[N,k,chi] = [1205,2,Mod(1,1205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1205, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1205.1");
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
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
\( p \)
Sign
\(5\)
\(1\)
\(241\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{15} - 2 T_{2}^{14} - 16 T_{2}^{13} + 31 T_{2}^{12} + 99 T_{2}^{11} - 184 T_{2}^{10} - 296 T_{2}^{9} + 519 T_{2}^{8} + 437 T_{2}^{7} - 699 T_{2}^{6} - 297 T_{2}^{5} + 394 T_{2}^{4} + 89 T_{2}^{3} - 57 T_{2}^{2} - 17 T_{2} - 1 \)
T2^15 - 2*T2^14 - 16*T2^13 + 31*T2^12 + 99*T2^11 - 184*T2^10 - 296*T2^9 + 519*T2^8 + 437*T2^7 - 699*T2^6 - 297*T2^5 + 394*T2^4 + 89*T2^3 - 57*T2^2 - 17*T2 - 1
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1205))\).
$p$
$F_p(T)$
$2$
\( T^{15} - 2 T^{14} - 16 T^{13} + 31 T^{12} + \cdots - 1 \)
T^15 - 2*T^14 - 16*T^13 + 31*T^12 + 99*T^11 - 184*T^10 - 296*T^9 + 519*T^8 + 437*T^7 - 699*T^6 - 297*T^5 + 394*T^4 + 89*T^3 - 57*T^2 - 17*T - 1
$3$
\( T^{15} + 7 T^{14} - T^{13} - 100 T^{12} + \cdots + 191 \)
T^15 + 7*T^14 - T^13 - 100*T^12 - 133*T^11 + 484*T^10 + 970*T^9 - 949*T^8 - 2749*T^7 + 499*T^6 + 3617*T^5 + 611*T^4 - 2131*T^3 - 721*T^2 + 447*T + 191
$5$
\( (T + 1)^{15} \)
(T + 1)^15
$7$
\( T^{15} + 3 T^{14} - 36 T^{13} - 110 T^{12} + \cdots + 241 \)
T^15 + 3*T^14 - 36*T^13 - 110*T^12 + 416*T^11 + 1298*T^10 - 1991*T^9 - 6226*T^8 + 5181*T^7 + 14084*T^6 - 8475*T^5 - 14696*T^4 + 8197*T^3 + 5244*T^2 - 3172*T + 241
$11$
\( T^{15} + 10 T^{14} - 25 T^{13} + \cdots + 20041 \)
T^15 + 10*T^14 - 25*T^13 - 512*T^12 - 527*T^11 + 8481*T^10 + 18797*T^9 - 50568*T^8 - 150635*T^7 + 74209*T^6 + 338234*T^5 - 7464*T^4 - 266533*T^3 - 46243*T^2 + 66359*T + 20041
$13$
\( T^{15} + 8 T^{14} - 60 T^{13} + \cdots + 17551 \)
T^15 + 8*T^14 - 60*T^13 - 600*T^12 + 533*T^11 + 13380*T^10 + 12886*T^9 - 97584*T^8 - 130656*T^7 + 295781*T^6 + 279384*T^5 - 515260*T^4 - 65839*T^3 + 338543*T^2 - 147967*T + 17551
$17$
\( T^{15} + T^{14} - 129 T^{13} + \cdots + 574859 \)
T^15 + T^14 - 129*T^13 + 31*T^12 + 6354*T^11 - 8325*T^10 - 142575*T^9 + 318770*T^8 + 1301136*T^7 - 4166233*T^6 - 1949876*T^5 + 13713675*T^4 - 4178942*T^3 - 8873889*T^2 - 320121*T + 574859
$19$
\( T^{15} + 30 T^{14} + 332 T^{13} + \cdots + 77731 \)
T^15 + 30*T^14 + 332*T^13 + 1245*T^12 - 5311*T^11 - 64268*T^10 - 163928*T^9 + 365509*T^8 + 2719031*T^7 + 4154115*T^6 - 2092049*T^5 - 8108117*T^4 - 517493*T^3 + 2324421*T^2 + 809612*T + 77731
$23$
\( T^{15} - 19 T^{14} + 1774 T^{12} + \cdots - 250937 \)
T^15 - 19*T^14 + 1774*T^12 - 6673*T^11 - 42191*T^10 + 190424*T^9 + 465578*T^8 - 1782908*T^7 - 2780084*T^6 + 5596003*T^5 + 7907583*T^4 - 2008070*T^3 - 5952482*T^2 - 2404746*T - 250937
$29$
\( T^{15} + 12 T^{14} - 96 T^{13} + \cdots + 116809 \)
T^15 + 12*T^14 - 96*T^13 - 1684*T^12 - 1474*T^11 + 50624*T^10 + 115385*T^9 - 587012*T^8 - 1479824*T^7 + 2834122*T^6 + 5846919*T^5 - 5690636*T^4 - 4967005*T^3 + 2156742*T^2 + 1444420*T + 116809
$31$
\( T^{15} + 22 T^{14} + \cdots - 21531245891 \)
T^15 + 22*T^14 - 25*T^13 - 3857*T^12 - 20949*T^11 + 199627*T^10 + 2111523*T^9 - 257675*T^8 - 69316528*T^7 - 224629774*T^6 + 527325092*T^5 + 4293208883*T^4 + 6580310295*T^3 - 7277745624*T^2 - 28963912421*T - 21531245891
$37$
\( T^{15} + 12 T^{14} + \cdots + 3773372033 \)
T^15 + 12*T^14 - 241*T^13 - 2804*T^12 + 22274*T^11 + 244893*T^10 - 931590*T^9 - 9871186*T^8 + 14263131*T^7 + 185888905*T^6 + 43867218*T^5 - 1411958548*T^4 - 1870786789*T^3 + 2738896827*T^2 + 6957819995*T + 3773372033
$41$
\( T^{15} + 13 T^{14} + \cdots - 8686116791 \)
T^15 + 13*T^14 - 274*T^13 - 3833*T^12 + 22080*T^11 + 364207*T^10 - 551203*T^9 - 13848678*T^8 + 3919347*T^7 + 241231053*T^6 - 1752142*T^5 - 1963303989*T^4 - 3946869*T^3 + 6982555384*T^2 + 25706705*T - 8686116791
$43$
\( T^{15} + 25 T^{14} + \cdots + 336240191 \)
T^15 + 25*T^14 + 19*T^13 - 3980*T^12 - 25875*T^11 + 148041*T^10 + 1675073*T^9 + 68267*T^8 - 35514290*T^7 - 69195241*T^6 + 227236965*T^5 + 700690434*T^4 - 230184430*T^3 - 1944490637*T^2 - 1118496962*T + 336240191
$47$
\( T^{15} - 16 T^{14} + \cdots + 176160469 \)
T^15 - 16*T^14 - 213*T^13 + 4829*T^12 - 1929*T^11 - 379125*T^10 + 1737709*T^9 + 6003789*T^8 - 53471588*T^7 + 62513397*T^6 + 254029038*T^5 - 627245293*T^4 + 39726897*T^3 + 932975726*T^2 - 773316628*T + 176160469
$53$
\( T^{15} + 4 T^{14} + \cdots + 27449089409 \)
T^15 + 4*T^14 - 331*T^13 - 1829*T^12 + 37343*T^11 + 258136*T^10 - 1710840*T^9 - 15487646*T^8 + 23802894*T^7 + 416308316*T^6 + 340626657*T^5 - 4460444524*T^4 - 9178972167*T^3 + 12947198063*T^2 + 46420605845*T + 27449089409
$59$
\( T^{15} + 50 T^{14} + \cdots + 4073980277 \)
T^15 + 50*T^14 + 771*T^13 - 209*T^12 - 110295*T^11 - 700368*T^10 + 4065108*T^9 + 47280836*T^8 - 7377568*T^7 - 1038795356*T^6 - 1353634751*T^5 + 6949750202*T^4 + 8771931181*T^3 - 13867347235*T^2 - 7992475897*T + 4073980277
$61$
\( T^{15} + 41 T^{14} + \cdots - 416282129347 \)
T^15 + 41*T^14 + 351*T^13 - 6871*T^12 - 131627*T^11 - 129888*T^10 + 11018266*T^9 + 63595228*T^8 - 243563701*T^7 - 2870977746*T^6 - 1842137971*T^5 + 43626834819*T^4 + 97067002349*T^3 - 185142686302*T^2 - 678098169085*T - 416282129347
$67$
\( T^{15} + 43 T^{14} + \cdots + 602438539877 \)
T^15 + 43*T^14 + 407*T^13 - 5722*T^12 - 100588*T^11 + 183371*T^10 + 8374646*T^9 + 5757617*T^8 - 343963403*T^7 - 488828325*T^6 + 7334025707*T^5 + 12123345606*T^4 - 76008374965*T^3 - 138660366045*T^2 + 300283480954*T + 602438539877
$71$
\( T^{15} + 14 T^{14} + \cdots + 236049251731 \)
T^15 + 14*T^14 - 351*T^13 - 4782*T^12 + 51027*T^11 + 630444*T^10 - 4077242*T^9 - 40133269*T^8 + 194332209*T^7 + 1238988951*T^6 - 5255528626*T^5 - 15674539666*T^4 + 66050096733*T^3 + 45530160689*T^2 - 307080454775*T + 236049251731
$73$
\( T^{15} + 10 T^{14} + \cdots - 416966690521 \)
T^15 + 10*T^14 - 601*T^13 - 6469*T^12 + 133258*T^11 + 1604984*T^10 - 12609062*T^9 - 186927996*T^8 + 350909714*T^7 + 9830267893*T^6 + 12624963431*T^5 - 172754523356*T^4 - 460395694842*T^3 + 421898973955*T^2 + 1397517898998*T - 416966690521
$79$
\( T^{15} + 44 T^{14} + \cdots - 1484049491 \)
T^15 + 44*T^14 + 575*T^13 - 1525*T^12 - 99115*T^11 - 694144*T^10 + 1853596*T^9 + 43678545*T^8 + 157654518*T^7 - 294317454*T^6 - 3309724089*T^5 - 7268360498*T^4 - 2271667975*T^3 + 6921356479*T^2 + 1325530706*T - 1484049491
$83$
\( T^{15} - 7 T^{14} + \cdots - 43476792713 \)
T^15 - 7*T^14 - 690*T^13 + 3235*T^12 + 176294*T^11 - 333119*T^10 - 20756309*T^9 - 13876991*T^8 + 1038480784*T^7 + 2411914424*T^6 - 16446086450*T^5 - 46672086160*T^4 + 63847221236*T^3 + 182476469251*T^2 - 47425742843*T - 43476792713
$89$
\( T^{15} - 4 T^{14} + \cdots + 2543711203 \)
T^15 - 4*T^14 - 570*T^13 + 1077*T^12 + 114463*T^11 - 51975*T^10 - 9999148*T^9 - 3341523*T^8 + 385699757*T^7 + 241273554*T^6 - 5920891733*T^5 - 1863608481*T^4 + 29431433868*T^3 - 11139090057*T^2 - 15039262155*T + 2543711203
$97$
\( T^{15} - 9 T^{14} + \cdots - 3592948942717 \)
T^15 - 9*T^14 - 544*T^13 + 3629*T^12 + 116371*T^11 - 543175*T^10 - 12734793*T^9 + 38426924*T^8 + 771986244*T^7 - 1294308146*T^6 - 25989696058*T^5 + 15308878746*T^4 + 452074119218*T^3 + 145904463222*T^2 - 3150077125084*T - 3592948942717
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