Properties

Label 784.2.m.b
Level $784$
Weight $2$
Character orbit 784.m
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
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.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000 1.00000i 2.00000i 1.00000 + 1.00000i −2.00000 0 2.00000 2.00000i 1.00000i 2.00000i
589.1 −1.00000 + 1.00000i 1.00000 + 1.00000i 2.00000i 1.00000 1.00000i −2.00000 0 2.00000 + 2.00000i 1.00000i 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.m.b 2
7.b odd 2 1 16.2.e.a 2
7.c even 3 2 784.2.x.c 4
7.d odd 6 2 784.2.x.f 4
16.e even 4 1 inner 784.2.m.b 2
21.c even 2 1 144.2.k.a 2
28.d even 2 1 64.2.e.a 2
35.c odd 2 1 400.2.l.c 2
35.f even 4 1 400.2.q.a 2
35.f even 4 1 400.2.q.b 2
56.e even 2 1 128.2.e.a 2
56.h odd 2 1 128.2.e.b 2
84.h odd 2 1 576.2.k.a 2
112.j even 4 1 64.2.e.a 2
112.j even 4 1 128.2.e.a 2
112.l odd 4 1 16.2.e.a 2
112.l odd 4 1 128.2.e.b 2
112.w even 12 2 784.2.x.c 4
112.x odd 12 2 784.2.x.f 4
140.c even 2 1 1600.2.l.a 2
140.j odd 4 1 1600.2.q.a 2
140.j odd 4 1 1600.2.q.b 2
168.e odd 2 1 1152.2.k.a 2
168.i even 2 1 1152.2.k.b 2
224.v odd 8 2 1024.2.a.b 2
224.v odd 8 2 1024.2.b.e 2
224.x even 8 2 1024.2.a.e 2
224.x even 8 2 1024.2.b.b 2
336.v odd 4 1 576.2.k.a 2
336.v odd 4 1 1152.2.k.a 2
336.y even 4 1 144.2.k.a 2
336.y even 4 1 1152.2.k.b 2
560.r even 4 1 400.2.q.a 2
560.u odd 4 1 1600.2.q.a 2
560.be even 4 1 1600.2.l.a 2
560.bf odd 4 1 400.2.l.c 2
560.bm odd 4 1 1600.2.q.b 2
560.bn even 4 1 400.2.q.b 2
672.bo even 8 2 9216.2.a.d 2
672.br odd 8 2 9216.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 7.b odd 2 1
16.2.e.a 2 112.l odd 4 1
64.2.e.a 2 28.d even 2 1
64.2.e.a 2 112.j even 4 1
128.2.e.a 2 56.e even 2 1
128.2.e.a 2 112.j even 4 1
128.2.e.b 2 56.h odd 2 1
128.2.e.b 2 112.l odd 4 1
144.2.k.a 2 21.c even 2 1
144.2.k.a 2 336.y even 4 1
400.2.l.c 2 35.c odd 2 1
400.2.l.c 2 560.bf odd 4 1
400.2.q.a 2 35.f even 4 1
400.2.q.a 2 560.r even 4 1
400.2.q.b 2 35.f even 4 1
400.2.q.b 2 560.bn even 4 1
576.2.k.a 2 84.h odd 2 1
576.2.k.a 2 336.v odd 4 1
784.2.m.b 2 1.a even 1 1 trivial
784.2.m.b 2 16.e even 4 1 inner
784.2.x.c 4 7.c even 3 2
784.2.x.c 4 112.w even 12 2
784.2.x.f 4 7.d odd 6 2
784.2.x.f 4 112.x odd 12 2
1024.2.a.b 2 224.v odd 8 2
1024.2.a.e 2 224.x even 8 2
1024.2.b.b 2 224.x even 8 2
1024.2.b.e 2 224.v odd 8 2
1152.2.k.a 2 168.e odd 2 1
1152.2.k.a 2 336.v odd 4 1
1152.2.k.b 2 168.i even 2 1
1152.2.k.b 2 336.y even 4 1
1600.2.l.a 2 140.c even 2 1
1600.2.l.a 2 560.be even 4 1
1600.2.q.a 2 140.j odd 4 1
1600.2.q.a 2 560.u odd 4 1
1600.2.q.b 2 140.j odd 4 1
1600.2.q.b 2 560.bm odd 4 1
9216.2.a.d 2 672.bo even 8 2
9216.2.a.s 2 672.br odd 8 2

Hecke kernels

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

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

Hecke characteristic polynomials

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