Properties

Label 18.42.a.b
Level 18
Weight 42
Character orbit 18.a
Self dual yes
Analytic conductor 191.649
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(191.649006822\)
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

\(f(q)\) \(=\) \( q + 1048576q^{2} + 1099511627776q^{4} + 48504195130650q^{5} - 119392445696650168q^{7} + 1152921504606846976q^{8} + O(q^{10}) \) \( q + 1048576q^{2} + 1099511627776q^{4} + 48504195130650q^{5} - 119392445696650168q^{7} + 1152921504606846976q^{8} + 50860334913316454400q^{10} - 3153808852281809358252q^{11} - 11410299686425943429074q^{13} - 125192053138810646560768q^{14} + 1208925819614629174706176q^{16} + 26723760622401267203746158q^{17} + 67975218671585815673353460q^{19} + 53330926542065714488934400q^{20} - 3307008271090250529638449152q^{22} + 13505073711965391061204062504q^{23} - 43122078143374240725182530625q^{25} - 11964566403993770057084698624q^{26} - 131273382312081512528103866368q^{28} - 136715473782261984486517330110q^{29} + 3061422960031285210493644903712q^{31} + 1267650600228229401496703205376q^{32} + 28021894018395031159435331371008q^{34} - 5791034483195853625722104449200q^{35} - 221949310451710778755633026909178q^{37} + 71277182893776768255502277672960q^{38} + 55921529629773098635948877414400q^{40} + 501985001900072034529449934630038q^{41} - 3118478207799237814439067265458484q^{43} - 3467649504866730539366166458007552q^{44} + 14161096172597821897393111044194304q^{46} - 13155518377394721101101992302953392q^{47} - 30313084236935636680507069400139783q^{49} - 45216776211266787842648997232640000q^{50} - 12545757181634171431377644944359424q^{52} + 323998598696161709213951685217879914q^{53} - 152972959975848202616098259795623800q^{55} - 137650118131273184080669039780691968q^{56} - 143356564636709142644934395937423360q^{58} - 3459574233993702447512816107676048220q^{59} - 978043389864200648530692366749996578q^{61} + 3210134641737764920878584198554714112q^{62} + 1329227995784915872903807060280344576q^{64} - 553447402489598467174323474438518100q^{65} + 16627547762843789524411885842685183652q^{67} + 29383085542232588193036062027686084608q^{68} - 6072339774251575411445181394924339200q^{70} - 116968857862178710569082285660379434632q^{71} + 190708537939407193015711554853622755466q^{73} - 232730720156213081544466656824318230528q^{74} + 74739543330024868550281556313201704960q^{76} + 376540952133670515087596878910227986336q^{77} - 561362454250092673715323867778422642480q^{79} + 58637973853068956675288730083681894400q^{80} + 526369425352369933678752494654626725888q^{82} + 605770926784301808468900622973886535884q^{83} + 1296214499853731758014333938302245542700q^{85} - 3269961405221293590513259396945395318784q^{86} - 3636094047215136842046417359871726845952q^{88} - 11915426835300121180468989518247227755290q^{89} + 1362303585694113890907809755352898184432q^{91} + 14848985580277933693880878806277086511104q^{92} - 13794560838095047073309122681061655969792q^{94} + 3297083270495201682097079345488329549000q^{95} - 63576010574535484047856854187104758743198q^{97} - 31785572616829022167899380803320973099008q^{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
1.04858e6 0 1.09951e12 4.85042e13 0 −1.19392e17 1.15292e18 0 5.08603e19
\(n\): 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 18.42.a.b 1
3.b odd 2 1 2.42.a.a 1
12.b even 2 1 16.42.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.42.a.a 1 3.b odd 2 1
16.42.a.a 1 12.b even 2 1
18.42.a.b 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - \)\(48\!\cdots\!50\)\( \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 1048576 T \)
$3$ 1
$5$ \( 1 - 48504195130650 T + \)\(45\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 + 119392445696650168 T + \)\(44\!\cdots\!07\)\( T^{2} \)
$11$ \( 1 + \)\(31\!\cdots\!52\)\( T + \)\(49\!\cdots\!11\)\( T^{2} \)
$13$ \( 1 + \)\(11\!\cdots\!74\)\( T + \)\(46\!\cdots\!13\)\( T^{2} \)
$17$ \( 1 - \)\(26\!\cdots\!58\)\( T + \)\(28\!\cdots\!17\)\( T^{2} \)
$19$ \( 1 - \)\(67\!\cdots\!60\)\( T + \)\(26\!\cdots\!19\)\( T^{2} \)
$23$ \( 1 - \)\(13\!\cdots\!04\)\( T + \)\(67\!\cdots\!23\)\( T^{2} \)
$29$ \( 1 + \)\(13\!\cdots\!10\)\( T + \)\(90\!\cdots\!29\)\( T^{2} \)
$31$ \( 1 - \)\(30\!\cdots\!12\)\( T + \)\(13\!\cdots\!31\)\( T^{2} \)
$37$ \( 1 + \)\(22\!\cdots\!78\)\( T + \)\(19\!\cdots\!37\)\( T^{2} \)
$41$ \( 1 - \)\(50\!\cdots\!38\)\( T + \)\(13\!\cdots\!41\)\( T^{2} \)
$43$ \( 1 + \)\(31\!\cdots\!84\)\( T + \)\(93\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 + \)\(13\!\cdots\!92\)\( T + \)\(35\!\cdots\!47\)\( T^{2} \)
$53$ \( 1 - \)\(32\!\cdots\!14\)\( T + \)\(49\!\cdots\!53\)\( T^{2} \)
$59$ \( 1 + \)\(34\!\cdots\!20\)\( T + \)\(40\!\cdots\!59\)\( T^{2} \)
$61$ \( 1 + \)\(97\!\cdots\!78\)\( T + \)\(15\!\cdots\!61\)\( T^{2} \)
$67$ \( 1 - \)\(16\!\cdots\!52\)\( T + \)\(73\!\cdots\!67\)\( T^{2} \)
$71$ \( 1 + \)\(11\!\cdots\!32\)\( T + \)\(79\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 - \)\(19\!\cdots\!66\)\( T + \)\(24\!\cdots\!73\)\( T^{2} \)
$79$ \( 1 + \)\(56\!\cdots\!80\)\( T + \)\(63\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 - \)\(60\!\cdots\!84\)\( T + \)\(48\!\cdots\!83\)\( T^{2} \)
$89$ \( 1 + \)\(11\!\cdots\!90\)\( T + \)\(84\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + \)\(63\!\cdots\!98\)\( T + \)\(28\!\cdots\!97\)\( T^{2} \)
show more
show less