Newform orbit 840.2.da.b

• Label: 840.2.da.b
• Level: $840$
• Weight: $2$
• Character orbit: 840.da
• Analytic conductor: $6.707$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.da (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.70743376979\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 3 q^{3} + 3 q^{5} - q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
89.1		0		−1.65811	−	0.500660i		0		2.06804	−	0.850426i		0		0.514896	+	2.59517i		0		2.49868	+	1.66030i		0
89.2		0		−1.65460	+	0.512140i		0		−2.23332	−	0.110759i		0		−1.74016	−	1.99295i		0		2.47543	−	1.69478i		0
89.3		0		−1.64704	+	0.535960i		0		0.720646	+	2.11676i		0		2.62364	−	0.341357i		0		2.42549	−	1.76550i		0
89.4		0		−1.52830	−	0.815040i		0		−1.18105	+	1.89872i		0		−0.693447	+	2.55326i		0		1.67142	+	2.49126i		0
89.5		0		−1.28768	+	1.15840i		0		2.19349	−	0.434282i		0		−2.62364	+	0.341357i		0		0.316218	−	2.98329i		0
89.6		0		−1.27083	+	1.17686i		0		−1.21258	−	1.87873i		0		1.74016	+	1.99295i		0		0.230007	−	2.99117i		0
89.7		0		−1.11699	−	1.32375i		0		2.19487	+	0.427254i		0		−0.571108	−	2.58338i		0		−0.504648	+	2.95725i		0
89.8		0		−1.03561	−	1.38835i		0		−1.55313	−	1.60866i		0		−2.47935	+	0.923490i		0		−0.855021	+	2.87558i		0
89.9		0		−0.833236	−	1.51846i		0		−1.98193	+	1.03534i		0		2.34610	−	1.22303i		0		−1.61144	+	2.53047i		0
89.10		0		−0.395473	+	1.68630i		0		0.297527	+	2.21619i		0		−0.514896	−	2.59517i		0		−2.68720	−	1.33377i		0
89.11		0		−0.0583061	+	1.73107i		0		1.05381	−	1.97218i		0		0.693447	−	2.55326i		0		−2.99320	−	0.201864i		0
89.12		0		0.204814	−	1.71990i		0		1.77181	+	1.36407i		0		2.55997	+	0.668258i		0		−2.91610	−	0.704519i		0
89.13		0		0.381836	−	1.68944i		0		0.955903	−	2.02145i		0		−1.96308	−	1.77379i		0		−2.70840	−	1.29018i		0
89.14		0		0.519994	−	1.65215i		0		−0.113055	+	2.23321i		0		−2.32178	+	1.26861i		0		−2.45921	−	1.71822i		0
89.15		0		0.587907	+	1.62922i		0		1.46745	+	1.68719i		0		0.571108	+	2.58338i		0		−2.30873	+	1.91566i		0
89.16		0		0.684539	+	1.59104i		0		−2.16971	−	0.540722i		0		2.47935	−	0.923490i		0		−2.06281	+	2.17826i		0
89.17		0		0.898406	+	1.48083i		0		−0.0943325	−	2.23408i		0		−2.34610	+	1.22303i		0		−1.38573	+	2.66078i		0
89.18		0		1.07500	−	1.35808i		0		−1.98178	−	1.03564i		0		1.54613	−	2.14697i		0		−0.688739	−	2.91987i		0
89.19		0		1.40058	−	1.01901i		0		1.36472	−	1.77131i		0		1.08470	+	2.41318i		0		0.923253	−	2.85440i		0
89.20		0		1.58278	−	0.703435i		0		−0.851635	+	2.06754i		0		−1.08470	−	2.41318i		0		2.01036	−	2.22676i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
15.d	odd	2	1	inner
105.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	840.2.da.b	yes	48
3.b	odd	2	1		840.2.da.a	✓	48
5.b	even	2	1		840.2.da.a	✓	48
7.d	odd	6	1	inner	840.2.da.b	yes	48
15.d	odd	2	1	inner	840.2.da.b	yes	48
21.g	even	6	1		840.2.da.a	✓	48
35.i	odd	6	1		840.2.da.a	✓	48
105.p	even	6	1	inner	840.2.da.b	yes	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.2.da.a	✓	48	3.b	odd	2	1
840.2.da.a	✓	48	5.b	even	2	1
840.2.da.a	✓	48	21.g	even	6	1
840.2.da.a	✓	48	35.i	odd	6	1
840.2.da.b	yes	48	1.a	even	1	1	trivial
840.2.da.b	yes	48	7.d	odd	6	1	inner
840.2.da.b	yes	48	15.d	odd	2	1	inner
840.2.da.b	yes	48	105.p	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T17^24 - 97*T17^22 + 5953*T17^20 - 1464*T17^19 - 220468*T17^18 + 108984*T17^17 + 5914686*T17^16 - 5511864*T17^15 - 109040712*T17^14 + 144729000*T17^13 + 1478896324*T17^12 - 2695613040*T17^11 - 13042753912*T17^10 + 28999424928*T17^9 + 83837291088*T17^8 - 222282256128*T17^7 - 316308049792*T17^6 + 1015879690752*T17^5 + 826024151808*T17^4 - 3092528474112*T17^3 - 229058189312*T17^2 + 2857742684160*T17 + 1695412326400