gp: [N,k,chi] = [616,4,Mod(1,616)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(616, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("616.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [7,0,0,0,6]
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 6 1,\beta_1,\ldots,\beta_{6} 1 , β 1 , … , β 6 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 7 − 145 x 5 − 10 x 4 + 4790 x 3 − 2452 x 2 − 1496 x + 320 x^{7} - 145x^{5} - 10x^{4} + 4790x^{3} - 2452x^{2} - 1496x + 320 x 7 − 1 4 5 x 5 − 1 0 x 4 + 4 7 9 0 x 3 − 2 4 5 2 x 2 − 1 4 9 6 x + 3 2 0
x^7 - 145*x^5 - 10*x^4 + 4790*x^3 - 2452*x^2 - 1496*x + 320
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( 41 ν 6 + 14 ν 5 − 5961 ν 4 − 2300 ν 3 + 196706 ν 2 − 44544 ν − 43900 ) / 2556 ( 41\nu^{6} + 14\nu^{5} - 5961\nu^{4} - 2300\nu^{3} + 196706\nu^{2} - 44544\nu - 43900 ) / 2556 ( 4 1 ν 6 + 1 4 ν 5 − 5 9 6 1 ν 4 − 2 3 0 0 ν 3 + 1 9 6 7 0 6 ν 2 − 4 4 5 4 4 ν − 4 3 9 0 0 ) / 2 5 5 6
(41*v^6 + 14*v^5 - 5961*v^4 - 2300*v^3 + 196706*v^2 - 44544*v - 43900) / 2556
β 3 \beta_{3} β 3 = = =
( 41 ν 6 + 14 ν 5 − 5961 ν 4 − 2300 ν 3 + 199262 ν 2 − 44544 ν − 151252 ) / 2556 ( 41\nu^{6} + 14\nu^{5} - 5961\nu^{4} - 2300\nu^{3} + 199262\nu^{2} - 44544\nu - 151252 ) / 2556 ( 4 1 ν 6 + 1 4 ν 5 − 5 9 6 1 ν 4 − 2 3 0 0 ν 3 + 1 9 9 2 6 2 ν 2 − 4 4 5 4 4 ν − 1 5 1 2 5 2 ) / 2 5 5 6
(41*v^6 + 14*v^5 - 5961*v^4 - 2300*v^3 + 199262*v^2 - 44544*v - 151252) / 2556
β 4 \beta_{4} β 4 = = =
( − 59 ν 6 − 34 ν 5 + 8391 ν 4 + 6032 ν 3 − 266094 ν 2 − 49036 ν + 236 ) / 2556 ( -59\nu^{6} - 34\nu^{5} + 8391\nu^{4} + 6032\nu^{3} - 266094\nu^{2} - 49036\nu + 236 ) / 2556 ( − 5 9 ν 6 − 3 4 ν 5 + 8 3 9 1 ν 4 + 6 0 3 2 ν 3 − 2 6 6 0 9 4 ν 2 − 4 9 0 3 6 ν + 2 3 6 ) / 2 5 5 6
(-59*v^6 - 34*v^5 + 8391*v^4 + 6032*v^3 - 266094*v^2 - 49036*v + 236) / 2556
β 5 \beta_{5} β 5 = = =
( 8 ν 6 + ν 5 − 1151 ν 4 − 357 ν 3 + 37663 ν 2 − 4906 ν − 14161 ) / 213 ( 8\nu^{6} + \nu^{5} - 1151\nu^{4} - 357\nu^{3} + 37663\nu^{2} - 4906\nu - 14161 ) / 213 ( 8 ν 6 + ν 5 − 1 1 5 1 ν 4 − 3 5 7 ν 3 + 3 7 6 6 3 ν 2 − 4 9 0 6 ν − 1 4 1 6 1 ) / 2 1 3
(8*v^6 + v^5 - 1151*v^4 - 357*v^3 + 37663*v^2 - 4906*v - 14161) / 213
β 6 \beta_{6} β 6 = = =
( − 23 ν 6 + 6 ν 5 + 3318 ν 4 − 580 ν 3 − 109000 ν 2 + 81040 ν + 23451 ) / 639 ( -23\nu^{6} + 6\nu^{5} + 3318\nu^{4} - 580\nu^{3} - 109000\nu^{2} + 81040\nu + 23451 ) / 639 ( − 2 3 ν 6 + 6 ν 5 + 3 3 1 8 ν 4 − 5 8 0 ν 3 − 1 0 9 0 0 0 ν 2 + 8 1 0 4 0 ν + 2 3 4 5 1 ) / 6 3 9
(-23*v^6 + 6*v^5 + 3318*v^4 - 580*v^3 - 109000*v^2 + 81040*v + 23451) / 639
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 3 − β 2 + 42 \beta_{3} - \beta_{2} + 42 β 3 − β 2 + 4 2
b3 - b2 + 42
ν 3 \nu^{3} ν 3 = = =
− β 6 − 2 β 5 − β 4 + 2 β 3 − β 2 + 79 β 1 + 5 -\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 79\beta _1 + 5 − β 6 − 2 β 5 − β 4 + 2 β 3 − β 2 + 7 9 β 1 + 5
-b6 - 2*b5 - b4 + 2*b3 - b2 + 79*b1 + 5
ν 4 \nu^{4} ν 4 = = =
− 7 β 6 − 8 β 5 − 16 β 4 + 94 β 3 − 114 β 2 + 48 β 1 + 3331 -7\beta_{6} - 8\beta_{5} - 16\beta_{4} + 94\beta_{3} - 114\beta_{2} + 48\beta _1 + 3331 − 7 β 6 − 8 β 5 − 1 6 β 4 + 9 4 β 3 − 1 1 4 β 2 + 4 8 β 1 + 3 3 3 1
-7*b6 - 8*b5 - 16*b4 + 94*b3 - 114*b2 + 48*b1 + 3331
ν 5 \nu^{5} ν 5 = = =
− 102 β 6 − 285 β 5 − 165 β 4 + 349 β 3 − 148 β 2 + 6709 β 1 + 2921 -102\beta_{6} - 285\beta_{5} - 165\beta_{4} + 349\beta_{3} - 148\beta_{2} + 6709\beta _1 + 2921 − 1 0 2 β 6 − 2 8 5 β 5 − 1 6 5 β 4 + 3 4 9 β 3 − 1 4 8 β 2 + 6 7 0 9 β 1 + 2 9 2 1
-102*b6 - 285*b5 - 165*b4 + 349*b3 - 148*b2 + 6709*b1 + 2921
ν 6 \nu^{6} ν 6 = = =
− 1039 β 6 − 1178 β 5 − 2326 β 4 + 8862 β 3 − 11720 β 2 + 10206 β 1 + 283145 -1039\beta_{6} - 1178\beta_{5} - 2326\beta_{4} + 8862\beta_{3} - 11720\beta_{2} + 10206\beta _1 + 283145 − 1 0 3 9 β 6 − 1 1 7 8 β 5 − 2 3 2 6 β 4 + 8 8 6 2 β 3 − 1 1 7 2 0 β 2 + 1 0 2 0 6 β 1 + 2 8 3 1 4 5
-1039*b6 - 1178*b5 - 2326*b4 + 8862*b3 - 11720*b2 + 10206*b1 + 283145
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
7 7 7
+ 1 +1 + 1
11 11 1 1
+ 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 4 n e w ( Γ 0 ( 616 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(616)) S 4 n e w ( Γ 0 ( 6 1 6 ) ) :
T 3 7 − 145 T 3 5 + 10 T 3 4 + 4790 T 3 3 + 2452 T 3 2 − 1496 T 3 − 320 T_{3}^{7} - 145T_{3}^{5} + 10T_{3}^{4} + 4790T_{3}^{3} + 2452T_{3}^{2} - 1496T_{3} - 320 T 3 7 − 1 4 5 T 3 5 + 1 0 T 3 4 + 4 7 9 0 T 3 3 + 2 4 5 2 T 3 2 − 1 4 9 6 T 3 − 3 2 0
T3^7 - 145*T3^5 + 10*T3^4 + 4790*T3^3 + 2452*T3^2 - 1496*T3 - 320
T 5 7 − 6 T 5 6 − 591 T 5 5 + 2228 T 5 4 + 107076 T 5 3 − 111752 T 5 2 − 6037280 T 5 − 14549536 T_{5}^{7} - 6T_{5}^{6} - 591T_{5}^{5} + 2228T_{5}^{4} + 107076T_{5}^{3} - 111752T_{5}^{2} - 6037280T_{5} - 14549536 T 5 7 − 6 T 5 6 − 5 9 1 T 5 5 + 2 2 2 8 T 5 4 + 1 0 7 0 7 6 T 5 3 − 1 1 1 7 5 2 T 5 2 − 6 0 3 7 2 8 0 T 5 − 1 4 5 4 9 5 3 6
T5^7 - 6*T5^6 - 591*T5^5 + 2228*T5^4 + 107076*T5^3 - 111752*T5^2 - 6037280*T5 - 14549536
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 7 T^{7} T 7
T^7
3 3 3
T 7 − 145 T 5 + ⋯ − 320 T^{7} - 145 T^{5} + \cdots - 320 T 7 − 1 4 5 T 5 + ⋯ − 3 2 0
T^7 - 145*T^5 + 10*T^4 + 4790*T^3 + 2452*T^2 - 1496*T - 320
5 5 5
T 7 − 6 T 6 + ⋯ − 14549536 T^{7} - 6 T^{6} + \cdots - 14549536 T 7 − 6 T 6 + ⋯ − 1 4 5 4 9 5 3 6
T^7 - 6*T^6 - 591*T^5 + 2228*T^4 + 107076*T^3 - 111752*T^2 - 6037280*T - 14549536
7 7 7
( T + 7 ) 7 (T + 7)^{7} ( T + 7 ) 7
(T + 7)^7
11 11 1 1
( T + 11 ) 7 (T + 11)^{7} ( T + 1 1 ) 7
(T + 11)^7
13 13 1 3
T 7 + ⋯ − 79161659136 T^{7} + \cdots - 79161659136 T 7 + ⋯ − 7 9 1 6 1 6 5 9 1 3 6
T^7 - 88*T^6 - 5570*T^5 + 444528*T^4 + 13868800*T^3 - 542018992*T^2 - 17733243584*T - 79161659136
17 17 1 7
T 7 + ⋯ + 151755555840 T^{7} + \cdots + 151755555840 T 7 + ⋯ + 1 5 1 7 5 5 5 5 5 8 4 0
T^7 - 134*T^6 - 7086*T^5 + 1464204*T^4 - 42464920*T^3 - 561585936*T^2 + 19509561728*T + 151755555840
19 19 1 9
T 7 + ⋯ − 15743217343488 T^{7} + \cdots - 15743217343488 T 7 + ⋯ − 1 5 7 4 3 2 1 7 3 4 3 4 8 8
T^7 - 14*T^6 - 32568*T^5 + 206040*T^4 + 350777040*T^3 + 864346304*T^2 - 1249215222656*T - 15743217343488
23 23 2 3
T 7 + ⋯ + 120191659904000 T^{7} + \cdots + 120191659904000 T 7 + ⋯ + 1 2 0 1 9 1 6 5 9 9 0 4 0 0 0
T^7 - 42*T^6 - 57327*T^5 + 3567936*T^4 + 867532456*T^3 - 70009628864*T^2 - 1338816653696*T + 120191659904000
29 29 2 9
T 7 + ⋯ + 2542483952512 T^{7} + \cdots + 2542483952512 T 7 + ⋯ + 2 5 4 2 4 8 3 9 5 2 5 1 2
T^7 - 482*T^6 + 57900*T^5 + 1297992*T^4 - 513498512*T^3 + 26714905632*T^2 - 491881428160*T + 2542483952512
31 31 3 1
T 7 + ⋯ − 5980457097664 T^{7} + \cdots - 5980457097664 T 7 + ⋯ − 5 9 8 0 4 5 7 0 9 7 6 6 4
T^7 - 50*T^6 - 43501*T^5 - 2969420*T^4 + 90935286*T^3 + 11551803088*T^2 + 102847772728*T - 5980457097664
37 37 3 7
T 7 + ⋯ − 16 ⋯ 72 T^{7} + \cdots - 16\!\cdots\!72 T 7 + ⋯ − 1 6 ⋯ 7 2
T^7 - 152*T^6 - 305927*T^5 + 49565910*T^4 + 24514937292*T^3 - 3954644636984*T^2 - 193615516281984*T - 1671990562943872
41 41 4 1
T 7 + ⋯ − 45 ⋯ 36 T^{7} + \cdots - 45\!\cdots\!36 T 7 + ⋯ − 4 5 ⋯ 3 6
T^7 - 234*T^6 - 238046*T^5 + 63952724*T^4 + 8307303464*T^3 - 3280282392624*T^2 + 230138923065344*T - 4515770199643136
43 43 4 3
T 7 + ⋯ + 32 ⋯ 28 T^{7} + \cdots + 32\!\cdots\!28 T 7 + ⋯ + 3 2 ⋯ 2 8
T^7 - 472*T^6 - 257300*T^5 + 84090640*T^4 + 24246280768*T^3 - 2720634231040*T^2 - 447023582428160*T + 32755184065302528
47 47 4 7
T 7 + ⋯ + 62 ⋯ 68 T^{7} + \cdots + 62\!\cdots\!68 T 7 + ⋯ + 6 2 ⋯ 6 8
T^7 - 728*T^6 - 195754*T^5 + 173861360*T^4 + 12356516440*T^3 - 10699476046848*T^2 - 524106396564480*T + 62191475628576768
53 53 5 3
T 7 + ⋯ + 47 ⋯ 96 T^{7} + \cdots + 47\!\cdots\!96 T 7 + ⋯ + 4 7 ⋯ 9 6
T^7 + 102*T^6 - 679992*T^5 - 32174800*T^4 + 106916628304*T^3 + 1798891594208*T^2 - 1305099795925504*T + 47251926943312896
59 59 5 9
T 7 + ⋯ − 11 ⋯ 76 T^{7} + \cdots - 11\!\cdots\!76 T 7 + ⋯ − 1 1 ⋯ 7 6
T^7 - 1704*T^6 + 424495*T^5 + 731489894*T^4 - 447265780354*T^3 - 14659981345524*T^2 + 61831625079816344*T - 11514526043687706176
61 61 6 1
T 7 + ⋯ + 49 ⋯ 96 T^{7} + \cdots + 49\!\cdots\!96 T 7 + ⋯ + 4 9 ⋯ 9 6
T^7 - 656*T^6 - 405366*T^5 + 313819168*T^4 - 19461087448*T^3 - 10924222065168*T^2 + 436042040186880*T + 4950023731442496
67 67 6 7
T 7 + ⋯ + 55 ⋯ 44 T^{7} + \cdots + 55\!\cdots\!44 T 7 + ⋯ + 5 5 ⋯ 4 4
T^7 - 1126*T^6 - 118599*T^5 + 474218504*T^4 - 103317315160*T^3 - 26330793140480*T^2 + 5828366912756736*T + 553875655986118144
71 71 7 1
T 7 + ⋯ + 10 ⋯ 44 T^{7} + \cdots + 10\!\cdots\!44 T 7 + ⋯ + 1 0 ⋯ 4 4
T^7 + 918*T^6 - 1095151*T^5 - 844184908*T^4 + 294726454424*T^3 + 151877544171936*T^2 - 29094938807414400*T + 1069588722849030144
73 73 7 3
T 7 + ⋯ − 90 ⋯ 40 T^{7} + \cdots - 90\!\cdots\!40 T 7 + ⋯ − 9 0 ⋯ 4 0
T^7 - 1094*T^6 - 1766614*T^5 + 2582868236*T^4 + 103860090776*T^3 - 1397318402776016*T^2 + 660825578565820992*T - 90325735881766352640
79 79 7 9
T 7 + ⋯ − 26 ⋯ 88 T^{7} + \cdots - 26\!\cdots\!88 T 7 + ⋯ − 2 6 ⋯ 8 8
T^7 - 672*T^6 - 1459984*T^5 + 982050048*T^4 + 373709766384*T^3 - 341342459482496*T^2 + 59837408496401408*T - 2680203356037644288
83 83 8 3
T 7 + ⋯ + 17 ⋯ 60 T^{7} + \cdots + 17\!\cdots\!60 T 7 + ⋯ + 1 7 ⋯ 6 0
T^7 - 782*T^6 - 1028748*T^5 + 1047555904*T^4 - 105538519584*T^3 - 95719017376640*T^2 + 18370436523516416*T + 178328409421148160
89 89 8 9
T 7 + ⋯ + 93 ⋯ 64 T^{7} + \cdots + 93\!\cdots\!64 T 7 + ⋯ + 9 3 ⋯ 6 4
T^7 + 464*T^6 - 2760631*T^5 - 1118220178*T^4 + 1818630900168*T^3 + 606485948970768*T^2 - 191100206519810032*T + 9335355776969482464
97 97 9 7
T 7 + ⋯ − 18 ⋯ 96 T^{7} + \cdots - 18\!\cdots\!96 T 7 + ⋯ − 1 8 ⋯ 9 6
T^7 - 3000*T^6 - 318799*T^5 + 7080114586*T^4 - 3584631472784*T^3 - 3148509436732448*T^2 + 2135876209171660144*T - 184305413897395723296
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