Properties

Label 1911.2.a.h
Level 19111911
Weight 22
Character orbit 1911.a
Self dual yes
Analytic conductor 15.25915.259
Analytic rank 11
Dimension 22
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] = [1911,2,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1911=37213 1911 = 3 \cdot 7^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1911.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: 15.259411826315.2594118263
Analytic rank: 11
Dimension: 22
Coefficient field: Q(2)\Q(\sqrt{2})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x22 x^{2} - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 39)
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)
 

Coefficients of the qq-expansion are expressed in terms of β=2\beta = \sqrt{2}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β1)q2q3+(2β+1)q4+2βq5+(β+1)q6+(β3)q8+q9+(2β+4)q102q11+(2β1)q12+q132βq15+2q99+O(q100) q + (\beta - 1) q^{2} - q^{3} + ( - 2 \beta + 1) q^{4} + 2 \beta q^{5} + ( - \beta + 1) q^{6} + (\beta - 3) q^{8} + q^{9} + ( - 2 \beta + 4) q^{10} - 2 q^{11} + (2 \beta - 1) q^{12} + q^{13} - 2 \beta q^{15} + \cdots - 2 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q22q3+2q4+2q66q8+2q9+8q104q112q12+2q13+6q164q172q1816q20+4q228q23+6q24+6q252q26+4q99+O(q100) 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 8 q^{10} - 4 q^{11} - 2 q^{12} + 2 q^{13} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 16 q^{20} + 4 q^{22} - 8 q^{23} + 6 q^{24} + 6 q^{25} - 2 q^{26}+ \cdots - 4 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
−1.41421
1.41421
−2.41421 −1.00000 3.82843 −2.82843 2.41421 0 −4.41421 1.00000 6.82843
1.2 0.414214 −1.00000 −1.82843 2.82843 −0.414214 0 −1.58579 1.00000 1.17157
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
77 1 -1
1313 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 1911.2.a.h 2
3.b odd 2 1 5733.2.a.u 2
7.b odd 2 1 39.2.a.b 2
21.c even 2 1 117.2.a.c 2
28.d even 2 1 624.2.a.k 2
35.c odd 2 1 975.2.a.l 2
35.f even 4 2 975.2.c.h 4
56.e even 2 1 2496.2.a.bi 2
56.h odd 2 1 2496.2.a.bf 2
63.l odd 6 2 1053.2.e.m 4
63.o even 6 2 1053.2.e.e 4
77.b even 2 1 4719.2.a.p 2
84.h odd 2 1 1872.2.a.w 2
91.b odd 2 1 507.2.a.h 2
91.i even 4 2 507.2.b.e 4
91.n odd 6 2 507.2.e.h 4
91.t odd 6 2 507.2.e.d 4
91.bc even 12 4 507.2.j.f 8
105.g even 2 1 2925.2.a.v 2
105.k odd 4 2 2925.2.c.u 4
168.e odd 2 1 7488.2.a.co 2
168.i even 2 1 7488.2.a.cl 2
273.g even 2 1 1521.2.a.f 2
273.o odd 4 2 1521.2.b.j 4
364.h even 2 1 8112.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 7.b odd 2 1
117.2.a.c 2 21.c even 2 1
507.2.a.h 2 91.b odd 2 1
507.2.b.e 4 91.i even 4 2
507.2.e.d 4 91.t odd 6 2
507.2.e.h 4 91.n odd 6 2
507.2.j.f 8 91.bc even 12 4
624.2.a.k 2 28.d even 2 1
975.2.a.l 2 35.c odd 2 1
975.2.c.h 4 35.f even 4 2
1053.2.e.e 4 63.o even 6 2
1053.2.e.m 4 63.l odd 6 2
1521.2.a.f 2 273.g even 2 1
1521.2.b.j 4 273.o odd 4 2
1872.2.a.w 2 84.h odd 2 1
1911.2.a.h 2 1.a even 1 1 trivial
2496.2.a.bf 2 56.h odd 2 1
2496.2.a.bi 2 56.e even 2 1
2925.2.a.v 2 105.g even 2 1
2925.2.c.u 4 105.k odd 4 2
4719.2.a.p 2 77.b even 2 1
5733.2.a.u 2 3.b odd 2 1
7488.2.a.cl 2 168.i even 2 1
7488.2.a.co 2 168.e odd 2 1
8112.2.a.bm 2 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(1911))S_{2}^{\mathrm{new}}(\Gamma_0(1911)):

T22+2T21 T_{2}^{2} + 2T_{2} - 1 Copy content Toggle raw display
T528 T_{5}^{2} - 8 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+2T1 T^{2} + 2T - 1 Copy content Toggle raw display
33 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
55 T28 T^{2} - 8 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
1313 (T1)2 (T - 1)^{2} Copy content Toggle raw display
1717 T2+4T28 T^{2} + 4T - 28 Copy content Toggle raw display
1919 T28 T^{2} - 8 Copy content Toggle raw display
2323 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
2929 (T2)2 (T - 2)^{2} Copy content Toggle raw display
3131 T28T+8 T^{2} - 8T + 8 Copy content Toggle raw display
3737 T2+4T28 T^{2} + 4T - 28 Copy content Toggle raw display
4141 T2+16T+56 T^{2} + 16T + 56 Copy content Toggle raw display
4343 T28T16 T^{2} - 8T - 16 Copy content Toggle raw display
4747 T212T+4 T^{2} - 12T + 4 Copy content Toggle raw display
5353 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
5959 T2+4T28 T^{2} + 4T - 28 Copy content Toggle raw display
6161 T2+4T124 T^{2} + 4T - 124 Copy content Toggle raw display
6767 T28T+8 T^{2} - 8T + 8 Copy content Toggle raw display
7171 (T2)2 (T - 2)^{2} Copy content Toggle raw display
7373 T2+12T+4 T^{2} + 12T + 4 Copy content Toggle raw display
7979 T2128 T^{2} - 128 Copy content Toggle raw display
8383 T24T28 T^{2} - 4T - 28 Copy content Toggle raw display
8989 T2+24T+136 T^{2} + 24T + 136 Copy content Toggle raw display
9797 T24T28 T^{2} - 4T - 28 Copy content Toggle raw display
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