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,-4,0,0,0,2,0,-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.585786 q^{5} -1.41421 q^{7} +1.00000 q^{9} +0.828427 q^{11} -4.82843 q^{13} -0.585786 q^{15} -0.828427 q^{17} -2.82843 q^{19} -1.41421 q^{21} -6.82843 q^{23} -4.65685 q^{25} +1.00000 q^{27} -4.58579 q^{29} +7.07107 q^{31} +0.828427 q^{33} +0.828427 q^{35} -0.343146 q^{37} -4.82843 q^{39} +6.48528 q^{41} +1.17157 q^{43} -0.585786 q^{45} +4.48528 q^{47} -5.00000 q^{49} -0.828427 q^{51} -10.2426 q^{53} -0.485281 q^{55} -2.82843 q^{57} -9.65685 q^{59} -11.6569 q^{61} -1.41421 q^{63} +2.82843 q^{65} +5.65685 q^{67} -6.82843 q^{69} -8.48528 q^{71} +11.3137 q^{73} -4.65685 q^{75} -1.17157 q^{77} +14.5858 q^{79} +1.00000 q^{81} -3.17157 q^{83} +0.485281 q^{85} -4.58579 q^{87} -17.3137 q^{89} +6.82843 q^{91} +7.07107 q^{93} +1.65685 q^{95} +3.65685 q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} + 4 q^{17} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 4 q^{33} - 4 q^{35} - 12 q^{37} - 4 q^{39} - 4 q^{41} + 8 q^{43} - 4 q^{45} - 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(7\) −1.41421 −0.534522 −0.267261 − 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 − 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) −4.82843 −1.33916 −0.669582 − 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 − 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 − 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 − 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.58579 −0.851559 −0.425780 − 0.904827i \(-0.640000\pi\)
−0.425780 + 0.904827i \(0.640000\pi\)
\(30\) 0 0
\(31\) 7.07107 1.27000 0.635001 − 0.772512i \(-0.281000\pi\)
0.635001 + 0.772512i \(0.281000\pi\)
\(32\) 0 0
\(33\) 0.828427 0.144211
\(34\) 0 0
\(35\) 0.828427 0.140030
\(36\) 0 0
\(37\) −0.343146 −0.0564128 −0.0282064 − 0.999602i \(-0.508980\pi\)
−0.0282064 + 0.999602i \(0.508980\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) 6.48528 1.01283 0.506415 − 0.862290i \(-0.330970\pi\)
0.506415 + 0.862290i \(0.330970\pi\)
\(42\) 0 0
\(43\) 1.17157 0.178663 0.0893316 − 0.996002i \(-0.471527\pi\)
0.0893316 + 0.996002i \(0.471527\pi\)
\(44\) 0 0
\(45\) −0.585786 −0.0873239
\(46\) 0 0
\(47\) 4.48528 0.654246 0.327123 − 0.944982i \(-0.393921\pi\)
0.327123 + 0.944982i \(0.393921\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) −0.828427 −0.116003
\(52\) 0 0
\(53\) −10.2426 −1.40693 −0.703467 − 0.710727i \(-0.748366\pi\)
−0.703467 + 0.710727i \(0.748366\pi\)
\(54\) 0 0
\(55\) −0.485281 −0.0654353
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) −9.65685 −1.25722 −0.628608 − 0.777723i \(-0.716375\pi\)
−0.628608 + 0.777723i \(0.716375\pi\)
\(60\) 0 0
\(61\) −11.6569 −1.49251 −0.746254 − 0.665662i \(-0.768149\pi\)
−0.746254 + 0.665662i \(0.768149\pi\)
\(62\) 0 0
\(63\) −1.41421 −0.178174
\(64\) 0 0
\(65\) 2.82843 0.350823
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 − 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 0 0
\(69\) −6.82843 −0.822046
\(70\) 0 0
\(71\) −8.48528 −1.00702 −0.503509 − 0.863990i \(-0.667958\pi\)
−0.503509 + 0.863990i \(0.667958\pi\)
\(72\) 0 0
\(73\) 11.3137 1.32417 0.662085 − 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0 0
\(75\) −4.65685 −0.537727
\(76\) 0 0
\(77\) −1.17157 −0.133513
\(78\) 0 0
\(79\) 14.5858 1.64103 0.820515 − 0.571626i \(-0.193687\pi\)
0.820515 + 0.571626i \(0.193687\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.17157 −0.348125 −0.174063 − 0.984735i \(-0.555690\pi\)
−0.174063 + 0.984735i \(0.555690\pi\)
\(84\) 0 0
\(85\) 0.485281 0.0526362
\(86\) 0 0
\(87\) −4.58579 −0.491648
\(88\) 0 0
\(89\) −17.3137 −1.83525 −0.917625 − 0.397448i \(-0.869896\pi\)
−0.917625 + 0.397448i \(0.869896\pi\)
\(90\) 0 0
\(91\) 6.82843 0.715814
\(92\) 0 0
\(93\) 7.07107 0.733236
\(94\) 0 0
\(95\) 1.65685 0.169990
\(96\) 0 0
\(97\) 3.65685 0.371297 0.185649 − 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) 0 0
\(99\) 0.828427 0.0832601
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.h.1.2 yes 2
3.2 odd 2 4608.2.a.q.1.1 2
4.3 odd 2 1536.2.a.a.1.2 ✓ 2
8.3 odd 2 1536.2.a.k.1.1 yes 2
8.5 even 2 1536.2.a.f.1.1 yes 2
12.11 even 2 4608.2.a.o.1.1 2
16.3 odd 4 1536.2.d.e.769.1 4
16.5 even 4 1536.2.d.b.769.2 4
16.11 odd 4 1536.2.d.e.769.4 4
16.13 even 4 1536.2.d.b.769.3 4
24.5 odd 2 4608.2.a.b.1.2 2
24.11 even 2 4608.2.a.d.1.2 2
48.5 odd 4 4608.2.d.f.2305.3 4
48.11 even 4 4608.2.d.h.2305.3 4
48.29 odd 4 4608.2.d.f.2305.2 4
48.35 even 4 4608.2.d.h.2305.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.a.1.2 ✓ 2 4.3 odd 2
1536.2.a.f.1.1 yes 2 8.5 even 2
1536.2.a.h.1.2 yes 2 1.1 even 1 trivial
1536.2.a.k.1.1 yes 2 8.3 odd 2
1536.2.d.b.769.2 4 16.5 even 4
1536.2.d.b.769.3 4 16.13 even 4
1536.2.d.e.769.1 4 16.3 odd 4
1536.2.d.e.769.4 4 16.11 odd 4
4608.2.a.b.1.2 2 24.5 odd 2
4608.2.a.d.1.2 2 24.11 even 2
4608.2.a.o.1.1 2 12.11 even 2
4608.2.a.q.1.1 2 3.2 odd 2
4608.2.d.f.2305.2 4 48.29 odd 4
4608.2.d.f.2305.3 4 48.5 odd 4
4608.2.d.h.2305.2 4 48.35 even 4
4608.2.d.h.2305.3 4 48.11 even 4