Properties

Label 240.2.bc.b
Level $240$
Weight $2$
Character orbit 240.bc
Analytic conductor $1.916$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.bc (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(1\)
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: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{2} + i q^{3} + 2 i q^{4} + ( -2 + i ) q^{5} + ( 1 - i ) q^{6} + ( -1 - i ) q^{7} + ( 2 - 2 i ) q^{8} - q^{9} +O(q^{10})\) \( q + ( -1 - i ) q^{2} + i q^{3} + 2 i q^{4} + ( -2 + i ) q^{5} + ( 1 - i ) q^{6} + ( -1 - i ) q^{7} + ( 2 - 2 i ) q^{8} - q^{9} + ( 3 + i ) q^{10} + ( -3 - 3 i ) q^{11} -2 q^{12} -4 q^{13} + 2 i q^{14} + ( -1 - 2 i ) q^{15} -4 q^{16} + ( -5 - 5 i ) q^{17} + ( 1 + i ) q^{18} + ( 5 + 5 i ) q^{19} + ( -2 - 4 i ) q^{20} + ( 1 - i ) q^{21} + 6 i q^{22} + ( -1 + i ) q^{23} + ( 2 + 2 i ) q^{24} + ( 3 - 4 i ) q^{25} + ( 4 + 4 i ) q^{26} -i q^{27} + ( 2 - 2 i ) q^{28} + ( -5 + 5 i ) q^{29} + ( -1 + 3 i ) q^{30} + 2 i q^{31} + ( 4 + 4 i ) q^{32} + ( 3 - 3 i ) q^{33} + 10 i q^{34} + ( 3 + i ) q^{35} -2 i q^{36} -8 q^{37} -10 i q^{38} -4 i q^{39} + ( -2 + 6 i ) q^{40} + 8 i q^{41} -2 q^{42} + 2 q^{43} + ( 6 - 6 i ) q^{44} + ( 2 - i ) q^{45} + 2 q^{46} + ( 5 - 5 i ) q^{47} -4 i q^{48} -5 i q^{49} + ( -7 + i ) q^{50} + ( 5 - 5 i ) q^{51} -8 i q^{52} + 6 i q^{53} + ( -1 + i ) q^{54} + ( 9 + 3 i ) q^{55} -4 q^{56} + ( -5 + 5 i ) q^{57} + 10 q^{58} + ( 5 - 5 i ) q^{59} + ( 4 - 2 i ) q^{60} + ( -1 - i ) q^{61} + ( 2 - 2 i ) q^{62} + ( 1 + i ) q^{63} -8 i q^{64} + ( 8 - 4 i ) q^{65} -6 q^{66} -2 q^{67} + ( 10 - 10 i ) q^{68} + ( -1 - i ) q^{69} + ( -2 - 4 i ) q^{70} + ( -2 + 2 i ) q^{72} + ( -5 - 5 i ) q^{73} + ( 8 + 8 i ) q^{74} + ( 4 + 3 i ) q^{75} + ( -10 + 10 i ) q^{76} + 6 i q^{77} + ( -4 + 4 i ) q^{78} -8 q^{79} + ( 8 - 4 i ) q^{80} + q^{81} + ( 8 - 8 i ) q^{82} + 12 i q^{83} + ( 2 + 2 i ) q^{84} + ( 15 + 5 i ) q^{85} + ( -2 - 2 i ) q^{86} + ( -5 - 5 i ) q^{87} -12 q^{88} + 2 q^{89} + ( -3 - i ) q^{90} + ( 4 + 4 i ) q^{91} + ( -2 - 2 i ) q^{92} -2 q^{93} -10 q^{94} + ( -15 - 5 i ) q^{95} + ( -4 + 4 i ) q^{96} + ( 1 + i ) q^{97} + ( -5 + 5 i ) q^{98} + ( 3 + 3 i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{5} + 2 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 4 q^{5} + 2 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} + 6 q^{10} - 6 q^{11} - 4 q^{12} - 8 q^{13} - 2 q^{15} - 8 q^{16} - 10 q^{17} + 2 q^{18} + 10 q^{19} - 4 q^{20} + 2 q^{21} - 2 q^{23} + 4 q^{24} + 6 q^{25} + 8 q^{26} + 4 q^{28} - 10 q^{29} - 2 q^{30} + 8 q^{32} + 6 q^{33} + 6 q^{35} - 16 q^{37} - 4 q^{40} - 4 q^{42} + 4 q^{43} + 12 q^{44} + 4 q^{45} + 4 q^{46} + 10 q^{47} - 14 q^{50} + 10 q^{51} - 2 q^{54} + 18 q^{55} - 8 q^{56} - 10 q^{57} + 20 q^{58} + 10 q^{59} + 8 q^{60} - 2 q^{61} + 4 q^{62} + 2 q^{63} + 16 q^{65} - 12 q^{66} - 4 q^{67} + 20 q^{68} - 2 q^{69} - 4 q^{70} - 4 q^{72} - 10 q^{73} + 16 q^{74} + 8 q^{75} - 20 q^{76} - 8 q^{78} - 16 q^{79} + 16 q^{80} + 2 q^{81} + 16 q^{82} + 4 q^{84} + 30 q^{85} - 4 q^{86} - 10 q^{87} - 24 q^{88} + 4 q^{89} - 6 q^{90} + 8 q^{91} - 4 q^{92} - 4 q^{93} - 20 q^{94} - 30 q^{95} - 8 q^{96} + 2 q^{97} - 10 q^{98} + 6 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(i\) \(1\) \(-i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.00000i
1.00000i
−1.00000 + 1.00000i 1.00000i 2.00000i −2.00000 1.00000i 1.00000 + 1.00000i −1.00000 + 1.00000i 2.00000 + 2.00000i −1.00000 3.00000 1.00000i
67.1 −1.00000 1.00000i 1.00000i 2.00000i −2.00000 + 1.00000i 1.00000 1.00000i −1.00000 1.00000i 2.00000 2.00000i −1.00000 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.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.bc.b yes 2
3.b odd 2 1 720.2.bd.c 2
4.b odd 2 1 960.2.bc.c 2
5.c odd 4 1 240.2.y.a 2
8.b even 2 1 1920.2.bc.c 2
8.d odd 2 1 1920.2.bc.d 2
15.e even 4 1 720.2.z.b 2
16.e even 4 1 960.2.y.b 2
16.e even 4 1 1920.2.y.b 2
16.f odd 4 1 240.2.y.a 2
16.f odd 4 1 1920.2.y.f 2
20.e even 4 1 960.2.y.b 2
40.i odd 4 1 1920.2.y.f 2
40.k even 4 1 1920.2.y.b 2
48.k even 4 1 720.2.z.b 2
80.i odd 4 1 1920.2.bc.d 2
80.j even 4 1 inner 240.2.bc.b yes 2
80.s even 4 1 1920.2.bc.c 2
80.t odd 4 1 960.2.bc.c 2
240.bd odd 4 1 720.2.bd.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.y.a 2 5.c odd 4 1
240.2.y.a 2 16.f odd 4 1
240.2.bc.b yes 2 1.a even 1 1 trivial
240.2.bc.b yes 2 80.j even 4 1 inner
720.2.z.b 2 15.e even 4 1
720.2.z.b 2 48.k even 4 1
720.2.bd.c 2 3.b odd 2 1
720.2.bd.c 2 240.bd odd 4 1
960.2.y.b 2 16.e even 4 1
960.2.y.b 2 20.e even 4 1
960.2.bc.c 2 4.b odd 2 1
960.2.bc.c 2 80.t odd 4 1
1920.2.y.b 2 16.e even 4 1
1920.2.y.b 2 40.k even 4 1
1920.2.y.f 2 16.f odd 4 1
1920.2.y.f 2 40.i odd 4 1
1920.2.bc.c 2 8.b even 2 1
1920.2.bc.c 2 80.s even 4 1
1920.2.bc.d 2 8.d odd 2 1
1920.2.bc.d 2 80.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{2} + 2 T_{7} + 2 \)
\( T_{11}^{2} + 6 T_{11} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 2 T + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 2 + 2 T + T^{2} \)
$11$ \( 18 + 6 T + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 50 + 10 T + T^{2} \)
$19$ \( 50 - 10 T + T^{2} \)
$23$ \( 2 + 2 T + T^{2} \)
$29$ \( 50 + 10 T + T^{2} \)
$31$ \( 4 + T^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 50 - 10 T + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( 50 - 10 T + T^{2} \)
$61$ \( 2 + 2 T + T^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 50 + 10 T + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( 2 - 2 T + T^{2} \)
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