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Properties

Label 1.18.a.a
Level 11
Weight 1818
Character orbit 1.a
Self dual yes
Analytic conductor 1.8321.832
Analytic rank 11
Dimension 11
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,18,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: N N == 1 1
Weight: k k == 18 18
Character orbit: [χ][\chi] == 1.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: 1.832220873451.83222087345
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q−528q2−4284q3+147712q4−1025850q5+2261952q6+3225992q7−8785920q8−110787507q9+541648800q10−753618228q11−632798208q12+2541064526q13+⋯+83 ⁣⋯ ⁣96q99+O(q100) q - 528 q^{2} - 4284 q^{3} + 147712 q^{4} - 1025850 q^{5} + 2261952 q^{6} + 3225992 q^{7} - 8785920 q^{8} - 110787507 q^{9} + 541648800 q^{10} - 753618228 q^{11} - 632798208 q^{12} + 2541064526 q^{13}+ \cdots + 83\!\cdots\!96 q^{99}+O(q^{100}) Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1.1
0
−528.000 −4284.00 147712. −1.02585e6 2.26195e6 3.22599e6 −8.78592e6 −1.10788e8 5.41649e8
nn: 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 1.18.a.a ✓ 1
3.b odd 2 1 9.18.a.b 1
4.b odd 2 1 16.18.a.b 1
5.b even 2 1 25.18.a.a 1
5.c odd 4 2 25.18.b.a 2
7.b odd 2 1 49.18.a.a 1
8.b even 2 1 64.18.a.d 1
8.d odd 2 1 64.18.a.b 1
11.b odd 2 1 121.18.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.18.a.a ✓ 1 1.a even 1 1 trivial
9.18.a.b 1 3.b odd 2 1
16.18.a.b 1 4.b odd 2 1
25.18.a.a 1 5.b even 2 1
25.18.b.a 2 5.c odd 4 2
49.18.a.a 1 7.b odd 2 1
64.18.a.b 1 8.d odd 2 1
64.18.a.d 1 8.b even 2 1
121.18.a.b 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace S18new(Γ0(1))S_{18}^{\mathrm{new}}(\Gamma_0(1)).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T+528 T + 528 Copy content Toggle raw display
33 T+4284 T + 4284 Copy content Toggle raw display
55 T+1025850 T + 1025850 Copy content Toggle raw display
77 T−3225992 T - 3225992 Copy content Toggle raw display
1111 T+753618228 T + 753618228 Copy content Toggle raw display
1313 T−2541064526 T - 2541064526 Copy content Toggle raw display
1717 T+5429742318 T + 5429742318 Copy content Toggle raw display
1919 T−1487499860 T - 1487499860 Copy content Toggle raw display
2323 T+317091823464 T + 317091823464 Copy content Toggle raw display
2929 T−2433410602590 T - 2433410602590 Copy content Toggle raw display
3131 T+8849722053088 T + 8849722053088 Copy content Toggle raw display
3737 T−12691652946662 T - 12691652946662 Copy content Toggle raw display
4141 T−48864151002282 T - 48864151002282 Copy content Toggle raw display
4343 T+91019974317844 T + 91019974317844 Copy content Toggle raw display
4747 T+49304994276048 T + 49304994276048 Copy content Toggle raw display
5353 T−22940453195766 T - 22940453195766 Copy content Toggle raw display
5959 T−32695090729980 T - 32695090729980 Copy content Toggle raw display
6161 T+1308285854869378 T + 1308285854869378 Copy content Toggle raw display
6767 T−5196143861984132 T - 5196143861984132 Copy content Toggle raw display
7171 T+3709489877412408 T + 3709489877412408 Copy content Toggle raw display
7373 T−3402372968272586 T - 3402372968272586 Copy content Toggle raw display
7979 T−2366533941308240 T - 2366533941308240 Copy content Toggle raw display
8383 T+29 ⁣⋯ ⁣04 T + 29\!\cdots\!04 Copy content Toggle raw display
8989 T−29 ⁣⋯ ⁣70 T - 29\!\cdots\!70 Copy content Toggle raw display
9797 T+63 ⁣⋯ ⁣98 T + 63\!\cdots\!98 Copy content Toggle raw display
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