Properties

Label 784.2.i.f
Level $784$
Weight $2$
Character orbit 784.i
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + 3 \zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{11} + 8 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + 2 q^{29} + 6 \zeta_{6} q^{37} + 12 q^{43} + ( 10 - 10 \zeta_{6} ) q^{53} + ( 4 - 4 \zeta_{6} ) q^{67} -16 q^{71} + 8 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{9} + 4q^{11} + 8q^{23} + 5q^{25} + 4q^{29} + 6q^{37} + 24q^{43} + 10q^{53} + 4q^{67} - 32q^{71} + 8q^{79} - 9q^{81} + 24q^{99} + O(q^{100}) \)

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 0 0 0 0 1.50000 2.59808i 0
753.1 0 0 0 0 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.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.i.f 2
4.b odd 2 1 49.2.c.a 2
7.b odd 2 1 CM 784.2.i.f 2
7.c even 3 1 784.2.a.f 1
7.c even 3 1 inner 784.2.i.f 2
7.d odd 6 1 784.2.a.f 1
7.d odd 6 1 inner 784.2.i.f 2
12.b even 2 1 441.2.e.d 2
21.g even 6 1 7056.2.a.bg 1
21.h odd 6 1 7056.2.a.bg 1
28.d even 2 1 49.2.c.a 2
28.f even 6 1 49.2.a.a 1
28.f even 6 1 49.2.c.a 2
28.g odd 6 1 49.2.a.a 1
28.g odd 6 1 49.2.c.a 2
56.j odd 6 1 3136.2.a.o 1
56.k odd 6 1 3136.2.a.n 1
56.m even 6 1 3136.2.a.n 1
56.p even 6 1 3136.2.a.o 1
84.h odd 2 1 441.2.e.d 2
84.j odd 6 1 441.2.a.c 1
84.j odd 6 1 441.2.e.d 2
84.n even 6 1 441.2.a.c 1
84.n even 6 1 441.2.e.d 2
140.p odd 6 1 1225.2.a.c 1
140.s even 6 1 1225.2.a.c 1
140.w even 12 2 1225.2.b.c 2
140.x odd 12 2 1225.2.b.c 2
308.m odd 6 1 5929.2.a.c 1
308.n even 6 1 5929.2.a.c 1
364.x even 6 1 8281.2.a.d 1
364.bl odd 6 1 8281.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 28.f even 6 1
49.2.a.a 1 28.g odd 6 1
49.2.c.a 2 4.b odd 2 1
49.2.c.a 2 28.d even 2 1
49.2.c.a 2 28.f even 6 1
49.2.c.a 2 28.g odd 6 1
441.2.a.c 1 84.j odd 6 1
441.2.a.c 1 84.n even 6 1
441.2.e.d 2 12.b even 2 1
441.2.e.d 2 84.h odd 2 1
441.2.e.d 2 84.j odd 6 1
441.2.e.d 2 84.n even 6 1
784.2.a.f 1 7.c even 3 1
784.2.a.f 1 7.d odd 6 1
784.2.i.f 2 1.a even 1 1 trivial
784.2.i.f 2 7.b odd 2 1 CM
784.2.i.f 2 7.c even 3 1 inner
784.2.i.f 2 7.d odd 6 1 inner
1225.2.a.c 1 140.p odd 6 1
1225.2.a.c 1 140.s even 6 1
1225.2.b.c 2 140.w even 12 2
1225.2.b.c 2 140.x odd 12 2
3136.2.a.n 1 56.k odd 6 1
3136.2.a.n 1 56.m even 6 1
3136.2.a.o 1 56.j odd 6 1
3136.2.a.o 1 56.p even 6 1
5929.2.a.c 1 308.m odd 6 1
5929.2.a.c 1 308.n even 6 1
7056.2.a.bg 1 21.g even 6 1
7056.2.a.bg 1 21.h odd 6 1
8281.2.a.d 1 364.x even 6 1
8281.2.a.d 1 364.bl odd 6 1

Hecke kernels

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

\( T_{3} \)
\( T_{5} \)
\( T_{11}^{2} - 4 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 - 19 T^{2} + 361 T^{4} \)
$23$ \( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 61 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 16 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 73 T^{2} + 5329 T^{4} \)
$79$ \( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 97 T^{2} )^{2} \)
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