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Properties

Label 784.2.w.d
Level $784$
Weight $2$
Character orbit 784.w
Analytic conductor $6.260$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,2,Mod(19,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.w (of order \(12\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 + 1) q^{2} + ( - 2 \beta_{4} + 2 \beta_1) q^{4} + \beta_{6} q^{5} + ( - 2 \beta_{4} + 2) q^{8} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 + 1) q^{2} + ( - 2 \beta_{4} + 2 \beta_1) q^{4} + \beta_{6} q^{5} + ( - 2 \beta_{4} + 2) q^{8} + 3 \beta_1 q^{9} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{10} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{7} + \beta_{5}) q^{13} + ( - 4 \beta_{2} + 4) q^{16} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{17} + ( - 3 \beta_{4} + 3 \beta_{2} + 3 \beta_1) q^{18} - 2 \beta_{6} q^{19} + (2 \beta_{7} - 2 \beta_{5}) q^{20} - 2 \beta_{4} q^{22} + (4 \beta_{2} - 4) q^{23} + ( - 7 \beta_{4} + 7 \beta_1) q^{25} + ( - \beta_{6} + \beta_{5}) q^{26} + ( - 3 \beta_{4} - 3) q^{29} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{31} + ( - 4 \beta_{4} - 4 \beta_{2} + 4 \beta_1) q^{32} + ( - 2 \beta_{7} + 2 \beta_{5}) q^{34} + 6 q^{36} + (5 \beta_{2} + 5 \beta_1 - 5) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{38} + (2 \beta_{6} - 2 \beta_{5}) q^{40} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{41} + ( - 5 \beta_{4} - 5) q^{43} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{44} + 3 \beta_{7} q^{45} + (4 \beta_{4} + 4 \beta_{2} - 4 \beta_1) q^{46} + (\beta_{6} - \beta_{5}) q^{47} + ( - 7 \beta_{4} + 7) q^{50} + ( - 2 \beta_{6} + 2 \beta_{3}) q^{52} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{53} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{55} - 6 \beta_1 q^{58} + 2 \beta_{7} q^{59} + \beta_{5} q^{61} - 2 \beta_{3} q^{62} - 8 \beta_{4} q^{64} + (12 \beta_{2} - 12) q^{65} + (5 \beta_{4} - 5 \beta_{2} - 5 \beta_1) q^{67} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{68} + 2 q^{71} + ( - 6 \beta_{2} + 6 \beta_1 + 6) q^{72} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3}) q^{73} + 10 \beta_{2} q^{74} + ( - 4 \beta_{7} + 4 \beta_{5}) q^{76} - 4 \beta_1 q^{79} + (4 \beta_{6} - 4 \beta_{3}) q^{80} + 9 \beta_{2} q^{81} + 2 \beta_{6} q^{82} + (12 \beta_{4} - 12) q^{85} - 10 \beta_1 q^{86} - 4 \beta_{2} q^{88} + (3 \beta_{7} - 3 \beta_{5} + 3 \beta_{3}) q^{90} + 8 \beta_{4} q^{92} + (2 \beta_{6} - 2 \beta_{3}) q^{94} + (24 \beta_{4} - 24 \beta_1) q^{95} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{97} + ( - 3 \beta_{4} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 16 q^{8} - 4 q^{11} + 16 q^{16} + 12 q^{18} - 16 q^{23} - 24 q^{29} - 16 q^{32} + 48 q^{36} - 20 q^{37} - 40 q^{43} + 8 q^{44} + 16 q^{46} + 56 q^{50} + 4 q^{53} - 48 q^{65} - 20 q^{67} + 16 q^{71} + 24 q^{72} + 40 q^{74} + 36 q^{81} - 96 q^{85} - 16 q^{88} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{5} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{7} + 4\zeta_{24}^{3} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + 2\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(-\beta_{4}\) \(-1\) \(1 - \beta_{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
1.36603 − 0.366025i 0 1.73205 − 1.00000i −3.34607 + 0.896575i 0 0 2.00000 − 2.00000i 2.59808 + 1.50000i −4.24264 + 2.44949i
19.2 1.36603 − 0.366025i 0 1.73205 − 1.00000i 3.34607 − 0.896575i 0 0 2.00000 − 2.00000i 2.59808 + 1.50000i 4.24264 − 2.44949i
227.1 −0.366025 + 1.36603i 0 −1.73205 − 1.00000i −0.896575 + 3.34607i 0 0 2.00000 − 2.00000i −2.59808 + 1.50000i −4.24264 − 2.44949i
227.2 −0.366025 + 1.36603i 0 −1.73205 − 1.00000i 0.896575 − 3.34607i 0 0 2.00000 − 2.00000i −2.59808 + 1.50000i 4.24264 + 2.44949i
411.1 −0.366025 − 1.36603i 0 −1.73205 + 1.00000i −0.896575 − 3.34607i 0 0 2.00000 + 2.00000i −2.59808 − 1.50000i −4.24264 + 2.44949i
411.2 −0.366025 − 1.36603i 0 −1.73205 + 1.00000i 0.896575 + 3.34607i 0 0 2.00000 + 2.00000i −2.59808 − 1.50000i 4.24264 − 2.44949i
619.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i −3.34607 − 0.896575i 0 0 2.00000 + 2.00000i 2.59808 − 1.50000i −4.24264 − 2.44949i
619.2 1.36603 + 0.366025i 0 1.73205 + 1.00000i 3.34607 + 0.896575i 0 0 2.00000 + 2.00000i 2.59808 − 1.50000i 4.24264 + 2.44949i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
16.f odd 4 1 inner
112.j even 4 1 inner
112.u odd 12 1 inner
112.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.w.d 8
7.b odd 2 1 inner 784.2.w.d 8
7.c even 3 1 112.2.j.a ✓ 4
7.c even 3 1 inner 784.2.w.d 8
7.d odd 6 1 112.2.j.a ✓ 4
7.d odd 6 1 inner 784.2.w.d 8
16.f odd 4 1 inner 784.2.w.d 8
28.f even 6 1 448.2.j.b 4
28.g odd 6 1 448.2.j.b 4
56.j odd 6 1 896.2.j.b 4
56.k odd 6 1 896.2.j.e 4
56.m even 6 1 896.2.j.e 4
56.p even 6 1 896.2.j.b 4
112.j even 4 1 inner 784.2.w.d 8
112.u odd 12 1 112.2.j.a ✓ 4
112.u odd 12 1 inner 784.2.w.d 8
112.u odd 12 1 896.2.j.b 4
112.v even 12 1 112.2.j.a ✓ 4
112.v even 12 1 inner 784.2.w.d 8
112.v even 12 1 896.2.j.b 4
112.w even 12 1 448.2.j.b 4
112.w even 12 1 896.2.j.e 4
112.x odd 12 1 448.2.j.b 4
112.x odd 12 1 896.2.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.j.a ✓ 4 7.c even 3 1
112.2.j.a ✓ 4 7.d odd 6 1
112.2.j.a ✓ 4 112.u odd 12 1
112.2.j.a ✓ 4 112.v even 12 1
448.2.j.b 4 28.f even 6 1
448.2.j.b 4 28.g odd 6 1
448.2.j.b 4 112.w even 12 1
448.2.j.b 4 112.x odd 12 1
784.2.w.d 8 1.a even 1 1 trivial
784.2.w.d 8 7.b odd 2 1 inner
784.2.w.d 8 7.c even 3 1 inner
784.2.w.d 8 7.d odd 6 1 inner
784.2.w.d 8 16.f odd 4 1 inner
784.2.w.d 8 112.j even 4 1 inner
784.2.w.d 8 112.u odd 12 1 inner
784.2.w.d 8 112.v even 12 1 inner
896.2.j.b 4 56.j odd 6 1
896.2.j.b 4 56.p even 6 1
896.2.j.b 4 112.u odd 12 1
896.2.j.b 4 112.v even 12 1
896.2.j.e 4 56.k odd 6 1
896.2.j.e 4 56.m even 6 1
896.2.j.e 4 112.w even 12 1
896.2.j.e 4 112.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{8} - 144T_{5}^{4} + 20736 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 2304 T^{4} + 5308416 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 18)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 10 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 50)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 2304 T^{4} + 5308416 \) Copy content Toggle raw display
$61$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$71$ \( (T - 2)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 96 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
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