Embedded newform 370.2.bd.a.187.3

• Label: 370.2.bd.a.187.3
• Level: $370$
• Weight: $2$
• Character: 370.187
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $108$
• 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:	\(108\)
Relative dimension:	\(9\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			187.3
Character	\(\chi\)	\(=\)	370.187
Dual form			370.2.bd.a.93.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 + 0.766044i) q^{2} +(-0.0763887 - 0.873127i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(2.23606 + 0.00416019i) q^{5} +(0.619753 - 0.619753i) q^{6} +(-0.122987 - 0.0573497i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(2.19791 - 0.387550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q + 6 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.642788	+	0.766044i	0.454519	+	0.541675i
							\(3\)	−0.0763887	−	0.873127i	−0.0441031	−	0.504100i	−0.985833	−	0.167730i	\(-0.946356\pi\)
0.941730	−	0.336370i	\(-0.109199\pi\)
\(5\)	2.23606	+	0.00416019i	0.999998	+	0.00186050i
							\(7\)	−0.122987	−	0.0573497i	−0.0464847	−	0.0216762i	0.399237	−	0.916848i	\(-0.369275\pi\)
−0.445722	+	0.895172i	\(0.647053\pi\)
\(11\)	3.47069	−	2.00380i	1.04645	−	0.604169i	0.124797	−	0.992182i	\(-0.460172\pi\)
							0.921654	+	0.388013i	\(0.126839\pi\)
\(13\)	−6.64449	−	1.17160i	−1.84285	−	0.324944i	−0.860134	−	0.510068i	\(-0.829620\pi\)
							−0.982715	+	0.185124i	\(0.940731\pi\)
\(17\)	0.834983	+	4.73542i	0.202513	+	1.14851i	0.901305	+	0.433184i	\(0.142610\pi\)
							−0.698792	+	0.715325i	\(0.746279\pi\)
\(19\)	−0.109749	−	1.25444i	−0.0251782	−	0.287788i	−0.998337	−	0.0576522i	\(-0.981639\pi\)
							0.973159	−	0.230136i	\(-0.0739170\pi\)
\(23\)	7.61015	+	4.39372i	1.58683	+	0.916154i	0.993826	+	0.110951i	\(0.0353898\pi\)
							0.593000	+	0.805203i	\(0.297944\pi\)
\(29\)	−2.73576	−	0.733044i	−0.508018	−	0.136123i	−0.00429905	−	0.999991i	\(-0.501368\pi\)
							−0.503718	+	0.863868i	\(0.668035\pi\)
\(31\)	1.83980	+	1.83980i	0.330438	+	0.330438i	0.852753	−	0.522315i	\(-0.174931\pi\)
							−0.522315	+	0.852753i	\(0.674931\pi\)
\(37\)	−4.92129	+	3.57503i	−0.809056	+	0.587732i
							\(41\)	−7.25700	−	1.27960i	−1.13335	−	0.199841i	−0.424657	−	0.905354i	\(-0.639605\pi\)
−0.708696	+	0.705514i	\(0.750716\pi\)
\(43\)		−	9.15337i		−	1.39588i	−0.716158	−	0.697938i	\(-0.754101\pi\)
							0.716158	−	0.697938i	\(-0.245899\pi\)
\(47\)	−0.530056	+	0.142028i	−0.0773165	+	0.0207169i	−0.297270	−	0.954793i	\(-0.596076\pi\)
							0.219953	+	0.975510i	\(0.429409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.bd.a.187.3	yes	108
5.3	odd	4		370.2.ba.a.113.7	✓	108
37.19	odd	36		370.2.ba.a.167.7	yes	108
185.93	even	36	inner	370.2.bd.a.93.3	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.a.113.7	✓	108	5.3	odd	4
370.2.ba.a.167.7	yes	108	37.19	odd	36
370.2.bd.a.93.3	yes	108	185.93	even	36	inner
370.2.bd.a.187.3	yes	108	1.1	even	1	trivial