↑

Properties

Label 36.9.d.d
Level $36$
Weight $9$
Character orbit 36.d
Analytic conductor $14.666$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,9,Mod(19,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 36.d (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: \(14.6656299622\)
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: \( x^{8} + 53x^{6} + 144x^{4} + 217088x^{2} + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} - 53) q^{4} + (\beta_{3} + 3 \beta_1) q^{5} - \beta_{7} q^{7} + (\beta_{4} - 2 \beta_{3} + \cdots - 54 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} - 53) q^{4} + (\beta_{3} + 3 \beta_1) q^{5} - \beta_{7} q^{7} + (\beta_{4} - 2 \beta_{3} + \cdots - 54 \beta_1) q^{8}+ \cdots + (15232 \beta_{4} + \cdots - 1652735 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 424 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 424 q^{4} + 5344 q^{10} + 54256 q^{13} + 80672 q^{16} + 516864 q^{22} + 1191672 q^{25} + 143808 q^{28} + 1695040 q^{34} - 1970672 q^{37} - 8654720 q^{40} + 13014528 q^{46} - 13221880 q^{49} - 38716592 q^{52} + 48451744 q^{58} + 13077520 q^{61} - 99487360 q^{64} + 240237312 q^{70} + 45249520 q^{73} - 230295168 q^{76} + 303942592 q^{82} + 36844544 q^{85} - 478006272 q^{88} + 684148224 q^{94} - 126751760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 53x^{6} + 144x^{4} + 217088x^{2} + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 53\nu^{5} + 144\nu^{3} + 217088\nu ) / 131072 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{7} - 689\nu^{5} - 1872\nu^{3} + 13955072\nu ) / 131072 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49\nu^{7} - 1499\nu^{5} + 52112\nu^{3} + 10309632\nu ) / 131072 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -73\nu^{7} + 4323\nu^{5} + 947952\nu^{3} + 2633728\nu ) / 131072 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 53\nu^{4} + 144\nu^{2} + 162816 ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 53\nu^{4} + 32912\nu^{2} + 596992 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{6} + 97\nu^{4} + 848\nu^{2} - 632832 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 13\beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 2\beta_{5} - 848 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{4} + 32\beta_{3} - 17\beta_{2} - 621\beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 64\beta_{7} - 5\beta_{6} + 778\beta_{5} + 40336 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 176\beta_{4} - 3744\beta_{3} - 267\beta_{2} + 192833\beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3392\beta_{7} + 121\beta_{6} + 24590\beta_{5} - 12435920 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -11632\beta_{4} + 193824\beta_{3} - 200489\beta_{2} + 3824347\beta_1 ) / 128 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
6.98289 − 3.90376i
6.98289 + 3.90376i
1.41043 − 7.87469i
1.41043 + 7.87469i
−1.41043 − 7.87469i
−1.41043 + 7.87469i
−6.98289 − 3.90376i
−6.98289 + 3.90376i
−13.9658 − 7.80751i 0 134.086 + 218.076i 112.921 0 1310.79i −169.979 − 4092.47i 0 −1577.03 − 881.631i
19.2 −13.9658 + 7.80751i 0 134.086 − 218.076i 112.921 0 − 1310.79i −169.979 + 4092.47i 0 −1577.03 + 881.631i
19.3 −2.82086 − 15.7494i 0 −240.086 + 88.8534i −1032.67 0 − 3621.73i 2076.63 + 3530.55i 0 2913.03 + 16264.0i
19.4 −2.82086 + 15.7494i 0 −240.086 − 88.8534i −1032.67 0 3621.73i 2076.63 − 3530.55i 0 2913.03 − 16264.0i
19.5 2.82086 − 15.7494i 0 −240.086 − 88.8534i 1032.67 0 3621.73i −2076.63 + 3530.55i 0 2913.03 − 16264.0i
19.6 2.82086 + 15.7494i 0 −240.086 + 88.8534i 1032.67 0 − 3621.73i −2076.63 − 3530.55i 0 2913.03 + 16264.0i
19.7 13.9658 − 7.80751i 0 134.086 − 218.076i −112.921 0 − 1310.79i 169.979 − 4092.47i 0 −1577.03 + 881.631i
19.8 13.9658 + 7.80751i 0 134.086 + 218.076i −112.921 0 1310.79i 169.979 + 4092.47i 0 −1577.03 − 881.631i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.9.d.d ✓ 8
3.b odd 2 1 inner 36.9.d.d ✓ 8
4.b odd 2 1 inner 36.9.d.d ✓ 8
12.b even 2 1 inner 36.9.d.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.9.d.d ✓ 8 1.a even 1 1 trivial
36.9.d.d ✓ 8 3.b odd 2 1 inner
36.9.d.d ✓ 8 4.b odd 2 1 inner
36.9.d.d ✓ 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 1079168T_{5}^{2} + 13598003200 \) acting on \(S_{9}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 1079168 T^{2} + 13598003200)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 22536976896000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13564 T - 527460860)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 15\!\cdots\!80)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 492668 T - 724381350140)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 25\!\cdots\!80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 22858640275964)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 485551901243900)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 47\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 38\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 905286067336700)^{4} \) Copy content Toggle raw display
show more
show less