↑

Properties

Label 3.25.b.b
Level $3$
Weight $25$
Character orbit 3.b
Analytic conductor $10.949$
Analytic rank $0$
Dimension $6$
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] = [3,25,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(10.9490145677\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{13}\cdot 3^{24}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 16 \beta_1 - 102807) q^{3} + (\beta_{5} - 8 \beta_{3} + \cdots - 10557800) q^{4}+ \cdots + (31836 \beta_{5} - 603 \beta_{4} + \cdots - 131864491647) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 16 \beta_1 - 102807) q^{3} + (\beta_{5} - 8 \beta_{3} + \cdots - 10557800) q^{4}+ \cdots + ( - 33\!\cdots\!72 \beta_{5} + \cdots - 30\!\cdots\!20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 616842 q^{3} - 63346800 q^{4} - 2673890352 q^{6} + 1988064876 q^{7} - 791186949882 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 616842 q^{3} - 63346800 q^{4} - 2673890352 q^{6} + 1988064876 q^{7} - 791186949882 q^{9} + 2770823311200 q^{10} - 18492861842352 q^{12} - 73542438063924 q^{13} + 227151045057600 q^{15} - 433335898904448 q^{16} - 12\!\cdots\!40 q^{18}+ \cdots - 18\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 253102x^{4} + 17425276096x^{2} + 250659115499520 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 18\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 927 \nu^{5} - 22464 \nu^{4} + 158192658 \nu^{3} - 8746814592 \nu^{2} + 4583765471040 \nu - 519220416540672 ) / 1041302528 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 927 \nu^{5} + 162432 \nu^{4} - 158192658 \nu^{3} + 11341271808 \nu^{2} + \cdots - 624717182509056 ) / 520651264 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 129501 \nu^{5} + 252288 \nu^{4} + 36839351862 \nu^{3} + 46328530176 \nu^{2} + \cdots + 14\!\cdots\!32 ) / 520651264 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 927 \nu^{5} + 2217024 \nu^{4} + 158192658 \nu^{3} + 370146519936 \nu^{2} + \cdots + 96\!\cdots\!28 ) / 1041302528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 8\beta_{3} - 17\beta_{2} + 8\beta _1 - 27335016 ) / 324 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -736\beta_{5} + 103\beta_{4} + 3393\beta_{3} - 21256\beta_{2} - 18690797\beta_1 ) / 2916 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -250237\beta_{5} + 18272248\beta_{3} + 36794733\beta_{2} - 18272248\beta _1 + 42588282369096 ) / 4374 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 103238176\beta_{5} - 8788481\beta_{4} - 481592855\beta_{3} + 3026837432\beta_{2} + 1194462231243\beta_1 ) / 1458 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 380.197i
− 298.476i
− 139.516i
139.516i
298.476i
380.197i
− 6843.54i 162008. − 506145.i −3.00569e7 3.79015e8i −3.46383e9 − 1.10871e9i −1.14171e10 9.08798e10i −2.29937e11 − 1.63999e11i 2.59380e12
2.2 − 5372.56i −24323.9 + 530884.i −1.20872e7 − 5.28531e7i 2.85221e9 + 1.30681e8i 2.07202e10 − 2.51975e10i −2.81246e11 − 2.58263e10i −2.83957e11
2.3 − 2511.29i −446105. − 288825.i 1.04706e7 − 3.68111e8i −7.25325e8 + 1.12030e9i −8.30906e9 − 6.84273e10i 1.15589e11 + 2.57693e11i −9.24434e11
2.4 2511.29i −446105. + 288825.i 1.04706e7 3.68111e8i −7.25325e8 − 1.12030e9i −8.30906e9 6.84273e10i 1.15589e11 − 2.57693e11i −9.24434e11
2.5 5372.56i −24323.9 − 530884.i −1.20872e7 5.28531e7i 2.85221e9 − 1.30681e8i 2.07202e10 2.51975e10i −2.81246e11 + 2.58263e10i −2.83957e11
2.6 6843.54i 162008. + 506145.i −3.00569e7 − 3.79015e8i −3.46383e9 + 1.10871e9i −1.14171e10 − 9.08798e10i −2.29937e11 + 1.63999e11i 2.59380e12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

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 3.25.b.b ✓ 6
3.b odd 2 1 inner 3.25.b.b ✓ 6
4.b odd 2 1 48.25.e.b 6
12.b even 2 1 48.25.e.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.25.b.b ✓ 6 1.a even 1 1 trivial
3.25.b.b ✓ 6 3.b odd 2 1 inner
48.25.e.b 6 4.b odd 2 1
48.25.e.b 6 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 82005048T_{2}^{4} + 1829235783453696T_{2}^{2} + 8525473984011546132480 \) acting on \(S_{25}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + \cdots + 85\!\cdots\!80 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 22\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{3} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots - 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 54\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots + 24\!\cdots\!72)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 26\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 53\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 18\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 25\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 29\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less