[N,k,chi] = [1980,2,Mod(1277,1980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1980, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1980.1277");
S:= CuspForms(chi, 2);
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 \(\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times\).
\(n\)
\(397\)
\(541\)
\(991\)
\(1541\)
\(\chi(n)\)
\(\beta_{5}\)
\(1\)
\(1\)
\(-1\)
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
\( T_{17}^{20} + 16 T_{17}^{19} + 128 T_{17}^{18} + 344 T_{17}^{17} + 432 T_{17}^{16} + 3680 T_{17}^{15} + 62752 T_{17}^{14} + 86800 T_{17}^{13} + 59256 T_{17}^{12} + 131488 T_{17}^{11} + 4040576 T_{17}^{10} + 4798240 T_{17}^{9} + \cdots + 984064 \)
T17^20 + 16*T17^19 + 128*T17^18 + 344*T17^17 + 432*T17^16 + 3680*T17^15 + 62752*T17^14 + 86800*T17^13 + 59256*T17^12 + 131488*T17^11 + 4040576*T17^10 + 4798240*T17^9 + 2710144*T17^8 - 1603072*T17^7 + 65081472*T17^6 + 71433664*T17^5 + 37581328*T17^4 - 79625856*T17^3 + 110707200*T17^2 + 14760960*T17 + 984064
acting on \(S_{2}^{\mathrm{new}}(1980, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{20} \)
T^20
$3$
\( T^{20} \)
T^20
$5$
\( T^{20} - 16 T^{17} + 21 T^{16} + \cdots + 9765625 \)
T^20 - 16*T^17 + 21*T^16 + 16*T^15 + 32*T^14 - 16*T^13 - 62*T^12 + 1168*T^11 - 2112*T^10 + 5840*T^9 - 1550*T^8 - 2000*T^7 + 20000*T^6 + 50000*T^5 + 328125*T^4 - 1250000*T^3 + 9765625
$7$
\( T^{20} + 40 T^{17} + 800 T^{16} + \cdots + 4096 \)
T^20 + 40*T^17 + 800*T^16 + 848*T^15 + 800*T^14 + 14224*T^13 + 148152*T^12 + 299584*T^11 + 288512*T^10 - 87328*T^9 + 1656000*T^8 + 2814272*T^7 + 2384000*T^6 - 991808*T^5 + 4977680*T^4 + 6598144*T^3 + 4333568*T^2 + 188416*T + 4096
$11$
\( (T^{2} + 1)^{10} \)
(T^2 + 1)^10
$13$
\( T^{20} + 4 T^{19} + \cdots + 1626347584 \)
T^20 + 4*T^19 + 8*T^18 + 48*T^17 + 1796*T^16 + 7968*T^15 + 18656*T^14 + 63152*T^13 + 1012664*T^12 + 4824704*T^11 + 12467136*T^10 + 25022688*T^9 + 189477664*T^8 + 920156928*T^7 + 2519988608*T^6 + 3164330048*T^5 + 1911763728*T^4 + 436961472*T^3 + 3454467200*T^2 + 3352063360*T + 1626347584
$17$
\( T^{20} + 16 T^{19} + 128 T^{18} + \cdots + 984064 \)
T^20 + 16*T^19 + 128*T^18 + 344*T^17 + 432*T^16 + 3680*T^15 + 62752*T^14 + 86800*T^13 + 59256*T^12 + 131488*T^11 + 4040576*T^10 + 4798240*T^9 + 2710144*T^8 - 1603072*T^7 + 65081472*T^6 + 71433664*T^5 + 37581328*T^4 - 79625856*T^3 + 110707200*T^2 + 14760960*T + 984064
$19$
\( T^{20} + 208 T^{18} + \cdots + 125440000 \)
T^20 + 208*T^18 + 17360*T^16 + 751328*T^14 + 18050720*T^12 + 235997056*T^10 + 1489571840*T^8 + 3259569664*T^6 + 2847457536*T^4 + 1036134400*T^2 + 125440000
$23$
\( T^{20} - 384 T^{17} + \cdots + 108720553984 \)
T^20 - 384*T^17 + 7168*T^16 - 24320*T^15 + 73728*T^14 - 1274880*T^13 + 17333376*T^12 - 90850304*T^11 + 256802816*T^10 - 346963968*T^9 + 2037100544*T^8 - 11799773184*T^7 + 36655333376*T^6 - 21708734464*T^5 - 27324837888*T^4 + 36461019136*T^3 + 336383705088*T^2 + 270450819072*T + 108720553984
$29$
\( (T^{10} + 8 T^{9} - 106 T^{8} - 896 T^{7} + \cdots - 1568)^{2} \)
(T^10 + 8*T^9 - 106*T^8 - 896*T^7 + 2768*T^6 + 28384*T^5 + 2368*T^4 - 219072*T^3 - 294448*T^2 - 46592*T - 1568)^2
$31$
\( (T^{10} - 8 T^{9} - 128 T^{8} + \cdots - 1252352)^{2} \)
(T^10 - 8*T^9 - 128*T^8 + 800*T^7 + 6392*T^6 - 21920*T^5 - 145472*T^4 + 105600*T^3 + 912656*T^2 - 92416*T - 1252352)^2
$37$
\( T^{20} + 4 T^{19} + \cdots + 3117569360896 \)
T^20 + 4*T^19 + 8*T^18 - 536*T^17 + 12404*T^16 + 5760*T^15 + 67456*T^14 - 2990336*T^13 + 41108256*T^12 - 71099264*T^11 + 169421568*T^10 - 2877713152*T^9 + 33752261760*T^8 - 83818520576*T^7 + 124181362688*T^6 - 392815435776*T^5 + 3570271440128*T^4 - 9317332478976*T^3 + 13041419061248*T^2 - 9017486178304*T + 3117569360896
$41$
\( T^{20} + 452 T^{18} + \cdots + 20879197029376 \)
T^20 + 452*T^18 + 80068*T^16 + 7158464*T^14 + 351852448*T^12 + 9942039680*T^10 + 164649931904*T^8 + 1581272767488*T^6 + 8378341570816*T^4 + 21786273940480*T^2 + 20879197029376
$43$
\( T^{20} - 216 T^{17} + \cdots + 29777364714496 \)
T^20 - 216*T^17 + 14432*T^16 - 17440*T^15 + 23328*T^14 - 1674768*T^13 + 63500408*T^12 - 134932320*T^11 + 177156992*T^10 - 2177276320*T^9 + 71755073472*T^8 - 149017415936*T^7 + 187868046464*T^6 - 1684390871232*T^5 + 14737338658576*T^4 - 45532317062528*T^3 + 77735087718912*T^2 - 68040371223552*T + 29777364714496
$47$
\( T^{20} + 3008 T^{16} + \cdots + 67108864 \)
T^20 + 3008*T^16 + 768*T^15 - 36864*T^13 + 2573824*T^12 + 897024*T^11 + 294912*T^10 - 60162048*T^9 + 508280832*T^8 - 494862336*T^7 + 481296384*T^6 - 4168089600*T^5 + 23676911616*T^4 - 39928725504*T^3 + 33294385152*T^2 + 2113929216*T + 67108864
$53$
\( T^{20} - 32 T^{19} + \cdots + 211393970176 \)
T^20 - 32*T^19 + 512*T^18 - 4416*T^17 + 24848*T^16 - 139392*T^15 + 1488896*T^14 - 12019200*T^13 + 60134496*T^12 - 172359680*T^11 + 522412032*T^10 - 2771891200*T^9 + 13860577536*T^8 - 36310042624*T^7 + 47217311744*T^6 - 6312902656*T^5 + 85365440768*T^4 - 322275254272*T^3 + 458836017152*T^2 + 440443338752*T + 211393970176
$59$
\( (T^{10} + 8 T^{9} - 232 T^{8} + \cdots - 21006464)^{2} \)
(T^10 + 8*T^9 - 232*T^8 - 1232*T^7 + 18152*T^6 + 55520*T^5 - 579200*T^4 - 851264*T^3 + 7074064*T^2 + 2658048*T - 21006464)^2
$61$
\( (T^{10} + 8 T^{9} - 216 T^{8} + \cdots - 1470976)^{2} \)
(T^10 + 8*T^9 - 216*T^8 - 928*T^7 + 13664*T^6 + 27584*T^5 - 261440*T^4 - 318464*T^3 + 1339840*T^2 + 805376*T - 1470976)^2
$67$
\( T^{20} + \cdots + 311673319849984 \)
T^20 + 16*T^19 + 128*T^18 - 128*T^17 + 35648*T^16 + 556096*T^15 + 4342784*T^14 - 2939648*T^13 + 370525312*T^12 + 5596343808*T^11 + 42300352512*T^10 - 7988584448*T^9 + 861401772032*T^8 + 12984891625472*T^7 + 97912186535936*T^6 - 5809655201792*T^5 + 268523673292800*T^4 + 4208546594586624*T^3 + 32964508517531648*T^2 - 4533024995934208*T + 311673319849984
$71$
\( T^{20} + \cdots + 360346088390656 \)
T^20 + 848*T^18 + 286096*T^16 + 49929216*T^14 + 4909490272*T^12 + 275971310592*T^10 + 8477078244608*T^8 + 123021789491200*T^6 + 543416994275584*T^4 + 809282336993280*T^2 + 360346088390656
$73$
\( T^{20} + 44 T^{19} + \cdots + 76\!\cdots\!24 \)
T^20 + 44*T^19 + 968*T^18 + 11488*T^17 + 123268*T^16 + 2224432*T^15 + 44538656*T^14 + 527931888*T^13 + 3838936312*T^12 + 21286717472*T^11 + 269264561600*T^10 + 3149653562016*T^9 + 22001255721248*T^8 + 66132820748224*T^7 + 329975613106304*T^6 + 3428177689683520*T^5 + 24988883898151952*T^4 + 59920462732360640*T^3 + 31553601832203392*T^2 - 219907148316308096*T + 766301643434818624
$79$
\( T^{20} + 608 T^{18} + \cdots + 1724536262656 \)
T^20 + 608*T^18 + 152016*T^16 + 20139552*T^14 + 1513675680*T^12 + 63715808384*T^10 + 1367200301056*T^8 + 11732226472448*T^6 + 27468427317504*T^4 + 16803244304384*T^2 + 1724536262656
$83$
\( T^{20} + \cdots + 697770398638336 \)
T^20 - 64*T^19 + 2048*T^18 - 40312*T^17 + 550208*T^16 - 5972816*T^15 + 67962912*T^14 - 888244336*T^13 + 10550366392*T^12 - 92259531648*T^11 + 547892926720*T^10 - 2020103404576*T^9 + 4285139435520*T^8 - 4851491941696*T^7 + 18451271301248*T^6 - 72672915085120*T^5 + 152286324289296*T^4 - 46775369414400*T^3 + 55755448328192*T^2 - 278942651475968*T + 697770398638336
$89$
\( (T^{10} + 16 T^{9} - 530 T^{8} + \cdots - 784056448)^{2} \)
(T^10 + 16*T^9 - 530*T^8 - 6848*T^7 + 115552*T^6 + 922880*T^5 - 12531648*T^4 - 32033792*T^3 + 553126336*T^2 - 938886144*T - 784056448)^2
$97$
\( T^{20} + \cdots + 320733140632576 \)
T^20 + 20*T^19 + 200*T^18 + 840*T^17 + 83188*T^16 + 1642240*T^15 + 16560000*T^14 + 62551040*T^13 + 1228572960*T^12 + 21184479360*T^11 + 201400697600*T^10 + 878074287360*T^9 + 3068180376704*T^8 + 20609003673600*T^7 + 183152636672000*T^6 + 805330640916480*T^5 + 1909648051137792*T^4 + 1291965270901760*T^3 + 240601067571200*T^2 - 392858081351680*T + 320733140632576
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