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Properties

Label 3.43.b.b
Level $3$
Weight $43$
Character orbit 3.b
Analytic conductor $33.518$
Analytic rank $0$
Dimension $12$
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] = [3,43,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, 43, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 43);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 43 \)
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: \(33.5183121516\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 33864221333 x^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{73}\cdot 3^{104}\cdot 5^{6}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 546 \beta_1 + 1415600433) q^{3} + (\beta_{3} + 78 \beta_{2} + \cdots - 2916625296824) q^{4}+ \cdots + (\beta_{11} - 4 \beta_{10} + \cdots - 87\!\cdots\!55) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 546 \beta_1 + 1415600433) q^{3} + (\beta_{3} + 78 \beta_{2} + \cdots - 2916625296824) q^{4}+ \cdots + ( - 76\!\cdots\!32 \beta_{11} + \cdots - 38\!\cdots\!80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16987205196 q^{3} - 34999503561888 q^{4} - 47\!\cdots\!88 q^{6}+ \cdots - 10\!\cdots\!60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 16987205196 q^{3} - 34999503561888 q^{4} - 47\!\cdots\!88 q^{6}+ \cdots - 45\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 33864221333 x^{10} + \cdots + 25\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 36\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 71\!\cdots\!25 \nu^{11} + \cdots - 19\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 93\!\cdots\!25 \nu^{11} + \cdots + 96\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!83 \nu^{11} + \cdots + 61\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!27 \nu^{11} + \cdots + 91\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 54\!\cdots\!37 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\!\cdots\!87 \nu^{11} + \cdots + 22\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 82\!\cdots\!25 \nu^{11} + \cdots + 25\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!45 \nu^{11} + \cdots - 71\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!89 \nu^{11} + \cdots - 49\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 35\!\cdots\!95 \nu^{11} + \cdots - 30\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 78\beta_{2} + 22\beta _1 - 7314671807928 ) / 1296 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - 22 \beta_{8} + 42 \beta_{7} - \beta_{6} - \beta_{5} + 2655 \beta_{4} + \cdots - 10893722619548 \beta_1 ) / 46656 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 981785 \beta_{10} - 58048401 \beta_{8} + 127239206 \beta_{7} - 17860621 \beta_{6} + \cdots + 39\!\cdots\!44 ) / 839808 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1748338560 \beta_{11} - 124887467840 \beta_{10} - 1123258478149 \beta_{9} + \cdots + 85\!\cdots\!88 \beta_1 ) / 3779136 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 44\!\cdots\!11 \beta_{10} + \cdots - 10\!\cdots\!28 ) / 22674816 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 18\!\cdots\!80 \beta_{11} + \cdots - 23\!\cdots\!16 \beta_1 ) / 102036672 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 15\!\cdots\!49 \beta_{10} + \cdots + 28\!\cdots\!36 ) / 612220032 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 93\!\cdots\!60 \beta_{11} + \cdots + 75\!\cdots\!88 \beta_1 ) / 306110016 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 18\!\cdots\!61 \beta_{10} + \cdots - 30\!\cdots\!16 ) / 612220032 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12\!\cdots\!20 \beta_{11} + \cdots - 80\!\cdots\!48 \beta_1 ) / 306110016 \) 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
− 104257.i
− 103520.i
− 74991.2i
− 59345.7i
− 52282.8i
− 19978.7i
19978.7i
52282.8i
59345.7i
74991.2i
103520.i
104257.i
− 3.75326e6i 1.02269e10 − 2.19746e9i −9.68891e12 5.14242e14i −8.24764e15 − 3.83843e16i 7.24611e17 1.98580e19i 9.97613e19 − 4.49466e19i 1.93008e21
2.2 − 3.72673e6i −8.44421e9 − 6.17368e9i −9.49045e12 − 9.25532e14i −2.30076e16 + 3.14693e16i 7.62151e16 1.89780e19i 3.31903e19 + 1.04264e20i −3.44920e21
2.3 − 2.69968e6i −4.62216e9 + 9.38374e9i −2.89024e12 1.56653e14i 2.53331e16 + 1.24784e16i 1.13686e17 − 4.07061e18i −6.66902e19 − 8.67463e19i 4.22913e20
2.4 − 2.13644e6i −3.48827e9 − 9.86159e9i −1.66350e11 7.27116e14i −2.10687e16 + 7.45250e15i −7.17356e17 − 9.04079e18i −8.50829e19 + 6.87998e19i 1.55344e21
2.5 − 1.88218e6i 9.19591e9 + 4.98540e9i 8.55436e11 − 4.92433e14i 9.38344e15 − 1.73084e16i −8.30626e17 − 9.88801e18i 5.97105e19 + 9.16906e19i −9.26849e20
2.6 − 719233.i 5.62540e9 − 8.81895e9i 3.88075e12 − 4.30398e14i −6.34288e15 − 4.04597e15i 6.12563e17 − 5.95439e18i −4.61287e19 − 9.92202e19i −3.09557e20
2.7 719233.i 5.62540e9 + 8.81895e9i 3.88075e12 4.30398e14i −6.34288e15 + 4.04597e15i 6.12563e17 5.95439e18i −4.61287e19 + 9.92202e19i −3.09557e20
2.8 1.88218e6i 9.19591e9 − 4.98540e9i 8.55436e11 4.92433e14i 9.38344e15 + 1.73084e16i −8.30626e17 9.88801e18i 5.97105e19 − 9.16906e19i −9.26849e20
2.9 2.13644e6i −3.48827e9 + 9.86159e9i −1.66350e11 − 7.27116e14i −2.10687e16 − 7.45250e15i −7.17356e17 9.04079e18i −8.50829e19 − 6.87998e19i 1.55344e21
2.10 2.69968e6i −4.62216e9 − 9.38374e9i −2.89024e12 − 1.56653e14i 2.53331e16 − 1.24784e16i 1.13686e17 4.07061e18i −6.66902e19 + 8.67463e19i 4.22913e20
2.11 3.72673e6i −8.44421e9 + 6.17368e9i −9.49045e12 9.25532e14i −2.30076e16 − 3.14693e16i 7.62151e16 − 1.89780e19i 3.31903e19 − 1.04264e20i −3.44920e21
2.12 3.75326e6i 1.02269e10 + 2.19746e9i −9.68891e12 − 5.14242e14i −8.24764e15 + 3.83843e16i 7.24611e17 − 1.98580e19i 9.97613e19 + 4.49466e19i 1.93008e21
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.12
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 3.43.b.b ✓ 12
3.b odd 2 1 inner 3.43.b.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.43.b.b ✓ 12 1.a even 1 1 trivial
3.43.b.b ✓ 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 43888030847568 T_{2}^{10} + \cdots + 11\!\cdots\!00 \) acting on \(S_{43}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 17\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 95\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 18\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
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