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Properties

Label 768.6.d.a
Level $768$
Weight $6$
Character orbit 768.d
Analytic conductor $123.175$
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] = [768,6,Mod(385,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.385"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-480,0,-162,0,0,0,0,0,-612] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(123.174773616\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q + 9 i q^{3} + 34 i q^{5} - 240 q^{7} - 81 q^{9} - 124 i q^{11} + 46 i q^{13} - 306 q^{15} + 1954 q^{17} + 1924 i q^{19} - 2160 i q^{21} + 2840 q^{23} + 1969 q^{25} - 729 i q^{27} - 8922 i q^{29} + 4648 q^{31} + \cdots + 10044 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 480 q^{7} - 162 q^{9} - 612 q^{15} + 3908 q^{17} + 5680 q^{23} + 3938 q^{25} + 9296 q^{31} + 2232 q^{33} - 828 q^{39} + 5772 q^{41} - 14016 q^{47} + 81586 q^{49} + 8432 q^{55} - 34632 q^{57} + 38880 q^{63}+ \cdots - 232316 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
− 1.00000i
1.00000i
0 − 9.00000i 0 − 34.0000i 0 −240.000 0 −81.0000 0
385.2 0 9.00000i 0 34.0000i 0 −240.000 0 −81.0000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.6.d.a 2
4.b odd 2 1 768.6.d.r 2
8.b even 2 1 inner 768.6.d.a 2
8.d odd 2 1 768.6.d.r 2
16.e even 4 1 48.6.a.d 1
16.e even 4 1 192.6.a.f 1
16.f odd 4 1 24.6.a.a ✓ 1
16.f odd 4 1 192.6.a.n 1
48.i odd 4 1 144.6.a.i 1
48.i odd 4 1 576.6.a.l 1
48.k even 4 1 72.6.a.e 1
48.k even 4 1 576.6.a.k 1
80.j even 4 1 600.6.f.f 2
80.k odd 4 1 600.6.a.i 1
80.s even 4 1 600.6.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.a.a ✓ 1 16.f odd 4 1
48.6.a.d 1 16.e even 4 1
72.6.a.e 1 48.k even 4 1
144.6.a.i 1 48.i odd 4 1
192.6.a.f 1 16.e even 4 1
192.6.a.n 1 16.f odd 4 1
576.6.a.k 1 48.k even 4 1
576.6.a.l 1 48.i odd 4 1
600.6.a.i 1 80.k odd 4 1
600.6.f.f 2 80.j even 4 1
600.6.f.f 2 80.s even 4 1
768.6.d.a 2 1.a even 1 1 trivial
768.6.d.a 2 8.b even 2 1 inner
768.6.d.r 2 4.b odd 2 1
768.6.d.r 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 1156 \) Copy content Toggle raw display
\( T_{7} + 240 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} + 1156 \) Copy content Toggle raw display
$7$ \( (T + 240)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 15376 \) Copy content Toggle raw display
$13$ \( T^{2} + 2116 \) Copy content Toggle raw display
$17$ \( (T - 1954)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3701776 \) Copy content Toggle raw display
$23$ \( (T - 2840)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 79602084 \) Copy content Toggle raw display
$31$ \( (T - 4648)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 19027044 \) Copy content Toggle raw display
$41$ \( (T - 2886)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 128414224 \) Copy content Toggle raw display
$47$ \( (T + 7008)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 510488836 \) Copy content Toggle raw display
$59$ \( T^{2} + 784 \) Copy content Toggle raw display
$61$ \( T^{2} + 40780996 \) Copy content Toggle raw display
$67$ \( T^{2} + 1526933776 \) Copy content Toggle raw display
$71$ \( (T + 54872)^{2} \) Copy content Toggle raw display
$73$ \( (T + 21034)^{2} \) Copy content Toggle raw display
$79$ \( (T + 26632)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3157091344 \) Copy content Toggle raw display
$89$ \( (T + 64410)^{2} \) Copy content Toggle raw display
$97$ \( (T + 116158)^{2} \) Copy content Toggle raw display
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