Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [576,3,Mod(65,576)] 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("576.65"); S:= CuspForms(chi, 3); 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, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-3,0,-6,0,2,0,-9] 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: \(15.6948632272\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 576.257
Dual form 576.3.q.a.65.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 2.59808i) q^{3} +(-3.00000 - 1.73205i) q^{5} +(1.00000 + 1.73205i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-1.50000 + 0.866025i) q^{11} +(-2.00000 + 3.46410i) q^{13} +10.3923i q^{15} +15.5885i q^{17} +11.0000 q^{19} +(3.00000 - 5.19615i) q^{21} +(24.0000 + 13.8564i) q^{23} +(-6.50000 - 11.2583i) q^{25} +27.0000 q^{27} +(-39.0000 + 22.5167i) q^{29} +(16.0000 - 27.7128i) q^{31} +(4.50000 + 2.59808i) q^{33} -6.92820i q^{35} +34.0000 q^{37} +12.0000 q^{39} +(-10.5000 - 6.06218i) q^{41} +(30.5000 + 52.8275i) q^{43} +(27.0000 - 15.5885i) q^{45} +(42.0000 - 24.2487i) q^{47} +(22.5000 - 38.9711i) q^{49} +(40.5000 - 23.3827i) q^{51} +6.00000 q^{55} +(-16.5000 - 28.5788i) q^{57} +(43.5000 + 25.1147i) q^{59} +(28.0000 + 48.4974i) q^{61} -18.0000 q^{63} +(12.0000 - 6.92820i) q^{65} +(15.5000 - 26.8468i) q^{67} -83.1384i q^{69} +31.1769i q^{71} +65.0000 q^{73} +(-19.5000 + 33.7750i) q^{75} +(-3.00000 - 1.73205i) q^{77} +(19.0000 + 32.9090i) q^{79} +(-40.5000 - 70.1481i) q^{81} +(-42.0000 + 24.2487i) q^{83} +(27.0000 - 46.7654i) q^{85} +(117.000 + 67.5500i) q^{87} +124.708i q^{89} -8.00000 q^{91} -96.0000 q^{93} +(-33.0000 - 19.0526i) q^{95} +(57.5000 + 99.5929i) q^{97} -15.5885i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 6 q^{5} + 2 q^{7} - 9 q^{9} - 3 q^{11} - 4 q^{13} + 22 q^{19} + 6 q^{21} + 48 q^{23} - 13 q^{25} + 54 q^{27} - 78 q^{29} + 32 q^{31} + 9 q^{33} + 68 q^{37} + 24 q^{39} - 21 q^{41} + 61 q^{43}+ \cdots + 115 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) −1.50000 − 2.59808i −0.500000 − 0.866025i
\(4\) 0 0
\(5\) −3.00000 − 1.73205i −0.600000 − 0.346410i 0.169042 − 0.985609i \(-0.445933\pi\)
−0.769042 + 0.639199i \(0.779266\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 0.142857 + 0.247436i 0.928571 − 0.371154i \(-0.121038\pi\)
−0.785714 + 0.618590i \(0.787704\pi\)
\(8\) 0 0
\(9\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) −1.50000 + 0.866025i −0.136364 + 0.0787296i −0.566630 − 0.823972i \(-0.691753\pi\)
0.430266 + 0.902702i \(0.358420\pi\)
\(12\) 0 0
\(13\) −2.00000 + 3.46410i −0.153846 + 0.266469i −0.932638 − 0.360813i \(-0.882499\pi\)
0.778792 + 0.627282i \(0.215833\pi\)
\(14\) 0 0
\(15\) 10.3923i 0.692820i
\(16\) 0 0
\(17\) 15.5885i 0.916968i 0.888703 + 0.458484i \(0.151607\pi\)
−0.888703 + 0.458484i \(0.848393\pi\)
\(18\) 0 0
\(19\) 11.0000 0.578947 0.289474 − 0.957186i \(-0.406520\pi\)
0.289474 + 0.957186i \(0.406520\pi\)
\(20\) 0 0
\(21\) 3.00000 − 5.19615i 0.142857 − 0.247436i
\(22\) 0 0
\(23\) 24.0000 + 13.8564i 1.04348 + 0.602452i 0.920817 − 0.389996i \(-0.127524\pi\)
0.122662 + 0.992449i \(0.460857\pi\)
\(24\) 0 0
\(25\) −6.50000 − 11.2583i −0.260000 − 0.450333i
\(26\) 0 0
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) −39.0000 + 22.5167i −1.34483 + 0.776437i −0.987511 − 0.157547i \(-0.949641\pi\)
−0.357316 + 0.933984i \(0.616308\pi\)
\(30\) 0 0
\(31\) 16.0000 − 27.7128i 0.516129 − 0.893962i −0.483696 − 0.875236i \(-0.660706\pi\)
0.999825 − 0.0187254i \(-0.00596084\pi\)
\(32\) 0 0
\(33\) 4.50000 + 2.59808i 0.136364 + 0.0787296i
\(34\) 0 0
\(35\) − 6.92820i − 0.197949i
\(36\) 0 0
\(37\) 34.0000 0.918919 0.459459 − 0.888199i \(-0.348043\pi\)
0.459459 + 0.888199i \(0.348043\pi\)
\(38\) 0 0
\(39\) 12.0000 0.307692
\(40\) 0 0
\(41\) −10.5000 − 6.06218i −0.256098 − 0.147858i 0.366456 − 0.930436i \(-0.380571\pi\)
−0.622553 + 0.782578i \(0.713905\pi\)
\(42\) 0 0
\(43\) 30.5000 + 52.8275i 0.709302 + 1.22855i 0.965116 + 0.261822i \(0.0843232\pi\)
−0.255814 + 0.966726i \(0.582343\pi\)
\(44\) 0 0
\(45\) 27.0000 − 15.5885i 0.600000 − 0.346410i
\(46\) 0 0
\(47\) 42.0000 − 24.2487i 0.893617 − 0.515930i 0.0184931 − 0.999829i \(-0.494113\pi\)
0.875124 + 0.483899i \(0.160780\pi\)
\(48\) 0 0
\(49\) 22.5000 − 38.9711i 0.459184 − 0.795329i
\(50\) 0 0
\(51\) 40.5000 − 23.3827i 0.794118 − 0.458484i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 6.00000 0.109091
\(56\) 0 0
\(57\) −16.5000 − 28.5788i −0.289474 − 0.501383i
\(58\) 0 0
\(59\) 43.5000 + 25.1147i 0.737288 + 0.425674i 0.821082 − 0.570810i \(-0.193371\pi\)
−0.0837943 + 0.996483i \(0.526704\pi\)
\(60\) 0 0
\(61\) 28.0000 + 48.4974i 0.459016 + 0.795040i 0.998909 − 0.0466940i \(-0.0148686\pi\)
−0.539893 + 0.841734i \(0.681535\pi\)
\(62\) 0 0
\(63\) −18.0000 −0.285714
\(64\) 0 0
\(65\) 12.0000 − 6.92820i 0.184615 − 0.106588i
\(66\) 0 0
\(67\) 15.5000 − 26.8468i 0.231343 − 0.400698i −0.726860 − 0.686785i \(-0.759021\pi\)
0.958204 + 0.286087i \(0.0923546\pi\)
\(68\) 0 0
\(69\) − 83.1384i − 1.20490i
\(70\) 0 0
\(71\) 31.1769i 0.439111i 0.975600 + 0.219556i \(0.0704608\pi\)
−0.975600 + 0.219556i \(0.929539\pi\)
\(72\) 0 0
\(73\) 65.0000 0.890411 0.445205 − 0.895428i \(-0.353131\pi\)
0.445205 + 0.895428i \(0.353131\pi\)
\(74\) 0 0
\(75\) −19.5000 + 33.7750i −0.260000 + 0.450333i
\(76\) 0 0
\(77\) −3.00000 − 1.73205i −0.0389610 − 0.0224942i
\(78\) 0 0
\(79\) 19.0000 + 32.9090i 0.240506 + 0.416569i 0.960859 − 0.277039i \(-0.0893532\pi\)
−0.720352 + 0.693608i \(0.756020\pi\)
\(80\) 0 0
\(81\) −40.5000 − 70.1481i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) −42.0000 + 24.2487i −0.506024 + 0.292153i −0.731198 − 0.682165i \(-0.761038\pi\)
0.225174 + 0.974319i \(0.427705\pi\)
\(84\) 0 0
\(85\) 27.0000 − 46.7654i 0.317647 − 0.550181i
\(86\) 0 0
\(87\) 117.000 + 67.5500i 1.34483 + 0.776437i
\(88\) 0 0
\(89\) 124.708i 1.40121i 0.713549 + 0.700605i \(0.247086\pi\)
−0.713549 + 0.700605i \(0.752914\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.0879121
\(92\) 0 0
\(93\) −96.0000 −1.03226
\(94\) 0 0
\(95\) −33.0000 − 19.0526i −0.347368 − 0.200553i
\(96\) 0 0
\(97\) 57.5000 + 99.5929i 0.592784 + 1.02673i 0.993856 + 0.110685i \(0.0353044\pi\)
−0.401072 + 0.916047i \(0.631362\pi\)
\(98\) 0 0
\(99\) − 15.5885i − 0.157459i
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 576.3.q.a.257.1 2
3.2 odd 2 1728.3.q.b.449.1 2
4.3 odd 2 576.3.q.b.257.1 2
8.3 odd 2 9.3.d.a.5.1 yes 2
8.5 even 2 144.3.q.a.113.1 2
9.2 odd 6 inner 576.3.q.a.65.1 2
9.7 even 3 1728.3.q.b.1601.1 2
12.11 even 2 1728.3.q.a.449.1 2
24.5 odd 2 432.3.q.a.17.1 2
24.11 even 2 27.3.d.a.17.1 2
36.7 odd 6 1728.3.q.a.1601.1 2
36.11 even 6 576.3.q.b.65.1 2
40.3 even 4 225.3.i.a.149.2 4
40.19 odd 2 225.3.j.a.176.1 2
40.27 even 4 225.3.i.a.149.1 4
56.3 even 6 441.3.j.b.275.1 2
56.11 odd 6 441.3.j.a.275.1 2
56.19 even 6 441.3.n.a.410.1 2
56.27 even 2 441.3.r.a.50.1 2
56.51 odd 6 441.3.n.b.410.1 2
72.5 odd 6 1296.3.e.a.161.2 2
72.11 even 6 9.3.d.a.2.1 ✓ 2
72.13 even 6 1296.3.e.a.161.1 2
72.29 odd 6 144.3.q.a.65.1 2
72.43 odd 6 27.3.d.a.8.1 2
72.59 even 6 81.3.b.a.80.2 2
72.61 even 6 432.3.q.a.305.1 2
72.67 odd 6 81.3.b.a.80.1 2
120.59 even 2 675.3.j.a.476.1 2
120.83 odd 4 675.3.i.a.449.1 4
120.107 odd 4 675.3.i.a.449.2 4
360.43 even 12 675.3.i.a.224.2 4
360.83 odd 12 225.3.i.a.74.1 4
360.187 even 12 675.3.i.a.224.1 4
360.227 odd 12 225.3.i.a.74.2 4
360.259 odd 6 675.3.j.a.251.1 2
360.299 even 6 225.3.j.a.101.1 2
504.11 even 6 441.3.n.b.128.1 2
504.83 odd 6 441.3.r.a.344.1 2
504.227 odd 6 441.3.n.a.128.1 2
504.299 odd 6 441.3.j.b.263.1 2
504.443 even 6 441.3.j.a.263.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.3.d.a.2.1 ✓ 2 72.11 even 6
9.3.d.a.5.1 yes 2 8.3 odd 2
27.3.d.a.8.1 2 72.43 odd 6
27.3.d.a.17.1 2 24.11 even 2
81.3.b.a.80.1 2 72.67 odd 6
81.3.b.a.80.2 2 72.59 even 6
144.3.q.a.65.1 2 72.29 odd 6
144.3.q.a.113.1 2 8.5 even 2
225.3.i.a.74.1 4 360.83 odd 12
225.3.i.a.74.2 4 360.227 odd 12
225.3.i.a.149.1 4 40.27 even 4
225.3.i.a.149.2 4 40.3 even 4
225.3.j.a.101.1 2 360.299 even 6
225.3.j.a.176.1 2 40.19 odd 2
432.3.q.a.17.1 2 24.5 odd 2
432.3.q.a.305.1 2 72.61 even 6
441.3.j.a.263.1 2 504.443 even 6
441.3.j.a.275.1 2 56.11 odd 6
441.3.j.b.263.1 2 504.299 odd 6
441.3.j.b.275.1 2 56.3 even 6
441.3.n.a.128.1 2 504.227 odd 6
441.3.n.a.410.1 2 56.19 even 6
441.3.n.b.128.1 2 504.11 even 6
441.3.n.b.410.1 2 56.51 odd 6
441.3.r.a.50.1 2 56.27 even 2
441.3.r.a.344.1 2 504.83 odd 6
576.3.q.a.65.1 2 9.2 odd 6 inner
576.3.q.a.257.1 2 1.1 even 1 trivial
576.3.q.b.65.1 2 36.11 even 6
576.3.q.b.257.1 2 4.3 odd 2
675.3.i.a.224.1 4 360.187 even 12
675.3.i.a.224.2 4 360.43 even 12
675.3.i.a.449.1 4 120.83 odd 4
675.3.i.a.449.2 4 120.107 odd 4
675.3.j.a.251.1 2 360.259 odd 6
675.3.j.a.476.1 2 120.59 even 2
1296.3.e.a.161.1 2 72.13 even 6
1296.3.e.a.161.2 2 72.5 odd 6
1728.3.q.a.449.1 2 12.11 even 2
1728.3.q.a.1601.1 2 36.7 odd 6
1728.3.q.b.449.1 2 3.2 odd 2
1728.3.q.b.1601.1 2 9.7 even 3