Properties

Label 6.28.a.d
Level $6$
Weight $28$
Character orbit 6.a
Self dual yes
Analytic conductor $27.711$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.7113344903\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 3386644380\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = 5184\sqrt{13546577521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8192 q^{2} -1594323 q^{3} + 67108864 q^{4} + ( 145720518 - 7 \beta ) q^{5} + 13060694016 q^{6} + ( 60823147664 - 415 \beta ) q^{7} -549755813888 q^{8} + 2541865828329 q^{9} +O(q^{10})\) \( q -8192 q^{2} -1594323 q^{3} +67108864 q^{4} +(145720518 - 7 \beta) q^{5} +13060694016 q^{6} +(60823147664 - 415 \beta) q^{7} -549755813888 q^{8} +2541865828329 q^{9} +(-1193742483456 + 57344 \beta) q^{10} +(-115903680883188 - 91630 \beta) q^{11} -106993205379072 q^{12} +(-577879635805882 - 2029370 \beta) q^{13} +(-498263225663488 + 3399680 \beta) q^{14} +(-232325573419314 + 11160261 \beta) q^{15} +4503599627370496 q^{16} +(-23062587888513726 - 84380030 \beta) q^{17} -20822964865671168 q^{18} +(-134259927376619500 + 313822810 \beta) q^{19} +(9779138424471552 - 469762048 \beta) q^{20} +(-96971743253111472 + 661644045 \beta) q^{21} +(949482953795076096 + 750632960 \beta) q^{22} +(649780963002964968 - 3529871870 \beta) q^{23} +876488338465357824 q^{24} +(10409043778459718023 - 2040087252 \beta) q^{25} +(4733989976521785344 + 16624599040 \beta) q^{26} -4052555153018976267 q^{27} +(4081772344635293696 - 27850178560 \beta) q^{28} +(30089181497892793950 + 57174757115 \beta) q^{29} +(1903211097451020288 - 91424858112 \beta) q^{30} +(\)\(10\!\cdots\!72\)\( - 55564821795 \beta) q^{31} -36893488147419103232 q^{32} +(\)\(18\!\cdots\!24\)\( + 146087816490 \beta) q^{33} +(\)\(18\!\cdots\!92\)\( + 691241205760 \beta) q^{34} +(\)\(10\!\cdots\!32\)\( - 486236048618 \beta) q^{35} +\)\(17\!\cdots\!56\)\( q^{36} +(\)\(15\!\cdots\!94\)\( - 1513397127760 \beta) q^{37} +(\)\(10\!\cdots\!00\)\( - 2570836459520 \beta) q^{38} +(\)\(92\!\cdots\!86\)\( + 3235471266510 \beta) q^{39} +(-80110701973270953984 + 3848290697216 \beta) q^{40} +(-\)\(54\!\cdots\!58\)\( + 7690294833830 \beta) q^{41} +(\)\(79\!\cdots\!24\)\( - 5420188016640 \beta) q^{42} +(-\)\(64\!\cdots\!72\)\( - 12471510586290 \beta) q^{43} +(-\)\(77\!\cdots\!32\)\( - 6149185208320 \beta) q^{44} +(\)\(37\!\cdots\!22\)\( - 17793060798303 \beta) q^{45} +(-\)\(53\!\cdots\!56\)\( + 28916710359040 \beta) q^{46} +(-\)\(49\!\cdots\!16\)\( + 38977125948390 \beta) q^{47} -\)\(71\!\cdots\!08\)\( q^{48} +(\)\(68\!\cdots\!53\)\( - 50483212561120 \beta) q^{49} +(-\)\(85\!\cdots\!16\)\( + 16712394768384 \beta) q^{50} +(\)\(36\!\cdots\!98\)\( + 134529022569690 \beta) q^{51} +(-\)\(38\!\cdots\!48\)\( - 136188715335680 \beta) q^{52} +(-\)\(17\!\cdots\!62\)\( - 403057629068185 \beta) q^{53} +\)\(33\!\cdots\!64\)\( q^{54} +(\)\(21\!\cdots\!76\)\( + 797973395117976 \beta) q^{55} +(-\)\(33\!\cdots\!32\)\( + 228148662763520 \beta) q^{56} +(\)\(21\!\cdots\!00\)\( - 500334923907630 \beta) q^{57} +(-\)\(24\!\cdots\!00\)\( - 468375610286080 \beta) q^{58} +(\)\(14\!\cdots\!40\)\( + 361350384652600 \beta) q^{59} +(-\)\(15\!\cdots\!96\)\( + 748952437653504 \beta) q^{60} +(\)\(10\!\cdots\!62\)\( - 1634587548610380 \beta) q^{61} +(-\)\(83\!\cdots\!24\)\( + 455187020144640 \beta) q^{62} +(\)\(15\!\cdots\!56\)\( - 1054874318756535 \beta) q^{63} +\)\(30\!\cdots\!44\)\( q^{64} +(\)\(50\!\cdots\!64\)\( + 3749436603027514 \beta) q^{65} +(-\)\(15\!\cdots\!08\)\( - 1196751392686080 \beta) q^{66} +(\)\(67\!\cdots\!04\)\( + 2654609626736760 \beta) q^{67} +(-\)\(15\!\cdots\!64\)\( - 5662647957585920 \beta) q^{68} +(-\)\(10\!\cdots\!64\)\( + 5627755909394010 \beta) q^{69} +(-\)\(87\!\cdots\!44\)\( + 3983245710278656 \beta) q^{70} +(-\)\(21\!\cdots\!68\)\( - 18704371070210630 \beta) q^{71} -\)\(13\!\cdots\!52\)\( q^{72} +(-\)\(10\!\cdots\!82\)\( - 5004210768948720 \beta) q^{73} +(-\)\(12\!\cdots\!48\)\( + 12397749270609920 \beta) q^{74} +(-\)\(16\!\cdots\!29\)\( + 3252558027870396 \beta) q^{75} +(-\)\(90\!\cdots\!00\)\( + 21060292276387840 \beta) q^{76} +(\)\(67\!\cdots\!68\)\( + 42526802546070700 \beta) q^{77} +(-\)\(75\!\cdots\!12\)\( - 26504980615249920 \beta) q^{78} +(\)\(28\!\cdots\!00\)\( - 52808725697193935 \beta) q^{79} +(\)\(65\!\cdots\!28\)\( - 31525197391593472 \beta) q^{80} +\)\(64\!\cdots\!41\)\( q^{81} +(\)\(44\!\cdots\!36\)\( - 62998895278735360 \beta) q^{82} +(-\)\(23\!\cdots\!52\)\( + 48107253314054970 \beta) q^{83} +(-\)\(65\!\cdots\!08\)\( + 44402180232314880 \beta) q^{84} +(\)\(21\!\cdots\!92\)\( + 149142213539140542 \beta) q^{85} +(\)\(53\!\cdots\!24\)\( + 102166614722887680 \beta) q^{86} +(-\)\(47\!\cdots\!50\)\( - 91155030287858145 \beta) q^{87} +(\)\(63\!\cdots\!44\)\( + 50374125226557440 \beta) q^{88} +(-\)\(95\!\cdots\!90\)\( - 499151220993490100 \beta) q^{89} +(-\)\(30\!\cdots\!24\)\( + 145760754059698176 \beta) q^{90} +(\)\(27\!\cdots\!52\)\( + 116387377684549350 \beta) q^{91} +(\)\(43\!\cdots\!52\)\( - 236885691261255680 \beta) q^{92} +(-\)\(16\!\cdots\!56\)\( + 88588273378669785 \beta) q^{93} +(\)\(40\!\cdots\!72\)\( - 319300615769210880 \beta) q^{94} +(-\)\(81\!\cdots\!20\)\( + 985549914069752080 \beta) q^{95} +\)\(58\!\cdots\!36\)\( q^{96} +(\)\(27\!\cdots\!34\)\( - 117489775290924260 \beta) q^{97} +(-\)\(56\!\cdots\!76\)\( + 413558477300695040 \beta) q^{98} +(-\)\(29\!\cdots\!52\)\( - 232911165849786270 \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16384q^{2} - 3188646q^{3} + 134217728q^{4} + 291441036q^{5} + 26121388032q^{6} + 121646295328q^{7} - 1099511627776q^{8} + 5083731656658q^{9} + O(q^{10}) \) \( 2q - 16384q^{2} - 3188646q^{3} + 134217728q^{4} + 291441036q^{5} + 26121388032q^{6} + 121646295328q^{7} - 1099511627776q^{8} + 5083731656658q^{9} - 2387484966912q^{10} - 231807361766376q^{11} - 213986410758144q^{12} - 1155759271611764q^{13} - 996526451326976q^{14} - 464651146838628q^{15} + 9007199254740992q^{16} - 46125175777027452q^{17} - 41645929731342336q^{18} - 268519854753239000q^{19} + 19558276848943104q^{20} - 193943486506222944q^{21} + 1898965907590152192q^{22} + 1299561926005929936q^{23} + 1752976676930715648q^{24} + 20818087556919436046q^{25} + 9467979953043570688q^{26} - 8105110306037952534q^{27} + 8163544689270587392q^{28} + 60178362995785587900q^{29} + 3806422194902040576q^{30} + \)\(20\!\cdots\!44\)\(q^{31} - 73786976294838206464q^{32} + \)\(36\!\cdots\!48\)\(q^{33} + \)\(37\!\cdots\!84\)\(q^{34} + \)\(21\!\cdots\!64\)\(q^{35} + \)\(34\!\cdots\!12\)\(q^{36} + \)\(30\!\cdots\!88\)\(q^{37} + \)\(21\!\cdots\!00\)\(q^{38} + \)\(18\!\cdots\!72\)\(q^{39} - \)\(16\!\cdots\!68\)\(q^{40} - \)\(10\!\cdots\!16\)\(q^{41} + \)\(15\!\cdots\!48\)\(q^{42} - \)\(12\!\cdots\!44\)\(q^{43} - \)\(15\!\cdots\!64\)\(q^{44} + \)\(74\!\cdots\!44\)\(q^{45} - \)\(10\!\cdots\!12\)\(q^{46} - \)\(99\!\cdots\!32\)\(q^{47} - \)\(14\!\cdots\!16\)\(q^{48} + \)\(13\!\cdots\!06\)\(q^{49} - \)\(17\!\cdots\!32\)\(q^{50} + \)\(73\!\cdots\!96\)\(q^{51} - \)\(77\!\cdots\!96\)\(q^{52} - \)\(35\!\cdots\!24\)\(q^{53} + \)\(66\!\cdots\!28\)\(q^{54} + \)\(43\!\cdots\!52\)\(q^{55} - \)\(66\!\cdots\!64\)\(q^{56} + \)\(42\!\cdots\!00\)\(q^{57} - \)\(49\!\cdots\!00\)\(q^{58} + \)\(29\!\cdots\!80\)\(q^{59} - \)\(31\!\cdots\!92\)\(q^{60} + \)\(21\!\cdots\!24\)\(q^{61} - \)\(16\!\cdots\!48\)\(q^{62} + \)\(30\!\cdots\!12\)\(q^{63} + \)\(60\!\cdots\!88\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{65} - \)\(30\!\cdots\!16\)\(q^{66} + \)\(13\!\cdots\!08\)\(q^{67} - \)\(30\!\cdots\!28\)\(q^{68} - \)\(20\!\cdots\!28\)\(q^{69} - \)\(17\!\cdots\!88\)\(q^{70} - \)\(43\!\cdots\!36\)\(q^{71} - \)\(27\!\cdots\!04\)\(q^{72} - \)\(20\!\cdots\!64\)\(q^{73} - \)\(24\!\cdots\!96\)\(q^{74} - \)\(33\!\cdots\!58\)\(q^{75} - \)\(18\!\cdots\!00\)\(q^{76} + \)\(13\!\cdots\!36\)\(q^{77} - \)\(15\!\cdots\!24\)\(q^{78} + \)\(56\!\cdots\!00\)\(q^{79} + \)\(13\!\cdots\!56\)\(q^{80} + \)\(12\!\cdots\!82\)\(q^{81} + \)\(89\!\cdots\!72\)\(q^{82} - \)\(46\!\cdots\!04\)\(q^{83} - \)\(13\!\cdots\!16\)\(q^{84} + \)\(42\!\cdots\!84\)\(q^{85} + \)\(10\!\cdots\!48\)\(q^{86} - \)\(95\!\cdots\!00\)\(q^{87} + \)\(12\!\cdots\!88\)\(q^{88} - \)\(19\!\cdots\!80\)\(q^{89} - \)\(60\!\cdots\!48\)\(q^{90} + \)\(54\!\cdots\!04\)\(q^{91} + \)\(87\!\cdots\!04\)\(q^{92} - \)\(32\!\cdots\!12\)\(q^{93} + \)\(81\!\cdots\!44\)\(q^{94} - \)\(16\!\cdots\!40\)\(q^{95} + \)\(11\!\cdots\!72\)\(q^{96} + \)\(54\!\cdots\!68\)\(q^{97} - \)\(11\!\cdots\!52\)\(q^{98} - \)\(58\!\cdots\!04\)\(q^{99} + O(q^{100}) \)

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
58195.4
−58194.4
−8192.00 −1.59432e6 6.71089e7 −4.07783e9 1.30607e10 −1.89573e11 −5.49756e11 2.54187e12 3.34056e13
1.2 −8192.00 −1.59432e6 6.71089e7 4.36927e9 1.30607e10 3.11219e11 −5.49756e11 2.54187e12 −3.57931e13
\(n\): 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 6.28.a.d 2
3.b odd 2 1 18.28.a.g 2
4.b odd 2 1 48.28.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.28.a.d 2 1.a even 1 1 trivial
18.28.a.g 2 3.b odd 2 1
48.28.a.g 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 291441036 T_{5} - \)\(17\!\cdots\!00\)\( \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8192 + T )^{2} \)
$3$ \( ( 1594323 + T )^{2} \)
$5$ \( -17817155436651169500 - 291441036 T + T^{2} \)
$7$ \( -\)\(58\!\cdots\!04\)\( - 121646295328 T + T^{2} \)
$11$ \( \)\(10\!\cdots\!44\)\( + 231807361766376 T + T^{2} \)
$13$ \( -\)\(11\!\cdots\!76\)\( + 1155759271611764 T + T^{2} \)
$17$ \( -\)\(20\!\cdots\!24\)\( + 46125175777027452 T + T^{2} \)
$19$ \( -\)\(17\!\cdots\!00\)\( + 268519854753239000 T + T^{2} \)
$23$ \( -\)\(41\!\cdots\!76\)\( - 1299561926005929936 T + T^{2} \)
$29$ \( -\)\(28\!\cdots\!00\)\( - 60178362995785587900 T + T^{2} \)
$31$ \( \)\(92\!\cdots\!84\)\( - \)\(20\!\cdots\!44\)\( T + T^{2} \)
$37$ \( \)\(14\!\cdots\!36\)\( - \)\(30\!\cdots\!88\)\( T + T^{2} \)
$41$ \( \)\(80\!\cdots\!64\)\( + \)\(10\!\cdots\!16\)\( T + T^{2} \)
$43$ \( -\)\(56\!\cdots\!16\)\( + \)\(12\!\cdots\!44\)\( T + T^{2} \)
$47$ \( \)\(19\!\cdots\!56\)\( + \)\(99\!\cdots\!32\)\( T + T^{2} \)
$53$ \( -\)\(58\!\cdots\!56\)\( + \)\(35\!\cdots\!24\)\( T + T^{2} \)
$59$ \( -\)\(25\!\cdots\!00\)\( - \)\(29\!\cdots\!80\)\( T + T^{2} \)
$61$ \( \)\(19\!\cdots\!44\)\( - \)\(21\!\cdots\!24\)\( T + T^{2} \)
$67$ \( \)\(43\!\cdots\!16\)\( - \)\(13\!\cdots\!08\)\( T + T^{2} \)
$71$ \( -\)\(12\!\cdots\!76\)\( + \)\(43\!\cdots\!36\)\( T + T^{2} \)
$73$ \( \)\(91\!\cdots\!24\)\( + \)\(20\!\cdots\!64\)\( T + T^{2} \)
$79$ \( -\)\(10\!\cdots\!00\)\( - \)\(56\!\cdots\!00\)\( T + T^{2} \)
$83$ \( -\)\(29\!\cdots\!96\)\( + \)\(46\!\cdots\!04\)\( T + T^{2} \)
$89$ \( -\)\(81\!\cdots\!00\)\( + \)\(19\!\cdots\!80\)\( T + T^{2} \)
$97$ \( \)\(69\!\cdots\!56\)\( - \)\(54\!\cdots\!68\)\( T + T^{2} \)
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