Properties

Label 18.34.a.c
Level 18
Weight 34
Character orbit 18.a
Self dual yes
Analytic conductor 124.169
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(124.169091986\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 65536q^{2} + 4294967296q^{4} - 538799132550q^{5} - 33347311051768q^{7} + 281474976710656q^{8} + O(q^{10}) \) \( q + 65536q^{2} + 4294967296q^{4} - 538799132550q^{5} - 33347311051768q^{7} + 281474976710656q^{8} - 35310739950796800q^{10} + 85882263625386228q^{11} + 1144054008875905166q^{13} - 2185449377088667648q^{14} + 18446744073709551616q^{16} + 139113675669385621998q^{17} + 80695000174130231060q^{19} - 2314124653415419084800q^{20} + 5628380028953311838208q^{22} + 14120372378143910765544q^{23} + 173889183409697655049375q^{25} + 74976723525691320958976q^{26} - 143225610376882922979328q^{28} + 1632686905195131326709090q^{29} - 1894078958241443951861728q^{31} + 1208925819614629174706176q^{32} + 9116953848668856123260928q^{34} + 17967502267567626543848400q^{35} - 96444218751358368990635098q^{37} + 5288427531411798822748160q^{38} - 151658473286232905141452800q^{40} - 641768233498553833164038442q^{41} - 817975597351211427387164884q^{43} + 368861513577484244628799488q^{44} + 925392724174039335930691584q^{46} + 6229246687280441243201826768q^{47} - 6618950565324076329761168583q^{49} + 11396001523937945521315840000q^{50} + 4913674552979706410367451136q^{52} + 21322120079333214208388446794q^{53} - 46273289142788517805116521400q^{55} - 9386433601659399240373239808q^{56} + 106999769018868126627206922240q^{58} - 298987905886407341741567881020q^{59} - 455881915835062287556960014658q^{61} - 124130358607311270829210206208q^{62} + 79228162514264337593543950336q^{64} - 616415307572687704036863753300q^{65} + 1172332419477563429554964377412q^{67} + 597488687426362154894028177408q^{68} + 1177518228607311973177648742400q^{70} - 2591524145775150288511661030472q^{71} - 2825174388069163226217247688374q^{73} - 6320568320089022070170261782528q^{74} + 346582386698603647647623413760q^{76} - 2863942558945695063591902251104q^{77} + 920688453939087595198198640720q^{79} - 9939089705286559671350250700800q^{80} - 42058922950561224010238423334912q^{82} + 16199219945453134166417678661804q^{83} - 74954327776507013723964597834900q^{85} - 53606848748008992105245237837824q^{86} + 24173708153814007455993003245568q^{88} + 203491630107372946965013220025510q^{89} - 38151124894006957908596484633488q^{91} + 60646537571469841919553803649024q^{92} + 408239910897610997314474919067648q^{94} - 43478396094943467445859067003000q^{95} + 226806680667600950875216250271842q^{97} - 433779544249078666347227944255488q^{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
65536.0 0 4.29497e9 −5.38799e11 0 −3.33473e13 2.81475e14 0 −3.53107e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 18.34.a.c 1
3.b odd 2 1 2.34.a.a 1
12.b even 2 1 16.34.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.34.a.a 1 3.b odd 2 1
16.34.a.a 1 12.b even 2 1
18.34.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 538799132550 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 65536 T \)
$3$ 1
$5$ \( 1 + 538799132550 T + \)\(11\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 + 33347311051768 T + \)\(77\!\cdots\!07\)\( T^{2} \)
$11$ \( 1 - 85882263625386228 T + \)\(23\!\cdots\!31\)\( T^{2} \)
$13$ \( 1 - 1144054008875905166 T + \)\(57\!\cdots\!53\)\( T^{2} \)
$17$ \( 1 - \)\(13\!\cdots\!98\)\( T + \)\(40\!\cdots\!37\)\( T^{2} \)
$19$ \( 1 - 80695000174130231060 T + \)\(15\!\cdots\!59\)\( T^{2} \)
$23$ \( 1 - \)\(14\!\cdots\!44\)\( T + \)\(86\!\cdots\!83\)\( T^{2} \)
$29$ \( 1 - \)\(16\!\cdots\!90\)\( T + \)\(18\!\cdots\!89\)\( T^{2} \)
$31$ \( 1 + \)\(18\!\cdots\!28\)\( T + \)\(16\!\cdots\!91\)\( T^{2} \)
$37$ \( 1 + \)\(96\!\cdots\!98\)\( T + \)\(56\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 + \)\(64\!\cdots\!42\)\( T + \)\(16\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 + \)\(81\!\cdots\!84\)\( T + \)\(80\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 - \)\(62\!\cdots\!68\)\( T + \)\(15\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 - \)\(21\!\cdots\!94\)\( T + \)\(79\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(27\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 + \)\(45\!\cdots\!58\)\( T + \)\(82\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 - \)\(11\!\cdots\!12\)\( T + \)\(18\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 + \)\(25\!\cdots\!72\)\( T + \)\(12\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 + \)\(28\!\cdots\!74\)\( T + \)\(30\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 - \)\(92\!\cdots\!20\)\( T + \)\(41\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(21\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 - \)\(20\!\cdots\!10\)\( T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 - \)\(22\!\cdots\!42\)\( T + \)\(36\!\cdots\!77\)\( T^{2} \)
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