Newform orbit 728.2.cy.a

• Label: 728.2.cy.a
• Level: $728$
• Weight: $2$
• Character orbit: 728.cy
• Analytic conductor: $5.813$
• Analytic rank: $0$
• Dimension: $24$
• CM discriminant: -104
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	728.cy (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.81310926715\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 36 q^{9} - 48 q^{16} + 60 q^{25} - 24 q^{30} - 72 q^{35} - 144 q^{36} + 96 q^{42} + 60 q^{51} + 192 q^{64} - 108 q^{81}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
467.1	−0.707107	+	1.22474i	−2.87940	+	1.66242i	−1.00000	−	1.73205i	3.16808	+	1.82909i		−	4.70203i	−1.53961	+	2.15165i	2.82843			4.02728	−	6.97545i	−4.48035	+	2.58673i
467.2	−0.707107	+	1.22474i	−2.41774	+	1.39588i	−1.00000	−	1.73205i	−2.74412	−	1.58432i		−	3.94816i	0.710858	+	2.54847i	2.82843			2.39698	−	4.15170i	3.88077	−	2.24056i
467.3	−0.707107	+	1.22474i	−0.329276	+	0.190107i	−1.00000	−	1.73205i	−0.994871	−	0.574389i		−	0.537705i	1.85161	−	1.88985i	2.82843			−1.42772	+	2.47288i	1.40696	−	0.812308i
467.4	−0.707107	+	1.22474i	0.710443	−	0.410175i	−1.00000	−	1.73205i	−3.51340	−	2.02846i			1.16015i	−1.09357	−	2.40917i	2.82843			−1.16351	+	2.01526i	4.96870	−	2.86868i
467.5	−0.707107	+	1.22474i	2.16895	−	1.25225i	−1.00000	−	1.73205i	0.345317	+	0.199369i			3.54188i	2.63319	+	0.257522i	2.82843			1.63624	−	2.83404i	−0.488352	+	0.281950i
467.6	−0.707107	+	1.22474i	2.74702	−	1.58599i	−1.00000	−	1.73205i	3.73899	+	2.15871i			4.48586i	−2.56247	−	0.658612i	2.82843			3.53074	−	6.11541i	−5.28773	+	3.05287i
467.7	0.707107	−	1.22474i	−2.87940	+	1.66242i	−1.00000	−	1.73205i	−3.16808	−	1.82909i			4.70203i	1.53961	−	2.15165i	−2.82843			4.02728	−	6.97545i	−4.48035	+	2.58673i
467.8	0.707107	−	1.22474i	−2.41774	+	1.39588i	−1.00000	−	1.73205i	2.74412	+	1.58432i			3.94816i	−0.710858	−	2.54847i	−2.82843			2.39698	−	4.15170i	3.88077	−	2.24056i
467.9	0.707107	−	1.22474i	−0.329276	+	0.190107i	−1.00000	−	1.73205i	0.994871	+	0.574389i			0.537705i	−1.85161	+	1.88985i	−2.82843			−1.42772	+	2.47288i	1.40696	−	0.812308i
467.10	0.707107	−	1.22474i	0.710443	−	0.410175i	−1.00000	−	1.73205i	3.51340	+	2.02846i		−	1.16015i	1.09357	+	2.40917i	−2.82843			−1.16351	+	2.01526i	4.96870	−	2.86868i
467.11	0.707107	−	1.22474i	2.16895	−	1.25225i	−1.00000	−	1.73205i	−0.345317	−	0.199369i		−	3.54188i	−2.63319	−	0.257522i	−2.82843			1.63624	−	2.83404i	−0.488352	+	0.281950i
467.12	0.707107	−	1.22474i	2.74702	−	1.58599i	−1.00000	−	1.73205i	−3.73899	−	2.15871i		−	4.48586i	2.56247	+	0.658612i	−2.82843			3.53074	−	6.11541i	−5.28773	+	3.05287i
675.1	−0.707107	−	1.22474i	−2.87940	−	1.66242i	−1.00000	+	1.73205i	3.16808	−	1.82909i			4.70203i	−1.53961	−	2.15165i	2.82843			4.02728	+	6.97545i	−4.48035	−	2.58673i
675.2	−0.707107	−	1.22474i	−2.41774	−	1.39588i	−1.00000	+	1.73205i	−2.74412	+	1.58432i			3.94816i	0.710858	−	2.54847i	2.82843			2.39698	+	4.15170i	3.88077	+	2.24056i
675.3	−0.707107	−	1.22474i	−0.329276	−	0.190107i	−1.00000	+	1.73205i	−0.994871	+	0.574389i			0.537705i	1.85161	+	1.88985i	2.82843			−1.42772	−	2.47288i	1.40696	+	0.812308i
675.4	−0.707107	−	1.22474i	0.710443	+	0.410175i	−1.00000	+	1.73205i	−3.51340	+	2.02846i		−	1.16015i	−1.09357	+	2.40917i	2.82843			−1.16351	−	2.01526i	4.96870	+	2.86868i
675.5	−0.707107	−	1.22474i	2.16895	+	1.25225i	−1.00000	+	1.73205i	0.345317	−	0.199369i		−	3.54188i	2.63319	−	0.257522i	2.82843			1.63624	+	2.83404i	−0.488352	−	0.281950i
675.6	−0.707107	−	1.22474i	2.74702	+	1.58599i	−1.00000	+	1.73205i	3.73899	−	2.15871i		−	4.48586i	−2.56247	+	0.658612i	2.82843			3.53074	+	6.11541i	−5.28773	−	3.05287i
675.7	0.707107	+	1.22474i	−2.87940	−	1.66242i	−1.00000	+	1.73205i	−3.16808	+	1.82909i		−	4.70203i	1.53961	+	2.15165i	−2.82843			4.02728	+	6.97545i	−4.48035	−	2.58673i
675.8	0.707107	+	1.22474i	−2.41774	−	1.39588i	−1.00000	+	1.73205i	2.74412	−	1.58432i		−	3.94816i	−0.710858	+	2.54847i	−2.82843			2.39698	+	4.15170i	3.88077	+	2.24056i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.h	odd	2	1	CM by \(\Q(\sqrt{-26}) \)
7.d	odd	6	1	inner
8.d	odd	2	1	inner
13.b	even	2	1	inner
56.m	even	6	1	inner
91.s	odd	6	1	inner
728.cy	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	728.2.cy.a	✓	24
7.d	odd	6	1	inner	728.2.cy.a	✓	24
8.d	odd	2	1	inner	728.2.cy.a	✓	24
13.b	even	2	1	inner	728.2.cy.a	✓	24
56.m	even	6	1	inner	728.2.cy.a	✓	24
91.s	odd	6	1	inner	728.2.cy.a	✓	24
104.h	odd	2	1	CM	728.2.cy.a	✓	24
728.cy	even	6	1	inner	728.2.cy.a	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
728.2.cy.a	✓	24	1.a	even	1	1	trivial
728.2.cy.a	✓	24	7.d	odd	6	1	inner
728.2.cy.a	✓	24	8.d	odd	2	1	inner
728.2.cy.a	✓	24	13.b	even	2	1	inner
728.2.cy.a	✓	24	56.m	even	6	1	inner
728.2.cy.a	✓	24	91.s	odd	6	1	inner
728.2.cy.a	✓	24	104.h	odd	2	1	CM
728.2.cy.a	✓	24	728.cy	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^12 - 18*T3^10 + 243*T3^8 - 54*T3^7 - 1400*T3^6 + 486*T3^5 + 6039*T3^4 - 4374*T3^3 - 891*T3^2 + 1242*T3 + 529