Embedded newform 98.4.e.a.15.3

• Label: 98.4.e.a.15.3
• Level: $98$
• Weight: $4$
• Character: 98.15
• Analytic conductor: $5.782$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.78218718056\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			15.3
Character	\(\chi\)	\(=\)	98.15
Dual form			98.4.e.a.85.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.445042 + 1.94986i) q^{2} +(-3.36657 + 4.22154i) q^{3} +(-3.60388 - 1.73553i) q^{4} +(-13.2280 + 16.5873i) q^{5} +(-6.73314 - 8.44309i) q^{6} +(10.2119 - 15.4505i) q^{7} +(4.98792 - 6.25465i) q^{8} +(-0.479583 - 2.10119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 14 q^{2} - 5 q^{3} - 28 q^{4} + 12 q^{5} - 10 q^{6} - 7 q^{7} - 56 q^{8} - 98 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.445042	+	1.94986i	−0.157346	+	0.689378i
							\(3\)	−3.36657	+	4.22154i	−0.647897	+	0.812437i	−0.991966	−	0.126508i	\(-0.959623\pi\)
0.344069	+	0.938944i	\(0.388195\pi\)
\(5\)	−13.2280	+	16.5873i	−1.18314	+	1.48362i	−0.344630	+	0.938739i	\(0.611996\pi\)
							−0.838515	+	0.544878i	\(0.816576\pi\)
\(7\)	10.2119	−	15.4505i	0.551389	−	0.834248i
							\(11\)	6.38556	−	27.9770i	0.175029	−	0.766852i	−0.808850	−	0.588015i	\(-0.799910\pi\)
0.983879	−	0.178837i	\(-0.0572333\pi\)
\(13\)	0.320311	−	1.40337i	0.00683371	−	0.0299404i	−0.971396	−	0.237465i	\(-0.923684\pi\)
							0.978230	+	0.207524i	\(0.0665407\pi\)
\(17\)	−18.1620	+	8.74637i	−0.259114	+	0.124783i	−0.558931	−	0.829214i	\(-0.688788\pi\)
							0.299817	+	0.953997i	\(0.403074\pi\)
\(19\)	−126.961			−1.53299			−0.766496	−	0.642249i	\(-0.778002\pi\)
							−0.766496	+	0.642249i	\(0.778002\pi\)
\(23\)	118.409	+	57.0229i	1.07348	+	0.516961i	0.885227	−	0.465159i	\(-0.154003\pi\)
							0.188254	+	0.982120i	\(0.439717\pi\)
\(29\)	−2.19854	+	1.05876i	−0.0140779	+	0.00677956i	−0.440909	−	0.897552i	\(-0.645344\pi\)
							0.426832	+	0.904331i	\(0.359630\pi\)
\(31\)	−68.4295			−0.396461			−0.198231	−	0.980155i	\(-0.563520\pi\)
							−0.198231	+	0.980155i	\(0.563520\pi\)
\(37\)	−264.503	+	127.378i	−1.17524	+	0.565967i	−0.916521	−	0.399986i	\(-0.869015\pi\)
							−0.258720	+	0.965952i	\(0.583301\pi\)
\(41\)	−227.004	+	284.654i	−0.864684	+	1.08428i	0.130992	+	0.991383i	\(0.458184\pi\)
							−0.995676	+	0.0928958i	\(0.970388\pi\)
\(43\)	154.665	+	193.943i	0.548515	+	0.687816i	0.976388	−	0.216023i	\(-0.0693085\pi\)
							−0.427873	+	0.903839i	\(0.640737\pi\)
\(47\)	22.1731	−	97.1465i	0.0688143	−	0.301495i	−0.928796	−	0.370592i	\(-0.879155\pi\)
							0.997610	+	0.0690972i	\(0.0220119\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.e.a.15.3	✓	42
49.36	even	7	inner	98.4.e.a.85.3	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.e.a.15.3	✓	42	1.1	even	1	trivial
98.4.e.a.85.3	yes	42	49.36	even	7	inner