Embedded newform 175.4.x.a.103.19

• Label: 175.4.x.a.103.19
• Level: $175$
• Weight: $4$
• Character: 175.103
• Analytic conductor: $10.325$
• Analytic rank: $0$
• Dimension: $928$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3253342510\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(58\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			103.19
Character	\(\chi\)	\(=\)	175.103
Dual form			175.4.x.a.17.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60067 - 0.136295i) q^{2} +(4.66596 + 5.76198i) q^{3} +(-1.21127 - 0.127309i) q^{4} +(1.25058 + 11.1102i) q^{5} +(-11.3493 - 15.6209i) q^{6} +(-1.27366 + 18.4764i) q^{7} +(23.7102 + 3.75532i) q^{8} +(-5.81561 + 27.3603i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 928 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 24 q^{7} + 84 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60067	−	0.136295i	−0.919476	−	0.0481877i	−0.413278	−	0.910605i	\(-0.635616\pi\)
							−0.506198	+	0.862417i	\(0.668949\pi\)
\(3\)	4.66596	+	5.76198i	0.897964	+	1.10889i	0.993541	+	0.113471i	\(0.0361969\pi\)
							−0.0955773	+	0.995422i	\(0.530470\pi\)
\(5\)	1.25058	+	11.1102i	0.111855	+	0.993725i
							\(7\)	−1.27366	+	18.4764i	−0.0687714	+	0.997632i
\(11\)	−17.0160	+	3.61685i	−0.466409	+	0.0991384i								−0.435121	−	0.900372i	\(-0.643294\pi\)
							−0.0312884	+	0.999510i	\(0.509961\pi\)
\(13\)	19.8203	+	10.0990i	0.422860	+	0.215458i	0.652453	−	0.757829i	\(-0.273740\pi\)
							−0.229594	+	0.973287i	\(0.573740\pi\)
\(17\)	40.0328	+	104.289i	0.571140	+	1.48787i	0.849603	+	0.527423i	\(0.176842\pi\)
							−0.278463	+	0.960447i	\(0.589825\pi\)
\(19\)	−10.2874	−	97.8778i	−0.124215	−	1.18183i	−0.862042	−	0.506837i	\(-0.830815\pi\)
							0.737827	−	0.674990i	\(-0.235852\pi\)
\(23\)	8.15245	−	155.558i	0.0739089	−	1.41027i	−0.670187	−	0.742192i	\(-0.733786\pi\)
							0.744096	−	0.668073i	\(-0.232881\pi\)
\(29\)	−6.80071	+	9.36037i	−0.0435469	+	0.0599371i	−0.830235	−	0.557413i	\(-0.811794\pi\)
							0.786688	+	0.617350i	\(0.211794\pi\)
\(31\)	−10.0475	+	22.5671i	−0.0582126	+	0.130748i	−0.940310	−	0.340320i	\(-0.889465\pi\)
							0.882097	+	0.471068i	\(0.156131\pi\)
\(37\)	−271.906	+	176.578i	−1.20814	+	0.784575i	−0.981882	−	0.189493i	\(-0.939316\pi\)
							−0.226257	+	0.974068i	\(0.572649\pi\)
\(41\)	−277.574	+	90.1893i	−1.05731	+	0.343542i	−0.785534	−	0.618818i	\(-0.787612\pi\)
							−0.271778	+	0.962360i	\(0.587612\pi\)
\(43\)	245.716	+	245.716i	0.871426	+	0.871426i	0.992628	−	0.121202i	\(-0.0386749\pi\)
							−0.121202	+	0.992628i	\(0.538675\pi\)
\(47\)	−181.708	−	69.7512i	−0.563933	−	0.216474i	0.0596656	−	0.998218i	\(-0.480997\pi\)
							−0.623598	+	0.781745i	\(0.714330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.x.a.103.19	yes	928
7.3	odd	6	inner	175.4.x.a.3.40	✓	928
25.17	odd	20	inner	175.4.x.a.117.40	yes	928
175.17	even	60	inner	175.4.x.a.17.19	yes	928

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.x.a.3.40	✓	928	7.3	odd	6	inner
175.4.x.a.17.19	yes	928	175.17	even	60	inner
175.4.x.a.103.19	yes	928	1.1	even	1	trivial
175.4.x.a.117.40	yes	928	25.17	odd	20	inner