gp: [N,k,chi] = [3364,2,Mod(1,3364)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3364, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3364.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,-4,0,-1,0,-2,0,4,0,-5,0,4,0,17,0,-15,0,-11,0,-19,0,-3,0,
3,0,-16,0,0,0,9,0,-16,0,9,0,2,0,-9,0,-38,0,3,0,3]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(45)] == 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 7 1,\beta_1,\ldots,\beta_{7} 1 , β 1 , … , β 7 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 8 − x 7 − 10 x 6 + 6 x 5 + 29 x 4 − 9 x 3 − 25 x 2 − x + 1 x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1 x 8 − x 7 − 1 0 x 6 + 6 x 5 + 2 9 x 4 − 9 x 3 − 2 5 x 2 − x + 1
x^8 - x^7 - 10*x^6 + 6*x^5 + 29*x^4 - 9*x^3 - 25*x^2 - x + 1
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 7 − ν 6 − 8 ν 5 + 5 ν 4 + 14 ν 3 − 11 ν 2 − 4 ν + 7 ) / 3 ( \nu^{7} - \nu^{6} - 8\nu^{5} + 5\nu^{4} + 14\nu^{3} - 11\nu^{2} - 4\nu + 7 ) / 3 ( ν 7 − ν 6 − 8 ν 5 + 5 ν 4 + 1 4 ν 3 − 1 1 ν 2 − 4 ν + 7 ) / 3
(v^7 - v^6 - 8*v^5 + 5*v^4 + 14*v^3 - 11*v^2 - 4*v + 7) / 3
β 3 \beta_{3} β 3 = = =
( ν 6 + ν 5 − 9 ν 4 − 13 ν 3 + 15 ν 2 + 25 ν + 1 ) / 3 ( \nu^{6} + \nu^{5} - 9\nu^{4} - 13\nu^{3} + 15\nu^{2} + 25\nu + 1 ) / 3 ( ν 6 + ν 5 − 9 ν 4 − 1 3 ν 3 + 1 5 ν 2 + 2 5 ν + 1 ) / 3
(v^6 + v^5 - 9*v^4 - 13*v^3 + 15*v^2 + 25*v + 1) / 3
β 4 \beta_{4} β 4 = = =
( ν 7 + ν 6 − 9 ν 5 − 13 ν 4 + 15 ν 3 + 28 ν 2 + ν − 6 ) / 3 ( \nu^{7} + \nu^{6} - 9\nu^{5} - 13\nu^{4} + 15\nu^{3} + 28\nu^{2} + \nu - 6 ) / 3 ( ν 7 + ν 6 − 9 ν 5 − 1 3 ν 4 + 1 5 ν 3 + 2 8 ν 2 + ν − 6 ) / 3
(v^7 + v^6 - 9*v^5 - 13*v^4 + 15*v^3 + 28*v^2 + v - 6) / 3
β 5 \beta_{5} β 5 = = =
( ν 7 − 2 ν 6 − 9 ν 5 + 14 ν 4 + 24 ν 3 − 23 ν 2 − 17 ν + 3 ) / 3 ( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 14\nu^{4} + 24\nu^{3} - 23\nu^{2} - 17\nu + 3 ) / 3 ( ν 7 − 2 ν 6 − 9 ν 5 + 1 4 ν 4 + 2 4 ν 3 − 2 3 ν 2 − 1 7 ν + 3 ) / 3
(v^7 - 2*v^6 - 9*v^5 + 14*v^4 + 24*v^3 - 23*v^2 - 17*v + 3) / 3
β 6 \beta_{6} β 6 = = =
( ν 7 − ν 6 − 11 ν 5 + 8 ν 4 + 35 ν 3 − 17 ν 2 − 31 ν + 1 ) / 3 ( \nu^{7} - \nu^{6} - 11\nu^{5} + 8\nu^{4} + 35\nu^{3} - 17\nu^{2} - 31\nu + 1 ) / 3 ( ν 7 − ν 6 − 1 1 ν 5 + 8 ν 4 + 3 5 ν 3 − 1 7 ν 2 − 3 1 ν + 1 ) / 3
(v^7 - v^6 - 11*v^5 + 8*v^4 + 35*v^3 - 17*v^2 - 31*v + 1) / 3
β 7 \beta_{7} β 7 = = =
( − 2 ν 7 + 20 ν 5 + 5 ν 4 − 50 ν 3 − 8 ν 2 + 30 ν − 4 ) / 3 ( -2\nu^{7} + 20\nu^{5} + 5\nu^{4} - 50\nu^{3} - 8\nu^{2} + 30\nu - 4 ) / 3 ( − 2 ν 7 + 2 0 ν 5 + 5 ν 4 − 5 0 ν 3 − 8 ν 2 + 3 0 ν − 4 ) / 3
(-2*v^7 + 20*v^5 + 5*v^4 - 50*v^3 - 8*v^2 + 30*v - 4) / 3
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 7 + β 6 + β 4 + 3 \beta_{7} + \beta_{6} + \beta_{4} + 3 β 7 + β 6 + β 4 + 3
b7 + b6 + b4 + 3
ν 3 \nu^{3} ν 3 = = =
β 7 + β 6 − β 5 + β 4 − β 3 + β 2 + 4 β 1 + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 + 2 β 7 + β 6 − β 5 + β 4 − β 3 + β 2 + 4 β 1 + 2
b7 + b6 - b5 + b4 - b3 + b2 + 4*b1 + 2
ν 4 \nu^{4} ν 4 = = =
7 β 7 + 8 β 6 − 2 β 5 + 6 β 4 + 2 β 2 + 2 β 1 + 16 7\beta_{7} + 8\beta_{6} - 2\beta_{5} + 6\beta_{4} + 2\beta_{2} + 2\beta _1 + 16 7 β 7 + 8 β 6 − 2 β 5 + 6 β 4 + 2 β 2 + 2 β 1 + 1 6
7*b7 + 8*b6 - 2*b5 + 6*b4 + 2*b2 + 2*b1 + 16
ν 5 \nu^{5} ν 5 = = =
12 β 7 + 12 β 6 − 9 β 5 + 11 β 4 − 7 β 3 + 10 β 2 + 21 β 1 + 22 12\beta_{7} + 12\beta_{6} - 9\beta_{5} + 11\beta_{4} - 7\beta_{3} + 10\beta_{2} + 21\beta _1 + 22 1 2 β 7 + 1 2 β 6 − 9 β 5 + 1 1 β 4 − 7 β 3 + 1 0 β 2 + 2 1 β 1 + 2 2
12*b7 + 12*b6 - 9*b5 + 11*b4 - 7*b3 + 10*b2 + 21*b1 + 22
ν 6 \nu^{6} ν 6 = = =
49 β 7 + 58 β 6 − 22 β 5 + 41 β 4 − 3 β 3 + 21 β 2 + 24 β 1 + 102 49\beta_{7} + 58\beta_{6} - 22\beta_{5} + 41\beta_{4} - 3\beta_{3} + 21\beta_{2} + 24\beta _1 + 102 4 9 β 7 + 5 8 β 6 − 2 2 β 5 + 4 1 β 4 − 3 β 3 + 2 1 β 2 + 2 4 β 1 + 1 0 2
49*b7 + 58*b6 - 22*b5 + 41*b4 - 3*b3 + 21*b2 + 24*b1 + 102
ν 7 \nu^{7} ν 7 = = =
107 β 7 + 111 β 6 − 70 β 5 + 96 β 4 − 45 β 3 + 80 β 2 + 130 β 1 + 196 107\beta_{7} + 111\beta_{6} - 70\beta_{5} + 96\beta_{4} - 45\beta_{3} + 80\beta_{2} + 130\beta _1 + 196 1 0 7 β 7 + 1 1 1 β 6 − 7 0 β 5 + 9 6 β 4 − 4 5 β 3 + 8 0 β 2 + 1 3 0 β 1 + 1 9 6
107*b7 + 111*b6 - 70*b5 + 96*b4 - 45*b3 + 80*b2 + 130*b1 + 196
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
29 29 2 9
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 2 n e w ( Γ 0 ( 3364 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(3364)) S 2 n e w ( Γ 0 ( 3 3 6 4 ) ) :
T 3 8 + 4 T 3 7 − 6 T 3 6 − 32 T 3 5 − T 3 4 + 41 T 3 3 − 14 T 3 2 − 3 T 3 + 1 T_{3}^{8} + 4T_{3}^{7} - 6T_{3}^{6} - 32T_{3}^{5} - T_{3}^{4} + 41T_{3}^{3} - 14T_{3}^{2} - 3T_{3} + 1 T 3 8 + 4 T 3 7 − 6 T 3 6 − 3 2 T 3 5 − T 3 4 + 4 1 T 3 3 − 1 4 T 3 2 − 3 T 3 + 1
T3^8 + 4*T3^7 - 6*T3^6 - 32*T3^5 - T3^4 + 41*T3^3 - 14*T3^2 - 3*T3 + 1
T 5 8 + T 5 7 − 21 T 5 6 + 7 T 5 5 + 109 T 5 4 − 156 T 5 3 + 51 T 5 2 + 18 T 5 − 9 T_{5}^{8} + T_{5}^{7} - 21T_{5}^{6} + 7T_{5}^{5} + 109T_{5}^{4} - 156T_{5}^{3} + 51T_{5}^{2} + 18T_{5} - 9 T 5 8 + T 5 7 − 2 1 T 5 6 + 7 T 5 5 + 1 0 9 T 5 4 − 1 5 6 T 5 3 + 5 1 T 5 2 + 1 8 T 5 − 9
T5^8 + T5^7 - 21*T5^6 + 7*T5^5 + 109*T5^4 - 156*T5^3 + 51*T5^2 + 18*T5 - 9
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 T^{8} T 8
T^8
3 3 3
T 8 + 4 T 7 + ⋯ + 1 T^{8} + 4 T^{7} + \cdots + 1 T 8 + 4 T 7 + ⋯ + 1
T^8 + 4*T^7 - 6*T^6 - 32*T^5 - T^4 + 41*T^3 - 14*T^2 - 3*T + 1
5 5 5
T 8 + T 7 − 21 T 6 + ⋯ − 9 T^{8} + T^{7} - 21 T^{6} + \cdots - 9 T 8 + T 7 − 2 1 T 6 + ⋯ − 9
T^8 + T^7 - 21*T^6 + 7*T^5 + 109*T^4 - 156*T^3 + 51*T^2 + 18*T - 9
7 7 7
T 8 + 2 T 7 + ⋯ + 211 T^{8} + 2 T^{7} + \cdots + 211 T 8 + 2 T 7 + ⋯ + 2 1 1
T^8 + 2*T^7 - 32*T^6 - 46*T^5 + 315*T^4 + 289*T^3 - 887*T^2 - 578*T + 211
11 11 1 1
T 8 + 5 T 7 + ⋯ + 5931 T^{8} + 5 T^{7} + \cdots + 5931 T 8 + 5 T 7 + ⋯ + 5 9 3 1
T^8 + 5*T^7 - 43*T^6 - 155*T^5 + 739*T^4 + 1230*T^3 - 5262*T^2 + 135*T + 5931
13 13 1 3
T 8 − 4 T 7 + ⋯ + 20851 T^{8} - 4 T^{7} + \cdots + 20851 T 8 − 4 T 7 + ⋯ + 2 0 8 5 1
T^8 - 4*T^7 - 65*T^6 + 239*T^5 + 1239*T^4 - 3716*T^3 - 8180*T^2 + 13966*T + 20851
17 17 1 7
T 8 + 15 T 7 + ⋯ + 81 T^{8} + 15 T^{7} + \cdots + 81 T 8 + 1 5 T 7 + ⋯ + 8 1
T^8 + 15*T^7 + 63*T^6 - 45*T^5 - 891*T^4 - 1890*T^3 - 1053*T^2 + 81
19 19 1 9
T 8 + 11 T 7 + ⋯ − 549 T^{8} + 11 T^{7} + \cdots - 549 T 8 + 1 1 T 7 + ⋯ − 5 4 9
T^8 + 11*T^7 - T^6 - 308*T^5 - 851*T^4 + 144*T^3 + 1821*T^2 - 117*T - 549
23 23 2 3
T 8 + 3 T 7 + ⋯ − 31599 T^{8} + 3 T^{7} + \cdots - 31599 T 8 + 3 T 7 + ⋯ − 3 1 5 9 9
T^8 + 3*T^7 - 79*T^6 - 276*T^5 + 1234*T^4 + 5487*T^3 - 2541*T^2 - 31059*T - 31599
29 29 2 9
T 8 T^{8} T 8
T^8
31 31 3 1
T 8 − 9 T 7 + ⋯ − 134009 T^{8} - 9 T^{7} + \cdots - 134009 T 8 − 9 T 7 + ⋯ − 1 3 4 0 0 9
T^8 - 9*T^7 - 65*T^6 + 604*T^5 + 944*T^4 - 12961*T^3 + 5715*T^2 + 90161*T - 134009
37 37 3 7
T 8 − 2 T 7 + ⋯ + 16651 T^{8} - 2 T^{7} + \cdots + 16651 T 8 − 2 T 7 + ⋯ + 1 6 6 5 1
T^8 - 2*T^7 - 124*T^6 + 244*T^5 + 4219*T^4 - 7363*T^3 - 24406*T^2 + 6401*T + 16651
41 41 4 1
T 8 + 38 T 7 + ⋯ − 1341909 T^{8} + 38 T^{7} + \cdots - 1341909 T 8 + 3 8 T 7 + ⋯ − 1 3 4 1 9 0 9
T^8 + 38*T^7 + 548*T^6 + 3301*T^5 - 185*T^4 - 114339*T^3 - 642072*T^2 - 1509642*T - 1341909
43 43 4 3
T 8 − 3 T 7 + ⋯ − 8009 T^{8} - 3 T^{7} + \cdots - 8009 T 8 − 3 T 7 + ⋯ − 8 0 0 9
T^8 - 3*T^7 - 135*T^6 + 372*T^5 + 4589*T^4 - 11517*T^3 - 3465*T^2 + 18708*T - 8009
47 47 4 7
T 8 − 7 T 7 + ⋯ + 8745291 T^{8} - 7 T^{7} + \cdots + 8745291 T 8 − 7 T 7 + ⋯ + 8 7 4 5 2 9 1
T^8 - 7*T^7 - 274*T^6 + 1714*T^5 + 24304*T^4 - 128733*T^3 - 790296*T^2 + 2745441*T + 8745291
53 53 5 3
T 8 + 5 T 7 + ⋯ + 5291775 T^{8} + 5 T^{7} + \cdots + 5291775 T 8 + 5 T 7 + ⋯ + 5 2 9 1 7 7 5
T^8 + 5*T^7 - 300*T^6 - 1240*T^5 + 28930*T^4 + 90300*T^3 - 950475*T^2 - 1959075*T + 5291775
59 59 5 9
T 8 + 7 T 7 + ⋯ − 36189 T^{8} + 7 T^{7} + \cdots - 36189 T 8 + 7 T 7 + ⋯ − 3 6 1 8 9
T^8 + 7*T^7 - 280*T^6 - 1423*T^5 + 22759*T^4 + 60663*T^3 - 406695*T^2 + 437508*T - 36189
61 61 6 1
T 8 + 39 T 7 + ⋯ + 15536221 T^{8} + 39 T^{7} + \cdots + 15536221 T 8 + 3 9 T 7 + ⋯ + 1 5 5 3 6 2 2 1
T^8 + 39*T^7 + 415*T^6 - 1499*T^5 - 50536*T^4 - 205894*T^3 + 924645*T^2 + 8264039*T + 15536221
67 67 6 7
T 8 + 26 T 7 + ⋯ − 4409 T^{8} + 26 T^{7} + \cdots - 4409 T 8 + 2 6 T 7 + ⋯ − 4 4 0 9
T^8 + 26*T^7 + 94*T^6 - 2758*T^5 - 33336*T^4 - 149186*T^3 - 288494*T^2 - 189842*T - 4409
71 71 7 1
T 8 − 52 T 7 + ⋯ − 5139 T^{8} - 52 T^{7} + \cdots - 5139 T 8 − 5 2 T 7 + ⋯ − 5 1 3 9
T^8 - 52*T^7 + 1055*T^6 - 10337*T^5 + 47104*T^4 - 60048*T^3 - 131370*T^2 + 84807*T - 5139
73 73 7 3
T 8 − 34 T 7 + ⋯ − 744749 T^{8} - 34 T^{7} + \cdots - 744749 T 8 − 3 4 T 7 + ⋯ − 7 4 4 7 4 9
T^8 - 34*T^7 + 280*T^6 + 2039*T^5 - 38196*T^4 + 124279*T^3 + 225760*T^2 - 807959*T - 744749
79 79 7 9
T 8 + 31 T 7 + ⋯ + 1438231 T^{8} + 31 T^{7} + \cdots + 1438231 T 8 + 3 1 T 7 + ⋯ + 1 4 3 8 2 3 1
T^8 + 31*T^7 + 219*T^6 - 2048*T^5 - 33331*T^4 - 109636*T^3 + 253831*T^2 + 1698093*T + 1438231
83 83 8 3
T 8 + 17 T 7 + ⋯ − 6356619 T^{8} + 17 T^{7} + \cdots - 6356619 T 8 + 1 7 T 7 + ⋯ − 6 3 5 6 6 1 9
T^8 + 17*T^7 - 294*T^6 - 7129*T^5 - 8006*T^4 + 659853*T^3 + 4915794*T^2 + 9409374*T - 6356619
89 89 8 9
T 8 + 24 T 7 + ⋯ + 1672641 T^{8} + 24 T^{7} + \cdots + 1672641 T 8 + 2 4 T 7 + ⋯ + 1 6 7 2 6 4 1
T^8 + 24*T^7 - 25*T^6 - 3879*T^5 - 16196*T^4 + 116376*T^3 + 350745*T^2 - 1868616*T + 1672641
97 97 9 7
T 8 + 22 T 7 + ⋯ + 343171 T^{8} + 22 T^{7} + \cdots + 343171 T 8 + 2 2 T 7 + ⋯ + 3 4 3 1 7 1
T^8 + 22*T^7 - 25*T^6 - 2333*T^5 - 1721*T^4 + 82523*T^3 + 34550*T^2 - 964897*T + 343171
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