Properties

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

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \sqrt{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - 2 \beta - 8) q^{5} + (\beta + 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - 2 \beta - 8) q^{5} + (\beta + 1) q^{7} + 9 q^{9} - 11 q^{11} + (4 \beta + 38) q^{13} + (6 \beta + 24) q^{15} + ( - 15 \beta - 13) q^{17} + ( - 9 \beta + 27) q^{19} + ( - 3 \beta - 3) q^{21} + 112 q^{23} + (32 \beta + 71) q^{25} - 27 q^{27} + (23 \beta - 111) q^{29} + ( - 52 \beta - 20) q^{31} + 33 q^{33} + ( - 10 \beta - 74) q^{35} + (22 \beta + 24) q^{37} + ( - 12 \beta - 114) q^{39} + ( - \beta - 247) q^{41} + ( - 43 \beta + 33) q^{43} + ( - 18 \beta - 72) q^{45} + (24 \beta - 32) q^{47} + (2 \beta - 309) q^{49} + (45 \beta + 39) q^{51} + ( - 64 \beta + 42) q^{53} + (22 \beta + 88) q^{55} + (27 \beta - 81) q^{57} - 196 q^{59} + ( - 26 \beta + 552) q^{61} + (9 \beta + 9) q^{63} + ( - 108 \beta - 568) q^{65} + (76 \beta - 464) q^{67} - 336 q^{69} + ( - 92 \beta + 228) q^{71} + (126 \beta - 296) q^{73} + ( - 96 \beta - 213) q^{75} + ( - 11 \beta - 11) q^{77} + (37 \beta - 115) q^{79} + 81 q^{81} + ( - 162 \beta - 174) q^{83} + (146 \beta + 1094) q^{85} + ( - 69 \beta + 333) q^{87} + ( - 4 \beta + 486) q^{89} + (42 \beta + 170) q^{91} + (156 \beta + 60) q^{93} + (18 \beta + 378) q^{95} + ( - 210 \beta - 592) q^{97} - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 16 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 16 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} + 76 q^{13} + 48 q^{15} - 26 q^{17} + 54 q^{19} - 6 q^{21} + 224 q^{23} + 142 q^{25} - 54 q^{27} - 222 q^{29} - 40 q^{31} + 66 q^{33} - 148 q^{35} + 48 q^{37} - 228 q^{39} - 494 q^{41} + 66 q^{43} - 144 q^{45} - 64 q^{47} - 618 q^{49} + 78 q^{51} + 84 q^{53} + 176 q^{55} - 162 q^{57} - 392 q^{59} + 1104 q^{61} + 18 q^{63} - 1136 q^{65} - 928 q^{67} - 672 q^{69} + 456 q^{71} - 592 q^{73} - 426 q^{75} - 22 q^{77} - 230 q^{79} + 162 q^{81} - 348 q^{83} + 2188 q^{85} + 666 q^{87} + 972 q^{89} + 340 q^{91} + 120 q^{93} + 756 q^{95} - 1184 q^{97} - 198 q^{99}+O(q^{100}) \) 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
3.37228
−2.37228
0 −3.00000 0 −19.4891 0 6.74456 0 9.00000 0
1.2 0 −3.00000 0 3.48913 0 −4.74456 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.ba 2
4.b odd 2 1 2112.4.a.bh 2
8.b even 2 1 33.4.a.d 2
8.d odd 2 1 528.4.a.o 2
24.f even 2 1 1584.4.a.x 2
24.h odd 2 1 99.4.a.e 2
40.f even 2 1 825.4.a.k 2
40.i odd 4 2 825.4.c.i 4
56.h odd 2 1 1617.4.a.j 2
88.b odd 2 1 363.4.a.j 2
120.i odd 2 1 2475.4.a.o 2
264.m even 2 1 1089.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 8.b even 2 1
99.4.a.e 2 24.h odd 2 1
363.4.a.j 2 88.b odd 2 1
528.4.a.o 2 8.d odd 2 1
825.4.a.k 2 40.f even 2 1
825.4.c.i 4 40.i odd 4 2
1089.4.a.t 2 264.m even 2 1
1584.4.a.x 2 24.f even 2 1
1617.4.a.j 2 56.h odd 2 1
2112.4.a.ba 2 1.a even 1 1 trivial
2112.4.a.bh 2 4.b odd 2 1
2475.4.a.o 2 120.i odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 16T_{5} - 68 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 32 \) 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
$5$ \( T^{2} + 16T - 68 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 76T + 916 \) Copy content Toggle raw display
$17$ \( T^{2} + 26T - 7256 \) Copy content Toggle raw display
$19$ \( T^{2} - 54T - 1944 \) Copy content Toggle raw display
$23$ \( (T - 112)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 222T - 5136 \) Copy content Toggle raw display
$31$ \( T^{2} + 40T - 88832 \) Copy content Toggle raw display
$37$ \( T^{2} - 48T - 15396 \) Copy content Toggle raw display
$41$ \( T^{2} + 494T + 60976 \) Copy content Toggle raw display
$43$ \( T^{2} - 66T - 59928 \) Copy content Toggle raw display
$47$ \( T^{2} + 64T - 17984 \) Copy content Toggle raw display
$53$ \( T^{2} - 84T - 133404 \) Copy content Toggle raw display
$59$ \( (T + 196)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 1104 T + 282396 \) Copy content Toggle raw display
$67$ \( T^{2} + 928T + 24688 \) Copy content Toggle raw display
$71$ \( T^{2} - 456T - 227328 \) Copy content Toggle raw display
$73$ \( T^{2} + 592T - 436292 \) Copy content Toggle raw display
$79$ \( T^{2} + 230T - 31952 \) Copy content Toggle raw display
$83$ \( T^{2} + 348T - 835776 \) Copy content Toggle raw display
$89$ \( T^{2} - 972T + 235668 \) Copy content Toggle raw display
$97$ \( T^{2} + 1184 T - 1104836 \) Copy content Toggle raw display
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