Embedded newform 103.4.e.a.8.11

• Label: 103.4.e.a.8.11
• Level: $103$
• Weight: $4$
• Character: 103.8
• Analytic conductor: $6.077$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	103.e (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.07719673059\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(25\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			8.11
Character	\(\chi\)	\(=\)	103.8
Dual form			103.4.e.a.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27733 + 0.238775i) q^{2} +(0.631381 + 6.81368i) q^{3} +(-5.88522 + 2.27994i) q^{4} +(10.8721 + 14.3970i) q^{5} +(-2.43342 - 8.55257i) q^{6} +(8.48745 + 5.25521i) q^{7} +(15.8115 - 9.79009i) q^{8} +(-19.4873 + 3.64282i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 15 q^{2} - 19 q^{3} - 111 q^{4} - 17 q^{5} - 19 q^{6} - 7 q^{7} - 41 q^{8} - 194 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{5}{17}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27733	+	0.238775i	−0.451605	+	0.0844196i	−0.404638	−	0.914477i	\(-0.632603\pi\)
							−0.0469667	+	0.998896i	\(0.514955\pi\)
\(3\)	0.631381	+	6.81368i	0.121509	+	1.31129i	0.810164	+	0.586204i	\(0.199378\pi\)
							−0.688654	+	0.725090i	\(0.741798\pi\)
\(5\)	10.8721	+	14.3970i	0.972434	+	1.28771i	0.957685	+	0.287820i	\(0.0929304\pi\)
							0.0147494	+	0.999891i	\(0.495305\pi\)
\(7\)	8.48745	+	5.25521i	0.458279	+	0.283754i	0.736093	−	0.676880i	\(-0.236669\pi\)
							−0.277814	+	0.960635i	\(0.589610\pi\)
\(11\)	−27.8678	+	5.20940i	−0.763860	+	0.142790i	−0.551242	−	0.834345i	\(-0.685846\pi\)
							−0.212618	+	0.977135i	\(0.568199\pi\)
\(13\)	−4.10782	−	2.54345i	−0.0876388	−	0.0542636i	0.481913	−	0.876219i	\(-0.339942\pi\)
							−0.569552	+	0.821955i	\(0.692883\pi\)
\(17\)	−18.1873	+	63.9216i	−0.259474	+	0.911957i	0.717135	+	0.696935i	\(0.245453\pi\)
							−0.976609	+	0.215023i	\(0.931017\pi\)
\(19\)	12.0279	−	129.802i	0.145231	−	1.56729i	−0.541701	−	0.840571i	\(-0.682219\pi\)
							0.686932	−	0.726722i	\(-0.258957\pi\)
\(23\)	41.9092	+	7.83419i	0.379942	+	0.0710236i	0.370257	−	0.928930i	\(-0.379270\pi\)
							0.00968592	+	0.999953i	\(0.496917\pi\)
\(29\)	−69.4891	−	92.0184i	−0.444959	−	0.589220i	0.519165	−	0.854674i	\(-0.326243\pi\)
							−0.964124	+	0.265454i	\(0.914478\pi\)
\(31\)	77.7307	−	70.8609i	0.450350	−	0.410548i	−0.416486	−	0.909142i	\(-0.636738\pi\)
							0.866836	+	0.498594i	\(0.166150\pi\)
\(37\)	34.1345	+	68.5512i	0.151667	+	0.304588i	0.957947	−	0.286946i	\(-0.0926401\pi\)
							−0.806280	+	0.591534i	\(0.798522\pi\)
\(41\)	108.518	−	143.701i	0.413357	−	0.547374i	−0.542939	−	0.839772i	\(-0.682689\pi\)
							0.956297	+	0.292398i	\(0.0944533\pi\)
\(43\)	−57.8032	+	116.084i	−0.204998	+	0.411691i	−0.973673	−	0.227948i	\(-0.926798\pi\)
							0.768676	+	0.639639i	\(0.220916\pi\)
\(47\)	636.087			1.97410			0.987052	−	0.160404i	\(-0.0512796\pi\)
							0.987052	+	0.160404i	\(0.0512796\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.4.e.a.8.11	✓	400
103.13	even	17	inner	103.4.e.a.13.11	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.4.e.a.8.11	✓	400	1.1	even	1	trivial
103.4.e.a.13.11	yes	400	103.13	even	17	inner