Properties

Label 275.4.a.d
Level $275$
Weight $4$
Character orbit 275.a
Self dual yes
Analytic conductor $16.226$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1 - 1) q^{2} + (3 \beta_{2} + 2) q^{3} + (2 \beta_1 + 7) q^{4} + ( - \beta_{2} - 8 \beta_1 + 22) q^{6} + ( - 5 \beta_{2} - 14 \beta_1 + 8) q^{7} + ( - 5 \beta_{2} - 5 \beta_1 - 11) q^{8}+ \cdots + ( - 33 \beta_{2} + 198 \beta_1 - 341) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{2} + 3 q^{3} + 23 q^{4} + 59 q^{6} + 15 q^{7} - 33 q^{8} + 72 q^{9} - 33 q^{11} - 5 q^{12} + 14 q^{13} + 111 q^{14} - 149 q^{16} - 91 q^{17} + 256 q^{18} - 19 q^{19} - 39 q^{21} + 55 q^{22} - 330 q^{23}+ \cdots - 792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
3.12489
−0.363328
−1.76156
−4.60975 0.545414 13.2498 0 −2.51422 −33.3241 −24.2001 −26.7025 0
1.2 −3.77801 −7.42401 6.27334 0 28.0480 28.7933 6.52332 28.1159 0
1.3 3.38776 9.87859 3.47689 0 33.4663 19.5308 −15.3232 70.5866 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.d 3
3.b odd 2 1 2475.4.a.ba 3
5.b even 2 1 55.4.a.c 3
5.c odd 4 2 275.4.b.d 6
15.d odd 2 1 495.4.a.f 3
20.d odd 2 1 880.4.a.x 3
55.d odd 2 1 605.4.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.c 3 5.b even 2 1
275.4.a.d 3 1.a even 1 1 trivial
275.4.b.d 6 5.c odd 4 2
495.4.a.f 3 15.d odd 2 1
605.4.a.h 3 55.d odd 2 1
880.4.a.x 3 20.d odd 2 1
2475.4.a.ba 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 5T_{2}^{2} - 11T_{2} - 59 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(275))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 5 T^{2} + \cdots - 59 \) Copy content Toggle raw display
$3$ \( T^{3} - 3 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 15 T^{2} + \cdots + 18740 \) Copy content Toggle raw display
$11$ \( (T + 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 14 T^{2} + \cdots - 50800 \) Copy content Toggle raw display
$17$ \( T^{3} + 91 T^{2} + \cdots - 1256 \) Copy content Toggle raw display
$19$ \( T^{3} + 19 T^{2} + \cdots - 549824 \) Copy content Toggle raw display
$23$ \( T^{3} + 330 T^{2} + \cdots - 1346224 \) Copy content Toggle raw display
$29$ \( T^{3} - 413 T^{2} + \cdots - 296164 \) Copy content Toggle raw display
$31$ \( T^{3} + 127 T^{2} + \cdots - 7744 \) Copy content Toggle raw display
$37$ \( T^{3} - 543 T^{2} + \cdots - 4866188 \) Copy content Toggle raw display
$41$ \( T^{3} - 58 T^{2} + \cdots + 8259848 \) Copy content Toggle raw display
$43$ \( T^{3} + 532 T^{2} + \cdots - 26232736 \) Copy content Toggle raw display
$47$ \( T^{3} + 478 T^{2} + \cdots - 907024 \) Copy content Toggle raw display
$53$ \( T^{3} + 261 T^{2} + \cdots + 581092 \) Copy content Toggle raw display
$59$ \( T^{3} + 630 T^{2} + \cdots - 215392 \) Copy content Toggle raw display
$61$ \( T^{3} - 1095 T^{2} + \cdots - 20829740 \) Copy content Toggle raw display
$67$ \( T^{3} - 976 T^{2} + \cdots - 20007488 \) Copy content Toggle raw display
$71$ \( T^{3} + 445 T^{2} + \cdots + 69739136 \) Copy content Toggle raw display
$73$ \( T^{3} - 1838 T^{2} + \cdots - 163030384 \) Copy content Toggle raw display
$79$ \( T^{3} - 318 T^{2} + \cdots + 502905248 \) Copy content Toggle raw display
$83$ \( T^{3} - 330 T^{2} + \cdots - 37348312 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 1186825660 \) Copy content Toggle raw display
$97$ \( T^{3} - 2744 T^{2} + \cdots + 146434160 \) Copy content Toggle raw display
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