Embedded newform 25.6.e.a.19.7

• Label: 25.6.e.a.19.7
• Level: $25$
• Weight: $6$
• Character: 25.19
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			19.7
Character	\(\chi\)	\(=\)	25.19
Dual form			25.6.e.a.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.835458 - 0.271457i) q^{2} +(2.21975 + 3.05523i) q^{3} +(-25.2642 + 18.3555i) q^{4} +(40.9701 + 38.0322i) q^{5} +(2.68387 + 1.94995i) q^{6} +153.044i q^{7} +(-32.6474 + 44.9353i) q^{8} +(70.6840 - 217.543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.835458	−	0.271457i	0.147690	−	0.0479872i	−0.234239	−	0.972179i	\(-0.575260\pi\)
							0.381929	+	0.924192i	\(0.375260\pi\)
\(3\)	2.21975	+	3.05523i	0.142397	+	0.195993i	0.874258	−	0.485461i	\(-0.161348\pi\)
							−0.731861	+	0.681454i	\(0.761348\pi\)
\(5\)	40.9701	+	38.0322i	0.732896	+	0.680341i
							\(7\)			153.044i			1.18051i	0.807216	+	0.590257i	\(0.200974\pi\)
−0.807216	+	0.590257i	\(0.799026\pi\)
\(11\)	174.896	+	538.275i	0.435811	+	1.34129i	0.892253	+	0.451536i	\(0.149124\pi\)
							−0.456442	+	0.889753i	\(0.650876\pi\)
\(13\)	−435.792	−	141.597i	−0.715188	−	0.232379i	−0.0712525	−	0.997458i	\(-0.522700\pi\)
							−0.643936	+	0.765080i	\(0.722700\pi\)
\(17\)	867.677	−	1194.25i	0.728175	−	1.00225i	−0.271038	−	0.962569i	\(-0.587367\pi\)
							0.999213	−	0.0396778i	\(-0.0126331\pi\)
\(19\)	−1214.56	−	882.430i	−0.771854	−	0.560785i	0.130669	−	0.991426i	\(-0.458287\pi\)
							−0.902523	+	0.430641i	\(0.858287\pi\)
\(23\)	3427.11	−	1113.54i	1.35085	−	0.438919i	0.457875	−	0.889017i	\(-0.348611\pi\)
							0.892979	+	0.450098i	\(0.148611\pi\)
\(29\)	−4037.33	+	2933.29i	−0.891455	+	0.647680i	−0.936257	−	0.351316i	\(-0.885734\pi\)
							0.0448017	+	0.998996i	\(0.485734\pi\)
\(31\)	6836.55	+	4967.04i	1.27771	+	0.928312i	0.999481	−	0.0322027i	\(-0.0102522\pi\)
							0.278230	+	0.960514i	\(0.410252\pi\)
\(37\)	9689.31	+	3148.25i	1.16356	+	0.378063i	0.826235	−	0.563325i	\(-0.190478\pi\)
							0.337324	+	0.941389i	\(0.390478\pi\)
\(41\)	−1176.48	+	3620.85i	−0.109302	+	0.336396i	−0.990716	−	0.135948i	\(-0.956592\pi\)
							0.881414	+	0.472344i	\(0.156592\pi\)
\(43\)		−	16262.5i		−	1.34127i	−0.741786	−	0.670637i	\(-0.766021\pi\)
							0.741786	−	0.670637i	\(-0.233979\pi\)
\(47\)	−2458.15	−	3383.35i	−0.162317	−	0.223410i	0.720110	−	0.693860i	\(-0.244091\pi\)
							−0.882426	+	0.470450i	\(0.844091\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.19.7	yes	48
25.2	odd	20		625.6.a.g.1.27		48
25.4	even	10	inner	25.6.e.a.4.7	✓	48
25.23	odd	20		625.6.a.g.1.22		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.7	✓	48	25.4	even	10	inner
25.6.e.a.19.7	yes	48	1.1	even	1	trivial
625.6.a.g.1.22		48	25.23	odd	20
625.6.a.g.1.27		48	25.2	odd	20