Properties

Label 784.2.p.d
Level 784
Weight 2
Character orbit 784.p
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.p (of order \(6\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + ( 3 + 3 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} + ( 10 - 5 \zeta_{6} ) q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} + 5 q^{27} -6 q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + ( -2 + \zeta_{6} ) q^{33} + 5 \zeta_{6} q^{37} + ( -4 + 8 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( -2 - 2 \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + ( 6 - 3 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + 3 q^{55} + 7 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + ( -10 + 5 \zeta_{6} ) q^{61} + ( 3 + 3 \zeta_{6} ) q^{67} + ( 5 - 10 \zeta_{6} ) q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( -1 - \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} + ( -6 + 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} -9 q^{85} + ( -6 + 6 \zeta_{6} ) q^{87} + ( 14 - 7 \zeta_{6} ) q^{89} -5 \zeta_{6} q^{93} + ( -7 - 7 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 3q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + q^{3} - 3q^{5} + 2q^{9} - 3q^{11} + 9q^{17} + 7q^{19} + 15q^{23} - 2q^{25} + 10q^{27} - 12q^{29} + 5q^{31} - 3q^{33} + 5q^{37} - 6q^{45} + 3q^{47} + 9q^{51} + 9q^{53} + 6q^{55} + 14q^{57} + 9q^{59} - 15q^{61} + 9q^{67} - 3q^{73} + 2q^{75} - 9q^{79} - q^{81} - 24q^{83} - 18q^{85} - 6q^{87} + 21q^{89} - 5q^{93} - 21q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 −1.50000 + 0.866025i 0 0 0 1.00000 + 1.73205i 0
607.1 0 0.500000 + 0.866025i 0 −1.50000 0.866025i 0 0 0 1.00000 1.73205i 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
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.p.d 2
4.b odd 2 1 784.2.p.c 2
7.b odd 2 1 112.2.p.a 2
7.c even 3 1 112.2.p.b yes 2
7.c even 3 1 784.2.f.a 2
7.d odd 6 1 784.2.f.b 2
7.d odd 6 1 784.2.p.c 2
21.c even 2 1 1008.2.cs.f 2
21.g even 6 1 7056.2.b.m 2
21.h odd 6 1 1008.2.cs.c 2
21.h odd 6 1 7056.2.b.b 2
28.d even 2 1 112.2.p.b yes 2
28.f even 6 1 784.2.f.a 2
28.f even 6 1 inner 784.2.p.d 2
28.g odd 6 1 112.2.p.a 2
28.g odd 6 1 784.2.f.b 2
56.e even 2 1 448.2.p.a 2
56.h odd 2 1 448.2.p.b 2
56.j odd 6 1 3136.2.f.a 2
56.k odd 6 1 448.2.p.b 2
56.k odd 6 1 3136.2.f.a 2
56.m even 6 1 3136.2.f.b 2
56.p even 6 1 448.2.p.a 2
56.p even 6 1 3136.2.f.b 2
84.h odd 2 1 1008.2.cs.c 2
84.j odd 6 1 7056.2.b.b 2
84.n even 6 1 1008.2.cs.f 2
84.n even 6 1 7056.2.b.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 7.b odd 2 1
112.2.p.a 2 28.g odd 6 1
112.2.p.b yes 2 7.c even 3 1
112.2.p.b yes 2 28.d even 2 1
448.2.p.a 2 56.e even 2 1
448.2.p.a 2 56.p even 6 1
448.2.p.b 2 56.h odd 2 1
448.2.p.b 2 56.k odd 6 1
784.2.f.a 2 7.c even 3 1
784.2.f.a 2 28.f even 6 1
784.2.f.b 2 7.d odd 6 1
784.2.f.b 2 28.g odd 6 1
784.2.p.c 2 4.b odd 2 1
784.2.p.c 2 7.d odd 6 1
784.2.p.d 2 1.a even 1 1 trivial
784.2.p.d 2 28.f even 6 1 inner
1008.2.cs.c 2 21.h odd 6 1
1008.2.cs.c 2 84.h odd 2 1
1008.2.cs.f 2 21.c even 2 1
1008.2.cs.f 2 84.n even 6 1
3136.2.f.a 2 56.j odd 6 1
3136.2.f.a 2 56.k odd 6 1
3136.2.f.b 2 56.m even 6 1
3136.2.f.b 2 56.p even 6 1
7056.2.b.b 2 21.h odd 6 1
7056.2.b.b 2 84.j odd 6 1
7056.2.b.m 2 21.g even 6 1
7056.2.b.m 2 84.n even 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{5}^{2} + 3 T_{5} + 3 \)
\( T_{11}^{2} + 3 T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \)
$5$ \( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( 1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \)
$37$ \( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 34 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 74 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( 1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 130 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \)
$79$ \( ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 21 T + 236 T^{2} - 1869 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 146 T^{2} + 9409 T^{4} \)
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