Properties

Label 3840.2.k.f
Level $3840$
Weight $2$
Character orbit 3840.k
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + i q^{3} -i q^{5} -4 q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} -i q^{5} -4 q^{7} - q^{9} -2 i q^{13} + q^{15} + 6 q^{17} -4 i q^{19} -4 i q^{21} - q^{25} -i q^{27} + 6 i q^{29} -8 q^{31} + 4 i q^{35} + 2 i q^{37} + 2 q^{39} + 6 q^{41} + 4 i q^{43} + i q^{45} + 9 q^{49} + 6 i q^{51} -6 i q^{53} + 4 q^{57} + 10 i q^{61} + 4 q^{63} -2 q^{65} -4 i q^{67} -2 q^{73} -i q^{75} -8 q^{79} + q^{81} + 12 i q^{83} -6 i q^{85} -6 q^{87} -18 q^{89} + 8 i q^{91} -8 i q^{93} -4 q^{95} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 8q^{7} - 2q^{9} + 2q^{15} + 12q^{17} - 2q^{25} - 16q^{31} + 4q^{39} + 12q^{41} + 18q^{49} + 8q^{57} + 8q^{63} - 4q^{65} - 4q^{73} - 16q^{79} + 2q^{81} - 12q^{87} - 36q^{89} - 8q^{95} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1921.1
1.00000i
1.00000i
0 1.00000i 0 1.00000i 0 −4.00000 0 −1.00000 0
1921.2 0 1.00000i 0 1.00000i 0 −4.00000 0 −1.00000 0
\(n\): 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 3840.2.k.f 2
4.b odd 2 1 3840.2.k.y 2
8.b even 2 1 inner 3840.2.k.f 2
8.d odd 2 1 3840.2.k.y 2
16.e even 4 1 240.2.a.b 1
16.e even 4 1 960.2.a.p 1
16.f odd 4 1 30.2.a.a 1
16.f odd 4 1 960.2.a.e 1
48.i odd 4 1 720.2.a.j 1
48.i odd 4 1 2880.2.a.q 1
48.k even 4 1 90.2.a.c 1
48.k even 4 1 2880.2.a.a 1
80.i odd 4 1 1200.2.f.e 2
80.i odd 4 1 4800.2.f.w 2
80.j even 4 1 150.2.c.a 2
80.j even 4 1 4800.2.f.p 2
80.k odd 4 1 150.2.a.b 1
80.k odd 4 1 4800.2.a.cq 1
80.q even 4 1 1200.2.a.k 1
80.q even 4 1 4800.2.a.d 1
80.s even 4 1 150.2.c.a 2
80.s even 4 1 4800.2.f.p 2
80.t odd 4 1 1200.2.f.e 2
80.t odd 4 1 4800.2.f.w 2
112.j even 4 1 1470.2.a.d 1
112.u odd 12 2 1470.2.i.o 2
112.v even 12 2 1470.2.i.q 2
144.u even 12 2 810.2.e.b 2
144.v odd 12 2 810.2.e.l 2
176.i even 4 1 3630.2.a.w 1
208.l even 4 1 5070.2.b.k 2
208.o odd 4 1 5070.2.a.w 1
208.s even 4 1 5070.2.b.k 2
240.t even 4 1 450.2.a.d 1
240.z odd 4 1 450.2.c.b 2
240.bb even 4 1 3600.2.f.i 2
240.bd odd 4 1 450.2.c.b 2
240.bf even 4 1 3600.2.f.i 2
240.bm odd 4 1 3600.2.a.f 1
272.k odd 4 1 8670.2.a.g 1
336.v odd 4 1 4410.2.a.z 1
560.be even 4 1 7350.2.a.ct 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 16.f odd 4 1
90.2.a.c 1 48.k even 4 1
150.2.a.b 1 80.k odd 4 1
150.2.c.a 2 80.j even 4 1
150.2.c.a 2 80.s even 4 1
240.2.a.b 1 16.e even 4 1
450.2.a.d 1 240.t even 4 1
450.2.c.b 2 240.z odd 4 1
450.2.c.b 2 240.bd odd 4 1
720.2.a.j 1 48.i odd 4 1
810.2.e.b 2 144.u even 12 2
810.2.e.l 2 144.v odd 12 2
960.2.a.e 1 16.f odd 4 1
960.2.a.p 1 16.e even 4 1
1200.2.a.k 1 80.q even 4 1
1200.2.f.e 2 80.i odd 4 1
1200.2.f.e 2 80.t odd 4 1
1470.2.a.d 1 112.j even 4 1
1470.2.i.o 2 112.u odd 12 2
1470.2.i.q 2 112.v even 12 2
2880.2.a.a 1 48.k even 4 1
2880.2.a.q 1 48.i odd 4 1
3600.2.a.f 1 240.bm odd 4 1
3600.2.f.i 2 240.bb even 4 1
3600.2.f.i 2 240.bf even 4 1
3630.2.a.w 1 176.i even 4 1
3840.2.k.f 2 1.a even 1 1 trivial
3840.2.k.f 2 8.b even 2 1 inner
3840.2.k.y 2 4.b odd 2 1
3840.2.k.y 2 8.d odd 2 1
4410.2.a.z 1 336.v odd 4 1
4800.2.a.d 1 80.q even 4 1
4800.2.a.cq 1 80.k odd 4 1
4800.2.f.p 2 80.j even 4 1
4800.2.f.p 2 80.s even 4 1
4800.2.f.w 2 80.i odd 4 1
4800.2.f.w 2 80.t odd 4 1
5070.2.a.w 1 208.o odd 4 1
5070.2.b.k 2 208.l even 4 1
5070.2.b.k 2 208.s even 4 1
7350.2.a.ct 1 560.be even 4 1
8670.2.a.g 1 272.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3840, [\chi])\):

\( T_{7} + 4 \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)
\( T_{17} - 6 \)
\( T_{23} \)
\( T_{31} + 8 \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 59 T^{2} )^{2} \)
$61$ \( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 2 T + 97 T^{2} )^{2} \)
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