LMFDB - Embedded newform 2394.2.e.b.1063.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1063,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2394.1063");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.e (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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{11} + 2x^{10} + 54x^{8} - 114x^{7} + 120x^{6} + 46x^{5} + 9x^{4} - 4x^{3} + 8x^{2} + 4x + 1 $ x^12 - 2x^11 + 2x^10 + 54x^8 - 114x^7 + 120x^6 + 46x^5 + 9x^4 - 4x^3 + 8x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 798)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1063.12
Root			$-1.93391 - 1.93391i$ of defining polynomial
Character	$\chi$	$=$	2394.1063
Dual form			2394.2.e.b.1063.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +3.86783i q^{5} +(1.61372 - 2.09664i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +3.86783i q^{5} +(1.61372 - 2.09664i) q^{7} -1.00000i q^{8} -3.86783 q^{10} -1.10913 q^{11} +5.86783 q^{13} +(2.09664 + 1.61372i) q^{14} +1.00000 q^{16} -5.87593i q^{17} +(0.266595 - 4.35074i) q^{19} -3.86783i q^{20} -1.10913i q^{22} +7.40717 q^{23} -9.96007 q^{25} +5.86783i q^{26} +(-1.61372 + 2.09664i) q^{28} -3.97501i q^{29} +3.04228 q^{31} +1.00000i q^{32} +5.87593 q^{34} +(8.10943 + 6.24161i) q^{35} -2.18517i q^{37} +(4.35074 + 0.266595i) q^{38} +3.86783 q^{40} +11.1371 q^{41} +1.61976 q^{43} +1.10913 q^{44} +7.40717i q^{46} -9.52549i q^{47} +(-1.79178 - 6.76679i) q^{49} -9.96007i q^{50} -5.86783 q^{52} -3.47403i q^{53} -4.28993i q^{55} +(-2.09664 - 1.61372i) q^{56} +3.97501 q^{58} -14.2408 q^{59} -7.08172i q^{61} +3.04228i q^{62} -1.00000 q^{64} +22.6957i q^{65} +4.38988i q^{67} +5.87593i q^{68} +(-6.24161 + 8.10943i) q^{70} +6.72647i q^{71} -3.51435i q^{73} +2.18517 q^{74} +(-0.266595 + 4.35074i) q^{76} +(-1.78984 + 2.32545i) q^{77} +14.8117i q^{79} +3.86783i q^{80} +11.1371i q^{82} +1.08960i q^{83} +22.7271 q^{85} +1.61976i q^{86} +1.10913i q^{88} -4.98419 q^{89} +(9.46905 - 12.3027i) q^{91} -7.40717 q^{92} +9.52549 q^{94} +(16.8279 + 1.03114i) q^{95} -17.2967 q^{97} +(6.76679 - 1.79178i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4}+O(q^{10}) $ 12 * q - 12 * q^4	$ 12 q - 12 q^{4} + 4 q^{10} - 12 q^{11} + 20 q^{13} + 4 q^{14} + 12 q^{16} - 8 q^{19} + 12 q^{23} - 12 q^{25} + 24 q^{31} + 4 q^{34} - 4 q^{35} - 4 q^{40} + 16 q^{41} + 12 q^{44} + 4 q^{49} - 20 q^{52} - 4 q^{56} + 8 q^{58} - 40 q^{59} - 12 q^{64} - 24 q^{70} + 8 q^{76} - 8 q^{77} + 8 q^{85} + 16 q^{89} + 24 q^{91} - 12 q^{92} + 44 q^{95} - 60 q^{97} + 16 q^{98}+O(q^{100}) $ 12 * q - 12 * q^4 + 4 * q^10 - 12 * q^11 + 20 * q^13 + 4 * q^14 + 12 * q^16 - 8 * q^19 + 12 * q^23 - 12 * q^25 + 24 * q^31 + 4 * q^34 - 4 * q^35 - 4 * q^40 + 16 * q^41 + 12 * q^44 + 4 * q^49 - 20 * q^52 - 4 * q^56 + 8 * q^58 - 40 * q^59 - 12 * q^64 - 24 * q^70 + 8 * q^76 - 8 * q^77 + 8 * q^85 + 16 * q^89 + 24 * q^91 - 12 * q^92 + 44 * q^95 - 60 * q^97 + 16 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$			3.86783i			1.72974i	0.501992	+	0.864872i	$0.332601\pi$
$5$			3.86783i			1.72974i	−0.501992	+	0.864872i	$0.667399\pi$
$6$		0			0
$7$	1.61372	−	2.09664i	0.609931	−	0.792455i
$7$	1.61372	−	2.09664i	0.609931	−	0.792455i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−3.86783			−1.22311
$11$	−1.10913			−0.334416			−0.167208	−	0.985922i	$-0.553475\pi$
$11$	−1.10913			−0.334416			−0.167208	+	0.985922i	$0.553475\pi$
$12$		0			0
$13$	5.86783			1.62744			0.813721	−	0.581256i	$-0.197438\pi$
$13$	5.86783			1.62744			0.813721	+	0.581256i	$0.197438\pi$
$14$	2.09664	+	1.61372i	0.560350	+	0.431286i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	5.87593i		−	1.42512i	−0.701610	−	0.712561i	$-0.747535\pi$
$17$		−	5.87593i		−	1.42512i	0.701610	−	0.712561i	$-0.252465\pi$
$18$		0			0
$19$	0.266595	−	4.35074i	0.0611612	−	0.998128i
$19$	0.266595	−	4.35074i	0.0611612	−	0.998128i
$20$		−	3.86783i		−	0.864872i
$21$		0			0
$22$		−	1.10913i		−	0.236468i
$23$	7.40717			1.54450			0.772251	−	0.635318i	$-0.219131\pi$
$23$	7.40717			1.54450			0.772251	+	0.635318i	$0.219131\pi$
$24$		0			0
$25$	−9.96007			−1.99201
$26$			5.86783i			1.15078i
$27$		0			0
$28$	−1.61372	+	2.09664i	−0.304965	+	0.396227i
$29$		−	3.97501i		−	0.738141i	−0.929401	−	0.369070i	$-0.879676\pi$
$29$		−	3.97501i		−	0.738141i	0.929401	−	0.369070i	$-0.120324\pi$
$30$		0			0
$31$	3.04228			0.546409			0.273204	−	0.961956i	$-0.411916\pi$
$31$	3.04228			0.546409			0.273204	+	0.961956i	$0.411916\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	5.87593			1.00771
$35$	8.10943	+	6.24161i	1.37074	+	1.05502i
$36$		0			0
$37$		−	2.18517i		−	0.359240i	−0.983736	−	0.179620i	$-0.942513\pi$
$37$		−	2.18517i		−	0.359240i	0.983736	−	0.179620i	$-0.0574869\pi$
$38$	4.35074	+	0.266595i	0.705783	+	0.0432475i
$39$		0			0
$40$	3.86783			0.611557
$41$	11.1371			1.73933			0.869665	−	0.493643i	$-0.164335\pi$
$41$	11.1371			1.73933			0.869665	+	0.493643i	$0.164335\pi$
$42$		0			0
$43$	1.61976			0.247011			0.123505	−	0.992344i	$-0.460586\pi$
$43$	1.61976			0.247011			0.123505	+	0.992344i	$0.460586\pi$
$44$	1.10913			0.167208
$45$		0			0
$46$			7.40717i			1.09213i
$47$		−	9.52549i		−	1.38943i	−0.719283	−	0.694717i	$-0.755529\pi$
$47$		−	9.52549i		−	1.38943i	0.719283	−	0.694717i	$-0.244471\pi$
$48$		0			0
$49$	−1.79178	−	6.76679i	−0.255969	−	0.966685i
$50$		−	9.96007i		−	1.40857i
$51$		0			0
$52$	−5.86783			−0.813721
$53$		−	3.47403i		−	0.477194i	−0.971119	−	0.238597i	$-0.923312\pi$
$53$		−	3.47403i		−	0.477194i	0.971119	−	0.238597i	$-0.0766876\pi$
$54$		0			0
$55$		−	4.28993i		−	0.578455i
$56$	−2.09664	−	1.61372i	−0.280175	−	0.215643i
$57$		0			0
$58$	3.97501			0.521944
$59$	−14.2408			−1.85400			−0.926999	−	0.375065i	$-0.877620\pi$
$59$	−14.2408			−1.85400			−0.926999	+	0.375065i	$0.877620\pi$
$60$		0			0
$61$		−	7.08172i		−	0.906721i	−0.891327	−	0.453361i	$-0.850225\pi$
$61$		−	7.08172i		−	0.906721i	0.891327	−	0.453361i	$-0.149775\pi$
$62$			3.04228i			0.386369i
$63$		0			0
$64$	−1.00000			−0.125000
$65$			22.6957i			2.81506i
$66$		0			0
$67$			4.38988i			0.536310i	0.963376	+	0.268155i	$0.0864139\pi$
$67$			4.38988i			0.536310i	−0.963376	+	0.268155i	$0.913586\pi$
$68$			5.87593i			0.712561i
$69$		0			0
$70$	−6.24161	+	8.10943i	−0.746015	+	0.969262i
$71$			6.72647i			0.798285i	0.916889	+	0.399142i	$0.130692\pi$
$71$			6.72647i			0.798285i	−0.916889	+	0.399142i	$0.869308\pi$
$72$		0			0
$73$		−	3.51435i		−	0.411324i	−0.978623	−	0.205662i	$-0.934065\pi$
$73$		−	3.51435i		−	0.411324i	0.978623	−	0.205662i	$-0.0659348\pi$
$74$	2.18517			0.254021
$75$		0			0
$76$	−0.266595	+	4.35074i	−0.0305806	+	0.499064i
$77$	−1.78984	+	2.32545i	−0.203971	+	0.265010i
$78$		0			0
$79$			14.8117i			1.66645i	0.552938	+	0.833223i	$0.313507\pi$
$79$			14.8117i			1.66645i	−0.552938	+	0.833223i	$0.686493\pi$
$80$			3.86783i			0.432436i
$81$		0			0
$82$			11.1371i			1.22989i
$83$			1.08960i			0.119599i	0.998210	+	0.0597994i	$0.0190461\pi$
$83$			1.08960i			0.119599i	−0.998210	+	0.0597994i	$0.980954\pi$
$84$		0			0
$85$	22.7271			2.46510
$86$			1.61976i			0.174663i
$87$		0			0
$88$			1.10913i			0.118234i
$89$	−4.98419			−0.528323			−0.264162	−	0.964478i	$-0.585095\pi$
$89$	−4.98419			−0.528323			−0.264162	+	0.964478i	$0.585095\pi$
$90$		0			0
$91$	9.46905	−	12.3027i	0.992627	−	1.28967i
$92$	−7.40717			−0.772251
$93$		0			0
$94$	9.52549			0.982479
$95$	16.8279	+	1.03114i	1.72651	+	0.105793i
$96$		0			0
$97$	−17.2967			−1.75621			−0.878105	−	0.478469i	$-0.841192\pi$
$97$	−17.2967			−1.75621			−0.878105	+	0.478469i	$0.841192\pi$
$98$	6.76679	−	1.79178i	0.683549	−	0.180998i
$99$		0			0
$100$	9.96007			0.996007
$101$		−	7.81784i		−	0.777905i	−0.921258	−	0.388952i	$-0.872837\pi$
$101$		−	7.81784i		−	0.777905i	0.921258	−	0.388952i	$-0.127163\pi$
$102$		0			0
$103$	17.1928			1.69406			0.847028	−	0.531548i	$-0.178390\pi$
$103$	17.1928			1.69406			0.847028	+	0.531548i	$0.178390\pi$
$104$		−	5.86783i		−	0.575388i
$105$		0			0
$106$	3.47403			0.337427
$107$			3.36530i			0.325336i	0.986681	+	0.162668i	$0.0520100\pi$
$107$			3.36530i			0.325336i	−0.986681	+	0.162668i	$0.947990\pi$
$108$		0			0
$109$			16.5386i			1.58411i	0.610451	+	0.792054i	$0.290988\pi$
$109$			16.5386i			1.58411i	−0.610451	+	0.792054i	$0.709012\pi$
$110$	4.28993			0.409029
$111$		0			0
$112$	1.61372	−	2.09664i	0.152483	−	0.198114i
$113$		−	8.62894i		−	0.811743i	−0.913930	−	0.405871i	$-0.866968\pi$
$113$		−	8.62894i		−	0.811743i	0.913930	−	0.405871i	$-0.133032\pi$
$114$		0			0
$115$			28.6496i			2.67159i
$116$			3.97501i			0.369070i
$117$		0			0
$118$		−	14.2408i		−	1.31097i
$119$	−12.3197	−	9.48213i	−1.12934	−	0.869225i
$120$		0			0
$121$	−9.76982			−0.888166
$122$	7.08172			0.641149
$123$		0			0
$124$	−3.04228			−0.273204
$125$		−	19.1847i		−	1.71593i
$126$		0			0
$127$			2.44619i			0.217064i	0.994093	+	0.108532i	$0.0346151\pi$
$127$			2.44619i			0.217064i	−0.994093	+	0.108532i	$0.965385\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−22.6957			−1.99055
$131$			17.0880i			1.49299i	0.665392	+	0.746494i	$0.268264\pi$
$131$			17.0880i			1.49299i	−0.665392	+	0.746494i	$0.731736\pi$
$132$		0			0
$133$	−8.69171	−	7.57985i	−0.753667	−	0.657256i
$134$	−4.38988			−0.379228
$135$		0			0
$136$	−5.87593			−0.503857
$137$	−0.617742			−0.0527773			−0.0263886	−	0.999652i	$-0.508401\pi$
$137$	−0.617742			−0.0527773			−0.0263886	+	0.999652i	$0.508401\pi$
$138$		0			0
$139$			11.9780i			1.01596i	0.861368	+	0.507982i	$0.169608\pi$
$139$			11.9780i			1.01596i	−0.861368	+	0.507982i	$0.830392\pi$
$140$	−8.10943	−	6.24161i	−0.685372	−	0.527512i
$141$		0			0
$142$	−6.72647			−0.564473
$143$	−6.50820			−0.544243
$144$		0			0
$145$	15.3746			1.27679
$146$	3.51435			0.290850
$147$		0			0
$148$			2.18517i			0.179620i
$149$	0.449524			0.0368264			0.0184132	−	0.999830i	$-0.494139\pi$
$149$	0.449524			0.0368264			0.0184132	+	0.999830i	$0.494139\pi$
$150$		0			0
$151$			10.3107i			0.839074i	0.907738	+	0.419537i	$0.137807\pi$
$151$			10.3107i			0.839074i	−0.907738	+	0.419537i	$0.862193\pi$
$152$	−4.35074	−	0.266595i	−0.352892	−	0.0216237i
$153$		0			0
$154$	−2.32545	−	1.78984i	−0.187390	−	0.144229i
$155$			11.7670i			0.945147i
$156$		0			0
$157$		−	16.1342i		−	1.28765i	−0.765172	−	0.643825i	$-0.777346\pi$
$157$		−	16.1342i		−	1.28765i	0.765172	−	0.643825i	$-0.222654\pi$
$158$	−14.8117			−1.17835
$159$		0			0
$160$	−3.86783			−0.305778
$161$	11.9531	−	15.5302i	0.942039	−	1.22395i
$162$		0			0
$163$	16.1060			1.26152			0.630760	−	0.775978i	$-0.282743\pi$
$163$	16.1060			1.26152			0.630760	+	0.775978i	$0.282743\pi$
$164$	−11.1371			−0.869665
$165$		0			0
$166$	−1.08960			−0.0845691
$167$	−3.37953			−0.261516			−0.130758	−	0.991414i	$-0.541741\pi$
$167$	−3.37953			−0.261516			−0.130758	+	0.991414i	$0.541741\pi$
$168$		0			0
$169$	21.4314			1.64857
$170$			22.7271i			1.74309i
$171$		0			0
$172$	−1.61976			−0.123505
$173$	17.8055			1.35372			0.676862	−	0.736110i	$-0.263339\pi$
$173$	17.8055			1.35372			0.676862	+	0.736110i	$0.263339\pi$
$174$		0			0
$175$	−16.0728	+	20.8827i	−1.21499	+	1.57858i
$176$	−1.10913			−0.0836041
$177$		0			0
$178$		−	4.98419i		−	0.373581i
$179$			18.4525i			1.37920i	0.724190	+	0.689601i	$0.242214\pi$
$179$			18.4525i			1.37920i	−0.724190	+	0.689601i	$0.757786\pi$
$180$		0			0
$181$	0.736995			0.0547804			0.0273902	−	0.999625i	$-0.491280\pi$
$181$	0.736995			0.0547804			0.0273902	+	0.999625i	$0.491280\pi$
$182$	12.3027	+	9.46905i	0.911937	+	0.701893i
$183$		0			0
$184$		−	7.40717i		−	0.546064i
$185$	8.45187			0.621394
$186$		0			0
$187$			6.51719i			0.476584i
$188$			9.52549i			0.694717i
$189$		0			0
$190$	−1.03114	+	16.8279i	−0.0748071	+	1.22082i
$191$	−12.9745			−0.938804			−0.469402	−	0.882984i	$-0.655531\pi$
$191$	−12.9745			−0.938804			−0.469402	+	0.882984i	$0.655531\pi$
$192$		0			0
$193$		−	14.5678i		−	1.04861i	−0.851530	−	0.524305i	$-0.824325\pi$
$193$		−	14.5678i		−	1.04861i	0.851530	−	0.524305i	$-0.175675\pi$
$194$		−	17.2967i		−	1.24183i
$195$		0			0
$196$	1.79178	+	6.76679i	0.127985	+	0.483342i
$197$	−9.09886			−0.648267			−0.324133	−	0.946011i	$-0.605073\pi$
$197$	−9.09886			−0.648267			−0.324133	+	0.946011i	$0.605073\pi$
$198$		0			0
$199$		−	14.9321i		−	1.05851i	−0.848464	−	0.529253i	$-0.822472\pi$
$199$		−	14.9321i		−	1.05851i	0.848464	−	0.529253i	$-0.177528\pi$
$200$			9.96007i			0.704283i
$201$		0			0
$202$	7.81784			0.550062
$203$	−8.33416	−	6.41457i	−0.584943	−	0.450215i
$204$		0			0
$205$			43.0765i			3.00859i
$206$			17.1928i			1.19788i
$207$		0			0
$208$	5.86783			0.406860
$209$	−0.295690	+	4.82555i	−0.0204533	+	0.333790i
$210$		0			0
$211$			2.52268i			0.173668i	0.996223	+	0.0868342i	$0.0276750\pi$
$211$			2.52268i			0.173668i	−0.996223	+	0.0868342i	$0.972325\pi$
$212$			3.47403i			0.238597i
$213$		0			0
$214$	−3.36530			−0.230047
$215$			6.26494i			0.427266i
$216$		0			0
$217$	4.90940	−	6.37855i	0.333271	−	0.433004i
$218$	−16.5386			−1.12013
$219$		0			0
$220$			4.28993i			0.289227i
$221$		−	34.4789i		−	2.31930i
$222$		0			0
$223$	15.6221			1.04614			0.523068	−	0.852291i	$-0.324787\pi$
$223$	15.6221			1.04614			0.523068	+	0.852291i	$0.324787\pi$
$224$	2.09664	+	1.61372i	0.140088	+	0.107822i
$225$		0			0
$226$	8.62894			0.573989
$227$	13.3153			0.883769			0.441884	−	0.897072i	$-0.354310\pi$
$227$	13.3153			0.883769			0.441884	+	0.897072i	$0.354310\pi$
$228$		0			0
$229$			15.1805i			1.00316i	0.865113	+	0.501578i	$0.167247\pi$
$229$			15.1805i			1.00316i	−0.865113	+	0.501578i	$0.832753\pi$
$230$	−28.6496			−1.88910
$231$		0			0
$232$	−3.97501			−0.260972
$233$	−26.2444			−1.71933			−0.859664	−	0.510860i	$-0.829327\pi$
$233$	−26.2444			−1.71933			−0.859664	+	0.510860i	$0.829327\pi$
$234$		0			0
$235$	36.8429			2.40337
$236$	14.2408			0.926999
$237$		0			0
$238$	9.48213	−	12.3197i	0.614635	−	0.798567i
$239$	−10.5130			−0.680028			−0.340014	−	0.940420i	$-0.610432\pi$
$239$	−10.5130			−0.680028			−0.340014	+	0.940420i	$0.610432\pi$
$240$		0			0
$241$	−13.8076			−0.889425			−0.444712	−	0.895673i	$-0.646694\pi$
$241$	−13.8076			−0.889425			−0.444712	+	0.895673i	$0.646694\pi$
$242$		−	9.76982i		−	0.628028i
$243$		0			0
$244$			7.08172i			0.453361i
$245$	26.1728	−	6.93031i	1.67212	−	0.442761i
$246$		0			0
$247$	1.56434	−	25.5294i	0.0995363	−	1.62440i
$248$		−	3.04228i		−	0.193185i
$249$		0			0
$250$	19.1847			1.21335
$251$			11.3936i			0.719159i	0.933115	+	0.359579i	$0.117080\pi$
$251$			11.3936i			0.719159i	−0.933115	+	0.359579i	$0.882920\pi$
$252$		0			0
$253$	−8.21554			−0.516507
$254$	−2.44619			−0.153488
$255$		0			0
$256$	1.00000			0.0625000
$257$	−12.0629			−0.752462			−0.376231	−	0.926526i	$-0.622780\pi$
$257$	−12.0629			−0.752462			−0.376231	+	0.926526i	$0.622780\pi$
$258$		0			0
$259$	−4.58152	−	3.52627i	−0.284682	−	0.219112i
$260$		−	22.6957i		−	1.40753i
$261$		0			0
$262$	−17.0880			−1.05570
$263$	10.2264			0.630584			0.315292	−	0.948995i	$-0.397897\pi$
$263$	10.2264			0.630584			0.315292	+	0.948995i	$0.397897\pi$
$264$		0			0
$265$	13.4369			0.825424
$266$	7.57985	−	8.69171i	0.464750	−	0.532923i
$267$		0			0
$268$		−	4.38988i		−	0.268155i
$269$	−3.72989			−0.227416			−0.113708	−	0.993514i	$-0.536273\pi$
$269$	−3.72989			−0.227416			−0.113708	+	0.993514i	$0.536273\pi$
$270$		0			0
$271$			13.1874i			0.801078i	0.916280	+	0.400539i	$0.131177\pi$
$271$			13.1874i			0.801078i	−0.916280	+	0.400539i	$0.868823\pi$
$272$		−	5.87593i		−	0.356280i
$273$		0			0
$274$		−	0.617742i		−	0.0373192i
$275$	11.0470			0.666162
$276$		0			0
$277$	11.5498			0.693960			0.346980	−	0.937873i	$-0.387207\pi$
$277$	11.5498			0.693960			0.346980	+	0.937873i	$0.387207\pi$
$278$	−11.9780			−0.718395
$279$		0			0
$280$	6.24161	−	8.10943i	0.373007	−	0.484631i
$281$			3.76367i			0.224522i	0.993679	+	0.112261i	$0.0358092\pi$
$281$			3.76367i			0.224522i	−0.993679	+	0.112261i	$0.964191\pi$
$282$		0			0
$283$			1.47490i			0.0876735i	0.999039	+	0.0438367i	$0.0139581\pi$
$283$			1.47490i			0.0876735i	−0.999039	+	0.0438367i	$0.986042\pi$
$284$		−	6.72647i		−	0.399142i
$285$		0			0
$286$		−	6.50820i		−	0.384838i
$287$	17.9723	−	23.3506i	1.06087	−	1.37834i
$288$		0			0
$289$	−17.5265			−1.03097
$290$			15.3746i			0.902830i
$291$		0			0
$292$			3.51435i			0.205662i
$293$	13.7598			0.803854			0.401927	−	0.915672i	$-0.368340\pi$
$293$	13.7598			0.803854			0.401927	+	0.915672i	$0.368340\pi$
$294$		0			0
$295$		−	55.0810i		−	3.20694i
$296$	−2.18517			−0.127011
$297$		0			0
$298$			0.449524i			0.0260402i
$299$	43.4640			2.51359
$300$		0			0
$301$	2.61384	−	3.39605i	0.150660	−	0.195745i
$302$	−10.3107			−0.593315
$303$		0			0
$304$	0.266595	−	4.35074i	0.0152903	−	0.249532i
$305$	27.3908			1.56840
$306$		0			0
$307$	−0.112863			−0.00644142			−0.00322071	−	0.999995i	$-0.501025\pi$
$307$	−0.112863			−0.00644142			−0.00322071	+	0.999995i	$0.501025\pi$
$308$	1.78984	−	2.32545i	0.101985	−	0.132505i
$309$		0			0
$310$	−11.7670			−0.668320
$311$			14.3115i			0.811532i	0.913977	+	0.405766i	$0.132995\pi$
$311$			14.3115i			0.811532i	−0.913977	+	0.405766i	$0.867005\pi$
$312$		0			0
$313$			10.6424i			0.601545i	0.953696	+	0.300773i	$0.0972445\pi$
$313$			10.6424i			0.601545i	−0.953696	+	0.300773i	$0.902755\pi$
$314$	16.1342			0.910507
$315$		0			0
$316$		−	14.8117i		−	0.833223i
$317$		−	10.3588i		−	0.581810i	−0.956752	−	0.290905i	$-0.906044\pi$
$317$		−	10.3588i		−	0.581810i	0.956752	−	0.290905i	$-0.0939564\pi$
$318$		0			0
$319$			4.40882i			0.246846i
$320$		−	3.86783i		−	0.216218i
$321$		0			0
$322$	15.5302	+	11.9531i	0.865462	+	0.666122i
$323$	−25.5646	−	1.56650i	−1.42245	−	0.0871621i
$324$		0			0
$325$	−58.4440			−3.24189
$326$			16.1060i			0.892029i
$327$		0			0
$328$		−	11.1371i		−	0.614946i
$329$	−19.9715	−	15.3715i	−1.10106	−	0.847459i
$330$		0			0
$331$		−	0.413713i		−	0.0227397i	−0.999935	−	0.0113699i	$-0.996381\pi$
$331$		−	0.413713i		−	0.0227397i	0.999935	−	0.0113699i	$-0.00361922\pi$
$332$		−	1.08960i		−	0.0597994i
$333$		0			0
$334$		−	3.37953i		−	0.184920i
$335$	−16.9793			−0.927679
$336$		0			0
$337$			7.89183i			0.429895i	0.976626	+	0.214948i	$0.0689581\pi$
$337$			7.89183i			0.429895i	−0.976626	+	0.214948i	$0.931042\pi$
$338$			21.4314i			1.16571i
$339$		0			0
$340$	−22.7271			−1.23255
$341$	−3.37429			−0.182728
$342$		0			0
$343$	−17.0790	−	7.16302i	−0.922178	−	0.386767i
$344$		−	1.61976i		−	0.0873315i
$345$		0			0
$346$			17.8055i			0.957227i
$347$	15.2726			0.819875			0.409937	−	0.912114i	$-0.365551\pi$
$347$	15.2726			0.819875			0.409937	+	0.912114i	$0.365551\pi$
$348$		0			0
$349$		−	17.0669i		−	0.913569i	−0.889577	−	0.456785i	$-0.849001\pi$
$349$		−	17.0669i		−	0.913569i	0.889577	−	0.456785i	$-0.150999\pi$
$350$	−20.8827	−	16.0728i	−1.11623	−	0.859128i
$351$		0			0
$352$		−	1.10913i		−	0.0591170i
$353$			18.8017i			1.00072i	0.865819	+	0.500358i	$0.166798\pi$
$353$			18.8017i			1.00072i	−0.865819	+	0.500358i	$0.833202\pi$
$354$		0			0
$355$	−26.0168			−1.38083
$356$	4.98419			0.264162
$357$		0			0
$358$	−18.4525			−0.975243
$359$	2.91401			0.153796			0.0768979	−	0.997039i	$-0.475498\pi$
$359$	2.91401			0.153796			0.0768979	+	0.997039i	$0.475498\pi$
$360$		0			0
$361$	−18.8579	−	2.31977i	−0.992519	−	0.122093i
$362$			0.736995i			0.0387356i
$363$		0			0
$364$	−9.46905	+	12.3027i	−0.496313	+	0.644837i
$365$	13.5929			0.711485
$366$		0			0
$367$		−	15.5472i		−	0.811558i	−0.913971	−	0.405779i	$-0.867000\pi$
$367$		−	15.5472i		−	0.811558i	0.913971	−	0.405779i	$-0.133000\pi$
$368$	7.40717			0.386125
$369$		0			0
$370$			8.45187i			0.439392i
$371$	−7.28378	−	5.60612i	−0.378155	−	0.291055i
$372$		0			0
$373$			8.51025i			0.440644i	0.975427	+	0.220322i	$0.0707108\pi$
$373$			8.51025i			0.440644i	−0.975427	+	0.220322i	$0.929289\pi$
$374$	−6.51719			−0.336996
$375$		0			0
$376$	−9.52549			−0.491239
$377$		−	23.3247i		−	1.20128i
$378$		0			0
$379$			18.0541i			0.927377i	0.885998	+	0.463689i	$0.153474\pi$
$379$			18.0541i			0.927377i	−0.885998	+	0.463689i	$0.846526\pi$
$380$	−16.8279	−	1.03114i	−0.863253	−	0.0528966i
$381$		0			0
$382$		−	12.9745i		−	0.663835i
$383$	−7.00586			−0.357983			−0.178991	−	0.983851i	$-0.557283\pi$
$383$	−7.00586			−0.357983			−0.178991	+	0.983851i	$0.557283\pi$
$384$		0			0
$385$	−8.99444	−	6.92277i	−0.458399	−	0.352817i
$386$	14.5678			0.741480
$387$		0			0
$388$	17.2967			0.878105
$389$	27.8125			1.41015			0.705074	−	0.709134i	$-0.250914\pi$
$389$	27.8125			1.41015			0.705074	+	0.709134i	$0.250914\pi$
$390$		0			0
$391$		−	43.5240i		−	2.20110i
$392$	−6.76679	+	1.79178i	−0.341775	+	0.0904988i
$393$		0			0
$394$		−	9.09886i		−	0.458394i
$395$	−57.2890			−2.88252
$396$		0			0
$397$		−	9.88942i		−	0.496336i	−0.968717	−	0.248168i	$-0.920172\pi$
$397$		−	9.88942i		−	0.496336i	0.968717	−	0.248168i	$-0.0798284\pi$
$398$	14.9321			0.748476
$399$		0			0
$400$	−9.96007			−0.498004
$401$			5.03716i			0.251544i	0.992059	+	0.125772i	$0.0401408\pi$
$401$			5.03716i			0.251544i	−0.992059	+	0.125772i	$0.959859\pi$
$402$		0			0
$403$	17.8515			0.889249
$404$			7.81784i			0.388952i
$405$		0			0
$406$	6.41457	−	8.33416i	0.318350	−	0.413617i
$407$			2.42365i			0.120136i
$408$		0			0
$409$	−9.92650			−0.490834			−0.245417	−	0.969418i	$-0.578925\pi$
$409$	−9.92650			−0.490834			−0.245417	+	0.969418i	$0.578925\pi$
$410$	−43.0765			−2.12740
$411$		0			0
$412$	−17.1928			−0.847028
$413$	−22.9808	+	29.8579i	−1.13081	+	1.46921i
$414$		0			0
$415$	−4.21437			−0.206875
$416$			5.86783i			0.287694i
$417$		0			0
$418$	−4.82555	−	0.295690i	−0.236025	−	0.0144627i
$419$		−	0.578669i		−	0.0282698i	−0.999900	−	0.0141349i	$-0.995501\pi$
$419$		−	0.578669i		−	0.0282698i	0.999900	−	0.0141349i	$-0.00449943\pi$
$420$		0			0
$421$			6.89896i			0.336235i	0.985767	+	0.168117i	$0.0537688\pi$
$421$			6.89896i			0.336235i	−0.985767	+	0.168117i	$0.946231\pi$
$422$	−2.52268			−0.122802
$423$		0			0
$424$	−3.47403			−0.168714
$425$			58.5247i			2.83886i
$426$		0			0
$427$	−14.8478	−	11.4279i	−0.718536	−	0.553037i
$428$		−	3.36530i		−	0.162668i
$429$		0			0
$430$	−6.26494			−0.302122
$431$		−	34.2796i		−	1.65119i	−0.564264	−	0.825595i	$-0.690840\pi$
$431$		−	34.2796i		−	1.65119i	0.564264	−	0.825595i	$-0.309160\pi$
$432$		0			0
$433$	−25.1600			−1.20911			−0.604555	−	0.796563i	$-0.706649\pi$
$433$	−25.1600			−1.20911			−0.604555	+	0.796563i	$0.706649\pi$
$434$	6.37855	+	4.90940i	0.306180	+	0.235659i
$435$		0			0
$436$		−	16.5386i		−	0.792054i
$437$	1.97472	−	32.2267i	0.0944635	−	1.54161i
$438$		0			0
$439$	−8.44013			−0.402826			−0.201413	−	0.979506i	$-0.564553\pi$
$439$	−8.44013			−0.402826			−0.201413	+	0.979506i	$0.564553\pi$
$440$	−4.28993			−0.204515
$441$		0			0
$442$	34.4789			1.63999
$443$	3.07126			0.145920			0.0729600	−	0.997335i	$-0.476755\pi$
$443$	3.07126			0.145920			0.0729600	+	0.997335i	$0.476755\pi$
$444$		0			0
$445$		−	19.2780i		−	0.913864i
$446$			15.6221i			0.739730i
$447$		0			0
$448$	−1.61372	+	2.09664i	−0.0762413	+	0.0990569i
$449$		−	15.8378i		−	0.747430i	−0.927544	−	0.373715i	$-0.878084\pi$
$449$		−	15.8378i		−	0.747430i	0.927544	−	0.373715i	$-0.121916\pi$
$450$		0			0
$451$	−12.3526			−0.581660
$452$			8.62894i			0.405871i
$453$		0			0
$454$			13.3153i			0.624919i
$455$	47.5847	+	36.6246i	2.23081	+	1.71699i
$456$		0			0
$457$	−7.56675			−0.353958			−0.176979	−	0.984215i	$-0.556632\pi$
$457$	−7.56675			−0.353958			−0.176979	+	0.984215i	$0.556632\pi$
$458$	−15.1805			−0.709338
$459$		0			0
$460$		−	28.6496i		−	1.33580i
$461$			20.3911i			0.949707i	0.880065	+	0.474853i	$0.157499\pi$
$461$			20.3911i			0.949707i	−0.880065	+	0.474853i	$0.842501\pi$
$462$		0			0
$463$	−30.1929			−1.40318			−0.701592	−	0.712579i	$-0.747527\pi$
$463$	−30.1929			−1.40318			−0.701592	+	0.712579i	$0.747527\pi$
$464$		−	3.97501i		−	0.184535i
$465$		0			0
$466$		−	26.2444i		−	1.21575i
$467$		−	6.19990i		−	0.286897i	−0.989658	−	0.143449i	$-0.954181\pi$
$467$		−	6.19990i		−	0.286897i	0.989658	−	0.143449i	$-0.0458192\pi$
$468$		0			0
$469$	9.20400	+	7.08407i	0.425001	+	0.327112i
$470$			36.8429i			1.69944i
$471$		0			0
$472$			14.2408i			0.655487i
$473$	−1.79653			−0.0826045
$474$		0			0
$475$	−2.65531	+	43.3337i	−0.121834	+	1.98828i
$476$	12.3197	+	9.48213i	0.564672	+	0.434613i
$477$		0			0
$478$		−	10.5130i		−	0.480852i
$479$		−	5.22893i		−	0.238916i	−0.992839	−	0.119458i	$-0.961884\pi$
$479$		−	5.22893i		−	0.238916i	0.992839	−	0.119458i	$-0.0381157\pi$
$480$		0			0
$481$		−	12.8222i		−	0.584643i
$482$		−	13.8076i		−	0.628918i
$483$		0			0
$484$	9.76982			0.444083
$485$		−	66.9004i		−	3.03779i
$486$		0			0
$487$		−	2.00481i		−	0.0908466i	−0.998968	−	0.0454233i	$-0.985536\pi$
$487$		−	2.00481i		−	0.0908466i	0.998968	−	0.0454233i	$-0.0144637\pi$
$488$	−7.08172			−0.320574
$489$		0			0
$490$	6.93031	+	26.1728i	0.313079	+	1.18237i
$491$	−1.16066			−0.0523799			−0.0261899	−	0.999657i	$-0.508337\pi$
$491$	−1.16066			−0.0523799			−0.0261899	+	0.999657i	$0.508337\pi$
$492$		0			0
$493$	−23.3569			−1.05194
$494$	25.5294	+	1.56434i	1.14862	+	0.0703828i
$495$		0			0
$496$	3.04228			0.136602
$497$	14.1030	+	10.8547i	0.632605	+	0.486898i
$498$		0			0
$499$	8.50028			0.380525			0.190262	−	0.981733i	$-0.439066\pi$
$499$	8.50028			0.380525			0.190262	+	0.981733i	$0.439066\pi$
$500$			19.1847i			0.857965i
$501$		0			0
$502$	−11.3936			−0.508522
$503$		−	6.62741i		−	0.295502i	−0.989025	−	0.147751i	$-0.952797\pi$
$503$		−	6.62741i		−	0.295502i	0.989025	−	0.147751i	$-0.0472034\pi$
$504$		0			0
$505$	30.2381			1.34558
$506$		−	8.21554i		−	0.365225i
$507$		0			0
$508$		−	2.44619i		−	0.108532i
$509$	−6.82964			−0.302718			−0.151359	−	0.988479i	$-0.548365\pi$
$509$	−6.82964			−0.302718			−0.151359	+	0.988479i	$0.548365\pi$
$510$		0			0
$511$	−7.36833	−	5.67120i	−0.325956	−	0.250879i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$		−	12.0629i		−	0.532071i
$515$			66.4987i			2.93028i
$516$		0			0
$517$			10.5650i			0.464650i
$518$	3.52627	−	4.58152i	0.154935	−	0.201300i
$519$		0			0
$520$	22.6957			0.995273
$521$	39.4529			1.72846			0.864232	−	0.503093i	$-0.167805\pi$
$521$	39.4529			1.72846			0.864232	+	0.503093i	$0.167805\pi$
$522$		0			0
$523$	18.8737			0.825288			0.412644	−	0.910892i	$-0.364605\pi$
$523$	18.8737			0.825288			0.412644	+	0.910892i	$0.364605\pi$
$524$		−	17.0880i		−	0.746494i
$525$		0			0
$526$			10.2264i			0.445890i
$527$		−	17.8762i		−	0.778699i
$528$		0			0
$529$	31.8662			1.38549
$530$			13.4369i			0.583663i
$531$		0			0
$532$	8.69171	+	7.57985i	0.376834	+	0.328628i
$533$	65.3508			2.83066
$534$		0			0
$535$	−13.0164			−0.562748
$536$	4.38988			0.189614
$537$		0			0
$538$		−	3.72989i		−	0.160807i
$539$	1.98733	+	7.50528i	0.0856003	+	0.323275i
$540$		0			0
$541$	−15.0585			−0.647418			−0.323709	−	0.946157i	$-0.604930\pi$
$541$	−15.0585			−0.647418			−0.323709	+	0.946157i	$0.604930\pi$
$542$	−13.1874			−0.566448
$543$		0			0
$544$	5.87593			0.251928
$545$	−63.9683			−2.74010
$546$		0			0
$547$		−	16.8460i		−	0.720282i	−0.932898	−	0.360141i	$-0.882729\pi$
$547$		−	16.8460i		−	0.720282i	0.932898	−	0.360141i	$-0.117271\pi$
$548$	0.617742			0.0263886
$549$		0			0
$550$			11.0470i			0.471048i
$551$	−17.2942	−	1.05972i	−0.736759	−	0.0451456i
$552$		0			0
$553$	31.0548	+	23.9020i	1.32058	+	1.01642i
$554$			11.5498i			0.490704i
$555$		0			0
$556$		−	11.9780i		−	0.507982i
$557$	−24.4103			−1.03430			−0.517149	−	0.855896i	$-0.673007\pi$
$557$	−24.4103			−1.03430			−0.517149	+	0.855896i	$0.673007\pi$
$558$		0			0
$559$	9.50446			0.401996
$560$	8.10943	+	6.24161i	0.342686	+	0.263756i
$561$		0			0
$562$	−3.76367			−0.158761
$563$	−17.8103			−0.750617			−0.375308	−	0.926900i	$-0.622463\pi$
$563$	−17.8103			−0.750617			−0.375308	+	0.926900i	$0.622463\pi$
$564$		0			0
$565$	33.3752			1.40411
$566$	−1.47490			−0.0619945
$567$		0			0
$568$	6.72647			0.282236
$569$		−	4.15926i		−	0.174365i	−0.996192	−	0.0871827i	$-0.972214\pi$
$569$		−	4.15926i		−	0.174365i	0.996192	−	0.0871827i	$-0.0277864\pi$
$570$		0			0
$571$	−30.0514			−1.25761			−0.628806	−	0.777562i	$-0.716456\pi$
$571$	−30.0514			−1.25761			−0.628806	+	0.777562i	$0.716456\pi$
$572$	6.50820			0.272122
$573$		0			0
$574$	23.3506	+	17.9723i	0.974634	+	0.750149i
$575$	−73.7759			−3.07667
$576$		0			0
$577$			29.4420i			1.22569i	0.790204	+	0.612844i	$0.209975\pi$
$577$			29.4420i			1.22569i	−0.790204	+	0.612844i	$0.790025\pi$
$578$		−	17.5265i		−	0.729007i
$579$		0			0
$580$	−15.3746			−0.638397
$581$	2.28449	+	1.75831i	0.0947766	+	0.0729470i
$582$		0			0
$583$			3.85316i			0.159582i
$584$	−3.51435			−0.145425
$585$		0			0
$586$			13.7598i			0.568411i
$587$			16.3838i			0.676231i	0.941105	+	0.338116i	$0.109790\pi$
$587$			16.3838i			0.676231i	−0.941105	+	0.338116i	$0.890210\pi$
$588$		0			0
$589$	0.811057	−	13.2361i	0.0334190	−	0.545386i
$590$	55.0810			2.26765
$591$		0			0
$592$		−	2.18517i		−	0.0898101i
$593$		−	41.6561i		−	1.71061i	−0.518124	−	0.855306i	$-0.673369\pi$
$593$		−	41.6561i		−	1.71061i	0.518124	−	0.855306i	$-0.326631\pi$
$594$		0			0
$595$	36.6752	−	47.6504i	1.50354	−	1.95348i
$596$	−0.449524			−0.0184132
$597$		0			0
$598$			43.4640i			1.77737i
$599$		−	7.37621i		−	0.301384i	−0.988581	−	0.150692i	$-0.951850\pi$
$599$		−	7.37621i		−	0.301384i	0.988581	−	0.150692i	$-0.0481501\pi$
$600$		0			0
$601$	−28.9390			−1.18045			−0.590223	−	0.807240i	$-0.700960\pi$
$601$	−28.9390			−1.18045			−0.590223	+	0.807240i	$0.700960\pi$
$602$	3.39605	+	2.61384i	0.138413	+	0.106532i
$603$		0			0
$604$		−	10.3107i		−	0.419537i
$605$		−	37.7880i		−	1.53630i
$606$		0			0
$607$	27.4786			1.11532			0.557661	−	0.830069i	$-0.311699\pi$
$607$	27.4786			1.11532			0.557661	+	0.830069i	$0.311699\pi$
$608$	4.35074	+	0.266595i	0.176446	+	0.0108119i
$609$		0			0
$610$			27.3908i			1.10902i
$611$		−	55.8939i		−	2.26122i
$612$		0			0
$613$	−10.6521			−0.430235			−0.215117	−	0.976588i	$-0.569013\pi$
$613$	−10.6521			−0.430235			−0.215117	+	0.976588i	$0.569013\pi$
$614$		−	0.112863i		−	0.00455477i
$615$		0			0
$616$	2.32545	+	1.78984i	0.0936951	+	0.0721145i
$617$	−7.45899			−0.300288			−0.150144	−	0.988664i	$-0.547974\pi$
$617$	−7.45899			−0.300288			−0.150144	+	0.988664i	$0.547974\pi$
$618$		0			0
$619$			8.92641i			0.358783i	0.983778	+	0.179391i	$0.0574128\pi$
$619$			8.92641i			0.358783i	−0.983778	+	0.179391i	$0.942587\pi$
$620$		−	11.7670i		−	0.472574i
$621$		0			0
$622$	−14.3115			−0.573839
$623$	−8.04312	+	10.4500i	−0.322241	+	0.418672i
$624$		0			0
$625$	24.4027			0.976106
$626$	−10.6424			−0.425357
$627$		0			0
$628$			16.1342i			0.643825i
$629$	−12.8399			−0.511961
$630$		0			0
$631$	−21.0230			−0.836913			−0.418457	−	0.908237i	$-0.637429\pi$
$631$	−21.0230			−0.836913			−0.418457	+	0.908237i	$0.637429\pi$
$632$	14.8117			0.589177
$633$		0			0
$634$	10.3588			0.411402
$635$	−9.46144			−0.375466
$636$		0			0
$637$	−10.5139	−	39.7064i	−0.416575	−	1.57322i
$638$	−4.40882			−0.174547
$639$		0			0
$640$	3.86783			0.152889
$641$			16.6806i			0.658845i	0.944183	+	0.329423i	$0.106854\pi$
$641$			16.6806i			0.658845i	−0.944183	+	0.329423i	$0.893146\pi$
$642$		0			0
$643$			6.28394i			0.247814i	0.992294	+	0.123907i	$0.0395425\pi$
$643$			6.28394i			0.247814i	−0.992294	+	0.123907i	$0.960458\pi$
$644$	−11.9531	+	15.5302i	−0.471019	+	0.611974i
$645$		0			0
$646$	1.56650	−	25.5646i	0.0616329	−	1.00583i
$647$			12.2836i			0.482918i	0.970411	+	0.241459i	$0.0776259\pi$
$647$			12.2836i			0.482918i	−0.970411	+	0.241459i	$0.922374\pi$
$648$		0			0
$649$	15.7950			0.620007
$650$		−	58.4440i		−	2.29236i
$651$		0			0
$652$	−16.1060			−0.630760
$653$	26.0792			1.02056			0.510278	−	0.860009i	$-0.329542\pi$
$653$	26.0792			1.02056			0.510278	+	0.860009i	$0.329542\pi$
$654$		0			0
$655$	−66.0935			−2.58249
$656$	11.1371			0.434832
$657$		0			0
$658$	15.3715	−	19.9715i	0.599244	−	0.778570i
$659$			16.6707i			0.649399i	0.945817	+	0.324700i	$0.105263\pi$
$659$			16.6707i			0.649399i	−0.945817	+	0.324700i	$0.894737\pi$
$660$		0			0
$661$	−20.3082			−0.789899			−0.394949	−	0.918703i	$-0.629238\pi$
$661$	−20.3082			−0.789899			−0.394949	+	0.918703i	$0.629238\pi$
$662$	0.413713			0.0160794
$663$		0			0
$664$	1.08960			0.0422846
$665$	29.3175	−	33.6180i	1.13689	−	1.30365i
$666$		0			0
$667$		−	29.4436i		−	1.14006i
$668$	3.37953			0.130758
$669$		0			0
$670$		−	16.9793i		−	0.655968i
$671$			7.85457i			0.303222i
$672$		0			0
$673$		−	18.2745i		−	0.704430i	−0.935919	−	0.352215i	$-0.885429\pi$
$673$		−	18.2745i		−	0.704430i	0.935919	−	0.352215i	$-0.114571\pi$
$674$	−7.89183			−0.303982
$675$		0			0
$676$	−21.4314			−0.824284
$677$	47.6774			1.83239			0.916195	−	0.400733i	$-0.131245\pi$
$677$	47.6774			1.83239			0.916195	+	0.400733i	$0.131245\pi$
$678$		0			0
$679$	−27.9120	+	36.2648i	−1.07117	+	1.39172i
$680$		−	22.7271i		−	0.871543i
$681$		0			0
$682$		−	3.37429i		−	0.129208i
$683$		−	35.9361i		−	1.37506i	−0.726158	−	0.687528i	$-0.758696\pi$
$683$		−	35.9361i		−	1.37506i	0.726158	−	0.687528i	$-0.241304\pi$
$684$		0			0
$685$		−	2.38932i		−	0.0912912i
$686$	7.16302	−	17.0790i	0.273485	−	0.652078i
$687$		0			0
$688$	1.61976			0.0617527
$689$		−	20.3850i		−	0.776606i
$690$		0			0
$691$		−	37.1441i		−	1.41303i	−0.707700	−	0.706513i	$-0.750267\pi$
$691$		−	37.1441i		−	1.41303i	0.707700	−	0.706513i	$-0.249733\pi$
$692$	−17.8055			−0.676862
$693$		0			0
$694$			15.2726i			0.579739i
$695$	−46.3290			−1.75736
$696$		0			0
$697$		−	65.4410i		−	2.47876i
$698$	17.0669			0.645991
$699$		0			0
$700$	16.0728	−	20.8827i	0.607495	−	0.789291i
$701$	−12.8872			−0.486745			−0.243372	−	0.969933i	$-0.578254\pi$
$701$	−12.8872			−0.486745			−0.243372	+	0.969933i	$0.578254\pi$
$702$		0			0
$703$	−9.50712	−	0.582557i	−0.358568	−	0.0219716i
$704$	1.10913			0.0418020
$705$		0			0
$706$	−18.8017			−0.707612
$707$	−16.3912	−	12.6159i	−0.616454	−	0.474468i
$708$		0			0
$709$	16.1876			0.607938			0.303969	−	0.952682i	$-0.401688\pi$
$709$	16.1876			0.607938			0.303969	+	0.952682i	$0.401688\pi$
$710$		−	26.0168i		−	0.976393i
$711$		0			0
$712$			4.98419i			0.186791i
$713$	22.5347			0.843929
$714$		0			0
$715$		−	25.1726i		−	0.941401i
$716$		−	18.4525i		−	0.689601i
$717$		0			0
$718$			2.91401i			0.108750i
$719$		−	20.1946i		−	0.753130i	−0.926390	−	0.376565i	$-0.877105\pi$
$719$		−	20.1946i		−	0.753130i	0.926390	−	0.376565i	$-0.122895\pi$
$720$		0			0
$721$	27.7444	−	36.0471i	1.03326	−	1.34246i
$722$	2.31977	−	18.8579i	0.0863330	−	0.701817i
$723$		0			0
$724$	−0.736995			−0.0273902
$725$			39.5914i			1.47039i
$726$		0			0
$727$			18.4856i			0.685592i	0.939410	+	0.342796i	$0.111374\pi$
$727$			18.4856i			0.685592i	−0.939410	+	0.342796i	$0.888626\pi$
$728$	−12.3027	−	9.46905i	−0.455969	−	0.350947i
$729$		0			0
$730$			13.5929i			0.503096i
$731$		−	9.51759i		−	0.352021i
$732$		0			0
$733$			21.3765i			0.789559i	0.918776	+	0.394780i	$0.129179\pi$
$733$			21.3765i			0.789559i	−0.918776	+	0.394780i	$0.870821\pi$
$734$	15.5472			0.573858
$735$		0			0
$736$			7.40717i			0.273032i
$737$		−	4.86897i		−	0.179351i
$738$		0			0
$739$	13.3776			0.492103			0.246052	−	0.969257i	$-0.420867\pi$
$739$	13.3776			0.492103			0.246052	+	0.969257i	$0.420867\pi$
$740$	−8.45187			−0.310697
$741$		0			0
$742$	5.60612	−	7.28378i	0.205807	−	0.267396i
$743$			33.7323i			1.23752i	0.785581	+	0.618759i	$0.212364\pi$
$743$			33.7323i			1.23752i	−0.785581	+	0.618759i	$0.787636\pi$
$744$		0			0
$745$			1.73868i			0.0637003i
$746$	−8.51025			−0.311582
$747$		0			0
$748$		−	6.51719i		−	0.238292i
$749$	7.05582	+	5.43067i	0.257814	+	0.198432i
$750$		0			0
$751$			26.5713i			0.969601i	0.874625	+	0.484801i	$0.161108\pi$
$751$			26.5713i			0.969601i	−0.874625	+	0.484801i	$0.838892\pi$
$752$		−	9.52549i		−	0.347359i
$753$		0			0
$754$	23.3247			0.849434
$755$	−39.8800			−1.45138
$756$		0			0
$757$	46.5039			1.69021			0.845107	−	0.534598i	$-0.179537\pi$
$757$	46.5039			1.69021			0.845107	+	0.534598i	$0.179537\pi$
$758$	−18.0541			−0.655755
$759$		0			0
$760$	1.03114	−	16.8279i	0.0374035	−	0.610412i
$761$			4.02018i			0.145732i	0.997342	+	0.0728658i	$0.0232145\pi$
$761$			4.02018i			0.145732i	−0.997342	+	0.0728658i	$0.976786\pi$
$762$		0			0
$763$	34.6754	+	26.6887i	1.25533	+	0.966196i
$764$	12.9745			0.469402
$765$		0			0
$766$		−	7.00586i		−	0.253132i
$767$	−83.5627			−3.01727
$768$		0			0
$769$		−	18.7051i		−	0.674523i	−0.941411	−	0.337262i	$-0.890499\pi$
$769$		−	18.7051i		−	0.674523i	0.941411	−	0.337262i	$-0.109501\pi$
$770$	6.92277	−	8.99444i	0.249479	−	0.324137i
$771$		0			0
$772$			14.5678i			0.524305i
$773$	51.1981			1.84147			0.920734	−	0.390191i	$-0.127591\pi$
$773$	51.1981			1.84147			0.920734	+	0.390191i	$0.127591\pi$
$774$		0			0
$775$	−30.3013			−1.08845
$776$			17.2967i			0.620914i
$777$		0			0
$778$			27.8125i			0.997125i
$779$	2.96911	−	48.4548i	0.106379	−	1.73607i
$780$		0			0
$781$		−	7.46055i		−	0.266959i
$782$	43.5240			1.55641
$783$		0			0
$784$	−1.79178	−	6.76679i	−0.0639923	−	0.241671i
$785$	62.4043			2.22731
$786$		0			0
$787$	21.8877			0.780213			0.390107	−	0.920770i	$-0.372438\pi$
$787$	21.8877			0.780213			0.390107	+	0.920770i	$0.372438\pi$
$788$	9.09886			0.324133
$789$		0			0
$790$		−	57.2890i		−	2.03825i
$791$	−18.0918	−	13.9247i	−0.643269	−	0.495107i
$792$		0			0
$793$		−	41.5543i		−	1.47564i
$794$	9.88942			0.350962
$795$		0			0
$796$			14.9321i			0.529253i
$797$	10.6309			0.376566			0.188283	−	0.982115i	$-0.439708\pi$
$797$	10.6309			0.376566			0.188283	+	0.982115i	$0.439708\pi$
$798$		0			0
$799$	−55.9711			−1.98011
$800$		−	9.96007i		−	0.352142i
$801$		0			0
$802$	−5.03716			−0.177868
$803$			3.89789i			0.137553i
$804$		0			0
$805$	60.0679	+	46.2326i	2.11712	+	1.62949i
$806$			17.8515i			0.628794i
$807$		0			0
$808$	−7.81784			−0.275031
$809$	25.7292			0.904592			0.452296	−	0.891868i	$-0.350605\pi$
$809$	25.7292			0.904592			0.452296	+	0.891868i	$0.350605\pi$
$810$		0			0
$811$	20.5010			0.719887			0.359943	−	0.932974i	$-0.382796\pi$
$811$	20.5010			0.719887			0.359943	+	0.932974i	$0.382796\pi$
$812$	8.33416	+	6.41457i	0.292472	+	0.225107i
$813$		0			0
$814$	−2.42365			−0.0849489
$815$			62.2952i			2.18211i
$816$		0			0
$817$	0.431820	−	7.04715i	0.0151075	−	0.246548i
$818$		−	9.92650i		−	0.347072i
$819$		0			0
$820$		−	43.0765i		−	1.50430i
$821$	20.0108			0.698382			0.349191	−	0.937051i	$-0.386456\pi$
$821$	20.0108			0.698382			0.349191	+	0.937051i	$0.386456\pi$
$822$		0			0
$823$	39.9945			1.39412			0.697060	−	0.717013i	$-0.254491\pi$
$823$	39.9945			1.39412			0.697060	+	0.717013i	$0.254491\pi$
$824$		−	17.1928i		−	0.598939i
$825$		0			0
$826$	−29.8579	−	22.9808i	−1.03889	−	0.799603i
$827$			23.3080i			0.810500i	0.914206	+	0.405250i	$0.132816\pi$
$827$			23.3080i			0.810500i	−0.914206	+	0.405250i	$0.867184\pi$
$828$		0			0
$829$	14.9746			0.520089			0.260044	−	0.965597i	$-0.416263\pi$
$829$	14.9746			0.520089			0.260044	+	0.965597i	$0.416263\pi$
$830$		−	4.21437i		−	0.146283i
$831$		0			0
$832$	−5.86783			−0.203430
$833$	−39.7612	+	10.5284i	−1.37764	+	0.364787i
$834$		0			0
$835$		−	13.0714i		−	0.452356i
$836$	0.295690	−	4.82555i	0.0102266	−	0.166895i
$837$		0			0
$838$	0.578669			0.0199898
$839$	30.8513			1.06510			0.532552	−	0.846397i	$-0.321233\pi$
$839$	30.8513			1.06510			0.532552	+	0.846397i	$0.321233\pi$
$840$		0			0
$841$	13.1993			0.455148
$842$	−6.89896			−0.237754
$843$		0			0
$844$		−	2.52268i		−	0.0868342i
$845$			82.8928i			2.85160i
$846$		0			0
$847$	−15.7658	+	20.4838i	−0.541720	+	0.703831i
$848$		−	3.47403i		−	0.119299i
$849$		0			0
$850$	−58.5247			−2.00738
$851$		−	16.1860i		−	0.554847i
$852$		0			0
$853$			51.2568i			1.75500i	0.479576	+	0.877500i	$0.340790\pi$
$853$			51.2568i			1.75500i	−0.479576	+	0.877500i	$0.659210\pi$
$854$	11.4279	−	14.8478i	0.391056	−	0.508081i
$855$		0			0
$856$	3.36530			0.115024
$857$	−57.6856			−1.97050			−0.985252	−	0.171111i	$-0.945264\pi$
$857$	−57.6856			−1.97050			−0.985252	+	0.171111i	$0.945264\pi$
$858$		0			0
$859$			37.9717i			1.29558i	0.761820	+	0.647788i	$0.224306\pi$
$859$			37.9717i			1.29558i	−0.761820	+	0.647788i	$0.775694\pi$
$860$		−	6.26494i		−	0.213633i
$861$		0			0
$862$	34.2796			1.16757
$863$			18.2430i			0.621000i	0.950573	+	0.310500i	$0.100496\pi$
$863$			18.2430i			0.621000i	−0.950573	+	0.310500i	$0.899504\pi$
$864$		0			0
$865$			68.8684i			2.34160i
$866$		−	25.1600i		−	0.854970i
$867$		0			0
$868$	−4.90940	+	6.37855i	−0.166636	+	0.216502i
$869$		−	16.4281i		−	0.557286i
$870$		0			0
$871$			25.7591i			0.872813i
$872$	16.5386			0.560067
$873$		0			0
$874$	32.2267	+	1.97472i	1.09008	+	0.0667958i
$875$	−40.2234	−	30.9588i	−1.35980	−	1.04660i
$876$		0			0
$877$			6.79450i			0.229434i	0.993398	+	0.114717i	$0.0365961\pi$
$877$			6.79450i			0.229434i	−0.993398	+	0.114717i	$0.963404\pi$
$878$		−	8.44013i		−	0.284841i
$879$		0			0
$880$		−	4.28993i		−	0.144614i
$881$			26.5975i			0.896092i	0.894010	+	0.448046i	$0.147880\pi$
$881$			26.5975i			0.896092i	−0.894010	+	0.448046i	$0.852120\pi$
$882$		0			0
$883$	0.248686			0.00836895			0.00418447	−	0.999991i	$-0.498668\pi$
$883$	0.248686			0.00836895			0.00418447	+	0.999991i	$0.498668\pi$
$884$			34.4789i			1.15965i
$885$		0			0
$886$			3.07126i			0.103181i
$887$	−3.87893			−0.130242			−0.0651208	−	0.997877i	$-0.520743\pi$
$887$	−3.87893			−0.130242			−0.0651208	+	0.997877i	$0.520743\pi$
$888$		0			0
$889$	5.12878	+	3.94748i	0.172014	+	0.132394i
$890$	19.2780			0.646200
$891$		0			0
$892$	−15.6221			−0.523068
$893$	−41.4429	−	2.53945i	−1.38683	−	0.0849795i
$894$		0			0
$895$	−71.3709			−2.38567
$896$	−2.09664	−	1.61372i	−0.0700438	−	0.0539108i
$897$		0			0
$898$	15.8378			0.528513
$899$		−	12.0931i		−	0.403327i
$900$		0			0
$901$	−20.4131			−0.680060
$902$		−	12.3526i		−	0.411296i
$903$		0			0
$904$	−8.62894			−0.286994
$905$			2.85057i			0.0947561i
$906$		0			0
$907$			28.2509i			0.938054i	0.883184	+	0.469027i	$0.155395\pi$
$907$			28.2509i			0.938054i	−0.883184	+	0.469027i	$0.844605\pi$
$908$	−13.3153			−0.441884
$909$		0			0
$910$	−36.6246	+	47.5847i	−1.21410	+	1.57742i
$911$			54.9552i			1.82075i	0.413787	+	0.910374i	$0.364206\pi$
$911$			54.9552i			1.82075i	−0.413787	+	0.910374i	$0.635794\pi$
$912$		0			0
$913$		−	1.20851i		−	0.0399958i
$914$		−	7.56675i		−	0.250286i
$915$		0			0
$916$		−	15.1805i		−	0.501578i
$917$	35.8274	+	27.5754i	1.18313	+	0.910619i
$918$		0			0
$919$	35.8350			1.18209			0.591044	−	0.806640i	$-0.298716\pi$
$919$	35.8350			1.18209			0.591044	+	0.806640i	$0.298716\pi$
$920$	28.6496			0.944551
$921$		0			0
$922$	−20.3911			−0.671544
$923$			39.4697i			1.29916i
$924$		0			0
$925$			21.7645i			0.715612i
$926$		−	30.1929i		−	0.992201i
$927$		0			0
$928$	3.97501			0.130486
$929$		−	52.2906i		−	1.71560i	−0.513983	−	0.857800i	$-0.671831\pi$
$929$		−	52.2906i		−	1.71560i	0.513983	−	0.857800i	$-0.328169\pi$
$930$		0			0
$931$	−29.9182	+	5.99159i	−0.980531	+	0.196366i
$932$	26.2444			0.859664
$933$		0			0
$934$	6.19990			0.202867
$935$	−25.2073			−0.824368
$936$		0			0
$937$			9.39120i			0.306797i	0.988164	+	0.153399i	$0.0490218\pi$
$937$			9.39120i			0.306797i	−0.988164	+	0.153399i	$0.950978\pi$
$938$	−7.08407	+	9.20400i	−0.231303	+	0.300521i
$939$		0			0
$940$	−36.8429			−1.20168
$941$	−32.0388			−1.04443			−0.522217	−	0.852813i	$-0.674895\pi$
$941$	−32.0388			−1.04443			−0.522217	+	0.852813i	$0.674895\pi$
$942$		0			0
$943$	82.4947			2.68640
$944$	−14.2408			−0.463499
$945$		0			0
$946$		−	1.79653i		−	0.0584102i
$947$	3.98436			0.129474			0.0647372	−	0.997902i	$-0.479379\pi$
$947$	3.98436			0.129474			0.0647372	+	0.997902i	$0.479379\pi$
$948$		0			0
$949$		−	20.6216i		−	0.669406i
$950$	−43.3337	−	2.65531i	−1.40593	−	0.0861496i
$951$		0			0
$952$	−9.48213	+	12.3197i	−0.307318	+	0.399284i
$953$		−	5.82769i		−	0.188778i	−0.995535	−	0.0943888i	$-0.969910\pi$
$953$		−	5.82769i		−	0.188778i	0.995535	−	0.0943888i	$-0.0300897\pi$
$954$		0			0
$955$		−	50.1832i		−	1.62389i
$956$	10.5130			0.340014
$957$		0			0
$958$	5.22893			0.168939
$959$	−0.996866	+	1.29518i	−0.0321905	+	0.0418236i
$960$		0			0
$961$	−21.7446			−0.701437
$962$	12.8222			0.413405
$963$		0			0
$964$	13.8076			0.444712
$965$	56.3456			1.81383
$966$		0			0
$967$	26.5332			0.853250			0.426625	−	0.904429i	$-0.359703\pi$
$967$	26.5332			0.853250			0.426625	+	0.904429i	$0.359703\pi$
$968$			9.76982i			0.314014i
$969$		0			0
$970$	66.9004			2.14804
$971$	36.0646			1.15737			0.578684	−	0.815552i	$-0.303566\pi$
$971$	36.0646			1.15737			0.578684	+	0.815552i	$0.303566\pi$
$972$		0			0
$973$	25.1136	+	19.3293i	0.805106	+	0.619668i
$974$	2.00481			0.0642382
$975$		0			0
$976$		−	7.08172i		−	0.226680i
$977$		−	4.88611i		−	0.156321i	−0.996941	−	0.0781603i	$-0.975095\pi$
$977$		−	4.88611i		−	0.156321i	0.996941	−	0.0781603i	$-0.0249046\pi$
$978$		0			0
$979$	5.52813			0.176680
$980$	−26.1728	+	6.93031i	−0.836059	+	0.221381i
$981$		0			0
$982$		−	1.16066i		−	0.0370382i
$983$	−48.2375			−1.53854			−0.769269	−	0.638925i	$-0.779379\pi$
$983$	−48.2375			−1.53854			−0.769269	+	0.638925i	$0.779379\pi$
$984$		0			0
$985$		−	35.1928i		−	1.12134i
$986$		−	23.3569i		−	0.743834i
$987$		0			0
$988$	−1.56434	+	25.5294i	−0.0497681	+	0.812198i
$989$	11.9978			0.381509
$990$		0			0
$991$		−	22.9110i		−	0.727793i	−0.931439	−	0.363897i	$-0.881446\pi$
$991$		−	22.9110i		−	0.727793i	0.931439	−	0.363897i	$-0.118554\pi$
$992$			3.04228i			0.0965923i
$993$		0			0
$994$	−10.8547	+	14.1030i	−0.344289	+	0.447319i
$995$	57.7546			1.83094
$996$		0			0
$997$			37.0200i			1.17243i	0.810154	+	0.586217i	$0.199384\pi$
$997$			37.0200i			1.17243i	−0.810154	+	0.586217i	$0.800616\pi$
$998$			8.50028i			0.269072i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.e.b.1063.12		12
3.2	odd	2		798.2.e.b.265.1	yes	12
7.6	odd	2		2394.2.e.a.1063.7		12
19.18	odd	2		2394.2.e.a.1063.6		12
21.20	even	2		798.2.e.a.265.6	✓	12
57.56	even	2		798.2.e.a.265.7	yes	12
133.132	even	2	inner	2394.2.e.b.1063.1		12
399.398	odd	2		798.2.e.b.265.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.e.a.265.6	✓	12	21.20	even	2
798.2.e.a.265.7	yes	12	57.56	even	2
798.2.e.b.265.1	yes	12	3.2	odd	2
798.2.e.b.265.12	yes	12	399.398	odd	2
2394.2.e.a.1063.6		12	19.18	odd	2
2394.2.e.a.1063.7		12	7.6	odd	2
2394.2.e.b.1063.1		12	133.132	even	2	inner
2394.2.e.b.1063.12		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.86783i q^{5} +(1.61372 - 2.09664i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.86783i q^{5} +(1.61372 - 2.09664i) q^{7} -1.00000i q^{8} -3.86783 q^{10} -1.10913 q^{11} +5.86783 q^{13} +(2.09664 + 1.61372i) q^{14} +1.00000 q^{16} -5.87593i q^{17} +(0.266595 - 4.35074i) q^{19} -3.86783i q^{20} -1.10913i q^{22} +7.40717 q^{23} -9.96007 q^{25} +5.86783i q^{26} +(-1.61372 + 2.09664i) q^{28} -3.97501i q^{29} +3.04228 q^{31} +1.00000i q^{32} +5.87593 q^{34} +(8.10943 + 6.24161i) q^{35} -2.18517i q^{37} +(4.35074 + 0.266595i) q^{38} +3.86783 q^{40} +11.1371 q^{41} +1.61976 q^{43} +1.10913 q^{44} +7.40717i q^{46} -9.52549i q^{47} +(-1.79178 - 6.76679i) q^{49} -9.96007i q^{50} -5.86783 q^{52} -3.47403i q^{53} -4.28993i q^{55} +(-2.09664 - 1.61372i) q^{56} +3.97501 q^{58} -14.2408 q^{59} -7.08172i q^{61} +3.04228i q^{62} -1.00000 q^{64} +22.6957i q^{65} +4.38988i q^{67} +5.87593i q^{68} +(-6.24161 + 8.10943i) q^{70} +6.72647i q^{71} -3.51435i q^{73} +2.18517 q^{74} +(-0.266595 + 4.35074i) q^{76} +(-1.78984 + 2.32545i) q^{77} +14.8117i q^{79} +3.86783i q^{80} +11.1371i q^{82} +1.08960i q^{83} +22.7271 q^{85} +1.61976i q^{86} +1.10913i q^{88} -4.98419 q^{89} +(9.46905 - 12.3027i) q^{91} -7.40717 q^{92} +9.52549 q^{94} +(16.8279 + 1.03114i) q^{95} -17.2967 q^{97} +(6.76679 - 1.79178i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4}+O(q^{10}) \) 12 * q - 12 * q^4	\( 12 q - 12 q^{4} + 4 q^{10} - 12 q^{11} + 20 q^{13} + 4 q^{14} + 12 q^{16} - 8 q^{19} + 12 q^{23} - 12 q^{25} + 24 q^{31} + 4 q^{34} - 4 q^{35} - 4 q^{40} + 16 q^{41} + 12 q^{44} + 4 q^{49} - 20 q^{52} - 4 q^{56} + 8 q^{58} - 40 q^{59} - 12 q^{64} - 24 q^{70} + 8 q^{76} - 8 q^{77} + 8 q^{85} + 16 q^{89} + 24 q^{91} - 12 q^{92} + 44 q^{95} - 60 q^{97} + 16 q^{98}+O(q^{100}) \) 12 * q - 12 * q^4 + 4 * q^10 - 12 * q^11 + 20 * q^13 + 4 * q^14 + 12 * q^16 - 8 * q^19 + 12 * q^23 - 12 * q^25 + 24 * q^31 + 4 * q^34 - 4 * q^35 - 4 * q^40 + 16 * q^41 + 12 * q^44 + 4 * q^49 - 20 * q^52 - 4 * q^56 + 8 * q^58 - 40 * q^59 - 12 * q^64 - 24 * q^70 + 8 * q^76 - 8 * q^77 + 8 * q^85 + 16 * q^89 + 24 * q^91 - 12 * q^92 + 44 * q^95 - 60 * q^97 + 16 * q^98

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