Properties

Label 56.3.h.c
Level 56
Weight 3
Character orbit 56.h
Self dual yes
Analytic conductor 1.526
Analytic rank 0
Dimension 2
CM discriminant -56
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 56.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.52588948042\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 2 \beta q^{3} + 4 q^{4} -6 \beta q^{5} + 4 \beta q^{6} -7 q^{7} + 8 q^{8} - q^{9} +O(q^{10})\) \( q + 2 q^{2} + 2 \beta q^{3} + 4 q^{4} -6 \beta q^{5} + 4 \beta q^{6} -7 q^{7} + 8 q^{8} - q^{9} -12 \beta q^{10} + 8 \beta q^{12} + 18 \beta q^{13} -14 q^{14} -24 q^{15} + 16 q^{16} -2 q^{18} -6 \beta q^{19} -24 \beta q^{20} -14 \beta q^{21} -10 q^{23} + 16 \beta q^{24} + 47 q^{25} + 36 \beta q^{26} -20 \beta q^{27} -28 q^{28} -48 q^{30} + 32 q^{32} + 42 \beta q^{35} -4 q^{36} -12 \beta q^{38} + 72 q^{39} -48 \beta q^{40} -28 \beta q^{42} + 6 \beta q^{45} -20 q^{46} + 32 \beta q^{48} + 49 q^{49} + 94 q^{50} + 72 \beta q^{52} -40 \beta q^{54} -56 q^{56} -24 q^{57} -54 \beta q^{59} -96 q^{60} -6 \beta q^{61} + 7 q^{63} + 64 q^{64} -216 q^{65} -20 \beta q^{69} + 84 \beta q^{70} -110 q^{71} -8 q^{72} + 94 \beta q^{75} -24 \beta q^{76} + 144 q^{78} + 130 q^{79} -96 \beta q^{80} -71 q^{81} + 18 \beta q^{83} -56 \beta q^{84} + 12 \beta q^{90} -126 \beta q^{91} -40 q^{92} + 72 q^{95} + 64 \beta q^{96} + 98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} - 14q^{7} + 16q^{8} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} - 14q^{7} + 16q^{8} - 2q^{9} - 28q^{14} - 48q^{15} + 32q^{16} - 4q^{18} - 20q^{23} + 94q^{25} - 56q^{28} - 96q^{30} + 64q^{32} - 8q^{36} + 144q^{39} - 40q^{46} + 98q^{49} + 188q^{50} - 112q^{56} - 48q^{57} - 192q^{60} + 14q^{63} + 128q^{64} - 432q^{65} - 220q^{71} - 16q^{72} + 288q^{78} + 260q^{79} - 142q^{81} - 80q^{92} + 144q^{95} + 196q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−1.41421
1.41421
2.00000 −2.82843 4.00000 8.48528 −5.65685 −7.00000 8.00000 −1.00000 16.9706
13.2 2.00000 2.82843 4.00000 −8.48528 5.65685 −7.00000 8.00000 −1.00000 −16.9706
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.3.h.c 2
3.b odd 2 1 504.3.l.a 2
4.b odd 2 1 224.3.h.c 2
7.b odd 2 1 inner 56.3.h.c 2
7.c even 3 2 392.3.j.a 4
7.d odd 6 2 392.3.j.a 4
8.b even 2 1 inner 56.3.h.c 2
8.d odd 2 1 224.3.h.c 2
12.b even 2 1 2016.3.l.c 2
21.c even 2 1 504.3.l.a 2
24.f even 2 1 2016.3.l.c 2
24.h odd 2 1 504.3.l.a 2
28.d even 2 1 224.3.h.c 2
56.e even 2 1 224.3.h.c 2
56.h odd 2 1 CM 56.3.h.c 2
56.j odd 6 2 392.3.j.a 4
56.p even 6 2 392.3.j.a 4
84.h odd 2 1 2016.3.l.c 2
168.e odd 2 1 2016.3.l.c 2
168.i even 2 1 504.3.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.c 2 1.a even 1 1 trivial
56.3.h.c 2 7.b odd 2 1 inner
56.3.h.c 2 8.b even 2 1 inner
56.3.h.c 2 56.h odd 2 1 CM
224.3.h.c 2 4.b odd 2 1
224.3.h.c 2 8.d odd 2 1
224.3.h.c 2 28.d even 2 1
224.3.h.c 2 56.e even 2 1
392.3.j.a 4 7.c even 3 2
392.3.j.a 4 7.d odd 6 2
392.3.j.a 4 56.j odd 6 2
392.3.j.a 4 56.p even 6 2
504.3.l.a 2 3.b odd 2 1
504.3.l.a 2 21.c even 2 1
504.3.l.a 2 24.h odd 2 1
504.3.l.a 2 168.i even 2 1
2016.3.l.c 2 12.b even 2 1
2016.3.l.c 2 24.f even 2 1
2016.3.l.c 2 84.h odd 2 1
2016.3.l.c 2 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{3}^{\mathrm{new}}(56, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )^{2} \)
$3$ \( 1 + 10 T^{2} + 81 T^{4} \)
$5$ \( 1 - 22 T^{2} + 625 T^{4} \)
$7$ \( ( 1 + 7 T )^{2} \)
$11$ \( ( 1 - 11 T )^{2}( 1 + 11 T )^{2} \)
$13$ \( 1 - 310 T^{2} + 28561 T^{4} \)
$17$ \( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \)
$19$ \( 1 + 650 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 + 10 T + 529 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T )^{2}( 1 + 29 T )^{2} \)
$31$ \( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \)
$37$ \( ( 1 - 37 T )^{2}( 1 + 37 T )^{2} \)
$41$ \( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \)
$43$ \( ( 1 - 43 T )^{2}( 1 + 43 T )^{2} \)
$47$ \( ( 1 - 47 T )^{2}( 1 + 47 T )^{2} \)
$53$ \( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \)
$59$ \( 1 + 1130 T^{2} + 12117361 T^{4} \)
$61$ \( 1 + 7370 T^{2} + 13845841 T^{4} \)
$67$ \( ( 1 - 67 T )^{2}( 1 + 67 T )^{2} \)
$71$ \( ( 1 + 110 T + 5041 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T )^{2}( 1 + 73 T )^{2} \)
$79$ \( ( 1 - 130 T + 6241 T^{2} )^{2} \)
$83$ \( 1 + 13130 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \)
$97$ \( ( 1 - 97 T )^{2}( 1 + 97 T )^{2} \)
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