Embedded newform 370.2.bd.b.313.8

• Label: 370.2.bd.b.313.8
• Level: $370$
• Weight: $2$
• Character: 370.313
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.bd (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95446487479\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			313.8
Character	\(\chi\)	\(=\)	370.313
Dual form			370.2.bd.b.357.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 + 0.173648i) q^{2} +(0.928067 + 1.32542i) q^{3} +(0.939693 - 0.342020i) q^{4} +(0.287938 + 2.21745i) q^{5} +(-1.14412 - 1.14412i) q^{6} +(1.27484 - 0.111534i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(0.130639 - 0.358928i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{3}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.984808	+	0.173648i	−0.696364	+	0.122788i
							\(3\)	0.928067	+	1.32542i	0.535820	+	0.765230i	0.992214	−	0.124548i	\(-0.0397481\pi\)
−0.456394	+	0.889778i	\(0.650859\pi\)
\(5\)	0.287938	+	2.21745i	0.128770	+	0.991674i
							\(7\)	1.27484	−	0.111534i	0.481843	−	0.0421558i	0.156353	−	0.987701i	\(-0.450026\pi\)
0.325490	+	0.945545i	\(0.394471\pi\)
\(11\)	3.41422	−	1.97120i	1.02942	−	0.594339i	0.112605	−	0.993640i	\(-0.464081\pi\)
							0.916820	+	0.399301i	\(0.130747\pi\)
\(13\)	2.06591	+	5.67603i	0.572979	+	1.57425i	0.799772	+	0.600304i	\(0.204954\pi\)
							−0.226793	+	0.973943i	\(0.572824\pi\)
\(17\)	−4.50077	−	1.63815i	−1.09160	−	0.397309i	−0.267384	−	0.963590i	\(-0.586159\pi\)
							−0.824212	+	0.566281i	\(0.808382\pi\)
\(19\)	1.17250	+	1.67451i	0.268991	+	0.384158i	0.930809	−	0.365506i	\(-0.119104\pi\)
							−0.661818	+	0.749664i	\(0.730215\pi\)
\(23\)	−4.52686	−	2.61359i	−0.943916	−	0.544970i	−0.0527304	−	0.998609i	\(-0.516792\pi\)
							−0.891186	+	0.453639i	\(0.850126\pi\)
\(29\)	−0.104271	+	0.389145i	−0.0193627	+	0.0722625i	−0.974931	−	0.222507i	\(-0.928576\pi\)
							0.955568	+	0.294769i	\(0.0952428\pi\)
\(31\)	−3.76456	+	3.76456i	−0.676134	+	0.676134i	−0.959123	−	0.282989i	\(-0.908674\pi\)
							0.282989	+	0.959123i	\(0.408674\pi\)
\(37\)	6.06743	+	0.431571i	0.997480	+	0.0709499i
							\(41\)	−3.22430	−	8.85869i	−0.503551	−	1.38349i	−0.887785	−	0.460259i	\(-0.847757\pi\)
0.384234	−	0.923236i	\(-0.374466\pi\)
\(43\)			2.78417i			0.424583i	0.977206	+	0.212291i	\(0.0680926\pi\)
							−0.977206	+	0.212291i	\(0.931907\pi\)
\(47\)	2.89511	+	10.8047i	0.422295	+	1.57603i	0.769759	+	0.638335i	\(0.220376\pi\)
							−0.347464	+	0.937693i	\(0.612957\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.bd.b.313.8	yes	120
5.2	odd	4		370.2.ba.b.17.8	✓	120
37.24	odd	36		370.2.ba.b.283.8	yes	120
185.172	even	36	inner	370.2.bd.b.357.8	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.b.17.8	✓	120	5.2	odd	4
370.2.ba.b.283.8	yes	120	37.24	odd	36
370.2.bd.b.313.8	yes	120	1.1	even	1	trivial
370.2.bd.b.357.8	yes	120	185.172	even	36	inner