Newform orbit 1320.2.w.f

• Label: 1320.2.w.f
• Level: $1320$
• Weight: $2$
• Character orbit: 1320.w
• Analytic conductor: $10.540$
• Analytic rank: $0$
• Dimension: $26$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1320.w (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5402530668\)
Analytic rank:	\(0\)
Dimension:	\(26\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 2 q^{4} - 2 q^{6} - 26 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
661.1	−1.39603	−	0.226081i		−	1.00000i	1.89777	+	0.631230i			1.00000i	−0.226081	+	1.39603i	3.33347			−2.50663	−	1.31026i	−1.00000			0.226081	−	1.39603i
661.2	−1.39603	+	0.226081i			1.00000i	1.89777	−	0.631230i		−	1.00000i	−0.226081	−	1.39603i	3.33347			−2.50663	+	1.31026i	−1.00000			0.226081	+	1.39603i
661.3	−1.34901	−	0.424482i		−	1.00000i	1.63963	+	1.14526i			1.00000i	−0.424482	+	1.34901i	−2.43629			−1.72573	−	2.24095i	−1.00000			0.424482	−	1.34901i
661.4	−1.34901	+	0.424482i			1.00000i	1.63963	−	1.14526i		−	1.00000i	−0.424482	−	1.34901i	−2.43629			−1.72573	+	2.24095i	−1.00000			0.424482	+	1.34901i
661.5	−1.17963	−	0.780052i			1.00000i	0.783038	+	1.84034i		−	1.00000i	0.780052	−	1.17963i	−4.54975			0.511869	−	2.78172i	−1.00000			−0.780052	+	1.17963i
661.6	−1.17963	+	0.780052i		−	1.00000i	0.783038	−	1.84034i			1.00000i	0.780052	+	1.17963i	−4.54975			0.511869	+	2.78172i	−1.00000			−0.780052	−	1.17963i
661.7	−0.812450	−	1.15755i			1.00000i	−0.679852	+	1.88090i		−	1.00000i	1.15755	−	0.812450i	3.29493			2.72959	−	0.741177i	−1.00000			−1.15755	+	0.812450i
661.8	−0.812450	+	1.15755i		−	1.00000i	−0.679852	−	1.88090i			1.00000i	1.15755	+	0.812450i	3.29493			2.72959	+	0.741177i	−1.00000			−1.15755	−	0.812450i
661.9	−0.658373	−	1.25162i		−	1.00000i	−1.13309	+	1.64806i			1.00000i	−1.25162	+	0.658373i	−0.525898			2.80874	+	0.333155i	−1.00000			1.25162	−	0.658373i
661.10	−0.658373	+	1.25162i			1.00000i	−1.13309	−	1.64806i		−	1.00000i	−1.25162	−	0.658373i	−0.525898			2.80874	−	0.333155i	−1.00000			1.25162	+	0.658373i
661.11	−0.151507	−	1.40607i			1.00000i	−1.95409	+	0.426059i		−	1.00000i	1.40607	−	0.151507i	−0.196844			0.895128	+	2.68305i	−1.00000			−1.40607	+	0.151507i
661.12	−0.151507	+	1.40607i		−	1.00000i	−1.95409	−	0.426059i			1.00000i	1.40607	+	0.151507i	−0.196844			0.895128	−	2.68305i	−1.00000			−1.40607	−	0.151507i
661.13	0.0534496	−	1.41320i		−	1.00000i	−1.99429	−	0.151070i			1.00000i	−1.41320	−	0.0534496i	4.19338			−0.320087	+	2.81026i	−1.00000			1.41320	+	0.0534496i
661.14	0.0534496	+	1.41320i			1.00000i	−1.99429	+	0.151070i		−	1.00000i	−1.41320	+	0.0534496i	4.19338			−0.320087	−	2.81026i	−1.00000			1.41320	−	0.0534496i
661.15	0.243769	−	1.39305i		−	1.00000i	−1.88115	−	0.679163i			1.00000i	−1.39305	−	0.243769i	−4.55351			−1.40467	+	2.45497i	−1.00000			1.39305	+	0.243769i
661.16	0.243769	+	1.39305i			1.00000i	−1.88115	+	0.679163i		−	1.00000i	−1.39305	+	0.243769i	−4.55351			−1.40467	−	2.45497i	−1.00000			1.39305	−	0.243769i
661.17	0.408364	−	1.35397i			1.00000i	−1.66648	−	1.10583i		−	1.00000i	1.35397	+	0.408364i	−0.893325			−2.17779	+	1.80478i	−1.00000			−1.35397	−	0.408364i
661.18	0.408364	+	1.35397i		−	1.00000i	−1.66648	+	1.10583i			1.00000i	1.35397	−	0.408364i	−0.893325			−2.17779	−	1.80478i	−1.00000			−1.35397	+	0.408364i
661.19	0.907577	−	1.08458i		−	1.00000i	−0.352608	−	1.96867i			1.00000i	−1.08458	−	0.907577i	4.28141			−2.45519	−	1.40429i	−1.00000			1.08458	+	0.907577i
661.20	0.907577	+	1.08458i			1.00000i	−0.352608	+	1.96867i		−	1.00000i	−1.08458	+	0.907577i	4.28141			−2.45519	+	1.40429i	−1.00000			1.08458	−	0.907577i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1320.2.w.f	✓	26
4.b	odd	2	1		5280.2.w.f		26
8.b	even	2	1	inner	1320.2.w.f	✓	26
8.d	odd	2	1		5280.2.w.f		26

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1320.2.w.f	✓	26	1.a	even	1	1	trivial
1320.2.w.f	✓	26	8.b	even	2	1	inner
5280.2.w.f		26	4.b	odd	2	1
5280.2.w.f		26	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1320, [\chi])\):

T7^13 - 58*T7^11 + 1220*T7^9 + 128*T7^8 - 11184*T7^7 - 3856*T7^6 + 41472*T7^5 + 28224*T7^4 - 31328*T7^3 - 35136*T7^2 - 10752*T7 - 1024

T23^13 - 136*T23^11 + 144*T23^10 + 6640*T23^9 - 11072*T23^8 - 150336*T23^7 + 316416*T23^6 + 1596672*T23^5 - 3996672*T23^4 - 6332416*T23^3 + 19709952*T23^2 - 442368*T23 - 15925248