Properties

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

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{5} -\beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{5} -\beta_{2} q^{9} + ( -2 - 2 \beta_{2} ) q^{11} + 4 q^{15} -\beta_{1} q^{17} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{19} + 4 \beta_{2} q^{23} + ( -3 - 3 \beta_{2} ) q^{25} -4 \beta_{3} q^{27} + 2 q^{29} -6 \beta_{1} q^{31} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{33} + 10 \beta_{2} q^{37} -7 \beta_{3} q^{41} -2 q^{43} -2 \beta_{1} q^{45} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} -2 \beta_{2} q^{51} + ( 2 + 2 \beta_{2} ) q^{53} + 4 \beta_{3} q^{55} + 10 q^{57} + \beta_{1} q^{59} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( 12 + 12 \beta_{2} ) q^{67} + 4 \beta_{3} q^{69} + 12 q^{71} -\beta_{1} q^{73} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{75} + 4 \beta_{2} q^{79} + ( 5 + 5 \beta_{2} ) q^{81} -7 \beta_{3} q^{83} -4 q^{85} + 2 \beta_{1} q^{87} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{89} -12 \beta_{2} q^{93} + ( -20 - 20 \beta_{2} ) q^{95} + 7 \beta_{3} q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{9} - 4 q^{11} + 16 q^{15} - 8 q^{23} - 6 q^{25} + 8 q^{29} - 20 q^{37} - 8 q^{43} + 4 q^{51} + 4 q^{53} + 40 q^{57} + 24 q^{67} + 48 q^{71} - 8 q^{79} + 10 q^{81} - 16 q^{85} + 24 q^{93} - 40 q^{95} - 8 q^{99} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −0.707107 1.22474i 0 −1.41421 + 2.44949i 0 0 0 0.500000 0.866025i 0
177.2 0 0.707107 + 1.22474i 0 1.41421 2.44949i 0 0 0 0.500000 0.866025i 0
753.1 0 −0.707107 + 1.22474i 0 −1.41421 2.44949i 0 0 0 0.500000 + 0.866025i 0
753.2 0 0.707107 1.22474i 0 1.41421 + 2.44949i 0 0 0 0.500000 + 0.866025i 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 inner
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.m 4
4.b odd 2 1 98.2.c.c 4
7.b odd 2 1 inner 784.2.i.m 4
7.c even 3 1 784.2.a.l 2
7.c even 3 1 inner 784.2.i.m 4
7.d odd 6 1 784.2.a.l 2
7.d odd 6 1 inner 784.2.i.m 4
12.b even 2 1 882.2.g.l 4
21.g even 6 1 7056.2.a.cl 2
21.h odd 6 1 7056.2.a.cl 2
28.d even 2 1 98.2.c.c 4
28.f even 6 1 98.2.a.b 2
28.f even 6 1 98.2.c.c 4
28.g odd 6 1 98.2.a.b 2
28.g odd 6 1 98.2.c.c 4
56.j odd 6 1 3136.2.a.bm 2
56.k odd 6 1 3136.2.a.bn 2
56.m even 6 1 3136.2.a.bn 2
56.p even 6 1 3136.2.a.bm 2
84.h odd 2 1 882.2.g.l 4
84.j odd 6 1 882.2.a.n 2
84.j odd 6 1 882.2.g.l 4
84.n even 6 1 882.2.a.n 2
84.n even 6 1 882.2.g.l 4
140.p odd 6 1 2450.2.a.bj 2
140.s even 6 1 2450.2.a.bj 2
140.w even 12 2 2450.2.c.v 4
140.x odd 12 2 2450.2.c.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 28.f even 6 1
98.2.a.b 2 28.g odd 6 1
98.2.c.c 4 4.b odd 2 1
98.2.c.c 4 28.d even 2 1
98.2.c.c 4 28.f even 6 1
98.2.c.c 4 28.g odd 6 1
784.2.a.l 2 7.c even 3 1
784.2.a.l 2 7.d odd 6 1
784.2.i.m 4 1.a even 1 1 trivial
784.2.i.m 4 7.b odd 2 1 inner
784.2.i.m 4 7.c even 3 1 inner
784.2.i.m 4 7.d odd 6 1 inner
882.2.a.n 2 84.j odd 6 1
882.2.a.n 2 84.n even 6 1
882.2.g.l 4 12.b even 2 1
882.2.g.l 4 84.h odd 2 1
882.2.g.l 4 84.j odd 6 1
882.2.g.l 4 84.n even 6 1
2450.2.a.bj 2 140.p odd 6 1
2450.2.a.bj 2 140.s even 6 1
2450.2.c.v 4 140.w even 12 2
2450.2.c.v 4 140.x odd 12 2
3136.2.a.bm 2 56.j odd 6 1
3136.2.a.bm 2 56.p even 6 1
3136.2.a.bn 2 56.k odd 6 1
3136.2.a.bn 2 56.m even 6 1
7056.2.a.cl 2 21.g even 6 1
7056.2.a.cl 2 21.h odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 2 T_{3}^{2} + 4 \)
\( T_{5}^{4} + 8 T_{5}^{2} + 64 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 + 2 T^{2} + T^{4} \)
$5$ \( 64 + 8 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 4 + 2 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 2500 + 50 T^{2} + T^{4} \)
$23$ \( ( 16 + 4 T + T^{2} )^{2} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( 5184 + 72 T^{2} + T^{4} \)
$37$ \( ( 100 + 10 T + T^{2} )^{2} \)
$41$ \( ( -98 + T^{2} )^{2} \)
$43$ \( ( 2 + T )^{4} \)
$47$ \( 64 + 8 T^{2} + T^{4} \)
$53$ \( ( 4 - 2 T + T^{2} )^{2} \)
$59$ \( 4 + 2 T^{2} + T^{4} \)
$61$ \( 64 + 8 T^{2} + T^{4} \)
$67$ \( ( 144 - 12 T + T^{2} )^{2} \)
$71$ \( ( -12 + T )^{4} \)
$73$ \( 4 + 2 T^{2} + T^{4} \)
$79$ \( ( 16 + 4 T + T^{2} )^{2} \)
$83$ \( ( -98 + T^{2} )^{2} \)
$89$ \( 2500 + 50 T^{2} + T^{4} \)
$97$ \( ( -98 + T^{2} )^{2} \)
show more
show less