Properties

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

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Newspace parameters

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

Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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

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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + (4 \zeta_{6} - 4) q^{11} + 2 q^{13} + ( - 6 \zeta_{6} + 6) q^{17} + 8 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{25} + 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + 2 \zeta_{6} q^{37} + 2 q^{41} + 4 q^{43} + ( - 6 \zeta_{6} + 6) q^{45} - 8 \zeta_{6} q^{47} + (6 \zeta_{6} - 6) q^{53} + 8 q^{55} + 6 \zeta_{6} q^{61} - 4 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} + 8 q^{71} + (10 \zeta_{6} - 10) q^{73} + 16 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} - 8 q^{83} - 12 q^{85} + 6 \zeta_{6} q^{89} + ( - 16 \zeta_{6} + 16) q^{95} - 6 q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 3 q^{9} - 4 q^{11} + 4 q^{13} + 6 q^{17} + 8 q^{19} + q^{25} + 12 q^{29} + 8 q^{31} + 2 q^{37} + 4 q^{41} + 8 q^{43} + 6 q^{45} - 8 q^{47} - 6 q^{53} + 16 q^{55} + 6 q^{61} - 4 q^{65} - 4 q^{67} + 16 q^{71} - 10 q^{73} + 16 q^{79} - 9 q^{81} - 16 q^{83} - 24 q^{85} + 6 q^{89} + 16 q^{95} - 12 q^{97} - 24 q^{99}+O(q^{100}) \) 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.

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 0 0 −1.00000 + 1.73205i 0 0 0 1.50000 2.59808i 0
753.1 0 0 0 −1.00000 1.73205i 0 0 0 1.50000 + 2.59808i 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.e 2
4.b odd 2 1 392.2.i.c 2
7.b odd 2 1 784.2.i.g 2
7.c even 3 1 112.2.a.b 1
7.c even 3 1 inner 784.2.i.e 2
7.d odd 6 1 784.2.a.e 1
7.d odd 6 1 784.2.i.g 2
12.b even 2 1 3528.2.s.t 2
21.g even 6 1 7056.2.a.bo 1
21.h odd 6 1 1008.2.a.d 1
28.d even 2 1 392.2.i.d 2
28.f even 6 1 392.2.a.d 1
28.f even 6 1 392.2.i.d 2
28.g odd 6 1 56.2.a.a 1
28.g odd 6 1 392.2.i.c 2
35.j even 6 1 2800.2.a.p 1
35.l odd 12 2 2800.2.g.p 2
56.j odd 6 1 3136.2.a.p 1
56.k odd 6 1 448.2.a.d 1
56.m even 6 1 3136.2.a.q 1
56.p even 6 1 448.2.a.e 1
84.h odd 2 1 3528.2.s.e 2
84.j odd 6 1 3528.2.a.x 1
84.j odd 6 1 3528.2.s.e 2
84.n even 6 1 504.2.a.c 1
84.n even 6 1 3528.2.s.t 2
112.u odd 12 2 1792.2.b.i 2
112.w even 12 2 1792.2.b.d 2
140.p odd 6 1 1400.2.a.g 1
140.s even 6 1 9800.2.a.u 1
140.w even 12 2 1400.2.g.g 2
168.s odd 6 1 4032.2.a.bk 1
168.v even 6 1 4032.2.a.bb 1
308.n even 6 1 6776.2.a.g 1
364.bl odd 6 1 9464.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 28.g odd 6 1
112.2.a.b 1 7.c even 3 1
392.2.a.d 1 28.f even 6 1
392.2.i.c 2 4.b odd 2 1
392.2.i.c 2 28.g odd 6 1
392.2.i.d 2 28.d even 2 1
392.2.i.d 2 28.f even 6 1
448.2.a.d 1 56.k odd 6 1
448.2.a.e 1 56.p even 6 1
504.2.a.c 1 84.n even 6 1
784.2.a.e 1 7.d odd 6 1
784.2.i.e 2 1.a even 1 1 trivial
784.2.i.e 2 7.c even 3 1 inner
784.2.i.g 2 7.b odd 2 1
784.2.i.g 2 7.d odd 6 1
1008.2.a.d 1 21.h odd 6 1
1400.2.a.g 1 140.p odd 6 1
1400.2.g.g 2 140.w even 12 2
1792.2.b.d 2 112.w even 12 2
1792.2.b.i 2 112.u odd 12 2
2800.2.a.p 1 35.j even 6 1
2800.2.g.p 2 35.l odd 12 2
3136.2.a.p 1 56.j odd 6 1
3136.2.a.q 1 56.m even 6 1
3528.2.a.x 1 84.j odd 6 1
3528.2.s.e 2 84.h odd 2 1
3528.2.s.e 2 84.j odd 6 1
3528.2.s.t 2 12.b even 2 1
3528.2.s.t 2 84.n even 6 1
4032.2.a.bb 1 168.v even 6 1
4032.2.a.bk 1 168.s odd 6 1
6776.2.a.g 1 308.n even 6 1
7056.2.a.bo 1 21.g even 6 1
9464.2.a.c 1 364.bl odd 6 1
9800.2.a.u 1 140.s even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$83$ \( (T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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