[N,k,chi] = [384,9,Mod(319,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.319");
S:= CuspForms(chi, 9);
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
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
\(n\)
\(127\)
\(133\)
\(257\)
\(\chi(n)\)
\(-1\)
\(-1\)
\(1\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 2071744T_{5}^{6} + 1116378502656T_{5}^{4} + 74558195297075200T_{5}^{2} + 1260034623602728960000 \)
T5^8 + 2071744*T5^6 + 1116378502656*T5^4 + 74558195297075200*T5^2 + 1260034623602728960000
acting on \(S_{9}^{\mathrm{new}}(384, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} \)
T^16
$3$
\( (T^{2} - 2187)^{8} \)
(T^2 - 2187)^8
$5$
\( (T^{8} + 2071744 T^{6} + \cdots + 12\!\cdots\!00)^{2} \)
(T^8 + 2071744*T^6 + 1116378502656*T^4 + 74558195297075200*T^2 + 1260034623602728960000)^2
$7$
\( (T^{8} + 35034432 T^{6} + \cdots + 30\!\cdots\!56)^{2} \)
(T^8 + 35034432*T^6 + 351334635886080*T^4 + 881037280713285255168*T^2 + 304532783553265188570464256)^2
$11$
\( (T^{8} - 672057024 T^{6} + \cdots + 13\!\cdots\!04)^{2} \)
(T^8 - 672057024*T^6 + 69087682981467648*T^4 - 563143738760619552718848*T^2 + 13109836962995430756180885504)^2
$13$
\( (T^{8} + 5452084224 T^{6} + \cdots + 43\!\cdots\!00)^{2} \)
(T^8 + 5452084224*T^6 + 9505188669604921344*T^4 + 5226069649146914830653849600*T^2 + 43849495717690583674408402944000000)^2
$17$
\( (T^{4} + 53832 T^{3} + \cdots + 15\!\cdots\!36)^{4} \)
(T^4 + 53832*T^3 - 8623326312*T^2 - 138735423106272*T + 15634781487832517136)^4
$19$
\( (T^{8} - 81692003520 T^{6} + \cdots + 12\!\cdots\!64)^{2} \)
(T^8 - 81692003520*T^6 + 2335332232016557438464*T^4 - 28193129722208452758081295269888*T^2 + 122273767101192392907511312366441126232064)^2
$23$
\( (T^{8} + 232142554368 T^{6} + \cdots + 32\!\cdots\!96)^{2} \)
(T^8 + 232142554368*T^6 + 4695728106321631862784*T^4 + 26521418522624042401448567242752*T^2 + 32262961853962231915858449327546534199296)^2
$29$
\( (T^{8} + 1006791742144 T^{6} + \cdots + 12\!\cdots\!64)^{2} \)
(T^8 + 1006791742144*T^6 + 326753067503320332301824*T^4 + 38982716159545224059954331732459520*T^2 + 1262108103492840539893667530171194423905419264)^2
$31$
\( (T^{8} + 2311568773440 T^{6} + \cdots + 13\!\cdots\!04)^{2} \)
(T^8 + 2311568773440*T^6 + 1288300642277541222127104*T^4 + 233890485632918431602820453902532608*T^2 + 13204169746032941774380863759312968320856162304)^2
$37$
\( (T^{8} + 7665737345280 T^{6} + \cdots + 20\!\cdots\!16)^{2} \)
(T^8 + 7665737345280*T^6 + 14382806597612602816094208*T^4 + 9589945250070724407091736842752491520*T^2 + 2094529140888324049775316658543252072184797986816)^2
$41$
\( (T^{4} - 102024 T^{3} + \cdots - 98\!\cdots\!64)^{4} \)
(T^4 - 102024*T^3 - 21004802435304*T^2 - 38861685327043669536*T - 9884414170262080206638064)^4
$43$
\( (T^{8} - 69312972148416 T^{6} + \cdots + 16\!\cdots\!76)^{2} \)
(T^8 - 69312972148416*T^6 + 1324658056998588413391115776*T^4 - 4294805310270896737792457846348533776384*T^2 + 1634818137535224830671148926526522245063408159358976)^2
$47$
\( (T^{8} + 76848245427456 T^{6} + \cdots + 51\!\cdots\!76)^{2} \)
(T^8 + 76848245427456*T^6 + 2007732382161298812411273216*T^4 + 19868423830922087172135357005123162210304*T^2 + 51445718260254376373115697143391287676277697256882176)^2
$53$
\( (T^{8} + 316051586123968 T^{6} + \cdots + 12\!\cdots\!00)^{2} \)
(T^8 + 316051586123968*T^6 + 29809799270622544758114080256*T^4 + 1070411679109233421233908825587780669849600*T^2 + 12567217866724711843175497260328072365903220577443840000)^2
$59$
\( (T^{8} - 587728140311232 T^{6} + \cdots + 14\!\cdots\!96)^{2} \)
(T^8 - 587728140311232*T^6 + 110414419128688670403788035584*T^4 - 7422067889698994158986098176104583163854848*T^2 + 143438185137333632459368660291659812132069957034243588096)^2
$61$
\( (T^{8} + 835172191745280 T^{6} + \cdots + 20\!\cdots\!24)^{2} \)
(T^8 + 835172191745280*T^6 + 226645520627097994318118608896*T^4 + 20953951078048596641094910474135668941389824*T^2 + 200542111104015480403915478625452901988045754208589185024)^2
$67$
\( (T^{8} - 741259303895232 T^{6} + \cdots + 43\!\cdots\!04)^{2} \)
(T^8 - 741259303895232*T^6 + 185450938637574472616746169856*T^4 - 17337396091009915223568329618992309136179200*T^2 + 436071189323324563180337540698905122475955626231666376704)^2
$71$
\( (T^{8} + \cdots + 87\!\cdots\!76)^{2} \)
(T^8 + 3364639161378048*T^6 + 3246716953973881030256477429760*T^4 + 812109989918379183572239233018825075606945792*T^2 + 8705378789361249311670065176732469601065062300884167294976)^2
$73$
\( (T^{4} + 20571784 T^{3} + \cdots - 79\!\cdots\!92)^{4} \)
(T^4 + 20571784*T^3 - 1519599821177064*T^2 + 20175694319384779042336*T - 79234735954120400062099319792)^4
$79$
\( (T^{8} + \cdots + 85\!\cdots\!76)^{2} \)
(T^8 + 7004624111012160*T^6 + 15406106505557268193153579095552*T^4 + 10599425882753380455871461296489166800420290560*T^2 + 852151338181494756486893345939807070619460215182663577829376)^2
$83$
\( (T^{8} + \cdots + 67\!\cdots\!76)^{2} \)
(T^8 - 5914408380930240*T^6 + 10733673793929491367810874455552*T^4 - 5870957467673954057767951475850993067344445440*T^2 + 679488161865237959564981301769789158907648691266796037144576)^2
$89$
\( (T^{4} + 66824520 T^{3} + \cdots - 44\!\cdots\!24)^{4} \)
(T^4 + 66824520*T^3 - 37380033499752*T^2 - 55620535789120372996320*T - 445681559743179452169736445424)^4
$97$
\( (T^{4} - 123233720 T^{3} + \cdots + 33\!\cdots\!00)^{4} \)
(T^4 - 123233720*T^3 - 9530019473284200*T^2 + 632141612110385859892000*T + 33913656309196795300235589610000)^4
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