Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1536,2,Mod(1,1536)] 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("1536.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1536, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-4,0,-4,0,2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.2650217505\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
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: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1536.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.585786 q^{5} -3.41421 q^{7} +1.00000 q^{9} +2.00000 q^{11} -2.82843 q^{13} -0.585786 q^{15} +3.65685 q^{17} -5.65685 q^{19} -3.41421 q^{21} -1.17157 q^{23} -4.65685 q^{25} +1.00000 q^{27} -0.585786 q^{29} -4.58579 q^{31} +2.00000 q^{33} +2.00000 q^{35} +9.65685 q^{37} -2.82843 q^{39} -11.6569 q^{41} +1.65685 q^{43} -0.585786 q^{45} -12.4853 q^{47} +4.65685 q^{49} +3.65685 q^{51} -11.8995 q^{53} -1.17157 q^{55} -5.65685 q^{57} -4.00000 q^{59} +9.65685 q^{61} -3.41421 q^{63} +1.65685 q^{65} -8.00000 q^{67} -1.17157 q^{69} -9.17157 q^{71} -1.65685 q^{73} -4.65685 q^{75} -6.82843 q^{77} -5.75736 q^{79} +1.00000 q^{81} +9.31371 q^{83} -2.14214 q^{85} -0.585786 q^{87} +2.00000 q^{89} +9.65685 q^{91} -4.58579 q^{93} +3.31371 q^{95} +13.3137 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{15} - 4 q^{17} - 4 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 12 q^{31} + 4 q^{33} + 4 q^{35} + 8 q^{37} - 12 q^{41} - 8 q^{43} - 4 q^{45}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.585786 −0.261972 −0.130986 − 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 0 0
\(7\) −3.41421 −1.29045 −0.645226 − 0.763992i \(-0.723237\pi\)
−0.645226 + 0.763992i \(0.723237\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −2.82843 −0.784465 −0.392232 − 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) 3.65685 0.886917 0.443459 − 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 − 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) −3.41421 −0.745042
\(22\) 0 0
\(23\) −1.17157 −0.244290 −0.122145 − 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.585786 −0.108778 −0.0543889 − 0.998520i \(-0.517321\pi\)
−0.0543889 + 0.998520i \(0.517321\pi\)
\(30\) 0 0
\(31\) −4.58579 −0.823632 −0.411816 − 0.911267i \(-0.635105\pi\)
−0.411816 + 0.911267i \(0.635105\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 9.65685 1.58758 0.793789 − 0.608194i \(-0.208106\pi\)
0.793789 + 0.608194i \(0.208106\pi\)
\(38\) 0 0
\(39\) −2.82843 −0.452911
\(40\) 0 0
\(41\) −11.6569 −1.82049 −0.910247 − 0.414065i \(-0.864109\pi\)
−0.910247 + 0.414065i \(0.864109\pi\)
\(42\) 0 0
\(43\) 1.65685 0.252668 0.126334 − 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(44\) 0 0
\(45\) −0.585786 −0.0873239
\(46\) 0 0
\(47\) −12.4853 −1.82117 −0.910583 − 0.413327i \(-0.864367\pi\)
−0.910583 + 0.413327i \(0.864367\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 3.65685 0.512062
\(52\) 0 0
\(53\) −11.8995 −1.63452 −0.817261 − 0.576268i \(-0.804508\pi\)
−0.817261 + 0.576268i \(0.804508\pi\)
\(54\) 0 0
\(55\) −1.17157 −0.157975
\(56\) 0 0
\(57\) −5.65685 −0.749269
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 9.65685 1.23643 0.618217 − 0.786008i \(-0.287855\pi\)
0.618217 + 0.786008i \(0.287855\pi\)
\(62\) 0 0
\(63\) −3.41421 −0.430150
\(64\) 0 0
\(65\) 1.65685 0.205507
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −1.17157 −0.141041
\(70\) 0 0
\(71\) −9.17157 −1.08847 −0.544233 − 0.838934i \(-0.683179\pi\)
−0.544233 + 0.838934i \(0.683179\pi\)
\(72\) 0 0
\(73\) −1.65685 −0.193920 −0.0969601 − 0.995288i \(-0.530912\pi\)
−0.0969601 + 0.995288i \(0.530912\pi\)
\(74\) 0 0
\(75\) −4.65685 −0.537727
\(76\) 0 0
\(77\) −6.82843 −0.778171
\(78\) 0 0
\(79\) −5.75736 −0.647754 −0.323877 − 0.946099i \(-0.604986\pi\)
−0.323877 + 0.946099i \(0.604986\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.31371 1.02231 0.511156 − 0.859488i \(-0.329217\pi\)
0.511156 + 0.859488i \(0.329217\pi\)
\(84\) 0 0
\(85\) −2.14214 −0.232347
\(86\) 0 0
\(87\) −0.585786 −0.0628029
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 9.65685 1.01231
\(92\) 0 0
\(93\) −4.58579 −0.475524
\(94\) 0 0
\(95\) 3.31371 0.339979
\(96\) 0 0
\(97\) 13.3137 1.35180 0.675901 − 0.736992i \(-0.263755\pi\)
0.675901 + 0.736992i \(0.263755\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
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 1536.2.a.g.1.2 yes 2
3.2 odd 2 4608.2.a.n.1.1 2
4.3 odd 2 1536.2.a.b.1.2 ✓ 2
8.3 odd 2 1536.2.a.l.1.1 yes 2
8.5 even 2 1536.2.a.e.1.1 yes 2
12.11 even 2 4608.2.a.r.1.1 2
16.3 odd 4 1536.2.d.a.769.1 4
16.5 even 4 1536.2.d.f.769.2 4
16.11 odd 4 1536.2.d.a.769.4 4
16.13 even 4 1536.2.d.f.769.3 4
24.5 odd 2 4608.2.a.a.1.2 2
24.11 even 2 4608.2.a.e.1.2 2
48.5 odd 4 4608.2.d.o.2305.3 4
48.11 even 4 4608.2.d.c.2305.3 4
48.29 odd 4 4608.2.d.o.2305.2 4
48.35 even 4 4608.2.d.c.2305.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.2 ✓ 2 4.3 odd 2
1536.2.a.e.1.1 yes 2 8.5 even 2
1536.2.a.g.1.2 yes 2 1.1 even 1 trivial
1536.2.a.l.1.1 yes 2 8.3 odd 2
1536.2.d.a.769.1 4 16.3 odd 4
1536.2.d.a.769.4 4 16.11 odd 4
1536.2.d.f.769.2 4 16.5 even 4
1536.2.d.f.769.3 4 16.13 even 4
4608.2.a.a.1.2 2 24.5 odd 2
4608.2.a.e.1.2 2 24.11 even 2
4608.2.a.n.1.1 2 3.2 odd 2
4608.2.a.r.1.1 2 12.11 even 2
4608.2.d.c.2305.2 4 48.35 even 4
4608.2.d.c.2305.3 4 48.11 even 4
4608.2.d.o.2305.2 4 48.29 odd 4
4608.2.d.o.2305.3 4 48.5 odd 4