Properties

Label 784.2.f.a
Level $784$
Weight $2$
Character orbit 784.f
Analytic conductor $6.260$
Analytic rank $1$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
783.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 0 1.73205i 0 0 0 −2.00000 0
783.2 0 −1.00000 0 1.73205i 0 0 0 −2.00000 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.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.f.a 2
3.b odd 2 1 7056.2.b.b 2
4.b odd 2 1 784.2.f.b 2
7.b odd 2 1 784.2.f.b 2
7.c even 3 1 112.2.p.b yes 2
7.c even 3 1 784.2.p.d 2
7.d odd 6 1 112.2.p.a 2
7.d odd 6 1 784.2.p.c 2
8.b even 2 1 3136.2.f.b 2
8.d odd 2 1 3136.2.f.a 2
12.b even 2 1 7056.2.b.m 2
21.c even 2 1 7056.2.b.m 2
21.g even 6 1 1008.2.cs.f 2
21.h odd 6 1 1008.2.cs.c 2
28.d even 2 1 inner 784.2.f.a 2
28.f even 6 1 112.2.p.b yes 2
28.f even 6 1 784.2.p.d 2
28.g odd 6 1 112.2.p.a 2
28.g odd 6 1 784.2.p.c 2
56.e even 2 1 3136.2.f.b 2
56.h odd 2 1 3136.2.f.a 2
56.j odd 6 1 448.2.p.b 2
56.k odd 6 1 448.2.p.b 2
56.m even 6 1 448.2.p.a 2
56.p even 6 1 448.2.p.a 2
84.h odd 2 1 7056.2.b.b 2
84.j odd 6 1 1008.2.cs.c 2
84.n even 6 1 1008.2.cs.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 7.d odd 6 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.f even 6 1
448.2.p.a 2 56.m even 6 1
448.2.p.a 2 56.p even 6 1
448.2.p.b 2 56.j odd 6 1
448.2.p.b 2 56.k odd 6 1
784.2.f.a 2 1.a even 1 1 trivial
784.2.f.a 2 28.d even 2 1 inner
784.2.f.b 2 4.b odd 2 1
784.2.f.b 2 7.b odd 2 1
784.2.p.c 2 7.d odd 6 1
784.2.p.c 2 28.g odd 6 1
784.2.p.d 2 7.c even 3 1
784.2.p.d 2 28.f even 6 1
1008.2.cs.c 2 21.h odd 6 1
1008.2.cs.c 2 84.j odd 6 1
1008.2.cs.f 2 21.g even 6 1
1008.2.cs.f 2 84.n even 6 1
3136.2.f.a 2 8.d odd 2 1
3136.2.f.a 2 56.h odd 2 1
3136.2.f.b 2 8.b even 2 1
3136.2.f.b 2 56.e even 2 1
7056.2.b.b 2 3.b odd 2 1
7056.2.b.b 2 84.h odd 2 1
7056.2.b.m 2 12.b even 2 1
7056.2.b.m 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\).