Properties

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

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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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{3} + 2 \zeta_{12}^{3} q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12} + 1) q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{6} + (2 \zeta_{12}^{3} - 2) q^{8} + ( - \zeta_{12}^{2} - \zeta_{12} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{3} + 2 \zeta_{12}^{3} q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12} + 1) q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{6} + (2 \zeta_{12}^{3} - 2) q^{8} + ( - \zeta_{12}^{2} - \zeta_{12} - 1) q^{9} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{10} + (\zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 3) q^{11} + ( - 2 \zeta_{12} + 2) q^{12} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{13} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 3) q^{15} - 4 q^{16} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 2 \zeta_{12}) q^{17} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2}) q^{18} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{19} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{20} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 6) q^{22} + (2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{23} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 6) q^{25} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 4) q^{26} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{27} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{29} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 5) q^{30} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{31} + ( - 4 \zeta_{12}^{3} - 4) q^{32} + (4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 2 \zeta_{12} + 5) q^{33} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - \zeta_{12} - 4) q^{34} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{36} + (5 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + \zeta_{12} + 1) q^{37} + ( - 6 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{38} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{39} + ( - 4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{40} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{41} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{43} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 6) q^{44} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{45} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} + 1) q^{46} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - \zeta_{12} + 6) q^{47} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4) q^{48} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 6) q^{50} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 1) q^{51} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{52} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 7 \zeta_{12} + 3) q^{53} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{54} + ( - 6 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 5) q^{55} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{57} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 4) q^{58} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 4 \zeta_{12} + 9) q^{59} + (6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{60} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - \zeta_{12} + 1) q^{61} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} - 2) q^{62} - 8 \zeta_{12}^{3} q^{64} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} - 3) q^{65} + (4 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{66} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 4 \zeta_{12} + 1) q^{67} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 6 \zeta_{12} - 8) q^{68} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 3) q^{69} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{71} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12} + 4) q^{72} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 6) q^{73} + ( - 5 \zeta_{12}^{2} + 7 \zeta_{12} - 5) q^{74} + (4 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{75} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{76} + (2 \zeta_{12} + 2) q^{78} + ( - 10 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 5 \zeta_{12} + 8) q^{79} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{80} + (5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 5 \zeta_{12}) q^{81} + ( - 2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 2) q^{82} + (3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} - 3) q^{83} + (3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{85} + (6 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{86} + (5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 5 \zeta_{12} + 6) q^{87} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12}) q^{88} + (3 \zeta_{12}^{2} + 3) q^{89} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{90} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{92} + ( - \zeta_{12}^{3} - \zeta_{12}^{2}) q^{93} + (2 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{94} + ( - 5 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 5 \zeta_{12}) q^{95} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{96} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 4) q^{97} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 2 q^{3} + 2 q^{6} - 8 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 2 q^{3} + 2 q^{6} - 8 q^{8} - 6 q^{9} - 6 q^{10} - 10 q^{11} + 8 q^{12} - 8 q^{13} + 12 q^{15} - 16 q^{16} + 6 q^{17} - 4 q^{18} - 10 q^{19} - 12 q^{20} - 18 q^{22} + 12 q^{23} + 12 q^{24} - 18 q^{25} - 16 q^{26} - 2 q^{27} - 8 q^{29} + 12 q^{30} + 4 q^{31} - 16 q^{32} + 10 q^{33} - 6 q^{34} + 4 q^{36} - 8 q^{37} - 2 q^{38} + 6 q^{39} - 12 q^{40} + 12 q^{43} - 16 q^{44} - 6 q^{45} + 10 q^{46} + 12 q^{47} + 8 q^{48} - 16 q^{50} - 16 q^{52} + 20 q^{53} - 16 q^{58} + 26 q^{59} - 4 q^{61} - 2 q^{62} - 6 q^{65} - 2 q^{66} - 6 q^{67} - 24 q^{68} - 10 q^{69} + 16 q^{72} - 18 q^{73} - 30 q^{74} - 2 q^{75} + 16 q^{76} + 8 q^{78} + 16 q^{79} - 4 q^{81} - 8 q^{82} - 20 q^{83} + 6 q^{85} + 18 q^{87} + 4 q^{88} + 18 q^{89} - 4 q^{92} - 2 q^{93} + 6 q^{94} + 12 q^{95} - 8 q^{96} - 16 q^{97} + 16 q^{99}+O(q^{100}) \) 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
1.00000 + 1.00000i −0.500000 1.86603i 2.00000i −0.866025 + 3.23205i 1.36603 2.36603i 0 −2.00000 + 2.00000i −0.633975 + 0.366025i −4.09808 + 2.36603i
373.1 1.00000 + 1.00000i −0.500000 0.133975i 2.00000i 0.866025 0.232051i −0.366025 0.633975i 0 −2.00000 + 2.00000i −2.36603 1.36603i 1.09808 + 0.633975i
557.1 1.00000 1.00000i −0.500000 + 0.133975i 2.00000i 0.866025 + 0.232051i −0.366025 + 0.633975i 0 −2.00000 2.00000i −2.36603 + 1.36603i 1.09808 0.633975i
765.1 1.00000 1.00000i −0.500000 + 1.86603i 2.00000i −0.866025 3.23205i 1.36603 + 2.36603i 0 −2.00000 2.00000i −0.633975 0.366025i −4.09808 2.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.h 4
7.b odd 2 1 112.2.w.b yes 4
7.c even 3 1 784.2.m.d 4
7.c even 3 1 784.2.x.a 4
7.d odd 6 1 112.2.w.a 4
7.d odd 6 1 784.2.m.e 4
16.e even 4 1 784.2.x.a 4
28.d even 2 1 448.2.ba.a 4
28.f even 6 1 448.2.ba.b 4
56.e even 2 1 896.2.ba.c 4
56.h odd 2 1 896.2.ba.b 4
56.j odd 6 1 896.2.ba.d 4
56.m even 6 1 896.2.ba.a 4
112.j even 4 1 448.2.ba.b 4
112.j even 4 1 896.2.ba.a 4
112.l odd 4 1 112.2.w.a 4
112.l odd 4 1 896.2.ba.d 4
112.v even 12 1 448.2.ba.a 4
112.v even 12 1 896.2.ba.c 4
112.w even 12 1 784.2.m.d 4
112.w even 12 1 inner 784.2.x.h 4
112.x odd 12 1 112.2.w.b yes 4
112.x odd 12 1 784.2.m.e 4
112.x odd 12 1 896.2.ba.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.a 4 7.d odd 6 1
112.2.w.a 4 112.l odd 4 1
112.2.w.b yes 4 7.b odd 2 1
112.2.w.b yes 4 112.x odd 12 1
448.2.ba.a 4 28.d even 2 1
448.2.ba.a 4 112.v even 12 1
448.2.ba.b 4 28.f even 6 1
448.2.ba.b 4 112.j even 4 1
784.2.m.d 4 7.c even 3 1
784.2.m.d 4 112.w even 12 1
784.2.m.e 4 7.d odd 6 1
784.2.m.e 4 112.x odd 12 1
784.2.x.a 4 7.c even 3 1
784.2.x.a 4 16.e even 4 1
784.2.x.h 4 1.a even 1 1 trivial
784.2.x.h 4 112.w even 12 1 inner
896.2.ba.a 4 56.m even 6 1
896.2.ba.a 4 112.j even 4 1
896.2.ba.b 4 56.h odd 2 1
896.2.ba.b 4 112.x odd 12 1
896.2.ba.c 4 56.e even 2 1
896.2.ba.c 4 112.v even 12 1
896.2.ba.d 4 56.j odd 6 1
896.2.ba.d 4 112.l odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} + 4T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 9T_{5}^{2} - 18T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 9 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 10 T^{3} + 41 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + 32 T^{2} + 16 T + 4 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + 41 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + 32 T^{2} + 16 T + 4 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + 137 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 72 T^{2} + 72 T + 36 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 111 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$53$ \( T^{4} - 20 T^{3} + 101 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$59$ \( T^{4} - 26 T^{3} + 365 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + 53 T^{2} + 14 T + 1 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} + 131 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} + 267 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$83$ \( T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
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