# Properties

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

# Related objects

## 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 60) 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} + ( 1 - \zeta_{12}^{2} ) q^{9} + 4 \zeta_{12}^{2} q^{11} + ( -2 + \zeta_{12}^{3} ) q^{15} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{17} -4 \zeta_{12} q^{23} + ( 4 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} -\zeta_{12}^{3} q^{27} + 6 q^{29} -4 \zeta_{12}^{2} q^{31} + 4 \zeta_{12} q^{33} -8 \zeta_{12} q^{37} -10 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{45} -4 \zeta_{12} q^{47} + ( -4 + 4 \zeta_{12}^{2} ) q^{51} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{53} + ( -4 - 8 \zeta_{12}^{3} ) q^{55} + 4 \zeta_{12}^{2} q^{59} + ( -2 + 2 \zeta_{12}^{2} ) q^{61} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{67} -4 q^{69} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{73} + ( 4 + 3 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{75} + ( -12 + 12 \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} + 4 \zeta_{12}^{3} q^{83} + ( 8 - 4 \zeta_{12}^{3} ) q^{85} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{87} + ( -10 + 10 \zeta_{12}^{2} ) q^{89} -4 \zeta_{12} q^{93} -8 \zeta_{12}^{3} q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{5} + 2q^{9} + 8q^{11} - 8q^{15} + 6q^{25} + 24q^{29} - 8q^{31} - 40q^{41} + 2q^{45} - 8q^{51} - 16q^{55} + 8q^{59} - 4q^{61} - 16q^{69} + 8q^{75} - 24q^{79} - 2q^{81} + 32q^{85} - 20q^{89} + 16q^{99} + O(q^{100})$$

## Character values

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

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

## 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 1.23205 1.86603i 0 0 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 −2.23205 + 0.133975i 0 0 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 1.23205 + 1.86603i 0 0 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 −2.23205 0.133975i 0 0 0 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d 4
5.b even 2 1 inner 2940.2.bb.d 4
7.b odd 2 1 2940.2.bb.e 4
7.c even 3 1 60.2.d.a 2
7.c even 3 1 inner 2940.2.bb.d 4
7.d odd 6 1 2940.2.k.c 2
7.d odd 6 1 2940.2.bb.e 4
21.h odd 6 1 180.2.d.a 2
28.g odd 6 1 240.2.f.b 2
35.c odd 2 1 2940.2.bb.e 4
35.i odd 6 1 2940.2.k.c 2
35.i odd 6 1 2940.2.bb.e 4
35.j even 6 1 60.2.d.a 2
35.j even 6 1 inner 2940.2.bb.d 4
35.l odd 12 1 300.2.a.a 1
35.l odd 12 1 300.2.a.d 1
56.k odd 6 1 960.2.f.c 2
56.p even 6 1 960.2.f.f 2
63.g even 3 1 1620.2.r.c 4
63.h even 3 1 1620.2.r.c 4
63.j odd 6 1 1620.2.r.d 4
63.n odd 6 1 1620.2.r.d 4
84.n even 6 1 720.2.f.c 2
105.o odd 6 1 180.2.d.a 2
105.x even 12 1 900.2.a.a 1
105.x even 12 1 900.2.a.h 1
112.u odd 12 1 3840.2.d.b 2
112.u odd 12 1 3840.2.d.be 2
112.w even 12 1 3840.2.d.o 2
112.w even 12 1 3840.2.d.r 2
140.p odd 6 1 240.2.f.b 2
140.w even 12 1 1200.2.a.a 1
140.w even 12 1 1200.2.a.s 1
168.s odd 6 1 2880.2.f.l 2
168.v even 6 1 2880.2.f.p 2
280.bf even 6 1 960.2.f.f 2
280.bi odd 6 1 960.2.f.c 2
280.br even 12 1 4800.2.a.bf 1
280.br even 12 1 4800.2.a.bk 1
280.bt odd 12 1 4800.2.a.bj 1
280.bt odd 12 1 4800.2.a.bn 1
315.r even 6 1 1620.2.r.c 4
315.v odd 6 1 1620.2.r.d 4
315.bo even 6 1 1620.2.r.c 4
315.br odd 6 1 1620.2.r.d 4
420.ba even 6 1 720.2.f.c 2
420.bp odd 12 1 3600.2.a.d 1
420.bp odd 12 1 3600.2.a.bm 1
560.cr even 12 1 3840.2.d.o 2
560.cr even 12 1 3840.2.d.r 2
560.cs odd 12 1 3840.2.d.b 2
560.cs odd 12 1 3840.2.d.be 2
840.cg odd 6 1 2880.2.f.l 2
840.cv even 6 1 2880.2.f.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 7.c even 3 1
60.2.d.a 2 35.j even 6 1
180.2.d.a 2 21.h odd 6 1
180.2.d.a 2 105.o odd 6 1
240.2.f.b 2 28.g odd 6 1
240.2.f.b 2 140.p odd 6 1
300.2.a.a 1 35.l odd 12 1
300.2.a.d 1 35.l odd 12 1
720.2.f.c 2 84.n even 6 1
720.2.f.c 2 420.ba even 6 1
900.2.a.a 1 105.x even 12 1
900.2.a.h 1 105.x even 12 1
960.2.f.c 2 56.k odd 6 1
960.2.f.c 2 280.bi odd 6 1
960.2.f.f 2 56.p even 6 1
960.2.f.f 2 280.bf even 6 1
1200.2.a.a 1 140.w even 12 1
1200.2.a.s 1 140.w even 12 1
1620.2.r.c 4 63.g even 3 1
1620.2.r.c 4 63.h even 3 1
1620.2.r.c 4 315.r even 6 1
1620.2.r.c 4 315.bo even 6 1
1620.2.r.d 4 63.j odd 6 1
1620.2.r.d 4 63.n odd 6 1
1620.2.r.d 4 315.v odd 6 1
1620.2.r.d 4 315.br odd 6 1
2880.2.f.l 2 168.s odd 6 1
2880.2.f.l 2 840.cg odd 6 1
2880.2.f.p 2 168.v even 6 1
2880.2.f.p 2 840.cv even 6 1
2940.2.k.c 2 7.d odd 6 1
2940.2.k.c 2 35.i odd 6 1
2940.2.bb.d 4 1.a even 1 1 trivial
2940.2.bb.d 4 5.b even 2 1 inner
2940.2.bb.d 4 7.c even 3 1 inner
2940.2.bb.d 4 35.j even 6 1 inner
2940.2.bb.e 4 7.b odd 2 1
2940.2.bb.e 4 7.d odd 6 1
2940.2.bb.e 4 35.c odd 2 1
2940.2.bb.e 4 35.i odd 6 1
3600.2.a.d 1 420.bp odd 12 1
3600.2.a.bm 1 420.bp odd 12 1
3840.2.d.b 2 112.u odd 12 1
3840.2.d.b 2 560.cs odd 12 1
3840.2.d.o 2 112.w even 12 1
3840.2.d.o 2 560.cr even 12 1
3840.2.d.r 2 112.w even 12 1
3840.2.d.r 2 560.cr even 12 1
3840.2.d.be 2 112.u odd 12 1
3840.2.d.be 2 560.cs odd 12 1
4800.2.a.bf 1 280.br even 12 1
4800.2.a.bj 1 280.bt odd 12 1
4800.2.a.bk 1 280.br even 12 1
4800.2.a.bn 1 280.bt odd 12 1

## Hecke kernels

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

 $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{13}$$ $$T_{19}$$ $$T_{31}^{2} + 4 T_{31} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 16 - 4 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$256 - 16 T^{2} + T^{4}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 16 + 4 T + T^{2} )^{2}$$
$37$ $$4096 - 64 T^{2} + T^{4}$$
$41$ $$( 10 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$256 - 16 T^{2} + T^{4}$$
$53$ $$20736 - 144 T^{2} + T^{4}$$
$59$ $$( 16 - 4 T + T^{2} )^{2}$$
$61$ $$( 4 + 2 T + T^{2} )^{2}$$
$67$ $$256 - 16 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$4096 - 64 T^{2} + T^{4}$$
$79$ $$( 144 + 12 T + T^{2} )^{2}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( 100 + 10 T + T^{2} )^{2}$$
$97$ $$( 64 + T^{2} )^{2}$$