[N,k,chi] = [46,14,Mod(1,46)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("46.1");
S:= CuspForms(chi, 14);
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
sage: f.q_expansion() # note that sage often uses an isomorphic number field
Coefficients of the q q q -expansion are expressed in terms of a basis 1 , β 1 , β 2 , β 3 , β 4 1,\beta_1,\beta_2,\beta_3,\beta_4 1 , β 1 , β 2 , β 3 , β 4 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 5 − x 4 − 3257825 x 3 + 1372618617 x 2 + 279354108456 x − 130815431589168 x^{5} - x^{4} - 3257825x^{3} + 1372618617x^{2} + 279354108456x - 130815431589168 x 5 − x 4 − 3 2 5 7 8 2 5 x 3 + 1 3 7 2 6 1 8 6 1 7 x 2 + 2 7 9 3 5 4 1 0 8 4 5 6 x − 1 3 0 8 1 5 4 3 1 5 8 9 1 6 8
x^5 - x^4 - 3257825*x^3 + 1372618617*x^2 + 279354108456*x - 130815431589168
:
β 1 \beta_{1} β 1 = = =
( − 12481 ν 4 − 12829265 ν 3 + 32956482977 ν 2 + 17271508397517 ν − 3307289103907668 ) / 4598290128456 ( -12481\nu^{4} - 12829265\nu^{3} + 32956482977\nu^{2} + 17271508397517\nu - 3307289103907668 ) / 4598290128456 ( − 1 2 4 8 1 ν 4 − 1 2 8 2 9 2 6 5 ν 3 + 3 2 9 5 6 4 8 2 9 7 7 ν 2 + 1 7 2 7 1 5 0 8 3 9 7 5 1 7 ν − 3 3 0 7 2 8 9 1 0 3 9 0 7 6 6 8 ) / 4 5 9 8 2 9 0 1 2 8 4 5 6
(-12481*v^4 - 12829265*v^3 + 32956482977*v^2 + 17271508397517*v - 3307289103907668) / 4598290128456
β 2 \beta_{2} β 2 = = =
( 259893 ν 4 − 582275233 ν 3 − 659872651633 ν 2 + ⋯ − 66 ⋯ 20 ) / 16860397137672 ( 259893 \nu^{4} - 582275233 \nu^{3} - 659872651633 \nu^{2} + \cdots - 66\!\cdots\!20 ) / 16860397137672 ( 2 5 9 8 9 3 ν 4 − 5 8 2 2 7 5 2 3 3 ν 3 − 6 5 9 8 7 2 6 5 1 6 3 3 ν 2 + ⋯ − 6 6 ⋯ 2 0 ) / 1 6 8 6 0 3 9 7 1 3 7 6 7 2
(259893*v^4 - 582275233*v^3 - 659872651633*v^2 + 2611849836959193*v - 664001303582951220) / 16860397137672
β 3 \beta_{3} β 3 = = =
( − 3998666 ν 4 − 1423823965 ν 3 + 12709676152996 ν 2 + ⋯ − 16 ⋯ 06 ) / 12645297853254 ( - 3998666 \nu^{4} - 1423823965 \nu^{3} + 12709676152996 \nu^{2} + \cdots - 16\!\cdots\!06 ) / 12645297853254 ( − 3 9 9 8 6 6 6 ν 4 − 1 4 2 3 8 2 3 9 6 5 ν 3 + 1 2 7 0 9 6 7 6 1 5 2 9 9 6 ν 2 + ⋯ − 1 6 ⋯ 0 6 ) / 1 2 6 4 5 2 9 7 8 5 3 2 5 4
(-3998666*v^4 - 1423823965*v^3 + 12709676152996*v^2 - 1501866419745993*v - 1654098019398291906) / 12645297853254
β 4 \beta_{4} β 4 = = =
( − 23195195 ν 4 − 8660318164 ν 3 + 72151790122993 ν 2 + ⋯ − 78 ⋯ 20 ) / 25290595706508 ( - 23195195 \nu^{4} - 8660318164 \nu^{3} + 72151790122993 \nu^{2} + \cdots - 78\!\cdots\!20 ) / 25290595706508 ( − 2 3 1 9 5 1 9 5 ν 4 − 8 6 6 0 3 1 8 1 6 4 ν 3 + 7 2 1 5 1 7 9 0 1 2 2 9 9 3 ν 2 + ⋯ − 7 8 ⋯ 2 0 ) / 2 5 2 9 0 5 9 5 7 0 6 5 0 8
(-23195195*v^4 - 8660318164*v^3 + 72151790122993*v^2 - 4644954616872354*v - 7874693169365665620) / 25290595706508
ν \nu ν = = =
( 11 β 4 − 28 β 3 + 13 β 2 − 381 β 1 + 385 ) / 1888 ( 11\beta_{4} - 28\beta_{3} + 13\beta_{2} - 381\beta _1 + 385 ) / 1888 ( 1 1 β 4 − 2 8 β 3 + 1 3 β 2 − 3 8 1 β 1 + 3 8 5 ) / 1 8 8 8
(11*b4 - 28*b3 + 13*b2 - 381*b1 + 385) / 1888
ν 2 \nu^{2} ν 2 = = =
( − 5815 β 4 + 20144 β 3 + 15655 β 2 − 293031 β 1 + 1230145423 ) / 944 ( -5815\beta_{4} + 20144\beta_{3} + 15655\beta_{2} - 293031\beta _1 + 1230145423 ) / 944 ( − 5 8 1 5 β 4 + 2 0 1 4 4 β 3 + 1 5 6 5 5 β 2 − 2 9 3 0 3 1 β 1 + 1 2 3 0 1 4 5 4 2 3 ) / 9 4 4
(-5815*b4 + 20144*b3 + 15655*b2 - 293031*b1 + 1230145423) / 944
ν 3 \nu^{3} ν 3 = = =
( 38119673 β 4 − 96700036 β 3 + 8974447 β 2 − 1563865983 β 1 − 1551179111813 ) / 1888 ( 38119673\beta_{4} - 96700036\beta_{3} + 8974447\beta_{2} - 1563865983\beta _1 - 1551179111813 ) / 1888 ( 3 8 1 1 9 6 7 3 β 4 − 9 6 7 0 0 0 3 6 β 3 + 8 9 7 4 4 4 7 β 2 − 1 5 6 3 8 6 5 9 8 3 β 1 − 1 5 5 1 1 7 9 1 1 1 8 1 3 ) / 1 8 8 8
(38119673*b4 - 96700036*b3 + 8974447*b2 - 1563865983*b1 - 1551179111813) / 1888
ν 4 \nu^{4} ν 4 = = =
( − 6833834341 β 4 + 20879125655 β 3 + 11429988907 β 2 − 145354014783 β 1 + 948897413820376 ) / 236 ( - 6833834341 \beta_{4} + 20879125655 \beta_{3} + 11429988907 \beta_{2} - 145354014783 \beta _1 + 948897413820376 ) / 236 ( − 6 8 3 3 8 3 4 3 4 1 β 4 + 2 0 8 7 9 1 2 5 6 5 5 β 3 + 1 1 4 2 9 9 8 8 9 0 7 β 2 − 1 4 5 3 5 4 0 1 4 7 8 3 β 1 + 9 4 8 8 9 7 4 1 3 8 2 0 3 7 6 ) / 2 3 6
(-6833834341*b4 + 20879125655*b3 + 11429988907*b2 - 145354014783*b1 + 948897413820376) / 236
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
2 2 2
+ 1 +1 + 1
23 23 2 3
+ 1 +1 + 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 3 5 − 690 T 3 4 − 5509108 T 3 3 + 924234162 T 3 2 + 6086476932219 T 3 + 681843043869048 T_{3}^{5} - 690T_{3}^{4} - 5509108T_{3}^{3} + 924234162T_{3}^{2} + 6086476932219T_{3} + 681843043869048 T 3 5 − 6 9 0 T 3 4 − 5 5 0 9 1 0 8 T 3 3 + 9 2 4 2 3 4 1 6 2 T 3 2 + 6 0 8 6 4 7 6 9 3 2 2 1 9 T 3 + 6 8 1 8 4 3 0 4 3 8 6 9 0 4 8
T3^5 - 690*T3^4 - 5509108*T3^3 + 924234162*T3^2 + 6086476932219*T3 + 681843043869048
acting on S 14 n e w ( Γ 0 ( 46 ) ) S_{14}^{\mathrm{new}}(\Gamma_0(46)) S 1 4 n e w ( Γ 0 ( 4 6 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T + 64 ) 5 (T + 64)^{5} ( T + 6 4 ) 5
(T + 64)^5
3 3 3
T 5 + ⋯ + 681843043869048 T^{5} + \cdots + 681843043869048 T 5 + ⋯ + 6 8 1 8 4 3 0 4 3 8 6 9 0 4 8
T^5 - 690*T^4 - 5509108*T^3 + 924234162*T^2 + 6086476932219*T + 681843043869048
5 5 5
T 5 + ⋯ + 40 ⋯ 00 T^{5} + \cdots + 40\!\cdots\!00 T 5 + ⋯ + 4 0 ⋯ 0 0
T^5 + 71798*T^4 - 2563687964*T^3 - 242197416112560*T^2 - 2565326163474880000*T + 40376328432605020800000
7 7 7
T 5 + ⋯ + 86 ⋯ 24 T^{5} + \cdots + 86\!\cdots\!24 T 5 + ⋯ + 8 6 ⋯ 2 4
T^5 - 31366*T^4 - 132178292016*T^3 - 13282398310785864*T^2 + 304971586094881384784*T + 8698711105689323287503424
11 11 1 1
T 5 + ⋯ − 25 ⋯ 40 T^{5} + \cdots - 25\!\cdots\!40 T 5 + ⋯ − 2 5 ⋯ 4 0
T^5 + 7592076*T^4 - 68774047084952*T^3 - 850794859148175893688*T^2 - 2715633250287228301501570176*T - 2565286902379741901201828353175040
13 13 1 3
T 5 + ⋯ − 14 ⋯ 74 T^{5} + \cdots - 14\!\cdots\!74 T 5 + ⋯ − 1 4 ⋯ 7 4
T^5 - 28315858*T^4 - 236865971782650*T^3 + 6789533801269105868616*T^2 + 28451027041180550453392258865*T - 149123588546843565736968082407635774
17 17 1 7
T 5 + ⋯ + 60 ⋯ 20 T^{5} + \cdots + 60\!\cdots\!20 T 5 + ⋯ + 6 0 ⋯ 2 0
T^5 + 148779352*T^4 - 4790970630743876*T^3 - 794440598565124315291656*T^2 + 14648275758500301245922280138752*T + 605224739159455769153961201368452674720
19 19 1 9
T 5 + ⋯ − 34 ⋯ 12 T^{5} + \cdots - 34\!\cdots\!12 T 5 + ⋯ − 3 4 ⋯ 1 2
T^5 - 208568226*T^4 - 134336979037531224*T^3 + 35362562655529119384379120*T^2 + 185347303993951437197489014475856*T - 344109665172508029419670126593982208554912
23 23 2 3
( T + 148035889 ) 5 (T + 148035889)^{5} ( T + 1 4 8 0 3 5 8 8 9 ) 5
(T + 148035889)^5
29 29 2 9
T 5 + ⋯ + 10 ⋯ 30 T^{5} + \cdots + 10\!\cdots\!30 T 5 + ⋯ + 1 0 ⋯ 3 0
T^5 + 2797854522*T^4 - 18764820002533108946*T^3 - 35886514031495067097696603776*T^2 + 85555645379331760718016659123681645577*T + 108655802072988210426949694991383767229884527630
31 31 3 1
T 5 + ⋯ − 15 ⋯ 00 T^{5} + \cdots - 15\!\cdots\!00 T 5 + ⋯ − 1 5 ⋯ 0 0
T^5 - 10251920410*T^4 - 6158128297456846324*T^3 + 247052087271918250797856009474*T^2 - 320057803850237290730525449756484342341*T - 153888476044387532054416577178288336200158956400
37 37 3 7
T 5 + ⋯ − 58 ⋯ 28 T^{5} + \cdots - 58\!\cdots\!28 T 5 + ⋯ − 5 8 ⋯ 2 8
T^5 - 15840804470*T^4 - 179587508618227148444*T^3 + 2453038756337383793124998010352*T^2 + 5115307831756294830398789718123352588800*T - 58747303744793197555639503641680729459083841302528
41 41 4 1
T 5 + ⋯ − 52 ⋯ 18 T^{5} + \cdots - 52\!\cdots\!18 T 5 + ⋯ − 5 2 ⋯ 1 8
T^5 + 63837021618*T^4 + 716474727501434440018*T^3 - 29345104054397432604340698920844*T^2 - 783162058121006334803967236107882966529291*T - 5222734335352029847433154059656611133735447715882718
43 43 4 3
T 5 + ⋯ + 16 ⋯ 00 T^{5} + \cdots + 16\!\cdots\!00 T 5 + ⋯ + 1 6 ⋯ 0 0
T^5 + 100763331186*T^4 + 3016332937910151700192*T^3 + 33825578977460401217120891394176*T^2 + 145184434585934536100294926616766777200640*T + 166043856808434908773005647370604007969112794726400
47 47 4 7
T 5 + ⋯ + 13 ⋯ 72 T^{5} + \cdots + 13\!\cdots\!72 T 5 + ⋯ + 1 3 ⋯ 7 2
T^5 + 174976267710*T^4 + 3086109022167707194828*T^3 - 477633771667832407131705654754038*T^2 - 8122053751959531407322482228103523848045173*T + 137283395076646892134334401085842262870350151067605472
53 53 5 3
T 5 + ⋯ − 27 ⋯ 08 T^{5} + \cdots - 27\!\cdots\!08 T 5 + ⋯ − 2 7 ⋯ 0 8
T^5 - 89802813568*T^4 - 42813239835089395476304*T^3 + 1910780589123909329638476805295120*T^2 + 152457252156002937904991587159012357838628784*T - 2744644541582859918724908705168951848156153596116249408
59 59 5 9
T 5 + ⋯ − 15 ⋯ 40 T^{5} + \cdots - 15\!\cdots\!40 T 5 + ⋯ − 1 5 ⋯ 4 0
T^5 + 1238416781664*T^4 + 440444067671416223482480*T^3 + 11678184461467717156748626256505792*T^2 - 15438223032529577088783544815112980160224983040*T - 1596099837938500615814174698339747154698285201170715991040
61 61 6 1
T 5 + ⋯ − 20 ⋯ 52 T^{5} + \cdots - 20\!\cdots\!52 T 5 + ⋯ − 2 0 ⋯ 5 2
T^5 + 1267556050856*T^4 + 271247889238154504440200*T^3 - 141955148261195578886396544774965184*T^2 - 48578423922114076098182042826997232145290132656*T - 2056560187715281777432305031976202775734625083497180596352
67 67 6 7
T 5 + ⋯ − 32 ⋯ 80 T^{5} + \cdots - 32\!\cdots\!80 T 5 + ⋯ − 3 2 ⋯ 8 0
T^5 + 1575419391340*T^4 + 612218737500645888743304*T^3 + 65366644794855121388937807746366904*T^2 - 907851532030370139104016340235103748754680320*T - 32809715607370391937274171065634080138051842564437217280
71 71 7 1
T 5 + ⋯ − 92 ⋯ 36 T^{5} + \cdots - 92\!\cdots\!36 T 5 + ⋯ − 9 2 ⋯ 3 6
T^5 + 1683993935490*T^4 - 2397841566294812431843304*T^3 - 6438165130586640509180221022223493290*T^2 - 4316000092386603036738238107896329465059862884033*T - 925497351079977898881612018768259464556885658084607252914736
73 73 7 3
T 5 + ⋯ + 63 ⋯ 46 T^{5} + \cdots + 63\!\cdots\!46 T 5 + ⋯ + 6 3 ⋯ 4 6
T^5 + 1127194561170*T^4 - 1234931191300824196987694*T^3 - 853622077914322350550944433496637500*T^2 + 415142932783808424220988870099351127636017114037*T + 63525416826732092799775156819913518992340879879611447136546
79 79 7 9
T 5 + ⋯ + 51 ⋯ 16 T^{5} + \cdots + 51\!\cdots\!16 T 5 + ⋯ + 5 1 ⋯ 1 6
T^5 + 1151083218244*T^4 - 13622784161305402409613464*T^3 - 17902512810929235013063200539625321632*T^2 + 21588284605675569880648411386766558193512274942160*T + 5182910070061633372698385477586234542842608918507793594436416
83 83 8 3
T 5 + ⋯ + 86 ⋯ 76 T^{5} + \cdots + 86\!\cdots\!76 T 5 + ⋯ + 8 6 ⋯ 7 6
T^5 + 846457405404*T^4 - 11433366991477630676383084*T^3 - 10157531043802504710142594869533509000*T^2 - 227184910359642962833157130263784657665266278880*T + 869052188478681349610044609698036278468660745746702160103776
89 89 8 9
T 5 + ⋯ + 94 ⋯ 04 T^{5} + \cdots + 94\!\cdots\!04 T 5 + ⋯ + 9 4 ⋯ 0 4
T^5 + 11559404717682*T^4 - 47879470017429411025569224*T^3 - 787093485655579401219892718310529335456*T^2 - 242145773712058127215370690123285682212927655386112*T + 9494307175698639826682579651907762695093080396263893888039215104
97 97 9 7
T 5 + ⋯ + 32 ⋯ 88 T^{5} + \cdots + 32\!\cdots\!88 T 5 + ⋯ + 3 2 ⋯ 8 8
T^5 + 2048127950232*T^4 - 135453993827329530413453900*T^3 - 210088234622029308719989739160695520472*T^2 + 2276009612801660826287753708107416672318542501028576*T + 3261161157309828620403135251343991007161173121711458659131329888
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