LMFDB - Embedded newform 3456.2.i.i.2305.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1153,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("3456.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
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} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} + \cdots + 729 $ x^12 - 2x^11 + 3x^10 - 8x^9 + 22x^8 - 42x^7 + 51x^6 - 126x^5 + 198x^4 - 216x^3 + 243x^2 - 486*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2305.2
Root			$0.952418 - 1.44669i$ of defining polynomial
Character	$\chi$	$=$	3456.2305
Dual form			3456.2.i.i.1153.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.24278 + 2.15256i) q^{5} +(-0.909142 - 1.57468i) q^{7} +O(q^{10})$	$q+(-1.24278 + 2.15256i) q^{5} +(-0.909142 - 1.57468i) q^{7} +(-0.598407 - 1.03647i) q^{11} +(2.83342 - 4.90762i) q^{13} +5.30021 q^{17} -4.55980 q^{19} +(-2.01328 + 3.48711i) q^{23} +(-0.589008 - 1.02019i) q^{25} +(-3.01513 - 5.22236i) q^{29} +(-2.81647 + 4.87827i) q^{31} +4.51946 q^{35} -5.18127 q^{37} +(-4.57620 + 7.92621i) q^{41} +(3.99129 + 6.91313i) q^{43} +(1.39470 + 2.41570i) q^{47} +(1.84692 - 3.19896i) q^{49} -1.54470 q^{53} +2.97475 q^{55} +(1.85725 - 3.21686i) q^{59} +(-4.01513 - 6.95441i) q^{61} +(7.04263 + 12.1982i) q^{65} +(-6.91372 + 11.9749i) q^{67} -11.1794 q^{71} +12.3969 q^{73} +(-1.08807 + 1.88460i) q^{77} +(-4.36480 - 7.56006i) q^{79} +(-8.89267 - 15.4025i) q^{83} +(-6.58700 + 11.4090i) q^{85} +0.455297 q^{89} -10.3039 q^{91} +(5.66683 - 9.81524i) q^{95} +(-1.01640 - 1.76045i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 12 * q - 2 * q^5 - 6 * q^7	$ 12 q - 2 q^{5} - 6 q^{7} - 4 q^{11} - 10 q^{13} - 4 q^{17} + 4 q^{19} + 8 q^{23} - 14 q^{25} - 2 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} - 14 q^{47} - 18 q^{49} + 24 q^{53} + 16 q^{55} - 6 q^{59} - 14 q^{61} + 8 q^{65} + 4 q^{67} - 28 q^{71} + 60 q^{73} + 2 q^{77} - 16 q^{79} - 24 q^{83} - 16 q^{85} + 48 q^{89} - 52 q^{91} - 20 q^{95} - 14 q^{97}+O(q^{100}) $ 12 * q - 2 * q^5 - 6 * q^7 - 4 * q^11 - 10 * q^13 - 4 * q^17 + 4 * q^19 + 8 * q^23 - 14 * q^25 - 2 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 - 14 * q^47 - 18 * q^49 + 24 * q^53 + 16 * q^55 - 6 * q^59 - 14 * q^61 + 8 * q^65 + 4 * q^67 - 28 * q^71 + 60 * q^73 + 2 * q^77 - 16 * q^79 - 24 * q^83 - 16 * q^85 + 48 * q^89 - 52 * q^91 - 20 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.24278	+	2.15256i	−0.555788	+	0.962654i	0.442053	+	0.896989i	$0.354250\pi$
$5$	−1.24278	+	2.15256i	−0.555788	+	0.962654i	−0.997842	+	0.0656650i	$0.979083\pi$
$6$		0			0
$7$	−0.909142	−	1.57468i	−0.343623	−	0.595173i	0.641479	−	0.767140i	$-0.278321\pi$
$7$	−0.909142	−	1.57468i	−0.343623	−	0.595173i	−0.985103	+	0.171967i	$0.944988\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.598407	−	1.03647i	−0.180426	−	0.312508i	0.761599	−	0.648048i	$-0.224414\pi$
$11$	−0.598407	−	1.03647i	−0.180426	−	0.312508i	−0.942026	+	0.335540i	$0.891081\pi$
$12$		0			0
$13$	2.83342	−	4.90762i	0.785848	−	1.36113i	−0.142643	−	0.989774i	$-0.545560\pi$
$13$	2.83342	−	4.90762i	0.785848	−	1.36113i	0.928491	−	0.371355i	$-0.121107\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.30021			1.28549			0.642745	−	0.766080i	$-0.277795\pi$
$17$	5.30021			1.28549			0.642745	+	0.766080i	$0.277795\pi$
$18$		0			0
$19$	−4.55980			−1.04609			−0.523045	−	0.852305i	$-0.675204\pi$
$19$	−4.55980			−1.04609			−0.523045	+	0.852305i	$0.675204\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.01328	+	3.48711i	−0.419798	+	0.727112i	−0.995919	−	0.0902526i	$-0.971233\pi$
$23$	−2.01328	+	3.48711i	−0.419798	+	0.727112i	0.576120	+	0.817365i	$0.304566\pi$
$24$		0			0
$25$	−0.589008	−	1.02019i	−0.117802	−	0.204038i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.01513	−	5.22236i	−0.559896	−	0.969768i	−0.997505	−	0.0706027i	$-0.977508\pi$
$29$	−3.01513	−	5.22236i	−0.559896	−	0.969768i	0.437609	−	0.899166i	$-0.355826\pi$
$30$		0			0
$31$	−2.81647	+	4.87827i	−0.505853	+	0.876163i	0.494124	+	0.869391i	$0.335489\pi$
$31$	−2.81647	+	4.87827i	−0.505853	+	0.876163i	−0.999977	+	0.00677135i	$0.997845\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.51946			0.763928
$36$		0			0
$37$	−5.18127			−0.851796			−0.425898	−	0.904771i	$-0.640042\pi$
$37$	−5.18127			−0.851796			−0.425898	+	0.904771i	$0.640042\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.57620	+	7.92621i	−0.714682	+	1.23787i	0.248400	+	0.968658i	$0.420095\pi$
$41$	−4.57620	+	7.92621i	−0.714682	+	1.23787i	−0.963082	+	0.269208i	$0.913238\pi$
$42$		0			0
$43$	3.99129	+	6.91313i	0.608667	+	1.05424i	0.991460	+	0.130408i	$0.0416287\pi$
$43$	3.99129	+	6.91313i	0.608667	+	1.05424i	−0.382794	+	0.923834i	$0.625038\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.39470	+	2.41570i	0.203438	+	0.352366i	0.949634	−	0.313361i	$-0.101455\pi$
$47$	1.39470	+	2.41570i	0.203438	+	0.352366i	−0.746196	+	0.665727i	$0.768122\pi$
$48$		0			0
$49$	1.84692	−	3.19896i	0.263846	−	0.456994i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.54470			−0.212181			−0.106091	−	0.994356i	$-0.533833\pi$
$53$	−1.54470			−0.212181			−0.106091	+	0.994356i	$0.533833\pi$
$54$		0			0
$55$	2.97475			0.401116
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.85725	−	3.21686i	0.241794	−	0.418799i	−0.719431	−	0.694563i	$-0.755598\pi$
$59$	1.85725	−	3.21686i	0.241794	−	0.418799i	0.961225	+	0.275764i	$0.0889309\pi$
$60$		0			0
$61$	−4.01513	−	6.95441i	−0.514085	−	0.890421i	−0.999866	−	0.0163411i	$-0.994798\pi$
$61$	−4.01513	−	6.95441i	−0.514085	−	0.890421i	0.485781	−	0.874080i	$-0.338535\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.04263	+	12.1982i	0.873531	+	1.51300i
$66$		0			0
$67$	−6.91372	+	11.9749i	−0.844645	+	1.46297i	0.0412836	+	0.999147i	$0.486855\pi$
$67$	−6.91372	+	11.9749i	−0.844645	+	1.46297i	−0.885929	+	0.463821i	$0.846478\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.1794			−1.32675			−0.663376	−	0.748287i	$-0.730877\pi$
$71$	−11.1794			−1.32675			−0.663376	+	0.748287i	$0.730877\pi$
$72$		0			0
$73$	12.3969			1.45095			0.725473	−	0.688251i	$-0.241621\pi$
$73$	12.3969			1.45095			0.725473	+	0.688251i	$0.241621\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.08807	+	1.88460i	−0.123998	+	0.214770i
$78$		0			0
$79$	−4.36480	−	7.56006i	−0.491079	−	0.850573i	0.508869	−	0.860844i	$-0.330064\pi$
$79$	−4.36480	−	7.56006i	−0.491079	−	0.850573i	−0.999947	+	0.0102710i	$0.996731\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.89267	−	15.4025i	−0.976097	−	1.69065i	−0.676267	−	0.736657i	$-0.736403\pi$
$83$	−8.89267	−	15.4025i	−0.976097	−	1.69065i	−0.299830	−	0.953993i	$-0.596930\pi$
$84$		0			0
$85$	−6.58700	+	11.4090i	−0.714461	+	1.23748i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.455297			0.0482614			0.0241307	−	0.999709i	$-0.492318\pi$
$89$	0.455297			0.0482614			0.0241307	+	0.999709i	$0.492318\pi$
$90$		0			0
$91$	−10.3039			−1.08014
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.66683	−	9.81524i	0.581405	−	1.00702i
$96$		0			0
$97$	−1.01640	−	1.76045i	−0.103199	−	0.178747i	0.809802	−	0.586704i	$-0.199575\pi$
$97$	−1.01640	−	1.76045i	−0.103199	−	0.178747i	−0.913001	+	0.407957i	$0.866241\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.06107	−	5.30192i	−0.304587	−	0.527561i	0.672582	−	0.740023i	$-0.265185\pi$
$101$	−3.06107	−	5.30192i	−0.304587	−	0.527561i	−0.977169	+	0.212462i	$0.931852\pi$
$102$		0			0
$103$	−3.09086	+	5.35352i	−0.304551	+	0.527498i	−0.977161	−	0.212499i	$-0.931840\pi$
$103$	−3.09086	+	5.35352i	−0.304551	+	0.527498i	0.672610	+	0.739997i	$0.265173\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.2193			−1.56798			−0.783990	−	0.620774i	$-0.786818\pi$
$107$	−16.2193			−1.56798			−0.783990	+	0.620774i	$0.786818\pi$
$108$		0			0
$109$	−2.08460			−0.199669			−0.0998344	−	0.995004i	$-0.531831\pi$
$109$	−2.08460			−0.199669			−0.0998344	+	0.995004i	$0.531831\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.45933	+	2.52763i	−0.137282	+	0.237779i	−0.926467	−	0.376376i	$-0.877170\pi$
$113$	−1.45933	+	2.52763i	−0.137282	+	0.237779i	0.789185	+	0.614156i	$0.210503\pi$
$114$		0			0
$115$	−5.00414	−	8.66742i	−0.466638	−	0.808241i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.81865	−	8.34614i	−0.441725	−	0.765090i
$120$		0			0
$121$	4.78382	−	8.28582i	0.434893	−	0.753256i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.49978			−0.849686
$126$		0			0
$127$	−15.0618			−1.33652			−0.668261	−	0.743926i	$-0.732961\pi$
$127$	−15.0618			−1.33652			−0.668261	+	0.743926i	$0.732961\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.89311	+	3.27896i	−0.165402	+	0.286484i	−0.936798	−	0.349871i	$-0.886225\pi$
$131$	−1.89311	+	3.27896i	−0.165402	+	0.286484i	0.771396	+	0.636355i	$0.219559\pi$
$132$		0			0
$133$	4.14551	+	7.18023i	0.359461	+	0.622605i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.39448	−	11.0756i	−0.546317	−	0.946250i	−0.998523	−	0.0543357i	$-0.982696\pi$
$137$	−6.39448	−	11.0756i	−0.546317	−	0.946250i	0.452205	−	0.891914i	$-0.350637\pi$
$138$		0			0
$139$	4.46539	−	7.73428i	0.378749	−	0.656013i	−0.612131	−	0.790756i	$-0.709688\pi$
$139$	4.46539	−	7.73428i	0.378749	−	0.656013i	0.990881	+	0.134743i	$0.0430209\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.78214			−0.567151
$144$		0			0
$145$	14.9886			1.24473
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.65170	+	8.05698i	−0.381082	+	0.660054i	−0.991217	−	0.132244i	$-0.957782\pi$
$149$	−4.65170	+	8.05698i	−0.381082	+	0.660054i	0.610135	+	0.792297i	$0.291115\pi$
$150$		0			0
$151$	−7.83527	−	13.5711i	−0.637625	−	1.10440i	−0.985953	−	0.167026i	$-0.946584\pi$
$151$	−7.83527	−	13.5711i	−0.637625	−	1.10440i	0.348328	−	0.937373i	$-0.386750\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.00051	−	12.1252i	−0.562294	−	0.973922i
$156$		0			0
$157$	−9.75491	+	16.8960i	−0.778526	+	1.34845i	0.154265	+	0.988030i	$0.450699\pi$
$157$	−9.75491	+	16.8960i	−0.778526	+	1.34845i	−0.932791	+	0.360418i	$0.882634\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.32144			0.577010
$162$		0			0
$163$	−5.02888			−0.393892			−0.196946	−	0.980414i	$-0.563102\pi$
$163$	−5.02888			−0.393892			−0.196946	+	0.980414i	$0.563102\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.65066	−	13.2513i	0.592026	−	1.02542i	−0.401933	−	0.915669i	$-0.631662\pi$
$167$	7.65066	−	13.2513i	0.592026	−	1.02542i	0.993959	−	0.109750i	$-0.0350050\pi$
$168$		0			0
$169$	−9.55650	−	16.5523i	−0.735115	−	1.27326i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.30192	−	9.18320i	−0.403098	−	0.698186i	0.591000	−	0.806671i	$-0.298733\pi$
$173$	−5.30192	−	9.18320i	−0.403098	−	0.698186i	−0.994098	+	0.108486i	$0.965400\pi$
$174$		0			0
$175$	−1.07098	+	1.85500i	−0.0809588	+	0.140225i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.27314			0.693107			0.346553	−	0.938030i	$-0.387352\pi$
$179$	9.27314			0.693107			0.346553	+	0.938030i	$0.387352\pi$
$180$		0			0
$181$	2.32975			0.173169			0.0865845	−	0.996245i	$-0.472405\pi$
$181$	2.32975			0.173169			0.0865845	+	0.996245i	$0.472405\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.43918	−	11.1530i	0.473418	−	0.819985i
$186$		0			0
$187$	−3.17169	−	5.49352i	−0.231937	−	0.401726i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.54208	−	14.7953i	−0.618083	−	1.07055i	−0.989835	−	0.142219i	$-0.954576\pi$
$191$	−8.54208	−	14.7953i	−0.618083	−	1.07055i	0.371752	−	0.928332i	$-0.378757\pi$
$192$		0			0
$193$	−12.1360	+	21.0202i	−0.873568	+	1.51306i	−0.0152882	+	0.999883i	$0.504867\pi$
$193$	−12.1360	+	21.0202i	−0.873568	+	1.51306i	−0.858280	+	0.513182i	$0.828467\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.6971			1.11837			0.559186	−	0.829042i	$-0.311114\pi$
$197$	15.6971			1.11837			0.559186	+	0.829042i	$0.311114\pi$
$198$		0			0
$199$	14.4764			1.02620			0.513101	−	0.858328i	$-0.328497\pi$
$199$	14.4764			1.02620			0.513101	+	0.858328i	$0.328497\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.48237	+	9.49574i	−0.384787	+	0.666470i
$204$		0			0
$205$	−11.3744	−	19.7011i	−0.794424	−	1.37598i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.72862	+	4.72610i	0.188742	+	0.326911i
$210$		0			0
$211$	3.21103	−	5.56167i	0.221056	−	0.382881i	−0.734073	−	0.679071i	$-0.762383\pi$
$211$	3.21103	−	5.56167i	0.221056	−	0.382881i	0.955129	+	0.296190i	$0.0957162\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−19.8412			−1.35316
$216$		0			0
$217$	10.2423			0.695291
$218$		0			0
$219$		0			0
$220$		0			0
$221$	15.0177	−	26.0114i	1.01020	−	1.74972i
$222$		0			0
$223$	7.90683	+	13.6950i	0.529481	+	0.917087i	0.999409	+	0.0343825i	$0.0109465\pi$
$223$	7.90683	+	13.6950i	0.529481	+	0.917087i	−0.469928	+	0.882705i	$0.655720\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.00928	+	5.21223i	0.199733	+	0.345948i	0.948442	−	0.316951i	$-0.102659\pi$
$227$	3.00928	+	5.21223i	0.199733	+	0.345948i	−0.748709	+	0.662899i	$0.769326\pi$
$228$		0			0
$229$	−13.8177	+	23.9329i	−0.913098	+	1.58153i	−0.103436	+	0.994636i	$0.532984\pi$
$229$	−13.8177	+	23.9329i	−0.913098	+	1.58153i	−0.809662	+	0.586896i	$0.800350\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.1858			0.798321			0.399161	−	0.916881i	$-0.369302\pi$
$233$	12.1858			0.798321			0.399161	+	0.916881i	$0.369302\pi$
$234$		0			0
$235$	−6.93324			−0.452275
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.82794	+	13.5584i	−0.506347	+	0.877019i	0.493626	+	0.869674i	$0.335671\pi$
$239$	−7.82794	+	13.5584i	−0.506347	+	0.877019i	−0.999973	+	0.00734451i	$0.997662\pi$
$240$		0			0
$241$	9.39281	+	16.2688i	0.605044	+	1.04797i	0.992045	+	0.125887i	$0.0401778\pi$
$241$	9.39281	+	16.2688i	0.605044	+	1.04797i	−0.387001	+	0.922079i	$0.626489\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.59064	+	7.95121i	0.293285	+	0.507984i
$246$		0			0
$247$	−12.9198	+	22.3778i	−0.822068	+	1.42386i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.1684			−0.704942			−0.352471	−	0.935823i	$-0.614658\pi$
$251$	−11.1684			−0.704942			−0.352471	+	0.935823i	$0.614658\pi$
$252$		0			0
$253$	4.81905			0.302971
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.19941	+	5.54154i	−0.199574	+	0.345672i	−0.948390	−	0.317106i	$-0.897289\pi$
$257$	−3.19941	+	5.54154i	−0.199574	+	0.345672i	0.748817	+	0.662777i	$0.230622\pi$
$258$		0			0
$259$	4.71051	+	8.15885i	0.292697	+	0.506966i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.71632	+	11.6330i	0.414146	+	0.717322i	0.995338	−	0.0964440i	$-0.0307469\pi$
$263$	6.71632	+	11.6330i	0.414146	+	0.717322i	−0.581192	+	0.813766i	$0.697414\pi$
$264$		0			0
$265$	1.91973	−	3.32507i	0.117928	−	0.204257i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−26.6295			−1.62363			−0.811814	−	0.583916i	$-0.801520\pi$
$269$	−26.6295			−1.62363			−0.811814	+	0.583916i	$0.801520\pi$
$270$		0			0
$271$	−14.9630			−0.908936			−0.454468	−	0.890763i	$-0.650171\pi$
$271$	−14.9630			−0.908936			−0.454468	+	0.890763i	$0.650171\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.704933	+	1.22098i	−0.0425091	+	0.0736278i
$276$		0			0
$277$	−8.36861	−	14.4949i	−0.502821	−	0.870912i	−0.999995	−	0.00326057i	$-0.998962\pi$
$277$	−8.36861	−	14.4949i	−0.502821	−	0.870912i	0.497174	−	0.867651i	$-0.334371\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.80547	+	4.85922i	0.167361	+	0.289877i	0.937491	−	0.348009i	$-0.113142\pi$
$281$	2.80547	+	4.85922i	0.167361	+	0.289877i	−0.770131	+	0.637886i	$0.779809\pi$
$282$		0			0
$283$	6.41074	−	11.1037i	0.381079	−	0.660048i	−0.610138	−	0.792295i	$-0.708886\pi$
$283$	6.41074	−	11.1037i	0.381079	−	0.660048i	0.991217	+	0.132247i	$0.0422194\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.6417			0.982326
$288$		0			0
$289$	11.0923			0.652487
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.24047	−	12.5409i	0.422993	−	0.732645i	−0.573238	−	0.819389i	$-0.694313\pi$
$293$	7.24047	−	12.5409i	0.422993	−	0.732645i	0.996231	+	0.0867441i	$0.0276463\pi$
$294$		0			0
$295$	4.61632	+	7.99570i	0.268772	+	0.465528i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.4089	+	19.7609i	0.659796	+	1.14280i
$300$		0			0
$301$	7.25731	−	12.5700i	0.418304	−	0.724524i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.9597			1.14289
$306$		0			0
$307$	−19.4320			−1.10905			−0.554523	−	0.832169i	$-0.687099\pi$
$307$	−19.4320			−1.10905			−0.554523	+	0.832169i	$0.687099\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.47547	−	6.01968i	0.197076	−	0.341345i	−0.750503	−	0.660867i	$-0.770189\pi$
$311$	3.47547	−	6.01968i	0.197076	−	0.341345i	0.947579	+	0.319522i	$0.103522\pi$
$312$		0			0
$313$	−2.19252	−	3.79756i	−0.123929	−	0.214651i	0.797385	−	0.603471i	$-0.206216\pi$
$313$	−2.19252	−	3.79756i	−0.123929	−	0.214651i	−0.921314	+	0.388820i	$0.872883\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.6742	−	27.1485i	−0.880350	−	1.52481i	−0.850952	−	0.525244i	$-0.823974\pi$
$317$	−15.6742	−	27.1485i	−0.880350	−	1.52481i	−0.0293983	−	0.999568i	$-0.509359\pi$
$318$		0			0
$319$	−3.60855	+	6.25020i	−0.202040	+	0.349944i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.1679			−1.34474
$324$		0			0
$325$	−6.67562			−0.370297
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.53597	−	4.39243i	0.139812	−	0.242162i
$330$		0			0
$331$	11.1515	+	19.3150i	0.612943	+	1.06165i	0.990742	+	0.135760i	$0.0433477\pi$
$331$	11.1515	+	19.3150i	0.612943	+	1.06165i	−0.377799	+	0.925888i	$0.623319\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.1845	−	29.7644i	−0.938888	−	1.62620i
$336$		0			0
$337$	−10.4077	+	18.0266i	−0.566943	+	0.981974i	0.429923	+	0.902866i	$0.358541\pi$
$337$	−10.4077	+	18.0266i	−0.566943	+	0.981974i	−0.996866	+	0.0791086i	$0.974793\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.74158			0.365077
$342$		0			0
$343$	−19.4444			−1.04990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.4912	−	25.0995i	0.777929	−	1.34741i	−0.155204	−	0.987882i	$-0.549604\pi$
$347$	14.4912	−	25.0995i	0.777929	−	1.34741i	0.933133	−	0.359530i	$-0.117063\pi$
$348$		0			0
$349$	7.60709	+	13.1759i	0.407198	+	0.705288i	0.994575	−	0.104026i	$-0.0331725\pi$
$349$	7.60709	+	13.1759i	0.407198	+	0.705288i	−0.587377	+	0.809314i	$0.699839\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.78818	+	11.7575i	0.361298	+	0.625787i	0.988175	−	0.153332i	$-0.0490003\pi$
$353$	6.78818	+	11.7575i	0.361298	+	0.625787i	−0.626877	+	0.779119i	$0.715667\pi$
$354$		0			0
$355$	13.8935	−	24.0643i	0.737393	−	1.27720i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.6623			1.67107			0.835536	−	0.549436i	$-0.185158\pi$
$359$	31.6623			1.67107			0.835536	+	0.549436i	$0.185158\pi$
$360$		0			0
$361$	1.79179			0.0943046
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.4066	+	26.6850i	−0.806419	+	1.39676i
$366$		0			0
$367$	−4.90625	−	8.49788i	−0.256104	−	0.443585i	0.709091	−	0.705117i	$-0.249106\pi$
$367$	−4.90625	−	8.49788i	−0.256104	−	0.443585i	−0.965195	+	0.261532i	$0.915772\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.40435	+	2.43241i	0.0729105	+	0.126285i
$372$		0			0
$373$	−4.98487	+	8.63404i	−0.258107	+	0.447054i	−0.965735	−	0.259531i	$-0.916432\pi$
$373$	−4.98487	+	8.63404i	−0.258107	+	0.447054i	0.707628	+	0.706585i	$0.249765\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−34.1725			−1.75997
$378$		0			0
$379$	−30.2351			−1.55307			−0.776537	−	0.630072i	$-0.783026\pi$
$379$	−30.2351			−1.55307			−0.776537	+	0.630072i	$0.783026\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.86835	+	17.0925i	−0.504249	+	0.873386i	0.495739	+	0.868472i	$0.334897\pi$
$383$	−9.86835	+	17.0925i	−0.504249	+	0.873386i	−0.999988	+	0.00491371i	$0.998436\pi$
$384$		0			0
$385$	−2.70447	−	4.68429i	−0.137833	−	0.238733i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.60577	+	6.24537i	0.182820	+	0.316653i	0.942840	−	0.333247i	$-0.108144\pi$
$389$	3.60577	+	6.24537i	0.182820	+	0.316653i	−0.760020	+	0.649900i	$0.774811\pi$
$390$		0			0
$391$	−10.6708	+	18.4824i	−0.539647	+	0.934696i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21.6980			1.09174
$396$		0			0
$397$	20.8930			1.04859			0.524296	−	0.851536i	$-0.324329\pi$
$397$	20.8930			1.04859			0.524296	+	0.851536i	$0.324329\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.59174	−	2.75697i	0.0794877	−	0.137677i	−0.823541	−	0.567256i	$-0.808005\pi$
$401$	1.59174	−	2.75697i	0.0794877	−	0.137677i	0.903029	+	0.429580i	$0.141338\pi$
$402$		0			0
$403$	15.9605	+	27.6443i	0.795047	+	1.37706i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.10051	+	5.37024i	0.153687	+	0.266193i
$408$		0			0
$409$	11.5046	−	19.9265i	0.568865	−	0.985302i	−0.427814	−	0.903867i	$-0.640716\pi$
$409$	11.5046	−	19.9265i	0.568865	−	0.985302i	0.996679	−	0.0814356i	$-0.0259505\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.75403			−0.332344
$414$		0			0
$415$	44.2065			2.17001
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.36640	−	2.36667i	0.0667528	−	0.115619i	−0.830717	−	0.556694i	$-0.812069\pi$
$419$	1.36640	−	2.36667i	0.0667528	−	0.115619i	0.897470	+	0.441075i	$0.145403\pi$
$420$		0			0
$421$	2.98079	+	5.16288i	0.145275	+	0.251624i	0.929475	−	0.368884i	$-0.120260\pi$
$421$	2.98079	+	5.16288i	0.145275	+	0.251624i	−0.784201	+	0.620507i	$0.786927\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.12187	−	5.40724i	−0.151433	−	0.262289i
$426$		0			0
$427$	−7.30065	+	12.6451i	−0.353303	+	0.611939i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.5125			0.988051			0.494026	−	0.869447i	$-0.335525\pi$
$431$	20.5125			0.988051			0.494026	+	0.869447i	$0.335525\pi$
$432$		0			0
$433$	−41.5464			−1.99659			−0.998295	−	0.0583639i	$-0.981412\pi$
$433$	−41.5464			−1.99659			−0.998295	+	0.0583639i	$0.981412\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.18017	−	15.9005i	0.439147	−	0.760625i
$438$		0			0
$439$	−15.0834	−	26.1253i	−0.719894	−	1.24689i	−0.961042	−	0.276404i	$-0.910857\pi$
$439$	−15.0834	−	26.1253i	−0.719894	−	1.24689i	0.241148	−	0.970488i	$-0.422476\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.67554	+	16.7585i	0.459699	+	0.796222i	0.998945	−	0.0459267i	$-0.0146241\pi$
$443$	9.67554	+	16.7585i	0.459699	+	0.796222i	−0.539246	+	0.842148i	$0.681291\pi$
$444$		0			0
$445$	−0.565834	+	0.980054i	−0.0268231	+	0.0464590i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.4220			−0.633424			−0.316712	−	0.948522i	$-0.602579\pi$
$449$	−13.4220			−0.633424			−0.316712	+	0.948522i	$0.602579\pi$
$450$		0			0
$451$	10.9537			0.515790
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.8055	−	22.1798i	0.600331	−	1.03980i
$456$		0			0
$457$	−3.06037	−	5.30072i	−0.143158	−	0.247957i	0.785526	−	0.618828i	$-0.212392\pi$
$457$	−3.06037	−	5.30072i	−0.143158	−	0.247957i	−0.928684	+	0.370871i	$0.879059\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.80084	+	10.0473i	0.270172	+	0.467952i	0.968906	−	0.247430i	$-0.0795861\pi$
$461$	5.80084	+	10.0473i	0.270172	+	0.467952i	−0.698734	+	0.715382i	$0.746253\pi$
$462$		0			0
$463$	−4.42830	+	7.67005i	−0.205801	+	0.356457i	−0.950388	−	0.311068i	$-0.899313\pi$
$463$	−4.42830	+	7.67005i	−0.205801	+	0.356457i	0.744587	+	0.667526i	$0.232647\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.20231			0.333283			0.166642	−	0.986018i	$-0.446708\pi$
$467$	7.20231			0.333283			0.166642	+	0.986018i	$0.446708\pi$
$468$		0			0
$469$	25.1422			1.16096
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.77684	−	8.27372i	0.219639	−	0.380426i
$474$		0			0
$475$	2.68576	+	4.65187i	0.123231	+	0.213443i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.2521	+	21.2213i	0.559815	+	0.969628i	0.997511	+	0.0705051i	$0.0224611\pi$
$479$	12.2521	+	21.2213i	0.559815	+	0.969628i	−0.437696	+	0.899123i	$0.644206\pi$
$480$		0			0
$481$	−14.6807	+	25.4277i	−0.669382	+	1.15940i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.05263			0.229428
$486$		0			0
$487$	37.7200			1.70926			0.854629	−	0.519239i	$-0.173785\pi$
$487$	37.7200			1.70926			0.854629	+	0.519239i	$0.173785\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.203651	−	0.352734i	0.00919063	−	0.0159186i	−0.861394	−	0.507938i	$-0.830408\pi$
$491$	0.203651	−	0.352734i	0.00919063	−	0.0159186i	0.870584	+	0.492020i	$0.163741\pi$
$492$		0			0
$493$	−15.9808	−	27.6796i	−0.719741	−	1.24663i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.1637	+	17.6040i	0.455903	+	0.789647i
$498$		0			0
$499$	−3.73644	+	6.47171i	−0.167266	+	0.289714i	−0.937458	−	0.348099i	$-0.886827\pi$
$499$	−3.73644	+	6.47171i	−0.167266	+	0.289714i	0.770192	+	0.637813i	$0.220161\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−34.5169			−1.53903			−0.769517	−	0.638626i	$-0.779503\pi$
$503$	−34.5169			−1.53903			−0.769517	+	0.638626i	$0.779503\pi$
$504$		0			0
$505$	15.2169			0.677145
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.41136	−	2.44454i	0.0625572	−	0.108352i	−0.833051	−	0.553197i	$-0.813408\pi$
$509$	1.41136	−	2.44454i	0.0625572	−	0.108352i	0.895608	+	0.444845i	$0.146741\pi$
$510$		0			0
$511$	−11.2705	−	19.5211i	−0.498579	−	0.863564i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.68252	−	13.3065i	−0.338532	−	0.586355i
$516$		0			0
$517$	1.66920	−	2.89114i	0.0734114	−	0.127152i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.11300			−0.399248			−0.199624	−	0.979873i	$-0.563972\pi$
$521$	−9.11300			−0.399248			−0.199624	+	0.979873i	$0.563972\pi$
$522$		0			0
$523$	1.96313			0.0858418			0.0429209	−	0.999078i	$-0.486334\pi$
$523$	1.96313			0.0858418			0.0429209	+	0.999078i	$0.486334\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.9279	+	25.8559i	−0.650269	+	1.12630i
$528$		0			0
$529$	3.39339	+	5.87752i	0.147539	+	0.255544i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	25.9325	+	44.9165i	1.12326	+	1.94555i
$534$		0			0
$535$	20.1570	−	34.9130i	0.871465	−	1.50942i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.42084			−0.190419
$540$		0			0
$541$	−3.46053			−0.148780			−0.0743899	−	0.997229i	$-0.523701\pi$
$541$	−3.46053			−0.148780			−0.0743899	+	0.997229i	$0.523701\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.59070	−	4.48723i	0.110974	−	0.192212i
$546$		0			0
$547$	−4.80884	−	8.32915i	−0.205611	−	0.356129i	0.744716	−	0.667381i	$-0.232585\pi$
$547$	−4.80884	−	8.32915i	−0.205611	−	0.356129i	−0.950327	+	0.311252i	$0.899252\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.7484	+	23.8129i	0.585702	+	1.01447i
$552$		0			0
$553$	−7.93645	+	13.7463i	−0.337492	+	0.584554i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.47602			0.401512			0.200756	−	0.979641i	$-0.435660\pi$
$557$	9.47602			0.401512			0.200756	+	0.979641i	$0.435660\pi$
$558$		0			0
$559$	45.2360			1.91328
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.8982	+	36.1967i	−0.880753	+	1.52551i	−0.0302481	+	0.999542i	$0.509630\pi$
$563$	−20.8982	+	36.1967i	−0.880753	+	1.52551i	−0.850505	+	0.525967i	$0.823704\pi$
$564$		0			0
$565$	−3.62725	−	6.28257i	−0.152599	−	0.264310i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.74041	−	16.8709i	−0.408339	−	0.707265i	0.586364	−	0.810047i	$-0.300559\pi$
$569$	−9.74041	−	16.8709i	−0.408339	−	0.707265i	−0.994704	+	0.102783i	$0.967225\pi$
$570$		0			0
$571$	9.27352	−	16.0622i	0.388085	−	0.672182i	−0.604107	−	0.796903i	$-0.706470\pi$
$571$	9.27352	−	16.0622i	0.388085	−	0.672182i	0.992192	+	0.124721i	$0.0398035\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.74336			0.197812
$576$		0			0
$577$	28.5458			1.18838			0.594188	−	0.804326i	$-0.297473\pi$
$577$	28.5458			1.18838			0.594188	+	0.804326i	$0.297473\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.1694	+	28.0062i	−0.670820	+	1.16189i
$582$		0			0
$583$	0.924361	+	1.60104i	0.0382831	+	0.0663083i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.49840	−	2.59531i	−0.0618457	−	0.107120i	0.833445	−	0.552603i	$-0.186365\pi$
$587$	−1.49840	−	2.59531i	−0.0618457	−	0.107120i	−0.895290	+	0.445483i	$0.853032\pi$
$588$		0			0
$589$	12.8425	−	22.2439i	0.529168	−	0.916545i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.7528			−1.01648			−0.508238	−	0.861216i	$-0.669703\pi$
$593$	−24.7528			−1.01648			−0.508238	+	0.861216i	$0.669703\pi$
$594$		0			0
$595$	23.9541			0.982022
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.2808	+	33.3954i	−0.787794	+	1.36450i	0.139522	+	0.990219i	$0.455443\pi$
$599$	−19.2808	+	33.3954i	−0.787794	+	1.36450i	−0.927316	+	0.374280i	$0.877890\pi$
$600$		0			0
$601$	−15.4398	−	26.7426i	−0.629804	−	1.09085i	−0.987591	−	0.157050i	$-0.949802\pi$
$601$	−15.4398	−	26.7426i	−0.629804	−	1.09085i	0.357786	−	0.933803i	$-0.383532\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.8905	+	20.5949i	0.483417	+	0.837302i
$606$		0			0
$607$	14.1298	−	24.4736i	0.573512	−	0.993352i	−0.422689	−	0.906275i	$-0.638914\pi$
$607$	14.1298	−	24.4736i	0.573512	−	0.993352i	0.996202	−	0.0870777i	$-0.0277528\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.8071			0.639487
$612$		0			0
$613$	28.6419			1.15684			0.578419	−	0.815740i	$-0.303670\pi$
$613$	28.6419			1.15684			0.578419	+	0.815740i	$0.303670\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.80372	+	3.12413i	−0.0726150	+	0.125773i	−0.900047	−	0.435794i	$-0.856468\pi$
$617$	−1.80372	+	3.12413i	−0.0726150	+	0.125773i	0.827432	+	0.561566i	$0.189801\pi$
$618$		0			0
$619$	4.76132	+	8.24686i	0.191374	+	0.331469i	0.945706	−	0.325024i	$-0.105372\pi$
$619$	4.76132	+	8.24686i	0.191374	+	0.331469i	−0.754332	+	0.656493i	$0.772039\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.413930	−	0.716947i	−0.0165837	−	0.0287239i
$624$		0			0
$625$	14.7512	−	25.5498i	0.590047	−	1.02199i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.4619			−1.09498
$630$		0			0
$631$	−30.5885			−1.21771			−0.608855	−	0.793281i	$-0.708371\pi$
$631$	−30.5885			−1.21771			−0.608855	+	0.793281i	$0.708371\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.7186	−	32.4215i	0.742824	−	1.28661i
$636$		0			0
$637$	−10.4662	−	18.1280i	−0.414686	−	0.718257i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.09007	−	7.08420i	−0.161548	−	0.279809i	0.773876	−	0.633337i	$-0.218315\pi$
$641$	−4.09007	−	7.08420i	−0.161548	−	0.279809i	−0.935424	+	0.353528i	$0.884982\pi$
$642$		0			0
$643$	13.0611	−	22.6225i	0.515079	−	0.892144i	−0.484768	−	0.874643i	$-0.661096\pi$
$643$	13.0611	−	22.6225i	0.515079	−	0.892144i	0.999847	−	0.0175005i	$-0.00557087\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	22.0170			0.865577			0.432789	−	0.901495i	$-0.357530\pi$
$647$	22.0170			0.865577			0.432789	+	0.901495i	$0.357530\pi$
$648$		0			0
$649$	−4.44557			−0.174504
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.14611	−	1.98512i	0.0448508	−	0.0776839i	−0.842729	−	0.538339i	$-0.819052\pi$
$653$	1.14611	−	1.98512i	0.0448508	−	0.0776839i	0.887579	+	0.460655i	$0.152385\pi$
$654$		0			0
$655$	−4.70544	−	8.15006i	−0.183857	−	0.318449i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.29926	−	5.71449i	−0.128521	−	0.222605i	0.794583	−	0.607156i	$-0.207690\pi$
$659$	−3.29926	−	5.71449i	−0.128521	−	0.222605i	−0.923104	+	0.384551i	$0.874356\pi$
$660$		0			0
$661$	−2.88305	+	4.99359i	−0.112138	+	0.194228i	−0.916632	−	0.399732i	$-0.869103\pi$
$661$	−2.88305	+	4.99359i	−0.112138	+	0.194228i	0.804494	+	0.593960i	$0.202436\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.6078			−0.799137
$666$		0			0
$667$	24.2813			0.940174
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.80537	+	8.32314i	−0.185509	+	0.321311i
$672$		0			0
$673$	−22.6226	−	39.1835i	−0.872038	−	1.51041i	−0.859885	−	0.510488i	$-0.829465\pi$
$673$	−22.6226	−	39.1835i	−0.872038	−	1.51041i	−0.0121528	−	0.999926i	$-0.503868\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.70089	+	6.41013i	0.142237	+	0.246361i	0.928339	−	0.371736i	$-0.121237\pi$
$677$	3.70089	+	6.41013i	0.142237	+	0.246361i	−0.786102	+	0.618097i	$0.787904\pi$
$678$		0			0
$679$	−1.84810	+	3.20100i	−0.0709235	+	0.122843i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.88608			−0.378281			−0.189140	−	0.981950i	$-0.560570\pi$
$683$	−9.88608			−0.378281			−0.189140	+	0.981950i	$0.560570\pi$
$684$		0			0
$685$	31.7878			1.21455
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.37679	+	7.58082i	−0.166742	+	0.288806i
$690$		0			0
$691$	−4.43367	−	7.67934i	−0.168665	−	0.292136i	0.769286	−	0.638905i	$-0.220612\pi$
$691$	−4.43367	−	7.67934i	−0.168665	−	0.292136i	−0.937951	+	0.346769i	$0.887279\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.0990	+	19.2240i	0.421009	+	0.729209i
$696$		0			0
$697$	−24.2548	+	42.0106i	−0.918717	+	1.59126i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.9889			1.47259			0.736296	−	0.676660i	$-0.236573\pi$
$701$	38.9889			1.47259			0.736296	+	0.676660i	$0.236573\pi$
$702$		0			0
$703$	23.6256			0.891055
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.56589	+	9.64040i	−0.209327	+	0.362564i
$708$		0			0
$709$	5.12709	+	8.88039i	0.192552	+	0.333510i	0.946095	−	0.323888i	$-0.104990\pi$
$709$	5.12709	+	8.88039i	0.192552	+	0.333510i	−0.753543	+	0.657398i	$0.771657\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.3407	−	19.6427i	−0.424712	−	0.735623i
$714$		0			0
$715$	8.42872	−	14.5990i	0.315216	−	0.545970i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.8003			−0.626545			−0.313272	−	0.949663i	$-0.601425\pi$
$719$	−16.8003			−0.626545			−0.313272	+	0.949663i	$0.601425\pi$
$720$		0			0
$721$	11.2401			0.418604
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.55187	+	6.15203i	−0.131913	+	0.228481i
$726$		0			0
$727$	−9.98309	−	17.2912i	−0.370252	−	0.641296i	0.619352	−	0.785113i	$-0.287395\pi$
$727$	−9.98309	−	17.2912i	−0.370252	−	0.641296i	−0.989604	+	0.143818i	$0.954062\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.1547	+	36.6410i	0.782436	+	1.35522i
$732$		0			0
$733$	2.98307	−	5.16683i	0.110182	−	0.190841i	−0.805661	−	0.592376i	$-0.798190\pi$
$733$	2.98307	−	5.16683i	0.110182	−	0.190841i	0.915844	+	0.401535i	$0.131523\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	16.5489			0.609585
$738$		0			0
$739$	0.673830			0.0247872			0.0123936	−	0.999923i	$-0.496055\pi$
$739$	0.673830			0.0247872			0.0123936	+	0.999923i	$0.496055\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.4737	−	26.8013i	0.567676	−	0.983244i	−0.429119	−	0.903248i	$-0.641176\pi$
$743$	15.4737	−	26.8013i	0.567676	−	0.983244i	0.996795	−	0.0799963i	$-0.0254909\pi$
$744$		0			0
$745$	−11.5621	−	20.0261i	−0.423602	−	0.733700i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	14.7457	+	25.5402i	0.538794	+	0.933219i
$750$		0			0
$751$	−0.0124745	+	0.0216064i	−0.000455200	+	0.000788429i	−0.866253	−	0.499606i	$-0.833478\pi$
$751$	−0.0124745	+	0.0216064i	−0.000455200	+	0.000788429i	0.865798	+	0.500394i	$0.166812\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.9501			1.41754
$756$		0			0
$757$	2.61883			0.0951829			0.0475914	−	0.998867i	$-0.484845\pi$
$757$	2.61883			0.0951829			0.0475914	+	0.998867i	$0.484845\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.19437	+	7.26486i	−0.152046	+	0.263351i	−0.931979	−	0.362511i	$-0.881919\pi$
$761$	−4.19437	+	7.26486i	−0.152046	+	0.263351i	0.779934	+	0.625862i	$0.215253\pi$
$762$		0			0
$763$	1.89520	+	3.28258i	0.0686109	+	0.118837i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.5247	−	18.2294i	−0.380027	−	0.658225i
$768$		0			0
$769$	−1.00513	+	1.74094i	−0.0362460	+	0.0627800i	−0.883579	−	0.468281i	$-0.844873\pi$
$769$	−1.00513	+	1.74094i	−0.0362460	+	0.0627800i	0.847333	+	0.531061i	$0.178207\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.2064			0.367099			0.183550	−	0.983010i	$-0.441241\pi$
$773$	10.2064			0.367099			0.183550	+	0.983010i	$0.441241\pi$
$774$		0			0
$775$	6.63569			0.238361
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.8666	−	36.1419i	0.747622	−	1.29492i
$780$		0			0
$781$	6.68983	+	11.5871i	0.239381	+	0.414620i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24.2464	−	41.9960i	−0.865392	−	1.49890i
$786$		0			0
$787$	13.6213	−	23.5928i	0.485547	−	0.840991i	−0.514315	−	0.857601i	$-0.671954\pi$
$787$	13.6213	−	23.5928i	0.485547	−	0.840991i	0.999862	+	0.0166097i	$0.00528727\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.30694			0.188693
$792$		0			0
$793$	−45.5062			−1.61597
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.73924	−	11.6727i	0.238716	−	0.413468i	−0.721630	−	0.692279i	$-0.756607\pi$
$797$	6.73924	−	11.6727i	0.238716	−	0.413468i	0.960346	+	0.278810i	$0.0899401\pi$
$798$		0			0
$799$	7.39223	+	12.8037i	0.261518	+	0.452963i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.41838	−	12.8490i	−0.261789	−	0.453432i
$804$		0			0
$805$	−9.09894	+	15.7598i	−0.320696	+	0.555461i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23.1457			−0.813758			−0.406879	−	0.913482i	$-0.633383\pi$
$809$	−23.1457			−0.813758			−0.406879	+	0.913482i	$0.633383\pi$
$810$		0			0
$811$	−3.56126			−0.125053			−0.0625263	−	0.998043i	$-0.519916\pi$
$811$	−3.56126			−0.125053			−0.0625263	+	0.998043i	$0.519916\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.24979	−	10.8250i	0.218921	−	0.379182i
$816$		0			0
$817$	−18.1995	−	31.5225i	−0.636720	−	1.10283i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.10855	+	8.84826i	0.178289	+	0.308806i	0.941295	−	0.337586i	$-0.109610\pi$
$821$	5.10855	+	8.84826i	0.178289	+	0.308806i	−0.763005	+	0.646392i	$0.776277\pi$
$822$		0			0
$823$	10.0949	−	17.4848i	0.351884	−	0.609482i	−0.634695	−	0.772763i	$-0.718874\pi$
$823$	10.0949	−	17.4848i	0.351884	−	0.609482i	0.986580	+	0.163281i	$0.0522077\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8.16401			−0.283891			−0.141945	−	0.989875i	$-0.545336\pi$
$827$	−8.16401			−0.283891			−0.141945	+	0.989875i	$0.545336\pi$
$828$		0			0
$829$	31.4145			1.09107			0.545536	−	0.838087i	$-0.316326\pi$
$829$	31.4145			1.09107			0.545536	+	0.838087i	$0.316326\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.78908	−	16.9552i	0.339171	−	0.587462i
$834$		0			0
$835$	19.0162	+	32.9370i	0.658082	+	1.13983i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.86279	−	10.1546i	−0.202406	−	0.350577i	0.746897	−	0.664939i	$-0.231543\pi$
$839$	−5.86279	−	10.1546i	−0.202406	−	0.350577i	−0.949303	+	0.314362i	$0.898209\pi$
$840$		0			0
$841$	−3.68205	+	6.37749i	−0.126967	+	0.219913i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	47.5065			1.63427
$846$		0			0
$847$	−17.3967			−0.597757
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.4314	−	18.0677i	0.357583	−	0.619351i
$852$		0			0
$853$	12.0413	+	20.8561i	0.412285	+	0.714098i	0.995139	−	0.0984785i	$-0.0313976\pi$
$853$	12.0413	+	20.8561i	0.412285	+	0.714098i	−0.582855	+	0.812577i	$0.698064\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.2545	+	31.6178i	0.623563	+	1.08004i	0.988817	+	0.149135i	$0.0476490\pi$
$857$	18.2545	+	31.6178i	0.623563	+	1.08004i	−0.365253	+	0.930908i	$0.619018\pi$
$858$		0			0
$859$	−17.9032	+	31.0092i	−0.610849	+	1.05802i	0.380248	+	0.924884i	$0.375838\pi$
$859$	−17.9032	+	31.0092i	−0.610849	+	1.05802i	−0.991098	+	0.133137i	$0.957495\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	38.2356			1.30156			0.650778	−	0.759268i	$-0.274443\pi$
$863$	38.2356			1.30156			0.650778	+	0.759268i	$0.274443\pi$
$864$		0			0
$865$	26.3565			0.896148
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.22386	+	9.04799i	−0.177207	+	0.306932i
$870$		0			0
$871$	39.1789	+	67.8598i	1.32753	+	2.29934i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.63665	+	14.9591i	0.291972	+	0.505710i
$876$		0			0
$877$	−6.01326	+	10.4153i	−0.203054	+	0.351699i	−0.949511	−	0.313734i	$-0.898420\pi$
$877$	−6.01326	+	10.4153i	−0.203054	+	0.351699i	0.746457	+	0.665433i	$0.231753\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.00744			0.303468			0.151734	−	0.988421i	$-0.451514\pi$
$881$	9.00744			0.303468			0.151734	+	0.988421i	$0.451514\pi$
$882$		0			0
$883$	2.49554			0.0839815			0.0419907	−	0.999118i	$-0.486630\pi$
$883$	2.49554			0.0839815			0.0419907	+	0.999118i	$0.486630\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.30538	−	2.26098i	0.0438303	−	0.0759163i	−0.843278	−	0.537478i	$-0.819377\pi$
$887$	1.30538	−	2.26098i	0.0438303	−	0.0759163i	0.887108	+	0.461561i	$0.152711\pi$
$888$		0			0
$889$	13.6934	+	23.7176i	0.459261	+	0.795463i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.35957	−	11.0151i	−0.212815	−	0.368606i
$894$		0			0
$895$	−11.5245	+	19.9610i	−0.385221	+	0.667222i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	33.9681			1.13290
$900$		0			0
$901$	−8.18726			−0.272757
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.89537	+	5.01493i	−0.0962453	+	0.166702i
$906$		0			0
$907$	−25.4329	−	44.0510i	−0.844485	−	1.46269i	−0.886068	−	0.463555i	$-0.846574\pi$
$907$	−25.4329	−	44.0510i	−0.844485	−	1.46269i	0.0415833	−	0.999135i	$-0.486760\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13.2421	+	22.9360i	0.438731	+	0.759905i	0.997592	−	0.0693569i	$-0.0220947\pi$
$911$	13.2421	+	22.9360i	0.438731	+	0.759905i	−0.558861	+	0.829262i	$0.688761\pi$
$912$		0			0
$913$	−10.6429	+	18.4340i	−0.352227	+	0.610076i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.88442			0.227344
$918$		0			0
$919$	52.6463			1.73664			0.868320	−	0.496004i	$-0.165200\pi$
$919$	52.6463			1.73664			0.868320	+	0.496004i	$0.165200\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−31.6759	+	54.8643i	−1.04263	+	1.80588i
$924$		0			0
$925$	3.05181	+	5.28589i	0.100343	+	0.173799i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.9141	−	48.3487i	−0.915832	−	1.58627i	−0.805678	−	0.592354i	$-0.798199\pi$
$929$	−27.9141	−	48.3487i	−0.915832	−	1.58627i	−0.110154	−	0.993914i	$-0.535135\pi$
$930$		0			0
$931$	−8.42159	+	14.5866i	−0.276007	+	0.478057i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	15.7668			0.515631
$936$		0			0
$937$	29.6597			0.968940			0.484470	−	0.874808i	$-0.339013\pi$
$937$	29.6597			0.968940			0.484470	+	0.874808i	$0.339013\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.15670	+	8.93166i	−0.168104	+	0.291164i	−0.937753	−	0.347303i	$-0.887098\pi$
$941$	−5.15670	+	8.93166i	−0.168104	+	0.291164i	0.769650	+	0.638467i	$0.220431\pi$
$942$		0			0
$943$	−18.4264	−	31.9154i	−0.600045	−	1.03931i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.34608	−	4.06353i	−0.0762375	−	0.132047i	0.825386	−	0.564568i	$-0.190957\pi$
$947$	−2.34608	−	4.06353i	−0.0762375	−	0.132047i	−0.901624	+	0.432521i	$0.857624\pi$
$948$		0			0
$949$	35.1255	−	60.8392i	1.14022	−	1.97492i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	52.8611			1.71234			0.856169	−	0.516696i	$-0.172838\pi$
$953$	52.8611			1.71234			0.856169	+	0.516696i	$0.172838\pi$
$954$		0			0
$955$	42.4637			1.37409
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.6270	+	20.1385i	−0.375455	+	0.650307i
$960$		0			0
$961$	−0.364991	−	0.632183i	−0.0117739	−	0.0203930i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−30.1648	−	52.2469i	−0.971038	−	1.68189i
$966$		0			0
$967$	−2.01885	+	3.49675i	−0.0649218	+	0.112448i	−0.896659	−	0.442721i	$-0.854013\pi$
$967$	−2.01885	+	3.49675i	−0.0649218	+	0.112448i	0.831737	+	0.555169i	$0.187346\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.1431			−0.582241			−0.291120	−	0.956686i	$-0.594028\pi$
$971$	−18.1431			−0.582241			−0.291120	+	0.956686i	$0.594028\pi$
$972$		0			0
$973$	−16.2387			−0.520589
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.77409	+	8.26896i	−0.152737	+	0.264547i	−0.932233	−	0.361860i	$-0.882142\pi$
$977$	−4.77409	+	8.26896i	−0.152737	+	0.264547i	0.779496	+	0.626407i	$0.215475\pi$
$978$		0			0
$979$	−0.272453	−	0.471902i	−0.00870763	−	0.0150821i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	26.2347	+	45.4398i	0.836757	+	1.44931i	0.892592	+	0.450866i	$0.148885\pi$
$983$	26.2347	+	45.4398i	0.836757	+	1.44931i	−0.0558348	+	0.998440i	$0.517782\pi$
$984$		0			0
$985$	−19.5081	+	33.7889i	−0.621578	+	1.07661i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.1424			−1.02207
$990$		0			0
$991$	1.87093			0.0594320			0.0297160	−	0.999558i	$-0.490540\pi$
$991$	1.87093			0.0594320			0.0297160	+	0.999558i	$0.490540\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17.9909	+	31.1612i	−0.570351	+	0.987877i
$996$		0			0
$997$	−24.4794	−	42.3996i	−0.775270	−	1.34281i	−0.934642	−	0.355589i	$-0.884280\pi$
$997$	−24.4794	−	42.3996i	−0.775270	−	1.34281i	0.159372	−	0.987219i	$-0.449053\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.i.i.2305.2		12
3.2	odd	2		1152.2.i.l.769.6	yes	12
4.3	odd	2		3456.2.i.j.2305.2		12
8.3	odd	2		3456.2.i.l.2305.5		12
8.5	even	2		3456.2.i.k.2305.5		12
9.2	odd	6		1152.2.i.l.385.6	yes	12
9.7	even	3	inner	3456.2.i.i.1153.2		12
12.11	even	2		1152.2.i.j.769.1	yes	12
24.5	odd	2		1152.2.i.i.769.1	yes	12
24.11	even	2		1152.2.i.k.769.6	yes	12
36.7	odd	6		3456.2.i.j.1153.2		12
36.11	even	6		1152.2.i.j.385.1	yes	12
72.11	even	6		1152.2.i.k.385.6	yes	12
72.29	odd	6		1152.2.i.i.385.1	✓	12
72.43	odd	6		3456.2.i.l.1153.5		12
72.61	even	6		3456.2.i.k.1153.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.i.i.385.1	✓	12	72.29	odd	6
1152.2.i.i.769.1	yes	12	24.5	odd	2
1152.2.i.j.385.1	yes	12	36.11	even	6
1152.2.i.j.769.1	yes	12	12.11	even	2
1152.2.i.k.385.6	yes	12	72.11	even	6
1152.2.i.k.769.6	yes	12	24.11	even	2
1152.2.i.l.385.6	yes	12	9.2	odd	6
1152.2.i.l.769.6	yes	12	3.2	odd	2
3456.2.i.i.1153.2		12	9.7	even	3	inner
3456.2.i.i.2305.2		12	1.1	even	1	trivial
3456.2.i.j.1153.2		12	36.7	odd	6
3456.2.i.j.2305.2		12	4.3	odd	2
3456.2.i.k.1153.5		12	72.61	even	6
3456.2.i.k.2305.5		12	8.5	even	2
3456.2.i.l.1153.5		12	72.43	odd	6
3456.2.i.l.2305.5		12	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.24278 + 2.15256i) q^{5} +(-0.909142 - 1.57468i) q^{7} +O(q^{10})\)	\(q+(-1.24278 + 2.15256i) q^{5} +(-0.909142 - 1.57468i) q^{7} +(-0.598407 - 1.03647i) q^{11} +(2.83342 - 4.90762i) q^{13} +5.30021 q^{17} -4.55980 q^{19} +(-2.01328 + 3.48711i) q^{23} +(-0.589008 - 1.02019i) q^{25} +(-3.01513 - 5.22236i) q^{29} +(-2.81647 + 4.87827i) q^{31} +4.51946 q^{35} -5.18127 q^{37} +(-4.57620 + 7.92621i) q^{41} +(3.99129 + 6.91313i) q^{43} +(1.39470 + 2.41570i) q^{47} +(1.84692 - 3.19896i) q^{49} -1.54470 q^{53} +2.97475 q^{55} +(1.85725 - 3.21686i) q^{59} +(-4.01513 - 6.95441i) q^{61} +(7.04263 + 12.1982i) q^{65} +(-6.91372 + 11.9749i) q^{67} -11.1794 q^{71} +12.3969 q^{73} +(-1.08807 + 1.88460i) q^{77} +(-4.36480 - 7.56006i) q^{79} +(-8.89267 - 15.4025i) q^{83} +(-6.58700 + 11.4090i) q^{85} +0.455297 q^{89} -10.3039 q^{91} +(5.66683 - 9.81524i) q^{95} +(-1.01640 - 1.76045i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 12 * q - 2 * q^5 - 6 * q^7	\( 12 q - 2 q^{5} - 6 q^{7} - 4 q^{11} - 10 q^{13} - 4 q^{17} + 4 q^{19} + 8 q^{23} - 14 q^{25} - 2 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} - 14 q^{47} - 18 q^{49} + 24 q^{53} + 16 q^{55} - 6 q^{59} - 14 q^{61} + 8 q^{65} + 4 q^{67} - 28 q^{71} + 60 q^{73} + 2 q^{77} - 16 q^{79} - 24 q^{83} - 16 q^{85} + 48 q^{89} - 52 q^{91} - 20 q^{95} - 14 q^{97}+O(q^{100}) \) 12 * q - 2 * q^5 - 6 * q^7 - 4 * q^11 - 10 * q^13 - 4 * q^17 + 4 * q^19 + 8 * q^23 - 14 * q^25 - 2 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 - 14 * q^47 - 18 * q^49 + 24 * q^53 + 16 * q^55 - 6 * q^59 - 14 * q^61 + 8 * q^65 + 4 * q^67 - 28 * q^71 + 60 * q^73 + 2 * q^77 - 16 * q^79 - 24 * q^83 - 16 * q^85 + 48 * q^89 - 52 * q^91 - 20 * q^95 - 14 * q^97

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