LMFDB - Embedded newform 2790.2.d.m.559.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(559,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2790.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.d (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:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 310)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.1
Root			$1.18254 + 1.18254i$ of defining polynomial
Character	$\chi$	$=$	2790.559
Dual form			2790.2.d.m.559.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(-1.60536 + 1.55654i) q^{5} +3.47817i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.60536 + 1.55654i) q^{5} +3.47817i q^{7} +1.00000i q^{8} +(1.55654 + 1.60536i) q^{10} +2.47063 q^{11} +7.00754i q^{13} +3.47817 q^{14} +1.00000 q^{16} +5.40399i q^{17} -0.480550 q^{19} +(1.60536 - 1.55654i) q^{20} -2.47063i q^{22} +0.365086i q^{23} +(0.154365 - 4.99762i) q^{25} +7.00754 q^{26} -3.47817i q^{28} -0.00991953 q^{29} -1.00000 q^{31} -1.00000i q^{32} +5.40399 q^{34} +(-5.41391 - 5.58371i) q^{35} -10.5270i q^{37} +0.480550i q^{38} +(-1.55654 - 1.60536i) q^{40} -9.63015 q^{41} +5.72264i q^{43} -2.47063 q^{44} +0.365086 q^{46} +2.94328i q^{47} -5.09764 q^{49} +(-4.99762 - 0.154365i) q^{50} -7.00754i q^{52} -6.60117i q^{53} +(-3.96625 + 3.84564i) q^{55} -3.47817 q^{56} +0.00991953i q^{58} -10.9563 q^{59} +8.31388 q^{61} +1.00000i q^{62} -1.00000 q^{64} +(-10.9075 - 11.2496i) q^{65} -7.01544i q^{67} -5.40399i q^{68} +(-5.58371 + 5.41391i) q^{70} -0.632531 q^{71} +3.20871i q^{73} -10.5270 q^{74} +0.480550 q^{76} +8.59326i q^{77} +13.5111 q^{79} +(-1.60536 + 1.55654i) q^{80} +9.63015i q^{82} +9.56589i q^{83} +(-8.41152 - 8.67535i) q^{85} +5.72264 q^{86} +2.47063i q^{88} +15.1516 q^{89} -24.3734 q^{91} -0.365086i q^{92} +2.94328 q^{94} +(0.771455 - 0.747995i) q^{95} -5.32981i q^{97} +5.09764i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 2 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 + 2 * q^5	$ 8 q - 8 q^{4} + 2 q^{5} + 2 q^{10} + 16 q^{11} - 12 q^{14} + 8 q^{16} - 12 q^{19} - 2 q^{20} + 12 q^{25} + 20 q^{26} - 12 q^{29} - 8 q^{31} + 8 q^{34} - 20 q^{35} - 2 q^{40} - 16 q^{44} - 16 q^{46} - 32 q^{49} + 8 q^{50} + 36 q^{55} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 8 q^{64} - 12 q^{65} - 20 q^{70} + 24 q^{71} - 40 q^{74} + 12 q^{76} + 32 q^{79} + 2 q^{80} + 4 q^{85} + 44 q^{86} + 24 q^{89} - 20 q^{91} + 4 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 + 2 * q^5 + 2 * q^10 + 16 * q^11 - 12 * q^14 + 8 * q^16 - 12 * q^19 - 2 * q^20 + 12 * q^25 + 20 * q^26 - 12 * q^29 - 8 * q^31 + 8 * q^34 - 20 * q^35 - 2 * q^40 - 16 * q^44 - 16 * q^46 - 32 * q^49 + 8 * q^50 + 36 * q^55 + 12 * q^56 - 8 * q^59 + 4 * q^61 - 8 * q^64 - 12 * q^65 - 20 * q^70 + 24 * q^71 - 40 * q^74 + 12 * q^76 + 32 * q^79 + 2 * q^80 + 4 * q^85 + 44 * q^86 + 24 * q^89 - 20 * q^91 + 4 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times$.

$n$	$1117$	$1801$	$2171$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−1.60536	+	1.55654i	−0.717939	+	0.696106i
$5$	−1.60536	+	1.55654i	−0.717939	+	0.696106i
$6$		0			0
$7$			3.47817i			1.31462i	0.753619	+	0.657312i	$0.228306\pi$
$7$			3.47817i			1.31462i	−0.753619	+	0.657312i	$0.771694\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	1.55654	+	1.60536i	0.492221	+	0.507660i
$11$	2.47063			0.744923			0.372462	−	0.928048i	$-0.378514\pi$
$11$	2.47063			0.744923			0.372462	+	0.928048i	$0.378514\pi$
$12$		0			0
$13$			7.00754i			1.94354i	0.235929	+	0.971770i	$0.424187\pi$
$13$			7.00754i			1.94354i	−0.235929	+	0.971770i	$0.575813\pi$
$14$	3.47817			0.929579
$15$		0			0
$16$	1.00000			0.250000
$17$			5.40399i			1.31066i	0.755343	+	0.655330i	$0.227470\pi$
$17$			5.40399i			1.31066i	−0.755343	+	0.655330i	$0.772530\pi$
$18$		0			0
$19$	−0.480550			−0.110246			−0.0551228	−	0.998480i	$-0.517555\pi$
$19$	−0.480550			−0.110246			−0.0551228	+	0.998480i	$0.517555\pi$
$20$	1.60536	−	1.55654i	0.358970	−	0.348053i
$21$		0			0
$22$		−	2.47063i		−	0.526740i
$23$			0.365086i			0.0761256i	0.999275	+	0.0380628i	$0.0121187\pi$
$23$			0.365086i			0.0761256i	−0.999275	+	0.0380628i	$0.987881\pi$
$24$		0			0
$25$	0.154365	−	4.99762i	0.0308729	−	0.999523i
$26$	7.00754			1.37429
$27$		0			0
$28$		−	3.47817i		−	0.657312i
$29$	−0.00991953			−0.00184201			−0.000921005	−	1.00000i	$-0.500293\pi$
$29$	−0.00991953			−0.00184201			−0.000921005		1.00000i	$0.500293\pi$
$30$		0			0
$31$	−1.00000			−0.179605
$31$	−1.00000			−0.179605
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	5.40399			0.926776
$35$	−5.41391	−	5.58371i	−0.915117	−	0.943819i
$36$		0			0
$37$		−	10.5270i		−	1.73063i	−0.501232	−	0.865313i	$-0.667120\pi$
$37$		−	10.5270i		−	1.73063i	0.501232	−	0.865313i	$-0.332880\pi$
$38$			0.480550i			0.0779554i
$39$		0			0
$40$	−1.55654	−	1.60536i	−0.246111	−	0.253830i
$41$	−9.63015			−1.50398			−0.751988	−	0.659177i	$-0.770905\pi$
$41$	−9.63015			−1.50398			−0.751988	+	0.659177i	$0.770905\pi$
$42$		0			0
$43$			5.72264i			0.872694i	0.899779	+	0.436347i	$0.143728\pi$
$43$			5.72264i			0.872694i	−0.899779	+	0.436347i	$0.856272\pi$
$44$	−2.47063			−0.372462
$45$		0			0
$46$	0.365086			0.0538289
$47$			2.94328i			0.429321i	0.976689	+	0.214660i	$0.0688645\pi$
$47$			2.94328i			0.429321i	−0.976689	+	0.214660i	$0.931136\pi$
$48$		0			0
$49$	−5.09764			−0.728234
$50$	−4.99762	−	0.154365i	−0.706770	−	0.0218305i
$51$		0			0
$52$		−	7.00754i		−	0.971770i
$53$		−	6.60117i		−	0.906740i	−0.891322	−	0.453370i	$-0.850222\pi$
$53$		−	6.60117i		−	0.906740i	0.891322	−	0.453370i	$-0.149778\pi$
$54$		0			0
$55$	−3.96625	+	3.84564i	−0.534809	+	0.518545i
$56$	−3.47817			−0.464790
$57$		0			0
$58$			0.00991953i			0.00130250i
$59$	−10.9563			−1.42639			−0.713196	−	0.700964i	$-0.752753\pi$
$59$	−10.9563			−1.42639			−0.713196	+	0.700964i	$0.752753\pi$
$60$		0			0
$61$	8.31388			1.06448			0.532242	−	0.846592i	$-0.321350\pi$
$61$	8.31388			1.06448			0.532242	+	0.846592i	$0.321350\pi$
$62$			1.00000i			0.127000i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−10.9075	−	11.2496i	−1.35291	−	1.39534i
$66$		0			0
$67$		−	7.01544i		−	0.857072i	−0.903525	−	0.428536i	$-0.859029\pi$
$67$		−	7.01544i		−	0.857072i	0.903525	−	0.428536i	$-0.140971\pi$
$68$		−	5.40399i		−	0.655330i
$69$		0			0
$70$	−5.58371	+	5.41391i	−0.667381	+	0.647086i
$71$	−0.632531			−0.0750676			−0.0375338	−	0.999295i	$-0.511950\pi$
$71$	−0.632531			−0.0750676			−0.0375338	+	0.999295i	$0.511950\pi$
$72$		0			0
$73$			3.20871i			0.375551i	0.982212	+	0.187775i	$0.0601277\pi$
$73$			3.20871i			0.375551i	−0.982212	+	0.187775i	$0.939872\pi$
$74$	−10.5270			−1.22374
$75$		0			0
$76$	0.480550			0.0551228
$77$			8.59326i			0.979293i
$78$		0			0
$79$	13.5111			1.52011			0.760057	−	0.649857i	$-0.225171\pi$
$79$	13.5111			1.52011			0.760057	+	0.649857i	$0.225171\pi$
$80$	−1.60536	+	1.55654i	−0.179485	+	0.174026i
$81$		0			0
$82$			9.63015i			1.06347i
$83$			9.56589i			1.04999i	0.851105	+	0.524996i	$0.175933\pi$
$83$			9.56589i			1.04999i	−0.851105	+	0.524996i	$0.824067\pi$
$84$		0			0
$85$	−8.41152	−	8.67535i	−0.912358	−	0.940973i
$86$	5.72264			0.617088
$87$		0			0
$88$			2.47063i			0.263370i
$89$	15.1516			1.60607			0.803034	−	0.595933i	$-0.203218\pi$
$89$	15.1516			1.60607			0.803034	+	0.595933i	$0.203218\pi$
$90$		0			0
$91$	−24.3734			−2.55502
$92$		−	0.365086i		−	0.0380628i
$93$		0			0
$94$	2.94328			0.303576
$95$	0.771455	−	0.747995i	0.0791497	−	0.0767427i
$96$		0			0
$97$		−	5.32981i		−	0.541160i	−0.962698	−	0.270580i	$-0.912785\pi$
$97$		−	5.32981i		−	0.541160i	0.962698	−	0.270580i	$-0.0872154\pi$
$98$			5.09764i			0.514939i
$99$		0			0
$100$	−0.154365	+	4.99762i	−0.0154365	+	0.499762i
$101$	0.300341			0.0298851			0.0149425	−	0.999888i	$-0.495243\pi$
$101$	0.300341			0.0298851			0.0149425	+	0.999888i	$0.495243\pi$
$102$		0			0
$103$			2.50401i			0.246727i	0.992362	+	0.123364i	$0.0393682\pi$
$103$			2.50401i			0.246727i	−0.992362	+	0.123364i	$0.960632\pi$
$104$	−7.00754			−0.687145
$105$		0			0
$106$	−6.60117			−0.641162
$107$			13.4369i			1.29899i	0.760365	+	0.649496i	$0.225020\pi$
$107$			13.4369i			1.29899i	−0.760365	+	0.649496i	$0.774980\pi$
$108$		0			0
$109$	0.706711			0.0676906			0.0338453	−	0.999427i	$-0.489225\pi$
$109$	0.706711			0.0676906			0.0338453	+	0.999427i	$0.489225\pi$
$110$	3.84564	+	3.96625i	0.366667	+	0.378167i
$111$		0			0
$112$			3.47817i			0.328656i
$113$			7.55597i			0.710806i	0.934713	+	0.355403i	$0.115656\pi$
$113$			7.55597i			0.710806i	−0.934713	+	0.355403i	$0.884344\pi$
$114$		0			0
$115$	−0.568271	−	0.586094i	−0.0529915	−	0.0546536i
$116$	0.00991953			0.000921005
$117$		0			0
$118$			10.9563i			1.00861i
$119$	−18.7960			−1.72302
$120$		0			0
$121$	−4.89599			−0.445090
$122$		−	8.31388i		−	0.752704i
$123$		0			0
$124$	1.00000			0.0898027
$125$	7.53118	+	8.26325i	0.673609	+	0.739088i
$126$		0			0
$127$		−	17.9389i		−	1.59182i	−0.605416	−	0.795909i	$-0.706993\pi$
$127$		−	17.9389i		−	1.59182i	0.605416	−	0.795909i	$-0.293007\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−11.2496	+	10.9075i	−0.986657	+	0.956652i
$131$	−2.15198			−0.188019			−0.0940097	−	0.995571i	$-0.529968\pi$
$131$	−2.15198			−0.188019			−0.0940097	+	0.995571i	$0.529968\pi$
$132$		0			0
$133$		−	1.67143i		−	0.144932i
$134$	−7.01544			−0.606042
$135$		0			0
$136$	−5.40399			−0.463388
$137$		−	10.3651i		−	0.885549i	−0.896633	−	0.442775i	$-0.853994\pi$
$137$		−	10.3651i		−	0.885549i	0.896633	−	0.442775i	$-0.146006\pi$
$138$		0			0
$139$	−14.0250			−1.18958			−0.594792	−	0.803880i	$-0.702766\pi$
$139$	−14.0250			−1.18958			−0.594792	+	0.803880i	$0.702766\pi$
$140$	5.41391	+	5.58371i	0.457559	+	0.471910i
$141$		0			0
$142$			0.632531i			0.0530808i
$143$			17.3130i			1.44779i
$144$		0			0
$145$	0.0159244	−	0.0154401i	0.00132245	−	0.00128223i
$146$	3.20871			0.265554
$147$		0			0
$148$			10.5270i			0.865313i
$149$	−20.5301			−1.68189			−0.840947	−	0.541118i	$-0.818001\pi$
$149$	−20.5301			−1.68189			−0.840947	+	0.541118i	$0.818001\pi$
$150$		0			0
$151$	−11.0849			−0.902073			−0.451036	−	0.892506i	$-0.648946\pi$
$151$	−11.0849			−0.902073			−0.451036	+	0.892506i	$0.648946\pi$
$152$		−	0.480550i		−	0.0389777i
$153$		0			0
$154$	8.59326			0.692465
$155$	1.60536	−	1.55654i	0.128946	−	0.125024i
$156$		0			0
$157$			15.3976i			1.22886i	0.788970	+	0.614432i	$0.210615\pi$
$157$			15.3976i			1.22886i	−0.788970	+	0.614432i	$0.789385\pi$
$158$		−	13.5111i		−	1.07488i
$159$		0			0
$160$	1.55654	+	1.60536i	0.123055	+	0.126915i
$161$	−1.26983			−0.100077
$162$		0			0
$163$			17.9325i			1.40458i	0.711890	+	0.702291i	$0.247839\pi$
$163$			17.9325i			1.40458i	−0.711890	+	0.702291i	$0.752161\pi$
$164$	9.63015			0.751988
$165$		0			0
$166$	9.56589			0.742457
$167$		−	12.9658i		−	1.00332i	−0.865065	−	0.501661i	$-0.832723\pi$
$167$		−	12.9658i		−	1.00332i	0.865065	−	0.501661i	$-0.167277\pi$
$168$		0			0
$169$	−36.1056			−2.77735
$170$	−8.67535	+	8.41152i	−0.665369	+	0.645134i
$171$		0			0
$172$		−	5.72264i		−	0.436347i
$173$			6.51908i			0.495637i	0.968807	+	0.247818i	$0.0797136\pi$
$173$			6.51908i			0.495637i	−0.968807	+	0.247818i	$0.920286\pi$
$174$		0			0
$175$	17.3825	+	0.536906i	1.31400	+	0.0405863i
$176$	2.47063			0.186231
$177$		0			0
$178$		−	15.1516i		−	1.13566i
$179$	14.2820			1.06749			0.533745	−	0.845646i	$-0.320784\pi$
$179$	14.2820			1.06749			0.533745	+	0.845646i	$0.320784\pi$
$180$		0			0
$181$	−6.92295			−0.514579			−0.257290	−	0.966334i	$-0.582829\pi$
$181$	−6.92295			−0.514579			−0.257290	+	0.966334i	$0.582829\pi$
$182$			24.3734i			1.80667i
$183$		0			0
$184$	−0.365086			−0.0269145
$185$	16.3857	+	16.8996i	1.20470	+	1.24248i
$186$		0			0
$187$			13.3513i			0.976340i
$188$		−	2.94328i		−	0.214660i
$189$		0			0
$190$	−0.747995	−	0.771455i	−0.0542653	−	0.0559673i
$191$	7.17783			0.519369			0.259685	−	0.965693i	$-0.416381\pi$
$191$	7.17783			0.519369			0.259685	+	0.965693i	$0.416381\pi$
$192$		0			0
$193$		−	9.63958i		−	0.693872i	−0.937889	−	0.346936i	$-0.887222\pi$
$193$		−	9.63958i		−	0.693872i	0.937889	−	0.346936i	$-0.112778\pi$
$194$	−5.32981			−0.382658
$195$		0			0
$196$	5.09764			0.364117
$197$		−	14.8309i		−	1.05666i	−0.849039	−	0.528331i	$-0.822818\pi$
$197$		−	14.8309i		−	1.05666i	0.849039	−	0.528331i	$-0.177182\pi$
$198$		0			0
$199$	−9.18726			−0.651268			−0.325634	−	0.945496i	$-0.605578\pi$
$199$	−9.18726			−0.651268			−0.325634	+	0.945496i	$0.605578\pi$
$200$	4.99762	+	0.154365i	0.353385	+	0.0109152i
$201$		0			0
$202$		−	0.300341i		−	0.0211319i
$203$		−	0.0345018i		−	0.00242155i
$204$		0			0
$205$	15.4599	−	14.9897i	1.07976	−	1.04693i
$206$	2.50401			0.174463
$207$		0			0
$208$			7.00754i			0.485885i
$209$	−1.18726			−0.0821245
$210$		0			0
$211$	7.97177			0.548799			0.274400	−	0.961616i	$-0.411521\pi$
$211$	7.97177			0.548799			0.274400	+	0.961616i	$0.411521\pi$
$212$			6.60117i			0.453370i
$213$		0			0
$214$	13.4369			0.918526
$215$	−8.90751	−	9.18689i	−0.607487	−	0.626541i
$216$		0			0
$217$		−	3.47817i		−	0.236113i
$218$		−	0.706711i		−	0.0478645i
$219$		0			0
$220$	3.96625	−	3.84564i	0.267405	−	0.259273i
$221$	−37.8686			−2.54732
$222$		0			0
$223$		−	23.2190i		−	1.55486i	−0.628969	−	0.777430i	$-0.716523\pi$
$223$		−	23.2190i		−	1.55486i	0.628969	−	0.777430i	$-0.283477\pi$
$224$	3.47817			0.232395
$225$		0			0
$226$	7.55597			0.502615
$227$		−	1.88252i		−	0.124947i	−0.998047	−	0.0624736i	$-0.980101\pi$
$227$		−	1.88252i		−	0.124947i	0.998047	−	0.0624736i	$-0.0198989\pi$
$228$		0			0
$229$	11.9472			0.789493			0.394746	−	0.918790i	$-0.370832\pi$
$229$	11.9472			0.789493			0.394746	+	0.918790i	$0.370832\pi$
$230$	−0.586094	+	0.568271i	−0.0386459	+	0.0374707i
$231$		0			0
$232$		−	0.00991953i		−	0.000651249i
$233$			10.4393i			0.683899i	0.939718	+	0.341950i	$0.111087\pi$
$233$			10.4393i			0.683899i	−0.939718	+	0.341950i	$0.888913\pi$
$234$		0			0
$235$	−4.58133	−	4.72502i	−0.298853	−	0.308226i
$236$	10.9563			0.713196
$237$		0			0
$238$			18.7960i			1.21836i
$239$	5.47615			0.354223			0.177111	−	0.984191i	$-0.443325\pi$
$239$	5.47615			0.354223			0.177111	+	0.984191i	$0.443325\pi$
$240$		0			0
$241$	5.73722			0.369567			0.184784	−	0.982779i	$-0.440842\pi$
$241$	5.73722			0.369567			0.184784	+	0.982779i	$0.440842\pi$
$242$			4.89599i			0.314726i
$243$		0			0
$244$	−8.31388			−0.532242
$245$	8.18355	−	7.93468i	0.522828	−	0.506928i
$246$		0			0
$247$		−	3.36747i		−	0.214267i
$248$		−	1.00000i		−	0.0635001i
$249$		0			0
$250$	8.26325	−	7.53118i	0.522614	−	0.476314i
$251$	−0.881767			−0.0556566			−0.0278283	−	0.999613i	$-0.508859\pi$
$251$	−0.881767			−0.0556566			−0.0278283	+	0.999613i	$0.508859\pi$
$252$		0			0
$253$			0.901992i			0.0567077i
$254$	−17.9389			−1.12559
$255$		0			0
$256$	1.00000			0.0625000
$257$			24.6079i			1.53500i	0.641049	+	0.767500i	$0.278500\pi$
$257$			24.6079i			1.53500i	−0.641049	+	0.767500i	$0.721500\pi$
$258$		0			0
$259$	36.6146			2.27512
$260$	10.9075	+	11.2496i	0.676455	+	0.697672i
$261$		0			0
$262$			2.15198i			0.132950i
$263$			12.9818i			0.800493i	0.916408	+	0.400246i	$0.131075\pi$
$263$			12.9818i			0.800493i	−0.916408	+	0.400246i	$0.868925\pi$
$264$		0			0
$265$	10.2750	+	10.5973i	0.631187	+	0.650984i
$266$	−1.67143			−0.102482
$267$		0			0
$268$			7.01544i			0.428536i
$269$	12.6611			0.771964			0.385982	−	0.922506i	$-0.373863\pi$
$269$	12.6611			0.771964			0.385982	+	0.922506i	$0.373863\pi$
$270$		0			0
$271$	−4.60165			−0.279530			−0.139765	−	0.990185i	$-0.544635\pi$
$271$	−4.60165			−0.279530			−0.139765	+	0.990185i	$0.544635\pi$
$272$			5.40399i			0.327665i
$273$		0			0
$274$	−10.3651			−0.626178
$275$	0.381378	−	12.3473i	0.0229980	−	0.744568i
$276$		0			0
$277$			11.7734i			0.707392i	0.935360	+	0.353696i	$0.115075\pi$
$277$			11.7734i			0.707392i	−0.935360	+	0.353696i	$0.884925\pi$
$278$			14.0250i			0.841163i
$279$		0			0
$280$	5.58371	−	5.41391i	0.333691	−	0.323543i
$281$	28.3293			1.68999			0.844993	−	0.534777i	$-0.179604\pi$
$281$	28.3293			1.68999			0.844993	+	0.534777i	$0.179604\pi$
$282$		0			0
$283$			22.2992i			1.32555i	0.748819	+	0.662775i	$0.230621\pi$
$283$			22.2992i			1.32555i	−0.748819	+	0.662775i	$0.769379\pi$
$284$	0.632531			0.0375338
$285$		0			0
$286$	17.3130			1.02374
$287$		−	33.4953i		−	1.97716i
$288$		0			0
$289$	−12.2031			−0.717828
$290$	−0.0154401	−	0.0159244i	−0.000906677	−	0.000935114i
$291$		0			0
$292$		−	3.20871i		−	0.187775i
$293$		−	10.5227i		−	0.614743i	−0.951590	−	0.307371i	$-0.900551\pi$
$293$		−	10.5227i		−	0.614743i	0.951590	−	0.307371i	$-0.0994494\pi$
$294$		0			0
$295$	17.5889	−	17.0540i	1.02406	−	0.992921i
$296$	10.5270			0.611869
$297$		0			0
$298$			20.5301i			1.18928i
$299$	−2.55835			−0.147953
$300$		0			0
$301$	−19.9043			−1.14726
$302$			11.0849i			0.637862i
$303$		0			0
$304$	−0.480550			−0.0275614
$305$	−13.3468	+	12.9409i	−0.764234	+	0.740993i
$306$		0			0
$307$		−	25.5955i		−	1.46082i	−0.683012	−	0.730408i	$-0.739330\pi$
$307$		−	25.5955i		−	1.46082i	0.683012	−	0.730408i	$-0.260670\pi$
$308$		−	8.59326i		−	0.489647i
$309$		0			0
$310$	−1.55654	−	1.60536i	−0.0884055	−	0.0911783i
$311$	−21.0492			−1.19359			−0.596795	−	0.802393i	$-0.703560\pi$
$311$	−21.0492			−1.19359			−0.596795	+	0.802393i	$0.703560\pi$
$312$		0			0
$313$			15.6658i			0.885483i	0.896649	+	0.442742i	$0.145994\pi$
$313$			15.6658i			0.885483i	−0.896649	+	0.442742i	$0.854006\pi$
$314$	15.3976			0.868938
$315$		0			0
$316$	−13.5111			−0.760057
$317$		−	23.8928i		−	1.34195i	−0.741478	−	0.670977i	$-0.765875\pi$
$317$		−	23.8928i		−	1.34195i	0.741478	−	0.670977i	$-0.234125\pi$
$318$		0			0
$319$	−0.0245075			−0.00137216
$320$	1.60536	−	1.55654i	0.0897424	−	0.0870132i
$321$		0			0
$322$			1.26983i			0.0707648i
$323$		−	2.59688i		−	0.144494i
$324$		0			0
$325$	35.0210	+	1.08172i	1.94261	+	0.0600028i
$326$	17.9325			0.993190
$327$		0			0
$328$		−	9.63015i		−	0.531736i
$329$	−10.2372			−0.564395
$330$		0			0
$331$	0.617845			0.0339598			0.0169799	−	0.999856i	$-0.494595\pi$
$331$	0.617845			0.0339598			0.0169799	+	0.999856i	$0.494595\pi$
$332$		−	9.56589i		−	0.524996i
$333$		0			0
$334$	−12.9658			−0.709455
$335$	10.9198	+	11.2623i	0.596613	+	0.615326i
$336$		0			0
$337$			20.6992i			1.12756i	0.825926	+	0.563778i	$0.190653\pi$
$337$			20.6992i			1.12756i	−0.825926	+	0.563778i	$0.809347\pi$
$338$			36.1056i			1.96388i
$339$		0			0
$340$	8.41152	+	8.67535i	0.456179	+	0.470487i
$341$	−2.47063			−0.133792
$342$		0			0
$343$			6.61672i			0.357269i
$344$	−5.72264			−0.308544
$345$		0			0
$346$	6.51908			0.350468
$347$		−	33.0611i		−	1.77482i	−0.460986	−	0.887408i	$-0.652504\pi$
$347$		−	33.0611i		−	1.77482i	0.460986	−	0.887408i	$-0.347496\pi$
$348$		0			0
$349$	−7.31198			−0.391401			−0.195701	−	0.980664i	$-0.562698\pi$
$349$	−7.31198			−0.391401			−0.195701	+	0.980664i	$0.562698\pi$
$350$	0.536906	−	17.3825i	0.0286988	−	0.929136i
$351$		0			0
$352$		−	2.47063i		−	0.131685i
$353$			21.7429i			1.15726i	0.815592	+	0.578628i	$0.196412\pi$
$353$			21.7429i			1.15726i	−0.815592	+	0.578628i	$0.803588\pi$
$354$		0			0
$355$	1.01544	−	0.984560i	0.0538940	−	0.0522550i
$356$	−15.1516			−0.803034
$357$		0			0
$358$		−	14.2820i		−	0.754829i
$359$	−7.72054			−0.407475			−0.203737	−	0.979026i	$-0.565309\pi$
$359$	−7.72054			−0.407475			−0.203737	+	0.979026i	$0.565309\pi$
$360$		0			0
$361$	−18.7691			−0.987846
$362$			6.92295i			0.363862i
$363$		0			0
$364$	24.3734			1.27751
$365$	−4.99448	−	5.15113i	−0.261423	−	0.269622i
$366$		0			0
$367$		−	23.0325i		−	1.20229i	−0.799141	−	0.601144i	$-0.794712\pi$
$367$		−	23.0325i		−	1.20229i	0.799141	−	0.601144i	$-0.205288\pi$
$368$			0.365086i			0.0190314i
$369$		0			0
$370$	16.8996	−	16.3857i	0.878569	−	0.851851i
$371$	22.9600			1.19202
$372$		0			0
$373$		−	7.35528i		−	0.380842i	−0.981703	−	0.190421i	$-0.939015\pi$
$373$		−	7.35528i		−	0.380842i	0.981703	−	0.190421i	$-0.0609853\pi$
$374$	13.3513			0.690377
$375$		0			0
$376$	−2.94328			−0.151788
$377$		−	0.0695115i		−	0.00358002i
$378$		0			0
$379$	−19.3928			−0.996144			−0.498072	−	0.867136i	$-0.665958\pi$
$379$	−19.3928			−0.996144			−0.498072	+	0.867136i	$0.665958\pi$
$380$	−0.771455	+	0.747995i	−0.0395748	+	0.0383713i
$381$		0			0
$382$		−	7.17783i		−	0.367249i
$383$			13.1111i			0.669944i	0.942228	+	0.334972i	$0.108727\pi$
$383$			13.1111i			0.669944i	−0.942228	+	0.334972i	$0.891273\pi$
$384$		0			0
$385$	−13.3758	−	13.7953i	−0.681692	−	0.703073i
$386$	−9.63958			−0.490642
$387$		0			0
$388$			5.32981i			0.270580i
$389$	−8.76355			−0.444330			−0.222165	−	0.975009i	$-0.571312\pi$
$389$	−8.76355			−0.444330			−0.222165	+	0.975009i	$0.571312\pi$
$390$		0			0
$391$	−1.97292			−0.0997748
$392$		−	5.09764i		−	0.257470i
$393$		0			0
$394$	−14.8309			−0.747172
$395$	−21.6901	+	21.0305i	−1.09135	+	1.05816i
$396$		0			0
$397$		−	22.2056i		−	1.11447i	−0.830356	−	0.557233i	$-0.811863\pi$
$397$		−	22.2056i		−	1.11447i	0.830356	−	0.557233i	$-0.188137\pi$
$398$			9.18726i			0.460516i
$399$		0			0
$400$	0.154365	−	4.99762i	0.00771823	−	0.249881i
$401$	5.65942			0.282618			0.141309	−	0.989966i	$-0.454869\pi$
$401$	5.65942			0.282618			0.141309	+	0.989966i	$0.454869\pi$
$402$		0			0
$403$		−	7.00754i		−	0.349070i
$404$	−0.300341			−0.0149425
$405$		0			0
$406$	−0.0345018			−0.00171229
$407$		−	26.0083i		−	1.28918i
$408$		0			0
$409$	13.2801			0.656660			0.328330	−	0.944563i	$-0.393514\pi$
$409$	13.2801			0.656660			0.328330	+	0.944563i	$0.393514\pi$
$410$	−14.9897	−	15.4599i	−0.740289	−	0.763508i
$411$		0			0
$412$		−	2.50401i		−	0.123364i
$413$		−	38.1079i		−	1.87517i
$414$		0			0
$415$	−14.8897	−	15.3567i	−0.730906	−	0.753831i
$416$	7.00754			0.343573
$417$		0			0
$418$			1.18726i			0.0580708i
$419$	18.8268			0.919751			0.459876	−	0.887983i	$-0.347894\pi$
$419$	18.8268			0.919751			0.459876	+	0.887983i	$0.347894\pi$
$420$		0			0
$421$	16.6627			0.812089			0.406045	−	0.913853i	$-0.366908\pi$
$421$	16.6627			0.812089			0.406045	+	0.913853i	$0.366908\pi$
$422$		−	7.97177i		−	0.388060i
$423$		0			0
$424$	6.60117			0.320581
$425$	27.0071	+	0.834185i	1.31003	+	0.0404639i
$426$		0			0
$427$			28.9171i			1.39939i
$428$		−	13.4369i		−	0.649496i
$429$		0			0
$430$	−9.18689	+	8.90751i	−0.443031	+	0.429558i
$431$	30.2936			1.45919			0.729595	−	0.683880i	$-0.239709\pi$
$431$	30.2936			1.45919			0.729595	+	0.683880i	$0.239709\pi$
$432$		0			0
$433$			10.7913i			0.518597i	0.965797	+	0.259298i	$0.0834913\pi$
$433$			10.7913i			0.518597i	−0.965797	+	0.259298i	$0.916509\pi$
$434$	−3.47817			−0.166957
$435$		0			0
$436$	−0.706711			−0.0338453
$437$		−	0.175442i		−	0.00839252i
$438$		0			0
$439$	−8.47303			−0.404396			−0.202198	−	0.979345i	$-0.564808\pi$
$439$	−8.47303			−0.404396			−0.202198	+	0.979345i	$0.564808\pi$
$440$	−3.84564	−	3.96625i	−0.183333	−	0.189084i
$441$		0			0
$442$			37.8686i			1.80123i
$443$		−	35.8921i		−	1.70528i	−0.522495	−	0.852642i	$-0.674999\pi$
$443$		−	35.8921i		−	1.70528i	0.522495	−	0.852642i	$-0.325001\pi$
$444$		0			0
$445$	−24.3238	+	23.5841i	−1.15306	+	1.11799i
$446$	−23.2190			−1.09945
$447$		0			0
$448$		−	3.47817i		−	0.164328i
$449$	−17.3271			−0.817714			−0.408857	−	0.912598i	$-0.634073\pi$
$449$	−17.3271			−0.817714			−0.408857	+	0.912598i	$0.634073\pi$
$450$		0			0
$451$	−23.7925			−1.12035
$452$		−	7.55597i		−	0.355403i
$453$		0			0
$454$	−1.88252			−0.0883511
$455$	39.1281	−	37.9381i	1.83435	−	1.77857i
$456$		0			0
$457$			20.0674i			0.938713i	0.883009	+	0.469357i	$0.155514\pi$
$457$			20.0674i			0.938713i	−0.883009	+	0.469357i	$0.844486\pi$
$458$		−	11.9472i		−	0.558256i
$459$		0			0
$460$	0.568271	+	0.586094i	0.0264958	+	0.0273268i
$461$	10.2586			0.477789			0.238895	−	0.971045i	$-0.423215\pi$
$461$	10.2586			0.477789			0.238895	+	0.971045i	$0.423215\pi$
$462$		0			0
$463$		−	0.766689i		−	0.0356310i	−0.999841	−	0.0178155i	$-0.994329\pi$
$463$		−	0.766689i		−	0.0356310i	0.999841	−	0.0178155i	$-0.00567116\pi$
$464$	−0.00991953			−0.000460503
$465$		0			0
$466$	10.4393			0.483590
$467$			21.4273i			0.991539i	0.868454	+	0.495770i	$0.165114\pi$
$467$			21.4273i			0.991539i	−0.868454	+	0.495770i	$0.834886\pi$
$468$		0			0
$469$	24.4009			1.12673
$470$	−4.72502	+	4.58133i	−0.217949	+	0.211321i
$471$		0			0
$472$		−	10.9563i		−	0.504306i
$473$			14.1385i			0.650090i
$474$		0			0
$475$	−0.0741799	+	2.40160i	−0.00340361	+	0.110193i
$476$	18.7960			0.861512
$477$		0			0
$478$		−	5.47615i		−	0.250473i
$479$	−29.6667			−1.35550			−0.677752	−	0.735290i	$-0.737046\pi$
$479$	−29.6667			−1.35550			−0.677752	+	0.735290i	$0.737046\pi$
$480$		0			0
$481$	73.7682			3.36354
$482$		−	5.73722i		−	0.261323i
$483$		0			0
$484$	4.89599			0.222545
$485$	8.29606	+	8.55626i	0.376705	+	0.388520i
$486$		0			0
$487$			25.8096i			1.16955i	0.811197	+	0.584773i	$0.198816\pi$
$487$			25.8096i			1.16955i	−0.811197	+	0.584773i	$0.801184\pi$
$488$			8.31388i			0.376352i
$489$		0			0
$490$	−7.93468	−	8.18355i	−0.358452	−	0.369695i
$491$	18.7250			0.845048			0.422524	−	0.906352i	$-0.361144\pi$
$491$	18.7250			0.845048			0.422524	+	0.906352i	$0.361144\pi$
$492$		0			0
$493$		−	0.0536050i		−	0.00241425i
$494$	−3.36747			−0.151510
$495$		0			0
$496$	−1.00000			−0.0449013
$497$		−	2.20005i		−	0.0986856i
$498$		0			0
$499$	−41.9722			−1.87893			−0.939466	−	0.342642i	$-0.888678\pi$
$499$	−41.9722			−1.87893			−0.939466	+	0.342642i	$0.888678\pi$
$500$	−7.53118	−	8.26325i	−0.336805	−	0.369544i
$501$		0			0
$502$			0.881767i			0.0393552i
$503$			30.3436i			1.35296i	0.736463	+	0.676478i	$0.236495\pi$
$503$			30.3436i			1.35296i	−0.736463	+	0.676478i	$0.763505\pi$
$504$		0			0
$505$	−0.482156	+	0.467493i	−0.0214556	+	0.0208032i
$506$	0.901992			0.0400984
$507$		0			0
$508$			17.9389i			0.795909i
$509$	−25.3183			−1.12221			−0.561107	−	0.827744i	$-0.689624\pi$
$509$	−25.3183			−1.12221			−0.561107	+	0.827744i	$0.689624\pi$
$510$		0			0
$511$	−11.1604			−0.493708
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	24.6079			1.08541
$515$	−3.89759	−	4.01984i	−0.171748	−	0.177135i
$516$		0			0
$517$			7.27175i			0.319811i
$518$		−	36.6146i		−	1.60875i
$519$		0			0
$520$	11.2496	−	10.9075i	0.493329	−	0.478326i
$521$	−7.15235			−0.313350			−0.156675	−	0.987650i	$-0.550078\pi$
$521$	−7.15235			−0.313350			−0.156675	+	0.987650i	$0.550078\pi$
$522$		0			0
$523$			20.5035i			0.896557i	0.893894	+	0.448278i	$0.147963\pi$
$523$			20.5035i			0.896557i	−0.893894	+	0.448278i	$0.852037\pi$
$524$	2.15198			0.0940097
$525$		0			0
$526$	12.9818			0.566034
$527$		−	5.40399i		−	0.235401i
$528$		0			0
$529$	22.8667			0.994205
$530$	10.5973	−	10.2750i	0.460315	−	0.446317i
$531$		0			0
$532$			1.67143i			0.0724658i
$533$		−	67.4836i		−	2.92304i
$534$		0			0
$535$	−20.9150	−	21.5710i	−0.904236	−	0.932597i
$536$	7.01544			0.303021
$537$		0			0
$538$		−	12.6611i		−	0.545861i
$539$	−12.5944			−0.542479
$540$		0			0
$541$	17.0504			0.733052			0.366526	−	0.930408i	$-0.380547\pi$
$541$	17.0504			0.733052			0.366526	+	0.930408i	$0.380547\pi$
$542$			4.60165i			0.197658i
$543$		0			0
$544$	5.40399			0.231694
$545$	−1.13453	+	1.10002i	−0.0485977	+	0.0471198i
$546$		0			0
$547$		−	19.3000i		−	0.825207i	−0.910911	−	0.412604i	$-0.864619\pi$
$547$		−	19.3000i		−	0.825207i	0.910911	−	0.412604i	$-0.135381\pi$
$548$			10.3651i			0.442775i
$549$		0			0
$550$	−12.3473	−	0.381378i	−0.526489	−	0.0162620i
$551$	0.00476683			0.000203074
$552$		0			0
$553$			46.9937i			1.99838i
$554$	11.7734			0.500202
$555$		0			0
$556$	14.0250			0.594792
$557$			12.9913i			0.550460i	0.961378	+	0.275230i	$0.0887540\pi$
$557$			12.9913i			0.550460i	−0.961378	+	0.275230i	$0.911246\pi$
$558$		0			0
$559$	−40.1016			−1.69612
$560$	−5.41391	−	5.58371i	−0.228779	−	0.235955i
$561$		0			0
$562$		−	28.3293i		−	1.19500i
$563$		−	17.6480i		−	0.743773i	−0.928278	−	0.371887i	$-0.878711\pi$
$563$		−	17.6480i		−	0.743773i	0.928278	−	0.371887i	$-0.121289\pi$
$564$		0			0
$565$	−11.7612	−	12.1301i	−0.494796	−	0.510315i
$566$	22.2992			0.937305
$567$		0			0
$568$		−	0.632531i		−	0.0265404i
$569$	16.5151			0.692347			0.346173	−	0.938171i	$-0.387481\pi$
$569$	16.5151			0.692347			0.346173	+	0.938171i	$0.387481\pi$
$570$		0			0
$571$	−34.8517			−1.45850			−0.729248	−	0.684249i	$-0.760130\pi$
$571$	−34.8517			−1.45850			−0.729248	+	0.684249i	$0.760130\pi$
$572$		−	17.3130i		−	0.723894i
$573$		0			0
$574$	−33.4953			−1.39806
$575$	1.82456	+	0.0563563i	0.0760893	+	0.00235022i
$576$		0			0
$577$			14.2440i			0.592985i	0.955035	+	0.296492i	$0.0958170\pi$
$577$			14.2440i			0.592985i	−0.955035	+	0.296492i	$0.904183\pi$
$578$			12.2031i			0.507581i
$579$		0			0
$580$	−0.0159244	+	0.0154401i	−0.000661226	+	0.000641117i
$581$	−33.2717			−1.38034
$582$		0			0
$583$		−	16.3090i		−	0.675451i
$584$	−3.20871			−0.132777
$585$		0			0
$586$	−10.5227			−0.434689
$587$		−	12.2766i		−	0.506711i	−0.967373	−	0.253355i	$-0.918466\pi$
$587$		−	12.2766i		−	0.506711i	0.967373	−	0.253355i	$-0.0815342\pi$
$588$		0			0
$589$	0.480550			0.0198007
$590$	−17.0540	−	17.5889i	−0.702101	−	0.724122i
$591$		0			0
$592$		−	10.5270i		−	0.432656i
$593$			42.1988i			1.73290i	0.499266	+	0.866449i	$0.333603\pi$
$593$			42.1988i			1.73290i	−0.499266	+	0.866449i	$0.666397\pi$
$594$		0			0
$595$	30.1743	−	29.2567i	1.23703	−	1.19941i
$596$	20.5301			0.840947
$597$		0			0
$598$			2.55835i			0.104619i
$599$	−8.14997			−0.332999			−0.166499	−	0.986042i	$-0.553246\pi$
$599$	−8.14997			−0.332999			−0.166499	+	0.986042i	$0.553246\pi$
$600$		0			0
$601$	14.6953			0.599432			0.299716	−	0.954028i	$-0.403108\pi$
$601$	14.6953			0.599432			0.299716	+	0.954028i	$0.403108\pi$
$602$			19.9043i			0.811238i
$603$		0			0
$604$	11.0849			0.451036
$605$	7.85982	−	7.62080i	0.319547	−	0.309830i
$606$		0			0
$607$		−	7.18249i		−	0.291528i	−0.989319	−	0.145764i	$-0.953436\pi$
$607$		−	7.18249i		−	0.291528i	0.989319	−	0.145764i	$-0.0465641\pi$
$608$			0.480550i			0.0194889i
$609$		0			0
$610$	12.9409	+	13.3468i	0.523961	+	0.540395i
$611$	−20.6251			−0.834403
$612$		0			0
$613$		−	4.55787i		−	0.184091i	−0.995755	−	0.0920453i	$-0.970660\pi$
$613$		−	4.55787i		−	0.184091i	0.995755	−	0.0920453i	$-0.0293404\pi$
$614$	−25.5955			−1.03295
$615$		0			0
$616$	−8.59326			−0.346232
$617$			12.5698i			0.506041i	0.967461	+	0.253021i	$0.0814240\pi$
$617$			12.5698i			0.506041i	−0.967461	+	0.253021i	$0.918576\pi$
$618$		0			0
$619$	12.3888			0.497948			0.248974	−	0.968510i	$-0.419907\pi$
$619$	12.3888			0.497948			0.248974	+	0.968510i	$0.419907\pi$
$620$	−1.60536	+	1.55654i	−0.0644728	+	0.0625122i
$621$		0			0
$622$			21.0492i			0.843996i
$623$			52.6998i			2.11137i
$624$		0			0
$625$	−24.9523	−	1.54291i	−0.998094	−	0.0617164i
$626$	15.6658			0.626131
$627$		0			0
$628$		−	15.3976i		−	0.614432i
$629$	56.8877			2.26826
$630$		0			0
$631$	24.3238			0.968315			0.484158	−	0.874981i	$-0.339126\pi$
$631$	24.3238			0.968315			0.484158	+	0.874981i	$0.339126\pi$
$632$			13.5111i			0.537441i
$633$		0			0
$634$	−23.8928			−0.948905
$635$	27.9226	+	28.7984i	1.10807	+	1.14283i
$636$		0			0
$637$		−	35.7219i		−	1.41535i
$638$			0.0245075i			0.000970261i
$639$		0			0
$640$	−1.55654	−	1.60536i	−0.0615277	−	0.0634574i
$641$	20.1238			0.794840			0.397420	−	0.917637i	$-0.369906\pi$
$641$	20.1238			0.794840			0.397420	+	0.917637i	$0.369906\pi$
$642$		0			0
$643$		−	14.7682i		−	0.582402i	−0.956662	−	0.291201i	$-0.905945\pi$
$643$		−	14.7682i		−	0.582402i	0.956662	−	0.291201i	$-0.0940548\pi$
$644$	1.26983			0.0500383
$645$		0			0
$646$	−2.59688			−0.102173
$647$			11.4557i			0.450369i	0.974316	+	0.225185i	$0.0722985\pi$
$647$			11.4557i			0.450369i	−0.974316	+	0.225185i	$0.927702\pi$
$648$		0			0
$649$	−27.0690			−1.06255
$650$	1.08172	−	35.0210i	0.0424284	−	1.37364i
$651$		0			0
$652$		−	17.9325i		−	0.702291i
$653$			6.14168i			0.240342i	0.992753	+	0.120171i	$0.0383443\pi$
$653$			6.14168i			0.240342i	−0.992753	+	0.120171i	$0.961656\pi$
$654$		0			0
$655$	3.45471	−	3.34965i	0.134987	−	0.130881i
$656$	−9.63015			−0.375994
$657$		0			0
$658$			10.2372i			0.399088i
$659$	−23.6164			−0.919963			−0.459981	−	0.887929i	$-0.652144\pi$
$659$	−23.6164			−0.919963			−0.459981	+	0.887929i	$0.652144\pi$
$660$		0			0
$661$	36.5330			1.42097			0.710485	−	0.703713i	$-0.248476\pi$
$661$	36.5330			1.42097			0.710485	+	0.703713i	$0.248476\pi$
$662$		−	0.617845i		−	0.0240132i
$663$		0			0
$664$	−9.56589			−0.371228
$665$	2.60165	+	2.68325i	0.100888	+	0.104052i
$666$		0			0
$667$		−	0.00362148i		−	0.000140224i
$668$			12.9658i			0.501661i
$669$		0			0
$670$	11.2623	−	10.9198i	0.435101	−	0.421869i
$671$	20.5405			0.792958
$672$		0			0
$673$		−	11.1721i		−	0.430652i	−0.976542	−	0.215326i	$-0.930919\pi$
$673$		−	11.1721i		−	0.430652i	0.976542	−	0.215326i	$-0.0690815\pi$
$674$	20.6992			0.797303
$675$		0			0
$676$	36.1056			1.38868
$677$			14.4172i			0.554096i	0.960856	+	0.277048i	$0.0893562\pi$
$677$			14.4172i			0.554096i	−0.960856	+	0.277048i	$0.910644\pi$
$678$		0			0
$679$	18.5380			0.711421
$680$	8.67535	−	8.41152i	0.332684	−	0.322567i
$681$		0			0
$682$			2.47063i			0.0946053i
$683$			42.5368i			1.62763i	0.581127	+	0.813813i	$0.302612\pi$
$683$			42.5368i			1.62763i	−0.581127	+	0.813813i	$0.697388\pi$
$684$		0			0
$685$	16.1337	+	16.6397i	0.616436	+	0.635770i
$686$	6.61672			0.252628
$687$		0			0
$688$			5.72264i			0.218173i
$689$	46.2579			1.76229
$690$		0			0
$691$	−44.3143			−1.68579			−0.842897	−	0.538075i	$-0.819152\pi$
$691$	−44.3143			−1.68579			−0.842897	+	0.538075i	$0.819152\pi$
$692$		−	6.51908i		−	0.247818i
$693$		0			0
$694$	−33.0611			−1.25498
$695$	22.5152	−	21.8305i	0.854049	−	0.828077i
$696$		0			0
$697$		−	52.0412i		−	1.97120i
$698$			7.31198i			0.276763i
$699$		0			0
$700$	−17.3825	−	0.536906i	−0.656998	−	0.0202931i
$701$	−23.8627			−0.901281			−0.450641	−	0.892706i	$-0.648804\pi$
$701$	−23.8627			−0.901281			−0.450641	+	0.892706i	$0.648804\pi$
$702$		0			0
$703$			5.05874i			0.190794i
$704$	−2.47063			−0.0931154
$705$		0			0
$706$	21.7429			0.818303
$707$			1.04464i			0.0392876i
$708$		0			0
$709$	46.3627			1.74119			0.870594	−	0.492002i	$-0.163735\pi$
$709$	46.3627			1.74119			0.870594	+	0.492002i	$0.163735\pi$
$710$	−0.984560	−	1.01544i	−0.0369499	−	0.0381088i
$711$		0			0
$712$			15.1516i			0.567831i
$713$		−	0.365086i		−	0.0136726i
$714$		0			0
$715$	−26.9484	−	27.7937i	−1.00781	−	1.03942i
$716$	−14.2820			−0.533745
$717$		0			0
$718$			7.72054i			0.288128i
$719$	−6.20252			−0.231315			−0.115658	−	0.993289i	$-0.536898\pi$
$719$	−6.20252			−0.231315			−0.115658	+	0.993289i	$0.536898\pi$
$720$		0			0
$721$	−8.70936			−0.324354
$722$			18.7691i			0.698513i
$723$		0			0
$724$	6.92295			0.257290
$725$	−0.00153122	+	0.0495740i	−5.68683e−5	+	0.00184113i
$726$		0			0
$727$		−	19.6556i		−	0.728987i	−0.931206	−	0.364493i	$-0.881242\pi$
$727$		−	19.6556i		−	0.728987i	0.931206	−	0.364493i	$-0.118758\pi$
$728$		−	24.3734i		−	0.903337i
$729$		0			0
$730$	−5.15113	+	4.99448i	−0.190652	+	0.184854i
$731$	−30.9250			−1.14380
$732$		0			0
$733$		−	26.8898i		−	0.993198i	−0.867980	−	0.496599i	$-0.834582\pi$
$733$		−	26.8898i		−	0.993198i	0.867980	−	0.496599i	$-0.165418\pi$
$734$	−23.0325			−0.850146
$735$		0			0
$736$	0.365086			0.0134572
$737$		−	17.3326i		−	0.638453i
$738$		0			0
$739$	31.8249			1.17070			0.585350	−	0.810781i	$-0.300957\pi$
$739$	31.8249			1.17070			0.585350	+	0.810781i	$0.300957\pi$
$740$	−16.3857	−	16.8996i	−0.602350	−	0.621242i
$741$		0			0
$742$		−	22.9600i		−	0.842886i
$743$		−	11.7373i		−	0.430601i	−0.976548	−	0.215300i	$-0.930927\pi$
$743$		−	11.7373i		−	0.430601i	0.976548	−	0.215300i	$-0.0690730\pi$
$744$		0			0
$745$	32.9582	−	31.9560i	1.20750	−	1.17078i
$746$	−7.35528			−0.269296
$747$		0			0
$748$		−	13.3513i		−	0.488170i
$749$	−46.7357			−1.70769
$750$		0			0
$751$	0.486363			0.0177477			0.00887383	−	0.999961i	$-0.497175\pi$
$751$	0.486363			0.0177477			0.00887383	+	0.999961i	$0.497175\pi$
$752$			2.94328i			0.107330i
$753$		0			0
$754$	−0.0695115			−0.00253146
$755$	17.7952	−	17.2540i	0.647633	−	0.627938i
$756$		0			0
$757$			30.9080i			1.12337i	0.827351	+	0.561685i	$0.189847\pi$
$757$			30.9080i			1.12337i	−0.827351	+	0.561685i	$0.810153\pi$
$758$			19.3928i			0.704380i
$759$		0			0
$760$	0.747995	+	0.771455i	0.0271326	+	0.0279836i
$761$	−22.4602			−0.814180			−0.407090	−	0.913388i	$-0.633456\pi$
$761$	−22.4602			−0.814180			−0.407090	+	0.913388i	$0.633456\pi$
$762$		0			0
$763$			2.45806i			0.0889877i
$764$	−7.17783			−0.259685
$765$		0			0
$766$	13.1111			0.473722
$767$		−	76.7769i		−	2.77225i
$768$		0			0
$769$	−4.70873			−0.169801			−0.0849005	−	0.996389i	$-0.527057\pi$
$769$	−4.70873			−0.169801			−0.0849005	+	0.996389i	$0.527057\pi$
$770$	−13.7953	+	13.3758i	−0.497148	+	0.482029i
$771$		0			0
$772$			9.63958i			0.346936i
$773$		−	23.9122i		−	0.860061i	−0.902814	−	0.430031i	$-0.858503\pi$
$773$		−	23.9122i		−	0.860061i	0.902814	−	0.430031i	$-0.141497\pi$
$774$		0			0
$775$	−0.154365	+	4.99762i	−0.00554494	+	0.179520i
$776$	5.32981			0.191329
$777$		0			0
$778$			8.76355i			0.314189i
$779$	4.62776			0.165807
$780$		0			0
$781$	−1.56275			−0.0559196
$782$			1.97292i			0.0705514i
$783$		0			0
$784$	−5.09764			−0.182059
$785$	−23.9670	−	24.7187i	−0.855419	−	0.882249i
$786$		0			0
$787$		−	12.6874i		−	0.452255i	−0.974098	−	0.226128i	$-0.927393\pi$
$787$		−	12.6874i		−	0.452255i	0.974098	−	0.226128i	$-0.0726067\pi$
$788$			14.8309i			0.528331i
$789$		0			0
$790$	21.0305	+	21.6901i	0.748232	+	0.771700i
$791$	−26.2809			−0.934442
$792$		0			0
$793$			58.2598i			2.06887i
$794$	−22.2056			−0.788047
$795$		0			0
$796$	9.18726			0.325634
$797$			30.6460i			1.08554i	0.839883	+	0.542768i	$0.182624\pi$
$797$			30.6460i			1.08554i	−0.839883	+	0.542768i	$0.817376\pi$
$798$		0			0
$799$	−15.9054			−0.562693
$800$	−4.99762	−	0.154365i	−0.176692	−	0.00545762i
$801$		0			0
$802$		−	5.65942i		−	0.199841i
$803$			7.92752i			0.279756i
$804$		0			0
$805$	2.03853	−	1.97654i	0.0718488	−	0.0696639i
$806$	−7.00754			−0.246830
$807$		0			0
$808$			0.300341i			0.0105660i
$809$	53.8848			1.89449			0.947245	−	0.320510i	$-0.103854\pi$
$809$	53.8848			1.89449			0.947245	+	0.320510i	$0.103854\pi$
$810$		0			0
$811$	−6.42547			−0.225629			−0.112814	−	0.993616i	$-0.535987\pi$
$811$	−6.42547			−0.225629			−0.112814	+	0.993616i	$0.535987\pi$
$812$			0.0345018i			0.00121077i
$813$		0			0
$814$	−26.0083			−0.911590
$815$	−27.9127	−	28.7881i	−0.977738	−	1.00840i
$816$		0			0
$817$		−	2.75001i		−	0.0962107i
$818$		−	13.2801i		−	0.464329i
$819$		0			0
$820$	−15.4599	+	14.9897i	−0.539882	+	0.523463i
$821$	12.2283			0.426769			0.213384	−	0.976968i	$-0.431551\pi$
$821$	12.2283			0.426769			0.213384	+	0.976968i	$0.431551\pi$
$822$		0			0
$823$			33.0905i			1.15346i	0.816934	+	0.576731i	$0.195672\pi$
$823$			33.0905i			1.15346i	−0.816934	+	0.576731i	$0.804328\pi$
$824$	−2.50401			−0.0872313
$825$		0			0
$826$	−38.1079			−1.32595
$827$			21.6099i			0.751451i	0.926731	+	0.375725i	$0.122606\pi$
$827$			21.6099i			0.751451i	−0.926731	+	0.375725i	$0.877394\pi$
$828$		0			0
$829$	2.19681			0.0762984			0.0381492	−	0.999272i	$-0.487854\pi$
$829$	2.19681			0.0762984			0.0381492	+	0.999272i	$0.487854\pi$
$830$	−15.3567	+	14.8897i	−0.533039	+	0.516829i
$831$		0			0
$832$		−	7.00754i		−	0.242943i
$833$		−	27.5476i		−	0.954467i
$834$		0			0
$835$	20.1817	+	20.8147i	0.698418	+	0.720323i
$836$	1.18726			0.0410623
$837$		0			0
$838$		−	18.8268i		−	0.650363i
$839$	23.3786			0.807120			0.403560	−	0.914953i	$-0.367773\pi$
$839$	23.3786			0.807120			0.403560	+	0.914953i	$0.367773\pi$
$840$		0			0
$841$	−28.9999			−0.999997
$842$		−	16.6627i		−	0.574234i
$843$		0			0
$844$	−7.97177			−0.274400
$845$	57.9624	−	56.1998i	1.99397	−	1.93333i
$846$		0			0
$847$		−	17.0291i		−	0.585125i
$848$		−	6.60117i		−	0.226685i
$849$		0			0
$850$	0.834185	−	27.0071i	0.0286123	−	0.926334i
$851$	3.84325			0.131745
$852$		0			0
$853$		−	13.8395i		−	0.473854i	−0.971527	−	0.236927i	$-0.923860\pi$
$853$		−	13.8395i		−	0.473854i	0.971527	−	0.236927i	$-0.0761402\pi$
$854$	28.9171			0.989522
$855$		0			0
$856$	−13.4369			−0.459263
$857$			4.45306i			0.152114i	0.997103	+	0.0760568i	$0.0242330\pi$
$857$			4.45306i			0.152114i	−0.997103	+	0.0760568i	$0.975767\pi$
$858$		0			0
$859$	−39.0357			−1.33188			−0.665940	−	0.746005i	$-0.731969\pi$
$859$	−39.0357			−1.33188			−0.665940	+	0.746005i	$0.731969\pi$
$860$	8.90751	+	9.18689i	0.303744	+	0.313270i
$861$		0			0
$862$		−	30.2936i		−	1.03180i
$863$			2.85709i			0.0972563i	0.998817	+	0.0486282i	$0.0154849\pi$
$863$			2.85709i			0.0972563i	−0.998817	+	0.0486282i	$0.984515\pi$
$864$		0			0
$865$	−10.1472	−	10.4655i	−0.345016	−	0.355837i
$866$	10.7913			0.366703
$867$		0			0
$868$			3.47817i			0.118057i
$869$	33.3808			1.13237
$870$		0			0
$871$	49.1610			1.66576
$872$			0.706711i			0.0239322i
$873$		0			0
$874$	−0.175442			−0.00593441
$875$	−28.7410	+	26.1947i	−0.971622	+	0.885542i
$876$		0			0
$877$		−	8.39130i		−	0.283354i	−0.989913	−	0.141677i	$-0.954751\pi$
$877$		−	8.39130i		−	0.283354i	0.989913	−	0.141677i	$-0.0452494\pi$
$878$			8.47303i			0.285951i
$879$		0			0
$880$	−3.96625	+	3.84564i	−0.133702	+	0.129636i
$881$	−11.8276			−0.398483			−0.199241	−	0.979950i	$-0.563848\pi$
$881$	−11.8276			−0.398483			−0.199241	+	0.979950i	$0.563848\pi$
$882$		0			0
$883$			13.6326i			0.458775i	0.973335	+	0.229388i	$0.0736723\pi$
$883$			13.6326i			0.458775i	−0.973335	+	0.229388i	$0.926328\pi$
$884$	37.8686			1.27366
$885$		0			0
$886$	−35.8921			−1.20582
$887$			0.478944i			0.0160814i	0.999968	+	0.00804068i	$0.00255945\pi$
$887$			0.478944i			0.0160814i	−0.999968	+	0.00804068i	$0.997441\pi$
$888$		0			0
$889$	62.3944			2.09264
$890$	23.5841	+	24.3238i	0.790541	+	0.815336i
$891$		0			0
$892$			23.2190i			0.777430i
$893$		−	1.41439i		−	0.0473308i
$894$		0			0
$895$	−22.9278	+	22.2306i	−0.766392	+	0.743086i
$896$	−3.47817			−0.116197
$897$		0			0
$898$			17.3271i			0.578211i
$899$	0.00991953			0.000330835
$900$		0			0
$901$	35.6726			1.18843
$902$			23.7925i			0.792205i
$903$		0			0
$904$	−7.55597			−0.251308
$905$	11.1138	−	10.7759i	0.369436	−	0.358202i
$906$		0			0
$907$		−	18.7958i		−	0.624104i	−0.950065	−	0.312052i	$-0.898984\pi$
$907$		−	18.7958i		−	0.624104i	0.950065	−	0.312052i	$-0.101016\pi$
$908$			1.88252i			0.0624736i
$909$		0			0
$910$	−37.9381	−	39.1281i	−1.25764	−	1.29708i
$911$	−9.54043			−0.316089			−0.158044	−	0.987432i	$-0.550519\pi$
$911$	−9.54043			−0.316089			−0.158044	+	0.987432i	$0.550519\pi$
$912$		0			0
$913$			23.6338i			0.782164i
$914$	20.0674			0.663771
$915$		0			0
$916$	−11.9472			−0.394746
$917$		−	7.48495i		−	0.247175i
$918$		0			0
$919$	−0.788134			−0.0259981			−0.0129991	−	0.999916i	$-0.504138\pi$
$919$	−0.788134			−0.0259981			−0.0129991	+	0.999916i	$0.504138\pi$
$920$	0.586094	−	0.568271i	0.0193230	−	0.0187353i
$921$		0			0
$922$		−	10.2586i		−	0.337848i
$923$		−	4.43248i		−	0.145897i
$924$		0			0
$925$	−52.6098	−	1.62499i	−1.72980	−	0.0534295i
$926$	−0.766689			−0.0251950
$927$		0			0
$928$			0.00991953i			0.000325625i
$929$	−25.3062			−0.830271			−0.415136	−	0.909760i	$-0.636266\pi$
$929$	−25.3062			−0.830271			−0.415136	+	0.909760i	$0.636266\pi$
$930$		0			0
$931$	2.44967			0.0802847
$932$		−	10.4393i		−	0.341950i
$933$		0			0
$934$	21.4273			0.701124
$935$	−20.7818	−	21.4336i	−0.679636	−	0.700953i
$936$		0			0
$937$		−	51.1730i		−	1.67175i	−0.548922	−	0.835874i	$-0.684961\pi$
$937$		−	51.1730i		−	1.67175i	0.548922	−	0.835874i	$-0.315039\pi$
$938$		−	24.4009i		−	0.796717i
$939$		0			0
$940$	4.58133	+	4.72502i	0.149426	+	0.154113i
$941$	37.4102			1.21954			0.609769	−	0.792579i	$-0.291262\pi$
$941$	37.4102			1.21954			0.609769	+	0.792579i	$0.291262\pi$
$942$		0			0
$943$		−	3.51583i		−	0.114491i
$944$	−10.9563			−0.356598
$945$		0			0
$946$	14.1385			0.459683
$947$			38.3159i			1.24510i	0.782580	+	0.622550i	$0.213903\pi$
$947$			38.3159i			1.24510i	−0.782580	+	0.622550i	$0.786097\pi$
$948$		0			0
$949$	−22.4851			−0.729898
$950$	2.40160	+	0.0741799i	0.0779183	+	0.00240671i
$951$		0			0
$952$		−	18.7960i		−	0.609181i
$953$			7.42383i			0.240481i	0.992745	+	0.120241i	$0.0383666\pi$
$953$			7.42383i			0.240481i	−0.992745	+	0.120241i	$0.961633\pi$
$954$		0			0
$955$	−11.5230	+	11.1726i	−0.372875	+	0.361536i
$956$	−5.47615			−0.177111
$957$		0			0
$958$			29.6667i			0.958486i
$959$	36.0515			1.16416
$960$		0			0
$961$	1.00000			0.0322581
$962$		−	73.7682i		−	2.37838i
$963$		0			0
$964$	−5.73722			−0.184784
$965$	15.0044	+	15.4750i	0.483009	+	0.498158i
$966$		0			0
$967$			7.13090i			0.229314i	0.993405	+	0.114657i	$0.0365770\pi$
$967$			7.13090i			0.229314i	−0.993405	+	0.114657i	$0.963423\pi$
$968$		−	4.89599i		−	0.157363i
$969$		0			0
$970$	8.55626	−	8.29606i	0.274725	−	0.266370i
$971$	−28.9403			−0.928738			−0.464369	−	0.885642i	$-0.653719\pi$
$971$	−28.9403			−0.928738			−0.464369	+	0.885642i	$0.653719\pi$
$972$		0			0
$973$		−	48.7813i		−	1.56386i
$974$	25.8096			0.826994
$975$		0			0
$976$	8.31388			0.266121
$977$		−	33.7944i		−	1.08118i	−0.841286	−	0.540590i	$-0.818201\pi$
$977$		−	33.7944i		−	1.08118i	0.841286	−	0.540590i	$-0.181799\pi$
$978$		0			0
$979$	37.4340			1.19640
$980$	−8.18355	+	7.93468i	−0.261414	+	0.253464i
$981$		0			0
$982$		−	18.7250i		−	0.597539i
$983$			19.8142i			0.631973i	0.948764	+	0.315987i	$0.102335\pi$
$983$			19.8142i			0.631973i	−0.948764	+	0.315987i	$0.897665\pi$
$984$		0			0
$985$	23.0850	+	23.8090i	0.735548	+	0.758618i
$986$	−0.0536050			−0.00170713
$987$		0			0
$988$			3.36747i			0.107133i
$989$	−2.08925			−0.0664344
$990$		0			0
$991$	−60.3261			−1.91632			−0.958161	−	0.286229i	$-0.907598\pi$
$991$	−60.3261			−1.91632			−0.958161	+	0.286229i	$0.907598\pi$
$992$			1.00000i			0.0317500i
$993$		0			0
$994$	−2.20005			−0.0697813
$995$	14.7489	−	14.3003i	0.467570	−	0.453351i
$996$		0			0
$997$			21.3759i			0.676980i	0.940970	+	0.338490i	$0.109916\pi$
$997$			21.3759i			0.676980i	−0.940970	+	0.338490i	$0.890084\pi$
$998$			41.9722i			1.32861i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.d.m.559.1		8
3.2	odd	2		310.2.b.a.249.7	yes	8
5.4	even	2	inner	2790.2.d.m.559.5		8
12.11	even	2		2480.2.d.d.1489.2		8
15.2	even	4		1550.2.a.o.1.3		4
15.8	even	4		1550.2.a.p.1.2		4
15.14	odd	2		310.2.b.a.249.2	✓	8
60.59	even	2		2480.2.d.d.1489.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.b.a.249.2	✓	8	15.14	odd	2
310.2.b.a.249.7	yes	8	3.2	odd	2
1550.2.a.o.1.3		4	15.2	even	4
1550.2.a.p.1.2		4	15.8	even	4
2480.2.d.d.1489.2		8	12.11	even	2
2480.2.d.d.1489.7		8	60.59	even	2
2790.2.d.m.559.1		8	1.1	even	1	trivial
2790.2.d.m.559.5		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.60536 + 1.55654i) q^{5} +3.47817i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.60536 + 1.55654i) q^{5} +3.47817i q^{7} +1.00000i q^{8} +(1.55654 + 1.60536i) q^{10} +2.47063 q^{11} +7.00754i q^{13} +3.47817 q^{14} +1.00000 q^{16} +5.40399i q^{17} -0.480550 q^{19} +(1.60536 - 1.55654i) q^{20} -2.47063i q^{22} +0.365086i q^{23} +(0.154365 - 4.99762i) q^{25} +7.00754 q^{26} -3.47817i q^{28} -0.00991953 q^{29} -1.00000 q^{31} -1.00000i q^{32} +5.40399 q^{34} +(-5.41391 - 5.58371i) q^{35} -10.5270i q^{37} +0.480550i q^{38} +(-1.55654 - 1.60536i) q^{40} -9.63015 q^{41} +5.72264i q^{43} -2.47063 q^{44} +0.365086 q^{46} +2.94328i q^{47} -5.09764 q^{49} +(-4.99762 - 0.154365i) q^{50} -7.00754i q^{52} -6.60117i q^{53} +(-3.96625 + 3.84564i) q^{55} -3.47817 q^{56} +0.00991953i q^{58} -10.9563 q^{59} +8.31388 q^{61} +1.00000i q^{62} -1.00000 q^{64} +(-10.9075 - 11.2496i) q^{65} -7.01544i q^{67} -5.40399i q^{68} +(-5.58371 + 5.41391i) q^{70} -0.632531 q^{71} +3.20871i q^{73} -10.5270 q^{74} +0.480550 q^{76} +8.59326i q^{77} +13.5111 q^{79} +(-1.60536 + 1.55654i) q^{80} +9.63015i q^{82} +9.56589i q^{83} +(-8.41152 - 8.67535i) q^{85} +5.72264 q^{86} +2.47063i q^{88} +15.1516 q^{89} -24.3734 q^{91} -0.365086i q^{92} +2.94328 q^{94} +(0.771455 - 0.747995i) q^{95} -5.32981i q^{97} +5.09764i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 2 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 + 2 * q^5	\( 8 q - 8 q^{4} + 2 q^{5} + 2 q^{10} + 16 q^{11} - 12 q^{14} + 8 q^{16} - 12 q^{19} - 2 q^{20} + 12 q^{25} + 20 q^{26} - 12 q^{29} - 8 q^{31} + 8 q^{34} - 20 q^{35} - 2 q^{40} - 16 q^{44} - 16 q^{46} - 32 q^{49} + 8 q^{50} + 36 q^{55} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 8 q^{64} - 12 q^{65} - 20 q^{70} + 24 q^{71} - 40 q^{74} + 12 q^{76} + 32 q^{79} + 2 q^{80} + 4 q^{85} + 44 q^{86} + 24 q^{89} - 20 q^{91} + 4 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 + 2 * q^5 + 2 * q^10 + 16 * q^11 - 12 * q^14 + 8 * q^16 - 12 * q^19 - 2 * q^20 + 12 * q^25 + 20 * q^26 - 12 * q^29 - 8 * q^31 + 8 * q^34 - 20 * q^35 - 2 * q^40 - 16 * q^44 - 16 * q^46 - 32 * q^49 + 8 * q^50 + 36 * q^55 + 12 * q^56 - 8 * q^59 + 4 * q^61 - 8 * q^64 - 12 * q^65 - 20 * q^70 + 24 * q^71 - 40 * q^74 + 12 * q^76 + 32 * q^79 + 2 * q^80 + 4 * q^85 + 44 * q^86 + 24 * q^89 - 20 * q^91 + 4 * q^94

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