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 769.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.f.769.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +0.585786i q^{5} +3.41421 q^{7} -1.00000 q^{9} -2.00000i q^{11} -2.82843i q^{13} -0.585786 q^{15} +3.65685 q^{17} -5.65685i q^{19} +3.41421i q^{21} +1.17157 q^{23} +4.65685 q^{25} -1.00000i q^{27} -0.585786i q^{29} -4.58579 q^{31} +2.00000 q^{33} +2.00000i q^{35} -9.65685i q^{37} +2.82843 q^{39} +11.6569 q^{41} -1.65685i q^{43} -0.585786i q^{45} -12.4853 q^{47} +4.65685 q^{49} +3.65685i q^{51} +11.8995i q^{53} +1.17157 q^{55} +5.65685 q^{57} +4.00000i q^{59} +9.65685i q^{61} -3.41421 q^{63} +1.65685 q^{65} -8.00000i q^{67} +1.17157i q^{69} +9.17157 q^{71} +1.65685 q^{73} +4.65685i q^{75} -6.82843i q^{77} -5.75736 q^{79} +1.00000 q^{81} +9.31371i q^{83} +2.14214i q^{85} +0.585786 q^{87} -2.00000 q^{89} -9.65685i q^{91} -4.58579i q^{93} +3.31371 q^{95} +13.3137 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 4 q^{9} - 8 q^{15} - 8 q^{17} + 16 q^{23} - 4 q^{25} - 24 q^{31} + 8 q^{33} + 24 q^{41} - 16 q^{47} - 4 q^{49} + 16 q^{55} - 8 q^{63} - 16 q^{65} + 48 q^{71} - 16 q^{73} - 40 q^{79} + 4 q^{81}+ \cdots + 8 q^{97}+O(q^{100}) \) 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
\(5\) 0.585786i 0.261972i 0.991384 + 0.130986i \(0.0418142\pi\)
−0.991384 + 0.130986i \(0.958186\pi\)
\(6\) 0 0
\(7\) 3.41421 1.29045 0.645226 − 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 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 \(-0.871703\pi\)
0.919866 − 0.392232i \(-0.128297\pi\)
\(14\) 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
\(19\) − 5.65685i − 1.29777i −0.760886 − 0.648886i \(-0.775235\pi\)
0.760886 − 0.648886i \(-0.224765\pi\)
\(20\) 0 0
\(21\) 3.41421i 0.745042i
\(22\) 0 0
\(23\) 1.17157 0.244290 0.122145 − 0.992512i \(-0.461023\pi\)
0.122145 + 0.992512i \(0.461023\pi\)
\(24\) 0 0
\(25\) 4.65685 0.931371
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 0.585786i − 0.108778i −0.998520 − 0.0543889i \(-0.982679\pi\)
0.998520 − 0.0543889i \(-0.0173211\pi\)
\(30\) 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
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) − 9.65685i − 1.58758i −0.608194 − 0.793789i \(-0.708106\pi\)
0.608194 − 0.793789i \(-0.291894\pi\)
\(38\) 0 0
\(39\) 2.82843 0.452911
\(40\) 0 0
\(41\) 11.6569 1.82049 0.910247 − 0.414065i \(-0.135891\pi\)
0.910247 + 0.414065i \(0.135891\pi\)
\(42\) 0 0
\(43\) − 1.65685i − 0.252668i −0.991988 − 0.126334i \(-0.959679\pi\)
0.991988 − 0.126334i \(-0.0403211\pi\)
\(44\) 0 0
\(45\) − 0.585786i − 0.0873239i
\(46\) 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
\(51\) 3.65685i 0.512062i
\(52\) 0 0
\(53\) 11.8995i 1.63452i 0.576268 + 0.817261i \(0.304508\pi\)
−0.576268 + 0.817261i \(0.695492\pi\)
\(54\) 0 0
\(55\) 1.17157 0.157975
\(56\) 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
\(61\) 9.65685i 1.23643i 0.786008 + 0.618217i \(0.212145\pi\)
−0.786008 + 0.618217i \(0.787855\pi\)
\(62\) 0 0
\(63\) −3.41421 −0.430150
\(64\) 0 0
\(65\) 1.65685 0.205507
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 1.17157i 0.141041i
\(70\) 0 0
\(71\) 9.17157 1.08847 0.544233 − 0.838934i \(-0.316821\pi\)
0.544233 + 0.838934i \(0.316821\pi\)
\(72\) 0 0
\(73\) 1.65685 0.193920 0.0969601 − 0.995288i \(-0.469088\pi\)
0.0969601 + 0.995288i \(0.469088\pi\)
\(74\) 0 0
\(75\) 4.65685i 0.537727i
\(76\) 0 0
\(77\) − 6.82843i − 0.778171i
\(78\) 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
\(83\) 9.31371i 1.02231i 0.859488 + 0.511156i \(0.170783\pi\)
−0.859488 + 0.511156i \(0.829217\pi\)
\(84\) 0 0
\(85\) 2.14214i 0.232347i
\(86\) 0 0
\(87\) 0.585786 0.0628029
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) − 9.65685i − 1.01231i
\(92\) 0 0
\(93\) − 4.58579i − 0.475524i
\(94\) 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
\(99\) 2.00000i 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.769.3 4
3.2 odd 2 4608.2.d.o.2305.2 4
4.3 odd 2 1536.2.d.a.769.1 4
8.3 odd 2 1536.2.d.a.769.4 4
8.5 even 2 inner 1536.2.d.f.769.2 4
12.11 even 2 4608.2.d.c.2305.2 4
16.3 odd 4 1536.2.a.l.1.1 yes 2
16.5 even 4 1536.2.a.g.1.2 yes 2
16.11 odd 4 1536.2.a.b.1.2 ✓ 2
16.13 even 4 1536.2.a.e.1.1 yes 2
24.5 odd 2 4608.2.d.o.2305.3 4
24.11 even 2 4608.2.d.c.2305.3 4
48.5 odd 4 4608.2.a.n.1.1 2
48.11 even 4 4608.2.a.r.1.1 2
48.29 odd 4 4608.2.a.a.1.2 2
48.35 even 4 4608.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.2 ✓ 2 16.11 odd 4
1536.2.a.e.1.1 yes 2 16.13 even 4
1536.2.a.g.1.2 yes 2 16.5 even 4
1536.2.a.l.1.1 yes 2 16.3 odd 4
1536.2.d.a.769.1 4 4.3 odd 2
1536.2.d.a.769.4 4 8.3 odd 2
1536.2.d.f.769.2 4 8.5 even 2 inner
1536.2.d.f.769.3 4 1.1 even 1 trivial
4608.2.a.a.1.2 2 48.29 odd 4
4608.2.a.e.1.2 2 48.35 even 4
4608.2.a.n.1.1 2 48.5 odd 4
4608.2.a.r.1.1 2 48.11 even 4
4608.2.d.c.2305.2 4 12.11 even 2
4608.2.d.c.2305.3 4 24.11 even 2
4608.2.d.o.2305.2 4 3.2 odd 2
4608.2.d.o.2305.3 4 24.5 odd 2