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 769.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.e.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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +0.585786i q^{5} -1.41421 q^{7} -1.00000 q^{9} +0.828427i q^{11} -4.82843i q^{13} +0.585786 q^{15} -0.828427 q^{17} +2.82843i q^{19} +1.41421i q^{21} -6.82843 q^{23} +4.65685 q^{25} +1.00000i q^{27} -4.58579i q^{29} -7.07107 q^{31} +0.828427 q^{33} -0.828427i q^{35} +0.343146i q^{37} -4.82843 q^{39} -6.48528 q^{41} +1.17157i q^{43} -0.585786i q^{45} -4.48528 q^{47} -5.00000 q^{49} +0.828427i q^{51} +10.2426i q^{53} -0.485281 q^{55} +2.82843 q^{57} -9.65685i q^{59} -11.6569i q^{61} +1.41421 q^{63} +2.82843 q^{65} -5.65685i q^{67} +6.82843i q^{69} -8.48528 q^{71} -11.3137 q^{73} -4.65685i q^{75} -1.17157i q^{77} -14.5858 q^{79} +1.00000 q^{81} +3.17157i q^{83} -0.485281i q^{85} -4.58579 q^{87} +17.3137 q^{89} +6.82843i q^{91} +7.07107i q^{93} -1.65685 q^{95} +3.65685 q^{97} -0.828427i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 8 q^{15} + 8 q^{17} - 16 q^{23} - 4 q^{25} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 16 q^{47} - 20 q^{49} + 32 q^{55} - 64 q^{79} + 4 q^{81} - 24 q^{87} + 24 q^{89} + 16 q^{95} - 8 q^{97}+O(q^{100}) \) 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
\(5\) 0.585786i 0.261972i 0.991384 + 0.130986i \(0.0418142\pi\)
−0.991384 + 0.130986i \(0.958186\pi\)
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 − 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.828427i 0.249780i 0.992171 + 0.124890i \(0.0398578\pi\)
−0.992171 + 0.124890i \(0.960142\pi\)
\(12\) 0 0
\(13\) − 4.82843i − 1.33916i −0.742738 − 0.669582i \(-0.766473\pi\)
0.742738 − 0.669582i \(-0.233527\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 − 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 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
\(23\) −6.82843 −1.42383 −0.711913 − 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) 4.65685 0.931371
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 4.58579i − 0.851559i −0.904827 − 0.425780i \(-0.860000\pi\)
0.904827 − 0.425780i \(-0.140000\pi\)
\(30\) 0 0
\(31\) −7.07107 −1.27000 −0.635001 − 0.772512i \(-0.719000\pi\)
−0.635001 + 0.772512i \(0.719000\pi\)
\(32\) 0 0
\(33\) 0.828427 0.144211
\(34\) 0 0
\(35\) − 0.828427i − 0.140030i
\(36\) 0 0
\(37\) 0.343146i 0.0564128i 0.999602 + 0.0282064i \(0.00897957\pi\)
−0.999602 + 0.0282064i \(0.991020\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) −6.48528 −1.01283 −0.506415 − 0.862290i \(-0.669030\pi\)
−0.506415 + 0.862290i \(0.669030\pi\)
\(42\) 0 0
\(43\) 1.17157i 0.178663i 0.996002 + 0.0893316i \(0.0284731\pi\)
−0.996002 + 0.0893316i \(0.971527\pi\)
\(44\) 0 0
\(45\) − 0.585786i − 0.0873239i
\(46\) 0 0
\(47\) −4.48528 −0.654246 −0.327123 − 0.944982i \(-0.606079\pi\)
−0.327123 + 0.944982i \(0.606079\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0.828427i 0.116003i
\(52\) 0 0
\(53\) 10.2426i 1.40693i 0.710727 + 0.703467i \(0.248366\pi\)
−0.710727 + 0.703467i \(0.751634\pi\)
\(54\) 0 0
\(55\) −0.485281 −0.0654353
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 0 0
\(59\) − 9.65685i − 1.25722i −0.777723 − 0.628608i \(-0.783625\pi\)
0.777723 − 0.628608i \(-0.216375\pi\)
\(60\) 0 0
\(61\) − 11.6569i − 1.49251i −0.665662 − 0.746254i \(-0.731851\pi\)
0.665662 − 0.746254i \(-0.268149\pi\)
\(62\) 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
\(69\) 6.82843i 0.822046i
\(70\) 0 0
\(71\) −8.48528 −1.00702 −0.503509 − 0.863990i \(-0.667958\pi\)
−0.503509 + 0.863990i \(0.667958\pi\)
\(72\) 0 0
\(73\) −11.3137 −1.32417 −0.662085 − 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) 0 0
\(75\) − 4.65685i − 0.537727i
\(76\) 0 0
\(77\) − 1.17157i − 0.133513i
\(78\) 0 0
\(79\) −14.5858 −1.64103 −0.820515 − 0.571626i \(-0.806313\pi\)
−0.820515 + 0.571626i \(0.806313\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.17157i 0.348125i 0.984735 + 0.174063i \(0.0556895\pi\)
−0.984735 + 0.174063i \(0.944310\pi\)
\(84\) 0 0
\(85\) − 0.485281i − 0.0526362i
\(86\) 0 0
\(87\) −4.58579 −0.491648
\(88\) 0 0
\(89\) 17.3137 1.83525 0.917625 − 0.397448i \(-0.130104\pi\)
0.917625 + 0.397448i \(0.130104\pi\)
\(90\) 0 0
\(91\) 6.82843i 0.715814i
\(92\) 0 0
\(93\) 7.07107i 0.733236i
\(94\) 0 0
\(95\) −1.65685 −0.169990
\(96\) 0 0
\(97\) 3.65685 0.371297 0.185649 − 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) 0 0
\(99\) − 0.828427i − 0.0832601i
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.769.1 4
3.2 odd 2 4608.2.d.h.2305.2 4
4.3 odd 2 1536.2.d.b.769.3 4
8.3 odd 2 1536.2.d.b.769.2 4
8.5 even 2 inner 1536.2.d.e.769.4 4
12.11 even 2 4608.2.d.f.2305.2 4
16.3 odd 4 1536.2.a.f.1.1 yes 2
16.5 even 4 1536.2.a.a.1.2 ✓ 2
16.11 odd 4 1536.2.a.h.1.2 yes 2
16.13 even 4 1536.2.a.k.1.1 yes 2
24.5 odd 2 4608.2.d.h.2305.3 4
24.11 even 2 4608.2.d.f.2305.3 4
48.5 odd 4 4608.2.a.o.1.1 2
48.11 even 4 4608.2.a.q.1.1 2
48.29 odd 4 4608.2.a.d.1.2 2
48.35 even 4 4608.2.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.a.1.2 ✓ 2 16.5 even 4
1536.2.a.f.1.1 yes 2 16.3 odd 4
1536.2.a.h.1.2 yes 2 16.11 odd 4
1536.2.a.k.1.1 yes 2 16.13 even 4
1536.2.d.b.769.2 4 8.3 odd 2
1536.2.d.b.769.3 4 4.3 odd 2
1536.2.d.e.769.1 4 1.1 even 1 trivial
1536.2.d.e.769.4 4 8.5 even 2 inner
4608.2.a.b.1.2 2 48.35 even 4
4608.2.a.d.1.2 2 48.29 odd 4
4608.2.a.o.1.1 2 48.5 odd 4
4608.2.a.q.1.1 2 48.11 even 4
4608.2.d.f.2305.2 4 12.11 even 2
4608.2.d.f.2305.3 4 24.11 even 2
4608.2.d.h.2305.2 4 3.2 odd 2
4608.2.d.h.2305.3 4 24.5 odd 2