gp: [N,k,chi] = [650,6,Mod(1,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
sage: traces = [4,-16,-20,64,0,80,-62]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 basis 1 , β 1 , β 2 , β 3 1,\beta_1,\beta_2,\beta_3 1 , β 1 , β 2 , β 3 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 4 − 762 x 2 − 4316 x + 63405 x^{4} - 762x^{2} - 4316x + 63405 x 4 − 7 6 2 x 2 − 4 3 1 6 x + 6 3 4 0 5
x^4 - 762*x^2 - 4316*x + 63405
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 3 − 13 ν 2 − 485 ν + 1713 ) / 6 ( \nu^{3} - 13\nu^{2} - 485\nu + 1713 ) / 6 ( ν 3 − 1 3 ν 2 − 4 8 5 ν + 1 7 1 3 ) / 6
(v^3 - 13*v^2 - 485*v + 1713) / 6
β 3 \beta_{3} β 3 = = =
( ν 3 − 19 ν 2 − 437 ν + 3999 ) / 6 ( \nu^{3} - 19\nu^{2} - 437\nu + 3999 ) / 6 ( ν 3 − 1 9 ν 2 − 4 3 7 ν + 3 9 9 9 ) / 6
(v^3 - 19*v^2 - 437*v + 3999) / 6
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
− β 3 + β 2 + 8 β 1 + 381 -\beta_{3} + \beta_{2} + 8\beta _1 + 381 − β 3 + β 2 + 8 β 1 + 3 8 1
-b3 + b2 + 8*b1 + 381
ν 3 \nu^{3} ν 3 = = =
− 13 β 3 + 19 β 2 + 589 β 1 + 3240 -13\beta_{3} + 19\beta_{2} + 589\beta _1 + 3240 − 1 3 β 3 + 1 9 β 2 + 5 8 9 β 1 + 3 2 4 0
-13*b3 + 19*b2 + 589*b1 + 3240
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
p p p
Sign
2 2 2
+ 1 +1 + 1
5 5 5
+ 1 +1 + 1
13 13 1 3
+ 1 +1 + 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 3 4 + 20 T 3 3 − 612 T 3 2 − 11436 T 3 + 23400 T_{3}^{4} + 20T_{3}^{3} - 612T_{3}^{2} - 11436T_{3} + 23400 T 3 4 + 2 0 T 3 3 − 6 1 2 T 3 2 − 1 1 4 3 6 T 3 + 2 3 4 0 0
T3^4 + 20*T3^3 - 612*T3^2 - 11436*T3 + 23400
acting on S 6 n e w ( Γ 0 ( 650 ) ) S_{6}^{\mathrm{new}}(\Gamma_0(650)) S 6 n e w ( Γ 0 ( 6 5 0 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T + 4 ) 4 (T + 4)^{4} ( T + 4 ) 4
(T + 4)^4
3 3 3
T 4 + 20 T 3 + ⋯ + 23400 T^{4} + 20 T^{3} + \cdots + 23400 T 4 + 2 0 T 3 + ⋯ + 2 3 4 0 0
T^4 + 20*T^3 - 612*T^2 - 11436*T + 23400
5 5 5
T 4 T^{4} T 4
T^4
7 7 7
T 4 + 62 T 3 + ⋯ + 108388800 T^{4} + 62 T^{3} + \cdots + 108388800 T 4 + 6 2 T 3 + ⋯ + 1 0 8 3 8 8 8 0 0
T^4 + 62*T^3 - 31620*T^2 - 1302120*T + 108388800
11 11 1 1
T 4 + ⋯ + 55844170200 T^{4} + \cdots + 55844170200 T 4 + ⋯ + 5 5 8 4 4 1 7 0 2 0 0
T^4 + 14*T^3 - 490968*T^2 + 5600772*T + 55844170200
13 13 1 3
( T + 169 ) 4 (T + 169)^{4} ( T + 1 6 9 ) 4
(T + 169)^4
17 17 1 7
T 4 + ⋯ − 605198227536 T^{4} + \cdots - 605198227536 T 4 + ⋯ − 6 0 5 1 9 8 2 2 7 5 3 6
T^4 + 1684*T^3 - 1581440*T^2 - 2298289776*T - 605198227536
19 19 1 9
T 4 + ⋯ − 1287459140200 T^{4} + \cdots - 1287459140200 T 4 + ⋯ − 1 2 8 7 4 5 9 1 4 0 2 0 0
T^4 - 2954*T^3 + 655680*T^2 + 2990525108*T - 1287459140200
23 23 2 3
T 4 + ⋯ − 1050893604696 T^{4} + \cdots - 1050893604696 T 4 + ⋯ − 1 0 5 0 8 9 3 6 0 4 6 9 6
T^4 + 3528*T^3 - 5025660*T^2 - 17777095948*T - 1050893604696
29 29 2 9
T 4 + ⋯ − 149094276505200 T^{4} + \cdots - 149094276505200 T 4 + ⋯ − 1 4 9 0 9 4 2 7 6 5 0 5 2 0 0
T^4 - 5156*T^3 - 25805360*T^2 + 138742732512*T - 149094276505200
31 31 3 1
T 4 + ⋯ − 14 ⋯ 80 T^{4} + \cdots - 14\!\cdots\!80 T 4 + ⋯ − 1 4 ⋯ 8 0
T^4 + 5642*T^3 - 85239648*T^2 - 731968400540*T - 1462530874343080
37 37 3 7
T 4 + ⋯ + 53 ⋯ 32 T^{4} + \cdots + 53\!\cdots\!32 T 4 + ⋯ + 5 3 ⋯ 3 2
T^4 + 15592*T^3 - 96176192*T^2 - 1151767227888*T + 5381709052846032
41 41 4 1
T 4 + ⋯ − 717263645799600 T^{4} + \cdots - 717263645799600 T 4 + ⋯ − 7 1 7 2 6 3 6 4 5 7 9 9 6 0 0
T^4 - 7472*T^3 - 179110280*T^2 + 1075446354720*T - 717263645799600
43 43 4 3
T 4 + ⋯ − 40 ⋯ 00 T^{4} + \cdots - 40\!\cdots\!00 T 4 + ⋯ − 4 0 ⋯ 0 0
T^4 - 8028*T^3 - 282621460*T^2 + 2984424309804*T - 4086302153614200
47 47 4 7
T 4 + ⋯ − 11 ⋯ 76 T^{4} + \cdots - 11\!\cdots\!76 T 4 + ⋯ − 1 1 ⋯ 7 6
T^4 - 6098*T^3 - 719509268*T^2 + 9225917921976*T - 11917363618584576
53 53 5 3
T 4 + ⋯ − 46 ⋯ 36 T^{4} + \cdots - 46\!\cdots\!36 T 4 + ⋯ − 4 6 ⋯ 3 6
T^4 - 21892*T^3 - 393424560*T^2 + 10903437406512*T - 46952713605480336
59 59 5 9
T 4 + ⋯ + 16 ⋯ 40 T^{4} + \cdots + 16\!\cdots\!40 T 4 + ⋯ + 1 6 ⋯ 4 0
T^4 + 13930*T^3 - 2621763808*T^2 - 17687283346468*T + 1650981854109446040
61 61 6 1
T 4 + ⋯ + 67 ⋯ 36 T^{4} + \cdots + 67\!\cdots\!36 T 4 + ⋯ + 6 7 ⋯ 3 6
T^4 - 4448*T^3 - 962395608*T^2 + 7423034489072*T + 67504407386765936
67 67 6 7
T 4 + ⋯ + 27 ⋯ 20 T^{4} + \cdots + 27\!\cdots\!20 T 4 + ⋯ + 2 7 ⋯ 2 0
T^4 + 29998*T^3 - 3637409788*T^2 - 31296997970200*T + 2762598294860368320
71 71 7 1
T 4 + ⋯ + 81 ⋯ 00 T^{4} + \cdots + 81\!\cdots\!00 T 4 + ⋯ + 8 1 ⋯ 0 0
T^4 - 14650*T^3 - 583440984*T^2 + 5524771017060*T + 81954238030673400
73 73 7 3
T 4 + ⋯ − 43 ⋯ 80 T^{4} + \cdots - 43\!\cdots\!80 T 4 + ⋯ − 4 3 ⋯ 8 0
T^4 - 9732*T^3 - 4489026872*T^2 + 105659721166656*T - 436241315859432080
79 79 7 9
T 4 + ⋯ − 20 ⋯ 20 T^{4} + \cdots - 20\!\cdots\!20 T 4 + ⋯ − 2 0 ⋯ 2 0
T^4 + 97500*T^3 + 1819114192*T^2 - 28750437485472*T - 206027074773211520
83 83 8 3
T 4 + ⋯ − 61 ⋯ 80 T^{4} + \cdots - 61\!\cdots\!80 T 4 + ⋯ − 6 1 ⋯ 8 0
T^4 - 107226*T^3 + 3667218948*T^2 - 37858592841480*T - 61047153602909280
89 89 8 9
T 4 + ⋯ − 62 ⋯ 00 T^{4} + \cdots - 62\!\cdots\!00 T 4 + ⋯ − 6 2 ⋯ 0 0
T^4 + 86952*T^3 - 9593559144*T^2 - 197226301704160*T - 627125667167113200
97 97 9 7
T 4 + ⋯ − 68 ⋯ 08 T^{4} + \cdots - 68\!\cdots\!08 T 4 + ⋯ − 6 8 ⋯ 0 8
T^4 - 142340*T^3 + 2008428416*T^2 + 317045226282896*T - 6875031017035265808
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