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Properties

Label 135.2.e.b
Level $135$
Weight $2$
Character orbit 135.e
Analytic conductor $1.078$
Analytic rank $0$
Dimension $6$
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] = [135,2,Mod(46,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.e (of order \(3\), degree \(2\), not 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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{3} + 1) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_{3}) q^{7} + (\beta_{4} - \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{3} + 1) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_{3}) q^{7} + (\beta_{4} - \beta_{2} - 1) q^{8} - \beta_{4} q^{10} + ( - \beta_{3} + \beta_1) q^{11} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{13} + ( - 2 \beta_{5} + 4 \beta_{3} + \cdots - 4) q^{14}+ \cdots + (2 \beta_{4} + 3 \beta_{2} + 15) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 5 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 5 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{13} - 9 q^{14} - 5 q^{16} + 4 q^{17} + 8 q^{19} + 5 q^{20} + 4 q^{22} + 3 q^{23} - 3 q^{25} + 4 q^{26} + 10 q^{28} - 7 q^{29} - 8 q^{31} + 17 q^{32} + 4 q^{34} - 10 q^{35} + 12 q^{37} - 2 q^{38} - 3 q^{40} - 13 q^{41} - 10 q^{43} + 44 q^{44} - 6 q^{46} + 13 q^{47} + 2 q^{49} + q^{50} + 12 q^{52} + 4 q^{53} - 4 q^{55} - 9 q^{56} + 17 q^{58} - 2 q^{59} - q^{61} - 84 q^{62} - 30 q^{64} + 4 q^{65} - 11 q^{67} - 22 q^{68} + 9 q^{70} + 20 q^{71} - 16 q^{73} - 16 q^{74} + 12 q^{76} - 2 q^{79} - 10 q^{80} - 58 q^{82} - 15 q^{83} + 2 q^{85} + 28 q^{86} + 24 q^{88} + 18 q^{89} + 20 q^{91} + 39 q^{92} + 31 q^{94} + 4 q^{95} + 18 q^{97} + 80 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 − 1.68443i
1.71903 − 0.211943i
−1.62241 − 0.606458i
0.403374 + 1.68443i
−1.04307 + 1.80664i 0 −1.17597 − 2.03684i 0.500000 + 0.866025i 0 −2.04307 + 3.53869i 0.734191 0 −2.08613
46.2 0.285997 − 0.495361i 0 0.836412 + 1.44871i 0.500000 + 0.866025i 0 −0.714003 + 1.23669i 2.10083 0 0.571993
46.3 1.25707 − 2.17731i 0 −2.16044 − 3.74200i 0.500000 + 0.866025i 0 0.257068 − 0.445256i −5.83502 0 2.51414
91.1 −1.04307 − 1.80664i 0 −1.17597 + 2.03684i 0.500000 − 0.866025i 0 −2.04307 − 3.53869i 0.734191 0 −2.08613
91.2 0.285997 + 0.495361i 0 0.836412 − 1.44871i 0.500000 − 0.866025i 0 −0.714003 − 1.23669i 2.10083 0 0.571993
91.3 1.25707 + 2.17731i 0 −2.16044 + 3.74200i 0.500000 − 0.866025i 0 0.257068 + 0.445256i −5.83502 0 2.51414
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.e.b 6
3.b odd 2 1 45.2.e.b ✓ 6
4.b odd 2 1 2160.2.q.k 6
5.b even 2 1 675.2.e.b 6
5.c odd 4 2 675.2.k.b 12
9.c even 3 1 inner 135.2.e.b 6
9.c even 3 1 405.2.a.i 3
9.d odd 6 1 45.2.e.b ✓ 6
9.d odd 6 1 405.2.a.j 3
12.b even 2 1 720.2.q.i 6
15.d odd 2 1 225.2.e.b 6
15.e even 4 2 225.2.k.b 12
36.f odd 6 1 2160.2.q.k 6
36.f odd 6 1 6480.2.a.bs 3
36.h even 6 1 720.2.q.i 6
36.h even 6 1 6480.2.a.bv 3
45.h odd 6 1 225.2.e.b 6
45.h odd 6 1 2025.2.a.n 3
45.j even 6 1 675.2.e.b 6
45.j even 6 1 2025.2.a.o 3
45.k odd 12 2 675.2.k.b 12
45.k odd 12 2 2025.2.b.m 6
45.l even 12 2 225.2.k.b 12
45.l even 12 2 2025.2.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b ✓ 6 3.b odd 2 1
45.2.e.b ✓ 6 9.d odd 6 1
135.2.e.b 6 1.a even 1 1 trivial
135.2.e.b 6 9.c even 3 1 inner
225.2.e.b 6 15.d odd 2 1
225.2.e.b 6 45.h odd 6 1
225.2.k.b 12 15.e even 4 2
225.2.k.b 12 45.l even 12 2
405.2.a.i 3 9.c even 3 1
405.2.a.j 3 9.d odd 6 1
675.2.e.b 6 5.b even 2 1
675.2.e.b 6 45.j even 6 1
675.2.k.b 12 5.c odd 4 2
675.2.k.b 12 45.k odd 12 2
720.2.q.i 6 12.b even 2 1
720.2.q.i 6 36.h even 6 1
2025.2.a.n 3 45.h odd 6 1
2025.2.a.o 3 45.j even 6 1
2025.2.b.l 6 45.l even 12 2
2025.2.b.m 6 45.k odd 12 2
2160.2.q.k 6 4.b odd 2 1
2160.2.q.k 6 36.f odd 6 1
6480.2.a.bs 3 36.f odd 6 1
6480.2.a.bv 3 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 6T_{2}^{4} - T_{2}^{3} + 28T_{2}^{2} - 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 6 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + \cdots + 13689 \) Copy content Toggle raw display
$29$ \( T^{6} + 7 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} + \cdots + 219024 \) Copy content Toggle raw display
$37$ \( (T^{3} - 6 T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 13 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{6} + 10 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{6} - 13 T^{5} + \cdots + 136161 \) Copy content Toggle raw display
$53$ \( (T^{3} - 2 T^{2} - 20 T + 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 2 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{6} + T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$67$ \( T^{6} + 11 T^{5} + \cdots + 257049 \) Copy content Toggle raw display
$71$ \( (T^{3} - 10 T^{2} + \cdots + 708)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 8 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 2 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$83$ \( T^{6} + 15 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$89$ \( (T - 3)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - 18 T^{5} + \cdots + 1700416 \) Copy content Toggle raw display
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