Properties

Label 855.1.z.a
Level $855$
Weight $1$
Character orbit 855.z
Analytic conductor $0.427$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -95
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,1,Mod(94,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.94");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 855.z (of order \(6\), degree \(2\), 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: \(0.426700585801\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.7695.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} - \zeta_{6}^{2} q^{6} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} - \zeta_{6}^{2} q^{6} - q^{8} + \zeta_{6}^{2} q^{9} + q^{10} + \zeta_{6} q^{11} + \zeta_{6}^{2} q^{13} - q^{15} + \zeta_{6} q^{16} + q^{18} + q^{19} - \zeta_{6}^{2} q^{22} - \zeta_{6} q^{24} - \zeta_{6} q^{25} + q^{26} - q^{27} + \zeta_{6} q^{30} + \zeta_{6}^{2} q^{33} + q^{37} - \zeta_{6} q^{38} - q^{39} - \zeta_{6}^{2} q^{40} - \zeta_{6} q^{45} + \zeta_{6}^{2} q^{48} + \zeta_{6}^{2} q^{49} + \zeta_{6}^{2} q^{50} + q^{53} + \zeta_{6} q^{54} - q^{55} + \zeta_{6} q^{57} - \zeta_{6} q^{61} + q^{64} - \zeta_{6} q^{65} + q^{66} - \zeta_{6}^{2} q^{67} - \zeta_{6}^{2} q^{72} - \zeta_{6} q^{74} - \zeta_{6}^{2} q^{75} + \zeta_{6} q^{78} - q^{80} - \zeta_{6} q^{81} - \zeta_{6} q^{88} + \zeta_{6}^{2} q^{90} + \zeta_{6}^{2} q^{95} + \zeta_{6} q^{97} + q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{5} + q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{5} + q^{6} - 2 q^{8} - q^{9} + 2 q^{10} + q^{11} - q^{13} - 2 q^{15} + q^{16} + 2 q^{18} + 2 q^{19} + q^{22} - q^{24} - q^{25} + 2 q^{26} - 2 q^{27} + q^{30} - q^{33} + 2 q^{37} - q^{38} - 2 q^{39} + q^{40} - q^{45} - q^{48} - q^{49} - q^{50} + 2 q^{53} + q^{54} - 2 q^{55} + q^{57} - 2 q^{61} + 2 q^{64} - q^{65} + 2 q^{66} + 2 q^{67} + q^{72} - q^{74} + q^{75} + q^{78} - 2 q^{80} - q^{81} - q^{88} - q^{90} - q^{95} + 2 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 −1.00000 −0.500000 + 0.866025i 1.00000
664.1 −0.500000 + 0.866025i 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 −1.00000 −0.500000 0.866025i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
9.c even 3 1 inner
855.z odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.1.z.a 2
3.b odd 2 1 2565.1.z.b 2
5.b even 2 1 855.1.z.b yes 2
9.c even 3 1 inner 855.1.z.a 2
9.d odd 6 1 2565.1.z.b 2
15.d odd 2 1 2565.1.z.a 2
19.b odd 2 1 855.1.z.b yes 2
45.h odd 6 1 2565.1.z.a 2
45.j even 6 1 855.1.z.b yes 2
57.d even 2 1 2565.1.z.a 2
95.d odd 2 1 CM 855.1.z.a 2
171.l even 6 1 2565.1.z.a 2
171.o odd 6 1 855.1.z.b yes 2
285.b even 2 1 2565.1.z.b 2
855.z odd 6 1 inner 855.1.z.a 2
855.bl even 6 1 2565.1.z.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.1.z.a 2 1.a even 1 1 trivial
855.1.z.a 2 9.c even 3 1 inner
855.1.z.a 2 95.d odd 2 1 CM
855.1.z.a 2 855.z odd 6 1 inner
855.1.z.b yes 2 5.b even 2 1
855.1.z.b yes 2 19.b odd 2 1
855.1.z.b yes 2 45.j even 6 1
855.1.z.b yes 2 171.o odd 6 1
2565.1.z.a 2 15.d odd 2 1
2565.1.z.a 2 45.h odd 6 1
2565.1.z.a 2 57.d even 2 1
2565.1.z.a 2 171.l even 6 1
2565.1.z.b 2 3.b odd 2 1
2565.1.z.b 2 9.d odd 6 1
2565.1.z.b 2 285.b even 2 1
2565.1.z.b 2 855.bl even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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