[N,k,chi] = [735,4,Mod(1,735)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("735.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
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.
Refresh table
p p p
Sign
3 3 3
− 1 -1 − 1
5 5 5
− 1 -1 − 1
7 7 7
+ 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 ( 735 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(735)) S 4 n e w ( Γ 0 ( 7 3 5 ) ) :
T 2 8 − 2 T 2 7 − 55 T 2 6 + 80 T 2 5 + 969 T 2 4 − 866 T 2 3 − 5783 T 2 2 + 2328 T 2 + 9992 T_{2}^{8} - 2T_{2}^{7} - 55T_{2}^{6} + 80T_{2}^{5} + 969T_{2}^{4} - 866T_{2}^{3} - 5783T_{2}^{2} + 2328T_{2} + 9992 T 2 8 − 2 T 2 7 − 5 5 T 2 6 + 8 0 T 2 5 + 9 6 9 T 2 4 − 8 6 6 T 2 3 − 5 7 8 3 T 2 2 + 2 3 2 8 T 2 + 9 9 9 2
T2^8 - 2*T2^7 - 55*T2^6 + 80*T2^5 + 969*T2^4 - 866*T2^3 - 5783*T2^2 + 2328*T2 + 9992
T 11 8 − 64 T 11 7 − 3258 T 11 6 + 304584 T 11 5 − 4199384 T 11 4 + ⋯ − 9346926848 T_{11}^{8} - 64 T_{11}^{7} - 3258 T_{11}^{6} + 304584 T_{11}^{5} - 4199384 T_{11}^{4} + \cdots - 9346926848 T 1 1 8 − 6 4 T 1 1 7 − 3 2 5 8 T 1 1 6 + 3 0 4 5 8 4 T 1 1 5 − 4 1 9 9 3 8 4 T 1 1 4 + ⋯ − 9 3 4 6 9 2 6 8 4 8
T11^8 - 64*T11^7 - 3258*T11^6 + 304584*T11^5 - 4199384*T11^4 - 89239488*T11^3 + 1290867456*T11^2 + 11889915392*T11 - 9346926848
T 13 8 − 10464 T 13 6 + 36160 T 13 5 + 22566928 T 13 4 + 176712704 T 13 3 + ⋯ + 211872661504 T_{13}^{8} - 10464 T_{13}^{6} + 36160 T_{13}^{5} + 22566928 T_{13}^{4} + 176712704 T_{13}^{3} + \cdots + 211872661504 T 1 3 8 − 1 0 4 6 4 T 1 3 6 + 3 6 1 6 0 T 1 3 5 + 2 2 5 6 6 9 2 8 T 1 3 4 + 1 7 6 7 1 2 7 0 4 T 1 3 3 + ⋯ + 2 1 1 8 7 2 6 6 1 5 0 4
T13^8 - 10464*T13^6 + 36160*T13^5 + 22566928*T13^4 + 176712704*T13^3 - 6448530432*T13^2 + 609058816*T13 + 211872661504
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 − 2 T 7 + ⋯ + 9992 T^{8} - 2 T^{7} + \cdots + 9992 T 8 − 2 T 7 + ⋯ + 9 9 9 2
T^8 - 2*T^7 - 55*T^6 + 80*T^5 + 969*T^4 - 866*T^3 - 5783*T^2 + 2328*T + 9992
3 3 3
( T − 3 ) 8 (T - 3)^{8} ( T − 3 ) 8
(T - 3)^8
5 5 5
( T − 5 ) 8 (T - 5)^{8} ( T − 5 ) 8
(T - 5)^8
7 7 7
T 8 T^{8} T 8
T^8
11 11 1 1
T 8 + ⋯ − 9346926848 T^{8} + \cdots - 9346926848 T 8 + ⋯ − 9 3 4 6 9 2 6 8 4 8
T^8 - 64*T^7 - 3258*T^6 + 304584*T^5 - 4199384*T^4 - 89239488*T^3 + 1290867456*T^2 + 11889915392*T - 9346926848
13 13 1 3
T 8 + ⋯ + 211872661504 T^{8} + \cdots + 211872661504 T 8 + ⋯ + 2 1 1 8 7 2 6 6 1 5 0 4
T^8 - 10464*T^6 + 36160*T^5 + 22566928*T^4 + 176712704*T^3 - 6448530432*T^2 + 609058816*T + 211872661504
17 17 1 7
T 8 + ⋯ − 6565556795392 T^{8} + \cdots - 6565556795392 T 8 + ⋯ − 6 5 6 5 5 5 6 7 9 5 3 9 2
T^8 - 48*T^7 - 19082*T^6 + 366224*T^5 + 121695024*T^4 + 494678400*T^3 - 257323178496*T^2 - 5780260792320*T - 6565556795392
19 19 1 9
T 8 + ⋯ − 129732409433664 T^{8} + \cdots - 129732409433664 T 8 + ⋯ − 1 2 9 7 3 2 4 0 9 4 3 3 6 6 4
T^8 - 80*T^7 - 26886*T^6 + 2943056*T^5 + 106255372*T^4 - 24461986752*T^3 + 1003969977624*T^2 - 9253374205248*T - 129732409433664
23 23 2 3
T 8 + ⋯ + 84 ⋯ 84 T^{8} + \cdots + 84\!\cdots\!84 T 8 + ⋯ + 8 4 ⋯ 8 4
T^8 - 120*T^7 - 53694*T^6 + 6097912*T^5 + 827746708*T^4 - 75143487616*T^3 - 4639922921832*T^2 + 186395987356768*T + 8425156207247584
29 29 2 9
T 8 + ⋯ + 17 ⋯ 96 T^{8} + \cdots + 17\!\cdots\!96 T 8 + ⋯ + 1 7 ⋯ 9 6
T^8 - 76*T^7 - 120742*T^6 + 11112680*T^5 + 3660533240*T^4 - 317483671808*T^3 - 21793545387424*T^2 + 485782861592704*T + 17282495932391296
31 31 3 1
T 8 + ⋯ + 17 ⋯ 04 T^{8} + \cdots + 17\!\cdots\!04 T 8 + ⋯ + 1 7 ⋯ 0 4
T^8 - 20*T^7 - 124092*T^6 + 479544*T^5 + 4080244352*T^4 + 3412190160*T^3 - 21935249679888*T^2 + 252630352929312*T + 17648421191019504
37 37 3 7
T 8 + ⋯ + 22 ⋯ 88 T^{8} + \cdots + 22\!\cdots\!88 T 8 + ⋯ + 2 2 ⋯ 8 8
T^8 - 348*T^7 - 181038*T^6 + 66890824*T^5 + 8726116456*T^4 - 3608598022528*T^3 - 98777762589728*T^2 + 44469914250484352*T + 2220449570653353088
41 41 4 1
T 8 + ⋯ − 60 ⋯ 12 T^{8} + \cdots - 60\!\cdots\!12 T 8 + ⋯ − 6 0 ⋯ 1 2
T^8 - 944*T^7 - 392*T^6 + 219484960*T^5 - 43885058432*T^4 - 10322980702720*T^3 + 2474688871839360*T^2 + 99021843002522112*T - 6060869008286150912
43 43 4 3
T 8 + ⋯ − 12 ⋯ 00 T^{8} + \cdots - 12\!\cdots\!00 T 8 + ⋯ − 1 2 ⋯ 0 0
T^8 - 1116*T^7 + 198114*T^6 + 195970544*T^5 - 105249720512*T^4 + 20852873218432*T^3 - 1928205972551552*T^2 + 81597990410905600*T - 1254772986077081600
47 47 4 7
T 8 + ⋯ + 28 ⋯ 48 T^{8} + \cdots + 28\!\cdots\!48 T 8 + ⋯ + 2 8 ⋯ 4 8
T^8 + 208*T^7 - 419802*T^6 - 43095848*T^5 + 51278536312*T^4 - 122343327552*T^3 - 1167727087978848*T^2 + 42186898578183552*T + 2852448017701938048
53 53 5 3
T 8 + ⋯ − 18 ⋯ 68 T^{8} + \cdots - 18\!\cdots\!68 T 8 + ⋯ − 1 8 ⋯ 6 8
T^8 - 1144*T^7 - 12718*T^6 + 382400728*T^5 - 85717441900*T^4 - 26862453878784*T^3 + 8010498272288760*T^2 + 509926299093237600*T - 189726556779865241568
59 59 5 9
T 8 + ⋯ + 13 ⋯ 16 T^{8} + \cdots + 13\!\cdots\!16 T 8 + ⋯ + 1 3 ⋯ 1 6
T^8 - 596*T^7 - 409148*T^6 + 231948192*T^5 + 61317501344*T^4 - 27252800983808*T^3 - 4468173729571072*T^2 + 904383648743184384*T + 137488515362392508416
61 61 6 1
T 8 + ⋯ + 76 ⋯ 04 T^{8} + \cdots + 76\!\cdots\!04 T 8 + ⋯ + 7 6 ⋯ 0 4
T^8 - 740*T^7 - 919586*T^6 + 685775128*T^5 + 196050482868*T^4 - 147501574178224*T^3 - 15185336184564056*T^2 + 8490441194547101216*T + 761282508029080647904
67 67 6 7
T 8 + ⋯ − 46 ⋯ 56 T^{8} + \cdots - 46\!\cdots\!56 T 8 + ⋯ − 4 6 ⋯ 5 6
T^8 - 1964*T^7 + 877210*T^6 + 420541872*T^5 - 382327659792*T^4 + 59895233936384*T^3 + 4418328215885696*T^2 - 827836451275359232*T - 46565771056767097856
71 71 7 1
T 8 + ⋯ − 79 ⋯ 44 T^{8} + \cdots - 79\!\cdots\!44 T 8 + ⋯ − 7 9 ⋯ 4 4
T^8 + 4*T^7 - 628732*T^6 + 62826272*T^5 + 47676458592*T^4 - 6864724619264*T^3 - 796565698791296*T^2 + 180442380699079680*T - 7987496752582073344
73 73 7 3
T 8 + ⋯ + 36 ⋯ 72 T^{8} + \cdots + 36\!\cdots\!72 T 8 + ⋯ + 3 6 ⋯ 7 2
T^8 + 1500*T^7 - 237644*T^6 - 1205882656*T^5 - 603058966064*T^4 - 73366900880448*T^3 + 18543007404714816*T^2 + 5389729058478306816*T + 361436718775318428672
79 79 7 9
T 8 + ⋯ + 50 ⋯ 64 T^{8} + \cdots + 50\!\cdots\!64 T 8 + ⋯ + 5 0 ⋯ 6 4
T^8 + 460*T^7 - 2196436*T^6 - 825171936*T^5 + 1512961486128*T^4 + 454627214110912*T^3 - 327756724935475648*T^2 - 81872035346077072384*T + 5002532184034419544064
83 83 8 3
T 8 + ⋯ + 24 ⋯ 32 T^{8} + \cdots + 24\!\cdots\!32 T 8 + ⋯ + 2 4 ⋯ 3 2
T^8 - 700*T^7 - 1578244*T^6 + 1780024768*T^5 - 393979690928*T^4 - 120407414100672*T^3 + 58254868395206208*T^2 - 7150678692595158528*T + 249717619588316447232
89 89 8 9
T 8 + ⋯ − 36 ⋯ 32 T^{8} + \cdots - 36\!\cdots\!32 T 8 + ⋯ − 3 6 ⋯ 3 2
T^8 - 2136080*T^6 - 205045376*T^5 + 715181542880*T^4 + 94318594663424*T^3 - 47769296491461888*T^2 - 8999033988289005568*T - 364840298355763058432
97 97 9 7
T 8 + ⋯ − 23 ⋯ 48 T^{8} + \cdots - 23\!\cdots\!48 T 8 + ⋯ − 2 3 ⋯ 4 8
T^8 + 2052*T^7 - 1879356*T^6 - 6107489152*T^5 - 3139581570064*T^4 + 98877211366464*T^3 + 304250859468431424*T^2 + 33037654387263178752*T - 2385256516316893352448
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