Properties

Label 8.16.a.b
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

\(f(q)\) \(=\) \( q + 2700q^{3} - 251890q^{5} + 1374072q^{7} - 7058907q^{9} + O(q^{10}) \) \( q + 2700q^{3} - 251890q^{5} + 1374072q^{7} - 7058907q^{9} - 43286716q^{11} - 323161466q^{13} - 680103000q^{15} - 191653646q^{17} - 6515456644q^{19} + 3709994400q^{21} + 23880801512q^{23} + 32930993975q^{25} - 57801097800q^{27} + 176820596982q^{29} - 152007193888q^{31} - 116874133200q^{33} - 346114996080q^{35} + 21581233902q^{37} - 872535958200q^{39} - 245334499686q^{41} + 2769961534756q^{43} + 1778068084230q^{45} + 2811771943248q^{47} - 2859487648759q^{49} - 517464844200q^{51} - 3491413730722q^{53} + 10903490893240q^{55} - 17591732938800q^{57} - 15827800893676q^{59} - 24609047974442q^{61} - 9699446459304q^{63} + 81401141670740q^{65} - 20706233653684q^{67} + 64478164082400q^{69} - 719982528200q^{71} + 29883036220282q^{73} + 88913683732500q^{75} - 59479064427552q^{77} - 148100908648400q^{79} - 54775363995351q^{81} - 302806756982468q^{83} + 48275636890940q^{85} + 477415611851400q^{87} - 496150966996374q^{89} - 444047121909552q^{91} - 410419423497600q^{93} + 1641178374057160q^{95} + 309183128990882q^{97} + 305556902579412q^{99} + O(q^{100}) \)

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 2700.00 0 −251890. 0 1.37407e6 0 −7.05891e6 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.b 1
3.b odd 2 1 72.16.a.c 1
4.b odd 2 1 16.16.a.b 1
8.b even 2 1 64.16.a.d 1
8.d odd 2 1 64.16.a.h 1
12.b even 2 1 144.16.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.b 1 1.a even 1 1 trivial
16.16.a.b 1 4.b odd 2 1
64.16.a.d 1 8.b even 2 1
64.16.a.h 1 8.d odd 2 1
72.16.a.c 1 3.b odd 2 1
144.16.a.m 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 \( T_{3} - 2700 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2700 T + 14348907 T^{2} \)
$5$ \( 1 + 251890 T + 30517578125 T^{2} \)
$7$ \( 1 - 1374072 T + 4747561509943 T^{2} \)
$11$ \( 1 + 43286716 T + 4177248169415651 T^{2} \)
$13$ \( 1 + 323161466 T + 51185893014090757 T^{2} \)
$17$ \( 1 + 191653646 T + 2862423051509815793 T^{2} \)
$19$ \( 1 + 6515456644 T + 15181127029874798299 T^{2} \)
$23$ \( 1 - 23880801512 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 - 176820596982 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 + 152007193888 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 - 21581233902 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 + 245334499686 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 - 2769961534756 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - 2811771943248 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 + 3491413730722 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 + 15827800893676 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 + 24609047974442 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 20706233653684 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 + 719982528200 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 - 29883036220282 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 + 148100908648400 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 + 302806756982468 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 + 496150966996374 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 - 309183128990882 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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