Properties

Label 18.8.a.b
Level 18
Weight 8
Character orbit 18.a
Self dual yes
Analytic conductor 5.623
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.62293045871\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 8q^{2} + 64q^{4} + 210q^{5} + 1016q^{7} + 512q^{8} + O(q^{10}) \) \( q + 8q^{2} + 64q^{4} + 210q^{5} + 1016q^{7} + 512q^{8} + 1680q^{10} - 1092q^{11} + 1382q^{13} + 8128q^{14} + 4096q^{16} - 14706q^{17} - 39940q^{19} + 13440q^{20} - 8736q^{22} - 68712q^{23} - 34025q^{25} + 11056q^{26} + 65024q^{28} + 102570q^{29} + 227552q^{31} + 32768q^{32} - 117648q^{34} + 213360q^{35} + 160526q^{37} - 319520q^{38} + 107520q^{40} - 10842q^{41} - 630748q^{43} - 69888q^{44} - 549696q^{46} - 472656q^{47} + 208713q^{49} - 272200q^{50} + 88448q^{52} + 1494018q^{53} - 229320q^{55} + 520192q^{56} + 820560q^{58} - 2640660q^{59} + 827702q^{61} + 1820416q^{62} + 262144q^{64} + 290220q^{65} - 126004q^{67} - 941184q^{68} + 1706880q^{70} + 1414728q^{71} + 980282q^{73} + 1284208q^{74} - 2556160q^{76} - 1109472q^{77} - 3566800q^{79} + 860160q^{80} - 86736q^{82} - 5672892q^{83} - 3088260q^{85} - 5045984q^{86} - 559104q^{88} + 11951190q^{89} + 1404112q^{91} - 4397568q^{92} - 3781248q^{94} - 8387400q^{95} + 8682146q^{97} + 1669704q^{98} + O(q^{100}) \)

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
8.00000 0 64.0000 210.000 0 1016.00 512.000 0 1680.00
\(n\): 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 18.8.a.b 1
3.b odd 2 1 2.8.a.a 1
4.b odd 2 1 144.8.a.i 1
5.b even 2 1 450.8.a.c 1
5.c odd 4 2 450.8.c.g 2
8.b even 2 1 576.8.a.g 1
8.d odd 2 1 576.8.a.f 1
9.c even 3 2 162.8.c.a 2
9.d odd 6 2 162.8.c.l 2
12.b even 2 1 16.8.a.b 1
15.d odd 2 1 50.8.a.g 1
15.e even 4 2 50.8.b.c 2
21.c even 2 1 98.8.a.a 1
21.g even 6 2 98.8.c.e 2
21.h odd 6 2 98.8.c.d 2
24.f even 2 1 64.8.a.e 1
24.h odd 2 1 64.8.a.c 1
33.d even 2 1 242.8.a.e 1
39.d odd 2 1 338.8.a.d 1
39.f even 4 2 338.8.b.d 2
48.i odd 4 2 256.8.b.b 2
48.k even 4 2 256.8.b.f 2
51.c odd 2 1 578.8.a.b 1
60.h even 2 1 400.8.a.l 1
60.l odd 4 2 400.8.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a 1 3.b odd 2 1
16.8.a.b 1 12.b even 2 1
18.8.a.b 1 1.a even 1 1 trivial
50.8.a.g 1 15.d odd 2 1
50.8.b.c 2 15.e even 4 2
64.8.a.c 1 24.h odd 2 1
64.8.a.e 1 24.f even 2 1
98.8.a.a 1 21.c even 2 1
98.8.c.d 2 21.h odd 6 2
98.8.c.e 2 21.g even 6 2
144.8.a.i 1 4.b odd 2 1
162.8.c.a 2 9.c even 3 2
162.8.c.l 2 9.d odd 6 2
242.8.a.e 1 33.d even 2 1
256.8.b.b 2 48.i odd 4 2
256.8.b.f 2 48.k even 4 2
338.8.a.d 1 39.d odd 2 1
338.8.b.d 2 39.f even 4 2
400.8.a.l 1 60.h even 2 1
400.8.c.j 2 60.l odd 4 2
450.8.a.c 1 5.b even 2 1
450.8.c.g 2 5.c odd 4 2
576.8.a.f 1 8.d odd 2 1
576.8.a.g 1 8.b even 2 1
578.8.a.b 1 51.c odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 210 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T \)
$3$ \( \)
$5$ \( 1 - 210 T + 78125 T^{2} \)
$7$ \( 1 - 1016 T + 823543 T^{2} \)
$11$ \( 1 + 1092 T + 19487171 T^{2} \)
$13$ \( 1 - 1382 T + 62748517 T^{2} \)
$17$ \( 1 + 14706 T + 410338673 T^{2} \)
$19$ \( 1 + 39940 T + 893871739 T^{2} \)
$23$ \( 1 + 68712 T + 3404825447 T^{2} \)
$29$ \( 1 - 102570 T + 17249876309 T^{2} \)
$31$ \( 1 - 227552 T + 27512614111 T^{2} \)
$37$ \( 1 - 160526 T + 94931877133 T^{2} \)
$41$ \( 1 + 10842 T + 194754273881 T^{2} \)
$43$ \( 1 + 630748 T + 271818611107 T^{2} \)
$47$ \( 1 + 472656 T + 506623120463 T^{2} \)
$53$ \( 1 - 1494018 T + 1174711139837 T^{2} \)
$59$ \( 1 + 2640660 T + 2488651484819 T^{2} \)
$61$ \( 1 - 827702 T + 3142742836021 T^{2} \)
$67$ \( 1 + 126004 T + 6060711605323 T^{2} \)
$71$ \( 1 - 1414728 T + 9095120158391 T^{2} \)
$73$ \( 1 - 980282 T + 11047398519097 T^{2} \)
$79$ \( 1 + 3566800 T + 19203908986159 T^{2} \)
$83$ \( 1 + 5672892 T + 27136050989627 T^{2} \)
$89$ \( 1 - 11951190 T + 44231334895529 T^{2} \)
$97$ \( 1 - 8682146 T + 80798284478113 T^{2} \)
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