Properties

Label 784.2.p.g
Level $784$
Weight $2$
Character orbit 784.p
Analytic conductor $6.260$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + 4 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + 4 \beta_{2} q^{9} + ( -\beta_{1} - 2 \beta_{3} ) q^{11} + ( 2 + 4 \beta_{2} ) q^{13} + ( 2 \beta_{1} + \beta_{3} ) q^{15} + ( -3 + 3 \beta_{2} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{3} ) q^{23} + ( -2 - 2 \beta_{2} ) q^{25} + \beta_{3} q^{27} -\beta_{1} q^{31} + ( 14 + 7 \beta_{2} ) q^{33} + 7 \beta_{2} q^{37} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{39} + ( -2 - 4 \beta_{2} ) q^{41} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{2} ) q^{45} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{47} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{51} + ( -3 - 3 \beta_{2} ) q^{53} -3 \beta_{3} q^{55} + 7 q^{57} -3 \beta_{1} q^{59} + ( -2 - \beta_{2} ) q^{61} + 6 \beta_{2} q^{65} + ( \beta_{1} + 2 \beta_{3} ) q^{67} + ( 7 + 14 \beta_{2} ) q^{69} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( 3 - 3 \beta_{2} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{75} + ( \beta_{1} - \beta_{3} ) q^{79} + ( 5 + 5 \beta_{2} ) q^{81} -9 q^{85} + ( -2 - \beta_{2} ) q^{89} -7 \beta_{2} q^{93} + ( -\beta_{1} - 2 \beta_{3} ) q^{95} + ( -2 - 4 \beta_{2} ) q^{97} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} - 8q^{9} + O(q^{10}) \) \( 4q + 6q^{5} - 8q^{9} - 18q^{17} - 4q^{25} + 42q^{33} - 14q^{37} - 24q^{45} - 6q^{53} + 28q^{57} - 6q^{61} - 12q^{65} + 18q^{73} + 10q^{81} - 36q^{85} - 6q^{89} + 14q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\)

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
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0 −1.32288 + 2.29129i 0 1.50000 0.866025i 0 0 0 −2.00000 3.46410i 0
31.2 0 1.32288 2.29129i 0 1.50000 0.866025i 0 0 0 −2.00000 3.46410i 0
607.1 0 −1.32288 2.29129i 0 1.50000 + 0.866025i 0 0 0 −2.00000 + 3.46410i 0
607.2 0 1.32288 + 2.29129i 0 1.50000 + 0.866025i 0 0 0 −2.00000 + 3.46410i 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
4.b odd 2 1 inner
7.d odd 6 1 inner
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.g 4
4.b odd 2 1 inner 784.2.p.g 4
7.b odd 2 1 112.2.p.c 4
7.c even 3 1 112.2.p.c 4
7.c even 3 1 784.2.f.d 4
7.d odd 6 1 784.2.f.d 4
7.d odd 6 1 inner 784.2.p.g 4
21.c even 2 1 1008.2.cs.q 4
21.g even 6 1 7056.2.b.s 4
21.h odd 6 1 1008.2.cs.q 4
21.h odd 6 1 7056.2.b.s 4
28.d even 2 1 112.2.p.c 4
28.f even 6 1 784.2.f.d 4
28.f even 6 1 inner 784.2.p.g 4
28.g odd 6 1 112.2.p.c 4
28.g odd 6 1 784.2.f.d 4
56.e even 2 1 448.2.p.c 4
56.h odd 2 1 448.2.p.c 4
56.j odd 6 1 3136.2.f.f 4
56.k odd 6 1 448.2.p.c 4
56.k odd 6 1 3136.2.f.f 4
56.m even 6 1 3136.2.f.f 4
56.p even 6 1 448.2.p.c 4
56.p even 6 1 3136.2.f.f 4
84.h odd 2 1 1008.2.cs.q 4
84.j odd 6 1 7056.2.b.s 4
84.n even 6 1 1008.2.cs.q 4
84.n even 6 1 7056.2.b.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 7.b odd 2 1
112.2.p.c 4 7.c even 3 1
112.2.p.c 4 28.d even 2 1
112.2.p.c 4 28.g odd 6 1
448.2.p.c 4 56.e even 2 1
448.2.p.c 4 56.h odd 2 1
448.2.p.c 4 56.k odd 6 1
448.2.p.c 4 56.p even 6 1
784.2.f.d 4 7.c even 3 1
784.2.f.d 4 7.d odd 6 1
784.2.f.d 4 28.f even 6 1
784.2.f.d 4 28.g odd 6 1
784.2.p.g 4 1.a even 1 1 trivial
784.2.p.g 4 4.b odd 2 1 inner
784.2.p.g 4 7.d odd 6 1 inner
784.2.p.g 4 28.f even 6 1 inner
1008.2.cs.q 4 21.c even 2 1
1008.2.cs.q 4 21.h odd 6 1
1008.2.cs.q 4 84.h odd 2 1
1008.2.cs.q 4 84.n even 6 1
3136.2.f.f 4 56.j odd 6 1
3136.2.f.f 4 56.k odd 6 1
3136.2.f.f 4 56.m even 6 1
3136.2.f.f 4 56.p even 6 1
7056.2.b.s 4 21.g even 6 1
7056.2.b.s 4 21.h odd 6 1
7056.2.b.s 4 84.j odd 6 1
7056.2.b.s 4 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 \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{4} + 7 T_{3}^{2} + 49 \)
\( T_{5}^{2} - 3 T_{5} + 3 \)
\( T_{11}^{4} - 21 T_{11}^{2} + 441 \)