↑

Properties

Label 243.9.b.f
Level $243$
Weight $9$
Character orbit 243.b
Analytic conductor $98.993$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [243,9,Mod(242,243)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("243.242"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 243.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-1234,0,0,5230] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(98.9930022449\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1641x^{6} + 754938x^{4} + 94603680x^{2} + 1736069760 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 154) q^{4} + ( - \beta_{2} + 11 \beta_1) q^{5} + (\beta_{5} - 3 \beta_{3} + 653) q^{7} + (\beta_{4} + 3 \beta_{2} - 208 \beta_1) q^{8} + (\beta_{7} - 2 \beta_{5} + 20 \beta_{3} - 4643) q^{10}+ \cdots + (3835 \beta_{6} - 13758 \beta_{4} + \cdots + 5969933 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1234 q^{4} + 5230 q^{7} - 37182 q^{10} - 26552 q^{13} + 369722 q^{16} + 279838 q^{19} + 461856 q^{22} - 383818 q^{25} - 4338872 q^{28} + 3390586 q^{31} + 4268628 q^{34} - 366650 q^{37} + 17672394 q^{40}+ \cdots - 342449126 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1641x^{6} + 754938x^{4} + 94603680x^{2} + 1736069760 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -55\nu^{7} - 91719\nu^{5} - 41780718\nu^{3} - 4283016912\nu ) / 17146368 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 410 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 55\nu^{7} + 91719\nu^{5} + 47496174\nu^{3} + 8398145232\nu ) / 5715456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} - 6741\nu^{4} - 2039850\nu^{2} - 76025520 ) / 35136 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -199\nu^{7} - 292887\nu^{5} - 105693390\nu^{3} - 6113461968\nu ) / 4286592 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{6} - 12951\nu^{4} - 6281550\nu^{2} - 472693392 ) / 35136 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 410 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_{2} - 720\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} + 14\beta_{5} - 975\beta_{3} + 295510 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 110\beta_{6} - 1167\beta_{4} - 5093\beta_{2} + 599452\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13482\beta_{7} - 25902\beta_{5} + 906525\beta_{3} - 246343986 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -183438\beta_{6} + 1186461\beta_{4} + 5902479\beta_{2} - 530582508\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
242.1
− 30.7845i
− 22.5290i
− 12.7951i
− 4.69533i
4.69533i
12.7951i
22.5290i
30.7845i
− 30.7845i 0 −691.683 − 787.156i 0 3933.22 13412.3i 0 −24232.2
242.2 − 22.5290i 0 −251.556 391.321i 0 −2569.03 − 100.109i 0 8816.09
242.3 − 12.7951i 0 92.2856 113.073i 0 2737.23 − 4456.35i 0 1446.77
242.4 − 4.69533i 0 233.954 − 984.315i 0 −1486.43 − 2300.50i 0 −4621.69
242.5 4.69533i 0 233.954 984.315i 0 −1486.43 2300.50i 0 −4621.69
242.6 12.7951i 0 92.2856 − 113.073i 0 2737.23 4456.35i 0 1446.77
242.7 22.5290i 0 −251.556 − 391.321i 0 −2569.03 100.109i 0 8816.09
242.8 30.7845i 0 −691.683 787.156i 0 3933.22 − 13412.3i 0 −24232.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 242.8
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 243.9.b.f ✓ 8
3.b odd 2 1 inner 243.9.b.f ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.9.b.f ✓ 8 1.a even 1 1 trivial
243.9.b.f ✓ 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(243, [\chi])\):

\( T_{2}^{8} + 1641T_{2}^{6} + 754938T_{2}^{4} + 94603680T_{2}^{2} + 1736069760 \) Copy content Toggle raw display
\( T_{7}^{4} - 2615T_{7}^{3} - 12466902T_{7}^{2} + 18189307444T_{7} + 41112308744920 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 1736069760 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 41112308744920)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 58\!\cdots\!70)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 81\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 26\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 17\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 68\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 29\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 18\!\cdots\!95)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 51\!\cdots\!75)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 29\!\cdots\!74)^{2} \) Copy content Toggle raw display
show more
show less