Properties

Label 1881.2.a.j
Level $1881$
Weight $2$
Character orbit 1881.a
Self dual yes
Analytic conductor $15.020$
Analytic rank $1$
Dimension $4$
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] = [1881,2,Mod(1,1881)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1881, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1881.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1881.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: \(15.0198606202\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.23377.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + 6x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 627)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_1 - 2) q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_1 - 2) q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{8} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{10} + q^{11} + ( - \beta_1 + 1) q^{13} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{14} + (\beta_{3} - \beta_1 + 1) q^{16} + (\beta_{3} + \beta_{2} + \beta_1 - 3) q^{17} - q^{19} + ( - \beta_{3} - \beta_{2} + \beta_1 - 7) q^{20} - \beta_1 q^{22} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{23} + (\beta_{3} - \beta_1 + 1) q^{25} + (\beta_{2} - \beta_1 + 4) q^{26} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 6) q^{28} + (\beta_{2} + \beta_1 - 6) q^{29} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{31} + ( - \beta_{2} + \beta_1 + 3) q^{32} + ( - 3 \beta_{3} + 2 \beta_1 - 5) q^{34} + (3 \beta_{2} - \beta_1 + 4) q^{35} + (3 \beta_{3} - 3 \beta_{2} + 2) q^{37} + \beta_1 q^{38} + (\beta_{3} + 4 \beta_1 - 3) q^{40} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{41}+ \cdots + (2 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 7 q^{4} - 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 7 q^{4} - 3 q^{5} - 5 q^{7} + q^{10} + 4 q^{11} + 3 q^{13} - 13 q^{14} + q^{16} - 14 q^{17} - 4 q^{19} - 24 q^{20} - q^{22} - 7 q^{23} + q^{25} + 14 q^{26} - 17 q^{28} - 24 q^{29} - 2 q^{31} + 14 q^{32} - 12 q^{34} + 12 q^{35} + 5 q^{37} + q^{38} - 10 q^{40} + 5 q^{41} + 2 q^{43} + 7 q^{44} - 28 q^{46} + 6 q^{47} + 17 q^{49} + 14 q^{50} + 5 q^{52} - 20 q^{53} - 3 q^{55} + 2 q^{56} - 9 q^{58} - q^{59} - 7 q^{61} - 15 q^{62} - 20 q^{64} - 2 q^{65} + 5 q^{67} + 3 q^{68} + 11 q^{70} - 14 q^{71} + 4 q^{73} - 2 q^{74} - 7 q^{76} - 5 q^{77} - 16 q^{79} - 9 q^{80} + 14 q^{82} - 15 q^{83} - 17 q^{85} - q^{86} - 8 q^{89} - 18 q^{91} - 22 q^{92} - 16 q^{94} + 3 q^{95} + 16 q^{97} + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 7x^{2} + 6x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 5\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 5\beta_1 \) 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
2.35438
1.76330
−0.696379
−2.42130
−2.35438 0 3.54311 −2.54311 0 −2.46743 −3.63308 0 5.98746
1.2 −1.76330 0 1.10923 −0.109229 0 3.98807 1.57070 0 0.192604
1.3 0.696379 0 −1.51506 2.51506 0 −2.32551 −2.44781 0 1.75143
1.4 2.42130 0 3.86271 −2.86271 0 −4.19513 4.51019 0 −6.93150
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(11\) \( -1 \)
\(19\) \( +1 \)

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 1881.2.a.j 4
3.b odd 2 1 627.2.a.h 4
33.d even 2 1 6897.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.a.h 4 3.b odd 2 1
1881.2.a.j 4 1.a even 1 1 trivial
6897.2.a.q 4 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1881))\):

\( T_{2}^{4} + T_{2}^{3} - 7T_{2}^{2} - 6T_{2} + 7 \) Copy content Toggle raw display
\( T_{5}^{4} + 3T_{5}^{3} - 6T_{5}^{2} - 19T_{5} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 7 T^{2} + \cdots + 7 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$7$ \( T^{4} + 5 T^{3} + \cdots - 96 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} + \cdots + 6 \) Copy content Toggle raw display
$17$ \( T^{4} + 14 T^{3} + \cdots - 678 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 7 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 24 T^{3} + \cdots + 722 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 368 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} + \cdots + 2278 \) Copy content Toggle raw display
$41$ \( T^{4} - 5 T^{3} + \cdots + 806 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots - 796 \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + \cdots - 24 \) Copy content Toggle raw display
$53$ \( T^{4} + 20 T^{3} + \cdots - 582 \) Copy content Toggle raw display
$59$ \( T^{4} + T^{3} + \cdots - 36 \) Copy content Toggle raw display
$61$ \( T^{4} + 7 T^{3} + \cdots + 3098 \) Copy content Toggle raw display
$67$ \( T^{4} - 5 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$71$ \( T^{4} + 14 T^{3} + \cdots - 744 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 17206 \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} + \cdots - 216 \) Copy content Toggle raw display
$83$ \( T^{4} + 15 T^{3} + \cdots - 1844 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} + \cdots + 8066 \) Copy content Toggle raw display
$97$ \( T^{4} - 16 T^{3} + \cdots - 7778 \) Copy content Toggle raw display
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