Embedded newform 149.2.d.a.5.2

• Label: 149.2.d.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	149.5
Dual form			149.2.d.a.30.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32476 + 1.79120i) q^{2} +(2.81711 + 1.27341i) q^{3} +(-0.867749 - 2.83350i) q^{4} +(0.938034 + 0.244245i) q^{5} +(-6.01294 + 3.35903i) q^{6} +(0.772686 - 0.623900i) q^{7} +(2.02371 + 0.715022i) q^{8} +(4.33249 + 4.92266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 396 q - 32 q^{2} - 35 q^{3} - 42 q^{4} - 27 q^{5} - 25 q^{6} - 27 q^{7} - 16 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32476	+	1.79120i	−0.936749	+	1.26657i	0.0269657	+	0.999636i	\(0.491416\pi\)
							−0.963715	+	0.266933i	\(0.913990\pi\)
\(3\)	2.81711	+	1.27341i	1.62646	+	0.735205i	0.999054	−	0.0434890i	\(-0.0138474\pi\)
							0.627404	+	0.778694i	\(0.284118\pi\)
\(5\)	0.938034	+	0.244245i	0.419502	+	0.109230i	0.455739	−	0.890113i	\(-0.349375\pi\)
							−0.0362372	+	0.999343i	\(0.511537\pi\)
\(7\)	0.772686	−	0.623900i	0.292048	−	0.235812i	−0.471032	−	0.882116i	\(-0.656119\pi\)
							0.763080	+	0.646304i	\(0.223686\pi\)
\(11\)	0.355662	−	1.16136i	0.107236	−	0.350163i	−0.886383	−	0.462954i	\(-0.846790\pi\)
							0.993619	+	0.112790i	\(0.0359789\pi\)
\(13\)	−4.41191	−	4.22847i	−1.22364	−	1.17277i	−0.979155	−	0.203114i	\(-0.934894\pi\)
							−0.244487	−	0.969653i	\(-0.578620\pi\)
\(17\)	0.371047	+	0.167724i	0.0899920	+	0.0406789i	0.456395	−	0.889777i	\(-0.349140\pi\)
							−0.366403	+	0.930456i	\(0.619411\pi\)
\(19\)	−7.69005	−	0.654520i	−1.76422	−	0.150157i	−0.842672	−	0.538427i	\(-0.819019\pi\)
							−0.921546	+	0.388270i	\(0.873073\pi\)
\(23\)	−2.24316	+	2.14989i	−0.467731	+	0.448284i	−0.888495	−	0.458886i	\(-0.848249\pi\)
							0.420764	+	0.907170i	\(0.361762\pi\)
\(29\)	1.72280	+	2.79801i	0.319916	+	0.519577i	0.970891	−	0.239522i	\(-0.0769908\pi\)
							−0.650974	+	0.759100i	\(0.725639\pi\)
\(31\)	5.88838	−	5.64356i	1.05758	−	1.01361i	0.0577017	−	0.998334i	\(-0.481623\pi\)
							0.999883	−	0.0152792i	\(-0.00486371\pi\)
\(37\)	1.16126	−	3.79191i	0.190910	−	0.623386i	−0.808482	−	0.588521i	\(-0.799711\pi\)
							0.999392	−	0.0348656i	\(-0.0111003\pi\)
\(41\)	0.181041	−	0.122362i	0.0282738	−	0.0191098i	−0.545756	−	0.837944i	\(-0.683758\pi\)
							0.574030	+	0.818834i	\(0.305379\pi\)
\(43\)	5.14640	+	10.2073i	0.784819	+	1.55660i	0.830096	+	0.557621i	\(0.188286\pi\)
							−0.0452769	+	0.998974i	\(0.514417\pi\)
\(47\)	10.5444	−	0.897458i	1.53805	−	0.130908i	0.715103	−	0.699019i	\(-0.246380\pi\)
							0.822952	+	0.568111i	\(0.192326\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	149.2.d.a.5.2	✓	396
149.30	even	37	inner	149.2.d.a.30.2	yes	396

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.d.a.5.2	✓	396	1.1	even	1	trivial
149.2.d.a.30.2	yes	396	149.30	even	37	inner