Newform orbit 2112.4.a.bl

• Label: 2112.4.a.bl
• 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:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
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 \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^3 + (b + 3) * q^5 + (b + 5) * q^7 + 9 * q^9 - 11 * q^11 + (-17*b + 1) * q^13 + (3*b + 9) * q^15 + (-18*b - 52) * q^17 + (30*b - 2) * q^19 + (3*b + 15) * q^21 + (13*b - 51) * q^23 + (6*b - 99) * q^25 + 27 * q^27 + (-26*b + 196) * q^29 + (40*b - 32) * q^31 - 33 * q^33 + (8*b + 32) * q^35 + (80*b + 82) * q^37 + (-51*b + 3) * q^39 + (20*b - 366) * q^41 + (-68*b - 84) * q^43 + (9*b + 27) * q^45 + (-45*b - 157) * q^47 + (10*b - 301) * q^49 + (-54*b - 156) * q^51 + (37*b + 191) * q^53 + (-11*b - 33) * q^55 + (90*b - 6) * q^57 + (-58*b - 254) * q^59 + (-151*b + 3) * q^61 + (9*b + 45) * q^63 + (-50*b - 286) * q^65 + (136*b - 108) * q^67 + (39*b - 153) * q^69 + (-63*b - 439) * q^71 + (-224*b + 130) * q^73 + (18*b - 297) * q^75 + (-11*b - 55) * q^77 + (-97*b + 59) * q^79 + 81 * q^81 + (-212*b - 248) * q^83 + (-106*b - 462) * q^85 + (-78*b + 588) * q^87 + (168*b - 878) * q^89 + (-84*b - 284) * q^91 + (120*b - 96) * q^93 + (88*b + 504) * q^95 + (38*b + 984) * q^97 - 99 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 + 6 * q^5 + 10 * q^7 + 18 * q^9 - 22 * q^11 + 2 * q^13 + 18 * q^15 - 104 * q^17 - 4 * q^19 + 30 * q^21 - 102 * q^23 - 198 * q^25 + 54 * q^27 + 392 * q^29 - 64 * q^31 - 66 * q^33 + 64 * q^35 + 164 * q^37 + 6 * q^39 - 732 * q^41 - 168 * q^43 + 54 * q^45 - 314 * q^47 - 602 * q^49 - 312 * q^51 + 382 * q^53 - 66 * q^55 - 12 * q^57 - 508 * q^59 + 6 * q^61 + 90 * q^63 - 572 * q^65 - 216 * q^67 - 306 * q^69 - 878 * q^71 + 260 * q^73 - 594 * q^75 - 110 * q^77 + 118 * q^79 + 162 * q^81 - 496 * q^83 - 924 * q^85 + 1176 * q^87 - 1756 * q^89 - 568 * q^91 - 192 * q^93 + 1008 * q^95 + 1968 * 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

−1.56155
2.56155

0

3.00000

0

−1.12311

0

0.876894

0

9.00000

0

1.2

0

3.00000

0

7.12311

0

9.12311

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.bl		2
4.b	odd	2	1		2112.4.a.be		2
8.b	even	2	1		264.4.a.e	✓	2
8.d	odd	2	1		528.4.a.r		2
24.f	even	2	1		1584.4.a.bf		2
24.h	odd	2	1		792.4.a.h		2

    

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

\( T_{5}^{2} - 6T_{5} - 8 \)

\( T_{7}^{2} - 10T_{7} + 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} - 6T - 8 \)
$7$	\( T^{2} - 10T + 8 \)
$11$	\( (T + 11)^{2} \)