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Properties

Label 405.2.e.k
Level $405$
Weight $2$
Character orbit 405.e
Analytic conductor $3.234$
Analytic rank $0$
Dimension $4$
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] = [405,2,Mod(136,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, 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("405.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} - 3 q^{8} + (\beta_{3} - 1) q^{10} - 2 \beta_1 q^{11} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{13} + 6 \beta_{2} q^{14} + (2 \beta_{2} - \beta_1 + 2) q^{16} + ( - 2 \beta_{3} - 1) q^{17} + ( - 2 \beta_{3} + 1) q^{19} + ( - \beta_{2} - \beta_1 - 1) q^{20} + ( - 2 \beta_{3} - 6 \beta_{2} + \cdots + 2) q^{22}+ \cdots + (5 \beta_{3} + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 3 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 3 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8} - 2 q^{10} - 2 q^{11} - 6 q^{13} - 12 q^{14} + 3 q^{16} - 8 q^{17} - 3 q^{20} + 14 q^{22} + 6 q^{23} - 2 q^{25} - 32 q^{26} - 20 q^{28} - 10 q^{29} + 4 q^{31} - 7 q^{32} + 11 q^{34} + 4 q^{35} + 8 q^{37} + 13 q^{38} + 6 q^{40} + 2 q^{41} + 6 q^{43} + 32 q^{44} + 6 q^{46} + 4 q^{47} - 14 q^{49} + q^{50} - 22 q^{52} - 8 q^{53} + 4 q^{55} + 6 q^{56} - 8 q^{58} + 10 q^{59} - 6 q^{61} + 30 q^{62} - 8 q^{64} - 6 q^{65} + 16 q^{67} - 7 q^{68} - 12 q^{70} + 44 q^{71} + 36 q^{73} + 2 q^{74} - 13 q^{76} + 24 q^{77} + 16 q^{79} - 6 q^{80} + 28 q^{82} - 6 q^{83} + 4 q^{85} + 10 q^{86} + 6 q^{88} - 12 q^{89} - 40 q^{91} + 9 q^{92} - 28 q^{94} - 16 q^{97} + 38 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
−0.651388 + 1.12824i
1.15139 − 1.99426i
−0.651388 − 1.12824i
1.15139 + 1.99426i
−0.651388 + 1.12824i 0 0.151388 + 0.262211i −0.500000 − 0.866025i 0 −2.30278 + 3.98852i −3.00000 0 1.30278
136.2 1.15139 − 1.99426i 0 −1.65139 − 2.86029i −0.500000 − 0.866025i 0 1.30278 − 2.25647i −3.00000 0 −2.30278
271.1 −0.651388 − 1.12824i 0 0.151388 − 0.262211i −0.500000 + 0.866025i 0 −2.30278 − 3.98852i −3.00000 0 1.30278
271.2 1.15139 + 1.99426i 0 −1.65139 + 2.86029i −0.500000 + 0.866025i 0 1.30278 + 2.25647i −3.00000 0 −2.30278
\(n\): e.g. 2-40 or 990-1000
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 405.2.e.k 4
3.b odd 2 1 405.2.e.j 4
9.c even 3 1 135.2.a.c ✓ 2
9.c even 3 1 inner 405.2.e.k 4
9.d odd 6 1 135.2.a.d yes 2
9.d odd 6 1 405.2.e.j 4
36.f odd 6 1 2160.2.a.ba 2
36.h even 6 1 2160.2.a.y 2
45.h odd 6 1 675.2.a.k 2
45.j even 6 1 675.2.a.p 2
45.k odd 12 2 675.2.b.i 4
45.l even 12 2 675.2.b.h 4
63.l odd 6 1 6615.2.a.p 2
63.o even 6 1 6615.2.a.v 2
72.j odd 6 1 8640.2.a.df 2
72.l even 6 1 8640.2.a.cy 2
72.n even 6 1 8640.2.a.cr 2
72.p odd 6 1 8640.2.a.ck 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c ✓ 2 9.c even 3 1
135.2.a.d yes 2 9.d odd 6 1
405.2.e.j 4 3.b odd 2 1
405.2.e.j 4 9.d odd 6 1
405.2.e.k 4 1.a even 1 1 trivial
405.2.e.k 4 9.c even 3 1 inner
675.2.a.k 2 45.h odd 6 1
675.2.a.p 2 45.j even 6 1
675.2.b.h 4 45.l even 12 2
675.2.b.i 4 45.k odd 12 2
2160.2.a.y 2 36.h even 6 1
2160.2.a.ba 2 36.f odd 6 1
6615.2.a.p 2 63.l odd 6 1
6615.2.a.v 2 63.o even 6 1
8640.2.a.ck 2 72.p odd 6 1
8640.2.a.cr 2 72.n even 6 1
8640.2.a.cy 2 72.l even 6 1
8640.2.a.df 2 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 16T_{7}^{2} - 24T_{7} + 144 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} + 16T_{11}^{2} - 24T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 4 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$37$ \( (T - 2)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$71$ \( (T^{2} - 22 T + 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 18 T + 68)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$83$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
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