[N,k,chi] = [4334,2,Mod(1,4334)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4334, 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("4334.1");
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
\(2\)
\(1\)
\(11\)
\(-1\)
\(197\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{15} + T_{3}^{14} - 23 T_{3}^{13} - 19 T_{3}^{12} + 194 T_{3}^{11} + 124 T_{3}^{10} - 761 T_{3}^{9} - 353 T_{3}^{8} + 1417 T_{3}^{7} + 465 T_{3}^{6} - 1128 T_{3}^{5} - 288 T_{3}^{4} + 316 T_{3}^{3} + 79 T_{3}^{2} - 20 T_{3} - 4 \)
T3^15 + T3^14 - 23*T3^13 - 19*T3^12 + 194*T3^11 + 124*T3^10 - 761*T3^9 - 353*T3^8 + 1417*T3^7 + 465*T3^6 - 1128*T3^5 - 288*T3^4 + 316*T3^3 + 79*T3^2 - 20*T3 - 4
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4334))\).
$p$
$F_p(T)$
$2$
\( (T + 1)^{15} \)
(T + 1)^15
$3$
\( T^{15} + T^{14} - 23 T^{13} - 19 T^{12} + \cdots - 4 \)
T^15 + T^14 - 23*T^13 - 19*T^12 + 194*T^11 + 124*T^10 - 761*T^9 - 353*T^8 + 1417*T^7 + 465*T^6 - 1128*T^5 - 288*T^4 + 316*T^3 + 79*T^2 - 20*T - 4
$5$
\( T^{15} + 7 T^{14} - 8 T^{13} - 146 T^{12} + \cdots + 4339 \)
T^15 + 7*T^14 - 8*T^13 - 146*T^12 - 118*T^11 + 1140*T^10 + 1791*T^9 - 4060*T^8 - 8913*T^7 + 6091*T^6 + 20552*T^5 - 868*T^4 - 22223*T^3 - 6310*T^2 + 9072*T + 4339
$7$
\( T^{15} - T^{14} - 37 T^{13} + 34 T^{12} + \cdots + 1 \)
T^15 - T^14 - 37*T^13 + 34*T^12 + 392*T^11 - 359*T^10 - 1508*T^9 + 1123*T^8 + 2290*T^7 - 1062*T^6 - 1135*T^5 + 395*T^4 + 199*T^3 - 54*T^2 - 8*T + 1
$11$
\( (T - 1)^{15} \)
(T - 1)^15
$13$
\( T^{15} + T^{14} - 81 T^{13} - 15 T^{12} + \cdots - 14284 \)
T^15 + T^14 - 81*T^13 - 15*T^12 + 2099*T^11 - 1547*T^10 - 21211*T^9 + 34204*T^8 + 63880*T^7 - 162987*T^6 - 7845*T^5 + 239704*T^4 - 149923*T^3 - 43893*T^2 + 62500*T - 14284
$17$
\( T^{15} + 6 T^{14} - 68 T^{13} + \cdots - 26093 \)
T^15 + 6*T^14 - 68*T^13 - 415*T^12 + 1587*T^11 + 10001*T^10 - 15619*T^9 - 105244*T^8 + 57512*T^7 + 470881*T^6 - 23539*T^5 - 678821*T^4 + 84510*T^3 + 258451*T^2 - 16519*T - 26093
$19$
\( T^{15} + 14 T^{14} + 15 T^{13} - 393 T^{12} + \cdots + 48 \)
T^15 + 14*T^14 + 15*T^13 - 393*T^12 - 808*T^11 + 2907*T^10 + 7105*T^9 - 6055*T^8 - 20740*T^7 - 292*T^6 + 21285*T^5 + 7806*T^4 - 5199*T^3 - 2501*T^2 - 72*T + 48
$23$
\( T^{15} - 2 T^{14} - 145 T^{13} + \cdots - 9538764 \)
T^15 - 2*T^14 - 145*T^13 + 96*T^12 + 8065*T^11 + 3013*T^10 - 208779*T^9 - 216002*T^8 + 2529934*T^7 + 3349242*T^6 - 13461620*T^5 - 17031994*T^4 + 29356811*T^3 + 29749903*T^2 - 19230828*T - 9538764
$29$
\( T^{15} - 8 T^{14} - 238 T^{13} + \cdots - 686143609 \)
T^15 - 8*T^14 - 238*T^13 + 1625*T^12 + 22611*T^11 - 115779*T^10 - 1119302*T^9 + 3383106*T^8 + 29823745*T^7 - 28205250*T^6 - 368570029*T^5 - 236382984*T^4 + 1135632878*T^3 + 1005736043*T^2 - 1022197129*T - 686143609
$31$
\( T^{15} + 33 T^{14} + 302 T^{13} + \cdots + 11274477 \)
T^15 + 33*T^14 + 302*T^13 - 1592*T^12 - 47250*T^11 - 286858*T^10 - 10171*T^9 + 6758714*T^8 + 24163452*T^7 - 3719574*T^6 - 173100080*T^5 - 286092994*T^4 + 32627318*T^3 + 365091902*T^2 + 188039427*T + 11274477
$37$
\( T^{15} + 9 T^{14} + \cdots - 2749788772 \)
T^15 + 9*T^14 - 264*T^13 - 2445*T^12 + 24099*T^11 + 235371*T^10 - 902382*T^9 - 9684766*T^8 + 14116819*T^7 + 172303271*T^6 - 129530484*T^5 - 1355781630*T^4 + 804710957*T^3 + 3848827721*T^2 - 1338111462*T - 2749788772
$41$
\( T^{15} + 10 T^{14} - 120 T^{13} + \cdots - 1025428 \)
T^15 + 10*T^14 - 120*T^13 - 996*T^12 + 5493*T^11 + 31872*T^10 - 109964*T^9 - 383157*T^8 + 815134*T^7 + 2164027*T^6 - 2338490*T^5 - 5793062*T^4 + 1810478*T^3 + 6091307*T^2 + 868366*T - 1025428
$43$
\( T^{15} + 6 T^{14} + \cdots - 34861703012 \)
T^15 + 6*T^14 - 312*T^13 - 1555*T^12 + 38547*T^11 + 152043*T^10 - 2411678*T^9 - 7181314*T^8 + 81394434*T^7 + 175008636*T^6 - 1468541149*T^5 - 2194751667*T^4 + 13025690858*T^3 + 13407010287*T^2 - 43111666508*T - 34861703012
$47$
\( T^{15} + T^{14} - 214 T^{13} + \cdots - 306948 \)
T^15 + T^14 - 214*T^13 - 469*T^12 + 14636*T^11 + 30704*T^10 - 460510*T^9 - 668215*T^8 + 7349344*T^7 + 3689143*T^6 - 58453002*T^5 + 29382362*T^4 + 175827335*T^3 - 253560679*T^2 + 90364002*T - 306948
$53$
\( T^{15} - 6 T^{14} + \cdots + 244181352116 \)
T^15 - 6*T^14 - 392*T^13 + 2027*T^12 + 61714*T^11 - 260562*T^10 - 5033783*T^9 + 16129211*T^8 + 225807933*T^7 - 499779953*T^6 - 5368753300*T^5 + 7552266843*T^4 + 58625555361*T^3 - 60593071691*T^2 - 220432118272*T + 244181352116
$59$
\( T^{15} + 15 T^{14} + \cdots - 735327198093 \)
T^15 + 15*T^14 - 306*T^13 - 4751*T^12 + 38317*T^11 + 614690*T^10 - 2485619*T^9 - 41548888*T^8 + 86001172*T^7 + 1552284969*T^6 - 1411826444*T^5 - 30910212181*T^4 + 5941268322*T^3 + 281538036383*T^2 + 44070261642*T - 735327198093
$61$
\( T^{15} + 25 T^{14} + \cdots + 448552684675 \)
T^15 + 25*T^14 - 211*T^13 - 10558*T^12 - 45859*T^11 + 1206361*T^10 + 12260577*T^9 - 15079414*T^8 - 595444394*T^7 - 1287149740*T^6 + 9455480313*T^5 + 30565354033*T^4 - 61475291364*T^3 - 209462440182*T^2 + 136171695820*T + 448552684675
$67$
\( T^{15} + 13 T^{14} + \cdots + 55312586988 \)
T^15 + 13*T^14 - 390*T^13 - 4621*T^12 + 59561*T^11 + 594149*T^10 - 4651736*T^9 - 34787267*T^8 + 194843358*T^7 + 947692459*T^6 - 4208154725*T^5 - 10307137267*T^4 + 40511166529*T^3 + 15359092067*T^2 - 103763751366*T + 55312586988
$71$
\( T^{15} + 4 T^{14} + \cdots + 75278625949 \)
T^15 + 4*T^14 - 433*T^13 - 3474*T^12 + 55812*T^11 + 727642*T^10 - 642644*T^9 - 45195501*T^8 - 191932894*T^7 + 334339693*T^6 + 3984686890*T^5 + 5292734895*T^4 - 20783822990*T^3 - 55644072324*T^2 + 2511425961*T + 75278625949
$73$
\( T^{15} + 4 T^{14} + \cdots - 2700639899 \)
T^15 + 4*T^14 - 561*T^13 - 3824*T^12 + 110382*T^11 + 1116706*T^10 - 7286132*T^9 - 127996611*T^8 - 206396702*T^7 + 4432116855*T^6 + 29940733870*T^5 + 77448990881*T^4 + 79925074688*T^3 + 8387210104*T^2 - 18727318151*T - 2700639899
$79$
\( T^{15} + 20 T^{14} + \cdots - 5866847775988 \)
T^15 + 20*T^14 - 540*T^13 - 12914*T^12 + 82331*T^11 + 3066962*T^10 + 1881621*T^9 - 321347445*T^8 - 1465976985*T^7 + 12571796018*T^6 + 110215827040*T^5 + 63697461965*T^4 - 1902856925916*T^3 - 7557043054027*T^2 - 11258093439186*T - 5866847775988
$83$
\( T^{15} - T^{14} - 807 T^{13} + \cdots - 15317447248 \)
T^15 - T^14 - 807*T^13 - 22*T^12 + 250236*T^11 + 239544*T^10 - 37257924*T^9 - 69644143*T^8 + 2665953552*T^7 + 7506242047*T^6 - 74564099958*T^5 - 284973832073*T^4 + 125320085096*T^3 + 975548614789*T^2 + 244068740188*T - 15317447248
$89$
\( T^{15} + 41 T^{14} + \cdots - 20211044090796 \)
T^15 + 41*T^14 + 48*T^13 - 19570*T^12 - 247193*T^11 + 1977978*T^10 + 55644583*T^9 + 162862949*T^8 - 3773615541*T^7 - 31716617496*T^6 + 8871350566*T^5 + 1056231415606*T^4 + 4311553352208*T^3 + 3101910015437*T^2 - 13629748019850*T - 20211044090796
$97$
\( T^{15} + 57 T^{14} + \cdots + 69305861104657 \)
T^15 + 57*T^14 + 548*T^13 - 26860*T^12 - 640218*T^11 + 643456*T^10 + 151403609*T^9 + 1246479776*T^8 - 8551733969*T^7 - 172650383487*T^6 - 604739608509*T^5 + 3747465264879*T^4 + 36927834057995*T^3 + 114741255424043*T^2 + 149986458963681*T + 69305861104657
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