[N,k,chi] = [105,4,Mod(16,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.16");
S:= CuspForms(chi, 4);
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 \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
\(n\)
\(22\)
\(31\)
\(71\)
\(\chi(n)\)
\(1\)
\(-1 + \beta_{4}\)
\(1\)
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
\( T_{2}^{10} - T_{2}^{9} + 31 T_{2}^{8} + 26 T_{2}^{7} + 738 T_{2}^{6} + 352 T_{2}^{5} + 5008 T_{2}^{4} + 5368 T_{2}^{3} + 26728 T_{2}^{2} + 13776 T_{2} + 7056 \)
T2^10 - T2^9 + 31*T2^8 + 26*T2^7 + 738*T2^6 + 352*T2^5 + 5008*T2^4 + 5368*T2^3 + 26728*T2^2 + 13776*T2 + 7056
acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} - T^{9} + 31 T^{8} + 26 T^{7} + \cdots + 7056 \)
T^10 - T^9 + 31*T^8 + 26*T^7 + 738*T^6 + 352*T^5 + 5008*T^4 + 5368*T^3 + 26728*T^2 + 13776*T + 7056
$3$
\( (T^{2} + 3 T + 9)^{5} \)
(T^2 + 3*T + 9)^5
$5$
\( (T^{2} - 5 T + 25)^{5} \)
(T^2 - 5*T + 25)^5
$7$
\( T^{10} - 56 T^{9} + \cdots + 4747561509943 \)
T^10 - 56*T^9 + 1761*T^8 - 46900*T^7 + 1033445*T^6 - 19485732*T^5 + 354471635*T^4 - 5517738100*T^3 + 71062701927*T^2 - 775112083256*T + 4747561509943
$11$
\( T^{10} - 33 T^{9} + \cdots + 1986036295824 \)
T^10 - 33*T^9 + 5985*T^8 - 102048*T^7 + 24418388*T^6 - 386384628*T^5 + 36524497140*T^4 + 528126494592*T^3 + 15040569179920*T^2 + 5499093388656*T + 1986036295824
$13$
\( (T^{5} + 23 T^{4} - 5870 T^{3} + \cdots - 25478281)^{2} \)
(T^5 + 23*T^4 - 5870*T^3 - 134666*T^2 + 4614873*T - 25478281)^2
$17$
\( T^{10} - 136 T^{9} + \cdots + 34\!\cdots\!00 \)
T^10 - 136*T^9 + 42192*T^8 - 3265696*T^7 + 867632660*T^6 - 58733646272*T^5 + 11195798876608*T^4 - 443293255910208*T^3 + 78587894771518864*T^2 - 2512440289366080000*T + 345968928057600000000
$19$
\( T^{10} - 39 T^{9} + \cdots + 22\!\cdots\!41 \)
T^10 - 39*T^9 + 24415*T^8 - 177938*T^7 + 469485405*T^6 - 7865529465*T^5 + 1957373559269*T^4 - 28216064109234*T^3 + 6507647996707239*T^2 - 113251636990501511*T + 2248247678314176841
$23$
\( T^{10} - 133 T^{9} + \cdots + 13\!\cdots\!00 \)
T^10 - 133*T^9 + 53649*T^8 - 8930056*T^7 + 2506253492*T^6 - 322949104388*T^5 + 36674046634324*T^4 - 1796800684433664*T^3 + 65138175671500624*T^2 - 1124927289361076880*T + 13945631776351203600
$29$
\( (T^{5} - 272 T^{4} + \cdots - 12806983008)^{2} \)
(T^5 - 272*T^4 - 32232*T^3 + 7948768*T^2 + 442539868*T - 12806983008)^2
$31$
\( T^{10} - 430 T^{9} + \cdots + 40\!\cdots\!64 \)
T^10 - 430*T^9 + 164862*T^8 - 16631636*T^7 + 2122801203*T^6 + 80012694626*T^5 + 15152305261486*T^4 + 89012168959896*T^3 + 8100260675968545*T^2 - 4058328655876698*T + 4081182415823420964
$37$
\( T^{10} + 3 T^{9} + \cdots + 16\!\cdots\!49 \)
T^10 + 3*T^9 + 55391*T^8 - 8735726*T^7 + 2879078833*T^6 - 251132774399*T^5 + 28102507313401*T^4 - 664854591272262*T^3 + 85471000055527311*T^2 - 2239673640559685253*T + 163357496064790953849
$41$
\( (T^{5} + 627 T^{4} + \cdots + 45007298820)^{2} \)
(T^5 + 627*T^4 - 46144*T^3 - 72632464*T^2 - 9167437684*T + 45007298820)^2
$43$
\( (T^{5} - 108 T^{4} + \cdots + 1217258521160)^{2} \)
(T^5 - 108*T^4 - 219178*T^3 - 4723972*T^2 + 11616976573*T + 1217258521160)^2
$47$
\( T^{10} - 553 T^{9} + \cdots + 15\!\cdots\!76 \)
T^10 - 553*T^9 + 362433*T^8 - 63008800*T^7 + 34488543800*T^6 - 8297878939184*T^5 + 1999408594849840*T^4 - 231043904540262528*T^3 + 21108532315580033920*T^2 - 656896785299909250816*T + 15944418822362471813376
$53$
\( T^{10} - 1135 T^{9} + \cdots + 10\!\cdots\!36 \)
T^10 - 1135*T^9 + 1088753*T^8 - 399432016*T^7 + 147382743040*T^6 - 5111172125120*T^5 + 8185807795063616*T^4 - 400503198869264384*T^3 + 177808360511736561664*T^2 + 9720251093168780537856*T + 1069662495502833221799936
$59$
\( T^{10} - 332 T^{9} + \cdots + 84\!\cdots\!36 \)
T^10 - 332*T^9 + 706424*T^8 + 39421152*T^7 + 294582123044*T^6 + 19848627769392*T^5 + 61320380672526368*T^4 + 17893448055423083776*T^3 + 6871379152165241193232*T^2 + 803071538578907948766144*T + 84842443551276906826772736
$61$
\( T^{10} - 584 T^{9} + \cdots + 29\!\cdots\!00 \)
T^10 - 584*T^9 + 450896*T^8 - 93117440*T^7 + 55278737600*T^6 - 7179571712000*T^5 + 5484852826240000*T^4 - 161599028160000000*T^3 + 141366041395776000000*T^2 - 4614691192934400000000*T + 2905710252687360000000000
$67$
\( T^{10} + 412 T^{9} + \cdots + 48\!\cdots\!00 \)
T^10 + 412*T^9 + 870530*T^8 - 148977392*T^7 + 377305734303*T^6 - 90460280046032*T^5 + 113840936931665258*T^4 - 40710452921667468180*T^3 + 18797046011427846818489*T^2 - 3127989960517862052618880*T + 481276769229452036751769600
$71$
\( (T^{5} + 142 T^{4} + \cdots - 27239555642904)^{2} \)
(T^5 + 142*T^4 - 769280*T^3 + 35683872*T^2 + 147258969164*T - 27239555642904)^2
$73$
\( T^{10} - 2074 T^{9} + \cdots + 38\!\cdots\!36 \)
T^10 - 2074*T^9 + 2951950*T^8 - 2443320492*T^7 + 1546069813175*T^6 - 608629944296906*T^5 + 198536825323918070*T^4 - 36200024742546821760*T^3 + 12242461297424388111193*T^2 - 1832571613572068445973702*T + 383304050465470768497925636
$79$
\( T^{10} + 28 T^{9} + \cdots + 35\!\cdots\!64 \)
T^10 + 28*T^9 + 751962*T^8 + 179251872*T^7 + 472267962083*T^6 + 88665849104232*T^5 + 80718492664113690*T^4 + 18675392318928123268*T^3 + 10865558123985378786049*T^2 + 1777592944752941736316680*T + 351566530691852983950796864
$83$
\( (T^{5} + 840 T^{4} + \cdots + 59475285699552)^{2} \)
(T^5 + 840*T^4 - 1464184*T^3 - 823734320*T^2 + 477301951676*T + 59475285699552)^2
$89$
\( T^{10} - 2978 T^{9} + \cdots + 17\!\cdots\!04 \)
T^10 - 2978*T^9 + 5716460*T^8 - 6655748880*T^7 + 5690822944292*T^6 - 3230615570349480*T^5 + 1342913936197150256*T^4 - 326531845329902438720*T^3 + 49765376272797332769424*T^2 + 2356846133373907978355808*T + 175306065139832199276515904
$97$
\( (T^{5} + 2168 T^{4} + \cdots + 701198534208512)^{2} \)
(T^5 + 2168*T^4 - 1646480*T^3 - 5467925696*T^2 - 1769182097088*T + 701198534208512)^2
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