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(769,1536)] 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("1536.769"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,8,0,-4,0,0,0,0,0,-8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.f.769.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +3.41421i q^{5} +0.585786 q^{7} -1.00000 q^{9} -2.00000i q^{11} +2.82843i q^{13} -3.41421 q^{15} -7.65685 q^{17} +5.65685i q^{19} +0.585786i q^{21} +6.82843 q^{23} -6.65685 q^{25} -1.00000i q^{27} -3.41421i q^{29} -7.41421 q^{31} +2.00000 q^{33} +2.00000i q^{35} +1.65685i q^{37} -2.82843 q^{39} +0.343146 q^{41} +9.65685i q^{43} -3.41421i q^{45} +4.48528 q^{47} -6.65685 q^{49} -7.65685i q^{51} -7.89949i q^{53} +6.82843 q^{55} -5.65685 q^{57} +4.00000i q^{59} -1.65685i q^{61} -0.585786 q^{63} -9.65685 q^{65} -8.00000i q^{67} +6.82843i q^{69} +14.8284 q^{71} -9.65685 q^{73} -6.65685i q^{75} -1.17157i q^{77} -14.2426 q^{79} +1.00000 q^{81} -13.3137i q^{83} -26.1421i q^{85} +3.41421 q^{87} -2.00000 q^{89} +1.65685i q^{91} -7.41421i q^{93} -19.3137 q^{95} -9.31371 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 4 q^{9} - 8 q^{15} - 8 q^{17} + 16 q^{23} - 4 q^{25} - 24 q^{31} + 8 q^{33} + 24 q^{41} - 16 q^{47} - 4 q^{49} + 16 q^{55} - 8 q^{63} - 16 q^{65} + 48 q^{71} - 16 q^{73} - 40 q^{79} + 4 q^{81}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 3.41421i 1.52688i 0.645877 + 0.763441i \(0.276492\pi\)
−0.645877 + 0.763441i \(0.723508\pi\)
\(6\) 0 0
\(7\) 0.585786 0.221406 0.110703 − 0.993854i \(-0.464690\pi\)
0.110703 + 0.993854i \(0.464690\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 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
\(19\) 5.65685i 1.29777i 0.760886 + 0.648886i \(0.224765\pi\)
−0.760886 + 0.648886i \(0.775235\pi\)
\(20\) 0 0
\(21\) 0.585786i 0.127829i
\(22\) 0 0
\(23\) 6.82843 1.42383 0.711913 − 0.702268i \(-0.247829\pi\)
0.711913 + 0.702268i \(0.247829\pi\)
\(24\) 0 0
\(25\) −6.65685 −1.33137
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 3.41421i − 0.634004i −0.948425 − 0.317002i \(-0.897324\pi\)
0.948425 − 0.317002i \(-0.102676\pi\)
\(30\) 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
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) 1.65685i 0.272385i 0.990682 + 0.136193i \(0.0434866\pi\)
−0.990682 + 0.136193i \(0.956513\pi\)
\(38\) 0 0
\(39\) −2.82843 −0.452911
\(40\) 0 0
\(41\) 0.343146 0.0535904 0.0267952 − 0.999641i \(-0.491470\pi\)
0.0267952 + 0.999641i \(0.491470\pi\)
\(42\) 0 0
\(43\) 9.65685i 1.47266i 0.676625 + 0.736328i \(0.263442\pi\)
−0.676625 + 0.736328i \(0.736558\pi\)
\(44\) 0 0
\(45\) − 3.41421i − 0.508961i
\(46\) 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
\(51\) − 7.65685i − 1.07217i
\(52\) 0 0
\(53\) − 7.89949i − 1.08508i −0.840030 − 0.542540i \(-0.817463\pi\)
0.840030 − 0.542540i \(-0.182537\pi\)
\(54\) 0 0
\(55\) 6.82843 0.920745
\(56\) 0 0
\(57\) −5.65685 −0.749269
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) − 1.65685i − 0.212138i −0.994359 − 0.106069i \(-0.966173\pi\)
0.994359 − 0.106069i \(-0.0338265\pi\)
\(62\) 0 0
\(63\) −0.585786 −0.0738022
\(64\) 0 0
\(65\) −9.65685 −1.19779
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 6.82843i 0.822046i
\(70\) 0 0
\(71\) 14.8284 1.75981 0.879905 − 0.475149i \(-0.157606\pi\)
0.879905 + 0.475149i \(0.157606\pi\)
\(72\) 0 0
\(73\) −9.65685 −1.13025 −0.565125 − 0.825006i \(-0.691172\pi\)
−0.565125 + 0.825006i \(0.691172\pi\)
\(74\) 0 0
\(75\) − 6.65685i − 0.768667i
\(76\) 0 0
\(77\) − 1.17157i − 0.133513i
\(78\) 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
\(83\) − 13.3137i − 1.46137i −0.682715 − 0.730685i \(-0.739201\pi\)
0.682715 − 0.730685i \(-0.260799\pi\)
\(84\) 0 0
\(85\) − 26.1421i − 2.83551i
\(86\) 0 0
\(87\) 3.41421 0.366042
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 1.65685i 0.173686i
\(92\) 0 0
\(93\) − 7.41421i − 0.768818i
\(94\) 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
\(99\) 2.00000i 0.201008i
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.d.f.769.4 4
3.2 odd 2 4608.2.d.o.2305.1 4
4.3 odd 2 1536.2.d.a.769.2 4
8.3 odd 2 1536.2.d.a.769.3 4
8.5 even 2 inner 1536.2.d.f.769.1 4
12.11 even 2 4608.2.d.c.2305.1 4
16.3 odd 4 1536.2.a.l.1.2 yes 2
16.5 even 4 1536.2.a.g.1.1 yes 2
16.11 odd 4 1536.2.a.b.1.1 ✓ 2
16.13 even 4 1536.2.a.e.1.2 yes 2
24.5 odd 2 4608.2.d.o.2305.4 4
24.11 even 2 4608.2.d.c.2305.4 4
48.5 odd 4 4608.2.a.n.1.2 2
48.11 even 4 4608.2.a.r.1.2 2
48.29 odd 4 4608.2.a.a.1.1 2
48.35 even 4 4608.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.1 ✓ 2 16.11 odd 4
1536.2.a.e.1.2 yes 2 16.13 even 4
1536.2.a.g.1.1 yes 2 16.5 even 4
1536.2.a.l.1.2 yes 2 16.3 odd 4
1536.2.d.a.769.2 4 4.3 odd 2
1536.2.d.a.769.3 4 8.3 odd 2
1536.2.d.f.769.1 4 8.5 even 2 inner
1536.2.d.f.769.4 4 1.1 even 1 trivial
4608.2.a.a.1.1 2 48.29 odd 4
4608.2.a.e.1.1 2 48.35 even 4
4608.2.a.n.1.2 2 48.5 odd 4
4608.2.a.r.1.2 2 48.11 even 4
4608.2.d.c.2305.1 4 12.11 even 2
4608.2.d.c.2305.4 4 24.11 even 2
4608.2.d.o.2305.1 4 3.2 odd 2
4608.2.d.o.2305.4 4 24.5 odd 2