Embedded newform 605.2.u.a.4.12

• Label: 605.2.u.a.4.12
• Level: $605$
• Weight: $2$
• Character: 605.4
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $2560$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.u (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(2560\)
Relative dimension:	\(64\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

Embedding invariants

Embedding label			4.12
Character	\(\chi\)	\(=\)	605.4
Dual form			605.2.u.a.454.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.112439 + 1.96633i) q^{2} +(-3.09691 - 1.00625i) q^{3} +(-1.86684 - 0.214200i) q^{4} +(0.326385 + 2.21212i) q^{5} +(2.32682 - 5.97640i) q^{6} +(-0.209014 + 0.494586i) q^{7} +(-0.0406069 + 0.234645i) q^{8} +(6.15126 + 4.46915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.112439	+	1.96633i	−0.0795062	+	1.39040i	0.676790	+	0.736176i	\(0.263370\pi\)
							−0.756297	+	0.654229i	\(0.772993\pi\)
\(3\)	−3.09691	−	1.00625i	−1.78800	−	0.580957i	−0.788578	−	0.614935i	\(-0.789182\pi\)
							−0.999423	+	0.0339779i	\(0.989182\pi\)
\(5\)	0.326385	+	2.21212i	0.145964	+	0.989290i
							\(7\)	−0.209014	+	0.494586i	−0.0789998	+	0.186936i	−0.955869	−	0.293793i	\(-0.905082\pi\)
0.876869	+	0.480729i	\(0.159628\pi\)
\(11\)	−3.29231	+	0.400829i	−0.992670	+	0.120854i
							\(13\)	2.02339	+	3.58296i	0.561187	+	0.993734i	0.995661	+	0.0930495i	\(0.0296615\pi\)
−0.434474	+	0.900684i	\(0.643066\pi\)
\(17\)	−0.248292	−	0.695886i	−0.0602196	−	0.168777i	0.907946	−	0.419086i	\(-0.137650\pi\)
							−0.968166	+	0.250309i	\(0.919468\pi\)
\(19\)	−0.165742	+	5.80174i	−0.0380239	+	1.33101i	0.726834	+	0.686813i	\(0.240991\pi\)
							−0.764858	+	0.644198i	\(0.777191\pi\)
\(23\)	1.27522	−	1.10499i	0.265903	−	0.230406i	−0.511697	−	0.859166i	\(-0.670983\pi\)
							0.777599	+	0.628760i	\(0.216437\pi\)
\(29\)	−4.40019	−	6.43508i	−0.817095	−	1.19496i	−0.978126	−	0.208014i	\(-0.933300\pi\)
							0.161031	−	0.986949i	\(-0.448518\pi\)
\(31\)	1.38960	−	6.85797i	0.249580	−	1.23173i	−0.637623	−	0.770349i	\(-0.720082\pi\)
							0.887203	−	0.461379i	\(-0.152645\pi\)
\(37\)	4.82982	+	5.26243i	0.794017	+	0.865138i	0.993390	−	0.114785i	\(-0.0366180\pi\)
							−0.199373	+	0.979924i	\(0.563891\pi\)
\(41\)	−0.749874	−	2.85281i	−0.117111	−	0.445535i	0.882586	−	0.470150i	\(-0.155800\pi\)
							−0.999697	+	0.0246154i	\(0.992164\pi\)
\(43\)	−2.06894	+	0.297468i	−0.315510	+	0.0453635i	−0.298251	−	0.954488i	\(-0.596403\pi\)
							−0.0172592	+	0.999851i	\(0.505494\pi\)
\(47\)	−5.77922	−	7.06769i	−0.842986	−	1.03093i	−0.999067	−	0.0431841i	\(-0.986250\pi\)
							0.156081	−	0.987744i	\(-0.450114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.u.a.4.12	✓	2560
5.4	even	2	inner	605.2.u.a.4.53	yes	2560
121.91	even	55	inner	605.2.u.a.454.53	yes	2560
605.454	even	110	inner	605.2.u.a.454.12	yes	2560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.u.a.4.12	✓	2560	1.1	even	1	trivial
605.2.u.a.4.53	yes	2560	5.4	even	2	inner
605.2.u.a.454.12	yes	2560	605.454	even	110	inner
605.2.u.a.454.53	yes	2560	121.91	even	55	inner