Newform orbit 806.2.p.a

• Label: 806.2.p.a
• Level: $806$
• Weight: $2$
• Character orbit: 806.p
• Analytic conductor: $6.436$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 806 = 2 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	806.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.43594240292\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 72 q^{4} - 32 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
25.1		−	1.00000i	−1.55265	−	2.68926i	−1.00000			−1.53397	−	0.885641i	−2.68926	+	1.55265i	0.110990	−	0.0640800i			1.00000i	−3.32142	+	5.75287i	−0.885641	+	1.53397i
25.2		−	1.00000i	−1.43304	−	2.48210i	−1.00000			3.06814	+	1.77139i	−2.48210	+	1.43304i	0.907135	−	0.523735i			1.00000i	−2.60720	+	4.51580i	1.77139	−	3.06814i
25.3		−	1.00000i	−1.12070	−	1.94111i	−1.00000			−0.653308	−	0.377188i	−1.94111	+	1.12070i	3.10111	−	1.79043i			1.00000i	−1.01195	+	1.75275i	−0.377188	+	0.653308i
25.4		−	1.00000i	−0.914439	−	1.58385i	−1.00000			0.898191	+	0.518571i	−1.58385	+	0.914439i	1.16748	−	0.674045i			1.00000i	−0.172397	+	0.298600i	0.518571	−	0.898191i
25.5		−	1.00000i	−0.912324	−	1.58019i	−1.00000			2.40386	+	1.38787i	−1.58019	+	0.912324i	−2.40655	+	1.38942i			1.00000i	−0.164669	+	0.285216i	1.38787	−	2.40386i
25.6		−	1.00000i	−0.881554	−	1.52690i	−1.00000			−3.66819	−	2.11783i	−1.52690	+	0.881554i	−4.08300	+	2.35732i			1.00000i	−0.0542761	+	0.0940089i	−2.11783	+	3.66819i
25.7		−	1.00000i	−0.603018	−	1.04446i	−1.00000			0.448331	+	0.258844i	−1.04446	+	0.603018i	−3.96469	+	2.28901i			1.00000i	0.772738	−	1.33842i	0.258844	−	0.448331i
25.8		−	1.00000i	−0.393646	−	0.681815i	−1.00000			−2.02594	−	1.16968i	−0.681815	+	0.393646i	2.89099	−	1.66912i			1.00000i	1.19009	−	2.06129i	−1.16968	+	2.02594i
25.9		−	1.00000i	0.0108365	+	0.0187694i	−1.00000			0.0881442	+	0.0508901i	0.0187694	−	0.0108365i	3.04253	−	1.75660i			1.00000i	1.49977	−	2.59767i	0.0508901	−	0.0881442i
25.10		−	1.00000i	0.0324824	+	0.0562611i	−1.00000			−3.06870	−	1.77172i	0.0562611	−	0.0324824i	−0.162307	+	0.0937079i			1.00000i	1.49789	−	2.59442i	−1.77172	+	3.06870i
25.11		−	1.00000i	0.420459	+	0.728257i	−1.00000			0.0804401	+	0.0464421i	0.728257	−	0.420459i	−1.61877	+	0.934598i			1.00000i	1.14643	−	1.98567i	0.0464421	−	0.0804401i
25.12		−	1.00000i	0.467052	+	0.808958i	−1.00000			0.189943	+	0.109664i	0.808958	−	0.467052i	−1.82514	+	1.05374i			1.00000i	1.06372	−	1.84242i	0.109664	−	0.189943i
25.13		−	1.00000i	0.588284	+	1.01894i	−1.00000			3.05170	+	1.76190i	1.01894	−	0.588284i	2.99583	−	1.72964i			1.00000i	0.807845	−	1.39923i	1.76190	−	3.05170i
25.14		−	1.00000i	1.09866	+	1.90293i	−1.00000			2.80546	+	1.61973i	1.90293	−	1.09866i	−3.53929	+	2.04341i			1.00000i	−0.914088	+	1.58325i	1.61973	−	2.80546i
25.15		−	1.00000i	1.11557	+	1.93223i	−1.00000			2.14566	+	1.23880i	1.93223	−	1.11557i	0.635666	−	0.367002i			1.00000i	−0.989008	+	1.71301i	1.23880	−	2.14566i
25.16		−	1.00000i	1.23318	+	2.13593i	−1.00000			−3.08536	−	1.78133i	2.13593	−	1.23318i	1.41150	−	0.814928i			1.00000i	−1.54148	+	2.66992i	−1.78133	+	3.08536i
25.17		−	1.00000i	1.25746	+	2.17799i	−1.00000			−1.80336	−	1.04117i	2.17799	−	1.25746i	−2.12616	+	1.22754i			1.00000i	−1.66242	+	2.87940i	−1.04117	+	1.80336i
25.18		−	1.00000i	1.58738	+	2.74943i	−1.00000			0.658964	+	0.380453i	2.74943	−	1.58738i	2.59665	−	1.49918i			1.00000i	−3.53957	+	6.13071i	0.380453	−	0.658964i
25.19			1.00000i	−1.55265	−	2.68926i	−1.00000			1.53397	+	0.885641i	2.68926	−	1.55265i	−0.110990	+	0.0640800i		−	1.00000i	−3.32142	+	5.75287i	−0.885641	+	1.53397i
25.20			1.00000i	−1.43304	−	2.48210i	−1.00000			−3.06814	−	1.77139i	2.48210	−	1.43304i	−0.907135	+	0.523735i		−	1.00000i	−2.60720	+	4.51580i	1.77139	−	3.06814i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
31.c	even	3	1	inner
403.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	806.2.p.a	✓	72
13.b	even	2	1	inner	806.2.p.a	✓	72
31.c	even	3	1	inner	806.2.p.a	✓	72
403.l	even	6	1	inner	806.2.p.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
806.2.p.a	✓	72	1.a	even	1	1	trivial
806.2.p.a	✓	72	13.b	even	2	1	inner
806.2.p.a	✓	72	31.c	even	3	1	inner
806.2.p.a	✓	72	403.l	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).