↑

Properties

Label 3.31.b.b
Level $3$
Weight $31$
Character orbit 3.b
Analytic conductor $17.104$
Analytic rank $0$
Dimension $8$
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] = [3,31,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, 31, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 31);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 31 \)
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: \(17.1042785708\)
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: \( x^{8} + 4973352x^{6} + 7969212867576x^{4} + 4062425696928972800x^{2} + 18794952951174896025600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{36}\cdot 3^{45}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 17 \beta_1 + 671625) q^{3} + (\beta_{3} - 25 \beta_{2} + \cdots - 537624224) q^{4}+ \cdots + (264 \beta_{7} + 2780 \beta_{6} + \cdots - 45325751837175) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 17 \beta_1 + 671625) q^{3} + (\beta_{3} - 25 \beta_{2} + \cdots - 537624224) q^{4}+ \cdots + (12\!\cdots\!92 \beta_{7} + \cdots - 21\!\cdots\!00) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5373000 q^{3} - 4300993792 q^{4} + 215014238208 q^{6} + 11739260009680 q^{7} - 362606014697400 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5373000 q^{3} - 4300993792 q^{4} + 215014238208 q^{6} + 11739260009680 q^{7} - 362606014697400 q^{9} - 990877236572160 q^{10} + 37\!\cdots\!80 q^{12}+ \cdots - 16\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4973352x^{6} + 7969212867576x^{4} + 4062425696928972800x^{2} + 18794952951174896025600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 36\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 592363155 \nu^{7} - 21642619776 \nu^{6} + \cdots - 79\!\cdots\!00 ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2961815775 \nu^{7} - 108213098880 \nu^{6} + \cdots + 17\!\cdots\!76 ) / 11\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15722606427 \nu^{7} - 1909961195232 \nu^{6} + \cdots + 26\!\cdots\!80 ) / 69\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 116695541535 \nu^{7} - 876279395190144 \nu^{6} + \cdots - 41\!\cdots\!60 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 513578855385 \nu^{7} + 315085584537216 \nu^{6} + \cdots - 61\!\cdots\!40 ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8806955181 \nu^{7} + 698596690944 \nu^{6} + \cdots - 15\!\cdots\!00 ) / 47\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 25\beta_{2} - 8\beta _1 - 1611366048 ) / 1296 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 54\beta_{7} + 32\beta_{6} + 7\beta_{5} + 82\beta_{4} + 243\beta_{3} + 40747\beta_{2} - 2292075412\beta_1 ) / 46656 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1955464 \beta_{6} - 374323 \beta_{5} - 158546855 \beta_{3} + 5806541925 \beta_{2} + \cdots + 23\!\cdots\!88 ) / 104976 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1685771622 \beta_{7} - 741456904 \beta_{6} - 178907663 \beta_{5} - 980240346 \beta_{4} + \cdots + 43\!\cdots\!92 \beta_1 ) / 472392 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7058354745064 \beta_{6} + 974547984503 \beta_{5} + 350459319841332 \beta_{3} + \cdots - 48\!\cdots\!24 ) / 118098 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 448323323723094 \beta_{7} + 313122944614496 \beta_{6} + 65415965468503 \beta_{5} + \cdots - 91\!\cdots\!40 \beta_1 ) / 52488 \) 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
− 1434.97i
− 1362.59i
− 1026.11i
− 68.3314i
68.3314i
1026.11i
1362.59i
1434.97i
− 51658.9i −1.11130e7 − 9.07701e6i −1.59490e9 2.97292e10i −4.68908e11 + 5.74086e11i 5.67527e12 2.69226e13i 4.11071e13 + 2.01746e14i 1.53578e15
2.2 − 49053.2i −2.11466e6 + 1.41922e7i −1.33247e9 − 5.17249e10i 6.96174e11 + 1.03731e11i −2.17688e12 1.26916e13i −1.96948e14 − 6.00236e13i −2.53727e15
2.3 − 36939.9i 1.37246e7 − 4.18633e6i −2.90813e8 1.12354e10i −1.54643e11 − 5.06987e11i −4.05781e12 − 2.89213e13i 1.70840e14 − 1.14912e14i 4.15033e14
2.4 − 2459.93i 2.18954e6 + 1.41809e7i 1.06769e9 3.70006e10i 3.48839e10 − 5.38611e9i 6.42905e12 − 5.26777e12i −1.96303e14 + 6.20991e13i 9.10188e13
2.5 2459.93i 2.18954e6 − 1.41809e7i 1.06769e9 − 3.70006e10i 3.48839e10 + 5.38611e9i 6.42905e12 5.26777e12i −1.96303e14 − 6.20991e13i 9.10188e13
2.6 36939.9i 1.37246e7 + 4.18633e6i −2.90813e8 − 1.12354e10i −1.54643e11 + 5.06987e11i −4.05781e12 2.89213e13i 1.70840e14 + 1.14912e14i 4.15033e14
2.7 49053.2i −2.11466e6 − 1.41922e7i −1.33247e9 5.17249e10i 6.96174e11 − 1.03731e11i −2.17688e12 − 1.26916e13i −1.96948e14 + 6.00236e13i −2.53727e15
2.8 51658.9i −1.11130e7 + 9.07701e6i −1.59490e9 − 2.97292e10i −4.68908e11 − 5.74086e11i 5.67527e12 − 2.69226e13i 4.11071e13 − 2.01746e14i 1.53578e15
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
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 3.31.b.b ✓ 8
3.b odd 2 1 inner 3.31.b.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.31.b.b ✓ 8 1.a even 1 1 trivial
3.31.b.b ✓ 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 6445464192 T_{2}^{6} + \cdots + 53\!\cdots\!00 \) acting on \(S_{31}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 27\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 50\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less