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 $8$

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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:

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

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