Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2112,4,Mod(1,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2112.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2112.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(124.612033932\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{97}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta + 7) q^{5} + ( - 2 \beta - 12) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta + 7) q^{5} + ( - 2 \beta - 12) q^{7} + 9 q^{9} - 11 q^{11} + (\beta - 15) q^{13} + (3 \beta - 21) q^{15} + (7 \beta + 53) q^{17} + (\beta + 25) q^{19} + (6 \beta + 36) q^{21} + ( - 5 \beta - 67) q^{23} + ( - 14 \beta + 21) q^{25} - 27 q^{27} + ( - 3 \beta + 99) q^{29} + (4 \beta - 180) q^{31} + 33 q^{33} + ( - 2 \beta + 110) q^{35} + (26 \beta + 164) q^{37} + ( - 3 \beta + 45) q^{39} + (7 \beta - 391) q^{41} + (13 \beta + 193) q^{43} + ( - 9 \beta + 63) q^{45} + (37 \beta - 133) q^{47} + (48 \beta + 189) q^{49} + ( - 21 \beta - 159) q^{51} + ( - 27 \beta + 261) q^{53} + (11 \beta - 77) q^{55} + ( - 3 \beta - 75) q^{57} + (50 \beta - 86) q^{59} + (17 \beta + 389) q^{61} + ( - 18 \beta - 108) q^{63} + (22 \beta - 202) q^{65} + (48 \beta - 388) q^{67} + (15 \beta + 201) q^{69} + (27 \beta - 315) q^{71} + (14 \beta + 648) q^{73} + (42 \beta - 63) q^{75} + (22 \beta + 132) q^{77} + (72 \beta - 326) q^{79} + 81 q^{81} + ( - 78 \beta - 162) q^{83} + ( - 4 \beta - 308) q^{85} + (9 \beta - 297) q^{87} + ( - 36 \beta - 378) q^{89} + (18 \beta - 14) q^{91} + ( - 12 \beta + 540) q^{93} + ( - 18 \beta + 78) q^{95} + ( - 96 \beta - 226) q^{97} - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 14 q^{5} - 24 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 14 q^{5} - 24 q^{7} + 18 q^{9} - 22 q^{11} - 30 q^{13} - 42 q^{15} + 106 q^{17} + 50 q^{19} + 72 q^{21} - 134 q^{23} + 42 q^{25} - 54 q^{27} + 198 q^{29} - 360 q^{31} + 66 q^{33} + 220 q^{35} + 328 q^{37} + 90 q^{39} - 782 q^{41} + 386 q^{43} + 126 q^{45} - 266 q^{47} + 378 q^{49} - 318 q^{51} + 522 q^{53} - 154 q^{55} - 150 q^{57} - 172 q^{59} + 778 q^{61} - 216 q^{63} - 404 q^{65} - 776 q^{67} + 402 q^{69} - 630 q^{71} + 1296 q^{73} - 126 q^{75} + 264 q^{77} - 652 q^{79} + 162 q^{81} - 324 q^{83} - 616 q^{85} - 594 q^{87} - 756 q^{89} - 28 q^{91} + 1080 q^{93} + 156 q^{95} - 452 q^{97} - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.42443
−4.42443
0 −3.00000 0 −2.84886 0 −31.6977 0 9.00000 0
1.2 0 −3.00000 0 16.8489 0 7.69772 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.bg 2
4.b odd 2 1 2112.4.a.bn 2
8.b even 2 1 528.4.a.p 2
8.d odd 2 1 33.4.a.c 2
24.f even 2 1 99.4.a.f 2
24.h odd 2 1 1584.4.a.bj 2
40.e odd 2 1 825.4.a.l 2
40.k even 4 2 825.4.c.h 4
56.e even 2 1 1617.4.a.k 2
88.g even 2 1 363.4.a.i 2
120.m even 2 1 2475.4.a.p 2
264.p odd 2 1 1089.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 8.d odd 2 1
99.4.a.f 2 24.f even 2 1
363.4.a.i 2 88.g even 2 1
528.4.a.p 2 8.b even 2 1
825.4.a.l 2 40.e odd 2 1
825.4.c.h 4 40.k even 4 2
1089.4.a.u 2 264.p odd 2 1
1584.4.a.bj 2 24.h odd 2 1
1617.4.a.k 2 56.e even 2 1
2112.4.a.bg 2 1.a even 1 1 trivial
2112.4.a.bn 2 4.b odd 2 1
2475.4.a.p 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5}^{2} - 14T_{5} - 48 \) Copy content Toggle raw display
\( T_{7}^{2} + 24T_{7} - 244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 14T - 48 \) Copy content Toggle raw display
$7$ \( T^{2} + 24T - 244 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 30T + 128 \) Copy content Toggle raw display
$17$ \( T^{2} - 106T - 1944 \) Copy content Toggle raw display
$19$ \( T^{2} - 50T + 528 \) Copy content Toggle raw display
$23$ \( T^{2} + 134T + 2064 \) Copy content Toggle raw display
$29$ \( T^{2} - 198T + 8928 \) Copy content Toggle raw display
$31$ \( T^{2} + 360T + 30848 \) Copy content Toggle raw display
$37$ \( T^{2} - 328T - 38676 \) Copy content Toggle raw display
$41$ \( T^{2} + 782T + 148128 \) Copy content Toggle raw display
$43$ \( T^{2} - 386T + 20856 \) Copy content Toggle raw display
$47$ \( T^{2} + 266T - 115104 \) Copy content Toggle raw display
$53$ \( T^{2} - 522T - 2592 \) Copy content Toggle raw display
$59$ \( T^{2} + 172T - 235104 \) Copy content Toggle raw display
$61$ \( T^{2} - 778T + 123288 \) Copy content Toggle raw display
$67$ \( T^{2} + 776T - 72944 \) Copy content Toggle raw display
$71$ \( T^{2} + 630T + 28512 \) Copy content Toggle raw display
$73$ \( T^{2} - 1296 T + 400892 \) Copy content Toggle raw display
$79$ \( T^{2} + 652T - 396572 \) Copy content Toggle raw display
$83$ \( T^{2} + 324T - 563904 \) Copy content Toggle raw display
$89$ \( T^{2} + 756T + 17172 \) Copy content Toggle raw display
$97$ \( T^{2} + 452T - 842876 \) Copy content Toggle raw display
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