.391753			0.195876	−	0.980629i	$-0.437245\pi$
$449$	175.897			0.391753			0.195876	+	0.980629i	$0.437245\pi$
$450$		0			0
$451$			767.177i			1.70106i
$452$	−101.350	+	80.1505i	−0.224225	+	0.177324i
$453$		0			0
$454$	−267.450	−	769.351i	−0.589097	−	1.69461i
$455$		0			0
$456$		0			0
$457$	−365.357			−0.799469			−0.399734	−	0.916631i	$-0.630898\pi$
$457$	−365.357			−0.799469			−0.399734	+	0.916631i	$0.630898\pi$
$458$	−14.9183	+	5.18607i	−0.0325728	+	0.0113233i
$459$		0			0
$460$		0			0
$461$	−308.350			−0.668873			−0.334437	−	0.942418i	$-0.608546\pi$
$461$	−308.350			−0.668873			−0.334437	+	0.942418i	$0.608546\pi$
$462$		0			0
$463$		−	92.6302i		−	0.200065i	−0.994984	−	0.100033i	$-0.968105\pi$
$463$		−	92.6302i		−	0.200065i	0.994984	−	0.100033i	$-0.0318947\pi$
$464$	−1.29238	+	5.45700i	−0.00278531	+	0.0117608i
$465$		0			0
$466$	53.1752	−	18.4854i	0.114110	−	0.0396681i
$467$			606.103i			1.29786i	0.760846	+	0.648932i	$0.224784\pi$
$467$			606.103i			1.29786i	−0.760846	+	0.648932i	$0.775216\pi$
$468$		0			0
$469$	285.196			0.608094
$470$		0			0
$471$		0			0
$472$	639.545	+	407.324i	1.35497	+	0.862975i
$473$	414.851			0.877063
$474$		0			0
$475$		0			0
$476$	407.547	+	515.339i	0.856190	+	1.08265i
$477$		0			0
$478$	194.390	+	559.185i	0.406673	+	1.16984i
$479$			138.947i			0.290078i	0.989426	+	0.145039i	$0.0463307\pi$
$479$			138.947i			0.290078i	−0.989426	+	0.145039i	$0.953669\pi$
$480$		0			0
$481$	171.299			0.356131
$482$	−879.935	+	305.893i	−1.82559	+	0.634632i
$483$		0			0
$484$	70.5876	−	55.8229i	0.145842	−	0.115337i
$485$		0			0
$486$		0			0
$487$			201.243i			0.413230i	0.978422	+	0.206615i	$0.0662447\pi$
$487$			201.243i			0.413230i	−0.978422	+	0.206615i	$0.933755\pi$
$488$	−163.307	+	256.411i	−0.334646	+	0.525433i
$489$		0			0
$490$		0			0
$491$			347.368i			0.707470i	0.935346	+	0.353735i	$0.115089\pi$
$491$			347.368i			0.707470i	−0.935346	+	0.353735i	$0.884911\pi$
$492$		0			0
$493$	6.02513			0.0122214
$494$	−129.897	−	373.664i	−0.262949	−	0.756404i
$495$		0			0
$496$	716.591	+	169.711i	1.44474	+	0.342159i
$497$	68.7610			0.138352
$498$		0			0
$499$		−	672.277i		−	1.34725i	−0.739074	−	0.673625i	$-0.764736\pi$
$499$		−	672.277i		−	1.34725i	0.739074	−	0.673625i	$-0.235264\pi$
$500$		0			0
$501$		0			0
$502$	93.0401	+	267.641i	0.185339	+	0.533149i
$503$		−	436.350i		−	0.867496i	−0.901034	−	0.433748i	$-0.857191\pi$
$503$		−	436.350i		−	0.867496i	0.901034	−	0.433748i	$-0.142809\pi$
$504$		0			0
$505$		0			0
$506$	31.3954	−	10.9140i	0.0620462	−	0.0215692i
$507$		0			0
$508$	476.523	+	602.559i	0.938037	+	1.18614i
$509$	−109.547			−0.215219			−0.107610	−	0.994193i	$-0.534320\pi$
$509$	−109.547			−0.215219			−0.107610	+	0.994193i	$0.534320\pi$
$510$		0			0
$511$			328.508i			0.642872i
$512$	507.644	+	66.6415i	0.991493	+	0.130159i
$513$		0			0
$514$	−78.8796	+	27.4210i	−0.153462	+	0.0533482i
$515$		0			0
$516$		0			0
$517$	−139.807			−0.270419
$518$	142.248	+	409.193i	0.274610	+	0.789947i
$519$		0			0
$520$		0			0
$521$	743.100			1.42629			0.713147	−	0.701014i	$-0.247269\pi$
$521$	743.100			1.42629			0.713147	+	0.701014i	$0.247269\pi$
$522$		0			0
$523$		−	366.211i		−	0.700212i	−0.936710	−	0.350106i	$-0.886146\pi$
$523$		−	366.211i		−	0.700212i	0.936710	−	0.350106i	$-0.113854\pi$
$524$	105.272	+	133.115i	0.200900	+	0.254037i
$525$		0			0
$526$	−133.498	−	384.023i	−0.253799	−	0.730083i
$527$		−	791.196i		−	1.50132i
$528$		0			0
$529$	526.196			0.994699
$530$		0			0
$531$		0			0
$532$	784.727	−	620.586i	1.47505	−	1.16652i
$533$	584.105			1.09588
$534$		0			0
$535$		0			0
$536$	−201.402	−	128.272i	−0.375750	−	0.239314i
$537$		0			0
$538$	461.033	−	160.269i	0.856939	−	0.297898i
$539$		−	419.809i		−	0.778867i
$540$		0			0
$541$	170.688			0.315504			0.157752	−	0.987479i	$-0.449575\pi$
$541$	170.688			0.315504			0.157752	+	0.987479i	$0.449575\pi$
$542$	306.412	+	881.430i	0.565336	+	1.62625i
$543$		0			0
$544$	−56.0208	−	547.228i	−0.102979	−	1.00593i
$545$		0			0
$546$		0			0
$547$		−	507.600i		−	0.927971i	−0.885843	−	0.463986i	$-0.846419\pi$
$547$		−	507.600i		−	0.927971i	0.885843	−	0.463986i	$-0.153581\pi$
$548$	648.997	−	513.247i	1.18430	−	0.936581i
$549$		0			0
$550$		0			0
$551$		−	9.17469i		−	0.0166510i
$552$		0			0
$553$	439.779			0.795261
$554$	933.561	−	324.534i	1.68513	−	0.585802i
$555$		0			0
$556$	−114.199	−	144.404i	−0.205394	−	0.259720i
$557$	−788.492			−1.41561			−0.707803	−	0.706410i	$-0.750314\pi$
$557$	−788.492			−1.41561			−0.707803	+	0.706410i	$0.750314\pi$
$558$		0			0
$559$		−	315.854i		−	0.565035i
$560$		0			0
$561$		0			0
$562$	−81.9910	+	28.5026i	−0.145892	+	0.0507164i
$563$			936.102i			1.66270i	0.555747	+	0.831351i	$0.312432\pi$
$563$			936.102i			1.66270i	−0.555747	+	0.831351i	$0.687568\pi$
$564$		0			0
$565$		0			0
$566$	204.096	+	587.107i	0.360594	+	1.03729i
$567$		0			0
$568$	−48.5582	−	30.9266i	−0.0854898	−	0.0544482i
$569$	−95.4983			−0.167835			−0.0839177	−	0.996473i	$-0.526743\pi$
$569$	−95.4983			−0.167835			−0.0839177	+	0.996473i	$0.526743\pi$
$570$		0			0
$571$		−	889.123i		−	1.55713i	−0.627562	−	0.778566i	$-0.715947\pi$
$571$		−	889.123i		−	1.55713i	0.627562	−	0.778566i	$-0.284053\pi$
$572$	186.080	+	235.297i	0.325315	+	0.411359i
$573$		0			0
$574$	485.045	+	1395.29i	0.845026	+	2.43081i
$575$		0			0
$576$		0			0
$577$	−522.147			−0.904934			−0.452467	−	0.891781i	$-0.649456\pi$
$577$	−522.147			−0.904934			−0.452467	+	0.891781i	$0.649456\pi$
$578$	−12.2886	+	4.27189i	−0.0212605	+	0.00739081i
$579$		0			0
$580$		0			0
$581$	−230.598			−0.396898
$582$		0			0
$583$			224.988i			0.385915i
$584$	147.752	−	231.988i	0.253001	−	0.397240i
$585$		0			0
$586$	−463.223	+	161.031i	−0.790484	+	0.274796i
$587$			95.8440i			0.163278i	0.996662	+	0.0816388i	$0.0260154\pi$
$587$			95.8440i			0.163278i	−0.996662	+	0.0816388i	$0.973985\pi$
$588$		0			0
$589$	−1204.78			−2.04547
$590$		0			0
$591$		0			0
$592$	83.5883	−	352.946i	0.141197	−	0.596192i
$593$	−878.624			−1.48166			−0.740829	−	0.671693i	$-0.765567\pi$
$593$	−878.624			−1.48166			−0.740829	+	0.671693i	$0.765567\pi$
$594$		0			0
$595$		0			0
$596$	36.5498	−	28.9047i	0.0613252	−	0.0484979i
$597$		0			0
$598$	−8.30957	−	23.9034i	−0.0138956	−	0.0399723i
$599$			967.652i			1.61545i	0.589563	+	0.807723i	$0.299300\pi$
$599$			967.652i			1.61545i	−0.589563	+	0.807723i	$0.700700\pi$
$600$		0			0
$601$	279.704			0.465398			0.232699	−	0.972549i	$-0.425244\pi$
$601$	279.704			0.465398			0.232699	+	0.972549i	$0.425244\pi$
$602$	754.501	−	262.288i	1.25332	−	0.435694i
$603$		0			0
$604$	311.203	+	393.513i	0.515236	+	0.651512i
$605$		0			0
$606$		0			0
$607$		−	747.564i		−	1.23157i	−0.787914	−	0.615786i	$-0.788839\pi$
$607$		−	747.564i		−	1.23157i	0.787914	−	0.615786i	$-0.211161\pi$
$608$	−833.285	+	85.3049i	−1.37053	+	0.140304i
$609$		0			0
$610$		0			0
$611$			106.444i			0.174213i
$612$		0			0
$613$	457.152			0.745761			0.372881	−	0.927879i	$-0.378370\pi$
$613$	457.152			0.745761			0.372881	+	0.927879i	$0.378370\pi$
$614$	221.650	+	637.600i	0.360993	+	1.03844i
$615$		0			0
$616$	−407.547	+	639.894i	−0.661601	+	1.03879i
$617$	−764.888			−1.23969			−0.619844	−	0.784725i	$-0.712804\pi$
$617$	−764.888			−1.23969			−0.619844	+	0.784725i	$0.712804\pi$
$618$		0			0
$619$			365.359i			0.590240i	0.955460	+	0.295120i	$0.0953597\pi$
$619$			365.359i			0.590240i	−0.955460	+	0.295120i	$0.904640\pi$
$620$		0			0
$621$		0			0
$622$	280.912	+	808.077i	0.451627	+	1.29916i
$623$		−	957.410i		−	1.53677i
$624$		0			0
$625$		0			0
$626$	−158.447	+	55.0810i	−0.253110	+	0.0879888i
$627$		0			0
$628$	618.771	−	489.343i	0.985304	−	0.779209i
$629$	−389.691			−0.619541
$630$		0			0
$631$			62.1578i			0.0985067i	0.998786	+	0.0492534i	$0.0156842\pi$
$631$			62.1578i			0.0985067i	−0.998786	+	0.0492534i	$0.984316\pi$
$632$	−310.567	−	197.799i	−0.491403	−	0.312973i
$633$		0			0
$634$	−211.966	+	73.6859i	−0.334331	+	0.116224i
$635$		0			0
$636$		0			0
$637$	−319.630			−0.501773
$638$	2.28444	+	6.57147i	0.00358063	+	0.0103001i
$639$		0			0
$640$		0			0
$641$	1111.69			1.73431			0.867154	−	0.498041i	$-0.165947\pi$
$641$	1111.69			1.73431			0.867154	+	0.498041i	$0.165947\pi$
$642$		0			0
$643$			468.983i			0.729367i	0.931132	+	0.364683i	$0.118823\pi$
$643$			468.983i			0.729367i	−0.931132	+	0.364683i	$0.881177\pi$
$644$	50.1993	−	39.6992i	0.0779493	−	0.0616447i
$645$		0			0
$646$	295.505	+	850.054i	0.457438	+	1.31587i
$647$		−	96.7647i		−	0.149559i	−0.997200	−	0.0747795i	$-0.976175\pi$
$647$		−	96.7647i		−	0.149559i	0.997200	−	0.0747795i	$-0.0238253\pi$
$648$		0			0
$649$	940.675			1.44942
$650$		0			0
$651$		0			0
$652$	−45.7027	−	57.7907i	−0.0700961	−	0.0886360i
$653$	920.353			1.40942			0.704712	−	0.709494i	$-0.251076\pi$
$653$	920.353			1.40942			0.704712	+	0.709494i	$0.251076\pi$
$654$		0			0
$655$		0			0
$656$	285.024	−	1203.49i	0.434488	−	1.83459i
$657$		0			0
$658$	−254.270	+	88.3921i	−0.386429	+	0.134335i
$659$		−	591.020i		−	0.896844i	−0.893822	−	0.448422i	$-0.851986\pi$
$659$		−	591.020i		−	0.896844i	0.893822	−	0.448422i	$-0.148014\pi$
$660$		0			0
$661$	−306.193			−0.463226			−0.231613	−	0.972808i	$-0.574400\pi$
$661$	−306.193			−0.463226			−0.231613	+	0.972808i	$0.574400\pi$
$662$	−87.0935	−	250.535i	−0.131561	−	0.378451i
$663$		0			0
$664$	162.846	+	103.716i	0.245249	+	0.156198i
$665$		0			0
$666$		0			0
$667$		−	0.586909i		−	0.000879924i
$668$	230.484	+	291.445i	0.345035	+	0.436295i
$669$		0			0
$670$		0			0
$671$			377.142i			0.562060i
$672$		0			0
$673$	−556.892			−0.827476			−0.413738	−	0.910396i	$-0.635777\pi$
$673$	−556.892			−0.827476			−0.413738	+	0.910396i	$0.635777\pi$
$674$	39.2442	−	13.6425i	0.0582258	−	0.0202411i
$675$		0			0
$676$	−351.083	+	277.647i	−0.519353	+	0.410721i
$677$	−58.1920			−0.0859557			−0.0429779	−	0.999076i	$-0.513684\pi$
$677$	−58.1920			−0.0859557			−0.0429779	+	0.999076i	$0.513684\pi$
$678$		0			0
$679$		−	1259.94i		−	1.85558i
$680$		0			0
$681$		0			0
$682$	862.940	−	299.984i	1.26531	−	0.439860i
$683$			357.274i			0.523096i	0.965191	+	0.261548i	$0.0842329\pi$
$683$			357.274i			0.523096i	−0.965191	+	0.261548i	$0.915767\pi$
$684$		0			0
$685$		0			0
$686$	42.0482	+	120.956i	0.0612947	+	0.176321i
$687$		0			0
$688$	−650.788	−	154.127i	−0.945913	−	0.224021i
$689$	171.299			0.248620
$690$		0			0
$691$		−	614.707i		−	0.889590i	−0.895632	−	0.444795i	$-0.853276\pi$
$691$		−	614.707i		−	0.889590i	0.895632	−	0.444795i	$-0.146724\pi$
$692$	368.655	−	291.544i	0.532739	−	0.421307i
$693$		0			0
$694$	−5.84222	−	16.8058i	−0.00841818	−	0.0242159i
$695$		0			0
$696$		0			0
$697$	−1328.79			−1.90644
$698$	36.6524	−	12.7415i	0.0525107	−	0.0182543i
$699$		0			0
$700$		0			0
$701$	886.028			1.26395			0.631975	−	0.774989i	$-0.282245\pi$
$701$	886.028			1.26395			0.631975	+	0.774989i	$0.282245\pi$
$702$		0			0
$703$			593.397i			0.844093i
$704$	575.609	−	268.584i	0.817626	−	0.381511i
$705$		0			0
$706$	151.677	−	52.7276i	0.214840	−	0.0746849i
$707$			281.398i			0.398017i
$708$		0			0
$709$	760.887			1.07318			0.536592	−	0.843842i	$-0.319712\pi$
$709$	760.887			1.07318			0.536592	+	0.843842i	$0.319712\pi$
$710$		0			0
$711$		0			0
$712$	−430.613	+	676.111i	−0.604794	+	0.949594i
$713$	−77.0706			−0.108093
$714$		0			0
$715$		0			0
$716$	575.609	+	727.853i	0.803923	+	1.01655i
$717$		0			0
$718$	206.283	+	593.397i	0.287302	+	0.826458i
$719$			575.877i			0.800942i	0.916309	+	0.400471i	$0.131154\pi$
$719$			575.877i			0.800942i	−0.916309	+	0.400471i	$0.868846\pi$
$720$		0			0
$721$	1373.88			1.90553
$722$	612.441	−	212.903i	0.848257	−	0.294880i
$723$		0			0
$724$	−684.268	+	541.141i	−0.945122	+	0.747432i
$725$		0			0
$726$		0			0
$727$			327.332i			0.450250i	0.974330	+	0.225125i	$0.0722790\pi$
$727$			327.332i			0.450250i	−0.974330	+	0.225125i	$0.927721\pi$
$728$	487.195	+	310.293i	0.669224	+	0.426227i
$729$		0			0
$730$		0			0
$731$			718.542i			0.982958i
$732$		0			0
$733$	−947.567			−1.29272			−0.646362	−	0.763031i	$-0.723710\pi$
$733$	−947.567			−1.29272			−0.646362	+	0.763031i	$0.723710\pi$
$734$	−313.120	−	900.727i	−0.426595	−	1.22715i
$735$		0			0
$736$	−53.3056	+	5.45700i	−0.0724261	+	0.00741440i
$737$	−296.232			−0.401943
$738$		0			0
$739$			183.234i			0.247948i	0.992285	+	0.123974i	$0.0395640\pi$
$739$			183.234i			0.247948i	−0.992285	+	0.123974i	$0.960436\pi$
$740$		0			0
$741$		0			0
$742$	142.248	+	409.193i	0.191709	+	0.551473i
$743$			183.712i			0.247258i	0.992329	+	0.123629i	$0.0394532\pi$
$743$			183.712i			0.247258i	−0.992329	+	0.123629i	$0.960547\pi$
$744$		0			0
$745$		0			0
$746$	162.715	−	56.5648i	0.218117	−	0.0758241i
$747$		0			0
$748$	−423.317	−	535.281i	−0.565932	−	0.715617i
$749$	−335.395			−0.447791
$750$		0			0
$751$			345.748i			0.460384i	0.973145	+	0.230192i	$0.0739354\pi$
$751$			345.748i			0.460384i	−0.973145	+	0.230192i	$0.926065\pi$
$752$	219.319	+	51.9414i	0.291647	+	0.0690710i
$753$		0			0
$754$	5.00331	−	1.73930i	0.00663569	−	0.00230677i
$755$		0			0
$756$		0			0
$757$	−549.335			−0.725674			−0.362837	−	0.931853i	$-0.618192\pi$
$757$	−549.335			−0.725674			−0.362837	+	0.931853i	$0.618192\pi$
$758$	−419.005	−	1205.32i	−0.552778	−	1.59013i
$759$		0			0
$760$		0			0
$761$	−251.485			−0.330467			−0.165233	−	0.986255i	$-0.552838\pi$
$761$	−251.485			−0.330467			−0.165233	+	0.986255i	$0.552838\pi$
$762$		0			0
$763$		−	1447.57i		−	1.89721i
$764$	340.440	+	430.484i	0.445602	+	0.563461i
$765$		0			0
$766$	142.337	+	409.450i	0.185819	+	0.534529i
$767$		−	716.200i		−	0.933768i
$768$		0			0
$769$	−583.691			−0.759026			−0.379513	−	0.925186i	$-0.623908\pi$
$769$	−583.691			−0.759026			−0.379513	+	0.925186i	$0.623908\pi$
$770$		0			0
$771$		0			0
$772$	−116.177	+	91.8764i	−0.150488	+	0.119011i
$773$	−1328.04			−1.71803			−0.859015	−	0.511951i	$-0.828923\pi$
$773$	−1328.04			−1.71803			−0.859015	+	0.511951i	$0.828923\pi$
$774$		0			0
$775$		0			0
$776$	−566.681	+	889.753i	−0.730259	+	1.14659i
$777$		0			0
$778$	−900.424	+	313.015i	−1.15736	+	0.402333i
$779$			2023.40i			2.59743i
$780$		0			0
$781$	−71.4219			−0.0914492
$782$	18.9036	+	54.3784i	0.0241734	+	0.0695376i
$783$		0			0
$784$	−155.969	+	658.567i	−0.198940	+	0.840009i
$785$		0			0
$786$		0			0
$787$			1318.83i			1.67577i	0.545850	+	0.837883i	$0.316207\pi$
$787$			1318.83i			1.67577i	−0.545850	+	0.837883i	$0.683793\pi$
$788$	−610.461	+	482.772i	−0.774697	+	0.612654i
$789$		0			0
$790$		0			0
$791$			308.658i			0.390212i
$792$		0			0
$793$	287.144			0.362099
$794$	81.3812	−	28.2906i	0.102495	−	0.0356305i
$795$		0			0
$796$	−436.784	−	552.310i	−0.548724	−	0.693857i
$797$	1277.40			1.60276			0.801381	−	0.598154i	$-0.204099\pi$
$797$	1277.40			1.60276			0.801381	+	0.598154i	$0.204099\pi$
$798$		0			0
$799$		−	242.152i		−	0.303069i
$800$		0			0
$801$		0			0
$802$	−318.682	+	110.784i	−0.397359	+	0.138134i
$803$		−	341.220i		−	0.424931i
$804$		0			0
$805$		0			0
$806$	−228.399	−	657.015i	−0.283373	−	0.815155i
$807$		0			0
$808$	126.564	−	198.720i	0.156639	−	0.245940i
$809$	321.093			0.396901			0.198451	−	0.980111i	$-0.436409\pi$
$809$	321.093			0.396901			0.198451	+	0.980111i	$0.436409\pi$
$810$		0			0
$811$			946.932i			1.16761i	0.811894	+	0.583805i	$0.198437\pi$
$811$			946.932i			1.16761i	−0.811894	+	0.583805i	$0.801563\pi$
$812$	8.30957	+	10.5074i	0.0102335	+	0.0129401i
$813$		0			0
$814$	−147.752	−	425.027i	−0.181514	−	0.522146i
$815$		0			0
$816$		0			0
$817$	1094.15			1.33923
$818$	706.319	−	245.538i	0.863471	−	0.300169i
$819$		0			0
$820$		0			0
$821$	−1169.34			−1.42429			−0.712144	−	0.702033i	$-0.752276\pi$
$821$	−1169.34			−1.42429			−0.712144	+	0.702033i	$0.752276\pi$
$822$		0			0
$823$		−	1251.71i		−	1.52091i	−0.649389	−	0.760457i	$-0.724975\pi$
$823$		−	1251.71i		−	1.52091i	0.649389	−	0.760457i	$-0.275025\pi$
$824$	−970.220	−	617.930i	−1.17745	−	0.749915i
$825$		0			0
$826$	1710.83	−	594.738i	2.07123	−	0.720022i
$827$		−	892.104i		−	1.07872i	−0.842074	−	0.539362i	$-0.818666\pi$
$827$		−	892.104i		−	1.07872i	0.842074	−	0.539362i	$-0.181334\pi$
$828$		0			0
$829$	−998.688			−1.20469			−0.602345	−	0.798236i	$-0.705767\pi$
$829$	−998.688			−1.20469			−0.602345	+	0.798236i	$0.705767\pi$
$830$		0			0
$831$		0			0
$832$	−204.491	−	438.251i	−0.245783	−	0.526743i
$833$	727.131			0.872906
$834$		0			0
$835$		0			0
$836$	−815.093	+	644.601i	−0.974992	+	0.771054i
$837$		0			0
$838$	57.5174	+	165.455i	0.0686365	+	0.197441i
$839$		−	610.359i		−	0.727484i	−0.931500	−	0.363742i	$-0.881499\pi$
$839$		−	610.359i		−	0.727484i	0.931500	−	0.363742i	$-0.118501\pi$
$840$		0			0
$841$	−840.877			−0.999854
$842$	−132.809	+	46.1684i	−0.157730	+	0.0548318i
$843$		0			0
$844$	−51.4086	−	65.0058i	−0.0609107	−	0.0770211i
$845$		0			0
$846$		0			0
$847$		−	214.973i		−	0.253805i
$848$	83.5883	−	352.946i	0.0985712	−	0.416209i
$849$		0			0
$850$		0			0
$851$			37.9599i			0.0446062i
$852$		0			0
$853$	−832.689			−0.976189			−0.488094	−	0.872791i	$-0.662308\pi$
$853$	−832.689			−0.976189			−0.488094	+	0.872791i	$0.662308\pi$
$854$	238.447	+	685.920i	0.279212	+	0.803185i
$855$		0			0
$856$	236.852	+	150.850i	0.276696	+	0.176227i
$857$	−149.415			−0.174347			−0.0871735	−	0.996193i	$-0.527783\pi$
$857$	−149.415			−0.174347			−0.0871735	+	0.996193i	$0.527783\pi$
$858$		0			0
$859$		−	394.144i		−	0.458840i	−0.973327	−	0.229420i	$-0.926317\pi$
$859$		−	394.144i		−	0.458840i	0.973327	−	0.229420i	$-0.0736830\pi$
$860$		0			0
$861$		0			0
$862$	−162.451	−	467.308i	−0.188458	−	0.542121i
$863$		−	1409.58i		−	1.63335i	−0.577095	−	0.816677i	$-0.695814\pi$
$863$		−	1409.58i		−	1.63335i	0.577095	−	0.816677i	$-0.304186\pi$
$864$		0			0
$865$		0			0
$866$	1201.94	−	417.831i	1.38792	−	0.482484i
$867$		0			0
$868$	1379.79	−	1091.18i	1.58962	−	1.25712i
$869$	−456.797			−0.525659
$870$		0			0
$871$			225.542i			0.258946i
$872$	−651.074	+	1022.26i	−0.746644	+	1.17232i
$873$		0			0
$874$	82.8040	−	28.7852i	0.0947414	−	0.0329350i
$875$		0			0
$876$		0			0
$877$	872.780			0.995189			0.497594	−	0.867410i	$-0.334217\pi$
$877$	872.780			0.995189			0.497594	+	0.867410i	$0.334217\pi$
$878$	−505.528	−	1454.21i	−0.575772	−	1.65627i
$879$		0			0
$880$		0			0
$881$	−1103.38			−1.25241			−0.626206	−	0.779657i	$-0.715393\pi$
$881$	−1103.38			−1.25241			−0.626206	+	0.779657i	$0.715393\pi$
$882$		0			0
$883$			536.884i			0.608023i	0.952668	+	0.304011i	$0.0983261\pi$
$883$			536.884i			0.608023i	−0.952668	+	0.304011i	$0.901674\pi$
$884$	−407.547	+	322.300i	−0.461025	+	0.364593i
$885$		0			0
$886$	−402.048	−	1156.54i	−0.453779	−	1.30535i
$887$			888.945i			1.00219i	0.865392	+	0.501096i	$0.167070\pi$
$887$			888.945i			1.00219i	−0.865392	+	0.501096i	$0.832930\pi$
$888$		0			0
$889$	1835.08			2.06421
$890$		0			0
$891$		0			0
$892$	240.742	+	304.417i	0.269890	+	0.341274i
$893$	−368.734			−0.412916
$894$		0			0
$895$		0			0
$896$	877.066	−	852.408i	0.978868	−	0.951348i
$897$		0			0
$898$	−332.288	+	115.514i	−0.370032	+	0.128634i
$899$		−	16.1319i		−	0.0179443i
$900$		0			0
$901$	−389.691			−0.432510
$902$	−503.814	−	1449.28i	−0.558553	−	1.60674i
$903$		0			0
$904$	138.825	−	217.971i	0.153567	−	0.241118i
$905$		0			0
$906$		0			0
$907$		−	1668.40i		−	1.83947i	−0.392543	−	0.919734i	$-0.628405\pi$
$907$		−	1668.40i		−	1.83947i	0.392543	−	0.919734i	$-0.371595\pi$
$908$	1010.48	+	1277.75i	1.11287	+	1.40721i
$909$		0			0
$910$		0			0
$911$		−	1498.13i		−	1.64449i	−0.569131	−	0.822247i	$-0.692720\pi$
$911$		−	1498.13i		−	1.64449i	0.569131	−	0.822247i	$-0.307280\pi$
$912$		0			0
$913$	239.521			0.262345
$914$	690.199	−	239.935i	0.755142	−	0.262510i
$915$		0			0
$916$	24.7766	−	19.5941i	0.0270487	−	0.0213909i
$917$	405.399			0.442093
$918$		0			0
$919$			1094.82i			1.19131i	0.803240	+	0.595656i	$0.203108\pi$
$919$			1094.82i			1.19131i	−0.803240	+	0.595656i	$0.796892\pi$
$920$		0			0
$921$		0			0
$922$	582.507	−	202.497i	0.631787	−	0.219629i
$923$			54.3784i			0.0589148i
$924$		0			0
$925$		0			0
$926$	60.8314	+	174.988i	0.0656926	+	0.188972i
$927$		0			0
$928$	−1.14222	−	11.1576i	−0.00123084	−	0.0120233i
$929$	1145.90			1.23348			0.616740	−	0.787167i	$-0.288453\pi$
$929$	1145.90			1.23348			0.616740	+	0.787167i	$0.288453\pi$
$930$		0			0
$931$		−	1107.23i		−	1.18929i
$932$	−88.3142	+	69.8417i	−0.0947578	+	0.0749374i
$933$		0			0
$934$	−398.035	−	1144.99i	−0.426162	−	1.22590i
$935$		0			0
$936$		0			0
$937$	−1272.49			−1.35805			−0.679025	−	0.734115i	$-0.737597\pi$
$937$	−1272.49			−1.35805			−0.679025	+	0.734115i	$0.737597\pi$
$938$	−538.766	+	187.292i	−0.574378	+	0.199671i
$939$		0			0
$940$		0			0
$941$	−707.360			−0.751711			−0.375856	−	0.926678i	$-0.622651\pi$
$941$	−707.360			−0.751711			−0.375856	+	0.926678i	$0.622651\pi$
$942$		0			0
$943$			129.438i			0.137262i
$944$	−1475.66	−	349.482i	−1.56320	−	0.370214i
$945$		0			0
$946$	−783.698	+	272.437i	−0.828433	+	0.287989i
$947$			284.977i			0.300926i	0.988616	+	0.150463i	$0.0480765\pi$
$947$			284.977i			0.300926i	−0.988616	+	0.150463i	$0.951924\pi$
$948$		0			0
$949$	−259.794			−0.273756
$950$		0			0
$951$		0			0
$952$	−1108.33	−	705.891i	−1.16421	−	0.741482i
$953$	295.247			0.309808			0.154904	−	0.987930i	$-0.450493\pi$
$953$	295.247			0.309808			0.154904	+	0.987930i	$0.450493\pi$
$954$		0			0
$955$		0			0
$956$	−734.447	−	928.702i	−0.768250	−	0.971446i
$957$		0			0
$958$	−91.2483	−	262.486i	−0.0952487	−	0.273994i
$959$		−	1976.50i		−	2.06100i
$960$		0			0
$961$	−1157.38			−1.20435
$962$	−323.602	+	112.494i	−0.336385	+	0.116938i
$963$		0			0
$964$	1461.41	−	1155.73i	1.51598	−	1.19889i
$965$		0			0
$966$		0			0
$967$		−	348.013i		−	0.359889i	−0.983677	−	0.179945i	$-0.942408\pi$
$967$		−	348.013i		−	0.359889i	0.983677	−	0.179945i	$-0.0575918\pi$
$968$	−96.6881	+	151.811i	−0.0998844	+	0.156830i
$969$		0			0
$970$		0			0
$971$		−	798.691i		−	0.822545i	−0.911513	−	0.411272i	$-0.865085\pi$
$971$		−	798.691i		−	0.822545i	0.911513	−	0.411272i	$-0.134915\pi$
$972$		0			0
$973$	−439.779			−0.451983
$974$	−132.159	−	380.170i	−0.135687	−	0.390318i
$975$		0			0
$976$	140.117	−	591.634i	0.143563	−	0.606183i
$977$	721.834			0.738827			0.369413	−	0.929265i	$-0.379559\pi$
$977$	721.834			0.738827			0.369413	+	0.929265i	$0.379559\pi$
$978$		0			0
$979$			994.458i			1.01579i
$980$		0			0
$981$		0			0
$982$	−228.121	−	656.216i	−0.232302	−	0.668244i
$983$			1717.25i			1.74695i	0.486870	+	0.873474i	$0.338139\pi$
$983$			1717.25i			1.74695i	−0.486870	+	0.873474i	$0.661861\pi$
$984$		0			0
$985$		0			0
$986$	−11.3821	+	3.95677i	−0.0115437	+	0.00401296i
$987$		0			0
$988$	490.779	+	620.586i	0.496740	+	0.628124i
$989$	69.9934			0.0707719
$990$		0			0
$991$		−	342.270i		−	0.345378i	−0.984976	−	0.172689i	$-0.944754\pi$
$991$		−	342.270i		−	0.345378i	0.984976	−	0.172689i	$-0.0552456\pi$
$992$	−1465.17	+	149.992i	−1.47699	+	0.151202i
$993$		0			0
$994$	−129.897	+	45.1562i	−0.130681	+	0.0454288i
$995$		0			0
$996$		0			0
$997$	1586.05			1.59082			0.795410	−	0.606072i	$-0.207256\pi$
$997$	1586.05			1.59082			0.795410	+	0.606072i	$0.207256\pi$
$998$	441.493	+	1270.00i	0.442377	+	1.27255i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.c.r.451.2		8
3.2	odd	2		300.3.c.f.151.7		8
4.3	odd	2	inner	900.3.c.r.451.1		8
5.2	odd	4		180.3.f.h.19.3		8
5.3	odd	4		180.3.f.h.19.6		8
5.4	even	2	inner	900.3.c.r.451.7		8
12.11	even	2		300.3.c.f.151.8		8
15.2	even	4		60.3.f.b.19.6	yes	8
15.8	even	4		60.3.f.b.19.3	✓	8
15.14	odd	2		300.3.c.f.151.2		8
20.3	even	4		180.3.f.h.19.4		8
20.7	even	4		180.3.f.h.19.5		8
20.19	odd	2	inner	900.3.c.r.451.8		8
60.23	odd	4		60.3.f.b.19.5	yes	8
60.47	odd	4		60.3.f.b.19.4	yes	8
60.59	even	2		300.3.c.f.151.1		8
120.53	even	4		960.3.j.e.319.8		8
120.77	even	4		960.3.j.e.319.3		8
120.83	odd	4		960.3.j.e.319.4		8
120.107	odd	4		960.3.j.e.319.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.f.b.19.3	✓	8	15.8	even	4
60.3.f.b.19.4	yes	8	60.47	odd	4
60.3.f.b.19.5	yes	8	60.23	odd	4
60.3.f.b.19.6	yes	8	15.2	even	4
180.3.f.h.19.3		8	5.2	odd	4
180.3.f.h.19.4		8	20.3	even	4
180.3.f.h.19.5		8	20.7	even	4
180.3.f.h.19.6		8	5.3	odd	4
300.3.c.f.151.1		8	60.59	even	2
300.3.c.f.151.2		8	15.14	odd	2
300.3.c.f.151.7		8	3.2	odd	2
300.3.c.f.151.8		8	12.11	even	2
900.3.c.r.451.1		8	4.3	odd	2	inner
900.3.c.r.451.2		8	1.1	even	1	trivial
900.3.c.r.451.7		8	5.4	even	2	inner
900.3.c.r.451.8		8	20.19	odd	2	inner
960.3.j.e.319.3		8	120.77	even	4
960.3.j.e.319.4		8	120.83	odd	4
960.3.j.e.319.7		8	120.107	odd	4
960.3.j.e.319.8		8	120.53	even	4

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.88911 + 0.656712i) q^{2} +(3.13746 - 2.48120i) q^{4} -9.55505i q^{7} +(-4.29756 + 6.74766i) q^{8} +O(q^{10})\)	\(q+(-1.88911 + 0.656712i) q^{2} +(3.13746 - 2.48120i) q^{4} -9.55505i q^{7} +(-4.29756 + 6.74766i) q^{8} +9.92480i q^{11} +7.55643 q^{13} +(6.27492 + 18.0505i) q^{14} +(3.68729 - 15.5693i) q^{16} -17.1903 q^{17} +26.1762i q^{19} +(-6.51774 - 18.7490i) q^{22} +1.67451i q^{23} +(-14.2749 + 4.96240i) q^{26} +(-23.7080 - 29.9786i) q^{28} -0.350497 q^{29} +46.0258i q^{31} +(3.25887 + 31.8336i) q^{32} +(32.4743 - 11.2890i) q^{34} +22.6693 q^{37} +(-17.1903 - 49.4498i) q^{38} +77.2990 q^{41} -41.7994i q^{43} +(24.6254 + 31.1386i) q^{44} +(-1.09967 - 3.16332i) q^{46} +14.0866i q^{47} -42.2990 q^{49} +(23.7080 - 18.7490i) q^{52} +22.6693 q^{53} +(64.4743 + 41.0634i) q^{56} +(0.662126 - 0.230175i) q^{58} -94.7802i q^{59} +38.0000 q^{61} +(-30.2257 - 86.9478i) q^{62} +(-27.0619 - 57.9970i) q^{64} +29.8477i q^{67} +(-53.9337 + 42.6525i) q^{68} +7.19630i q^{71} -34.3805 q^{73} +(-42.8248 + 14.8872i) q^{74} +(64.9485 + 82.1269i) q^{76} +94.8320 q^{77} +46.0258i q^{79} +(-146.026 + 50.7632i) q^{82} -24.1336i q^{83} +(27.4502 + 78.9636i) q^{86} +(-66.9692 - 42.6525i) q^{88} +100.199 q^{89} -72.2021i q^{91} +(4.15479 + 5.25370i) q^{92} +(-9.25083 - 26.6111i) q^{94} +131.861 q^{97} +(79.9074 - 27.7783i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{4}+O(q^{10}) \) 8 * q + 10 * q^4	\( 8 q + 10 q^{4} + 20 q^{14} - 46 q^{16} - 84 q^{26} - 184 q^{29} - 12 q^{34} + 256 q^{41} + 348 q^{44} + 112 q^{46} + 24 q^{49} + 244 q^{56} + 304 q^{61} + 10 q^{64} - 252 q^{74} - 24 q^{76} + 280 q^{86} + 560 q^{89} - 376 q^{94}+O(q^{100}) \) 8 * q + 10 * q^4 + 20 * q^14 - 46 * q^16 - 84 * q^26 - 184 * q^29 - 12 * q^34 + 256 * q^41 + 348 * q^44 + 112 * q^46 + 24 * q^49 + 244 * q^56 + 304 * q^61 + 10 * q^64 - 252 * q^74 - 24 * q^76 + 280 * q^86 + 560 * q^89 - 376 * q^94

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