Properties

Label 5.34.b.a
Level $5$
Weight $34$
Character orbit 5.b
Analytic conductor $34.491$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5,34,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), 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: \(34.4914144405\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 26286285043 x^{14} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{26}\cdot 5^{53}\cdot 7^{4}\cdot 11^{8} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 52 \beta_1) q^{3} + (\beta_{2} - 4553207930) q^{4} + ( - \beta_{4} - 70 \beta_{3} + 3 \beta_{2} + 105885 \beta_1 - 14510510019) q^{5} + (\beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 176 \beta_{2} + \cdots + 687611084267) q^{6}+ \cdots + ( - \beta_{9} - 295 \beta_{5} - 3119 \beta_{4} + \cdots - 17\!\cdots\!57) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 52 \beta_1) q^{3} + (\beta_{2} - 4553207930) q^{4} + ( - \beta_{4} - 70 \beta_{3} + 3 \beta_{2} + 105885 \beta_1 - 14510510019) q^{5} + (\beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 176 \beta_{2} + \cdots + 687611084267) q^{6}+ \cdots + ( - 53188529362764 \beta_{14} + \cdots + 21\!\cdots\!11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 72851326872 q^{4} - 232168160280 q^{5} + 11001777346872 q^{6} - 27\!\cdots\!68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 72851326872 q^{4} - 232168160280 q^{5} + 11001777346872 q^{6} - 27\!\cdots\!68 q^{9}+ \cdots + 34\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 26286285043 x^{14} + \cdots + 16\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 13143142522 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37\!\cdots\!75 \nu^{15} + \cdots - 74\!\cdots\!40 \nu ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30\!\cdots\!91 \nu^{15} + \cdots + 86\!\cdots\!88 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 60\!\cdots\!57 \nu^{15} + \cdots + 13\!\cdots\!24 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42\!\cdots\!31 \nu^{15} + \cdots + 12\!\cdots\!92 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!81 \nu^{15} + \cdots + 30\!\cdots\!08 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45\!\cdots\!93 \nu^{15} + \cdots - 17\!\cdots\!76 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!37 \nu^{15} + \cdots - 10\!\cdots\!16 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!09 \nu^{15} + \cdots - 37\!\cdots\!88 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!49 \nu^{15} + \cdots + 41\!\cdots\!68 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 81\!\cdots\!53 \nu^{15} + \cdots - 21\!\cdots\!96 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 37\!\cdots\!61 \nu^{15} + \cdots - 86\!\cdots\!48 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 53\!\cdots\!73 \nu^{15} + \cdots - 10\!\cdots\!64 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!99 \nu^{15} + \cdots - 85\!\cdots\!08 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 13143142522 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} - \beta_{8} - 2\beta_{5} + 473\beta_{4} + 1699774\beta_{3} - 90\beta_{2} - 20788434785\beta _1 + 46 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{14} + 4 \beta_{13} - \beta_{12} - 420 \beta_{11} + 2628 \beta_{10} + 1101 \beta_{9} + 4876 \beta_{8} - 1265 \beta_{7} + 349 \beta_{6} + 153948 \beta_{5} - 18509684 \beta_{4} - 7294339 \beta_{3} + \cdots + 27\!\cdots\!42 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 339944 \beta_{15} + 143566 \beta_{14} + 53300 \beta_{13} + 11091950 \beta_{11} - 8112256005 \beta_{10} + 266500 \beta_{9} + 12443321181 \beta_{8} + 82506078 \beta_{7} + \cdots - 461840191346 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 38691410076 \beta_{14} - 38691410076 \beta_{13} + 13223250699 \beta_{12} + 4162009207020 \beta_{11} - 32986056615036 \beta_{10} + \cdots - 15\!\cdots\!10 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 31\!\cdots\!88 \beta_{15} + \cdots + 33\!\cdots\!42 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 26\!\cdots\!96 \beta_{14} + \cdots + 96\!\cdots\!14 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 22\!\cdots\!04 \beta_{15} + \cdots - 21\!\cdots\!78 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15\!\cdots\!40 \beta_{14} + \cdots - 59\!\cdots\!06 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15\!\cdots\!40 \beta_{15} + \cdots + 13\!\cdots\!10 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 87\!\cdots\!60 \beta_{14} + \cdots + 37\!\cdots\!78 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 98\!\cdots\!80 \beta_{15} + \cdots - 80\!\cdots\!46 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 45\!\cdots\!64 \beta_{14} + \cdots - 23\!\cdots\!42 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 62\!\cdots\!04 \beta_{15} + \cdots + 47\!\cdots\!66 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
81004.3i
78261.9i
78034.2i
57471.0i
51228.2i
31348.8i
20202.3i
13863.7i
13863.7i
20202.3i
31348.8i
51228.2i
57471.0i
78034.2i
78261.9i
81004.3i
162009.i 1.50582e7i −1.76568e10 3.29148e11 + 8.98715e10i −2.43956e12 1.33679e14i 1.46891e15i 5.33231e15 1.45600e16 5.33248e16i
4.2 156524.i 1.41317e8i −1.59098e10 −2.87690e11 1.83439e11i 2.21195e13 8.25005e13i 1.14573e15i −1.44114e16 −2.87126e16 + 4.50303e16i
4.3 156068.i 6.96239e7i −1.57674e10 −2.14154e11 2.65619e11i −1.08661e13 1.55193e14i 1.12018e15i 7.11573e14 −4.14548e16 + 3.34226e16i
4.4 114942.i 6.13071e7i −4.62174e9 1.22948e11 + 3.18275e11i 7.04677e12 1.47106e14i 4.56113e14i 1.80050e15 3.65832e16 1.41318e16i
4.5 102456.i 8.77372e7i −1.90737e9 −2.51871e11 + 2.30165e11i −8.98924e12 1.11815e14i 6.84671e14i −2.13876e15 2.35819e16 + 2.58058e16i
4.6 62697.7i 3.24591e7i 4.65894e9 1.34579e11 3.13534e11i 2.03511e12 2.03055e13i 8.30673e14i 4.50547e15 −1.96579e16 8.43781e15i
4.7 40404.6i 1.28550e8i 6.95740e9 3.40199e11 2.60749e10i −5.19400e12 4.50276e13i 6.28184e14i −1.09659e16 −1.05355e15 1.37456e16i
4.8 27727.3i 6.45007e7i 7.82113e9 −2.89243e11 + 1.80980e11i 1.78843e12 3.09117e13i 4.55035e14i 1.39871e15 5.01809e15 + 8.01994e15i
4.9 27727.3i 6.45007e7i 7.82113e9 −2.89243e11 1.80980e11i 1.78843e12 3.09117e13i 4.55035e14i 1.39871e15 5.01809e15 8.01994e15i
4.10 40404.6i 1.28550e8i 6.95740e9 3.40199e11 + 2.60749e10i −5.19400e12 4.50276e13i 6.28184e14i −1.09659e16 −1.05355e15 + 1.37456e16i
4.11 62697.7i 3.24591e7i 4.65894e9 1.34579e11 + 3.13534e11i 2.03511e12 2.03055e13i 8.30673e14i 4.50547e15 −1.96579e16 + 8.43781e15i
4.12 102456.i 8.77372e7i −1.90737e9 −2.51871e11 2.30165e11i −8.98924e12 1.11815e14i 6.84671e14i −2.13876e15 2.35819e16 2.58058e16i
4.13 114942.i 6.13071e7i −4.62174e9 1.22948e11 3.18275e11i 7.04677e12 1.47106e14i 4.56113e14i 1.80050e15 3.65832e16 + 1.41318e16i
4.14 156068.i 6.96239e7i −1.57674e10 −2.14154e11 + 2.65619e11i −1.08661e13 1.55193e14i 1.12018e15i 7.11573e14 −4.14548e16 3.34226e16i
4.15 156524.i 1.41317e8i −1.59098e10 −2.87690e11 + 1.83439e11i 2.21195e13 8.25005e13i 1.14573e15i −1.44114e16 −2.87126e16 4.50303e16i
4.16 162009.i 1.50582e7i −1.76568e10 3.29148e11 8.98715e10i −2.43956e12 1.33679e14i 1.46891e15i 5.33231e15 1.45600e16 + 5.33248e16i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
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 5.34.b.a 16
5.b even 2 1 inner 5.34.b.a 16
5.c odd 4 2 25.34.a.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.34.b.a 16 1.a even 1 1 trivial
5.34.b.a 16 5.b even 2 1 inner
25.34.a.f 16 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{34}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 105145140172 T^{14} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{16} + 232168160280 T^{15} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 26\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 77\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 28\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 88\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 85\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
show more
show less