Properties

Label 3840.2.d.x
Level $3840$
Weight $2$
Character orbit 3840.d
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.d (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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta - 1) q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta - 1) q^{5} + \beta q^{7} + q^{9} + \beta q^{11} + 6 q^{13} + (\beta - 1) q^{15} + \beta q^{17} + \beta q^{21} + 2 \beta q^{23} + ( - 2 \beta - 3) q^{25} + q^{27} + 8 q^{31} + \beta q^{33} + ( - \beta - 4) q^{35} + 2 q^{37} + 6 q^{39} - 2 q^{41} - 4 q^{43} + (\beta - 1) q^{45} + 4 \beta q^{47} + 3 q^{49} + \beta q^{51} - 6 q^{53} + ( - \beta - 4) q^{55} - 5 \beta q^{59} + \beta q^{61} + \beta q^{63} + (6 \beta - 6) q^{65} - 8 q^{67} + 2 \beta q^{69} + 12 q^{71} - 2 \beta q^{73} + ( - 2 \beta - 3) q^{75} - 4 q^{77} + q^{81} + 4 q^{83} + ( - \beta - 4) q^{85} - 10 q^{89} + 6 \beta q^{91} + 8 q^{93} - 4 \beta q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 12 q^{13} - 2 q^{15} - 6 q^{25} + 2 q^{27} + 16 q^{31} - 8 q^{35} + 4 q^{37} + 12 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} + 6 q^{49} - 12 q^{53} - 8 q^{55} - 12 q^{65} - 16 q^{67} + 24 q^{71} - 6 q^{75} - 8 q^{77} + 2 q^{81} + 8 q^{83} - 8 q^{85} - 20 q^{89} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2689.1
1.00000i
1.00000i
0 1.00000 0 −1.00000 2.00000i 0 2.00000i 0 1.00000 0
2689.2 0 1.00000 0 −1.00000 + 2.00000i 0 2.00000i 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
40.f 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.d.x 2
4.b odd 2 1 3840.2.d.g 2
5.b even 2 1 3840.2.d.j 2
8.b even 2 1 3840.2.d.j 2
8.d odd 2 1 3840.2.d.y 2
16.e even 4 1 240.2.f.a 2
16.e even 4 1 960.2.f.i 2
16.f odd 4 1 30.2.c.a 2
16.f odd 4 1 960.2.f.h 2
20.d odd 2 1 3840.2.d.y 2
40.e odd 2 1 3840.2.d.g 2
40.f even 2 1 inner 3840.2.d.x 2
48.i odd 4 1 720.2.f.f 2
48.i odd 4 1 2880.2.f.c 2
48.k even 4 1 90.2.c.a 2
48.k even 4 1 2880.2.f.e 2
80.i odd 4 1 1200.2.a.g 1
80.i odd 4 1 4800.2.a.m 1
80.j even 4 1 150.2.a.a 1
80.j even 4 1 4800.2.a.l 1
80.k odd 4 1 30.2.c.a 2
80.k odd 4 1 960.2.f.h 2
80.q even 4 1 240.2.f.a 2
80.q even 4 1 960.2.f.i 2
80.s even 4 1 150.2.a.c 1
80.s even 4 1 4800.2.a.cg 1
80.t odd 4 1 1200.2.a.m 1
80.t odd 4 1 4800.2.a.cj 1
112.j even 4 1 1470.2.g.g 2
112.u odd 12 2 1470.2.n.h 4
112.v even 12 2 1470.2.n.a 4
144.u even 12 2 810.2.i.b 4
144.v odd 12 2 810.2.i.e 4
240.t even 4 1 90.2.c.a 2
240.t even 4 1 2880.2.f.e 2
240.z odd 4 1 450.2.a.b 1
240.bb even 4 1 3600.2.a.bg 1
240.bd odd 4 1 450.2.a.f 1
240.bf even 4 1 3600.2.a.o 1
240.bm odd 4 1 720.2.f.f 2
240.bm odd 4 1 2880.2.f.c 2
560.u odd 4 1 7350.2.a.cc 1
560.be even 4 1 1470.2.g.g 2
560.bm odd 4 1 7350.2.a.bg 1
560.co even 12 2 1470.2.n.a 4
560.cs odd 12 2 1470.2.n.h 4
720.cz odd 12 2 810.2.i.e 4
720.da even 12 2 810.2.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 16.f odd 4 1
30.2.c.a 2 80.k odd 4 1
90.2.c.a 2 48.k even 4 1
90.2.c.a 2 240.t even 4 1
150.2.a.a 1 80.j even 4 1
150.2.a.c 1 80.s even 4 1
240.2.f.a 2 16.e even 4 1
240.2.f.a 2 80.q even 4 1
450.2.a.b 1 240.z odd 4 1
450.2.a.f 1 240.bd odd 4 1
720.2.f.f 2 48.i odd 4 1
720.2.f.f 2 240.bm odd 4 1
810.2.i.b 4 144.u even 12 2
810.2.i.b 4 720.da even 12 2
810.2.i.e 4 144.v odd 12 2
810.2.i.e 4 720.cz odd 12 2
960.2.f.h 2 16.f odd 4 1
960.2.f.h 2 80.k odd 4 1
960.2.f.i 2 16.e even 4 1
960.2.f.i 2 80.q even 4 1
1200.2.a.g 1 80.i odd 4 1
1200.2.a.m 1 80.t odd 4 1
1470.2.g.g 2 112.j even 4 1
1470.2.g.g 2 560.be even 4 1
1470.2.n.a 4 112.v even 12 2
1470.2.n.a 4 560.co even 12 2
1470.2.n.h 4 112.u odd 12 2
1470.2.n.h 4 560.cs odd 12 2
2880.2.f.c 2 48.i odd 4 1
2880.2.f.c 2 240.bm odd 4 1
2880.2.f.e 2 48.k even 4 1
2880.2.f.e 2 240.t even 4 1
3600.2.a.o 1 240.bf even 4 1
3600.2.a.bg 1 240.bb even 4 1
3840.2.d.g 2 4.b odd 2 1
3840.2.d.g 2 40.e odd 2 1
3840.2.d.j 2 5.b even 2 1
3840.2.d.j 2 8.b even 2 1
3840.2.d.x 2 1.a even 1 1 trivial
3840.2.d.x 2 40.f even 2 1 inner
3840.2.d.y 2 8.d odd 2 1
3840.2.d.y 2 20.d odd 2 1
4800.2.a.l 1 80.j even 4 1
4800.2.a.m 1 80.i odd 4 1
4800.2.a.cg 1 80.s even 4 1
4800.2.a.cj 1 80.t odd 4 1
7350.2.a.bg 1 560.bm odd 4 1
7350.2.a.cc 1 560.u odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{31} - 8 \) Copy content Toggle raw display
\( T_{37} - 2 \) Copy content Toggle raw display
\( T_{43} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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