# Properties

 Label 578.2.c.e Level $578$ Weight $2$ Character orbit 578.c Analytic conductor $4.615$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{2} + 2 \zeta_{8} q^{3} - q^{4} -2 \zeta_{8}^{3} q^{6} + 4 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{2} + 2 \zeta_{8} q^{3} - q^{4} -2 \zeta_{8}^{3} q^{6} + 4 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} + 6 \zeta_{8}^{3} q^{11} -2 \zeta_{8} q^{12} -2 q^{13} + 4 \zeta_{8} q^{14} + q^{16} + q^{18} + 4 \zeta_{8}^{2} q^{19} -8 q^{21} + 6 \zeta_{8} q^{22} + 2 \zeta_{8}^{3} q^{24} -5 \zeta_{8}^{2} q^{25} + 2 \zeta_{8}^{2} q^{26} -4 \zeta_{8}^{3} q^{27} -4 \zeta_{8}^{3} q^{28} + 4 \zeta_{8} q^{31} -\zeta_{8}^{2} q^{32} -12 q^{33} -\zeta_{8}^{2} q^{36} + 4 \zeta_{8} q^{37} + 4 q^{38} -4 \zeta_{8} q^{39} -6 \zeta_{8}^{3} q^{41} + 8 \zeta_{8}^{2} q^{42} + 8 \zeta_{8}^{2} q^{43} -6 \zeta_{8}^{3} q^{44} + 2 \zeta_{8} q^{48} -9 \zeta_{8}^{2} q^{49} -5 q^{50} + 2 q^{52} + 6 \zeta_{8}^{2} q^{53} -4 \zeta_{8} q^{54} -4 \zeta_{8} q^{56} + 8 \zeta_{8}^{3} q^{57} + 4 \zeta_{8}^{3} q^{61} -4 \zeta_{8}^{3} q^{62} -4 \zeta_{8} q^{63} - q^{64} + 12 \zeta_{8}^{2} q^{66} + 8 q^{67} - q^{72} + 2 \zeta_{8} q^{73} -4 \zeta_{8}^{3} q^{74} -10 \zeta_{8}^{3} q^{75} -4 \zeta_{8}^{2} q^{76} -24 \zeta_{8}^{2} q^{77} + 4 \zeta_{8}^{3} q^{78} + 8 \zeta_{8}^{3} q^{79} + 11 q^{81} -6 \zeta_{8} q^{82} + 8 q^{84} + 8 q^{86} -6 \zeta_{8} q^{88} + 6 q^{89} -8 \zeta_{8}^{3} q^{91} + 8 \zeta_{8}^{2} q^{93} -2 \zeta_{8}^{3} q^{96} + 14 \zeta_{8} q^{97} -9 q^{98} -6 \zeta_{8} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} - 8q^{13} + 4q^{16} + 4q^{18} - 32q^{21} - 48q^{33} + 16q^{38} - 20q^{50} + 8q^{52} - 4q^{64} + 32q^{67} - 4q^{72} + 44q^{81} + 32q^{84} + 32q^{86} + 24q^{89} - 36q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
1.00000i −1.41421 1.41421i −1.00000 0 −1.41421 + 1.41421i 2.82843 2.82843i 1.00000i 1.00000i 0
251.2 1.00000i 1.41421 + 1.41421i −1.00000 0 1.41421 1.41421i −2.82843 + 2.82843i 1.00000i 1.00000i 0
327.1 1.00000i −1.41421 + 1.41421i −1.00000 0 −1.41421 1.41421i 2.82843 + 2.82843i 1.00000i 1.00000i 0
327.2 1.00000i 1.41421 1.41421i −1.00000 0 1.41421 + 1.41421i −2.82843 2.82843i 1.00000i 1.00000i 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
17.c even 4 2 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(578, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$16 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$256 + T^{4}$$
$11$ $$1296 + T^{4}$$
$13$ $$( 2 + T )^{4}$$
$17$ $$T^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$256 + T^{4}$$
$37$ $$256 + T^{4}$$
$41$ $$1296 + T^{4}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$256 + T^{4}$$
$67$ $$( -8 + T )^{4}$$
$71$ $$T^{4}$$
$73$ $$16 + T^{4}$$
$79$ $$4096 + T^{4}$$
$83$ $$T^{4}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$38416 + T^{4}$$