Embedded newform 460.2.q.a.11.11

• Label: 460.2.q.a.11.11
• Level: $460$
• Weight: $2$
• Character: 460.11
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.q (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(48\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.11
Character	\(\chi\)	\(=\)	460.11
Dual form			460.2.q.a.251.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18308 - 0.774803i) q^{2} +(3.11182 + 0.447412i) q^{3} +(0.799362 + 1.83331i) q^{4} +(-0.281733 - 0.959493i) q^{5} +(-3.33488 - 2.94037i) q^{6} +(1.26361 + 1.45829i) q^{7} +(0.474742 - 2.78830i) q^{8} +(6.60476 + 1.93933i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18308	−	0.774803i	−0.836565	−	0.547868i
							\(3\)	3.11182	+	0.447412i	1.79661	+	0.258313i	0.958071	−	0.286530i	\(-0.0925017\pi\)
0.838538	+	0.544843i	\(0.183411\pi\)
\(5\)	−0.281733	−	0.959493i	−0.125995	−	0.429098i
							\(7\)	1.26361	+	1.45829i	0.477601	+	0.551181i	0.942510	−	0.334177i	\(-0.108458\pi\)
−0.464909	+	0.885358i	\(0.653913\pi\)
\(11\)	−0.236180	−	0.151784i	−0.0712109	−	0.0457645i	0.504551	−	0.863382i	\(-0.331658\pi\)
							−0.575762	+	0.817617i	\(0.695294\pi\)
\(13\)	−2.33114	+	2.69028i	−0.646542	+	0.746150i	−0.980517	−	0.196432i	\(-0.937064\pi\)
							0.333975	+	0.942582i	\(0.391610\pi\)
\(17\)	3.37331	−	1.54054i	0.818149	−	0.373636i	0.0380370	−	0.999276i	\(-0.487890\pi\)
							0.780112	+	0.625640i	\(0.215162\pi\)
\(19\)	0.695216	−	1.52231i	0.159493	−	0.349242i	−0.812967	−	0.582310i	\(-0.802149\pi\)
							0.972461	+	0.233068i	\(0.0748764\pi\)
\(23\)	−1.57040	−	4.53143i	−0.327451	−	0.944868i
							\(29\)	1.75509	+	3.84310i	0.325911	+	0.713646i	0.999680	−	0.0252963i	\(-0.00805291\pi\)
−0.673769	+	0.738942i	\(0.735326\pi\)
\(31\)	−8.31174	+	1.19505i	−1.49283	+	0.214637i	−0.839884	−	0.542765i	\(-0.817377\pi\)
							−0.652948	+	0.757402i	\(0.726468\pi\)
\(37\)	−0.508667	+	1.73236i	−0.0836243	+	0.284798i	−0.990680	−	0.136213i	\(-0.956507\pi\)
							0.907055	+	0.421012i	\(0.138325\pi\)
\(41\)	−1.23623	+	0.362988i	−0.193066	+	0.0566893i	−0.376836	−	0.926280i	\(-0.622988\pi\)
							0.183770	+	0.982969i	\(0.441170\pi\)
\(43\)	0.904775	−	6.29285i	0.137977	−	0.959651i	−0.796756	−	0.604301i	\(-0.793453\pi\)
							0.934733	−	0.355350i	\(-0.115638\pi\)
\(47\)			8.65171i			1.26198i	0.775790	+	0.630991i	\(0.217352\pi\)
							−0.775790	+	0.630991i	\(0.782648\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.q.a.11.11	✓	480
4.3	odd	2	inner	460.2.q.a.11.28	yes	480
23.21	odd	22	inner	460.2.q.a.251.28	yes	480
92.67	even	22	inner	460.2.q.a.251.11	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.q.a.11.11	✓	480	1.1	even	1	trivial
460.2.q.a.11.28	yes	480	4.3	odd	2	inner
460.2.q.a.251.11	yes	480	92.67	even	22	inner
460.2.q.a.251.28	yes	480	23.21	odd	22	inner