Properties

Label 18.12.a.d
Level 18
Weight 12
Character orbit 18.a
Self dual yes
Analytic conductor 13.830
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \( q + 32q^{2} + 1024q^{4} - 5280q^{5} - 49036q^{7} + 32768q^{8} + O(q^{10}) \) \( q + 32q^{2} + 1024q^{4} - 5280q^{5} - 49036q^{7} + 32768q^{8} - 168960q^{10} - 414336q^{11} - 522982q^{13} - 1569152q^{14} + 1048576q^{16} - 9499968q^{17} + 13053944q^{19} - 5406720q^{20} - 13258752q^{22} - 58755840q^{23} - 20949725q^{25} - 16735424q^{26} - 50212864q^{28} + 117142944q^{29} + 142907156q^{31} + 33554432q^{32} - 303998976q^{34} + 258910080q^{35} + 718521806q^{37} + 417726208q^{38} - 173015040q^{40} + 668055360q^{41} + 141575864q^{43} - 424280064q^{44} - 1880186880q^{46} - 729235200q^{47} + 427202553q^{49} - 670391200q^{50} - 535533568q^{52} - 4917225312q^{53} + 2187694080q^{55} - 1606811648q^{56} + 3748574208q^{58} - 1408015104q^{59} - 3223327018q^{61} + 4573028992q^{62} + 1073741824q^{64} + 2761344960q^{65} - 2358681328q^{67} - 9727967232q^{68} + 8285122560q^{70} + 22245092352q^{71} - 28036594330q^{73} + 22992697792q^{74} + 13367238656q^{76} + 20317380096q^{77} - 20685045676q^{79} - 5536481280q^{80} + 21377771520q^{82} - 37818604416q^{83} + 50159831040q^{85} + 4530427648q^{86} - 13576962048q^{88} - 11288711808q^{89} + 25644945352q^{91} - 60165980160q^{92} - 23335526400q^{94} - 68924824320q^{95} - 115724393266q^{97} + 13670481696q^{98} + O(q^{100}) \)

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 −5280.00 0 −49036.0 32768.0 0 −168960.
\(n\): 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.d yes 1
3.b odd 2 1 18.12.a.b 1
4.b odd 2 1 144.12.a.c 1
9.c even 3 2 162.12.c.c 2
9.d odd 6 2 162.12.c.h 2
12.b even 2 1 144.12.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.12.a.b 1 3.b odd 2 1
18.12.a.d yes 1 1.a even 1 1 trivial
144.12.a.c 1 4.b odd 2 1
144.12.a.k 1 12.b even 2 1
162.12.c.c 2 9.c even 3 2
162.12.c.h 2 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 5280 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 32 T \)
$3$ 1
$5$ \( 1 + 5280 T + 48828125 T^{2} \)
$7$ \( 1 + 49036 T + 1977326743 T^{2} \)
$11$ \( 1 + 414336 T + 285311670611 T^{2} \)
$13$ \( 1 + 522982 T + 1792160394037 T^{2} \)
$17$ \( 1 + 9499968 T + 34271896307633 T^{2} \)
$19$ \( 1 - 13053944 T + 116490258898219 T^{2} \)
$23$ \( 1 + 58755840 T + 952809757913927 T^{2} \)
$29$ \( 1 - 117142944 T + 12200509765705829 T^{2} \)
$31$ \( 1 - 142907156 T + 25408476896404831 T^{2} \)
$37$ \( 1 - 718521806 T + 177917621779460413 T^{2} \)
$41$ \( 1 - 668055360 T + 550329031716248441 T^{2} \)
$43$ \( 1 - 141575864 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 729235200 T + 2472159215084012303 T^{2} \)
$53$ \( 1 + 4917225312 T + 9269035929372191597 T^{2} \)
$59$ \( 1 + 1408015104 T + 30155888444737842659 T^{2} \)
$61$ \( 1 + 3223327018 T + 43513917611435838661 T^{2} \)
$67$ \( 1 + 2358681328 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 22245092352 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 28036594330 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 + 20685045676 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 37818604416 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 + 11288711808 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + 115724393266 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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