# Properties

 Label 2940.2.k.c 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 60) 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.

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

## 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 −1.00000 2.00000i 0 0 0 −1.00000 0
589.2 0 1.00000i 0 −1.00000 + 2.00000i 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.c 2
5.b even 2 1 inner 2940.2.k.c 2
7.b odd 2 1 60.2.d.a 2
7.c even 3 2 2940.2.bb.e 4
7.d odd 6 2 2940.2.bb.d 4
21.c even 2 1 180.2.d.a 2
28.d even 2 1 240.2.f.b 2
35.c odd 2 1 60.2.d.a 2
35.f even 4 1 300.2.a.a 1
35.f even 4 1 300.2.a.d 1
35.i odd 6 2 2940.2.bb.d 4
35.j even 6 2 2940.2.bb.e 4
56.e even 2 1 960.2.f.c 2
56.h odd 2 1 960.2.f.f 2
63.l odd 6 2 1620.2.r.c 4
63.o even 6 2 1620.2.r.d 4
84.h odd 2 1 720.2.f.c 2
105.g even 2 1 180.2.d.a 2
105.k odd 4 1 900.2.a.a 1
105.k odd 4 1 900.2.a.h 1
112.j even 4 1 3840.2.d.b 2
112.j even 4 1 3840.2.d.be 2
112.l odd 4 1 3840.2.d.o 2
112.l odd 4 1 3840.2.d.r 2
140.c even 2 1 240.2.f.b 2
140.j odd 4 1 1200.2.a.a 1
140.j odd 4 1 1200.2.a.s 1
168.e odd 2 1 2880.2.f.p 2
168.i even 2 1 2880.2.f.l 2
280.c odd 2 1 960.2.f.f 2
280.n even 2 1 960.2.f.c 2
280.s even 4 1 4800.2.a.bj 1
280.s even 4 1 4800.2.a.bn 1
280.y odd 4 1 4800.2.a.bf 1
280.y odd 4 1 4800.2.a.bk 1
315.z even 6 2 1620.2.r.d 4
315.bg odd 6 2 1620.2.r.c 4
420.o odd 2 1 720.2.f.c 2
420.w even 4 1 3600.2.a.d 1
420.w even 4 1 3600.2.a.bm 1
560.be even 4 1 3840.2.d.b 2
560.be even 4 1 3840.2.d.be 2
560.bf odd 4 1 3840.2.d.o 2
560.bf odd 4 1 3840.2.d.r 2
840.b odd 2 1 2880.2.f.p 2
840.u even 2 1 2880.2.f.l 2

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

## Hecke kernels

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

 $$T_{11} + 4$$ $$T_{13}$$ $$T_{19}$$

## Hecke characteristic polynomials

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