Properties

Label 784.2.x.c
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: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{5} -2 q^{6} + ( 2 - 2 \zeta_{12}^{3} ) q^{8} -\zeta_{12} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{5} -2 q^{6} + ( 2 - 2 \zeta_{12}^{3} ) q^{8} -\zeta_{12} q^{9} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{10} + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{11} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{12} + ( 1 - \zeta_{12}^{3} ) q^{13} + 2 q^{15} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} -2 \zeta_{12}^{2} q^{17} + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{18} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{19} + ( -2 + 2 \zeta_{12}^{3} ) q^{20} -2 \zeta_{12}^{3} q^{22} -6 \zeta_{12} q^{23} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{24} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{25} + ( 2 - 2 \zeta_{12}^{2} ) q^{26} + ( 4 + 4 \zeta_{12}^{3} ) q^{27} + ( 3 - 3 \zeta_{12}^{3} ) q^{29} + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{30} -8 \zeta_{12}^{2} q^{31} + ( 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{32} + ( -2 + 2 \zeta_{12}^{2} ) q^{33} + ( -2 - 2 \zeta_{12}^{3} ) q^{34} -2 q^{36} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{37} -6 \zeta_{12}^{2} q^{38} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{39} + ( -4 + 4 \zeta_{12}^{2} ) q^{40} + ( 5 + 5 \zeta_{12}^{3} ) q^{43} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{44} + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{45} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{46} + ( 8 - 8 \zeta_{12}^{2} ) q^{47} + ( -4 + 4 \zeta_{12}^{3} ) q^{48} + ( -3 + 3 \zeta_{12}^{3} ) q^{50} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{51} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( -5 \zeta_{12} + 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{53} + 8 \zeta_{12} q^{54} + 2 \zeta_{12}^{3} q^{55} + 6 \zeta_{12}^{3} q^{57} + ( 6 - 6 \zeta_{12}^{2} ) q^{58} + ( 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{59} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{60} + ( -9 + 9 \zeta_{12} + 9 \zeta_{12}^{2} ) q^{61} + ( -8 - 8 \zeta_{12}^{3} ) q^{62} -8 \zeta_{12}^{3} q^{64} + ( -2 + 2 \zeta_{12}^{2} ) q^{65} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{66} + ( 5 \zeta_{12} + 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} -4 \zeta_{12} q^{68} + ( 6 + 6 \zeta_{12}^{3} ) q^{69} -10 \zeta_{12}^{3} q^{71} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{72} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{74} + ( 3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{75} + ( -6 - 6 \zeta_{12}^{3} ) q^{76} + ( -2 + 2 \zeta_{12}^{3} ) q^{78} + ( -4 \zeta_{12} + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{80} -5 \zeta_{12}^{2} q^{81} + ( 1 - \zeta_{12}^{3} ) q^{83} + ( 2 + 2 \zeta_{12}^{3} ) q^{85} + 10 \zeta_{12} q^{86} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{87} -4 \zeta_{12}^{2} q^{88} + 4 \zeta_{12} q^{89} + 2 q^{90} -12 q^{92} + ( -8 + 8 \zeta_{12} + 8 \zeta_{12}^{2} ) q^{93} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{94} + 6 \zeta_{12}^{2} q^{95} + ( -8 + 8 \zeta_{12}^{2} ) q^{96} + 2 q^{97} + ( -1 + \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{3} - 2q^{5} - 8q^{6} + 8q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{3} - 2q^{5} - 8q^{6} + 8q^{8} - 2q^{11} - 4q^{12} + 4q^{13} + 8q^{15} + 8q^{16} - 4q^{17} - 2q^{18} + 6q^{19} - 8q^{20} + 4q^{26} + 16q^{27} + 12q^{29} + 4q^{30} - 16q^{31} - 8q^{32} - 4q^{33} - 8q^{34} - 8q^{36} - 6q^{37} - 12q^{38} - 8q^{40} + 20q^{43} + 4q^{44} + 2q^{45} - 12q^{46} + 16q^{47} - 16q^{48} - 12q^{50} - 4q^{51} - 4q^{52} + 10q^{53} + 12q^{58} - 6q^{59} - 18q^{61} - 32q^{62} - 4q^{65} + 4q^{66} + 10q^{67} + 24q^{69} - 4q^{72} + 6q^{75} - 24q^{76} - 8q^{78} + 8q^{80} - 10q^{81} + 4q^{83} + 8q^{85} - 8q^{88} + 8q^{90} - 48q^{92} - 16q^{93} - 16q^{94} + 12q^{95} - 16q^{96} + 8q^{97} - 4q^{99} + O(q^{100}) \)

Character values

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

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

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
−0.366025 + 1.36603i 0.366025 + 1.36603i −1.73205 1.00000i 0.366025 1.36603i −2.00000 0 2.00000 2.00000i 0.866025 0.500000i 1.73205 + 1.00000i
373.1 1.36603 0.366025i −1.36603 0.366025i 1.73205 1.00000i −1.36603 + 0.366025i −2.00000 0 2.00000 2.00000i −0.866025 0.500000i −1.73205 + 1.00000i
557.1 1.36603 + 0.366025i −1.36603 + 0.366025i 1.73205 + 1.00000i −1.36603 0.366025i −2.00000 0 2.00000 + 2.00000i −0.866025 + 0.500000i −1.73205 1.00000i
765.1 −0.366025 1.36603i 0.366025 1.36603i −1.73205 + 1.00000i 0.366025 + 1.36603i −2.00000 0 2.00000 + 2.00000i 0.866025 + 0.500000i 1.73205 1.00000i
\(n\): 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.c 4
7.b odd 2 1 784.2.x.f 4
7.c even 3 1 784.2.m.b 2
7.c even 3 1 inner 784.2.x.c 4
7.d odd 6 1 16.2.e.a 2
7.d odd 6 1 784.2.x.f 4
16.e even 4 1 inner 784.2.x.c 4
21.g even 6 1 144.2.k.a 2
28.f even 6 1 64.2.e.a 2
35.i odd 6 1 400.2.l.c 2
35.k even 12 1 400.2.q.a 2
35.k even 12 1 400.2.q.b 2
56.j odd 6 1 128.2.e.b 2
56.m even 6 1 128.2.e.a 2
84.j odd 6 1 576.2.k.a 2
112.l odd 4 1 784.2.x.f 4
112.v even 12 1 64.2.e.a 2
112.v even 12 1 128.2.e.a 2
112.w even 12 1 784.2.m.b 2
112.w even 12 1 inner 784.2.x.c 4
112.x odd 12 1 16.2.e.a 2
112.x odd 12 1 128.2.e.b 2
112.x odd 12 1 784.2.x.f 4
140.s even 6 1 1600.2.l.a 2
140.x odd 12 1 1600.2.q.a 2
140.x odd 12 1 1600.2.q.b 2
168.ba even 6 1 1152.2.k.b 2
168.be odd 6 1 1152.2.k.a 2
224.bc odd 24 2 1024.2.a.b 2
224.bc odd 24 2 1024.2.b.e 2
224.be even 24 2 1024.2.a.e 2
224.be even 24 2 1024.2.b.b 2
336.bo even 12 1 144.2.k.a 2
336.bo even 12 1 1152.2.k.b 2
336.br odd 12 1 576.2.k.a 2
336.br odd 12 1 1152.2.k.a 2
560.ce odd 12 1 1600.2.q.b 2
560.ch even 12 1 400.2.q.b 2
560.cn odd 12 1 400.2.l.c 2
560.co even 12 1 1600.2.l.a 2
560.cz even 12 1 400.2.q.a 2
560.da odd 12 1 1600.2.q.a 2
672.ci odd 24 2 9216.2.a.s 2
672.cl even 24 2 9216.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 7.d odd 6 1
16.2.e.a 2 112.x odd 12 1
64.2.e.a 2 28.f even 6 1
64.2.e.a 2 112.v even 12 1
128.2.e.a 2 56.m even 6 1
128.2.e.a 2 112.v even 12 1
128.2.e.b 2 56.j odd 6 1
128.2.e.b 2 112.x odd 12 1
144.2.k.a 2 21.g even 6 1
144.2.k.a 2 336.bo even 12 1
400.2.l.c 2 35.i odd 6 1
400.2.l.c 2 560.cn odd 12 1
400.2.q.a 2 35.k even 12 1
400.2.q.a 2 560.cz even 12 1
400.2.q.b 2 35.k even 12 1
400.2.q.b 2 560.ch even 12 1
576.2.k.a 2 84.j odd 6 1
576.2.k.a 2 336.br odd 12 1
784.2.m.b 2 7.c even 3 1
784.2.m.b 2 112.w even 12 1
784.2.x.c 4 1.a even 1 1 trivial
784.2.x.c 4 7.c even 3 1 inner
784.2.x.c 4 16.e even 4 1 inner
784.2.x.c 4 112.w even 12 1 inner
784.2.x.f 4 7.b odd 2 1
784.2.x.f 4 7.d odd 6 1
784.2.x.f 4 112.l odd 4 1
784.2.x.f 4 112.x odd 12 1
1024.2.a.b 2 224.bc odd 24 2
1024.2.a.e 2 224.be even 24 2
1024.2.b.b 2 224.be even 24 2
1024.2.b.e 2 224.bc odd 24 2
1152.2.k.a 2 168.be odd 6 1
1152.2.k.a 2 336.br odd 12 1
1152.2.k.b 2 168.ba even 6 1
1152.2.k.b 2 336.bo even 12 1
1600.2.l.a 2 140.s even 6 1
1600.2.l.a 2 560.co even 12 1
1600.2.q.a 2 140.x odd 12 1
1600.2.q.a 2 560.da odd 12 1
1600.2.q.b 2 140.x odd 12 1
1600.2.q.b 2 560.ce odd 12 1
9216.2.a.d 2 672.cl even 24 2
9216.2.a.s 2 672.ci odd 24 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( ( 2 - 2 T + T^{2} )^{2} \)
$17$ \( ( 4 + 2 T + T^{2} )^{2} \)
$19$ \( 324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 1296 - 36 T^{2} + T^{4} \)
$29$ \( ( 18 - 6 T + T^{2} )^{2} \)
$31$ \( ( 64 + 8 T + T^{2} )^{2} \)
$37$ \( 324 + 108 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 50 - 10 T + T^{2} )^{2} \)
$47$ \( ( 64 - 8 T + T^{2} )^{2} \)
$53$ \( 2500 - 500 T + 50 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 324 + 108 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$61$ \( 26244 + 2916 T + 162 T^{2} + 18 T^{3} + T^{4} \)
$67$ \( 2500 - 500 T + 50 T^{2} - 10 T^{3} + T^{4} \)
$71$ \( ( 100 + T^{2} )^{2} \)
$73$ \( 256 - 16 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 2 - 2 T + T^{2} )^{2} \)
$89$ \( 256 - 16 T^{2} + T^{4} \)
$97$ \( ( -2 + T )^{4} \)
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