Properties

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

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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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + 4 \zeta_{12}^{3} q^{6} + (2 \zeta_{12}^{3} + 2) q^{8} + 5 \zeta_{12} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + 4 \zeta_{12}^{3} q^{6} + (2 \zeta_{12}^{3} + 2) q^{8} + 5 \zeta_{12} q^{9} + 4 \zeta_{12}^{2} q^{10} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{12} + 8 q^{15} + ( - 4 \zeta_{12}^{2} + 4) q^{16} - 6 \zeta_{12}^{2} q^{17} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12}) q^{18} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{19} + ( - 4 \zeta_{12}^{3} + 4) q^{20} + 6 q^{22} - 2 \zeta_{12} q^{23} - 8 \zeta_{12}^{2} q^{24} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{25} + ( - 4 \zeta_{12}^{3} - 4) q^{27} + (\zeta_{12}^{3} - 1) q^{29} + ( - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 8) q^{30} + 4 \zeta_{12}^{2} q^{31} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{32} + ( - 12 \zeta_{12}^{2} + 12) q^{33} + (6 \zeta_{12}^{3} - 6) q^{34} - 10 q^{36} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{37} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{38} - 8 \zeta_{12} q^{40} - 2 \zeta_{12}^{3} q^{41} + ( - 5 \zeta_{12}^{3} - 5) q^{43} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{44} + (10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 10 \zeta_{12}) q^{45} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{46} + (8 \zeta_{12}^{2} - 8) q^{47} + (8 \zeta_{12}^{3} - 8) q^{48} + ( - 3 \zeta_{12}^{3} - 3) q^{50} + (12 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{51} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12}) q^{53} + (8 \zeta_{12}^{2} - 8) q^{54} - 12 \zeta_{12}^{3} q^{55} - 16 \zeta_{12}^{3} q^{57} + 2 \zeta_{12} q^{58} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{59} + (16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{60} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{61} + ( - 4 \zeta_{12}^{3} + 4) q^{62} + 8 \zeta_{12}^{3} q^{64} + (12 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 12 \zeta_{12}) q^{66} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12}) q^{67} + 12 \zeta_{12} q^{68} + (4 \zeta_{12}^{3} + 4) q^{69} - 8 \zeta_{12}^{3} q^{71} + (10 \zeta_{12}^{2} + 10 \zeta_{12} - 10) q^{72} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{73} + 6 \zeta_{12}^{2} q^{74} + (6 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{75} + ( - 8 \zeta_{12}^{3} - 8) q^{76} + ( - 14 \zeta_{12}^{2} + 14) q^{79} + (8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{80} + \zeta_{12}^{2} q^{81} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{82} + (12 \zeta_{12}^{3} + 12) q^{85} + (10 \zeta_{12}^{2} - 10) q^{86} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{87} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{88} + 6 \zeta_{12} q^{89} + 20 \zeta_{12}^{3} q^{90} + 4 q^{92} + ( - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 8) q^{93} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12}) q^{94} - 16 \zeta_{12}^{2} q^{95} + 16 \zeta_{12} q^{96} + 6 q^{97} + (15 \zeta_{12}^{3} - 15) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{3} - 4 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{3} - 4 q^{5} + 8 q^{8} + 8 q^{10} + 6 q^{11} + 8 q^{12} + 32 q^{15} + 8 q^{16} - 12 q^{17} - 10 q^{18} - 8 q^{19} + 16 q^{20} + 24 q^{22} - 16 q^{24} - 16 q^{27} - 4 q^{29} + 16 q^{30} + 8 q^{31} - 8 q^{32} + 24 q^{33} - 24 q^{34} - 40 q^{36} - 6 q^{37} - 20 q^{43} + 12 q^{44} - 20 q^{45} + 4 q^{46} - 16 q^{47} - 32 q^{48} - 12 q^{50} - 24 q^{51} - 14 q^{53} - 16 q^{54} + 4 q^{59} + 12 q^{61} + 16 q^{62} - 24 q^{66} + 10 q^{67} + 16 q^{69} - 20 q^{72} + 12 q^{74} - 12 q^{75} - 32 q^{76} + 28 q^{79} + 16 q^{80} + 2 q^{81} - 4 q^{82} + 48 q^{85} - 20 q^{86} + 16 q^{92} + 16 q^{93} + 16 q^{94} - 32 q^{95} + 24 q^{97} - 60 q^{99}+O(q^{100}) \) 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 0.732051 + 2.73205i 1.73205 + 1.00000i 0.732051 2.73205i 4.00000i 0 2.00000 + 2.00000i −4.33013 + 2.50000i 2.00000 3.46410i
373.1 −0.366025 1.36603i −2.73205 0.732051i −1.73205 + 1.00000i −2.73205 + 0.732051i 4.00000i 0 2.00000 + 2.00000i 4.33013 + 2.50000i 2.00000 + 3.46410i
557.1 −0.366025 + 1.36603i −2.73205 + 0.732051i −1.73205 1.00000i −2.73205 0.732051i 4.00000i 0 2.00000 2.00000i 4.33013 2.50000i 2.00000 3.46410i
765.1 1.36603 0.366025i 0.732051 2.73205i 1.73205 1.00000i 0.732051 + 2.73205i 4.00000i 0 2.00000 2.00000i −4.33013 2.50000i 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.b 4
7.b odd 2 1 784.2.x.g 4
7.c even 3 1 784.2.m.c 2
7.c even 3 1 inner 784.2.x.b 4
7.d odd 6 1 112.2.m.a 2
7.d odd 6 1 784.2.x.g 4
16.e even 4 1 inner 784.2.x.b 4
28.f even 6 1 448.2.m.b 2
56.j odd 6 1 896.2.m.d 2
56.m even 6 1 896.2.m.a 2
112.l odd 4 1 784.2.x.g 4
112.v even 12 1 448.2.m.b 2
112.v even 12 1 896.2.m.a 2
112.w even 12 1 784.2.m.c 2
112.w even 12 1 inner 784.2.x.b 4
112.x odd 12 1 112.2.m.a 2
112.x odd 12 1 784.2.x.g 4
112.x odd 12 1 896.2.m.d 2
224.bc odd 24 2 7168.2.a.q 2
224.be even 24 2 7168.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.a 2 7.d odd 6 1
112.2.m.a 2 112.x odd 12 1
448.2.m.b 2 28.f even 6 1
448.2.m.b 2 112.v even 12 1
784.2.m.c 2 7.c even 3 1
784.2.m.c 2 112.w even 12 1
784.2.x.b 4 1.a even 1 1 trivial
784.2.x.b 4 7.c even 3 1 inner
784.2.x.b 4 16.e even 4 1 inner
784.2.x.b 4 112.w even 12 1 inner
784.2.x.g 4 7.b odd 2 1
784.2.x.g 4 7.d odd 6 1
784.2.x.g 4 112.l odd 4 1
784.2.x.g 4 112.x odd 12 1
896.2.m.a 2 56.m even 6 1
896.2.m.a 2 112.v even 12 1
896.2.m.d 2 56.j odd 6 1
896.2.m.d 2 112.x odd 12 1
7168.2.a.i 2 224.be even 24 2
7168.2.a.q 2 224.bc odd 24 2

Hecke kernels

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

\( T_{3}^{4} + 4T_{3}^{3} + 8T_{3}^{2} + 32T_{3} + 64 \) 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
$3$ \( T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64 \) 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
$11$ \( T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 9604 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + 8 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$71$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$97$ \( (T - 6)^{4} \) Copy content Toggle raw display
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