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 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.41421 q^{5} -0.585786 q^{7} +1.00000 q^{9} +2.00000 q^{11} +2.82843 q^{13} -3.41421 q^{15} -7.65685 q^{17} +5.65685 q^{19} -0.585786 q^{21} -6.82843 q^{23} +6.65685 q^{25} +1.00000 q^{27} -3.41421 q^{29} -7.41421 q^{31} +2.00000 q^{33} +2.00000 q^{35} -1.65685 q^{37} +2.82843 q^{39} -0.343146 q^{41} -9.65685 q^{43} -3.41421 q^{45} +4.48528 q^{47} -6.65685 q^{49} -7.65685 q^{51} +7.89949 q^{53} -6.82843 q^{55} +5.65685 q^{57} -4.00000 q^{59} -1.65685 q^{61} -0.585786 q^{63} -9.65685 q^{65} -8.00000 q^{67} -6.82843 q^{69} -14.8284 q^{71} +9.65685 q^{73} +6.65685 q^{75} -1.17157 q^{77} -14.2426 q^{79} +1.00000 q^{81} -13.3137 q^{83} +26.1421 q^{85} -3.41421 q^{87} +2.00000 q^{89} -1.65685 q^{91} -7.41421 q^{93} -19.3137 q^{95} -9.31371 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{15} - 4 q^{17} - 4 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 12 q^{31} + 4 q^{33} + 4 q^{35} + 8 q^{37} - 12 q^{41} - 8 q^{43} - 4 q^{45}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(5\) −3.41421 −1.52688 −0.763441 − 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) −0.585786 −0.221406 −0.110703 − 0.993854i \(-0.535310\pi\)
−0.110703 + 0.993854i \(0.535310\pi\)
\(8\) 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 \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) −7.65685 −1.85706 −0.928530 − 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) 5.65685 1.29777 0.648886 − 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) −0.585786 −0.127829
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 − 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.41421 −0.634004 −0.317002 − 0.948425i \(-0.602676\pi\)
−0.317002 + 0.948425i \(0.602676\pi\)
\(30\) 0 0
\(31\) −7.41421 −1.33163 −0.665816 − 0.746116i \(-0.731916\pi\)
−0.665816 + 0.746116i \(0.731916\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −1.65685 −0.272385 −0.136193 − 0.990682i \(-0.543487\pi\)
−0.136193 + 0.990682i \(0.543487\pi\)
\(38\) 0 0
\(39\) 2.82843 0.452911
\(40\) 0 0
\(41\) −0.343146 −0.0535904 −0.0267952 − 0.999641i \(-0.508530\pi\)
−0.0267952 + 0.999641i \(0.508530\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 − 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 0 0
\(47\) 4.48528 0.654246 0.327123 − 0.944982i \(-0.393921\pi\)
0.327123 + 0.944982i \(0.393921\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) −7.65685 −1.07217
\(52\) 0 0
\(53\) 7.89949 1.08508 0.542540 − 0.840030i \(-0.317463\pi\)
0.542540 + 0.840030i \(0.317463\pi\)
\(54\) 0 0
\(55\) −6.82843 −0.920745
\(56\) 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
\(61\) −1.65685 −0.212138 −0.106069 − 0.994359i \(-0.533827\pi\)
−0.106069 + 0.994359i \(0.533827\pi\)
\(62\) 0 0
\(63\) −0.585786 −0.0738022
\(64\) 0 0
\(65\) −9.65685 −1.19779
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −6.82843 −0.822046
\(70\) 0 0
\(71\) −14.8284 −1.75981 −0.879905 − 0.475149i \(-0.842394\pi\)
−0.879905 + 0.475149i \(0.842394\pi\)
\(72\) 0 0
\(73\) 9.65685 1.13025 0.565125 − 0.825006i \(-0.308828\pi\)
0.565125 + 0.825006i \(0.308828\pi\)
\(74\) 0 0
\(75\) 6.65685 0.768667
\(76\) 0 0
\(77\) −1.17157 −0.133513
\(78\) 0 0
\(79\) −14.2426 −1.60242 −0.801211 − 0.598382i \(-0.795811\pi\)
−0.801211 + 0.598382i \(0.795811\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.3137 −1.46137 −0.730685 − 0.682715i \(-0.760799\pi\)
−0.730685 + 0.682715i \(0.760799\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) 0 0
\(87\) −3.41421 −0.366042
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −1.65685 −0.173686
\(92\) 0 0
\(93\) −7.41421 −0.768818
\(94\) 0 0
\(95\) −19.3137 −1.98154
\(96\) 0 0
\(97\) −9.31371 −0.945664 −0.472832 − 0.881153i \(-0.656768\pi\)
−0.472832 + 0.881153i \(0.656768\pi\)
\(98\) 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.1.1 yes 2
3.2 odd 2 4608.2.a.n.1.2 2
4.3 odd 2 1536.2.a.b.1.1 ✓ 2
8.3 odd 2 1536.2.a.l.1.2 yes 2
8.5 even 2 1536.2.a.e.1.2 yes 2
12.11 even 2 4608.2.a.r.1.2 2
16.3 odd 4 1536.2.d.a.769.2 4
16.5 even 4 1536.2.d.f.769.1 4
16.11 odd 4 1536.2.d.a.769.3 4
16.13 even 4 1536.2.d.f.769.4 4
24.5 odd 2 4608.2.a.a.1.1 2
24.11 even 2 4608.2.a.e.1.1 2
48.5 odd 4 4608.2.d.o.2305.4 4
48.11 even 4 4608.2.d.c.2305.4 4
48.29 odd 4 4608.2.d.o.2305.1 4
48.35 even 4 4608.2.d.c.2305.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.1 ✓ 2 4.3 odd 2
1536.2.a.e.1.2 yes 2 8.5 even 2
1536.2.a.g.1.1 yes 2 1.1 even 1 trivial
1536.2.a.l.1.2 yes 2 8.3 odd 2
1536.2.d.a.769.2 4 16.3 odd 4
1536.2.d.a.769.3 4 16.11 odd 4
1536.2.d.f.769.1 4 16.5 even 4
1536.2.d.f.769.4 4 16.13 even 4
4608.2.a.a.1.1 2 24.5 odd 2
4608.2.a.e.1.1 2 24.11 even 2
4608.2.a.n.1.2 2 3.2 odd 2
4608.2.a.r.1.2 2 12.11 even 2
4608.2.d.c.2305.1 4 48.35 even 4
4608.2.d.c.2305.4 4 48.11 even 4
4608.2.d.o.2305.1 4 48.29 odd 4
4608.2.d.o.2305.4 4 48.5 odd 4