Newform orbit 760.2.p.i

• Label: 760.2.p.i
• Level: $760$
• Weight: $2$
• Character orbit: 760.p
• Analytic conductor: $6.069$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.06863055362\)
Analytic rank:	\(0\)
Dimension:	\(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 16 q^{4} - 8 q^{6} + 88 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
379.1	−1.39575	−	0.227746i	1.92999			1.89626	+	0.635755i	−0.518150	−	2.17521i	−2.69379	−	0.439547i	−3.45334			−2.50193	−	1.31922i	0.724849			0.227817	+	3.15406i
379.2	−1.39575	−	0.227746i	1.92999			1.89626	+	0.635755i	0.518150	−	2.17521i	−2.69379	−	0.439547i	3.45334			−2.50193	−	1.31922i	0.724849			−1.21861	+	2.91805i
379.3	−1.39575	+	0.227746i	1.92999			1.89626	−	0.635755i	−0.518150	+	2.17521i	−2.69379	+	0.439547i	−3.45334			−2.50193	+	1.31922i	0.724849			0.227817	−	3.15406i
379.4	−1.39575	+	0.227746i	1.92999			1.89626	−	0.635755i	0.518150	+	2.17521i	−2.69379	+	0.439547i	3.45334			−2.50193	+	1.31922i	0.724849			−1.21861	−	2.91805i
379.5	−1.39359	−	0.240666i	−0.258346			1.88416	+	0.670777i	−1.58207	+	1.58020i	0.360027	+	0.0621750i	1.26592			−2.46430	−	1.38824i	−2.93326			2.58506	−	1.82140i
379.6	−1.39359	−	0.240666i	−0.258346			1.88416	+	0.670777i	1.58207	+	1.58020i	0.360027	+	0.0621750i	−1.26592			−2.46430	−	1.38824i	−2.93326			−1.82445	−	2.58290i
379.7	−1.39359	+	0.240666i	−0.258346			1.88416	−	0.670777i	−1.58207	−	1.58020i	0.360027	−	0.0621750i	1.26592			−2.46430	+	1.38824i	−2.93326			2.58506	+	1.82140i
379.8	−1.39359	+	0.240666i	−0.258346			1.88416	−	0.670777i	1.58207	−	1.58020i	0.360027	−	0.0621750i	−1.26592			−2.46430	+	1.38824i	−2.93326			−1.82445	+	2.58290i
379.9	−1.31112	−	0.530067i	−2.23972			1.43806	+	1.38996i	−1.59332	−	1.56886i	2.93653	+	1.18720i	2.42178			−1.14869	−	2.58467i	2.01633			1.25743	+	2.90153i
379.10	−1.31112	−	0.530067i	−2.23972			1.43806	+	1.38996i	1.59332	−	1.56886i	2.93653	+	1.18720i	−2.42178			−1.14869	−	2.58467i	2.01633			−2.92063	+	1.21239i
379.11	−1.31112	+	0.530067i	−2.23972			1.43806	−	1.38996i	−1.59332	+	1.56886i	2.93653	−	1.18720i	2.42178			−1.14869	+	2.58467i	2.01633			1.25743	−	2.90153i
379.12	−1.31112	+	0.530067i	−2.23972			1.43806	−	1.38996i	1.59332	+	1.56886i	2.93653	−	1.18720i	−2.42178			−1.14869	+	2.58467i	2.01633			−2.92063	−	1.21239i
379.13	−1.23558	−	0.687996i	2.17214			1.05332	+	1.70015i	−2.19004	−	0.451368i	−2.68385	−	1.49442i	−2.03346			−0.131770	−	2.82536i	1.71818			2.39543	+	2.06444i
379.14	−1.23558	−	0.687996i	2.17214			1.05332	+	1.70015i	2.19004	−	0.451368i	−2.68385	−	1.49442i	2.03346			−0.131770	−	2.82536i	1.71818			−3.01651	−	0.949035i
379.15	−1.23558	+	0.687996i	2.17214			1.05332	−	1.70015i	−2.19004	+	0.451368i	−2.68385	+	1.49442i	−2.03346			−0.131770	+	2.82536i	1.71818			2.39543	−	2.06444i
379.16	−1.23558	+	0.687996i	2.17214			1.05332	−	1.70015i	2.19004	+	0.451368i	−2.68385	+	1.49442i	2.03346			−0.131770	+	2.82536i	1.71818			−3.01651	+	0.949035i
379.17	−0.757052	−	1.19452i	−2.97713			−0.853744	+	1.80862i	−2.06882	+	0.848517i	2.25384	+	3.55623i	1.42544			2.80676	−	0.349410i	5.86328			2.57977	+	1.82887i
379.18	−0.757052	−	1.19452i	−2.97713			−0.853744	+	1.80862i	2.06882	+	0.848517i	2.25384	+	3.55623i	−1.42544			2.80676	−	0.349410i	5.86328			−0.552636	−	3.11361i
379.19	−0.757052	+	1.19452i	−2.97713			−0.853744	−	1.80862i	−2.06882	−	0.848517i	2.25384	−	3.55623i	1.42544			2.80676	+	0.349410i	5.86328			2.57977	−	1.82887i
379.20	−0.757052	+	1.19452i	−2.97713			−0.853744	−	1.80862i	2.06882	−	0.848517i	2.25384	−	3.55623i	−1.42544			2.80676	+	0.349410i	5.86328			−0.552636	+	3.11361i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.d	odd	2	1	inner
19.b	odd	2	1	inner
40.e	odd	2	1	inner
95.d	odd	2	1	inner
152.b	even	2	1	inner
760.p	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	760.2.p.i	✓	56
5.b	even	2	1	inner	760.2.p.i	✓	56
8.d	odd	2	1	inner	760.2.p.i	✓	56
19.b	odd	2	1	inner	760.2.p.i	✓	56
40.e	odd	2	1	inner	760.2.p.i	✓	56
95.d	odd	2	1	inner	760.2.p.i	✓	56
152.b	even	2	1	inner	760.2.p.i	✓	56
760.p	even	2	1	inner	760.2.p.i	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
760.2.p.i	✓	56	1.a	even	1	1	trivial
760.2.p.i	✓	56	5.b	even	2	1	inner
760.2.p.i	✓	56	8.d	odd	2	1	inner
760.2.p.i	✓	56	19.b	odd	2	1	inner
760.2.p.i	✓	56	40.e	odd	2	1	inner
760.2.p.i	✓	56	95.d	odd	2	1	inner
760.2.p.i	✓	56	152.b	even	2	1	inner
760.2.p.i	✓	56	760.p	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3}^{14} - 32T_{3}^{12} + 410T_{3}^{10} - 2684T_{3}^{8} + 9440T_{3}^{6} - 16868T_{3}^{4} + 12112T_{3}^{2} - 736 \)

\( T_{7}^{14} - 40T_{7}^{12} + 605T_{7}^{10} - 4394T_{7}^{8} + 16400T_{7}^{6} - 31216T_{7}^{4} + 27968T_{7}^{2} - 8912 \)

T29^14 - 204*T29^12 + 16376*T29^10 - 657408*T29^8 + 13968384*T29^6 - 155406336*T29^4 + 837550080*T29^2 - 1679163392