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Properties

Label 576.2.c.a.575.1
Level $576$
Weight $2$
Character 576.575
Analytic conductor $4.599$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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(575,576)] 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("576.575"); 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([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 575.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 576.575
Dual form 576.2.c.a.575.2

$q$-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)
 
\(f(q)\) \(=\) \(q-4.24264i q^{5} -4.00000 q^{13} -4.24264i q^{17} -13.0000 q^{25} +4.24264i q^{29} -2.00000 q^{37} -12.7279i q^{41} +7.00000 q^{49} -12.7279i q^{53} +10.0000 q^{61} +16.9706i q^{65} +16.0000 q^{73} -18.0000 q^{85} +4.24264i q^{89} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{13} - 26 q^{25} - 4 q^{37} + 14 q^{49} + 20 q^{61} + 32 q^{73} - 36 q^{85} - 16 q^{97}+O(q^{100}) \) 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
\(5\) − 4.24264i − 1.89737i −0.316228 − 0.948683i \(-0.602416\pi\)
0.316228 − 0.948683i \(-0.397584\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 − 0.514496i \(-0.827979\pi\)
0.857493 − 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −13.0000 −2.60000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264i 0.787839i 0.919145 + 0.393919i \(0.128881\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 12.7279i − 1.98777i −0.110432 − 0.993884i \(-0.535223\pi\)
0.110432 − 0.993884i \(-0.464777\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 12.7279i − 1.74831i −0.485643 − 0.874157i \(-0.661414\pi\)
0.485643 − 0.874157i \(-0.338586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.9706i 2.10494i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −18.0000 −1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 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.c.a.575.1 2
3.2 odd 2 inner 576.2.c.a.575.2 2
4.3 odd 2 CM 576.2.c.a.575.1 2
8.3 odd 2 144.2.c.a.143.2 yes 2
8.5 even 2 144.2.c.a.143.2 yes 2
12.11 even 2 inner 576.2.c.a.575.2 2
16.3 odd 4 2304.2.f.f.1151.1 4
16.5 even 4 2304.2.f.f.1151.4 4
16.11 odd 4 2304.2.f.f.1151.4 4
16.13 even 4 2304.2.f.f.1151.1 4
24.5 odd 2 144.2.c.a.143.1 ✓ 2
24.11 even 2 144.2.c.a.143.1 ✓ 2
40.3 even 4 3600.2.o.a.3599.3 4
40.13 odd 4 3600.2.o.a.3599.3 4
40.19 odd 2 3600.2.h.b.1151.2 2
40.27 even 4 3600.2.o.a.3599.2 4
40.29 even 2 3600.2.h.b.1151.2 2
40.37 odd 4 3600.2.o.a.3599.2 4
48.5 odd 4 2304.2.f.f.1151.2 4
48.11 even 4 2304.2.f.f.1151.2 4
48.29 odd 4 2304.2.f.f.1151.3 4
48.35 even 4 2304.2.f.f.1151.3 4
56.13 odd 2 7056.2.h.b.4607.1 2
56.27 even 2 7056.2.h.b.4607.1 2
72.5 odd 6 1296.2.s.h.431.1 4
72.11 even 6 1296.2.s.h.863.2 4
72.13 even 6 1296.2.s.h.431.2 4
72.29 odd 6 1296.2.s.h.863.2 4
72.43 odd 6 1296.2.s.h.863.1 4
72.59 even 6 1296.2.s.h.431.1 4
72.61 even 6 1296.2.s.h.863.1 4
72.67 odd 6 1296.2.s.h.431.2 4
120.29 odd 2 3600.2.h.b.1151.1 2
120.53 even 4 3600.2.o.a.3599.4 4
120.59 even 2 3600.2.h.b.1151.1 2
120.77 even 4 3600.2.o.a.3599.1 4
120.83 odd 4 3600.2.o.a.3599.4 4
120.107 odd 4 3600.2.o.a.3599.1 4
168.83 odd 2 7056.2.h.b.4607.2 2
168.125 even 2 7056.2.h.b.4607.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.c.a.143.1 ✓ 2 24.5 odd 2
144.2.c.a.143.1 ✓ 2 24.11 even 2
144.2.c.a.143.2 yes 2 8.3 odd 2
144.2.c.a.143.2 yes 2 8.5 even 2
576.2.c.a.575.1 2 1.1 even 1 trivial
576.2.c.a.575.1 2 4.3 odd 2 CM
576.2.c.a.575.2 2 3.2 odd 2 inner
576.2.c.a.575.2 2 12.11 even 2 inner
1296.2.s.h.431.1 4 72.5 odd 6
1296.2.s.h.431.1 4 72.59 even 6
1296.2.s.h.431.2 4 72.13 even 6
1296.2.s.h.431.2 4 72.67 odd 6
1296.2.s.h.863.1 4 72.43 odd 6
1296.2.s.h.863.1 4 72.61 even 6
1296.2.s.h.863.2 4 72.11 even 6
1296.2.s.h.863.2 4 72.29 odd 6
2304.2.f.f.1151.1 4 16.3 odd 4
2304.2.f.f.1151.1 4 16.13 even 4
2304.2.f.f.1151.2 4 48.5 odd 4
2304.2.f.f.1151.2 4 48.11 even 4
2304.2.f.f.1151.3 4 48.29 odd 4
2304.2.f.f.1151.3 4 48.35 even 4
2304.2.f.f.1151.4 4 16.5 even 4
2304.2.f.f.1151.4 4 16.11 odd 4
3600.2.h.b.1151.1 2 120.29 odd 2
3600.2.h.b.1151.1 2 120.59 even 2
3600.2.h.b.1151.2 2 40.19 odd 2
3600.2.h.b.1151.2 2 40.29 even 2
3600.2.o.a.3599.1 4 120.77 even 4
3600.2.o.a.3599.1 4 120.107 odd 4
3600.2.o.a.3599.2 4 40.27 even 4
3600.2.o.a.3599.2 4 40.37 odd 4
3600.2.o.a.3599.3 4 40.3 even 4
3600.2.o.a.3599.3 4 40.13 odd 4
3600.2.o.a.3599.4 4 120.53 even 4
3600.2.o.a.3599.4 4 120.83 odd 4
7056.2.h.b.4607.1 2 56.13 odd 2
7056.2.h.b.4607.1 2 56.27 even 2
7056.2.h.b.4607.2 2 168.83 odd 2
7056.2.h.b.4607.2 2 168.125 even 2