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Properties

Label 196.14.a.a
Level 196196
Weight 1414
Character orbit 196.a
Self dual yes
Analytic conductor 210.173210.173
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] = [196,14,Mod(1,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: N N == 196=22⋅72 196 = 2^{2} \cdot 7^{2}
Weight: k k == 14 14
Character orbit: [χ][\chi] == 196.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: 210.172620746210.172620746
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 4)
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−468q3−56214q5−1375299q9−6397380q11−15199742q13+26308152q15−43114194q17+365115484q19−57226824q23+1939310671q25+1389783096q27−46418994q29+⋯+8798310316620q99+O(q100) q - 468 q^{3} - 56214 q^{5} - 1375299 q^{9} - 6397380 q^{11} - 15199742 q^{13} + 26308152 q^{15} - 43114194 q^{17} + 365115484 q^{19} - 57226824 q^{23} + 1939310671 q^{25} + 1389783096 q^{27} - 46418994 q^{29}+ \cdots + 8798310316620 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
0 −468.000 0 −56214.0 0 0 0 −1.37530e6 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 −1 -1
77 −1 -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 196.14.a.a 1
7.b odd 2 1 4.14.a.a ✓ 1
7.c even 3 2 196.14.e.b 2
7.d odd 6 2 196.14.e.a 2
21.c even 2 1 36.14.a.a 1
28.d even 2 1 16.14.a.b 1
35.c odd 2 1 100.14.a.a 1
35.f even 4 2 100.14.c.a 2
56.e even 2 1 64.14.a.g 1
56.h odd 2 1 64.14.a.c 1
84.h odd 2 1 144.14.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.14.a.a ✓ 1 7.b odd 2 1
16.14.a.b 1 28.d even 2 1
36.14.a.a 1 21.c even 2 1
64.14.a.c 1 56.h odd 2 1
64.14.a.g 1 56.e even 2 1
100.14.a.a 1 35.c odd 2 1
100.14.c.a 2 35.f even 4 2
144.14.a.a 1 84.h odd 2 1
196.14.a.a 1 1.a even 1 1 trivial
196.14.e.a 2 7.d odd 6 2
196.14.e.b 2 7.c even 3 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3+468 T_{3} + 468 acting on S14new(Γ0(196))S_{14}^{\mathrm{new}}(\Gamma_0(196)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T T Copy content Toggle raw display
33 T+468 T + 468 Copy content Toggle raw display
55 T+56214 T + 56214 Copy content Toggle raw display
77 T T Copy content Toggle raw display
1111 T+6397380 T + 6397380 Copy content Toggle raw display
1313 T+15199742 T + 15199742 Copy content Toggle raw display
1717 T+43114194 T + 43114194 Copy content Toggle raw display
1919 T−365115484 T - 365115484 Copy content Toggle raw display
2323 T+57226824 T + 57226824 Copy content Toggle raw display
2929 T+46418994 T + 46418994 Copy content Toggle raw display
3131 T−5682185824 T - 5682185824 Copy content Toggle raw display
3737 T+1887185098 T + 1887185098 Copy content Toggle raw display
4141 T−7336802934 T - 7336802934 Copy content Toggle raw display
4343 T+26886674980 T + 26886674980 Copy content Toggle raw display
4747 T+101839834224 T + 101839834224 Copy content Toggle raw display
5353 T−278731884294 T - 278731884294 Copy content Toggle raw display
5959 T+59573945772 T + 59573945772 Copy content Toggle raw display
6161 T−27484470418 T - 27484470418 Copy content Toggle raw display
6767 T−784410054932 T - 784410054932 Copy content Toggle raw display
7171 T+360365227992 T + 360365227992 Copy content Toggle raw display
7373 T−1592635413718 T - 1592635413718 Copy content Toggle raw display
7979 T+23161184752 T + 23161184752 Copy content Toggle raw display
8383 T+2050158110436 T + 2050158110436 Copy content Toggle raw display
8989 T−3485391237126 T - 3485391237126 Copy content Toggle raw display
9797 T+6706667416802 T + 6706667416802 Copy content Toggle raw display
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