Properties

Label 784.2.x.d
Level $784$
Weight $2$
Character orbit 784.x
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.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + 2) q^{8} - 3 \zeta_{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + 2) q^{8} - 3 \zeta_{12} q^{9} + 4 \zeta_{12}^{2} q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + 2 \zeta_{12}^{2} q^{17} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{18} + (2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{19} + ( - 4 \zeta_{12}^{3} + 4) q^{20} - 2 q^{22} + 6 \zeta_{12} q^{23} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{25} + ( - 7 \zeta_{12}^{3} + 7) q^{29} + 8 \zeta_{12}^{2} q^{31} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{32} + ( - 2 \zeta_{12}^{3} + 2) q^{34} + 6 q^{36} + ( - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{37} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{38} - 8 \zeta_{12} q^{40} + 10 \zeta_{12}^{3} q^{41} + ( - \zeta_{12}^{3} - 1) q^{43} + (2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{44} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12}) q^{45} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 6 \zeta_{12}) q^{46} + ( - 12 \zeta_{12}^{2} + 12) q^{47} + ( - 3 \zeta_{12}^{3} - 3) q^{50} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{53} + 4 \zeta_{12}^{3} q^{55} - 14 \zeta_{12} q^{58} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12}) q^{59} + (6 \zeta_{12}^{2} + 6 \zeta_{12} - 6) q^{61} + ( - 8 \zeta_{12}^{3} + 8) q^{62} + 8 \zeta_{12}^{3} q^{64} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{67} - 4 \zeta_{12} q^{68} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{72} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{73} - 10 \zeta_{12}^{2} q^{74} + ( - 4 \zeta_{12}^{3} - 4) q^{76} + (10 \zeta_{12}^{2} - 10) q^{79} + (8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{80} + 9 \zeta_{12}^{2} q^{81} + ( - 10 \zeta_{12}^{2} + 10 \zeta_{12} + 10) q^{82} + (10 \zeta_{12}^{3} - 10) q^{83} + ( - 4 \zeta_{12}^{3} - 4) q^{85} + (2 \zeta_{12}^{2} - 2) q^{86} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{88} + 14 \zeta_{12} q^{89} - 12 \zeta_{12}^{3} q^{90} - 12 q^{92} + (12 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 12 \zeta_{12}) q^{94} - 8 \zeta_{12}^{2} q^{95} - 2 q^{97} + (3 \zeta_{12}^{3} - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{5} + 8 q^{8} + 8 q^{10} - 2 q^{11} + 8 q^{16} + 4 q^{17} + 6 q^{18} - 4 q^{19} + 16 q^{20} - 8 q^{22} + 28 q^{29} + 16 q^{31} - 8 q^{32} + 8 q^{34} + 24 q^{36} + 10 q^{37} - 4 q^{43} - 4 q^{44} + 12 q^{45} - 12 q^{46} + 24 q^{47} - 12 q^{50} + 2 q^{53} - 16 q^{59} - 12 q^{61} + 32 q^{62} - 6 q^{67} + 12 q^{72} - 20 q^{74} - 16 q^{76} - 20 q^{79} + 16 q^{80} + 18 q^{81} + 20 q^{82} - 40 q^{83} - 16 q^{85} - 4 q^{86} - 48 q^{92} - 24 q^{94} - 16 q^{95} - 8 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
1.36603 + 0.366025i 0 1.73205 + 1.00000i 0.732051 2.73205i 0 0 2.00000 + 2.00000i 2.59808 1.50000i 2.00000 3.46410i
373.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −2.73205 + 0.732051i 0 0 2.00000 + 2.00000i −2.59808 1.50000i 2.00000 + 3.46410i
557.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i −2.73205 0.732051i 0 0 2.00000 2.00000i −2.59808 + 1.50000i 2.00000 3.46410i
765.1 1.36603 0.366025i 0 1.73205 1.00000i 0.732051 + 2.73205i 0 0 2.00000 2.00000i 2.59808 + 1.50000i 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.d 4
7.b odd 2 1 784.2.x.e 4
7.c even 3 1 112.2.m.b 2
7.c even 3 1 inner 784.2.x.d 4
7.d odd 6 1 784.2.m.a 2
7.d odd 6 1 784.2.x.e 4
16.e even 4 1 inner 784.2.x.d 4
28.g odd 6 1 448.2.m.a 2
56.k odd 6 1 896.2.m.c 2
56.p even 6 1 896.2.m.b 2
112.l odd 4 1 784.2.x.e 4
112.u odd 12 1 448.2.m.a 2
112.u odd 12 1 896.2.m.c 2
112.w even 12 1 112.2.m.b 2
112.w even 12 1 inner 784.2.x.d 4
112.w even 12 1 896.2.m.b 2
112.x odd 12 1 784.2.m.a 2
112.x odd 12 1 784.2.x.e 4
224.bd even 24 2 7168.2.a.b 2
224.bf odd 24 2 7168.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 7.c even 3 1
112.2.m.b 2 112.w even 12 1
448.2.m.a 2 28.g odd 6 1
448.2.m.a 2 112.u odd 12 1
784.2.m.a 2 7.d odd 6 1
784.2.m.a 2 112.x odd 12 1
784.2.x.d 4 1.a even 1 1 trivial
784.2.x.d 4 7.c even 3 1 inner
784.2.x.d 4 16.e even 4 1 inner
784.2.x.d 4 112.w even 12 1 inner
784.2.x.e 4 7.b odd 2 1
784.2.x.e 4 7.d odd 6 1
784.2.x.e 4 112.l odd 4 1
784.2.x.e 4 112.x odd 12 1
896.2.m.b 2 56.p even 6 1
896.2.m.b 2 112.w even 12 1
896.2.m.c 2 56.k odd 6 1
896.2.m.c 2 112.u odd 12 1
7168.2.a.b 2 224.bd even 24 2
7168.2.a.k 2 224.bf odd 24 2

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} + 4T_{5}^{3} + 8T_{5}^{2} + 32T_{5} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$41$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$59$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 20 T + 200)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 196 T^{2} + 38416 \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
show more
show less