# Properties

 Label 56.2.i.a Level $56$ Weight $2$ Character orbit 56.i Analytic conductor $0.447$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 56.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.447162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{6} q^{3} + ( 1 - \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -3 \zeta_{6} q^{3} + ( 1 - \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{9} + \zeta_{6} q^{11} + 2 q^{13} -3 q^{15} -3 \zeta_{6} q^{17} + ( -5 + 5 \zeta_{6} ) q^{19} + ( 6 - 9 \zeta_{6} ) q^{21} + ( 3 - 3 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + 9 q^{27} -6 q^{29} + \zeta_{6} q^{31} + ( 3 - 3 \zeta_{6} ) q^{33} + ( 3 - \zeta_{6} ) q^{35} + ( 5 - 5 \zeta_{6} ) q^{37} -6 \zeta_{6} q^{39} -10 q^{41} -4 q^{43} + 6 \zeta_{6} q^{45} + ( -1 + \zeta_{6} ) q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -9 + 9 \zeta_{6} ) q^{51} + 9 \zeta_{6} q^{53} + q^{55} + 15 q^{57} -3 \zeta_{6} q^{59} + ( -3 + 3 \zeta_{6} ) q^{61} + ( -18 + 6 \zeta_{6} ) q^{63} + ( 2 - 2 \zeta_{6} ) q^{65} -11 \zeta_{6} q^{67} -9 q^{69} + 16 q^{71} -7 \zeta_{6} q^{73} + ( 12 - 12 \zeta_{6} ) q^{75} + ( -2 + 3 \zeta_{6} ) q^{77} + ( 11 - 11 \zeta_{6} ) q^{79} -9 \zeta_{6} q^{81} -4 q^{83} -3 q^{85} + 18 \zeta_{6} q^{87} + ( 9 - 9 \zeta_{6} ) q^{89} + ( 2 + 4 \zeta_{6} ) q^{91} + ( 3 - 3 \zeta_{6} ) q^{93} + 5 \zeta_{6} q^{95} + 6 q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} + q^{5} + 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 3q^{3} + q^{5} + 4q^{7} - 6q^{9} + q^{11} + 4q^{13} - 6q^{15} - 3q^{17} - 5q^{19} + 3q^{21} + 3q^{23} + 4q^{25} + 18q^{27} - 12q^{29} + q^{31} + 3q^{33} + 5q^{35} + 5q^{37} - 6q^{39} - 20q^{41} - 8q^{43} + 6q^{45} - q^{47} + 2q^{49} - 9q^{51} + 9q^{53} + 2q^{55} + 30q^{57} - 3q^{59} - 3q^{61} - 30q^{63} + 2q^{65} - 11q^{67} - 18q^{69} + 32q^{71} - 7q^{73} + 12q^{75} - q^{77} + 11q^{79} - 9q^{81} - 8q^{83} - 6q^{85} + 18q^{87} + 9q^{89} + 8q^{91} + 3q^{93} + 5q^{95} + 12q^{97} - 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/56\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$29$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 2.59808i 0 0.500000 0.866025i 0 2.00000 + 1.73205i 0 −3.00000 + 5.19615i 0
25.1 0 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 2.00000 1.73205i 0 −3.00000 5.19615i 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 56.2.i.a 2
3.b odd 2 1 504.2.s.e 2
4.b odd 2 1 112.2.i.c 2
5.b even 2 1 1400.2.q.g 2
5.c odd 4 2 1400.2.bh.f 4
7.b odd 2 1 392.2.i.f 2
7.c even 3 1 inner 56.2.i.a 2
7.c even 3 1 392.2.a.f 1
7.d odd 6 1 392.2.a.a 1
7.d odd 6 1 392.2.i.f 2
8.b even 2 1 448.2.i.f 2
8.d odd 2 1 448.2.i.a 2
12.b even 2 1 1008.2.s.e 2
21.c even 2 1 3528.2.s.o 2
21.g even 6 1 3528.2.a.k 1
21.g even 6 1 3528.2.s.o 2
21.h odd 6 1 504.2.s.e 2
21.h odd 6 1 3528.2.a.r 1
28.d even 2 1 784.2.i.a 2
28.f even 6 1 784.2.a.j 1
28.f even 6 1 784.2.i.a 2
28.g odd 6 1 112.2.i.c 2
28.g odd 6 1 784.2.a.a 1
35.i odd 6 1 9800.2.a.bp 1
35.j even 6 1 1400.2.q.g 2
35.j even 6 1 9800.2.a.b 1
35.l odd 12 2 1400.2.bh.f 4
56.j odd 6 1 3136.2.a.bb 1
56.k odd 6 1 448.2.i.a 2
56.k odd 6 1 3136.2.a.bc 1
56.m even 6 1 3136.2.a.a 1
56.p even 6 1 448.2.i.f 2
56.p even 6 1 3136.2.a.b 1
84.j odd 6 1 7056.2.a.s 1
84.n even 6 1 1008.2.s.e 2
84.n even 6 1 7056.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 1.a even 1 1 trivial
56.2.i.a 2 7.c even 3 1 inner
112.2.i.c 2 4.b odd 2 1
112.2.i.c 2 28.g odd 6 1
392.2.a.a 1 7.d odd 6 1
392.2.a.f 1 7.c even 3 1
392.2.i.f 2 7.b odd 2 1
392.2.i.f 2 7.d odd 6 1
448.2.i.a 2 8.d odd 2 1
448.2.i.a 2 56.k odd 6 1
448.2.i.f 2 8.b even 2 1
448.2.i.f 2 56.p even 6 1
504.2.s.e 2 3.b odd 2 1
504.2.s.e 2 21.h odd 6 1
784.2.a.a 1 28.g odd 6 1
784.2.a.j 1 28.f even 6 1
784.2.i.a 2 28.d even 2 1
784.2.i.a 2 28.f even 6 1
1008.2.s.e 2 12.b even 2 1
1008.2.s.e 2 84.n even 6 1
1400.2.q.g 2 5.b even 2 1
1400.2.q.g 2 35.j even 6 1
1400.2.bh.f 4 5.c odd 4 2
1400.2.bh.f 4 35.l odd 12 2
3136.2.a.a 1 56.m even 6 1
3136.2.a.b 1 56.p even 6 1
3136.2.a.bb 1 56.j odd 6 1
3136.2.a.bc 1 56.k odd 6 1
3528.2.a.k 1 21.g even 6 1
3528.2.a.r 1 21.h odd 6 1
3528.2.s.o 2 21.c even 2 1
3528.2.s.o 2 21.g even 6 1
7056.2.a.s 1 84.j odd 6 1
7056.2.a.bi 1 84.n even 6 1
9800.2.a.b 1 35.j even 6 1
9800.2.a.bp 1 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(56, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$1 - T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$1 + T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$9 + 3 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$49 + 7 T + T^{2}$$
$79$ $$121 - 11 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$81 - 9 T + T^{2}$$
$97$ $$( -6 + T )^{2}$$