Embedded newform 441.2.bn.a.164.4

• Label: 441.2.bn.a.164.4
• Level: $441$
• Weight: $2$
• Character: 441.164
• 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.bn (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			164.4
Character	\(\chi\)	\(=\)	441.164
Dual form			441.2.bn.a.320.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05598 + 2.19277i) q^{2} +(1.37134 - 1.05803i) q^{3} +(-2.44617 - 3.06740i) q^{4} +(-3.22503 - 0.994791i) q^{5} +(0.871911 + 4.12430i) q^{6} +(-0.531380 + 2.59184i) q^{7} +(4.56367 - 1.04163i) q^{8} +(0.761139 - 2.90184i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05598	+	2.19277i	−0.746693	+	1.55052i	0.0856891	+	0.996322i	\(0.472691\pi\)
							−0.832382	+	0.554202i	\(0.813023\pi\)
\(3\)	1.37134	−	1.05803i	0.791743	−	0.610855i
							\(5\)	−3.22503	−	0.994791i	−1.44228	−	0.444884i	−0.527748	−	0.849401i	\(-0.676964\pi\)
−0.914531	+	0.404517i	\(0.867440\pi\)
\(7\)	−0.531380	+	2.59184i	−0.200843	+	0.979623i
							\(11\)	2.38848	−	0.178992i	0.720154	−	0.0539681i	0.290389	−	0.956909i	\(-0.406215\pi\)
0.429765	+	0.902941i	\(0.358596\pi\)
\(13\)	5.81737	−	0.435952i	1.61345	−	0.120911i	0.762796	−	0.646640i	\(-0.223826\pi\)
							0.850652	+	0.525728i	\(0.176207\pi\)
\(17\)	−1.47076	−	3.74743i	−0.356712	−	0.908886i	−0.990764	−	0.135596i	\(-0.956705\pi\)
							0.634053	−	0.773290i	\(-0.281390\pi\)
\(19\)	3.32577	−	1.92013i	0.762984	−	0.440509i	−0.0673821	−	0.997727i	\(-0.521465\pi\)
							0.830366	+	0.557218i	\(0.188131\pi\)
\(23\)	0.156184	+	1.03621i	0.0325667	+	0.216066i	0.999195	−	0.0401172i	\(-0.0127731\pi\)
							−0.966628	+	0.256183i	\(0.917535\pi\)
\(29\)	5.17115	−	2.02953i	0.960258	−	0.376874i	0.167135	−	0.985934i	\(-0.446549\pi\)
							0.793124	+	0.609060i	\(0.208453\pi\)
\(31\)		−	2.94815i		−	0.529504i	−0.964317	−	0.264752i	\(-0.914710\pi\)
							0.964317	−	0.264752i	\(-0.0852900\pi\)
\(37\)	11.0077	+	1.65914i	1.80965	+	0.272761i	0.964942	−	0.262464i	\(-0.0845352\pi\)
							0.844709	+	0.535225i	\(0.179773\pi\)
\(41\)	−3.77830	+	3.50575i	−0.590071	+	0.547506i	−0.917687	−	0.397305i	\(-0.869945\pi\)
							0.327616	+	0.944811i	\(0.393755\pi\)
\(43\)	−2.94822	−	2.73554i	−0.449599	−	0.417167i	0.422602	−	0.906316i	\(-0.361117\pi\)
							−0.872200	+	0.489149i	\(0.837307\pi\)
\(47\)	−2.68897	−	1.29494i	−0.392226	−	0.188886i	0.227367	−	0.973809i	\(-0.426988\pi\)
							−0.619593	+	0.784923i	\(0.712702\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.164.4	yes	648
9.5	odd	6		441.2.bd.a.311.51	yes	648
49.26	odd	42		441.2.bd.a.173.51	✓	648
441.320	even	42	inner	441.2.bn.a.320.4	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.51	✓	648	49.26	odd	42
441.2.bd.a.311.51	yes	648	9.5	odd	6
441.2.bn.a.164.4	yes	648	1.1	even	1	trivial
441.2.bn.a.320.4	yes	648	441.320	even	42	inner