# Properties

 Label 578.2.d.e Level $578$ Weight $2$ Character orbit 578.d Analytic conductor $4.615$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$578 = 2 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 578.d (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.61535323683$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 34) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{6} q^{2} + 2 \zeta_{16}^{5} q^{3} -\zeta_{16}^{4} q^{4} -2 \zeta_{16}^{3} q^{6} + 4 \zeta_{16}^{7} q^{7} + \zeta_{16}^{2} q^{8} -\zeta_{16}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{16}^{6} q^{2} + 2 \zeta_{16}^{5} q^{3} -\zeta_{16}^{4} q^{4} -2 \zeta_{16}^{3} q^{6} + 4 \zeta_{16}^{7} q^{7} + \zeta_{16}^{2} q^{8} -\zeta_{16}^{2} q^{9} -6 \zeta_{16}^{3} q^{11} + 2 \zeta_{16} q^{12} + 2 \zeta_{16}^{4} q^{13} -4 \zeta_{16}^{5} q^{14} - q^{16} + q^{18} -4 \zeta_{16}^{6} q^{19} -8 \zeta_{16}^{4} q^{21} + 6 \zeta_{16} q^{22} + 2 \zeta_{16}^{7} q^{24} -5 \zeta_{16}^{2} q^{25} -2 \zeta_{16}^{2} q^{26} + 4 \zeta_{16}^{7} q^{27} + 4 \zeta_{16}^{3} q^{28} -4 \zeta_{16}^{5} q^{31} -\zeta_{16}^{6} q^{32} + 12 q^{33} + \zeta_{16}^{6} q^{36} + 4 \zeta_{16}^{5} q^{37} + 4 \zeta_{16}^{4} q^{38} -4 \zeta_{16} q^{39} -6 \zeta_{16}^{7} q^{41} + 8 \zeta_{16}^{2} q^{42} -8 \zeta_{16}^{2} q^{43} + 6 \zeta_{16}^{7} q^{44} -2 \zeta_{16}^{5} q^{48} -9 \zeta_{16}^{6} q^{49} + 5 q^{50} + 2 q^{52} -6 \zeta_{16}^{6} q^{53} -4 \zeta_{16}^{5} q^{54} -4 \zeta_{16} q^{56} + 8 \zeta_{16}^{3} q^{57} -4 \zeta_{16}^{7} q^{61} + 4 \zeta_{16}^{3} q^{62} + 4 \zeta_{16} q^{63} + \zeta_{16}^{4} q^{64} + 12 \zeta_{16}^{6} q^{66} -8 q^{67} -\zeta_{16}^{4} q^{72} + 2 \zeta_{16} q^{73} -4 \zeta_{16}^{3} q^{74} -10 \zeta_{16}^{7} q^{75} -4 \zeta_{16}^{2} q^{76} + 24 \zeta_{16}^{2} q^{77} -4 \zeta_{16}^{7} q^{78} -8 \zeta_{16}^{3} q^{79} -11 \zeta_{16}^{4} q^{81} + 6 \zeta_{16}^{5} q^{82} -8 q^{84} + 8 q^{86} -6 \zeta_{16}^{5} q^{88} + 6 \zeta_{16}^{4} q^{89} -8 \zeta_{16}^{3} q^{91} + 8 \zeta_{16}^{2} q^{93} + 2 \zeta_{16}^{3} q^{96} -14 \zeta_{16} q^{97} + 9 \zeta_{16}^{4} q^{98} + 6 \zeta_{16}^{5} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{16} + 8q^{18} + 96q^{33} + 40q^{50} + 16q^{52} - 64q^{67} - 64q^{84} + 64q^{86} + O(q^{100})$$

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{16}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1
 −0.382683 − 0.923880i 0.382683 + 0.923880i −0.382683 + 0.923880i 0.382683 − 0.923880i 0.923880 + 0.382683i −0.923880 − 0.382683i 0.923880 − 0.382683i −0.923880 + 0.382683i
0.707107 + 0.707107i −1.84776 + 0.765367i 1.00000i 0 −1.84776 0.765367i 1.53073 3.69552i −0.707107 + 0.707107i 0.707107 0.707107i 0
155.2 0.707107 + 0.707107i 1.84776 0.765367i 1.00000i 0 1.84776 + 0.765367i −1.53073 + 3.69552i −0.707107 + 0.707107i 0.707107 0.707107i 0
179.1 0.707107 0.707107i −1.84776 0.765367i 1.00000i 0 −1.84776 + 0.765367i 1.53073 + 3.69552i −0.707107 0.707107i 0.707107 + 0.707107i 0
179.2 0.707107 0.707107i 1.84776 + 0.765367i 1.00000i 0 1.84776 0.765367i −1.53073 3.69552i −0.707107 0.707107i 0.707107 + 0.707107i 0
399.1 −0.707107 + 0.707107i −0.765367 + 1.84776i 1.00000i 0 −0.765367 1.84776i −3.69552 + 1.53073i 0.707107 + 0.707107i −0.707107 0.707107i 0
399.2 −0.707107 + 0.707107i 0.765367 1.84776i 1.00000i 0 0.765367 + 1.84776i 3.69552 1.53073i 0.707107 + 0.707107i −0.707107 0.707107i 0
423.1 −0.707107 0.707107i −0.765367 1.84776i 1.00000i 0 −0.765367 + 1.84776i −3.69552 1.53073i 0.707107 0.707107i −0.707107 + 0.707107i 0
423.2 −0.707107 0.707107i 0.765367 + 1.84776i 1.00000i 0 0.765367 1.84776i 3.69552 + 1.53073i 0.707107 0.707107i −0.707107 + 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 423.2 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
17.c even 4 2 inner
17.d even 8 4 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 256$$ acting on $$S_{2}^{\mathrm{new}}(578, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$256 + T^{8}$$
$5$ $$T^{8}$$
$7$ $$65536 + T^{8}$$
$11$ $$1679616 + T^{8}$$
$13$ $$( 4 + T^{2} )^{4}$$
$17$ $$T^{8}$$
$19$ $$( 256 + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$65536 + T^{8}$$
$37$ $$65536 + T^{8}$$
$41$ $$1679616 + T^{8}$$
$43$ $$( 4096 + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$( 1296 + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$65536 + T^{8}$$
$67$ $$( 8 + T )^{8}$$
$71$ $$T^{8}$$
$73$ $$256 + T^{8}$$
$79$ $$16777216 + T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 36 + T^{2} )^{4}$$
$97$ $$1475789056 + T^{8}$$