Properties

Label 8.16.a.a
Level 8
Weight 16
Character orbit 8.a
Self dual yes
Analytic conductor 11.415
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.4154804080\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3444q^{3} + 313358q^{5} - 2324616q^{7} - 2487771q^{9} + O(q^{10}) \) \( q - 3444q^{3} + 313358q^{5} - 2324616q^{7} - 2487771q^{9} - 55249084q^{11} - 110259578q^{13} - 1079204952q^{15} - 2601428750q^{17} + 1952124284q^{19} + 8005977504q^{21} - 25430340376q^{23} + 67675658039q^{25} + 57985519032q^{27} - 2277224202q^{29} - 190667257120q^{31} + 190277845296q^{33} - 728437020528q^{35} - 288229450002q^{37} + 379733986632q^{39} + 756412456602q^{41} - 354186592988q^{43} - 779562945018q^{45} + 6035922573648q^{47} + 656278037513q^{49} + 8959320615000q^{51} - 12198920684962q^{53} - 17312742464072q^{55} - 6723116034096q^{57} - 4090911936748q^{59} + 17565907389910q^{61} + 5783112270936q^{63} - 34550720842924q^{65} - 3931246965172q^{67} + 87582092254944q^{69} + 58825436072248q^{71} + 107571519617914q^{73} - 233074966286316q^{75} + 128432904651744q^{77} + 61543860115504q^{79} - 164005332829911q^{81} + 13432070277436q^{83} - 815178510242500q^{85} + 7842760151688q^{87} + 269696339030634q^{89} + 256311179172048q^{91} + 656658033521280q^{93} + 611713761385672q^{95} - 793796744596318q^{97} + 137447068951764q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3444.00 0 313358. 0 −2.32462e6 0 −2.48777e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.16.a.a 1
3.b odd 2 1 72.16.a.a 1
4.b odd 2 1 16.16.a.e 1
8.b even 2 1 64.16.a.j 1
8.d odd 2 1 64.16.a.b 1
12.b even 2 1 144.16.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.a 1 1.a even 1 1 trivial
16.16.a.e 1 4.b odd 2 1
64.16.a.b 1 8.d odd 2 1
64.16.a.j 1 8.b even 2 1
72.16.a.a 1 3.b odd 2 1
144.16.a.a 1 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 3444 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3444 T + 14348907 T^{2} \)
$5$ \( 1 - 313358 T + 30517578125 T^{2} \)
$7$ \( 1 + 2324616 T + 4747561509943 T^{2} \)
$11$ \( 1 + 55249084 T + 4177248169415651 T^{2} \)
$13$ \( 1 + 110259578 T + 51185893014090757 T^{2} \)
$17$ \( 1 + 2601428750 T + 2862423051509815793 T^{2} \)
$19$ \( 1 - 1952124284 T + 15181127029874798299 T^{2} \)
$23$ \( 1 + 25430340376 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 + 2277224202 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 + 190667257120 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 + 288229450002 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 - 756412456602 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 + 354186592988 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - 6035922573648 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 + 12198920684962 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 + 4090911936748 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 - 17565907389910 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 3931246965172 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 - 58825436072248 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 - 107571519617914 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 - 61543860115504 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 - 13432070277436 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 - 269696339030634 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 + 793796744596318 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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