Embedded newform 149.2.d.a.5.5

• Label: 149.2.d.a.5.5
• 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			5.5
Character	\(\chi\)	\(=\)	149.5
Dual form			149.2.d.a.30.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.418691 + 0.566108i) q^{2} +(1.04210 + 0.471061i) q^{3} +(0.440470 + 1.43828i) q^{4} +(1.34433 + 0.350037i) q^{5} +(-0.702991 + 0.392714i) q^{6} +(-0.836535 + 0.675455i) q^{7} +(-2.32644 - 0.821982i) q^{8} +(-1.11794 - 1.27023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 396 q - 32 q^{2} - 35 q^{3} - 42 q^{4} - 27 q^{5} - 25 q^{6} - 27 q^{7} - 16 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.418691	+	0.566108i	−0.296059	+	0.400299i	−0.926716	−	0.375764i	\(-0.877380\pi\)
							0.630656	+	0.776062i	\(0.282786\pi\)
\(3\)	1.04210	+	0.471061i	0.601659	+	0.271967i	0.689637	−	0.724155i	\(-0.257770\pi\)
							−0.0879778	+	0.996122i	\(0.528040\pi\)
\(5\)	1.34433	+	0.350037i	0.601204	+	0.156541i	0.540115	−	0.841591i	\(-0.318381\pi\)
							0.0610885	+	0.998132i	\(0.480543\pi\)
\(7\)	−0.836535	+	0.675455i	−0.316180	+	0.255298i	−0.773205	−	0.634156i	\(-0.781348\pi\)
							0.457024	+	0.889454i	\(0.348915\pi\)
\(11\)	0.574677	−	1.87652i	0.173272	−	0.565791i	−0.826723	−	0.562609i	\(-0.809798\pi\)
							0.999995	−	0.00318264i	\(-0.00101307\pi\)
\(13\)	2.08649	+	1.99973i	0.578687	+	0.554627i	0.923281	−	0.384125i	\(-0.125497\pi\)
							−0.344594	+	0.938752i	\(0.611983\pi\)
\(17\)	2.62576	+	1.18692i	0.636840	+	0.287870i	0.704354	−	0.709848i	\(-0.251237\pi\)
							−0.0675144	+	0.997718i	\(0.521507\pi\)
\(19\)	−2.57672	−	0.219311i	−0.591140	−	0.0503134i	−0.214581	−	0.976706i	\(-0.568839\pi\)
							−0.376559	+	0.926393i	\(0.622893\pi\)
\(23\)	4.52378	−	4.33569i	0.943273	−	0.904054i	−0.0522442	−	0.998634i	\(-0.516637\pi\)
							0.995517	+	0.0945803i	\(0.0301509\pi\)
\(29\)	1.05772	+	1.71784i	0.196413	+	0.318996i	0.934665	−	0.355530i	\(-0.115700\pi\)
							−0.738252	+	0.674526i	\(0.764348\pi\)
\(31\)	1.36043	−	1.30387i	0.244340	−	0.234181i	−0.559790	−	0.828635i	\(-0.689118\pi\)
							0.804130	+	0.594453i	\(0.202632\pi\)
\(37\)	0.169893	−	0.554758i	0.0279302	−	0.0912017i	−0.941114	−	0.338090i	\(-0.890219\pi\)
							0.969044	+	0.246889i	\(0.0794082\pi\)
\(41\)	1.15924	−	0.783507i	0.181042	−	0.122363i	−0.466101	−	0.884731i	\(-0.654342\pi\)
							0.647144	+	0.762368i	\(0.275963\pi\)
\(43\)	−2.68428	−	5.32395i	−0.409349	−	0.811895i	−0.999988	−	0.00495874i	\(-0.998422\pi\)
							0.590639	−	0.806936i	\(-0.298876\pi\)
\(47\)	−6.70933	+	0.571048i	−0.978656	+	0.0832959i	−0.563203	−	0.826319i	\(-0.690431\pi\)
							−0.415453	+	0.909614i	\(0.636377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	149.2.d.a.5.5	✓	396
149.30	even	37	inner	149.2.d.a.30.5	yes	396

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.d.a.5.5	✓	396	1.1	even	1	trivial
149.2.d.a.30.5	yes	396	149.30	even	37	inner