Embedded newform 31.9.e.a.6.20

• Label: 31.9.e.a.6.20
• Level: $31$
• Weight: $9$
• Character: 31.6
• Analytic conductor: $12.629$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	31.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.6287369119\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			6.20
Character	\(\chi\)	\(=\)	31.6
Dual form			31.9.e.a.26.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+31.4642 q^{2} +(56.0387 - 32.3540i) q^{3} +733.997 q^{4} +(-236.886 + 410.298i) q^{5} +(1763.21 - 1017.99i) q^{6} +(-513.570 - 889.530i) q^{7} +15039.8 q^{8} +(-1186.94 + 2055.84i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 4 q^{2} - 84 q^{3} + 4128 q^{4} + 146 q^{5} + 7482 q^{6} + 272 q^{7} + 17656 q^{8} + 45088 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	31.4642			1.96651			0.983257	−	0.182224i	\(-0.0583297\pi\)
							0.983257	+	0.182224i	\(0.0583297\pi\)
\(3\)	56.0387	−	32.3540i	0.691836	−	0.399432i	−0.112463	−	0.993656i	\(-0.535874\pi\)
							0.804299	+	0.594224i	\(0.202541\pi\)
\(5\)	−236.886	+	410.298i	−0.379017	+	0.656477i	−0.990920	−	0.134456i	\(-0.957071\pi\)
							0.611902	+	0.790933i	\(0.290405\pi\)
\(7\)	−513.570	−	889.530i	−0.213898	−	0.370483i	0.739033	−	0.673669i	\(-0.235283\pi\)
							−0.952931	+	0.303186i	\(0.901950\pi\)
\(11\)	−6307.97	−	3641.91i	−0.430843	−	0.248747i	0.268863	−	0.963178i	\(-0.413352\pi\)
							−0.699706	+	0.714431i	\(0.746685\pi\)
\(13\)	−19409.9	−	11206.3i	−0.679596	−	0.392365i	0.120107	−	0.992761i	\(-0.461676\pi\)
							−0.799703	+	0.600396i	\(0.795010\pi\)
\(17\)	−4762.29	+	2749.51i	−0.0570190	+	0.0329200i	−0.528239	−	0.849096i	\(-0.677147\pi\)
							0.471219	+	0.882016i	\(0.343814\pi\)
\(19\)	−119729.	−	207377.i	−0.918725	−	1.59128i	−0.801354	−	0.598190i	\(-0.795887\pi\)
							−0.117370	−	0.993088i	\(-0.537446\pi\)
\(23\)			379220.i			1.35513i	0.735464	+	0.677563i	\(0.236964\pi\)
							−0.735464	+	0.677563i	\(0.763036\pi\)
\(29\)		−	330677.i		−	0.467533i	−0.972293	−	0.233767i	\(-0.924895\pi\)
							0.972293	−	0.233767i	\(-0.0751052\pi\)
\(31\)	714980.	−	584546.i	0.774189	−	0.632954i
							\(37\)	−2.10122e6	+	1.21314e6i	−1.12115	+	0.647299i	−0.941695	−	0.336467i	\(-0.890768\pi\)
−0.179459	+	0.983765i	\(0.557435\pi\)
\(41\)	741787.	−	1.28481e6i	0.262509	−	0.454679i	−0.704399	−	0.709804i	\(-0.748783\pi\)
							0.966908	+	0.255125i	\(0.0821168\pi\)
\(43\)	−4.63302e6	+	2.67487e6i	−1.35516	+	0.782401i	−0.988967	−	0.148139i	\(-0.952672\pi\)
							−0.366192	+	0.930539i	\(0.619339\pi\)
\(47\)	809541.			0.165900			0.0829502	−	0.996554i	\(-0.473566\pi\)
							0.0829502	+	0.996554i	\(0.473566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.9.e.a.6.20	✓	40
31.26	odd	6	inner	31.9.e.a.26.20	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.9.e.a.6.20	✓	40	1.1	even	1	trivial
31.9.e.a.26.20	yes	40	31.26	odd	6	inner