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Properties

Label 864.2.i.c.289.1
Level $864$
Weight $2$
Character 864.289
Analytic conductor $6.899$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [864,2,Mod(289,864)] 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("864.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-2,0,-2] 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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 864.289
Dual form 864.2.i.c.577.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(-1.72474 + 2.98735i) q^{7} +(0.724745 - 1.25529i) q^{11} +(-2.94949 - 5.10867i) q^{13} -4.89898 q^{17} +4.00000 q^{19} +(-2.72474 - 4.71940i) q^{23} +(2.00000 - 3.46410i) q^{25} +(0.0505103 - 0.0874863i) q^{29} +(-1.27526 - 2.20881i) q^{31} +3.44949 q^{35} -0.898979 q^{37} +(-5.94949 - 10.3048i) q^{41} +(1.17423 - 2.03383i) q^{43} +(-3.17423 + 5.49794i) q^{47} +(-2.44949 - 4.24264i) q^{49} -8.89898 q^{53} -1.44949 q^{55} +(7.17423 + 12.4261i) q^{59} +(3.94949 - 6.84072i) q^{61} +(-2.94949 + 5.10867i) q^{65} +(-6.17423 - 10.6941i) q^{67} -7.79796 q^{71} -4.89898 q^{73} +(2.50000 + 4.33013i) q^{77} +(-6.72474 + 11.6476i) q^{79} +(-0.275255 + 0.476756i) q^{83} +(2.44949 + 4.24264i) q^{85} +12.8990 q^{89} +20.3485 q^{91} +(-2.00000 - 3.46410i) q^{95} +(1.94949 - 3.37662i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 2 q^{7} - 2 q^{11} - 2 q^{13} + 16 q^{19} - 6 q^{23} + 8 q^{25} + 10 q^{29} - 10 q^{31} + 4 q^{35} + 16 q^{37} - 14 q^{41} - 10 q^{43} + 2 q^{47} - 16 q^{53} + 4 q^{55} + 14 q^{59} + 6 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i 0.732294 − 0.680989i \(-0.238450\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) −1.72474 + 2.98735i −0.651892 + 1.12911i 0.330771 + 0.943711i \(0.392691\pi\)
−0.982663 + 0.185399i \(0.940642\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.724745 − 1.25529i 0.218519 − 0.378486i −0.735837 − 0.677159i \(-0.763211\pi\)
0.954355 + 0.298674i \(0.0965442\pi\)
\(12\) 0 0
\(13\) −2.94949 − 5.10867i −0.818041 − 1.41689i −0.907123 − 0.420865i \(-0.861727\pi\)
0.0890821 − 0.996024i \(-0.471607\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 − 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.72474 − 4.71940i −0.568149 − 0.984062i −0.996749 − 0.0805681i \(-0.974327\pi\)
0.428601 − 0.903494i \(-0.359007\pi\)
\(24\) 0 0
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.0505103 − 0.0874863i 0.00937952 − 0.0162458i −0.861298 − 0.508101i \(-0.830348\pi\)
0.870677 + 0.491855i \(0.163681\pi\)
\(30\) 0 0
\(31\) −1.27526 − 2.20881i −0.229043 − 0.396713i 0.728482 − 0.685065i \(-0.240226\pi\)
−0.957525 + 0.288352i \(0.906893\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.44949 0.583070
\(36\) 0 0
\(37\) −0.898979 −0.147791 −0.0738957 − 0.997266i \(-0.523543\pi\)
−0.0738957 + 0.997266i \(0.523543\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.94949 − 10.3048i −0.929154 − 1.60934i −0.784740 − 0.619825i \(-0.787204\pi\)
−0.144414 − 0.989517i \(-0.546130\pi\)
\(42\) 0 0
\(43\) 1.17423 − 2.03383i 0.179069 − 0.310157i −0.762493 − 0.646997i \(-0.776025\pi\)
0.941562 + 0.336840i \(0.109358\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.17423 + 5.49794i −0.463010 + 0.801956i −0.999109 − 0.0421984i \(-0.986564\pi\)
0.536100 + 0.844155i \(0.319897\pi\)
\(48\) 0 0
\(49\) −2.44949 − 4.24264i −0.349927 − 0.606092i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.89898 −1.22237 −0.611184 − 0.791488i \(-0.709307\pi\)
−0.611184 + 0.791488i \(0.709307\pi\)
\(54\) 0 0
\(55\) −1.44949 −0.195449
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.17423 + 12.4261i 0.934006 + 1.61775i 0.776397 + 0.630244i \(0.217045\pi\)
0.157609 + 0.987502i \(0.449622\pi\)
\(60\) 0 0
\(61\) 3.94949 − 6.84072i 0.505680 − 0.875864i −0.494298 − 0.869292i \(-0.664575\pi\)
0.999978 − 0.00657156i \(-0.00209181\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.94949 + 5.10867i −0.365839 + 0.633652i
\(66\) 0 0
\(67\) −6.17423 − 10.6941i −0.754303 − 1.30649i −0.945720 − 0.324982i \(-0.894642\pi\)
0.191417 − 0.981509i \(-0.438692\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.79796 −0.925447 −0.462724 − 0.886503i \(-0.653128\pi\)
−0.462724 + 0.886503i \(0.653128\pi\)
\(72\) 0 0
\(73\) −4.89898 −0.573382 −0.286691 − 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.50000 + 4.33013i 0.284901 + 0.493464i
\(78\) 0 0
\(79\) −6.72474 + 11.6476i −0.756593 + 1.31046i 0.187986 + 0.982172i \(0.439804\pi\)
−0.944579 + 0.328286i \(0.893529\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.275255 + 0.476756i −0.0302132 + 0.0523308i −0.880737 − 0.473606i \(-0.842952\pi\)
0.850523 + 0.525937i \(0.176285\pi\)
\(84\) 0 0
\(85\) 2.44949 + 4.24264i 0.265684 + 0.460179i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.8990 1.36729 0.683645 − 0.729815i \(-0.260394\pi\)
0.683645 + 0.729815i \(0.260394\pi\)
\(90\) 0 0
\(91\) 20.3485 2.13310
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 − 3.46410i −0.205196 − 0.355409i
\(96\) 0 0
\(97\) 1.94949 − 3.37662i 0.197941 − 0.342843i −0.749920 − 0.661529i \(-0.769908\pi\)
0.947861 + 0.318685i \(0.103241\pi\)
\(98\) 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 864.2.i.c.289.1 4
3.2 odd 2 288.2.i.e.97.2 yes 4
4.3 odd 2 864.2.i.e.289.2 4
8.3 odd 2 1728.2.i.m.1153.2 4
8.5 even 2 1728.2.i.k.1153.1 4
9.2 odd 6 2592.2.a.n.1.2 2
9.4 even 3 inner 864.2.i.c.577.1 4
9.5 odd 6 288.2.i.e.193.1 yes 4
9.7 even 3 2592.2.a.s.1.2 2
12.11 even 2 288.2.i.c.97.1 ✓ 4
24.5 odd 2 576.2.i.i.385.1 4
24.11 even 2 576.2.i.m.385.2 4
36.7 odd 6 2592.2.a.o.1.1 2
36.11 even 6 2592.2.a.j.1.1 2
36.23 even 6 288.2.i.c.193.2 yes 4
36.31 odd 6 864.2.i.e.577.2 4
72.5 odd 6 576.2.i.i.193.2 4
72.11 even 6 5184.2.a.bu.1.1 2
72.13 even 6 1728.2.i.k.577.1 4
72.29 odd 6 5184.2.a.by.1.2 2
72.43 odd 6 5184.2.a.bj.1.1 2
72.59 even 6 576.2.i.m.193.1 4
72.61 even 6 5184.2.a.bn.1.2 2
72.67 odd 6 1728.2.i.m.577.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.c.97.1 ✓ 4 12.11 even 2
288.2.i.c.193.2 yes 4 36.23 even 6
288.2.i.e.97.2 yes 4 3.2 odd 2
288.2.i.e.193.1 yes 4 9.5 odd 6
576.2.i.i.193.2 4 72.5 odd 6
576.2.i.i.385.1 4 24.5 odd 2
576.2.i.m.193.1 4 72.59 even 6
576.2.i.m.385.2 4 24.11 even 2
864.2.i.c.289.1 4 1.1 even 1 trivial
864.2.i.c.577.1 4 9.4 even 3 inner
864.2.i.e.289.2 4 4.3 odd 2
864.2.i.e.577.2 4 36.31 odd 6
1728.2.i.k.577.1 4 72.13 even 6
1728.2.i.k.1153.1 4 8.5 even 2
1728.2.i.m.577.2 4 72.67 odd 6
1728.2.i.m.1153.2 4 8.3 odd 2
2592.2.a.j.1.1 2 36.11 even 6
2592.2.a.n.1.2 2 9.2 odd 6
2592.2.a.o.1.1 2 36.7 odd 6
2592.2.a.s.1.2 2 9.7 even 3
5184.2.a.bj.1.1 2 72.43 odd 6
5184.2.a.bn.1.2 2 72.61 even 6
5184.2.a.bu.1.1 2 72.11 even 6
5184.2.a.by.1.2 2 72.29 odd 6