gp: [N,k,chi] = [69,2,Mod(4,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [10,4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the q q q -expansion are expressed in terms of a primitive root of unity ζ 22 \zeta_{22} ζ 2 2 .
We also show the integral q q q -expansion of the trace form .
Character values
We give the values of χ \chi χ on generators for ( Z / 69 Z ) × \left(\mathbb{Z}/69\mathbb{Z}\right)^\times ( Z / 6 9 Z ) × .
n n n
28 28 2 8
47 47 4 7
χ ( n ) \chi(n) χ ( n )
− ζ 22 -\zeta_{22} − ζ 2 2
1 1 1
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
T 2 10 − 4 T 2 9 + 16 T 2 8 − 31 T 2 7 + 47 T 2 6 − 23 T 2 5 − 18 T 2 4 + 39 T 2 3 + 31 T 2 2 + 8 T 2 + 1 T_{2}^{10} - 4T_{2}^{9} + 16T_{2}^{8} - 31T_{2}^{7} + 47T_{2}^{6} - 23T_{2}^{5} - 18T_{2}^{4} + 39T_{2}^{3} + 31T_{2}^{2} + 8T_{2} + 1 T 2 1 0 − 4 T 2 9 + 1 6 T 2 8 − 3 1 T 2 7 + 4 7 T 2 6 − 2 3 T 2 5 − 1 8 T 2 4 + 3 9 T 2 3 + 3 1 T 2 2 + 8 T 2 + 1
T2^10 - 4*T2^9 + 16*T2^8 - 31*T2^7 + 47*T2^6 - 23*T2^5 - 18*T2^4 + 39*T2^3 + 31*T2^2 + 8*T2 + 1
acting on S 2 n e w ( 69 , [ χ ] ) S_{2}^{\mathrm{new}}(69, [\chi]) S 2 n e w ( 6 9 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 10 − 4 T 9 + ⋯ + 1 T^{10} - 4 T^{9} + \cdots + 1 T 1 0 − 4 T 9 + ⋯ + 1
T^10 - 4*T^9 + 16*T^8 - 31*T^7 + 47*T^6 - 23*T^5 - 18*T^4 + 39*T^3 + 31*T^2 + 8*T + 1
3 3 3
T 10 − T 9 + ⋯ + 1 T^{10} - T^{9} + \cdots + 1 T 1 0 − T 9 + ⋯ + 1
T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
5 5 5
T 10 + 3 T 9 + ⋯ + 11881 T^{10} + 3 T^{9} + \cdots + 11881 T 1 0 + 3 T 9 + ⋯ + 1 1 8 8 1
T^10 + 3*T^9 + 9*T^8 + 71*T^7 + 290*T^6 + 782*T^5 + 3204*T^4 + 8336*T^3 + 6737*T^2 + 763*T + 11881
7 7 7
T 10 − 6 T 9 + ⋯ + 1849 T^{10} - 6 T^{9} + \cdots + 1849 T 1 0 − 6 T 9 + ⋯ + 1 8 4 9
T^10 - 6*T^9 + 36*T^8 - 95*T^7 + 240*T^6 - 472*T^5 + 93*T^4 + 597*T^3 + 576*T^2 - 1806*T + 1849
11 11 1 1
T 10 + 15 T 9 + ⋯ + 1 T^{10} + 15 T^{9} + \cdots + 1 T 1 0 + 1 5 T 9 + ⋯ + 1
T^10 + 15*T^9 + 137*T^8 + 746*T^7 + 2676*T^6 + 5754*T^5 + 7209*T^4 + 4856*T^3 + 1384*T^2 + 14*T + 1
13 13 1 3
T 10 − 8 T 9 + ⋯ + 1 T^{10} - 8 T^{9} + \cdots + 1 T 1 0 − 8 T 9 + ⋯ + 1
T^10 - 8*T^9 + 31*T^8 - 39*T^7 - 18*T^6 + 23*T^5 + 47*T^4 + 31*T^3 + 16*T^2 + 4*T + 1
17 17 1 7
T 10 − T 9 + ⋯ + 39601 T^{10} - T^{9} + \cdots + 39601 T 1 0 − T 9 + ⋯ + 3 9 6 0 1
T^10 - T^9 + T^8 - 111*T^7 + 111*T^6 + 615*T^5 + 2190*T^4 - 2465*T^3 + 24740*T^2 - 35223*T + 39601
19 19 1 9
T 10 + 9 T 9 + ⋯ + 109561 T^{10} + 9 T^{9} + \cdots + 109561 T 1 0 + 9 T 9 + ⋯ + 1 0 9 5 6 1
T^10 + 9*T^9 + 48*T^8 + 124*T^7 - 28*T^6 + 529*T^5 + 3573*T^4 - 7718*T^3 + 42056*T^2 - 96652*T + 109561
23 23 2 3
T 10 − 21 T 9 + ⋯ + 6436343 T^{10} - 21 T^{9} + \cdots + 6436343 T 1 0 − 2 1 T 9 + ⋯ + 6 4 3 6 3 4 3
T^10 - 21*T^9 + 221*T^8 - 1627*T^7 + 9505*T^6 - 47937*T^5 + 218615*T^4 - 860683*T^3 + 2688907*T^2 - 5876661*T + 6436343
29 29 2 9
T 10 + 8 T 9 + ⋯ + 109561 T^{10} + 8 T^{9} + \cdots + 109561 T 1 0 + 8 T 9 + ⋯ + 1 0 9 5 6 1
T^10 + 8*T^9 + 108*T^8 + 743*T^7 + 4272*T^6 + 27103*T^5 + 115052*T^4 + 270569*T^3 + 375303*T^2 + 293597*T + 109561
31 31 3 1
T 10 + 23 T 9 + ⋯ + 11881 T^{10} + 23 T^{9} + \cdots + 11881 T 1 0 + 2 3 T 9 + ⋯ + 1 1 8 8 1
T^10 + 23*T^9 + 298*T^8 + 2487*T^7 + 14697*T^6 + 52383*T^5 + 95085*T^4 - 99824*T^3 + 12035*T^2 - 12099*T + 11881
37 37 3 7
T 10 − 3 T 9 + ⋯ + 4190209 T^{10} - 3 T^{9} + \cdots + 4190209 T 1 0 − 3 T 9 + ⋯ + 4 1 9 0 2 0 9
T^10 - 3*T^9 - 57*T^8 - 137*T^7 + 2919*T^6 - 3268*T^5 + 88102*T^4 - 144516*T^3 + 1033785*T^2 - 2034718*T + 4190209
41 41 4 1
T 10 + ⋯ + 203889841 T^{10} + \cdots + 203889841 T 1 0 + ⋯ + 2 0 3 8 8 9 8 4 1
T^10 + 15*T^9 + 137*T^8 + 372*T^7 + 2896*T^6 + 4038*T^5 + 13094*T^4 - 1042685*T^3 + 3215309*T^2 - 34826481*T + 203889841
43 43 4 3
T 10 − 22 T 9 + ⋯ + 1437601 T^{10} - 22 T^{9} + \cdots + 1437601 T 1 0 − 2 2 T 9 + ⋯ + 1 4 3 7 6 0 1
T^10 - 22*T^9 + 220*T^8 - 759*T^7 - 660*T^6 - 6633*T^5 + 55176*T^4 + 254947*T^3 + 735317*T^2 + 1147443*T + 1437601
47 47 4 7
( T 5 − 2 T 4 + ⋯ − 13133 ) 2 (T^{5} - 2 T^{4} + \cdots - 13133)^{2} ( T 5 − 2 T 4 + ⋯ − 1 3 1 3 3 ) 2
(T^5 - 2*T^4 - 148*T^3 + 365*T^2 + 5064*T - 13133)^2
53 53 5 3
T 10 − 29 T 9 + ⋯ + 55965361 T^{10} - 29 T^{9} + \cdots + 55965361 T 1 0 − 2 9 T 9 + ⋯ + 5 5 9 6 5 3 6 1
T^10 - 29*T^9 + 434*T^8 - 5634*T^7 + 73296*T^6 - 747889*T^5 + 4803407*T^4 - 16284428*T^3 + 43238776*T^2 - 69678034*T + 55965361
59 59 5 9
T 10 + ⋯ + 474411961 T^{10} + \cdots + 474411961 T 1 0 + ⋯ + 4 7 4 4 1 1 9 6 1
T^10 + 54*T^9 + 1431*T^8 + 23847*T^7 + 270832*T^6 + 2150587*T^5 + 11903068*T^4 + 45244121*T^3 + 133825286*T^2 + 314321611*T + 474411961
61 61 6 1
T 10 + 30 T 9 + ⋯ + 591361 T^{10} + 30 T^{9} + \cdots + 591361 T 1 0 + 3 0 T 9 + ⋯ + 5 9 1 3 6 1
T^10 + 30*T^9 + 504*T^8 + 5440*T^7 + 46193*T^6 + 186856*T^5 + 921198*T^4 + 2908798*T^3 + 15157235*T^2 - 5842093*T + 591361
67 67 6 7
T 10 − T 9 + ⋯ + 6285049 T^{10} - T^{9} + \cdots + 6285049 T 1 0 − T 9 + ⋯ + 6 2 8 5 0 4 9
T^10 - T^9 - 32*T^8 + 131*T^7 + 5479*T^6 - 103357*T^5 + 496189*T^4 - 1085206*T^3 + 11220727*T^2 + 10757537*T + 6285049
71 71 7 1
T 10 + 3 T 9 + ⋯ + 21743569 T^{10} + 3 T^{9} + \cdots + 21743569 T 1 0 + 3 T 9 + ⋯ + 2 1 7 4 3 5 6 9
T^10 + 3*T^9 + 163*T^8 - 941*T^7 + 10443*T^6 - 41381*T^5 + 40406*T^4 + 748405*T^3 - 1903248*T^2 - 1916493*T + 21743569
73 73 7 3
T 10 + 47 T 9 + ⋯ + 58081 T^{10} + 47 T^{9} + \cdots + 58081 T 1 0 + 4 7 T 9 + ⋯ + 5 8 0 8 1
T^10 + 47*T^9 + 1197*T^8 + 18892*T^7 + 201700*T^6 + 1422433*T^5 + 6224023*T^4 + 15303594*T^3 + 16654951*T^2 - 228950*T + 58081
79 79 7 9
T 10 − 18 T 9 + ⋯ + 34774609 T^{10} - 18 T^{9} + \cdots + 34774609 T 1 0 − 1 8 T 9 + ⋯ + 3 4 7 7 4 6 0 9
T^10 - 18*T^9 + 16*T^8 - 783*T^7 + 21189*T^6 + 100464*T^5 + 965980*T^4 + 1218145*T^3 + 7610920*T^2 - 29237326*T + 34774609
83 83 8 3
T 10 + ⋯ + 141681409 T^{10} + \cdots + 141681409 T 1 0 + ⋯ + 1 4 1 6 8 1 4 0 9
T^10 - 18*T^9 + 335*T^8 - 5535*T^7 + 79346*T^6 - 810501*T^5 + 6243582*T^4 - 33988191*T^3 + 127482474*T^2 - 222026659*T + 141681409
89 89 8 9
T 10 − 25 T 9 + ⋯ + 69169 T^{10} - 25 T^{9} + \cdots + 69169 T 1 0 − 2 5 T 9 + ⋯ + 6 9 1 6 9
T^10 - 25*T^9 + 438*T^8 - 4009*T^7 + 23621*T^6 - 73646*T^5 + 151704*T^4 - 189693*T^3 + 110555*T^2 + 238278*T + 69169
97 97 9 7
T 10 − 21 T 9 + ⋯ + 39601 T^{10} - 21 T^{9} + \cdots + 39601 T 1 0 − 2 1 T 9 + ⋯ + 3 9 6 0 1
T^10 - 21*T^9 + 210*T^8 - 1440*T^7 + 6755*T^6 - 17940*T^5 + 43055*T^4 + 57432*T^3 + 359844*T^2 - 203378*T + 39601
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