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Properties

Label 2600.1.r.a
Level $2600$
Weight $1$
Character orbit 2600.r
Analytic conductor $1.298$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
RM discriminant 40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2600,1,Mod(2101,2600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2600.2101");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2600.r (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: \(1.29756903285\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 520)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.439400.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{3} - \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{2} - 1) q^{6} - \zeta_{8} q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{3} - \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{2} - 1) q^{6} - \zeta_{8} q^{8} - q^{9} + (\zeta_{8}^{3} - \zeta_{8}) q^{12} - \zeta_{8} q^{13} - q^{16} + \zeta_{8}^{3} q^{18} + (\zeta_{8}^{2} - 1) q^{24} - q^{26} + (\zeta_{8}^{2} + 1) q^{31} + \zeta_{8}^{3} q^{32} + \zeta_{8}^{2} q^{36} - \zeta_{8} q^{37} + (\zeta_{8}^{2} - 1) q^{39} + ( - \zeta_{8}^{2} - 1) q^{41} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{43} + (\zeta_{8}^{3} + \zeta_{8}) q^{48} + \zeta_{8}^{2} q^{49} + \zeta_{8}^{3} q^{52} + (\zeta_{8}^{3} + \zeta_{8}) q^{53} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{62} + \zeta_{8}^{2} q^{64} - \zeta_{8}^{3} q^{67} + (\zeta_{8}^{2} + 1) q^{71} + \zeta_{8} q^{72} - 2 q^{74} + (\zeta_{8}^{3} + \zeta_{8}) q^{78} - q^{81} + (\zeta_{8}^{3} - \zeta_{8}) q^{82} - \zeta_{8}^{3} q^{83} + ( - \zeta_{8}^{2} + 1) q^{86} + ( - \zeta_{8}^{2} + 1) q^{89} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{93} + (\zeta_{8}^{2} + 1) q^{96} + \zeta_{8} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} - 4 q^{9} - 4 q^{16} - 4 q^{24} - 4 q^{26} + 4 q^{31} - 4 q^{39} - 4 q^{41} + 4 q^{71} - 8 q^{74} - 4 q^{81} + 4 q^{86} + 4 q^{89} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1301\) \(1601\) \(1951\) \(1977\)
\(\chi(n)\) \(-1\) \(\zeta_{8}^{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2101.1
−0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i 1.41421i − 1.00000i 0 −1.00000 − 1.00000i 0 0.707107 + 0.707107i −1.00000 0
2101.2 0.707107 − 0.707107i − 1.41421i − 1.00000i 0 −1.00000 − 1.00000i 0 −0.707107 − 0.707107i −1.00000 0
2501.1 −0.707107 − 0.707107i − 1.41421i 1.00000i 0 −1.00000 + 1.00000i 0 0.707107 − 0.707107i −1.00000 0
2501.2 0.707107 + 0.707107i 1.41421i 1.00000i 0 −1.00000 + 1.00000i 0 −0.707107 + 0.707107i −1.00000 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.f even 2 1 RM by \(\Q(\sqrt{10}) \)
5.b even 2 1 inner
8.b even 2 1 inner
13.d odd 4 1 inner
65.g odd 4 1 inner
104.j odd 4 1 inner
520.bo odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.1.r.a 4
5.b even 2 1 inner 2600.1.r.a 4
5.c odd 4 2 520.1.bo.a ✓ 4
8.b even 2 1 inner 2600.1.r.a 4
13.d odd 4 1 inner 2600.1.r.a 4
20.e even 4 2 2080.1.cs.a 4
40.f even 2 1 RM 2600.1.r.a 4
40.i odd 4 2 520.1.bo.a ✓ 4
40.k even 4 2 2080.1.cs.a 4
65.f even 4 1 520.1.bo.a ✓ 4
65.g odd 4 1 inner 2600.1.r.a 4
65.k even 4 1 520.1.bo.a ✓ 4
104.j odd 4 1 inner 2600.1.r.a 4
260.l odd 4 1 2080.1.cs.a 4
260.s odd 4 1 2080.1.cs.a 4
520.x odd 4 1 2080.1.cs.a 4
520.y even 4 1 520.1.bo.a ✓ 4
520.bj even 4 1 520.1.bo.a ✓ 4
520.bk odd 4 1 2080.1.cs.a 4
520.bo odd 4 1 inner 2600.1.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.1.bo.a ✓ 4 5.c odd 4 2
520.1.bo.a ✓ 4 40.i odd 4 2
520.1.bo.a ✓ 4 65.f even 4 1
520.1.bo.a ✓ 4 65.k even 4 1
520.1.bo.a ✓ 4 520.y even 4 1
520.1.bo.a ✓ 4 520.bj even 4 1
2080.1.cs.a 4 20.e even 4 2
2080.1.cs.a 4 40.k even 4 2
2080.1.cs.a 4 260.l odd 4 1
2080.1.cs.a 4 260.s odd 4 1
2080.1.cs.a 4 520.x odd 4 1
2080.1.cs.a 4 520.bk odd 4 1
2600.1.r.a 4 1.a even 1 1 trivial
2600.1.r.a 4 5.b even 2 1 inner
2600.1.r.a 4 8.b even 2 1 inner
2600.1.r.a 4 13.d odd 4 1 inner
2600.1.r.a 4 40.f even 2 1 RM
2600.1.r.a 4 65.g odd 4 1 inner
2600.1.r.a 4 104.j odd 4 1 inner
2600.1.r.a 4 520.bo odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 16 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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