Embedded newform 98.4.e.b.15.2

• Label: 98.4.e.b.15.2
• 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			15.2
Character	\(\chi\)	\(=\)	98.15
Dual form			98.4.e.b.85.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.445042 - 1.94986i) q^{2} +(-2.69998 + 3.38567i) q^{3} +(-3.60388 - 1.73553i) q^{4} +(4.53800 - 5.69048i) q^{5} +(5.39996 + 6.77133i) q^{6} +(18.2297 + 3.26763i) q^{7} +(-4.98792 + 6.25465i) q^{8} +(1.83522 + 8.04061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 14 q^{2} - q^{3} - 28 q^{4} + 14 q^{5} + 2 q^{6} + 7 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.69998	+	3.38567i	−0.519611	+	0.651572i	−0.970526	−	0.240995i	\(-0.922526\pi\)
0.450915	+	0.892567i	\(0.351098\pi\)
\(5\)	4.53800	−	5.69048i	0.405891	−	0.508972i	−0.536309	−	0.844022i	\(-0.680182\pi\)
							0.942200	+	0.335050i	\(0.108753\pi\)
\(7\)	18.2297	+	3.26763i	0.984312	+	0.176435i
							\(11\)	14.6559	−	64.2118i	0.401721	−	1.76005i	−0.218708	−	0.975790i	\(-0.570184\pi\)
0.620429	−	0.784263i	\(-0.286959\pi\)
\(13\)	9.14784	−	40.0793i	0.195166	−	0.855076i	−0.778600	−	0.627521i	\(-0.784070\pi\)
							0.973765	−	0.227555i	\(-0.0730732\pi\)
\(17\)	68.4451	−	32.9614i	0.976492	−	0.470254i	0.123595	−	0.992333i	\(-0.460558\pi\)
							0.852897	+	0.522079i	\(0.174843\pi\)
\(19\)	60.1638			0.726448			0.363224	−	0.931702i	\(-0.381676\pi\)
							0.363224	+	0.931702i	\(0.381676\pi\)
\(23\)	28.7435	+	13.8421i	0.260584	+	0.125491i	0.559615	−	0.828753i	\(-0.310949\pi\)
							−0.299031	+	0.954243i	\(0.596663\pi\)
\(29\)	−121.625	+	58.5717i	−0.778802	+	0.375051i	−0.780667	−	0.624947i	\(-0.785121\pi\)
							0.00186496	+	0.999998i	\(0.499406\pi\)
\(31\)	71.4037			0.413693			0.206846	−	0.978373i	\(-0.433680\pi\)
							0.206846	+	0.978373i	\(0.433680\pi\)
\(37\)	−268.860	+	129.476i	−1.19460	+	0.575290i	−0.922132	−	0.386875i	\(-0.873555\pi\)
							−0.272469	+	0.962164i	\(0.587840\pi\)
\(41\)	−213.688	+	267.956i	−0.813962	+	1.02068i	0.185316	+	0.982679i	\(0.440669\pi\)
							−0.999278	+	0.0379971i	\(0.987902\pi\)
\(43\)	160.171	+	200.848i	0.568043	+	0.712303i	0.980022	−	0.198890i	\(-0.0637337\pi\)
							−0.411979	+	0.911193i	\(0.635162\pi\)
\(47\)	125.852	−	551.394i	0.390583	−	1.71126i	−0.272025	−	0.962290i	\(-0.587693\pi\)
							0.662608	−	0.748966i	\(-0.269450\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.e.b.15.2	✓	42
49.36	even	7	inner	98.4.e.b.85.2	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.e.b.15.2	✓	42	1.1	even	1	trivial
98.4.e.b.85.2	yes	42	49.36	even	7	inner