Properties

Label 6.30.a.d
Level $6$
Weight $30$
Character orbit 6.a
Self dual yes
Analytic conductor $31.967$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 2722854711072\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 5184\sqrt{10891418844289}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 16384 q^{2} -4782969 q^{3} + 268435456 q^{4} + ( -3278973378 - \beta ) q^{5} -78364164096 q^{6} + ( -554181573424 - 115 \beta ) q^{7} + 4398046511104 q^{8} + 22876792454961 q^{9} +O(q^{10})\) \( q +16384 q^{2} -4782969 q^{3} +268435456 q^{4} +(-3278973378 - \beta) q^{5} -78364164096 q^{6} +(-554181573424 - 115 \beta) q^{7} +4398046511104 q^{8} +22876792454961 q^{9} +(-53722699825152 - 16384 \beta) q^{10} +(167669699068740 + 116930 \beta) q^{11} -1283918464548864 q^{12} +(-8797553457222898 - 497990 \beta) q^{13} +(-9079710898978816 - 1884160 \beta) q^{14} +(15683228018799282 + 4782969 \beta) q^{15} +72057594037927936 q^{16} +(794644941971971746 + 17551990 \beta) q^{17} +374813367582081024 q^{18} +(3401510598710157116 - 114604550 \beta) q^{19} +(-880192713935290368 - 268435456 \beta) q^{20} +(2650633286058215856 + 550041435 \beta) q^{21} +(2747100349542236160 + 1915781120 \beta) q^{22} +(-6254931092292849000 - 3982823750 \beta) q^{23} -21035720123168587776 q^{24} +(\)\(11\!\cdots\!43\)\( + 6557946756 \beta) q^{25} +(-\)\(14\!\cdots\!32\)\( - 8159068160 \beta) q^{26} -\)\(10\!\cdots\!09\)\( q^{27} +(-\)\(14\!\cdots\!44\)\( - 30870077440 \beta) q^{28} +(\)\(55\!\cdots\!34\)\( + 72073925765 \beta) q^{29} +(\)\(25\!\cdots\!88\)\( + 78364164096 \beta) q^{30} +(\)\(38\!\cdots\!88\)\( - 119728399935 \beta) q^{31} +\)\(11\!\cdots\!24\)\( q^{32} +(-\)\(80\!\cdots\!60\)\( - 559272565170 \beta) q^{33} +(\)\(13\!\cdots\!64\)\( + 287571804160 \beta) q^{34} +(\)\(35\!\cdots\!32\)\( + 931263511894 \beta) q^{35} +\)\(61\!\cdots\!16\)\( q^{36} +(\)\(49\!\cdots\!02\)\( + 616677638960 \beta) q^{37} +(\)\(55\!\cdots\!44\)\( - 1877680947200 \beta) q^{38} +(\)\(42\!\cdots\!62\)\( + 2381870732310 \beta) q^{39} +(-\)\(14\!\cdots\!12\)\( - 4398046511104 \beta) q^{40} +(\)\(14\!\cdots\!14\)\( - 12680981058190 \beta) q^{41} +(\)\(43\!\cdots\!04\)\( + 9011878871040 \beta) q^{42} +(\)\(67\!\cdots\!16\)\( + 22913193173790 \beta) q^{43} +(\)\(45\!\cdots\!40\)\( + 31388157870080 \beta) q^{44} +(-\)\(75\!\cdots\!58\)\( - 22876792454961 \beta) q^{45} +(-\)\(10\!\cdots\!00\)\( - 65254584320000 \beta) q^{46} +(\)\(15\!\cdots\!20\)\( - 47374483292610 \beta) q^{47} -\)\(34\!\cdots\!84\)\( q^{48} +(\)\(95\!\cdots\!69\)\( + 127461761887520 \beta) q^{49} +(\)\(19\!\cdots\!12\)\( + 107445399650304 \beta) q^{50} +(-\)\(38\!\cdots\!74\)\( - 83950624058310 \beta) q^{51} +(-\)\(23\!\cdots\!88\)\( - 133678172733440 \beta) q^{52} +(-\)\(10\!\cdots\!06\)\( + 287228427432785 \beta) q^{53} -\)\(17\!\cdots\!56\)\( q^{54} +(-\)\(34\!\cdots\!40\)\( - 551080056158280 \beta) q^{55} +(-\)\(24\!\cdots\!96\)\( - 505775348776960 \beta) q^{56} +(-\)\(16\!\cdots\!04\)\( + 548150009908950 \beta) q^{57} +(\)\(90\!\cdots\!56\)\( + 1180859199733760 \beta) q^{58} +(\)\(20\!\cdots\!40\)\( - 1602463563453320 \beta) q^{59} +(\)\(42\!\cdots\!92\)\( + 1283918464548864 \beta) q^{60} +(-\)\(24\!\cdots\!90\)\( + 1811893801386540 \beta) q^{61} +(\)\(63\!\cdots\!92\)\( - 1961630104535040 \beta) q^{62} +(-\)\(12\!\cdots\!64\)\( - 2630831132320515 \beta) q^{63} +\)\(19\!\cdots\!16\)\( q^{64} +(\)\(17\!\cdots\!04\)\( + 10430449409733118 \beta) q^{65} +(-\)\(13\!\cdots\!40\)\( - 9163121707745280 \beta) q^{66} +(\)\(69\!\cdots\!56\)\( - 15699933931731720 \beta) q^{67} +(\)\(21\!\cdots\!76\)\( + 4711576439357440 \beta) q^{68} +(\)\(29\!\cdots\!00\)\( + 19049722528713750 \beta) q^{69} +(\)\(58\!\cdots\!88\)\( + 15257821378871296 \beta) q^{70} +(-\)\(27\!\cdots\!80\)\( - 8991372114904910 \beta) q^{71} +\)\(10\!\cdots\!44\)\( q^{72} +(-\)\(23\!\cdots\!18\)\( - 13046828493056400 \beta) q^{73} +(\)\(81\!\cdots\!68\)\( + 10103646436720640 \beta) q^{74} +(-\)\(56\!\cdots\!67\)\( - 31366456037598564 \beta) q^{75} +(\)\(91\!\cdots\!96\)\( - 30763924638924800 \beta) q^{76} +(-\)\(40\!\cdots\!60\)\( - 84082466773373420 \beta) q^{77} +(\)\(68\!\cdots\!08\)\( + 39024570078167040 \beta) q^{78} +(-\)\(22\!\cdots\!72\)\( + 320351382523826485 \beta) q^{79} +(-\)\(23\!\cdots\!08\)\( - 72057594037927936 \beta) q^{80} +\)\(52\!\cdots\!21\)\( q^{81} +(\)\(23\!\cdots\!76\)\( - 207765193657384960 \beta) q^{82} +(-\)\(62\!\cdots\!48\)\( + 124481994504701850 \beta) q^{83} +(\)\(71\!\cdots\!36\)\( + 147650623423119360 \beta) q^{84} +(-\)\(77\!\cdots\!48\)\( - 852197449912893966 \beta) q^{85} +(\)\(11\!\cdots\!44\)\( + 375409756959375360 \beta) q^{86} +(-\)\(26\!\cdots\!46\)\( - 344727352642296285 \beta) q^{87} +(\)\(73\!\cdots\!60\)\( + 514263578543390720 \beta) q^{88} +(-\)\(16\!\cdots\!82\)\( + 695475888024279460 \beta) q^{89} +(-\)\(12\!\cdots\!72\)\( - 374813367582081024 \beta) q^{90} +(\)\(21\!\cdots\!52\)\( + 1287695529330051030 \beta) q^{91} +(-\)\(16\!\cdots\!00\)\( - 1069131109498880000 \beta) q^{92} +(-\)\(18\!\cdots\!72\)\( + 572657225308707015 \beta) q^{93} +(\)\(25\!\cdots\!80\)\( - 776183534266122240 \beta) q^{94} +(\)\(22\!\cdots\!52\)\( - 3025725330262487216 \beta) q^{95} -\)\(56\!\cdots\!56\)\( q^{96} +(\)\(90\!\cdots\!62\)\( - 41843299807633100 \beta) q^{97} +(\)\(15\!\cdots\!96\)\( + 2088333506765127680 \beta) q^{98} +(\)\(38\!\cdots\!40\)\( + 2674983341758589730 \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 32768q^{2} - 9565938q^{3} + 536870912q^{4} - 6557946756q^{5} - 156728328192q^{6} - 1108363146848q^{7} + 8796093022208q^{8} + 45753584909922q^{9} + O(q^{10}) \) \( 2q + 32768q^{2} - 9565938q^{3} + 536870912q^{4} - 6557946756q^{5} - 156728328192q^{6} - 1108363146848q^{7} + 8796093022208q^{8} + 45753584909922q^{9} - 107445399650304q^{10} + 335339398137480q^{11} - 2567836929097728q^{12} - 17595106914445796q^{13} - 18159421797957632q^{14} + 31366456037598564q^{15} + 144115188075855872q^{16} + 1589289883943943492q^{17} + 749626735164162048q^{18} + 6803021197420314232q^{19} - 1760385427870580736q^{20} + 5301266572116431712q^{21} + 5494200699084472320q^{22} - 12509862184585698000q^{23} - 42071440246337175552q^{24} + \)\(23\!\cdots\!86\)\(q^{25} - \)\(28\!\cdots\!64\)\(q^{26} - \)\(21\!\cdots\!18\)\(q^{27} - \)\(29\!\cdots\!88\)\(q^{28} + \)\(11\!\cdots\!68\)\(q^{29} + \)\(51\!\cdots\!76\)\(q^{30} + \)\(77\!\cdots\!76\)\(q^{31} + \)\(23\!\cdots\!48\)\(q^{32} - \)\(16\!\cdots\!20\)\(q^{33} + \)\(26\!\cdots\!28\)\(q^{34} + \)\(70\!\cdots\!64\)\(q^{35} + \)\(12\!\cdots\!32\)\(q^{36} + \)\(99\!\cdots\!04\)\(q^{37} + \)\(11\!\cdots\!88\)\(q^{38} + \)\(84\!\cdots\!24\)\(q^{39} - \)\(28\!\cdots\!24\)\(q^{40} + \)\(28\!\cdots\!28\)\(q^{41} + \)\(86\!\cdots\!08\)\(q^{42} + \)\(13\!\cdots\!32\)\(q^{43} + \)\(90\!\cdots\!80\)\(q^{44} - \)\(15\!\cdots\!16\)\(q^{45} - \)\(20\!\cdots\!00\)\(q^{46} + \)\(30\!\cdots\!40\)\(q^{47} - \)\(68\!\cdots\!68\)\(q^{48} + \)\(19\!\cdots\!38\)\(q^{49} + \)\(38\!\cdots\!24\)\(q^{50} - \)\(76\!\cdots\!48\)\(q^{51} - \)\(47\!\cdots\!76\)\(q^{52} - \)\(20\!\cdots\!12\)\(q^{53} - \)\(35\!\cdots\!12\)\(q^{54} - \)\(69\!\cdots\!80\)\(q^{55} - \)\(48\!\cdots\!92\)\(q^{56} - \)\(32\!\cdots\!08\)\(q^{57} + \)\(18\!\cdots\!12\)\(q^{58} + \)\(41\!\cdots\!80\)\(q^{59} + \)\(84\!\cdots\!84\)\(q^{60} - \)\(48\!\cdots\!80\)\(q^{61} + \)\(12\!\cdots\!84\)\(q^{62} - \)\(25\!\cdots\!28\)\(q^{63} + \)\(38\!\cdots\!32\)\(q^{64} + \)\(34\!\cdots\!08\)\(q^{65} - \)\(26\!\cdots\!80\)\(q^{66} + \)\(13\!\cdots\!12\)\(q^{67} + \)\(42\!\cdots\!52\)\(q^{68} + \)\(59\!\cdots\!00\)\(q^{69} + \)\(11\!\cdots\!76\)\(q^{70} - \)\(54\!\cdots\!60\)\(q^{71} + \)\(20\!\cdots\!88\)\(q^{72} - \)\(47\!\cdots\!36\)\(q^{73} + \)\(16\!\cdots\!36\)\(q^{74} - \)\(11\!\cdots\!34\)\(q^{75} + \)\(18\!\cdots\!92\)\(q^{76} - \)\(80\!\cdots\!20\)\(q^{77} + \)\(13\!\cdots\!16\)\(q^{78} - \)\(45\!\cdots\!44\)\(q^{79} - \)\(47\!\cdots\!16\)\(q^{80} + \)\(10\!\cdots\!42\)\(q^{81} + \)\(46\!\cdots\!52\)\(q^{82} - \)\(12\!\cdots\!96\)\(q^{83} + \)\(14\!\cdots\!72\)\(q^{84} - \)\(15\!\cdots\!96\)\(q^{85} + \)\(22\!\cdots\!88\)\(q^{86} - \)\(52\!\cdots\!92\)\(q^{87} + \)\(14\!\cdots\!20\)\(q^{88} - \)\(33\!\cdots\!64\)\(q^{89} - \)\(24\!\cdots\!44\)\(q^{90} + \)\(43\!\cdots\!04\)\(q^{91} - \)\(33\!\cdots\!00\)\(q^{92} - \)\(37\!\cdots\!44\)\(q^{93} + \)\(50\!\cdots\!60\)\(q^{94} + \)\(44\!\cdots\!04\)\(q^{95} - \)\(11\!\cdots\!12\)\(q^{96} + \)\(18\!\cdots\!24\)\(q^{97} + \)\(31\!\cdots\!92\)\(q^{98} + \)\(76\!\cdots\!80\)\(q^{99} + O(q^{100}) \)

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
1.65011e6
−1.65011e6
16384.0 −4.78297e6 2.68435e8 −2.03873e10 −7.83642e10 −2.52164e12 4.39805e12 2.28768e13 −3.34025e14
1.2 16384.0 −4.78297e6 2.68435e8 1.38293e10 −7.83642e10 1.41327e12 4.39805e12 2.28768e13 2.26580e14
\(n\): 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.d 2
3.b odd 2 1 18.30.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.30.a.d 2 1.a even 1 1 trivial
18.30.a.f 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6557946756 T_{5} - \)\(28\!\cdots\!00\)\( \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -16384 + T )^{2} \)
$3$ \( ( 4782969 + T )^{2} \)
$5$ \( -\)\(28\!\cdots\!00\)\( + 6557946756 T + T^{2} \)
$7$ \( -\)\(35\!\cdots\!24\)\( + 1108363146848 T + T^{2} \)
$11$ \( -\)\(39\!\cdots\!00\)\( - 335339398137480 T + T^{2} \)
$13$ \( \)\(48\!\cdots\!04\)\( + 17595106914445796 T + T^{2} \)
$17$ \( \)\(54\!\cdots\!16\)\( - 1589289883943943492 T + T^{2} \)
$19$ \( \)\(77\!\cdots\!56\)\( - 6803021197420314232 T + T^{2} \)
$23$ \( -\)\(46\!\cdots\!00\)\( + 12509862184585698000 T + T^{2} \)
$29$ \( -\)\(12\!\cdots\!44\)\( - \)\(11\!\cdots\!68\)\( T + T^{2} \)
$31$ \( \)\(10\!\cdots\!44\)\( - \)\(77\!\cdots\!76\)\( T + T^{2} \)
$37$ \( \)\(23\!\cdots\!04\)\( - \)\(99\!\cdots\!04\)\( T + T^{2} \)
$41$ \( -\)\(27\!\cdots\!04\)\( - \)\(28\!\cdots\!28\)\( T + T^{2} \)
$43$ \( -\)\(14\!\cdots\!44\)\( - \)\(13\!\cdots\!32\)\( T + T^{2} \)
$47$ \( \)\(16\!\cdots\!00\)\( - \)\(30\!\cdots\!40\)\( T + T^{2} \)
$53$ \( \)\(82\!\cdots\!36\)\( + \)\(20\!\cdots\!12\)\( T + T^{2} \)
$59$ \( -\)\(31\!\cdots\!00\)\( - \)\(41\!\cdots\!80\)\( T + T^{2} \)
$61$ \( -\)\(36\!\cdots\!00\)\( + \)\(48\!\cdots\!80\)\( T + T^{2} \)
$67$ \( -\)\(67\!\cdots\!64\)\( - \)\(13\!\cdots\!12\)\( T + T^{2} \)
$71$ \( \)\(50\!\cdots\!00\)\( + \)\(54\!\cdots\!60\)\( T + T^{2} \)
$73$ \( \)\(68\!\cdots\!24\)\( + \)\(47\!\cdots\!36\)\( T + T^{2} \)
$79$ \( -\)\(29\!\cdots\!16\)\( + \)\(45\!\cdots\!44\)\( T + T^{2} \)
$83$ \( \)\(34\!\cdots\!04\)\( + \)\(12\!\cdots\!96\)\( T + T^{2} \)
$89$ \( \)\(14\!\cdots\!24\)\( + \)\(33\!\cdots\!64\)\( T + T^{2} \)
$97$ \( \)\(82\!\cdots\!44\)\( - \)\(18\!\cdots\!24\)\( T + T^{2} \)
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