Embedded newform 25.6.d.a.11.10

• Label: 25.6.d.a.11.10
• Level: $25$
• Weight: $6$
• Character: 25.11
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			11.10
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.08950 - 5.15083i) q^{2} +(5.03780 - 15.5047i) q^{3} +(13.8415 - 42.5998i) q^{4} +(5.66616 + 55.6138i) q^{5} +(-44.1468 - 135.870i) q^{6} -72.9673 q^{7} +(-34.6403 - 106.612i) q^{8} +(-18.4266 - 13.3877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.08950	−	5.15083i	1.25326	−	0.910546i	0.254853	−	0.966980i	\(-0.417973\pi\)
							0.998406	+	0.0564337i	\(0.0179730\pi\)
\(3\)	5.03780	−	15.5047i	0.323175	−	0.994630i	−0.649083	−	0.760718i	\(-0.724847\pi\)
							0.972258	−	0.233912i	\(-0.0751528\pi\)
\(5\)	5.66616	+	55.6138i	0.101359	+	0.994850i
							\(7\)	−72.9673			−0.562838			−0.281419	−	0.959585i	\(-0.590805\pi\)
−0.281419	+	0.959585i	\(0.590805\pi\)
\(11\)	142.406	−	103.464i	0.354852	−	0.257815i	−0.396050	−	0.918229i	\(-0.629619\pi\)
							0.750902	+	0.660414i	\(0.229619\pi\)
\(13\)	4.89891	+	3.55927i	0.00803972	+	0.00584120i	0.591798	−	0.806086i	\(-0.298418\pi\)
							−0.583758	+	0.811928i	\(0.698418\pi\)
\(17\)	−533.079	−	1640.65i	−0.447373	−	1.37687i	−0.879861	−	0.475232i	\(-0.842364\pi\)
							0.432488	−	0.901640i	\(-0.357636\pi\)
\(19\)	753.289	+	2318.38i	0.478716	+	1.47334i	0.840880	+	0.541221i	\(0.182038\pi\)
							−0.362165	+	0.932114i	\(0.617962\pi\)
\(23\)	−3830.65	+	2783.13i	−1.50991	+	1.09702i	−0.543700	+	0.839280i	\(0.682977\pi\)
							−0.966215	+	0.257738i	\(0.917023\pi\)
\(29\)	388.108	−	1194.47i	0.0856954	−	0.263743i	−0.899022	−	0.437904i	\(-0.855721\pi\)
							0.984717	+	0.174160i	\(0.0557211\pi\)
\(31\)	−1006.41	−	3097.40i	−0.188091	−	0.578885i	0.811897	−	0.583801i	\(-0.198435\pi\)
							−0.999988	+	0.00491585i	\(0.998435\pi\)
\(37\)	−12425.9	−	9027.96i	−1.49219	−	1.08414i	−0.973365	−	0.229260i	\(-0.926370\pi\)
							−0.518825	−	0.854880i	\(-0.673630\pi\)
\(41\)	−4218.31	−	3064.78i	−0.391903	−	0.284734i	0.374332	−	0.927295i	\(-0.377872\pi\)
							−0.766235	+	0.642561i	\(0.777872\pi\)
\(43\)	14637.4			1.20724			0.603619	−	0.797273i	\(-0.293725\pi\)
							0.603619	+	0.797273i	\(0.293725\pi\)
\(47\)	−2273.25	+	6996.33i	−0.150107	+	0.461983i	−0.997632	−	0.0687726i	\(-0.978092\pi\)
							0.847525	+	0.530755i	\(0.178092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.10	✓	44
25.4	even	10		625.6.a.c.1.20		22
25.16	even	5	inner	25.6.d.a.16.10	yes	44
25.21	even	5		625.6.a.d.1.3		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.10	✓	44	1.1	even	1	trivial
25.6.d.a.16.10	yes	44	25.16	even	5	inner
625.6.a.c.1.20		22	25.4	even	10
625.6.a.d.1.3		22	25.21	even	5