# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2940.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.4760181943$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} - q^{13} - q^{15} + ( -4 + 4 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + q^{27} + ( -5 + 5 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} + 5 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} -2 q^{41} -9 q^{43} + ( 1 - \zeta_{6} ) q^{45} -2 \zeta_{6} q^{47} -4 \zeta_{6} q^{51} + ( -12 + 12 \zeta_{6} ) q^{53} + 2 q^{55} + q^{57} + ( -8 + 8 \zeta_{6} ) q^{59} -14 \zeta_{6} q^{61} -\zeta_{6} q^{65} + ( -9 + 9 \zeta_{6} ) q^{67} + 4 q^{69} + 2 q^{71} + ( 1 - \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 18 q^{83} -4 q^{85} -4 \zeta_{6} q^{89} -5 \zeta_{6} q^{93} + ( 1 - \zeta_{6} ) q^{95} -10 q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} - q^{9} + O(q^{10})$$ $$2 q - q^{3} + q^{5} - q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 4 q^{17} - q^{19} - 4 q^{23} - q^{25} + 2 q^{27} - 5 q^{31} + 2 q^{33} + 5 q^{37} + q^{39} - 4 q^{41} - 18 q^{43} + q^{45} - 2 q^{47} - 4 q^{51} - 12 q^{53} + 4 q^{55} + 2 q^{57} - 8 q^{59} - 14 q^{61} - q^{65} - 9 q^{67} + 8 q^{69} + 4 q^{71} + q^{73} - q^{75} + 3 q^{79} - q^{81} + 36 q^{83} - 8 q^{85} - 4 q^{89} - 5 q^{93} + q^{95} - 20 q^{97} - 4 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$1081$$ $$1177$$ $$1471$$ $$1961$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 0.500000 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.q.h 2
7.b odd 2 1 420.2.q.a 2
7.c even 3 1 2940.2.a.h 1
7.c even 3 1 inner 2940.2.q.h 2
7.d odd 6 1 420.2.q.a 2
7.d odd 6 1 2940.2.a.d 1
21.c even 2 1 1260.2.s.d 2
21.g even 6 1 1260.2.s.d 2
21.g even 6 1 8820.2.a.j 1
21.h odd 6 1 8820.2.a.y 1
28.d even 2 1 1680.2.bg.a 2
28.f even 6 1 1680.2.bg.a 2
35.c odd 2 1 2100.2.q.a 2
35.f even 4 2 2100.2.bc.c 4
35.i odd 6 1 2100.2.q.a 2
35.k even 12 2 2100.2.bc.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 7.b odd 2 1
420.2.q.a 2 7.d odd 6 1
1260.2.s.d 2 21.c even 2 1
1260.2.s.d 2 21.g even 6 1
1680.2.bg.a 2 28.d even 2 1
1680.2.bg.a 2 28.f even 6 1
2100.2.q.a 2 35.c odd 2 1
2100.2.q.a 2 35.i odd 6 1
2100.2.bc.c 4 35.f even 4 2
2100.2.bc.c 4 35.k even 12 2
2940.2.a.d 1 7.d odd 6 1
2940.2.a.h 1 7.c even 3 1
2940.2.q.h 2 1.a even 1 1 trivial
2940.2.q.h 2 7.c even 3 1 inner
8820.2.a.j 1 21.g even 6 1
8820.2.a.y 1 21.h odd 6 1

## Hecke kernels

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

 $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13} + 1$$ $$T_{17}^{2} + 4 T_{17} + 16$$ $$T_{31}^{2} + 5 T_{31} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$( 9 + T )^{2}$$
$47$ $$4 + 2 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$64 + 8 T + T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$81 + 9 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$1 - T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$( -18 + T )^{2}$$
$89$ $$16 + 4 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$