[N,k,chi] = [78,16,Mod(1,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.1");
S:= CuspForms(chi, 16);
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 \(\iota_m\) 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
\( p \)
Sign
\(2\)
\(-1\)
\(3\)
\(1\)
\(13\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 19180T_{5}^{3} - 99032909880T_{5}^{2} + 6419103628128800T_{5} + 1106094956596202320000 \)
T5^4 - 19180*T5^3 - 99032909880*T5^2 + 6419103628128800*T5 + 1106094956596202320000
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(78))\).
$p$
$F_p(T)$
$2$
\( (T - 128)^{4} \)
(T - 128)^4
$3$
\( (T + 2187)^{4} \)
(T + 2187)^4
$5$
\( T^{4} - 19180 T^{3} + \cdots + 11\!\cdots\!00 \)
T^4 - 19180*T^3 - 99032909880*T^2 + 6419103628128800*T + 1106094956596202320000
$7$
\( T^{4} - 2510980 T^{3} + \cdots - 10\!\cdots\!68 \)
T^4 - 2510980*T^3 - 12563851566504*T^2 + 31117045235130113504*T - 10803507844324269463587968
$11$
\( T^{4} + 41108920 T^{3} + \cdots + 24\!\cdots\!00 \)
T^4 + 41108920*T^3 - 8175013593417480*T^2 + 4423057358558720015200*T + 2407488131185304485249246810000
$13$
\( (T + 62748517)^{4} \)
(T + 62748517)^4
$17$
\( T^{4} + 2487508320 T^{3} + \cdots - 15\!\cdots\!00 \)
T^4 + 2487508320*T^3 - 6362408722247237880*T^2 - 22183743228948774106062552000*T - 15141623536920206761643115883839150000
$19$
\( T^{4} + 11351292132 T^{3} + \cdots - 24\!\cdots\!00 \)
T^4 + 11351292132*T^3 + 41268270273258915000*T^2 + 41407689545706772119546156960*T - 24178087529640506118461347656066211200
$23$
\( T^{4} + 6738531488 T^{3} + \cdots + 61\!\cdots\!00 \)
T^4 + 6738531488*T^3 - 561635172326248597920*T^2 - 3062341766609343279992675453440*T + 61067707820403326214738956384949941459200
$29$
\( T^{4} - 205829454856 T^{3} + \cdots - 17\!\cdots\!00 \)
T^4 - 205829454856*T^3 + 5617654030256493705432*T^2 + 460224164691364093320566008393952*T - 17121341792787230715788568958763020735554800
$31$
\( T^{4} - 245508625876 T^{3} + \cdots - 11\!\cdots\!32 \)
T^4 - 245508625876*T^3 - 13033779919191578837208*T^2 + 3586210552537829877576899150342240*T - 116077299623467641158917906031619950979708032
$37$
\( T^{4} + 1174927152528 T^{3} + \cdots - 98\!\cdots\!96 \)
T^4 + 1174927152528*T^3 - 194105845798846850318040*T^2 - 456276571742437826440662251271975168*T - 98723700122234879983874152591832128350014427696
$41$
\( T^{4} - 1786321433340 T^{3} + \cdots - 16\!\cdots\!88 \)
T^4 - 1786321433340*T^3 - 2835510737467933110289464*T^2 + 4898770511046112078619416035808958496*T - 1660748888154620372911871121190007814973349397888
$43$
\( T^{4} + 2478301019624 T^{3} + \cdots - 61\!\cdots\!72 \)
T^4 + 2478301019624*T^3 - 4323529425963849867304608*T^2 - 13683069777329592530597873389149934720*T - 6184348031885624206195923486336287047977249711872
$47$
\( T^{4} + 4173552490816 T^{3} + \cdots - 20\!\cdots\!00 \)
T^4 + 4173552490816*T^3 - 30129466330125301646834328*T^2 - 174449479947077789604891930785584430912*T - 204156715475102792194618964042717821773175505418800
$53$
\( T^{4} + 5700182621112 T^{3} + \cdots - 31\!\cdots\!24 \)
T^4 + 5700182621112*T^3 - 148495492986461728298334216*T^2 - 1416174700988500361412777946388531735712*T - 3198203826292382646901610948680968434947545705888624
$59$
\( T^{4} + 19103066746688 T^{3} + \cdots - 35\!\cdots\!00 \)
T^4 + 19103066746688*T^3 - 235353390858779111031052632*T^2 - 1040867526992504420705232888011754900544*T - 359661516664445781679937823879441881218766790916400
$61$
\( T^{4} + 35026221391592 T^{3} + \cdots - 47\!\cdots\!72 \)
T^4 + 35026221391592*T^3 - 736373290322176466786601672*T^2 - 14139248650449881005326945308089343772640*T - 47925665819028646523400638715459883325386311705410672
$67$
\( T^{4} + 146670006946988 T^{3} + \cdots - 39\!\cdots\!00 \)
T^4 + 146670006946988*T^3 - 2353227174881249623988459160*T^2 - 1105601313533141635279532229895126585060000*T - 39267136511219999058248749791364734460967168522763920000
$71$
\( T^{4} - 54469169758856 T^{3} + \cdots - 18\!\cdots\!00 \)
T^4 - 54469169758856*T^3 - 5549498829903222232120665768*T^2 - 20592835108379719176119190228525779145248*T - 18988074578332383220230513297926149862180366999010800
$73$
\( T^{4} + 280227940264048 T^{3} + \cdots - 22\!\cdots\!84 \)
T^4 + 280227940264048*T^3 + 9766737250761139438332148584*T^2 - 1532868578594918986828079775803395052398208*T - 2225885831766582613011997241966127609060258916254830384
$79$
\( T^{4} + 76381735869712 T^{3} + \cdots - 12\!\cdots\!96 \)
T^4 + 76381735869712*T^3 - 98062166999537558867753385600*T^2 - 13275153067895146646947971259213084765279232*T - 128983628374772358820059207861788158086894662061477687296
$83$
\( T^{4} + 899056113684240 T^{3} + \cdots + 62\!\cdots\!32 \)
T^4 + 899056113684240*T^3 + 246848319179554842737063564616*T^2 + 22497663129782543871273656741377194824323584*T + 622581519440468571911486107635740761787840114085924286032
$89$
\( T^{4} - 191408718717108 T^{3} + \cdots + 76\!\cdots\!64 \)
T^4 - 191408718717108*T^3 - 188086668029839069035888862440*T^2 + 20796291628899180455014911827258514716330208*T + 7649862172374097441304461075323253115360367291519032379264
$97$
\( T^{4} + \cdots - 93\!\cdots\!28 \)
T^4 + 2235163266483040*T^3 + 430561042958019900903761018184*T^2 - 1749297368530963292314749983156377091027089472*T - 937094484344236522509653360578772247247462150860642106532528
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