Properties

Label 560.2.bb.b
Level 560
Weight 2
Character orbit 560.bb
Analytic conductor 4.472
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bb (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{2} + ( 1 + i ) q^{3} + 2 i q^{4} + ( 2 - i ) q^{5} + 2 i q^{6} - q^{7} + ( -2 + 2 i ) q^{8} -i q^{9} +O(q^{10})\) \( q + ( 1 + i ) q^{2} + ( 1 + i ) q^{3} + 2 i q^{4} + ( 2 - i ) q^{5} + 2 i q^{6} - q^{7} + ( -2 + 2 i ) q^{8} -i q^{9} + ( 3 + i ) q^{10} + ( 1 + i ) q^{11} + ( -2 + 2 i ) q^{12} + ( 3 + 3 i ) q^{13} + ( -1 - i ) q^{14} + ( 3 + i ) q^{15} -4 q^{16} + ( 1 - i ) q^{18} + ( -3 + 3 i ) q^{19} + ( 2 + 4 i ) q^{20} + ( -1 - i ) q^{21} + 2 i q^{22} -4 q^{24} + ( 3 - 4 i ) q^{25} + 6 i q^{26} + ( 4 - 4 i ) q^{27} -2 i q^{28} + ( -1 + i ) q^{29} + ( 2 + 4 i ) q^{30} -8 q^{31} + ( -4 - 4 i ) q^{32} + 2 i q^{33} + ( -2 + i ) q^{35} + 2 q^{36} + ( 5 - 5 i ) q^{37} -6 q^{38} + 6 i q^{39} + ( -2 + 6 i ) q^{40} -12 i q^{41} -2 i q^{42} + ( 1 - i ) q^{43} + ( -2 + 2 i ) q^{44} + ( -1 - 2 i ) q^{45} -10 i q^{47} + ( -4 - 4 i ) q^{48} + q^{49} + ( 7 - i ) q^{50} + ( -6 + 6 i ) q^{52} + ( -3 + 3 i ) q^{53} + 8 q^{54} + ( 3 + i ) q^{55} + ( 2 - 2 i ) q^{56} -6 q^{57} -2 q^{58} + ( -1 - i ) q^{59} + ( -2 + 6 i ) q^{60} + ( -7 + 7 i ) q^{61} + ( -8 - 8 i ) q^{62} + i q^{63} -8 i q^{64} + ( 9 + 3 i ) q^{65} + ( -2 + 2 i ) q^{66} + ( 3 + 3 i ) q^{67} + ( -3 - i ) q^{70} + 6 i q^{71} + ( 2 + 2 i ) q^{72} + 10 q^{73} + 10 q^{74} + ( 7 - i ) q^{75} + ( -6 - 6 i ) q^{76} + ( -1 - i ) q^{77} + ( -6 + 6 i ) q^{78} -8 q^{79} + ( -8 + 4 i ) q^{80} + 5 q^{81} + ( 12 - 12 i ) q^{82} + ( 5 + 5 i ) q^{83} + ( 2 - 2 i ) q^{84} + 2 q^{86} -2 q^{87} -4 q^{88} + ( 1 - 3 i ) q^{90} + ( -3 - 3 i ) q^{91} + ( -8 - 8 i ) q^{93} + ( 10 - 10 i ) q^{94} + ( -3 + 9 i ) q^{95} -8 i q^{96} -8 i q^{97} + ( 1 + i ) q^{98} + ( 1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 4q^{5} - 2q^{7} - 4q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 4q^{5} - 2q^{7} - 4q^{8} + 6q^{10} + 2q^{11} - 4q^{12} + 6q^{13} - 2q^{14} + 6q^{15} - 8q^{16} + 2q^{18} - 6q^{19} + 4q^{20} - 2q^{21} - 8q^{24} + 6q^{25} + 8q^{27} - 2q^{29} + 4q^{30} - 16q^{31} - 8q^{32} - 4q^{35} + 4q^{36} + 10q^{37} - 12q^{38} - 4q^{40} + 2q^{43} - 4q^{44} - 2q^{45} - 8q^{48} + 2q^{49} + 14q^{50} - 12q^{52} - 6q^{53} + 16q^{54} + 6q^{55} + 4q^{56} - 12q^{57} - 4q^{58} - 2q^{59} - 4q^{60} - 14q^{61} - 16q^{62} + 18q^{65} - 4q^{66} + 6q^{67} - 6q^{70} + 4q^{72} + 20q^{73} + 20q^{74} + 14q^{75} - 12q^{76} - 2q^{77} - 12q^{78} - 16q^{79} - 16q^{80} + 10q^{81} + 24q^{82} + 10q^{83} + 4q^{84} + 4q^{86} - 4q^{87} - 8q^{88} + 2q^{90} - 6q^{91} - 16q^{93} + 20q^{94} - 6q^{95} + 2q^{98} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000 1.00000i 2.00000i 2.00000 + 1.00000i 2.00000i −1.00000 −2.00000 2.00000i 1.00000i 3.00000 1.00000i
309.1 1.00000 + 1.00000i 1.00000 + 1.00000i 2.00000i 2.00000 1.00000i 2.00000i −1.00000 −2.00000 + 2.00000i 1.00000i 3.00000 + 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
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bb.b yes 2
5.b even 2 1 560.2.bb.a 2
16.e even 4 1 560.2.bb.a 2
80.q even 4 1 inner 560.2.bb.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bb.a 2 5.b even 2 1
560.2.bb.a 2 16.e even 4 1
560.2.bb.b yes 2 1.a even 1 1 trivial
560.2.bb.b yes 2 80.q even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} \)
$3$ \( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 17 T^{2} )^{2} \)
$19$ \( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 + 2 T + 2 T^{2} + 58 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 2 T + 37 T^{2} ) \)
$41$ \( 1 + 62 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 2 T + 2 T^{2} - 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 6 T + 18 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 2 T + 2 T^{2} + 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 14 T + 98 T^{2} + 854 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 6 T + 18 T^{2} - 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 106 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 10 T + 50 T^{2} - 830 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 89 T^{2} )^{2} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \)
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