Properties

Label 280.2.b.a
Level $280$
Weight $2$
Character orbit 280.b
Analytic conductor $2.236$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{2} + 2 i q^{3} + 2 i q^{4} -i q^{5} + ( -2 + 2 i ) q^{6} - q^{7} + ( -2 + 2 i ) q^{8} - q^{9} +O(q^{10})\) \( q + ( 1 + i ) q^{2} + 2 i q^{3} + 2 i q^{4} -i q^{5} + ( -2 + 2 i ) q^{6} - q^{7} + ( -2 + 2 i ) q^{8} - q^{9} + ( 1 - i ) q^{10} + 4 i q^{11} -4 q^{12} -2 i q^{13} + ( -1 - i ) q^{14} + 2 q^{15} -4 q^{16} + 4 q^{17} + ( -1 - i ) q^{18} -4 i q^{19} + 2 q^{20} -2 i q^{21} + ( -4 + 4 i ) q^{22} + ( -4 - 4 i ) q^{24} - q^{25} + ( 2 - 2 i ) q^{26} + 4 i q^{27} -2 i q^{28} -8 i q^{29} + ( 2 + 2 i ) q^{30} + 8 q^{31} + ( -4 - 4 i ) q^{32} -8 q^{33} + ( 4 + 4 i ) q^{34} + i q^{35} -2 i q^{36} + 6 i q^{37} + ( 4 - 4 i ) q^{38} + 4 q^{39} + ( 2 + 2 i ) q^{40} + 6 q^{41} + ( 2 - 2 i ) q^{42} + 4 i q^{43} -8 q^{44} + i q^{45} -8 i q^{48} + q^{49} + ( -1 - i ) q^{50} + 8 i q^{51} + 4 q^{52} + 6 i q^{53} + ( -4 + 4 i ) q^{54} + 4 q^{55} + ( 2 - 2 i ) q^{56} + 8 q^{57} + ( 8 - 8 i ) q^{58} -8 i q^{59} + 4 i q^{60} -10 i q^{61} + ( 8 + 8 i ) q^{62} + q^{63} -8 i q^{64} -2 q^{65} + ( -8 - 8 i ) q^{66} + 4 i q^{67} + 8 i q^{68} + ( -1 + i ) q^{70} + 10 q^{71} + ( 2 - 2 i ) q^{72} -16 q^{73} + ( -6 + 6 i ) q^{74} -2 i q^{75} + 8 q^{76} -4 i q^{77} + ( 4 + 4 i ) q^{78} -10 q^{79} + 4 i q^{80} -11 q^{81} + ( 6 + 6 i ) q^{82} -14 i q^{83} + 4 q^{84} -4 i q^{85} + ( -4 + 4 i ) q^{86} + 16 q^{87} + ( -8 - 8 i ) q^{88} -6 q^{89} + ( -1 + i ) q^{90} + 2 i q^{91} + 16 i q^{93} -4 q^{95} + ( 8 - 8 i ) q^{96} -12 q^{97} + ( 1 + i ) q^{98} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + 2q^{10} - 8q^{12} - 2q^{14} + 4q^{15} - 8q^{16} + 8q^{17} - 2q^{18} + 4q^{20} - 8q^{22} - 8q^{24} - 2q^{25} + 4q^{26} + 4q^{30} + 16q^{31} - 8q^{32} - 16q^{33} + 8q^{34} + 8q^{38} + 8q^{39} + 4q^{40} + 12q^{41} + 4q^{42} - 16q^{44} + 2q^{49} - 2q^{50} + 8q^{52} - 8q^{54} + 8q^{55} + 4q^{56} + 16q^{57} + 16q^{58} + 16q^{62} + 2q^{63} - 4q^{65} - 16q^{66} - 2q^{70} + 20q^{71} + 4q^{72} - 32q^{73} - 12q^{74} + 16q^{76} + 8q^{78} - 20q^{79} - 22q^{81} + 12q^{82} + 8q^{84} - 8q^{86} + 32q^{87} - 16q^{88} - 12q^{89} - 2q^{90} - 8q^{95} + 16q^{96} - 24q^{97} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1
1.00000i
1.00000i
1.00000 1.00000i 2.00000i 2.00000i 1.00000i −2.00000 2.00000i −1.00000 −2.00000 2.00000i −1.00000 1.00000 + 1.00000i
141.2 1.00000 + 1.00000i 2.00000i 2.00000i 1.00000i −2.00000 + 2.00000i −1.00000 −2.00000 + 2.00000i −1.00000 1.00000 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
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 280.2.b.a 2
4.b odd 2 1 1120.2.b.b 2
8.b even 2 1 inner 280.2.b.a 2
8.d odd 2 1 1120.2.b.b 2
16.e even 4 1 8960.2.a.d 1
16.e even 4 1 8960.2.a.r 1
16.f odd 4 1 8960.2.a.g 1
16.f odd 4 1 8960.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.a 2 1.a even 1 1 trivial
280.2.b.a 2 8.b even 2 1 inner
1120.2.b.b 2 4.b odd 2 1
1120.2.b.b 2 8.d odd 2 1
8960.2.a.d 1 16.e even 4 1
8960.2.a.g 1 16.f odd 4 1
8960.2.a.m 1 16.f odd 4 1
8960.2.a.r 1 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 - 2 T + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( 100 + T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( ( 16 + T )^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( ( 12 + T )^{2} \)
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