Embedded newform 25.6.d.a.11.7

• Label: 25.6.d.a.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.06882 - 1.50308i) q^{2} +(-3.25658 + 10.0227i) q^{3} +(-7.86780 + 24.2146i) q^{4} +(-54.0584 + 14.2370i) q^{5} +(8.32772 + 25.6301i) q^{6} -36.6142 q^{7} +(45.4065 + 139.747i) q^{8} +(106.742 + 77.5524i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.06882	−	1.50308i	0.365719	−	0.265710i	−0.389715	−	0.920936i	\(-0.627426\pi\)
							0.755433	+	0.655225i	\(0.227426\pi\)
\(3\)	−3.25658	+	10.0227i	−0.208909	+	0.642957i	0.790621	+	0.612306i	\(0.209758\pi\)
							−0.999530	+	0.0306509i	\(0.990242\pi\)
\(5\)	−54.0584	+	14.2370i	−0.967026	+	0.254679i
							\(7\)	−36.6142			−0.282426			−0.141213	−	0.989979i	\(-0.545100\pi\)
−0.141213	+	0.989979i	\(0.545100\pi\)
\(11\)	55.3269	−	40.1974i	0.137865	−	0.100165i	−0.516715	−	0.856157i	\(-0.672845\pi\)
							0.654580	+	0.755992i	\(0.272845\pi\)
\(13\)	278.848	+	202.595i	0.457624	+	0.332483i	0.792599	−	0.609744i	\(-0.208728\pi\)
							−0.334974	+	0.942227i	\(0.608728\pi\)
\(17\)	−201.510	−	620.182i	−0.169112	−	0.520472i	0.830204	−	0.557459i	\(-0.188224\pi\)
							−0.999316	+	0.0369876i	\(0.988224\pi\)
\(19\)	322.638	+	992.978i	0.205037	+	0.631038i	0.999712	+	0.0240033i	\(0.00764123\pi\)
							−0.794675	+	0.607035i	\(0.792359\pi\)
\(23\)	1655.07	−	1202.48i	0.652375	−	0.473978i	−0.211704	−	0.977334i	\(-0.567901\pi\)
							0.864079	+	0.503355i	\(0.167901\pi\)
\(29\)	−1240.01	+	3816.37i	−0.273799	+	0.842666i	0.715736	+	0.698371i	\(0.246091\pi\)
							−0.989535	+	0.144295i	\(0.953909\pi\)
\(31\)	2940.31	+	9049.34i	0.549527	+	1.69127i	0.709977	+	0.704225i	\(0.248705\pi\)
							−0.160450	+	0.987044i	\(0.551295\pi\)
\(37\)	−3171.57	−	2304.28i	−0.380864	−	0.276714i	0.380837	−	0.924642i	\(-0.375636\pi\)
							−0.761702	+	0.647928i	\(0.775636\pi\)
\(41\)	11804.6	+	8576.56i	1.09671	+	0.796807i	0.980520	−	0.196419i	\(-0.0629314\pi\)
							0.116191	+	0.993227i	\(0.462931\pi\)
\(43\)	11969.7			0.987214			0.493607	−	0.869685i	\(-0.335678\pi\)
							0.493607	+	0.869685i	\(0.335678\pi\)
\(47\)	7139.63	−	21973.5i	0.471445	−	1.45096i	−0.379247	−	0.925295i	\(-0.623817\pi\)
							0.850692	−	0.525664i	\(-0.176183\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.7	✓	44
25.4	even	10		625.6.a.c.1.14		22
25.16	even	5	inner	25.6.d.a.16.7	yes	44
25.21	even	5		625.6.a.d.1.9		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.7	✓	44	1.1	even	1	trivial
25.6.d.a.16.7	yes	44	25.16	even	5	inner
625.6.a.c.1.14		22	25.4	even	10
625.6.a.d.1.9		22	25.21	even	5