# Properties

 Label 2940.2.k.d Level $2940$ Weight $2$ Character orbit 2940.k 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.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$1081$$ $$1177$$ $$1471$$ $$1961$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 − 1.00000i 1.00000i
0 1.00000i 0 2.00000 + 1.00000i 0 0 0 −1.00000 0
589.2 0 1.00000i 0 2.00000 1.00000i 0 0 0 −1.00000 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

## Twists

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

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