Newform orbit 2112.4.a.bj

• Label: 2112.4.a.bj
• Level: $2112$
• Weight: $4$
• Character orbit: 2112.a
• Self dual: yes
• Analytic conductor: $124.612$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{73}) \)
Defining polynomial:	\( x^{2} - x - 18 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 1056)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{73}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^3 + (-b - 3) * q^5 + (2*b + 8) * q^7 + 9 * q^9 + 11 * q^11 + (b + 7) * q^13 + (-3*b - 9) * q^15 + (-5*b - 49) * q^17 + (7*b - 59) * q^19 + (6*b + 24) * q^21 + (-15*b - 3) * q^23 + (6*b - 43) * q^25 + 27 * q^27 + (9*b + 125) * q^29 + (-8*b - 44) * q^31 + 33 * q^33 + (-14*b - 170) * q^35 + (26*b - 80) * q^37 + (3*b + 21) * q^39 + (-21*b - 133) * q^41 + (-9*b - 39) * q^43 + (-9*b - 27) * q^45 + (7*b - 125) * q^47 + (32*b + 13) * q^49 + (-15*b - 147) * q^51 + (5*b + 127) * q^53 + (-11*b - 33) * q^55 + (21*b - 177) * q^57 + (-38*b - 194) * q^59 + (b - 37) * q^61 + (18*b + 72) * q^63 + (-10*b - 94) * q^65 + (32*b - 268) * q^67 + (-45*b - 9) * q^69 + (b - 11) * q^71 + (22*b - 476) * q^73 + (18*b - 129) * q^75 + (22*b + 88) * q^77 + (-8*b + 262) * q^79 + 81 * q^81 + (-70*b - 694) * q^83 + (64*b + 512) * q^85 + (27*b + 375) * q^87 + (-48*b - 190) * q^89 + (22*b + 202) * q^91 + (-24*b - 132) * q^93 + (38*b - 334) * q^95 + (136*b - 458) * q^97 + 99 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 - 6 * q^5 + 16 * q^7 + 18 * q^9 + 22 * q^11 + 14 * q^13 - 18 * q^15 - 98 * q^17 - 118 * q^19 + 48 * q^21 - 6 * q^23 - 86 * q^25 + 54 * q^27 + 250 * q^29 - 88 * q^31 + 66 * q^33 - 340 * q^35 - 160 * q^37 + 42 * q^39 - 266 * q^41 - 78 * q^43 - 54 * q^45 - 250 * q^47 + 26 * q^49 - 294 * q^51 + 254 * q^53 - 66 * q^55 - 354 * q^57 - 388 * q^59 - 74 * q^61 + 144 * q^63 - 188 * q^65 - 536 * q^67 - 18 * q^69 - 22 * q^71 - 952 * q^73 - 258 * q^75 + 176 * q^77 + 524 * q^79 + 162 * q^81 - 1388 * q^83 + 1024 * q^85 + 750 * q^87 - 380 * q^89 + 404 * q^91 - 264 * q^93 - 668 * q^95 - 916 * 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

4.77200
−3.77200

0

3.00000

0

−11.5440

0

25.0880

0

9.00000

0

1.2

0

3.00000

0

5.54400

0

−9.08801

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.bj		2
4.b	odd	2	1		2112.4.a.bc		2
8.b	even	2	1		1056.4.a.c	✓	2
8.d	odd	2	1		1056.4.a.d	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1056.4.a.c	✓	2	8.b	even	2	1
1056.4.a.d	yes	2	8.d	odd	2	1
2112.4.a.bc		2	4.b	odd	2	1
2112.4.a.bj		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5}^{2} + 6T_{5} - 64 \)

\( T_{7}^{2} - 16T_{7} - 228 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} + 6T - 64 \)
$7$	\( T^{2} - 16T - 228 \)
$11$	\( (T - 11)^{2} \)