Properties

Label 784.2.i.b
Level $784$
Weight $2$
Character orbit 784.i
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,2,Mod(177,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.i (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} - 4 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{3} - 4 \zeta_{6} q^{5} - \zeta_{6} q^{9} + 8 q^{15} + (2 \zeta_{6} - 2) q^{17} + 2 \zeta_{6} q^{19} + 8 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} - 4 q^{27} + 2 q^{29} + (4 \zeta_{6} - 4) q^{31} + 6 \zeta_{6} q^{37} + 2 q^{41} - 8 q^{43} + (4 \zeta_{6} - 4) q^{45} + 4 \zeta_{6} q^{47} - 4 \zeta_{6} q^{51} + ( - 10 \zeta_{6} + 10) q^{53} - 4 q^{57} + (6 \zeta_{6} - 6) q^{59} + 4 \zeta_{6} q^{61} + (12 \zeta_{6} - 12) q^{67} - 16 q^{69} + (14 \zeta_{6} - 14) q^{73} - 22 \zeta_{6} q^{75} - 8 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 6 q^{83} + 8 q^{85} + (4 \zeta_{6} - 4) q^{87} + 10 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + ( - 8 \zeta_{6} + 8) q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{5} - q^{9} + 16 q^{15} - 2 q^{17} + 2 q^{19} + 8 q^{23} - 11 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} + 6 q^{37} + 4 q^{41} - 16 q^{43} - 4 q^{45} + 4 q^{47} - 4 q^{51} + 10 q^{53} - 8 q^{57} - 6 q^{59} + 4 q^{61} - 12 q^{67} - 32 q^{69} - 14 q^{73} - 22 q^{75} - 8 q^{79} + 11 q^{81} + 12 q^{83} + 16 q^{85} - 4 q^{87} + 10 q^{89} - 8 q^{93} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 1.73205i 0 −2.00000 + 3.46410i 0 0 0 −0.500000 + 0.866025i 0
753.1 0 −1.00000 + 1.73205i 0 −2.00000 3.46410i 0 0 0 −0.500000 0.866025i 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
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 784.2.i.b 2
4.b odd 2 1 392.2.i.e 2
7.b odd 2 1 784.2.i.j 2
7.c even 3 1 784.2.a.i 1
7.c even 3 1 inner 784.2.i.b 2
7.d odd 6 1 112.2.a.a 1
7.d odd 6 1 784.2.i.j 2
12.b even 2 1 3528.2.s.ba 2
21.g even 6 1 1008.2.a.m 1
21.h odd 6 1 7056.2.a.c 1
28.d even 2 1 392.2.i.a 2
28.f even 6 1 56.2.a.b 1
28.f even 6 1 392.2.i.a 2
28.g odd 6 1 392.2.a.b 1
28.g odd 6 1 392.2.i.e 2
35.i odd 6 1 2800.2.a.bd 1
35.k even 12 2 2800.2.g.g 2
56.j odd 6 1 448.2.a.h 1
56.k odd 6 1 3136.2.a.w 1
56.m even 6 1 448.2.a.c 1
56.p even 6 1 3136.2.a.c 1
84.h odd 2 1 3528.2.s.a 2
84.j odd 6 1 504.2.a.h 1
84.j odd 6 1 3528.2.s.a 2
84.n even 6 1 3528.2.a.b 1
84.n even 6 1 3528.2.s.ba 2
112.v even 12 2 1792.2.b.a 2
112.x odd 12 2 1792.2.b.h 2
140.p odd 6 1 9800.2.a.bj 1
140.s even 6 1 1400.2.a.a 1
140.x odd 12 2 1400.2.g.b 2
168.ba even 6 1 4032.2.a.a 1
168.be odd 6 1 4032.2.a.d 1
308.m odd 6 1 6776.2.a.h 1
364.x even 6 1 9464.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 28.f even 6 1
112.2.a.a 1 7.d odd 6 1
392.2.a.b 1 28.g odd 6 1
392.2.i.a 2 28.d even 2 1
392.2.i.a 2 28.f even 6 1
392.2.i.e 2 4.b odd 2 1
392.2.i.e 2 28.g odd 6 1
448.2.a.c 1 56.m even 6 1
448.2.a.h 1 56.j odd 6 1
504.2.a.h 1 84.j odd 6 1
784.2.a.i 1 7.c even 3 1
784.2.i.b 2 1.a even 1 1 trivial
784.2.i.b 2 7.c even 3 1 inner
784.2.i.j 2 7.b odd 2 1
784.2.i.j 2 7.d odd 6 1
1008.2.a.m 1 21.g even 6 1
1400.2.a.a 1 140.s even 6 1
1400.2.g.b 2 140.x odd 12 2
1792.2.b.a 2 112.v even 12 2
1792.2.b.h 2 112.x odd 12 2
2800.2.a.bd 1 35.i odd 6 1
2800.2.g.g 2 35.k even 12 2
3136.2.a.c 1 56.p even 6 1
3136.2.a.w 1 56.k odd 6 1
3528.2.a.b 1 84.n even 6 1
3528.2.s.a 2 84.h odd 2 1
3528.2.s.a 2 84.j odd 6 1
3528.2.s.ba 2 12.b even 2 1
3528.2.s.ba 2 84.n even 6 1
4032.2.a.a 1 168.ba even 6 1
4032.2.a.d 1 168.be odd 6 1
6776.2.a.h 1 308.m odd 6 1
7056.2.a.c 1 21.h odd 6 1
9464.2.a.h 1 364.x even 6 1
9800.2.a.bj 1 140.p odd 6 1

Hecke kernels

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

\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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