Embedded newform 175.3.v.a.31.3

• Label: 175.3.v.a.31.3
• 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			31.3
Character	\(\chi\)	\(=\)	175.31
Dual form			175.3.v.a.96.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40308 + 2.66889i) q^{2} +(-0.888631 + 0.0933988i) q^{3} +(-0.930071 - 8.84903i) q^{4} +(2.13906 + 4.51934i) q^{5} +(1.88618 - 2.59610i) q^{6} +(-0.0558604 + 6.99978i) q^{7} +(14.2303 + 10.3389i) q^{8} +(-8.02239 + 1.70521i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 304 q - 3 q^{2} - 9 q^{3} + 69 q^{4} - 15 q^{5} - 18 q^{7} - 46 q^{8} - 105 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40308	+	2.66889i	−1.20154	+	1.33445i	−0.273542	+	0.961860i	\(0.588195\pi\)
							−0.927998	+	0.372586i	\(0.878471\pi\)
\(3\)	−0.888631	+	0.0933988i	−0.296210	+	0.0311329i	−0.251468	−	0.967866i	\(-0.580913\pi\)
							−0.0447422	+	0.998999i	\(0.514247\pi\)
\(5\)	2.13906	+	4.51934i	0.427812	+	0.903868i
							\(7\)	−0.0558604	+	6.99978i	−0.00798006	+	0.999968i
\(11\)	−1.37321	−	0.291885i	−0.124837	−	0.0265350i								0.145069	−	0.989421i	\(-0.453659\pi\)
							−0.269907	+	0.962887i	\(0.586993\pi\)
\(13\)	−1.54371	−	0.501581i	−0.118747	−	0.0385831i	0.249041	−	0.968493i	\(-0.419885\pi\)
							−0.367788	+	0.929910i	\(0.619885\pi\)
\(17\)	−0.339848	−	0.763311i	−0.0199911	−	0.0449007i	0.903274	−	0.429064i	\(-0.141157\pi\)
							−0.923265	+	0.384163i	\(0.874490\pi\)
\(19\)	−28.9680	−	3.04466i	−1.52463	−	0.160245i	−0.695185	−	0.718831i	\(-0.744678\pi\)
							−0.829447	+	0.558585i	\(0.811344\pi\)
\(23\)	10.1785	−	11.3043i	0.442543	−	0.491493i	−0.480065	−	0.877233i	\(-0.659387\pi\)
							0.922608	+	0.385740i	\(0.126054\pi\)
\(29\)	41.5910	−	30.2176i	1.43417	−	1.04199i	0.444951	−	0.895555i	\(-0.353221\pi\)
							0.989221	−	0.146432i	\(-0.0467789\pi\)
\(31\)	−7.97048	−	17.9020i	−0.257112	−	0.577484i	0.738158	−	0.674628i	\(-0.235696\pi\)
							−0.995270	+	0.0971441i	\(0.969029\pi\)
\(37\)	−42.5145	+	9.03674i	−1.14904	+	0.244236i	−0.742790	−	0.669525i	\(-0.766498\pi\)
							−0.406252	+	0.913761i	\(0.633164\pi\)
\(41\)	−53.0539	−	17.2383i	−1.29400	−	0.420445i	−0.420508	−	0.907289i	\(-0.638148\pi\)
							−0.873489	+	0.486844i	\(0.838148\pi\)
\(43\)	−44.9541			−1.04544			−0.522722	−	0.852503i	\(-0.675083\pi\)
							−0.522722	+	0.852503i	\(0.675083\pi\)
\(47\)	1.06791	−	2.39857i	0.0227215	−	0.0510333i	−0.901828	−	0.432095i	\(-0.857775\pi\)
							0.924550	+	0.381061i	\(0.124441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.3.v.a.31.3	✓	304
7.5	odd	6	inner	175.3.v.a.131.36	yes	304
25.21	even	5	inner	175.3.v.a.171.36	yes	304
175.96	odd	30	inner	175.3.v.a.96.3	yes	304

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.3.v.a.31.3	✓	304	1.1	even	1	trivial
175.3.v.a.96.3	yes	304	175.96	odd	30	inner
175.3.v.a.131.36	yes	304	7.5	odd	6	inner
175.3.v.a.171.36	yes	304	25.21	even	5	inner