[N,k,chi] = [966,2,Mod(85,966)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("966.85");
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 \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).
\(n\)
\(323\)
\(829\)
\(925\)
\(\chi(n)\)
\(1\)
\(1\)
\(-\beta_{7}\)
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
\( T_{5}^{20} - 12 T_{5}^{19} + 69 T_{5}^{18} - 236 T_{5}^{17} + 462 T_{5}^{16} - 186 T_{5}^{15} - 1695 T_{5}^{14} + 4623 T_{5}^{13} - 1380 T_{5}^{12} - 14652 T_{5}^{11} + 30119 T_{5}^{10} - 2937 T_{5}^{9} - 46066 T_{5}^{8} + \cdots + 7921 \)
T5^20 - 12*T5^19 + 69*T5^18 - 236*T5^17 + 462*T5^16 - 186*T5^15 - 1695*T5^14 + 4623*T5^13 - 1380*T5^12 - 14652*T5^11 + 30119*T5^10 - 2937*T5^9 - 46066*T5^8 + 50829*T5^7 + 123272*T5^6 - 86030*T5^5 + 151272*T5^4 - 58419*T5^3 + 74710*T5^2 + 12282*T5 + 7921
acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \)
(T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$3$
\( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \)
(T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$5$
\( T^{20} - 12 T^{19} + 69 T^{18} + \cdots + 7921 \)
T^20 - 12*T^19 + 69*T^18 - 236*T^17 + 462*T^16 - 186*T^15 - 1695*T^14 + 4623*T^13 - 1380*T^12 - 14652*T^11 + 30119*T^10 - 2937*T^9 - 46066*T^8 + 50829*T^7 + 123272*T^6 - 86030*T^5 + 151272*T^4 - 58419*T^3 + 74710*T^2 + 12282*T + 7921
$7$
\( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \)
(T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$11$
\( T^{20} + 7 T^{19} + 60 T^{18} + 277 T^{17} + \cdots + 121 \)
T^20 + 7*T^19 + 60*T^18 + 277*T^17 + 1125*T^16 + 3662*T^15 + 10652*T^14 + 24096*T^13 + 50136*T^12 + 77657*T^11 + 96647*T^10 + 31735*T^9 - 94688*T^8 - 174702*T^7 - 12870*T^6 + 163570*T^5 + 35090*T^4 - 57112*T^3 + 18150*T^2 - 2057*T + 121
$13$
\( T^{20} + 8 T^{19} + 7 T^{18} - 246 T^{17} + \cdots + 529 \)
T^20 + 8*T^19 + 7*T^18 - 246*T^17 - 1300*T^16 - 844*T^15 + 23986*T^14 + 125310*T^13 + 333756*T^12 + 655677*T^11 + 1180114*T^10 + 2090033*T^9 + 3859689*T^8 + 6167387*T^7 + 6158919*T^6 + 2699914*T^5 + 1049404*T^4 + 212800*T^3 + 48784*T^2 + 5911*T + 529
$17$
\( T^{20} - 13 T^{19} + 39 T^{18} + \cdots + 19562929 \)
T^20 - 13*T^19 + 39*T^18 + 237*T^17 - 880*T^16 - 10680*T^15 + 64810*T^14 - 9410*T^13 - 116561*T^12 - 1471041*T^11 + 1226446*T^10 + 20677613*T^9 + 57551760*T^8 + 119900524*T^7 + 187173537*T^6 + 188359238*T^5 + 219335941*T^4 + 130865946*T^3 + 119397241*T^2 + 53960600*T + 19562929
$19$
\( T^{20} + 8 T^{19} + 81 T^{18} + \cdots + 591361 \)
T^20 + 8*T^19 + 81*T^18 + 608*T^17 + 3425*T^16 + 8718*T^15 + 71055*T^14 + 269430*T^13 + 1137789*T^12 + 2198017*T^11 + 3360544*T^10 + 6182579*T^9 + 10148645*T^8 + 6458447*T^7 - 4161279*T^6 + 4978463*T^5 + 27918735*T^4 + 3961021*T^3 + 25172064*T^2 - 7541583*T + 591361
$23$
\( T^{20} - 32 T^{18} + \cdots + 41426511213649 \)
T^20 - 32*T^18 - 330*T^17 + 628*T^16 + 9471*T^15 + 37423*T^14 - 191697*T^13 - 1119029*T^12 + 517242*T^11 + 29715973*T^10 + 11896566*T^9 - 591966341*T^8 - 2332377399*T^7 + 10472489743*T^6 + 60958604553*T^5 + 92966538292*T^4 - 1123592397510*T^3 - 2505951528992*T^2 + 41426511213649
$29$
\( T^{20} - 3 T^{19} + 29 T^{18} + \cdots + 4280761 \)
T^20 - 3*T^19 + 29*T^18 + 84*T^17 + 2110*T^16 + 2258*T^15 + 15098*T^14 - 43716*T^13 + 105176*T^12 + 64856*T^11 + 286551*T^10 + 1061874*T^9 + 256925*T^8 + 557255*T^7 + 754971*T^6 - 4756271*T^5 + 9586009*T^4 - 9643431*T^3 + 16461807*T^2 - 14489207*T + 4280761
$31$
\( T^{20} - 9 T^{19} + 95 T^{18} + \cdots + 113401201 \)
T^20 - 9*T^19 + 95*T^18 - 409*T^17 + 2503*T^16 - 13018*T^15 + 61443*T^14 - 273184*T^13 + 921694*T^12 - 2458467*T^11 + 6256207*T^10 - 14149091*T^9 + 24673454*T^8 - 36016777*T^7 + 55674728*T^6 - 108953653*T^5 + 279499480*T^4 - 564855207*T^3 + 649069676*T^2 - 382192610*T + 113401201
$37$
\( T^{20} + T^{19} + 9 T^{18} + \cdots + 73462139521 \)
T^20 + T^19 + 9*T^18 + 39*T^17 + 1464*T^16 + 21422*T^15 + 150746*T^14 + 544531*T^13 + 2379358*T^12 + 25170330*T^11 + 330052074*T^10 - 686486752*T^9 + 6521551822*T^8 - 17506861206*T^7 + 36698021093*T^6 - 132723526700*T^5 + 214804395446*T^4 - 82460746501*T^3 + 46726794247*T^2 + 133435752168*T + 73462139521
$41$
\( T^{20} - 10 T^{19} + 98 T^{18} + \cdots + 529 \)
T^20 - 10*T^19 + 98*T^18 - 740*T^17 + 4685*T^16 - 26043*T^15 + 117938*T^14 - 436021*T^13 + 1311579*T^12 - 3001306*T^11 + 5910928*T^10 - 8144301*T^9 + 6401602*T^8 - 3853742*T^7 + 9167072*T^6 - 9961429*T^5 + 1718895*T^4 + 1119504*T^3 + 302750*T^2 + 20378*T + 529
$43$
\( T^{20} + 15 T^{19} + 158 T^{18} + \cdots + 982081 \)
T^20 + 15*T^19 + 158*T^18 + 166*T^17 - 682*T^16 - 40052*T^15 - 14227*T^14 - 221324*T^13 + 5802103*T^12 - 1435467*T^11 + 32811329*T^10 - 338711241*T^9 + 221180574*T^8 + 703434816*T^7 + 246120574*T^6 + 848870912*T^5 + 9484425533*T^4 + 5859702157*T^3 + 1290866801*T^2 - 5904378*T + 982081
$47$
\( (T^{10} + 14 T^{9} - 189 T^{8} + \cdots + 2025893)^{2} \)
(T^10 + 14*T^9 - 189*T^8 - 3064*T^7 + 4635*T^6 + 160689*T^5 + 226251*T^4 - 1511270*T^3 - 1514838*T^2 + 2511315*T + 2025893)^2
$53$
\( T^{20} - 38 T^{19} + \cdots + 47475035185369 \)
T^20 - 38*T^19 + 652*T^18 - 7077*T^17 + 66581*T^16 - 617222*T^15 + 5583231*T^14 - 45637232*T^13 + 294487967*T^12 - 1461632588*T^11 + 9632933311*T^10 + 581944264*T^9 + 225975833215*T^8 + 730885691217*T^7 + 2930068649937*T^6 + 5081923758285*T^5 - 1126313357455*T^4 - 7923063388031*T^3 + 12725550394007*T^2 + 49521125276997*T + 47475035185369
$59$
\( T^{20} + 10 T^{19} + \cdots + 195345436441 \)
T^20 + 10*T^19 - 100*T^18 - 2659*T^17 + 8040*T^16 + 119719*T^15 + 913535*T^14 - 7141967*T^13 + 176907725*T^12 + 485737780*T^11 + 4761919790*T^10 + 39308970481*T^9 + 229059899616*T^8 + 756909982996*T^7 + 4572618168098*T^6 + 11929269153499*T^5 + 16711752840532*T^4 + 14363280604782*T^3 + 6666962707028*T^2 + 766616111332*T + 195345436441
$61$
\( T^{20} + 6 T^{19} + \cdots + 381272905729 \)
T^20 + 6*T^19 + 108*T^18 + 1443*T^17 + 9416*T^16 + 92478*T^15 + 640965*T^14 + 2575780*T^13 + 24680185*T^12 + 62710362*T^11 + 352981564*T^10 + 1323597165*T^9 + 3735146674*T^8 + 11268223816*T^7 + 40462491981*T^6 + 68784591030*T^5 - 3013353893*T^4 - 208168934099*T^3 - 194396433063*T^2 + 147961967625*T + 381272905729
$67$
\( T^{20} - 36 T^{19} + \cdots + 43\!\cdots\!21 \)
T^20 - 36*T^19 + 884*T^18 - 17234*T^17 + 256689*T^16 - 3306242*T^15 + 38380513*T^14 - 377135191*T^13 + 3517490673*T^12 - 29613472430*T^11 + 231153704848*T^10 - 1602143037829*T^9 + 10137651978543*T^8 - 58528980402392*T^7 + 294979884152260*T^6 - 1099603174739789*T^5 + 2350138696056644*T^4 - 317280008623800*T^3 - 7670828615341128*T^2 - 4100237577136552*T + 43574897108412121
$71$
\( T^{20} + T^{19} + \cdots + 603311046361 \)
T^20 + T^19 + 150*T^18 - 1186*T^17 + 12001*T^16 - 107436*T^15 + 966219*T^14 - 5826440*T^13 + 45410631*T^12 - 268093276*T^11 + 1477921490*T^10 - 7689966680*T^9 + 34888388002*T^8 - 138364114223*T^7 + 457761876504*T^6 - 1114591512759*T^5 + 1689916611086*T^4 - 936824739988*T^3 - 877743788465*T^2 + 1065802315884*T + 603311046361
$73$
\( T^{20} - 65 T^{19} + \cdots + 29094577935721 \)
T^20 - 65*T^19 + 1880*T^18 - 31826*T^17 + 364044*T^16 - 3284848*T^15 + 26703816*T^14 - 188895754*T^13 + 1019172951*T^12 - 3767687946*T^11 + 10080679874*T^10 - 34879693486*T^9 + 182682912729*T^8 - 861015302490*T^7 + 3633678075491*T^6 - 10353045355251*T^5 + 25464215507601*T^4 - 42398933570421*T^3 + 68690294210458*T^2 - 45796160291700*T + 29094577935721
$79$
\( T^{20} - 10 T^{19} + \cdots + 2135456521 \)
T^20 - 10*T^19 + 210*T^18 - 1649*T^17 + 19207*T^16 - 102343*T^15 + 912396*T^14 - 2104244*T^13 + 29669883*T^12 + 69225630*T^11 + 659808469*T^10 + 2591841395*T^9 + 1265596574*T^8 + 10330309660*T^7 + 67086584356*T^6 + 85391575401*T^5 + 44606132967*T^4 - 12026205609*T^3 - 14385881543*T^2 - 1084248693*T + 2135456521
$83$
\( T^{20} + \cdots + 272358943896601 \)
T^20 + 14*T^19 + 354*T^18 + 2988*T^17 + 54545*T^16 + 507996*T^15 + 7730117*T^14 + 58486467*T^13 + 431062349*T^12 + 600092097*T^11 - 5200696929*T^10 - 49361345759*T^9 + 120587037740*T^8 + 2583183983326*T^7 + 1844903949952*T^6 - 37091328339950*T^5 - 35745505223719*T^4 + 152422589044163*T^3 + 382179948715506*T^2 + 233528343477002*T + 272358943896601
$89$
\( T^{20} + 18 T^{19} + \cdots + 21708582607009 \)
T^20 + 18*T^19 + 105*T^18 - 160*T^17 + 1383*T^16 + 100920*T^15 + 1108344*T^14 + 5145463*T^13 + 17343965*T^12 + 207333104*T^11 + 2399618695*T^10 + 13473281360*T^9 + 28743924793*T^8 - 91472153040*T^7 - 406476967517*T^6 - 1494144611702*T^5 + 6279870968797*T^4 - 11146407965401*T^3 + 58548277937936*T^2 + 5704912485716*T + 21708582607009
$97$
\( (T^{10} + 10 T^{9} + 243 T^{8} + \cdots + 3659569)^{2} \)
(T^10 + 10*T^9 + 243*T^8 - 243*T^7 + 8086*T^6 - 43308*T^5 + 64461*T^4 + 159477*T^3 - 306536*T^2 - 1513183*T + 3659569)^2
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