Properties

Label 1875.2.b.d
Level 18751875
Weight 22
Character orbit 1875.b
Analytic conductor 14.97214.972
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1875,2,Mod(1249,1875)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1875.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1875=354 1875 = 3 \cdot 5^{4}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1875.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 14.971950379014.9719503790
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.324000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8+9x6+26x4+24x2+1 x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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+(β7β3)q2+β5q3β4q4+β2q6+(β7+2β5+2β3)q7+(β7+β5)q8q9+(β6β4β22)q11++(β6+β4+β2+2)q99+O(q100) q + ( - \beta_{7} - \beta_{3}) q^{2} + \beta_{5} q^{3} - \beta_{4} q^{4} + \beta_{2} q^{6} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{7} + ( - \beta_{7} + \beta_{5}) q^{8} - q^{9} + (\beta_{6} - \beta_{4} - \beta_{2} - 2) q^{11}+ \cdots + ( - \beta_{6} + \beta_{4} + \beta_{2} + 2) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q2q42q68q912q11+20q1418q16+18q1910q216q244q26+56q2920q3114q34+2q36+14q39+18q44+10q46+6q49++12q99+O(q100) 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+9x6+26x4+24x2+1 x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2+2 \nu^{2} + 2 Copy content Toggle raw display
β3\beta_{3}== ν3+3ν \nu^{3} + 3\nu Copy content Toggle raw display
β4\beta_{4}== ν4+4ν2+2 \nu^{4} + 4\nu^{2} + 2 Copy content Toggle raw display
β5\beta_{5}== ν5+5ν3+5ν \nu^{5} + 5\nu^{3} + 5\nu Copy content Toggle raw display
β6\beta_{6}== ν6+6ν4+9ν2+2 \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 Copy content Toggle raw display
β7\beta_{7}== ν7+7ν5+14ν3+7ν \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β22 \beta_{2} - 2 Copy content Toggle raw display
ν3\nu^{3}== β33β1 \beta_{3} - 3\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β44β2+6 \beta_{4} - 4\beta_{2} + 6 Copy content Toggle raw display
ν5\nu^{5}== β55β3+10β1 \beta_{5} - 5\beta_{3} + 10\beta_1 Copy content Toggle raw display
ν6\nu^{6}== β66β4+15β220 \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 Copy content Toggle raw display
ν7\nu^{7}== β77β5+21β335β1 \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1875Z)×\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times.

nn 626626 12521252
χ(n)\chi(n) 11 1-1

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
1249.1
0.209057i
1.95630i
1.82709i
1.33826i
1.33826i
1.82709i
1.95630i
0.209057i
1.95630i 1.00000i −1.82709 0 1.95630 4.57433i 0.338261i −1.00000 0
1249.2 1.82709i 1.00000i −1.33826 0 −1.82709 1.44512i 1.20906i −1.00000 0
1249.3 1.33826i 1.00000i 0.209057 0 −1.33826 1.27977i 2.95630i −1.00000 0
1249.4 0.209057i 1.00000i 1.95630 0 0.209057 0.591023i 0.827091i −1.00000 0
1249.5 0.209057i 1.00000i 1.95630 0 0.209057 0.591023i 0.827091i −1.00000 0
1249.6 1.33826i 1.00000i 0.209057 0 −1.33826 1.27977i 2.95630i −1.00000 0
1249.7 1.82709i 1.00000i −1.33826 0 −1.82709 1.44512i 1.20906i −1.00000 0
1249.8 1.95630i 1.00000i −1.82709 0 1.95630 4.57433i 0.338261i −1.00000 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
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 1875.2.b.d 8
5.b even 2 1 inner 1875.2.b.d 8
5.c odd 4 1 1875.2.a.f 4
5.c odd 4 1 1875.2.a.g yes 4
15.e even 4 1 5625.2.a.j 4
15.e even 4 1 5625.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.f 4 5.c odd 4 1
1875.2.a.g yes 4 5.c odd 4 1
1875.2.b.d 8 1.a even 1 1 trivial
1875.2.b.d 8 5.b even 2 1 inner
5625.2.a.j 4 15.e even 4 1
5625.2.a.m 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T28+9T26+26T24+24T22+1 T_{2}^{8} + 9T_{2}^{6} + 26T_{2}^{4} + 24T_{2}^{2} + 1 acting on S2new(1875,[χ])S_{2}^{\mathrm{new}}(1875, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+9T6++1 T^{8} + 9 T^{6} + \cdots + 1 Copy content Toggle raw display
33 (T2+1)4 (T^{2} + 1)^{4} Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 T8+25T6++25 T^{8} + 25 T^{6} + \cdots + 25 Copy content Toggle raw display
1111 (T4+6T3+6T2+9)2 (T^{4} + 6 T^{3} + 6 T^{2} + \cdots - 9)^{2} Copy content Toggle raw display
1313 T8+61T6++7921 T^{8} + 61 T^{6} + \cdots + 7921 Copy content Toggle raw display
1717 T8+81T6++128881 T^{8} + 81 T^{6} + \cdots + 128881 Copy content Toggle raw display
1919 (T49T3+6T2+9)2 (T^{4} - 9 T^{3} + 6 T^{2} + \cdots - 9)^{2} Copy content Toggle raw display
2323 T8+60T6++25 T^{8} + 60 T^{6} + \cdots + 25 Copy content Toggle raw display
2929 (T428T3++1801)2 (T^{4} - 28 T^{3} + \cdots + 1801)^{2} Copy content Toggle raw display
3131 (T4+10T3+125)2 (T^{4} + 10 T^{3} + \cdots - 125)^{2} Copy content Toggle raw display
3737 T8+280T6++2175625 T^{8} + 280 T^{6} + \cdots + 2175625 Copy content Toggle raw display
4141 (T470T2++145)2 (T^{4} - 70 T^{2} + \cdots + 145)^{2} Copy content Toggle raw display
4343 T8+159T6++175561 T^{8} + 159 T^{6} + \cdots + 175561 Copy content Toggle raw display
4747 T8+351T6++37075921 T^{8} + 351 T^{6} + \cdots + 37075921 Copy content Toggle raw display
5353 T8+290T6++8970025 T^{8} + 290 T^{6} + \cdots + 8970025 Copy content Toggle raw display
5959 (T4+4T3++1531)2 (T^{4} + 4 T^{3} + \cdots + 1531)^{2} Copy content Toggle raw display
6161 (T4+43T3++10261)2 (T^{4} + 43 T^{3} + \cdots + 10261)^{2} Copy content Toggle raw display
6767 T8+186T6++22801 T^{8} + 186 T^{6} + \cdots + 22801 Copy content Toggle raw display
7171 (T4+27T3++271)2 (T^{4} + 27 T^{3} + \cdots + 271)^{2} Copy content Toggle raw display
7373 T8+345T6++7535025 T^{8} + 345 T^{6} + \cdots + 7535025 Copy content Toggle raw display
7979 (T4+10T3+3155)2 (T^{4} + 10 T^{3} + \cdots - 3155)^{2} Copy content Toggle raw display
8383 T8+501T6++182007081 T^{8} + 501 T^{6} + \cdots + 182007081 Copy content Toggle raw display
8989 (T49T34T2++61)2 (T^{4} - 9 T^{3} - 4 T^{2} + \cdots + 61)^{2} Copy content Toggle raw display
9797 T8+601T6++216119401 T^{8} + 601 T^{6} + \cdots + 216119401 Copy content Toggle raw display
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