[N,k,chi] = [231,2,Mod(64,231)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("231.64");
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 \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).
\(n\)
\(155\)
\(199\)
\(211\)
\(\chi(n)\)
\(1\)
\(1\)
\(-\beta_{9}\)
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
\( T_{2}^{20} + 12 T_{2}^{18} + 3 T_{2}^{17} + 94 T_{2}^{16} + 10 T_{2}^{15} + 662 T_{2}^{14} + 153 T_{2}^{13} + 4638 T_{2}^{12} + 1174 T_{2}^{11} + 15808 T_{2}^{10} + 3393 T_{2}^{9} + 26062 T_{2}^{8} + 15494 T_{2}^{7} + 11660 T_{2}^{6} + \cdots + 121 \)
T2^20 + 12*T2^18 + 3*T2^17 + 94*T2^16 + 10*T2^15 + 662*T2^14 + 153*T2^13 + 4638*T2^12 + 1174*T2^11 + 15808*T2^10 + 3393*T2^9 + 26062*T2^8 + 15494*T2^7 + 11660*T2^6 + 33295*T2^5 + 67756*T2^4 + 43696*T2^3 + 13084*T2^2 + 1155*T2 + 121
acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{20} + 12 T^{18} + 3 T^{17} + 94 T^{16} + \cdots + 121 \)
T^20 + 12*T^18 + 3*T^17 + 94*T^16 + 10*T^15 + 662*T^14 + 153*T^13 + 4638*T^12 + 1174*T^11 + 15808*T^10 + 3393*T^9 + 26062*T^8 + 15494*T^7 + 11660*T^6 + 33295*T^5 + 67756*T^4 + 43696*T^3 + 13084*T^2 + 1155*T + 121
$3$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{5} \)
(T^4 + T^3 + T^2 + T + 1)^5
$5$
\( T^{20} + 5 T^{19} + 29 T^{18} + \cdots + 18800896 \)
T^20 + 5*T^19 + 29*T^18 + 93*T^17 + 485*T^16 + 1810*T^15 + 9535*T^14 + 31906*T^13 + 138723*T^12 + 427609*T^11 + 1264706*T^10 + 2674714*T^9 + 6602789*T^8 + 8908158*T^7 + 12690116*T^6 + 7065848*T^5 + 10263648*T^4 - 165024*T^3 + 18833728*T^2 - 15540224*T + 18800896
$7$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{5} \)
(T^4 - T^3 + T^2 - T + 1)^5
$11$
\( T^{20} + T^{19} - T^{18} + \cdots + 25937424601 \)
T^20 + T^19 - T^18 + 25*T^17 + 246*T^16 + 17*T^15 - 807*T^14 + 1153*T^13 + 25424*T^12 - 22607*T^11 - 174843*T^10 - 248677*T^9 + 3076304*T^8 + 1534643*T^7 - 11815287*T^6 + 2737867*T^5 + 435804006*T^4 + 487179275*T^3 - 214358881*T^2 + 2357947691*T + 25937424601
$13$
\( T^{20} - 13 T^{19} + 131 T^{18} + \cdots + 7929856 \)
T^20 - 13*T^19 + 131*T^18 - 893*T^17 + 5797*T^16 - 26324*T^15 + 119084*T^14 - 279864*T^13 + 941856*T^12 - 1219296*T^11 + 12951296*T^10 + 18971264*T^9 + 161489664*T^8 + 234243584*T^7 + 1078514688*T^6 + 1556865024*T^5 + 2955522048*T^4 + 4062216192*T^3 + 5323653120*T^2 + 330530816*T + 7929856
$17$
\( T^{20} + T^{19} + 51 T^{18} + \cdots + 69488896 \)
T^20 + T^19 + 51*T^18 - 109*T^17 + 1185*T^16 + 1280*T^15 + 35691*T^14 + 109007*T^13 + 414024*T^12 + 1286774*T^11 + 4277628*T^10 + 7296323*T^9 + 20039333*T^8 + 53200892*T^7 + 114419352*T^6 + 205492640*T^5 + 565284288*T^4 + 1014974272*T^3 + 968550464*T^2 + 224471808*T + 69488896
$19$
\( T^{20} - 10 T^{19} + \cdots + 2595291136 \)
T^20 - 10*T^19 + 93*T^18 - 512*T^17 + 4566*T^16 - 42046*T^15 + 422088*T^14 - 3159236*T^13 + 21361241*T^12 - 115433580*T^11 + 526596732*T^10 - 1764500936*T^9 + 4228297792*T^8 - 4635873696*T^7 + 4394613952*T^6 - 12345912960*T^5 + 43376079616*T^4 + 70070199296*T^3 + 167586840576*T^2 - 11972247552*T + 2595291136
$23$
\( (T^{10} - 106 T^{8} + 38 T^{7} + \cdots - 600380)^{2} \)
(T^10 - 106*T^8 + 38*T^7 + 4029*T^6 - 2484*T^5 - 64464*T^4 + 43860*T^3 + 382275*T^2 - 132050*T - 600380)^2
$29$
\( T^{20} - 3 T^{19} + \cdots + 12760212756736 \)
T^20 - 3*T^19 + 141*T^18 + 117*T^17 + 9947*T^16 - 3174*T^15 + 817489*T^14 - 1378139*T^13 + 45786026*T^12 - 32081936*T^11 + 1770281036*T^10 + 988377929*T^9 + 36843393649*T^8 + 21007766204*T^7 + 447009342708*T^6 - 420224180496*T^5 + 2936877994768*T^4 - 14257735240768*T^3 + 32074578191680*T^2 - 30699348461824*T + 12760212756736
$31$
\( T^{20} + 13 T^{19} + \cdots + 283449760000 \)
T^20 + 13*T^19 + 220*T^18 + 2564*T^17 + 27110*T^16 + 214501*T^15 + 1611537*T^14 + 9245112*T^13 + 48527429*T^12 + 171837421*T^11 + 516192113*T^10 + 1966235319*T^9 + 18262864101*T^8 + 70149979180*T^7 + 162440092220*T^6 + 276645380400*T^5 + 503880952400*T^4 + 627022440000*T^3 + 474340328000*T^2 + 178034560000*T + 283449760000
$37$
\( T^{20} + 32 T^{19} + \cdots + 2807916867856 \)
T^20 + 32*T^19 + 681*T^18 + 10964*T^17 + 150316*T^16 + 1678020*T^15 + 15844807*T^14 + 121882966*T^13 + 783479867*T^12 + 4169538566*T^11 + 18812606947*T^10 + 72743307690*T^9 + 243495714976*T^8 + 667945044586*T^7 + 1484503124253*T^6 + 2728176267016*T^5 + 4333961281961*T^4 + 4231452092442*T^3 + 313858716748*T^2 - 3620533120920*T + 2807916867856
$41$
\( T^{20} + 3 T^{19} + 54 T^{18} + \cdots + 101929216 \)
T^20 + 3*T^19 + 54*T^18 - 4*T^17 + 3083*T^16 + 19378*T^15 + 276021*T^14 + 1218771*T^13 + 8691401*T^12 + 20141397*T^11 + 122112709*T^10 + 82205046*T^9 + 928769061*T^8 - 984647428*T^7 + 3945345996*T^6 - 8030571016*T^5 + 16139525920*T^4 - 457274528*T^3 - 778062656*T^2 + 17526656*T + 101929216
$43$
\( (T^{10} - 6 T^{9} - 118 T^{8} + 494 T^{7} + \cdots - 21296)^{2} \)
(T^10 - 6*T^9 - 118*T^8 + 494*T^7 + 4706*T^6 - 9046*T^5 - 69175*T^4 - 3128*T^3 + 213700*T^2 + 140448*T - 21296)^2
$47$
\( T^{20} - 20 T^{19} + \cdots + 7259791360000 \)
T^20 - 20*T^19 + 376*T^18 - 3557*T^17 + 33271*T^16 - 287814*T^15 + 3845560*T^14 - 45140016*T^13 + 468854848*T^12 - 3629768096*T^11 + 24047895744*T^10 - 128395164032*T^9 + 586807220736*T^8 - 2109596613120*T^7 + 5778074373120*T^6 - 9281606758400*T^5 + 11118341734400*T^4 - 13310964736000*T^3 + 20160278528000*T^2 + 20365352960000*T + 7259791360000
$53$
\( T^{20} - 3 T^{19} + \cdots + 57838326016 \)
T^20 - 3*T^19 + 139*T^18 + 1153*T^17 + 9383*T^16 + 7780*T^15 + 2139907*T^14 + 23099748*T^13 + 213464263*T^12 + 835253157*T^11 + 2886672388*T^10 + 2493024726*T^9 + 77727446689*T^8 + 368061185728*T^7 + 2570636440036*T^6 + 3527735721440*T^5 + 6913017243520*T^4 + 3068214093824*T^3 + 2939916758656*T^2 - 681819627776*T + 57838326016
$59$
\( T^{20} + 9 T^{19} + \cdots + 25\!\cdots\!56 \)
T^20 + 9*T^19 + 284*T^18 + 1472*T^17 + 36121*T^16 + 175770*T^15 + 3974168*T^14 - 2199472*T^13 + 494631312*T^12 + 875753472*T^11 + 19772923968*T^10 + 91351262080*T^9 + 889285181696*T^8 + 494032148992*T^7 + 24106163331072*T^6 + 12148671019008*T^5 + 150681061195776*T^4 + 98380815048704*T^3 + 2473677057441792*T^2 - 9818087030456320*T + 25314978897657856
$61$
\( T^{20} + 15 T^{19} + \cdots + 77\!\cdots\!36 \)
T^20 + 15*T^19 + 215*T^18 + 495*T^17 + 12135*T^16 + 203398*T^15 + 6375840*T^14 + 18079640*T^13 + 275318320*T^12 + 3072164960*T^11 + 23766702464*T^10 + 51214999680*T^9 + 1657540724480*T^8 + 13769341731840*T^7 + 72808850206720*T^6 + 238098377818112*T^5 + 1084428053381120*T^4 + 2205054561157120*T^3 + 5837164888719360*T^2 + 5623171965255680*T + 7727158222913536
$67$
\( (T^{10} - 38 T^{9} + 269 T^{8} + \cdots + 52960256)^{2} \)
(T^10 - 38*T^9 + 269*T^8 + 5994*T^7 - 93167*T^6 + 98118*T^5 + 4585076*T^4 - 22705600*T^3 + 2490688*T^2 + 98937344*T + 52960256)^2
$71$
\( T^{20} + \cdots + 291518744166400 \)
T^20 - 37*T^19 + 870*T^18 - 11936*T^17 + 113680*T^16 - 468069*T^15 + 1729897*T^14 - 7064488*T^13 + 568422389*T^12 - 3825168729*T^11 + 28304964343*T^10 + 165397761229*T^9 + 1048690057781*T^8 + 5177216897220*T^7 + 106020424943040*T^6 + 239525708213440*T^5 + 316556179344640*T^4 + 258211396531200*T^3 + 949145725952000*T^2 - 228473636044800*T + 291518744166400
$73$
\( T^{20} - 27 T^{19} + \cdots + 57999556673536 \)
T^20 - 27*T^19 + 482*T^18 - 3986*T^17 + 12541*T^16 + 74734*T^15 + 2607116*T^14 - 28215920*T^13 + 108554864*T^12 + 174897536*T^11 + 9334274112*T^10 - 15237816064*T^9 + 201523640832*T^8 - 1190261111808*T^7 + 8916845292544*T^6 - 24222582466560*T^5 + 81994904301568*T^4 - 21422348476416*T^3 + 176044730580992*T^2 - 53510532530176*T + 57999556673536
$79$
\( T^{20} - 5 T^{19} + \cdots + 3187281807616 \)
T^20 - 5*T^19 + 169*T^18 - 787*T^17 + 16285*T^16 - 28982*T^15 + 979795*T^14 + 1338440*T^13 + 76833191*T^12 + 40585347*T^11 + 3305384702*T^10 + 5560650836*T^9 + 131444274473*T^8 + 331015477672*T^7 + 4817269074516*T^6 + 14661696682176*T^5 + 135294385712336*T^4 + 58605779613952*T^3 + 112228546126144*T^2 - 30978056285696*T + 3187281807616
$83$
\( T^{20} + 42 T^{19} + \cdots + 607252381696 \)
T^20 + 42*T^19 + 1076*T^18 + 17232*T^17 + 183552*T^16 + 1278176*T^15 + 8633984*T^14 + 98599296*T^13 + 1363611136*T^12 + 13498345984*T^11 + 92290523136*T^10 + 410152873984*T^9 + 1364600295424*T^8 + 3882435354624*T^7 + 11277998522368*T^6 + 11904074645504*T^5 + 30060575981568*T^4 - 2555430043648*T^3 + 36935292682240*T^2 - 7627760205824*T + 607252381696
$89$
\( (T^{10} + 9 T^{9} - 250 T^{8} + \cdots + 106384)^{2} \)
(T^10 + 9*T^9 - 250*T^8 - 3159*T^7 - 90*T^6 + 100529*T^5 + 212045*T^4 - 488456*T^3 - 1679912*T^2 - 1115496*T + 106384)^2
$97$
\( T^{20} + 27 T^{19} + \cdots + 231384088576 \)
T^20 + 27*T^19 + 339*T^18 + 2239*T^17 + 14713*T^16 + 99192*T^15 + 541696*T^14 + 1100144*T^13 + 15298976*T^12 + 103619328*T^11 + 619474944*T^10 + 645640704*T^9 + 6611766016*T^8 - 32986340352*T^7 + 207142537216*T^6 - 502036496384*T^5 + 1095419596800*T^4 - 680382595072*T^3 + 855174529024*T^2 + 5048827904*T + 231384088576
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