Newform orbit 1368.2.e.g

• Label: 1368.2.e.g
• Level: $1368$
• Weight: $2$
• Character orbit: 1368.e
• Analytic conductor: $10.924$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9235349965\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
379.1	−1.41195	−	0.0800606i		0		1.98718	+	0.226082i		−	1.12499i		0				4.22432i	−2.78769	−	0.478311i		0		−0.0900673	+	1.58842i
379.2	−1.41195	+	0.0800606i		0		1.98718	−	0.226082i			1.12499i		0			−	4.22432i	−2.78769	+	0.478311i		0		−0.0900673	−	1.58842i
379.3	−1.38840	−	0.268989i		0		1.85529	+	0.746926i			3.66131i		0			−	0.477575i	−2.37496	−	1.53608i		0		0.984850	−	5.08335i
379.4	−1.38840	+	0.268989i		0		1.85529	−	0.746926i		−	3.66131i		0				0.477575i	−2.37496	+	1.53608i		0		0.984850	+	5.08335i
379.5	−1.33983	−	0.452615i		0		1.59028	+	1.21285i		−	0.393257i		0				1.35370i	−1.58175	−	2.34480i		0		−0.177994	+	0.526897i
379.6	−1.33983	+	0.452615i		0		1.59028	−	1.21285i			0.393257i		0			−	1.35370i	−1.58175	+	2.34480i		0		−0.177994	−	0.526897i
379.7	−1.10595	−	0.881408i		0		0.446239	+	1.94958i			0.946513i		0			−	1.89799i	1.22486	−	2.54945i		0		0.834264	−	1.04679i
379.8	−1.10595	+	0.881408i		0		0.446239	−	1.94958i		−	0.946513i		0				1.89799i	1.22486	+	2.54945i		0		0.834264	+	1.04679i
379.9	−0.939794	−	1.05678i		0		−0.233575	+	1.98631i			1.70737i		0				4.15645i	2.31861	−	1.61989i		0		1.80431	−	1.60457i
379.10	−0.939794	+	1.05678i		0		−0.233575	−	1.98631i		−	1.70737i		0			−	4.15645i	2.31861	+	1.61989i		0		1.80431	+	1.60457i
379.11	−0.906554	−	1.08543i		0		−0.356321	+	1.96800i		−	3.19828i		0				0.607439i	2.45916	−	1.39734i		0		−3.47151	+	2.89941i
379.12	−0.906554	+	1.08543i		0		−0.356321	−	1.96800i			3.19828i		0			−	0.607439i	2.45916	+	1.39734i		0		−3.47151	−	2.89941i
379.13	−0.612280	−	1.27480i		0		−1.25023	+	1.56107i			4.23483i		0			−	3.84958i	2.75554	+	0.637979i		0		5.39856	−	2.59290i
379.14	−0.612280	+	1.27480i		0		−1.25023	−	1.56107i		−	4.23483i		0				3.84958i	2.75554	−	0.637979i		0		5.39856	+	2.59290i
379.15	−0.558288	−	1.29935i		0		−1.37663	+	1.45082i		−	1.71991i		0				0.342554i	2.65369	+	0.978748i		0		−2.23477	+	0.960208i
379.16	−0.558288	+	1.29935i		0		−1.37663	−	1.45082i			1.71991i		0			−	0.342554i	2.65369	−	0.978748i		0		−2.23477	−	0.960208i
379.17	−0.388306	−	1.35986i		0		−1.69844	+	1.05608i			1.84435i		0			−	2.17959i	2.09564	+	1.89955i		0		2.50805	−	0.716171i
379.18	−0.388306	+	1.35986i		0		−1.69844	−	1.05608i		−	1.84435i		0				2.17959i	2.09564	−	1.89955i		0		2.50805	+	0.716171i
379.19	−0.134537	−	1.40780i		0		−1.96380	+	0.378802i			2.61555i		0				4.81248i	0.797481	+	2.71367i		0		3.68217	−	0.351888i
379.20	−0.134537	+	1.40780i		0		−1.96380	−	0.378802i		−	2.61555i		0			−	4.81248i	0.797481	−	2.71367i		0		3.68217	+	0.351888i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
19.b	odd	2	1	inner
152.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1368.2.e.g		40
3.b	odd	2	1		456.2.e.a	✓	40
4.b	odd	2	1		5472.2.e.g		40
8.b	even	2	1		5472.2.e.g		40
8.d	odd	2	1	inner	1368.2.e.g		40
12.b	even	2	1		1824.2.e.a		40
19.b	odd	2	1	inner	1368.2.e.g		40
24.f	even	2	1		456.2.e.a	✓	40
24.h	odd	2	1		1824.2.e.a		40
57.d	even	2	1		456.2.e.a	✓	40
76.d	even	2	1		5472.2.e.g		40
152.b	even	2	1	inner	1368.2.e.g		40
152.g	odd	2	1		5472.2.e.g		40
228.b	odd	2	1		1824.2.e.a		40
456.l	odd	2	1		456.2.e.a	✓	40
456.p	even	2	1		1824.2.e.a		40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
456.2.e.a	✓	40	3.b	odd	2	1
456.2.e.a	✓	40	24.f	even	2	1
456.2.e.a	✓	40	57.d	even	2	1
456.2.e.a	✓	40	456.l	odd	2	1
1368.2.e.g		40	1.a	even	1	1	trivial
1368.2.e.g		40	8.d	odd	2	1	inner
1368.2.e.g		40	19.b	odd	2	1	inner
1368.2.e.g		40	152.b	even	2	1	inner
1824.2.e.a		40	12.b	even	2	1
1824.2.e.a		40	24.h	odd	2	1
1824.2.e.a		40	228.b	odd	2	1
1824.2.e.a		40	456.p	even	2	1
5472.2.e.g		40	4.b	odd	2	1
5472.2.e.g		40	8.b	even	2	1
5472.2.e.g		40	76.d	even	2	1
5472.2.e.g		40	152.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

T5^20 + 60*T5^18 + 1458*T5^16 + 18700*T5^14 + 138793*T5^12 + 617504*T5^10 + 1652556*T5^8 + 2583248*T5^6 + 2183776*T5^4 + 841216*T5^2 + 86528

T7^20 + 84*T7^18 + 2822*T7^16 + 48372*T7^14 + 448065*T7^12 + 2213664*T7^10 + 5604704*T7^8 + 6599936*T7^6 + 3033344*T7^4 + 552960*T7^2 + 32768