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Properties

Label 4725.2.a.cf
Level $4725$
Weight $2$
Character orbit 4725.a
Self dual yes
Analytic conductor $37.729$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4725,2,Mod(1,4725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4725, 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("4725.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4725 = 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4725.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: \(37.7293149551\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 15x^{6} + 68x^{4} - 91x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 945)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + q^{7} + (\beta_{4} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + q^{7} + (\beta_{4} + 2 \beta_1) q^{8} + (\beta_{6} - \beta_{4} + \beta_{2}) q^{11} + (\beta_{5} + 1) q^{13} + \beta_1 q^{14} + (\beta_{5} + 2 \beta_{3} + 4) q^{16} + ( - \beta_{6} + \beta_1) q^{17} + ( - \beta_{7} + 2) q^{19} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{22} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{23} + (\beta_{6} + \beta_{4} - 2 \beta_{2}) q^{26} + (\beta_{3} + 2) q^{28} + ( - \beta_{4} - \beta_1) q^{29} + (\beta_{7} + \beta_{3} + 3) q^{31} + (\beta_{6} + \beta_{4} + \cdots + 3 \beta_1) q^{32}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 14 q^{4} + 8 q^{7} + 10 q^{13} + 30 q^{16} + 18 q^{19} + 2 q^{22} + 14 q^{28} + 20 q^{31} + 22 q^{34} - 8 q^{37} + 20 q^{46} + 8 q^{49} + 2 q^{52} - 28 q^{58} + 38 q^{61} + 52 q^{64} + 10 q^{67} + 24 q^{73} + 10 q^{76} + 26 q^{79} + 14 q^{82} - 72 q^{88} + 10 q^{91} + 70 q^{94} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 15x^{6} + 68x^{4} - 91x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 10\nu^{5} - 18\nu^{3} - 19\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} - 8\nu^{2} + 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 15\nu^{5} - 63\nu^{3} + 56\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} - 12\nu^{4} + 38\nu^{2} - 23 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 8\beta_{3} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + 9\beta_{4} - 2\beta_{2} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 12\beta_{5} + 58\beta_{3} + 159 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{6} + 72\beta_{4} - 30\beta_{2} + 263\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.73099
−2.35037
−1.28439
−0.606483
0.606483
1.28439
2.35037
2.73099
−2.73099 0 5.45830 0 0 1.00000 −9.44459 0 0
1.2 −2.35037 0 3.52423 0 0 1.00000 −3.58249 0 0
1.3 −1.28439 0 −0.350351 0 0 1.00000 3.01876 0 0
1.4 −0.606483 0 −1.63218 0 0 1.00000 2.20285 0 0
1.5 0.606483 0 −1.63218 0 0 1.00000 −2.20285 0 0
1.6 1.28439 0 −0.350351 0 0 1.00000 −3.01876 0 0
1.7 2.35037 0 3.52423 0 0 1.00000 3.58249 0 0
1.8 2.73099 0 5.45830 0 0 1.00000 9.44459 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4725.2.a.cf 8
3.b odd 2 1 inner 4725.2.a.cf 8
5.b even 2 1 4725.2.a.ce 8
5.c odd 4 2 945.2.d.f ✓ 16
15.d odd 2 1 4725.2.a.ce 8
15.e even 4 2 945.2.d.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.d.f ✓ 16 5.c odd 4 2
945.2.d.f ✓ 16 15.e even 4 2
4725.2.a.ce 8 5.b even 2 1
4725.2.a.ce 8 15.d odd 2 1
4725.2.a.cf 8 1.a even 1 1 trivial
4725.2.a.cf 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4725))\):

\( T_{2}^{8} - 15T_{2}^{6} + 68T_{2}^{4} - 91T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{8} - 72T_{11}^{6} + 1590T_{11}^{4} - 11464T_{11}^{2} + 5625 \) Copy content Toggle raw display
\( T_{13}^{4} - 5T_{13}^{3} - 31T_{13}^{2} + 112T_{13} + 212 \) Copy content Toggle raw display
\( T_{37}^{4} + 4T_{37}^{3} - 52T_{37}^{2} - 180T_{37} - 85 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 15 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 72 T^{6} + \cdots + 5625 \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} + \cdots + 212)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 45 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$19$ \( (T^{4} - 9 T^{3} + \cdots + 381)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 71 T^{6} + \cdots + 7569 \) Copy content Toggle raw display
$29$ \( T^{8} - 73 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} + \cdots + 325)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} - 52 T^{2} + \cdots - 85)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 167 T^{6} + \cdots + 265225 \) Copy content Toggle raw display
$43$ \( (T^{4} - 49 T^{2} + \cdots + 232)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 146 T^{6} + \cdots + 876096 \) Copy content Toggle raw display
$53$ \( T^{8} - 61 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$59$ \( T^{8} - 106 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{4} - 19 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{3} - 7 T^{2} + \cdots + 20)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 439 T^{6} + \cdots + 994009 \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} + \cdots + 1032)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 13 T^{3} + \cdots - 7880)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 202 T^{6} + \cdots + 115600 \) Copy content Toggle raw display
$89$ \( T^{8} - 308 T^{6} + \cdots + 4601025 \) Copy content Toggle raw display
$97$ \( (T^{4} + 9 T^{3} + \cdots + 12016)^{2} \) Copy content Toggle raw display
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