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Properties

Label 405.3.l.c
Level $405$
Weight $3$
Character orbit 405.l
Analytic conductor $11.035$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,3,Mod(28,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.28");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.l (of order \(12\), degree \(4\), 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{2} + ( - \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{4} - 5 \zeta_{12}^{2} q^{5} + (7 \zeta_{12}^{3} + 7 \zeta_{12}^{2} + \cdots - 4) q^{7} + ( - 4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + \cdots + 4) q^{8} + \cdots + (105 \zeta_{12}^{3} + 65 \zeta_{12}^{2} + \cdots - 105) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 6 q^{4} - 10 q^{5} - 2 q^{7} + 6 q^{8} - 20 q^{10} - 2 q^{11} - 38 q^{13} + 42 q^{14} - 2 q^{16} - 58 q^{17} - 22 q^{22} + 62 q^{23} - 50 q^{25} - 80 q^{26} - 16 q^{28} - 48 q^{29} - 52 q^{31}+ \cdots - 290 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(-1 + \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
0.500000 + 0.133975i 0 −3.23205 − 1.86603i −2.50000 − 4.33013i 0 2.09808 + 0.562178i −2.83013 − 2.83013i 0 −0.669873 − 2.50000i
217.1 0.500000 − 0.133975i 0 −3.23205 + 1.86603i −2.50000 + 4.33013i 0 2.09808 − 0.562178i −2.83013 + 2.83013i 0 −0.669873 + 2.50000i
298.1 0.500000 + 1.86603i 0 0.232051 − 0.133975i −2.50000 + 4.33013i 0 −3.09808 − 11.5622i 5.83013 + 5.83013i 0 −9.33013 − 2.50000i
352.1 0.500000 − 1.86603i 0 0.232051 + 0.133975i −2.50000 − 4.33013i 0 −3.09808 + 11.5622i 5.83013 − 5.83013i 0 −9.33013 + 2.50000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.l.c 4
3.b odd 2 1 405.3.l.b 4
5.c odd 4 1 405.3.l.a 4
9.c even 3 1 405.3.g.b yes 4
9.c even 3 1 405.3.l.a 4
9.d odd 6 1 405.3.g.a ✓ 4
9.d odd 6 1 405.3.l.d 4
15.e even 4 1 405.3.l.d 4
45.k odd 12 1 405.3.g.b yes 4
45.k odd 12 1 inner 405.3.l.c 4
45.l even 12 1 405.3.g.a ✓ 4
45.l even 12 1 405.3.l.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.3.g.a ✓ 4 9.d odd 6 1
405.3.g.a ✓ 4 45.l even 12 1
405.3.g.b yes 4 9.c even 3 1
405.3.g.b yes 4 45.k odd 12 1
405.3.l.a 4 5.c odd 4 1
405.3.l.a 4 9.c even 3 1
405.3.l.b 4 3.b odd 2 1
405.3.l.b 4 45.l even 12 1
405.3.l.c 4 1.a even 1 1 trivial
405.3.l.c 4 45.k odd 12 1 inner
405.3.l.d 4 9.d odd 6 1
405.3.l.d 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$13$ \( T^{4} + 38 T^{3} + \cdots + 64009 \) Copy content Toggle raw display
$17$ \( T^{4} + 58 T^{3} + \cdots + 89401 \) Copy content Toggle raw display
$19$ \( T^{4} + 936T^{2} + 324 \) Copy content Toggle raw display
$23$ \( T^{4} - 62 T^{3} + \cdots + 81796 \) Copy content Toggle raw display
$29$ \( T^{4} + 48 T^{3} + \cdots + 1089 \) Copy content Toggle raw display
$31$ \( T^{4} + 52 T^{3} + \cdots + 87616 \) Copy content Toggle raw display
$37$ \( T^{4} + 26 T^{3} + \cdots + 1018081 \) Copy content Toggle raw display
$41$ \( T^{4} - 52 T^{3} + \cdots + 59536 \) Copy content Toggle raw display
$43$ \( T^{4} + 26 T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$47$ \( T^{4} - 44 T^{3} + \cdots + 238144 \) Copy content Toggle raw display
$53$ \( T^{4} + 88 T^{3} + \cdots + 145924 \) Copy content Toggle raw display
$59$ \( T^{4} + 156 T^{3} + \cdots + 3968064 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + \cdots + 942841 \) Copy content Toggle raw display
$67$ \( T^{4} - 70 T^{3} + \cdots + 4169764 \) Copy content Toggle raw display
$71$ \( (T^{2} - 134 T + 3166)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 158 T^{3} + \cdots + 703921 \) Copy content Toggle raw display
$79$ \( T^{4} - 138 T^{3} + \cdots + 2762244 \) Copy content Toggle raw display
$83$ \( T^{4} + 16 T^{3} + \cdots + 46895104 \) Copy content Toggle raw display
$89$ \( T^{4} + 30582 T^{2} + 9865881 \) Copy content Toggle raw display
$97$ \( T^{4} + 134 T^{3} + \cdots + 19909444 \) Copy content Toggle raw display
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