LMFDB - Embedded newform 945.2.d.c.379.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(379,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("945.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			379.8
Root			$-1.09445 + 0.895644i$ of defining polynomial
Character	$\chi$	$=$	945.379
Dual form			945.2.d.c.379.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18890i q^{2} -2.79129 q^{4} +(2.18890 + 0.456850i) q^{5} +1.00000i q^{7} -1.73205i q^{8} +O(q^{10})$	\(q+2.18890i q^{2} -2.79129 q^{4} +(2.18890 + 0.456850i) q^{5} +1.00000i q^{7} -1.73205i q^{8} +(-1.00000 + 4.79129i) q^{10} +1.73205 q^{11} +4.79129i q^{13} -2.18890 q^{14} -1.79129 q^{16} -5.65300i q^{17} +6.79129 q^{19} +(-6.10985 - 1.27520i) q^{20} +3.79129i q^{22} +4.83465i q^{23} +(4.58258 + 2.00000i) q^{25} -10.4877 q^{26} -2.79129i q^{28} +3.00725 q^{29} -8.58258 q^{31} -7.38505i q^{32} +12.3739 q^{34} +(-0.456850 + 2.18890i) q^{35} +10.1652i q^{37} +14.8655i q^{38} +(0.791288 - 3.79129i) q^{40} -9.11710 q^{41} -1.00000i q^{43} -4.83465 q^{44} -10.5826 q^{46} +1.73205i q^{47} -1.00000 q^{49} +(-4.37780 + 10.0308i) q^{50} -13.3739i q^{52} -6.56670i q^{53} +(3.79129 + 0.791288i) q^{55} +1.73205 q^{56} +6.58258i q^{58} -7.74655 q^{59} +4.79129 q^{61} -18.7864i q^{62} +12.5826 q^{64} +(-2.18890 + 10.4877i) q^{65} -1.37386i q^{67} +15.7792i q^{68} +(-4.79129 - 1.00000i) q^{70} +1.17985 q^{71} -4.58258i q^{73} -22.2505 q^{74} -18.9564 q^{76} +1.73205i q^{77} -3.37386 q^{79} +(-3.92095 - 0.818350i) q^{80} -19.9564i q^{82} -17.4159i q^{83} +(2.58258 - 12.3739i) q^{85} +2.18890 q^{86} -3.00000i q^{88} +8.66025 q^{89} -4.79129 q^{91} -13.4949i q^{92} -3.79129 q^{94} +(14.8655 + 3.10260i) q^{95} +1.37386i q^{97} -2.18890i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4}+O(q^{10}) $ 8 * q - 4 * q^4	$ 8 q - 4 q^{4} - 8 q^{10} + 4 q^{16} + 36 q^{19} - 32 q^{31} + 44 q^{34} - 12 q^{40} - 48 q^{46} - 8 q^{49} + 12 q^{55} + 20 q^{61} + 64 q^{64} - 20 q^{70} - 60 q^{76} + 28 q^{79} - 16 q^{85} - 20 q^{91} - 12 q^{94}+O(q^{100}) $ 8 * q - 4 * q^4 - 8 * q^10 + 4 * q^16 + 36 * q^19 - 32 * q^31 + 44 * q^34 - 12 * q^40 - 48 * q^46 - 8 * q^49 + 12 * q^55 + 20 * q^61 + 64 * q^64 - 20 * q^70 - 60 * q^76 + 28 * q^79 - 16 * q^85 - 20 * q^91 - 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/945\mathbb{Z}\right)^\times$.

$n$	$136$	$596$	$757$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$			2.18890i			1.54779i	0.633316	+	0.773893i	$0.281693\pi$
$2$			2.18890i			1.54779i	−0.633316	+	0.773893i	$0.718307\pi$
$3$		0			0
$3$		0			0
$4$	−2.79129			−1.39564
$5$	2.18890	+	0.456850i	0.978906	+	0.204310i
$5$	2.18890	+	0.456850i	0.978906	+	0.204310i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		−	1.73205i		−	0.612372i
$9$		0			0
$10$	−1.00000	+	4.79129i	−0.316228	+	1.51514i
$11$	1.73205			0.522233			0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205			0.522233			0.261116	+	0.965307i	$0.415909\pi$
$12$		0			0
$13$			4.79129i			1.32886i	0.747349	+	0.664432i	$0.231327\pi$
$13$			4.79129i			1.32886i	−0.747349	+	0.664432i	$0.768673\pi$
$14$	−2.18890			−0.585008
$15$		0			0
$16$	−1.79129			−0.447822
$17$		−	5.65300i		−	1.37105i	−0.728047	−	0.685527i	$-0.759572\pi$
$17$		−	5.65300i		−	1.37105i	0.728047	−	0.685527i	$-0.240428\pi$
$18$		0			0
$19$	6.79129			1.55803			0.779014	−	0.627006i	$-0.215720\pi$
$19$	6.79129			1.55803			0.779014	+	0.627006i	$0.215720\pi$
$20$	−6.10985	−	1.27520i	−1.36620	−	0.285144i
$21$		0			0
$22$			3.79129i			0.808305i
$23$			4.83465i			1.00809i	0.863676	+	0.504047i	$0.168156\pi$
$23$			4.83465i			1.00809i	−0.863676	+	0.504047i	$0.831844\pi$
$24$		0			0
$25$	4.58258	+	2.00000i	0.916515	+	0.400000i
$26$	−10.4877			−2.05680
$27$		0			0
$28$		−	2.79129i		−	0.527504i
$29$	3.00725			0.558433			0.279216	−	0.960228i	$-0.409925\pi$
$29$	3.00725			0.558433			0.279216	+	0.960228i	$0.409925\pi$
$30$		0			0
$31$	−8.58258			−1.54148			−0.770738	−	0.637152i	$-0.780112\pi$
$31$	−8.58258			−1.54148			−0.770738	+	0.637152i	$0.780112\pi$
$32$		−	7.38505i		−	1.30551i
$33$		0			0
$34$	12.3739			2.12210
$35$	−0.456850	+	2.18890i	−0.0772218	+	0.369992i
$36$		0			0
$37$			10.1652i			1.67114i	0.549384	+	0.835570i	$0.314863\pi$
$37$			10.1652i			1.67114i	−0.549384	+	0.835570i	$0.685137\pi$
$38$			14.8655i			2.41150i
$39$		0			0
$40$	0.791288	−	3.79129i	0.125114	−	0.599455i
$41$	−9.11710			−1.42385			−0.711926	−	0.702254i	$-0.752177\pi$
$41$	−9.11710			−1.42385			−0.711926	+	0.702254i	$0.752177\pi$
$42$		0			0
$43$		−	1.00000i		−	0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$		−	1.00000i		−	0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	−4.83465			−0.728851
$45$		0			0
$46$	−10.5826			−1.56032
$47$			1.73205i			0.252646i	0.991989	+	0.126323i	$0.0403175\pi$
$47$			1.73205i			0.252646i	−0.991989	+	0.126323i	$0.959682\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	−4.37780	+	10.0308i	−0.619115	+	1.41857i
$51$		0			0
$52$		−	13.3739i		−	1.85462i
$53$		−	6.56670i		−	0.902006i	−0.892522	−	0.451003i	$-0.851066\pi$
$53$		−	6.56670i		−	0.902006i	0.892522	−	0.451003i	$-0.148934\pi$
$54$		0			0
$55$	3.79129	+	0.791288i	0.511217	+	0.106697i
$56$	1.73205			0.231455
$57$		0			0
$58$			6.58258i			0.864335i
$59$	−7.74655			−1.00852			−0.504258	−	0.863553i	$-0.668234\pi$
$59$	−7.74655			−1.00852			−0.504258	+	0.863553i	$0.668234\pi$
$60$		0			0
$61$	4.79129			0.613462			0.306731	−	0.951796i	$-0.400765\pi$
$61$	4.79129			0.613462			0.306731	+	0.951796i	$0.400765\pi$
$62$		−	18.7864i		−	2.38588i
$63$		0			0
$64$	12.5826			1.57282
$65$	−2.18890	+	10.4877i	−0.271500	+	1.30083i
$66$		0			0
$67$		−	1.37386i		−	0.167844i	−0.996472	−	0.0839221i	$-0.973255\pi$
$67$		−	1.37386i		−	0.167844i	0.996472	−	0.0839221i	$-0.0267447\pi$
$68$			15.7792i			1.91350i
$69$		0			0
$70$	−4.79129	−	1.00000i	−0.572668	−	0.119523i
$71$	1.17985			0.140022			0.0700112	−	0.997546i	$-0.477696\pi$
$71$	1.17985			0.140022			0.0700112	+	0.997546i	$0.477696\pi$
$72$		0			0
$73$		−	4.58258i		−	0.536350i	−0.963370	−	0.268175i	$-0.913579\pi$
$73$		−	4.58258i		−	0.536350i	0.963370	−	0.268175i	$-0.0864205\pi$
$74$	−22.2505			−2.58657
$75$		0			0
$76$	−18.9564			−2.17445
$77$			1.73205i			0.197386i
$78$		0			0
$79$	−3.37386			−0.379589			−0.189795	−	0.981824i	$-0.560782\pi$
$79$	−3.37386			−0.379589			−0.189795	+	0.981824i	$0.560782\pi$
$80$	−3.92095	−	0.818350i	−0.438376	−	0.0914943i
$81$		0			0
$82$		−	19.9564i		−	2.20382i
$83$		−	17.4159i		−	1.91164i	−0.293955	−	0.955819i	$-0.594971\pi$
$83$		−	17.4159i		−	1.91164i	0.293955	−	0.955819i	$-0.405029\pi$
$84$		0			0
$85$	2.58258	−	12.3739i	0.280120	−	1.34213i
$86$	2.18890			0.236035
$87$		0			0
$88$		−	3.00000i		−	0.319801i
$89$	8.66025			0.917985			0.458993	−	0.888440i	$-0.348210\pi$
$89$	8.66025			0.917985			0.458993	+	0.888440i	$0.348210\pi$
$90$		0			0
$91$	−4.79129			−0.502263
$92$		−	13.4949i		−	1.40694i
$93$		0			0
$94$	−3.79129			−0.391041
$95$	14.8655	+	3.10260i	1.52516	+	0.318320i
$96$		0			0
$97$			1.37386i			0.139495i	0.997565	+	0.0697474i	$0.0222193\pi$
$97$			1.37386i			0.139495i	−0.997565	+	0.0697474i	$0.977781\pi$
$98$		−	2.18890i		−	0.221112i
$99$		0			0
$100$	−12.7913	−	5.58258i	−1.27913	−	0.558258i
$101$	15.8745			1.57957			0.789786	−	0.613382i	$-0.210191\pi$
$101$	15.8745			1.57957			0.789786	+	0.613382i	$0.210191\pi$
$102$		0			0
$103$			10.3739i			1.02217i	0.859531	+	0.511084i	$0.170756\pi$
$103$			10.3739i			1.02217i	−0.859531	+	0.511084i	$0.829244\pi$
$104$	8.29875			0.813760
$105$		0			0
$106$	14.3739			1.39611
$107$		−	3.55945i		−	0.344105i	−0.985088	−	0.172053i	$-0.944960\pi$
$107$		−	3.55945i		−	0.344105i	0.985088	−	0.172053i	$-0.0550399\pi$
$108$		0			0
$109$	0.208712			0.0199910			0.00999550	−	0.999950i	$-0.496818\pi$
$109$	0.208712			0.0199910			0.00999550	+	0.999950i	$0.496818\pi$
$110$	−1.73205	+	8.29875i	−0.165145	+	0.791255i
$111$		0			0
$112$		−	1.79129i		−	0.169261i
$113$		−	1.37055i		−	0.128931i	−0.997920	−	0.0644653i	$-0.979466\pi$
$113$		−	1.37055i		−	0.128931i	0.997920	−	0.0644653i	$-0.0205342\pi$
$114$		0			0
$115$	−2.20871	+	10.5826i	−0.205963	+	0.986830i
$116$	−8.39410			−0.779373
$117$		0			0
$118$		−	16.9564i		−	1.56097i
$119$	5.65300			0.518210
$120$		0			0
$121$	−8.00000			−0.727273
$122$			10.4877i			0.949508i
$123$		0			0
$124$	23.9564			2.15135
$125$	9.11710	+	6.47135i	0.815459	+	0.578815i
$126$		0			0
$127$			15.3739i			1.36421i	0.731254	+	0.682105i	$0.238935\pi$
$127$			15.3739i			1.36421i	−0.731254	+	0.682105i	$0.761065\pi$
$128$			12.7719i			1.12889i
$129$		0			0
$130$	−22.9564	−	4.79129i	−2.01341	−	0.420224i
$131$	11.8582			1.03606			0.518028	−	0.855364i	$-0.326666\pi$
$131$	11.8582			1.03606			0.518028	+	0.855364i	$0.326666\pi$
$132$		0			0
$133$			6.79129i			0.588879i
$134$	3.00725			0.259787
$135$		0			0
$136$	−9.79129			−0.839596
$137$		−	15.8745i		−	1.35625i	−0.734946	−	0.678125i	$-0.762793\pi$
$137$		−	15.8745i		−	1.35625i	0.734946	−	0.678125i	$-0.237207\pi$
$138$		0			0
$139$	11.4174			0.968413			0.484207	−	0.874954i	$-0.339108\pi$
$139$	11.4174			0.968413			0.484207	+	0.874954i	$0.339108\pi$
$140$	1.27520	−	6.10985i	0.107774	−	0.516377i
$141$		0			0
$142$			2.58258i			0.216725i
$143$			8.29875i			0.693977i
$144$		0			0
$145$	6.58258	+	1.37386i	0.546653	+	0.114093i
$146$	10.0308			0.830155
$147$		0			0
$148$		−	28.3739i		−	2.33232i
$149$	13.6856			1.12117			0.560584	−	0.828097i	$-0.310577\pi$
$149$	13.6856			1.12117			0.560584	+	0.828097i	$0.310577\pi$
$150$		0			0
$151$	−22.1652			−1.80377			−0.901887	−	0.431972i	$-0.857818\pi$
$151$	−22.1652			−1.80377			−0.901887	+	0.431972i	$0.857818\pi$
$152$		−	11.7629i		−	0.954094i
$153$		0			0
$154$	−3.79129			−0.305511
$155$	−18.7864	−	3.92095i	−1.50896	−	0.314938i
$156$		0			0
$157$		−	10.7477i		−	0.857762i	−0.903361	−	0.428881i	$-0.858908\pi$
$157$		−	10.7477i		−	0.857762i	0.903361	−	0.428881i	$-0.141092\pi$
$158$		−	7.38505i		−	0.587523i
$159$		0			0
$160$	3.37386	−	16.1652i	0.266727	−	1.27797i
$161$	−4.83465			−0.381024
$162$		0			0
$163$		−	12.7477i		−	0.998479i	−0.866464	−	0.499240i	$-0.833613\pi$
$163$		−	12.7477i		−	0.998479i	0.866464	−	0.499240i	$-0.166387\pi$
$164$	25.4485			1.98719
$165$		0			0
$166$	38.1216			2.95881
$167$		−	4.47315i		−	0.346143i	−0.984909	−	0.173071i	$-0.944631\pi$
$167$		−	4.47315i		−	0.346143i	0.984909	−	0.173071i	$-0.0553692\pi$
$168$		0			0
$169$	−9.95644			−0.765880
$170$	27.0852	+	5.65300i	2.07734	+	0.433566i
$171$		0			0
$172$			2.79129i			0.212834i
$173$		−	10.6784i		−	0.811860i	−0.913904	−	0.405930i	$-0.866948\pi$
$173$		−	10.6784i		−	0.811860i	0.913904	−	0.405930i	$-0.133052\pi$
$174$		0			0
$175$	−2.00000	+	4.58258i	−0.151186	+	0.346410i
$176$	−3.10260			−0.233867
$177$		0			0
$178$			18.9564i			1.42085i
$179$	−9.76465			−0.729845			−0.364922	−	0.931038i	$-0.618904\pi$
$179$	−9.76465			−0.729845			−0.364922	+	0.931038i	$0.618904\pi$
$180$		0			0
$181$	21.3303			1.58547			0.792734	−	0.609567i	$-0.208657\pi$
$181$	21.3303			1.58547			0.792734	+	0.609567i	$0.208657\pi$
$182$		−	10.4877i		−	0.777397i
$183$		0			0
$184$	8.37386			0.617329
$185$	−4.64395	+	22.2505i	−0.341430	+	1.63589i
$186$		0			0
$187$		−	9.79129i		−	0.716010i
$188$		−	4.83465i		−	0.352603i
$189$		0			0
$190$	−6.79129	+	32.5390i	−0.492692	+	2.36063i
$191$	−5.55765			−0.402138			−0.201069	−	0.979577i	$-0.564441\pi$
$191$	−5.55765			−0.402138			−0.201069	+	0.979577i	$0.564441\pi$
$192$		0			0
$193$			10.9564i			0.788662i	0.918969	+	0.394331i	$0.129024\pi$
$193$			10.9564i			0.788662i	−0.918969	+	0.394331i	$0.870976\pi$
$194$	−3.00725			−0.215908
$195$		0			0
$196$	2.79129			0.199378
$197$			5.00545i			0.356624i	0.983974	+	0.178312i	$0.0570636\pi$
$197$			5.00545i			0.356624i	−0.983974	+	0.178312i	$0.942936\pi$
$198$		0			0
$199$	18.2087			1.29078			0.645391	−	0.763853i	$-0.276695\pi$
$199$	18.2087			1.29078			0.645391	+	0.763853i	$0.276695\pi$
$200$	3.46410	−	7.93725i	0.244949	−	0.561249i
$201$		0			0
$202$			34.7477i			2.44484i
$203$			3.00725i			0.211068i
$204$		0			0
$205$	−19.9564	−	4.16515i	−1.39382	−	0.290907i
$206$	−22.7074			−1.58210
$207$		0			0
$208$		−	8.58258i		−	0.595095i
$209$	11.7629			0.813654
$210$		0			0
$211$	−0.834849			−0.0574733			−0.0287367	−	0.999587i	$-0.509148\pi$
$211$	−0.834849			−0.0574733			−0.0287367	+	0.999587i	$0.509148\pi$
$212$			18.3296i			1.25888i
$213$		0			0
$214$	7.79129			0.532601
$215$	0.456850	−	2.18890i	0.0311569	−	0.149282i
$216$		0			0
$217$		−	8.58258i		−	0.582623i
$218$			0.456850i			0.0309418i
$219$		0			0
$220$	−10.5826	−	2.20871i	−0.713477	−	0.148911i
$221$	27.0852			1.82195
$222$		0			0
$223$			9.00000i			0.602685i	0.953516	+	0.301342i	$0.0974347\pi$
$223$			9.00000i			0.602685i	−0.953516	+	0.301342i	$0.902565\pi$
$224$	7.38505			0.493435
$225$		0			0
$226$	3.00000			0.199557
$227$			9.11710i			0.605123i	0.953130	+	0.302562i	$0.0978418\pi$
$227$			9.11710i			0.605123i	−0.953130	+	0.302562i	$0.902158\pi$
$228$		0			0
$229$	19.0000			1.25556			0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000			1.25556			0.627778	+	0.778393i	$0.283965\pi$
$230$	−23.1642	−	4.83465i	−1.52740	−	0.318788i
$231$		0			0
$232$		−	5.20871i		−	0.341969i
$233$		−	8.48945i		−	0.556163i	−0.960558	−	0.278081i	$-0.910302\pi$
$233$		−	8.48945i		−	0.556163i	0.960558	−	0.278081i	$-0.0896985\pi$
$234$		0			0
$235$	−0.791288	+	3.79129i	−0.0516179	+	0.247316i
$236$	21.6229			1.40753
$237$		0			0
$238$			12.3739i			0.802078i
$239$	−1.73205			−0.112037			−0.0560185	−	0.998430i	$-0.517841\pi$
$239$	−1.73205			−0.112037			−0.0560185	+	0.998430i	$0.517841\pi$
$240$		0			0
$241$	3.37386			0.217330			0.108665	−	0.994078i	$-0.465342\pi$
$241$	3.37386			0.217330			0.108665	+	0.994078i	$0.465342\pi$
$242$		−	17.5112i		−	1.12566i
$243$		0			0
$244$	−13.3739			−0.856174
$245$	−2.18890	−	0.456850i	−0.139844	−	0.0291871i
$246$		0			0
$247$			32.5390i			2.07041i
$248$			14.8655i			0.943957i
$249$		0			0
$250$	−14.1652	+	19.9564i	−0.895883	+	1.26216i
$251$	−15.8745			−1.00199			−0.500995	−	0.865450i	$-0.667033\pi$
$251$	−15.8745			−1.00199			−0.500995	+	0.865450i	$0.667033\pi$
$252$		0			0
$253$			8.37386i			0.526460i
$254$	−33.6519			−2.11151
$255$		0			0
$256$	−2.79129			−0.174455
$257$		−	8.75560i		−	0.546160i	−0.961991	−	0.273080i	$-0.911958\pi$
$257$		−	8.75560i		−	0.546160i	0.961991	−	0.273080i	$-0.0880423\pi$
$258$		0			0
$259$	−10.1652			−0.631632
$260$	6.10985	−	29.2741i	0.378917	−	1.81550i
$261$		0			0
$262$			25.9564i			1.60359i
$263$			14.2179i			0.876714i	0.898801	+	0.438357i	$0.144439\pi$
$263$			14.2179i			0.876714i	−0.898801	+	0.438357i	$0.855561\pi$
$264$		0			0
$265$	3.00000	−	14.3739i	0.184289	−	0.882979i
$266$	−14.8655			−0.911460
$267$		0			0
$268$			3.83485i			0.234251i
$269$	28.1896			1.71875			0.859374	−	0.511348i	$-0.170853\pi$
$269$	28.1896			1.71875			0.859374	+	0.511348i	$0.170853\pi$
$270$		0			0
$271$	−11.7913			−0.716270			−0.358135	−	0.933670i	$-0.616587\pi$
$271$	−11.7913			−0.716270			−0.358135	+	0.933670i	$0.616587\pi$
$272$			10.1262i			0.613988i
$273$		0			0
$274$	34.7477			2.09919
$275$	7.93725	+	3.46410i	0.478634	+	0.208893i
$276$		0			0
$277$		−	5.79129i		−	0.347965i	−0.984749	−	0.173982i	$-0.944336\pi$
$277$		−	5.79129i		−	0.347965i	0.984749	−	0.173982i	$-0.0556636\pi$
$278$			24.9916i			1.49890i
$279$		0			0
$280$	3.79129	+	0.791288i	0.226573	+	0.0472885i
$281$	29.9017			1.78379			0.891893	−	0.452246i	$-0.149377\pi$
$281$	29.9017			1.78379			0.891893	+	0.452246i	$0.149377\pi$
$282$		0			0
$283$		−	10.6261i		−	0.631658i	−0.948816	−	0.315829i	$-0.897717\pi$
$283$		−	10.6261i		−	0.631658i	0.948816	−	0.315829i	$-0.102283\pi$
$284$	−3.29330			−0.195422
$285$		0			0
$286$	−18.1652			−1.07413
$287$		−	9.11710i		−	0.538166i
$288$		0			0
$289$	−14.9564			−0.879791
$290$	−3.00725	+	14.4086i	−0.176592	+	0.846103i
$291$		0			0
$292$			12.7913i			0.748554i
$293$			17.3205i			1.01187i	0.862570	+	0.505937i	$0.168853\pi$
$293$			17.3205i			1.01187i	−0.862570	+	0.505937i	$0.831147\pi$
$294$		0			0
$295$	−16.9564	−	3.53901i	−0.987242	−	0.206049i
$296$	17.6066			1.02336
$297$		0			0
$298$			29.9564i			1.73533i
$299$	−23.1642			−1.33962
$300$		0			0
$301$	1.00000			0.0576390
$302$		−	48.5173i		−	2.79186i
$303$		0			0
$304$	−12.1652			−0.697719
$305$	10.4877	+	2.18890i	0.600521	+	0.125336i
$306$		0			0
$307$		−	14.4174i		−	0.822846i	−0.911445	−	0.411423i	$-0.865032\pi$
$307$		−	14.4174i		−	0.822846i	0.911445	−	0.411423i	$-0.134968\pi$
$308$		−	4.83465i		−	0.275480i
$309$		0			0
$310$	8.58258	−	41.1216i	0.487458	−	2.33555i
$311$	−9.02175			−0.511577			−0.255788	−	0.966733i	$-0.582335\pi$
$311$	−9.02175			−0.511577			−0.255788	+	0.966733i	$0.582335\pi$
$312$		0			0
$313$			16.5826i			0.937303i	0.883383	+	0.468651i	$0.155260\pi$
$313$			16.5826i			0.937303i	−0.883383	+	0.468651i	$0.844740\pi$
$314$	23.5257			1.32763
$315$		0			0
$316$	9.41742			0.529772
$317$		−	9.38325i		−	0.527016i	−0.964657	−	0.263508i	$-0.915120\pi$
$317$		−	9.38325i		−	0.527016i	0.964657	−	0.263508i	$-0.0848795\pi$
$318$		0			0
$319$	5.20871			0.291632
$320$	27.5420	+	5.74835i	1.53965	+	0.321343i
$321$		0			0
$322$		−	10.5826i		−	0.589744i
$323$		−	38.3912i		−	2.13614i
$324$		0			0
$325$	−9.58258	+	21.9564i	−0.531546	+	1.21792i
$326$	27.9035			1.54543
$327$		0			0
$328$			15.7913i			0.871928i
$329$	−1.73205			−0.0954911
$330$		0			0
$331$	−9.62614			−0.529100			−0.264550	−	0.964372i	$-0.585223\pi$
$331$	−9.62614			−0.529100			−0.264550	+	0.964372i	$0.585223\pi$
$332$			48.6127i			2.66797i
$333$		0			0
$334$	9.79129			0.535755
$335$	0.627650	−	3.00725i	0.0342922	−	0.164304i
$336$		0			0
$337$		−	32.7042i		−	1.78151i	−0.454484	−	0.890755i	$-0.650176\pi$
$337$		−	32.7042i		−	1.78151i	0.454484	−	0.890755i	$-0.349824\pi$
$338$		−	21.7937i		−	1.18542i
$339$		0			0
$340$	−7.20871	+	34.5390i	−0.390947	+	1.87314i
$341$	−14.8655			−0.805010
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$	−1.73205			−0.0933859
$345$		0			0
$346$	23.3739			1.25659
$347$			25.5438i			1.37126i	0.727948	+	0.685632i	$0.240474\pi$
$347$			25.5438i			1.37126i	−0.727948	+	0.685632i	$0.759526\pi$
$348$		0			0
$349$	21.7477			1.16413			0.582065	−	0.813143i	$-0.302245\pi$
$349$	21.7477			1.16413			0.582065	+	0.813143i	$0.302245\pi$
$350$	−10.0308	−	4.37780i	−0.536169	−	0.234003i
$351$		0			0
$352$		−	12.7913i		−	0.681778i
$353$		−	17.0544i		−	0.907712i	−0.891075	−	0.453856i	$-0.850048\pi$
$353$		−	17.0544i		−	0.907712i	0.891075	−	0.453856i	$-0.149952\pi$
$354$		0			0
$355$	2.58258	+	0.539015i	0.137069	+	0.0286079i
$356$	−24.1733			−1.28118
$357$		0			0
$358$		−	21.3739i		−	1.12964i
$359$	11.5921			0.611805			0.305903	−	0.952063i	$-0.401042\pi$
$359$	11.5921			0.611805			0.305903	+	0.952063i	$0.401042\pi$
$360$		0			0
$361$	27.1216			1.42745
$362$			46.6899i			2.45397i
$363$		0			0
$364$	13.3739			0.700981
$365$	2.09355	−	10.0308i	0.109581	−	0.525036i
$366$		0			0
$367$		−	18.9564i		−	0.989518i	−0.869030	−	0.494759i	$-0.835256\pi$
$367$		−	18.9564i		−	0.989518i	0.869030	−	0.494759i	$-0.164744\pi$
$368$		−	8.66025i		−	0.451447i
$369$		0			0
$370$	−48.7042	−	10.1652i	−2.53201	−	0.528461i
$371$	6.56670			0.340926
$372$		0			0
$373$		−	9.95644i		−	0.515525i	−0.966208	−	0.257762i	$-0.917015\pi$
$373$		−	9.95644i		−	0.515525i	0.966208	−	0.257762i	$-0.0829852\pi$
$374$	21.4322			1.10823
$375$		0			0
$376$	3.00000			0.154713
$377$			14.4086i			0.742081i
$378$		0			0
$379$	5.58258			0.286758			0.143379	−	0.989668i	$-0.454203\pi$
$379$	5.58258			0.286758			0.143379	+	0.989668i	$0.454203\pi$
$380$	−41.4938	−	8.66025i	−2.12859	−	0.444262i
$381$		0			0
$382$		−	12.1652i		−	0.622423i
$383$		−	36.5638i		−	1.86832i	−0.356853	−	0.934161i	$-0.616150\pi$
$383$		−	36.5638i		−	1.86832i	0.356853	−	0.934161i	$-0.383850\pi$
$384$		0			0
$385$	−0.791288	+	3.79129i	−0.0403278	+	0.193222i
$386$	−23.9826			−1.22068
$387$		0			0
$388$		−	3.83485i		−	0.194685i
$389$	−36.4883			−1.85003			−0.925016	−	0.379929i	$-0.875948\pi$
$389$	−36.4883			−1.85003			−0.925016	+	0.379929i	$0.875948\pi$
$390$		0			0
$391$	27.3303			1.38215
$392$			1.73205i			0.0874818i
$393$		0			0
$394$	−10.9564			−0.551977
$395$	−7.38505	−	1.54135i	−0.371582	−	0.0775538i
$396$		0			0
$397$		−	25.1652i		−	1.26300i	−0.775375	−	0.631501i	$-0.782439\pi$
$397$		−	25.1652i		−	1.26300i	0.775375	−	0.631501i	$-0.217561\pi$
$398$			39.8571i			1.99785i
$399$		0			0
$400$	−8.20871	−	3.58258i	−0.410436	−	0.179129i
$401$	4.11165			0.205326			0.102663	−	0.994716i	$-0.467264\pi$
$401$	4.11165			0.205326			0.102663	+	0.994716i	$0.467264\pi$
$402$		0			0
$403$		−	41.1216i		−	2.04841i
$404$	−44.3103			−2.20452
$405$		0			0
$406$	−6.58258			−0.326688
$407$			17.6066i			0.872725i
$408$		0			0
$409$	1.16515			0.0576130			0.0288065	−	0.999585i	$-0.490829\pi$
$409$	1.16515			0.0576130			0.0288065	+	0.999585i	$0.490829\pi$
$410$	9.11710	−	43.6827i	0.450262	−	2.15733i
$411$		0			0
$412$		−	28.9564i		−	1.42658i
$413$		−	7.74655i		−	0.381183i
$414$		0			0
$415$	7.95644	−	38.1216i	0.390566	−	1.87132i
$416$	35.3839			1.73484
$417$		0			0
$418$			25.7477i			1.25936i
$419$	22.5167			1.10001			0.550005	−	0.835161i	$-0.314626\pi$
$419$	22.5167			1.10001			0.550005	+	0.835161i	$0.314626\pi$
$420$		0			0
$421$	−24.7913			−1.20825			−0.604127	−	0.796888i	$-0.706478\pi$
$421$	−24.7913			−1.20825			−0.604127	+	0.796888i	$0.706478\pi$
$422$		−	1.82740i		−	0.0889565i
$423$		0			0
$424$	−11.3739			−0.552364
$425$	11.3060	−	25.9053i	0.548422	−	1.25659i
$426$		0			0
$427$			4.79129i			0.231867i
$428$			9.93545i			0.480248i
$429$		0			0
$430$	4.79129	+	1.00000i	0.231056	+	0.0482243i
$431$	−14.4086			−0.694038			−0.347019	−	0.937858i	$-0.612806\pi$
$431$	−14.4086			−0.694038			−0.347019	+	0.937858i	$0.612806\pi$
$432$		0			0
$433$		−	18.5390i		−	0.890928i	−0.895300	−	0.445464i	$-0.853039\pi$
$433$		−	18.5390i		−	0.890928i	0.895300	−	0.445464i	$-0.146961\pi$
$434$	18.7864			0.901776
$435$		0			0
$436$	−0.582576			−0.0279003
$437$			32.8335i			1.57064i
$438$		0			0
$439$	−33.3303			−1.59077			−0.795384	−	0.606105i	$-0.792731\pi$
$439$	−33.3303			−1.59077			−0.795384	+	0.606105i	$0.792731\pi$
$440$	1.37055	−	6.56670i	0.0653384	−	0.313055i
$441$		0			0
$442$			59.2867i			2.81998i
$443$			4.30235i			0.204411i	0.994763	+	0.102205i	$0.0325899\pi$
$443$			4.30235i			0.204411i	−0.994763	+	0.102205i	$0.967410\pi$
$444$		0			0
$445$	18.9564	+	3.95644i	0.898621	+	0.187553i
$446$	−19.7001			−0.932827
$447$		0			0
$448$			12.5826i			0.594471i
$449$	24.4394			1.15337			0.576684	−	0.816968i	$-0.304347\pi$
$449$	24.4394			1.15337			0.576684	+	0.816968i	$0.304347\pi$
$450$		0			0
$451$	−15.7913			−0.743583
$452$			3.82560i			0.179941i
$453$		0			0
$454$	−19.9564			−0.936602
$455$	−10.4877	−	2.18890i	−0.491669	−	0.102617i
$456$		0			0
$457$			28.8693i			1.35045i	0.737612	+	0.675225i	$0.235953\pi$
$457$			28.8693i			1.35045i	−0.737612	+	0.675225i	$0.764047\pi$
$458$			41.5891i			1.94333i
$459$		0			0
$460$	6.16515	−	29.5390i	0.287452	−	1.37726i
$461$	−17.2451			−0.803182			−0.401591	−	0.915819i	$-0.631543\pi$
$461$	−17.2451			−0.803182			−0.401591	+	0.915819i	$0.631543\pi$
$462$		0			0
$463$		−	38.3739i		−	1.78338i	−0.452642	−	0.891692i	$-0.649518\pi$
$463$		−	38.3739i		−	1.78338i	0.452642	−	0.891692i	$-0.350482\pi$
$464$	−5.38685			−0.250078
$465$		0			0
$466$	18.5826			0.860821
$467$			15.1515i			0.701128i	0.936539	+	0.350564i	$0.114010\pi$
$467$			15.1515i			0.701128i	−0.936539	+	0.350564i	$0.885990\pi$
$468$		0			0
$469$	1.37386			0.0634391
$470$	−8.29875	−	1.73205i	−0.382793	−	0.0798935i
$471$		0			0
$472$			13.4174i			0.617587i
$473$		−	1.73205i		−	0.0796398i
$474$		0			0
$475$	31.1216	+	13.5826i	1.42796	+	0.623211i
$476$	−15.7792			−0.723237
$477$		0			0
$478$		−	3.79129i		−	0.173409i
$479$	−1.08450			−0.0495521			−0.0247760	−	0.999693i	$-0.507887\pi$
$479$	−1.08450			−0.0495521			−0.0247760	+	0.999693i	$0.507887\pi$
$480$		0			0
$481$	−48.7042			−2.22072
$482$			7.38505i			0.336380i
$483$		0			0
$484$	22.3303			1.01501
$485$	−0.627650	+	3.00725i	−0.0285001	+	0.136552i
$486$		0			0
$487$		−	31.9129i		−	1.44611i	−0.690790	−	0.723055i	$-0.742737\pi$
$487$		−	31.9129i		−	1.44611i	0.690790	−	0.723055i	$-0.257263\pi$
$488$		−	8.29875i		−	0.375667i
$489$		0			0
$490$	1.00000	−	4.79129i	0.0451754	−	0.216448i
$491$	−39.9524			−1.80303			−0.901514	−	0.432751i	$-0.857543\pi$
$491$	−39.9524			−1.80303			−0.901514	+	0.432751i	$0.857543\pi$
$492$		0			0
$493$		−	17.0000i		−	0.765641i
$494$	−71.2247			−3.20455
$495$		0			0
$496$	15.3739			0.690307
$497$			1.17985i			0.0529235i
$498$		0			0
$499$	20.1652			0.902716			0.451358	−	0.892343i	$-0.350940\pi$
$499$	20.1652			0.902716			0.451358	+	0.892343i	$0.350940\pi$
$500$	−25.4485	−	18.0634i	−1.13809	−	0.807820i
$501$		0			0
$502$		−	34.7477i		−	1.55087i
$503$		−	14.5040i		−	0.646699i	−0.946280	−	0.323350i	$-0.895191\pi$
$503$		−	14.5040i		−	0.646699i	0.946280	−	0.323350i	$-0.104809\pi$
$504$		0			0
$505$	34.7477	+	7.25227i	1.54625	+	0.322722i
$506$	−18.3296			−0.814848
$507$		0			0
$508$		−	42.9129i		−	1.90395i
$509$	−41.1323			−1.82316			−0.911578	−	0.411127i	$-0.865135\pi$
$509$	−41.1323			−1.82316			−0.911578	+	0.411127i	$0.865135\pi$
$510$		0			0
$511$	4.58258			0.202721
$512$			19.4340i			0.858868i
$513$		0			0
$514$	19.1652			0.845339
$515$	−4.73930	+	22.7074i	−0.208839	+	1.00061i
$516$		0			0
$517$			3.00000i			0.131940i
$518$		−	22.2505i		−	0.977631i
$519$		0			0
$520$	18.1652	+	3.79129i	0.796595	+	0.166259i
$521$	−14.6748			−0.642913			−0.321456	−	0.946924i	$-0.604172\pi$
$521$	−14.6748			−0.642913			−0.321456	+	0.946924i	$0.604172\pi$
$522$		0			0
$523$			10.0436i			0.439174i	0.975593	+	0.219587i	$0.0704710\pi$
$523$			10.0436i			0.439174i	−0.975593	+	0.219587i	$0.929529\pi$
$524$	−33.0997			−1.44597
$525$		0			0
$526$	−31.1216			−1.35697
$527$			48.5173i			2.11345i
$528$		0			0
$529$	−0.373864			−0.0162549
$530$	31.4630	+	6.56670i	1.36666	+	0.285239i
$531$		0			0
$532$		−	18.9564i		−	0.821866i
$533$		−	43.6827i		−	1.89211i
$534$		0			0
$535$	1.62614	−	7.79129i	0.0703040	−	0.336847i
$536$	−2.37960			−0.102783
$537$		0			0
$538$			61.7042i			2.66026i
$539$	−1.73205			−0.0746047
$540$		0			0
$541$	−35.7042			−1.53504			−0.767521	−	0.641024i	$-0.778510\pi$
$541$	−35.7042			−1.53504			−0.767521	+	0.641024i	$0.778510\pi$
$542$		−	25.8100i		−	1.10863i
$543$		0			0
$544$	−41.7477			−1.78992
$545$	0.456850	+	0.0953502i	0.0195693	+	0.00408435i
$546$		0			0
$547$			8.58258i			0.366964i	0.983023	+	0.183482i	$0.0587370\pi$
$547$			8.58258i			0.366964i	−0.983023	+	0.183482i	$0.941263\pi$
$548$			44.3103i			1.89284i
$549$		0			0
$550$	−7.58258	+	17.3739i	−0.323322	+	0.740824i
$551$	20.4231			0.870054
$552$		0			0
$553$		−	3.37386i		−	0.143471i
$554$	12.6766			0.538575
$555$		0			0
$556$	−31.8693			−1.35156
$557$			39.8571i			1.68880i	0.535715	+	0.844399i	$0.320042\pi$
$557$			39.8571i			1.68880i	−0.535715	+	0.844399i	$0.679958\pi$
$558$		0			0
$559$	4.79129			0.202650
$560$	0.818350	−	3.92095i	0.0345816	−	0.165690i
$561$		0			0
$562$			65.4519i			2.76092i
$563$		−	4.54860i		−	0.191701i	−0.995396	−	0.0958504i	$-0.969443\pi$
$563$		−	4.54860i		−	0.191701i	0.995396	−	0.0958504i	$-0.0305570\pi$
$564$		0			0
$565$	0.626136	−	3.00000i	0.0263418	−	0.126211i
$566$	23.2596			0.977672
$567$		0			0
$568$		−	2.04356i		−	0.0857459i
$569$	−10.6784			−0.447660			−0.223830	−	0.974628i	$-0.571856\pi$
$569$	−10.6784			−0.447660			−0.223830	+	0.974628i	$0.571856\pi$
$570$		0			0
$571$	−2.20871			−0.0924317			−0.0462159	−	0.998931i	$-0.514716\pi$
$571$	−2.20871			−0.0924317			−0.0462159	+	0.998931i	$0.514716\pi$
$572$		−	23.1642i		−	0.968544i
$573$		0			0
$574$	19.9564			0.832966
$575$	−9.66930	+	22.1552i	−0.403238	+	0.923934i
$576$		0			0
$577$		−	17.3303i		−	0.721470i	−0.932668	−	0.360735i	$-0.882526\pi$
$577$		−	17.3303i		−	0.721470i	0.932668	−	0.360735i	$-0.117474\pi$
$578$		−	32.7382i		−	1.36173i
$579$		0			0
$580$	−18.3739	−	3.83485i	−0.762933	−	0.159233i
$581$	17.4159			0.722532
$582$		0			0
$583$		−	11.3739i		−	0.471057i
$584$	−7.93725			−0.328446
$585$		0			0
$586$	−37.9129			−1.56617
$587$			3.38865i			0.139865i	0.997552	+	0.0699323i	$0.0222783\pi$
$587$			3.38865i			0.139865i	−0.997552	+	0.0699323i	$0.977722\pi$
$588$		0			0
$589$	−58.2867			−2.40166
$590$	7.74655	−	37.1160i	0.318921	−	1.52804i
$591$		0			0
$592$		−	18.2087i		−	0.748373i
$593$		−	4.28245i		−	0.175859i	−0.996127	−	0.0879296i	$-0.971975\pi$
$593$		−	4.28245i		−	0.175859i	0.996127	−	0.0879296i	$-0.0280251\pi$
$594$		0			0
$595$	12.3739	+	2.58258i	0.507279	+	0.105875i
$596$	−38.2005			−1.56475
$597$		0			0
$598$		−	50.7042i		−	2.07345i
$599$	−44.6917			−1.82605			−0.913027	−	0.407899i	$-0.866262\pi$
$599$	−44.6917			−1.82605			−0.913027	+	0.407899i	$0.866262\pi$
$600$		0			0
$601$	−48.6170			−1.98313			−0.991565	−	0.129608i	$-0.958628\pi$
$601$	−48.6170			−1.98313			−0.991565	+	0.129608i	$0.958628\pi$
$602$			2.18890i			0.0892129i
$603$		0			0
$604$	61.8693			2.51743
$605$	−17.5112	−	3.65480i	−0.711932	−	0.148589i
$606$		0			0
$607$			29.8348i			1.21096i	0.795861	+	0.605480i	$0.207019\pi$
$607$			29.8348i			1.21096i	−0.795861	+	0.605480i	$0.792981\pi$
$608$		−	50.1540i		−	2.03401i
$609$		0			0
$610$	−4.79129	+	22.9564i	−0.193994	+	0.929479i
$611$	−8.29875			−0.335732
$612$		0			0
$613$			13.3303i			0.538406i	0.963083	+	0.269203i	$0.0867602\pi$
$613$			13.3303i			0.538406i	−0.963083	+	0.269203i	$0.913240\pi$
$614$	31.5583			1.27359
$615$		0			0
$616$	3.00000			0.120873
$617$		−	16.0453i		−	0.645960i	−0.946406	−	0.322980i	$-0.895315\pi$
$617$		−	16.0453i		−	0.645960i	0.946406	−	0.322980i	$-0.104685\pi$
$618$		0			0
$619$	44.4955			1.78842			0.894212	−	0.447644i	$-0.147737\pi$
$619$	44.4955			1.78842			0.894212	+	0.447644i	$0.147737\pi$
$620$	52.4383	+	10.9445i	2.10597	+	0.439542i
$621$		0			0
$622$		−	19.7477i		−	0.791812i
$623$			8.66025i			0.346966i
$624$		0			0
$625$	17.0000	+	18.3303i	0.680000	+	0.733212i
$626$	−36.2976			−1.45074
$627$		0			0
$628$			30.0000i			1.19713i
$629$	57.4636			2.29122
$630$		0			0
$631$	−37.0780			−1.47605			−0.738027	−	0.674772i	$-0.764242\pi$
$631$	−37.0780			−1.47605			−0.738027	+	0.674772i	$0.764242\pi$
$632$			5.84370i			0.232450i
$633$		0			0
$634$	20.5390			0.815708
$635$	−7.02355	+	33.6519i	−0.278721	+	1.33543i
$636$		0			0
$637$		−	4.79129i		−	0.189838i
$638$			11.4014i			0.451384i
$639$		0			0
$640$	−5.83485	+	27.9564i	−0.230643	+	1.10508i
$641$	26.0007			1.02696			0.513482	−	0.858100i	$-0.328355\pi$
$641$	26.0007			1.02696			0.513482	+	0.858100i	$0.328355\pi$
$642$		0			0
$643$			4.79129i			0.188950i	0.995527	+	0.0944750i	$0.0301172\pi$
$643$			4.79129i			0.188950i	−0.995527	+	0.0944750i	$0.969883\pi$
$644$	13.4949			0.531774
$645$		0			0
$646$	84.0345			3.30629
$647$			6.39590i			0.251449i	0.992065	+	0.125724i	$0.0401255\pi$
$647$			6.39590i			0.251449i	−0.992065	+	0.125724i	$0.959874\pi$
$648$		0			0
$649$	−13.4174			−0.526680
$650$	−48.0605	−	20.9753i	−1.88509	−	0.822719i
$651$		0			0
$652$			35.5826i			1.39352i
$653$		−	12.8673i		−	0.503535i	−0.967788	−	0.251767i	$-0.918988\pi$
$653$		−	12.8673i		−	0.503535i	0.967788	−	0.251767i	$-0.0810118\pi$
$654$		0			0
$655$	25.9564	+	5.41742i	1.01420	+	0.211676i
$656$	16.3314			0.637632
$657$		0			0
$658$		−	3.79129i		−	0.147800i
$659$	−12.4104			−0.483441			−0.241720	−	0.970346i	$-0.577712\pi$
$659$	−12.4104			−0.483441			−0.241720	+	0.970346i	$0.577712\pi$
$660$		0			0
$661$	17.5390			0.682189			0.341094	−	0.940029i	$-0.389202\pi$
$661$	17.5390			0.682189			0.341094	+	0.940029i	$0.389202\pi$
$662$		−	21.0707i		−	0.818934i
$663$		0			0
$664$	−30.1652			−1.17063
$665$	−3.10260	+	14.8655i	−0.120314	+	0.576458i
$666$		0			0
$667$			14.5390i			0.562953i
$668$			12.4859i			0.483092i
$669$		0			0
$670$	6.58258	+	1.37386i	0.254307	+	0.0530770i
$671$	8.29875			0.320370
$672$		0			0
$673$			7.91288i			0.305019i	0.988302	+	0.152510i	$0.0487355\pi$
$673$			7.91288i			0.305019i	−0.988302	+	0.152510i	$0.951265\pi$
$674$	71.5862			2.75740
$675$		0			0
$676$	27.7913			1.06890
$677$		−	11.5921i		−	0.445519i	−0.974873	−	0.222759i	$-0.928494\pi$
$677$		−	11.5921i		−	0.445519i	0.974873	−	0.222759i	$-0.0715064\pi$
$678$		0			0
$679$	−1.37386			−0.0527240
$680$	−21.4322	−	4.47315i	−0.821886	−	0.171538i
$681$		0			0
$682$		−	32.5390i		−	1.24598i
$683$		−	21.0508i		−	0.805485i	−0.915313	−	0.402742i	$-0.868057\pi$
$683$		−	21.0508i		−	0.805485i	0.915313	−	0.402742i	$-0.131943\pi$
$684$		0			0
$685$	7.25227	−	34.7477i	0.277095	−	1.32764i
$686$	2.18890			0.0835726
$687$		0			0
$688$			1.79129i			0.0682922i
$689$	31.4630			1.19864
$690$		0			0
$691$	7.33030			0.278858			0.139429	−	0.990232i	$-0.455473\pi$
$691$	7.33030			0.278858			0.139429	+	0.990232i	$0.455473\pi$
$692$			29.8064i			1.13307i
$693$		0			0
$694$	−55.9129			−2.12242
$695$	24.9916	+	5.21605i	0.947986	+	0.197856i
$696$		0			0
$697$			51.5390i			1.95218i
$698$			47.6036i			1.80182i
$699$		0			0
$700$	5.58258	−	12.7913i	0.211002	−	0.483465i
$701$	8.85095			0.334296			0.167148	−	0.985932i	$-0.446544\pi$
$701$	8.85095			0.334296			0.167148	+	0.985932i	$0.446544\pi$
$702$		0			0
$703$			69.0345i			2.60368i
$704$	21.7937			0.821379
$705$		0			0
$706$	37.3303			1.40494
$707$			15.8745i			0.597022i
$708$		0			0
$709$	25.4955			0.957502			0.478751	−	0.877951i	$-0.341090\pi$
$709$	25.4955			0.957502			0.478751	+	0.877951i	$0.341090\pi$
$710$	−1.17985	+	5.65300i	−0.0442790	+	0.212153i
$711$		0			0
$712$		−	15.0000i		−	0.562149i
$713$		−	41.4938i		−	1.55395i
$714$		0			0
$715$	−3.79129	+	18.1652i	−0.141786	+	0.679338i
$716$	27.2560			1.01860
$717$		0			0
$718$			25.3739i			0.946944i
$719$	−38.3157			−1.42894			−0.714468	−	0.699669i	$-0.753331\pi$
$719$	−38.3157			−1.42894			−0.714468	+	0.699669i	$0.753331\pi$
$720$		0			0
$721$	−10.3739			−0.386343
$722$			59.3665i			2.20939i
$723$		0			0
$724$	−59.5390			−2.21275
$725$	13.7810	+	6.01450i	0.511812	+	0.223373i
$726$		0			0
$727$			35.4519i			1.31484i	0.753525	+	0.657419i	$0.228352\pi$
$727$			35.4519i			1.31484i	−0.753525	+	0.657419i	$0.771648\pi$
$728$			8.29875i			0.307572i
$729$		0			0
$730$	21.9564	+	4.58258i	0.812644	+	0.169609i
$731$	−5.65300			−0.209084
$732$		0			0
$733$			36.2087i			1.33740i	0.743533	+	0.668700i	$0.233149\pi$
$733$			36.2087i			1.33740i	−0.743533	+	0.668700i	$0.766851\pi$
$734$	41.4938			1.53156
$735$		0			0
$736$	35.7042			1.31607
$737$		−	2.37960i		−	0.0876537i
$738$		0			0
$739$	23.4955			0.864294			0.432147	−	0.901803i	$-0.357756\pi$
$739$	23.4955			0.864294			0.432147	+	0.901803i	$0.357756\pi$
$740$	12.9626	−	62.1076i	0.476515	−	2.28312i
$741$		0			0
$742$			14.3739i			0.527681i
$743$			35.3839i			1.29811i	0.760741	+	0.649055i	$0.224836\pi$
$743$			35.3839i			1.29811i	−0.760741	+	0.649055i	$0.775164\pi$
$744$		0			0
$745$	29.9564	+	6.25227i	1.09752	+	0.229066i
$746$	21.7937			0.797923
$747$		0			0
$748$			27.3303i			0.999295i
$749$	3.55945			0.130060
$750$		0			0
$751$	−36.6606			−1.33776			−0.668882	−	0.743368i	$-0.733227\pi$
$751$	−36.6606			−1.33776			−0.668882	+	0.743368i	$0.733227\pi$
$752$		−	3.10260i		−	0.113140i
$753$		0			0
$754$	−31.5390			−1.14858
$755$	−48.5173	−	10.1262i	−1.76573	−	0.368529i
$756$		0			0
$757$			15.3739i			0.558773i	0.960179	+	0.279386i	$0.0901310\pi$
$757$			15.3739i			0.558773i	−0.960179	+	0.279386i	$0.909869\pi$
$758$			12.2197i			0.443840i
$759$		0			0
$760$	5.37386	−	25.7477i	0.194931	−	0.933968i
$761$	43.8535			1.58969			0.794844	−	0.606814i	$-0.207553\pi$
$761$	43.8535			1.58969			0.794844	+	0.606814i	$0.207553\pi$
$762$		0			0
$763$			0.208712i			0.00755589i
$764$	15.5130			0.561241
$765$		0			0
$766$	80.0345			2.89176
$767$		−	37.1160i		−	1.34018i
$768$		0			0
$769$	7.74773			0.279390			0.139695	−	0.990195i	$-0.455388\pi$
$769$	7.74773			0.279390			0.139695	+	0.990195i	$0.455388\pi$
$770$	−8.29875	−	1.73205i	−0.299066	−	0.0624188i
$771$		0			0
$772$		−	30.5826i		−	1.10069i
$773$			17.9681i			0.646266i	0.946354	+	0.323133i	$0.104736\pi$
$773$			17.9681i			0.646266i	−0.946354	+	0.323133i	$0.895264\pi$
$774$		0			0
$775$	−39.3303	−	17.1652i	−1.41279	−	0.616590i
$776$	2.37960			0.0854227
$777$		0			0
$778$		−	79.8693i		−	2.86345i
$779$	−61.9169			−2.21840
$780$		0			0
$781$	2.04356			0.0731244
$782$			59.8233i			2.13928i
$783$		0			0
$784$	1.79129			0.0639746
$785$	4.91010	−	23.5257i	0.175249	−	0.839669i
$786$		0			0
$787$		−	11.1652i		−	0.397995i	−0.980000	−	0.198997i	$-0.936231\pi$
$787$		−	11.1652i		−	0.397995i	0.980000	−	0.198997i	$-0.0637685\pi$
$788$		−	13.9717i		−	0.497720i
$789$		0			0
$790$	3.37386	−	16.1652i	0.120037	−	0.575130i
$791$	1.37055			0.0487312
$792$		0			0
$793$			22.9564i			0.815207i
$794$	55.0840			1.95486
$795$		0			0
$796$	−50.8258			−1.80147
$797$		−	34.4702i		−	1.22100i	−0.792017	−	0.610499i	$-0.790969\pi$
$797$		−	34.4702i		−	1.22100i	0.792017	−	0.610499i	$-0.209031\pi$
$798$		0			0
$799$	9.79129			0.346391
$800$	14.7701	−	33.8426i	0.522202	−	1.19652i
$801$		0			0
$802$			9.00000i			0.317801i
$803$		−	7.93725i		−	0.280100i
$804$		0			0
$805$	−10.5826	−	2.20871i	−0.372987	−	0.0778469i
$806$	90.0111			3.17051
$807$		0			0
$808$		−	27.4955i		−	0.967287i
$809$	0.286051			0.0100570			0.00502850	−	0.999987i	$-0.498399\pi$
$809$	0.286051			0.0100570			0.00502850	+	0.999987i	$0.498399\pi$
$810$		0			0
$811$	33.8693			1.18931			0.594656	−	0.803980i	$-0.297288\pi$
$811$	33.8693			1.18931			0.594656	+	0.803980i	$0.297288\pi$
$812$		−	8.39410i		−	0.294575i
$813$		0			0
$814$	−38.5390			−1.35079
$815$	5.82380	−	27.9035i	0.203999	−	0.977417i
$816$		0			0
$817$		−	6.79129i		−	0.237597i
$818$			2.55040i			0.0891727i
$819$		0			0
$820$	55.7042	+	11.6261i	1.94527	+	0.406002i
$821$	−30.9307			−1.07949			−0.539744	−	0.841829i	$-0.681479\pi$
$821$	−30.9307			−1.07949			−0.539744	+	0.841829i	$0.681479\pi$
$822$		0			0
$823$		−	19.0000i		−	0.662298i	−0.943578	−	0.331149i	$-0.892564\pi$
$823$		−	19.0000i		−	0.662298i	0.943578	−	0.331149i	$-0.107436\pi$
$824$	17.9681			0.625947
$825$		0			0
$826$	16.9564			0.589990
$827$		−	17.6066i		−	0.612240i	−0.951993	−	0.306120i	$-0.900969\pi$
$827$		−	17.6066i		−	0.612240i	0.951993	−	0.306120i	$-0.0990309\pi$
$828$		0			0
$829$	−4.91288			−0.170631			−0.0853157	−	0.996354i	$-0.527190\pi$
$829$	−4.91288			−0.170631			−0.0853157	+	0.996354i	$0.527190\pi$
$830$	83.4444	+	17.4159i	2.89640	+	0.604513i
$831$		0			0
$832$			60.2867i			2.09007i
$833$			5.65300i			0.195865i
$834$		0			0
$835$	2.04356	−	9.79129i	0.0707203	−	0.338841i
$836$	−32.8335			−1.13557
$837$		0			0
$838$			49.2867i			1.70258i
$839$	−3.82560			−0.132074			−0.0660372	−	0.997817i	$-0.521036\pi$
$839$	−3.82560			−0.132074			−0.0660372	+	0.997817i	$0.521036\pi$
$840$		0			0
$841$	−19.9564			−0.688153
$842$		−	54.2657i		−	1.87012i
$843$		0			0
$844$	2.33030			0.0802123
$845$	−21.7937	−	4.54860i	−0.749725	−	0.156477i
$846$		0			0
$847$		−	8.00000i		−	0.274883i
$848$			11.7629i			0.403938i
$849$		0			0
$850$	56.7042	+	24.7477i	1.94494	+	0.848840i
$851$	−49.1450			−1.68467
$852$		0			0
$853$		−	27.6606i		−	0.947081i	−0.880772	−	0.473541i	$-0.842976\pi$
$853$		−	27.6606i		−	0.947081i	0.880772	−	0.473541i	$-0.157024\pi$
$854$	−10.4877			−0.358880
$855$		0			0
$856$	−6.16515			−0.210721
$857$			10.8293i			0.369920i	0.982746	+	0.184960i	$0.0592156\pi$
$857$			10.8293i			0.369920i	−0.982746	+	0.184960i	$0.940784\pi$
$858$		0			0
$859$	−2.41742			−0.0824815			−0.0412407	−	0.999149i	$-0.513131\pi$
$859$	−2.41742			−0.0824815			−0.0412407	+	0.999149i	$0.513131\pi$
$860$	−1.27520	+	6.10985i	−0.0434840	+	0.208344i
$861$		0			0
$862$		−	31.5390i		−	1.07422i
$863$			34.6410i			1.17919i	0.807698	+	0.589597i	$0.200713\pi$
$863$			34.6410i			1.17919i	−0.807698	+	0.589597i	$0.799287\pi$
$864$		0			0
$865$	4.87841	−	23.3739i	0.165871	−	0.794735i
$866$	40.5801			1.37897
$867$		0			0
$868$			23.9564i			0.813135i
$869$	−5.84370			−0.198234
$870$		0			0
$871$	6.58258			0.223042
$872$		−	0.361500i		−	0.0122419i
$873$		0			0
$874$	−71.8693			−2.43102
$875$	−6.47135	+	9.11710i	−0.218772	+	0.308214i
$876$		0			0
$877$		−	38.4955i		−	1.29990i	−0.759977	−	0.649950i	$-0.774790\pi$
$877$		−	38.4955i		−	1.29990i	0.759977	−	0.649950i	$-0.225210\pi$
$878$		−	72.9567i		−	2.46217i
$879$		0			0
$880$	−6.79129	−	1.41742i	−0.228934	−	0.0477814i
$881$	−36.4684			−1.22865			−0.614326	−	0.789052i	$-0.710572\pi$
$881$	−36.4684			−1.22865			−0.614326	+	0.789052i	$0.710572\pi$
$882$		0			0
$883$			27.6606i			0.930853i	0.885086	+	0.465427i	$0.154099\pi$
$883$			27.6606i			0.930853i	−0.885086	+	0.465427i	$0.845901\pi$
$884$	−75.6025			−2.54279
$885$		0			0
$886$	−9.41742			−0.316385
$887$		−	52.7243i		−	1.77031i	−0.465297	−	0.885155i	$-0.654052\pi$
$887$		−	52.7243i		−	1.77031i	0.465297	−	0.885155i	$-0.345948\pi$
$888$		0			0
$889$	−15.3739			−0.515623
$890$	−8.66025	+	41.4938i	−0.290292	+	1.39087i
$891$		0			0
$892$		−	25.1216i		−	0.841133i
$893$			11.7629i			0.393629i
$894$		0			0
$895$	−21.3739	−	4.46099i	−0.714449	−	0.149114i
$896$	−12.7719			−0.426679
$897$		0			0
$898$			53.4955i			1.78517i
$899$	−25.8100			−0.860810
$900$		0			0
$901$	−37.1216			−1.23670
$902$		−	34.5656i		−	1.15091i
$903$		0			0
$904$	−2.37386			−0.0789535
$905$	46.6899	+	9.74475i	1.55203	+	0.323927i
$906$		0			0
$907$		−	0.373864i		−	0.0124139i	−0.999981	−	0.00620697i	$-0.998024\pi$
$907$		−	0.373864i		−	0.0124139i	0.999981	−	0.00620697i	$-0.00197575\pi$
$908$		−	25.4485i		−	0.844537i
$909$		0			0
$910$	4.79129	−	22.9564i	0.158830	−	0.760999i
$911$	−53.9796			−1.78842			−0.894212	−	0.447643i	$-0.852264\pi$
$911$	−53.9796			−1.78842			−0.894212	+	0.447643i	$0.852264\pi$
$912$		0			0
$913$		−	30.1652i		−	0.998321i
$914$	−63.1921			−2.09021
$915$		0			0
$916$	−53.0345			−1.75231
$917$			11.8582i			0.391592i
$918$		0			0
$919$	4.20871			0.138833			0.0694163	−	0.997588i	$-0.477886\pi$
$919$	4.20871			0.138833			0.0694163	+	0.997588i	$0.477886\pi$
$920$	18.3296	+	3.82560i	0.604308	+	0.126126i
$921$		0			0
$922$		−	37.7477i		−	1.24316i
$923$			5.65300i			0.186071i
$924$		0			0
$925$	−20.3303	+	46.5826i	−0.668456	+	1.53163i
$926$	83.9966			2.76030
$927$		0			0
$928$		−	22.2087i		−	0.729037i
$929$	6.18530			0.202933			0.101467	−	0.994839i	$-0.467647\pi$
$929$	6.18530			0.202933			0.101467	+	0.994839i	$0.467647\pi$
$930$		0			0
$931$	−6.79129			−0.222575
$932$			23.6965i			0.776205i
$933$		0			0
$934$	−33.1652			−1.08520
$935$	4.47315	−	21.4322i	0.146288	−	0.700907i
$936$		0			0
$937$			28.3303i			0.925511i	0.886486	+	0.462755i	$0.153139\pi$
$937$			28.3303i			0.925511i	−0.886486	+	0.462755i	$0.846861\pi$
$938$			3.00725i			0.0981902i
$939$		0			0
$940$	2.20871	−	10.5826i	0.0720402	−	0.345166i
$941$	27.9989			0.912737			0.456368	−	0.889791i	$-0.349150\pi$
$941$	27.9989			0.912737			0.456368	+	0.889791i	$0.349150\pi$
$942$		0			0
$943$		−	44.0780i		−	1.43538i
$944$	13.8763			0.451635
$945$		0			0
$946$	3.79129			0.123265
$947$			21.7182i			0.705747i	0.935671	+	0.352874i	$0.114795\pi$
$947$			21.7182i			0.705747i	−0.935671	+	0.352874i	$0.885205\pi$
$948$		0			0
$949$	21.9564			0.712736
$950$	−29.7309	+	68.1221i	−0.964598	+	2.21017i
$951$		0			0
$952$		−	9.79129i		−	0.317337i
$953$			20.4986i			0.664013i	0.943277	+	0.332007i	$0.107726\pi$
$953$			20.4986i			0.664013i	−0.943277	+	0.332007i	$0.892274\pi$
$954$		0			0
$955$	−12.1652	−	2.53901i	−0.393655	−	0.0821606i
$956$	4.83465			0.156364
$957$		0			0
$958$		−	2.37386i		−	0.0766960i
$959$	15.8745			0.512615
$960$		0			0
$961$	42.6606			1.37615
$962$		−	106.609i		−	3.43720i
$963$		0			0
$964$	−9.41742			−0.303315
$965$	−5.00545	+	23.9826i	−0.161131	+	0.772026i
$966$		0			0
$967$			44.1652i			1.42026i	0.704073	+	0.710128i	$0.251363\pi$
$967$			44.1652i			1.42026i	−0.704073	+	0.710128i	$0.748637\pi$
$968$			13.8564i			0.445362i
$969$		0			0
$970$	−6.58258	−	1.37386i	−0.211354	−	0.0441121i
$971$	−6.39590			−0.205254			−0.102627	−	0.994720i	$-0.532725\pi$
$971$	−6.39590			−0.205254			−0.102627	+	0.994720i	$0.532725\pi$
$972$		0			0
$973$			11.4174i			0.366026i
$974$	69.8541			2.23827
$975$		0			0
$976$	−8.58258			−0.274722
$977$			57.4437i			1.83779i	0.394505	+	0.918894i	$0.370916\pi$
$977$			57.4437i			1.83779i	−0.394505	+	0.918894i	$0.629084\pi$
$978$		0			0
$979$	15.0000			0.479402
$980$	6.10985	+	1.27520i	0.195172	+	0.0407348i
$981$		0			0
$982$		−	87.4519i		−	2.79070i
$983$		−	9.59386i		−	0.305996i	−0.988226	−	0.152998i	$-0.951107\pi$
$983$		−	9.59386i		−	0.305996i	0.988226	−	0.152998i	$-0.0488929\pi$
$984$		0			0
$985$	−2.28674	+	10.9564i	−0.0728617	+	0.349101i
$986$	37.2113			1.18505
$987$		0			0
$988$		−	90.8258i		−	2.88955i
$989$	4.83465			0.153733
$990$		0			0
$991$	42.0780			1.33665			0.668326	−	0.743868i	$-0.267011\pi$
$991$	42.0780			1.33665			0.668326	+	0.743868i	$0.267011\pi$
$992$			63.3828i			2.01241i
$993$		0			0
$994$	−2.58258			−0.0819143
$995$	39.8571	+	8.31865i	1.26355	+	0.263719i
$996$		0			0
$997$			33.2432i			1.05282i	0.850230	+	0.526411i	$0.176463\pi$
$997$			33.2432i			1.05282i	−0.850230	+	0.526411i	$0.823537\pi$
$998$			44.1395i			1.39721i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.d.c.379.8	yes	8
3.2	odd	2	inner	945.2.d.c.379.1	✓	8
5.2	odd	4		4725.2.a.br.1.1		4
5.3	odd	4		4725.2.a.bs.1.4		4
5.4	even	2	inner	945.2.d.c.379.2	yes	8
15.2	even	4		4725.2.a.br.1.4		4
15.8	even	4		4725.2.a.bs.1.1		4
15.14	odd	2	inner	945.2.d.c.379.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.d.c.379.1	✓	8	3.2	odd	2	inner
945.2.d.c.379.2	yes	8	5.4	even	2	inner
945.2.d.c.379.7	yes	8	15.14	odd	2	inner
945.2.d.c.379.8	yes	8	1.1	even	1	trivial
4725.2.a.br.1.1		4	5.2	odd	4
4725.2.a.br.1.4		4	15.2	even	4
4725.2.a.bs.1.1		4	15.8	even	4
4725.2.a.bs.1.4		4	5.3	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.18890i q^{2} -2.79129 q^{4} +(2.18890 + 0.456850i) q^{5} +1.00000i q^{7} -1.73205i q^{8} +O(q^{10})\)	\(q+2.18890i q^{2} -2.79129 q^{4} +(2.18890 + 0.456850i) q^{5} +1.00000i q^{7} -1.73205i q^{8} +(-1.00000 + 4.79129i) q^{10} +1.73205 q^{11} +4.79129i q^{13} -2.18890 q^{14} -1.79129 q^{16} -5.65300i q^{17} +6.79129 q^{19} +(-6.10985 - 1.27520i) q^{20} +3.79129i q^{22} +4.83465i q^{23} +(4.58258 + 2.00000i) q^{25} -10.4877 q^{26} -2.79129i q^{28} +3.00725 q^{29} -8.58258 q^{31} -7.38505i q^{32} +12.3739 q^{34} +(-0.456850 + 2.18890i) q^{35} +10.1652i q^{37} +14.8655i q^{38} +(0.791288 - 3.79129i) q^{40} -9.11710 q^{41} -1.00000i q^{43} -4.83465 q^{44} -10.5826 q^{46} +1.73205i q^{47} -1.00000 q^{49} +(-4.37780 + 10.0308i) q^{50} -13.3739i q^{52} -6.56670i q^{53} +(3.79129 + 0.791288i) q^{55} +1.73205 q^{56} +6.58258i q^{58} -7.74655 q^{59} +4.79129 q^{61} -18.7864i q^{62} +12.5826 q^{64} +(-2.18890 + 10.4877i) q^{65} -1.37386i q^{67} +15.7792i q^{68} +(-4.79129 - 1.00000i) q^{70} +1.17985 q^{71} -4.58258i q^{73} -22.2505 q^{74} -18.9564 q^{76} +1.73205i q^{77} -3.37386 q^{79} +(-3.92095 - 0.818350i) q^{80} -19.9564i q^{82} -17.4159i q^{83} +(2.58258 - 12.3739i) q^{85} +2.18890 q^{86} -3.00000i q^{88} +8.66025 q^{89} -4.79129 q^{91} -13.4949i q^{92} -3.79129 q^{94} +(14.8655 + 3.10260i) q^{95} +1.37386i q^{97} -2.18890i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4}+O(q^{10}) \) 8 * q - 4 * q^4	\( 8 q - 4 q^{4} - 8 q^{10} + 4 q^{16} + 36 q^{19} - 32 q^{31} + 44 q^{34} - 12 q^{40} - 48 q^{46} - 8 q^{49} + 12 q^{55} + 20 q^{61} + 64 q^{64} - 20 q^{70} - 60 q^{76} + 28 q^{79} - 16 q^{85} - 20 q^{91} - 12 q^{94}+O(q^{100}) \) 8 * q - 4 * q^4 - 8 * q^10 + 4 * q^16 + 36 * q^19 - 32 * q^31 + 44 * q^34 - 12 * q^40 - 48 * q^46 - 8 * q^49 + 12 * q^55 + 20 * q^61 + 64 * q^64 - 20 * q^70 - 60 * q^76 + 28 * q^79 - 16 * q^85 - 20 * q^91 - 12 * q^94

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