LMFDB - Embedded newform 900.3.c.u.451.1

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:	8.0.85100625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 $ x^8 - x^7 - 2x^6 + x^5 + 3x^4 + 2x^3 - 8x^2 - 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			451.1
Root			$-1.34966 + 0.422403i$ of defining polynomial
Character	$\chi$	$=$	900.451
Dual form			900.3.c.u.451.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.99281 - 0.169449i) q^{2} +(3.94257 + 0.675358i) q^{4} +12.3959i q^{7} +(-7.74236 - 2.01392i) q^{8} +11.0403i q^{11} -2.82009 q^{13} +(2.10047 - 24.7027i) q^{14} +(15.0878 + 5.32529i) q^{16} +6.52606 q^{17} +27.9928i q^{19} +(1.87077 - 22.0012i) q^{22} +7.90421i q^{23} +(5.61989 + 0.477860i) q^{26} +(-8.37167 + 48.8718i) q^{28} -50.7169 q^{29} -36.3467i q^{31} +(-29.1647 - 13.1689i) q^{32} +(-13.0052 - 1.10583i) q^{34} +18.9279 q^{37} +(4.74333 - 55.7842i) q^{38} -5.30410 q^{41} -45.5870i q^{43} +(-7.45616 + 43.5273i) q^{44} +(1.33936 - 15.7516i) q^{46} -11.7246i q^{47} -104.658 q^{49} +(-11.1184 - 1.90457i) q^{52} +41.1680 q^{53} +(24.9644 - 95.9735i) q^{56} +(101.069 + 8.59391i) q^{58} +10.7008i q^{59} +56.1297 q^{61} +(-6.15889 + 72.4319i) q^{62} +(55.8882 + 31.1850i) q^{64} -16.1709i q^{67} +(25.7295 + 4.40743i) q^{68} +66.1617i q^{71} -15.6330 q^{73} +(-37.7198 - 3.20731i) q^{74} +(-18.9051 + 110.363i) q^{76} -136.855 q^{77} -123.057i q^{79} +(10.5701 + 0.898773i) q^{82} +99.6700i q^{83} +(-7.72465 + 90.8461i) q^{86} +(22.2343 - 85.4781i) q^{88} -101.083 q^{89} -34.9575i q^{91} +(-5.33817 + 31.1629i) q^{92} +(-1.98672 + 23.3649i) q^{94} -127.293 q^{97} +(208.564 + 17.7342i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 10 q^{4} - 20 q^{8} - 16 q^{13} + 20 q^{14} + 34 q^{16} - 68 q^{22} + 36 q^{26} - 28 q^{28} - 64 q^{29} - 76 q^{32} - 92 q^{34} + 112 q^{37} - 40 q^{38} + 16 q^{41} - 172 q^{44} + 152 q^{46}+ \cdots + 660 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 10 * q^4 - 20 * q^8 - 16 * q^13 + 20 * q^14 + 34 * q^16 - 68 * q^22 + 36 * q^26 - 28 * q^28 - 64 * q^29 - 76 * q^32 - 92 * q^34 + 112 * q^37 - 40 * q^38 + 16 * q^41 - 172 * q^44 + 152 * q^46 - 56 * q^49 + 128 * q^52 + 352 * q^53 - 116 * q^56 + 204 * q^58 - 176 * q^61 - 56 * q^62 - 110 * q^64 - 184 * q^68 + 240 * q^73 - 132 * q^74 - 24 * q^76 - 288 * q^77 - 40 * q^82 + 200 * q^86 - 140 * q^88 - 80 * q^89 + 144 * q^92 - 96 * q^94 - 432 * q^97 + 660 * q^98

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 $

$2$	−1.99281	−	0.169449i	−0.996404	−	0.0847243i
$2$	−1.99281	−	0.169449i	−0.996404	−	0.0847243i
$3$		0			0
$3$		0			0
$4$	3.94257	+	0.675358i	0.985644	+	0.168839i
$5$		0			0
$5$		0			0
$6$		0			0
$7$			12.3959i			1.77084i	0.464789	+	0.885422i	$0.346130\pi$
$7$			12.3959i			1.77084i	−0.464789	+	0.885422i	$0.653870\pi$
$8$	−7.74236	−	2.01392i	−0.967795	−	0.251740i
$9$		0			0
$10$		0			0
$11$			11.0403i			1.00366i	0.864965	+	0.501832i	$0.167341\pi$
$11$			11.0403i			1.00366i	−0.864965	+	0.501832i	$0.832659\pi$
$12$		0			0
$13$	−2.82009			−0.216930			−0.108465	−	0.994100i	$-0.534593\pi$
$13$	−2.82009			−0.216930			−0.108465	+	0.994100i	$0.534593\pi$
$14$	2.10047	−	24.7027i	0.150034	−	1.76448i
$15$		0			0
$16$	15.0878	+	5.32529i	0.942987	+	0.332831i
$17$	6.52606			0.383886			0.191943	−	0.981406i	$-0.438521\pi$
$17$	6.52606			0.383886			0.191943	+	0.981406i	$0.438521\pi$
$18$		0			0
$19$			27.9928i			1.47330i	0.676273	+	0.736651i	$0.263594\pi$
$19$			27.9928i			1.47330i	−0.676273	+	0.736651i	$0.736406\pi$
$20$		0			0
$21$		0			0
$22$	1.87077	−	22.0012i	0.0850348	−	1.00006i
$23$			7.90421i			0.343661i	0.985126	+	0.171831i	$0.0549682\pi$
$23$			7.90421i			0.343661i	−0.985126	+	0.171831i	$0.945032\pi$
$24$		0			0
$25$		0			0
$26$	5.61989	+	0.477860i	0.216150	+	0.0183792i
$27$		0			0
$28$	−8.37167	+	48.8718i	−0.298988	+	1.74542i
$29$	−50.7169			−1.74886			−0.874429	−	0.485153i	$-0.838764\pi$
$29$	−50.7169			−1.74886			−0.874429	+	0.485153i	$0.838764\pi$
$30$		0			0
$31$		−	36.3467i		−	1.17247i	−0.810140	−	0.586236i	$-0.800609\pi$
$31$		−	36.3467i		−	1.17247i	0.810140	−	0.586236i	$-0.199391\pi$
$32$	−29.1647	−	13.1689i	−0.911397	−	0.411528i
$33$		0			0
$34$	−13.0052	−	1.10583i	−0.382506	−	0.0325245i
$35$		0			0
$36$		0			0
$37$	18.9279			0.511566			0.255783	−	0.966734i	$-0.417667\pi$
$37$	18.9279			0.511566			0.255783	+	0.966734i	$0.417667\pi$
$38$	4.74333	−	55.7842i	0.124825	−	1.46801i
$39$		0			0
$40$		0			0
$41$	−5.30410			−0.129368			−0.0646842	−	0.997906i	$-0.520604\pi$
$41$	−5.30410			−0.129368			−0.0646842	+	0.997906i	$0.520604\pi$
$42$		0			0
$43$		−	45.5870i		−	1.06016i	−0.847947	−	0.530081i	$-0.822162\pi$
$43$		−	45.5870i		−	1.06016i	0.847947	−	0.530081i	$-0.177838\pi$
$44$	−7.45616	+	43.5273i	−0.169458	+	0.989256i
$45$		0			0
$46$	1.33936	−	15.7516i	0.0291165	−	0.342426i
$47$		−	11.7246i		−	0.249460i	−0.992191	−	0.124730i	$-0.960194\pi$
$47$		−	11.7246i		−	0.249460i	0.992191	−	0.124730i	$-0.0398064\pi$
$48$		0			0
$49$	−104.658			−2.13589
$50$		0			0
$51$		0			0
$52$	−11.1184	−	1.90457i	−0.213815	−	0.0366263i
$53$	41.1680			0.776755			0.388378	−	0.921500i	$-0.373036\pi$
$53$	41.1680			0.776755			0.388378	+	0.921500i	$0.373036\pi$
$54$		0			0
$55$		0			0
$56$	24.9644	−	95.9735i	0.445793	−	1.71381i
$57$		0			0
$58$	101.069	+	8.59391i	1.74257	+	0.148171i
$59$			10.7008i			0.181370i	0.995880	+	0.0906848i	$0.0289056\pi$
$59$			10.7008i			0.181370i	−0.995880	+	0.0906848i	$0.971094\pi$
$60$		0			0
$61$	56.1297			0.920159			0.460080	−	0.887878i	$-0.347821\pi$
$61$	56.1297			0.920159			0.460080	+	0.887878i	$0.347821\pi$
$62$	−6.15889	+	72.4319i	−0.0993370	+	1.16826i
$63$		0			0
$64$	55.8882	+	31.1850i	0.873254	+	0.487266i
$65$		0			0
$66$		0			0
$67$		−	16.1709i		−	0.241357i	−0.992692	−	0.120679i	$-0.961493\pi$
$67$		−	16.1709i		−	0.241357i	0.992692	−	0.120679i	$-0.0385071\pi$
$68$	25.7295	+	4.40743i	0.378375	+	0.0648151i
$69$		0			0
$70$		0			0
$71$			66.1617i			0.931855i	0.884823	+	0.465928i	$0.154279\pi$
$71$			66.1617i			0.931855i	−0.884823	+	0.465928i	$0.845721\pi$
$72$		0			0
$73$	−15.6330			−0.214150			−0.107075	−	0.994251i	$-0.534149\pi$
$73$	−15.6330			−0.214150			−0.107075	+	0.994251i	$0.534149\pi$
$74$	−37.7198	−	3.20731i	−0.509727	−	0.0433421i
$75$		0			0
$76$	−18.9051	+	110.363i	−0.248752	+	1.45215i
$77$	−136.855			−1.77733
$78$		0			0
$79$		−	123.057i		−	1.55768i	−0.627223	−	0.778840i	$-0.715809\pi$
$79$		−	123.057i		−	1.55768i	0.627223	−	0.778840i	$-0.284191\pi$
$80$		0			0
$81$		0			0
$82$	10.5701	+	0.898773i	0.128903	+	0.0109606i
$83$			99.6700i			1.20084i	0.799684	+	0.600422i	$0.205001\pi$
$83$			99.6700i			1.20084i	−0.799684	+	0.600422i	$0.794999\pi$
$84$		0			0
$85$		0			0
$86$	−7.72465	+	90.8461i	−0.0898215	+	1.05635i
$87$		0			0
$88$	22.2343	−	85.4781i	0.252663	−	0.971342i
$89$	−101.083			−1.13576			−0.567881	−	0.823110i	$-0.692237\pi$
$89$	−101.083			−1.13576			−0.567881	+	0.823110i	$0.692237\pi$
$90$		0			0
$91$		−	34.9575i		−	0.384148i
$92$	−5.33817	+	31.1629i	−0.0580236	+	0.338728i
$93$		0			0
$94$	−1.98672	+	23.3649i	−0.0211353	+	0.248563i
$95$		0			0
$96$		0			0
$97$	−127.293			−1.31230			−0.656151	−	0.754630i	$-0.727817\pi$
$97$	−127.293			−1.31230			−0.656151	+	0.754630i	$0.727817\pi$
$98$	208.564	+	17.7342i	2.12821	+	0.180962i
$99$		0			0
$100$		0			0
$101$	94.3535			0.934193			0.467096	−	0.884206i	$-0.345300\pi$
$101$	94.3535			0.934193			0.467096	+	0.884206i	$0.345300\pi$
$102$		0			0
$103$		−	31.8455i		−	0.309180i	−0.987979	−	0.154590i	$-0.950594\pi$
$103$		−	31.8455i		−	0.309180i	0.987979	−	0.154590i	$-0.0494056\pi$
$104$	21.8341	+	5.67943i	0.209943	+	0.0546099i
$105$		0			0
$106$	−82.0400	−	6.97587i	−0.773962	−	0.0658101i
$107$		−	33.7912i		−	0.315805i	−0.987455	−	0.157903i	$-0.949527\pi$
$107$		−	33.7912i		−	0.315805i	0.987455	−	0.157903i	$-0.0504732\pi$
$108$		0			0
$109$	−83.4266			−0.765382			−0.382691	−	0.923876i	$-0.625003\pi$
$109$	−83.4266			−0.765382			−0.382691	+	0.923876i	$0.625003\pi$
$110$		0			0
$111$		0			0
$112$	−66.0118	+	187.027i	−0.589391	+	1.66988i
$113$	−111.796			−0.989342			−0.494671	−	0.869080i	$-0.664711\pi$
$113$	−111.796			−0.989342			−0.494671	+	0.869080i	$0.664711\pi$
$114$		0			0
$115$		0			0
$116$	−199.955	−	34.2520i	−1.72375	−	0.295276i
$117$		0			0
$118$	1.81324	−	21.3247i	0.0153664	−	0.180717i
$119$			80.8964i			0.679802i
$120$		0			0
$121$	−0.888544			−0.00734334
$122$	−111.856	−	9.51110i	−0.916851	−	0.0779599i
$123$		0			0
$124$	24.5470	−	143.299i	0.197960	−	1.15564i
$125$		0			0
$126$		0			0
$127$			16.6855i			0.131382i	0.997840	+	0.0656909i	$0.0209251\pi$
$127$			16.6855i			0.131382i	−0.997840	+	0.0656909i	$0.979075\pi$
$128$	−106.090	−	71.6160i	−0.828831	−	0.559500i
$129$		0			0
$130$		0			0
$131$		−	196.418i		−	1.49937i	−0.661794	−	0.749686i	$-0.730204\pi$
$131$		−	196.418i		−	1.49937i	0.661794	−	0.749686i	$-0.269796\pi$
$132$		0			0
$133$	−346.995			−2.60899
$134$	−2.74015	+	32.2256i	−0.0204488	+	0.240490i
$135$		0			0
$136$	−50.5271	−	13.1430i	−0.371523	−	0.0966396i
$137$	−117.127			−0.854942			−0.427471	−	0.904029i	$-0.640595\pi$
$137$	−117.127			−0.854942			−0.427471	+	0.904029i	$0.640595\pi$
$138$		0			0
$139$			187.238i			1.34704i	0.739170	+	0.673519i	$0.235218\pi$
$139$			187.238i			1.34704i	−0.739170	+	0.673519i	$0.764782\pi$
$140$		0			0
$141$		0			0
$142$	11.2110	−	131.848i	0.0789508	−	0.928505i
$143$		−	31.1346i		−	0.217725i
$144$		0			0
$145$		0			0
$146$	31.1535	+	2.64899i	0.213380	+	0.0181437i
$147$		0			0
$148$	74.6248	+	12.7831i	0.504222	+	0.0863725i
$149$	50.2274			0.337096			0.168548	−	0.985693i	$-0.446092\pi$
$149$	50.2274			0.337096			0.168548	+	0.985693i	$0.446092\pi$
$150$		0			0
$151$			213.160i			1.41166i	0.708382	+	0.705829i	$0.249425\pi$
$151$			213.160i			1.41166i	−0.708382	+	0.705829i	$0.750575\pi$
$152$	56.3752	−	216.730i	0.370890	−	1.42585i
$153$		0			0
$154$	272.725	+	23.1898i	1.77094	+	0.150583i
$155$		0			0
$156$		0			0
$157$	−203.918			−1.29884			−0.649419	−	0.760431i	$-0.724988\pi$
$157$	−203.918			−1.29884			−0.649419	+	0.760431i	$0.724988\pi$
$158$	−20.8518	+	245.228i	−0.131973	+	1.55208i
$159$		0			0
$160$		0			0
$161$	−97.9798			−0.608570
$162$		0			0
$163$			215.898i			1.32452i	0.749272	+	0.662262i	$0.230404\pi$
$163$			215.898i			1.32452i	−0.749272	+	0.662262i	$0.769596\pi$
$164$	−20.9118	−	3.58216i	−0.127511	−	0.0218425i
$165$		0			0
$166$	16.8889	−	198.623i	0.101741	−	1.19653i
$167$			255.029i			1.52712i	0.645737	+	0.763560i	$0.276550\pi$
$167$			255.029i			1.52712i	−0.645737	+	0.763560i	$0.723450\pi$
$168$		0			0
$169$	−161.047			−0.952942
$170$		0			0
$171$		0			0
$172$	30.7875	−	179.730i	0.178997	−	1.04494i
$173$	−235.426			−1.36084			−0.680421	−	0.732822i	$-0.738203\pi$
$173$	−235.426			−1.36084			−0.680421	+	0.732822i	$0.738203\pi$
$174$		0			0
$175$		0			0
$176$	−58.7929	+	166.574i	−0.334051	+	0.946443i
$177$		0			0
$178$	201.439	+	17.1284i	1.13168	+	0.0962267i
$179$		−	102.669i		−	0.573572i	−0.957995	−	0.286786i	$-0.907413\pi$
$179$		−	102.669i		−	0.573572i	0.957995	−	0.286786i	$-0.0925869\pi$
$180$		0			0
$181$	−56.8222			−0.313935			−0.156967	−	0.987604i	$-0.550172\pi$
$181$	−56.8222			−0.313935			−0.156967	+	0.987604i	$0.550172\pi$
$182$	−5.92350	+	69.6636i	−0.0325467	+	0.382767i
$183$		0			0
$184$	15.9185	−	61.1972i	0.0865134	−	0.332594i
$185$		0			0
$186$		0			0
$187$			72.0498i			0.385293i
$188$	7.91830	−	46.2251i	0.0421186	−	0.245878i
$189$		0			0
$190$		0			0
$191$			158.493i			0.829808i	0.909865	+	0.414904i	$0.136185\pi$
$191$			158.493i			0.829808i	−0.909865	+	0.414904i	$0.863815\pi$
$192$		0			0
$193$	156.732			0.812084			0.406042	−	0.913854i	$-0.366909\pi$
$193$	156.732			0.812084			0.406042	+	0.913854i	$0.366909\pi$
$194$	253.671	+	21.5697i	1.30758	+	0.111184i
$195$		0			0
$196$	−412.624	−	70.6819i	−2.10522	−	0.360622i
$197$	260.127			1.32044			0.660221	−	0.751072i	$-0.270463\pi$
$197$	260.127			1.32044			0.660221	+	0.751072i	$0.270463\pi$
$198$		0			0
$199$			14.0326i			0.0705157i	0.999378	+	0.0352579i	$0.0112253\pi$
$199$			14.0326i			0.0705157i	−0.999378	+	0.0352579i	$0.988775\pi$
$200$		0			0
$201$		0			0
$202$	−188.028	−	15.9881i	−0.930834	−	0.0791489i
$203$		−	628.682i		−	3.09696i
$204$		0			0
$205$		0			0
$206$	−5.39618	+	63.4620i	−0.0261950	+	0.308068i
$207$		0			0
$208$	−42.5488	−	15.0178i	−0.204562	−	0.0722009i
$209$	−309.049			−1.47870
$210$		0			0
$211$		−	74.4941i		−	0.353052i	−0.984296	−	0.176526i	$-0.943514\pi$
$211$		−	74.4941i		−	0.353052i	0.984296	−	0.176526i	$-0.0564860\pi$
$212$	162.308	+	27.8031i	0.765604	+	0.131147i
$213$		0			0
$214$	−5.72587	+	67.3393i	−0.0267564	+	0.314670i
$215$		0			0
$216$		0			0
$217$	450.550			2.07627
$218$	166.253	+	14.1365i	0.762630	+	0.0648465i
$219$		0			0
$220$		0			0
$221$	−18.4041			−0.0832763
$222$		0			0
$223$			159.996i			0.717471i	0.933439	+	0.358736i	$0.116792\pi$
$223$			159.996i			0.717471i	−0.933439	+	0.358736i	$0.883208\pi$
$224$	163.240	−	361.523i	0.728752	−	1.61394i
$225$		0			0
$226$	222.787	+	18.9436i	0.985784	+	0.0838213i
$227$		−	175.978i		−	0.775236i	−0.921820	−	0.387618i	$-0.873298\pi$
$227$		−	175.978i		−	0.775236i	0.921820	−	0.387618i	$-0.126702\pi$
$228$		0			0
$229$	−114.170			−0.498560			−0.249280	−	0.968431i	$-0.580194\pi$
$229$	−114.170			−0.498560			−0.249280	+	0.968431i	$0.580194\pi$
$230$		0			0
$231$		0			0
$232$	392.668	+	102.140i	1.69254	+	0.440258i
$233$	260.062			1.11615			0.558073	−	0.829792i	$-0.311541\pi$
$233$	260.062			1.11615			0.558073	+	0.829792i	$0.311541\pi$
$234$		0			0
$235$		0			0
$236$	−7.22687	+	42.1887i	−0.0306223	+	0.178766i
$237$		0			0
$238$	13.7078	−	161.211i	0.0575958	−	0.677358i
$239$		−	140.089i		−	0.586147i	−0.956090	−	0.293073i	$-0.905322\pi$
$239$		−	140.089i		−	0.586147i	0.956090	−	0.293073i	$-0.0946780\pi$
$240$		0			0
$241$	105.920			0.439503			0.219752	−	0.975556i	$-0.429475\pi$
$241$	105.920			0.439503			0.219752	+	0.975556i	$0.429475\pi$
$242$	1.77070	+	0.150563i	0.00731693	+	0.000622159i
$243$		0			0
$244$	221.296	+	37.9076i	0.906949	+	0.155359i
$245$		0			0
$246$		0			0
$247$		−	78.9419i		−	0.319603i
$248$	−73.1993	+	281.409i	−0.295159	+	1.13471i
$249$		0			0
$250$		0			0
$251$		−	167.879i		−	0.668839i	−0.942424	−	0.334420i	$-0.891460\pi$
$251$		−	167.879i		−	0.668839i	0.942424	−	0.334420i	$-0.108540\pi$
$252$		0			0
$253$	−87.2650			−0.344921
$254$	2.82733	−	33.2510i	0.0111312	−	0.130909i
$255$		0			0
$256$	199.282	+	160.694i	0.778447	+	0.627710i
$257$	198.849			0.773732			0.386866	−	0.922136i	$-0.373558\pi$
$257$	198.849			0.773732			0.386866	+	0.922136i	$0.373558\pi$
$258$		0			0
$259$			234.629i			0.905903i
$260$		0			0
$261$		0			0
$262$	−33.2827	+	391.423i	−0.127033	+	1.49398i
$263$			480.528i			1.82710i	0.406722	+	0.913552i	$0.366672\pi$
$263$			480.528i			1.82710i	−0.406722	+	0.913552i	$0.633328\pi$
$264$		0			0
$265$		0			0
$266$	691.496	+	58.7979i	2.59961	+	0.221045i
$267$		0			0
$268$	10.9212	−	63.7552i	0.0407506	−	0.237892i
$269$	291.496			1.08363			0.541815	−	0.840498i	$-0.317737\pi$
$269$	291.496			1.08363			0.541815	+	0.840498i	$0.317737\pi$
$270$		0			0
$271$			174.063i			0.642299i	0.947029	+	0.321150i	$0.104069\pi$
$271$			174.063i			0.642299i	−0.947029	+	0.321150i	$0.895931\pi$
$272$	98.4638	+	34.7532i	0.361999	+	0.127769i
$273$		0			0
$274$	233.412	+	19.8470i	0.851868	+	0.0724344i
$275$		0			0
$276$		0			0
$277$	50.5203			0.182384			0.0911918	−	0.995833i	$-0.470932\pi$
$277$	50.5203			0.182384			0.0911918	+	0.995833i	$0.470932\pi$
$278$	31.7273	−	373.130i	0.114127	−	1.34219i
$279$		0			0
$280$		0			0
$281$	66.0514			0.235058			0.117529	−	0.993069i	$-0.462503\pi$
$281$	66.0514			0.235058			0.117529	+	0.993069i	$0.462503\pi$
$282$		0			0
$283$			116.934i			0.413196i	0.978426	+	0.206598i	$0.0662392\pi$
$283$			116.934i			0.413196i	−0.978426	+	0.206598i	$0.933761\pi$
$284$	−44.6828	+	260.848i	−0.157334	+	0.918477i
$285$		0			0
$286$	−5.27572	+	62.0454i	−0.0184466	+	0.216942i
$287$		−	65.7491i		−	0.229091i
$288$		0			0
$289$	−246.411			−0.852631
$290$		0			0
$291$		0			0
$292$	−61.6341	−	10.5578i	−0.211076	−	0.0361570i
$293$	−68.3732			−0.233356			−0.116678	−	0.993170i	$-0.537224\pi$
$293$	−68.3732			−0.233356			−0.116678	+	0.993170i	$0.537224\pi$
$294$		0			0
$295$		0			0
$296$	−146.547	−	38.1194i	−0.495091	−	0.128782i
$297$		0			0
$298$	−100.094	−	8.51096i	−0.335884	−	0.0285603i
$299$		−	22.2905i		−	0.0745503i
$300$		0			0
$301$	565.092			1.87738
$302$	36.1197	−	424.788i	0.119602	−	1.40658i
$303$		0			0
$304$	−149.070	+	422.349i	−0.490361	+	1.38930i
$305$		0			0
$306$		0			0
$307$		−	369.497i		−	1.20357i	−0.798657	−	0.601786i	$-0.794456\pi$
$307$		−	369.497i		−	1.20357i	0.798657	−	0.601786i	$-0.205544\pi$
$308$	−539.560	−	92.4258i	−1.75182	−	0.300084i
$309$		0			0
$310$		0			0
$311$		−	303.446i		−	0.975712i	−0.872924	−	0.487856i	$-0.837779\pi$
$311$		−	303.446i		−	0.975712i	0.872924	−	0.487856i	$-0.162221\pi$
$312$		0			0
$313$	−297.693			−0.951097			−0.475549	−	0.879689i	$-0.657750\pi$
$313$	−297.693			−0.951097			−0.475549	+	0.879689i	$0.657750\pi$
$314$	406.369	+	34.5536i	1.29417	+	0.110043i
$315$		0			0
$316$	83.1072	−	485.160i	0.262998	−	1.53532i
$317$	264.678			0.834948			0.417474	−	0.908689i	$-0.362916\pi$
$317$	264.678			0.834948			0.417474	+	0.908689i	$0.362916\pi$
$318$		0			0
$319$		−	559.931i		−	1.75527i
$320$		0			0
$321$		0			0
$322$	195.255	+	16.6026i	0.606382	+	0.0515607i
$323$			182.682i			0.565580i
$324$		0			0
$325$		0			0
$326$	36.5835	−	430.243i	0.112219	−	1.31976i
$327$		0			0
$328$	41.0663	+	10.6820i	0.125202	+	0.0325672i
$329$	145.337			0.441754
$330$		0			0
$331$		−	473.426i		−	1.43029i	−0.698976	−	0.715145i	$-0.746361\pi$
$331$		−	473.426i		−	1.43029i	0.698976	−	0.715145i	$-0.253639\pi$
$332$	−67.3129	+	392.956i	−0.202750	+	1.18360i
$333$		0			0
$334$	43.2143	−	508.224i	0.129384	−	1.52163i
$335$		0			0
$336$		0			0
$337$	−29.7588			−0.0883051			−0.0441526	−	0.999025i	$-0.514059\pi$
$337$	−29.7588			−0.0883051			−0.0441526	+	0.999025i	$0.514059\pi$
$338$	320.936	+	27.2892i	0.949515	+	0.0807373i
$339$		0			0
$340$		0			0
$341$	401.278			1.17677
$342$		0			0
$343$		−	689.937i		−	2.01148i
$344$	−91.8086	+	352.951i	−0.266885	+	1.02602i
$345$		0			0
$346$	469.158	+	39.8925i	1.35595	+	0.115296i
$347$		−	306.190i		−	0.882391i	−0.897411	−	0.441195i	$-0.854555\pi$
$347$		−	306.190i		−	0.882391i	0.897411	−	0.441195i	$-0.145445\pi$
$348$		0			0
$349$	649.149			1.86002			0.930012	−	0.367528i	$-0.119796\pi$
$349$	649.149			1.86002			0.930012	+	0.367528i	$0.119796\pi$
$350$		0			0
$351$		0			0
$352$	145.389	−	321.988i	0.413036	−	0.914737i
$353$	−275.547			−0.780587			−0.390293	−	0.920691i	$-0.627626\pi$
$353$	−275.547			−0.780587			−0.390293	+	0.920691i	$0.627626\pi$
$354$		0			0
$355$		0			0
$356$	−398.527	−	68.2671i	−1.11946	−	0.191761i
$357$		0			0
$358$	−17.3972	+	204.600i	−0.0485955	+	0.571510i
$359$			507.672i			1.41413i	0.707149	+	0.707065i	$0.249981\pi$
$359$			507.672i			1.41413i	−0.707149	+	0.707065i	$0.750019\pi$
$360$		0			0
$361$	−422.594			−1.17062
$362$	113.236	+	9.62845i	0.312806	+	0.0265979i
$363$		0			0
$364$	23.6088	−	137.823i	0.0648594	−	0.378633i
$365$		0			0
$366$		0			0
$367$		−	62.7671i		−	0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$367$		−	62.7671i		−	0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$368$	−42.0923	+	119.257i	−0.114381	+	0.324068i
$369$		0			0
$370$		0			0
$371$			510.315i			1.37551i
$372$		0			0
$373$	272.776			0.731302			0.365651	−	0.930752i	$-0.380846\pi$
$373$	272.776			0.731302			0.365651	+	0.930752i	$0.380846\pi$
$374$	12.2087	−	143.581i	0.0326437	−	0.383908i
$375$		0			0
$376$	−23.6124	+	90.7761i	−0.0627990	+	0.241426i
$377$	143.026			0.379379
$378$		0			0
$379$		−	376.828i		−	0.994270i	−0.867673	−	0.497135i	$-0.834385\pi$
$379$		−	376.828i		−	0.994270i	0.867673	−	0.497135i	$-0.165615\pi$
$380$		0			0
$381$		0			0
$382$	26.8565	−	315.847i	0.0703049	−	0.826825i
$383$			412.206i			1.07625i	0.842864	+	0.538127i	$0.180868\pi$
$383$			412.206i			1.07625i	−0.842864	+	0.538127i	$0.819132\pi$
$384$		0			0
$385$		0			0
$386$	−312.337	−	26.5581i	−0.809164	−	0.0688032i
$387$		0			0
$388$	−501.863	−	85.9685i	−1.29346	−	0.221568i
$389$	161.289			0.414623			0.207312	−	0.978275i	$-0.433529\pi$
$389$	161.289			0.414623			0.207312	+	0.978275i	$0.433529\pi$
$390$		0			0
$391$			51.5834i			0.131927i
$392$	810.303	+	210.774i	2.06710	+	0.537689i
$393$		0			0
$394$	−518.383	−	44.0782i	−1.31569	−	0.111874i
$395$		0			0
$396$		0			0
$397$	186.505			0.469785			0.234893	−	0.972021i	$-0.424526\pi$
$397$	186.505			0.469785			0.234893	+	0.972021i	$0.424526\pi$
$398$	2.37781	−	27.9643i	0.00597440	−	0.0702622i
$399$		0			0
$400$		0			0
$401$	−239.061			−0.596162			−0.298081	−	0.954541i	$-0.596347\pi$
$401$	−239.061			−0.596162			−0.298081	+	0.954541i	$0.596347\pi$
$402$		0			0
$403$			102.501i			0.254344i
$404$	371.996	+	63.7223i	0.920781	+	0.157729i
$405$		0			0
$406$	−106.529	+	1252.84i	−0.262387	+	3.08582i
$407$			208.970i			0.513441i
$408$		0			0
$409$	47.8016			0.116874			0.0584372	−	0.998291i	$-0.481388\pi$
$409$	47.8016			0.116874			0.0584372	+	0.998291i	$0.481388\pi$
$410$		0			0
$411$		0			0
$412$	21.5071	−	125.553i	0.0522017	−	0.304741i
$413$	−132.646			−0.321177
$414$		0			0
$415$		0			0
$416$	82.2470	+	37.1374i	0.197709	+	0.0892726i
$417$		0			0
$418$	615.875	+	52.3679i	1.47339	+	0.125282i
$419$		−	239.009i		−	0.570428i	−0.958464	−	0.285214i	$-0.907935\pi$
$419$		−	239.009i		−	0.570428i	0.958464	−	0.285214i	$-0.0920647\pi$
$420$		0			0
$421$	−257.592			−0.611857			−0.305929	−	0.952054i	$-0.598967\pi$
$421$	−257.592			−0.611857			−0.305929	+	0.952054i	$0.598967\pi$
$422$	−12.6229	+	148.452i	−0.0299121	+	0.351783i
$423$		0			0
$424$	−318.738	−	82.9092i	−0.751740	−	0.195541i
$425$		0			0
$426$		0			0
$427$			695.779i			1.62946i
$428$	22.8211	−	133.224i	0.0533204	−	0.311271i
$429$		0			0
$430$		0			0
$431$			343.164i			0.796205i	0.917341	+	0.398103i	$0.130331\pi$
$431$			343.164i			0.796205i	−0.917341	+	0.398103i	$0.869669\pi$
$432$		0			0
$433$	234.760			0.542171			0.271085	−	0.962555i	$-0.412617\pi$
$433$	234.760			0.542171			0.271085	+	0.962555i	$0.412617\pi$
$434$	−897.859	−	76.3450i	−2.06880	−	0.175910i
$435$		0			0
$436$	−328.916	−	56.3428i	−0.754394	−	0.129227i
$437$	−221.261			−0.506317
$438$		0			0
$439$			374.473i			0.853013i	0.904484	+	0.426507i	$0.140256\pi$
$439$			374.473i			0.853013i	−0.904484	+	0.426507i	$0.859744\pi$
$440$		0			0
$441$		0			0
$442$	36.6758	+	3.11854i	0.0829768	+	0.00705553i
$443$		−	108.557i		−	0.245050i	−0.992465	−	0.122525i	$-0.960901\pi$
$443$		−	108.557i		−	0.245050i	0.992465	−	0.122525i	$-0.0390992\pi$
$444$		0			0
$445$		0			0
$446$	27.1111	−	318.842i	0.0607873	−	0.714891i
$447$		0			0
$448$	−386.566	+	692.785i	−0.862872	+	1.54640i
$449$	431.511			0.961050			0.480525	−	0.876981i	$-0.340446\pi$
$449$	431.511			0.961050			0.480525	+	0.876981i	$0.340446\pi$
$450$		0			0
$451$		−	58.5589i		−	0.129842i
$452$	−440.762	−	75.5020i	−0.975138	−	0.167040i
$453$		0			0
$454$	−29.8193	+	350.692i	−0.0656813	+	0.772448i
$455$		0			0
$456$		0			0
$457$	−219.747			−0.480847			−0.240424	−	0.970668i	$-0.577286\pi$
$457$	−219.747			−0.480847			−0.240424	+	0.970668i	$0.577286\pi$
$458$	227.520	+	19.3460i	0.496768	+	0.0422402i
$459$		0			0
$460$		0			0
$461$	−223.434			−0.484673			−0.242337	−	0.970192i	$-0.577914\pi$
$461$	−223.434			−0.484673			−0.242337	+	0.970192i	$0.577914\pi$
$462$		0			0
$463$		−	740.855i		−	1.60012i	−0.599921	−	0.800059i	$-0.704802\pi$
$463$		−	740.855i		−	1.60012i	0.599921	−	0.800059i	$-0.295198\pi$
$464$	−765.206	−	270.082i	−1.64915	−	0.582074i
$465$		0			0
$466$	−518.254	−	44.0672i	−1.11213	−	0.0945648i
$467$			249.381i			0.534007i	0.963696	+	0.267004i	$0.0860336\pi$
$467$			249.381i			0.534007i	−0.963696	+	0.267004i	$0.913966\pi$
$468$		0			0
$469$	200.454			0.427406
$470$		0			0
$471$		0			0
$472$	21.5506	−	82.8495i	0.0456580	−	0.175529i
$473$	503.294			1.06405
$474$		0			0
$475$		0			0
$476$	−54.6340	+	318.940i	−0.114777	+	0.670043i
$477$		0			0
$478$	−23.7379	+	279.171i	−0.0496609	+	0.584039i
$479$			210.915i			0.440324i	0.975463	+	0.220162i	$0.0706587\pi$
$479$			210.915i			0.440324i	−0.975463	+	0.220162i	$0.929341\pi$
$480$		0			0
$481$	−53.3784			−0.110974
$482$	−211.079	−	17.9480i	−0.437923	−	0.0372366i
$483$		0			0
$484$	−3.50315	−	0.600085i	−0.00723791	−	0.00123984i
$485$		0			0
$486$		0			0
$487$			710.541i			1.45902i	0.683972	+	0.729508i	$0.260251\pi$
$487$			710.541i			1.45902i	−0.683972	+	0.729508i	$0.739749\pi$
$488$	−434.576	−	113.041i	−0.890525	−	0.231641i
$489$		0			0
$490$		0			0
$491$			697.876i			1.42134i	0.703528	+	0.710668i	$0.251607\pi$
$491$			697.876i			1.42134i	−0.703528	+	0.710668i	$0.748393\pi$
$492$		0			0
$493$	−330.982			−0.671363
$494$	−13.3766	+	157.316i	−0.0270781	+	0.318454i
$495$		0			0
$496$	193.557	−	548.390i	0.390235	−	1.10563i
$497$	−820.135			−1.65017
$498$		0			0
$499$			875.602i			1.75471i	0.479838	+	0.877357i	$0.340695\pi$
$499$			875.602i			1.75471i	−0.479838	+	0.877357i	$0.659305\pi$
$500$		0			0
$501$		0			0
$502$	−28.4468	+	334.550i	−0.0566670	+	0.666434i
$503$		−	142.849i		−	0.283995i	−0.989867	−	0.141997i	$-0.954648\pi$
$503$		−	142.849i		−	0.283995i	0.989867	−	0.141997i	$-0.0453524\pi$
$504$		0			0
$505$		0			0
$506$	173.902	+	14.7869i	0.343681	+	0.0292232i
$507$		0			0
$508$	−11.2687	+	65.7837i	−0.0221824	+	0.129496i
$509$	147.662			0.290102			0.145051	−	0.989424i	$-0.453665\pi$
$509$	147.662			0.290102			0.145051	+	0.989424i	$0.453665\pi$
$510$		0			0
$511$		−	193.785i		−	0.379227i
$512$	−369.903	−	354.000i	−0.722466	−	0.691407i
$513$		0			0
$514$	−396.268	−	33.6947i	−0.770950	−	0.0655539i
$515$		0			0
$516$		0			0
$517$	129.443			0.250374
$518$	39.7576	−	467.571i	0.0767520	−	0.902646i
$519$		0			0
$520$		0			0
$521$	348.592			0.669082			0.334541	−	0.942381i	$-0.391419\pi$
$521$	348.592			0.669082			0.334541	+	0.942381i	$0.391419\pi$
$522$		0			0
$523$			370.317i			0.708063i	0.935233	+	0.354032i	$0.115189\pi$
$523$			370.317i			0.708063i	−0.935233	+	0.354032i	$0.884811\pi$
$524$	132.652	−	774.392i	0.253153	−	1.47785i
$525$		0			0
$526$	81.4249	−	957.601i	0.154800	−	1.82053i
$527$		−	237.201i		−	0.450096i
$528$		0			0
$529$	466.523			0.881897
$530$		0			0
$531$		0			0
$532$	−1368.06	−	234.346i	−2.57153	−	0.440500i
$533$	14.9580			0.0280638
$534$		0			0
$535$		0			0
$536$	−32.5670	+	125.201i	−0.0607594	+	0.233584i
$537$		0			0
$538$	−580.897	−	49.3937i	−1.07973	−	0.0918098i
$539$		−	1155.46i		−	2.14371i
$540$		0			0
$541$	−279.719			−0.517041			−0.258520	−	0.966006i	$-0.583235\pi$
$541$	−279.719			−0.517041			−0.258520	+	0.966006i	$0.583235\pi$
$542$	29.4947	−	346.874i	0.0544183	−	0.639990i
$543$		0			0
$544$	−190.331	−	85.9411i	−0.349873	−	0.157980i
$545$		0			0
$546$		0			0
$547$		−	387.716i		−	0.708804i	−0.935093	−	0.354402i	$-0.884685\pi$
$547$		−	387.716i		−	0.708804i	0.935093	−	0.354402i	$-0.115315\pi$
$548$	−461.782	−	79.1026i	−0.842668	−	0.144348i
$549$		0			0
$550$		0			0
$551$		−	1419.71i		−	2.57660i
$552$		0			0
$553$	1525.40			2.75841
$554$	−100.677	−	8.56059i	−0.181728	−	0.0154523i
$555$		0			0
$556$	−126.453	+	738.201i	−0.227433	+	1.32770i
$557$	43.5564			0.0781983			0.0390991	−	0.999235i	$-0.487551\pi$
$557$	43.5564			0.0781983			0.0390991	+	0.999235i	$0.487551\pi$
$558$		0			0
$559$			128.559i			0.229981i
$560$		0			0
$561$		0			0
$562$	−131.628	−	11.1923i	−0.234213	−	0.0199152i
$563$			361.646i			0.642355i	0.947019	+	0.321178i	$0.104079\pi$
$563$			361.646i			0.642355i	−0.947019	+	0.321178i	$0.895921\pi$
$564$		0			0
$565$		0			0
$566$	19.8144	−	233.028i	0.0350078	−	0.411710i
$567$		0			0
$568$	133.245	−	512.248i	0.234586	−	0.901845i
$569$	−888.559			−1.56161			−0.780807	−	0.624772i	$-0.785192\pi$
$569$	−888.559			−1.56161			−0.780807	+	0.624772i	$0.785192\pi$
$570$		0			0
$571$			447.745i			0.784142i	0.919935	+	0.392071i	$0.128241\pi$
$571$			447.745i			0.784142i	−0.919935	+	0.392071i	$0.871759\pi$
$572$	21.0270	−	122.751i	0.0367605	−	0.214599i
$573$		0			0
$574$	−11.1411	+	131.025i	−0.0194096	+	0.228267i
$575$		0			0
$576$		0			0
$577$	−1069.90			−1.85425			−0.927124	−	0.374756i	$-0.877727\pi$
$577$	−1069.90			−1.85425			−0.927124	+	0.374756i	$0.877727\pi$
$578$	491.049	+	41.7539i	0.849566	+	0.0722386i
$579$		0			0
$580$		0			0
$581$	−1235.50			−2.12651
$582$		0			0
$583$			454.508i			0.779602i
$584$	121.036	+	31.4836i	0.207253	+	0.0539102i
$585$		0			0
$586$	136.255	+	11.5857i	0.232517	+	0.0197709i
$587$			129.637i			0.220847i	0.993885	+	0.110424i	$0.0352208\pi$
$587$			129.637i			0.220847i	−0.993885	+	0.110424i	$0.964779\pi$
$588$		0			0
$589$	1017.44			1.72741
$590$		0			0
$591$		0			0
$592$	285.581	+	100.797i	0.482400	+	0.170265i
$593$	892.757			1.50549			0.752746	−	0.658311i	$-0.228729\pi$
$593$	892.757			1.50549			0.752746	+	0.658311i	$0.228729\pi$
$594$		0			0
$595$		0			0
$596$	198.025	+	33.9214i	0.332257	+	0.0569151i
$597$		0			0
$598$	−3.77710	+	44.4208i	−0.00631623	+	0.0742823i
$599$			1030.62i			1.72057i	0.509816	+	0.860284i	$0.329714\pi$
$599$			1030.62i			1.72057i	−0.509816	+	0.860284i	$0.670286\pi$
$600$		0			0
$601$	−815.961			−1.35767			−0.678836	−	0.734289i	$-0.737515\pi$
$601$	−815.961			−1.35767			−0.678836	+	0.734289i	$0.737515\pi$
$602$	−1126.12	−	95.7540i	−1.87063	−	0.159060i
$603$		0			0
$604$	−143.959	+	840.401i	−0.238343	+	1.39139i
$605$		0			0
$606$		0			0
$607$		−	842.678i		−	1.38827i	−0.719847	−	0.694133i	$-0.755788\pi$
$607$		−	842.678i		−	1.38827i	0.719847	−	0.694133i	$-0.244212\pi$
$608$	368.634	−	816.400i	0.606305	−	1.34276i
$609$		0			0
$610$		0			0
$611$			33.0644i			0.0541152i
$612$		0			0
$613$	731.088			1.19264			0.596320	−	0.802747i	$-0.296629\pi$
$613$	731.088			1.19264			0.596320	+	0.802747i	$0.296629\pi$
$614$	−62.6107	+	736.336i	−0.101972	+	1.19924i
$615$		0			0
$616$	1059.58	+	275.615i	1.72009	+	0.447426i
$617$	−919.609			−1.49045			−0.745226	−	0.666812i	$-0.767658\pi$
$617$	−919.609			−1.49045			−0.745226	+	0.666812i	$0.767658\pi$
$618$		0			0
$619$		−	688.974i		−	1.11304i	−0.830833	−	0.556522i	$-0.812135\pi$
$619$		−	688.974i		−	1.11304i	0.830833	−	0.556522i	$-0.187865\pi$
$620$		0			0
$621$		0			0
$622$	−51.4186	+	604.711i	−0.0826665	+	0.972203i
$623$		−	1253.01i		−	2.01126i
$624$		0			0
$625$		0			0
$626$	593.246	+	50.4438i	0.947678	+	0.0805811i
$627$		0			0
$628$	−803.960	−	137.717i	−1.28019	−	0.219295i
$629$	123.525			0.196383
$630$		0			0
$631$		−	418.968i		−	0.663975i	−0.943284	−	0.331987i	$-0.892281\pi$
$631$		−	418.968i		−	0.663975i	0.943284	−	0.331987i	$-0.107719\pi$
$632$	−247.827	+	952.749i	−0.392131	+	1.50751i
$633$		0			0
$634$	−527.454	−	44.8494i	−0.831946	−	0.0707404i
$635$		0			0
$636$		0			0
$637$	295.146			0.463337
$638$	−94.8795	+	1115.83i	−0.148714	+	1.74896i
$639$		0			0
$640$		0			0
$641$	47.2426			0.0737014			0.0368507	−	0.999321i	$-0.488267\pi$
$641$	47.2426			0.0737014			0.0368507	+	0.999321i	$0.488267\pi$
$642$		0			0
$643$			710.880i			1.10557i	0.833325	+	0.552784i	$0.186435\pi$
$643$			710.880i			1.10557i	−0.833325	+	0.552784i	$0.813565\pi$
$644$	−386.293	−	66.1714i	−0.599834	−	0.102751i
$645$		0			0
$646$	30.9553	−	364.051i	0.0479184	−	0.563547i
$647$		−	468.195i		−	0.723641i	−0.932248	−	0.361820i	$-0.882155\pi$
$647$		−	468.195i		−	0.723641i	0.932248	−	0.361820i	$-0.117845\pi$
$648$		0			0
$649$	−118.140			−0.182034
$650$		0			0
$651$		0			0
$652$	−145.808	+	851.192i	−0.223632	+	1.30551i
$653$	551.066			0.843900			0.421950	−	0.906619i	$-0.361346\pi$
$653$	551.066			0.843900			0.421950	+	0.906619i	$0.361346\pi$
$654$		0			0
$655$		0			0
$656$	−80.0271	−	28.2459i	−0.121993	−	0.0430578i
$657$		0			0
$658$	−289.629	−	24.6272i	−0.440165	−	0.0374273i
$659$		−	158.259i		−	0.240151i	−0.992765	−	0.120075i	$-0.961686\pi$
$659$		−	158.259i		−	0.240151i	0.992765	−	0.120075i	$-0.0383136\pi$
$660$		0			0
$661$	92.4953			0.139932			0.0699662	−	0.997549i	$-0.477711\pi$
$661$	92.4953			0.139932			0.0699662	+	0.997549i	$0.477711\pi$
$662$	−80.2214	+	943.447i	−0.121180	+	1.42515i
$663$		0			0
$664$	200.728	−	771.681i	0.302301	−	1.16217i
$665$		0			0
$666$		0			0
$667$		−	400.877i		−	0.601015i
$668$	−172.236	+	1005.47i	−0.257838	+	1.50520i
$669$		0			0
$670$		0			0
$671$			619.690i			0.923532i
$672$		0			0
$673$	−956.062			−1.42060			−0.710299	−	0.703900i	$-0.751440\pi$
$673$	−956.062			−1.42060			−0.710299	+	0.703900i	$0.751440\pi$
$674$	59.3037	+	5.04259i	0.0879876	+	0.00748159i
$675$		0			0
$676$	−634.940	−	108.764i	−0.939261	−	0.160894i
$677$	1116.67			1.64944			0.824719	−	0.565543i	$-0.191333\pi$
$677$	1116.67			1.64944			0.824719	+	0.565543i	$0.191333\pi$
$678$		0			0
$679$		−	1577.92i		−	2.32388i
$680$		0			0
$681$		0			0
$682$	−799.671	−	67.9961i	−1.17254	−	0.0997010i
$683$		−	826.776i		−	1.21051i	−0.796033	−	0.605254i	$-0.793072\pi$
$683$		−	826.776i		−	1.21051i	0.796033	−	0.605254i	$-0.206928\pi$
$684$		0			0
$685$		0			0
$686$	−116.909	+	1374.91i	−0.170421	+	2.00424i
$687$		0			0
$688$	242.764	−	687.806i	0.352855	−	0.999718i
$689$	−116.097			−0.168501
$690$		0			0
$691$			965.432i			1.39715i	0.715536	+	0.698576i	$0.246182\pi$
$691$			965.432i			1.39715i	−0.715536	+	0.698576i	$0.753818\pi$
$692$	−928.183	−	158.996i	−1.34130	−	0.229764i
$693$		0			0
$694$	−51.8834	+	610.177i	−0.0747600	+	0.879218i
$695$		0			0
$696$		0			0
$697$	−34.6149			−0.0496627
$698$	−1293.63	−	109.997i	−1.85334	−	0.157589i
$699$		0			0
$700$		0			0
$701$	1109.94			1.58337			0.791686	−	0.610928i	$-0.209203\pi$
$701$	1109.94			1.58337			0.791686	+	0.610928i	$0.209203\pi$
$702$		0			0
$703$			529.845i			0.753691i
$704$	−344.292	+	617.024i	−0.489052	+	0.876454i
$705$		0			0
$706$	549.113	+	46.6911i	0.777780	+	0.0661347i
$707$			1169.60i			1.65431i
$708$		0			0
$709$	−964.244			−1.36001			−0.680003	−	0.733210i	$-0.738021\pi$
$709$	−964.244			−1.36001			−0.680003	+	0.733210i	$0.738021\pi$
$710$		0			0
$711$		0			0
$712$	782.620	+	203.573i	1.09919	+	0.285917i
$713$	287.292			0.402934
$714$		0			0
$715$		0			0
$716$	69.3385	−	404.782i	0.0968415	−	0.565338i
$717$		0			0
$718$	86.0244	−	1011.69i	0.119811	−	1.40904i
$719$		−	190.820i		−	0.265396i	−0.991157	−	0.132698i	$-0.957636\pi$
$719$		−	190.820i		−	0.265396i	0.991157	−	0.132698i	$-0.0423641\pi$
$720$		0			0
$721$	394.754			0.547509
$722$	842.149	+	71.6080i	1.16641	+	0.0991801i
$723$		0			0
$724$	−224.026	−	38.3753i	−0.309428	−	0.0530046i
$725$		0			0
$726$		0			0
$727$		−	202.134i		−	0.278039i	−0.990290	−	0.139019i	$-0.955605\pi$
$727$		−	202.134i		−	0.278039i	0.990290	−	0.139019i	$-0.0443951\pi$
$728$	−70.4017	+	270.654i	−0.0967056	+	0.371777i
$729$		0			0
$730$		0			0
$731$		−	297.503i		−	0.406981i
$732$		0			0
$733$	962.435			1.31301			0.656504	−	0.754322i	$-0.272034\pi$
$733$	962.435			1.31301			0.656504	+	0.754322i	$0.272034\pi$
$734$	−10.6358	+	125.083i	−0.0144902	+	0.170413i
$735$		0			0
$736$	104.090	−	230.524i	0.141426	−	0.313212i
$737$	178.532			0.242242
$738$		0			0
$739$			932.112i			1.26132i	0.776061	+	0.630658i	$0.217215\pi$
$739$			932.112i			1.26132i	−0.776061	+	0.630658i	$0.782785\pi$
$740$		0			0
$741$		0			0
$742$	86.4722	−	1016.96i	0.116539	−	1.37057i
$743$			1153.70i			1.55276i	0.630266	+	0.776379i	$0.282946\pi$
$743$			1153.70i			1.55276i	−0.630266	+	0.776379i	$0.717054\pi$
$744$		0			0
$745$		0			0
$746$	−543.590	−	46.2215i	−0.728672	−	0.0619591i
$747$		0			0
$748$	−48.6594	+	284.062i	−0.0650526	+	0.379762i
$749$	418.872			0.559242
$750$		0			0
$751$			204.359i			0.272116i	0.990701	+	0.136058i	$0.0434434\pi$
$751$			204.359i			0.272116i	−0.990701	+	0.136058i	$0.956557\pi$
$752$	62.4369	−	176.898i	0.0830279	−	0.235237i
$753$		0			0
$754$	−285.023	−	24.2356i	−0.378015	−	0.0321427i
$755$		0			0
$756$		0			0
$757$	216.739			0.286314			0.143157	−	0.989700i	$-0.454275\pi$
$757$	216.739			0.286314			0.143157	+	0.989700i	$0.454275\pi$
$758$	−63.8530	+	750.947i	−0.0842388	+	0.990695i
$759$		0			0
$760$		0			0
$761$	−1324.78			−1.74085			−0.870424	−	0.492303i	$-0.836155\pi$
$761$	−1324.78			−1.74085			−0.870424	+	0.492303i	$0.836155\pi$
$762$		0			0
$763$		−	1034.15i		−	1.35537i
$764$	−107.040	+	624.872i	−0.140104	+	0.817895i
$765$		0			0
$766$	69.8477	−	821.447i	0.0911849	−	1.07238i
$767$		−	30.1772i		−	0.0393444i
$768$		0			0
$769$	444.088			0.577488			0.288744	−	0.957406i	$-0.406762\pi$
$769$	444.088			0.577488			0.288744	+	0.957406i	$0.406762\pi$
$770$		0			0
$771$		0			0
$772$	617.928	+	105.850i	0.800425	+	0.137112i
$773$	−751.987			−0.972817			−0.486408	−	0.873732i	$-0.661693\pi$
$773$	−751.987			−0.972817			−0.486408	+	0.873732i	$0.661693\pi$
$774$		0			0
$775$		0			0
$776$	985.550	+	256.359i	1.27004	+	0.330359i
$777$		0			0
$778$	−321.417	−	27.3301i	−0.413133	−	0.0351287i
$779$		−	148.476i		−	0.190599i
$780$		0			0
$781$	−730.446			−0.935271
$782$	8.74073	−	102.796i	0.0111774	−	0.131452i
$783$		0			0
$784$	−1579.06	−	557.337i	−2.01411	−	0.710889i
$785$		0			0
$786$		0			0
$787$			442.296i			0.562002i	0.959707	+	0.281001i	$0.0906665\pi$
$787$			442.296i			0.562002i	−0.959707	+	0.281001i	$0.909334\pi$
$788$	1025.57	+	175.679i	1.30148	+	0.222943i
$789$		0			0
$790$		0			0
$791$		−	1385.81i		−	1.75197i
$792$		0			0
$793$	−158.291			−0.199610
$794$	−371.668	−	31.6030i	−0.468096	−	0.0398022i
$795$		0			0
$796$	−9.47704	+	55.3247i	−0.0119058	+	0.0695034i
$797$	−56.2072			−0.0705235			−0.0352618	−	0.999378i	$-0.511226\pi$
$797$	−56.2072			−0.0705235			−0.0352618	+	0.999378i	$0.511226\pi$
$798$		0			0
$799$		−	76.5155i		−	0.0957641i
$800$		0			0
$801$		0			0
$802$	476.403	+	40.5086i	0.594019	+	0.0505094i
$803$		−	172.593i		−	0.214935i
$804$		0			0
$805$		0			0
$806$	17.3686	−	204.264i	0.0215491	−	0.253430i
$807$		0			0
$808$	−730.518	−	190.021i	−0.904107	−	0.235174i
$809$	−1522.16			−1.88153			−0.940765	−	0.339060i	$-0.889891\pi$
$809$	−1522.16			−1.88153			−0.940765	+	0.339060i	$0.889891\pi$
$810$		0			0
$811$			930.734i			1.14764i	0.818982	+	0.573819i	$0.194539\pi$
$811$			930.734i			1.14764i	−0.818982	+	0.573819i	$0.805461\pi$
$812$	424.585	−	2478.63i	0.522888	−	3.05249i
$813$		0			0
$814$	35.4098	−	416.438i	0.0435009	−	0.511595i
$815$		0			0
$816$		0			0
$817$	1276.10			1.56194
$818$	−95.2595	−	8.09992i	−0.116454	−	0.00990210i
$819$		0			0
$820$		0			0
$821$	−349.814			−0.426083			−0.213041	−	0.977043i	$-0.568337\pi$
$821$	−349.814			−0.426083			−0.213041	+	0.977043i	$0.568337\pi$
$822$		0			0
$823$		−	61.2187i		−	0.0743849i	−0.999308	−	0.0371924i	$-0.988159\pi$
$823$		−	61.2187i		−	0.0743849i	0.999308	−	0.0371924i	$-0.0118414\pi$
$824$	−64.1344	+	246.559i	−0.0778330	+	0.299222i
$825$		0			0
$826$	264.338	+	22.4767i	0.320022	+	0.0272115i
$827$			46.2063i			0.0558721i	0.999610	+	0.0279361i	$0.00889348\pi$
$827$			46.2063i			0.0558721i	−0.999610	+	0.0279361i	$0.991107\pi$
$828$		0			0
$829$	223.832			0.270002			0.135001	−	0.990845i	$-0.456896\pi$
$829$	223.832			0.270002			0.135001	+	0.990845i	$0.456896\pi$
$830$		0			0
$831$		0			0
$832$	−157.610	−	87.9444i	−0.189435	−	0.105702i
$833$	−683.007			−0.819937
$834$		0			0
$835$		0			0
$836$	−1218.45	−	208.718i	−1.45747	−	0.249663i
$837$		0			0
$838$	−40.4998	+	476.300i	−0.0483291	+	0.568377i
$839$			361.794i			0.431220i	0.976480	+	0.215610i	$0.0691740\pi$
$839$			361.794i			0.431220i	−0.976480	+	0.215610i	$0.930826\pi$
$840$		0			0
$841$	1731.20			2.05851
$842$	513.332	+	43.6486i	0.609657	+	0.0518392i
$843$		0			0
$844$	50.3101	−	293.698i	0.0596092	−	0.347984i
$845$		0			0
$846$		0			0
$847$		−	11.0143i		−	0.0130039i
$848$	621.134	+	219.232i	0.732470	+	0.258528i
$849$		0			0
$850$		0			0
$851$			149.610i			0.175805i
$852$		0			0
$853$	844.503			0.990039			0.495019	−	0.868882i	$-0.335161\pi$
$853$	844.503			0.990039			0.495019	+	0.868882i	$0.335161\pi$
$854$	117.899	−	1386.55i	0.138055	−	1.62360i
$855$		0			0
$856$	−68.0528	+	261.623i	−0.0795009	+	0.305635i
$857$	−1389.51			−1.62137			−0.810685	−	0.585482i	$-0.800905\pi$
$857$	−1389.51			−1.62137			−0.810685	+	0.585482i	$0.800905\pi$
$858$		0			0
$859$			1205.45i			1.40332i	0.712512	+	0.701660i	$0.247558\pi$
$859$			1205.45i			1.40332i	−0.712512	+	0.701660i	$0.752442\pi$
$860$		0			0
$861$		0			0
$862$	58.1487	−	683.861i	0.0674579	−	0.793342i
$863$			258.868i			0.299963i	0.988689	+	0.149981i	$0.0479214\pi$
$863$			258.868i			0.299963i	−0.988689	+	0.149981i	$0.952079\pi$
$864$		0			0
$865$		0			0
$866$	−467.832	−	39.7798i	−0.540222	−	0.0459351i
$867$		0			0
$868$	1776.33	+	304.282i	2.04646	+	0.350555i
$869$	1358.58			1.56339
$870$		0			0
$871$			45.6035i			0.0523576i
$872$	645.919	+	168.015i	0.740732	+	0.192677i
$873$		0			0
$874$	440.930	+	37.4923i	0.504497	+	0.0428974i
$875$		0			0
$876$		0			0
$877$	156.268			0.178185			0.0890926	−	0.996023i	$-0.471603\pi$
$877$	156.268			0.178185			0.0890926	+	0.996023i	$0.471603\pi$
$878$	63.4539	−	746.253i	0.0722710	−	0.849946i
$879$		0			0
$880$		0			0
$881$	1343.58			1.52507			0.762533	−	0.646950i	$-0.223956\pi$
$881$	1343.58			1.52507			0.762533	+	0.646950i	$0.223956\pi$
$882$		0			0
$883$		−	149.478i		−	0.169284i	−0.996411	−	0.0846420i	$-0.973025\pi$
$883$		−	149.478i		−	0.169284i	0.996411	−	0.0846420i	$-0.0269747\pi$
$884$	−72.5594	−	12.4293i	−0.0820807	−	0.0140603i
$885$		0			0
$886$	−18.3949	+	216.334i	−0.0207617	+	0.244169i
$887$		−	1532.07i		−	1.72725i	−0.504134	−	0.863626i	$-0.668188\pi$
$887$		−	1532.07i		−	1.72725i	0.504134	−	0.863626i	$-0.331812\pi$
$888$		0			0
$889$	−206.832			−0.232656
$890$		0			0
$891$		0			0
$892$	−108.055	+	630.796i	−0.121137	+	0.707171i
$893$	328.204			0.367529
$894$		0			0
$895$		0			0
$896$	887.745	−	1315.09i	0.990786	−	1.46773i
$897$		0			0
$898$	−859.919	−	73.1190i	−0.957594	−	0.0814243i
$899$			1843.39i			2.05049i
$900$		0			0
$901$	268.665			0.298186
$902$	−9.92273	+	116.697i	−0.0110008	+	0.129376i
$903$		0			0
$904$	865.562	+	225.148i	0.957480	+	0.249057i
$905$		0			0
$906$		0			0
$907$			1245.02i			1.37268i	0.727280	+	0.686341i	$0.240784\pi$
$907$			1245.02i			1.37268i	−0.727280	+	0.686341i	$0.759216\pi$
$908$	118.848	−	693.808i	0.130890	−	0.764106i
$909$		0			0
$910$		0			0
$911$			173.681i			0.190649i	0.995446	+	0.0953245i	$0.0303889\pi$
$911$			173.681i			0.190649i	−0.995446	+	0.0953245i	$0.969611\pi$
$912$		0			0
$913$	−1100.39			−1.20524
$914$	437.914	+	37.2359i	0.479118	+	0.0407395i
$915$		0			0
$916$	−450.125	−	77.1058i	−0.491403	−	0.0841766i
$917$	2434.78			2.65515
$918$		0			0
$919$			874.426i			0.951498i	0.879581	+	0.475749i	$0.157823\pi$
$919$			874.426i			0.951498i	−0.879581	+	0.475749i	$0.842177\pi$
$920$		0			0
$921$		0			0
$922$	445.262	+	37.8607i	0.482931	+	0.0410636i
$923$		−	186.582i		−	0.202147i
$924$		0			0
$925$		0			0
$926$	−125.537	+	1476.38i	−0.135569	+	1.59436i
$927$		0			0
$928$	1479.14	+	667.886i	1.59390	+	0.719705i
$929$	1564.05			1.68358			0.841792	−	0.539803i	$-0.181501\pi$
$929$	1564.05			1.68358			0.841792	+	0.539803i	$0.181501\pi$
$930$		0			0
$931$		−	2929.68i		−	3.14681i
$932$	1025.31	+	175.635i	1.10012	+	0.188450i
$933$		0			0
$934$	42.2573	−	496.970i	0.0452434	−	0.532087i
$935$		0			0
$936$		0			0
$937$	−958.621			−1.02308			−0.511538	−	0.859261i	$-0.670924\pi$
$937$	−958.621			−1.02308			−0.511538	+	0.859261i	$0.670924\pi$
$938$	−399.466	−	33.9666i	−0.425869	−	0.0362117i
$939$		0			0
$940$		0			0
$941$	752.357			0.799529			0.399765	−	0.916618i	$-0.369092\pi$
$941$	752.357			0.799529			0.399765	+	0.916618i	$0.369092\pi$
$942$		0			0
$943$		−	41.9247i		−	0.0444589i
$944$	−56.9849	+	161.451i	−0.0603654	+	0.171029i
$945$		0			0
$946$	−1002.97	−	85.2825i	−1.06022	−	0.0901507i
$947$			1013.16i			1.06986i	0.844895	+	0.534932i	$0.179663\pi$
$947$			1013.16i			1.06986i	−0.844895	+	0.534932i	$0.820337\pi$
$948$		0			0
$949$	44.0863			0.0464555
$950$		0			0
$951$		0			0
$952$	162.919	−	626.329i	0.171134	−	0.657909i
$953$	−21.5482			−0.0226109			−0.0113054	−	0.999936i	$-0.503599\pi$
$953$	−21.5482			−0.0226109			−0.0113054	+	0.999936i	$0.503599\pi$
$954$		0			0
$955$		0			0
$956$	94.6102	−	552.312i	0.0989646	−	0.577732i
$957$		0			0
$958$	35.7393	−	420.314i	0.0373062	−	0.438741i
$959$		−	1451.90i		−	1.51397i
$960$		0			0
$961$	−360.079			−0.374692
$962$	106.373	+	9.04490i	0.110575	+	0.00940218i
$963$		0			0
$964$	417.599	+	71.5340i	0.433193	+	0.0742054i
$965$		0			0
$966$		0			0
$967$		−	303.965i		−	0.314338i	−0.987572	−	0.157169i	$-0.949763\pi$
$967$		−	303.965i		−	0.314338i	0.987572	−	0.157169i	$-0.0502367\pi$
$968$	6.87942	+	1.78946i	0.00710684	+	0.00184861i
$969$		0			0
$970$		0			0
$971$			356.162i			0.366799i	0.983038	+	0.183399i	$0.0587101\pi$
$971$			356.162i			0.366799i	−0.983038	+	0.183399i	$0.941290\pi$
$972$		0			0
$973$	−2320.99			−2.38539
$974$	120.400	−	1415.97i	0.123614	−	1.45377i
$975$		0			0
$976$	846.873	+	298.907i	0.867698	+	0.306257i
$977$	−1845.09			−1.88852			−0.944262	−	0.329194i	$-0.893223\pi$
$977$	−1845.09			−1.88852			−0.944262	+	0.329194i	$0.893223\pi$
$978$		0			0
$979$		−	1115.99i		−	1.13993i
$980$		0			0
$981$		0			0
$982$	118.254	−	1390.73i	0.120422	−	1.41623i
$983$		−	289.444i		−	0.294449i	−0.989103	−	0.147225i	$-0.952966\pi$
$983$		−	289.444i		−	0.294449i	0.989103	−	0.147225i	$-0.0470340\pi$
$984$		0			0
$985$		0			0
$986$	659.583	+	56.0844i	0.668949	+	0.0568807i
$987$		0			0
$988$	53.3140	−	311.234i	0.0539616	−	0.315015i
$989$	360.329			0.364337
$990$		0			0
$991$		−	441.980i		−	0.445994i	−0.974819	−	0.222997i	$-0.928416\pi$
$991$		−	441.980i		−	0.445994i	0.974819	−	0.222997i	$-0.0715840\pi$
$992$	−478.645	+	1060.04i	−0.482505	+	1.06859i
$993$		0			0
$994$	1634.37	+	138.971i	1.64424	+	0.139810i
$995$		0			0
$996$		0			0
$997$	1045.42			1.04856			0.524282	−	0.851545i	$-0.324334\pi$
$997$	1045.42			1.04856			0.524282	+	0.851545i	$0.324334\pi$
$998$	148.370	−	1744.91i	0.148667	−	1.74841i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.c.u.451.1		8
3.2	odd	2		300.3.c.d.151.8		8
4.3	odd	2	inner	900.3.c.u.451.2		8
5.2	odd	4		900.3.f.f.199.9		16
5.3	odd	4		900.3.f.f.199.8		16
5.4	even	2		180.3.c.b.91.8		8
12.11	even	2		300.3.c.d.151.7		8
15.2	even	4		300.3.f.b.199.8		16
15.8	even	4		300.3.f.b.199.9		16
15.14	odd	2		60.3.c.a.31.1	✓	8
20.3	even	4		900.3.f.f.199.10		16
20.7	even	4		900.3.f.f.199.7		16
20.19	odd	2		180.3.c.b.91.7		8
40.19	odd	2		2880.3.e.j.2431.4		8
40.29	even	2		2880.3.e.j.2431.1		8
60.23	odd	4		300.3.f.b.199.7		16
60.47	odd	4		300.3.f.b.199.10		16
60.59	even	2		60.3.c.a.31.2	yes	8
120.29	odd	2		960.3.e.c.511.7		8
120.59	even	2		960.3.e.c.511.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.c.a.31.1	✓	8	15.14	odd	2
60.3.c.a.31.2	yes	8	60.59	even	2
180.3.c.b.91.7		8	20.19	odd	2
180.3.c.b.91.8		8	5.4	even	2
300.3.c.d.151.7		8	12.11	even	2
300.3.c.d.151.8		8	3.2	odd	2
300.3.f.b.199.7		16	60.23	odd	4
300.3.f.b.199.8		16	15.2	even	4
300.3.f.b.199.9		16	15.8	even	4
300.3.f.b.199.10		16	60.47	odd	4
900.3.c.u.451.1		8	1.1	even	1	trivial
900.3.c.u.451.2		8	4.3	odd	2	inner
900.3.f.f.199.7		16	20.7	even	4
900.3.f.f.199.8		16	5.3	odd	4
900.3.f.f.199.9		16	5.2	odd	4
900.3.f.f.199.10		16	20.3	even	4
960.3.e.c.511.4		8	120.59	even	2
960.3.e.c.511.7		8	120.29	odd	2
2880.3.e.j.2431.1		8	40.29	even	2
2880.3.e.j.2431.4		8	40.19	odd	2

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)

\(f(q)\)	\(=\)	\(q+(-1.99281 - 0.169449i) q^{2} +(3.94257 + 0.675358i) q^{4} +12.3959i q^{7} +(-7.74236 - 2.01392i) q^{8} +11.0403i q^{11} -2.82009 q^{13} +(2.10047 - 24.7027i) q^{14} +(15.0878 + 5.32529i) q^{16} +6.52606 q^{17} +27.9928i q^{19} +(1.87077 - 22.0012i) q^{22} +7.90421i q^{23} +(5.61989 + 0.477860i) q^{26} +(-8.37167 + 48.8718i) q^{28} -50.7169 q^{29} -36.3467i q^{31} +(-29.1647 - 13.1689i) q^{32} +(-13.0052 - 1.10583i) q^{34} +18.9279 q^{37} +(4.74333 - 55.7842i) q^{38} -5.30410 q^{41} -45.5870i q^{43} +(-7.45616 + 43.5273i) q^{44} +(1.33936 - 15.7516i) q^{46} -11.7246i q^{47} -104.658 q^{49} +(-11.1184 - 1.90457i) q^{52} +41.1680 q^{53} +(24.9644 - 95.9735i) q^{56} +(101.069 + 8.59391i) q^{58} +10.7008i q^{59} +56.1297 q^{61} +(-6.15889 + 72.4319i) q^{62} +(55.8882 + 31.1850i) q^{64} -16.1709i q^{67} +(25.7295 + 4.40743i) q^{68} +66.1617i q^{71} -15.6330 q^{73} +(-37.7198 - 3.20731i) q^{74} +(-18.9051 + 110.363i) q^{76} -136.855 q^{77} -123.057i q^{79} +(10.5701 + 0.898773i) q^{82} +99.6700i q^{83} +(-7.72465 + 90.8461i) q^{86} +(22.2343 - 85.4781i) q^{88} -101.083 q^{89} -34.9575i q^{91} +(-5.33817 + 31.1629i) q^{92} +(-1.98672 + 23.3649i) q^{94} -127.293 q^{97} +(208.564 + 17.7342i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 10 q^{4} - 20 q^{8} - 16 q^{13} + 20 q^{14} + 34 q^{16} - 68 q^{22} + 36 q^{26} - 28 q^{28} - 64 q^{29} - 76 q^{32} - 92 q^{34} + 112 q^{37} - 40 q^{38} + 16 q^{41} - 172 q^{44} + 152 q^{46}+ \cdots + 660 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 10 * q^4 - 20 * q^8 - 16 * q^13 + 20 * q^14 + 34 * q^16 - 68 * q^22 + 36 * q^26 - 28 * q^28 - 64 * q^29 - 76 * q^32 - 92 * q^34 + 112 * q^37 - 40 * q^38 + 16 * q^41 - 172 * q^44 + 152 * q^46 - 56 * q^49 + 128 * q^52 + 352 * q^53 - 116 * q^56 + 204 * q^58 - 176 * q^61 - 56 * q^62 - 110 * q^64 - 184 * q^68 + 240 * q^73 - 132 * q^74 - 24 * q^76 - 288 * q^77 - 40 * q^82 + 200 * q^86 - 140 * q^88 - 80 * q^89 + 144 * q^92 - 96 * q^94 - 432 * q^97 + 660 * q^98

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