Properties

Label 275.4.a.e
Level $275$
Weight $4$
Character orbit 275.a
Self dual yes
Analytic conductor $16.226$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,4,Mod(1,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(16.2255252516\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1539480.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 25x^{2} + 9x + 96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 2) q^{3} + (\beta_{3} + \beta_1 + 5) q^{4} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{6} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{7} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots - 10) q^{8}+ \cdots + ( - 33 \beta_{2} - 66 \beta_1 + 143) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 9 q^{3} + 19 q^{4} - 19 q^{6} - 9 q^{7} - 33 q^{8} + 49 q^{9} + 44 q^{11} - 75 q^{12} - 70 q^{13} - 49 q^{14} - 37 q^{16} - 103 q^{17} + 356 q^{18} - 205 q^{19} - 181 q^{21} - 11 q^{22} + 56 q^{23}+ \cdots + 539 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 25x^{2} + 9x + 96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 15\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_{2} + 17\beta _1 + 10 \) 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
4.90807
2.30365
−2.03085
−4.18087
−4.90807 1.10798 16.0892 0 −5.43806 17.0703 −39.7022 −25.7724 0
1.2 −2.30365 −6.23583 −2.69320 0 14.3652 −13.1506 24.6334 11.8856 0
1.3 2.03085 5.45953 −3.87566 0 11.0875 −27.2109 −24.1176 2.80648 0
1.4 4.18087 −9.33168 9.47970 0 −39.0146 14.2911 6.18645 60.0803 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 275.4.a.e 4
3.b odd 2 1 2475.4.a.bc 4
5.b even 2 1 55.4.a.d 4
5.c odd 4 2 275.4.b.e 8
15.d odd 2 1 495.4.a.n 4
20.d odd 2 1 880.4.a.z 4
55.d odd 2 1 605.4.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.d 4 5.b even 2 1
275.4.a.e 4 1.a even 1 1 trivial
275.4.b.e 8 5.c odd 4 2
495.4.a.n 4 15.d odd 2 1
605.4.a.j 4 55.d odd 2 1
880.4.a.z 4 20.d odd 2 1
2475.4.a.bc 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} - 25T_{2}^{2} - 9T_{2} + 96 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(275))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + \cdots + 96 \) Copy content Toggle raw display
$3$ \( T^{4} + 9 T^{3} + \cdots + 352 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 9 T^{3} + \cdots + 87296 \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 70 T^{3} + \cdots - 2887232 \) Copy content Toggle raw display
$17$ \( T^{4} + 103 T^{3} + \cdots + 199152 \) Copy content Toggle raw display
$19$ \( T^{4} + 205 T^{3} + \cdots - 5224000 \) Copy content Toggle raw display
$23$ \( T^{4} - 56 T^{3} + \cdots - 221568 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 1106175480 \) Copy content Toggle raw display
$31$ \( T^{4} - 49 T^{3} + \cdots + 126259200 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 4092453032 \) Copy content Toggle raw display
$41$ \( T^{4} - 736 T^{3} + \cdots + 329735184 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 2589511936 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 19971136128 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 20698646424 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 123367943040 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 4578287464 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 316737807616 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 1139751168 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 138483587488 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 289419632640 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 340395563136 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 27515045400 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 144669247168 \) Copy content Toggle raw display
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