↑

Properties

Label 2900.1.be.c
Level $2900$
Weight $1$
Character orbit 2900.be
Analytic conductor $1.447$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -4
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2900,1,Mod(51,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.51"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 0, 13])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2900.be (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.44728853664\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 580)
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + \zeta_{28}^{5} q^{2} + \zeta_{28}^{10} q^{4} - \zeta_{28} q^{8} - \zeta_{28}^{8} q^{9} + ( - \zeta_{28}^{9} - \zeta_{28}^{3}) q^{13} - \zeta_{28}^{6} q^{16} + ( - \zeta_{28}^{9} - \zeta_{28}^{5}) q^{17} + \cdots + \zeta_{28}^{13} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} + 2 q^{9} - 2 q^{16} + 14 q^{26} - 2 q^{29} + 10 q^{34} - 2 q^{36} - 2 q^{49} + 2 q^{64} - 4 q^{74} - 2 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\) \(1451\)
\(\chi(n)\) \(\zeta_{28}^{10}\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.781831 − 0.623490i
−0.781831 + 0.623490i
−0.433884 + 0.900969i
0.433884 − 0.900969i
−0.974928 + 0.222521i
0.974928 − 0.222521i
0.781831 + 0.623490i
−0.781831 − 0.623490i
−0.433884 − 0.900969i
0.433884 + 0.900969i
−0.974928 − 0.222521i
0.974928 + 0.222521i
−0.974928 + 0.222521i 0 0.900969 − 0.433884i 0 0 0 −0.781831 + 0.623490i −0.623490 − 0.781831i 0
51.2 0.974928 − 0.222521i 0 0.900969 − 0.433884i 0 0 0 0.781831 − 0.623490i −0.623490 − 0.781831i 0
151.1 −0.781831 − 0.623490i 0 0.222521 + 0.974928i 0 0 0 0.433884 − 0.900969i 0.900969 + 0.433884i 0
151.2 0.781831 + 0.623490i 0 0.222521 + 0.974928i 0 0 0 −0.433884 + 0.900969i 0.900969 + 0.433884i 0
651.1 −0.433884 + 0.900969i 0 −0.623490 − 0.781831i 0 0 0 0.974928 − 0.222521i 0.222521 + 0.974928i 0
651.2 0.433884 − 0.900969i 0 −0.623490 − 0.781831i 0 0 0 −0.974928 + 0.222521i 0.222521 + 0.974928i 0
1251.1 −0.974928 − 0.222521i 0 0.900969 + 0.433884i 0 0 0 −0.781831 − 0.623490i −0.623490 + 0.781831i 0
1251.2 0.974928 + 0.222521i 0 0.900969 + 0.433884i 0 0 0 0.781831 + 0.623490i −0.623490 + 0.781831i 0
2151.1 −0.781831 + 0.623490i 0 0.222521 − 0.974928i 0 0 0 0.433884 + 0.900969i 0.900969 − 0.433884i 0
2151.2 0.781831 − 0.623490i 0 0.222521 − 0.974928i 0 0 0 −0.433884 − 0.900969i 0.900969 − 0.433884i 0
2851.1 −0.433884 − 0.900969i 0 −0.623490 + 0.781831i 0 0 0 0.974928 + 0.222521i 0.222521 − 0.974928i 0
2851.2 0.433884 + 0.900969i 0 −0.623490 + 0.781831i 0 0 0 −0.974928 − 0.222521i 0.222521 − 0.974928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner
29.e even 14 1 inner
116.h odd 14 1 inner
145.l even 14 1 inner
580.y odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2900.1.be.c 12
4.b odd 2 1 CM 2900.1.be.c 12
5.b even 2 1 inner 2900.1.be.c 12
5.c odd 4 1 580.1.y.a ✓ 6
5.c odd 4 1 580.1.y.b yes 6
20.d odd 2 1 inner 2900.1.be.c 12
20.e even 4 1 580.1.y.a ✓ 6
20.e even 4 1 580.1.y.b yes 6
29.e even 14 1 inner 2900.1.be.c 12
116.h odd 14 1 inner 2900.1.be.c 12
145.l even 14 1 inner 2900.1.be.c 12
145.q odd 28 1 580.1.y.a ✓ 6
145.q odd 28 1 580.1.y.b yes 6
580.y odd 14 1 inner 2900.1.be.c 12
580.bh even 28 1 580.1.y.a ✓ 6
580.bh even 28 1 580.1.y.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.1.y.a ✓ 6 5.c odd 4 1
580.1.y.a ✓ 6 20.e even 4 1
580.1.y.a ✓ 6 145.q odd 28 1
580.1.y.a ✓ 6 580.bh even 28 1
580.1.y.b yes 6 5.c odd 4 1
580.1.y.b yes 6 20.e even 4 1
580.1.y.b yes 6 145.q odd 28 1
580.1.y.b yes 6 580.bh even 28 1
2900.1.be.c 12 1.a even 1 1 trivial
2900.1.be.c 12 4.b odd 2 1 CM
2900.1.be.c 12 5.b even 2 1 inner
2900.1.be.c 12 20.d odd 2 1 inner
2900.1.be.c 12 29.e even 14 1 inner
2900.1.be.c 12 116.h odd 14 1 inner
2900.1.be.c 12 145.l even 14 1 inner
2900.1.be.c 12 580.y odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(2900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 35 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + 7 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( (T^{6} + 7 T^{3} - 7 T + 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( (T^{6} + 7 T^{3} - 7 T + 7)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
show more
show less