Embedded newform 37.7.i.a.2.12

• Label: 37.7.i.a.2.12
• Level: $37$
• Weight: $7$
• Character: 37.2
• 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			2.12
Character	\(\chi\)	\(=\)	37.2
Dual form			37.7.i.a.19.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.48890 + 0.480216i) q^{2} +(-25.0783 + 29.8871i) q^{3} +(-33.1303 - 5.84176i) q^{4} +(69.6929 - 149.457i) q^{5} +(-152.004 + 152.004i) q^{6} +(37.2020 - 13.5404i) q^{7} +(-519.659 - 139.242i) q^{8} +(-137.731 - 781.110i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 12 q^{2} - 12 q^{3} - 282 q^{4} + 186 q^{5} - 12 q^{6} - 12 q^{7} + 1782 q^{8} - 972 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.48890	+	0.480216i	0.686112	+	0.0600271i	0.424881	−	0.905249i	\(-0.360316\pi\)
							0.261231	+	0.965276i	\(0.415871\pi\)
\(3\)	−25.0783	+	29.8871i	−0.928825	+	1.10693i	0.0652103	+	0.997872i	\(0.479228\pi\)
							−0.994035	+	0.109059i	\(0.965216\pi\)
\(5\)	69.6929	−	149.457i	0.557543	−	1.19565i	−0.402401	−	0.915463i	\(-0.631824\pi\)
							0.959944	−	0.280191i	\(-0.0903979\pi\)
\(7\)	37.2020	−	13.5404i	0.108461	−	0.0394765i	−0.287220	−	0.957865i	\(-0.592731\pi\)
							0.395680	+	0.918388i	\(0.370509\pi\)
\(11\)	1899.57	−	1096.72i	1.42717	−	0.823980i	0.430278	−	0.902697i	\(-0.358416\pi\)
							0.996897	+	0.0787168i	\(0.0250823\pi\)
\(13\)	−1702.03	−	2430.76i	−0.774708	−	1.10640i	−0.991491	−	0.130178i	\(-0.958445\pi\)
							0.216783	−	0.976220i	\(-0.430444\pi\)
\(17\)	−5034.43	−	3525.15i	−1.02472	−	0.717514i	−0.0648277	−	0.997896i	\(-0.520650\pi\)
							−0.959888	+	0.280383i	\(0.909539\pi\)
\(19\)	−7922.62	+	693.139i	−1.15507	+	0.101055i	−0.648549	−	0.761173i	\(-0.724624\pi\)
							−0.506520	+	0.862228i	\(0.669068\pi\)
\(23\)	3827.19	+	14283.3i	0.314555	+	1.17394i	0.924403	+	0.381417i	\(0.124564\pi\)
							−0.609848	+	0.792518i	\(0.708770\pi\)
\(29\)	3341.94	−	12472.3i	0.137027	−	0.511391i	−0.862955	−	0.505281i	\(-0.831389\pi\)
							0.999981	−	0.00610916i	\(-0.00194462\pi\)
\(31\)	−15048.2	−	15048.2i	−0.505125	−	0.505125i	0.407901	−	0.913026i	\(-0.366261\pi\)
							−0.913026	+	0.407901i	\(0.866261\pi\)
\(37\)	−43277.5	−	26320.8i	−0.854391	−	0.519630i
							\(41\)	113086.	+	19940.1i	1.64080	+	0.289318i	0.916461	−	0.400123i	\(-0.131033\pi\)
0.724342	+	0.689441i	\(0.242144\pi\)
\(43\)	3208.76	−	3208.76i	0.0403582	−	0.0403582i	−0.686640	−	0.726998i	\(-0.740915\pi\)
							0.726998	+	0.686640i	\(0.240915\pi\)
\(47\)	67988.6	−	117760.i	0.654851	−	1.13424i	−0.327080	−	0.944997i	\(-0.606065\pi\)
							0.981931	−	0.189238i	\(-0.0606019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.i.a.2.12	✓	216
37.19	odd	36	inner	37.7.i.a.19.12	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.i.a.2.12	✓	216	1.1	even	1	trivial
37.7.i.a.19.12	yes	216	37.19	odd	36	inner