Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,8,Mod(109,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("324.109"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,508] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(101.212748257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 109.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 324.109
Dual form 324.8.e.c.217.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(254.000 + 439.941i) q^{7} +(7307.00 - 12656.1i) q^{13} -57448.0 q^{19} +(39062.5 + 67658.2i) q^{25} +(-89458.0 + 154946. i) q^{31} +279710. q^{37} +(-517612. - 896530. i) q^{43} +(282740. - 489719. i) q^{49} +(1.76777e6 + 3.06187e6i) q^{61} +(192536. - 333482. i) q^{67} -6.27481e6 q^{73} +(-4.38152e6 - 7.58902e6i) q^{79} +7.42391e6 q^{91} +(-6.12260e6 - 1.06047e7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 508 q^{7} + 14614 q^{13} - 114896 q^{19} + 78125 q^{25} - 178916 q^{31} + 559420 q^{37} - 1035224 q^{43} + 565479 q^{49} + 3535546 q^{61} + 385072 q^{67} - 12549620 q^{73} - 8763044 q^{79} + 14847824 q^{91}+ \cdots - 12245198 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 254.000 + 439.941i 0.279892 + 0.484787i 0.971358 − 0.237622i \(-0.0763680\pi\)
−0.691466 + 0.722409i \(0.743035\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 7307.00 − 12656.1i 0.922438 − 1.59771i 0.126808 − 0.991927i \(-0.459527\pi\)
0.795630 − 0.605783i \(-0.207140\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −57448.0 −1.92149 −0.960743 − 0.277439i \(-0.910514\pi\)
−0.960743 + 0.277439i \(0.910514\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 39062.5 + 67658.2i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) −89458.0 + 154946.i −0.539328 + 0.934144i 0.459612 + 0.888120i \(0.347988\pi\)
−0.998940 + 0.0460243i \(0.985345\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 279710. 0.907825 0.453912 − 0.891046i \(-0.350028\pi\)
0.453912 + 0.891046i \(0.350028\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) −517612. − 896530.i −0.992807 − 1.71959i −0.600090 − 0.799932i \(-0.704869\pi\)
−0.392716 − 0.919660i \(-0.628465\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 282740. − 489719.i 0.343321 − 0.594649i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 1.76777e6 + 3.06187e6i 0.997177 + 1.72716i 0.563620 + 0.826034i \(0.309408\pi\)
0.433556 + 0.901126i \(0.357258\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 192536. − 333482.i 0.0782078 − 0.135460i −0.824269 − 0.566198i \(-0.808414\pi\)
0.902477 + 0.430739i \(0.141747\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.27481e6 −1.88786 −0.943932 − 0.330141i \(-0.892904\pi\)
−0.943932 + 0.330141i \(0.892904\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.38152e6 − 7.58902e6i −0.999839 − 1.73177i −0.515448 − 0.856921i \(-0.672374\pi\)
−0.484392 − 0.874851i \(-0.660959\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 7.42391e6 1.03273
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.12260e6 − 1.06047e7i −0.681137 − 1.17976i −0.974634 − 0.223805i \(-0.928152\pi\)
0.293497 − 0.955960i \(-0.405181\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 324.8.e.c.109.1 2
3.2 odd 2 CM 324.8.e.c.109.1 2
9.2 odd 6 inner 324.8.e.c.217.1 2
9.4 even 3 36.8.a.b.1.1 ✓ 1
9.5 odd 6 36.8.a.b.1.1 ✓ 1
9.7 even 3 inner 324.8.e.c.217.1 2
36.23 even 6 144.8.a.e.1.1 1
36.31 odd 6 144.8.a.e.1.1 1
72.5 odd 6 576.8.a.q.1.1 1
72.13 even 6 576.8.a.q.1.1 1
72.59 even 6 576.8.a.r.1.1 1
72.67 odd 6 576.8.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.8.a.b.1.1 ✓ 1 9.4 even 3
36.8.a.b.1.1 ✓ 1 9.5 odd 6
144.8.a.e.1.1 1 36.23 even 6
144.8.a.e.1.1 1 36.31 odd 6
324.8.e.c.109.1 2 1.1 even 1 trivial
324.8.e.c.109.1 2 3.2 odd 2 CM
324.8.e.c.217.1 2 9.2 odd 6 inner
324.8.e.c.217.1 2 9.7 even 3 inner
576.8.a.q.1.1 1 72.5 odd 6
576.8.a.q.1.1 1 72.13 even 6
576.8.a.r.1.1 1 72.59 even 6
576.8.a.r.1.1 1 72.67 odd 6