Embedded newform 25.6.d.a.6.10

• Label: 25.6.d.a.6.10
• Level: $25$
• Weight: $6$
• Character: 25.6
• 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			6.10
Character	\(\chi\)	\(=\)	25.6
Dual form			25.6.d.a.21.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.71092 + 8.34335i) q^{2} +(-5.61116 + 4.07675i) q^{3} +(-36.3739 + 26.4272i) q^{4} +(-55.5748 + 6.03628i) q^{5} +(-49.2252 - 35.7642i) q^{6} +102.277 q^{7} +(-91.9847 - 66.8308i) q^{8} +(-60.2259 + 185.356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.71092	+	8.34335i	0.479227	+	1.47491i	0.840171	+	0.542322i	\(0.182455\pi\)
							−0.360943	+	0.932588i	\(0.617545\pi\)
\(3\)	−5.61116	+	4.07675i	−0.359956	+	0.261524i	−0.753034	−	0.657982i	\(-0.771410\pi\)
							0.393078	+	0.919505i	\(0.371410\pi\)
\(5\)	−55.5748	+	6.03628i	−0.994153	+	0.107980i
							\(7\)	102.277			0.788921			0.394461	−	0.918913i	\(-0.370931\pi\)
0.394461	+	0.918913i	\(0.370931\pi\)
\(11\)	−24.8961	−	76.6222i	−0.0620367	−	0.190929i	0.915235	−	0.402921i	\(-0.132005\pi\)
							−0.977271	+	0.211992i	\(0.932005\pi\)
\(13\)	−82.4625	+	253.793i	−0.135331	+	0.416507i	−0.995641	−	0.0932643i	\(-0.970270\pi\)
							0.860310	+	0.509771i	\(0.170270\pi\)
\(17\)	653.462	+	474.768i	0.548400	+	0.398436i	0.827195	−	0.561914i	\(-0.189935\pi\)
							−0.278795	+	0.960351i	\(0.589935\pi\)
\(19\)	2243.33	+	1629.88i	1.42564	+	1.03579i	0.990808	+	0.135277i	\(0.0431923\pi\)
							0.434832	+	0.900512i	\(0.356808\pi\)
\(23\)	−1104.92	−	3400.60i	−0.435524	−	1.34040i	−0.892549	−	0.450951i	\(-0.851085\pi\)
							0.457025	−	0.889454i	\(-0.348915\pi\)
\(29\)	−436.050	+	316.809i	−0.0962812	+	0.0699524i	−0.634884	−	0.772607i	\(-0.718952\pi\)
							0.538603	+	0.842560i	\(0.318952\pi\)
\(31\)	1420.56	+	1032.10i	0.265495	+	0.192893i	0.712566	−	0.701605i	\(-0.247533\pi\)
							−0.447071	+	0.894498i	\(0.647533\pi\)
\(37\)	949.410	−	2921.98i	0.114012	−	0.350892i	−0.877728	−	0.479159i	\(-0.840942\pi\)
							0.991740	+	0.128267i	\(0.0409416\pi\)
\(41\)	5288.69	−	16276.9i	0.491347	−	1.51221i	−0.331226	−	0.943551i	\(-0.607462\pi\)
							0.822573	−	0.568660i	\(-0.192538\pi\)
\(43\)	16170.7			1.33370			0.666848	−	0.745194i	\(-0.267643\pi\)
							0.666848	+	0.745194i	\(0.267643\pi\)
\(47\)	−14180.5	+	10302.7i	−0.936368	+	0.680311i	−0.947544	−	0.319626i	\(-0.896443\pi\)
							0.0111757	+	0.999938i	\(0.496443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.6.10	✓	44
25.11	even	5		625.6.a.d.1.20		22
25.14	even	10		625.6.a.c.1.3		22
25.21	even	5	inner	25.6.d.a.21.10	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.6.10	✓	44	1.1	even	1	trivial
25.6.d.a.21.10	yes	44	25.21	even	5	inner
625.6.a.c.1.3		22	25.14	even	10
625.6.a.d.1.20		22	25.11	even	5