[N,k,chi] = [1620,3,Mod(161,1620)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1620.161");
S:= CuspForms(chi, 3);
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 \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).
\(n\)
\(811\)
\(1297\)
\(1541\)
\(\chi(n)\)
\(1\)
\(1\)
\(-1\)
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
\( T_{7}^{8} - 4T_{7}^{7} - 242T_{7}^{6} + 740T_{7}^{5} + 15175T_{7}^{4} - 51028T_{7}^{3} - 253790T_{7}^{2} + 1202828T_{7} - 1228634 \)
T7^8 - 4*T7^7 - 242*T7^6 + 740*T7^5 + 15175*T7^4 - 51028*T7^3 - 253790*T7^2 + 1202828*T7 - 1228634
acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} \)
T^16
$3$
\( T^{16} \)
T^16
$5$
\( (T^{2} + 5)^{8} \)
(T^2 + 5)^8
$7$
\( (T^{8} - 4 T^{7} - 242 T^{6} + \cdots - 1228634)^{2} \)
(T^8 - 4*T^7 - 242*T^6 + 740*T^5 + 15175*T^4 - 51028*T^3 - 253790*T^2 + 1202828*T - 1228634)^2
$11$
\( T^{16} + 960 T^{14} + \cdots + 822424079376 \)
T^16 + 960*T^14 + 348570*T^12 + 58044168*T^10 + 4129654833*T^8 + 79960714920*T^6 + 591949533816*T^4 + 1630411859232*T^2 + 822424079376
$13$
\( (T^{8} + 20 T^{7} - 632 T^{6} + \cdots - 122845736)^{2} \)
(T^8 + 20*T^7 - 632*T^6 - 8728*T^5 + 167305*T^4 + 650228*T^3 - 18200708*T^2 + 85187504*T - 122845736)^2
$17$
\( T^{16} + 2148 T^{14} + \cdots + 39\!\cdots\!96 \)
T^16 + 2148*T^14 + 1699056*T^12 + 639616392*T^10 + 122909485860*T^8 + 12154456410048*T^6 + 588231615200256*T^4 + 11700711883487232*T^2 + 39893804977360896
$19$
\( (T^{8} - 28 T^{7} - 752 T^{6} + \cdots + 14011609)^{2} \)
(T^8 - 28*T^7 - 752*T^6 + 16220*T^5 + 130678*T^4 - 2486548*T^3 - 103040*T^2 + 50054516*T + 14011609)^2
$23$
\( T^{16} + 3228 T^{14} + \cdots + 10\!\cdots\!16 \)
T^16 + 3228*T^14 + 3824118*T^12 + 2168424324*T^10 + 651720416913*T^8 + 110193682949928*T^6 + 10536115275755712*T^4 + 531152464436483904*T^2 + 10944316233714494016
$29$
\( T^{16} + 6072 T^{14} + \cdots + 15\!\cdots\!36 \)
T^16 + 6072*T^14 + 13806450*T^12 + 15071059800*T^10 + 8586881626113*T^8 + 2532778308505440*T^6 + 340992913216730400*T^4 + 13422262621305245184*T^2 + 152829642274689966336
$31$
\( (T^{8} + 32 T^{7} - 4022 T^{6} + \cdots - 2745236384)^{2} \)
(T^8 + 32*T^7 - 4022*T^6 - 127096*T^5 + 3812545*T^4 + 114001544*T^3 - 523496312*T^2 - 19068747808*T - 2745236384)^2
$37$
\( (T^{8} + 44 T^{7} - 3842 T^{6} + \cdots - 42707709224)^{2} \)
(T^8 + 44*T^7 - 3842*T^6 - 177820*T^5 + 2504770*T^4 + 125492528*T^3 - 483118640*T^2 - 21369202768*T - 42707709224)^2
$41$
\( T^{16} + 12960 T^{14} + \cdots + 52\!\cdots\!01 \)
T^16 + 12960*T^14 + 67529700*T^12 + 181626232608*T^10 + 269590569609798*T^8 + 219087278552700000*T^6 + 90033731552179719396*T^4 + 15028208682063322993632*T^2 + 524426447536613004003201
$43$
\( (T^{8} - 64 T^{7} + \cdots - 1225653312416)^{2} \)
(T^8 - 64*T^7 - 6818*T^6 + 531020*T^5 + 1792438*T^4 - 622180144*T^3 + 4064084560*T^2 + 183972419072*T - 1225653312416)^2
$47$
\( T^{16} + 14064 T^{14} + \cdots + 19\!\cdots\!56 \)
T^16 + 14064*T^14 + 73771398*T^12 + 180676037052*T^10 + 213077708054325*T^8 + 111095798681489364*T^6 + 18604290833023706280*T^4 + 409644378676723745016*T^2 + 1909194567305212696356
$53$
\( T^{16} + 28680 T^{14} + \cdots + 20\!\cdots\!96 \)
T^16 + 28680*T^14 + 314965206*T^12 + 1656482274828*T^10 + 4271704265636901*T^8 + 5037075813918810204*T^6 + 2325781432031037380712*T^4 + 392839169734580444266248*T^2 + 20443723425845181557487396
$59$
\( T^{16} + 34512 T^{14} + \cdots + 45\!\cdots\!81 \)
T^16 + 34512*T^14 + 473917716*T^12 + 3329259430896*T^10 + 12854473345985814*T^8 + 27584016925679016048*T^6 + 32382660071171136861684*T^4 + 19269678045194075811384144*T^2 + 4515086384036576403084439281
$61$
\( (T^{8} - 4 T^{7} + \cdots + 217364537682304)^{2} \)
(T^8 - 4*T^7 - 19784*T^6 + 78536*T^5 + 116123140*T^4 - 341025088*T^3 - 269218458656*T^2 + 425588616320*T + 217364537682304)^2
$67$
\( (T^{8} + 20 T^{7} + \cdots + 2578090965376)^{2} \)
(T^8 + 20*T^7 - 11660*T^6 - 186208*T^5 + 39290728*T^4 + 393228800*T^3 - 36917347904*T^2 + 44371682048*T + 2578090965376)^2
$71$
\( T^{16} + 37080 T^{14} + \cdots + 36\!\cdots\!36 \)
T^16 + 37080*T^14 + 503856450*T^12 + 3237213105576*T^10 + 10396892558997873*T^8 + 16263637290030044880*T^6 + 11531705105777709383424*T^4 + 3453592331572386245190144*T^2 + 360361003931304736893428736
$73$
\( (T^{8} - 28 T^{7} + \cdots + 276269820962176)^{2} \)
(T^8 - 28*T^7 - 33110*T^6 + 768692*T^5 + 347763670*T^4 - 9041518096*T^3 - 1220516678528*T^2 + 38963943631232*T + 276269820962176)^2
$79$
\( (T^{8} + 68 T^{7} - 23816 T^{6} + \cdots - 1341375104)^{2} \)
(T^8 + 68*T^7 - 23816*T^6 - 976840*T^5 + 134635780*T^4 - 718054240*T^3 - 40870248416*T^2 + 212805915008*T - 1341375104)^2
$83$
\( T^{16} + 73644 T^{14} + \cdots + 13\!\cdots\!36 \)
T^16 + 73644*T^14 + 2149098480*T^12 + 32113308755736*T^10 + 266653992245635044*T^8 + 1240661479328386842816*T^6 + 3068946806845346797777920*T^4 + 3527709092894320090458937344*T^2 + 1385689656922514714035160027136
$89$
\( T^{16} + 77472 T^{14} + \cdots + 49\!\cdots\!16 \)
T^16 + 77472*T^14 + 2371179258*T^12 + 37870611123816*T^10 + 345139457439799953*T^8 + 1825673641679728153512*T^6 + 5424478723578430779102072*T^4 + 8248447767399791299332623136*T^2 + 4919291879967404561082590681616
$97$
\( (T^{8} - 64 T^{7} + \cdots + 1694320236544)^{2} \)
(T^8 - 64*T^7 - 29570*T^6 + 728732*T^5 + 159647110*T^4 - 2038555936*T^3 - 162311358464*T^2 - 439292357632*T + 1694320236544)^2
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