↑

Properties

Label 616.2.q.c
Level $616$
Weight $2$
Character orbit 616.q
Analytic conductor $4.919$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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] = [616,2,Mod(177,616)] 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("616.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(616, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.91878476451\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - \beta_1) q^{3} + ( - \beta_{5} - 2 \beta_{4} + \beta_1 + 1) q^{5} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots - 1) q^{7} + ( - \beta_{5} - 3 \beta_{4} - 4 \beta_1 + 1) q^{9}+ \cdots + ( - 3 \beta_{3} + 4 \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 5 q^{3} + 2 q^{5} - 4 q^{9} - 3 q^{11} - 6 q^{13} + 2 q^{15} + 11 q^{17} - 7 q^{19} + 14 q^{21} + 6 q^{23} - 19 q^{25} - 16 q^{27} - 6 q^{29} - 19 q^{31} + 5 q^{33} + 35 q^{35} - 12 q^{37} - 5 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/616\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(309\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.222521 − 0.385418i
−0.623490 + 1.07992i
0.900969 − 1.56052i
0.222521 + 0.385418i
−0.623490 − 1.07992i
0.900969 + 1.56052i
0 0.0990311 + 0.171527i 0 1.96950 − 3.41127i 0 0.167563 + 2.64044i 0 1.48039 − 2.56410i 0
177.2 0 0.777479 + 1.34663i 0 −1.92543 + 3.33494i 0 −2.37047 − 1.17511i 0 0.291053 − 0.504118i 0
177.3 0 1.62349 + 2.81197i 0 0.955927 − 1.65571i 0 2.20291 − 1.46533i 0 −3.77144 + 6.53232i 0
529.1 0 0.0990311 − 0.171527i 0 1.96950 + 3.41127i 0 0.167563 − 2.64044i 0 1.48039 + 2.56410i 0
529.2 0 0.777479 − 1.34663i 0 −1.92543 − 3.33494i 0 −2.37047 + 1.17511i 0 0.291053 + 0.504118i 0
529.3 0 1.62349 − 2.81197i 0 0.955927 + 1.65571i 0 2.20291 + 1.46533i 0 −3.77144 − 6.53232i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 616.2.q.c ✓ 6
4.b odd 2 1 1232.2.q.i 6
7.c even 3 1 inner 616.2.q.c ✓ 6
7.c even 3 1 4312.2.a.v 3
7.d odd 6 1 4312.2.a.x 3
28.f even 6 1 8624.2.a.cf 3
28.g odd 6 1 1232.2.q.i 6
28.g odd 6 1 8624.2.a.cq 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.c ✓ 6 1.a even 1 1 trivial
616.2.q.c ✓ 6 7.c even 3 1 inner
1232.2.q.i 6 4.b odd 2 1
1232.2.q.i 6 28.g odd 6 1
4312.2.a.v 3 7.c even 3 1
4312.2.a.x 3 7.d odd 6 1
8624.2.a.cf 3 28.f even 6 1
8624.2.a.cq 3 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 5T_{3}^{5} + 19T_{3}^{4} - 28T_{3}^{3} + 31T_{3}^{2} - 6T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(616, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$7$ \( T^{6} + 7T^{3} + 343 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 11 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$19$ \( T^{6} + 7 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} - 46 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 19 T^{5} + \cdots + 57121 \) Copy content Toggle raw display
$37$ \( T^{6} + 12 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( (T^{3} + 17 T^{2} + \cdots - 167)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 12 T^{2} + \cdots + 83)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 17 T^{5} + \cdots + 851929 \) Copy content Toggle raw display
$53$ \( T^{6} - 3 T^{5} + \cdots + 9409 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots + 1042441 \) Copy content Toggle raw display
$61$ \( T^{6} + 84 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{6} + 84 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3 T^{2} - 130 T - 41)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 26 T^{5} + \cdots + 12769 \) Copy content Toggle raw display
$79$ \( T^{6} + 23 T^{5} + \cdots + 187489 \) Copy content Toggle raw display
$83$ \( (T^{3} + 9 T^{2} + \cdots - 673)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 13 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$97$ \( (T^{3} - T^{2} - 184 T + 911)^{2} \) Copy content Toggle raw display
show more
show less