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 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.41421 q^{5} +1.41421 q^{7} +1.00000 q^{9} -4.82843 q^{11} +0.828427 q^{13} -3.41421 q^{15} +4.82843 q^{17} +2.82843 q^{19} +1.41421 q^{21} -1.17157 q^{23} +6.65685 q^{25} +1.00000 q^{27} -7.41421 q^{29} -7.07107 q^{31} -4.82843 q^{33} -4.82843 q^{35} -11.6569 q^{37} +0.828427 q^{39} -10.4853 q^{41} +6.82843 q^{43} -3.41421 q^{45} -12.4853 q^{47} -5.00000 q^{49} +4.82843 q^{51} -1.75736 q^{53} +16.4853 q^{55} +2.82843 q^{57} +1.65685 q^{59} -0.343146 q^{61} +1.41421 q^{63} -2.82843 q^{65} -5.65685 q^{67} -1.17157 q^{69} +8.48528 q^{71} -11.3137 q^{73} +6.65685 q^{75} -6.82843 q^{77} +17.4142 q^{79} +1.00000 q^{81} -8.82843 q^{83} -16.4853 q^{85} -7.41421 q^{87} +5.31371 q^{89} +1.17157 q^{91} -7.07107 q^{93} -9.65685 q^{95} -7.65685 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} + 4 q^{17} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 4 q^{33} - 4 q^{35} - 12 q^{37} - 4 q^{39} - 4 q^{41} + 8 q^{43} - 4 q^{45} - 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(5\) −3.41421 −1.52688 −0.763441 − 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 − 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 − 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) 0.828427 0.229764 0.114882 − 0.993379i \(-0.463351\pi\)
0.114882 + 0.993379i \(0.463351\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 4.82843 1.17107 0.585533 − 0.810649i \(-0.300885\pi\)
0.585533 + 0.810649i \(0.300885\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 1.41421 0.308607
\(22\) 0 0
\(23\) −1.17157 −0.244290 −0.122145 − 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.41421 −1.37678 −0.688392 − 0.725338i \(-0.741683\pi\)
−0.688392 + 0.725338i \(0.741683\pi\)
\(30\) 0 0
\(31\) −7.07107 −1.27000 −0.635001 − 0.772512i \(-0.719000\pi\)
−0.635001 + 0.772512i \(0.719000\pi\)
\(32\) 0 0
\(33\) −4.82843 −0.840521
\(34\) 0 0
\(35\) −4.82843 −0.816153
\(36\) 0 0
\(37\) −11.6569 −1.91638 −0.958188 − 0.286141i \(-0.907627\pi\)
−0.958188 + 0.286141i \(0.907627\pi\)
\(38\) 0 0
\(39\) 0.828427 0.132655
\(40\) 0 0
\(41\) −10.4853 −1.63753 −0.818763 − 0.574132i \(-0.805340\pi\)
−0.818763 + 0.574132i \(0.805340\pi\)
\(42\) 0 0
\(43\) 6.82843 1.04133 0.520663 − 0.853762i \(-0.325685\pi\)
0.520663 + 0.853762i \(0.325685\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 0 0
\(47\) −12.4853 −1.82117 −0.910583 − 0.413327i \(-0.864367\pi\)
−0.910583 + 0.413327i \(0.864367\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 4.82843 0.676115
\(52\) 0 0
\(53\) −1.75736 −0.241392 −0.120696 − 0.992690i \(-0.538513\pi\)
−0.120696 + 0.992690i \(0.538513\pi\)
\(54\) 0 0
\(55\) 16.4853 2.22287
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 0 0
\(59\) 1.65685 0.215704 0.107852 − 0.994167i \(-0.465603\pi\)
0.107852 + 0.994167i \(0.465603\pi\)
\(60\) 0 0
\(61\) −0.343146 −0.0439353 −0.0219677 − 0.999759i \(-0.506993\pi\)
−0.0219677 + 0.999759i \(0.506993\pi\)
\(62\) 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 \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) −1.17157 −0.141041
\(70\) 0 0
\(71\) 8.48528 1.00702 0.503509 − 0.863990i \(-0.332042\pi\)
0.503509 + 0.863990i \(0.332042\pi\)
\(72\) 0 0
\(73\) −11.3137 −1.32417 −0.662085 − 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) 0 0
\(75\) 6.65685 0.768667
\(76\) 0 0
\(77\) −6.82843 −0.778171
\(78\) 0 0
\(79\) 17.4142 1.95925 0.979626 − 0.200830i \(-0.0643640\pi\)
0.979626 + 0.200830i \(0.0643640\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.82843 −0.969046 −0.484523 − 0.874779i \(-0.661007\pi\)
−0.484523 + 0.874779i \(0.661007\pi\)
\(84\) 0 0
\(85\) −16.4853 −1.78808
\(86\) 0 0
\(87\) −7.41421 −0.794887
\(88\) 0 0
\(89\) 5.31371 0.563252 0.281626 − 0.959524i \(-0.409126\pi\)
0.281626 + 0.959524i \(0.409126\pi\)
\(90\) 0 0
\(91\) 1.17157 0.122814
\(92\) 0 0
\(93\) −7.07107 −0.733236
\(94\) 0 0
\(95\) −9.65685 −0.990772
\(96\) 0 0
\(97\) −7.65685 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(98\) 0 0
\(99\) −4.82843 −0.485275
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.1.1 yes 2
3.2 odd 2 4608.2.a.q.1.2 2
4.3 odd 2 1536.2.a.a.1.1 ✓ 2
8.3 odd 2 1536.2.a.k.1.2 yes 2
8.5 even 2 1536.2.a.f.1.2 yes 2
12.11 even 2 4608.2.a.o.1.2 2
16.3 odd 4 1536.2.d.e.769.2 4
16.5 even 4 1536.2.d.b.769.1 4
16.11 odd 4 1536.2.d.e.769.3 4
16.13 even 4 1536.2.d.b.769.4 4
24.5 odd 2 4608.2.a.b.1.1 2
24.11 even 2 4608.2.a.d.1.1 2
48.5 odd 4 4608.2.d.f.2305.4 4
48.11 even 4 4608.2.d.h.2305.4 4
48.29 odd 4 4608.2.d.f.2305.1 4
48.35 even 4 4608.2.d.h.2305.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.a.1.1 ✓ 2 4.3 odd 2
1536.2.a.f.1.2 yes 2 8.5 even 2
1536.2.a.h.1.1 yes 2 1.1 even 1 trivial
1536.2.a.k.1.2 yes 2 8.3 odd 2
1536.2.d.b.769.1 4 16.5 even 4
1536.2.d.b.769.4 4 16.13 even 4
1536.2.d.e.769.2 4 16.3 odd 4
1536.2.d.e.769.3 4 16.11 odd 4
4608.2.a.b.1.1 2 24.5 odd 2
4608.2.a.d.1.1 2 24.11 even 2
4608.2.a.o.1.2 2 12.11 even 2
4608.2.a.q.1.2 2 3.2 odd 2
4608.2.d.f.2305.1 4 48.29 odd 4
4608.2.d.f.2305.4 4 48.5 odd 4
4608.2.d.h.2305.1 4 48.35 even 4
4608.2.d.h.2305.4 4 48.11 even 4