Embedded newform 75.9.k.a.37.34

• Label: 75.9.k.a.37.34
• Level: $75$
• Weight: $9$
• Character: 75.37
• Analytic conductor: $30.553$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.k (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(30.5533957546\)
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			37.34
Character	\(\chi\)	\(=\)	75.37
Dual form			75.9.k.a.73.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.74538 + 23.6474i) q^{2} +(-41.6683 - 21.2310i) q^{3} +(-301.702 + 98.0288i) q^{4} +(-47.5920 + 623.185i) q^{5} +(345.995 - 1064.86i) q^{6} +(1029.85 + 1029.85i) q^{7} +(-665.517 - 1306.15i) q^{8} +(1285.49 + 1769.32i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 444 q^{5} - 4540 q^{7} - 17460 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.74538	+	23.6474i	0.234086	+	1.47796i	0.772356	+	0.635190i	\(0.219078\pi\)
							−0.538269	+	0.842773i	\(0.680922\pi\)
\(3\)	−41.6683	−	21.2310i	−0.514423	−	0.262112i
							\(5\)	−47.5920	+	623.185i	−0.0761472	+	0.997097i
\(7\)	1029.85	+	1029.85i	0.428924	+	0.428924i								0.888262	−	0.459338i	\(-0.151913\pi\)
							−0.459338	+	0.888262i	\(0.651913\pi\)
\(11\)	−19957.0	−	14499.6i	−1.36309	−	0.990345i	−0.998241	−	0.0592791i	\(-0.981120\pi\)
							−0.364851	−	0.931066i	\(-0.618880\pi\)
\(13\)	−4436.01	+	28007.9i	−0.155317	+	0.980633i	0.779732	+	0.626114i	\(0.215356\pi\)
							−0.935049	+	0.354519i	\(0.884644\pi\)
\(17\)	−93471.8	+	47626.3i	−1.11914	+	0.570231i	−0.912865	−	0.408262i	\(-0.866135\pi\)
							−0.206277	+	0.978494i	\(0.566135\pi\)
\(19\)	−19515.7	−	6341.03i	−0.149751	−	0.0486570i	0.233182	−	0.972433i	\(-0.425086\pi\)
							−0.382933	+	0.923776i	\(0.625086\pi\)
\(23\)	452277.	−	71633.7i	1.61619	−	0.255980i	0.718151	−	0.695887i	\(-0.244989\pi\)
							0.898043	+	0.439907i	\(0.144989\pi\)
\(29\)	480360.	−	156078.i	0.679165	−	0.220674i	0.0509350	−	0.998702i	\(-0.483780\pi\)
							0.628230	+	0.778028i	\(0.283780\pi\)
\(31\)	390297.	−	1.20121e6i	0.422619	−	1.30069i	−0.482638	−	0.875820i	\(-0.660321\pi\)
							0.905256	−	0.424866i	\(-0.139679\pi\)
\(37\)	51523.2	+	8160.47i	0.0274913	+	0.00435420i	0.170165	−	0.985416i	\(-0.445570\pi\)
							−0.142674	+	0.989770i	\(0.545570\pi\)
\(41\)	−1.62107e6	+	1.17778e6i	−0.573677	+	0.416800i	−0.836439	−	0.548060i	\(-0.815366\pi\)
							0.262762	+	0.964861i	\(0.415366\pi\)
\(43\)	4.21747e6	−	4.21747e6i	1.23361	−	1.23361i	0.271045	−	0.962567i	\(-0.412631\pi\)
							0.962567	−	0.271045i	\(-0.0873691\pi\)
\(47\)	−3.09611e6	+	6.07646e6i	−0.634491	+	1.24526i	0.320113	+	0.947379i	\(0.396279\pi\)
							−0.954604	+	0.297879i	\(0.903721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.k.a.37.34	✓	320
25.23	odd	20	inner	75.9.k.a.73.34	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.k.a.37.34	✓	320	1.1	even	1	trivial
75.9.k.a.73.34	yes	320	25.23	odd	20	inner