LMFDB - Embedded newform 960.3.c.j.449.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(449,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.c (of order $2$, degree $1$, 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:	$26.1581053786$
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} + 34x^{10} + 305x^{8} + 616x^{6} + 305x^{4} + 34x^{2} + 1 $ x^12 + 34x^10 + 305x^8 + 616x^6 + 305x^4 + 34*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.9
Root			$-3.28615i$ of defining polynomial
Character	$\chi$	$=$	960.449
Dual form			960.3.c.j.449.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.49147 - 1.67109i) q^{3} +(-4.19906 - 2.71439i) q^{5} -12.7692i q^{7} +(3.41489 - 8.32698i) q^{9} +O(q^{10})$	\(q+(2.49147 - 1.67109i) q^{3} +(-4.19906 - 2.71439i) q^{5} -12.7692i q^{7} +(3.41489 - 8.32698i) q^{9} -12.6296i q^{11} +7.44085i q^{13} +(-14.9978 + 0.254179i) q^{15} +14.0550 q^{17} -31.0176 q^{19} +(-21.3386 - 31.8142i) q^{21} +7.50423 q^{23} +(10.2641 + 22.7958i) q^{25} +(-5.40707 - 26.4530i) q^{27} +15.7298i q^{29} +20.4893 q^{31} +(-21.1053 - 31.4663i) q^{33} +(-34.6607 + 53.6186i) q^{35} +12.9261i q^{37} +(12.4344 + 18.5387i) q^{39} +13.8451i q^{41} +30.0797i q^{43} +(-36.9420 + 25.6961i) q^{45} +20.2570 q^{47} -114.053 q^{49} +(35.0176 - 23.4872i) q^{51} +29.1185 q^{53} +(-34.2817 + 53.0324i) q^{55} +(-77.2795 + 51.8333i) q^{57} -47.6333i q^{59} -43.0176 q^{61} +(-106.329 - 43.6054i) q^{63} +(20.1974 - 31.2445i) q^{65} +0.630153i q^{67} +(18.6966 - 12.5403i) q^{69} -90.4047i q^{71} +46.2193i q^{73} +(63.6667 + 39.6427i) q^{75} -161.270 q^{77} -37.9610 q^{79} +(-57.6771 - 56.8713i) q^{81} +80.2267 q^{83} +(-59.0176 - 38.1507i) q^{85} +(26.2860 + 39.1904i) q^{87} -140.923i q^{89} +95.0138 q^{91} +(51.0486 - 34.2396i) q^{93} +(130.245 + 84.1939i) q^{95} -10.3429i q^{97} +(-105.166 - 43.1286i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{9}+O(q^{10}) $ 12 * q + 8 * q^9	$ 12 q + 8 q^{9} - 16 q^{15} - 4 q^{21} + 36 q^{25} + 48 q^{31} + 128 q^{39} + 68 q^{45} - 252 q^{49} + 48 q^{51} + 48 q^{55} - 144 q^{61} - 268 q^{69} + 304 q^{75} - 432 q^{79} - 188 q^{81} - 336 q^{85} - 560 q^{99}+O(q^{100}) $ 12 * q + 8 * q^9 - 16 * q^15 - 4 * q^21 + 36 * q^25 + 48 * q^31 + 128 * q^39 + 68 * q^45 - 252 * q^49 + 48 * q^51 + 48 * q^55 - 144 * q^61 - 268 * q^69 + 304 * q^75 - 432 * q^79 - 188 * q^81 - 336 * q^85 - 560 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	2.49147	−	1.67109i	0.830491	−	0.557032i
$3$	2.49147	−	1.67109i	0.830491	−	0.557032i
$4$		0			0
$5$	−4.19906	−	2.71439i	−0.839811	−	0.542879i
$5$	−4.19906	−	2.71439i	−0.839811	−	0.542879i
$6$		0			0
$7$		−	12.7692i		−	1.82417i	−0.409998	−	0.912086i	$-0.634471\pi$
$7$		−	12.7692i		−	1.82417i	0.409998	−	0.912086i	$-0.365529\pi$
$8$		0			0
$9$	3.41489	−	8.32698i	0.379432	−	0.925220i
$10$		0			0
$11$		−	12.6296i		−	1.14815i	−0.818804	−	0.574073i	$-0.805363\pi$
$11$		−	12.6296i		−	1.14815i	0.818804	−	0.574073i	$-0.194637\pi$
$12$		0			0
$13$			7.44085i			0.572373i	0.958174	+	0.286187i	$0.0923877\pi$
$13$			7.44085i			0.572373i	−0.958174	+	0.286187i	$0.907612\pi$
$14$		0			0
$15$	−14.9978	+	0.254179i	−0.999856	+	0.0169452i
$16$		0			0
$17$	14.0550			0.826763			0.413381	−	0.910558i	$-0.364348\pi$
$17$	14.0550			0.826763			0.413381	+	0.910558i	$0.364348\pi$
$18$		0			0
$19$	−31.0176			−1.63250			−0.816252	−	0.577696i	$-0.803952\pi$
$19$	−31.0176			−1.63250			−0.816252	+	0.577696i	$0.803952\pi$
$20$		0			0
$21$	−21.3386	−	31.8142i	−1.01612	−	1.51496i
$22$		0			0
$23$	7.50423			0.326271			0.163135	−	0.986604i	$-0.447839\pi$
$23$	7.50423			0.326271			0.163135	+	0.986604i	$0.447839\pi$
$24$		0			0
$25$	10.2641	+	22.7958i	0.410565	+	0.911831i
$26$		0			0
$27$	−5.40707	−	26.4530i	−0.200262	−	0.979742i
$28$		0			0
$29$			15.7298i			0.542407i	0.962522	+	0.271204i	$0.0874217\pi$
$29$			15.7298i			0.542407i	−0.962522	+	0.271204i	$0.912578\pi$
$30$		0			0
$31$	20.4893			0.660945			0.330473	−	0.943816i	$-0.392792\pi$
$31$	20.4893			0.660945			0.330473	+	0.943816i	$0.392792\pi$
$32$		0			0
$33$	−21.1053	−	31.4663i	−0.639553	−	0.953525i
$34$		0			0
$35$	−34.6607	+	53.6186i	−0.990305	+	1.53196i
$36$		0			0
$37$			12.9261i			0.349355i	0.984626	+	0.174677i	$0.0558883\pi$
$37$			12.9261i			0.349355i	−0.984626	+	0.174677i	$0.944112\pi$
$38$		0			0
$39$	12.4344	+	18.5387i	0.318830	+	0.475351i
$40$		0			0
$41$			13.8451i			0.337685i	0.985643	+	0.168843i	$0.0540029\pi$
$41$			13.8451i			0.337685i	−0.985643	+	0.168843i	$0.945997\pi$
$42$		0			0
$43$			30.0797i			0.699528i	0.936838	+	0.349764i	$0.113738\pi$
$43$			30.0797i			0.699528i	−0.936838	+	0.349764i	$0.886262\pi$
$44$		0			0
$45$	−36.9420	+	25.6961i	−0.820933	+	0.571024i
$46$		0			0
$47$	20.2570			0.431000			0.215500	−	0.976504i	$-0.430862\pi$
$47$	20.2570			0.431000			0.215500	+	0.976504i	$0.430862\pi$
$48$		0			0
$49$	−114.053			−2.32761
$50$		0			0
$51$	35.0176	−	23.4872i	0.686619	−	0.460533i
$52$		0			0
$53$	29.1185			0.549406			0.274703	−	0.961529i	$-0.411420\pi$
$53$	29.1185			0.549406			0.274703	+	0.961529i	$0.411420\pi$
$54$		0			0
$55$	−34.2817	+	53.0324i	−0.623304	+	0.964226i
$56$		0			0
$57$	−77.2795	+	51.8333i	−1.35578	+	0.909356i
$58$		0			0
$59$		−	47.6333i		−	0.807344i	−0.914904	−	0.403672i	$-0.867734\pi$
$59$		−	47.6333i		−	0.807344i	0.914904	−	0.403672i	$-0.132266\pi$
$60$		0			0
$61$	−43.0176			−0.705206			−0.352603	−	0.935773i	$-0.614703\pi$
$61$	−43.0176			−0.705206			−0.352603	+	0.935773i	$0.614703\pi$
$62$		0			0
$63$	−106.329	−	43.6054i	−1.68776	−	0.692149i
$64$		0			0
$65$	20.1974	−	31.2445i	0.310729	−	0.480685i
$66$		0			0
$67$			0.630153i			0.00940527i	0.999989	+	0.00470264i	$0.00149690\pi$
$67$			0.630153i			0.00940527i	−0.999989	+	0.00470264i	$0.998503\pi$
$68$		0			0
$69$	18.6966	−	12.5403i	0.270965	−	0.181743i
$70$		0			0
$71$		−	90.4047i		−	1.27330i	−0.771151	−	0.636652i	$-0.780319\pi$
$71$		−	90.4047i		−	1.27330i	0.771151	−	0.636652i	$-0.219681\pi$
$72$		0			0
$73$			46.2193i			0.633140i	0.948569	+	0.316570i	$0.102531\pi$
$73$			46.2193i			0.633140i	−0.948569	+	0.316570i	$0.897469\pi$
$74$		0			0
$75$	63.6667	+	39.6427i	0.848890	+	0.528570i
$76$		0			0
$77$	−161.270			−2.09442
$78$		0			0
$79$	−37.9610			−0.480519			−0.240260	−	0.970709i	$-0.577233\pi$
$79$	−37.9610			−0.480519			−0.240260	+	0.970709i	$0.577233\pi$
$80$		0			0
$81$	−57.6771	−	56.8713i	−0.712063	−	0.702115i
$82$		0			0
$83$	80.2267			0.966587			0.483294	−	0.875458i	$-0.339440\pi$
$83$	80.2267			0.966587			0.483294	+	0.875458i	$0.339440\pi$
$84$		0			0
$85$	−59.0176	−	38.1507i	−0.694324	−	0.448832i
$86$		0			0
$87$	26.2860	+	39.1904i	0.302138	+	0.450465i
$88$		0			0
$89$		−	140.923i		−	1.58341i	−0.610907	−	0.791703i	$-0.709195\pi$
$89$		−	140.923i		−	1.58341i	0.610907	−	0.791703i	$-0.290805\pi$
$90$		0			0
$91$	95.0138			1.04411
$92$		0			0
$93$	51.0486	−	34.2396i	0.548909	−	0.368168i
$94$		0			0
$95$	130.245	+	84.1939i	1.37100	+	0.886252i
$96$		0			0
$97$		−	10.3429i		−	0.106628i	−0.998578	−	0.0533138i	$-0.983022\pi$
$97$		−	10.3429i		−	0.106628i	0.998578	−	0.0533138i	$-0.0169784\pi$
$98$		0			0
$99$	−105.166	−	43.1286i	−1.06229	−	0.435643i
$100$		0			0
$101$			19.2739i			0.190830i	0.995438	+	0.0954152i	$0.0304179\pi$
$101$			19.2739i			0.190830i	−0.995438	+	0.0954152i	$0.969582\pi$
$102$		0			0
$103$		−	6.97008i		−	0.0676707i	−0.999427	−	0.0338353i	$-0.989228\pi$
$103$		−	6.97008i		−	0.0676707i	0.999427	−	0.0338353i	$-0.0107722\pi$
$104$		0			0
$105$	3.24566	+	191.511i	0.0309110	+	1.82391i
$106$		0			0
$107$	73.7731			0.689468			0.344734	−	0.938700i	$-0.387969\pi$
$107$	73.7731			0.689468			0.344734	+	0.938700i	$0.387969\pi$
$108$		0			0
$109$	−74.0314			−0.679187			−0.339593	−	0.940572i	$-0.610289\pi$
$109$	−74.0314			−0.679187			−0.339593	+	0.940572i	$0.610289\pi$
$110$		0			0
$111$	21.6008	+	32.2051i	0.194602	+	0.290136i
$112$		0			0
$113$	−147.215			−1.30279			−0.651394	−	0.758739i	$-0.725816\pi$
$113$	−147.215			−1.30279			−0.651394	+	0.758739i	$0.725816\pi$
$114$		0			0
$115$	−31.5107	−	20.3694i	−0.274006	−	0.177126i
$116$		0			0
$117$	61.9598	+	25.4096i	0.529571	+	0.217176i
$118$		0			0
$119$		−	179.471i		−	1.50816i
$120$		0			0
$121$	−38.5069			−0.318239
$122$		0			0
$123$	23.1364	+	34.4947i	0.188101	+	0.280444i
$124$		0			0
$125$	18.7770	−	123.582i	0.150216	−	0.988653i
$126$		0			0
$127$			90.0171i			0.708796i	0.935095	+	0.354398i	$0.115314\pi$
$127$			90.0171i			0.708796i	−0.935095	+	0.354398i	$0.884686\pi$
$128$		0			0
$129$	50.2660	+	74.9428i	0.389659	+	0.580952i
$130$		0			0
$131$			11.2911i			0.0861917i	0.999071	+	0.0430958i	$0.0137221\pi$
$131$			11.2911i			0.0861917i	−0.999071	+	0.0430958i	$0.986278\pi$
$132$		0			0
$133$			396.070i			2.97797i
$134$		0			0
$135$	−49.0994	+	125.755i	−0.363699	+	0.931516i
$136$		0			0
$137$	22.4905			0.164164			0.0820822	−	0.996626i	$-0.473843\pi$
$137$	22.4905			0.164164			0.0820822	+	0.996626i	$0.473843\pi$
$138$		0			0
$139$	91.0955			0.655363			0.327682	−	0.944788i	$-0.393733\pi$
$139$	91.0955			0.655363			0.327682	+	0.944788i	$0.393733\pi$
$140$		0			0
$141$	50.4698	−	33.8514i	0.357942	−	0.240081i
$142$		0			0
$143$	93.9750			0.657168
$144$		0			0
$145$	42.6969	−	66.0504i	0.294461	−	0.455520i
$146$		0			0
$147$	−284.159	+	190.593i	−1.93306	+	1.29655i
$148$		0			0
$149$		−	228.330i		−	1.53242i	−0.642593	−	0.766208i	$-0.722141\pi$
$149$		−	228.330i		−	1.53242i	0.642593	−	0.766208i	$-0.277859\pi$
$150$		0			0
$151$	74.0390			0.490324			0.245162	−	0.969482i	$-0.421159\pi$
$151$	74.0390			0.490324			0.245162	+	0.969482i	$0.421159\pi$
$152$		0			0
$153$	47.9961	−	117.035i	0.313700	−	0.764937i
$154$		0			0
$155$	−86.0358	−	55.6161i	−0.555069	−	0.358813i
$156$		0			0
$157$		−	245.742i		−	1.56523i	−0.622504	−	0.782616i	$-0.713885\pi$
$157$		−	245.742i		−	1.56523i	0.622504	−	0.782616i	$-0.286115\pi$
$158$		0			0
$159$	72.5481	−	48.6598i	0.456277	−	0.306037i
$160$		0			0
$161$		−	95.8231i		−	0.595175i
$162$		0			0
$163$			60.1570i			0.369061i	0.982827	+	0.184531i	$0.0590765\pi$
$163$			60.1570i			0.369061i	−0.982827	+	0.184531i	$0.940924\pi$
$164$		0			0
$165$	3.21017	+	189.417i	0.0194556	+	1.14798i
$166$		0			0
$167$	−81.5664			−0.488421			−0.244211	−	0.969722i	$-0.578529\pi$
$167$	−81.5664			−0.488421			−0.244211	+	0.969722i	$0.578529\pi$
$168$		0			0
$169$	113.634			0.672389
$170$		0			0
$171$	−105.921	+	258.283i	−0.619424	+	1.51043i
$172$		0			0
$173$	167.064			0.965687			0.482843	−	0.875707i	$-0.339604\pi$
$173$	167.064			0.965687			0.482843	+	0.875707i	$0.339604\pi$
$174$		0			0
$175$	291.084	−	131.065i	1.66334	−	0.748942i
$176$		0			0
$177$	−79.5997	−	118.677i	−0.449716	−	0.670492i
$178$		0			0
$179$		−	270.104i		−	1.50896i	−0.656322	−	0.754481i	$-0.727889\pi$
$179$		−	270.104i		−	1.50896i	0.656322	−	0.754481i	$-0.272111\pi$
$180$		0			0
$181$	−86.9786			−0.480545			−0.240272	−	0.970705i	$-0.577237\pi$
$181$	−86.9786			−0.480545			−0.240272	+	0.970705i	$0.577237\pi$
$182$		0			0
$183$	−107.177	+	71.8865i	−0.585668	+	0.392822i
$184$		0			0
$185$	35.0866	−	54.2776i	0.189657	−	0.293392i
$186$		0			0
$187$		−	177.509i		−	0.949244i
$188$		0			0
$189$	−337.785	+	69.0440i	−1.78722	+	0.365312i
$190$		0			0
$191$			302.223i			1.58232i	0.611610	+	0.791159i	$0.290522\pi$
$191$			302.223i			1.58232i	−0.611610	+	0.791159i	$0.709478\pi$
$192$		0			0
$193$			306.780i			1.58953i	0.606916	+	0.794766i	$0.292407\pi$
$193$			306.780i			1.58953i	−0.606916	+	0.794766i	$0.707593\pi$
$194$		0			0
$195$	−1.89130	−	111.597i	−0.00969900	−	0.572291i
$196$		0			0
$197$	−289.956			−1.47186			−0.735930	−	0.677058i	$-0.763255\pi$
$197$	−289.956			−1.47186			−0.735930	+	0.677058i	$0.763255\pi$
$198$		0			0
$199$	−382.595			−1.92259			−0.961293	−	0.275527i	$-0.911148\pi$
$199$	−382.595			−1.92259			−0.961293	+	0.275527i	$0.911148\pi$
$200$		0			0
$201$	1.05305	+	1.57001i	0.00523903	+	0.00781100i
$202$		0			0
$203$	200.857			0.989445
$204$		0			0
$205$	37.5810	−	58.1363i	0.183322	−	0.283592i
$206$		0			0
$207$	25.6261	−	62.4876i	0.123798	−	0.301872i
$208$		0			0
$209$			391.740i			1.87435i
$210$		0			0
$211$	−321.115			−1.52187			−0.760937	−	0.648826i	$-0.775260\pi$
$211$	−321.115			−1.52187			−0.760937	+	0.648826i	$0.775260\pi$
$212$		0			0
$213$	−151.075	−	225.241i	−0.709271	−	1.05747i
$214$		0			0
$215$	81.6482	−	126.306i	0.379759	−	0.587471i
$216$		0			0
$217$		−	261.632i		−	1.20568i
$218$		0			0
$219$	77.2368	+	115.154i	0.352679	+	0.525818i
$220$		0			0
$221$			104.581i			0.473217i
$222$		0			0
$223$		−	292.432i		−	1.31135i	−0.755042	−	0.655676i	$-0.772384\pi$
$223$		−	292.432i		−	1.31135i	0.755042	−	0.655676i	$-0.227616\pi$
$224$		0			0
$225$	224.871	−	7.62426i	0.999426	−	0.0338856i
$226$		0			0
$227$	370.155			1.63064			0.815319	−	0.579013i	$-0.196562\pi$
$227$	370.155			1.63064			0.815319	+	0.579013i	$0.196562\pi$
$228$		0			0
$229$	381.985			1.66806			0.834028	−	0.551722i	$-0.186029\pi$
$229$	381.985			1.66806			0.834028	+	0.551722i	$0.186029\pi$
$230$		0			0
$231$	−401.800	+	269.498i	−1.73939	+	1.16666i
$232$		0			0
$233$	144.262			0.619150			0.309575	−	0.950875i	$-0.399813\pi$
$233$	144.262			0.619150			0.309575	+	0.950875i	$0.399813\pi$
$234$		0			0
$235$	−85.0603	−	54.9855i	−0.361959	−	0.233981i
$236$		0			0
$237$	−94.5789	+	63.4365i	−0.399067	+	0.267665i
$238$		0			0
$239$		−	249.478i		−	1.04384i	−0.852994	−	0.521921i	$-0.825216\pi$
$239$		−	249.478i		−	1.04384i	0.852994	−	0.521921i	$-0.174784\pi$
$240$		0			0
$241$	30.4541			0.126366			0.0631829	−	0.998002i	$-0.479875\pi$
$241$	30.4541			0.126366			0.0631829	+	0.998002i	$0.479875\pi$
$242$		0			0
$243$	−238.738	−	45.3095i	−0.982463	−	0.186459i
$244$		0			0
$245$	478.914	+	309.584i	1.95475	+	1.26361i
$246$		0			0
$247$		−	230.797i		−	0.934401i
$248$		0			0
$249$	199.883	−	134.066i	0.802742	−	0.538420i
$250$		0			0
$251$			68.9183i			0.274575i	0.990531	+	0.137287i	$0.0438384\pi$
$251$			68.9183i			0.274575i	−0.990531	+	0.137287i	$0.956162\pi$
$252$		0			0
$253$		−	94.7755i		−	0.374607i
$254$		0			0
$255$	−210.794	+	3.57247i	−0.826644	+	0.0140097i
$256$		0			0
$257$	−362.374			−1.41002			−0.705009	−	0.709199i	$-0.749057\pi$
$257$	−362.374			−1.41002			−0.705009	+	0.709199i	$0.749057\pi$
$258$		0			0
$259$	165.057			0.637284
$260$		0			0
$261$	130.982	+	53.7155i	0.501846	+	0.205807i
$262$		0			0
$263$	−23.4485			−0.0891580			−0.0445790	−	0.999006i	$-0.514195\pi$
$263$	−23.4485			−0.0891580			−0.0445790	+	0.999006i	$0.514195\pi$
$264$		0			0
$265$	−122.270	−	79.0391i	−0.461397	−	0.298261i
$266$		0			0
$267$	−235.496	−	351.106i	−0.882007	−	1.31500i
$268$		0			0
$269$			396.738i			1.47486i	0.675421	+	0.737432i	$0.263962\pi$
$269$			396.738i			1.47486i	−0.675421	+	0.737432i	$0.736038\pi$
$270$		0			0
$271$	143.926			0.531092			0.265546	−	0.964098i	$-0.414448\pi$
$271$	143.926			0.531092			0.265546	+	0.964098i	$0.414448\pi$
$272$		0			0
$273$	236.724	−	158.777i	0.867122	−	0.581601i
$274$		0			0
$275$	287.902	−	129.632i	1.04692	−	0.471389i
$276$		0			0
$277$		−	360.884i		−	1.30283i	−0.758722	−	0.651415i	$-0.774176\pi$
$277$		−	360.884i		−	1.30283i	0.758722	−	0.651415i	$-0.225824\pi$
$278$		0			0
$279$	69.9686	−	170.614i	0.250784	−	0.611520i
$280$		0			0
$281$			531.655i			1.89201i	0.324149	+	0.946006i	$0.394922\pi$
$281$			531.655i			1.89201i	−0.324149	+	0.946006i	$0.605078\pi$
$282$		0			0
$283$		−	257.400i		−	0.909541i	−0.890609	−	0.454770i	$-0.849721\pi$
$283$		−	257.400i		−	0.909541i	0.890609	−	0.454770i	$-0.150279\pi$
$284$		0			0
$285$	465.197	−	7.88400i	1.63227	−	0.0276632i
$286$		0			0
$287$	176.791			0.615996
$288$		0			0
$289$	−91.4579			−0.316463
$290$		0			0
$291$	−17.2839	−	25.7690i	−0.0593949	−	0.0885533i
$292$		0			0
$293$	310.456			1.05958			0.529789	−	0.848130i	$-0.322271\pi$
$293$	310.456			1.05958			0.529789	+	0.848130i	$0.322271\pi$
$294$		0			0
$295$	−129.295	+	200.015i	−0.438290	+	0.678016i
$296$		0			0
$297$	−334.091	+	68.2892i	−1.12489	+	0.229930i
$298$		0			0
$299$			55.8379i			0.186749i
$300$		0			0
$301$	384.094			1.27606
$302$		0			0
$303$	32.2085	+	48.0204i	0.106299	+	0.158483i
$304$		0			0
$305$	180.633	+	116.767i	0.592240	+	0.382841i
$306$		0			0
$307$			266.169i			0.866999i	0.901154	+	0.433499i	$0.142721\pi$
$307$			266.169i			0.866999i	−0.901154	+	0.433499i	$0.857279\pi$
$308$		0			0
$309$	−11.6477	−	17.3658i	−0.0376947	−	0.0561999i
$310$		0			0
$311$		−	186.580i		−	0.599934i	−0.953950	−	0.299967i	$-0.903024\pi$
$311$		−	186.580i		−	0.599934i	0.953950	−	0.299967i	$-0.0969757\pi$
$312$		0			0
$313$			20.5021i			0.0655019i	0.999464	+	0.0327510i	$0.0104268\pi$
$313$			20.5021i			0.0655019i	−0.999464	+	0.0327510i	$0.989573\pi$
$314$		0			0
$315$	328.119	+	471.720i	1.04165	+	1.49752i
$316$		0			0
$317$	18.9792			0.0598711			0.0299356	−	0.999552i	$-0.490470\pi$
$317$	18.9792			0.0598711			0.0299356	+	0.999552i	$0.490470\pi$
$318$		0			0
$319$	198.661			0.622763
$320$		0			0
$321$	183.804	−	123.282i	0.572597	−	0.384055i
$322$		0			0
$323$	−435.951			−1.34969
$324$		0			0
$325$	−169.620	+	76.3739i	−0.521908	+	0.234997i
$326$		0			0
$327$	−184.447	+	123.713i	−0.564059	+	0.378328i
$328$		0			0
$329$		−	258.666i		−	0.786219i
$330$		0			0
$331$	413.193			1.24832			0.624159	−	0.781297i	$-0.285442\pi$
$331$	413.193			1.24832			0.624159	+	0.781297i	$0.285442\pi$
$332$		0			0
$333$	107.636	+	44.1413i	0.323230	+	0.132556i
$334$		0			0
$335$	1.71048	−	2.64605i	0.00510592	−	0.00789865i
$336$		0			0
$337$		−	484.733i		−	1.43838i	−0.694816	−	0.719188i	$-0.744514\pi$
$337$		−	484.733i		−	1.43838i	0.694816	−	0.719188i	$-0.255486\pi$
$338$		0			0
$339$	−366.783	+	246.010i	−1.08195	+	0.725694i
$340$		0			0
$341$		−	258.772i		−	0.758862i
$342$		0			0
$343$			830.672i			2.42178i
$344$		0			0
$345$	−112.547	+	1.90741i	−0.326224	+	0.00552874i
$346$		0			0
$347$	336.214			0.968915			0.484457	−	0.874815i	$-0.339017\pi$
$347$	336.214			0.968915			0.484457	+	0.874815i	$0.339017\pi$
$348$		0			0
$349$	428.261			1.22711			0.613555	−	0.789652i	$-0.289739\pi$
$349$	428.261			1.22711			0.613555	+	0.789652i	$0.289739\pi$
$350$		0			0
$351$	196.833	−	40.2332i	0.560778	−	0.114625i
$352$		0			0
$353$	558.817			1.58305			0.791525	−	0.611137i	$-0.209287\pi$
$353$	558.817			1.58305			0.791525	+	0.611137i	$0.209287\pi$
$354$		0			0
$355$	−245.394	+	379.614i	−0.691250	+	1.06934i
$356$		0			0
$357$	−299.913	−	447.147i	−0.840092	−	1.25251i
$358$		0			0
$359$			206.915i			0.576365i	0.957576	+	0.288182i	$0.0930509\pi$
$359$			206.915i			0.576365i	−0.957576	+	0.288182i	$0.906949\pi$
$360$		0			0
$361$	601.090			1.66507
$362$		0			0
$363$	−95.9389	+	64.3487i	−0.264295	+	0.177269i
$364$		0			0
$365$	125.457	−	194.077i	0.343718	−	0.531718i
$366$		0			0
$367$			185.425i			0.505246i	0.967565	+	0.252623i	$0.0812932\pi$
$367$			185.425i			0.505246i	−0.967565	+	0.252623i	$0.918707\pi$
$368$		0			0
$369$	115.288	+	47.2794i	0.312433	+	0.128128i
$370$		0			0
$371$		−	371.821i		−	1.00221i
$372$		0			0
$373$		−	427.345i		−	1.14570i	−0.819661	−	0.572848i	$-0.805838\pi$
$373$		−	427.345i		−	1.14570i	0.819661	−	0.572848i	$-0.194162\pi$
$374$		0			0
$375$	−159.734	−	339.279i	−0.425958	−	0.904743i
$376$		0			0
$377$	−117.043			−0.310459
$378$		0			0
$379$	117.727			0.310626			0.155313	−	0.987865i	$-0.450361\pi$
$379$	117.727			0.310626			0.155313	+	0.987865i	$0.450361\pi$
$380$		0			0
$381$	150.427	+	224.275i	0.394822	+	0.588649i
$382$		0			0
$383$	470.016			1.22720			0.613598	−	0.789619i	$-0.289722\pi$
$383$	470.016			1.22720			0.613598	+	0.789619i	$0.289722\pi$
$384$		0			0
$385$	677.182	+	437.750i	1.75891	+	1.13701i
$386$		0			0
$387$	250.473	+	102.719i	0.647217	+	0.265423i
$388$		0			0
$389$			128.160i			0.329461i	0.986339	+	0.164730i	$0.0526754\pi$
$389$			128.160i			0.329461i	−0.986339	+	0.164730i	$0.947325\pi$
$390$		0			0
$391$	105.472			0.269749
$392$		0			0
$393$	18.8685	+	28.1315i	0.0480115	+	0.0715814i
$394$		0			0
$395$	159.401	+	103.041i	0.403546	+	0.260864i
$396$		0			0
$397$			276.316i			0.696011i	0.937493	+	0.348005i	$0.113141\pi$
$397$			276.316i			0.696011i	−0.937493	+	0.348005i	$0.886859\pi$
$398$		0			0
$399$	661.871	+	986.798i	1.65882	+	2.47318i
$400$		0			0
$401$		−	549.912i		−	1.37135i	−0.727907	−	0.685675i	$-0.759507\pi$
$401$		−	549.912i		−	1.37135i	0.727907	−	0.685675i	$-0.240493\pi$
$402$		0			0
$403$			152.458i			0.378307i
$404$		0			0
$405$	87.8182	+	395.364i	0.216835	+	0.976208i
$406$		0			0
$407$	163.252			0.401110
$408$		0			0
$409$	−219.166			−0.535858			−0.267929	−	0.963439i	$-0.586339\pi$
$409$	−219.166			−0.535858			−0.267929	+	0.963439i	$0.586339\pi$
$410$		0			0
$411$	56.0346	−	37.5838i	0.136337	−	0.0914448i
$412$		0			0
$413$	−608.240			−1.47273
$414$		0			0
$415$	−336.877	−	217.767i	−0.811751	−	0.524740i
$416$		0			0
$417$	226.962	−	152.229i	0.544274	−	0.365058i
$418$		0			0
$419$		−	204.522i		−	0.488119i	−0.969760	−	0.244060i	$-0.921521\pi$
$419$		−	204.522i		−	0.488119i	0.969760	−	0.244060i	$-0.0784792\pi$
$420$		0			0
$421$	−577.186			−1.37099			−0.685494	−	0.728078i	$-0.740414\pi$
$421$	−577.186			−1.37099			−0.685494	+	0.728078i	$0.740414\pi$
$422$		0			0
$423$	69.1754	−	168.680i	0.163535	−	0.398770i
$424$		0			0
$425$	144.262	+	320.394i	0.339440	+	0.753868i
$426$		0			0
$427$			549.301i			1.28642i
$428$		0			0
$429$	234.136	−	157.041i	0.545772	−	0.366063i
$430$		0			0
$431$		−	663.363i		−	1.53913i	−0.638571	−	0.769563i	$-0.720474\pi$
$431$		−	663.363i		−	1.53913i	0.638571	−	0.769563i	$-0.279526\pi$
$432$		0			0
$433$		−	226.876i		−	0.523964i	−0.965073	−	0.261982i	$-0.915624\pi$
$433$		−	226.876i		−	0.523964i	0.965073	−	0.261982i	$-0.0843760\pi$
$434$		0			0
$435$	−3.99818	−	235.913i	−0.00919122	−	0.542329i
$436$		0			0
$437$	−232.763			−0.532639
$438$		0			0
$439$	−282.524			−0.643564			−0.321782	−	0.946814i	$-0.604282\pi$
$439$	−282.524			−0.643564			−0.321782	+	0.946814i	$0.604282\pi$
$440$		0			0
$441$	−389.477	+	949.715i	−0.883168	+	2.15355i
$442$		0			0
$443$	455.605			1.02845			0.514227	−	0.857654i	$-0.328079\pi$
$443$	455.605			1.02845			0.514227	+	0.857654i	$0.328079\pi$
$444$		0			0
$445$	−382.521	+	591.744i	−0.859597	+	1.32976i
$446$		0			0
$447$	−381.561	−	568.878i	−0.853604	−	1.27266i
$448$		0			0
$449$			157.206i			0.350124i	0.984557	+	0.175062i	$0.0560127\pi$
$449$			157.206i			0.350124i	−0.984557	+	0.175062i	$0.943987\pi$
$450$		0			0
$451$	174.858			0.387712
$452$		0			0
$453$	184.466	−	123.726i	0.407210	−	0.273126i
$454$		0			0
$455$	−398.968	−	257.905i	−0.876853	−	0.566824i
$456$		0			0
$457$		−	397.152i		−	0.869041i	−0.900662	−	0.434520i	$-0.856918\pi$
$457$		−	397.152i		−	0.869041i	0.900662	−	0.434520i	$-0.143082\pi$
$458$		0			0
$459$	−75.9962	−	371.797i	−0.165569	−	0.810014i
$460$		0			0
$461$			350.730i			0.760803i	0.924821	+	0.380401i	$0.124214\pi$
$461$			350.730i			0.760803i	−0.924821	+	0.380401i	$0.875786\pi$
$462$		0			0
$463$		−	308.385i		−	0.666058i	−0.942917	−	0.333029i	$-0.891929\pi$
$463$		−	308.385i		−	0.666058i	0.942917	−	0.333029i	$-0.108071\pi$
$464$		0			0
$465$	−307.296	+	5.20794i	−0.660851	+	0.0111999i
$466$		0			0
$467$	−800.129			−1.71334			−0.856669	−	0.515866i	$-0.827470\pi$
$467$	−800.129			−1.71334			−0.856669	+	0.515866i	$0.827470\pi$
$468$		0			0
$469$	8.04656			0.0171568
$470$		0			0
$471$	−410.657	−	612.259i	−0.871884	−	1.29991i
$472$		0			0
$473$	379.895			0.803160
$474$		0			0
$475$	−318.369	−	707.070i	−0.670250	−	1.48857i
$476$		0			0
$477$	99.4364	−	242.469i	0.208462	−	0.508321i
$478$		0			0
$479$			630.135i			1.31552i	0.753227	+	0.657761i	$0.228496\pi$
$479$			630.135i			1.31552i	−0.753227	+	0.657761i	$0.771504\pi$
$480$		0			0
$481$	−96.1814			−0.199961
$482$		0			0
$483$	−160.130	−	238.741i	−0.331531	−	0.494287i
$484$		0			0
$485$	−28.0746	+	43.4303i	−0.0578858	+	0.0895470i
$486$		0			0
$487$			103.952i			0.213454i	0.994288	+	0.106727i	$0.0340372\pi$
$487$			103.952i			0.213454i	−0.994288	+	0.106727i	$0.965963\pi$
$488$		0			0
$489$	100.528	+	149.879i	0.205579	+	0.306502i
$490$		0			0
$491$		−	286.049i		−	0.582585i	−0.956634	−	0.291292i	$-0.905915\pi$
$491$		−	286.049i		−	0.582585i	0.956634	−	0.291292i	$-0.0940853\pi$
$492$		0			0
$493$			221.082i			0.448442i
$494$		0			0
$495$	324.532	+	466.563i	0.655619	+	0.942551i
$496$		0			0
$497$	−1154.40			−2.32273
$498$		0			0
$499$	−528.285			−1.05869			−0.529344	−	0.848407i	$-0.677562\pi$
$499$	−528.285			−1.05869			−0.529344	+	0.848407i	$0.677562\pi$
$500$		0			0
$501$	−203.220	+	136.305i	−0.405630	+	0.272066i
$502$		0			0
$503$	733.418			1.45809			0.729044	−	0.684467i	$-0.239965\pi$
$503$	733.418			1.45809			0.729044	+	0.684467i	$0.239965\pi$
$504$		0			0
$505$	52.3169	−	80.9321i	0.103598	−	0.160262i
$506$		0			0
$507$	283.116	−	189.893i	0.558413	−	0.374542i
$508$		0			0
$509$		−	312.619i		−	0.614183i	−0.951680	−	0.307092i	$-0.900644\pi$
$509$		−	312.619i		−	0.614183i	0.951680	−	0.307092i	$-0.0993558\pi$
$510$		0			0
$511$	590.183			1.15496
$512$		0			0
$513$	167.714	+	820.509i	0.326928	+	1.59943i
$514$		0			0
$515$	−18.9195	+	29.2677i	−0.0367370	+	0.0568306i
$516$		0			0
$517$		−	255.838i		−	0.494851i
$518$		0			0
$519$	416.235	−	279.179i	0.801995	−	0.537918i
$520$		0			0
$521$		−	715.719i		−	1.37374i	−0.726780	−	0.686870i	$-0.758984\pi$
$521$		−	715.719i		−	1.37374i	0.726780	−	0.686870i	$-0.241016\pi$
$522$		0			0
$523$		−	109.080i		−	0.208567i	−0.994548	−	0.104283i	$-0.966745\pi$
$523$		−	109.080i		−	0.208567i	0.994548	−	0.104283i	$-0.0332549\pi$
$524$		0			0
$525$	506.207	−	812.974i	0.964203	−	1.54852i
$526$		0			0
$527$	287.977			0.546445
$528$		0			0
$529$	−472.686			−0.893547
$530$		0			0
$531$	−396.641	−	162.662i	−0.746971	−	0.306332i
$532$		0			0
$533$	−103.019			−0.193282
$534$		0			0
$535$	−309.777	−	200.249i	−0.579023	−	0.374297i
$536$		0			0
$537$	−451.370	−	672.958i	−0.840540	−	1.25318i
$538$		0			0
$539$			1440.44i			2.67243i
$540$		0			0
$541$	14.9710			0.0276729			0.0138364	−	0.999904i	$-0.495596\pi$
$541$	14.9710			0.0276729			0.0138364	+	0.999904i	$0.495596\pi$
$542$		0			0
$543$	−216.705	+	145.350i	−0.399088	+	0.267679i
$544$		0			0
$545$	310.862	+	200.950i	0.570389	+	0.368716i
$546$		0			0
$547$		−	842.765i		−	1.54070i	−0.637618	−	0.770352i	$-0.720080\pi$
$547$		−	842.765i		−	1.54070i	0.637618	−	0.770352i	$-0.279920\pi$
$548$		0			0
$549$	−146.900	+	358.206i	−0.267578	+	0.652471i
$550$		0			0
$551$		−	487.901i		−	0.885482i
$552$		0			0
$553$			484.733i			0.876551i
$554$		0			0
$555$	−3.28555	−	193.864i	−0.00591990	−	0.349305i
$556$		0			0
$557$	−845.989			−1.51883			−0.759415	−	0.650606i	$-0.774515\pi$
$557$	−845.989			−1.51883			−0.759415	+	0.650606i	$0.774515\pi$
$558$		0			0
$559$	−223.819			−0.400391
$560$		0			0
$561$	−296.634	−	442.258i	−0.528759	−	0.788339i
$562$		0			0
$563$	−336.509			−0.597707			−0.298853	−	0.954299i	$-0.596604\pi$
$563$	−336.509			−0.597707			−0.298853	+	0.954299i	$0.596604\pi$
$564$		0			0
$565$	618.164	+	399.600i	1.09410	+	0.707256i
$566$		0			0
$567$	−726.202	+	736.491i	−1.28078	+	1.29893i
$568$		0			0
$569$		−	552.736i		−	0.971416i	−0.874121	−	0.485708i	$-0.838562\pi$
$569$		−	552.736i		−	0.971416i	0.874121	−	0.485708i	$-0.161438\pi$
$570$		0			0
$571$	772.222			1.35240			0.676202	−	0.736717i	$-0.263625\pi$
$571$	772.222			1.35240			0.676202	+	0.736717i	$0.263625\pi$
$572$		0			0
$573$	505.043	+	752.980i	0.881401	+	1.31410i
$574$		0			0
$575$	77.0245	+	171.065i	0.133956	+	0.297504i
$576$		0			0
$577$			848.557i			1.47064i	0.677722	+	0.735318i	$0.262967\pi$
$577$			848.557i			1.47064i	−0.677722	+	0.735318i	$0.737033\pi$
$578$		0			0
$579$	512.658	+	764.334i	0.885420	+	1.32009i
$580$		0			0
$581$		−	1024.43i		−	1.76322i
$582$		0			0
$583$		−	367.755i		−	0.630798i
$584$		0			0
$585$	−191.201	−	274.880i	−0.326839	−	0.469880i
$586$		0			0
$587$	−575.536			−0.980470			−0.490235	−	0.871590i	$-0.663089\pi$
$587$	−575.536			−0.980470			−0.490235	+	0.871590i	$0.663089\pi$
$588$		0			0
$589$	−635.529			−1.07900
$590$		0			0
$591$	−722.419	+	484.544i	−1.22237	+	0.819872i
$592$		0			0
$593$	−156.935			−0.264646			−0.132323	−	0.991207i	$-0.542244\pi$
$593$	−156.935			−0.264646			−0.132323	+	0.991207i	$0.542244\pi$
$594$		0			0
$595$	−487.154	+	753.608i	−0.818747	+	1.26657i
$596$		0			0
$597$	−953.225	+	639.352i	−1.59669	+	1.07094i
$598$		0			0
$599$			517.564i			0.864047i	0.901862	+	0.432024i	$0.142200\pi$
$599$			517.564i			0.864047i	−0.901862	+	0.432024i	$0.857800\pi$
$600$		0			0
$601$	−883.588			−1.47020			−0.735098	−	0.677961i	$-0.762864\pi$
$601$	−883.588			−1.47020			−0.735098	+	0.677961i	$0.762864\pi$
$602$		0			0
$603$	5.24727	+	2.15190i	0.00870194	+	0.00356866i
$604$		0			0
$605$	161.693	+	104.523i	0.267260	+	0.172765i
$606$		0			0
$607$			242.929i			0.400212i	0.979774	+	0.200106i	$0.0641287\pi$
$607$			242.929i			0.400212i	−0.979774	+	0.200106i	$0.935871\pi$
$608$		0			0
$609$	500.431	−	335.652i	0.821725	−	0.551152i
$610$		0			0
$611$			150.729i			0.246693i
$612$		0			0
$613$		−	157.263i		−	0.256546i	−0.991739	−	0.128273i	$-0.959057\pi$
$613$		−	157.263i		−	0.256546i	0.991739	−	0.128273i	$-0.0409434\pi$
$614$		0			0
$615$	−3.51912	−	207.646i	−0.00572215	−	0.337637i
$616$		0			0
$617$	−471.040			−0.763436			−0.381718	−	0.924279i	$-0.624667\pi$
$617$	−471.040			−0.763436			−0.381718	+	0.924279i	$0.624667\pi$
$618$		0			0
$619$	550.320			0.889047			0.444524	−	0.895767i	$-0.353373\pi$
$619$	550.320			0.889047			0.444524	+	0.895767i	$0.353373\pi$
$620$		0			0
$621$	−40.5759	−	198.510i	−0.0653396	−	0.319662i
$622$		0			0
$623$	−1799.48			−2.88841
$624$		0			0
$625$	−414.295	+	467.958i	−0.662872	+	0.748733i
$626$		0			0
$627$	654.634	+	976.009i	1.04407	+	1.55663i
$628$		0			0
$629$			181.676i			0.288834i
$630$		0			0
$631$	347.362			0.550495			0.275248	−	0.961373i	$-0.411240\pi$
$631$	347.362			0.550495			0.275248	+	0.961373i	$0.411240\pi$
$632$		0			0
$633$	−800.051	+	536.614i	−1.26390	+	0.847732i
$634$		0			0
$635$	244.342	−	377.987i	0.384790	−	0.595255i
$636$		0			0
$637$		−	848.649i		−	1.33226i
$638$		0			0
$639$	−752.798	−	308.721i	−1.17809	−	0.483132i
$640$		0			0
$641$			333.802i			0.520752i	0.965507	+	0.260376i	$0.0838465\pi$
$641$			333.802i			0.520752i	−0.965507	+	0.260376i	$0.916153\pi$
$642$		0			0
$643$		−	495.512i		−	0.770625i	−0.922786	−	0.385313i	$-0.874094\pi$
$643$		−	495.512i		−	0.770625i	0.922786	−	0.385313i	$-0.125906\pi$
$644$		0			0
$645$	−7.64561	−	451.131i	−0.0118537	−	0.699428i
$646$		0			0
$647$	−149.493			−0.231056			−0.115528	−	0.993304i	$-0.536856\pi$
$647$	−149.493			−0.231056			−0.115528	+	0.993304i	$0.536856\pi$
$648$		0			0
$649$	−601.590			−0.926948
$650$		0			0
$651$	−437.212	−	651.850i	−0.671601	−	1.00131i
$652$		0			0
$653$	261.846			0.400990			0.200495	−	0.979695i	$-0.435745\pi$
$653$	261.846			0.400990			0.200495	+	0.979695i	$0.435745\pi$
$654$		0			0
$655$	30.6485	−	47.4120i	0.0467916	−	0.0723847i
$656$		0			0
$657$	384.867	+	157.833i	0.585794	+	0.240234i
$658$		0			0
$659$		−	148.718i		−	0.225672i	−0.993614	−	0.112836i	$-0.964006\pi$
$659$		−	148.718i		−	0.225672i	0.993614	−	0.112836i	$-0.0359935\pi$
$660$		0			0
$661$	535.548			0.810209			0.405104	−	0.914270i	$-0.367235\pi$
$661$	535.548			0.810209			0.405104	+	0.914270i	$0.367235\pi$
$662$		0			0
$663$	174.765	+	260.561i	0.263597	+	0.393002i
$664$		0			0
$665$	1075.09	−	1663.12i	1.61668	−	2.50093i
$666$		0			0
$667$			118.040i			0.176972i
$668$		0			0
$669$	−488.681	−	728.586i	−0.730465	−	1.08907i
$670$		0			0
$671$			543.295i			0.809680i
$672$		0			0
$673$		−	678.388i		−	1.00801i	−0.863702	−	0.504003i	$-0.831860\pi$
$673$		−	678.388i		−	1.00801i	0.863702	−	0.504003i	$-0.168140\pi$
$674$		0			0
$675$	547.519	−	394.776i	0.811139	−	0.584853i
$676$		0			0
$677$	809.743			1.19608			0.598038	−	0.801468i	$-0.295947\pi$
$677$	809.743			1.19608			0.598038	+	0.801468i	$0.295947\pi$
$678$		0			0
$679$	−132.070			−0.194507
$680$		0			0
$681$	922.231	−	618.563i	1.35423	−	0.908316i
$682$		0			0
$683$	150.099			0.219764			0.109882	−	0.993945i	$-0.464953\pi$
$683$	150.099			0.219764			0.109882	+	0.993945i	$0.464953\pi$
$684$		0			0
$685$	−94.4390	−	61.0481i	−0.137867	−	0.0891214i
$686$		0			0
$687$	951.705	−	638.333i	1.38531	−	0.929160i
$688$		0			0
$689$			216.667i			0.314465i
$690$		0			0
$691$	−334.001			−0.483359			−0.241679	−	0.970356i	$-0.577698\pi$
$691$	−334.001			−0.483359			−0.241679	+	0.970356i	$0.577698\pi$
$692$		0			0
$693$	−550.719	+	1342.89i	−0.794688	+	1.93780i
$694$		0			0
$695$	−382.515	−	247.269i	−0.550381	−	0.355783i
$696$		0			0
$697$			194.592i			0.279185i
$698$		0			0
$699$	359.425	−	241.076i	0.514199	−	0.344886i
$700$		0			0
$701$			924.471i			1.31879i	0.751797	+	0.659394i	$0.229187\pi$
$701$			924.471i			1.31879i	−0.751797	+	0.659394i	$0.770813\pi$
$702$		0			0
$703$		−	400.937i		−	0.570324i
$704$		0			0
$705$	−303.812	+	5.14890i	−0.430938	+	0.00730340i
$706$		0			0
$707$	246.112			0.348108
$708$		0			0
$709$	−797.459			−1.12477			−0.562383	−	0.826877i	$-0.690115\pi$
$709$	−797.459			−1.12477			−0.562383	+	0.826877i	$0.690115\pi$
$710$		0			0
$711$	−129.633	+	316.101i	−0.182324	+	0.444586i
$712$		0			0
$713$	153.757			0.215647
$714$		0			0
$715$	−394.606	−	255.085i	−0.551897	−	0.356762i
$716$		0			0
$717$	−416.902	−	621.568i	−0.581453	−	0.866902i
$718$		0			0
$719$		−	907.966i		−	1.26282i	−0.775450	−	0.631409i	$-0.782477\pi$
$719$		−	907.966i		−	1.26282i	0.775450	−	0.631409i	$-0.217523\pi$
$720$		0			0
$721$	−89.0024			−0.123443
$722$		0			0
$723$	75.8757	−	50.8918i	0.104946	−	0.0703897i
$724$		0			0
$725$	−358.573	+	161.453i	−0.494584	+	0.222694i
$726$		0			0
$727$		−	1038.16i		−	1.42801i	−0.700140	−	0.714005i	$-0.746879\pi$
$727$		−	1038.16i		−	1.42801i	0.700140	−	0.714005i	$-0.253121\pi$
$728$		0			0
$729$	−670.527	+	286.067i	−0.919790	+	0.392410i
$730$		0			0
$731$			422.769i			0.578344i
$732$		0			0
$733$			399.107i			0.544485i	0.962229	+	0.272242i	$0.0877652\pi$
$733$			399.107i			0.544485i	−0.962229	+	0.272242i	$0.912235\pi$
$734$		0			0
$735$	1710.55	−	28.9898i	2.32727	−	0.0394418i
$736$		0			0
$737$	7.95858			0.0107986
$738$		0			0
$739$	452.504			0.612319			0.306159	−	0.951980i	$-0.400956\pi$
$739$	452.504			0.612319			0.306159	+	0.951980i	$0.400956\pi$
$740$		0			0
$741$	−385.684	−	575.025i	−0.520491	−	0.776012i
$742$		0			0
$743$	486.909			0.655328			0.327664	−	0.944794i	$-0.393739\pi$
$743$	486.909			0.655328			0.327664	+	0.944794i	$0.393739\pi$
$744$		0			0
$745$	−619.777	+	958.770i	−0.831916	+	1.28694i
$746$		0			0
$747$	273.965	−	668.046i	0.366754	−	0.894306i
$748$		0			0
$749$		−	942.024i		−	1.25771i
$750$		0			0
$751$	470.899			0.627029			0.313515	−	0.949583i	$-0.398494\pi$
$751$	470.899			0.627029			0.313515	+	0.949583i	$0.398494\pi$
$752$		0			0
$753$	115.169	+	171.708i	0.152947	+	0.228032i
$754$		0			0
$755$	−310.894	−	200.971i	−0.411780	−	0.266187i
$756$		0			0
$757$			886.264i			1.17076i	0.810760	+	0.585379i	$0.199054\pi$
$757$			886.264i			1.17076i	−0.810760	+	0.585379i	$0.800946\pi$
$758$		0			0
$759$	−158.379	−	236.131i	−0.208668	−	0.311108i
$760$		0			0
$761$			1022.30i			1.34336i	0.740841	+	0.671680i	$0.234427\pi$
$761$			1022.30i			1.34336i	−0.740841	+	0.671680i	$0.765573\pi$
$762$		0			0
$763$			945.322i			1.23895i
$764$		0			0
$765$	−519.218	+	361.158i	−0.678717	+	0.472102i
$766$		0			0
$767$	354.432			0.462102
$768$		0			0
$769$	1051.96			1.36796			0.683982	−	0.729499i	$-0.260247\pi$
$769$	1051.96			1.36796			0.683982	+	0.729499i	$0.260247\pi$
$770$		0			0
$771$	−902.847	+	605.562i	−1.17101	+	0.785424i
$772$		0			0
$773$	983.554			1.27238			0.636192	−	0.771530i	$-0.280508\pi$
$773$	983.554			1.27238			0.636192	+	0.771530i	$0.280508\pi$
$774$		0			0
$775$	210.305	+	467.070i	0.271361	+	0.602671i
$776$		0			0
$777$	411.234	−	275.825i	0.529259	−	0.354987i
$778$		0			0
$779$		−	429.441i		−	0.551272i
$780$		0			0
$781$	−1141.77			−1.46194
$782$		0			0
$783$	416.101	−	85.0522i	0.531419	−	0.108624i
$784$		0			0
$785$	−667.039	+	1031.88i	−0.849731	+	1.31450i
$786$		0			0
$787$		−	984.726i		−	1.25124i	−0.780128	−	0.625620i	$-0.784846\pi$
$787$		−	984.726i		−	1.25124i	0.780128	−	0.625620i	$-0.215154\pi$
$788$		0			0
$789$	−58.4214	+	39.1847i	−0.0740449	+	0.0496638i
$790$		0			0
$791$			1879.82i			2.37651i
$792$		0			0
$793$		−	320.087i		−	0.403641i
$794$		0			0
$795$	−436.715	+	7.40130i	−0.549327	+	0.00930982i
$796$		0			0
$797$	−359.710			−0.451330			−0.225665	−	0.974205i	$-0.572456\pi$
$797$	−359.710			−0.451330			−0.225665	+	0.974205i	$0.572456\pi$
$798$		0			0
$799$	284.712			0.356335
$800$		0			0
$801$	−1173.46	−	481.236i	−1.46500	−	0.600794i
$802$		0			0
$803$	583.731			0.726938
$804$		0			0
$805$	−260.102	+	402.367i	−0.323108	+	0.499834i
$806$		0			0
$807$	662.987	+	988.463i	0.821546	+	1.22486i
$808$		0			0
$809$			1174.75i			1.45211i	0.687638	+	0.726053i	$0.258648\pi$
$809$			1174.75i			1.45211i	−0.687638	+	0.726053i	$0.741352\pi$
$810$		0			0
$811$	−1000.55			−1.23373			−0.616863	−	0.787071i	$-0.711597\pi$
$811$	−1000.55			−1.23373			−0.616863	+	0.787071i	$0.711597\pi$
$812$		0			0
$813$	358.588	−	240.514i	0.441067	−	0.295835i
$814$		0			0
$815$	163.290	−	252.602i	0.200355	−	0.309942i
$816$		0			0
$817$		−	933.000i		−	1.14198i
$818$		0			0
$819$	324.461	−	791.178i	0.396167	−	0.966029i
$820$		0			0
$821$		−	536.992i		−	0.654071i	−0.945012	−	0.327036i	$-0.893950\pi$
$821$		−	536.992i		−	0.654071i	0.945012	−	0.327036i	$-0.106050\pi$
$822$		0			0
$823$			563.712i			0.684948i	0.939527	+	0.342474i	$0.111265\pi$
$823$			563.712i			0.684948i	−0.939527	+	0.342474i	$0.888735\pi$
$824$		0			0
$825$	500.672	−	804.086i	0.606875	−	0.974649i
$826$		0			0
$827$	−976.487			−1.18076			−0.590379	−	0.807126i	$-0.701022\pi$
$827$	−976.487			−1.18076			−0.590379	+	0.807126i	$0.701022\pi$
$828$		0			0
$829$	358.250			0.432147			0.216074	−	0.976377i	$-0.430675\pi$
$829$	358.250			0.432147			0.216074	+	0.976377i	$0.430675\pi$
$830$		0			0
$831$	−603.071	−	899.133i	−0.725718	−	1.08199i
$832$		0			0
$833$	−1603.01			−1.92438
$834$		0			0
$835$	342.502	+	221.403i	0.410182	+	0.265154i
$836$		0			0
$837$	−110.787	−	542.005i	−0.132362	−	0.647556i
$838$		0			0
$839$			98.9138i			0.117895i	0.998261	+	0.0589474i	$0.0187744\pi$
$839$			98.9138i			0.117895i	−0.998261	+	0.0589474i	$0.981226\pi$
$840$		0			0
$841$	593.573			0.705794
$842$		0			0
$843$	888.447	+	1324.61i	1.05391	+	1.57130i
$844$		0			0
$845$	−477.154	−	308.447i	−0.564680	−	0.365026i
$846$		0			0
$847$			491.703i			0.580523i
$848$		0			0
$849$	−430.140	−	641.305i	−0.506643	−	0.755366i
$850$		0			0
$851$			97.0007i			0.113984i
$852$		0			0
$853$		−	262.197i		−	0.307383i	−0.988119	−	0.153691i	$-0.950884\pi$
$853$		−	262.197i		−	0.307383i	0.988119	−	0.153691i	$-0.0491161\pi$
$854$		0			0
$855$	1145.85	−	797.031i	1.34018	−	0.932200i
$856$		0			0
$857$	41.1542			0.0480212			0.0240106	−	0.999712i	$-0.492356\pi$
$857$	41.1542			0.0480212			0.0240106	+	0.999712i	$0.492356\pi$
$858$		0			0
$859$	−414.736			−0.482813			−0.241406	−	0.970424i	$-0.577609\pi$
$859$	−414.736			−0.482813			−0.241406	+	0.970424i	$0.577609\pi$
$860$		0			0
$861$	440.470	−	295.434i	0.511579	−	0.343129i
$862$		0			0
$863$	−1253.21			−1.45216			−0.726078	−	0.687613i	$-0.758659\pi$
$863$	−1253.21			−1.45216			−0.726078	+	0.687613i	$0.758659\pi$
$864$		0			0
$865$	−701.510	−	453.477i	−0.810995	−	0.524251i
$866$		0			0
$867$	−227.865	+	152.835i	−0.262820	+	0.176280i
$868$		0			0
$869$			479.433i			0.551706i
$870$		0			0
$871$	−4.68888			−0.00538332
$872$		0			0
$873$	−86.1249	−	35.3197i	−0.0986539	−	0.0404579i
$874$		0			0
$875$	−1578.04	−	239.768i	−1.80347	−	0.274021i
$876$		0			0
$877$			861.957i			0.982847i	0.870921	+	0.491424i	$0.163523\pi$
$877$			861.957i			0.982847i	−0.870921	+	0.491424i	$0.836477\pi$
$878$		0			0
$879$	773.494	−	518.802i	0.879970	−	0.590218i
$880$		0			0
$881$			87.0009i			0.0987524i	0.998780	+	0.0493762i	$0.0157233\pi$
$881$			87.0009i			0.0987524i	−0.998780	+	0.0493762i	$0.984277\pi$
$882$		0			0
$883$			1022.42i			1.15789i	0.815366	+	0.578945i	$0.196536\pi$
$883$			1022.42i			1.15789i	−0.815366	+	0.578945i	$0.803464\pi$
$884$		0			0
$885$	12.1074	+	714.397i	0.0136806	+	0.807228i
$886$		0			0
$887$	−669.924			−0.755269			−0.377634	−	0.925955i	$-0.623262\pi$
$887$	−669.924			−0.755269			−0.377634	+	0.925955i	$0.623262\pi$
$888$		0			0
$889$	1149.45			1.29297
$890$		0			0
$891$	−718.263	+	728.439i	−0.806131	+	0.817552i
$892$		0			0
$893$	−628.324			−0.703610
$894$		0			0
$895$	−733.169	+	1134.18i	−0.819183	+	1.26724i
$896$		0			0
$897$	93.3104	+	139.119i	0.104025	+	0.155093i
$898$		0			0
$899$			322.293i			0.358502i
$900$		0			0
$901$	409.260			0.454229
$902$		0			0
$903$	956.960	−	641.858i	1.05976	−	0.710806i
$904$		0			0
$905$	365.228	+	236.094i	0.403567	+	0.260878i
$906$		0			0
$907$			1296.52i			1.42946i	0.699402	+	0.714729i	$0.253450\pi$
$907$			1296.52i			1.42946i	−0.699402	+	0.714729i	$0.746550\pi$
$908$		0			0
$909$	160.493	+	65.8181i	0.176560	+	0.0724071i
$910$		0			0
$911$			926.622i			1.01715i	0.861018	+	0.508574i	$0.169827\pi$
$911$			926.622i			1.01715i	−0.861018	+	0.508574i	$0.830173\pi$
$912$		0			0
$913$		−	1013.23i		−	1.10978i
$914$		0			0
$915$	645.171	−	10.9341i	0.705105	−	0.0119499i
$916$		0			0
$917$	144.179			0.157229
$918$		0			0
$919$	478.391			0.520556			0.260278	−	0.965534i	$-0.416186\pi$
$919$	478.391			0.520556			0.260278	+	0.965534i	$0.416186\pi$
$920$		0			0
$921$	444.793	+	663.152i	0.482946	+	0.720035i
$922$		0			0
$923$	672.687			0.728805
$924$		0			0
$925$	−294.661	+	132.676i	−0.318553	+	0.143433i
$926$		0			0
$927$	−58.0397	−	23.8020i	−0.0626102	−	0.0256764i
$928$		0			0
$929$		−	1452.00i		−	1.56297i	−0.623923	−	0.781486i	$-0.714462\pi$
$929$		−	1452.00i		−	1.56297i	0.623923	−	0.781486i	$-0.285538\pi$
$930$		0			0
$931$	3537.64			3.79983
$932$		0			0
$933$	−311.792	−	464.858i	−0.334182	−	0.498240i
$934$		0			0
$935$	−481.828	+	745.369i	−0.515324	+	0.797186i
$936$		0			0
$937$			1535.86i			1.63912i	0.572991	+	0.819562i	$0.305783\pi$
$937$			1535.86i			1.63912i	−0.572991	+	0.819562i	$0.694217\pi$
$938$		0			0
$939$	34.2610	+	51.0805i	0.0364866	+	0.0543988i
$940$		0			0
$941$			1237.50i			1.31509i	0.753415	+	0.657545i	$0.228405\pi$
$941$			1237.50i			1.31509i	−0.753415	+	0.657545i	$0.771595\pi$
$942$		0			0
$943$			103.897i			0.110177i
$944$		0			0
$945$	1605.79	+	626.960i	1.69925	+	0.663450i
$946$		0			0
$947$	603.800			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$947$	603.800			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$948$		0			0
$949$	−343.911			−0.362393
$950$		0			0
$951$	47.2861	−	31.7160i	0.0497225	−	0.0333501i
$952$		0			0
$953$	534.553			0.560916			0.280458	−	0.959866i	$-0.409514\pi$
$953$	534.553			0.560916			0.280458	+	0.959866i	$0.409514\pi$
$954$		0			0
$955$	820.352	−	1269.05i	0.859007	−	1.32885i
$956$		0			0
$957$	494.959	−	331.982i	0.517199	−	0.346899i
$958$		0			0
$959$		−	287.186i		−	0.299464i
$960$		0			0
$961$	−541.188			−0.563151
$962$		0			0
$963$	251.927	−	614.307i	0.261606	−	0.637909i
$964$		0			0
$965$	832.721	−	1288.19i	0.862923	−	1.33491i
$966$		0			0
$967$		−	271.601i		−	0.280870i	−0.990090	−	0.140435i	$-0.955150\pi$
$967$		−	271.601i		−	0.280870i	0.990090	−	0.140435i	$-0.0448500\pi$
$968$		0			0
$969$	−1086.16	+	728.515i	−1.12091	+	0.751822i
$970$		0			0
$971$			912.176i			0.939419i	0.882821	+	0.469709i	$0.155641\pi$
$971$			912.176i			0.939419i	−0.882821	+	0.469709i	$0.844359\pi$
$972$		0			0
$973$		−	1163.22i		−	1.19550i
$974$		0			0
$975$	−294.976	+	473.735i	−0.302539	+	0.485882i
$976$		0			0
$977$	333.628			0.341482			0.170741	−	0.985316i	$-0.445384\pi$
$977$	333.628			0.341482			0.170741	+	0.985316i	$0.445384\pi$
$978$		0			0
$979$	−1779.80			−1.81798
$980$		0			0
$981$	−252.809	+	616.458i	−0.257705	+	0.628397i
$982$		0			0
$983$	−1600.35			−1.62803			−0.814013	−	0.580847i	$-0.802722\pi$
$983$	−1600.35			−1.62803			−0.814013	+	0.580847i	$0.802722\pi$
$984$		0			0
$985$	1217.54	+	787.056i	1.23608	+	0.799041i
$986$		0			0
$987$	−432.256	−	644.460i	−0.437949	−	0.652948i
$988$		0			0
$989$			225.725i			0.228236i
$990$		0			0
$991$	69.3757			0.0700057			0.0350029	−	0.999387i	$-0.488856\pi$
$991$	69.3757			0.0700057			0.0350029	+	0.999387i	$0.488856\pi$
$992$		0			0
$993$	1029.46	−	690.485i	1.03672	−	0.695353i
$994$		0			0
$995$	1606.54	+	1038.51i	1.61461	+	1.04373i
$996$		0			0
$997$		−	482.497i		−	0.483949i	−0.970283	−	0.241974i	$-0.922205\pi$
$997$		−	482.497i		−	0.483949i	0.970283	−	0.241974i	$-0.0777950\pi$
$998$		0			0
$999$	341.936	−	69.8925i	0.342278	−	0.0699625i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.c.j.449.9		12
3.2	odd	2	inner	960.3.c.j.449.3		12
4.3	odd	2		960.3.c.k.449.4		12
5.4	even	2	inner	960.3.c.j.449.4		12
8.3	odd	2		120.3.c.a.89.9	yes	12
8.5	even	2		240.3.c.e.209.4		12
12.11	even	2		960.3.c.k.449.10		12
15.14	odd	2	inner	960.3.c.j.449.10		12
20.19	odd	2		960.3.c.k.449.9		12
24.5	odd	2		240.3.c.e.209.10		12
24.11	even	2		120.3.c.a.89.3	✓	12
40.3	even	4		600.3.l.g.401.10		12
40.13	odd	4		1200.3.l.y.401.3		12
40.19	odd	2		120.3.c.a.89.4	yes	12
40.27	even	4		600.3.l.g.401.3		12
40.29	even	2		240.3.c.e.209.9		12
40.37	odd	4		1200.3.l.y.401.10		12
60.59	even	2		960.3.c.k.449.3		12
120.29	odd	2		240.3.c.e.209.3		12
120.53	even	4		1200.3.l.y.401.4		12
120.59	even	2		120.3.c.a.89.10	yes	12
120.77	even	4		1200.3.l.y.401.9		12
120.83	odd	4		600.3.l.g.401.9		12
120.107	odd	4		600.3.l.g.401.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.c.a.89.3	✓	12	24.11	even	2
120.3.c.a.89.4	yes	12	40.19	odd	2
120.3.c.a.89.9	yes	12	8.3	odd	2
120.3.c.a.89.10	yes	12	120.59	even	2
240.3.c.e.209.3		12	120.29	odd	2
240.3.c.e.209.4		12	8.5	even	2
240.3.c.e.209.9		12	40.29	even	2
240.3.c.e.209.10		12	24.5	odd	2
600.3.l.g.401.3		12	40.27	even	4
600.3.l.g.401.4		12	120.107	odd	4
600.3.l.g.401.9		12	120.83	odd	4
600.3.l.g.401.10		12	40.3	even	4
960.3.c.j.449.3		12	3.2	odd	2	inner
960.3.c.j.449.4		12	5.4	even	2	inner
960.3.c.j.449.9		12	1.1	even	1	trivial
960.3.c.j.449.10		12	15.14	odd	2	inner
960.3.c.k.449.3		12	60.59	even	2
960.3.c.k.449.4		12	4.3	odd	2
960.3.c.k.449.9		12	20.19	odd	2
960.3.c.k.449.10		12	12.11	even	2
1200.3.l.y.401.3		12	40.13	odd	4
1200.3.l.y.401.4		12	120.53	even	4
1200.3.l.y.401.9		12	120.77	even	4
1200.3.l.y.401.10		12	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.49147 - 1.67109i) q^{3} +(-4.19906 - 2.71439i) q^{5} -12.7692i q^{7} +(3.41489 - 8.32698i) q^{9} +O(q^{10})\)	\(q+(2.49147 - 1.67109i) q^{3} +(-4.19906 - 2.71439i) q^{5} -12.7692i q^{7} +(3.41489 - 8.32698i) q^{9} -12.6296i q^{11} +7.44085i q^{13} +(-14.9978 + 0.254179i) q^{15} +14.0550 q^{17} -31.0176 q^{19} +(-21.3386 - 31.8142i) q^{21} +7.50423 q^{23} +(10.2641 + 22.7958i) q^{25} +(-5.40707 - 26.4530i) q^{27} +15.7298i q^{29} +20.4893 q^{31} +(-21.1053 - 31.4663i) q^{33} +(-34.6607 + 53.6186i) q^{35} +12.9261i q^{37} +(12.4344 + 18.5387i) q^{39} +13.8451i q^{41} +30.0797i q^{43} +(-36.9420 + 25.6961i) q^{45} +20.2570 q^{47} -114.053 q^{49} +(35.0176 - 23.4872i) q^{51} +29.1185 q^{53} +(-34.2817 + 53.0324i) q^{55} +(-77.2795 + 51.8333i) q^{57} -47.6333i q^{59} -43.0176 q^{61} +(-106.329 - 43.6054i) q^{63} +(20.1974 - 31.2445i) q^{65} +0.630153i q^{67} +(18.6966 - 12.5403i) q^{69} -90.4047i q^{71} +46.2193i q^{73} +(63.6667 + 39.6427i) q^{75} -161.270 q^{77} -37.9610 q^{79} +(-57.6771 - 56.8713i) q^{81} +80.2267 q^{83} +(-59.0176 - 38.1507i) q^{85} +(26.2860 + 39.1904i) q^{87} -140.923i q^{89} +95.0138 q^{91} +(51.0486 - 34.2396i) q^{93} +(130.245 + 84.1939i) q^{95} -10.3429i q^{97} +(-105.166 - 43.1286i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{9}+O(q^{10}) \) 12 * q + 8 * q^9	\( 12 q + 8 q^{9} - 16 q^{15} - 4 q^{21} + 36 q^{25} + 48 q^{31} + 128 q^{39} + 68 q^{45} - 252 q^{49} + 48 q^{51} + 48 q^{55} - 144 q^{61} - 268 q^{69} + 304 q^{75} - 432 q^{79} - 188 q^{81} - 336 q^{85} - 560 q^{99}+O(q^{100}) \) 12 * q + 8 * q^9 - 16 * q^15 - 4 * q^21 + 36 * q^25 + 48 * q^31 + 128 * q^39 + 68 * q^45 - 252 * q^49 + 48 * q^51 + 48 * q^55 - 144 * q^61 - 268 * q^69 + 304 * q^75 - 432 * q^79 - 188 * q^81 - 336 * q^85 - 560 * q^99

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