gp: [N,k,chi] = [396,2,Mod(23,396)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(396, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("396.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [60,0,0,0,0,13]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
The algebraic q q q -expansion of this newform has not been computed, but we have computed the trace expansion .
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 7 60 − 120 T 7 58 + 8109 T 7 56 − 2352 T 7 55 − 375352 T 7 54 + ⋯ + 96 ⋯ 04 T_{7}^{60} - 120 T_{7}^{58} + 8109 T_{7}^{56} - 2352 T_{7}^{55} - 375352 T_{7}^{54} + \cdots + 96\!\cdots\!04 T 7 6 0 − 1 2 0 T 7 5 8 + 8 1 0 9 T 7 5 6 − 2 3 5 2 T 7 5 5 − 3 7 5 3 5 2 T 7 5 4 + ⋯ + 9 6 ⋯ 0 4
T7^60 - 120*T7^58 + 8109*T7^56 - 2352*T7^55 - 375352*T7^54 + 238776*T7^53 + 13149573*T7^52 - 13856688*T7^51 - 361475802*T7^50 + 546135144*T7^49 + 7989561586*T7^48 - 16191257592*T7^47 - 142464811470*T7^46 + 373200275592*T7^45 + 2044460159655*T7^44 - 6869563648524*T7^43 - 23038476726598*T7^42 + 101438825534064*T7^41 + 194167251434460*T7^40 - 1207402965070380*T7^39 - 1031662137974418*T7^38 + 11457029421945408*T7^37 + 497404444390465*T7^36 - 86071575795527796*T7^35 + 54967125363559266*T7^34 + 498719591686904400*T7^33 - 623147030657157024*T7^32 - 2206990365358436484*T7^31 + 4213205068891594664*T7^30 + 7077154884578970528*T7^29 - 19811227046388032532*T7^28 - 15551230210311763872*T7^27 + 69979630752821430192*T7^26 + 15931768836041492400*T7^25 - 187429141392744498560*T7^24 + 26621351851542429600*T7^23 + 389284684800414551376*T7^22 - 166483441471720792896*T7^21 - 617100376274459718768*T7^20 + 404993683247986296096*T7^19 + 750819669144355970624*T7^18 - 654065515004639632896*T7^17 - 670443181721896767744*T7^16 + 735870277099089897984*T7^15 + 436592843031093954048*T7^14 - 608322643598785329408*T7^13 - 180107701452914149376*T7^12 + 353108232197933457408*T7^11 + 43483920606634303488*T7^10 - 148727234277148299264*T7^9 + 1950252540546130944*T7^8 + 40852436502554578944*T7^7 - 2435643876581425152*T7^6 - 7968727176645574656*T7^5 + 740555598938701824*T7^4 + 921605969368055808*T7^3 - 65600106475290624*T7^2 - 53898077087465472*T7 + 9684222637768704
acting on S 2 n e w ( 396 , [ χ ] ) S_{2}^{\mathrm{new}}(396, [\chi]) S 2 n e w ( 3 9 6 , [ χ ] ) .