[N,k,chi] = [855,2,Mod(226,855)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 0, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("855.226");
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 \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).
\(n\)
\(172\)
\(191\)
\(496\)
\(\chi(n)\)
\(1\)
\(1\)
\(\beta_{4} + \beta_{8}\)
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
\( T_{2}^{18} + 3 T_{2}^{17} + 6 T_{2}^{16} + 5 T_{2}^{15} - 15 T_{2}^{14} - 24 T_{2}^{13} + 70 T_{2}^{12} + 96 T_{2}^{11} - 30 T_{2}^{10} + 52 T_{2}^{9} + 282 T_{2}^{8} - 357 T_{2}^{7} + 553 T_{2}^{6} + 1890 T_{2}^{5} + 750 T_{2}^{4} + 36 T_{2}^{3} + \cdots + 361 \)
T2^18 + 3*T2^17 + 6*T2^16 + 5*T2^15 - 15*T2^14 - 24*T2^13 + 70*T2^12 + 96*T2^11 - 30*T2^10 + 52*T2^9 + 282*T2^8 - 357*T2^7 + 553*T2^6 + 1890*T2^5 + 750*T2^4 + 36*T2^3 + 444*T2^2 - 342*T2 + 361
acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{18} + 3 T^{17} + 6 T^{16} + 5 T^{15} + \cdots + 361 \)
T^18 + 3*T^17 + 6*T^16 + 5*T^15 - 15*T^14 - 24*T^13 + 70*T^12 + 96*T^11 - 30*T^10 + 52*T^9 + 282*T^8 - 357*T^7 + 553*T^6 + 1890*T^5 + 750*T^4 + 36*T^3 + 444*T^2 - 342*T + 361
$3$
\( T^{18} \)
T^18
$5$
\( (T^{6} + T^{3} + 1)^{3} \)
(T^6 + T^3 + 1)^3
$7$
\( T^{18} + 33 T^{16} + 20 T^{15} + \cdots + 117649 \)
T^18 + 33*T^16 + 20*T^15 + 762*T^14 + 558*T^13 + 9069*T^12 + 10749*T^11 + 78999*T^10 + 89607*T^9 + 329889*T^8 + 452469*T^7 + 1009604*T^6 + 982548*T^5 + 1114554*T^4 + 473683*T^3 + 374556*T^2 + 50421*T + 117649
$11$
\( T^{18} + 36 T^{16} + 80 T^{15} + \cdots + 361 \)
T^18 + 36*T^16 + 80*T^15 + 1014*T^14 + 1761*T^13 + 10718*T^12 + 11724*T^11 + 73617*T^10 + 53088*T^9 + 234795*T^8 - 109761*T^7 + 265407*T^6 - 36192*T^5 + 87558*T^4 - 34226*T^3 + 16173*T^2 - 2565*T + 361
$13$
\( T^{18} + 3 T^{17} + 12 T^{16} + \cdots + 98743969 \)
T^18 + 3*T^17 + 12*T^16 + 3*T^15 - 96*T^14 - 2508*T^13 + 2247*T^12 + 53346*T^11 + 305688*T^10 + 1816837*T^9 + 6452754*T^8 + 14256555*T^7 + 30785622*T^6 + 50728344*T^5 + 122419266*T^4 + 533065431*T^3 + 1021694559*T^2 + 478168440*T + 98743969
$17$
\( T^{18} - 24 T^{17} + 306 T^{16} - 2641 T^{15} + \cdots + 1 \)
T^18 - 24*T^17 + 306*T^16 - 2641*T^15 + 16179*T^14 - 69303*T^13 + 200112*T^12 - 360561*T^11 + 349512*T^10 - 127500*T^9 + 119493*T^8 - 62436*T^7 + 73013*T^6 - 54948*T^5 + 21036*T^4 - 5138*T^3 + 963*T^2 - 54*T + 1
$19$
\( T^{18} + 12 T^{17} + \cdots + 322687697779 \)
T^18 + 12*T^17 + 69*T^16 + 387*T^15 + 1893*T^14 + 4533*T^13 - 1521*T^12 - 75264*T^11 - 718563*T^10 - 4160945*T^9 - 13652697*T^8 - 27170304*T^7 - 10432539*T^6 + 590745093*T^5 + 4687255407*T^4 + 18206755947*T^3 + 61677149991*T^2 + 203802756492*T + 322687697779
$23$
\( T^{18} + 21 T^{17} + \cdots + 3993860809 \)
T^18 + 21*T^17 + 144*T^16 + 107*T^15 - 3726*T^14 - 30126*T^13 - 26309*T^12 + 1041381*T^11 + 4954518*T^10 + 847174*T^9 - 11211516*T^8 + 65874675*T^7 + 412663312*T^6 + 97232817*T^5 + 2056342638*T^4 - 2053998366*T^3 + 3699987315*T^2 - 3215084178*T + 3993860809
$29$
\( T^{18} - 9 T^{17} + \cdots + 492805404001 \)
T^18 - 9*T^17 - 57*T^16 + 1172*T^15 - 498*T^14 - 49062*T^13 + 367698*T^12 - 2040480*T^11 + 5045091*T^10 + 49420881*T^9 + 229365279*T^8 + 1146500886*T^7 + 2323039067*T^6 + 1540393764*T^5 + 35058825963*T^4 + 151712699077*T^3 + 358547261748*T^2 + 437427353115*T + 492805404001
$31$
\( T^{18} - 30 T^{17} + \cdots + 993257404129 \)
T^18 - 30*T^17 + 615*T^16 - 8470*T^15 + 97092*T^14 - 916485*T^13 + 7814137*T^12 - 57705795*T^11 + 376894374*T^10 - 2045754194*T^9 + 9299087217*T^8 - 33396452925*T^7 + 96900041116*T^6 - 211841159529*T^5 + 384241234950*T^4 - 527206017759*T^3 + 779687237190*T^2 - 815591415165*T + 993257404129
$37$
\( (T^{9} + 30 T^{8} + 267 T^{7} + \cdots + 27721)^{2} \)
(T^9 + 30*T^8 + 267*T^7 - 358*T^6 - 19953*T^5 - 126639*T^4 - 334715*T^3 - 356199*T^2 - 101946*T + 27721)^2
$41$
\( T^{18} - 6 T^{17} + \cdots + 132479256529 \)
T^18 - 6*T^17 + 171*T^16 - 1093*T^15 + 7287*T^14 + 26790*T^13 - 286846*T^12 + 2742000*T^11 + 9830937*T^10 - 131376204*T^9 + 737739546*T^8 - 962838900*T^7 + 3272812086*T^6 + 872614581*T^5 - 138736737*T^4 + 153509333179*T^3 + 417325836498*T^2 + 394868455944*T + 132479256529
$43$
\( T^{18} + 6 T^{17} + 186 T^{16} + \cdots + 101989801 \)
T^18 + 6*T^17 + 186*T^16 + 968*T^15 + 11904*T^14 + 45009*T^13 + 227592*T^12 + 116961*T^11 - 519963*T^10 - 10694622*T^9 + 25879197*T^8 + 74453274*T^7 - 247131028*T^6 - 120203658*T^5 + 1239717402*T^4 - 1537291658*T^3 + 284832954*T^2 + 523865427*T + 101989801
$47$
\( T^{18} + 33 T^{17} + \cdots + 32737816559809 \)
T^18 + 33*T^17 + 561*T^16 + 5880*T^15 + 30456*T^14 - 102141*T^13 - 2475773*T^12 - 8756832*T^11 + 129998277*T^10 + 1826494087*T^9 + 9886932228*T^8 + 2450850135*T^7 - 67038566051*T^6 + 1949033852754*T^5 + 21678514175436*T^4 + 88154625013001*T^3 + 147611231178531*T^2 + 43990334747889*T + 32737816559809
$53$
\( T^{18} + 24 T^{17} + \cdots + 19250147475001 \)
T^18 + 24*T^17 + 177*T^16 - 920*T^15 - 24375*T^14 - 123795*T^13 + 1017442*T^12 + 24328677*T^11 + 263877942*T^10 + 2080924547*T^9 + 12982558779*T^8 + 65941225440*T^7 + 281573420083*T^6 + 1024137334884*T^5 + 3121812627873*T^4 + 7721468605068*T^3 + 15881074591848*T^2 + 24397859651733*T + 19250147475001
$59$
\( T^{18} + 18 T^{17} + \cdots + 333749999521 \)
T^18 + 18*T^17 + 57*T^16 + 429*T^15 + 19551*T^14 + 159801*T^13 + 1721647*T^12 + 16052535*T^11 + 75900219*T^10 + 223054966*T^9 + 207917142*T^8 - 5664299076*T^7 - 5925722927*T^6 + 30703238430*T^5 + 141182906871*T^4 - 344989830412*T^3 + 398118220722*T^2 - 199053791316*T + 333749999521
$61$
\( T^{18} - 6 T^{17} + \cdots + 3468418292161 \)
T^18 - 6*T^17 + 306*T^16 - 882*T^15 + 24411*T^14 - 132264*T^13 + 1627723*T^12 - 23832345*T^11 + 164147079*T^10 - 659732348*T^9 + 5386522683*T^8 + 1278112725*T^7 - 129100461785*T^6 + 386661300792*T^5 + 3584784109038*T^4 + 5586817220009*T^3 - 617962952385*T^2 - 3413051924160*T + 3468418292161
$67$
\( T^{18} + 24 T^{17} + \cdots + 36\!\cdots\!81 \)
T^18 + 24*T^17 + 477*T^16 + 5047*T^15 + 53754*T^14 + 441522*T^13 + 10338556*T^12 + 139444560*T^11 + 1463846511*T^10 + 8116074182*T^9 - 23792730639*T^8 - 643972235106*T^7 - 2796194578520*T^6 + 9397140493683*T^5 + 445957181411103*T^4 + 2619845469341829*T^3 + 7237735375106982*T^2 + 8412970770274476*T + 3606994474111081
$71$
\( T^{18} + 24 T^{17} + \cdots + 8590138489 \)
T^18 + 24*T^17 + 357*T^16 + 3062*T^15 + 22308*T^14 + 90459*T^13 + 1192790*T^12 + 13785708*T^11 + 81171726*T^10 + 336238917*T^9 + 1252430772*T^8 + 3241000569*T^7 + 6391590090*T^6 + 9106972983*T^5 + 6055026060*T^4 + 128539843*T^3 + 6299174574*T^2 + 15193987605*T + 8590138489
$73$
\( T^{18} - 6 T^{17} + \cdots + 1047660743809 \)
T^18 - 6*T^17 + 261*T^16 - 1924*T^15 + 7728*T^14 - 122445*T^13 - 1110911*T^12 + 9283833*T^11 + 260272686*T^10 - 188073812*T^9 - 8088356112*T^8 + 15268575909*T^7 + 156917847355*T^6 - 956979682572*T^5 + 2538807263307*T^4 - 3895731864450*T^3 + 3785651251221*T^2 - 2416931052195*T + 1047660743809
$79$
\( T^{18} + \cdots + 316957384070209 \)
T^18 - 9*T^17 + 81*T^16 - 2931*T^15 + 30141*T^14 - 356232*T^13 + 8605117*T^12 - 90091038*T^11 + 763469097*T^10 - 5984555122*T^9 + 46585978533*T^8 - 540987206631*T^7 + 5669900116861*T^6 - 35358197402004*T^5 + 125652544829676*T^4 - 240489842463323*T^3 + 308457054068937*T^2 - 378738937221816*T + 316957384070209
$83$
\( T^{18} + 171 T^{16} + \cdots + 96609241 \)
T^18 + 171*T^16 + 1014*T^15 + 26091*T^14 + 91485*T^13 + 750273*T^12 + 50985*T^11 + 8450838*T^10 - 9731293*T^9 + 94596957*T^8 - 181199070*T^7 + 427332993*T^6 - 416415537*T^5 + 504618588*T^4 - 285719709*T^3 + 336302856*T^2 - 149941395*T + 96609241
$89$
\( T^{18} - 6 T^{17} - 63 T^{16} + \cdots + 143304841 \)
T^18 - 6*T^17 - 63*T^16 - 1876*T^15 + 561*T^14 + 397758*T^13 + 9356410*T^12 + 18879318*T^11 + 496929534*T^10 + 7135408126*T^9 + 37686601374*T^8 - 103343714457*T^7 + 12554167736425*T^6 - 48437163821268*T^5 + 463604835861969*T^4 - 629866247484495*T^3 + 257371597921167*T^2 - 294174372378*T + 143304841
$97$
\( T^{18} + 87 T^{17} + \cdots + 36\!\cdots\!09 \)
T^18 + 87*T^17 + 3717*T^16 + 101302*T^15 + 1961043*T^14 + 28308798*T^13 + 307215832*T^12 + 2405174310*T^11 + 11512621515*T^10 + 1686674630*T^9 - 470761652373*T^8 - 3321434248437*T^7 + 3379845128947*T^6 + 247598196402720*T^5 + 2378140726713561*T^4 + 13054542504561801*T^3 + 41783246330958471*T^2 + 57610985066636475*T + 36609425480794609
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