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 = [4,0,0,0,0,0,0,0,-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.e.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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -3.41421i q^{5} +1.41421 q^{7} -1.00000 q^{9} +4.82843i q^{11} -0.828427i q^{13} +3.41421 q^{15} +4.82843 q^{17} +2.82843i q^{19} +1.41421i q^{21} -1.17157 q^{23} -6.65685 q^{25} -1.00000i q^{27} +7.41421i q^{29} +7.07107 q^{31} -4.82843 q^{33} -4.82843i q^{35} -11.6569i q^{37} +0.828427 q^{39} +10.4853 q^{41} -6.82843i q^{43} +3.41421i q^{45} +12.4853 q^{47} -5.00000 q^{49} +4.82843i q^{51} -1.75736i q^{53} +16.4853 q^{55} -2.82843 q^{57} -1.65685i q^{59} +0.343146i q^{61} -1.41421 q^{63} -2.82843 q^{65} -5.65685i q^{67} -1.17157i q^{69} +8.48528 q^{71} +11.3137 q^{73} -6.65685i q^{75} +6.82843i q^{77} -17.4142 q^{79} +1.00000 q^{81} -8.82843i q^{83} -16.4853i q^{85} -7.41421 q^{87} -5.31371 q^{89} -1.17157i q^{91} +7.07107i q^{93} +9.65685 q^{95} -7.65685 q^{97} -4.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 8 q^{15} + 8 q^{17} - 16 q^{23} - 4 q^{25} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 16 q^{47} - 20 q^{49} + 32 q^{55} - 64 q^{79} + 4 q^{81} - 24 q^{87} + 24 q^{89} + 16 q^{95} - 8 q^{97}+O(q^{100}) \) 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 \(-0.723508\pi\)
0.645877 − 0.763441i \(-0.276492\pi\)
\(6\) 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
\(11\) 4.82843i 1.45583i 0.685670 + 0.727913i \(0.259509\pi\)
−0.685670 + 0.727913i \(0.740491\pi\)
\(12\) 0 0
\(13\) − 0.828427i − 0.229764i −0.993379 − 0.114882i \(-0.963351\pi\)
0.993379 − 0.114882i \(-0.0366490\pi\)
\(14\) 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
\(19\) 2.82843i 0.648886i 0.945905 + 0.324443i \(0.105177\pi\)
−0.945905 + 0.324443i \(0.894823\pi\)
\(20\) 0 0
\(21\) 1.41421i 0.308607i
\(22\) 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
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 7.41421i 1.37678i 0.725338 + 0.688392i \(0.241683\pi\)
−0.725338 + 0.688392i \(0.758317\pi\)
\(30\) 0 0
\(31\) 7.07107 1.27000 0.635001 − 0.772512i \(-0.281000\pi\)
0.635001 + 0.772512i \(0.281000\pi\)
\(32\) 0 0
\(33\) −4.82843 −0.840521
\(34\) 0 0
\(35\) − 4.82843i − 0.816153i
\(36\) 0 0
\(37\) − 11.6569i − 1.91638i −0.286141 − 0.958188i \(-0.592373\pi\)
0.286141 − 0.958188i \(-0.407627\pi\)
\(38\) 0 0
\(39\) 0.828427 0.132655
\(40\) 0 0
\(41\) 10.4853 1.63753 0.818763 − 0.574132i \(-0.194660\pi\)
0.818763 + 0.574132i \(0.194660\pi\)
\(42\) 0 0
\(43\) − 6.82843i − 1.04133i −0.853762 − 0.520663i \(-0.825685\pi\)
0.853762 − 0.520663i \(-0.174315\pi\)
\(44\) 0 0
\(45\) 3.41421i 0.508961i
\(46\) 0 0
\(47\) 12.4853 1.82117 0.910583 − 0.413327i \(-0.135633\pi\)
0.910583 + 0.413327i \(0.135633\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 4.82843i 0.676115i
\(52\) 0 0
\(53\) − 1.75736i − 0.241392i −0.992690 − 0.120696i \(-0.961487\pi\)
0.992690 − 0.120696i \(-0.0385126\pi\)
\(54\) 0 0
\(55\) 16.4853 2.22287
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) − 1.65685i − 0.215704i −0.994167 − 0.107852i \(-0.965603\pi\)
0.994167 − 0.107852i \(-0.0343973\pi\)
\(60\) 0 0
\(61\) 0.343146i 0.0439353i 0.999759 + 0.0219677i \(0.00699308\pi\)
−0.999759 + 0.0219677i \(0.993007\pi\)
\(62\) 0 0
\(63\) −1.41421 −0.178174
\(64\) 0 0
\(65\) −2.82843 −0.350823
\(66\) 0 0
\(67\) − 5.65685i − 0.691095i −0.938401 − 0.345547i \(-0.887693\pi\)
0.938401 − 0.345547i \(-0.112307\pi\)
\(68\) 0 0
\(69\) − 1.17157i − 0.141041i
\(70\) 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 \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0 0
\(75\) − 6.65685i − 0.768667i
\(76\) 0 0
\(77\) 6.82843i 0.778171i
\(78\) 0 0
\(79\) −17.4142 −1.95925 −0.979626 − 0.200830i \(-0.935636\pi\)
−0.979626 + 0.200830i \(0.935636\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 8.82843i − 0.969046i −0.874779 − 0.484523i \(-0.838993\pi\)
0.874779 − 0.484523i \(-0.161007\pi\)
\(84\) 0 0
\(85\) − 16.4853i − 1.78808i
\(86\) 0 0
\(87\) −7.41421 −0.794887
\(88\) 0 0
\(89\) −5.31371 −0.563252 −0.281626 − 0.959524i \(-0.590874\pi\)
−0.281626 + 0.959524i \(0.590874\pi\)
\(90\) 0 0
\(91\) − 1.17157i − 0.122814i
\(92\) 0 0
\(93\) 7.07107i 0.733236i
\(94\) 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
\(99\) − 4.82843i − 0.485275i
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.e.769.3 4
3.2 odd 2 4608.2.d.h.2305.4 4
4.3 odd 2 1536.2.d.b.769.1 4
8.3 odd 2 1536.2.d.b.769.4 4
8.5 even 2 inner 1536.2.d.e.769.2 4
12.11 even 2 4608.2.d.f.2305.4 4
16.3 odd 4 1536.2.a.h.1.1 yes 2
16.5 even 4 1536.2.a.k.1.2 yes 2
16.11 odd 4 1536.2.a.f.1.2 yes 2
16.13 even 4 1536.2.a.a.1.1 ✓ 2
24.5 odd 2 4608.2.d.h.2305.1 4
24.11 even 2 4608.2.d.f.2305.1 4
48.5 odd 4 4608.2.a.d.1.1 2
48.11 even 4 4608.2.a.b.1.1 2
48.29 odd 4 4608.2.a.o.1.2 2
48.35 even 4 4608.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.a.1.1 ✓ 2 16.13 even 4
1536.2.a.f.1.2 yes 2 16.11 odd 4
1536.2.a.h.1.1 yes 2 16.3 odd 4
1536.2.a.k.1.2 yes 2 16.5 even 4
1536.2.d.b.769.1 4 4.3 odd 2
1536.2.d.b.769.4 4 8.3 odd 2
1536.2.d.e.769.2 4 8.5 even 2 inner
1536.2.d.e.769.3 4 1.1 even 1 trivial
4608.2.a.b.1.1 2 48.11 even 4
4608.2.a.d.1.1 2 48.5 odd 4
4608.2.a.o.1.2 2 48.29 odd 4
4608.2.a.q.1.2 2 48.35 even 4
4608.2.d.f.2305.1 4 24.11 even 2
4608.2.d.f.2305.4 4 12.11 even 2
4608.2.d.h.2305.1 4 24.5 odd 2
4608.2.d.h.2305.4 4 3.2 odd 2