Embedded newform 37.10.c.a.10.19

• Label: 37.10.c.a.10.19
• Level: $37$
• Weight: $10$
• Character: 37.10
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.19
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.0930 - 19.2137i) q^{2} +(11.3281 + 19.6208i) q^{3} +(9.88860 + 17.1276i) q^{4} +(-837.494 - 1450.58i) q^{5} +502.652 q^{6} +(1133.95 + 1964.06i) q^{7} +11798.1 q^{8} +(9584.85 - 16601.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.0930	−	19.2137i	0.490248	−	0.849135i	−0.509689	−	0.860359i	\(-0.670240\pi\)
							0.999937	+	0.0112242i	\(0.00357285\pi\)
\(3\)	11.3281	+	19.6208i	0.0807441	+	0.139853i	0.903570	−	0.428441i	\(-0.140937\pi\)
							−0.822826	+	0.568294i	\(0.807604\pi\)
\(5\)	−837.494	−	1450.58i	−0.599262	−	1.03795i	−0.992930	−	0.118700i	\(-0.962127\pi\)
							0.393668	−	0.919253i	\(-0.371206\pi\)
\(7\)	1133.95	+	1964.06i	0.178506	+	0.309182i	0.941369	−	0.337379i	\(-0.109540\pi\)
							−0.762863	+	0.646560i	\(0.776207\pi\)
\(11\)	−76326.4			−1.57184			−0.785918	−	0.618330i	\(-0.787809\pi\)
							−0.785918	+	0.618330i	\(0.787809\pi\)
\(13\)	−27512.3	−	47652.7i	−0.267166	−	0.462746i	0.700963	−	0.713198i	\(-0.252754\pi\)
							−0.968129	+	0.250452i	\(0.919421\pi\)
\(17\)	−48889.0	+	84678.2i	−0.141968	+	0.245896i	−0.928238	−	0.371988i	\(-0.878676\pi\)
							0.786270	+	0.617884i	\(0.212010\pi\)
\(19\)	−467313.	−	809409.i	−0.822652	−	1.42488i	−0.903700	−	0.428165i	\(-0.859160\pi\)
							0.0810480	−	0.996710i	\(-0.474173\pi\)
\(23\)	−491714.			−0.366385			−0.183192	−	0.983077i	\(-0.558643\pi\)
							−0.183192	+	0.983077i	\(0.558643\pi\)
\(29\)	−6.25586e6			−1.64246			−0.821232	−	0.570594i	\(-0.806713\pi\)
							−0.821232	+	0.570594i	\(0.806713\pi\)
\(31\)	−1.29925e6			−0.252677			−0.126339	−	0.991987i	\(-0.540323\pi\)
							−0.126339	+	0.991987i	\(0.540323\pi\)
\(37\)	6.82166e6	+	9.13382e6i	0.598388	+	0.801207i
							\(41\)	−7.44106e6	−	1.28883e7i	−0.411252	−	0.712309i	0.583775	−	0.811915i	\(-0.301575\pi\)
−0.995027	+	0.0996065i	\(0.968242\pi\)
\(43\)	1.76223e7			0.786059			0.393029	−	0.919526i	\(-0.371427\pi\)
							0.393029	+	0.919526i	\(0.371427\pi\)
\(47\)	5.08884e7			1.52117			0.760586	−	0.649237i	\(-0.224912\pi\)
							0.760586	+	0.649237i	\(0.224912\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.19	✓	54
37.26	even	3	inner	37.10.c.a.26.19	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.19	✓	54	1.1	even	1	trivial
37.10.c.a.26.19	yes	54	37.26	even	3	inner