[N,k,chi] = [37,6,Mod(1,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.1");
S:= CuspForms(chi, 6);
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
\(37\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 8T_{2}^{6} - 133T_{2}^{5} - 906T_{2}^{4} + 4326T_{2}^{3} + 19928T_{2}^{2} - 40920T_{2} - 109600 \)
T2^7 + 8*T2^6 - 133*T2^5 - 906*T2^4 + 4326*T2^3 + 19928*T2^2 - 40920*T2 - 109600
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(37))\).
$p$
$F_p(T)$
$2$
\( T^{7} + 8 T^{6} - 133 T^{5} + \cdots - 109600 \)
T^7 + 8*T^6 - 133*T^5 - 906*T^4 + 4326*T^3 + 19928*T^2 - 40920*T - 109600
$3$
\( T^{7} + 47 T^{6} + 108 T^{5} + \cdots + 175797 \)
T^7 + 47*T^6 + 108*T^5 - 18352*T^4 - 194360*T^3 - 161292*T^2 + 1144131*T + 175797
$5$
\( T^{7} + 64 T^{6} + \cdots - 330953386760 \)
T^7 + 64*T^6 - 15005*T^5 - 863300*T^4 + 53962745*T^3 + 2655054546*T^2 - 10937871316*T - 330953386760
$7$
\( T^{7} + 293 T^{6} + \cdots - 8969445254224 \)
T^7 + 293*T^6 - 1309*T^5 - 7387993*T^4 - 892551644*T^3 - 45169970404*T^2 - 1044013638896*T - 8969445254224
$11$
\( T^{7} + 1457 T^{6} + \cdots - 74\!\cdots\!65 \)
T^7 + 1457*T^6 + 449952*T^5 - 232901324*T^4 - 166961096620*T^3 - 34674892815112*T^2 - 2854039831714653*T - 74213620727384165
$13$
\( T^{7} + 536 T^{6} + \cdots - 60\!\cdots\!84 \)
T^7 + 536*T^6 - 1471949*T^5 - 836103196*T^4 + 445968452201*T^3 + 299314175490794*T^2 + 26359228976309772*T - 604013534162646184
$17$
\( T^{7} + 3068 T^{6} + \cdots - 26\!\cdots\!68 \)
T^7 + 3068*T^6 - 615816*T^5 - 11993312544*T^4 - 17072446996720*T^3 - 10272914032250688*T^2 - 2774906167432439808*T - 266183254044928954368
$19$
\( T^{7} + 1900 T^{6} + \cdots + 15\!\cdots\!88 \)
T^7 + 1900*T^6 - 4963880*T^5 - 11648859168*T^4 + 1954540397200*T^3 + 16819528950484416*T^2 + 10650797547651984384*T + 1565115567488898674688
$23$
\( T^{7} + 3986 T^{6} + \cdots - 21\!\cdots\!12 \)
T^7 + 3986*T^6 - 20140823*T^5 - 97608475638*T^4 - 2054127175419*T^3 + 341328888548300294*T^2 + 187084045164026171356*T - 214762171413502925757112
$29$
\( T^{7} + 7436 T^{6} + \cdots + 54\!\cdots\!08 \)
T^7 + 7436*T^6 - 84122933*T^5 - 816010344448*T^4 + 418621014481441*T^3 + 21037490292638374878*T^2 + 63934938977599603116540*T + 54699942984882655505903208
$31$
\( T^{7} - 5776 T^{6} + \cdots - 15\!\cdots\!52 \)
T^7 - 5776*T^6 - 35979679*T^5 + 274998417656*T^4 - 271074417222383*T^3 - 1211964909507929836*T^2 + 2740780805588774729584*T - 1502215804475485096121152
$37$
\( (T + 1369)^{7} \)
(T + 1369)^7
$41$
\( T^{7} + 25089 T^{6} + \cdots + 31\!\cdots\!45 \)
T^7 + 25089*T^6 + 95419212*T^5 - 1949379900830*T^4 - 15993750446506246*T^3 + 948523390957630440*T^2 + 241272475194498130687149*T + 312210226493738812828107945
$43$
\( T^{7} + 22538 T^{6} + \cdots - 26\!\cdots\!96 \)
T^7 + 22538*T^6 - 170478128*T^5 - 5703685810368*T^4 - 15466480774562688*T^3 + 133562723428641203584*T^2 + 106108857912540832507328*T - 260944845448755662050580096
$47$
\( T^{7} + 60861 T^{6} + \cdots - 26\!\cdots\!76 \)
T^7 + 60861*T^6 + 1295715859*T^5 + 11614793308927*T^4 + 42133195067528452*T^3 + 22579165223963411196*T^2 - 177589230521749197449904*T - 262926608469323161086148176
$53$
\( T^{7} + 15681 T^{6} + \cdots - 80\!\cdots\!12 \)
T^7 + 15681*T^6 - 1490140093*T^5 - 26050748635109*T^4 + 435739691702682312*T^3 + 7344299595372380239608*T^2 - 8235523381559548048082064*T - 8030117343400339923091720912
$59$
\( T^{7} + 54536 T^{6} + \cdots - 27\!\cdots\!04 \)
T^7 + 54536*T^6 - 1100791964*T^5 - 64483212950624*T^4 + 463266567559853840*T^3 + 24014921193078744831488*T^2 - 73131985012602856071568384*T - 2780082546640830584816776085504
$61$
\( T^{7} - 48694 T^{6} + \cdots + 41\!\cdots\!40 \)
T^7 - 48694*T^6 - 1427161005*T^5 + 115088668854362*T^4 - 2218404766034679471*T^3 + 16523516855988580834872*T^2 - 45794842416053133036255168*T + 41631352867727358809031216640
$67$
\( T^{7} + 39724 T^{6} + \cdots + 34\!\cdots\!52 \)
T^7 + 39724*T^6 - 4347581347*T^5 - 72397538427348*T^4 + 6112012900785106185*T^3 - 46772483927949249767752*T^2 - 199812475455411345517503040*T + 349605990163496051243822346752
$71$
\( T^{7} + 92187 T^{6} + \cdots - 58\!\cdots\!00 \)
T^7 + 92187*T^6 + 49418223*T^5 - 168451784367227*T^4 - 3430283536039423576*T^3 + 5938521533880444173256*T^2 + 209843942047838016888394800*T - 58847104733223730903908030000
$73$
\( T^{7} - 73251 T^{6} + \cdots + 25\!\cdots\!45 \)
T^7 - 73251*T^6 - 4978957472*T^5 + 365392047807642*T^4 + 1421534929931989642*T^3 - 210213925071455610072524*T^2 - 251358156665223157623770503*T + 25832995218483806442061456213945
$79$
\( T^{7} - 78604 T^{6} + \cdots + 54\!\cdots\!80 \)
T^7 - 78604*T^6 - 13500723803*T^5 + 985757477459628*T^4 + 60809206756391750241*T^3 - 4037643189885491406829576*T^2 - 89929000318107835821959164912*T + 5490335179560759499350699927911680
$83$
\( T^{7} + 82223 T^{6} + \cdots - 19\!\cdots\!68 \)
T^7 + 82223*T^6 - 8095558673*T^5 - 548005120432767*T^4 + 18925764797002158832*T^3 + 822449722227511866702192*T^2 - 3486691218614363374455722880*T - 192318490806658225382304837312768
$89$
\( T^{7} - 181680 T^{6} + \cdots - 14\!\cdots\!48 \)
T^7 - 181680*T^6 - 2299926336*T^5 + 1709122863549312*T^4 - 52622231458286980096*T^3 - 2418317168038458712420352*T^2 + 127891214567817805491890049024*T - 1465482555787921770992390610485248
$97$
\( T^{7} - 39092 T^{6} + \cdots - 49\!\cdots\!20 \)
T^7 - 39092*T^6 - 42242406296*T^5 + 182311390910880*T^4 + 531505121174469950352*T^3 + 10906980063231124614801344*T^2 - 1639641471494463103602145345792*T - 49379134041282449589243719103339520
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