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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(13.8301772501\)
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

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

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.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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 32 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5766 \) Copy content Toggle raw display
$7$ \( T - 72464 \) Copy content Toggle raw display
$11$ \( T - 408948 \) Copy content Toggle raw display
$13$ \( T - 1367558 \) Copy content Toggle raw display
$17$ \( T + 5422914 \) Copy content Toggle raw display
$19$ \( T - 15166100 \) Copy content Toggle raw display
$23$ \( T - 52194072 \) Copy content Toggle raw display
$29$ \( T + 118581150 \) Copy content Toggle raw display
$31$ \( T + 57652408 \) Copy content Toggle raw display
$37$ \( T + 375985186 \) Copy content Toggle raw display
$41$ \( T + 856316202 \) Copy content Toggle raw display
$43$ \( T + 1245189172 \) Copy content Toggle raw display
$47$ \( T - 1306762656 \) Copy content Toggle raw display
$53$ \( T + 409556358 \) Copy content Toggle raw display
$59$ \( T - 2882866260 \) Copy content Toggle raw display
$61$ \( T - 5731767302 \) Copy content Toggle raw display
$67$ \( T - 3893272244 \) Copy content Toggle raw display
$71$ \( T - 9075890088 \) Copy content Toggle raw display
$73$ \( T + 15571822822 \) Copy content Toggle raw display
$79$ \( T + 30196762600 \) Copy content Toggle raw display
$83$ \( T + 23135252628 \) Copy content Toggle raw display
$89$ \( T - 25614819990 \) Copy content Toggle raw display
$97$ \( T + 61937553406 \) Copy content Toggle raw display
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