# Properties

 Label 2940.2.q.i 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -\zeta_{6} q^{9} + ( -6 + 6 \zeta_{6} ) q^{11} + 4 q^{13} - q^{15} + ( 6 - 6 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} + ( -1 + \zeta_{6} ) q^{25} - q^{27} + 6 q^{29} + ( -10 + 10 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{33} -2 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{39} + 6 q^{41} -4 q^{43} + ( -1 + \zeta_{6} ) q^{45} -6 \zeta_{6} q^{51} + ( 12 - 12 \zeta_{6} ) q^{53} + 6 q^{55} + 2 q^{57} + 14 \zeta_{6} q^{61} -4 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} + 6 q^{71} + ( -4 + 4 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + 16 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} -6 q^{85} + ( 6 - 6 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} + 10 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} + 16 q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{5} - q^{9} + O(q^{10})$$ $$2 q + q^{3} - q^{5} - q^{9} - 6 q^{11} + 8 q^{13} - 2 q^{15} + 6 q^{17} + 2 q^{19} - q^{25} - 2 q^{27} + 12 q^{29} - 10 q^{31} + 6 q^{33} - 2 q^{37} + 4 q^{39} + 12 q^{41} - 8 q^{43} - q^{45} - 6 q^{51} + 12 q^{53} + 12 q^{55} + 4 q^{57} + 14 q^{61} - 4 q^{65} + 4 q^{67} + 12 q^{71} - 4 q^{73} + q^{75} + 16 q^{79} - q^{81} + 24 q^{83} - 12 q^{85} + 6 q^{87} + 6 q^{89} + 10 q^{93} + 2 q^{95} + 32 q^{97} + 12 q^{99} + O(q^{100})$$

## 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.i 2
7.b odd 2 1 2940.2.q.e 2
7.c even 3 1 2940.2.a.f 1
7.c even 3 1 inner 2940.2.q.i 2
7.d odd 6 1 420.2.a.c 1
7.d odd 6 1 2940.2.q.e 2
21.g even 6 1 1260.2.a.i 1
21.h odd 6 1 8820.2.a.b 1
28.f even 6 1 1680.2.a.a 1
35.i odd 6 1 2100.2.a.d 1
35.k even 12 2 2100.2.k.j 2
56.j odd 6 1 6720.2.a.x 1
56.m even 6 1 6720.2.a.ch 1
84.j odd 6 1 5040.2.a.bc 1
105.p even 6 1 6300.2.a.a 1
105.w odd 12 2 6300.2.k.a 2
140.s even 6 1 8400.2.a.cj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 7.d odd 6 1
1260.2.a.i 1 21.g even 6 1
1680.2.a.a 1 28.f even 6 1
2100.2.a.d 1 35.i odd 6 1
2100.2.k.j 2 35.k even 12 2
2940.2.a.f 1 7.c even 3 1
2940.2.q.e 2 7.b odd 2 1
2940.2.q.e 2 7.d odd 6 1
2940.2.q.i 2 1.a even 1 1 trivial
2940.2.q.i 2 7.c even 3 1 inner
5040.2.a.bc 1 84.j odd 6 1
6300.2.a.a 1 105.p even 6 1
6300.2.k.a 2 105.w odd 12 2
6720.2.a.x 1 56.j odd 6 1
6720.2.a.ch 1 56.m even 6 1
8400.2.a.cj 1 140.s even 6 1
8820.2.a.b 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 $$S_{2}^{\mathrm{new}}(2940, [\chi])$$:

 $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{13} - 4$$ $$T_{17}^{2} - 6 T_{17} + 36$$ $$T_{31}^{2} + 10 T_{31} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$4 - 2 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$100 + 10 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$16 + 4 T + T^{2}$$
$79$ $$256 - 16 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -16 + T )^{2}$$