Properties

Label 4851.2.a.ca
Level $4851$
Weight $2$
Character orbit 4851.a
Self dual yes
Analytic conductor $38.735$
Analytic rank $0$
Dimension $5$
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] = [4851,2,Mod(1,4851)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4851, 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("4851.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.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: \(38.7354300205\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.3676752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 14x^{2} + 11x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{4} + 1) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{4} + 1) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{8} + (2 \beta_{4} + \beta_{3} - \beta_1) q^{10} - q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{13} + (\beta_{4} + \beta_{2} + 3) q^{16} - \beta_1 q^{17} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{19} + ( - 3 \beta_{4} - 2 \beta_{3} + 3) q^{20} + \beta_1 q^{22} + ( - \beta_{4} - \beta_{2} - 3) q^{23} + ( - 2 \beta_{4} + 2 \beta_1 + 1) q^{25} + (3 \beta_{4} + 2 \beta_{2} - 4 \beta_1 + 3) q^{26} + ( - 2 \beta_{4} + \beta_{3}) q^{29} + ( - \beta_{2} + 1) q^{31} + ( - 2 \beta_{4} - \beta_1) q^{32} + (\beta_{2} + 4) q^{34} + ( - \beta_{4} - 2 \beta_1 + 4) q^{37} + (3 \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{38} + (4 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{40} + (2 \beta_{4} + \beta_{3} - \beta_1 + 4) q^{41} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{43} + ( - \beta_{2} - 2) q^{44} + (2 \beta_{4} + 2 \beta_{3} + 5 \beta_1) q^{46} + ( - \beta_{2} - 4 \beta_1 + 2) q^{47} + (4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{50} + ( - 4 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 12) q^{52} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{53} + (\beta_{4} - 1) q^{55} + (3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{58} + ( - \beta_{4} - \beta_{2} - 3) q^{59} + (3 \beta_{3} + 3 \beta_1) q^{61} + (\beta_{3} + \beta_1) q^{62} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2) q^{64} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{65} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{67} + ( - \beta_{3} - 4 \beta_1) q^{68} + (3 \beta_{2} - 2 \beta_1 - 4) q^{71} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 2) q^{73} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 8) q^{74} + ( - 4 \beta_{4} - 2 \beta_{3} - \beta_1 + 8) q^{76} + ( - 3 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{79} + ( - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{80} + ( - 5 \beta_{4} - 2 \beta_{3} - 4 \beta_1 + 5) q^{82} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 2) q^{83} + (2 \beta_{4} + \beta_{3} - \beta_1) q^{85} + (4 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2) q^{86} + (\beta_{3} + 2 \beta_1) q^{88} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{89} + ( - 4 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 12) q^{92} + (\beta_{3} + 4 \beta_{2} + 16) q^{94} + ( - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{95} + (2 \beta_{4} + 4 \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 10 q^{4} + 4 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 10 q^{4} + 4 q^{5} - 6 q^{8} + 2 q^{10} - 5 q^{11} + 5 q^{13} + 16 q^{16} - 2 q^{17} - 3 q^{19} + 8 q^{20} + 2 q^{22} - 16 q^{23} + 7 q^{25} + 10 q^{26} + 5 q^{31} - 4 q^{32} + 20 q^{34} + 15 q^{37} - 6 q^{38} - 6 q^{40} + 22 q^{41} + 3 q^{43} - 10 q^{44} + 16 q^{46} + 2 q^{47} - 34 q^{50} + 40 q^{52} - 6 q^{53} - 4 q^{55} + 12 q^{58} - 16 q^{59} + 12 q^{61} + 4 q^{62} - 4 q^{64} + 28 q^{65} + 7 q^{67} - 10 q^{68} - 24 q^{71} + 17 q^{73} + 36 q^{74} + 30 q^{76} + 7 q^{79} - 16 q^{80} + 8 q^{82} + 12 q^{83} + 2 q^{85} + 18 q^{86} + 6 q^{88} + 6 q^{89} - 68 q^{92} + 82 q^{94} + 18 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 8x^{3} + 14x^{2} + 11x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 23 \) 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
2.59450
2.35232
0.614936
−1.06884
−2.49291
−2.59450 0 4.73141 −2.19202 0 0 −7.08664 0 5.68719
1.2 −2.35232 0 3.53341 4.11523 0 0 −3.60708 0 −9.68034
1.3 −0.614936 0 −1.62185 −1.49597 0 0 2.22721 0 0.919926
1.4 1.06884 0 −0.857576 2.69184 0 0 −3.05430 0 2.87715
1.5 2.49291 0 4.21460 0.880926 0 0 5.52081 0 2.19607
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 4851.2.a.ca 5
3.b odd 2 1 1617.2.a.ba 5
7.b odd 2 1 4851.2.a.bz 5
7.c even 3 2 693.2.i.j 10
21.c even 2 1 1617.2.a.bb 5
21.h odd 6 2 231.2.i.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.f 10 21.h odd 6 2
693.2.i.j 10 7.c even 3 2
1617.2.a.ba 5 3.b odd 2 1
1617.2.a.bb 5 21.c even 2 1
4851.2.a.bz 5 7.b odd 2 1
4851.2.a.ca 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4851))\):

\( T_{2}^{5} + 2T_{2}^{4} - 8T_{2}^{3} - 14T_{2}^{2} + 11T_{2} + 10 \) Copy content Toggle raw display
\( T_{5}^{5} - 4T_{5}^{4} - 8T_{5}^{3} + 28T_{5}^{2} + 20T_{5} - 32 \) Copy content Toggle raw display
\( T_{13}^{5} - 5T_{13}^{4} - 38T_{13}^{3} + 80T_{13}^{2} + 524T_{13} + 476 \) Copy content Toggle raw display
\( T_{17}^{5} + 2T_{17}^{4} - 8T_{17}^{3} - 14T_{17}^{2} + 11T_{17} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2 T^{4} - 8 T^{3} - 14 T^{2} + \cdots + 10 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 4 T^{4} - 8 T^{3} + 28 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( (T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 5 T^{4} - 38 T^{3} + 80 T^{2} + \cdots + 476 \) Copy content Toggle raw display
$17$ \( T^{5} + 2 T^{4} - 8 T^{3} - 14 T^{2} + \cdots + 10 \) Copy content Toggle raw display
$19$ \( T^{5} + 3 T^{4} - 42 T^{3} - 108 T^{2} + \cdots + 603 \) Copy content Toggle raw display
$23$ \( T^{5} + 16 T^{4} + 70 T^{3} + \cdots - 196 \) Copy content Toggle raw display
$29$ \( T^{5} - 96 T^{3} - 24 T^{2} + \cdots + 1290 \) Copy content Toggle raw display
$31$ \( T^{5} - 5 T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$37$ \( T^{5} - 15 T^{4} + 30 T^{3} + \cdots + 201 \) Copy content Toggle raw display
$41$ \( T^{5} - 22 T^{4} + 124 T^{3} + \cdots + 11536 \) Copy content Toggle raw display
$43$ \( T^{5} - 3 T^{4} - 108 T^{3} + \cdots - 2973 \) Copy content Toggle raw display
$47$ \( T^{5} - 2 T^{4} - 194 T^{3} + \cdots - 27994 \) Copy content Toggle raw display
$53$ \( T^{5} + 6 T^{4} - 96 T^{3} + \cdots + 6504 \) Copy content Toggle raw display
$59$ \( T^{5} + 16 T^{4} + 70 T^{3} + \cdots - 196 \) Copy content Toggle raw display
$61$ \( T^{5} - 12 T^{4} - 180 T^{3} + \cdots - 48600 \) Copy content Toggle raw display
$67$ \( T^{5} - 7 T^{4} - 92 T^{3} + \cdots - 3044 \) Copy content Toggle raw display
$71$ \( T^{5} + 24 T^{4} + 66 T^{3} + \cdots - 5232 \) Copy content Toggle raw display
$73$ \( T^{5} - 17 T^{4} - 98 T^{3} + \cdots + 3044 \) Copy content Toggle raw display
$79$ \( T^{5} - 7 T^{4} - 254 T^{3} + \cdots - 94868 \) Copy content Toggle raw display
$83$ \( T^{5} - 12 T^{4} - 120 T^{3} + \cdots - 2688 \) Copy content Toggle raw display
$89$ \( T^{5} - 6 T^{4} - 168 T^{3} + \cdots - 17760 \) Copy content Toggle raw display
$97$ \( T^{5} + 14 T^{4} - 326 T^{3} + \cdots - 38906 \) Copy content Toggle raw display
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