[N,k,chi] = [88,5,Mod(3,88)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(88, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 8]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("88.3");
S:= CuspForms(chi, 5);
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 \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).
\(n\)
\(23\)
\(45\)
\(57\)
\(\chi(n)\)
\(-1\)
\(-1\)
\(-1 + \beta_{2} - \beta_{4} + \beta_{6}\)
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
\( T_{3}^{8} - 28 T_{3}^{7} + 588 T_{3}^{6} - 5306 T_{3}^{5} + 19550 T_{3}^{4} + 264124 T_{3}^{3} + 2058753 T_{3}^{2} - 1484938 T_{3} + 66569281 \)
T3^8 - 28*T3^7 + 588*T3^6 - 5306*T3^5 + 19550*T3^4 + 264124*T3^3 + 2058753*T3^2 - 1484938*T3 + 66569281
acting on \(S_{5}^{\mathrm{new}}(88, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2} \)
(T^4 + 4*T^3 + 16*T^2 + 64*T + 256)^2
$3$
\( T^{8} - 28 T^{7} + 588 T^{6} + \cdots + 66569281 \)
T^8 - 28*T^7 + 588*T^6 - 5306*T^5 + 19550*T^4 + 264124*T^3 + 2058753*T^2 - 1484938*T + 66569281
$5$
\( T^{8} \)
T^8
$7$
\( T^{8} \)
T^8
$11$
\( T^{8} - 46 T^{7} + \cdots + 45\!\cdots\!61 \)
T^8 - 46*T^7 - 12525*T^6 + 1249636*T^5 + 125895269*T^4 + 18295920676*T^3 - 2684844984525*T^2 - 144367705329166*T + 45949729863572161
$13$
\( T^{8} \)
T^8
$17$
\( T^{8} - 1148 T^{7} + \cdots + 34\!\cdots\!01 \)
T^8 - 1148*T^7 + 988428*T^6 - 516771626*T^5 + 199757272350*T^4 - 52350428882276*T^3 + 8848621222808273*T^2 - 809365795979563578*T + 34133600380335033601
$19$
\( T^{8} - 1302 T^{7} + \cdots + 54\!\cdots\!21 \)
T^8 - 1302*T^7 + 1216673*T^6 - 647339644*T^5 + 292569341790*T^4 - 72694009165414*T^3 + 39540408991171788*T^2 - 748959052133606692*T + 5465839248256806721
$23$
\( T^{8} \)
T^8
$29$
\( T^{8} \)
T^8
$31$
\( T^{8} \)
T^8
$37$
\( T^{8} \)
T^8
$41$
\( T^{8} - 2492 T^{7} + \cdots + 41\!\cdots\!61 \)
T^8 - 2492*T^7 + 4657548*T^6 + 9866751286*T^5 + 26095194472350*T^4 - 29511371123449124*T^3 + 597116387407354936433*T^2 + 280203048887197653592998*T + 416149239170787553778655361
$43$
\( (T^{4} - 3502 T^{3} + \cdots - 794150195999)^{2} \)
(T^4 - 3502*T^3 - 4830001*T^2 + 16914663502*T - 794150195999)^2
$47$
\( T^{8} \)
T^8
$53$
\( T^{8} \)
T^8
$59$
\( T^{8} + 714 T^{7} + \cdots + 53\!\cdots\!41 \)
T^8 + 714*T^7 + 60756737*T^6 + 28852800452*T^5 + 1471736255025630*T^4 + 523368638682781178*T^3 + 290320179589833975372*T^2 - 21044015508385405538694244*T + 533956859147296008810946050241
$61$
\( T^{8} \)
T^8
$67$
\( (T^{4} - 5134 T^{3} + \cdots + 69368428937761)^{2} \)
(T^4 - 5134*T^3 - 74397649*T^2 + 381957529966*T + 69368428937761)^2
$71$
\( T^{8} \)
T^8
$73$
\( T^{8} + 19012 T^{7} + \cdots + 40\!\cdots\!41 \)
T^8 + 19012*T^7 + 271092108*T^6 + 2086233710134*T^5 + 10400200849077150*T^4 + 21241554796325820124*T^3 + 36591517016764810608113*T^2 - 233068772803174068118925658*T + 400611649390955802320459495041
$79$
\( T^{8} \)
T^8
$83$
\( T^{8} - 33558 T^{7} + \cdots + 76\!\cdots\!81 \)
T^8 - 33558*T^7 + 612671393*T^6 - 6708353889916*T^5 + 52214413114416990*T^4 - 220918552606771523686*T^3 + 880173013233282308588748*T^2 - 3077573523060794567424808228*T + 7691551236616323656199295337281
$89$
\( (T^{4} + 5474 T^{3} + \cdots + 11\!\cdots\!01)^{2} \)
(T^4 + 5474*T^3 - 283746529*T^2 - 1553228499746*T + 11180571220900801)^2
$97$
\( T^{8} + 29946 T^{7} + \cdots + 25\!\cdots\!21 \)
T^8 + 29946*T^7 + 741567377*T^6 + 9831602543588*T^5 + 122479767155404830*T^4 + 832341751904766514922*T^3 + 12721382419731042443489292*T^2 - 29323945776310231388792143876*T + 25099407894058170248701653342721
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