Embedded newform 37.8.f.a.16.12

• Label: 37.8.f.a.16.12
• Level: $37$
• Weight: $8$
• Character: 37.16
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			16.12
Character	\(\chi\)	\(=\)	37.16
Dual form			37.8.f.a.7.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.323158 - 0.117620i) q^{2} +(-38.0853 + 13.8619i) q^{3} +(-97.9631 - 82.2008i) q^{4} +(13.0043 - 73.7511i) q^{5} +13.9380 q^{6} +(126.462 - 717.201i) q^{7} +(43.9985 + 76.2077i) q^{8} +(-417.005 + 349.909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.323158	−	0.117620i	−0.0285634	−	0.0103962i	0.327699	−	0.944782i	\(-0.393727\pi\)
							−0.356262	+	0.934386i	\(0.615949\pi\)
\(3\)	−38.0853	+	13.8619i	−0.814390	+	0.296414i	−0.715436	−	0.698678i	\(-0.753772\pi\)
							−0.0989540	+	0.995092i	\(0.531550\pi\)
\(5\)	13.0043	−	73.7511i	0.0465256	−	0.263860i	−0.952668	−	0.304013i	\(-0.901673\pi\)
							0.999194	+	0.0401530i	\(0.0127845\pi\)
\(7\)	126.462	−	717.201i	0.139353	−	0.790311i	−0.832376	−	0.554212i	\(-0.813020\pi\)
							0.971729	−	0.236099i	\(-0.0758690\pi\)
\(11\)	3587.91	+	6214.44i	0.812769	+	1.40776i	0.910919	+	0.412585i	\(0.135374\pi\)
							−0.0981505	+	0.995172i	\(0.531293\pi\)
\(13\)	−1112.82	−	933.770i	−0.140483	−	0.117880i	0.569838	−	0.821757i	\(-0.307006\pi\)
							−0.710321	+	0.703878i	\(0.751450\pi\)
\(17\)	−19294.0	+	16189.6i	−0.952469	+	0.799217i	−0.979712	−	0.200412i	\(-0.935772\pi\)
							0.0272424	+	0.999629i	\(0.491327\pi\)
\(19\)	51336.6	−	18685.0i	1.71708	−	0.624965i	0.719497	−	0.694495i	\(-0.244372\pi\)
							0.997580	+	0.0695304i	\(0.0221501\pi\)
\(23\)	−4514.46	+	7819.27i	−0.0773674	+	0.134004i	−0.902113	−	0.431499i	\(-0.857985\pi\)
							0.824746	+	0.565503i	\(0.191318\pi\)
\(29\)	105296.	+	182379.i	0.801716	+	1.38861i	0.918486	+	0.395453i	\(0.129412\pi\)
							−0.116771	+	0.993159i	\(0.537254\pi\)
\(31\)	−191400.			−1.15392			−0.576962	−	0.816771i	\(-0.695762\pi\)
							−0.576962	+	0.816771i	\(0.695762\pi\)
\(37\)	253930.	−	174504.i	0.824152	−	0.566369i
							\(41\)	−463992.	−	389336.i	−1.05140	−	0.882228i	−0.0581580	−	0.998307i	\(-0.518523\pi\)
−0.993240	+	0.116080i	\(0.962967\pi\)
\(43\)	−105795.			−0.202921			−0.101460	−	0.994840i	\(-0.532351\pi\)
							−0.101460	+	0.994840i	\(0.532351\pi\)
\(47\)	−597654.	+	1.03517e6i	−0.839668	+	1.45435i	0.0505051	+	0.998724i	\(0.483917\pi\)
							−0.890173	+	0.455623i	\(0.849416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.16.12	yes	132
37.7	even	9	inner	37.8.f.a.7.12	✓	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.12	✓	132	37.7	even	9	inner
37.8.f.a.16.12	yes	132	1.1	even	1	trivial