[N,k,chi] = [275,2,Mod(26,275)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(275, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.26");
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 \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).
\(n\)
\(101\)
\(177\)
\(\chi(n)\)
\(-1 - \beta_{6} + \beta_{7} - \beta_{9}\)
\(1\)
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
\( T_{2}^{16} + 5 T_{2}^{14} + 12 T_{2}^{13} + 32 T_{2}^{12} - 30 T_{2}^{11} + 259 T_{2}^{10} + 193 T_{2}^{9} + 584 T_{2}^{8} + 833 T_{2}^{7} + 1629 T_{2}^{6} + 659 T_{2}^{5} + 759 T_{2}^{4} - 391 T_{2}^{3} + 96 T_{2}^{2} - 11 T_{2} + 1 \)
T2^16 + 5*T2^14 + 12*T2^13 + 32*T2^12 - 30*T2^11 + 259*T2^10 + 193*T2^9 + 584*T2^8 + 833*T2^7 + 1629*T2^6 + 659*T2^5 + 759*T2^4 - 391*T2^3 + 96*T2^2 - 11*T2 + 1
acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} + 5 T^{14} + 12 T^{13} + 32 T^{12} + \cdots + 1 \)
T^16 + 5*T^14 + 12*T^13 + 32*T^12 - 30*T^11 + 259*T^10 + 193*T^9 + 584*T^8 + 833*T^7 + 1629*T^6 + 659*T^5 + 759*T^4 - 391*T^3 + 96*T^2 - 11*T + 1
$3$
\( T^{16} - 2 T^{15} + 4 T^{14} - 3 T^{13} + \cdots + 16 \)
T^16 - 2*T^15 + 4*T^14 - 3*T^13 + 69*T^12 - 152*T^11 + 731*T^10 - 1186*T^9 + 3199*T^8 - 3639*T^7 + 5341*T^6 - 465*T^5 + 1787*T^4 + 3834*T^3 + 9000*T^2 + 240*T + 16
$5$
\( T^{16} \)
T^16
$7$
\( T^{16} + 4 T^{15} + 14 T^{14} + 50 T^{13} + \cdots + 961 \)
T^16 + 4*T^15 + 14*T^14 + 50*T^13 + 213*T^12 + 8*T^11 - 162*T^10 + 412*T^9 + 5137*T^8 - 7216*T^7 + 18582*T^6 - 22720*T^5 + 29485*T^4 - 24924*T^3 + 14570*T^2 - 3844*T + 961
$11$
\( T^{16} + 5 T^{15} + 17 T^{14} + \cdots + 214358881 \)
T^16 + 5*T^15 + 17*T^14 + 15*T^13 - 22*T^12 - 225*T^11 - 241*T^10 + 4405*T^9 + 21290*T^8 + 48455*T^7 - 29161*T^6 - 299475*T^5 - 322102*T^4 + 2415765*T^3 + 30116537*T^2 + 97435855*T + 214358881
$13$
\( T^{16} + 7 T^{15} + 28 T^{14} + \cdots + 12075625 \)
T^16 + 7*T^15 + 28*T^14 + 69*T^13 + 615*T^12 + 2299*T^11 + 14023*T^10 + 52727*T^9 + 217236*T^8 + 656915*T^7 + 1787555*T^6 + 3271500*T^5 + 6717300*T^4 + 9623750*T^3 + 11033625*T^2 + 6950000*T + 12075625
$17$
\( T^{16} + 12 T^{15} + 75 T^{14} + \cdots + 192721 \)
T^16 + 12*T^15 + 75*T^14 + 326*T^13 + 2753*T^12 + 8736*T^11 + 21644*T^10 + 97629*T^9 + 834585*T^8 + 808113*T^7 + 7262913*T^6 + 24505316*T^5 + 42180277*T^4 + 24792366*T^3 + 16639907*T^2 - 854733*T + 192721
$19$
\( T^{16} + 13 T^{15} + \cdots + 109830400 \)
T^16 + 13*T^15 + 133*T^14 + 761*T^13 + 3355*T^12 + 9916*T^11 + 35573*T^10 + 62888*T^9 + 194441*T^8 - 38375*T^7 + 1480120*T^6 - 3276970*T^5 + 21143185*T^4 - 57269700*T^3 + 138466800*T^2 - 172500800*T + 109830400
$23$
\( (T^{8} - 2 T^{7} - 62 T^{6} + 252 T^{5} + \cdots - 121)^{2} \)
(T^8 - 2*T^7 - 62*T^6 + 252*T^5 + 131*T^4 - 1430*T^3 + 778*T^2 + 1078*T - 121)^2
$29$
\( T^{16} + 11 T^{15} + \cdots + 1259895025 \)
T^16 + 11*T^15 + 120*T^14 + 769*T^13 + 5119*T^12 + 24039*T^11 + 130305*T^10 + 350811*T^9 + 1995746*T^8 + 7943855*T^7 + 32825665*T^6 + 105852160*T^5 + 415911210*T^4 + 995811450*T^3 + 2098086975*T^2 + 2332731400*T + 1259895025
$31$
\( T^{16} + 10 T^{15} + \cdots + 1126877761 \)
T^16 + 10*T^15 + 125*T^14 + 502*T^13 + 2922*T^12 + 16890*T^11 + 195039*T^10 + 1094903*T^9 + 7039364*T^8 + 27585323*T^7 + 122411369*T^6 + 289147879*T^5 + 618987259*T^4 - 1135969611*T^3 + 2721954406*T^2 - 2484408121*T + 1126877761
$37$
\( T^{16} - 4 T^{15} + 96 T^{14} + \cdots + 19811401 \)
T^16 - 4*T^15 + 96*T^14 - 184*T^13 + 3384*T^12 - 7224*T^11 + 61574*T^10 + 81752*T^9 + 1705914*T^8 + 4307852*T^7 + 12277444*T^6 - 10727000*T^5 + 25785237*T^4 - 14362272*T^3 + 165499990*T^2 + 92758840*T + 19811401
$41$
\( T^{16} - 25 T^{15} + \cdots + 195873515776 \)
T^16 - 25*T^15 + 511*T^14 - 6637*T^13 + 70817*T^12 - 621144*T^11 + 4557832*T^10 - 25657160*T^9 + 109517152*T^8 - 346008448*T^7 + 1337387008*T^6 - 9079973952*T^5 + 55443833536*T^4 - 194256948416*T^3 + 365046114048*T^2 - 265779280128*T + 195873515776
$43$
\( (T^{8} - 14 T^{7} - 28 T^{6} + 732 T^{5} + \cdots - 101)^{2} \)
(T^8 - 14*T^7 - 28*T^6 + 732*T^5 + 676*T^4 - 10950*T^3 - 20940*T^2 - 9376*T - 101)^2
$47$
\( T^{16} + 8 T^{15} + \cdots + 211111761961 \)
T^16 + 8*T^15 + 95*T^14 + 330*T^13 + 5470*T^12 + 6024*T^11 + 337577*T^10 - 681355*T^9 + 16559480*T^8 + 11728045*T^7 + 407453867*T^6 - 2733690479*T^5 + 23183121205*T^4 - 90676914155*T^3 + 221644924280*T^2 - 215365525963*T + 211111761961
$53$
\( T^{16} - 22 T^{15} + \cdots + 2390818816 \)
T^16 - 22*T^15 + 464*T^14 - 4613*T^13 + 41419*T^12 - 233912*T^11 + 2565051*T^10 - 25186296*T^9 + 292088939*T^8 - 1678603669*T^7 + 11682527531*T^6 - 63804472765*T^5 + 233150253337*T^4 - 83021579296*T^3 + 24613601280*T^2 - 7209226240*T + 2390818816
$59$
\( T^{16} - 14 T^{15} + \cdots + 6178746025 \)
T^16 - 14*T^15 + 139*T^14 + 282*T^13 + 8674*T^12 - 55726*T^11 + 3951331*T^10 - 10011897*T^9 + 136029696*T^8 + 614560775*T^7 + 1053198275*T^6 - 21667175345*T^5 + 97963845685*T^4 + 36874744475*T^3 + 153220406500*T^2 - 17193664675*T + 6178746025
$61$
\( T^{16} - 16 T^{15} + \cdots + 11020667028121 \)
T^16 - 16*T^15 + 360*T^14 - 5326*T^13 + 71375*T^12 - 678460*T^11 + 5571042*T^10 - 42590984*T^9 + 451227447*T^8 - 4818457752*T^7 + 43500650258*T^6 - 287115248048*T^5 + 1397137439403*T^4 - 4615424853270*T^3 + 11051474804084*T^2 - 15664507166444*T + 11020667028121
$67$
\( (T^{8} - 7 T^{7} - 323 T^{6} + \cdots + 23994961)^{2} \)
(T^8 - 7*T^7 - 323*T^6 + 1940*T^5 + 34476*T^4 - 150560*T^3 - 1516857*T^2 + 3514469*T + 23994961)^2
$71$
\( T^{16} + 46 T^{15} + \cdots + 923250017881 \)
T^16 + 46*T^15 + 1161*T^14 + 19866*T^13 + 255879*T^12 + 2598256*T^11 + 21862844*T^10 + 153555257*T^9 + 865710269*T^8 + 3482328057*T^7 + 8935796369*T^6 + 12565232820*T^5 + 23372027117*T^4 + 8165747368*T^3 + 245958880075*T^2 + 270150312145*T + 923250017881
$73$
\( T^{16} + 7 T^{15} + \cdots + 217828491841 \)
T^16 + 7*T^15 + 101*T^14 + 1250*T^13 + 26633*T^12 - 176416*T^11 + 2072367*T^10 + 1065034*T^9 + 21485832*T^8 - 511491238*T^7 + 12339658053*T^6 - 57108357785*T^5 + 140764483895*T^4 - 167763797477*T^3 + 255293290760*T^2 - 79469984833*T + 217828491841
$79$
\( T^{16} + 39 T^{15} + \cdots + 275304843025 \)
T^16 + 39*T^15 + 680*T^14 + 5931*T^13 + 38999*T^12 + 437601*T^11 + 6424945*T^10 + 63729649*T^9 + 611592146*T^8 + 4768384355*T^7 + 32003156325*T^6 + 154248256190*T^5 + 565497644610*T^4 + 932536218350*T^3 + 1610050035525*T^2 - 268313282150*T + 275304843025
$83$
\( T^{16} + 28 T^{15} + 467 T^{14} + \cdots + 69372241 \)
T^16 + 28*T^15 + 467*T^14 + 6764*T^13 + 95496*T^12 + 770198*T^11 + 2104395*T^10 - 10906523*T^9 + 16278514*T^8 + 460574171*T^7 + 1469027385*T^6 - 1633074109*T^5 + 7150305061*T^4 - 3897014973*T^3 + 3815943038*T^2 - 366067879*T + 69372241
$89$
\( (T^{8} + 11 T^{7} - 258 T^{6} + \cdots - 278125)^{2} \)
(T^8 + 11*T^7 - 258*T^6 - 3101*T^5 + 11416*T^4 + 195325*T^3 + 211750*T^2 - 1576875*T - 278125)^2
$97$
\( T^{16} - 39 T^{15} + \cdots + 7330099456 \)
T^16 - 39*T^15 + 1075*T^14 - 18927*T^13 + 245301*T^12 - 2417364*T^11 + 18850332*T^10 - 119857668*T^9 + 644783608*T^8 - 2883902112*T^7 + 10077245472*T^6 - 24926777856*T^5 + 42947603456*T^4 - 61615957248*T^3 + 104169438080*T^2 - 43138135296*T + 7330099456
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