Properties

Label 512.2.g.h
Level $512$
Weight $2$
Character orbit 512.g
Analytic conductor $4.088$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} - \beta_{4} + \beta_{7} ) q^{3} + ( 1 + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{7} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} - \beta_{4} + \beta_{7} ) q^{3} + ( 1 + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{7} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{9} + ( 2 - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( 1 - 2 \beta_{2} - \beta_{6} + 2 \beta_{7} ) q^{13} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{15} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -1 + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{21} + ( 1 + \beta_{1} + \beta_{3} + \beta_{7} ) q^{23} + ( 1 - 3 \beta_{4} - \beta_{7} ) q^{25} + ( 2 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{27} + ( 1 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{29} + ( 4 - 2 \beta_{4} - 2 \beta_{6} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{33} + ( 2 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{35} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} ) q^{37} + ( -1 - \beta_{1} + \beta_{3} - 6 \beta_{4} + 3 \beta_{7} ) q^{39} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{41} + ( -2 + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{43} + ( -2 + 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{45} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{49} + ( 4 + 2 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{51} + ( 3 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{53} + ( 3 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{55} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{57} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + 4 \beta_{6} - 3 \beta_{7} ) q^{59} + ( -3 + 2 \beta_{1} - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{61} + ( -5 - \beta_{1} + \beta_{2} + 4 \beta_{4} + 4 \beta_{6} ) q^{63} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{65} + ( -3 - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{67} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{69} + ( 3 - 3 \beta_{2} + 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{71} + ( 5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 5 \beta_{7} ) q^{73} + ( 2 - \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{75} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} ) q^{77} + ( -4 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} ) q^{79} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{81} + ( -3 - 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 9 \beta_{6} - \beta_{7} ) q^{83} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{85} + ( -7 - 3 \beta_{1} - 3 \beta_{3} - 4 \beta_{6} - 7 \beta_{7} ) q^{87} + ( 3 - \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{89} + ( -3 \beta_{2} - 3 \beta_{3} - 8 \beta_{6} + 5 \beta_{7} ) q^{91} + ( 4 + 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{93} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{95} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{97} + ( -3 + 3 \beta_{1} + \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + \beta_{6} - 6 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} + 8q^{5} + 8q^{7} + O(q^{10}) \) \( 8q + 4q^{3} + 8q^{5} + 8q^{7} + 12q^{11} - 4q^{19} - 16q^{21} + 8q^{23} + 8q^{25} + 16q^{27} - 8q^{29} + 32q^{31} - 16q^{33} + 8q^{35} - 16q^{37} - 16q^{39} - 8q^{41} - 20q^{43} - 8q^{45} + 32q^{51} + 16q^{53} + 16q^{55} - 16q^{57} - 28q^{59} - 40q^{63} - 12q^{67} + 24q^{71} + 32q^{73} + 20q^{75} + 16q^{77} - 36q^{83} + 8q^{85} - 56q^{87} + 16q^{89} + 32q^{93} - 8q^{95} + 32q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 27 \nu^{4} + 41 \nu^{3} - 37 \nu^{2} + 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 28 \nu^{4} + 43 \nu^{3} - 44 \nu^{2} + 30 \nu - 10 \)
\(\beta_{3}\)\(=\)\( -5 \nu^{7} + 17 \nu^{6} - 59 \nu^{5} + 102 \nu^{4} - 146 \nu^{3} + 121 \nu^{2} - 66 \nu + 15 \)
\(\beta_{4}\)\(=\)\( 5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19 \)
\(\beta_{5}\)\(=\)\( -5 \nu^{7} + 18 \nu^{6} - 62 \nu^{5} + 113 \nu^{4} - 163 \nu^{3} + 145 \nu^{2} - 82 \nu + 20 \)
\(\beta_{6}\)\(=\)\( -5 \nu^{7} + 18 \nu^{6} - 63 \nu^{5} + 115 \nu^{4} - 170 \nu^{3} + 152 \nu^{2} - 89 \nu + 23 \)
\(\beta_{7}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{6} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{1} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_{1} - 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{7} - 19 \beta_{6} + 3 \beta_{5} - \beta_{4} - 5 \beta_{3} - 4 \beta_{2} - 8 \beta_{1} + 12\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-13 \beta_{7} + 2 \beta_{6} + 15 \beta_{5} - 16 \beta_{4} - 10 \beta_{3} + 13 \beta_{2} - 2 \beta_{1} + 23\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-67 \beta_{7} + 90 \beta_{6} + 4 \beta_{5} - 10 \beta_{4} + 16 \beta_{3} + 31 \beta_{2} + 33 \beta_{1} - 28\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{7} + 87 \beta_{6} - 68 \beta_{5} + 71 \beta_{4} + 65 \beta_{3} - 26 \beta_{2} + 37 \beta_{1} - 125\)\()/2\)

Character values

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

\(n\) \(5\) \(511\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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
0.500000 0.0297061i
0.500000 + 1.44392i
0.500000 + 2.10607i
0.500000 0.691860i
0.500000 2.10607i
0.500000 + 0.691860i
0.500000 + 0.0297061i
0.500000 1.44392i
0 −0.943920 + 2.27882i 0 1.70711 0.707107i 0 0.665096 0.665096i 0 −2.18073 2.18073i 0
65.2 0 0.529706 1.27882i 0 1.70711 0.707107i 0 2.74912 2.74912i 0 0.766519 + 0.766519i 0
193.1 0 −0.191860 + 0.0794708i 0 0.292893 0.707107i 0 2.27133 + 2.27133i 0 −2.09083 + 2.09083i 0
193.2 0 2.60607 1.07947i 0 0.292893 0.707107i 0 −1.68554 1.68554i 0 3.50504 3.50504i 0
321.1 0 −0.191860 0.0794708i 0 0.292893 + 0.707107i 0 2.27133 2.27133i 0 −2.09083 2.09083i 0
321.2 0 2.60607 + 1.07947i 0 0.292893 + 0.707107i 0 −1.68554 + 1.68554i 0 3.50504 + 3.50504i 0
449.1 0 −0.943920 2.27882i 0 1.70711 + 0.707107i 0 0.665096 + 0.665096i 0 −2.18073 + 2.18073i 0
449.2 0 0.529706 + 1.27882i 0 1.70711 + 0.707107i 0 2.74912 + 2.74912i 0 0.766519 0.766519i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.2
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.h 8
4.b odd 2 1 512.2.g.f 8
8.b even 2 1 512.2.g.e 8
8.d odd 2 1 512.2.g.g 8
16.e even 4 1 32.2.g.b 8
16.e even 4 1 256.2.g.d 8
16.f odd 4 1 128.2.g.b 8
16.f odd 4 1 256.2.g.c 8
32.g even 8 1 32.2.g.b 8
32.g even 8 1 256.2.g.d 8
32.g even 8 1 512.2.g.e 8
32.g even 8 1 inner 512.2.g.h 8
32.h odd 8 1 128.2.g.b 8
32.h odd 8 1 256.2.g.c 8
32.h odd 8 1 512.2.g.f 8
32.h odd 8 1 512.2.g.g 8
48.i odd 4 1 288.2.v.b 8
48.k even 4 1 1152.2.v.b 8
64.i even 16 2 4096.2.a.k 8
64.j odd 16 2 4096.2.a.q 8
80.i odd 4 1 800.2.ba.c 8
80.q even 4 1 800.2.y.b 8
80.t odd 4 1 800.2.ba.d 8
96.o even 8 1 1152.2.v.b 8
96.p odd 8 1 288.2.v.b 8
160.v odd 8 1 800.2.ba.d 8
160.z even 8 1 800.2.y.b 8
160.bb odd 8 1 800.2.ba.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b 8 16.e even 4 1
32.2.g.b 8 32.g even 8 1
128.2.g.b 8 16.f odd 4 1
128.2.g.b 8 32.h odd 8 1
256.2.g.c 8 16.f odd 4 1
256.2.g.c 8 32.h odd 8 1
256.2.g.d 8 16.e even 4 1
256.2.g.d 8 32.g even 8 1
288.2.v.b 8 48.i odd 4 1
288.2.v.b 8 96.p odd 8 1
512.2.g.e 8 8.b even 2 1
512.2.g.e 8 32.g even 8 1
512.2.g.f 8 4.b odd 2 1
512.2.g.f 8 32.h odd 8 1
512.2.g.g 8 8.d odd 2 1
512.2.g.g 8 32.h odd 8 1
512.2.g.h 8 1.a even 1 1 trivial
512.2.g.h 8 32.g even 8 1 inner
800.2.y.b 8 80.q even 4 1
800.2.y.b 8 160.z even 8 1
800.2.ba.c 8 80.i odd 4 1
800.2.ba.c 8 160.bb odd 8 1
800.2.ba.d 8 80.t odd 4 1
800.2.ba.d 8 160.v odd 8 1
1152.2.v.b 8 48.k even 4 1
1152.2.v.b 8 96.o even 8 1
4096.2.a.k 8 64.i even 16 2
4096.2.a.q 8 64.j odd 16 2

Hecke kernels

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

\(T_{3}^{8} - \cdots\)
\( T_{5}^{4} - 4 T_{5}^{3} + 6 T_{5}^{2} - 4 T_{5} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 4 + 32 T + 64 T^{2} - 56 T^{3} + 64 T^{4} - 24 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$5$ \( ( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$7$ \( 784 - 1344 T + 1152 T^{2} - 224 T^{3} + 56 T^{4} - 48 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$11$ \( 4 - 32 T + 256 T^{2} - 584 T^{3} + 576 T^{4} - 264 T^{5} + 72 T^{6} - 12 T^{7} + T^{8} \)
$13$ \( 6724 - 1968 T + 3176 T^{2} + 1648 T^{3} + 8 T^{4} - 88 T^{5} - 4 T^{6} + T^{8} \)
$17$ \( 256 + 5120 T^{2} + 1056 T^{4} + 64 T^{6} + T^{8} \)
$19$ \( 196 + 896 T + 1504 T^{2} + 1016 T^{3} + 384 T^{4} + 104 T^{5} + 24 T^{6} + 4 T^{7} + T^{8} \)
$23$ \( 16 + 64 T + 128 T^{2} + 32 T^{3} - 8 T^{4} - 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$29$ \( 188356 - 93744 T + 6856 T^{2} + 1856 T^{3} + 392 T^{4} + 152 T^{5} + 44 T^{6} + 8 T^{7} + T^{8} \)
$31$ \( ( 8 - 8 T + T^{2} )^{4} \)
$37$ \( 64516 + 107696 T + 65416 T^{2} + 16880 T^{3} + 4296 T^{4} + 1208 T^{5} + 204 T^{6} + 16 T^{7} + T^{8} \)
$41$ \( 26896 - 7872 T + 1152 T^{2} + 416 T^{3} + 968 T^{4} - 240 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$43$ \( 31684 + 54112 T + 42912 T^{2} + 20824 T^{3} + 6848 T^{4} + 1496 T^{5} + 216 T^{6} + 20 T^{7} + T^{8} \)
$47$ \( 256 + 1024 T^{2} + 544 T^{4} + 64 T^{6} + T^{8} \)
$53$ \( 158404 - 54128 T + 42088 T^{2} + 6160 T^{3} - 888 T^{4} + 232 T^{5} + 60 T^{6} - 16 T^{7} + T^{8} \)
$59$ \( 643204 + 718592 T + 391360 T^{2} + 133928 T^{3} + 30720 T^{4} + 4664 T^{5} + 456 T^{6} + 28 T^{7} + T^{8} \)
$61$ \( 42436 + 103824 T + 99240 T^{2} + 44976 T^{3} + 12168 T^{4} + 1992 T^{5} + 156 T^{6} + T^{8} \)
$67$ \( 1285956 + 489888 T + 69984 T^{2} - 9720 T^{3} - 5184 T^{4} - 216 T^{5} + 216 T^{6} + 12 T^{7} + T^{8} \)
$71$ \( 21196816 - 9060672 T + 1936512 T^{2} - 173472 T^{3} + 10232 T^{4} - 1200 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} \)
$73$ \( 38416 - 112896 T + 165888 T^{2} - 88192 T^{3} + 26504 T^{4} - 4672 T^{5} + 512 T^{6} - 32 T^{7} + T^{8} \)
$79$ \( 99361024 + 4775936 T^{2} + 78880 T^{4} + 512 T^{6} + T^{8} \)
$83$ \( 138250564 + 54933376 T + 10390560 T^{2} + 1285336 T^{3} + 122240 T^{4} + 9512 T^{5} + 664 T^{6} + 36 T^{7} + T^{8} \)
$89$ \( 17007376 - 7786112 T + 1782272 T^{2} - 209472 T^{3} + 14024 T^{4} - 672 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} \)
$97$ \( ( -992 + 288 T + 40 T^{2} - 16 T^{3} + T^{4} )^{2} \)
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