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Properties

Label 576.2.r.f.97.6
Level $576$
Weight $2$
Character 576.97
Analytic conductor $4.599$
Analytic rank $0$
Dimension $12$
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] = [576,2,Mod(97,576)] 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("576.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,18,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 97.6
Root \(2.17840 + 0.583700i\) of defining polynomial
Character \(\chi\) \(=\) 576.97
Dual form 576.2.r.f.481.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.59470 + 0.675970i) q^{3} +(1.50000 - 0.866025i) q^{5} +(1.80664 - 3.12920i) q^{7} +(2.08613 + 2.15594i) q^{9} +(0.635828 + 0.367095i) q^{11} +(-0.527909 + 0.304788i) q^{13} +(2.97746 - 0.367095i) q^{15} -5.52420 q^{17} -2.00000i q^{19} +(4.99629 - 3.76889i) q^{21} +(-2.36788 - 4.10129i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(1.86940 + 4.84823i) q^{27} +(6.78630 + 3.91807i) q^{29} +(-4.70951 - 8.15710i) q^{31} +(0.765809 + 1.01521i) q^{33} -6.25839i q^{35} +2.34163i q^{37} +(-1.04788 + 0.129195i) q^{39} +(4.26210 + 7.38217i) q^{41} +(8.88403 + 5.12920i) q^{43} +(4.99629 + 1.42726i) q^{45} +(-5.88032 + 10.1850i) q^{47} +(-3.02791 - 5.24449i) q^{49} +(-8.80944 - 3.73419i) q^{51} +13.0323i q^{53} +1.27166 q^{55} +(1.35194 - 3.18940i) q^{57} +(1.04788 - 0.604996i) q^{59} +(-9.78630 - 5.65012i) q^{61} +(10.5152 - 2.63290i) q^{63} +(-0.527909 + 0.914365i) q^{65} +(5.46826 - 3.15710i) q^{67} +(-1.00371 - 8.14093i) q^{69} -2.63999 q^{71} -2.05582 q^{73} +(-2.76551 + 2.08613i) q^{75} +(2.29743 - 1.32642i) q^{77} +(1.24540 - 2.15710i) q^{79} +(-0.296122 + 8.99513i) q^{81} +(-10.6161 - 6.12920i) q^{83} +(-8.28630 + 4.78410i) q^{85} +(8.17361 + 10.8355i) q^{87} -1.94418 q^{89} +2.20257i q^{91} +(-1.99629 - 16.1916i) q^{93} +(-1.73205 - 3.00000i) q^{95} +(7.78630 - 13.4863i) q^{97} +(0.534986 + 2.13661i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{5} - 4 q^{9} + 6 q^{13} + 6 q^{21} - 12 q^{25} - 18 q^{29} + 30 q^{33} + 18 q^{41} + 6 q^{45} - 24 q^{49} + 8 q^{57} - 18 q^{61} + 6 q^{65} - 66 q^{69} + 90 q^{77} - 20 q^{81} - 48 q^{89} + 30 q^{93}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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
\(3\) 1.59470 + 0.675970i 0.920700 + 0.390271i
\(4\) 0 0
\(5\) 1.50000 − 0.866025i 0.670820 − 0.387298i −0.125567 − 0.992085i \(-0.540075\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) 1.80664 − 3.12920i 0.682846 − 1.18272i −0.291262 − 0.956643i \(-0.594075\pi\)
0.974108 − 0.226081i \(-0.0725915\pi\)
\(8\) 0 0
\(9\) 2.08613 + 2.15594i 0.695377 + 0.718645i
\(10\) 0 0
\(11\) 0.635828 + 0.367095i 0.191709 + 0.110683i 0.592783 − 0.805363i \(-0.298029\pi\)
−0.401073 + 0.916046i \(0.631363\pi\)
\(12\) 0 0
\(13\) −0.527909 + 0.304788i −0.146416 + 0.0845331i −0.571418 − 0.820659i \(-0.693607\pi\)
0.425003 + 0.905192i \(0.360273\pi\)
\(14\) 0 0
\(15\) 2.97746 − 0.367095i 0.768776 − 0.0947837i
\(16\) 0 0
\(17\) −5.52420 −1.33982 −0.669908 − 0.742444i \(-0.733666\pi\)
−0.669908 + 0.742444i \(0.733666\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 4.99629 − 3.76889i 1.09028 − 0.822439i
\(22\) 0 0
\(23\) −2.36788 − 4.10129i −0.493737 − 0.855177i 0.506237 − 0.862394i \(-0.331036\pi\)
−0.999974 + 0.00721700i \(0.997703\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 1.86940 + 4.84823i 0.359767 + 0.933042i
\(28\) 0 0
\(29\) 6.78630 + 3.91807i 1.26018 + 0.727568i 0.973110 − 0.230341i \(-0.0739841\pi\)
0.287074 + 0.957908i \(0.407317\pi\)
\(30\) 0 0
\(31\) −4.70951 − 8.15710i −0.845852 − 1.46506i −0.884879 − 0.465821i \(-0.845759\pi\)
0.0390267 − 0.999238i \(-0.487574\pi\)
\(32\) 0 0
\(33\) 0.765809 + 1.01521i 0.133310 + 0.176725i
\(34\) 0 0
\(35\) − 6.25839i − 1.05786i
\(36\) 0 0
\(37\) 2.34163i 0.384961i 0.981301 + 0.192481i \(0.0616532\pi\)
−0.981301 + 0.192481i \(0.938347\pi\)
\(38\) 0 0
\(39\) −1.04788 + 0.129195i −0.167796 + 0.0206878i
\(40\) 0 0
\(41\) 4.26210 + 7.38217i 0.665628 + 1.15290i 0.979115 + 0.203309i \(0.0651696\pi\)
−0.313486 + 0.949593i \(0.601497\pi\)
\(42\) 0 0
\(43\) 8.88403 + 5.12920i 1.35480 + 0.782195i 0.988918 − 0.148466i \(-0.0474334\pi\)
0.365884 + 0.930661i \(0.380767\pi\)
\(44\) 0 0
\(45\) 4.99629 + 1.42726i 0.744803 + 0.212764i
\(46\) 0 0
\(47\) −5.88032 + 10.1850i −0.857733 + 1.48564i 0.0163535 + 0.999866i \(0.494794\pi\)
−0.874086 + 0.485771i \(0.838539\pi\)
\(48\) 0 0
\(49\) −3.02791 − 5.24449i −0.432558 − 0.749213i
\(50\) 0 0
\(51\) −8.80944 − 3.73419i −1.23357 − 0.522891i
\(52\) 0 0
\(53\) 13.0323i 1.79012i 0.445942 + 0.895062i \(0.352869\pi\)
−0.445942 + 0.895062i \(0.647131\pi\)
\(54\) 0 0
\(55\) 1.27166 0.171470
\(56\) 0 0
\(57\) 1.35194 − 3.18940i 0.179069 − 0.422446i
\(58\) 0 0
\(59\) 1.04788 − 0.604996i 0.136423 − 0.0787637i −0.430235 − 0.902717i \(-0.641569\pi\)
0.566658 + 0.823953i \(0.308236\pi\)
\(60\) 0 0
\(61\) −9.78630 − 5.65012i −1.25301 − 0.723424i −0.281302 − 0.959619i \(-0.590766\pi\)
−0.971706 + 0.236195i \(0.924099\pi\)
\(62\) 0 0
\(63\) 10.5152 − 2.63290i 1.32480 − 0.331715i
\(64\) 0 0
\(65\) −0.527909 + 0.914365i −0.0654790 + 0.113413i
\(66\) 0 0
\(67\) 5.46826 − 3.15710i 0.668055 − 0.385702i −0.127284 − 0.991866i \(-0.540626\pi\)
0.795339 + 0.606165i \(0.207293\pi\)
\(68\) 0 0
\(69\) −1.00371 − 8.14093i −0.120832 − 0.980053i
\(70\) 0 0
\(71\) −2.63999 −0.313309 −0.156655 − 0.987653i \(-0.550071\pi\)
−0.156655 + 0.987653i \(0.550071\pi\)
\(72\) 0 0
\(73\) −2.05582 −0.240615 −0.120308 − 0.992737i \(-0.538388\pi\)
−0.120308 + 0.992737i \(0.538388\pi\)
\(74\) 0 0
\(75\) −2.76551 + 2.08613i −0.319334 + 0.240886i
\(76\) 0 0
\(77\) 2.29743 − 1.32642i 0.261816 − 0.151160i
\(78\) 0 0
\(79\) 1.24540 − 2.15710i 0.140119 − 0.242693i −0.787422 − 0.616414i \(-0.788585\pi\)
0.927541 + 0.373721i \(0.121918\pi\)
\(80\) 0 0
\(81\) −0.296122 + 8.99513i −0.0329024 + 0.999459i
\(82\) 0 0
\(83\) −10.6161 − 6.12920i −1.16527 − 0.672767i −0.212706 − 0.977116i \(-0.568228\pi\)
−0.952560 + 0.304350i \(0.901561\pi\)
\(84\) 0 0
\(85\) −8.28630 + 4.78410i −0.898775 + 0.518908i
\(86\) 0 0
\(87\) 8.17361 + 10.8355i 0.876303 + 1.16169i
\(88\) 0 0
\(89\) −1.94418 −0.206083 −0.103041 − 0.994677i \(-0.532857\pi\)
−0.103041 + 0.994677i \(0.532857\pi\)
\(90\) 0 0
\(91\) 2.20257i 0.230892i
\(92\) 0 0
\(93\) −1.99629 − 16.1916i −0.207006 − 1.67899i
\(94\) 0 0
\(95\) −1.73205 − 3.00000i −0.177705 − 0.307794i
\(96\) 0 0
\(97\) 7.78630 − 13.4863i 0.790579 − 1.36932i −0.135030 − 0.990842i \(-0.543113\pi\)
0.925609 − 0.378481i \(-0.123554\pi\)
\(98\) 0 0
\(99\) 0.534986 + 2.13661i 0.0537681 + 0.214738i
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 576.2.r.f.97.6 yes 12
3.2 odd 2 1728.2.r.e.289.5 12
4.3 odd 2 inner 576.2.r.f.97.1 yes 12
8.3 odd 2 576.2.r.e.97.6 yes 12
8.5 even 2 576.2.r.e.97.1 ✓ 12
9.2 odd 6 5184.2.d.r.2593.2 12
9.4 even 3 576.2.r.e.481.1 yes 12
9.5 odd 6 1728.2.r.f.1441.5 12
9.7 even 3 5184.2.d.q.2593.8 12
12.11 even 2 1728.2.r.e.289.2 12
24.5 odd 2 1728.2.r.f.289.5 12
24.11 even 2 1728.2.r.f.289.2 12
36.7 odd 6 5184.2.d.q.2593.11 12
36.11 even 6 5184.2.d.r.2593.5 12
36.23 even 6 1728.2.r.f.1441.2 12
36.31 odd 6 576.2.r.e.481.6 yes 12
72.5 odd 6 1728.2.r.e.1441.5 12
72.11 even 6 5184.2.d.r.2593.11 12
72.13 even 6 inner 576.2.r.f.481.6 yes 12
72.29 odd 6 5184.2.d.r.2593.8 12
72.43 odd 6 5184.2.d.q.2593.5 12
72.59 even 6 1728.2.r.e.1441.2 12
72.61 even 6 5184.2.d.q.2593.2 12
72.67 odd 6 inner 576.2.r.f.481.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.e.97.1 ✓ 12 8.5 even 2
576.2.r.e.97.6 yes 12 8.3 odd 2
576.2.r.e.481.1 yes 12 9.4 even 3
576.2.r.e.481.6 yes 12 36.31 odd 6
576.2.r.f.97.1 yes 12 4.3 odd 2 inner
576.2.r.f.97.6 yes 12 1.1 even 1 trivial
576.2.r.f.481.1 yes 12 72.67 odd 6 inner
576.2.r.f.481.6 yes 12 72.13 even 6 inner
1728.2.r.e.289.2 12 12.11 even 2
1728.2.r.e.289.5 12 3.2 odd 2
1728.2.r.e.1441.2 12 72.59 even 6
1728.2.r.e.1441.5 12 72.5 odd 6
1728.2.r.f.289.2 12 24.11 even 2
1728.2.r.f.289.5 12 24.5 odd 2
1728.2.r.f.1441.2 12 36.23 even 6
1728.2.r.f.1441.5 12 9.5 odd 6
5184.2.d.q.2593.2 12 72.61 even 6
5184.2.d.q.2593.5 12 72.43 odd 6
5184.2.d.q.2593.8 12 9.7 even 3
5184.2.d.q.2593.11 12 36.7 odd 6
5184.2.d.r.2593.2 12 9.2 odd 6
5184.2.d.r.2593.5 12 36.11 even 6
5184.2.d.r.2593.8 12 72.29 odd 6
5184.2.d.r.2593.11 12 72.11 even 6