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Properties

Label 2600.1.cy.b
Level $2600$
Weight $1$
Character orbit 2600.cy
Analytic conductor $1.298$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2600,1,Mod(843,2600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2600, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 9, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2600.843");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2600.cy (of order \(12\), degree \(4\), 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: \(1.29756903285\)
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
Projective image: \(D_{12}\)
Projective 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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{5} q^{9} + (\zeta_{12}^{3} + \zeta_{12}^{2}) q^{11} + \zeta_{12}^{2} q^{13} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{14} + \zeta_{12}^{4} q^{16} - q^{18} + ( - \zeta_{12}^{4} + \zeta_{12}^{3}) q^{19} + (\zeta_{12}^{4} + \zeta_{12}^{3}) q^{22} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{23} + \zeta_{12}^{3} q^{26} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{28} + \zeta_{12}^{5} q^{32} - \zeta_{12} q^{36} + \zeta_{12}^{4} q^{37} + ( - \zeta_{12}^{5} + \zeta_{12}^{4}) q^{38} + (\zeta_{12}^{4} - \zeta_{12}) q^{41} + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{44} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{46} + q^{47} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{49} + \zeta_{12}^{4} q^{52} + ( - \zeta_{12}^{2} + \zeta_{12}) q^{53} + ( - \zeta_{12}^{4} + 1) q^{56} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{59} + (\zeta_{12}^{2} + 1) q^{63} - q^{64} - \zeta_{12}^{2} q^{72} + \zeta_{12}^{5} q^{74} + (\zeta_{12}^{5} + 1) q^{76} + ( - \zeta_{12}^{5} - \zeta_{12}^{4} + \cdots + 1) q^{77} + \cdots + ( - \zeta_{12}^{2} - \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{11} + 2 q^{13} - 2 q^{16} - 4 q^{18} + 2 q^{19} - 2 q^{22} + 2 q^{23} - 2 q^{37} - 2 q^{38} - 2 q^{41} - 2 q^{44} + 2 q^{46} + 4 q^{47} - 4 q^{49} - 2 q^{52} - 2 q^{53} + 6 q^{56} - 2 q^{59} + 6 q^{63} - 4 q^{64} - 2 q^{72} + 4 q^{76} + 6 q^{77} + 2 q^{81} - 2 q^{82} - 4 q^{88} + 2 q^{89} + 4 q^{92} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times\).

\(n\) \(1301\) \(1601\) \(1951\) \(1977\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{5}\) \(-1\) \(-\zeta_{12}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
843.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 0.866025 − 1.50000i 1.00000i 0.866025 + 0.500000i 0
1107.1 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 −0.866025 + 1.50000i − 1.00000i −0.866025 − 0.500000i 0
2243.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −0.866025 − 1.50000i 1.00000i −0.866025 + 0.500000i 0
2307.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 0 0.866025 + 1.50000i − 1.00000i 0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
65.t even 12 1 inner
520.ci odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.1.cy.b yes 4
5.b even 2 1 2600.1.cy.a ✓ 4
5.c odd 4 1 2600.1.dl.a yes 4
5.c odd 4 1 2600.1.dl.b yes 4
8.d odd 2 1 2600.1.cy.a ✓ 4
13.f odd 12 1 2600.1.dl.b yes 4
40.e odd 2 1 CM 2600.1.cy.b yes 4
40.k even 4 1 2600.1.dl.a yes 4
40.k even 4 1 2600.1.dl.b yes 4
65.o even 12 1 2600.1.cy.a ✓ 4
65.s odd 12 1 2600.1.dl.a yes 4
65.t even 12 1 inner 2600.1.cy.b yes 4
104.u even 12 1 2600.1.dl.a yes 4
520.ci odd 12 1 inner 2600.1.cy.b yes 4
520.cv odd 12 1 2600.1.cy.a ✓ 4
520.cz even 12 1 2600.1.dl.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2600.1.cy.a ✓ 4 5.b even 2 1
2600.1.cy.a ✓ 4 8.d odd 2 1
2600.1.cy.a ✓ 4 65.o even 12 1
2600.1.cy.a ✓ 4 520.cv odd 12 1
2600.1.cy.b yes 4 1.a even 1 1 trivial
2600.1.cy.b yes 4 40.e odd 2 1 CM
2600.1.cy.b yes 4 65.t even 12 1 inner
2600.1.cy.b yes 4 520.ci odd 12 1 inner
2600.1.dl.a yes 4 5.c odd 4 1
2600.1.dl.a yes 4 40.k even 4 1
2600.1.dl.a yes 4 65.s odd 12 1
2600.1.dl.a yes 4 104.u even 12 1
2600.1.dl.b yes 4 5.c odd 4 1
2600.1.dl.b yes 4 13.f odd 12 1
2600.1.dl.b yes 4 40.k even 4 1
2600.1.dl.b yes 4 520.cz even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{4} - 2T_{23}^{3} + 2T_{23}^{2} - 4T_{23} + 4 \) acting on \(S_{1}^{\mathrm{new}}(2600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T - 1)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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