Properties

Label 6.30.a.c
Level $6$
Weight $30$
Character orbit 6.a
Self dual yes
Analytic conductor $31.967$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9668254298\)
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

\(f(q)\) \(=\) \( q + 16384q^{2} + 4782969q^{3} + 268435456q^{4} - 21003872250q^{5} + 78364164096q^{6} + 1540629019832q^{7} + 4398046511104q^{8} + 22876792454961q^{9} + O(q^{10}) \) \( q + 16384q^{2} + 4782969q^{3} + 268435456q^{4} - 21003872250q^{5} + 78364164096q^{6} + 1540629019832q^{7} + 4398046511104q^{8} + 22876792454961q^{9} - 344127442944000q^{10} - 1765121303482548q^{11} + 1283918464548864q^{12} - 2817533925589474q^{13} + 25241665860927488q^{14} - 100460869851710250q^{15} + 72057594037927936q^{16} - 1217344130484395598q^{17} + 374813367582081024q^{18} + 2696824417300672340q^{19} - 5638184025194496000q^{20} + 7368780842356841208q^{21} - 28919747436258066432q^{22} - 83091607940596084104q^{23} + 21035720123168587776q^{24} + 254898134571224359375q^{25} - 46162475836857942016q^{26} + 109418989131512359209q^{27} + 413559453465435963392q^{28} - 373821742435215352290q^{29} - 1645950891650420736000q^{30} - 2503240865137999551088q^{31} + 1180591620717411303424q^{32} - 8442520475796619125012q^{33} - 19944966233856337477632q^{34} - 32359175117194044462000q^{35} + 6140942214464815497216q^{36} - 55452778541712818086858q^{37} + 44184771253054215618560q^{38} - 13476177422542760868306q^{39} - 92376007068786622464000q^{40} - 397795591315092632336598q^{41} + 120730105321174486351872q^{42} + 210619358285038091878316q^{43} - 473821141995652160421888q^{44} - 480501226213764722732250q^{45} - 1361372904498726241959936q^{46} + 2594313197853807893435472q^{47} + 344649238497994142121984q^{48} - 846367979064670677529383q^{49} + 4176251036814939904000000q^{50} - 5822519238438819128970462q^{51} - 756326004111080521990144q^{52} + 5777440175609105519260566q^{53} + 1792720717930698493280256q^{54} + 37074382364100918296493000q^{55} + 6775758085577702824214528q^{56} + 12898827586392179481377460q^{57} - 6124695428058568331919360q^{58} + 64902007109801538344613660q^{59} - 26967259408800493338624000q^{60} - 112174996267136890127951698q^{61} - 41013098334420984645025792q^{62} + 35244650336786658435786552q^{63} + 19342813113834066795298816q^{64} + 59179122633122317840696500q^{65} - 138322255475451807744196608q^{66} - 248737931110802576032407388q^{67} - 326778326775502233233522688q^{68} - 397424584940024911790824776q^{69} - 530172725120107224465408000q^{70} + 612934098996643793966308392q^{71} + 100613197241791537106386944q^{72} + 1455538645154676278929109546q^{73} - 908538323627422811535081472q^{74} + 1219169875811994402935484375q^{75} + 723923292210040268694487040q^{76} - 2719397103668900133357891936q^{77} - 220793690890940594066325504q^{78} + 2435982470924032282819775360q^{79} - 1513488499815000022450176000q^{80} + 523347633027360537213511521q^{81} - 6517482968106477688202821632q^{82} + 5283721488999468732467326356q^{83} + 1978042045582122784389070848q^{84} + 25568940600981575758854355500q^{85} + 3450787566142064097334329344q^{86} - 1787977805593619538327149010q^{87} - 7763085590456764996352212992q^{88} + 6175849698987958280301400890q^{89} - 7872532090286321217245184000q^{90} - 4340774530124318551436448368q^{91} - 22304733667307130748271591424q^{92} - 11972923457488232574867820272q^{93} + 42505227433636788526046773248q^{94} - 56643755541664011668468565000q^{95} + 5646733123551136024526585856q^{96} - 62517784184796470636199850078q^{97} - 13866892968995564380641411072q^{98} - 40380313717600479579739520628q^{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
16384.0 4.78297e6 2.68435e8 −2.10039e10 7.83642e10 1.54063e12 4.39805e12 2.28768e13 −3.44127e14
\(n\): 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 6.30.a.c 1
3.b odd 2 1 18.30.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.30.a.c 1 1.a even 1 1 trivial
18.30.a.b 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 21003872250 \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -16384 + T \)
$3$ \( -4782969 + T \)
$5$ \( 21003872250 + T \)
$7$ \( -1540629019832 + T \)
$11$ \( 1765121303482548 + T \)
$13$ \( 2817533925589474 + T \)
$17$ \( 1217344130484395598 + T \)
$19$ \( -2696824417300672340 + T \)
$23$ \( 83091607940596084104 + T \)
$29$ \( \)\(37\!\cdots\!90\)\( + T \)
$31$ \( \)\(25\!\cdots\!88\)\( + T \)
$37$ \( \)\(55\!\cdots\!58\)\( + T \)
$41$ \( \)\(39\!\cdots\!98\)\( + T \)
$43$ \( -\)\(21\!\cdots\!16\)\( + T \)
$47$ \( -\)\(25\!\cdots\!72\)\( + T \)
$53$ \( -\)\(57\!\cdots\!66\)\( + T \)
$59$ \( -\)\(64\!\cdots\!60\)\( + T \)
$61$ \( \)\(11\!\cdots\!98\)\( + T \)
$67$ \( \)\(24\!\cdots\!88\)\( + T \)
$71$ \( -\)\(61\!\cdots\!92\)\( + T \)
$73$ \( -\)\(14\!\cdots\!46\)\( + T \)
$79$ \( -\)\(24\!\cdots\!60\)\( + T \)
$83$ \( -\)\(52\!\cdots\!56\)\( + T \)
$89$ \( -\)\(61\!\cdots\!90\)\( + T \)
$97$ \( \)\(62\!\cdots\!78\)\( + T \)
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