Embedded newform 37.7.i.a.5.1

• Label: 37.7.i.a.5.1
• Level: $37$
• Weight: $7$
• Character: 37.5
• 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			5.1
Character	\(\chi\)	\(=\)	37.5
Dual form			37.7.i.a.15.1

$q$-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 \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{23}{36}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.i.a.5.1	✓	216	1.1	even	1	trivial
37.7.i.a.15.1	yes	216	37.15	odd	36	inner