gp: [N,k,chi] = [162,10,Mod(55,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 10, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.55");
S:= CuspForms(chi, 10);
N := Newforms(S);
Newform invariants
sage: traces = [4,-32,0,-512,-1704]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 − x 3 + 76 x 2 + 75 x + 5625 x^{4} - x^{3} + 76x^{2} + 75x + 5625 x 4 − x 3 + 7 6 x 2 + 7 5 x + 5 6 2 5
x^4 - x^3 + 76*x^2 + 75*x + 5625
:
β 1 \beta_{1} β 1 = = =
( − ν 3 + 76 ν 2 − 76 ν + 5625 ) / 5700 ( -\nu^{3} + 76\nu^{2} - 76\nu + 5625 ) / 5700 ( − ν 3 + 7 6 ν 2 − 7 6 ν + 5 6 2 5 ) / 5 7 0 0
(-v^3 + 76*v^2 - 76*v + 5625) / 5700
β 2 \beta_{2} β 2 = = =
( 9 ν 3 − 684 ν 2 + 103284 ν − 50625 ) / 475 ( 9\nu^{3} - 684\nu^{2} + 103284\nu - 50625 ) / 475 ( 9 ν 3 − 6 8 4 ν 2 + 1 0 3 2 8 4 ν − 5 0 6 2 5 ) / 4 7 5
(9*v^3 - 684*v^2 + 103284*v - 50625) / 475
β 3 \beta_{3} β 3 = = =
( 54 ν 3 + 6102 ) / 19 ( 54\nu^{3} + 6102 ) / 19 ( 5 4 ν 3 + 6 1 0 2 ) / 1 9
(54*v^3 + 6102) / 19
ν \nu ν = = =
( β 2 + 108 β 1 ) / 216 ( \beta_{2} + 108\beta_1 ) / 216 ( β 2 + 1 0 8 β 1 ) / 2 1 6
(b2 + 108*b1) / 216
ν 2 \nu^{2} ν 2 = = =
( β 3 + β 2 + 16308 β 1 − 16308 ) / 216 ( \beta_{3} + \beta_{2} + 16308\beta _1 - 16308 ) / 216 ( β 3 + β 2 + 1 6 3 0 8 β 1 − 1 6 3 0 8 ) / 2 1 6
(b3 + b2 + 16308*b1 - 16308) / 216
ν 3 \nu^{3} ν 3 = = =
( 19 β 3 − 6102 ) / 54 ( 19\beta_{3} - 6102 ) / 54 ( 1 9 β 3 − 6 1 0 2 ) / 5 4
(19*b3 - 6102) / 54
Character values
We give the values of χ \chi χ on generators for ( Z / 162 Z ) × \left(\mathbb{Z}/162\mathbb{Z}\right)^\times ( Z / 1 6 2 Z ) × .
n n n
83 83 8 3
χ ( n ) \chi(n) χ ( n )
− β 1 -\beta_{1} − β 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 5 4 + 1704 T 5 3 + 5688576 T 5 2 − 4745571840 T 5 + 7756002201600 T_{5}^{4} + 1704T_{5}^{3} + 5688576T_{5}^{2} - 4745571840T_{5} + 7756002201600 T 5 4 + 1 7 0 4 T 5 3 + 5 6 8 8 5 7 6 T 5 2 − 4 7 4 5 5 7 1 8 4 0 T 5 + 7 7 5 6 0 0 2 2 0 1 6 0 0
T5^4 + 1704*T5^3 + 5688576*T5^2 - 4745571840*T5 + 7756002201600
acting on S 10 n e w ( 162 , [ χ ] ) S_{10}^{\mathrm{new}}(162, [\chi]) S 1 0 n e w ( 1 6 2 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 16 T + 256 ) 2 (T^{2} + 16 T + 256)^{2} ( T 2 + 1 6 T + 2 5 6 ) 2
(T^2 + 16*T + 256)^2
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
T 4 + ⋯ + 7756002201600 T^{4} + \cdots + 7756002201600 T 4 + ⋯ + 7 7 5 6 0 0 2 2 0 1 6 0 0
T^4 + 1704*T^3 + 5688576*T^2 - 4745571840*T + 7756002201600
7 7 7
T 4 + ⋯ + 20 ⋯ 69 T^{4} + \cdots + 20\!\cdots\!69 T 4 + ⋯ + 2 0 ⋯ 6 9
T^4 + 6538*T^3 + 88232907*T^2 - 297397033094*T + 2069109290176369
11 11 1 1
T 4 + ⋯ + 32 ⋯ 00 T^{4} + \cdots + 32\!\cdots\!00 T 4 + ⋯ + 3 2 ⋯ 0 0
T^4 + 14568*T^3 + 2016417024*T^2 - 26283445747200*T + 3255102999452160000
13 13 1 3
T 4 + ⋯ + 83 ⋯ 25 T^{4} + \cdots + 83\!\cdots\!25 T 4 + ⋯ + 8 3 ⋯ 2 5
T^4 + 37138*T^3 + 10527799539*T^2 - 339759536767310*T + 83696305507706565025
17 17 1 7
( T 2 + 87912 T − 49470429888 ) 2 (T^{2} + 87912 T - 49470429888)^{2} ( T 2 + 8 7 9 1 2 T − 4 9 4 7 0 4 2 9 8 8 8 ) 2
(T^2 + 87912*T - 49470429888)^2
19 19 1 9
( T 2 − 709150 T + 81683152609 ) 2 (T^{2} - 709150 T + 81683152609)^{2} ( T 2 − 7 0 9 1 5 0 T + 8 1 6 8 3 1 5 2 6 0 9 ) 2
(T^2 - 709150*T + 81683152609)^2
23 23 2 3
T 4 + ⋯ + 92 ⋯ 76 T^{4} + \cdots + 92\!\cdots\!76 T 4 + ⋯ + 9 2 ⋯ 7 6
T^4 + 144552*T^3 + 30504860928*T^2 - 1389084040539648*T + 92344032081491890176
29 29 2 9
T 4 + ⋯ + 67 ⋯ 44 T^{4} + \cdots + 67\!\cdots\!44 T 4 + ⋯ + 6 7 ⋯ 4 4
T^4 + 10307328*T^3 + 80288333955072*T^2 + 267502749622191783936*T + 673541419824063324717318144
31 31 3 1
T 4 + ⋯ + 10 ⋯ 00 T^{4} + \cdots + 10\!\cdots\!00 T 4 + ⋯ + 1 0 ⋯ 0 0
T^4 - 2968520*T^3 + 40785237764400*T^2 + 94912866291154480000*T + 1022280835706275647076000000
37 37 3 7
( T 2 + ⋯ + 204253599323425 ) 2 (T^{2} + \cdots + 204253599323425)^{2} ( T 2 + ⋯ + 2 0 4 2 5 3 5 9 9 3 2 3 4 2 5 ) 2
(T^2 + 29227298*T + 204253599323425)^2
41 41 4 1
T 4 + ⋯ + 53 ⋯ 00 T^{4} + \cdots + 53\!\cdots\!00 T 4 + ⋯ + 5 3 ⋯ 0 0
T^4 + 17735808*T^3 + 545611297849344*T^2 - 4097901224910221475840*T + 53385217292717257770034790400
43 43 4 3
T 4 + ⋯ + 49 ⋯ 36 T^{4} + \cdots + 49\!\cdots\!36 T 4 + ⋯ + 4 9 ⋯ 3 6
T^4 - 30013232*T^3 + 678072836952768*T^2 - 6684584791679296596992*T + 49604758824371363426599899136
47 47 4 7
T 4 + ⋯ + 22 ⋯ 64 T^{4} + \cdots + 22\!\cdots\!64 T 4 + ⋯ + 2 2 ⋯ 6 4
T^4 + 44124936*T^3 + 1474484097017088*T^2 + 20850174212770408831488*T + 223280707257496287534248792064
53 53 5 3
( T 2 + ⋯ + 16 ⋯ 00 ) 2 (T^{2} + \cdots + 16\!\cdots\!00)^{2} ( T 2 + ⋯ + 1 6 ⋯ 0 0 ) 2
(T^2 - 93166128*T + 1676311667078400)^2
59 59 5 9
T 4 + ⋯ + 19 ⋯ 04 T^{4} + \cdots + 19\!\cdots\!04 T 4 + ⋯ + 1 9 ⋯ 0 4
T^4 + 133725288*T^3 + 13439187998606592*T^2 + 594176845259129969189376*T + 19742600768391185389424837627904
61 61 6 1
T 4 + ⋯ + 20 ⋯ 81 T^{4} + \cdots + 20\!\cdots\!81 T 4 + ⋯ + 2 0 ⋯ 8 1
T^4 - 3590330*T^3 + 4581006074260059*T^2 + 16401042499206228692470*T + 20867680178371047114534311843281
67 67 6 7
T 4 + ⋯ + 20 ⋯ 41 T^{4} + \cdots + 20\!\cdots\!41 T 4 + ⋯ + 2 0 ⋯ 4 1
T^4 + 132828790*T^3 + 22218849169888971*T^2 - 607739760684736014836090*T + 20933934841616775731527632566641
71 71 7 1
( T 2 + ⋯ − 10 ⋯ 12 ) 2 (T^{2} + \cdots - 10\!\cdots\!12)^{2} ( T 2 + ⋯ − 1 0 ⋯ 1 2 ) 2
(T^2 - 85960656*T - 100384459450682112)^2
73 73 7 3
( T 2 + ⋯ + 74 ⋯ 45 ) 2 (T^{2} + \cdots + 74\!\cdots\!45)^{2} ( T 2 + ⋯ + 7 4 ⋯ 4 5 ) 2
(T^2 + 440277278*T + 7455683527525345)^2
79 79 7 9
T 4 + ⋯ + 33 ⋯ 49 T^{4} + \cdots + 33\!\cdots\!49 T 4 + ⋯ + 3 3 ⋯ 4 9
T^4 - 376775834*T^3 + 160185225897623163*T^2 + 6866813728364422191249238*T + 332157798807980934462174358946449
83 83 8 3
T 4 + ⋯ + 16 ⋯ 76 T^{4} + \cdots + 16\!\cdots\!76 T 4 + ⋯ + 1 6 ⋯ 7 6
T^4 - 1285892400*T^3 + 1242771033541579776*T^2 - 528178028345689795071897600*T + 168714109135051994637204407008690176
89 89 8 9
( T 2 + ⋯ − 79 ⋯ 92 ) 2 (T^{2} + \cdots - 79\!\cdots\!92)^{2} ( T 2 + ⋯ − 7 9 ⋯ 9 2 ) 2
(T^2 + 205421256*T - 797669539998720192)^2
97 97 9 7
T 4 + ⋯ + 25 ⋯ 25 T^{4} + \cdots + 25\!\cdots\!25 T 4 + ⋯ + 2 5 ⋯ 2 5
T^4 - 1672271798*T^3 + 2294957716466710539*T^2 - 838703254143165071648742470*T + 251537606911757412481172048668330225
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