Properties

Label 4840.2.a.bf
Level $4840$
Weight $2$
Character orbit 4840.a
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45753625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - q^{5} + ( - \beta_{5} - \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{4} - 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - q^{5} + ( - \beta_{5} - \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{4} - 2 \beta_{2}) q^{9} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{13} + \beta_{3} q^{15} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} + 1) q^{17} + ( - \beta_{5} - \beta_{2} + \beta_1 - 2) q^{19} + ( - \beta_{5} - \beta_1 + 1) q^{21} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 + 2) q^{23} + q^{25} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_1 - 1) q^{27} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{29} + (\beta_{5} - 2 \beta_{3} - 1) q^{31} + (\beta_{5} + \beta_{2} - 1) q^{35} + (\beta_{4} - 3 \beta_{2} + 2 \beta_1 - 1) q^{37} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{39} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{41} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 1) q^{43} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{45} + (2 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{47} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{49} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{51} + (\beta_{4} - \beta_{2} + 2) q^{53} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - \beta_1 + 3) q^{57} + (2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{59} + (3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{61} + (2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 4) q^{63} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{65} + (2 \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{67} + ( - \beta_{4} - 5 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 2) q^{69} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 - 2) q^{71} + (2 \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{73} - \beta_{3} q^{75} + ( - \beta_{4} - \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 8) q^{79} + ( - 3 \beta_{4} - 2 \beta_{3} - 10 \beta_{2} - \beta_1 - 1) q^{81} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{2} + \beta_1 - 5) q^{83} + (\beta_{5} + \beta_{3} + 2 \beta_{2} - 1) q^{85} + (\beta_{5} - \beta_{4} + 3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 1) q^{87} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 3) q^{89} + ( - 4 \beta_{4} + 6 \beta_{2} + 1) q^{91} + (3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 5) q^{93} + (\beta_{5} + \beta_{2} - \beta_1 + 2) q^{95} + ( - 3 \beta_{5} + \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} - 6 q^{13} - 2 q^{15} + 11 q^{17} - 11 q^{19} + 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} - 6 q^{29} + q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} - 4 q^{41} + 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} + 31 q^{51} + 14 q^{53} - 5 q^{57} + 2 q^{59} - 4 q^{61} - 16 q^{63} + 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} - 9 q^{73} + 2 q^{75} + 36 q^{79} + 30 q^{81} - 45 q^{83} - 11 q^{85} + 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} + 11 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} - 13\nu^{3} - 9\nu^{2} + 35\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 10\nu^{3} + 3\nu^{2} - 15\nu - 10 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 12\nu^{3} - 8\nu^{2} - 30\nu + 10 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - \nu^{4} + 13\nu^{3} + 14\nu^{2} - 35\nu - 25 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 2\beta_{3} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} + 3\beta_{4} - \beta_{3} + 22\beta_{2} + \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{5} + 10\beta_{4} - 25\beta_{3} + 6\beta_{2} + 55\beta _1 - 5 \) 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
1.87511
−2.80540
−0.444728
1.36422
3.05921
−2.04842
0 −3.03399 0 −1.00000 0 0.865927 0 6.20511 0
1.2 0 −1.73383 0 −1.00000 0 −1.25225 0 0.00617996 0
1.3 0 0.719585 0 −1.00000 0 4.18418 0 −2.48220 0
1.4 0 0.843136 0 −1.00000 0 4.75693 0 −2.28912 0
1.5 0 1.89070 0 −1.00000 0 −2.74075 0 0.574737 0
1.6 0 3.31441 0 −1.00000 0 0.185958 0 7.98529 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 4840.2.a.bf 6
4.b odd 2 1 9680.2.a.cx 6
11.b odd 2 1 4840.2.a.be 6
11.c even 5 2 440.2.y.b 12
44.c even 2 1 9680.2.a.cy 6
44.h odd 10 2 880.2.bo.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.b 12 11.c even 5 2
880.2.bo.j 12 44.h odd 10 2
4840.2.a.be 6 11.b odd 2 1
4840.2.a.bf 6 1.a even 1 1 trivial
9680.2.a.cx 6 4.b odd 2 1
9680.2.a.cy 6 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4840))\):

\( T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 23T_{3}^{3} + 21T_{3}^{2} - 50T_{3} + 20 \) Copy content Toggle raw display
\( T_{7}^{6} - 6T_{7}^{5} - 7T_{7}^{4} + 61T_{7}^{3} + 15T_{7}^{2} - 64T_{7} + 11 \) Copy content Toggle raw display
\( T_{13}^{6} + 6T_{13}^{5} - 25T_{13}^{4} - 141T_{13}^{3} + 177T_{13}^{2} + 738T_{13} - 151 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} - 12 T^{4} + 23 T^{3} + \cdots + 20 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} - 7 T^{4} + 61 T^{3} + \cdots + 11 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} - 25 T^{4} - 141 T^{3} + \cdots - 151 \) Copy content Toggle raw display
$17$ \( T^{6} - 11 T^{5} + 6 T^{4} + 216 T^{3} + \cdots - 484 \) Copy content Toggle raw display
$19$ \( T^{6} + 11 T^{5} + 15 T^{4} - 127 T^{3} + \cdots + 55 \) Copy content Toggle raw display
$23$ \( T^{6} - 18 T^{5} + 50 T^{4} + \cdots + 9505 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} - 36 T^{4} - 19 T^{3} + \cdots - 36 \) Copy content Toggle raw display
$31$ \( T^{6} - T^{5} - 75 T^{4} - 93 T^{3} + \cdots + 2980 \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} - 90 T^{4} + \cdots - 14821 \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} - 176 T^{4} + \cdots - 37441 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} - 163 T^{4} + \cdots + 75284 \) Copy content Toggle raw display
$47$ \( T^{6} - 14 T^{5} - 5 T^{4} + 503 T^{3} + \cdots + 995 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} + 57 T^{4} + \cdots - 225 \) Copy content Toggle raw display
$59$ \( T^{6} - 2 T^{5} - 219 T^{4} + \cdots - 319 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} - 237 T^{4} + \cdots + 70256 \) Copy content Toggle raw display
$67$ \( T^{6} - 11 T^{5} - 66 T^{4} + \cdots - 19900 \) Copy content Toggle raw display
$71$ \( T^{6} - 7 T^{5} - 158 T^{4} + \cdots - 64476 \) Copy content Toggle raw display
$73$ \( T^{6} + 9 T^{5} - 80 T^{4} + \cdots + 3244 \) Copy content Toggle raw display
$79$ \( T^{6} - 36 T^{5} + 344 T^{4} + \cdots + 14864 \) Copy content Toggle raw display
$83$ \( T^{6} + 45 T^{5} + 679 T^{4} + \cdots - 339676 \) Copy content Toggle raw display
$89$ \( T^{6} - T^{5} - 419 T^{4} + \cdots + 179771 \) Copy content Toggle raw display
$97$ \( T^{6} - 20 T^{5} - 330 T^{4} + \cdots + 1270064 \) Copy content Toggle raw display
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