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Properties

Label 2800.1.z.b
Level $2800$
Weight $1$
Character orbit 2800.z
Analytic conductor $1.397$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2800,1,Mod(1301,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.1301");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2800.z (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: \(1.39738203537\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{3} q^{9} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{11} + \zeta_{12}^{2} q^{14} - \zeta_{12}^{2} q^{16} - \zeta_{12}^{2} q^{18} + (\zeta_{12}^{4} + \zeta_{12}^{3}) q^{22} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{23} - \zeta_{12} q^{28} + (\zeta_{12}^{5} - \zeta_{12}^{4}) q^{29} + \zeta_{12} q^{32} + \zeta_{12} q^{36} + (\zeta_{12}^{2} - \zeta_{12}) q^{37} + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{43} + ( - \zeta_{12}^{3} - \zeta_{12}^{2}) q^{44} + (\zeta_{12}^{3} + \zeta_{12}) q^{46} - q^{49} + ( - \zeta_{12}^{3} + 1) q^{53} + q^{56} + ( - \zeta_{12}^{4} + \zeta_{12}^{3}) q^{58} + q^{63} - q^{64} + ( - \zeta_{12}^{5} + \zeta_{12}^{4}) q^{67} - \zeta_{12}^{3} q^{71} - q^{72} + ( - \zeta_{12} + 1) q^{74} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{77} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{79} - q^{81} + ( - \zeta_{12}^{4} - \zeta_{12}^{3}) q^{86} + (\zeta_{12}^{2} + \zeta_{12}) q^{88} + ( - \zeta_{12}^{2} - 1) q^{92} - \zeta_{12}^{5} q^{98} + (\zeta_{12}^{2} + \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{11} + 2 q^{14} - 2 q^{16} - 2 q^{18} - 2 q^{22} + 2 q^{29} + 2 q^{37} - 2 q^{43} - 2 q^{44} - 4 q^{49} + 4 q^{53} + 4 q^{56} + 2 q^{58} + 4 q^{63} - 4 q^{64} - 2 q^{67} - 4 q^{72} + 4 q^{74} - 2 q^{77} - 4 q^{81} + 2 q^{86} + 2 q^{88} - 6 q^{92} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
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 1.00000i − 1.00000i − 1.00000i 0
1301.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 1.00000i − 1.00000i − 1.00000i 0
2701.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 − 1.00000i 1.00000i 1.00000i 0
2701.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 − 1.00000i 1.00000i 1.00000i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.1.z.b ✓ 4
5.b even 2 1 2800.1.z.c yes 4
5.c odd 4 1 2800.1.bf.c 4
5.c odd 4 1 2800.1.bf.d 4
7.b odd 2 1 CM 2800.1.z.b ✓ 4
16.e even 4 1 inner 2800.1.z.b ✓ 4
35.c odd 2 1 2800.1.z.c yes 4
35.f even 4 1 2800.1.bf.c 4
35.f even 4 1 2800.1.bf.d 4
80.i odd 4 1 2800.1.bf.d 4
80.q even 4 1 2800.1.z.c yes 4
80.t odd 4 1 2800.1.bf.c 4
112.l odd 4 1 inner 2800.1.z.b ✓ 4
560.r even 4 1 2800.1.bf.c 4
560.bf odd 4 1 2800.1.z.c yes 4
560.bn even 4 1 2800.1.bf.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2800.1.z.b ✓ 4 1.a even 1 1 trivial
2800.1.z.b ✓ 4 7.b odd 2 1 CM
2800.1.z.b ✓ 4 16.e even 4 1 inner
2800.1.z.b ✓ 4 112.l odd 4 1 inner
2800.1.z.c yes 4 5.b even 2 1
2800.1.z.c yes 4 35.c odd 2 1
2800.1.z.c yes 4 80.q even 4 1
2800.1.z.c yes 4 560.bf odd 4 1
2800.1.bf.c 4 5.c odd 4 1
2800.1.bf.c 4 35.f even 4 1
2800.1.bf.c 4 80.t odd 4 1
2800.1.bf.c 4 560.r even 4 1
2800.1.bf.d 4 5.c odd 4 1
2800.1.bf.d 4 35.f even 4 1
2800.1.bf.d 4 80.i odd 4 1
2800.1.bf.d 4 560.bn even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2800, [\chi])\):

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