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Properties

Label 588.3.c.e
Level $588$
Weight $3$
Character orbit 588.c
Analytic conductor $16.022$
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] = [588,3,Mod(197,588)] 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("588.197"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,0,0,0,0,-17,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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 \(\beta = \frac{1}{2}(1 + \sqrt{-35})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} + ( - 2 \beta + 1) q^{5} + ( - \beta - 8) q^{9} + (2 \beta - 1) q^{11} - 6 q^{13} + (\beta + 17) q^{15} + (2 \beta - 1) q^{17} + 23 q^{19} + (14 \beta - 7) q^{23} - 10 q^{25} + ( - 8 \beta + 17) q^{27}+ \cdots + ( - 17 \beta + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 17 q^{9} - 12 q^{13} + 35 q^{15} + 46 q^{19} - 20 q^{25} + 26 q^{27} + 78 q^{31} - 35 q^{33} + 94 q^{37} + 6 q^{39} - 44 q^{43} - 35 q^{45} - 35 q^{51} + 70 q^{55} - 23 q^{57} - 162 q^{61}+ \cdots + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0.500000 − 2.95804i
0.500000 + 2.95804i
0 −0.500000 − 2.95804i 0 5.91608i 0 0 0 −8.50000 + 2.95804i 0
197.2 0 −0.500000 + 2.95804i 0 − 5.91608i 0 0 0 −8.50000 − 2.95804i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.c.e 2
3.b odd 2 1 inner 588.3.c.e 2
7.b odd 2 1 588.3.c.f 2
7.c even 3 2 84.3.p.c ✓ 4
7.d odd 6 2 588.3.p.d 4
21.c even 2 1 588.3.c.f 2
21.g even 6 2 588.3.p.d 4
21.h odd 6 2 84.3.p.c ✓ 4
28.g odd 6 2 336.3.bn.d 4
84.n even 6 2 336.3.bn.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.p.c ✓ 4 7.c even 3 2
84.3.p.c ✓ 4 21.h odd 6 2
336.3.bn.d 4 28.g odd 6 2
336.3.bn.d 4 84.n even 6 2
588.3.c.e 2 1.a even 1 1 trivial
588.3.c.e 2 3.b odd 2 1 inner
588.3.c.f 2 7.b odd 2 1
588.3.c.f 2 21.c even 2 1
588.3.p.d 4 7.d odd 6 2
588.3.p.d 4 21.g even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{2} + 35 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 35 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 35 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 35 \) Copy content Toggle raw display
$19$ \( (T - 23)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1715 \) Copy content Toggle raw display
$29$ \( T^{2} + 2240 \) Copy content Toggle raw display
$31$ \( (T - 39)^{2} \) Copy content Toggle raw display
$37$ \( (T - 47)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 22)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2835 \) Copy content Toggle raw display
$53$ \( T^{2} + 2835 \) Copy content Toggle raw display
$59$ \( T^{2} + 10115 \) Copy content Toggle raw display
$61$ \( (T + 81)^{2} \) Copy content Toggle raw display
$67$ \( (T - 31)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8960 \) Copy content Toggle raw display
$73$ \( (T + 17)^{2} \) Copy content Toggle raw display
$79$ \( (T + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2240 \) Copy content Toggle raw display
$89$ \( T^{2} + 7875 \) Copy content Toggle raw display
$97$ \( (T - 82)^{2} \) Copy content Toggle raw display
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