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Properties

Label 510.2.c.a
Level $510$
Weight $2$
Character orbit 510.c
Analytic conductor $4.072$
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] = [510,2,Mod(271,510)] 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("510.271"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(510, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] 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: \(4.07237050309\)
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.

\(f(q)\) \(=\) \( q - q^{2} + i q^{3} + q^{4} - i q^{5} - i q^{6} - q^{8} - q^{9} + i q^{10} + 4 i q^{11} + i q^{12} + q^{15} + q^{16} + (4 i - 1) q^{17} + q^{18} + 8 q^{19} - i q^{20} - 4 i q^{22} - 2 i q^{23} - i q^{24} + \cdots - 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 16 q^{19} - 2 q^{25} - 2 q^{30} - 2 q^{32} - 8 q^{33} + 2 q^{34} - 2 q^{36} - 16 q^{38} + 12 q^{43} - 16 q^{47}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(307\) \(341\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
− 1.00000i
1.00000i
−1.00000 − 1.00000i 1.00000 1.00000i 1.00000i 0 −1.00000 −1.00000 − 1.00000i
271.2 −1.00000 1.00000i 1.00000 − 1.00000i − 1.00000i 0 −1.00000 −1.00000 1.00000i
\(n\): e.g. 2-40 or 80-90
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 510.2.c.a ✓ 2
3.b odd 2 1 1530.2.c.e 2
4.b odd 2 1 4080.2.h.h 2
5.b even 2 1 2550.2.c.h 2
5.c odd 4 1 2550.2.f.d 2
5.c odd 4 1 2550.2.f.k 2
17.b even 2 1 inner 510.2.c.a ✓ 2
17.c even 4 1 8670.2.a.s 1
17.c even 4 1 8670.2.a.w 1
51.c odd 2 1 1530.2.c.e 2
68.d odd 2 1 4080.2.h.h 2
85.c even 2 1 2550.2.c.h 2
85.g odd 4 1 2550.2.f.d 2
85.g odd 4 1 2550.2.f.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.c.a ✓ 2 1.a even 1 1 trivial
510.2.c.a ✓ 2 17.b even 2 1 inner
1530.2.c.e 2 3.b odd 2 1
1530.2.c.e 2 51.c odd 2 1
2550.2.c.h 2 5.b even 2 1
2550.2.c.h 2 85.c even 2 1
2550.2.f.d 2 5.c odd 4 1
2550.2.f.d 2 85.g odd 4 1
2550.2.f.k 2 5.c odd 4 1
2550.2.f.k 2 85.g odd 4 1
4080.2.h.h 2 4.b odd 2 1
4080.2.h.h 2 68.d odd 2 1
8670.2.a.s 1 17.c even 4 1
8670.2.a.w 1 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\). 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
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 4 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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