Properties

Label 1148.2.a.e
Level $1148$
Weight $2$
Character orbit 1148.a
Self dual yes
Analytic conductor $9.167$
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] = [1148,2,Mod(1,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, 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("1148.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1935333.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 13x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{4} + 1) q^{5} + q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{4} + 1) q^{5} + q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} - \beta_{2} q^{11} + (\beta_{3} + 1) q^{13} + (\beta_{4} + \beta_{2} + 1) q^{15} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{3} - \beta_1 + 2) q^{19} + \beta_1 q^{21} + (\beta_1 + 2) q^{23} + (\beta_{4} + 2 \beta_{3} - \beta_{2}) q^{25} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{27} + ( - \beta_{4} - \beta_{2} - 1) q^{29} + ( - \beta_{4} + 1) q^{31} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 1) q^{33} + (\beta_{4} + 1) q^{35} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{37} + ( - \beta_{2} + \beta_1 + 1) q^{39} + q^{41} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{43} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{45} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{47} + q^{49} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{51} + ( - \beta_{4} - 1) q^{53} + (\beta_{4} - 2 \beta_1 + 3) q^{55} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{57} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 1) q^{59} + ( - 2 \beta_{4} - 4 \beta_{3} + \beta_{2} + 4) q^{61} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{63} + (3 \beta_{4} + 2 \beta_{3} + 1) q^{65} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 5) q^{67} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 4) q^{69} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{71} + ( - 2 \beta_{4} + \beta_{2}) q^{73} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{75} - \beta_{2} q^{77} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{79} + (3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 1) q^{81} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{83} + ( - \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{85} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{87} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} - 4 \beta_1) q^{89} + (\beta_{3} + 1) q^{91} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{93} + (3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{95} + ( - 3 \beta_1 + 2) q^{97} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{13} + 3 q^{15} - 3 q^{17} + 10 q^{19} + 2 q^{21} + 12 q^{23} + 2 q^{25} + 14 q^{27} - 3 q^{29} + 7 q^{31} - 3 q^{33} + 3 q^{35} + q^{37} + 7 q^{39} + 5 q^{41} + 13 q^{43} - 3 q^{45} + 9 q^{47} + 5 q^{49} + 9 q^{51} - 3 q^{53} + 9 q^{55} - 11 q^{57} - 3 q^{59} + 16 q^{61} + 5 q^{63} + 3 q^{65} + 19 q^{67} + 24 q^{69} - 12 q^{71} + 4 q^{73} + 8 q^{75} + 28 q^{79} + 5 q^{81} + 18 q^{83} - 15 q^{85} - 6 q^{87} + 7 q^{91} + q^{93} + 3 q^{95} + 4 q^{97} - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 13x - 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 9\nu - 8 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 5\nu^{2} + 12\nu + 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 11\nu^{2} - 3\nu + 16 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 3\beta_{3} + 2\beta_{2} + 7\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{4} - 11\beta_{3} + 11\beta_{2} + 14\beta _1 + 28 \) 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
−2.10409
−1.43444
0.704110
1.60064
3.23378
0 −2.10409 0 −1.26228 0 1.00000 0 1.42721 0
1.2 0 −1.43444 0 1.63444 0 1.00000 0 −0.942381 0
1.3 0 0.704110 0 3.89333 0 1.00000 0 −2.50423 0
1.4 0 1.60064 0 −2.47347 0 1.00000 0 −0.437946 0
1.5 0 3.23378 0 1.20798 0 1.00000 0 7.45735 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
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

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 1148.2.a.e 5
4.b odd 2 1 4592.2.a.bd 5
7.b odd 2 1 8036.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.e 5 1.a even 1 1 trivial
4592.2.a.bd 5 4.b odd 2 1
8036.2.a.j 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 2T_{3}^{4} - 8T_{3}^{3} + 10T_{3}^{2} + 13T_{3} - 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1148))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} - 8 T^{3} + 10 T^{2} + \cdots - 11 \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} - 9 T^{3} + 20 T^{2} + \cdots - 24 \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 21 T^{3} + 4 T^{2} + 84 T - 72 \) Copy content Toggle raw display
$13$ \( T^{5} - 7 T^{4} + 7 T^{3} + 38 T^{2} + \cdots + 29 \) Copy content Toggle raw display
$17$ \( T^{5} + 3 T^{4} - 45 T^{3} - 72 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{5} - 10 T^{4} + 22 T^{3} + 30 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$23$ \( T^{5} - 12 T^{4} + 48 T^{3} - 70 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$29$ \( T^{5} + 3 T^{4} - 21 T^{3} - 64 T^{2} + \cdots + 264 \) Copy content Toggle raw display
$31$ \( T^{5} - 7 T^{4} + 7 T^{3} + 26 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$37$ \( T^{5} - T^{4} - 56 T^{3} - 145 T^{2} + \cdots + 188 \) Copy content Toggle raw display
$41$ \( (T - 1)^{5} \) Copy content Toggle raw display
$43$ \( T^{5} - 13 T^{4} + 19 T^{3} + \cdots - 593 \) Copy content Toggle raw display
$47$ \( T^{5} - 9 T^{4} - 66 T^{3} + 107 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$53$ \( T^{5} + 3 T^{4} - 9 T^{3} - 20 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$59$ \( T^{5} + 3 T^{4} - 87 T^{3} - 178 T^{2} + \cdots - 792 \) Copy content Toggle raw display
$61$ \( T^{5} - 16 T^{4} - 131 T^{3} + \cdots + 35992 \) Copy content Toggle raw display
$67$ \( T^{5} - 19 T^{4} + 25 T^{3} + \cdots - 904 \) Copy content Toggle raw display
$71$ \( T^{5} + 12 T^{4} - 69 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$73$ \( T^{5} - 4 T^{4} - 83 T^{3} + \cdots - 3832 \) Copy content Toggle raw display
$79$ \( T^{5} - 28 T^{4} + 184 T^{3} + \cdots + 8352 \) Copy content Toggle raw display
$83$ \( T^{5} - 18 T^{4} + 3 T^{3} + \cdots + 9000 \) Copy content Toggle raw display
$89$ \( T^{5} - 228 T^{3} + 1284 T^{2} + \cdots - 3753 \) Copy content Toggle raw display
$97$ \( T^{5} - 4 T^{4} - 80 T^{3} + 226 T^{2} + \cdots + 127 \) Copy content Toggle raw display
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