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Properties

Label 4925.2.a.j
Level 49254925
Weight 22
Character orbit 4925.a
Self dual yes
Analytic conductor 39.32639.326
Analytic rank 11
Dimension 1010
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] = [4925,2,Mod(1,4925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 4925=52⋅197 4925 = 5^{2} \cdot 197
Weight: k k == 2 2
Character orbit: [χ][\chi] == 4925.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: 39.326322995539.3263229955
Analytic rank: 11
Dimension: 1010
Coefficient field: 10.10.21886214112361.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x10−2x9−9x8+16x7+26x6−38x5−27x4+32x3+6x2−7x+1 x^{10} - 2x^{9} - 9x^{8} + 16x^{7} + 26x^{6} - 38x^{5} - 27x^{4} + 32x^{3} + 6x^{2} - 7x + 1 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 985)
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 a basis 1,β1,…,β91,\beta_1,\ldots,\beta_{9} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+β5q3+(β4+β3+β1)q4+(−β4−β3+β2−1)q6+(−β9−β7−β6+⋯+1)q7+(β9−β8−β6+⋯+β1)q8+⋯+(−β9−β7+β6+⋯+1)q99+O(q100) q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{4} + \beta_{3} + \beta_1) q^{4} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \cdots + 1) q^{7} + (\beta_{9} - \beta_{8} - \beta_{6} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{9} - \beta_{7} + \beta_{6} + \cdots + 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 10q+2q2+3q3+2q4−9q6+6q7+6q8−5q9−11q11+5q13−9q14−2q16+4q17−15q18−28q19−7q21+12q22+24q23−3q24+7q26+⋯−16q99+O(q100) 10 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 9 q^{6} + 6 q^{7} + 6 q^{8} - 5 q^{9} - 11 q^{11} + 5 q^{13} - 9 q^{14} - 2 q^{16} + 4 q^{17} - 15 q^{18} - 28 q^{19} - 7 q^{21} + 12 q^{22} + 24 q^{23} - 3 q^{24} + 7 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x10−2x9−9x8+16x7+26x6−38x5−27x4+32x3+6x2−7x+1 x^{10} - 2x^{9} - 9x^{8} + 16x^{7} + 26x^{6} - 38x^{5} - 27x^{4} + 32x^{3} + 6x^{2} - 7x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν9−2ν8−8ν7+14ν6+19ν5−25ν4−15ν3+13ν2+2ν−2 \nu^{9} - 2\nu^{8} - 8\nu^{7} + 14\nu^{6} + 19\nu^{5} - 25\nu^{4} - 15\nu^{3} + 13\nu^{2} + 2\nu - 2 Copy content Toggle raw display
β3\beta_{3}== −ν9+2ν8+9ν7−16ν6−26ν5+38ν4+27ν3−31ν2−7ν+4 -\nu^{9} + 2\nu^{8} + 9\nu^{7} - 16\nu^{6} - 26\nu^{5} + 38\nu^{4} + 27\nu^{3} - 31\nu^{2} - 7\nu + 4 Copy content Toggle raw display
β4\beta_{4}== ν9−2ν8−9ν7+16ν6+26ν5−38ν4−27ν3+32ν2+6ν−6 \nu^{9} - 2\nu^{8} - 9\nu^{7} + 16\nu^{6} + 26\nu^{5} - 38\nu^{4} - 27\nu^{3} + 32\nu^{2} + 6\nu - 6 Copy content Toggle raw display
β5\beta_{5}== ν9−ν8−11ν7+8ν6+40ν5−19ν4−52ν3+17ν2+18ν−4 \nu^{9} - \nu^{8} - 11\nu^{7} + 8\nu^{6} + 40\nu^{5} - 19\nu^{4} - 52\nu^{3} + 17\nu^{2} + 18\nu - 4 Copy content Toggle raw display
β6\beta_{6}== −3ν9+5ν8+29ν7−39ν6−93ν5+87ν4+112ν3−64ν2−38ν+10 -3\nu^{9} + 5\nu^{8} + 29\nu^{7} - 39\nu^{6} - 93\nu^{5} + 87\nu^{4} + 112\nu^{3} - 64\nu^{2} - 38\nu + 10 Copy content Toggle raw display
β7\beta_{7}== 4ν9−7ν8−38ν7+55ν6+120ν5−125ν4−146ν3+95ν2+54ν−16 4\nu^{9} - 7\nu^{8} - 38\nu^{7} + 55\nu^{6} + 120\nu^{5} - 125\nu^{4} - 146\nu^{3} + 95\nu^{2} + 54\nu - 16 Copy content Toggle raw display
β8\beta_{8}== −4ν9+7ν8+38ν7−55ν6−120ν5+126ν4+145ν3−100ν2−51ν+19 -4\nu^{9} + 7\nu^{8} + 38\nu^{7} - 55\nu^{6} - 120\nu^{5} + 126\nu^{4} + 145\nu^{3} - 100\nu^{2} - 51\nu + 19 Copy content Toggle raw display
β9\beta_{9}== −8ν9+14ν8+75ν7−108ν6−232ν5+238ν4+273ν3−177ν2−96ν+31 -8\nu^{9} + 14\nu^{8} + 75\nu^{7} - 108\nu^{6} - 232\nu^{5} + 238\nu^{4} + 273\nu^{3} - 177\nu^{2} - 96\nu + 31 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β4+β3+β1+2 \beta_{4} + \beta_{3} + \beta _1 + 2 Copy content Toggle raw display
ν3\nu^{3}== β9−β8−β6+β2+5β1 \beta_{9} - \beta_{8} - \beta_{6} + \beta_{2} + 5\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β9+β7−β6+5β4+5β3+β2+7β1+7 \beta_{9} + \beta_{7} - \beta_{6} + 5\beta_{4} + 5\beta_{3} + \beta_{2} + 7\beta _1 + 7 Copy content Toggle raw display
ν5\nu^{5}== 7β9−7β8+β7−6β6+β3+7β2+26β1+2 7\beta_{9} - 7\beta_{8} + \beta_{7} - 6\beta_{6} + \beta_{3} + 7\beta_{2} + 26\beta _1 + 2 Copy content Toggle raw display
ν6\nu^{6}== 9β9−β8+9β7−7β6+β5+25β4+24β3+9β2+43β1+30 9\beta_{9} - \beta_{8} + 9\beta_{7} - 7\beta_{6} + \beta_{5} + 25\beta_{4} + 24\beta_{3} + 9\beta_{2} + 43\beta _1 + 30 Copy content Toggle raw display
ν7\nu^{7}== 42β9−39β8+12β7−31β6+2β5+3β4+9β3+⋯+17 42 \beta_{9} - 39 \beta_{8} + 12 \beta_{7} - 31 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 9 \beta_{3} + \cdots + 17 Copy content Toggle raw display
ν8\nu^{8}== 64β9−13β8+63β7−40β6+13β5+125β4+116β3+⋯+141 64 \beta_{9} - 13 \beta_{8} + 63 \beta_{7} - 40 \beta_{6} + 13 \beta_{5} + 125 \beta_{4} + 116 \beta_{3} + \cdots + 141 Copy content Toggle raw display
ν9\nu^{9}== 245β9−206β8+102β7−156β6+28β5+36β4+61β3+⋯+111 245 \beta_{9} - 206 \beta_{8} + 102 \beta_{7} - 156 \beta_{6} + 28 \beta_{5} + 36 \beta_{4} + 61 \beta_{3} + \cdots + 111 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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
−2.12691
−1.39693
−1.23959
−0.638386
0.231201
0.285611
0.888606
1.33777
2.24278
2.41585
−2.12691 2.07716 2.52376 0 −4.41794 2.06572 −1.11399 1.31459 0
1.2 −1.39693 −0.475994 −0.0485756 0 0.664932 3.39303 2.86172 −2.77343 0
1.3 −1.23959 2.93211 −0.463416 0 −3.63462 −2.16985 3.05363 5.59727 0
1.4 −0.638386 −1.45932 −1.59246 0 0.931607 2.37873 2.29338 −0.870397 0
1.5 0.231201 0.400645 −1.94655 0 0.0926295 −1.70963 −0.912444 −2.83948 0
1.6 0.285611 1.26844 −1.91843 0 0.362280 2.73433 −1.11915 −1.39107 0
1.7 0.888606 −1.66962 −1.21038 0 −1.48364 −1.49037 −2.85276 −0.212359 0
1.8 1.33777 1.51312 −0.210373 0 2.02421 1.25982 −2.95697 −0.710457 0
1.9 2.24278 −1.69515 3.03007 0 −3.80186 3.86416 2.31023 −0.126460 0
1.10 2.41585 0.108610 3.83635 0 0.262386 −4.32596 4.43635 −2.98820 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 +1 +1
197197 +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 4925.2.a.j 10
5.b even 2 1 985.2.a.e ✓ 10
15.d odd 2 1 8865.2.a.t 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
985.2.a.e ✓ 10 5.b even 2 1
4925.2.a.j 10 1.a even 1 1 trivial
8865.2.a.t 10 15.d odd 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(4925))S_{2}^{\mathrm{new}}(\Gamma_0(4925)):

T210−2T29−9T28+16T27+26T26−38T25−27T24+32T23+6T22−7T2+1 T_{2}^{10} - 2T_{2}^{9} - 9T_{2}^{8} + 16T_{2}^{7} + 26T_{2}^{6} - 38T_{2}^{5} - 27T_{2}^{4} + 32T_{2}^{3} + 6T_{2}^{2} - 7T_{2} + 1 Copy content Toggle raw display
T310−3T39−8T38+24T37+22T36−64T35−21T34+61T33+T32−10T3+1 T_{3}^{10} - 3T_{3}^{9} - 8T_{3}^{8} + 24T_{3}^{7} + 22T_{3}^{6} - 64T_{3}^{5} - 21T_{3}^{4} + 61T_{3}^{3} + T_{3}^{2} - 10T_{3} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10−2T9+⋯+1 T^{10} - 2 T^{9} + \cdots + 1 Copy content Toggle raw display
33 T10−3T9+⋯+1 T^{10} - 3 T^{9} + \cdots + 1 Copy content Toggle raw display
55 T10 T^{10} Copy content Toggle raw display
77 T10−6T9+⋯+5308 T^{10} - 6 T^{9} + \cdots + 5308 Copy content Toggle raw display
1111 T10+11T9+⋯+876 T^{10} + 11 T^{9} + \cdots + 876 Copy content Toggle raw display
1313 T10−5T9+⋯+17477 T^{10} - 5 T^{9} + \cdots + 17477 Copy content Toggle raw display
1717 T10−4T9+⋯−137988 T^{10} - 4 T^{9} + \cdots - 137988 Copy content Toggle raw display
1919 T10+28T9+⋯−25811 T^{10} + 28 T^{9} + \cdots - 25811 Copy content Toggle raw display
2323 T10−24T9+⋯−2075712 T^{10} - 24 T^{9} + \cdots - 2075712 Copy content Toggle raw display
2929 T10+16T9+⋯+98764 T^{10} + 16 T^{9} + \cdots + 98764 Copy content Toggle raw display
3131 T10+34T9+⋯−73173232 T^{10} + 34 T^{9} + \cdots - 73173232 Copy content Toggle raw display
3737 T10+5T9+⋯+2426304 T^{10} + 5 T^{9} + \cdots + 2426304 Copy content Toggle raw display
4141 T10+15T9+⋯+337193 T^{10} + 15 T^{9} + \cdots + 337193 Copy content Toggle raw display
4343 T10−5T9+⋯+2284 T^{10} - 5 T^{9} + \cdots + 2284 Copy content Toggle raw display
4747 T10−14T9+⋯+28474764 T^{10} - 14 T^{9} + \cdots + 28474764 Copy content Toggle raw display
5353 T10−15T9+⋯+317244 T^{10} - 15 T^{9} + \cdots + 317244 Copy content Toggle raw display
5959 T10+49T9+⋯+12406847 T^{10} + 49 T^{9} + \cdots + 12406847 Copy content Toggle raw display
6161 T10+⋯−1693803719 T^{10} + \cdots - 1693803719 Copy content Toggle raw display
6767 T10−2T9+⋯−6547 T^{10} - 2 T^{9} + \cdots - 6547 Copy content Toggle raw display
7171 T10+⋯+150197316 T^{10} + \cdots + 150197316 Copy content Toggle raw display
7373 T10+18T9+⋯+506257 T^{10} + 18 T^{9} + \cdots + 506257 Copy content Toggle raw display
7979 T10+11T9+⋯+74241116 T^{10} + 11 T^{9} + \cdots + 74241116 Copy content Toggle raw display
8383 T10+4T9+⋯+132 T^{10} + 4 T^{9} + \cdots + 132 Copy content Toggle raw display
8989 T10+5T9+⋯−10810148 T^{10} + 5 T^{9} + \cdots - 10810148 Copy content Toggle raw display
9797 T10+⋯−2022756524 T^{10} + \cdots - 2022756524 Copy content Toggle raw display
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