Properties

Label 1872.4.a.v
Level $1872$
Weight $4$
Character orbit 1872.a
Self dual yes
Analytic conductor $110.452$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,4,Mod(1,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, 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("1872.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.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: \(110.451575531\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{43}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
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 \(\beta = 2\sqrt{43}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 6) q^{5} + (\beta - 22) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 6) q^{5} + (\beta - 22) q^{7} + 26 q^{11} - 13 q^{13} + ( - 2 \beta + 10) q^{17} + ( - 5 \beta + 30) q^{19} + (12 \beta - 4) q^{23} + ( - 12 \beta + 83) q^{25} + ( - 12 \beta - 66) q^{29} + ( - 15 \beta + 70) q^{31} + ( - 28 \beta + 304) q^{35} + (14 \beta + 34) q^{37} + ( - 7 \beta - 14) q^{41} + 14 \beta q^{43} + (6 \beta + 18) q^{47} + ( - 44 \beta + 313) q^{49} + ( - 4 \beta - 334) q^{53} + (26 \beta - 156) q^{55} + (22 \beta - 254) q^{59} + (8 \beta + 170) q^{61} + ( - 13 \beta + 78) q^{65} + ( - 43 \beta + 470) q^{67} + ( - 68 \beta + 150) q^{71} + (6 \beta + 562) q^{73} + (26 \beta - 572) q^{77} + ( - 44 \beta + 760) q^{79} + ( - 42 \beta + 262) q^{83} + (22 \beta - 404) q^{85} + ( - \beta - 950) q^{89} + ( - 13 \beta + 286) q^{91} + (60 \beta - 1040) q^{95} + (18 \beta - 718) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{5} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{5} - 44 q^{7} + 52 q^{11} - 26 q^{13} + 20 q^{17} + 60 q^{19} - 8 q^{23} + 166 q^{25} - 132 q^{29} + 140 q^{31} + 608 q^{35} + 68 q^{37} - 28 q^{41} + 36 q^{47} + 626 q^{49} - 668 q^{53} - 312 q^{55} - 508 q^{59} + 340 q^{61} + 156 q^{65} + 940 q^{67} + 300 q^{71} + 1124 q^{73} - 1144 q^{77} + 1520 q^{79} + 524 q^{83} - 808 q^{85} - 1900 q^{89} + 572 q^{91} - 2080 q^{95} - 1436 q^{97}+O(q^{100}) \) 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
−6.55744
6.55744
0 0 0 −19.1149 0 −35.1149 0 0 0
1.2 0 0 0 7.11488 0 −8.88512 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(13\) \( +1 \)

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 1872.4.a.v 2
3.b odd 2 1 624.4.a.l 2
4.b odd 2 1 936.4.a.c 2
12.b even 2 1 312.4.a.f 2
24.f even 2 1 2496.4.a.u 2
24.h odd 2 1 2496.4.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.f 2 12.b even 2 1
624.4.a.l 2 3.b odd 2 1
936.4.a.c 2 4.b odd 2 1
1872.4.a.v 2 1.a even 1 1 trivial
2496.4.a.u 2 24.f even 2 1
2496.4.a.bd 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1872))\):

\( T_{5}^{2} + 12T_{5} - 136 \) Copy content Toggle raw display
\( T_{7}^{2} + 44T_{7} + 312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12T - 136 \) Copy content Toggle raw display
$7$ \( T^{2} + 44T + 312 \) Copy content Toggle raw display
$11$ \( (T - 26)^{2} \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 20T - 588 \) Copy content Toggle raw display
$19$ \( T^{2} - 60T - 3400 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 24752 \) Copy content Toggle raw display
$29$ \( T^{2} + 132T - 20412 \) Copy content Toggle raw display
$31$ \( T^{2} - 140T - 33800 \) Copy content Toggle raw display
$37$ \( T^{2} - 68T - 32556 \) Copy content Toggle raw display
$41$ \( T^{2} + 28T - 8232 \) Copy content Toggle raw display
$43$ \( T^{2} - 33712 \) Copy content Toggle raw display
$47$ \( T^{2} - 36T - 5868 \) Copy content Toggle raw display
$53$ \( T^{2} + 668T + 108804 \) Copy content Toggle raw display
$59$ \( T^{2} + 508T - 18732 \) Copy content Toggle raw display
$61$ \( T^{2} - 340T + 17892 \) Copy content Toggle raw display
$67$ \( T^{2} - 940T - 97128 \) Copy content Toggle raw display
$71$ \( T^{2} - 300T - 772828 \) Copy content Toggle raw display
$73$ \( T^{2} - 1124 T + 309652 \) Copy content Toggle raw display
$79$ \( T^{2} - 1520 T + 244608 \) Copy content Toggle raw display
$83$ \( T^{2} - 524T - 234764 \) Copy content Toggle raw display
$89$ \( T^{2} + 1900 T + 902328 \) Copy content Toggle raw display
$97$ \( T^{2} + 1436 T + 459796 \) Copy content Toggle raw display
show more
show less