Properties

Label 2940.2.bb.h
Level $2940$
Weight $2$
Character orbit 2940.bb
Analytic conductor $23.476$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{3} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + \zeta_{12}^{2} q^{9} + ( 4 - 4 \zeta_{12}^{2} ) q^{11} -6 \zeta_{12}^{3} q^{13} + ( 1 + 2 \zeta_{12}^{3} ) q^{15} + 2 \zeta_{12} q^{17} + 6 \zeta_{12}^{2} q^{19} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{23} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + \zeta_{12}^{3} q^{27} -6 q^{29} + ( 2 - 2 \zeta_{12}^{2} ) q^{31} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{33} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{37} + ( 6 - 6 \zeta_{12}^{2} ) q^{39} + 8 q^{41} -4 \zeta_{12}^{3} q^{43} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{45} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + 2 \zeta_{12}^{2} q^{51} -6 \zeta_{12} q^{53} + ( 8 - 4 \zeta_{12}^{3} ) q^{55} + 6 \zeta_{12}^{3} q^{57} + ( 4 - 4 \zeta_{12}^{2} ) q^{59} -14 \zeta_{12}^{2} q^{61} + ( 12 \zeta_{12} - 6 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{65} + 4 \zeta_{12} q^{67} + 2 q^{69} + 10 \zeta_{12} q^{73} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{75} + ( -1 + \zeta_{12}^{2} ) q^{81} -16 \zeta_{12}^{3} q^{83} + ( 2 + 4 \zeta_{12}^{3} ) q^{85} -6 \zeta_{12} q^{87} + 8 \zeta_{12}^{2} q^{89} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{93} + ( -12 + 6 \zeta_{12} + 12 \zeta_{12}^{2} ) q^{95} -10 \zeta_{12}^{3} q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 4q + 4q^{5} + 2q^{9} + 8q^{11} + 4q^{15} + 12q^{19} - 6q^{25} - 24q^{29} + 4q^{31} + 12q^{39} + 32q^{41} - 4q^{45} + 4q^{51} + 32q^{55} + 8q^{59} - 28q^{61} - 12q^{65} + 8q^{69} + 8q^{75} - 2q^{81} + 8q^{85} + 16q^{89} - 24q^{95} + 16q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −0.866025 0.500000i 0 0.133975 + 2.23205i 0 0 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 1.86603 + 1.23205i 0 0 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0.133975 2.23205i 0 0 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 1.86603 1.23205i 0 0 0 0.500000 0.866025i 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.bb.h 4
5.b even 2 1 inner 2940.2.bb.h 4
7.b odd 2 1 2940.2.bb.c 4
7.c even 3 1 420.2.k.a 2
7.c even 3 1 inner 2940.2.bb.h 4
7.d odd 6 1 2940.2.k.d 2
7.d odd 6 1 2940.2.bb.c 4
21.h odd 6 1 1260.2.k.d 2
28.g odd 6 1 1680.2.t.a 2
35.c odd 2 1 2940.2.bb.c 4
35.i odd 6 1 2940.2.k.d 2
35.i odd 6 1 2940.2.bb.c 4
35.j even 6 1 420.2.k.a 2
35.j even 6 1 inner 2940.2.bb.h 4
35.l odd 12 1 2100.2.a.e 1
35.l odd 12 1 2100.2.a.j 1
84.n even 6 1 5040.2.t.o 2
105.o odd 6 1 1260.2.k.d 2
105.x even 12 1 6300.2.a.n 1
105.x even 12 1 6300.2.a.bc 1
140.p odd 6 1 1680.2.t.a 2
140.w even 12 1 8400.2.a.bh 1
140.w even 12 1 8400.2.a.cd 1
420.ba even 6 1 5040.2.t.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.a 2 7.c even 3 1
420.2.k.a 2 35.j even 6 1
1260.2.k.d 2 21.h odd 6 1
1260.2.k.d 2 105.o odd 6 1
1680.2.t.a 2 28.g odd 6 1
1680.2.t.a 2 140.p odd 6 1
2100.2.a.e 1 35.l odd 12 1
2100.2.a.j 1 35.l odd 12 1
2940.2.k.d 2 7.d odd 6 1
2940.2.k.d 2 35.i odd 6 1
2940.2.bb.c 4 7.b odd 2 1
2940.2.bb.c 4 7.d odd 6 1
2940.2.bb.c 4 35.c odd 2 1
2940.2.bb.c 4 35.i odd 6 1
2940.2.bb.h 4 1.a even 1 1 trivial
2940.2.bb.h 4 5.b even 2 1 inner
2940.2.bb.h 4 7.c even 3 1 inner
2940.2.bb.h 4 35.j even 6 1 inner
5040.2.t.o 2 84.n even 6 1
5040.2.t.o 2 420.ba even 6 1
6300.2.a.n 1 105.x even 12 1
6300.2.a.bc 1 105.x even 12 1
8400.2.a.bh 1 140.w even 12 1
8400.2.a.cd 1 140.w even 12 1

Hecke kernels

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

\( T_{11}^{2} - 4 T_{11} + 16 \)
\( T_{13}^{2} + 36 \)
\( T_{19}^{2} - 6 T_{19} + 36 \)
\( T_{31}^{2} - 2 T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 16 - 4 T + T^{2} )^{2} \)
$13$ \( ( 36 + T^{2} )^{2} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 36 - 6 T + T^{2} )^{2} \)
$23$ \( 16 - 4 T^{2} + T^{4} \)
$29$ \( ( 6 + T )^{4} \)
$31$ \( ( 4 - 2 T + T^{2} )^{2} \)
$37$ \( 256 - 16 T^{2} + T^{4} \)
$41$ \( ( -8 + T )^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( 256 - 16 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( ( 16 - 4 T + T^{2} )^{2} \)
$61$ \( ( 196 + 14 T + T^{2} )^{2} \)
$67$ \( 256 - 16 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 10000 - 100 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 256 + T^{2} )^{2} \)
$89$ \( ( 64 - 8 T + T^{2} )^{2} \)
$97$ \( ( 100 + T^{2} )^{2} \)
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