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Properties

Label 2550.2.c.a
Level $2550$
Weight $2$
Character orbit 2550.c
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2550,2,Mod(1801,2550)] 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("2550.1801"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,0,0,0,-2,-2,0,0,0,-8,0,0,2,-8,2,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + i q^{3} + q^{4} - i q^{6} + 2 i q^{7} - q^{8} - q^{9} + i q^{12} - 4 q^{13} - 2 i q^{14} + q^{16} + (i - 4) q^{17} + q^{18} - 2 q^{21} - i q^{23} - i q^{24} + 4 q^{26} - i q^{27} + 2 i q^{28} + \cdots - 3 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 8 q^{13} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 4 q^{21} + 8 q^{26} - 2 q^{32} + 8 q^{34} - 2 q^{36} + 4 q^{42} + 12 q^{43} - 16 q^{47} + 6 q^{49} - 2 q^{51} - 8 q^{52}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(851\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1801.1
− 1.00000i
1.00000i
−1.00000 − 1.00000i 1.00000 0 1.00000i − 2.00000i −1.00000 −1.00000 0
1801.2 −1.00000 1.00000i 1.00000 0 − 1.00000i 2.00000i −1.00000 −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
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2550.2.c.a ✓ 2
5.b even 2 1 2550.2.c.k yes 2
5.c odd 4 1 2550.2.f.f 2
5.c odd 4 1 2550.2.f.i 2
17.b even 2 1 inner 2550.2.c.a ✓ 2
85.c even 2 1 2550.2.c.k yes 2
85.g odd 4 1 2550.2.f.f 2
85.g odd 4 1 2550.2.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2550.2.c.a ✓ 2 1.a even 1 1 trivial
2550.2.c.a ✓ 2 17.b even 2 1 inner
2550.2.c.k yes 2 5.b even 2 1
2550.2.c.k yes 2 85.c even 2 1
2550.2.f.f 2 5.c odd 4 1
2550.2.f.f 2 85.g odd 4 1
2550.2.f.i 2 5.c odd 4 1
2550.2.f.i 2 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2550, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 25 \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( (T + 5)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 25 \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 16 \) Copy content Toggle raw display
$83$ \( (T + 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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