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,-4,0,0,0,0,0,0,0,-24] 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 769.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.c.769.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.41421i q^{5} -4.24264 q^{7} -1.00000 q^{9} -6.00000i q^{11} +5.65685i q^{13} -1.41421 q^{15} -6.00000 q^{17} +4.00000i q^{19} +4.24264i q^{21} +2.82843 q^{23} +3.00000 q^{25} +1.00000i q^{27} +1.41421i q^{29} +1.41421 q^{31} -6.00000 q^{33} +6.00000i q^{35} +8.48528i q^{37} +5.65685 q^{39} +2.00000 q^{41} +1.41421i q^{45} -2.82843 q^{47} +11.0000 q^{49} +6.00000i q^{51} +9.89949i q^{53} -8.48528 q^{55} +4.00000 q^{57} -4.00000i q^{59} -8.48528i q^{61} +4.24264 q^{63} +8.00000 q^{65} +8.00000i q^{67} -2.82843i q^{69} -2.82843 q^{71} -8.00000 q^{73} -3.00000i q^{75} +25.4558i q^{77} -12.7279 q^{79} +1.00000 q^{81} +2.00000i q^{83} +8.48528i q^{85} +1.41421 q^{87} -2.00000 q^{89} -24.0000i q^{91} -1.41421i q^{93} +5.65685 q^{95} +2.00000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 24 q^{17} + 12 q^{25} - 24 q^{33} + 8 q^{41} + 44 q^{49} + 16 q^{57} + 32 q^{65} - 32 q^{73} + 4 q^{81} - 8 q^{89} + 8 q^{97}+O(q^{100}) \) 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
\(5\) − 1.41421i − 0.632456i −0.948683 − 0.316228i \(-0.897584\pi\)
0.948683 − 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) −4.24264 −1.60357 −0.801784 − 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 6.00000i − 1.80907i −0.426401 − 0.904534i \(-0.640219\pi\)
0.426401 − 0.904534i \(-0.359781\pi\)
\(12\) 0 0
\(13\) 5.65685i 1.56893i 0.620174 + 0.784465i \(0.287062\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 4.24264i 0.925820i
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 − 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 1.41421i 0.262613i 0.991342 + 0.131306i \(0.0419172\pi\)
−0.991342 + 0.131306i \(0.958083\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 − 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 6.00000i 1.01419i
\(36\) 0 0
\(37\) 8.48528i 1.39497i 0.716599 + 0.697486i \(0.245698\pi\)
−0.716599 + 0.697486i \(0.754302\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 1.41421i 0.210819i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 − 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) 9.89949i 1.35980i 0.733305 + 0.679900i \(0.237977\pi\)
−0.733305 + 0.679900i \(0.762023\pi\)
\(54\) 0 0
\(55\) −8.48528 −1.14416
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 − 0.260378i \(-0.916153\pi\)
0.965507 − 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) − 8.48528i − 1.08643i −0.839594 − 0.543214i \(-0.817207\pi\)
0.839594 − 0.543214i \(-0.182793\pi\)
\(62\) 0 0
\(63\) 4.24264 0.534522
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) − 2.82843i − 0.340503i
\(70\) 0 0
\(71\) −2.82843 −0.335673 −0.167836 − 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) − 3.00000i − 0.346410i
\(76\) 0 0
\(77\) 25.4558i 2.90096i
\(78\) 0 0
\(79\) −12.7279 −1.43200 −0.716002 − 0.698099i \(-0.754030\pi\)
−0.716002 + 0.698099i \(0.754030\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 8.48528i 0.920358i
\(86\) 0 0
\(87\) 1.41421 0.151620
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) − 24.0000i − 2.51588i
\(92\) 0 0
\(93\) − 1.41421i − 0.146647i
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
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.c.769.1 4
3.2 odd 2 4608.2.d.n.2305.3 4
4.3 odd 2 inner 1536.2.d.c.769.3 4
8.3 odd 2 inner 1536.2.d.c.769.2 4
8.5 even 2 inner 1536.2.d.c.769.4 4
12.11 even 2 4608.2.d.n.2305.4 4
16.3 odd 4 1536.2.a.d.1.1 ✓ 2
16.5 even 4 1536.2.a.d.1.2 yes 2
16.11 odd 4 1536.2.a.i.1.2 yes 2
16.13 even 4 1536.2.a.i.1.1 yes 2
24.5 odd 2 4608.2.d.n.2305.1 4
24.11 even 2 4608.2.d.n.2305.2 4
48.5 odd 4 4608.2.a.g.1.1 2
48.11 even 4 4608.2.a.l.1.1 2
48.29 odd 4 4608.2.a.l.1.2 2
48.35 even 4 4608.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.d.1.1 ✓ 2 16.3 odd 4
1536.2.a.d.1.2 yes 2 16.5 even 4
1536.2.a.i.1.1 yes 2 16.13 even 4
1536.2.a.i.1.2 yes 2 16.11 odd 4
1536.2.d.c.769.1 4 1.1 even 1 trivial
1536.2.d.c.769.2 4 8.3 odd 2 inner
1536.2.d.c.769.3 4 4.3 odd 2 inner
1536.2.d.c.769.4 4 8.5 even 2 inner
4608.2.a.g.1.1 2 48.5 odd 4
4608.2.a.g.1.2 2 48.35 even 4
4608.2.a.l.1.1 2 48.11 even 4
4608.2.a.l.1.2 2 48.29 odd 4
4608.2.d.n.2305.1 4 24.5 odd 2
4608.2.d.n.2305.2 4 24.11 even 2
4608.2.d.n.2305.3 4 3.2 odd 2
4608.2.d.n.2305.4 4 12.11 even 2