Properties

Label 285.2.k.a
Level $285$
Weight $2$
Character orbit 285.k
Analytic conductor $2.276$
Analytic rank $1$
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] = [285,2,Mod(77,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.k (of order \(4\), 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: \(2.27573645761\)
Analytic rank: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - 1) q^{2} + ( - 2 \zeta_{12}^{2} + 1) q^{3} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 1) q^{6}+ \cdots - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - 1) q^{2} + ( - 2 \zeta_{12}^{2} + 1) q^{3} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 1) q^{6}+ \cdots + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{5} - 6 q^{7} - 8 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{5} - 6 q^{7} - 8 q^{8} - 12 q^{9} + 2 q^{10} - 8 q^{13} + 12 q^{14} + 12 q^{15} + 16 q^{16} - 10 q^{17} + 12 q^{18} - 6 q^{21} + 8 q^{22} - 12 q^{23} - 6 q^{25} - 6 q^{30} - 28 q^{31} + 12 q^{33} + 12 q^{35} - 4 q^{37} + 4 q^{38} + 12 q^{39} + 12 q^{40} - 10 q^{43} + 12 q^{45} + 24 q^{46} - 10 q^{47} - 2 q^{50} - 18 q^{51} + 16 q^{53} - 8 q^{55} + 20 q^{59} - 24 q^{61} + 28 q^{62} + 18 q^{63} - 14 q^{65} - 12 q^{66} + 12 q^{67} - 24 q^{69} - 6 q^{70} + 24 q^{72} + 18 q^{73} + 8 q^{74} - 18 q^{75} - 6 q^{77} - 24 q^{78} - 16 q^{80} + 36 q^{81} + 4 q^{82} - 8 q^{83} + 32 q^{85} - 16 q^{88} - 36 q^{89} - 6 q^{90} + 36 q^{91} + 2 q^{95} - 12 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−1.00000 + 1.00000i 1.73205i 0 −0.133975 + 2.23205i 1.73205 + 1.73205i −2.36603 2.36603i −2.00000 2.00000i −3.00000 −2.09808 2.36603i
77.2 −1.00000 + 1.00000i 1.73205i 0 −1.86603 1.23205i −1.73205 1.73205i −0.633975 0.633975i −2.00000 2.00000i −3.00000 3.09808 0.633975i
248.1 −1.00000 1.00000i 1.73205i 0 −1.86603 + 1.23205i −1.73205 + 1.73205i −0.633975 + 0.633975i −2.00000 + 2.00000i −3.00000 3.09808 + 0.633975i
248.2 −1.00000 1.00000i 1.73205i 0 −0.133975 2.23205i 1.73205 1.73205i −2.36603 + 2.36603i −2.00000 + 2.00000i −3.00000 −2.09808 + 2.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.k.a 4
3.b odd 2 1 285.2.k.b yes 4
5.c odd 4 1 285.2.k.b yes 4
15.e even 4 1 inner 285.2.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.k.a 4 1.a even 1 1 trivial
285.2.k.a 4 15.e even 4 1 inner
285.2.k.b yes 4 3.b odd 2 1
285.2.k.b yes 4 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 14 T + 46)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$41$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{4} + 10 T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 33)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$73$ \( T^{4} - 18 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$79$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 78)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
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