Properties

Label 6.24.a.b
Level $6$
Weight $24$
Character orbit 6.a
Self dual yes
Analytic conductor $20.112$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.1122422407\)
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

\(f(q)\) \(=\) \( q - 2048q^{2} + 177147q^{3} + 4194304q^{4} - 35483250q^{5} - 362797056q^{6} - 2385847912q^{7} - 8589934592q^{8} + 31381059609q^{9} + O(q^{10}) \) \( q - 2048q^{2} + 177147q^{3} + 4194304q^{4} - 35483250q^{5} - 362797056q^{6} - 2385847912q^{7} - 8589934592q^{8} + 31381059609q^{9} + 72669696000q^{10} + 427835351460q^{11} + 743008370688q^{12} + 4303510800614q^{13} + 4886216523776q^{14} - 6285751287750q^{15} + 17592186044416q^{16} - 211566679094862q^{17} - 64268410079232q^{18} - 303299666491876q^{19} - 148827537408000q^{20} - 422645800067064q^{21} - 876206799790080q^{22} - 4084826356392600q^{23} - 1521681143169024q^{24} - 10661867924515625q^{25} - 8813590119657472q^{26} + 5559060566555523q^{27} - 10006971440693248q^{28} - 76724512266210954q^{29} + 12873218637312000q^{30} - 95662499637633472q^{31} - 36028797018963968q^{32} + 75789749005084620q^{33} + 433288558786277376q^{34} + 84657637923474000q^{35} + 131621703842267136q^{36} + 1916787087325361486q^{37} + 621157716975362048q^{38} + 762354027796368258q^{39} + 304798796611584000q^{40} - 3821928337631245926q^{41} + 865578598537347072q^{42} - 5028833488465187068q^{43} + 1794471525970083840q^{44} - 1113501983371049250q^{45} + 8365724377892044800q^{46} - 20587597004644658160q^{47} + 3116402981210161152q^{48} - 21676477080886156599q^{49} + 21835505509408000000q^{50} - 37478402501617518714q^{51} + 18050232565058502656q^{52} - 17205347518114927842q^{53} - 11384956040305711104q^{54} - 15180988734693045000q^{55} + 20494277510539771904q^{56} - 53728626020036357772q^{57} + 157131801121200033792q^{58} + 109297555349407820340q^{59} - 26364351769214976000q^{60} + 475260275455205930870q^{61} + 195916799257873350656q^{62} - 74870435544480186408q^{63} + 73786976294838206464q^{64} - 152702549615886715500q^{65} - 155217405962413301760q^{66} + 472131632539902962348q^{67} - 887374968394296066048q^{68} - 723614734555879912200q^{69} - 173378842467274752000q^{70} - 3017521662759223717320q^{71} - 269561249468963094528q^{72} + 4697126809344265031354q^{73} - 3925579954842340323328q^{74} - 1888717917224169421875q^{75} - 1272131004365541474304q^{76} - 1020750079960627151520q^{77} - 1561301048926962192384q^{78} + 9688757824459562759312q^{79} - 624227935460524032000q^{80} + 984770902183611232881q^{81} + 7827309235468791656448q^{82} + 9020535128755560906204q^{83} - 1772704969804486803456q^{84} + 7507073365992762061500q^{85} + 10299050984376703115264q^{86} - 13591517174422471868238q^{87} - 3675077685186731704320q^{88} - 7813973672159946207318q^{89} + 2280452061943908864000q^{90} - 10267522257914360217968q^{91} - 17133003525922907750400q^{92} - 16946324823307856664384q^{93} + 42163398665512259911680q^{94} + 10762057891047859077000q^{95} - 6382393305518410039296q^{96} - 57431905133561269660894q^{97} + 44393425061654848714752q^{98} + 13425926667003725179140q^{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
−2048.00 177147. 4.19430e6 −3.54832e7 −3.62797e8 −2.38585e9 −8.58993e9 3.13811e10 7.26697e10
\(n\): 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 6.24.a.b 1
3.b odd 2 1 18.24.a.c 1
4.b odd 2 1 48.24.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.24.a.b 1 1.a even 1 1 trivial
18.24.a.c 1 3.b odd 2 1
48.24.a.b 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 35483250 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2048 + T \)
$3$ \( -177147 + T \)
$5$ \( 35483250 + T \)
$7$ \( 2385847912 + T \)
$11$ \( -427835351460 + T \)
$13$ \( -4303510800614 + T \)
$17$ \( 211566679094862 + T \)
$19$ \( 303299666491876 + T \)
$23$ \( 4084826356392600 + T \)
$29$ \( 76724512266210954 + T \)
$31$ \( 95662499637633472 + T \)
$37$ \( -1916787087325361486 + T \)
$41$ \( 3821928337631245926 + T \)
$43$ \( 5028833488465187068 + T \)
$47$ \( 20587597004644658160 + T \)
$53$ \( 17205347518114927842 + T \)
$59$ \( -\)\(10\!\cdots\!40\)\( + T \)
$61$ \( -\)\(47\!\cdots\!70\)\( + T \)
$67$ \( -\)\(47\!\cdots\!48\)\( + T \)
$71$ \( \)\(30\!\cdots\!20\)\( + T \)
$73$ \( -\)\(46\!\cdots\!54\)\( + T \)
$79$ \( -\)\(96\!\cdots\!12\)\( + T \)
$83$ \( -\)\(90\!\cdots\!04\)\( + T \)
$89$ \( \)\(78\!\cdots\!18\)\( + T \)
$97$ \( \)\(57\!\cdots\!94\)\( + T \)
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