Properties

Label 8880.2.a.cl
Level $8880$
Weight $2$
Character orbit 8880.a
Self dual yes
Analytic conductor $70.907$
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] = [8880,2,Mod(1,8880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8880.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8880.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: \(70.9071569949\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.20193189.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 21x^{3} + 10x^{2} + 72x - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4440)
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + \beta_{4} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + \beta_{4} q^{7} + q^{9} + ( - \beta_{4} - \beta_1) q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{13} - q^{15} + (\beta_{3} - \beta_1) q^{17} + (\beta_{3} - \beta_{2} - 2) q^{19} + \beta_{4} q^{21} + (\beta_{4} - \beta_{2} - \beta_1 + 2) q^{23} + q^{25} + q^{27} + (\beta_{4} + \beta_{2} + 1) q^{29} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{31} + ( - \beta_{4} - \beta_1) q^{33} - \beta_{4} q^{35} - q^{37} + ( - \beta_{2} - \beta_1 - 1) q^{39} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{41} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{43} - q^{45} + ( - \beta_{3} + 2 \beta_1 + 1) q^{47} + ( - \beta_{4} + 2 \beta_{2} + 3) q^{49} + (\beta_{3} - \beta_1) q^{51} + (\beta_{4} + \beta_{3} + 1) q^{53} + (\beta_{4} + \beta_1) q^{55} + (\beta_{3} - \beta_{2} - 2) q^{57} + (\beta_{3} + \beta_1 + 1) q^{59} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{61} + \beta_{4} q^{63} + (\beta_{2} + \beta_1 + 1) q^{65} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{67} + (\beta_{4} - \beta_{2} - \beta_1 + 2) q^{69} + ( - 2 \beta_{2} - \beta_1 + 2) q^{71} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 4) q^{73} + q^{75} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 8) q^{77} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{79} + q^{81} + ( - 2 \beta_{3} + \beta_{2} + 3) q^{83} + ( - \beta_{3} + \beta_1) q^{85} + (\beta_{4} + \beta_{2} + 1) q^{87} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{89} + ( - 3 \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{91} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{3} + \beta_{2} + 2) q^{95} + ( - \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{97} + ( - \beta_{4} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} - 5 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} - 5 q^{5} - 2 q^{7} + 5 q^{9} + q^{11} - 4 q^{13} - 5 q^{15} - q^{17} - 8 q^{19} - 2 q^{21} + 9 q^{23} + 5 q^{25} + 5 q^{27} + q^{29} + 9 q^{31} + q^{33} + 2 q^{35} - 5 q^{37} - 4 q^{39} + q^{41} - 5 q^{43} - 5 q^{45} + 7 q^{47} + 13 q^{49} - q^{51} + 3 q^{53} - q^{55} - 8 q^{57} + 6 q^{59} + 9 q^{61} - 2 q^{63} + 4 q^{65} + 11 q^{67} + 9 q^{69} + 13 q^{71} + 22 q^{73} + 5 q^{75} - 37 q^{77} - 5 q^{79} + 5 q^{81} + 13 q^{83} + q^{85} + q^{87} + 14 q^{89} - 5 q^{91} + 9 q^{93} + 8 q^{95} + 7 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 21x^{3} + 10x^{2} + 72x - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - \nu^{3} - 21\nu^{2} + 2\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + \nu^{3} - 23\nu^{2} - 24\nu + 68 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 20\nu^{2} - 13\nu + 46 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 6\beta_{2} + 14\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 22\beta_{4} - 20\beta_{3} - 40\beta_{2} + 33\beta _1 + 114 \) 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
1.41902
−2.41446
4.52713
−3.65928
1.12759
0 1.00000 0 −1.00000 0 −4.33241 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −2.61002 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 −1.35439 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 2.53251 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 3.76431 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(37\) \(1\)

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 8880.2.a.cl 5
4.b odd 2 1 4440.2.a.w 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4440.2.a.w 5 4.b odd 2 1
8880.2.a.cl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8880))\):

\( T_{7}^{5} + 2T_{7}^{4} - 22T_{7}^{3} - 36T_{7}^{2} + 101T_{7} + 146 \) Copy content Toggle raw display
\( T_{11}^{5} - T_{11}^{4} - 34T_{11}^{3} + 33T_{11}^{2} + 233T_{11} - 256 \) Copy content Toggle raw display
\( T_{13}^{5} + 4T_{13}^{4} - 36T_{13}^{3} - 94T_{13}^{2} + 339T_{13} + 298 \) Copy content Toggle raw display
\( T_{23}^{5} - 9T_{23}^{4} - 27T_{23}^{3} + 301T_{23}^{2} - 402T_{23} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 2 T^{4} - 22 T^{3} - 36 T^{2} + \cdots + 146 \) Copy content Toggle raw display
$11$ \( T^{5} - T^{4} - 34 T^{3} + 33 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} - 36 T^{3} - 94 T^{2} + \cdots + 298 \) Copy content Toggle raw display
$17$ \( T^{5} + T^{4} - 32 T^{3} - 53 T^{2} + \cdots + 298 \) Copy content Toggle raw display
$19$ \( T^{5} + 8 T^{4} - 22 T^{3} + \cdots + 1004 \) Copy content Toggle raw display
$23$ \( T^{5} - 9 T^{4} - 27 T^{3} + 301 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{5} - T^{4} - 66 T^{3} - 89 T^{2} + \cdots + 674 \) Copy content Toggle raw display
$31$ \( T^{5} - 9 T^{4} - 33 T^{3} + \cdots - 3188 \) Copy content Toggle raw display
$37$ \( (T + 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} - T^{4} - 65 T^{3} - 59 T^{2} + \cdots + 1984 \) Copy content Toggle raw display
$43$ \( T^{5} + 5 T^{4} - 116 T^{3} + \cdots + 16004 \) Copy content Toggle raw display
$47$ \( T^{5} - 7 T^{4} - 61 T^{3} + 279 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$53$ \( T^{5} - 3 T^{4} - 45 T^{3} - 21 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{5} - 6 T^{4} - 50 T^{3} + 421 T^{2} + \cdots + 436 \) Copy content Toggle raw display
$61$ \( T^{5} - 9 T^{4} - 77 T^{3} + 25 T^{2} + \cdots - 236 \) Copy content Toggle raw display
$67$ \( T^{5} - 11 T^{4} - 201 T^{3} + \cdots - 67328 \) Copy content Toggle raw display
$71$ \( T^{5} - 13 T^{4} - 47 T^{3} + \cdots - 13056 \) Copy content Toggle raw display
$73$ \( T^{5} - 22 T^{4} + 72 T^{3} + \cdots + 2776 \) Copy content Toggle raw display
$79$ \( T^{5} + 5 T^{4} - 155 T^{3} + \cdots - 7808 \) Copy content Toggle raw display
$83$ \( T^{5} - 13 T^{4} - 56 T^{3} + \cdots + 538 \) Copy content Toggle raw display
$89$ \( T^{5} - 14 T^{4} - 134 T^{3} + \cdots - 34366 \) Copy content Toggle raw display
$97$ \( T^{5} - 7 T^{4} - 195 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
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