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,-2,0,0,0,0,0,2,0,12] 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: \(0\)
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 1.2
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.41421 q^{5} +4.24264 q^{7} +1.00000 q^{9} +6.00000 q^{11} +5.65685 q^{13} -1.41421 q^{15} -6.00000 q^{17} +4.00000 q^{19} -4.24264 q^{21} -2.82843 q^{23} -3.00000 q^{25} -1.00000 q^{27} +1.41421 q^{29} +1.41421 q^{31} -6.00000 q^{33} +6.00000 q^{35} -8.48528 q^{37} -5.65685 q^{39} -2.00000 q^{41} +1.41421 q^{45} -2.82843 q^{47} +11.0000 q^{49} +6.00000 q^{51} -9.89949 q^{53} +8.48528 q^{55} -4.00000 q^{57} +4.00000 q^{59} -8.48528 q^{61} +4.24264 q^{63} +8.00000 q^{65} +8.00000 q^{67} +2.82843 q^{69} +2.82843 q^{71} +8.00000 q^{73} +3.00000 q^{75} +25.4558 q^{77} -12.7279 q^{79} +1.00000 q^{81} +2.00000 q^{83} -8.48528 q^{85} -1.41421 q^{87} +2.00000 q^{89} +24.0000 q^{91} -1.41421 q^{93} +5.65685 q^{95} +2.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9} + 12 q^{11} - 12 q^{17} + 8 q^{19} - 6 q^{25} - 2 q^{27} - 12 q^{33} + 12 q^{35} - 4 q^{41} + 22 q^{49} + 12 q^{51} - 8 q^{57} + 8 q^{59} + 16 q^{65} + 16 q^{67} + 16 q^{73} + 6 q^{75}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(5\) 1.41421 0.632456 0.316228 − 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 4.24264 1.60357 0.801784 − 0.597614i \(-0.203885\pi\)
0.801784 + 0.597614i \(0.203885\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 − 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 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
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −4.24264 −0.925820
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 − 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 − 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 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
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 − 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 0 0
\(39\) −5.65685 −0.905822
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.41421 0.210819
\(46\) 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
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −9.89949 −1.35980 −0.679900 − 0.733305i \(-0.737977\pi\)
−0.679900 + 0.733305i \(0.737977\pi\)
\(54\) 0 0
\(55\) 8.48528 1.14416
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −8.48528 −1.08643 −0.543214 − 0.839594i \(-0.682793\pi\)
−0.543214 + 0.839594i \(0.682793\pi\)
\(62\) 0 0
\(63\) 4.24264 0.534522
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 − 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 25.4558 2.90096
\(78\) 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
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) −8.48528 −0.920358
\(86\) 0 0
\(87\) −1.41421 −0.151620
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) 0 0
\(93\) −1.41421 −0.146647
\(94\) 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
\(99\) 6.00000 0.603023
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.d.1.2 yes 2
3.2 odd 2 4608.2.a.g.1.1 2
4.3 odd 2 1536.2.a.i.1.2 yes 2
8.3 odd 2 inner 1536.2.a.d.1.1 ✓ 2
8.5 even 2 1536.2.a.i.1.1 yes 2
12.11 even 2 4608.2.a.l.1.1 2
16.3 odd 4 1536.2.d.c.769.3 4
16.5 even 4 1536.2.d.c.769.4 4
16.11 odd 4 1536.2.d.c.769.2 4
16.13 even 4 1536.2.d.c.769.1 4
24.5 odd 2 4608.2.a.l.1.2 2
24.11 even 2 4608.2.a.g.1.2 2
48.5 odd 4 4608.2.d.n.2305.1 4
48.11 even 4 4608.2.d.n.2305.2 4
48.29 odd 4 4608.2.d.n.2305.3 4
48.35 even 4 4608.2.d.n.2305.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.d.1.1 ✓ 2 8.3 odd 2 inner
1536.2.a.d.1.2 yes 2 1.1 even 1 trivial
1536.2.a.i.1.1 yes 2 8.5 even 2
1536.2.a.i.1.2 yes 2 4.3 odd 2
1536.2.d.c.769.1 4 16.13 even 4
1536.2.d.c.769.2 4 16.11 odd 4
1536.2.d.c.769.3 4 16.3 odd 4
1536.2.d.c.769.4 4 16.5 even 4
4608.2.a.g.1.1 2 3.2 odd 2
4608.2.a.g.1.2 2 24.11 even 2
4608.2.a.l.1.1 2 12.11 even 2
4608.2.a.l.1.2 2 24.5 odd 2
4608.2.d.n.2305.1 4 48.5 odd 4
4608.2.d.n.2305.2 4 48.11 even 4
4608.2.d.n.2305.3 4 48.29 odd 4
4608.2.d.n.2305.4 4 48.35 even 4