Properties

Label 2925.2.c.r
Level 29252925
Weight 22
Character orbit 2925.c
Analytic conductor 23.35623.356
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2925,2,Mod(2224,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2925.2224");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2925=325213 2925 = 3^{2} \cdot 5^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2925.c (of order 22, degree 11, not minimal)

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: no
Analytic conductor: 23.356242591223.3562425912
Analytic rank: 00
Dimension: 44
Coefficient field: Q(ζ8)\Q(\zeta_{8})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+1 x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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 a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β2+β1)q2+(2β31)q4+(2β2+2β1)q7+(β23β1)q8+(β32)q11+β1q13+(4β36)q14++(13β221β1)q98+O(q100) q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 1) q^{4} + (2 \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} + ( - \beta_{3} - 2) q^{11} + \beta_1 q^{13} + ( - 4 \beta_{3} - 6) q^{14}+ \cdots + ( - 13 \beta_{2} - 21 \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q48q1124q14+12q168q194q26+24q31+8q34+24q41+24q448q4620q49+40q56+24q5932q61+28q648q71+48q74+24q94+O(q100) 4 q - 4 q^{4} - 8 q^{11} - 24 q^{14} + 12 q^{16} - 8 q^{19} - 4 q^{26} + 24 q^{31} + 8 q^{34} + 24 q^{41} + 24 q^{44} - 8 q^{46} - 20 q^{49} + 40 q^{56} + 24 q^{59} - 32 q^{61} + 28 q^{64} - 8 q^{71} + 48 q^{74}+ \cdots - 24 q^{94}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring

β1\beta_{1}== ζ82 \zeta_{8}^{2} Copy content Toggle raw display
β2\beta_{2}== ζ83+ζ8 \zeta_{8}^{3} + \zeta_{8} Copy content Toggle raw display
β3\beta_{3}== ζ83+ζ8 -\zeta_{8}^{3} + \zeta_{8} Copy content Toggle raw display
ζ8\zeta_{8}== (β3+β2)/2 ( \beta_{3} + \beta_{2} ) / 2 Copy content Toggle raw display
ζ82\zeta_{8}^{2}== β1 \beta_1 Copy content Toggle raw display
ζ83\zeta_{8}^{3}== (β3+β2)/2 ( -\beta_{3} + \beta_{2} ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2925Z)×\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times.

nn 326326 352352 22512251
χ(n)\chi(n) 11 1-1 11

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}
2224.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 0 −3.82843 0 0 4.82843i 4.41421i 0 0
2224.2 0.414214i 0 1.82843 0 0 0.828427i 1.58579i 0 0
2224.3 0.414214i 0 1.82843 0 0 0.828427i 1.58579i 0 0
2224.4 2.41421i 0 −3.82843 0 0 4.82843i 4.41421i 0 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2925.2.c.r 4
3.b odd 2 1 325.2.b.f 4
5.b even 2 1 inner 2925.2.c.r 4
5.c odd 4 1 585.2.a.m 2
5.c odd 4 1 2925.2.a.u 2
15.d odd 2 1 325.2.b.f 4
15.e even 4 1 65.2.a.b 2
15.e even 4 1 325.2.a.i 2
20.e even 4 1 9360.2.a.cd 2
60.l odd 4 1 1040.2.a.j 2
60.l odd 4 1 5200.2.a.bu 2
65.h odd 4 1 7605.2.a.x 2
105.k odd 4 1 3185.2.a.j 2
120.q odd 4 1 4160.2.a.z 2
120.w even 4 1 4160.2.a.bf 2
165.l odd 4 1 7865.2.a.j 2
195.j odd 4 1 845.2.c.b 4
195.s even 4 1 845.2.a.g 2
195.s even 4 1 4225.2.a.r 2
195.u odd 4 1 845.2.c.b 4
195.bc odd 12 2 845.2.m.f 8
195.bf even 12 2 845.2.e.c 4
195.bl even 12 2 845.2.e.h 4
195.bn odd 12 2 845.2.m.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 15.e even 4 1
325.2.a.i 2 15.e even 4 1
325.2.b.f 4 3.b odd 2 1
325.2.b.f 4 15.d odd 2 1
585.2.a.m 2 5.c odd 4 1
845.2.a.g 2 195.s even 4 1
845.2.c.b 4 195.j odd 4 1
845.2.c.b 4 195.u odd 4 1
845.2.e.c 4 195.bf even 12 2
845.2.e.h 4 195.bl even 12 2
845.2.m.f 8 195.bc odd 12 2
845.2.m.f 8 195.bn odd 12 2
1040.2.a.j 2 60.l odd 4 1
2925.2.a.u 2 5.c odd 4 1
2925.2.c.r 4 1.a even 1 1 trivial
2925.2.c.r 4 5.b even 2 1 inner
3185.2.a.j 2 105.k odd 4 1
4160.2.a.z 2 120.q odd 4 1
4160.2.a.bf 2 120.w even 4 1
4225.2.a.r 2 195.s even 4 1
5200.2.a.bu 2 60.l odd 4 1
7605.2.a.x 2 65.h odd 4 1
7865.2.a.j 2 165.l odd 4 1
9360.2.a.cd 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(2925,[χ])S_{2}^{\mathrm{new}}(2925, [\chi]):

T24+6T22+1 T_{2}^{4} + 6T_{2}^{2} + 1 Copy content Toggle raw display
T74+24T72+16 T_{7}^{4} + 24T_{7}^{2} + 16 Copy content Toggle raw display
T112+4T11+2 T_{11}^{2} + 4T_{11} + 2 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+6T2+1 T^{4} + 6T^{2} + 1 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 T4+24T2+16 T^{4} + 24T^{2} + 16 Copy content Toggle raw display
1111 (T2+4T+2)2 (T^{2} + 4 T + 2)^{2} Copy content Toggle raw display
1313 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
1717 T4+24T2+16 T^{4} + 24T^{2} + 16 Copy content Toggle raw display
1919 (T2+4T+2)2 (T^{2} + 4 T + 2)^{2} Copy content Toggle raw display
2323 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
2929 (T232)2 (T^{2} - 32)^{2} Copy content Toggle raw display
3131 (T212T+18)2 (T^{2} - 12 T + 18)^{2} Copy content Toggle raw display
3737 (T2+72)2 (T^{2} + 72)^{2} Copy content Toggle raw display
4141 (T212T+28)2 (T^{2} - 12 T + 28)^{2} Copy content Toggle raw display
4343 T4+132T2+1156 T^{4} + 132T^{2} + 1156 Copy content Toggle raw display
4747 T4+24T2+16 T^{4} + 24T^{2} + 16 Copy content Toggle raw display
5353 T4+216T2+1296 T^{4} + 216T^{2} + 1296 Copy content Toggle raw display
5959 (T212T+18)2 (T^{2} - 12 T + 18)^{2} Copy content Toggle raw display
6161 (T+8)4 (T + 8)^{4} Copy content Toggle raw display
6767 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
7171 (T2+4T94)2 (T^{2} + 4 T - 94)^{2} Copy content Toggle raw display
7373 (T2+72)2 (T^{2} + 72)^{2} Copy content Toggle raw display
7979 (T272)2 (T^{2} - 72)^{2} Copy content Toggle raw display
8383 T4+88T2+784 T^{4} + 88T^{2} + 784 Copy content Toggle raw display
8989 (T6)4 (T - 6)^{4} Copy content Toggle raw display
9797 T4+72T2+784 T^{4} + 72T^{2} + 784 Copy content Toggle raw display
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