Embedded newform 37.7.i.a.15.1

• Label: 37.7.i.a.15.1
• Level: $37$
• Weight: $7$
• Character: 37.15
• Analytic conductor: $8.512$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	37.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.51200109393\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(18\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			15.1
Character	\(\chi\)	\(=\)	37.15
Dual form			37.7.i.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.52282 - 13.9882i) q^{2} +(4.44480 + 12.2120i) q^{3} +(-111.985 + 133.458i) q^{4} +(-123.914 - 176.967i) q^{5} +(141.831 - 141.831i) q^{6} +(-94.3880 + 535.301i) q^{7} +(1643.16 + 440.285i) q^{8} +(429.070 - 360.033i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 12 q^{2} - 12 q^{3} - 282 q^{4} + 186 q^{5} - 12 q^{6} - 12 q^{7} + 1782 q^{8} - 972 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.52282	−	13.9882i	−0.815352	−	1.74853i	−0.646541	−	0.762879i	\(-0.723785\pi\)
							−0.168811	−	0.985648i	\(-0.553993\pi\)
\(3\)	4.44480	+	12.2120i	0.164622	+	0.452296i	0.994385	−	0.105820i	\(-0.0337468\pi\)
							−0.829763	+	0.558116i	\(0.811525\pi\)
\(5\)	−123.914	−	176.967i	−0.991310	−	1.41574i	−0.907883	−	0.419223i	\(-0.862302\pi\)
							−0.0834267	−	0.996514i	\(-0.526586\pi\)
\(7\)	−94.3880	+	535.301i	−0.275184	+	1.56064i	0.463194	+	0.886257i	\(0.346703\pi\)
							−0.738378	+	0.674387i	\(0.764408\pi\)
\(11\)	73.1432	−	42.2292i	0.0549535	−	0.0317274i	−0.472272	−	0.881453i	\(-0.656566\pi\)
							0.527225	+	0.849726i	\(0.323232\pi\)
\(13\)	752.738	−	65.8561i	0.342621	−	0.0299755i	0.0854531	−	0.996342i	\(-0.472766\pi\)
							0.257168	+	0.966367i	\(0.417211\pi\)
\(17\)	−337.108	+	3853.16i	−0.0686155	+	0.784279i	0.881460	+	0.472259i	\(0.156561\pi\)
							−0.950075	+	0.312020i	\(0.898994\pi\)
\(19\)	−481.243	+	1032.03i	−0.0701623	+	0.150464i	−0.938261	−	0.345928i	\(-0.887564\pi\)
							0.868099	+	0.496392i	\(0.165342\pi\)
\(23\)	1120.98	+	4183.54i	0.0921325	+	0.343843i	0.996569	−	0.0827648i	\(-0.0263750\pi\)
							−0.904437	+	0.426608i	\(0.859708\pi\)
\(29\)	−1013.09	+	3780.92i	−0.0415390	+	0.155026i	−0.983580	−	0.180472i	\(-0.942238\pi\)
							0.942041	+	0.335497i	\(0.108904\pi\)
\(31\)	36328.5	+	36328.5i	1.21945	+	1.21945i	0.967826	+	0.251621i	\(0.0809636\pi\)
							0.251621	+	0.967826i	\(0.419036\pi\)
\(37\)	−18758.7	+	47051.4i	−0.370338	+	0.928897i
							\(41\)	−24113.9	+	28737.8i	−0.349878	+	0.416968i	−0.912067	−	0.410040i	\(-0.865515\pi\)
0.562190	+	0.827008i	\(0.309959\pi\)
\(43\)	−26113.5	+	26113.5i	−0.328443	+	0.328443i	−0.851994	−	0.523551i	\(-0.824607\pi\)
							0.523551	+	0.851994i	\(0.324607\pi\)
\(47\)	−12926.4	+	22389.2i	−0.124504	+	0.215647i	−0.921539	−	0.388286i	\(-0.873067\pi\)
							0.797035	+	0.603933i	\(0.206401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.i.a.15.1	yes	216
37.5	odd	36	inner	37.7.i.a.5.1	✓	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.i.a.5.1	✓	216	37.5	odd	36	inner
37.7.i.a.15.1	yes	216	1.1	even	1	trivial