# Properties

 Label 57.3.c.a Level $57$ Weight $3$ Character orbit 57.c Analytic conductor $1.553$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 57.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.55313750685$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + q^{4} + 4 q^{5} -3 q^{6} -10 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + q^{4} + 4 q^{5} -3 q^{6} -10 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} -3 q^{9} + ( 4 - 8 \zeta_{6} ) q^{10} + 10 q^{11} + ( 1 - 2 \zeta_{6} ) q^{12} + ( -14 + 28 \zeta_{6} ) q^{13} + ( -10 + 20 \zeta_{6} ) q^{14} + ( 4 - 8 \zeta_{6} ) q^{15} -11 q^{16} + 10 q^{17} + ( -3 + 6 \zeta_{6} ) q^{18} + 19 q^{19} + 4 q^{20} + ( -10 + 20 \zeta_{6} ) q^{21} + ( 10 - 20 \zeta_{6} ) q^{22} -20 q^{23} -15 q^{24} -9 q^{25} + 42 q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} -10 q^{28} + ( -20 + 40 \zeta_{6} ) q^{29} -12 q^{30} + ( 10 - 20 \zeta_{6} ) q^{31} + ( 9 - 18 \zeta_{6} ) q^{32} + ( 10 - 20 \zeta_{6} ) q^{33} + ( 10 - 20 \zeta_{6} ) q^{34} -40 q^{35} -3 q^{36} + ( -6 + 12 \zeta_{6} ) q^{37} + ( 19 - 38 \zeta_{6} ) q^{38} + 42 q^{39} + ( 20 - 40 \zeta_{6} ) q^{40} + ( 20 - 40 \zeta_{6} ) q^{41} + 30 q^{42} -10 q^{43} + 10 q^{44} -12 q^{45} + ( -20 + 40 \zeta_{6} ) q^{46} -80 q^{47} + ( -11 + 22 \zeta_{6} ) q^{48} + 51 q^{49} + ( -9 + 18 \zeta_{6} ) q^{50} + ( 10 - 20 \zeta_{6} ) q^{51} + ( -14 + 28 \zeta_{6} ) q^{52} + ( 24 - 48 \zeta_{6} ) q^{53} + 9 q^{54} + 40 q^{55} + ( -50 + 100 \zeta_{6} ) q^{56} + ( 19 - 38 \zeta_{6} ) q^{57} + 60 q^{58} + ( 20 - 40 \zeta_{6} ) q^{59} + ( 4 - 8 \zeta_{6} ) q^{60} -10 q^{61} -30 q^{62} + 30 q^{63} -71 q^{64} + ( -56 + 112 \zeta_{6} ) q^{65} -30 q^{66} + ( -44 + 88 \zeta_{6} ) q^{67} + 10 q^{68} + ( -20 + 40 \zeta_{6} ) q^{69} + ( -40 + 80 \zeta_{6} ) q^{70} + ( 60 - 120 \zeta_{6} ) q^{71} + ( -15 + 30 \zeta_{6} ) q^{72} -10 q^{73} + 18 q^{74} + ( -9 + 18 \zeta_{6} ) q^{75} + 19 q^{76} -100 q^{77} + ( 42 - 84 \zeta_{6} ) q^{78} + ( -10 + 20 \zeta_{6} ) q^{79} -44 q^{80} + 9 q^{81} -60 q^{82} + 70 q^{83} + ( -10 + 20 \zeta_{6} ) q^{84} + 40 q^{85} + ( -10 + 20 \zeta_{6} ) q^{86} + 60 q^{87} + ( 50 - 100 \zeta_{6} ) q^{88} + ( -60 + 120 \zeta_{6} ) q^{89} + ( -12 + 24 \zeta_{6} ) q^{90} + ( 140 - 280 \zeta_{6} ) q^{91} -20 q^{92} -30 q^{93} + ( -80 + 160 \zeta_{6} ) q^{94} + 76 q^{95} -27 q^{96} + ( 44 - 88 \zeta_{6} ) q^{97} + ( 51 - 102 \zeta_{6} ) q^{98} -30 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 8q^{5} - 6q^{6} - 20q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{4} + 8q^{5} - 6q^{6} - 20q^{7} - 6q^{9} + 20q^{11} - 22q^{16} + 20q^{17} + 38q^{19} + 8q^{20} - 40q^{23} - 30q^{24} - 18q^{25} + 84q^{26} - 20q^{28} - 24q^{30} - 80q^{35} - 6q^{36} + 84q^{39} + 60q^{42} - 20q^{43} + 20q^{44} - 24q^{45} - 160q^{47} + 102q^{49} + 18q^{54} + 80q^{55} + 120q^{58} - 20q^{61} - 60q^{62} + 60q^{63} - 142q^{64} - 60q^{66} + 20q^{68} - 20q^{73} + 36q^{74} + 38q^{76} - 200q^{77} - 88q^{80} + 18q^{81} - 120q^{82} + 140q^{83} + 80q^{85} + 120q^{87} - 40q^{92} - 60q^{93} + 152q^{95} - 54q^{96} - 60q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 1.73205i 1.00000 4.00000 −3.00000 −10.0000 8.66025i −3.00000 6.92820i
37.2 1.73205i 1.73205i 1.00000 4.00000 −3.00000 −10.0000 8.66025i −3.00000 6.92820i
 $$n$$: 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.3.c.a 2
3.b odd 2 1 171.3.c.c 2
4.b odd 2 1 912.3.o.a 2
12.b even 2 1 2736.3.o.e 2
19.b odd 2 1 inner 57.3.c.a 2
57.d even 2 1 171.3.c.c 2
76.d even 2 1 912.3.o.a 2
228.b odd 2 1 2736.3.o.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.a 2 1.a even 1 1 trivial
57.3.c.a 2 19.b odd 2 1 inner
171.3.c.c 2 3.b odd 2 1
171.3.c.c 2 57.d even 2 1
912.3.o.a 2 4.b odd 2 1
912.3.o.a 2 76.d even 2 1
2736.3.o.e 2 12.b even 2 1
2736.3.o.e 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{3}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$( -4 + T )^{2}$$
$7$ $$( 10 + T )^{2}$$
$11$ $$( -10 + T )^{2}$$
$13$ $$588 + T^{2}$$
$17$ $$( -10 + T )^{2}$$
$19$ $$( -19 + T )^{2}$$
$23$ $$( 20 + T )^{2}$$
$29$ $$1200 + T^{2}$$
$31$ $$300 + T^{2}$$
$37$ $$108 + T^{2}$$
$41$ $$1200 + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$( 80 + T )^{2}$$
$53$ $$1728 + T^{2}$$
$59$ $$1200 + T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$5808 + T^{2}$$
$71$ $$10800 + T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$300 + T^{2}$$
$83$ $$( -70 + T )^{2}$$
$89$ $$10800 + T^{2}$$
$97$ $$5808 + T^{2}$$