Properties

Label 825.2.n.l
Level 825825
Weight 22
Character orbit 825.n
Analytic conductor 6.5886.588
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(301,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 825=35211 825 = 3 \cdot 5^{2} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 825.n (of order 55, degree 44, 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: 6.587658166766.58765816676
Analytic rank: 00
Dimension: 88
Relative dimension: 22 over Q(ζ5)\Q(\zeta_{5})
Coefficient field: 8.0.819390625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x83x7+10x613x5+29x47x3+80x2+143x+121 x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C5]\mathrm{SU}(2)[C_{5}]

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,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β6β3β21)q3+(β6β4+β3+1)q4+β7q6+(β7β5++2β1)q7+(2β6β4+2β3+1)q8++(β6+β3+β2+3)q99+O(q100) q + \beta_1 q^{2} + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{3} + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots - 1) q^{4} + \beta_{7} q^{6} + (\beta_{7} - \beta_{5} + \cdots + 2 \beta_1) q^{7} + ( - 2 \beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{8}+ \cdots + (\beta_{6} + \beta_{3} + \beta_{2} + 3) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+3q22q37q4+3q6+7q716q82q9+2q11+18q125q132q1411q16+8q172q185q198q218q222q23+19q24++22q99+O(q100) 8 q + 3 q^{2} - 2 q^{3} - 7 q^{4} + 3 q^{6} + 7 q^{7} - 16 q^{8} - 2 q^{9} + 2 q^{11} + 18 q^{12} - 5 q^{13} - 2 q^{14} - 11 q^{16} + 8 q^{17} - 2 q^{18} - 5 q^{19} - 8 q^{21} - 8 q^{22} - 2 q^{23} + 19 q^{24}+ \cdots + 22 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x83x7+10x613x5+29x47x3+80x2+143x+121 x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (256ν7+341ν6+3310ν516865ν4+32996ν359433ν2+33270ν118459)/171589 ( -256\nu^{7} + 341\nu^{6} + 3310\nu^{5} - 16865\nu^{4} + 32996\nu^{3} - 59433\nu^{2} + 33270\nu - 118459 ) / 171589 Copy content Toggle raw display
β3\beta_{3}== (620ν7+2532ν69045ν5+18870ν441955ν3+81515ν2102225ν16104)/171589 ( -620\nu^{7} + 2532\nu^{6} - 9045\nu^{5} + 18870\nu^{4} - 41955\nu^{3} + 81515\nu^{2} - 102225\nu - 16104 ) / 171589 Copy content Toggle raw display
β4\beta_{4}== (672ν7+2845ν610810ν5+23975ν477175ν3+52625ν272556ν75020)/171589 ( -672\nu^{7} + 2845\nu^{6} - 10810\nu^{5} + 23975\nu^{4} - 77175\nu^{3} + 52625\nu^{2} - 72556\nu - 75020 ) / 171589 Copy content Toggle raw display
β5\beta_{5}== (1187ν7+4445ν617191ν5+14238ν412015ν332510ν2+143748)/171589 ( - 1187 \nu^{7} + 4445 \nu^{6} - 17191 \nu^{5} + 14238 \nu^{4} - 12015 \nu^{3} - 32510 \nu^{2} + \cdots - 143748 ) / 171589 Copy content Toggle raw display
β6\beta_{6}== (1188ν74751ν6+16325ν532635ν4+48690ν320331ν2++144881)/171589 ( 1188 \nu^{7} - 4751 \nu^{6} + 16325 \nu^{5} - 32635 \nu^{4} + 48690 \nu^{3} - 20331 \nu^{2} + \cdots + 144881 ) / 171589 Copy content Toggle raw display
β7\beta_{7}== (1432ν7+1420ν67808ν52207ν427965ν333635ν2+249744)/171589 ( - 1432 \nu^{7} + 1420 \nu^{6} - 7808 \nu^{5} - 2207 \nu^{4} - 27965 \nu^{3} - 33635 \nu^{2} + \cdots - 249744 ) / 171589 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6β4+3β3+β11 \beta_{6} - \beta_{4} + 3\beta_{3} + \beta _1 - 1 Copy content Toggle raw display
ν3\nu^{3}== 2β65β4+2β3β21 -2\beta_{6} - 5\beta_{4} + 2\beta_{3} - \beta_{2} - 1 Copy content Toggle raw display
ν4\nu^{4}== β715β68β58β45β35β2 \beta_{7} - 15\beta_{6} - 8\beta_{5} - 8\beta_{4} - 5\beta_{3} - 5\beta_{2} Copy content Toggle raw display
ν5\nu^{5}== 2β725β628β5+25β22β1+12 -2\beta_{7} - 25\beta_{6} - 28\beta_{5} + 25\beta_{2} - 2\beta _1 + 12 Copy content Toggle raw display
ν6\nu^{6}== 55β7+55β4+22β3+110β245β1+22 -55\beta_{7} + 55\beta_{4} + 22\beta_{3} + 110\beta_{2} - 45\beta _1 + 22 Copy content Toggle raw display
ν7\nu^{7}== 165β7+175β6+165β5+188β480β3188β1175 -165\beta_{7} + 175\beta_{6} + 165\beta_{5} + 188\beta_{4} - 80\beta_{3} - 188\beta _1 - 175 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/825Z)×\left(\mathbb{Z}/825\mathbb{Z}\right)^\times.

nn 376376 551551 727727
χ(n)\chi(n) β3\beta_{3} 11 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}
301.1
−0.755243 0.548716i
2.06426 + 1.49977i
−0.575405 + 1.77091i
0.766388 2.35870i
−0.575405 1.77091i
0.766388 + 2.35870i
−0.755243 + 0.548716i
2.06426 1.49977i
−0.755243 0.548716i 0.309017 0.951057i −0.348732 1.07329i 0 −0.755243 + 0.548716i −1.37328 4.22651i −0.902506 + 2.77763i −0.809017 0.587785i 0
301.2 2.06426 + 1.49977i 0.309017 0.951057i 1.39382 + 4.28973i 0 2.06426 1.49977i 1.44623 + 4.45102i −1.97946 + 6.09215i −0.809017 0.587785i 0
526.1 −0.575405 + 1.77091i −0.809017 + 0.587785i −1.18701 0.862413i 0 −0.575405 1.77091i 1.04263 + 0.757515i −0.802588 + 0.583114i 0.309017 0.951057i 0
526.2 0.766388 2.35870i −0.809017 + 0.587785i −3.35808 2.43978i 0 0.766388 + 2.35870i 2.38442 + 1.73238i −4.31545 + 3.13535i 0.309017 0.951057i 0
676.1 −0.575405 1.77091i −0.809017 0.587785i −1.18701 + 0.862413i 0 −0.575405 + 1.77091i 1.04263 0.757515i −0.802588 0.583114i 0.309017 + 0.951057i 0
676.2 0.766388 + 2.35870i −0.809017 0.587785i −3.35808 + 2.43978i 0 0.766388 2.35870i 2.38442 1.73238i −4.31545 3.13535i 0.309017 + 0.951057i 0
751.1 −0.755243 + 0.548716i 0.309017 + 0.951057i −0.348732 + 1.07329i 0 −0.755243 0.548716i −1.37328 + 4.22651i −0.902506 2.77763i −0.809017 + 0.587785i 0
751.2 2.06426 1.49977i 0.309017 + 0.951057i 1.39382 4.28973i 0 2.06426 + 1.49977i 1.44623 4.45102i −1.97946 6.09215i −0.809017 + 0.587785i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.l yes 8
5.b even 2 1 825.2.n.h 8
5.c odd 4 2 825.2.bx.i 16
11.c even 5 1 inner 825.2.n.l yes 8
11.c even 5 1 9075.2.a.cp 4
11.d odd 10 1 9075.2.a.dh 4
55.h odd 10 1 9075.2.a.cn 4
55.j even 10 1 825.2.n.h 8
55.j even 10 1 9075.2.a.de 4
55.k odd 20 2 825.2.bx.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.h 8 5.b even 2 1
825.2.n.h 8 55.j even 10 1
825.2.n.l yes 8 1.a even 1 1 trivial
825.2.n.l yes 8 11.c even 5 1 inner
825.2.bx.i 16 5.c odd 4 2
825.2.bx.i 16 55.k odd 20 2
9075.2.a.cn 4 55.h odd 10 1
9075.2.a.cp 4 11.c even 5 1
9075.2.a.de 4 55.j even 10 1
9075.2.a.dh 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(825,[χ])S_{2}^{\mathrm{new}}(825, [\chi]):

T283T27+10T2613T25+29T247T23+80T22+143T2+121 T_{2}^{8} - 3T_{2}^{7} + 10T_{2}^{6} - 13T_{2}^{5} + 29T_{2}^{4} - 7T_{2}^{3} + 80T_{2}^{2} + 143T_{2} + 121 Copy content Toggle raw display
T138+5T137+38T136+255T135+1069T134+2185T133+2722T132+2015T13+961 T_{13}^{8} + 5T_{13}^{7} + 38T_{13}^{6} + 255T_{13}^{5} + 1069T_{13}^{4} + 2185T_{13}^{3} + 2722T_{13}^{2} + 2015T_{13} + 961 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T83T7++121 T^{8} - 3 T^{7} + \cdots + 121 Copy content Toggle raw display
33 (T4+T3+T2++1)2 (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 T87T7++6241 T^{8} - 7 T^{7} + \cdots + 6241 Copy content Toggle raw display
1111 (T4T39T2++121)2 (T^{4} - T^{3} - 9 T^{2} + \cdots + 121)^{2} Copy content Toggle raw display
1313 T8+5T7++961 T^{8} + 5 T^{7} + \cdots + 961 Copy content Toggle raw display
1717 T88T7++9801 T^{8} - 8 T^{7} + \cdots + 9801 Copy content Toggle raw display
1919 T8+5T7++6561 T^{8} + 5 T^{7} + \cdots + 6561 Copy content Toggle raw display
2323 (T4+T339T2++341)2 (T^{4} + T^{3} - 39 T^{2} + \cdots + 341)^{2} Copy content Toggle raw display
2929 T8+3T7++10201 T^{8} + 3 T^{7} + \cdots + 10201 Copy content Toggle raw display
3131 T85T7++1 T^{8} - 5 T^{7} + \cdots + 1 Copy content Toggle raw display
3737 T8+19T7++116281 T^{8} + 19 T^{7} + \cdots + 116281 Copy content Toggle raw display
4141 T8+18T7++32761 T^{8} + 18 T^{7} + \cdots + 32761 Copy content Toggle raw display
4343 (T426T3+1089)2 (T^{4} - 26 T^{3} + \cdots - 1089)^{2} Copy content Toggle raw display
4747 T818T7++17161 T^{8} - 18 T^{7} + \cdots + 17161 Copy content Toggle raw display
5353 T831T7++101761 T^{8} - 31 T^{7} + \cdots + 101761 Copy content Toggle raw display
5959 T8+2T7++9801 T^{8} + 2 T^{7} + \cdots + 9801 Copy content Toggle raw display
6161 T8+22T7++8202496 T^{8} + 22 T^{7} + \cdots + 8202496 Copy content Toggle raw display
6767 (T4+19T3++2351)2 (T^{4} + 19 T^{3} + \cdots + 2351)^{2} Copy content Toggle raw display
7171 T8+303T6++138039001 T^{8} + 303 T^{6} + \cdots + 138039001 Copy content Toggle raw display
7373 T8+18T7++1234321 T^{8} + 18 T^{7} + \cdots + 1234321 Copy content Toggle raw display
7979 T8+18T7++60543961 T^{8} + 18 T^{7} + \cdots + 60543961 Copy content Toggle raw display
8383 T8+28T7++83156161 T^{8} + 28 T^{7} + \cdots + 83156161 Copy content Toggle raw display
8989 (T417T3++99)2 (T^{4} - 17 T^{3} + \cdots + 99)^{2} Copy content Toggle raw display
9797 T8+11T7++19321 T^{8} + 11 T^{7} + \cdots + 19321 Copy content Toggle raw display
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