Embedded newform 441.2.bh.a.104.38

• Label: 441.2.bh.a.104.38
• Level: $441$
• Weight: $2$
• Character: 441.104
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bh (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			104.38
Character	\(\chi\)	\(=\)	441.104
Dual form			441.2.bh.a.335.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.167777 - 1.11313i) q^{2} +(1.73188 - 0.0242936i) q^{3} +(0.700245 + 0.215997i) q^{4} +(-2.64308 + 1.80202i) q^{5} +(0.263527 - 1.93188i) q^{6} +(2.49896 + 0.869024i) q^{7} +(1.33476 - 2.77166i) q^{8} +(2.99882 - 0.0841471i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.167777	−	1.11313i	0.118636	−	0.787099i	−0.847823	−	0.530279i	\(-0.822087\pi\)
							0.966459	−	0.256820i	\(-0.0826747\pi\)
\(3\)	1.73188	−	0.0242936i	0.999902	−	0.0140259i
							\(5\)	−2.64308	+	1.80202i	−1.18202	+	0.805889i	−0.984667	−	0.174444i	\(-0.944187\pi\)
−0.197354	+	0.980332i	\(0.563235\pi\)
\(7\)	2.49896	+	0.869024i	0.944518	+	0.328460i
							\(11\)	−0.594925	+	3.94707i	−0.179377	+	1.19009i	0.700065	+	0.714079i	\(0.253154\pi\)
−0.879442	+	0.476007i	\(0.842084\pi\)
\(13\)	−1.28779	+	0.505421i	−0.357169	+	0.140179i	−0.537140	−	0.843493i	\(-0.680495\pi\)
							0.179970	+	0.983672i	\(0.442400\pi\)
\(17\)	1.65298	−	7.24216i	0.400906	−	1.75648i	−0.222840	−	0.974855i	\(-0.571533\pi\)
							0.623746	−	0.781627i	\(-0.285610\pi\)
\(19\)			4.92307i			1.12943i	0.825286	+	0.564715i	\(0.191014\pi\)
							−0.825286	+	0.564715i	\(0.808986\pi\)
\(23\)	0.200246	−	0.649181i	0.0417541	−	0.135364i	−0.932296	−	0.361697i	\(-0.882198\pi\)
							0.974050	+	0.226333i	\(0.0726738\pi\)
\(29\)	−1.36654	−	4.43021i	−0.253760	−	0.822670i	−0.989390	−	0.145286i	\(-0.953590\pi\)
							0.735630	−	0.677384i	\(-0.236886\pi\)
\(31\)	−1.10338	−	0.637037i	−0.198173	−	0.114415i	0.397630	−	0.917546i	\(-0.369833\pi\)
							−0.595803	+	0.803131i	\(0.703166\pi\)
\(37\)	1.68999	−	7.40433i	0.277833	−	1.21727i	−0.622694	−	0.782465i	\(-0.713962\pi\)
							0.900527	−	0.434800i	\(-0.143181\pi\)
\(41\)	−9.37338	+	6.39066i	−1.46388	+	0.998053i	−0.470591	+	0.882351i	\(0.655959\pi\)
							−0.993284	+	0.115702i	\(0.963088\pi\)
\(43\)	−2.02570	−	1.38110i	−0.308916	−	0.210616i	0.398931	−	0.916981i	\(-0.369381\pi\)
							−0.707847	+	0.706365i	\(0.750334\pi\)
\(47\)	−9.31846	−	1.40453i	−1.35924	−	0.204872i	−0.571357	−	0.820701i	\(-0.693583\pi\)
							−0.787880	+	0.615829i	\(0.788821\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.104.38	yes	648
9.2	odd	6	inner	441.2.bh.a.398.38	yes	648
49.41	odd	14	inner	441.2.bh.a.41.38	✓	648
441.335	even	42	inner	441.2.bh.a.335.38	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.41.38	✓	648	49.41	odd	14	inner
441.2.bh.a.104.38	yes	648	1.1	even	1	trivial
441.2.bh.a.335.38	yes	648	441.335	even	42	inner
441.2.bh.a.398.38	yes	648	9.2	odd	6	inner