Embedded newform 825.2.v.a.379.2

• Label: 825.2.v.a.379.2
• Level: $825$
• Weight: $2$
• Character: 825.379
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.v (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			379.2
Character	\(\chi\)	\(=\)	825.379
Dual form			825.2.v.a.394.59

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.70542i q^{2} +(0.951057 - 0.309017i) q^{3} -5.31930 q^{4} +(-1.72411 - 1.42388i) q^{5} +(-0.836021 - 2.57301i) q^{6} +(-2.63011 - 3.62003i) q^{7} +8.98009i q^{8} +(0.809017 - 0.587785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 240 q^{4} - 4 q^{5} + 4 q^{6} - 20 q^{7} + 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.70542i		−	1.91302i	−0.291698	−	0.956510i	\(-0.594220\pi\)
							0.291698	−	0.956510i	\(-0.405780\pi\)
\(3\)	0.951057	−	0.309017i	0.549093	−	0.178411i
							\(5\)	−1.72411	−	1.42388i	−0.771045	−	0.636781i
\(7\)	−2.63011	−	3.62003i	−0.994087	−	1.36824i								−0.928884	−	0.370371i	\(-0.879231\pi\)
							−0.0652030	−	0.997872i	\(-0.520769\pi\)
\(11\)	−2.91702	−	1.57829i	−0.879515	−	0.475872i
							\(13\)	0.891762	+	1.22740i	0.247330	+	0.340421i	0.914574	−	0.404419i	\(-0.132526\pi\)
−0.667244	+	0.744839i	\(0.732526\pi\)
\(17\)	4.63842	−	1.50711i	1.12498	−	0.365529i	0.313315	−	0.949649i	\(-0.398560\pi\)
							0.811667	+	0.584120i	\(0.198560\pi\)
\(19\)	5.11028			1.17238			0.586190	−	0.810174i	\(-0.300627\pi\)
							0.586190	+	0.810174i	\(0.300627\pi\)
\(23\)	0.512286	+	0.166452i	0.106819	+	0.0347076i	0.361939	−	0.932202i	\(-0.382115\pi\)
							−0.255120	+	0.966909i	\(0.582115\pi\)
\(29\)	−7.00553			−1.30089			−0.650447	−	0.759551i	\(-0.725419\pi\)
							−0.650447	+	0.759551i	\(0.725419\pi\)
\(31\)	−2.16918	−	1.57600i	−0.389596	−	0.283058i	0.375694	−	0.926744i	\(-0.377404\pi\)
							−0.765290	+	0.643686i	\(0.777404\pi\)
\(37\)	−2.29110	+	3.15343i	−0.376655	+	0.518421i	−0.954694	−	0.297588i	\(-0.903818\pi\)
							0.578040	+	0.816009i	\(0.303818\pi\)
\(41\)	2.14648	−	6.60618i	0.335224	−	1.03171i	−0.631388	−	0.775467i	\(-0.717514\pi\)
							0.966612	−	0.256246i	\(-0.0824856\pi\)
\(43\)			7.24053i			1.10417i	0.833788	+	0.552085i	\(0.186168\pi\)
							−0.833788	+	0.552085i	\(0.813832\pi\)
\(47\)	1.61413	−	0.524464i	0.235445	−	0.0765009i	−0.188918	−	0.981993i	\(-0.560498\pi\)
							0.424363	+	0.905492i	\(0.360498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.v.a.379.2	✓	240
11.9	even	5		825.2.bv.a.229.2	yes	240
25.19	even	10		825.2.bv.a.544.2	yes	240
275.119	even	10	inner	825.2.v.a.394.59	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.v.a.379.2	✓	240	1.1	even	1	trivial
825.2.v.a.394.59	yes	240	275.119	even	10	inner
825.2.bv.a.229.2	yes	240	11.9	even	5
825.2.bv.a.544.2	yes	240	25.19	even	10