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Properties

Label 2592.2.s.g.863.1
Level $2592$
Weight $2$
Character 2592.863
Analytic conductor $20.697$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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 = [8,0,0,0,0,0,12,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,12] 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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 863.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2592.863
Dual form 2592.2.s.g.1727.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.34607 - 1.93185i) q^{5} +(3.23205 - 1.86603i) q^{7} +(0.517638 + 0.896575i) q^{11} +(-2.23205 + 3.86603i) q^{13} -1.79315i q^{17} -1.73205i q^{19} +(4.38134 - 7.58871i) q^{23} +(4.96410 + 8.59808i) q^{25} +(6.69213 - 3.86370i) q^{29} +(6.46410 + 3.73205i) q^{31} -14.4195 q^{35} +0.464102 q^{37} +(-6.69213 - 3.86370i) q^{41} +(-0.464102 + 0.267949i) q^{43} +(-2.31079 - 4.00240i) q^{47} +(3.46410 - 6.00000i) q^{49} -3.58630i q^{53} -4.00000i q^{55} +(-6.17449 + 10.6945i) q^{59} +(-5.69615 - 9.86603i) q^{61} +(14.9372 - 8.62398i) q^{65} +(-5.42820 - 3.13397i) q^{67} -11.3137 q^{71} -3.92820 q^{73} +(3.34607 + 1.93185i) q^{77} +(4.16025 - 2.40192i) q^{79} +(1.03528 + 1.79315i) q^{83} +(-3.46410 + 6.00000i) q^{85} -1.79315i q^{89} +16.6603i q^{91} +(-3.34607 + 5.79555i) q^{95} +(-3.50000 - 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{7} - 4 q^{13} + 12 q^{25} + 24 q^{31} - 24 q^{37} + 24 q^{43} - 4 q^{61} + 12 q^{67} + 24 q^{73} - 36 q^{79} - 28 q^{97}+O(q^{100}) \) 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
\(5\) −3.34607 − 1.93185i −1.49641 − 0.863950i −0.496414 − 0.868086i \(-0.665350\pi\)
−0.999991 + 0.00413535i \(0.998684\pi\)
\(6\) 0 0
\(7\) 3.23205 − 1.86603i 1.22160 − 0.705291i 0.256341 − 0.966586i \(-0.417483\pi\)
0.965259 + 0.261295i \(0.0841495\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.517638 + 0.896575i 0.156074 + 0.270328i 0.933449 − 0.358709i \(-0.116783\pi\)
−0.777376 + 0.629037i \(0.783450\pi\)
\(12\) 0 0
\(13\) −2.23205 + 3.86603i −0.619060 + 1.07224i 0.370598 + 0.928793i \(0.379153\pi\)
−0.989658 + 0.143449i \(0.954181\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.79315i − 0.434903i −0.976071 − 0.217451i \(-0.930226\pi\)
0.976071 − 0.217451i \(-0.0697744\pi\)
\(18\) 0 0
\(19\) − 1.73205i − 0.397360i −0.980064 − 0.198680i \(-0.936335\pi\)
0.980064 − 0.198680i \(-0.0636654\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.38134 − 7.58871i 0.913573 − 1.58235i 0.104595 − 0.994515i \(-0.466645\pi\)
0.808977 − 0.587840i \(-0.200021\pi\)
\(24\) 0 0
\(25\) 4.96410 + 8.59808i 0.992820 + 1.71962i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.69213 − 3.86370i 1.24270 − 0.717472i 0.273055 − 0.961998i \(-0.411966\pi\)
0.969643 + 0.244527i \(0.0786326\pi\)
\(30\) 0 0
\(31\) 6.46410 + 3.73205i 1.16099 + 0.670296i 0.951540 − 0.307524i \(-0.0995004\pi\)
0.209447 + 0.977820i \(0.432834\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −14.4195 −2.43735
\(36\) 0 0
\(37\) 0.464102 0.0762978 0.0381489 − 0.999272i \(-0.487854\pi\)
0.0381489 + 0.999272i \(0.487854\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.69213 − 3.86370i −1.04514 − 0.603409i −0.123852 − 0.992301i \(-0.539525\pi\)
−0.921283 + 0.388892i \(0.872858\pi\)
\(42\) 0 0
\(43\) −0.464102 + 0.267949i −0.0707748 + 0.0408619i −0.534970 − 0.844871i \(-0.679677\pi\)
0.464195 + 0.885733i \(0.346344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.31079 − 4.00240i −0.337063 − 0.583811i 0.646816 − 0.762646i \(-0.276100\pi\)
−0.983879 + 0.178836i \(0.942767\pi\)
\(48\) 0 0
\(49\) 3.46410 − 6.00000i 0.494872 − 0.857143i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.58630i − 0.492616i −0.969192 − 0.246308i \(-0.920782\pi\)
0.969192 − 0.246308i \(-0.0792175\pi\)
\(54\) 0 0
\(55\) − 4.00000i − 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.17449 + 10.6945i −0.803850 + 1.39231i 0.113214 + 0.993571i \(0.463885\pi\)
−0.917064 + 0.398739i \(0.869448\pi\)
\(60\) 0 0
\(61\) −5.69615 − 9.86603i −0.729318 − 1.26322i −0.957172 − 0.289520i \(-0.906504\pi\)
0.227854 − 0.973695i \(-0.426829\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.9372 − 8.62398i 1.85273 − 1.06967i
\(66\) 0 0
\(67\) −5.42820 − 3.13397i −0.663161 − 0.382876i 0.130320 − 0.991472i \(-0.458400\pi\)
−0.793480 + 0.608596i \(0.791733\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 − 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) −3.92820 −0.459761 −0.229881 − 0.973219i \(-0.573834\pi\)
−0.229881 + 0.973219i \(0.573834\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.34607 + 1.93185i 0.381320 + 0.220155i
\(78\) 0 0
\(79\) 4.16025 − 2.40192i 0.468065 − 0.270238i −0.247364 − 0.968923i \(-0.579564\pi\)
0.715429 + 0.698685i \(0.246231\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.03528 + 1.79315i 0.113636 + 0.196824i 0.917234 − 0.398349i \(-0.130417\pi\)
−0.803597 + 0.595173i \(0.797083\pi\)
\(84\) 0 0
\(85\) −3.46410 + 6.00000i −0.375735 + 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.79315i − 0.190074i −0.995474 − 0.0950368i \(-0.969703\pi\)
0.995474 − 0.0950368i \(-0.0302969\pi\)
\(90\) 0 0
\(91\) 16.6603i 1.74647i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.34607 + 5.79555i −0.343299 + 0.594611i
\(96\) 0 0
\(97\) −3.50000 − 6.06218i −0.355371 − 0.615521i 0.631810 − 0.775123i \(-0.282312\pi\)
−0.987181 + 0.159602i \(0.948979\pi\)
\(98\) 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.g.863.1 8
3.2 odd 2 inner 2592.2.s.g.863.4 8
4.3 odd 2 2592.2.s.c.863.1 8
9.2 odd 6 2592.2.s.c.1727.1 8
9.4 even 3 864.2.c.b.863.8 yes 8
9.5 odd 6 864.2.c.b.863.2 yes 8
9.7 even 3 2592.2.s.c.1727.4 8
12.11 even 2 2592.2.s.c.863.4 8
36.7 odd 6 inner 2592.2.s.g.1727.4 8
36.11 even 6 inner 2592.2.s.g.1727.1 8
36.23 even 6 864.2.c.b.863.1 ✓ 8
36.31 odd 6 864.2.c.b.863.7 yes 8
72.5 odd 6 1728.2.c.f.1727.8 8
72.13 even 6 1728.2.c.f.1727.2 8
72.59 even 6 1728.2.c.f.1727.7 8
72.67 odd 6 1728.2.c.f.1727.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.c.b.863.1 ✓ 8 36.23 even 6
864.2.c.b.863.2 yes 8 9.5 odd 6
864.2.c.b.863.7 yes 8 36.31 odd 6
864.2.c.b.863.8 yes 8 9.4 even 3
1728.2.c.f.1727.1 8 72.67 odd 6
1728.2.c.f.1727.2 8 72.13 even 6
1728.2.c.f.1727.7 8 72.59 even 6
1728.2.c.f.1727.8 8 72.5 odd 6
2592.2.s.c.863.1 8 4.3 odd 2
2592.2.s.c.863.4 8 12.11 even 2
2592.2.s.c.1727.1 8 9.2 odd 6
2592.2.s.c.1727.4 8 9.7 even 3
2592.2.s.g.863.1 8 1.1 even 1 trivial
2592.2.s.g.863.4 8 3.2 odd 2 inner
2592.2.s.g.1727.1 8 36.11 even 6 inner
2592.2.s.g.1727.4 8 36.7 odd 6 inner