Properties

Label 512.2.g.b
Level $512$
Weight $2$
Character orbit 512.g
Analytic conductor $4.088$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.g (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{8}^{3} ) q^{3} + ( 2 + 2 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} + ( 1 + \zeta_{8}^{2} ) q^{7} + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{8}^{3} ) q^{3} + ( 2 + 2 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} + ( 1 + \zeta_{8}^{2} ) q^{7} + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} + ( -1 + \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{13} + ( -3 \zeta_{8} + 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{15} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{17} + ( -3 + 2 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{19} + ( -1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{21} + ( 3 - 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( -1 + 5 \zeta_{8} - \zeta_{8}^{2} ) q^{25} + ( -1 - \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{27} + ( 2 + \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{29} + 4 q^{31} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{33} + ( 3 + \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{35} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{37} + ( 1 - 2 \zeta_{8} + \zeta_{8}^{2} ) q^{39} + ( 3 - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{41} + ( -3 + 3 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{43} + ( 3 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{45} + ( 4 \zeta_{8} - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{47} -5 \zeta_{8}^{2} q^{49} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} + ( 4 - 4 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{53} + ( -5 + 5 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{55} + ( 5 - 4 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{57} + ( 1 + \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{59} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{61} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{63} + ( -2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{65} + ( -3 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{67} + ( -3 - 3 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{69} + ( -3 - 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{71} + ( -7 + 7 \zeta_{8}^{2} ) q^{73} + ( 6 - 6 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{75} + ( -3 - \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{77} + 6 \zeta_{8}^{2} q^{79} + ( -5 \zeta_{8} + 3 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{81} + ( -1 - 4 \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{83} + ( -6 + 6 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{85} + ( -1 + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{87} + ( -3 - 8 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{89} + ( 1 + \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{91} + ( -4 - 4 \zeta_{8}^{3} ) q^{93} + ( -16 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{95} + ( -10 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{97} + ( 2 + 5 \zeta_{8} + 5 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 8q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 8q^{5} + 4q^{7} + 4q^{9} - 4q^{11} - 12q^{19} - 4q^{21} + 12q^{23} - 4q^{25} - 4q^{27} + 8q^{29} + 16q^{31} + 8q^{33} + 12q^{35} + 4q^{39} + 12q^{41} - 12q^{43} + 12q^{45} + 8q^{51} + 16q^{53} - 20q^{55} + 20q^{57} + 4q^{59} + 8q^{63} - 8q^{65} - 12q^{67} - 12q^{69} - 12q^{71} - 28q^{73} + 24q^{75} - 12q^{77} - 4q^{83} - 24q^{85} - 4q^{87} - 12q^{89} + 4q^{91} - 16q^{93} - 64q^{95} - 40q^{97} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(\zeta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 −0.292893 + 0.707107i 0 2.70711 1.12132i 0 1.00000 1.00000i 0 1.70711 + 1.70711i 0
193.1 0 −1.70711 + 0.707107i 0 1.29289 3.12132i 0 1.00000 + 1.00000i 0 0.292893 0.292893i 0
321.1 0 −1.70711 0.707107i 0 1.29289 + 3.12132i 0 1.00000 1.00000i 0 0.292893 + 0.292893i 0
449.1 0 −0.292893 0.707107i 0 2.70711 + 1.12132i 0 1.00000 + 1.00000i 0 1.70711 1.70711i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.g.b 4
4.b odd 2 1 512.2.g.d 4
8.b even 2 1 512.2.g.c 4
8.d odd 2 1 512.2.g.a 4
16.e even 4 1 128.2.g.a 4
16.e even 4 1 256.2.g.a 4
16.f odd 4 1 32.2.g.a 4
16.f odd 4 1 256.2.g.b 4
32.g even 8 1 128.2.g.a 4
32.g even 8 1 256.2.g.a 4
32.g even 8 1 inner 512.2.g.b 4
32.g even 8 1 512.2.g.c 4
32.h odd 8 1 32.2.g.a 4
32.h odd 8 1 256.2.g.b 4
32.h odd 8 1 512.2.g.a 4
32.h odd 8 1 512.2.g.d 4
48.i odd 4 1 1152.2.v.a 4
48.k even 4 1 288.2.v.a 4
64.i even 16 2 4096.2.a.f 4
64.j odd 16 2 4096.2.a.e 4
80.j even 4 1 800.2.ba.b 4
80.k odd 4 1 800.2.y.a 4
80.s even 4 1 800.2.ba.a 4
96.o even 8 1 288.2.v.a 4
96.p odd 8 1 1152.2.v.a 4
160.u even 8 1 800.2.ba.b 4
160.y odd 8 1 800.2.y.a 4
160.ba even 8 1 800.2.ba.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 16.f odd 4 1
32.2.g.a 4 32.h odd 8 1
128.2.g.a 4 16.e even 4 1
128.2.g.a 4 32.g even 8 1
256.2.g.a 4 16.e even 4 1
256.2.g.a 4 32.g even 8 1
256.2.g.b 4 16.f odd 4 1
256.2.g.b 4 32.h odd 8 1
288.2.v.a 4 48.k even 4 1
288.2.v.a 4 96.o even 8 1
512.2.g.a 4 8.d odd 2 1
512.2.g.a 4 32.h odd 8 1
512.2.g.b 4 1.a even 1 1 trivial
512.2.g.b 4 32.g even 8 1 inner
512.2.g.c 4 8.b even 2 1
512.2.g.c 4 32.g even 8 1
512.2.g.d 4 4.b odd 2 1
512.2.g.d 4 32.h odd 8 1
800.2.y.a 4 80.k odd 4 1
800.2.y.a 4 160.y odd 8 1
800.2.ba.a 4 80.s even 4 1
800.2.ba.a 4 160.ba even 8 1
800.2.ba.b 4 80.j even 4 1
800.2.ba.b 4 160.u even 8 1
1152.2.v.a 4 48.i odd 4 1
1152.2.v.a 4 96.p odd 8 1
4096.2.a.e 4 64.j odd 16 2
4096.2.a.f 4 64.i even 16 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(512, [\chi])\):

\( T_{3}^{4} + 4 T_{3}^{3} + 6 T_{3}^{2} + 4 T_{3} + 2 \)
\( T_{5}^{4} - 8 T_{5}^{3} + 34 T_{5}^{2} - 84 T_{5} + 98 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( 98 - 84 T + 34 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( ( 2 - 2 T + T^{2} )^{2} \)
$11$ \( 2 + 12 T + 22 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( 2 - 4 T + 2 T^{2} + T^{4} \)
$17$ \( ( 8 + T^{2} )^{2} \)
$19$ \( 578 + 340 T + 86 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( 98 - 84 T + 34 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( 2 - 4 T + 2 T^{2} + T^{4} \)
$41$ \( 4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( 1922 + 124 T + 38 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( 16 + 136 T^{2} + T^{4} \)
$53$ \( 98 - 84 T + 82 T^{2} - 16 T^{3} + T^{4} \)
$59$ \( 1058 - 276 T + 22 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( 2 - 4 T + 2 T^{2} + T^{4} \)
$67$ \( 578 + 340 T + 86 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$73$ \( ( 98 + 14 T + T^{2} )^{2} \)
$79$ \( ( 36 + T^{2} )^{2} \)
$83$ \( 1058 + 276 T + 22 T^{2} + 4 T^{3} + T^{4} \)
$89$ \( 2116 - 552 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( ( 28 + 20 T + T^{2} )^{2} \)
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