Embedded newform 149.2.d.a.5.7

• Label: 149.2.d.a.5.7
• Level: $149$
• Weight: $2$
• Character: 149.5
• Analytic conductor: $1.190$
• Analytic rank: $0$
• Dimension: $396$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	149.d (of order \(37\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.18977099012\)
Analytic rank:	\(0\)
Dimension:	\(396\)
Relative dimension:	\(11\) over \(\Q(\zeta_{37})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{37}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	149.5
Dual form			149.2.d.a.30.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.214479 - 0.289995i) q^{2} +(2.27274 + 1.02735i) q^{3} +(0.547550 + 1.78794i) q^{4} +(-2.84494 - 0.740764i) q^{5} +(0.785382 - 0.438741i) q^{6} +(0.796898 - 0.643450i) q^{7} +(1.31611 + 0.465011i) q^{8} +(2.12791 + 2.41777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 396 q - 32 q^{2} - 35 q^{3} - 42 q^{4} - 27 q^{5} - 25 q^{6} - 27 q^{7} - 16 q^{8} - 34 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/149\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{26}{37}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.214479	−	0.289995i	0.151660	−	0.205058i	−0.721603	−	0.692307i	\(-0.756594\pi\)
							0.873263	+	0.487249i	\(0.162000\pi\)
\(3\)	2.27274	+	1.02735i	1.31217	+	0.593138i	0.941933	−	0.335800i	\(-0.109006\pi\)
							0.370236	+	0.928938i	\(0.379277\pi\)
\(5\)	−2.84494	−	0.740764i	−1.27229	−	0.331280i	−0.446262	−	0.894902i	\(-0.647245\pi\)
							−0.826032	+	0.563623i	\(0.809407\pi\)
\(7\)	0.796898	−	0.643450i	0.301199	−	0.243201i	−0.465739	−	0.884922i	\(-0.654212\pi\)
							0.766938	+	0.641721i	\(0.221779\pi\)
\(11\)	0.286596	−	0.935834i	0.0864119	−	0.282165i	−0.902495	−	0.430700i	\(-0.858267\pi\)
							0.988907	+	0.148535i	\(0.0474559\pi\)
\(13\)	−2.81353	−	2.69655i	−0.780333	−	0.747889i	0.192023	−	0.981390i	\(-0.438495\pi\)
							−0.972356	+	0.233502i	\(0.924982\pi\)
\(17\)	−3.15742	−	1.42724i	−0.765786	−	0.346157i	−0.00911977	−	0.999958i	\(-0.502903\pi\)
							−0.756666	+	0.653801i	\(0.773173\pi\)
\(19\)	8.49570	+	0.723090i	1.94905	+	0.165888i	0.992196	−	0.124692i	\(-0.0397942\pi\)
							0.956851	+	0.290580i	\(0.0938482\pi\)
\(23\)	1.07497	−	1.03028i	0.224148	−	0.214828i	−0.571477	−	0.820618i	\(-0.693629\pi\)
							0.795625	+	0.605790i	\(0.207143\pi\)
\(29\)	−4.29620	−	6.97747i	−0.797784	−	1.29568i	−0.951524	−	0.307573i	\(-0.900483\pi\)
							0.153740	−	0.988111i	\(-0.450868\pi\)
\(31\)	−0.690023	+	0.661334i	−0.123932	+	0.118779i	−0.751351	−	0.659902i	\(-0.770598\pi\)
							0.627420	+	0.778681i	\(0.284111\pi\)
\(37\)	−2.60341	+	8.50103i	−0.427998	+	1.39756i	0.438657	+	0.898654i	\(0.355454\pi\)
							−0.866656	+	0.498907i	\(0.833735\pi\)
\(41\)	−5.08755	+	3.43859i	−0.794542	+	0.537017i	−0.888672	−	0.458543i	\(-0.848372\pi\)
							0.0941298	+	0.995560i	\(0.469993\pi\)
\(43\)	−1.27501	−	2.52883i	−0.194437	−	0.385642i	0.774640	−	0.632403i	\(-0.217931\pi\)
							−0.969077	+	0.246760i	\(0.920634\pi\)
\(47\)	2.33986	−	0.199151i	0.341303	−	0.0290492i	0.0870987	−	0.996200i	\(-0.472240\pi\)
							0.254204	+	0.967151i	\(0.418186\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	149.2.d.a.5.7	✓	396
149.30	even	37	inner	149.2.d.a.30.7	yes	396

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.d.a.5.7	✓	396	1.1	even	1	trivial
149.2.d.a.30.7	yes	396	149.30	even	37	inner