Properties

Label 18.12.a.c
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

\(f(q)\) \(=\) \( q + 32q^{2} + 1024q^{4} - 5766q^{5} + 72464q^{7} + 32768q^{8} + O(q^{10}) \) \( q + 32q^{2} + 1024q^{4} - 5766q^{5} + 72464q^{7} + 32768q^{8} - 184512q^{10} + 408948q^{11} + 1367558q^{13} + 2318848q^{14} + 1048576q^{16} - 5422914q^{17} + 15166100q^{19} - 5904384q^{20} + 13086336q^{22} + 52194072q^{23} - 15581369q^{25} + 43761856q^{26} + 74203136q^{28} - 118581150q^{29} - 57652408q^{31} + 33554432q^{32} - 173533248q^{34} - 417827424q^{35} - 375985186q^{37} + 485315200q^{38} - 188940288q^{40} - 856316202q^{41} - 1245189172q^{43} + 418762752q^{44} + 1670210304q^{46} + 1306762656q^{47} + 3273704553q^{49} - 498603808q^{50} + 1400379392q^{52} - 409556358q^{53} - 2357994168q^{55} + 2374500352q^{56} - 3794596800q^{58} + 2882866260q^{59} + 5731767302q^{61} - 1844877056q^{62} + 1073741824q^{64} - 7885339428q^{65} + 3893272244q^{67} - 5553063936q^{68} - 13370477568q^{70} + 9075890088q^{71} - 15571822822q^{73} - 12031525952q^{74} + 15530086400q^{76} + 29634007872q^{77} - 30196762600q^{79} - 6046089216q^{80} - 27402118464q^{82} - 23135252628q^{83} + 31268522124q^{85} - 39846053504q^{86} + 13400408064q^{88} + 25614819990q^{89} + 99098722912q^{91} + 53446729728q^{92} + 41816404992q^{94} - 87447732600q^{95} - 61937553406q^{97} + 104758545696q^{98} + O(q^{100}) \)

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 −5766.00 0 72464.0 32768.0 0 −184512.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.c 1
3.b odd 2 1 6.12.a.a 1
4.b odd 2 1 144.12.a.b 1
9.c even 3 2 162.12.c.d 2
9.d odd 6 2 162.12.c.g 2
12.b even 2 1 48.12.a.h 1
15.d odd 2 1 150.12.a.g 1
15.e even 4 2 150.12.c.f 2
24.f even 2 1 192.12.a.b 1
24.h odd 2 1 192.12.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.a 1 3.b odd 2 1
18.12.a.c 1 1.a even 1 1 trivial
48.12.a.h 1 12.b even 2 1
144.12.a.b 1 4.b odd 2 1
150.12.a.g 1 15.d odd 2 1
150.12.c.f 2 15.e even 4 2
162.12.c.d 2 9.c even 3 2
162.12.c.g 2 9.d odd 6 2
192.12.a.b 1 24.f even 2 1
192.12.a.l 1 24.h odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 5766 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 32 T \)
$3$ \( \)
$5$ \( 1 + 5766 T + 48828125 T^{2} \)
$7$ \( 1 - 72464 T + 1977326743 T^{2} \)
$11$ \( 1 - 408948 T + 285311670611 T^{2} \)
$13$ \( 1 - 1367558 T + 1792160394037 T^{2} \)
$17$ \( 1 + 5422914 T + 34271896307633 T^{2} \)
$19$ \( 1 - 15166100 T + 116490258898219 T^{2} \)
$23$ \( 1 - 52194072 T + 952809757913927 T^{2} \)
$29$ \( 1 + 118581150 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 57652408 T + 25408476896404831 T^{2} \)
$37$ \( 1 + 375985186 T + 177917621779460413 T^{2} \)
$41$ \( 1 + 856316202 T + 550329031716248441 T^{2} \)
$43$ \( 1 + 1245189172 T + 929293739471222707 T^{2} \)
$47$ \( 1 - 1306762656 T + 2472159215084012303 T^{2} \)
$53$ \( 1 + 409556358 T + 9269035929372191597 T^{2} \)
$59$ \( 1 - 2882866260 T + 30155888444737842659 T^{2} \)
$61$ \( 1 - 5731767302 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 3893272244 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 9075890088 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 15571822822 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 + 30196762600 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 23135252628 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 - 25614819990 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + 61937553406 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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