gp: [N,k,chi] = [605,4,Mod(1,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,-24,18]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == 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 − 35 x 6 + 376 x 4 − 1452 x 2 + 1764 x^{8} - 35x^{6} + 376x^{4} - 1452x^{2} + 1764 x 8 − 3 5 x 6 + 3 7 6 x 4 − 1 4 5 2 x 2 + 1 7 6 4
x^8 - 35*x^6 + 376*x^4 - 1452*x^2 + 1764
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( − 2 ν 7 + 49 ν 5 − 143 ν 3 − 1086 ν ) / 126 ( -2\nu^{7} + 49\nu^{5} - 143\nu^{3} - 1086\nu ) / 126 ( − 2 ν 7 + 4 9 ν 5 − 1 4 3 ν 3 − 1 0 8 6 ν ) / 1 2 6
(-2*v^7 + 49*v^5 - 143*v^3 - 1086*v) / 126
β 3 \beta_{3} β 3 = = =
( − 5 ν 7 + 154 ν 5 − 1271 ν 3 + 2640 ν ) / 378 ( -5\nu^{7} + 154\nu^{5} - 1271\nu^{3} + 2640\nu ) / 378 ( − 5 ν 7 + 1 5 4 ν 5 − 1 2 7 1 ν 3 + 2 6 4 0 ν ) / 3 7 8
(-5*v^7 + 154*v^5 - 1271*v^3 + 2640*v) / 378
β 4 \beta_{4} β 4 = = =
( − ν 6 + 29 ν 4 − 202 ν 2 + 276 ) / 18 ( -\nu^{6} + 29\nu^{4} - 202\nu^{2} + 276 ) / 18 ( − ν 6 + 2 9 ν 4 − 2 0 2 ν 2 + 2 7 6 ) / 1 8
(-v^6 + 29*v^4 - 202*v^2 + 276) / 18
β 5 \beta_{5} β 5 = = =
( ν 6 − 29 ν 4 + 220 ν 2 − 438 ) / 18 ( \nu^{6} - 29\nu^{4} + 220\nu^{2} - 438 ) / 18 ( ν 6 − 2 9 ν 4 + 2 2 0 ν 2 − 4 3 8 ) / 1 8
(v^6 - 29*v^4 + 220*v^2 - 438) / 18
β 6 \beta_{6} β 6 = = =
( − 19 ν 7 + 623 ν 5 − 5548 ν 3 + 11544 ν ) / 378 ( -19\nu^{7} + 623\nu^{5} - 5548\nu^{3} + 11544\nu ) / 378 ( − 1 9 ν 7 + 6 2 3 ν 5 − 5 5 4 8 ν 3 + 1 1 5 4 4 ν ) / 3 7 8
(-19*v^7 + 623*v^5 - 5548*v^3 + 11544*v) / 378
β 7 \beta_{7} β 7 = = =
( − 2 ν 6 + 67 ν 4 − 620 ν 2 + 1380 ) / 9 ( -2\nu^{6} + 67\nu^{4} - 620\nu^{2} + 1380 ) / 9 ( − 2 ν 6 + 6 7 ν 4 − 6 2 0 ν 2 + 1 3 8 0 ) / 9
(-2*v^6 + 67*v^4 - 620*v^2 + 1380) / 9
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 5 + β 4 + 9 \beta_{5} + \beta_{4} + 9 β 5 + β 4 + 9
b5 + b4 + 9
ν 3 \nu^{3} ν 3 = = =
β 6 − 5 β 3 + β 2 + 13 β 1 \beta_{6} - 5\beta_{3} + \beta_{2} + 13\beta_1 β 6 − 5 β 3 + β 2 + 1 3 β 1
b6 - 5*b3 + b2 + 13*b1
ν 4 \nu^{4} ν 4 = = =
β 7 + 24 β 5 + 20 β 4 + 124 \beta_{7} + 24\beta_{5} + 20\beta_{4} + 124 β 7 + 2 4 β 5 + 2 0 β 4 + 1 2 4
b7 + 24*b5 + 20*b4 + 124
ν 5 \nu^{5} ν 5 = = =
29 β 6 − 133 β 3 + 19 β 2 + 207 β 1 29\beta_{6} - 133\beta_{3} + 19\beta_{2} + 207\beta_1 2 9 β 6 − 1 3 3 β 3 + 1 9 β 2 + 2 0 7 β 1
29*b6 - 133*b3 + 19*b2 + 207*b1
ν 6 \nu^{6} ν 6 = = =
29 β 7 + 494 β 5 + 360 β 4 + 2054 29\beta_{7} + 494\beta_{5} + 360\beta_{4} + 2054 2 9 β 7 + 4 9 4 β 5 + 3 6 0 β 4 + 2 0 5 4
29*b7 + 494*b5 + 360*b4 + 2054
ν 7 \nu^{7} ν 7 = = =
639 β 6 − 2901 β 3 + 331 β 2 + 3599 β 1 639\beta_{6} - 2901\beta_{3} + 331\beta_{2} + 3599\beta_1 6 3 9 β 6 − 2 9 0 1 β 3 + 3 3 1 β 2 + 3 5 9 9 β 1
639*b6 - 2901*b3 + 331*b2 + 3599*b1
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
5 5 5
− 1 -1 − 1
11 11 1 1
+ 1 +1 + 1
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 4 n e w ( Γ 0 ( 605 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(605)) S 4 n e w ( Γ 0 ( 6 0 5 ) ) :
T 2 8 − 41 T 2 6 + 487 T 2 4 − 1755 T 2 2 + 144 T_{2}^{8} - 41T_{2}^{6} + 487T_{2}^{4} - 1755T_{2}^{2} + 144 T 2 8 − 4 1 T 2 6 + 4 8 7 T 2 4 − 1 7 5 5 T 2 2 + 1 4 4
T2^8 - 41*T2^6 + 487*T2^4 - 1755*T2^2 + 144
T 3 4 + 12 T 3 3 − 8 T 3 2 − 282 T 3 − 11 T_{3}^{4} + 12T_{3}^{3} - 8T_{3}^{2} - 282T_{3} - 11 T 3 4 + 1 2 T 3 3 − 8 T 3 2 − 2 8 2 T 3 − 1 1
T3^4 + 12*T3^3 - 8*T3^2 - 282*T3 - 11
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 − 41 T 6 + ⋯ + 144 T^{8} - 41 T^{6} + \cdots + 144 T 8 − 4 1 T 6 + ⋯ + 1 4 4
T^8 - 41*T^6 + 487*T^4 - 1755*T^2 + 144
3 3 3
( T 4 + 12 T 3 + ⋯ − 11 ) 2 (T^{4} + 12 T^{3} + \cdots - 11)^{2} ( T 4 + 1 2 T 3 + ⋯ − 1 1 ) 2
(T^4 + 12*T^3 - 8*T^2 - 282*T - 11)^2
5 5 5
( T − 5 ) 8 (T - 5)^{8} ( T − 5 ) 8
(T - 5)^8
7 7 7
T 8 + ⋯ + 1597041369 T^{8} + \cdots + 1597041369 T 8 + ⋯ + 1 5 9 7 0 4 1 3 6 9
T^8 - 980*T^6 + 302842*T^4 - 37360104*T^2 + 1597041369
11 11 1 1
T 8 T^{8} T 8
T^8
13 13 1 3
T 8 + ⋯ + 6219956192256 T^{8} + \cdots + 6219956192256 T 8 + ⋯ + 6 2 1 9 9 5 6 1 9 2 2 5 6
T^8 - 9176*T^6 + 25232176*T^4 - 22546112688*T^2 + 6219956192256
17 17 1 7
T 8 + ⋯ + 20351214157824 T^{8} + \cdots + 20351214157824 T 8 + ⋯ + 2 0 3 5 1 2 1 4 1 5 7 8 2 4
T^8 - 25680*T^6 + 174033792*T^4 - 210902379264*T^2 + 20351214157824
19 19 1 9
T 8 + ⋯ + 11606940354816 T^{8} + \cdots + 11606940354816 T 8 + ⋯ + 1 1 6 0 6 9 4 0 3 5 4 8 1 6
T^8 - 14964*T^6 + 42096096*T^4 - 40252740912*T^2 + 11606940354816
23 23 2 3
( T 4 + 286 T 3 + ⋯ + 5778720 ) 2 (T^{4} + 286 T^{3} + \cdots + 5778720)^{2} ( T 4 + 2 8 6 T 3 + ⋯ + 5 7 7 8 7 2 0 ) 2
(T^4 + 286*T^3 + 26068*T^2 + 828084*T + 5778720)^2
29 29 2 9
T 8 + ⋯ + 13 ⋯ 00 T^{8} + \cdots + 13\!\cdots\!00 T 8 + ⋯ + 1 3 ⋯ 0 0
T^8 - 154320*T^6 + 8261411904*T^4 - 178937906934528*T^2 + 1329414165445017600
31 31 3 1
( T 4 + 246 T 3 + ⋯ − 84685544 ) 2 (T^{4} + 246 T^{3} + \cdots - 84685544)^{2} ( T 4 + 2 4 6 T 3 + ⋯ − 8 4 6 8 5 5 4 4 ) 2
(T^4 + 246*T^3 - 43064*T^2 - 10687452*T - 84685544)^2
37 37 3 7
( T 4 + 16 T 3 + ⋯ − 446406528 ) 2 (T^{4} + 16 T^{3} + \cdots - 446406528)^{2} ( T 4 + 1 6 T 3 + ⋯ − 4 4 6 4 0 6 5 2 8 ) 2
(T^4 + 16*T^3 - 81032*T^2 - 11898000*T - 446406528)^2
41 41 4 1
T 8 + ⋯ + 21 ⋯ 49 T^{8} + \cdots + 21\!\cdots\!49 T 8 + ⋯ + 2 1 ⋯ 4 9
T^8 - 265764*T^6 + 19723734198*T^4 - 459712194389460*T^2 + 2189264711863589649
43 43 4 3
T 8 + ⋯ + 10 ⋯ 25 T^{8} + \cdots + 10\!\cdots\!25 T 8 + ⋯ + 1 0 ⋯ 2 5
T^8 - 356460*T^6 + 20262603834*T^4 - 128395535974128*T^2 + 100336039182236025
47 47 4 7
( T 4 + 236 T 3 + ⋯ + 6087784941 ) 2 (T^{4} + 236 T^{3} + \cdots + 6087784941)^{2} ( T 4 + 2 3 6 T 3 + ⋯ + 6 0 8 7 7 8 4 9 4 1 ) 2
(T^4 + 236*T^3 - 143192*T^2 - 18780822*T + 6087784941)^2
53 53 5 3
( T 4 + 756 T 3 + ⋯ − 8094368016 ) 2 (T^{4} + 756 T^{3} + \cdots - 8094368016)^{2} ( T 4 + 7 5 6 T 3 + ⋯ − 8 0 9 4 3 6 8 0 1 6 ) 2
(T^4 + 756*T^3 + 79692*T^2 - 44113860*T - 8094368016)^2
59 59 5 9
( T 4 − 54 T 3 + ⋯ + 2928268152 ) 2 (T^{4} - 54 T^{3} + \cdots + 2928268152)^{2} ( T 4 − 5 4 T 3 + ⋯ + 2 9 2 8 2 6 8 1 5 2 ) 2
(T^4 - 54*T^3 - 208596*T^2 + 2343924*T + 2928268152)^2
61 61 6 1
T 8 + ⋯ + 61 ⋯ 25 T^{8} + \cdots + 61\!\cdots\!25 T 8 + ⋯ + 6 1 ⋯ 2 5
T^8 - 828860*T^6 + 216615870094*T^4 - 20410605440351868*T^2 + 612483955670450108025
67 67 6 7
( T 4 + 2392 T 3 + ⋯ + 91567885245 ) 2 (T^{4} + 2392 T^{3} + \cdots + 91567885245)^{2} ( T 4 + 2 3 9 2 T 3 + ⋯ + 9 1 5 6 7 8 8 5 2 4 5 ) 2
(T^4 + 2392*T^3 + 2033884*T^2 + 724565694*T + 91567885245)^2
71 71 7 1
( T 4 + 1454 T 3 + ⋯ + 903758616 ) 2 (T^{4} + 1454 T^{3} + \cdots + 903758616)^{2} ( T 4 + 1 4 5 4 T 3 + ⋯ + 9 0 3 7 5 8 6 1 6 ) 2
(T^4 + 1454*T^3 + 626764*T^2 + 76177548*T + 903758616)^2
73 73 7 3
T 8 + ⋯ + 68 ⋯ 00 T^{8} + \cdots + 68\!\cdots\!00 T 8 + ⋯ + 6 8 ⋯ 0 0
T^8 - 1538288*T^6 + 643420404736*T^4 - 42021515368415232*T^2 + 684947253127289241600
79 79 7 9
T 8 + ⋯ + 56 ⋯ 96 T^{8} + \cdots + 56\!\cdots\!96 T 8 + ⋯ + 5 6 ⋯ 9 6
T^8 - 2068448*T^6 + 488018414368*T^4 - 24757282884070656*T^2 + 567901601797591296
83 83 8 3
T 8 + ⋯ + 19 ⋯ 96 T^{8} + \cdots + 19\!\cdots\!96 T 8 + ⋯ + 1 9 ⋯ 9 6
T^8 - 1344612*T^6 + 400749293808*T^4 - 15910729375158000*T^2 + 19959715457991429696
89 89 8 9
( T 4 + 160 T 3 + ⋯ + 339659726877 ) 2 (T^{4} + 160 T^{3} + \cdots + 339659726877)^{2} ( T 4 + 1 6 0 T 3 + ⋯ + 3 3 9 6 5 9 7 2 6 8 7 7 ) 2
(T^4 + 160*T^3 - 2057846*T^2 - 285075192*T + 339659726877)^2
97 97 9 7
( T 4 + 932 T 3 + ⋯ − 237654925680 ) 2 (T^{4} + 932 T^{3} + \cdots - 237654925680)^{2} ( T 4 + 9 3 2 T 3 + ⋯ − 2 3 7 6 5 4 9 2 5 6 8 0 ) 2
(T^4 + 932*T^3 - 2411156*T^2 - 1579326516*T - 237654925680)^2
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