Properties

Label 18.12.a.e
Level 18
Weight 12
Character orbit 18.a
Self dual yes
Analytic conductor 13.830
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.8301772501\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 32q^{2} + 1024q^{4} + 11730q^{5} - 50008q^{7} + 32768q^{8} + O(q^{10}) \) \( q + 32q^{2} + 1024q^{4} + 11730q^{5} - 50008q^{7} + 32768q^{8} + 375360q^{10} + 531420q^{11} + 1332566q^{13} - 1600256q^{14} + 1048576q^{16} + 5109678q^{17} + 2901404q^{19} + 12011520q^{20} + 17005440q^{22} - 30597000q^{23} + 88764775q^{25} + 42642112q^{26} - 51208192q^{28} + 77006634q^{29} - 239418352q^{31} + 33554432q^{32} + 163509696q^{34} - 586593840q^{35} - 785041666q^{37} + 92844928q^{38} + 384368640q^{40} - 411252954q^{41} + 351233348q^{43} + 544174080q^{44} - 979104000q^{46} - 95821680q^{47} + 523473321q^{49} + 2840472800q^{50} + 1364547584q^{52} + 1465857378q^{53} + 6233556600q^{55} - 1638662144q^{56} + 2464212288q^{58} - 5621152020q^{59} - 10473587770q^{61} - 7661387264q^{62} + 1073741824q^{64} + 15630999180q^{65} + 4515307532q^{67} + 5232310272q^{68} - 18771002880q^{70} + 8509579560q^{71} + 2012496986q^{73} - 25121333312q^{74} + 2971037696q^{76} - 26575251360q^{77} - 22238409568q^{79} + 12299796480q^{80} - 13160094528q^{82} - 6328647516q^{83} + 59936522940q^{85} + 11239467136q^{86} + 17413570560q^{88} + 50123706678q^{89} - 66638960528q^{91} - 31331328000q^{92} - 3066293760q^{94} + 34033468920q^{95} + 94805961314q^{97} + 16751146272q^{98} + O(q^{100}) \)

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
32.0000 0 1024.00 11730.0 0 −50008.0 32768.0 0 375360.
\(n\): 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 18.12.a.e 1
3.b odd 2 1 6.12.a.b 1
4.b odd 2 1 144.12.a.o 1
9.c even 3 2 162.12.c.a 2
9.d odd 6 2 162.12.c.j 2
12.b even 2 1 48.12.a.a 1
15.d odd 2 1 150.12.a.f 1
15.e even 4 2 150.12.c.b 2
24.f even 2 1 192.12.a.t 1
24.h odd 2 1 192.12.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.b 1 3.b odd 2 1
18.12.a.e 1 1.a even 1 1 trivial
48.12.a.a 1 12.b even 2 1
144.12.a.o 1 4.b odd 2 1
150.12.a.f 1 15.d odd 2 1
150.12.c.b 2 15.e even 4 2
162.12.c.a 2 9.c even 3 2
162.12.c.j 2 9.d odd 6 2
192.12.a.j 1 24.h odd 2 1
192.12.a.t 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 11730 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 32 T \)
$3$ 1
$5$ \( 1 - 11730 T + 48828125 T^{2} \)
$7$ \( 1 + 50008 T + 1977326743 T^{2} \)
$11$ \( 1 - 531420 T + 285311670611 T^{2} \)
$13$ \( 1 - 1332566 T + 1792160394037 T^{2} \)
$17$ \( 1 - 5109678 T + 34271896307633 T^{2} \)
$19$ \( 1 - 2901404 T + 116490258898219 T^{2} \)
$23$ \( 1 + 30597000 T + 952809757913927 T^{2} \)
$29$ \( 1 - 77006634 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 239418352 T + 25408476896404831 T^{2} \)
$37$ \( 1 + 785041666 T + 177917621779460413 T^{2} \)
$41$ \( 1 + 411252954 T + 550329031716248441 T^{2} \)
$43$ \( 1 - 351233348 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 95821680 T + 2472159215084012303 T^{2} \)
$53$ \( 1 - 1465857378 T + 9269035929372191597 T^{2} \)
$59$ \( 1 + 5621152020 T + 30155888444737842659 T^{2} \)
$61$ \( 1 + 10473587770 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 4515307532 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 8509579560 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 - 2012496986 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 + 22238409568 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 6328647516 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 - 50123706678 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 - 94805961314 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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