Embedded newform 525.2.bh.a.13.34

• Label: 525.2.bh.a.13.34
• Level: $525$
• Weight: $2$
• Character: 525.13
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bh (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.34
Character	\(\chi\)	\(=\)	525.13
Dual form			525.2.bh.a.202.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.94527 - 0.308100i) q^{2} +(0.891007 + 0.453990i) q^{3} +(1.78703 - 0.580641i) q^{4} +(1.99948 + 1.00103i) q^{5} +(1.87312 + 0.608614i) q^{6} +(2.48889 - 0.897452i) q^{7} +(-0.212346 + 0.108196i) q^{8} +(0.587785 + 0.809017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.94527	−	0.308100i	1.37551	−	0.217860i	0.575480	−	0.817816i	\(-0.304815\pi\)
							0.800033	+	0.599956i	\(0.204815\pi\)
\(3\)	0.891007	+	0.453990i	0.514423	+	0.262112i
							\(5\)	1.99948	+	1.00103i	0.894196	+	0.447676i
\(7\)	2.48889	−	0.897452i	0.940712	−	0.339205i
							\(11\)	−4.54320	−	3.30083i	−1.36983	−	0.995237i	−0.997751	−	0.0670342i	\(-0.978646\pi\)
−0.372075	−	0.928203i	\(-0.621354\pi\)
\(13\)	−0.426435	+	2.69241i	−0.118272	+	0.746739i	0.855262	+	0.518196i	\(0.173396\pi\)
							−0.973534	+	0.228543i	\(0.926604\pi\)
\(17\)	−5.55853	+	2.83221i	−1.34814	+	0.686912i	−0.970963	−	0.239229i	\(-0.923105\pi\)
							−0.377178	+	0.926141i	\(0.623105\pi\)
\(19\)	1.53500	−	4.72425i	0.352153	−	1.08382i	−0.605489	−	0.795854i	\(-0.707022\pi\)
							0.957642	−	0.287962i	\(-0.0929778\pi\)
\(23\)	−0.912851	−	5.76351i	−0.190343	−	1.20178i	−0.879048	−	0.476734i	\(-0.841821\pi\)
							0.688705	−	0.725041i	\(-0.258179\pi\)
\(29\)	6.41675	−	2.08493i	1.19156	−	0.387161i	0.354911	−	0.934900i	\(-0.384511\pi\)
							0.836649	+	0.547739i	\(0.184511\pi\)
\(31\)	−2.33013	−	0.757105i	−0.418503	−	0.135980i	0.0921930	−	0.995741i	\(-0.470612\pi\)
							−0.510696	+	0.859761i	\(0.670612\pi\)
\(37\)	−0.502989	+	3.17575i	−0.0826909	+	0.522090i	0.911221	+	0.411917i	\(0.135141\pi\)
							−0.993912	+	0.110173i	\(0.964859\pi\)
\(41\)	2.26692	+	3.12015i	0.354034	+	0.487285i	0.948474	−	0.316854i	\(-0.102627\pi\)
							−0.594441	+	0.804139i	\(0.702627\pi\)
\(43\)	0.217118	+	0.217118i	0.0331102	+	0.0331102i	0.723468	−	0.690358i	\(-0.242547\pi\)
							−0.690358	+	0.723468i	\(0.742547\pi\)
\(47\)	2.01262	−	3.95000i	0.293571	−	0.576166i	−0.696364	−	0.717689i	\(-0.745200\pi\)
							0.989935	+	0.141523i	\(0.0451999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.13.34	yes	320
7.6	odd	2	inner	525.2.bh.a.13.33	✓	320
25.2	odd	20	inner	525.2.bh.a.202.33	yes	320
175.27	even	20	inner	525.2.bh.a.202.34	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.13.33	✓	320	7.6	odd	2	inner
525.2.bh.a.13.34	yes	320	1.1	even	1	trivial
525.2.bh.a.202.33	yes	320	25.2	odd	20	inner
525.2.bh.a.202.34	yes	320	175.27	even	20	inner