# Properties

 Label 56.2.b.a Level $56$ Weight $2$ Character orbit 56.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.447162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + \beta q^{3} -2 q^{4} -\beta q^{5} -2 q^{6} + q^{7} -2 \beta q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + \beta q^{3} -2 q^{4} -\beta q^{5} -2 q^{6} + q^{7} -2 \beta q^{8} + q^{9} + 2 q^{10} + 2 \beta q^{11} -2 \beta q^{12} -3 \beta q^{13} + \beta q^{14} + 2 q^{15} + 4 q^{16} -6 q^{17} + \beta q^{18} -3 \beta q^{19} + 2 \beta q^{20} + \beta q^{21} -4 q^{22} -6 q^{23} + 4 q^{24} + 3 q^{25} + 6 q^{26} + 4 \beta q^{27} -2 q^{28} + 2 \beta q^{29} + 2 \beta q^{30} -4 q^{31} + 4 \beta q^{32} -4 q^{33} -6 \beta q^{34} -\beta q^{35} -2 q^{36} + 6 \beta q^{37} + 6 q^{38} + 6 q^{39} -4 q^{40} + 6 q^{41} -2 q^{42} -6 \beta q^{43} -4 \beta q^{44} -\beta q^{45} -6 \beta q^{46} + 4 \beta q^{48} + q^{49} + 3 \beta q^{50} -6 \beta q^{51} + 6 \beta q^{52} -4 \beta q^{53} -8 q^{54} + 4 q^{55} -2 \beta q^{56} + 6 q^{57} -4 q^{58} -\beta q^{59} -4 q^{60} + 9 \beta q^{61} -4 \beta q^{62} + q^{63} -8 q^{64} -6 q^{65} -4 \beta q^{66} + 12 q^{68} -6 \beta q^{69} + 2 q^{70} -2 \beta q^{72} + 2 q^{73} -12 q^{74} + 3 \beta q^{75} + 6 \beta q^{76} + 2 \beta q^{77} + 6 \beta q^{78} + 8 q^{79} -4 \beta q^{80} -5 q^{81} + 6 \beta q^{82} + 11 \beta q^{83} -2 \beta q^{84} + 6 \beta q^{85} + 12 q^{86} -4 q^{87} + 8 q^{88} + 6 q^{89} + 2 q^{90} -3 \beta q^{91} + 12 q^{92} -4 \beta q^{93} -6 q^{95} -8 q^{96} -10 q^{97} + \beta q^{98} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 4q^{6} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{4} - 4q^{6} + 2q^{7} + 2q^{9} + 4q^{10} + 4q^{15} + 8q^{16} - 12q^{17} - 8q^{22} - 12q^{23} + 8q^{24} + 6q^{25} + 12q^{26} - 4q^{28} - 8q^{31} - 8q^{33} - 4q^{36} + 12q^{38} + 12q^{39} - 8q^{40} + 12q^{41} - 4q^{42} + 2q^{49} - 16q^{54} + 8q^{55} + 12q^{57} - 8q^{58} - 8q^{60} + 2q^{63} - 16q^{64} - 12q^{65} + 24q^{68} + 4q^{70} + 4q^{73} - 24q^{74} + 16q^{79} - 10q^{81} + 24q^{86} - 8q^{87} + 16q^{88} + 12q^{89} + 4q^{90} + 24q^{92} - 12q^{95} - 16q^{96} - 20q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$15$$ $$17$$ $$29$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 1.41421i 1.41421i
1.41421i 1.41421i −2.00000 1.41421i −2.00000 1.00000 2.82843i 1.00000 2.00000
29.2 1.41421i 1.41421i −2.00000 1.41421i −2.00000 1.00000 2.82843i 1.00000 2.00000
 $$n$$: 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.2.b.a 2
3.b odd 2 1 504.2.c.a 2
4.b odd 2 1 224.2.b.a 2
7.b odd 2 1 392.2.b.b 2
7.c even 3 2 392.2.p.a 4
7.d odd 6 2 392.2.p.b 4
8.b even 2 1 inner 56.2.b.a 2
8.d odd 2 1 224.2.b.a 2
12.b even 2 1 2016.2.c.a 2
16.e even 4 2 1792.2.a.n 2
16.f odd 4 2 1792.2.a.p 2
24.f even 2 1 2016.2.c.a 2
24.h odd 2 1 504.2.c.a 2
28.d even 2 1 1568.2.b.a 2
28.f even 6 2 1568.2.t.b 4
28.g odd 6 2 1568.2.t.c 4
56.e even 2 1 1568.2.b.a 2
56.h odd 2 1 392.2.b.b 2
56.j odd 6 2 392.2.p.b 4
56.k odd 6 2 1568.2.t.c 4
56.m even 6 2 1568.2.t.b 4
56.p even 6 2 392.2.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.a 2 1.a even 1 1 trivial
56.2.b.a 2 8.b even 2 1 inner
224.2.b.a 2 4.b odd 2 1
224.2.b.a 2 8.d odd 2 1
392.2.b.b 2 7.b odd 2 1
392.2.b.b 2 56.h odd 2 1
392.2.p.a 4 7.c even 3 2
392.2.p.a 4 56.p even 6 2
392.2.p.b 4 7.d odd 6 2
392.2.p.b 4 56.j odd 6 2
504.2.c.a 2 3.b odd 2 1
504.2.c.a 2 24.h odd 2 1
1568.2.b.a 2 28.d even 2 1
1568.2.b.a 2 56.e even 2 1
1568.2.t.b 4 28.f even 6 2
1568.2.t.b 4 56.m even 6 2
1568.2.t.c 4 28.g odd 6 2
1568.2.t.c 4 56.k odd 6 2
1792.2.a.n 2 16.e even 4 2
1792.2.a.p 2 16.f odd 4 2
2016.2.c.a 2 12.b even 2 1
2016.2.c.a 2 24.f even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$2 + T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$8 + T^{2}$$
$13$ $$18 + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$18 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$8 + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$72 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$72 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$32 + T^{2}$$
$59$ $$2 + T^{2}$$
$61$ $$162 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$242 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( 10 + T )^{2}$$