[N,k,chi] = [810,2,Mod(91,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([8, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.91");
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 \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
\(n\)
\(487\)
\(731\)
\(\chi(n)\)
\(1\)
\(-\beta_{2}\)
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
\( T_{7}^{18} + 3 T_{7}^{17} - 21 T_{7}^{16} - 57 T_{7}^{15} + 249 T_{7}^{14} + 291 T_{7}^{13} + 852 T_{7}^{12} - 3282 T_{7}^{11} + 6918 T_{7}^{10} - 28483 T_{7}^{9} + 292062 T_{7}^{8} - 416307 T_{7}^{7} + 282075 T_{7}^{6} + \cdots + 982081 \)
T7^18 + 3*T7^17 - 21*T7^16 - 57*T7^15 + 249*T7^14 + 291*T7^13 + 852*T7^12 - 3282*T7^11 + 6918*T7^10 - 28483*T7^9 + 292062*T7^8 - 416307*T7^7 + 282075*T7^6 + 865113*T7^5 - 256776*T7^4 + 1856067*T7^3 + 3485199*T7^2 - 261624*T7 + 982081
acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{6} - T^{3} + 1)^{3} \)
(T^6 - T^3 + 1)^3
$3$
\( T^{18} \)
T^18
$5$
\( (T^{6} + T^{3} + 1)^{3} \)
(T^6 + T^3 + 1)^3
$7$
\( T^{18} + 3 T^{17} - 21 T^{16} + \cdots + 982081 \)
T^18 + 3*T^17 - 21*T^16 - 57*T^15 + 249*T^14 + 291*T^13 + 852*T^12 - 3282*T^11 + 6918*T^10 - 28483*T^9 + 292062*T^8 - 416307*T^7 + 282075*T^6 + 865113*T^5 - 256776*T^4 + 1856067*T^3 + 3485199*T^2 - 261624*T + 982081
$11$
\( T^{18} + 9 T^{17} + 63 T^{16} + \cdots + 2985984 \)
T^18 + 9*T^17 + 63*T^16 + 270*T^15 + 648*T^14 - 702*T^13 - 10935*T^12 - 41796*T^11 + 29403*T^10 + 1158192*T^9 + 6205491*T^8 + 17211204*T^7 + 28274265*T^6 + 29323296*T^5 + 20307024*T^4 - 2099520*T^3 + 6718464*T + 2985984
$13$
\( T^{18} + 15 T^{17} + 120 T^{16} + \cdots + 87616 \)
T^18 + 15*T^17 + 120*T^16 + 861*T^15 + 4611*T^14 + 10167*T^13 - 13116*T^12 - 169536*T^11 - 479022*T^10 + 718004*T^9 + 9464946*T^8 + 30315207*T^7 + 53315745*T^6 + 57065304*T^5 + 39041448*T^4 + 17309184*T^3 + 5854272*T^2 + 1019424*T + 87616
$17$
\( T^{18} - 3 T^{17} + 90 T^{16} + \cdots + 3779136 \)
T^18 - 3*T^17 + 90*T^16 - 249*T^15 + 6048*T^14 - 16308*T^13 + 144045*T^12 - 305613*T^11 + 2221344*T^10 - 3971646*T^9 + 15908643*T^8 - 4551876*T^7 + 28990953*T^6 + 40525110*T^5 + 84566916*T^4 + 68619312*T^3 + 46294416*T^2 + 15116544*T + 3779136
$19$
\( T^{18} - 9 T^{17} + 135 T^{16} + \cdots + 19855936 \)
T^18 - 9*T^17 + 135*T^16 - 588*T^15 + 7110*T^14 - 25380*T^13 + 245013*T^12 - 442890*T^11 + 3819555*T^10 - 3333454*T^9 + 43981857*T^8 - 7055208*T^7 + 266867529*T^6 + 154377558*T^5 + 1206029628*T^4 + 570424416*T^3 + 459227088*T^2 - 77962176*T + 19855936
$23$
\( T^{18} - 24 T^{17} + \cdots + 297666009 \)
T^18 - 24*T^17 + 288*T^16 - 2478*T^15 + 17262*T^14 - 95634*T^13 + 413730*T^12 - 1434996*T^11 + 4193613*T^10 - 11451780*T^9 + 34680717*T^8 - 110242296*T^7 + 297978750*T^6 - 609372558*T^5 + 945827199*T^4 - 1166230872*T^3 + 1153463166*T^2 - 813340926*T + 297666009
$29$
\( T^{18} - 9 T^{17} + 45 T^{16} + \cdots + 3674889 \)
T^18 - 9*T^17 + 45*T^16 - 486*T^15 + 1809*T^14 + 8343*T^13 + 10125*T^12 - 236763*T^11 + 21465*T^10 + 535275*T^9 + 1803060*T^8 + 14691051*T^7 + 36638811*T^6 + 49187088*T^5 + 102743073*T^4 + 87545610*T^3 + 30176226*T^2 - 17546301*T + 3674889
$31$
\( T^{18} + 3 T^{17} + 60 T^{16} + \cdots + 4700224 \)
T^18 + 3*T^17 + 60*T^16 - 66*T^15 - 1569*T^14 + 1497*T^13 + 14505*T^12 + 5835*T^11 + 245805*T^10 - 1166506*T^9 - 181518*T^8 + 13211295*T^7 + 1442607*T^6 - 26308506*T^5 + 60812436*T^4 - 57518400*T^3 + 123603168*T^2 - 36344352*T + 4700224
$37$
\( T^{18} + \cdots + 578992548805696 \)
T^18 + 12*T^17 + 366*T^16 + 3174*T^15 + 71250*T^14 + 552360*T^13 + 8636379*T^12 + 52170402*T^11 + 663931284*T^10 + 3428641622*T^9 + 35131522890*T^8 + 134376297126*T^7 + 1119650363241*T^6 + 3341542589520*T^5 + 23398263835500*T^4 + 35915434414512*T^3 + 164650316941200*T^2 - 141858884913888*T + 578992548805696
$41$
\( T^{18} + 6 T^{17} + \cdots + 269887523049 \)
T^18 + 6*T^17 + 189*T^16 + 237*T^15 + 12087*T^14 - 115938*T^13 + 1237293*T^12 - 14205213*T^11 + 109070550*T^10 - 341569116*T^9 + 2360306709*T^8 - 37596709953*T^7 + 265015412820*T^6 - 938342350179*T^5 + 1880363179440*T^4 - 2383028639946*T^3 + 2199349149192*T^2 - 1185100926921*T + 269887523049
$43$
\( T^{18} + 33 T^{17} + \cdots + 669790017649 \)
T^18 + 33*T^17 + 651*T^16 + 9285*T^15 + 107688*T^14 + 835278*T^13 + 3206490*T^12 - 8676858*T^11 - 148529823*T^10 - 600716752*T^9 + 1564464462*T^8 + 32277960489*T^7 + 149377698429*T^6 + 220718978121*T^5 - 83415390780*T^4 - 46426262250*T^3 + 883679149041*T^2 + 233773867515*T + 669790017649
$47$
\( T^{18} - 33 T^{17} + \cdots + 27\!\cdots\!61 \)
T^18 - 33*T^17 + 729*T^16 - 12657*T^15 + 177984*T^14 - 2248128*T^13 + 27646650*T^12 - 331517529*T^11 + 3871437768*T^10 - 42449896503*T^9 + 401330999376*T^8 - 3072805393212*T^7 + 18885959529876*T^6 - 94932960248115*T^5 + 390237739832961*T^4 - 1249315173129834*T^3 + 2848050066455064*T^2 - 4062674469601998*T + 2775853478404161
$53$
\( (T^{9} + 12 T^{8} - 198 T^{7} + \cdots + 2346408)^{2} \)
(T^9 + 12*T^8 - 198*T^7 - 2604*T^6 + 10341*T^5 + 176472*T^4 - 10665*T^3 - 3843288*T^2 - 6876900*T + 2346408)^2
$59$
\( T^{18} - 12 T^{17} + \cdots + 1719926784 \)
T^18 - 12*T^17 + 225*T^16 - 3336*T^15 + 19116*T^14 - 35181*T^13 - 11070*T^12 + 4356585*T^11 + 3407751*T^10 - 91899414*T^9 - 168768198*T^8 + 748152045*T^7 + 3315140217*T^6 + 3677909976*T^5 + 255068352*T^4 - 1143631872*T^3 + 4246069248*T^2 - 2364899328*T + 1719926784
$61$
\( T^{18} - 21 T^{17} + \cdots + 19813208147209 \)
T^18 - 21*T^17 + 138*T^16 - 804*T^15 + 20694*T^14 - 302988*T^13 + 4707780*T^12 - 32125089*T^11 + 198364341*T^10 - 1227750037*T^9 + 29802767055*T^8 - 205627698384*T^7 + 1553320694958*T^6 - 4940894064273*T^5 - 66757721367*T^4 + 13617617655741*T^3 + 23968183673256*T^2 + 30672184937889*T + 19813208147209
$67$
\( T^{18} - 27 T^{17} + \cdots + 5542008097609 \)
T^18 - 27*T^17 + 261*T^16 - 606*T^15 - 24588*T^14 + 407286*T^13 + 906378*T^12 - 43546698*T^11 + 99818109*T^10 + 299489141*T^9 + 4096284210*T^8 + 22757550660*T^7 + 502684891161*T^6 - 758887398621*T^5 + 71653139127*T^4 + 3712785047295*T^3 + 11939035232439*T^2 - 1263611943720*T + 5542008097609
$71$
\( T^{18} - 36 T^{17} + \cdots + 68\!\cdots\!84 \)
T^18 - 36*T^17 + 1296*T^16 - 27864*T^15 + 641682*T^14 - 11115063*T^13 + 197764578*T^12 - 2718755928*T^11 + 37147104792*T^10 - 403994124666*T^9 + 4489732289295*T^8 - 39765797533377*T^7 + 358489714708053*T^6 - 2404884201174684*T^5 + 16934347882462032*T^4 - 83307156868575360*T^3 + 501916439034978048*T^2 - 1671645201518444544*T + 6847616448843485184
$73$
\( T^{18} + 12 T^{17} + \cdots + 3169339909696 \)
T^18 + 12*T^17 + 429*T^16 + 1950*T^15 + 87243*T^14 + 285366*T^13 + 11344857*T^12 + 4953972*T^11 + 854994138*T^10 - 100842676*T^9 + 39741820599*T^8 - 104303433243*T^7 + 577355229609*T^6 - 693382311120*T^5 + 2771495462460*T^4 - 3837731688912*T^3 + 8888379862224*T^2 - 5491979139936*T + 3169339909696
$79$
\( T^{18} + 75 T^{17} + \cdots + 22\!\cdots\!36 \)
T^18 + 75*T^17 + 2715*T^16 + 63429*T^15 + 1081878*T^14 + 14418957*T^13 + 156516162*T^12 + 1426099692*T^11 + 11262771618*T^10 + 80215467536*T^9 + 538026185217*T^8 + 3533280457092*T^7 + 22946031053961*T^6 + 139549288335594*T^5 + 720335922332700*T^4 + 2895371538700992*T^3 + 8609512185294528*T^2 + 18119285759863968*T + 22244870563356736
$83$
\( T^{18} + 33 T^{17} + \cdots + 51843191481 \)
T^18 + 33*T^17 + 450*T^16 + 2271*T^15 - 12285*T^14 - 247185*T^13 - 1153035*T^12 + 7033392*T^11 + 154485954*T^10 + 1262375064*T^9 + 6441808176*T^8 + 22795388955*T^7 + 60283775178*T^6 + 126296605188*T^5 + 212250897855*T^4 + 251728896312*T^3 + 168172772346*T^2 + 61968382560*T + 51843191481
$89$
\( T^{18} - 33 T^{17} + \cdots + 90\!\cdots\!41 \)
T^18 - 33*T^17 + 999*T^16 - 16158*T^15 + 279387*T^14 - 3208707*T^13 + 44991117*T^12 - 387481401*T^11 + 3989842272*T^10 - 18546327432*T^9 + 157024928484*T^8 - 281409933996*T^7 + 5100714702288*T^6 + 1258843909473*T^5 + 112464469373742*T^4 + 262890402652251*T^3 + 2039120989158204*T^2 + 3627765253487619*T + 9017862921580041
$97$
\( T^{18} + 18 T^{17} + \cdots + 679384765504 \)
T^18 + 18*T^17 + 189*T^16 + 5244*T^15 + 31212*T^14 - 668583*T^13 + 4542666*T^12 - 55429425*T^11 + 20622267*T^10 + 3718918226*T^9 + 34649853966*T^8 + 77437393443*T^7 + 100896999963*T^6 - 102538649034*T^5 + 479752055784*T^4 - 950723889312*T^3 + 4016619170496*T^2 - 2989725533568*T + 679384765504
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