Newform orbit 2112.4.a.bk

• Label: 2112.4.a.bk
• Level: $2112$
• Weight: $4$
• Character orbit: 2112.a
• Self dual: yes
• Analytic conductor: $124.612$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(124.612033932\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{185}) \)
Defining polynomial:	\( x^{2} - x - 46 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{185}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^3 + (b + 3) * q^5 + (b - 11) * q^7 + 9 * q^9 - 11 * q^11 + (-b - 47) * q^13 + (3*b + 9) * q^15 + (6*b + 28) * q^17 + (6*b + 38) * q^19 + (3*b - 33) * q^21 + (5*b - 27) * q^23 + (6*b + 69) * q^25 + 27 * q^27 + (-2*b - 52) * q^29 + (-8*b - 112) * q^31 - 33 * q^33 + (-8*b + 152) * q^35 + (16*b + 34) * q^37 + (-3*b - 141) * q^39 + (-4*b - 150) * q^41 + (4*b + 228) * q^43 + (9*b + 27) * q^45 + (27*b + 11) * q^47 + (-22*b - 37) * q^49 + (18*b + 84) * q^51 + (-35*b + 55) * q^53 + (-11*b - 33) * q^55 + (18*b + 114) * q^57 + (22*b + 34) * q^59 + (b + 27) * q^61 + (9*b - 99) * q^63 + (-50*b - 326) * q^65 + (-8*b + 780) * q^67 + (15*b - 81) * q^69 + (-15*b + 137) * q^71 + (32*b + 274) * q^73 + (18*b + 207) * q^75 + (-11*b + 121) * q^77 + (23*b - 461) * q^79 + 81 * q^81 + (44*b + 592) * q^83 + (46*b + 1194) * q^85 + (-6*b - 156) * q^87 + (72*b - 478) * q^89 + (-36*b + 332) * q^91 + (-24*b - 336) * q^93 + (56*b + 1224) * q^95 + (70*b - 336) * q^97 - 99 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 + 6 * q^5 - 22 * q^7 + 18 * q^9 - 22 * q^11 - 94 * q^13 + 18 * q^15 + 56 * q^17 + 76 * q^19 - 66 * q^21 - 54 * q^23 + 138 * q^25 + 54 * q^27 - 104 * q^29 - 224 * q^31 - 66 * q^33 + 304 * q^35 + 68 * q^37 - 282 * q^39 - 300 * q^41 + 456 * q^43 + 54 * q^45 + 22 * q^47 - 74 * q^49 + 168 * q^51 + 110 * q^53 - 66 * q^55 + 228 * q^57 + 68 * q^59 + 54 * q^61 - 198 * q^63 - 652 * q^65 + 1560 * q^67 - 162 * q^69 + 274 * q^71 + 548 * q^73 + 414 * q^75 + 242 * q^77 - 922 * q^79 + 162 * q^81 + 1184 * q^83 + 2388 * q^85 - 312 * q^87 - 956 * q^89 + 664 * q^91 - 672 * q^93 + 2448 * q^95 - 672 * q^97 - 198 * 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

−6.30074
7.30074

0

3.00000

0

−10.6015

0

−24.6015

0

9.00000

0

1.2

0

3.00000

0

16.6015

0

2.60147

0

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(11\)	\( +1 \)

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	2112.4.a.bk		2
4.b	odd	2	1		2112.4.a.bf		2
8.b	even	2	1		528.4.a.m		2
8.d	odd	2	1		264.4.a.g	✓	2
24.f	even	2	1		792.4.a.j		2
24.h	odd	2	1		1584.4.a.bd		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
264.4.a.g	✓	2	8.d	odd	2	1
528.4.a.m		2	8.b	even	2	1
792.4.a.j		2	24.f	even	2	1
1584.4.a.bd		2	24.h	odd	2	1
2112.4.a.bf		2	4.b	odd	2	1
2112.4.a.bk		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5}^{2} - 6T_{5} - 176 \)

\( T_{7}^{2} + 22T_{7} - 64 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} - 6T - 176 \)
$7$	\( T^{2} + 22T - 64 \)
$11$	\( (T + 11)^{2} \)