Embedded newform 460.2.w.a.167.61

• Label: 460.2.w.a.167.61
• Level: $460$
• Weight: $2$
• Character: 460.167
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $1360$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.w (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(1360\)
Relative dimension:	\(68\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			167.61
Character	\(\chi\)	\(=\)	460.167
Dual form			460.2.w.a.303.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32993 - 0.480928i) q^{2} +(0.789091 + 1.44511i) q^{3} +(1.53742 - 1.27920i) q^{4} +(-1.67600 + 1.48021i) q^{5} +(1.74443 + 1.54240i) q^{6} +(1.83326 + 2.44895i) q^{7} +(1.42945 - 2.44063i) q^{8} +(0.156239 - 0.243112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1360 q - 18 q^{2} - 36 q^{5} - 44 q^{6} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32993	−	0.480928i	0.940401	−	0.340067i
							\(3\)	0.789091	+	1.44511i	0.455582	+	0.834336i	0.999990	−	0.00448481i	\(-0.00142756\pi\)
−0.544408	+	0.838820i	\(0.683246\pi\)
\(5\)	−1.67600	+	1.48021i	−0.749530	+	0.661971i
							\(7\)	1.83326	+	2.44895i	0.692907	+	0.925616i	0.999657	−	0.0261741i	\(-0.00833244\pi\)
−0.306750	+	0.951790i	\(0.599242\pi\)
\(11\)	−0.0939173	+	0.0428906i	−0.0283171	+	0.0129320i	−0.429523	−	0.903056i	\(-0.641318\pi\)
							0.401206	+	0.915988i	\(0.368591\pi\)
\(13\)	−0.494316	−	0.370040i	−0.137099	−	0.102631i	0.528527	−	0.848916i	\(-0.322744\pi\)
							−0.665626	+	0.746286i	\(0.731835\pi\)
\(17\)	−3.62892	+	0.259545i	−0.880142	+	0.0629490i	−0.504092	−	0.863650i	\(-0.668173\pi\)
							−0.376050	+	0.926599i	\(0.622718\pi\)
\(19\)	3.72389	+	4.29760i	0.854319	+	0.985937i	0.999994	−	0.00339572i	\(-0.00108089\pi\)
							−0.145675	+	0.989332i	\(0.546535\pi\)
\(23\)	−4.71812	+	0.859828i	−0.983797	+	0.179287i
							\(29\)	6.58288	+	5.70410i	1.22241	+	1.05922i	0.996372	+	0.0851058i	\(0.0271228\pi\)
0.226038	+	0.974118i	\(0.427423\pi\)
\(31\)	−2.31627	−	7.88851i	−0.416015	−	1.41682i	−0.855143	−	0.518393i	\(-0.826531\pi\)
							0.439128	−	0.898425i	\(-0.355288\pi\)
\(37\)	−1.66089	−	7.63500i	−0.273049	−	1.25519i	−0.885485	−	0.464667i	\(-0.846174\pi\)
							0.612436	−	0.790520i	\(-0.290190\pi\)
\(41\)	5.13316	−	3.29888i	0.801664	−	0.515198i	−0.0744941	−	0.997221i	\(-0.523734\pi\)
							0.876159	+	0.482023i	\(0.160098\pi\)
\(43\)	−5.60374	−	10.2625i	−0.854562	−	1.56501i	−0.824778	−	0.565457i	\(-0.808700\pi\)
							−0.0297844	−	0.999556i	\(-0.509482\pi\)
\(47\)	−8.44949	+	8.44949i	−1.23249	+	1.23249i	−0.269479	+	0.963006i	\(0.586851\pi\)
							−0.963006	+	0.269479i	\(0.913149\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.w.a.167.61	yes	1360
4.3	odd	2	inner	460.2.w.a.167.60	yes	1360
5.3	odd	4	inner	460.2.w.a.443.26	yes	1360
20.3	even	4	inner	460.2.w.a.443.24	yes	1360
23.4	even	11	inner	460.2.w.a.27.24	✓	1360
92.27	odd	22	inner	460.2.w.a.27.26	yes	1360
115.73	odd	44	inner	460.2.w.a.303.60	yes	1360
460.303	even	44	inner	460.2.w.a.303.61	yes	1360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.w.a.27.24	✓	1360	23.4	even	11	inner
460.2.w.a.27.26	yes	1360	92.27	odd	22	inner
460.2.w.a.167.60	yes	1360	4.3	odd	2	inner
460.2.w.a.167.61	yes	1360	1.1	even	1	trivial
460.2.w.a.303.60	yes	1360	115.73	odd	44	inner
460.2.w.a.303.61	yes	1360	460.303	even	44	inner
460.2.w.a.443.24	yes	1360	20.3	even	4	inner
460.2.w.a.443.26	yes	1360	5.3	odd	4	inner