Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,6,Mod(145,288)] 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("288.145"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.1905401061\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-54 +2 \sqrt{73}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 145.4
Root \(-1.88600 + 2.10784i\) of defining polynomial
Character \(\chi\) \(=\) 288.145
Dual form 288.6.d.b.145.1

$q$-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)
 
\(f(q)\) \(=\) \(q+73.9600i q^{5} +112.704 q^{7} +575.407i q^{11} -117.735i q^{13} +223.408 q^{17} +1752.26i q^{19} +2361.15 q^{23} -2345.08 q^{25} -3865.52i q^{29} +1591.55 q^{31} +8335.59i q^{35} -4736.44i q^{37} -8153.88 q^{41} +4920.23i q^{43} -21062.0 q^{47} -4104.79 q^{49} -12709.0i q^{53} -42557.1 q^{55} +14111.1i q^{59} +42030.6i q^{61} +8707.71 q^{65} +54153.4i q^{67} +43879.9 q^{71} -31290.6 q^{73} +64850.7i q^{77} +50211.5 q^{79} +43707.0i q^{83} +16523.3i q^{85} -64418.7 q^{89} -13269.3i q^{91} -129597. q^{95} -62350.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 96 q^{7} - 200 q^{17} + 2336 q^{23} + 1556 q^{25} + 12928 q^{31} + 4568 q^{41} - 54720 q^{47} + 9828 q^{49} - 85472 q^{55} + 19520 q^{65} + 206688 q^{71} + 39976 q^{73} + 247872 q^{79} + 84632 q^{89}+ \cdots - 99576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 73.9600i 1.32304i 0.749929 + 0.661518i \(0.230088\pi\)
−0.749929 + 0.661518i \(0.769912\pi\)
\(6\) 0 0
\(7\) 112.704 0.869350 0.434675 − 0.900587i \(-0.356863\pi\)
0.434675 + 0.900587i \(0.356863\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 575.407i 1.43382i 0.697167 + 0.716909i \(0.254443\pi\)
−0.697167 + 0.716909i \(0.745557\pi\)
\(12\) 0 0
\(13\) − 117.735i − 0.193219i −0.995322 − 0.0966093i \(-0.969200\pi\)
0.995322 − 0.0966093i \(-0.0307997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 223.408 0.187489 0.0937447 − 0.995596i \(-0.470116\pi\)
0.0937447 + 0.995596i \(0.470116\pi\)
\(18\) 0 0
\(19\) 1752.26i 1.11356i 0.830660 + 0.556781i \(0.187964\pi\)
−0.830660 + 0.556781i \(0.812036\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2361.15 0.930689 0.465344 − 0.885130i \(-0.345930\pi\)
0.465344 + 0.885130i \(0.345930\pi\)
\(24\) 0 0
\(25\) −2345.08 −0.750426
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3865.52i − 0.853518i −0.904365 − 0.426759i \(-0.859655\pi\)
0.904365 − 0.426759i \(-0.140345\pi\)
\(30\) 0 0
\(31\) 1591.55 0.297452 0.148726 − 0.988878i \(-0.452483\pi\)
0.148726 + 0.988878i \(0.452483\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8335.59i 1.15018i
\(36\) 0 0
\(37\) − 4736.44i − 0.568785i −0.958708 − 0.284392i \(-0.908208\pi\)
0.958708 − 0.284392i \(-0.0917918\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8153.88 −0.757538 −0.378769 − 0.925491i \(-0.623653\pi\)
−0.378769 + 0.925491i \(0.623653\pi\)
\(42\) 0 0
\(43\) 4920.23i 0.405802i 0.979199 + 0.202901i \(0.0650370\pi\)
−0.979199 + 0.202901i \(0.934963\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −21062.0 −1.39077 −0.695385 − 0.718637i \(-0.744766\pi\)
−0.695385 + 0.718637i \(0.744766\pi\)
\(48\) 0 0
\(49\) −4104.79 −0.244231
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 12709.0i − 0.621470i −0.950497 − 0.310735i \(-0.899425\pi\)
0.950497 − 0.310735i \(-0.100575\pi\)
\(54\) 0 0
\(55\) −42557.1 −1.89699
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14111.1i 0.527752i 0.964557 + 0.263876i \(0.0850010\pi\)
−0.964557 + 0.263876i \(0.914999\pi\)
\(60\) 0 0
\(61\) 42030.6i 1.44624i 0.690721 + 0.723121i \(0.257293\pi\)
−0.690721 + 0.723121i \(0.742707\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8707.71 0.255635
\(66\) 0 0
\(67\) 54153.4i 1.47380i 0.676001 + 0.736901i \(0.263711\pi\)
−0.676001 + 0.736901i \(0.736289\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 43879.9 1.03305 0.516523 − 0.856273i \(-0.327226\pi\)
0.516523 + 0.856273i \(0.327226\pi\)
\(72\) 0 0
\(73\) −31290.6 −0.687238 −0.343619 − 0.939109i \(-0.611653\pi\)
−0.343619 + 0.939109i \(0.611653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 64850.7i 1.24649i
\(78\) 0 0
\(79\) 50211.5 0.905180 0.452590 − 0.891719i \(-0.350500\pi\)
0.452590 + 0.891719i \(0.350500\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 43707.0i 0.696395i 0.937421 + 0.348197i \(0.113206\pi\)
−0.937421 + 0.348197i \(0.886794\pi\)
\(84\) 0 0
\(85\) 16523.3i 0.248055i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −64418.7 −0.862059 −0.431030 − 0.902338i \(-0.641850\pi\)
−0.431030 + 0.902338i \(0.641850\pi\)
\(90\) 0 0
\(91\) − 13269.3i − 0.167974i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −129597. −1.47328
\(96\) 0 0
\(97\) −62350.9 −0.672843 −0.336421 − 0.941712i \(-0.609217\pi\)
−0.336421 + 0.941712i \(0.609217\pi\)
\(98\) 0 0
\(99\) 0 0
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 288.6.d.b.145.4 4
3.2 odd 2 32.6.b.a.17.3 4
4.3 odd 2 72.6.d.b.37.1 4
8.3 odd 2 72.6.d.b.37.2 4
8.5 even 2 inner 288.6.d.b.145.1 4
12.11 even 2 8.6.b.a.5.4 yes 4
15.2 even 4 800.6.f.a.49.6 8
15.8 even 4 800.6.f.a.49.3 8
15.14 odd 2 800.6.d.a.401.2 4
24.5 odd 2 32.6.b.a.17.2 4
24.11 even 2 8.6.b.a.5.3 ✓ 4
48.5 odd 4 256.6.a.n.1.3 4
48.11 even 4 256.6.a.k.1.2 4
48.29 odd 4 256.6.a.n.1.2 4
48.35 even 4 256.6.a.k.1.3 4
60.23 odd 4 200.6.f.a.149.7 8
60.47 odd 4 200.6.f.a.149.2 8
60.59 even 2 200.6.d.a.101.1 4
120.29 odd 2 800.6.d.a.401.3 4
120.53 even 4 800.6.f.a.49.5 8
120.59 even 2 200.6.d.a.101.2 4
120.77 even 4 800.6.f.a.49.4 8
120.83 odd 4 200.6.f.a.149.1 8
120.107 odd 4 200.6.f.a.149.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.6.b.a.5.3 ✓ 4 24.11 even 2
8.6.b.a.5.4 yes 4 12.11 even 2
32.6.b.a.17.2 4 24.5 odd 2
32.6.b.a.17.3 4 3.2 odd 2
72.6.d.b.37.1 4 4.3 odd 2
72.6.d.b.37.2 4 8.3 odd 2
200.6.d.a.101.1 4 60.59 even 2
200.6.d.a.101.2 4 120.59 even 2
200.6.f.a.149.1 8 120.83 odd 4
200.6.f.a.149.2 8 60.47 odd 4
200.6.f.a.149.7 8 60.23 odd 4
200.6.f.a.149.8 8 120.107 odd 4
256.6.a.k.1.2 4 48.11 even 4
256.6.a.k.1.3 4 48.35 even 4
256.6.a.n.1.2 4 48.29 odd 4
256.6.a.n.1.3 4 48.5 odd 4
288.6.d.b.145.1 4 8.5 even 2 inner
288.6.d.b.145.4 4 1.1 even 1 trivial
800.6.d.a.401.2 4 15.14 odd 2
800.6.d.a.401.3 4 120.29 odd 2
800.6.f.a.49.3 8 15.8 even 4
800.6.f.a.49.4 8 120.77 even 4
800.6.f.a.49.5 8 120.53 even 4
800.6.f.a.49.6 8 15.2 even 4