Properties

Label 2940.2.k.c
Level $2940$
Weight $2$
Character orbit 2940.k
Analytic conductor $23.476$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.4760181943\)
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 60)
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.

\(f(q)\) \(=\) \( q + i q^{3} + ( -1 + 2 i ) q^{5} - q^{9} +O(q^{10})\) \( q + i q^{3} + ( -1 + 2 i ) q^{5} - q^{9} -4 q^{11} + ( -2 - i ) q^{15} -4 i q^{17} -4 i q^{23} + ( -3 - 4 i ) q^{25} -i q^{27} + 6 q^{29} -4 q^{31} -4 i q^{33} -8 i q^{37} + 10 q^{41} -4 i q^{43} + ( 1 - 2 i ) q^{45} + 4 i q^{47} + 4 q^{51} + 12 i q^{53} + ( 4 - 8 i ) q^{55} + 4 q^{59} -2 q^{61} -4 i q^{67} + 4 q^{69} -8 i q^{73} + ( 4 - 3 i ) q^{75} + 12 q^{79} + q^{81} + 4 i q^{83} + ( 8 + 4 i ) q^{85} + 6 i q^{87} -10 q^{89} -4 i q^{93} -8 i q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{5} - 2q^{9} - 8q^{11} - 4q^{15} - 6q^{25} + 12q^{29} - 8q^{31} + 20q^{41} + 2q^{45} + 8q^{51} + 8q^{55} + 8q^{59} - 4q^{61} + 8q^{69} + 8q^{75} + 24q^{79} + 2q^{81} + 16q^{85} - 20q^{89} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
1.00000i
1.00000i
0 1.00000i 0 −1.00000 2.00000i 0 0 0 −1.00000 0
589.2 0 1.00000i 0 −1.00000 + 2.00000i 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.k.c 2
5.b even 2 1 inner 2940.2.k.c 2
7.b odd 2 1 60.2.d.a 2
7.c even 3 2 2940.2.bb.e 4
7.d odd 6 2 2940.2.bb.d 4
21.c even 2 1 180.2.d.a 2
28.d even 2 1 240.2.f.b 2
35.c odd 2 1 60.2.d.a 2
35.f even 4 1 300.2.a.a 1
35.f even 4 1 300.2.a.d 1
35.i odd 6 2 2940.2.bb.d 4
35.j even 6 2 2940.2.bb.e 4
56.e even 2 1 960.2.f.c 2
56.h odd 2 1 960.2.f.f 2
63.l odd 6 2 1620.2.r.c 4
63.o even 6 2 1620.2.r.d 4
84.h odd 2 1 720.2.f.c 2
105.g even 2 1 180.2.d.a 2
105.k odd 4 1 900.2.a.a 1
105.k odd 4 1 900.2.a.h 1
112.j even 4 1 3840.2.d.b 2
112.j even 4 1 3840.2.d.be 2
112.l odd 4 1 3840.2.d.o 2
112.l odd 4 1 3840.2.d.r 2
140.c even 2 1 240.2.f.b 2
140.j odd 4 1 1200.2.a.a 1
140.j odd 4 1 1200.2.a.s 1
168.e odd 2 1 2880.2.f.p 2
168.i even 2 1 2880.2.f.l 2
280.c odd 2 1 960.2.f.f 2
280.n even 2 1 960.2.f.c 2
280.s even 4 1 4800.2.a.bj 1
280.s even 4 1 4800.2.a.bn 1
280.y odd 4 1 4800.2.a.bf 1
280.y odd 4 1 4800.2.a.bk 1
315.z even 6 2 1620.2.r.d 4
315.bg odd 6 2 1620.2.r.c 4
420.o odd 2 1 720.2.f.c 2
420.w even 4 1 3600.2.a.d 1
420.w even 4 1 3600.2.a.bm 1
560.be even 4 1 3840.2.d.b 2
560.be even 4 1 3840.2.d.be 2
560.bf odd 4 1 3840.2.d.o 2
560.bf odd 4 1 3840.2.d.r 2
840.b odd 2 1 2880.2.f.p 2
840.u even 2 1 2880.2.f.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 7.b odd 2 1
60.2.d.a 2 35.c odd 2 1
180.2.d.a 2 21.c even 2 1
180.2.d.a 2 105.g even 2 1
240.2.f.b 2 28.d even 2 1
240.2.f.b 2 140.c even 2 1
300.2.a.a 1 35.f even 4 1
300.2.a.d 1 35.f even 4 1
720.2.f.c 2 84.h odd 2 1
720.2.f.c 2 420.o odd 2 1
900.2.a.a 1 105.k odd 4 1
900.2.a.h 1 105.k odd 4 1
960.2.f.c 2 56.e even 2 1
960.2.f.c 2 280.n even 2 1
960.2.f.f 2 56.h odd 2 1
960.2.f.f 2 280.c odd 2 1
1200.2.a.a 1 140.j odd 4 1
1200.2.a.s 1 140.j odd 4 1
1620.2.r.c 4 63.l odd 6 2
1620.2.r.c 4 315.bg odd 6 2
1620.2.r.d 4 63.o even 6 2
1620.2.r.d 4 315.z even 6 2
2880.2.f.l 2 168.i even 2 1
2880.2.f.l 2 840.u even 2 1
2880.2.f.p 2 168.e odd 2 1
2880.2.f.p 2 840.b odd 2 1
2940.2.k.c 2 1.a even 1 1 trivial
2940.2.k.c 2 5.b even 2 1 inner
2940.2.bb.d 4 7.d odd 6 2
2940.2.bb.d 4 35.i odd 6 2
2940.2.bb.e 4 7.c even 3 2
2940.2.bb.e 4 35.j even 6 2
3600.2.a.d 1 420.w even 4 1
3600.2.a.bm 1 420.w even 4 1
3840.2.d.b 2 112.j even 4 1
3840.2.d.b 2 560.be even 4 1
3840.2.d.o 2 112.l odd 4 1
3840.2.d.o 2 560.bf odd 4 1
3840.2.d.r 2 112.l odd 4 1
3840.2.d.r 2 560.bf odd 4 1
3840.2.d.be 2 112.j even 4 1
3840.2.d.be 2 560.be even 4 1
4800.2.a.bf 1 280.y odd 4 1
4800.2.a.bj 1 280.s even 4 1
4800.2.a.bk 1 280.y odd 4 1
4800.2.a.bn 1 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2940, [\chi])\):

\( T_{11} + 4 \)
\( T_{13} \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 64 + T^{2} \)
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