Properties

Label 25.34.a.e
Level $25$
Weight $34$
Character orbit 25.a
Self dual yes
Analytic conductor $172.457$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,34,Mod(1,25)]
 
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([0]))
 
N = Newforms(chi, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.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: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 71909733210 x^{9} - 392361971317440 x^{8} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{47}\cdot 3^{12}\cdot 5^{30}\cdot 7^{2}\cdot 11^{3} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 854) q^{2} + ( - \beta_{2} + 124 \beta_1 + 7302793) q^{3} + (\beta_{3} - 8 \beta_{2} + 6477 \beta_1 + 4485290926) q^{4} + (\beta_{4} - 201 \beta_{3} + 5750 \beta_{2} + \cdots - 1616724083345) q^{6}+ \cdots + (\beta_{7} + 2 \beta_{6} + \beta_{5} - 223 \beta_{4} + \cdots + 22\!\cdots\!62) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 854) q^{2} + ( - \beta_{2} + 124 \beta_1 + 7302793) q^{3} + (\beta_{3} - 8 \beta_{2} + 6477 \beta_1 + 4485290926) q^{4} + (\beta_{4} - 201 \beta_{3} + 5750 \beta_{2} + \cdots - 1616724083345) q^{6}+ \cdots + (23802223156536 \beta_{10} + \cdots + 11\!\cdots\!38) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 9393 q^{2} + 80330849 q^{3} + 49338206677 q^{4} - 17783977676723 q^{6} + 21344025107658 q^{7} - 970199832420105 q^{8} + 24\!\cdots\!58 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 9393 q^{2} + 80330849 q^{3} + 49338206677 q^{4} - 17783977676723 q^{6} + 21344025107658 q^{7} - 970199832420105 q^{8} + 24\!\cdots\!58 q^{9}+ \cdots + 13\!\cdots\!06 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - x^{10} - 71909733210 x^{9} - 392361971317440 x^{8} + \cdots + 13\!\cdots\!48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\!\cdots\!81 \nu^{10} + \cdots - 15\!\cdots\!48 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29\!\cdots\!81 \nu^{10} + \cdots - 51\!\cdots\!08 ) / 38\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{10} + \cdots - 27\!\cdots\!28 ) / 30\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 65\!\cdots\!73 \nu^{10} + \cdots - 33\!\cdots\!80 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!23 \nu^{10} + \cdots + 13\!\cdots\!16 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 83\!\cdots\!47 \nu^{10} + \cdots - 61\!\cdots\!84 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17\!\cdots\!37 \nu^{10} + \cdots - 60\!\cdots\!04 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26\!\cdots\!99 \nu^{10} + \cdots - 61\!\cdots\!84 ) / 61\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 52\!\cdots\!87 \nu^{10} + \cdots + 14\!\cdots\!56 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 8\beta_{2} + 8185\beta _1 + 13074496202 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{6} + \beta_{5} - 21\beta_{4} + 18185\beta_{3} - 1311661\beta_{2} + 22152168592\beta _1 + 107025408036041 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 26 \beta_{10} + 57 \beta_{9} - 259 \beta_{8} - 1245 \beta_{7} + 75517 \beta_{6} - 7678 \beta_{5} - 1361202 \beta_{4} + 30371982236 \beta_{3} - 480354168056 \beta_{2} + \cdots + 28\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 522454 \beta_{10} + 820053 \beta_{9} - 12720239 \beta_{8} + 720283927 \beta_{7} + 139649806935 \beta_{6} + 36933048324 \beta_{5} - 1327607518252 \beta_{4} + \cdots + 59\!\cdots\!62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1021661423510 \beta_{10} + 2937492549339 \beta_{9} - 12067437892993 \beta_{8} - 56702122654215 \beta_{7} + \cdots + 74\!\cdots\!70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 18\!\cdots\!78 \beta_{10} + \cdots + 23\!\cdots\!66 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32\!\cdots\!82 \beta_{10} + \cdots + 20\!\cdots\!22 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 50\!\cdots\!06 \beta_{10} + \cdots + 80\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 97\!\cdots\!18 \beta_{10} + \cdots + 58\!\cdots\!54 \) 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
178021.
154848.
110093.
62096.8
42069.3
211.095
−33894.6
−99441.3
−103767.
−139697.
−170539.
−177167. 1.20911e8 2.27982e10 0 −2.14214e13 −1.11259e14 −2.51723e15 9.06036e15 0
1.2 −153994. −7.94763e7 1.51243e10 0 1.22389e13 4.85062e13 −1.00626e15 7.57418e14 0
1.3 −109239. 1.96178e7 3.34318e9 0 −2.14302e12 −7.24227e12 5.73150e14 −5.17420e15 0
1.4 −61242.8 1.16397e8 −4.83925e9 0 −7.12847e12 1.52684e14 8.22441e14 7.98915e15 0
1.5 −41215.3 −1.10688e8 −6.89123e9 0 4.56203e12 −7.02715e13 6.38061e14 6.69270e15 0
1.6 642.905 5.85306e7 −8.58952e9 0 3.76296e10 −1.03914e14 −1.10448e13 −2.13323e15 0
1.7 34748.6 −4.98225e7 −7.38247e9 0 −1.73126e12 1.47813e14 −5.55018e14 −3.07678e15 0
1.8 100295. 1.18201e8 1.46921e9 0 1.18550e13 1.07417e13 −7.14175e14 8.41244e15 0
1.9 104621. −2.08087e7 2.35554e9 0 −2.17702e12 −5.34780e13 −6.52247e14 −5.12606e15 0
1.10 140551. −1.29136e8 1.11647e10 0 −1.81502e13 −4.00085e13 3.61884e14 1.11170e16 0
1.11 171393. 3.66048e7 2.07856e10 0 6.27381e12 4.77732e13 2.09024e15 −4.21915e15 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

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

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 25.34.a.e yes 11
5.b even 2 1 25.34.a.d 11
5.c odd 4 2 25.34.b.d 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.34.a.d 11 5.b even 2 1
25.34.a.e yes 11 1.a even 1 1 trivial
25.34.b.d 22 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - 9393 T_{2}^{10} - 71869629370 T_{2}^{9} + 944957445671160 T_{2}^{8} + \cdots + 42\!\cdots\!32 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} - 9393 T^{10} + \cdots + 42\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{11} - 80330849 T^{10} + \cdots + 82\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{11} \) Copy content Toggle raw display
$7$ \( T^{11} - 21344025107658 T^{10} + \cdots - 70\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 13\!\cdots\!57 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 39\!\cdots\!13 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 26\!\cdots\!75 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots - 24\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 23\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 20\!\cdots\!53 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 18\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 26\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 46\!\cdots\!13 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 61\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 80\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 92\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 17\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 12\!\cdots\!75 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 85\!\cdots\!72 \) Copy content Toggle raw display
show more
show less