Properties

Label 784.2.x.f
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 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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{5} + 2 q^{6} + ( - 2 \zeta_{12}^{3} + 2) q^{8} - \zeta_{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{5} + 2 q^{6} + ( - 2 \zeta_{12}^{3} + 2) q^{8} - \zeta_{12} q^{9} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{11} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{12} + (\zeta_{12}^{3} - 1) q^{13} + 2 q^{15} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + 2 \zeta_{12}^{2} q^{17} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{18} + (3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{19} + ( - 2 \zeta_{12}^{3} + 2) q^{20} - 2 \zeta_{12}^{3} q^{22} - 6 \zeta_{12} q^{23} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{24} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{25} + (2 \zeta_{12}^{2} - 2) q^{26} + ( - 4 \zeta_{12}^{3} - 4) q^{27} + ( - 3 \zeta_{12}^{3} + 3) q^{29} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{30} + 8 \zeta_{12}^{2} q^{31} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{32} + ( - 2 \zeta_{12}^{2} + 2) q^{33} + (2 \zeta_{12}^{3} + 2) q^{34} - 2 q^{36} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{37} + 6 \zeta_{12}^{2} q^{38} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{39} + ( - 4 \zeta_{12}^{2} + 4) q^{40} + (5 \zeta_{12}^{3} + 5) q^{43} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{44} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{45} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{46} + (8 \zeta_{12}^{2} - 8) q^{47} + ( - 4 \zeta_{12}^{3} + 4) q^{48} + (3 \zeta_{12}^{3} - 3) q^{50} + (2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{51} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{52} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12}) q^{53} - 8 \zeta_{12} q^{54} - 2 \zeta_{12}^{3} q^{55} + 6 \zeta_{12}^{3} q^{57} + ( - 6 \zeta_{12}^{2} + 6) q^{58} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{59} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{60} + ( - 9 \zeta_{12}^{2} - 9 \zeta_{12} + 9) q^{61} + (8 \zeta_{12}^{3} + 8) q^{62} - 8 \zeta_{12}^{3} q^{64} + (2 \zeta_{12}^{2} - 2) q^{65} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{66} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12}) q^{67} + 4 \zeta_{12} q^{68} + ( - 6 \zeta_{12}^{3} - 6) q^{69} - 10 \zeta_{12}^{3} q^{71} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{72} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{73} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{74} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{75} + (6 \zeta_{12}^{3} + 6) q^{76} + (2 \zeta_{12}^{3} - 2) q^{78} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{80} - 5 \zeta_{12}^{2} q^{81} + (\zeta_{12}^{3} - 1) q^{83} + (2 \zeta_{12}^{3} + 2) q^{85} + 10 \zeta_{12} q^{86} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{87} - 4 \zeta_{12}^{2} q^{88} - 4 \zeta_{12} q^{89} - 2 q^{90} - 12 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} + 6 \zeta_{12}^{2} q^{95} + ( - 8 \zeta_{12}^{2} + 8) q^{96} - 2 q^{97} + (\zeta_{12}^{3} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} + 2 q^{5} + 8 q^{6} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{3} + 2 q^{5} + 8 q^{6} + 8 q^{8} - 2 q^{11} + 4 q^{12} - 4 q^{13} + 8 q^{15} + 8 q^{16} + 4 q^{17} - 2 q^{18} - 6 q^{19} + 8 q^{20} - 4 q^{26} - 16 q^{27} + 12 q^{29} + 4 q^{30} + 16 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{34} - 8 q^{36} - 6 q^{37} + 12 q^{38} + 8 q^{40} + 20 q^{43} + 4 q^{44} - 2 q^{45} - 12 q^{46} - 16 q^{47} + 16 q^{48} - 12 q^{50} - 4 q^{51} + 4 q^{52} + 10 q^{53} + 12 q^{58} + 6 q^{59} + 18 q^{61} + 32 q^{62} - 4 q^{65} - 4 q^{66} + 10 q^{67} - 24 q^{69} - 4 q^{72} - 6 q^{75} + 24 q^{76} - 8 q^{78} - 8 q^{80} - 10 q^{81} - 4 q^{83} + 8 q^{85} - 8 q^{88} - 8 q^{90} - 48 q^{92} - 16 q^{93} + 16 q^{94} + 12 q^{95} + 16 q^{96} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−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.f 4
7.b odd 2 1 784.2.x.c 4
7.c even 3 1 16.2.e.a 2
7.c even 3 1 inner 784.2.x.f 4
7.d odd 6 1 784.2.m.b 2
7.d odd 6 1 784.2.x.c 4
16.e even 4 1 inner 784.2.x.f 4
21.h odd 6 1 144.2.k.a 2
28.g odd 6 1 64.2.e.a 2
35.j even 6 1 400.2.l.c 2
35.l odd 12 1 400.2.q.a 2
35.l odd 12 1 400.2.q.b 2
56.k odd 6 1 128.2.e.a 2
56.p even 6 1 128.2.e.b 2
84.n even 6 1 576.2.k.a 2
112.l odd 4 1 784.2.x.c 4
112.u odd 12 1 64.2.e.a 2
112.u odd 12 1 128.2.e.a 2
112.w even 12 1 16.2.e.a 2
112.w even 12 1 128.2.e.b 2
112.w even 12 1 inner 784.2.x.f 4
112.x odd 12 1 784.2.m.b 2
112.x odd 12 1 784.2.x.c 4
140.p odd 6 1 1600.2.l.a 2
140.w even 12 1 1600.2.q.a 2
140.w even 12 1 1600.2.q.b 2
168.s odd 6 1 1152.2.k.b 2
168.v even 6 1 1152.2.k.a 2
224.bd even 24 2 1024.2.a.b 2
224.bd even 24 2 1024.2.b.e 2
224.bf odd 24 2 1024.2.a.e 2
224.bf odd 24 2 1024.2.b.b 2
336.bt odd 12 1 144.2.k.a 2
336.bt odd 12 1 1152.2.k.b 2
336.bu even 12 1 576.2.k.a 2
336.bu even 12 1 1152.2.k.a 2
560.cf even 12 1 1600.2.q.a 2
560.cg odd 12 1 400.2.q.a 2
560.cr even 12 1 400.2.l.c 2
560.cs odd 12 1 1600.2.l.a 2
560.cy odd 12 1 400.2.q.b 2
560.db even 12 1 1600.2.q.b 2
672.ce odd 24 2 9216.2.a.d 2
672.ch even 24 2 9216.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 7.c even 3 1
16.2.e.a 2 112.w even 12 1
64.2.e.a 2 28.g odd 6 1
64.2.e.a 2 112.u odd 12 1
128.2.e.a 2 56.k odd 6 1
128.2.e.a 2 112.u odd 12 1
128.2.e.b 2 56.p even 6 1
128.2.e.b 2 112.w even 12 1
144.2.k.a 2 21.h odd 6 1
144.2.k.a 2 336.bt odd 12 1
400.2.l.c 2 35.j even 6 1
400.2.l.c 2 560.cr even 12 1
400.2.q.a 2 35.l odd 12 1
400.2.q.a 2 560.cg odd 12 1
400.2.q.b 2 35.l odd 12 1
400.2.q.b 2 560.cy odd 12 1
576.2.k.a 2 84.n even 6 1
576.2.k.a 2 336.bu even 12 1
784.2.m.b 2 7.d odd 6 1
784.2.m.b 2 112.x odd 12 1
784.2.x.c 4 7.b odd 2 1
784.2.x.c 4 7.d odd 6 1
784.2.x.c 4 112.l odd 4 1
784.2.x.c 4 112.x odd 12 1
784.2.x.f 4 1.a even 1 1 trivial
784.2.x.f 4 7.c even 3 1 inner
784.2.x.f 4 16.e even 4 1 inner
784.2.x.f 4 112.w even 12 1 inner
1024.2.a.b 2 224.bd even 24 2
1024.2.a.e 2 224.bf odd 24 2
1024.2.b.b 2 224.bf odd 24 2
1024.2.b.e 2 224.bd even 24 2
1152.2.k.a 2 168.v even 6 1
1152.2.k.a 2 336.bu even 12 1
1152.2.k.b 2 168.s odd 6 1
1152.2.k.b 2 336.bt odd 12 1
1600.2.l.a 2 140.p odd 6 1
1600.2.l.a 2 560.cs odd 12 1
1600.2.q.a 2 140.w even 12 1
1600.2.q.a 2 560.cf even 12 1
1600.2.q.b 2 140.w even 12 1
1600.2.q.b 2 560.db even 12 1
9216.2.a.d 2 672.ce odd 24 2
9216.2.a.s 2 672.ch even 24 2

Hecke kernels

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

\( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} - 2T_{5}^{3} + 2T_{5}^{2} - 4T_{5} + 4 \) 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
$3$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) 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
$13$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$41$ \( T^{4} \) 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
$53$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$61$ \( T^{4} - 18 T^{3} + 162 T^{2} + \cdots + 26244 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$71$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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