Newform orbit 648.7.e.c

• Label: 648.7.e.c
• Level: $648$
• Weight: $7$
• Character orbit: 648.e
• Analytic conductor: $149.075$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	648.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(149.075046186\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Twist minimal:	no (minimal twist has level 72)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
161.1		0			0			0			−	244.807i		0		397.030				0			0			0
161.2		0			0			0			−	208.701i		0		−347.432				0			0			0
161.3		0			0			0			−	201.898i		0		387.828				0			0			0
161.4		0			0			0			−	183.121i		0		−198.208				0			0			0
161.5		0			0			0			−	182.017i		0		−399.532				0			0			0
161.6		0			0			0			−	176.491i		0		−305.068				0			0			0
161.7		0			0			0			−	143.962i		0		−106.579				0			0			0
161.8		0			0			0			−	130.791i		0		287.184				0			0			0
161.9		0			0			0			−	107.663i		0		64.9668				0			0			0
161.10		0			0			0			−	105.872i		0		−476.866				0			0			0
161.11		0			0			0			−	100.993i		0		42.3502				0			0			0
161.12		0			0			0			−	99.7374i		0		415.322				0			0			0
161.13		0			0			0			−	87.7969i		0		575.780				0			0			0
161.14		0			0			0			−	56.6696i		0		452.974				0			0			0
161.15		0			0			0			−	50.0519i		0		−94.5066				0			0			0
161.16		0			0			0			−	30.2276i		0		−602.813				0			0			0
161.17		0			0			0			−	13.7535i		0		−348.446				0			0			0
161.18		0			0			0			−	11.4688i		0		256.017				0			0			0
161.19		0			0			0				11.4688i		0		256.017				0			0			0
161.20		0			0			0				13.7535i		0		−348.446				0			0			0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.7.e.c		36
3.b	odd	2	1	inner	648.7.e.c		36
9.c	even	3	1		72.7.m.a	✓	36
9.c	even	3	1		216.7.m.a		36
9.d	odd	6	1		72.7.m.a	✓	36
9.d	odd	6	1		216.7.m.a		36
36.f	odd	6	1		144.7.q.d		36
36.f	odd	6	1		432.7.q.d		36
36.h	even	6	1		144.7.q.d		36
36.h	even	6	1		432.7.q.d		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.7.m.a	✓	36	9.c	even	3	1
72.7.m.a	✓	36	9.d	odd	6	1
144.7.q.d		36	36.f	odd	6	1
144.7.q.d		36	36.h	even	6	1
216.7.m.a		36	9.c	even	3	1
216.7.m.a		36	9.d	odd	6	1
432.7.q.d		36	36.f	odd	6	1
432.7.q.d		36	36.h	even	6	1
648.7.e.c		36	1.a	even	1	1	trivial
648.7.e.c		36	3.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^36 + 337500*T5^34 + 51150720402*T5^32 + 4612027100780820*T5^30 + 276279170516509076847*T5^28 + 11630040421180913265019680*T5^26 + 354892649208479661004232304876*T5^24 + 7983899198236592627599887674718600*T5^22 + 133371525278599451570611106544415179375*T5^20 + 1653464461579429122690544985663952546437500*T5^18 + 15091286824652719545684041867019712353944531250*T5^16 + 99754094919513007542876585320092423342778789062500*T5^14 + 464771256234196441348024018530863366747872730712890625*T5^12 + 1463526352664413287727566064720844948201112241845703125000*T5^10 + 2920326489961002436268132964190004009160945721267089843750000*T5^8 + 3337774163383998258167069596180306264733790064057617187500000000*T5^6 + 1841182077022735191865548955213319257945089308299584960937500000000*T5^4 + 375107127779547368374093253071558997465630725313247070312500000000000*T5^2 + 24261596347363433327205864665367423340282869085646728515625000000000000