Embedded newform 175.3.v.a.31.1

• Label: 175.3.v.a.31.1
• Level: $175$
• Weight: $3$
• Character: 175.31
• Analytic conductor: $4.768$
• Analytic rank: $0$
• Dimension: $304$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	175.v (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.76840462631\)
Analytic rank:	\(0\)
Dimension:	\(304\)
Relative dimension:	\(38\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			31.1
Character	\(\chi\)	\(=\)	175.31
Dual form			175.3.v.a.96.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49483 + 2.77079i) q^{2} +(-4.88519 + 0.513454i) q^{3} +(-1.03498 - 9.84720i) q^{4} +(0.0370158 - 4.99986i) q^{5} +(10.7650 - 14.8168i) q^{6} +(-6.93002 + 0.987337i) q^{7} +(17.8011 + 12.9332i) q^{8} +(14.7981 - 3.14543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 304 q - 3 q^{2} - 9 q^{3} + 69 q^{4} - 15 q^{5} - 18 q^{7} - 46 q^{8} - 105 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49483	+	2.77079i	−1.24741	+	1.38539i	−0.354570	+	0.935030i	\(0.615373\pi\)
							−0.892845	+	0.450365i	\(0.851294\pi\)
\(3\)	−4.88519	+	0.513454i	−1.62840	+	0.171151i	−0.874223	−	0.485525i	\(-0.838629\pi\)
							−0.754173	+	0.656676i	\(0.771962\pi\)
\(5\)	0.0370158	−	4.99986i	0.00740317	−	0.999973i
							\(7\)	−6.93002	+	0.987337i	−0.990003	+	0.141048i
\(11\)	−14.4473	−	3.07086i	−1.31339	−	0.279169i								−0.502604	−	0.864517i	\(-0.667625\pi\)
							−0.810782	+	0.585348i	\(0.800958\pi\)
\(13\)	6.05830	+	1.96846i	0.466023	+	0.151420i	0.532610	−	0.846361i	\(-0.321211\pi\)
							−0.0665874	+	0.997781i	\(0.521211\pi\)
\(17\)	−9.06881	−	20.3689i	−0.533459	−	1.19817i	−0.956413	−	0.292018i	\(-0.905673\pi\)
							0.422954	−	0.906151i	\(-0.360993\pi\)
\(19\)	8.32547	+	0.875042i	0.438183	+	0.0460549i	0.321052	−	0.947062i	\(-0.395964\pi\)
							0.117131	+	0.993116i	\(0.462630\pi\)
\(23\)	−8.99906	+	9.99447i	−0.391264	+	0.434542i	−0.906306	−	0.422623i	\(-0.861110\pi\)
							0.515042	+	0.857165i	\(0.327776\pi\)
\(29\)	−13.9420	+	10.1295i	−0.480759	+	0.349292i	−0.801619	−	0.597835i	\(-0.796028\pi\)
							0.320861	+	0.947126i	\(0.396028\pi\)
\(31\)	21.3730	+	48.0046i	0.689453	+	1.54854i	0.829571	+	0.558402i	\(0.188585\pi\)
							−0.140118	+	0.990135i	\(0.544748\pi\)
\(37\)	12.3425	−	2.62348i	0.333581	−	0.0709048i	−0.0380764	−	0.999275i	\(-0.512123\pi\)
							0.371657	+	0.928370i	\(0.378790\pi\)
\(41\)	6.54459	+	2.12647i	0.159624	+	0.0518651i	0.387739	−	0.921769i	\(-0.373256\pi\)
							−0.228115	+	0.973634i	\(0.573256\pi\)
\(43\)	1.34859			0.0313625			0.0156813	−	0.999877i	\(-0.495008\pi\)
							0.0156813	+	0.999877i	\(0.495008\pi\)
\(47\)	−11.2304	+	25.2238i	−0.238944	+	0.536676i	−0.992719	−	0.120456i	\(-0.961564\pi\)
							0.753775	+	0.657133i	\(0.228231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.3.v.a.31.1	✓	304
7.5	odd	6	inner	175.3.v.a.131.38	yes	304
25.21	even	5	inner	175.3.v.a.171.38	yes	304
175.96	odd	30	inner	175.3.v.a.96.1	yes	304

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.3.v.a.31.1	✓	304	1.1	even	1	trivial
175.3.v.a.96.1	yes	304	175.96	odd	30	inner
175.3.v.a.131.38	yes	304	7.5	odd	6	inner
175.3.v.a.171.38	yes	304	25.21	even	5	inner