Embedded newform 350.2.q.b.81.6

• Label: 350.2.q.b.81.6
• Level: $350$
• Weight: $2$
• Character: 350.81
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			81.6
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.b.121.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669131 + 0.743145i) q^{2} +(0.0698073 + 0.664172i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(-1.87316 + 1.22117i) q^{5} +(-0.540286 - 0.392541i) q^{6} +(2.57149 - 0.622454i) q^{7} +(0.809017 + 0.587785i) q^{8} +(2.49819 - 0.531007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.669131	+	0.743145i	−0.473147	+	0.525483i
							\(3\)	0.0698073	+	0.664172i	0.0403033	+	0.383460i	0.996017	+	0.0891623i	\(0.0284190\pi\)
−0.955714	+	0.294298i	\(0.904914\pi\)
\(5\)	−1.87316	+	1.22117i	−0.837705	+	0.546124i
							\(7\)	2.57149	−	0.622454i	0.971931	−	0.235265i
\(11\)	2.69149	+	0.572095i	0.811516	+	0.172493i								0.594937	−	0.803772i	\(-0.297177\pi\)
							0.216579	+	0.976265i	\(0.430510\pi\)
\(13\)	−1.13444	+	3.49144i	−0.314636	+	0.968352i	0.661267	+	0.750150i	\(0.270019\pi\)
							−0.975904	+	0.218201i	\(0.929981\pi\)
\(17\)	−5.85908	+	2.60863i	−1.42104	+	0.632686i	−0.966177	−	0.257878i	\(-0.916977\pi\)
							−0.454858	+	0.890564i	\(0.650310\pi\)
\(19\)	−0.520538	+	4.95259i	−0.119420	+	1.13620i	0.756584	+	0.653897i	\(0.226867\pi\)
							−0.876003	+	0.482305i	\(0.839800\pi\)
\(23\)	−1.50889	+	1.67579i	−0.314626	+	0.349427i	−0.879628	−	0.475662i	\(-0.842209\pi\)
							0.565002	+	0.825089i	\(0.308875\pi\)
\(29\)	−2.79726	+	2.03233i	−0.519438	+	0.377394i	−0.816392	−	0.577498i	\(-0.804029\pi\)
							0.296954	+	0.954892i	\(0.404029\pi\)
\(31\)	3.92903	−	1.74932i	0.705675	−	0.314187i	−0.0223378	−	0.999750i	\(-0.507111\pi\)
							0.728013	+	0.685564i	\(0.240444\pi\)
\(37\)	11.4292	−	2.42935i	1.87895	−	0.399383i	0.881589	−	0.472018i	\(-0.156474\pi\)
							0.997359	+	0.0726358i	\(0.0231411\pi\)
\(41\)	0.869484	−	2.67600i	0.135791	−	0.417921i	−0.859921	−	0.510426i	\(-0.829488\pi\)
							0.995712	+	0.0925056i	\(0.0294876\pi\)
\(43\)	−10.4334			−1.59107			−0.795537	−	0.605904i	\(-0.792811\pi\)
							−0.795537	+	0.605904i	\(0.792811\pi\)
\(47\)	7.77238	+	3.46049i	1.13372	+	0.504764i	0.885823	−	0.464023i	\(-0.153595\pi\)
							0.247895	+	0.968787i	\(0.420261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.81.6	✓	72
7.2	even	3	inner	350.2.q.b.331.4	yes	72
25.21	even	5	inner	350.2.q.b.221.4	yes	72
175.121	even	15	inner	350.2.q.b.121.6	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.81.6	✓	72	1.1	even	1	trivial
350.2.q.b.121.6	yes	72	175.121	even	15	inner
350.2.q.b.221.4	yes	72	25.21	even	5	inner
350.2.q.b.331.4	yes	72	7.2	even	3	inner