Properties

Label 6.16.a.c
Level $6$
Weight $16$
Character orbit 6.a
Self dual yes
Analytic conductor $8.562$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.56161030600\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 128q^{2} + 2187q^{3} + 16384q^{4} + 77646q^{5} + 279936q^{6} + 762104q^{7} + 2097152q^{8} + 4782969q^{9} + O(q^{10}) \) \( q + 128q^{2} + 2187q^{3} + 16384q^{4} + 77646q^{5} + 279936q^{6} + 762104q^{7} + 2097152q^{8} + 4782969q^{9} + 9938688q^{10} + 48011172q^{11} + 35831808q^{12} + 285130118q^{13} + 97549312q^{14} + 169811802q^{15} + 268435456q^{16} - 3173671566q^{17} + 612220032q^{18} - 5895116260q^{19} + 1272152064q^{20} + 1666721448q^{21} + 6145430016q^{22} - 333010392q^{23} + 4586471424q^{24} - 24488676809q^{25} + 36496655104q^{26} + 10460353203q^{27} + 12486311936q^{28} + 117285392310q^{29} + 21735910656q^{30} - 225821452768q^{31} + 34359738368q^{32} + 105000433164q^{33} - 406229960448q^{34} + 59174327184q^{35} + 78364164096q^{36} - 477657973906q^{37} - 754574881280q^{38} + 623579568066q^{39} + 162835464192q^{40} + 1196721561882q^{41} + 213340345344q^{42} + 1066802913668q^{43} + 786615042048q^{44} + 371378410974q^{45} - 42625330176q^{46} + 1324913565264q^{47} + 587068342272q^{48} - 4166759003127q^{49} - 3134550631552q^{50} - 6940819714842q^{51} + 4671571853312q^{52} - 6573181204962q^{53} + 1338925209984q^{54} + 3727875461112q^{55} + 1598247927808q^{56} - 12892619260620q^{57} + 15012530215680q^{58} + 7973946241140q^{59} + 2782196563968q^{60} + 14311350203222q^{61} - 28905145954304q^{62} + 3645119806776q^{63} + 4398046511104q^{64} + 22139213142228q^{65} + 13440055444992q^{66} + 41052380998124q^{67} - 51997434937344q^{68} - 728293727304q^{69} + 7574313879552q^{70} + 67253761134072q^{71} + 10030613004288q^{72} - 156200366359942q^{73} - 61140220659968q^{74} - 53556736181283q^{75} - 96585584803840q^{76} + 36589506225888q^{77} + 79818184712448q^{78} - 138004701018640q^{79} + 20842939416576q^{80} + 22876792454961q^{81} + 153180359920896q^{82} + 469396029824988q^{83} + 27307564204032q^{84} - 246422902413636q^{85} + 136550772949504q^{86} + 256503152981970q^{87} + 100686725382144q^{88} - 422649074576790q^{89} + 47536436604672q^{90} + 217298803448272q^{91} - 5456042262528q^{92} - 493871517203616q^{93} + 169588936353792q^{94} - 457732197123960q^{95} + 75144747810816q^{96} - 201862519502686q^{97} - 533345152400256q^{98} + 229635947329668q^{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
128.000 2187.00 16384.0 77646.0 279936. 762104. 2.09715e6 4.78297e6 9.93869e6
\(n\): 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 6.16.a.c 1
3.b odd 2 1 18.16.a.a 1
4.b odd 2 1 48.16.a.b 1
5.b even 2 1 150.16.a.a 1
5.c odd 4 2 150.16.c.h 2
12.b even 2 1 144.16.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.c 1 1.a even 1 1 trivial
18.16.a.a 1 3.b odd 2 1
48.16.a.b 1 4.b odd 2 1
144.16.a.e 1 12.b even 2 1
150.16.a.a 1 5.b even 2 1
150.16.c.h 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 77646 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -128 + T \)
$3$ \( -2187 + T \)
$5$ \( -77646 + T \)
$7$ \( -762104 + T \)
$11$ \( -48011172 + T \)
$13$ \( -285130118 + T \)
$17$ \( 3173671566 + T \)
$19$ \( 5895116260 + T \)
$23$ \( 333010392 + T \)
$29$ \( -117285392310 + T \)
$31$ \( 225821452768 + T \)
$37$ \( 477657973906 + T \)
$41$ \( -1196721561882 + T \)
$43$ \( -1066802913668 + T \)
$47$ \( -1324913565264 + T \)
$53$ \( 6573181204962 + T \)
$59$ \( -7973946241140 + T \)
$61$ \( -14311350203222 + T \)
$67$ \( -41052380998124 + T \)
$71$ \( -67253761134072 + T \)
$73$ \( 156200366359942 + T \)
$79$ \( 138004701018640 + T \)
$83$ \( -469396029824988 + T \)
$89$ \( 422649074576790 + T \)
$97$ \( 201862519502686 + T \)
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