Properties

Label 27.9.b.b
Level $27$
Weight $9$
Character orbit 27.b
Analytic conductor $10.999$
Analytic rank $0$
Dimension $2$
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] = [27,9,Mod(26,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(\beta = 12\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 608 q^{4} - 28 \beta q^{5} + 1967 q^{7} - 352 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 608 q^{4} - 28 \beta q^{5} + 1967 q^{7} - 352 \beta q^{8} + 24192 q^{10} - 428 \beta q^{11} - 45505 q^{13} + 1967 \beta q^{14} + 148480 q^{16} - 2028 \beta q^{17} + 152399 q^{19} + 17024 \beta q^{20} + 369792 q^{22} - 4468 \beta q^{23} - 286751 q^{25} - 45505 \beta q^{26} - 1195936 q^{28} - 20024 \beta q^{29} - 164350 q^{31} + 58368 \beta q^{32} + 1752192 q^{34} - 55076 \beta q^{35} - 663937 q^{37} + 152399 \beta q^{38} - 8515584 q^{40} - 31912 \beta q^{41} + 575330 q^{43} + 260224 \beta q^{44} + 3860352 q^{46} - 314156 \beta q^{47} - 1895712 q^{49} - 286751 \beta q^{50} + 27667040 q^{52} + 353016 \beta q^{53} - 10354176 q^{55} - 692384 \beta q^{56} + 17300736 q^{58} - 171460 \beta q^{59} - 19212961 q^{61} - 164350 \beta q^{62} - 12419072 q^{64} + 1274140 \beta q^{65} - 598033 q^{67} + 1233024 \beta q^{68} + 47585664 q^{70} + 995856 \beta q^{71} + 12850175 q^{73} - 663937 \beta q^{74} - 92658592 q^{76} - 841876 \beta q^{77} - 23584657 q^{79} - 4157440 \beta q^{80} + 27571968 q^{82} + 1138024 \beta q^{83} - 49061376 q^{85} + 575330 \beta q^{86} - 130166784 q^{88} - 962268 \beta q^{89} - 89508335 q^{91} + 2716544 \beta q^{92} + 271430784 q^{94} - 4267172 \beta q^{95} + 136489631 q^{97} - 1895712 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1216 q^{4} + 3934 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1216 q^{4} + 3934 q^{7} + 48384 q^{10} - 91010 q^{13} + 296960 q^{16} + 304798 q^{19} + 739584 q^{22} - 573502 q^{25} - 2391872 q^{28} - 328700 q^{31} + 3504384 q^{34} - 1327874 q^{37} - 17031168 q^{40} + 1150660 q^{43} + 7720704 q^{46} - 3791424 q^{49} + 55334080 q^{52} - 20708352 q^{55} + 34601472 q^{58} - 38425922 q^{61} - 24838144 q^{64} - 1196066 q^{67} + 95171328 q^{70} + 25700350 q^{73} - 185317184 q^{76} - 47169314 q^{79} + 55143936 q^{82} - 98122752 q^{85} - 260333568 q^{88} - 179016670 q^{91} + 542861568 q^{94} + 272979262 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
2.44949i
2.44949i
29.3939i 0 −608.000 823.029i 0 1967.00 10346.6i 0 24192.0
26.2 29.3939i 0 −608.000 823.029i 0 1967.00 10346.6i 0 24192.0
\(n\): e.g. 2-40 or 990-1000
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 27.9.b.b 2
3.b odd 2 1 inner 27.9.b.b 2
4.b odd 2 1 432.9.e.d 2
9.c even 3 2 81.9.d.e 4
9.d odd 6 2 81.9.d.e 4
12.b even 2 1 432.9.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.b 2 1.a even 1 1 trivial
27.9.b.b 2 3.b odd 2 1 inner
81.9.d.e 4 9.c even 3 2
81.9.d.e 4 9.d odd 6 2
432.9.e.d 2 4.b odd 2 1
432.9.e.d 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 864 \) acting on \(S_{9}^{\mathrm{new}}(27, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 864 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 677376 \) Copy content Toggle raw display
$7$ \( (T - 1967)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 158270976 \) Copy content Toggle raw display
$13$ \( (T + 45505)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3553445376 \) Copy content Toggle raw display
$19$ \( (T - 152399)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 17248052736 \) Copy content Toggle raw display
$29$ \( T^{2} + 346429937664 \) Copy content Toggle raw display
$31$ \( (T + 164350)^{2} \) Copy content Toggle raw display
$37$ \( (T + 663937)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 879876642816 \) Copy content Toggle raw display
$43$ \( (T - 575330)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 85271609378304 \) Copy content Toggle raw display
$53$ \( T^{2} + 107671935965184 \) Copy content Toggle raw display
$59$ \( T^{2} + 25400331302400 \) Copy content Toggle raw display
$61$ \( (T + 19212961)^{2} \) Copy content Toggle raw display
$67$ \( (T + 598033)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 856854005243904 \) Copy content Toggle raw display
$73$ \( (T - 12850175)^{2} \) Copy content Toggle raw display
$79$ \( (T + 23584657)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + 800029184103936 \) Copy content Toggle raw display
$97$ \( (T - 136489631)^{2} \) Copy content Toggle raw display
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