Properties

Label 18.14.a.c
Level $18$
Weight $14$
Character orbit 18.a
Self dual yes
Analytic conductor $19.302$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,14,Mod(1,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.3015672113\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 64 q^{2} + 4096 q^{4} + 57450 q^{5} + 64232 q^{7} - 262144 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 64 q^{2} + 4096 q^{4} + 57450 q^{5} + 64232 q^{7} - 262144 q^{8} - 3676800 q^{10} - 2464572 q^{11} + 8032766 q^{13} - 4110848 q^{14} + 16777216 q^{16} - 71112402 q^{17} + 136337060 q^{19} + 235315200 q^{20} + 157732608 q^{22} + 1186563144 q^{23} + 2079799375 q^{25} - 514097024 q^{26} + 263094272 q^{28} + 890583090 q^{29} + 4595552672 q^{31} - 1073741824 q^{32} + 4551193728 q^{34} + 3690128400 q^{35} - 19585053898 q^{37} - 8725571840 q^{38} - 15060172800 q^{40} + 2724170358 q^{41} + 51762321116 q^{43} - 10094886912 q^{44} - 75940041216 q^{46} + 53572833168 q^{47} - 92763260583 q^{49} - 133107160000 q^{50} + 32902209536 q^{52} - 82633440006 q^{53} - 141589661400 q^{55} - 16838033408 q^{56} - 56997317760 q^{58} + 394266352980 q^{59} + 671061772142 q^{61} - 294115371008 q^{62} + 68719476736 q^{64} + 461482406700 q^{65} + 388156449812 q^{67} - 291276398592 q^{68} - 236168217600 q^{70} + 388772243928 q^{71} + 1540972938026 q^{73} + 1253443449472 q^{74} + 558436597760 q^{76} - 158304388704 q^{77} - 3306509559280 q^{79} + 963851059200 q^{80} - 174346902912 q^{82} - 4931756967396 q^{83} - 4085407494900 q^{85} - 3312788551424 q^{86} + 646072762368 q^{88} - 3502949738490 q^{89} + 515960625712 q^{91} + 4860162637824 q^{92} - 3428661322752 q^{94} + 7832564097000 q^{95} - 388932598558 q^{97} + 5936848677312 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
−64.0000 0 4096.00 57450.0 0 64232.0 −262144. 0 −3.67680e6
\(n\): 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 18.14.a.c 1
3.b odd 2 1 2.14.a.b 1
4.b odd 2 1 144.14.a.l 1
12.b even 2 1 16.14.a.a 1
15.d odd 2 1 50.14.a.a 1
15.e even 4 2 50.14.b.d 2
21.c even 2 1 98.14.a.c 1
21.g even 6 2 98.14.c.d 2
21.h odd 6 2 98.14.c.a 2
24.f even 2 1 64.14.a.h 1
24.h odd 2 1 64.14.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.14.a.b 1 3.b odd 2 1
16.14.a.a 1 12.b even 2 1
18.14.a.c 1 1.a even 1 1 trivial
50.14.a.a 1 15.d odd 2 1
50.14.b.d 2 15.e even 4 2
64.14.a.b 1 24.h odd 2 1
64.14.a.h 1 24.f even 2 1
98.14.a.c 1 21.c even 2 1
98.14.c.a 2 21.h odd 6 2
98.14.c.d 2 21.g even 6 2
144.14.a.l 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 57450 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(18))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 64 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 57450 \) Copy content Toggle raw display
$7$ \( T - 64232 \) Copy content Toggle raw display
$11$ \( T + 2464572 \) Copy content Toggle raw display
$13$ \( T - 8032766 \) Copy content Toggle raw display
$17$ \( T + 71112402 \) Copy content Toggle raw display
$19$ \( T - 136337060 \) Copy content Toggle raw display
$23$ \( T - 1186563144 \) Copy content Toggle raw display
$29$ \( T - 890583090 \) Copy content Toggle raw display
$31$ \( T - 4595552672 \) Copy content Toggle raw display
$37$ \( T + 19585053898 \) Copy content Toggle raw display
$41$ \( T - 2724170358 \) Copy content Toggle raw display
$43$ \( T - 51762321116 \) Copy content Toggle raw display
$47$ \( T - 53572833168 \) Copy content Toggle raw display
$53$ \( T + 82633440006 \) Copy content Toggle raw display
$59$ \( T - 394266352980 \) Copy content Toggle raw display
$61$ \( T - 671061772142 \) Copy content Toggle raw display
$67$ \( T - 388156449812 \) Copy content Toggle raw display
$71$ \( T - 388772243928 \) Copy content Toggle raw display
$73$ \( T - 1540972938026 \) Copy content Toggle raw display
$79$ \( T + 3306509559280 \) Copy content Toggle raw display
$83$ \( T + 4931756967396 \) Copy content Toggle raw display
$89$ \( T + 3502949738490 \) Copy content Toggle raw display
$97$ \( T + 388932598558 \) Copy content Toggle raw display
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