Embedded newform 464.2.bc.a.11.5

• Label: 464.2.bc.a.11.5
• Level: $464$
• Weight: $2$
• Character: 464.11
• Analytic conductor: $3.705$
• Analytic rank: $0$
• Dimension: $696$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	464.bc (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.70505865379\)
Analytic rank:	\(0\)
Dimension:	\(696\)
Relative dimension:	\(58\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	464.11
Dual form			464.2.bc.a.211.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39307 - 0.243616i) q^{2} +(-0.160604 + 0.201391i) q^{3} +(1.88130 + 0.678750i) q^{4} +(-0.652184 + 0.228209i) q^{5} +(0.272795 - 0.241427i) q^{6} +(0.0366070 - 0.0459037i) q^{7} +(-2.45544 - 1.40386i) q^{8} +(0.652798 + 2.86010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 696 q - 10 q^{2} - 10 q^{3} - 14 q^{4} - 14 q^{5} - 6 q^{6} - 20 q^{7} - 46 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\)	\(117\)	\(175\)	\(321\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(e\left(\frac{25}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39307	−	0.243616i	−0.985051	−	0.172263i
							\(3\)	−0.160604	+	0.201391i	−0.0927248	+	0.116273i	−0.826029	−	0.563627i	\(-0.809406\pi\)
0.733305	+	0.679900i	\(0.237977\pi\)
\(5\)	−0.652184	+	0.228209i	−0.291666	+	0.102058i	−0.472146	−	0.881520i	\(-0.656520\pi\)
							0.180480	+	0.983579i	\(0.442235\pi\)
\(7\)	0.0366070	−	0.0459037i	0.0138361	−	0.0173500i	−0.774865	−	0.632126i	\(-0.782182\pi\)
							0.788701	+	0.614776i	\(0.210754\pi\)
\(11\)	−4.26112	−	0.972574i	−1.28478	−	0.293242i	−0.475003	−	0.879984i	\(-0.657553\pi\)
							−0.809774	+	0.586742i	\(0.800410\pi\)
\(13\)	−1.63459	+	1.02708i	−0.453354	+	0.284861i	−0.739290	−	0.673387i	\(-0.764839\pi\)
							0.285935	+	0.958249i	\(0.407696\pi\)
\(17\)	0.613031	−	0.613031i	0.148682	−	0.148682i	−0.628847	−	0.777529i	\(-0.716473\pi\)
							0.777529	+	0.628847i	\(0.216473\pi\)
\(19\)	−1.05861	−	1.32746i	−0.242862	−	0.304540i	0.645429	−	0.763820i	\(-0.276679\pi\)
							−0.888291	+	0.459281i	\(0.848107\pi\)
\(23\)	−0.859876	+	0.414095i	−0.179297	+	0.0863447i	−0.521379	−	0.853325i	\(-0.674582\pi\)
							0.342082	+	0.939670i	\(0.388868\pi\)
\(29\)	−3.92655	−	3.68541i	−0.729142	−	0.684363i
							\(31\)	1.13812	+	3.25255i	0.204412	+	0.584176i	0.999758	−	0.0219793i	\(-0.00699678\pi\)
−0.795346	+	0.606155i	\(0.792711\pi\)
\(37\)	−3.26586	+	0.745411i	−0.536904	+	0.122545i	−0.482371	−	0.875967i	\(-0.660224\pi\)
							−0.0545333	+	0.998512i	\(0.517367\pi\)
\(41\)	−7.83251	+	7.83251i	−1.22323	+	1.22323i	−0.256757	+	0.966476i	\(0.582654\pi\)
							−0.966476	+	0.256757i	\(0.917346\pi\)
\(43\)	0.601383	+	1.24878i	0.0917100	+	0.190438i	0.941778	−	0.336235i	\(-0.109154\pi\)
							−0.850068	+	0.526673i	\(0.823439\pi\)
\(47\)	−5.23433	−	8.33038i	−0.763505	−	1.21511i	−0.971514	−	0.236984i	\(-0.923841\pi\)
							0.208009	−	0.978127i	\(-0.433302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.2.bc.a.11.5	✓	696
16.3	odd	4		464.2.bm.a.243.17	yes	696
29.8	odd	28		464.2.bm.a.443.17	yes	696
464.211	even	28	inner	464.2.bc.a.211.5	yes	696

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
464.2.bc.a.11.5	✓	696	1.1	even	1	trivial
464.2.bc.a.211.5	yes	696	464.211	even	28	inner
464.2.bm.a.243.17	yes	696	16.3	odd	4
464.2.bm.a.443.17	yes	696	29.8	odd	28