Embedded newform 25.8.e.a.4.16

• Label: 25.8.e.a.4.16
• Level: $25$
• Weight: $8$
• Character: 25.4
• Analytic conductor: $7.810$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.80962563710\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(19.7814 + 6.42736i) q^{2} +(47.2089 - 64.9775i) q^{3} +(246.438 + 179.048i) q^{4} +(-275.144 - 49.2038i) q^{5} +(1351.49 - 981.916i) q^{6} +664.316i q^{7} +(2159.21 + 2971.90i) q^{8} +(-1317.57 - 4055.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	19.7814	+	6.42736i	1.74844	+	0.568104i	0.995902	−	0.0904402i	\(-0.0288274\pi\)
							0.752542	+	0.658544i	\(0.228827\pi\)
\(3\)	47.2089	−	64.9775i	1.00948	−	1.38944i	0.0901626	−	0.995927i	\(-0.471261\pi\)
							0.919321	−	0.393508i	\(-0.128739\pi\)
\(5\)	−275.144	−	49.2038i	−0.984384	−	0.176037i
							\(7\)			664.316i			0.732034i	0.930608	+	0.366017i	\(0.119279\pi\)
−0.930608	+	0.366017i	\(0.880721\pi\)
\(11\)	320.430	−	986.182i	0.0725869	−	0.223400i	−0.908181	−	0.418578i	\(-0.862529\pi\)
							0.980768	+	0.195178i	\(0.0625286\pi\)
\(13\)	−6484.12	+	2106.82i	−0.818558	+	0.265965i	−0.688218	−	0.725504i	\(-0.741607\pi\)
							−0.130340	+	0.991469i	\(0.541607\pi\)
\(17\)	17913.9	+	24656.3i	0.884337	+	1.21719i	0.975201	+	0.221323i	\(0.0710374\pi\)
							−0.0908633	+	0.995863i	\(0.528963\pi\)
\(19\)	−4851.77	+	3525.02i	−0.162279	+	0.117903i	−0.665961	−	0.745986i	\(-0.731978\pi\)
							0.503682	+	0.863889i	\(0.331978\pi\)
\(23\)	−72622.7	−	23596.6i	−1.24459	−	0.404391i	−0.388609	−	0.921403i	\(-0.627044\pi\)
							−0.855978	+	0.517012i	\(0.827044\pi\)
\(29\)	−132371.	−	96173.5i	−1.00786	−	0.732255i	−0.0441028	−	0.999027i	\(-0.514043\pi\)
							−0.963760	+	0.266772i	\(0.914043\pi\)
\(31\)	142811.	−	103758.i	0.860984	−	0.625542i	−0.0671684	−	0.997742i	\(-0.521396\pi\)
							0.928153	+	0.372200i	\(0.121396\pi\)
\(37\)	36604.9	−	11893.7i	0.118805	−	0.0386020i	−0.249011	−	0.968501i	\(-0.580106\pi\)
							0.367816	+	0.929899i	\(0.380106\pi\)
\(41\)	75151.3	+	231292.i	0.170291	+	0.524103i	0.999387	−	0.0350034i	\(-0.0111442\pi\)
							−0.829096	+	0.559107i	\(0.811144\pi\)
\(43\)		−	485890.i		−	0.931962i	−0.884795	−	0.465981i	\(-0.845701\pi\)
							0.884795	−	0.465981i	\(-0.154299\pi\)
\(47\)	510842.	−	703114.i	0.717702	−	0.987832i	−0.281895	−	0.959445i	\(-0.590963\pi\)
							0.999597	−	0.0283868i	\(-0.00903700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.16	✓	64
25.12	odd	20		625.8.a.g.1.4		64
25.13	odd	20		625.8.a.g.1.61		64
25.19	even	10	inner	25.8.e.a.19.16	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.16	✓	64	1.1	even	1	trivial
25.8.e.a.19.16	yes	64	25.19	even	10	inner
625.8.a.g.1.4		64	25.12	odd	20
625.8.a.g.1.61		64	25.13	odd	20