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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [576,2,Mod(289,576)] 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("576.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 576.289
Dual form 576.2.d.b.289.3

$q$-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)
 
\(f(q)\) \(=\) \(q-3.46410i q^{5} -3.46410 q^{7} -6.00000 q^{17} +4.00000i q^{19} -6.92820 q^{23} -7.00000 q^{25} -3.46410i q^{29} +3.46410 q^{31} +12.0000i q^{35} -6.92820i q^{37} +6.00000 q^{41} -4.00000i q^{43} -6.92820 q^{47} +5.00000 q^{49} +3.46410i q^{53} -12.0000i q^{59} -6.92820i q^{61} -4.00000i q^{67} +6.92820 q^{71} +2.00000 q^{73} -10.3923 q^{79} +20.7846i q^{85} -6.00000 q^{89} +13.8564 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{17} - 28 q^{25} + 24 q^{41} + 20 q^{49} + 8 q^{73} - 24 q^{89} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 3.46410i − 1.54919i −0.632456 − 0.774597i \(-0.717953\pi\)
0.632456 − 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 − 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 − 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.46410i − 0.643268i −0.946864 − 0.321634i \(-0.895768\pi\)
0.946864 − 0.321634i \(-0.104232\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 − 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000i 2.02837i
\(36\) 0 0
\(37\) − 6.92820i − 1.13899i −0.821995 − 0.569495i \(-0.807139\pi\)
0.821995 − 0.569495i \(-0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 − 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.46410i 0.475831i 0.971286 + 0.237915i \(0.0764641\pi\)
−0.971286 + 0.237915i \(0.923536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 − 0.781133i \(-0.714642\pi\)
0.624364 − 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) − 6.92820i − 0.887066i −0.896258 − 0.443533i \(-0.853725\pi\)
0.896258 − 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 − 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 − 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 20.7846i 2.25441i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.8564 1.42164
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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 576.2.d.b.289.1 4
3.2 odd 2 192.2.d.a.97.4 yes 4
4.3 odd 2 inner 576.2.d.b.289.2 4
8.3 odd 2 inner 576.2.d.b.289.4 4
8.5 even 2 inner 576.2.d.b.289.3 4
12.11 even 2 192.2.d.a.97.2 yes 4
15.2 even 4 4800.2.d.o.1249.2 4
15.8 even 4 4800.2.d.j.1249.3 4
15.14 odd 2 4800.2.k.j.2401.2 4
16.3 odd 4 2304.2.a.u.1.1 2
16.5 even 4 2304.2.a.u.1.2 2
16.11 odd 4 2304.2.a.s.1.2 2
16.13 even 4 2304.2.a.s.1.1 2
24.5 odd 2 192.2.d.a.97.1 ✓ 4
24.11 even 2 192.2.d.a.97.3 yes 4
48.5 odd 4 768.2.a.k.1.1 2
48.11 even 4 768.2.a.j.1.1 2
48.29 odd 4 768.2.a.j.1.2 2
48.35 even 4 768.2.a.k.1.2 2
60.23 odd 4 4800.2.d.o.1249.1 4
60.47 odd 4 4800.2.d.j.1249.4 4
60.59 even 2 4800.2.k.j.2401.3 4
120.29 odd 2 4800.2.k.j.2401.4 4
120.53 even 4 4800.2.d.o.1249.3 4
120.59 even 2 4800.2.k.j.2401.1 4
120.77 even 4 4800.2.d.j.1249.2 4
120.83 odd 4 4800.2.d.j.1249.1 4
120.107 odd 4 4800.2.d.o.1249.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.d.a.97.1 ✓ 4 24.5 odd 2
192.2.d.a.97.2 yes 4 12.11 even 2
192.2.d.a.97.3 yes 4 24.11 even 2
192.2.d.a.97.4 yes 4 3.2 odd 2
576.2.d.b.289.1 4 1.1 even 1 trivial
576.2.d.b.289.2 4 4.3 odd 2 inner
576.2.d.b.289.3 4 8.5 even 2 inner
576.2.d.b.289.4 4 8.3 odd 2 inner
768.2.a.j.1.1 2 48.11 even 4
768.2.a.j.1.2 2 48.29 odd 4
768.2.a.k.1.1 2 48.5 odd 4
768.2.a.k.1.2 2 48.35 even 4
2304.2.a.s.1.1 2 16.13 even 4
2304.2.a.s.1.2 2 16.11 odd 4
2304.2.a.u.1.1 2 16.3 odd 4
2304.2.a.u.1.2 2 16.5 even 4
4800.2.d.j.1249.1 4 120.83 odd 4
4800.2.d.j.1249.2 4 120.77 even 4
4800.2.d.j.1249.3 4 15.8 even 4
4800.2.d.j.1249.4 4 60.47 odd 4
4800.2.d.o.1249.1 4 60.23 odd 4
4800.2.d.o.1249.2 4 15.2 even 4
4800.2.d.o.1249.3 4 120.53 even 4
4800.2.d.o.1249.4 4 120.107 odd 4
4800.2.k.j.2401.1 4 120.59 even 2
4800.2.k.j.2401.2 4 15.14 odd 2
4800.2.k.j.2401.3 4 60.59 even 2
4800.2.k.j.2401.4 4 120.29 odd 2