Properties

Label 825.4.a.l
Level 825825
Weight 44
Character orbit 825.a
Self dual yes
Analytic conductor 48.67748.677
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 825=35211 825 = 3 \cdot 5^{2} \cdot 11
Weight: k k == 4 4
Character orbit: [χ][\chi] == 825.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: 48.676575754748.6765757547
Analytic rank: 11
Dimension: 22
Coefficient field: Q(97)\Q(\sqrt{97})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x24 x^{2} - x - 24 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 33)
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=12(1+97)\beta = \frac{1}{2}(1 + \sqrt{97}). We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβq2+3q3+(β+16)q43βq6+(4β14)q7+(9β24)q8+9q911q11+(3β+48)q12+(2β14)q13+(10β96)q14+99q99+O(q100) q - \beta q^{2} + 3 q^{3} + (\beta + 16) q^{4} - 3 \beta q^{6} + (4 \beta - 14) q^{7} + ( - 9 \beta - 24) q^{8} + 9 q^{9} - 11 q^{11} + (3 \beta + 48) q^{12} + ( - 2 \beta - 14) q^{13} + (10 \beta - 96) q^{14} + \cdots - 99 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2qq2+6q3+33q43q624q757q8+18q922q11+99q1230q13182q14+201q16106q179q18+50q1972q21+11q22134q23+198q99+O(q100) 2 q - q^{2} + 6 q^{3} + 33 q^{4} - 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9} - 22 q^{11} + 99 q^{12} - 30 q^{13} - 182 q^{14} + 201 q^{16} - 106 q^{17} - 9 q^{18} + 50 q^{19} - 72 q^{21} + 11 q^{22} - 134 q^{23}+ \cdots - 198 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
5.42443
−4.42443
−5.42443 3.00000 21.4244 0 −16.2733 7.69772 −72.8199 9.00000 0
1.2 4.42443 3.00000 11.5756 0 13.2733 −31.6977 15.8199 9.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
55 +1 +1
1111 +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 825.4.a.l 2
3.b odd 2 1 2475.4.a.p 2
5.b even 2 1 33.4.a.c 2
5.c odd 4 2 825.4.c.h 4
15.d odd 2 1 99.4.a.f 2
20.d odd 2 1 528.4.a.p 2
35.c odd 2 1 1617.4.a.k 2
40.e odd 2 1 2112.4.a.bg 2
40.f even 2 1 2112.4.a.bn 2
55.d odd 2 1 363.4.a.i 2
60.h even 2 1 1584.4.a.bj 2
165.d even 2 1 1089.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 5.b even 2 1
99.4.a.f 2 15.d odd 2 1
363.4.a.i 2 55.d odd 2 1
528.4.a.p 2 20.d odd 2 1
825.4.a.l 2 1.a even 1 1 trivial
825.4.c.h 4 5.c odd 4 2
1089.4.a.u 2 165.d even 2 1
1584.4.a.bj 2 60.h even 2 1
1617.4.a.k 2 35.c odd 2 1
2112.4.a.bg 2 40.e odd 2 1
2112.4.a.bn 2 40.f even 2 1
2475.4.a.p 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(825))S_{4}^{\mathrm{new}}(\Gamma_0(825)):

T22+T224 T_{2}^{2} + T_{2} - 24 Copy content Toggle raw display
T72+24T7244 T_{7}^{2} + 24T_{7} - 244 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+T24 T^{2} + T - 24 Copy content Toggle raw display
33 (T3)2 (T - 3)^{2} Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+24T244 T^{2} + 24T - 244 Copy content Toggle raw display
1111 (T+11)2 (T + 11)^{2} Copy content Toggle raw display
1313 T2+30T+128 T^{2} + 30T + 128 Copy content Toggle raw display
1717 T2+106T1944 T^{2} + 106T - 1944 Copy content Toggle raw display
1919 T250T+528 T^{2} - 50T + 528 Copy content Toggle raw display
2323 T2+134T+2064 T^{2} + 134T + 2064 Copy content Toggle raw display
2929 T2+198T+8928 T^{2} + 198T + 8928 Copy content Toggle raw display
3131 T2360T+30848 T^{2} - 360T + 30848 Copy content Toggle raw display
3737 T2328T38676 T^{2} - 328T - 38676 Copy content Toggle raw display
4141 T2+782T+148128 T^{2} + 782T + 148128 Copy content Toggle raw display
4343 T2+386T+20856 T^{2} + 386T + 20856 Copy content Toggle raw display
4747 T2+266T115104 T^{2} + 266T - 115104 Copy content Toggle raw display
5353 T2522T2592 T^{2} - 522T - 2592 Copy content Toggle raw display
5959 T2+172T235104 T^{2} + 172T - 235104 Copy content Toggle raw display
6161 T2+778T+123288 T^{2} + 778T + 123288 Copy content Toggle raw display
6767 T2776T72944 T^{2} - 776T - 72944 Copy content Toggle raw display
7171 T2630T+28512 T^{2} - 630T + 28512 Copy content Toggle raw display
7373 T2+1296T+400892 T^{2} + 1296 T + 400892 Copy content Toggle raw display
7979 T2652T396572 T^{2} - 652T - 396572 Copy content Toggle raw display
8383 T2324T563904 T^{2} - 324T - 563904 Copy content Toggle raw display
8989 T2+756T+17172 T^{2} + 756T + 17172 Copy content Toggle raw display
9797 T2452T842876 T^{2} - 452T - 842876 Copy content Toggle raw display
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