Embedded newform 25.8.d.a.6.13

• Label: 25.8.d.a.6.13
• Level: $25$
• Weight: $8$
• Character: 25.6
• Analytic conductor: $7.810$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.80962563710\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(17\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			6.13
Character	\(\chi\)	\(=\)	25.6
Dual form			25.8.d.a.21.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.26313 + 13.1206i) q^{2} +(-1.52497 + 1.10796i) q^{3} +(-50.4211 + 36.6331i) q^{4} +(18.8507 + 278.872i) q^{5} +(-21.0382 - 15.2852i) q^{6} -1238.47 q^{7} +(733.013 + 532.565i) q^{8} +(-674.722 + 2076.58i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 11 q^{2} + 23 q^{3} - 1091 q^{4} + 475 q^{5} - 179 q^{6} + 1734 q^{7} + 220 q^{8} - 14554 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.26313	+	13.1206i	0.376811	+	1.15971i	0.942248	+	0.334915i	\(0.108708\pi\)
							−0.565437	+	0.824791i	\(0.691292\pi\)
\(3\)	−1.52497	+	1.10796i	−0.0326090	+	0.0236918i	−0.603970	−	0.797007i	\(-0.706415\pi\)
							0.571361	+	0.820699i	\(0.306415\pi\)
\(5\)	18.8507	+	278.872i	0.0674423	+	0.997723i
							\(7\)	−1238.47			−1.36472			−0.682361	−	0.731016i	\(-0.739047\pi\)
−0.682361	+	0.731016i	\(0.739047\pi\)
\(11\)	−1266.91	−	3899.16i	−0.286994	−	0.883277i	−0.985794	−	0.167959i	\(-0.946282\pi\)
							0.698800	−	0.715317i	\(-0.253718\pi\)
\(13\)	565.743	−	1741.18i	0.0714196	−	0.219807i	−0.908975	−	0.416850i	\(-0.863134\pi\)
							0.980395	+	0.197043i	\(0.0631339\pi\)
\(17\)	−15679.4	−	11391.8i	−0.774033	−	0.562368i	0.129149	−	0.991625i	\(-0.458775\pi\)
							−0.903182	+	0.429257i	\(0.858775\pi\)
\(19\)	44280.5	+	32171.7i	1.48107	+	1.07606i	0.977211	+	0.212272i	\(0.0680861\pi\)
							0.503857	+	0.863787i	\(0.331914\pi\)
\(23\)	29623.3	+	91171.0i	0.507675	+	1.56246i	0.796227	+	0.604998i	\(0.206826\pi\)
							−0.288553	+	0.957464i	\(0.593174\pi\)
\(29\)	159673.	−	116009.i	1.21573	−	0.883281i	0.219994	−	0.975501i	\(-0.429396\pi\)
							0.995739	+	0.0922201i	\(0.0293963\pi\)
\(31\)	205176.	+	149069.i	1.23698	+	0.898716i	0.997393	−	0.0721603i	\(-0.0229893\pi\)
							0.239583	+	0.970876i	\(0.422989\pi\)
\(37\)	22351.5	−	68790.7i	0.0725437	−	0.223267i	−0.908210	−	0.418514i	\(-0.862551\pi\)
							0.980754	+	0.195248i	\(0.0625510\pi\)
\(41\)	12178.3	−	37481.1i	0.0275959	−	0.0849314i	−0.936310	−	0.351174i	\(-0.885782\pi\)
							0.963906	+	0.266243i	\(0.0857824\pi\)
\(43\)	−197.277			−0.000378387			−0.000189194	−	1.00000i	\(-0.500060\pi\)
							−0.000189194		1.00000i	\(0.500060\pi\)
\(47\)	−443040.	+	321887.i	−0.622444	+	0.452232i	−0.853774	−	0.520643i	\(-0.825692\pi\)
							0.231330	+	0.972875i	\(0.425692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.d.a.6.13	✓	68
25.11	even	5		625.8.a.d.1.26		34
25.14	even	10		625.8.a.c.1.9		34
25.21	even	5	inner	25.8.d.a.21.13	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.d.a.6.13	✓	68	1.1	even	1	trivial
25.8.d.a.21.13	yes	68	25.21	even	5	inner
625.8.a.c.1.9		34	25.14	even	10
625.8.a.d.1.26		34	25.11	even	5