Embedded newform 98.4.e.a.15.4

• Label: 98.4.e.a.15.4
• 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			15.4
Character	\(\chi\)	\(=\)	98.15
Dual form			98.4.e.a.85.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.445042 + 1.94986i) q^{2} +(-0.145956 + 0.183024i) q^{3} +(-3.60388 - 1.73553i) q^{4} +(2.48380 - 3.11459i) q^{5} +(-0.291913 - 0.366047i) q^{6} +(16.9271 + 7.51489i) q^{7} +(4.98792 - 6.25465i) q^{8} +(5.99587 + 26.2696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 14 q^{2} - 5 q^{3} - 28 q^{4} + 12 q^{5} - 10 q^{6} - 7 q^{7} - 56 q^{8} - 98 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.145956	+	0.183024i	−0.0280893	+	0.0352229i	−0.795678	−	0.605720i	\(-0.792885\pi\)
0.767588	+	0.640943i	\(0.221457\pi\)
\(5\)	2.48380	−	3.11459i	0.222158	−	0.278578i	−0.658245	−	0.752804i	\(-0.728701\pi\)
							0.880403	+	0.474226i	\(0.157272\pi\)
\(7\)	16.9271	+	7.51489i	0.913977	+	0.405766i
							\(11\)	−7.45889	+	32.6795i	−0.204449	+	0.895750i	0.763739	+	0.645525i	\(0.223362\pi\)
−0.968188	+	0.250225i	\(0.919496\pi\)
\(13\)	−5.10166	+	22.3518i	−0.108842	+	0.476868i	0.890901	+	0.454197i	\(0.150074\pi\)
							−0.999743	+	0.0226703i	\(0.992783\pi\)
\(17\)	−5.52738	+	2.66184i	−0.0788580	+	0.0379760i	−0.472897	−	0.881118i	\(-0.656792\pi\)
							0.394039	+	0.919094i	\(0.371077\pi\)
\(19\)	8.33095			0.100592			0.0502961	−	0.998734i	\(-0.483984\pi\)
							0.0502961	+	0.998734i	\(0.483984\pi\)
\(23\)	31.7976	+	15.3129i	0.288272	+	0.138824i	0.572431	−	0.819953i	\(-0.306000\pi\)
							−0.284160	+	0.958777i	\(0.591715\pi\)
\(29\)	43.2931	−	20.8488i	0.277218	−	0.133501i	−0.290109	−	0.956993i	\(-0.593692\pi\)
							0.567327	+	0.823492i	\(0.307977\pi\)
\(31\)	58.6718			0.339928			0.169964	−	0.985450i	\(-0.445635\pi\)
							0.169964	+	0.985450i	\(0.445635\pi\)
\(37\)	151.348	−	72.8855i	0.672473	−	0.323846i	−0.0663005	−	0.997800i	\(-0.521120\pi\)
							0.738774	+	0.673954i	\(0.235405\pi\)
\(41\)	69.4252	−	87.0564i	0.264449	−	0.331608i	−0.631824	−	0.775112i	\(-0.717693\pi\)
							0.896272	+	0.443504i	\(0.146265\pi\)
\(43\)	−155.358	−	194.813i	−0.550973	−	0.690899i	0.425887	−	0.904776i	\(-0.359962\pi\)
							−0.976860	+	0.213878i	\(0.931391\pi\)
\(47\)	113.865	−	498.876i	0.353382	−	1.54827i	−0.415931	−	0.909396i	\(-0.636544\pi\)
							0.769313	−	0.638872i	\(-0.220598\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.e.a.15.4	✓	42
49.36	even	7	inner	98.4.e.a.85.4	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.e.a.15.4	✓	42	1.1	even	1	trivial
98.4.e.a.85.4	yes	42	49.36	even	7	inner