Embedded newform 825.2.s.a.479.98

• Label: 825.2.s.a.479.98
• Level: $825$
• Weight: $2$
• Character: 825.479
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $464$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(116\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			479.98
Character	\(\chi\)	\(=\)	825.479
Dual form			825.2.s.a.794.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.11925i q^{2} +(1.69678 + 0.347748i) q^{3} -2.49121 q^{4} +(-0.713406 + 2.11921i) q^{5} +(-0.736965 + 3.59590i) q^{6} +(-4.07113 + 2.95785i) q^{7} -1.04100i q^{8} +(2.75814 + 1.18011i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 460 q^{4} + 5 q^{6} + 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.11925i			1.49853i	0.662268	+	0.749267i	\(0.269594\pi\)
							−0.662268	+	0.749267i	\(0.730406\pi\)
\(3\)	1.69678	+	0.347748i	0.979638	+	0.200773i
							\(5\)	−0.713406	+	2.11921i	−0.319045	+	0.947740i
\(7\)	−4.07113	+	2.95785i	−1.53874	+	1.11796i								−0.587622	+	0.809136i	\(0.699936\pi\)
							−0.951119	+	0.308825i	\(0.900064\pi\)
\(11\)	3.05198	−	1.29825i	0.920205	−	0.391436i
							\(13\)	1.02337	−	0.743519i	0.283831	−	0.206215i	−0.436756	−	0.899580i	\(-0.643873\pi\)
0.720587	+	0.693365i	\(0.243873\pi\)
\(17\)	−1.21991	+	0.396374i	−0.295873	+	0.0961348i	−0.453192	−	0.891413i	\(-0.649715\pi\)
							0.157319	+	0.987548i	\(0.449715\pi\)
\(19\)		−	3.25014i		−	0.745633i	−0.927905	−	0.372817i	\(-0.878392\pi\)
							0.927905	−	0.372817i	\(-0.121608\pi\)
\(23\)	0.243965	−	0.750847i	0.0508702	−	0.156562i	−0.922394	−	0.386250i	\(-0.873770\pi\)
							0.973264	+	0.229687i	\(0.0737703\pi\)
\(29\)	0.339242			0.0629956			0.0314978	−	0.999504i	\(-0.489972\pi\)
							0.0314978	+	0.999504i	\(0.489972\pi\)
\(31\)	7.05670	+	5.12699i	1.26742	+	0.920835i	0.999097	−	0.0424944i	\(-0.0135305\pi\)
							0.268323	+	0.963329i	\(0.413530\pi\)
\(37\)	2.14684	−	2.95488i	0.352939	−	0.485779i	−0.595225	−	0.803559i	\(-0.702937\pi\)
							0.948165	+	0.317780i	\(0.102937\pi\)
\(41\)	−0.721915	+	2.22182i	−0.112744	+	0.346991i	−0.991470	−	0.130336i	\(-0.958394\pi\)
							0.878726	+	0.477327i	\(0.158394\pi\)
\(43\)	−7.35902			−1.12224			−0.561120	−	0.827734i	\(-0.689629\pi\)
							−0.561120	+	0.827734i	\(0.689629\pi\)
\(47\)	3.64021	+	11.2034i	0.530979	+	1.63418i	0.752181	+	0.658956i	\(0.229002\pi\)
							−0.221203	+	0.975228i	\(0.570998\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.s.a.479.98	yes	464
3.2	odd	2	inner	825.2.s.a.479.19	✓	464
11.2	odd	10		825.2.bu.a.629.19	yes	464
25.19	even	10		825.2.bu.a.644.98	yes	464
33.2	even	10		825.2.bu.a.629.98	yes	464
75.44	odd	10		825.2.bu.a.644.19	yes	464
275.244	odd	10	inner	825.2.s.a.794.98	yes	464
825.794	even	10	inner	825.2.s.a.794.19	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.s.a.479.19	✓	464	3.2	odd	2	inner
825.2.s.a.479.98	yes	464	1.1	even	1	trivial
825.2.s.a.794.19	yes	464	825.794	even	10	inner
825.2.s.a.794.98	yes	464	275.244	odd	10	inner
825.2.bu.a.629.19	yes	464	11.2	odd	10
825.2.bu.a.629.98	yes	464	33.2	even	10
825.2.bu.a.644.19	yes	464	75.44	odd	10
825.2.bu.a.644.98	yes	464	25.19	even	10