# Properties

 Label 578.2.b.a Level $578$ Weight $2$ Character orbit 578.b Analytic conductor $4.615$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$578 = 2 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 578.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.61535323683$$ 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: $$2$$ Twist minimal: no (minimal twist has level 34) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + 2 i q^{3} + q^{4} -2 i q^{6} -4 i q^{7} - q^{8} - q^{9} +O(q^{10})$$ $$q - q^{2} + 2 i q^{3} + q^{4} -2 i q^{6} -4 i q^{7} - q^{8} - q^{9} + 6 i q^{11} + 2 i q^{12} + 2 q^{13} + 4 i q^{14} + q^{16} + q^{18} + 4 q^{19} + 8 q^{21} -6 i q^{22} -2 i q^{24} + 5 q^{25} -2 q^{26} + 4 i q^{27} -4 i q^{28} + 4 i q^{31} - q^{32} -12 q^{33} - q^{36} + 4 i q^{37} -4 q^{38} + 4 i q^{39} + 6 i q^{41} -8 q^{42} -8 q^{43} + 6 i q^{44} + 2 i q^{48} -9 q^{49} -5 q^{50} + 2 q^{52} + 6 q^{53} -4 i q^{54} + 4 i q^{56} + 8 i q^{57} -4 i q^{61} -4 i q^{62} + 4 i q^{63} + q^{64} + 12 q^{66} + 8 q^{67} + q^{72} -2 i q^{73} -4 i q^{74} + 10 i q^{75} + 4 q^{76} + 24 q^{77} -4 i q^{78} + 8 i q^{79} -11 q^{81} -6 i q^{82} + 8 q^{84} + 8 q^{86} -6 i q^{88} -6 q^{89} -8 i q^{91} -8 q^{93} -2 i q^{96} -14 i q^{97} + 9 q^{98} -6 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + 4q^{13} + 2q^{16} + 2q^{18} + 8q^{19} + 16q^{21} + 10q^{25} - 4q^{26} - 2q^{32} - 24q^{33} - 2q^{36} - 8q^{38} - 16q^{42} - 16q^{43} - 18q^{49} - 10q^{50} + 4q^{52} + 12q^{53} + 2q^{64} + 24q^{66} + 16q^{67} + 2q^{72} + 8q^{76} + 48q^{77} - 22q^{81} + 16q^{84} + 16q^{86} - 12q^{89} - 16q^{93} + 18q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/578\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
−1.00000 2.00000i 1.00000 0 2.00000i 4.00000i −1.00000 −1.00000 0
577.2 −1.00000 2.00000i 1.00000 0 2.00000i 4.00000i −1.00000 −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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 578.2.b.a 2
17.b even 2 1 inner 578.2.b.a 2
17.c even 4 1 34.2.a.a 1
17.c even 4 1 578.2.a.a 1
17.d even 8 4 578.2.c.e 4
17.e odd 16 8 578.2.d.e 8
51.f odd 4 1 306.2.a.a 1
51.f odd 4 1 5202.2.a.d 1
68.f odd 4 1 272.2.a.d 1
68.f odd 4 1 4624.2.a.a 1
85.f odd 4 1 850.2.c.b 2
85.i odd 4 1 850.2.c.b 2
85.j even 4 1 850.2.a.e 1
119.f odd 4 1 1666.2.a.m 1
136.i even 4 1 1088.2.a.l 1
136.j odd 4 1 1088.2.a.d 1
187.f odd 4 1 4114.2.a.a 1
204.l even 4 1 2448.2.a.k 1
221.k even 4 1 5746.2.a.b 1
255.i odd 4 1 7650.2.a.ci 1
340.n odd 4 1 6800.2.a.b 1
408.q even 4 1 9792.2.a.bj 1
408.t odd 4 1 9792.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 17.c even 4 1
272.2.a.d 1 68.f odd 4 1
306.2.a.a 1 51.f odd 4 1
578.2.a.a 1 17.c even 4 1
578.2.b.a 2 1.a even 1 1 trivial
578.2.b.a 2 17.b even 2 1 inner
578.2.c.e 4 17.d even 8 4
578.2.d.e 8 17.e odd 16 8
850.2.a.e 1 85.j even 4 1
850.2.c.b 2 85.f odd 4 1
850.2.c.b 2 85.i odd 4 1
1088.2.a.d 1 136.j odd 4 1
1088.2.a.l 1 136.i even 4 1
1666.2.a.m 1 119.f odd 4 1
2448.2.a.k 1 204.l even 4 1
4114.2.a.a 1 187.f odd 4 1
4624.2.a.a 1 68.f odd 4 1
5202.2.a.d 1 51.f odd 4 1
5746.2.a.b 1 221.k even 4 1
6800.2.a.b 1 340.n odd 4 1
7650.2.a.ci 1 255.i odd 4 1
9792.2.a.y 1 408.t odd 4 1
9792.2.a.bj 1 408.q even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(578, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$36 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$16 + T^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$64 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$196 + T^{2}$$