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Properties

Label 25.36.b.b
Level $25$
Weight $36$
Character orbit 25.b
Analytic conductor $193.988$
Analytic rank $0$
Dimension $10$
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] = [25,36,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 36, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 36);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(193.987826584\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 64066974145 x^{8} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{12}\cdot 5^{24}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + 15 \beta_{5}) q^{2} + ( - \beta_{7} - 204 \beta_{6} - 8312 \beta_{5}) q^{3} + (\beta_{3} + 27 \beta_1 - 16893840948) q^{4} + (249 \beta_{4} - 1104 \beta_{3} + \cdots + 13487977581552) q^{6}+ \cdots + (168468 \beta_{4} + \cdots + 45\!\cdots\!43) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + 15 \beta_{5}) q^{2} + ( - \beta_{7} - 204 \beta_{6} - 8312 \beta_{5}) q^{3} + (\beta_{3} + 27 \beta_1 - 16893840948) q^{4} + (249 \beta_{4} - 1104 \beta_{3} + \cdots + 13487977581552) q^{6}+ \cdots + (43\!\cdots\!97 \beta_{4} + \cdots + 13\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 168938409480 q^{4} + 134879775815520 q^{6} + 45\!\cdots\!30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 168938409480 q^{4} + 134879775815520 q^{6} + 45\!\cdots\!30 q^{9}+ \cdots + 13\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 64066974145 x^{8} + \cdots + 24\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1555749796697 \nu^{8} + \cdots + 12\!\cdots\!12 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!35 \nu^{8} + \cdots + 19\!\cdots\!04 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 42005244510819 \nu^{8} + \cdots - 16\!\cdots\!40 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79\!\cdots\!17 \nu^{8} + \cdots - 10\!\cdots\!80 ) / 16\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!03 \nu^{9} + \cdots - 17\!\cdots\!88 \nu ) / 25\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 82\!\cdots\!15 \nu^{9} + \cdots + 10\!\cdots\!32 \nu ) / 85\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55\!\cdots\!25 \nu^{9} + \cdots - 11\!\cdots\!72 \nu ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 88\!\cdots\!35 \nu^{9} + \cdots - 66\!\cdots\!80 \nu ) / 35\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66\!\cdots\!77 \nu^{9} + \cdots - 17\!\cdots\!88 \nu ) / 11\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 15\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 27\beta _1 - 51253579316 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2078\beta_{9} + 19147\beta_{8} - 22151755\beta_{7} - 86818862097\beta_{6} - 1928246348922\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1159721196 \beta_{4} - 34283639793 \beta_{3} + 8276808600 \beta_{2} - 994978524083 \beta _1 + 12\!\cdots\!84 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 113246215352542 \beta_{9} - 704828671092811 \beta_{8} + \cdots + 60\!\cdots\!98 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 70\!\cdots\!12 \beta_{4} + \cdots - 34\!\cdots\!96 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 44\!\cdots\!82 \beta_{9} + \cdots - 19\!\cdots\!58 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 29\!\cdots\!76 \beta_{4} + \cdots + 10\!\cdots\!44 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 15\!\cdots\!18 \beta_{9} + \cdots + 63\!\cdots\!54 \beta_{5} ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 183732.i
− 126980.i
− 97796.2i
− 55659.7i
− 39032.3i
39032.3i
55659.7i
97796.2i
126980.i
183732.i
− 367464.i 2.64714e8i −1.00670e11 0 9.72731e13 − 7.92623e14i 2.43667e16i −2.00422e16 0
24.2 − 253960.i − 1.81015e8i −3.01358e10 0 −4.59705e13 4.79980e14i − 1.07272e15i 1.72651e16 0
24.3 − 195592.i 1.75923e8i −3.89669e9 0 3.44092e13 4.08781e14i − 5.95834e15i 1.90827e16 0
24.4 − 111319.i 4.77965e7i 2.19677e10 0 5.32068e12 − 4.16245e14i − 6.27034e15i 4.77470e16 0
24.5 − 78064.7i − 3.02218e8i 2.82656e10 0 −2.35925e13 − 4.58916e14i − 4.88883e15i −4.13040e16 0
24.6 78064.7i 3.02218e8i 2.82656e10 0 −2.35925e13 4.58916e14i 4.88883e15i −4.13040e16 0
24.7 111319.i − 4.77965e7i 2.19677e10 0 5.32068e12 4.16245e14i 6.27034e15i 4.77470e16 0
24.8 195592.i − 1.75923e8i −3.89669e9 0 3.44092e13 − 4.08781e14i 5.95834e15i 1.90827e16 0
24.9 253960.i 1.81015e8i −3.01358e10 0 −4.59705e13 − 4.79980e14i 1.07272e15i 1.72651e16 0
24.10 367464.i − 2.64714e8i −1.00670e11 0 9.72731e13 7.92623e14i − 2.43667e16i −2.00422e16 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.36.b.b 10
5.b even 2 1 inner 25.36.b.b 10
5.c odd 4 1 5.36.a.a ✓ 5
5.c odd 4 1 25.36.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.36.a.a ✓ 5 5.c odd 4 1
25.36.a.b 5 5.c odd 4 1
25.36.b.b 10 1.a even 1 1 trivial
25.36.b.b 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 256267896580 T_{2}^{8} + \cdots + 25\!\cdots\!76 \) acting on \(S_{36}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 57\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 45\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 85\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 42\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
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