Properties

Label 8.13.d.a
Level 8
Weight 13
Character orbit 8.d
Self dual yes
Analytic conductor 7.312
Analytic rank 0
Dimension 1
CM discriminant -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 13 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(7.31195053821\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 64q^{2} + 658q^{3} + 4096q^{4} + 42112q^{6} + 262144q^{8} - 98477q^{9} + O(q^{10}) \) \( q + 64q^{2} + 658q^{3} + 4096q^{4} + 42112q^{6} + 262144q^{8} - 98477q^{9} + 1923122q^{11} + 2695168q^{12} + 16777216q^{16} - 45296062q^{17} - 6302528q^{18} - 87931438q^{19} + 123079808q^{22} + 172490752q^{24} + 244140625q^{25} - 414486044q^{27} + 1073741824q^{32} + 1265414276q^{33} - 2898947968q^{34} - 403361792q^{36} - 5627612032q^{38} + 8628259682q^{41} - 7030618702q^{43} + 7877107712q^{44} + 11039408128q^{48} + 13841287201q^{49} + 15625000000q^{50} - 29804808796q^{51} - 26527106816q^{54} - 57858886204q^{57} + 8638314482q^{59} + 68719476736q^{64} + 80986513664q^{66} + 175045819538q^{67} - 185532669952q^{68} - 25815154688q^{72} + 49139489378q^{73} + 160644531250q^{75} - 360167170048q^{76} - 220397101595q^{81} + 552208619648q^{82} - 192940233262q^{83} - 449959596928q^{86} + 504134893568q^{88} - 866326445278q^{89} + 706522120192q^{96} + 1656488134658q^{97} + 885842380864q^{98} - 189383285194q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
64.0000 658.000 4096.00 0 42112.0 0 262144. −98477.0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.13.d.a 1
3.b odd 2 1 72.13.b.a 1
4.b odd 2 1 32.13.d.a 1
8.b even 2 1 32.13.d.a 1
8.d odd 2 1 CM 8.13.d.a 1
24.f even 2 1 72.13.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.13.d.a 1 1.a even 1 1 trivial
8.13.d.a 1 8.d odd 2 1 CM
32.13.d.a 1 4.b odd 2 1
32.13.d.a 1 8.b even 2 1
72.13.b.a 1 3.b odd 2 1
72.13.b.a 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 658 \) acting on \(S_{13}^{\mathrm{new}}(8, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 64 T \)
$3$ \( 1 - 658 T + 531441 T^{2} \)
$5$ \( ( 1 - 15625 T )( 1 + 15625 T ) \)
$7$ \( ( 1 - 117649 T )( 1 + 117649 T ) \)
$11$ \( 1 - 1923122 T + 3138428376721 T^{2} \)
$13$ \( ( 1 - 4826809 T )( 1 + 4826809 T ) \)
$17$ \( 1 + 45296062 T + 582622237229761 T^{2} \)
$19$ \( 1 + 87931438 T + 2213314919066161 T^{2} \)
$23$ \( ( 1 - 148035889 T )( 1 + 148035889 T ) \)
$29$ \( ( 1 - 594823321 T )( 1 + 594823321 T ) \)
$31$ \( ( 1 - 887503681 T )( 1 + 887503681 T ) \)
$37$ \( ( 1 - 2565726409 T )( 1 + 2565726409 T ) \)
$41$ \( 1 - 8628259682 T + 22563490300366186081 T^{2} \)
$43$ \( 1 + 7030618702 T + 39959630797262576401 T^{2} \)
$47$ \( ( 1 - 10779215329 T )( 1 + 10779215329 T ) \)
$53$ \( ( 1 - 22164361129 T )( 1 + 22164361129 T ) \)
$59$ \( 1 - 8638314482 T + \)\(17\!\cdots\!81\)\( T^{2} \)
$61$ \( ( 1 - 51520374361 T )( 1 + 51520374361 T ) \)
$67$ \( 1 - 175045819538 T + \)\(81\!\cdots\!61\)\( T^{2} \)
$71$ \( ( 1 - 128100283921 T )( 1 + 128100283921 T ) \)
$73$ \( 1 - 49139489378 T + \)\(22\!\cdots\!21\)\( T^{2} \)
$79$ \( ( 1 - 243087455521 T )( 1 + 243087455521 T ) \)
$83$ \( 1 + 192940233262 T + \)\(10\!\cdots\!61\)\( T^{2} \)
$89$ \( 1 + 866326445278 T + \)\(24\!\cdots\!21\)\( T^{2} \)
$97$ \( 1 - 1656488134658 T + \)\(69\!\cdots\!41\)\( T^{2} \)
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