Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2592,2,Mod(863,2592)] 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("2592.863"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 863.2
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2592.863
Dual form 2592.2.s.d.1727.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.22474 - 0.707107i) q^{5} +(3.46410 - 2.00000i) q^{7} +(2.82843 + 4.89898i) q^{11} +(-2.00000 + 3.46410i) q^{13} +4.24264i q^{17} +(-2.82843 + 4.89898i) q^{23} +(-1.50000 - 2.59808i) q^{25} +(-1.22474 + 0.707107i) q^{29} +(-3.46410 - 2.00000i) q^{31} -5.65685 q^{35} -6.00000 q^{37} +(8.57321 + 4.94975i) q^{41} +(-6.92820 + 4.00000i) q^{43} +(-2.82843 - 4.89898i) q^{47} +(4.50000 - 7.79423i) q^{49} +4.24264i q^{53} -8.00000i q^{55} +(-5.65685 + 9.79796i) q^{59} +(1.00000 + 1.73205i) q^{61} +(4.89898 - 2.82843i) q^{65} +(6.92820 + 4.00000i) q^{67} -5.65685 q^{71} +(19.5959 + 11.3137i) q^{77} +(-3.46410 + 2.00000i) q^{79} +(-2.82843 - 4.89898i) q^{83} +(3.00000 - 5.19615i) q^{85} -4.24264i q^{89} +16.0000i q^{91} +(4.00000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{13} - 12 q^{25} - 48 q^{37} + 36 q^{49} + 8 q^{61} + 24 q^{85} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(5\) −1.22474 − 0.707107i −0.547723 − 0.316228i 0.200480 − 0.979698i \(-0.435750\pi\)
−0.748203 + 0.663470i \(0.769083\pi\)
\(6\) 0 0
\(7\) 3.46410 − 2.00000i 1.30931 − 0.755929i 0.327327 − 0.944911i \(-0.393852\pi\)
0.981981 + 0.188982i \(0.0605189\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843 + 4.89898i 0.852803 + 1.47710i 0.878668 + 0.477432i \(0.158432\pi\)
−0.0258656 + 0.999665i \(0.508234\pi\)
\(12\) 0 0
\(13\) −2.00000 + 3.46410i −0.554700 + 0.960769i 0.443227 + 0.896410i \(0.353834\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.24264i 1.02899i 0.857493 + 0.514496i \(0.172021\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 + 4.89898i −0.589768 + 1.02151i 0.404495 + 0.914540i \(0.367447\pi\)
−0.994263 + 0.106967i \(0.965886\pi\)
\(24\) 0 0
\(25\) −1.50000 − 2.59808i −0.300000 − 0.519615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.22474 + 0.707107i −0.227429 + 0.131306i −0.609386 − 0.792874i \(-0.708584\pi\)
0.381956 + 0.924180i \(0.375251\pi\)
\(30\) 0 0
\(31\) −3.46410 − 2.00000i −0.622171 − 0.359211i 0.155543 − 0.987829i \(-0.450287\pi\)
−0.777714 + 0.628619i \(0.783621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.65685 −0.956183
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.57321 + 4.94975i 1.33891 + 0.773021i 0.986646 − 0.162880i \(-0.0520782\pi\)
0.352265 + 0.935900i \(0.385412\pi\)
\(42\) 0 0
\(43\) −6.92820 + 4.00000i −1.05654 + 0.609994i −0.924473 − 0.381246i \(-0.875495\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82843 − 4.89898i −0.412568 − 0.714590i 0.582601 − 0.812758i \(-0.302035\pi\)
−0.995170 + 0.0981685i \(0.968702\pi\)
\(48\) 0 0
\(49\) 4.50000 − 7.79423i 0.642857 − 1.11346i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264i 0.582772i 0.956606 + 0.291386i \(0.0941163\pi\)
−0.956606 + 0.291386i \(0.905884\pi\)
\(54\) 0 0
\(55\) − 8.00000i − 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.65685 + 9.79796i −0.736460 + 1.27559i 0.217620 + 0.976034i \(0.430171\pi\)
−0.954080 + 0.299552i \(0.903163\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 − 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.89898 − 2.82843i 0.607644 − 0.350823i
\(66\) 0 0
\(67\) 6.92820 + 4.00000i 0.846415 + 0.488678i 0.859440 − 0.511237i \(-0.170813\pi\)
−0.0130248 + 0.999915i \(0.504146\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 − 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 19.5959 + 11.3137i 2.23316 + 1.28932i
\(78\) 0 0
\(79\) −3.46410 + 2.00000i −0.389742 + 0.225018i −0.682048 − 0.731307i \(-0.738911\pi\)
0.292306 + 0.956325i \(0.405577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.82843 − 4.89898i −0.310460 − 0.537733i 0.668002 − 0.744160i \(-0.267150\pi\)
−0.978462 + 0.206427i \(0.933816\pi\)
\(84\) 0 0
\(85\) 3.00000 − 5.19615i 0.325396 − 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.24264i − 0.449719i −0.974391 − 0.224860i \(-0.927808\pi\)
0.974391 − 0.224860i \(-0.0721923\pi\)
\(90\) 0 0
\(91\) 16.0000i 1.67726i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.00000 + 6.92820i 0.406138 + 0.703452i 0.994453 − 0.105180i \(-0.0335417\pi\)
−0.588315 + 0.808632i \(0.700208\pi\)
\(98\) 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 2592.2.s.d.863.2 8
3.2 odd 2 inner 2592.2.s.d.863.4 8
4.3 odd 2 inner 2592.2.s.d.863.1 8
9.2 odd 6 inner 2592.2.s.d.1727.1 8
9.4 even 3 288.2.c.a.287.4 yes 4
9.5 odd 6 288.2.c.a.287.2 yes 4
9.7 even 3 inner 2592.2.s.d.1727.3 8
12.11 even 2 inner 2592.2.s.d.863.3 8
36.7 odd 6 inner 2592.2.s.d.1727.4 8
36.11 even 6 inner 2592.2.s.d.1727.2 8
36.23 even 6 288.2.c.a.287.1 ✓ 4
36.31 odd 6 288.2.c.a.287.3 yes 4
45.4 even 6 7200.2.h.d.1151.1 4
45.13 odd 12 7200.2.o.n.7199.2 4
45.14 odd 6 7200.2.h.d.1151.2 4
45.22 odd 12 7200.2.o.a.7199.1 4
45.23 even 12 7200.2.o.n.7199.4 4
45.32 even 12 7200.2.o.a.7199.3 4
72.5 odd 6 576.2.c.c.575.4 4
72.13 even 6 576.2.c.c.575.2 4
72.59 even 6 576.2.c.c.575.3 4
72.67 odd 6 576.2.c.c.575.1 4
144.5 odd 12 2304.2.f.e.1151.3 4
144.13 even 12 2304.2.f.c.1151.3 4
144.59 even 12 2304.2.f.c.1151.4 4
144.67 odd 12 2304.2.f.e.1151.4 4
144.77 odd 12 2304.2.f.c.1151.1 4
144.85 even 12 2304.2.f.e.1151.1 4
144.131 even 12 2304.2.f.e.1151.2 4
144.139 odd 12 2304.2.f.c.1151.2 4
180.23 odd 12 7200.2.o.a.7199.2 4
180.59 even 6 7200.2.h.d.1151.3 4
180.67 even 12 7200.2.o.n.7199.3 4
180.103 even 12 7200.2.o.a.7199.4 4
180.139 odd 6 7200.2.h.d.1151.4 4
180.167 odd 12 7200.2.o.n.7199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.c.a.287.1 ✓ 4 36.23 even 6
288.2.c.a.287.2 yes 4 9.5 odd 6
288.2.c.a.287.3 yes 4 36.31 odd 6
288.2.c.a.287.4 yes 4 9.4 even 3
576.2.c.c.575.1 4 72.67 odd 6
576.2.c.c.575.2 4 72.13 even 6
576.2.c.c.575.3 4 72.59 even 6
576.2.c.c.575.4 4 72.5 odd 6
2304.2.f.c.1151.1 4 144.77 odd 12
2304.2.f.c.1151.2 4 144.139 odd 12
2304.2.f.c.1151.3 4 144.13 even 12
2304.2.f.c.1151.4 4 144.59 even 12
2304.2.f.e.1151.1 4 144.85 even 12
2304.2.f.e.1151.2 4 144.131 even 12
2304.2.f.e.1151.3 4 144.5 odd 12
2304.2.f.e.1151.4 4 144.67 odd 12
2592.2.s.d.863.1 8 4.3 odd 2 inner
2592.2.s.d.863.2 8 1.1 even 1 trivial
2592.2.s.d.863.3 8 12.11 even 2 inner
2592.2.s.d.863.4 8 3.2 odd 2 inner
2592.2.s.d.1727.1 8 9.2 odd 6 inner
2592.2.s.d.1727.2 8 36.11 even 6 inner
2592.2.s.d.1727.3 8 9.7 even 3 inner
2592.2.s.d.1727.4 8 36.7 odd 6 inner
7200.2.h.d.1151.1 4 45.4 even 6
7200.2.h.d.1151.2 4 45.14 odd 6
7200.2.h.d.1151.3 4 180.59 even 6
7200.2.h.d.1151.4 4 180.139 odd 6
7200.2.o.a.7199.1 4 45.22 odd 12
7200.2.o.a.7199.2 4 180.23 odd 12
7200.2.o.a.7199.3 4 45.32 even 12
7200.2.o.a.7199.4 4 180.103 even 12
7200.2.o.n.7199.1 4 180.167 odd 12
7200.2.o.n.7199.2 4 45.13 odd 12
7200.2.o.n.7199.3 4 180.67 even 12
7200.2.o.n.7199.4 4 45.23 even 12