Newform orbit 2496.4.a.bf

• Label: 2496.4.a.bf
• Level: $2496$
• Weight: $4$
• Character orbit: 2496.a
• Self dual: yes
• Analytic conductor: $147.269$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(147.268767374\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{22}) \)
Defining polynomial:	\( x^{2} - 22 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^3 + b * q^5 + (3*b + 4) * q^7 + 9 * q^9 + (2*b + 30) * q^11 + 13 * q^13 + 3*b * q^15 + (-6*b - 54) * q^17 + (3*b + 108) * q^19 + (9*b + 12) * q^21 + (8*b - 108) * q^23 - 37 * q^25 + 27 * q^27 + (-20*b + 54) * q^29 + (15*b + 40) * q^31 + (6*b + 90) * q^33 + (4*b + 264) * q^35 + (18*b - 54) * q^37 + 39 * q^39 + (15*b - 24) * q^41 + (54*b - 4) * q^43 + 9*b * q^45 + (-44*b - 114) * q^47 + (24*b + 465) * q^49 + (-18*b - 162) * q^51 + (-12*b - 270) * q^53 + (30*b + 176) * q^55 + (9*b + 324) * q^57 + (40*b + 426) * q^59 + (-12*b - 154) * q^61 + (27*b + 36) * q^63 + 13*b * q^65 + (-39*b + 152) * q^67 + (24*b - 324) * q^69 + (82*b - 114) * q^71 + (6*b + 710) * q^73 - 111 * q^75 + (98*b + 648) * q^77 + (-60*b + 248) * q^79 + 81 * q^81 + (-12*b - 618) * q^83 + (-54*b - 528) * q^85 + (-60*b + 162) * q^87 + (-63*b - 708) * q^89 + (39*b + 52) * q^91 + (45*b + 120) * q^93 + (108*b + 264) * q^95 + (-30*b + 302) * q^97 + (18*b + 270) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 + 8 * q^7 + 18 * q^9 + 60 * q^11 + 26 * q^13 - 108 * q^17 + 216 * q^19 + 24 * q^21 - 216 * q^23 - 74 * q^25 + 54 * q^27 + 108 * q^29 + 80 * q^31 + 180 * q^33 + 528 * q^35 - 108 * q^37 + 78 * q^39 - 48 * q^41 - 8 * q^43 - 228 * q^47 + 930 * q^49 - 324 * q^51 - 540 * q^53 + 352 * q^55 + 648 * q^57 + 852 * q^59 - 308 * q^61 + 72 * q^63 + 304 * q^67 - 648 * q^69 - 228 * q^71 + 1420 * q^73 - 222 * q^75 + 1296 * q^77 + 496 * q^79 + 162 * q^81 - 1236 * q^83 - 1056 * q^85 + 324 * q^87 - 1416 * q^89 + 104 * q^91 + 240 * q^93 + 528 * q^95 + 604 * q^97 + 540 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−4.69042
4.69042

0

3.00000

0

−9.38083

0

−24.1425

0

9.00000

0

1.2

0

3.00000

0

9.38083

0

32.1425

0

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(13\)	\( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2496.4.a.bf		2
4.b	odd	2	1		2496.4.a.w		2
8.b	even	2	1		156.4.a.c	✓	2
8.d	odd	2	1		624.4.a.p		2
24.f	even	2	1		1872.4.a.y		2
24.h	odd	2	1		468.4.a.g		2
104.e	even	2	1		2028.4.a.d		2
104.j	odd	4	2		2028.4.b.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.4.a.c	✓	2	8.b	even	2	1
468.4.a.g		2	24.h	odd	2	1
624.4.a.p		2	8.d	odd	2	1
1872.4.a.y		2	24.f	even	2	1
2028.4.a.d		2	104.e	even	2	1
2028.4.b.e		4	104.j	odd	4	2
2496.4.a.w		2	4.b	odd	2	1
2496.4.a.bf		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2496))\):

\( T_{5}^{2} - 88 \)

\( T_{7}^{2} - 8T_{7} - 776 \)

\( T_{11}^{2} - 60T_{11} + 548 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} - 88 \)
$7$	\( T^{2} - 8T - 776 \)
$11$	\( T^{2} - 60T + 548 \)