# Properties

 Label 2940.2.bb.f 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 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 2 + \zeta_{12} - 2 \zeta_{12}^{2} ) q^{5} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 2 + \zeta_{12} - 2 \zeta_{12}^{2} ) q^{5} + ( 1 - \zeta_{12}^{2} ) q^{9} -4 \zeta_{12}^{2} q^{11} -2 \zeta_{12}^{3} q^{13} + ( 1 - 2 \zeta_{12}^{3} ) q^{15} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( 2 - 2 \zeta_{12}^{2} ) q^{19} + 6 \zeta_{12} q^{23} + ( 4 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} -\zeta_{12}^{3} q^{27} -6 q^{29} + 6 \zeta_{12}^{2} q^{31} -4 \zeta_{12} q^{33} -4 \zeta_{12} q^{37} -2 \zeta_{12}^{2} q^{39} -4 \zeta_{12}^{3} q^{43} + ( \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{45} -4 \zeta_{12} q^{47} + ( 2 - 2 \zeta_{12}^{2} ) q^{51} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{53} + ( -8 - 4 \zeta_{12}^{3} ) q^{55} -2 \zeta_{12}^{3} q^{57} -4 \zeta_{12}^{2} q^{59} + ( -2 + 2 \zeta_{12}^{2} ) q^{61} + ( 2 - 4 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{65} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{67} + 6 q^{69} -8 q^{71} + ( -14 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{73} + ( 4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{75} + ( 16 - 16 \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} -16 \zeta_{12}^{3} q^{83} + ( 2 - 4 \zeta_{12}^{3} ) q^{85} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{87} + ( -16 + 16 \zeta_{12}^{2} ) q^{89} + 6 \zeta_{12} q^{93} + ( 2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{95} -14 \zeta_{12}^{3} q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{5} + 2q^{9} + O(q^{10})$$ $$4q + 4q^{5} + 2q^{9} - 8q^{11} + 4q^{15} + 4q^{19} - 6q^{25} - 24q^{29} + 12q^{31} - 4q^{39} - 4q^{45} + 4q^{51} - 32q^{55} - 8q^{59} - 4q^{61} + 4q^{65} + 24q^{69} - 32q^{71} + 8q^{75} + 32q^{79} - 2q^{81} + 8q^{85} - 32q^{89} - 8q^{95} - 16q^{99} + O(q^{100})$$

## Character values

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

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

## 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 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: 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.f 4
5.b even 2 1 inner 2940.2.bb.f 4
7.b odd 2 1 2940.2.bb.a 4
7.c even 3 1 2940.2.k.b 2
7.c even 3 1 inner 2940.2.bb.f 4
7.d odd 6 1 420.2.k.b 2
7.d odd 6 1 2940.2.bb.a 4
21.g even 6 1 1260.2.k.a 2
28.f even 6 1 1680.2.t.g 2
35.c odd 2 1 2940.2.bb.a 4
35.i odd 6 1 420.2.k.b 2
35.i odd 6 1 2940.2.bb.a 4
35.j even 6 1 2940.2.k.b 2
35.j even 6 1 inner 2940.2.bb.f 4
35.k even 12 1 2100.2.a.i 1
35.k even 12 1 2100.2.a.n 1
84.j odd 6 1 5040.2.t.d 2
105.p even 6 1 1260.2.k.a 2
105.w odd 12 1 6300.2.a.b 1
105.w odd 12 1 6300.2.a.r 1
140.s even 6 1 1680.2.t.g 2
140.x odd 12 1 8400.2.a.o 1
140.x odd 12 1 8400.2.a.bm 1
420.be odd 6 1 5040.2.t.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.b 2 7.d odd 6 1
420.2.k.b 2 35.i odd 6 1
1260.2.k.a 2 21.g even 6 1
1260.2.k.a 2 105.p even 6 1
1680.2.t.g 2 28.f even 6 1
1680.2.t.g 2 140.s even 6 1
2100.2.a.i 1 35.k even 12 1
2100.2.a.n 1 35.k even 12 1
2940.2.k.b 2 7.c even 3 1
2940.2.k.b 2 35.j even 6 1
2940.2.bb.a 4 7.b odd 2 1
2940.2.bb.a 4 7.d odd 6 1
2940.2.bb.a 4 35.c odd 2 1
2940.2.bb.a 4 35.i odd 6 1
2940.2.bb.f 4 1.a even 1 1 trivial
2940.2.bb.f 4 5.b even 2 1 inner
2940.2.bb.f 4 7.c even 3 1 inner
2940.2.bb.f 4 35.j even 6 1 inner
5040.2.t.d 2 84.j odd 6 1
5040.2.t.d 2 420.be odd 6 1
6300.2.a.b 1 105.w odd 12 1
6300.2.a.r 1 105.w odd 12 1
8400.2.a.o 1 140.x odd 12 1
8400.2.a.bm 1 140.x odd 12 1

## Hecke kernels

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

 $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13}^{2} + 4$$ $$T_{19}^{2} - 2 T_{19} + 4$$ $$T_{31}^{2} - 6 T_{31} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 16 + 4 T + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 4 - 2 T + T^{2} )^{2}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$( 36 - 6 T + T^{2} )^{2}$$
$37$ $$256 - 16 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$256 - 16 T^{2} + T^{4}$$
$53$ $$16 - 4 T^{2} + T^{4}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$( 4 + 2 T + T^{2} )^{2}$$
$67$ $$20736 - 144 T^{2} + T^{4}$$
$71$ $$( 8 + T )^{4}$$
$73$ $$38416 - 196 T^{2} + T^{4}$$
$79$ $$( 256 - 16 T + T^{2} )^{2}$$
$83$ $$( 256 + T^{2} )^{2}$$
$89$ $$( 256 + 16 T + T^{2} )^{2}$$
$97$ $$( 196 + T^{2} )^{2}$$