Embedded newform 441.2.bd.a.47.17

• Label: 441.2.bd.a.47.17
• Level: $441$
• Weight: $2$
• Character: 441.47
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bd (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			47.17
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.932976 - 1.00551i) q^{2} +(-1.41782 + 0.994877i) q^{3} +(0.00885622 - 0.118178i) q^{4} +(0.700700 + 0.878650i) q^{5} +(2.32315 + 0.497436i) q^{6} +(-1.28096 + 2.31498i) q^{7} +(-2.27193 + 1.81180i) q^{8} +(1.02044 - 2.82112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.932976	−	1.00551i	−0.659713	−	0.711002i	0.311224	−	0.950337i	\(-0.399261\pi\)
							−0.970937	+	0.239335i	\(0.923071\pi\)
\(3\)	−1.41782	+	0.994877i	−0.818580	+	0.574393i
							\(5\)	0.700700	+	0.878650i	0.313363	+	0.392944i	0.913424	−	0.407010i	\(-0.133429\pi\)
−0.600061	+	0.799954i	\(0.704857\pi\)
\(7\)	−1.28096	+	2.31498i	−0.484158	+	0.874981i
							\(11\)	3.25355	−	0.742602i	0.980983	−	0.223903i	0.298184	−	0.954509i	\(-0.403619\pi\)
0.682800	+	0.730606i	\(0.260762\pi\)
\(13\)	−4.37787	−	4.71822i	−1.21420	−	1.30860i	−0.937642	−	0.347602i	\(-0.886996\pi\)
							−0.276560	−	0.960997i	\(-0.589195\pi\)
\(17\)	0.0530108	+	0.707380i	0.0128570	+	0.171565i	0.999936	+	0.0113294i	\(0.00360633\pi\)
							−0.987079	+	0.160236i	\(0.948775\pi\)
\(19\)	−2.38042	−	1.37434i	−0.546107	−	0.315295i	0.201444	−	0.979500i	\(-0.435437\pi\)
							−0.747550	+	0.664205i	\(0.768770\pi\)
\(23\)	−0.168790	−	0.350496i	−0.0351952	−	0.0730835i	0.882630	−	0.470068i	\(-0.155771\pi\)
							−0.917825	+	0.396985i	\(0.870057\pi\)
\(29\)	−3.07828	−	4.51501i	−0.571622	−	0.838416i	0.426183	−	0.904637i	\(-0.359858\pi\)
							−0.997805	+	0.0662214i	\(0.978906\pi\)
\(31\)	−8.01820	−	4.62931i	−1.44011	−	0.831449i	−0.442255	−	0.896889i	\(-0.645821\pi\)
							−0.997856	+	0.0654404i	\(0.979155\pi\)
\(37\)	−7.00707	+	4.77734i	−1.15196	+	0.785390i	−0.979859	−	0.199690i	\(-0.936006\pi\)
							−0.172096	+	0.985080i	\(0.555054\pi\)
\(41\)	0.536673	−	1.36742i	0.0838142	−	0.213555i	−0.882822	−	0.469707i	\(-0.844360\pi\)
							0.966637	+	0.256152i	\(0.0824548\pi\)
\(43\)	−1.78598	−	4.55060i	−0.272360	−	0.693961i	−0.999968	−	0.00796834i	\(-0.997464\pi\)
							0.727609	−	0.685992i	\(-0.240632\pi\)
\(47\)	−7.86721	+	7.29970i	−1.14755	+	1.06477i	−0.150454	+	0.988617i	\(0.548074\pi\)
							−0.997096	+	0.0761540i	\(0.975736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.17	✓	648
9.5	odd	6		441.2.bn.a.194.17	yes	648
49.24	odd	42		441.2.bn.a.416.17	yes	648
441.122	even	42	inner	441.2.bd.a.122.17	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.17	✓	648	1.1	even	1	trivial
441.2.bd.a.122.17	yes	648	441.122	even	42	inner
441.2.bn.a.194.17	yes	648	9.5	odd	6
441.2.bn.a.416.17	yes	648	49.24	odd	42