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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,5,Mod(127,288)] 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("288.127"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 288.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.7705493681\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 288.127
Dual form 288.5.g.c.127.2

$q$-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)
 
\(f(q)\) \(=\) \(q-42.7846 q^{5} -16.7846i q^{7} -67.2154i q^{11} +27.5692 q^{13} -451.261 q^{17} -550.046i q^{19} +874.831i q^{23} +1205.52 q^{25} +937.615 q^{29} +1547.92i q^{31} +718.123i q^{35} +497.200 q^{37} +585.600 q^{41} +1605.74i q^{43} +632.770i q^{47} +2119.28 q^{49} +3847.77 q^{53} +2875.78i q^{55} -4383.18i q^{59} -4738.31 q^{61} -1179.54 q^{65} +7100.81i q^{67} -440.060i q^{71} +4504.28 q^{73} -1128.18 q^{77} -2956.32i q^{79} +2664.01i q^{83} +19307.0 q^{85} +8123.85 q^{89} -462.739i q^{91} +23533.5i q^{95} -8179.72 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 88 q^{5} - 56 q^{13} + 24 q^{17} + 1164 q^{25} + 1672 q^{29} + 6312 q^{37} + 4504 q^{41} + 7812 q^{49} + 3336 q^{53} + 1000 q^{61} - 2224 q^{65} + 17352 q^{73} + 3136 q^{77} + 37488 q^{85} + 20856 q^{89}+ \cdots - 11768 q^{97}+O(q^{100}) \) 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
\(5\) −42.7846 −1.71138 −0.855692 − 0.517485i \(-0.826868\pi\)
−0.855692 + 0.517485i \(0.826868\pi\)
\(6\) 0 0
\(7\) − 16.7846i − 0.342543i −0.985224 − 0.171272i \(-0.945212\pi\)
0.985224 − 0.171272i \(-0.0547875\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 67.2154i − 0.555499i −0.960654 − 0.277750i \(-0.910411\pi\)
0.960654 − 0.277750i \(-0.0895885\pi\)
\(12\) 0 0
\(13\) 27.5692 0.163131 0.0815657 − 0.996668i \(-0.474008\pi\)
0.0815657 + 0.996668i \(0.474008\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −451.261 −1.56146 −0.780729 − 0.624870i \(-0.785152\pi\)
−0.780729 + 0.624870i \(0.785152\pi\)
\(18\) 0 0
\(19\) − 550.046i − 1.52367i −0.647769 − 0.761837i \(-0.724298\pi\)
0.647769 − 0.761837i \(-0.275702\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 874.831i 1.65374i 0.562390 + 0.826872i \(0.309882\pi\)
−0.562390 + 0.826872i \(0.690118\pi\)
\(24\) 0 0
\(25\) 1205.52 1.92884
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 937.615 1.11488 0.557441 − 0.830217i \(-0.311783\pi\)
0.557441 + 0.830217i \(0.311783\pi\)
\(30\) 0 0
\(31\) 1547.92i 1.61074i 0.592771 + 0.805371i \(0.298034\pi\)
−0.592771 + 0.805371i \(0.701966\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 718.123i 0.586223i
\(36\) 0 0
\(37\) 497.200 0.363185 0.181593 − 0.983374i \(-0.441875\pi\)
0.181593 + 0.983374i \(0.441875\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 585.600 0.348364 0.174182 − 0.984713i \(-0.444272\pi\)
0.174182 + 0.984713i \(0.444272\pi\)
\(42\) 0 0
\(43\) 1605.74i 0.868436i 0.900808 + 0.434218i \(0.142975\pi\)
−0.900808 + 0.434218i \(0.857025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 632.770i 0.286451i 0.989690 + 0.143225i \(0.0457474\pi\)
−0.989690 + 0.143225i \(0.954253\pi\)
\(48\) 0 0
\(49\) 2119.28 0.882664
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3847.77 1.36980 0.684900 − 0.728637i \(-0.259846\pi\)
0.684900 + 0.728637i \(0.259846\pi\)
\(54\) 0 0
\(55\) 2875.78i 0.950672i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4383.18i − 1.25917i −0.776930 − 0.629587i \(-0.783224\pi\)
0.776930 − 0.629587i \(-0.216776\pi\)
\(60\) 0 0
\(61\) −4738.31 −1.27340 −0.636698 − 0.771113i \(-0.719700\pi\)
−0.636698 + 0.771113i \(0.719700\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1179.54 −0.279181
\(66\) 0 0
\(67\) 7100.81i 1.58183i 0.611929 + 0.790913i \(0.290394\pi\)
−0.611929 + 0.790913i \(0.709606\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 440.060i − 0.0872962i −0.999047 − 0.0436481i \(-0.986102\pi\)
0.999047 − 0.0436481i \(-0.0138980\pi\)
\(72\) 0 0
\(73\) 4504.28 0.845239 0.422619 − 0.906307i \(-0.361111\pi\)
0.422619 + 0.906307i \(0.361111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1128.18 −0.190282
\(78\) 0 0
\(79\) − 2956.32i − 0.473694i −0.971547 − 0.236847i \(-0.923886\pi\)
0.971547 − 0.236847i \(-0.0761139\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2664.01i 0.386705i 0.981129 + 0.193353i \(0.0619362\pi\)
−0.981129 + 0.193353i \(0.938064\pi\)
\(84\) 0 0
\(85\) 19307.0 2.67226
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8123.85 1.02561 0.512804 − 0.858506i \(-0.328607\pi\)
0.512804 + 0.858506i \(0.328607\pi\)
\(90\) 0 0
\(91\) − 462.739i − 0.0558796i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 23533.5i 2.60759i
\(96\) 0 0
\(97\) −8179.72 −0.869351 −0.434675 − 0.900587i \(-0.643137\pi\)
−0.434675 + 0.900587i \(0.643137\pi\)
\(98\) 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 288.5.g.c.127.1 4
3.2 odd 2 96.5.g.b.31.2 ✓ 4
4.3 odd 2 inner 288.5.g.c.127.2 4
8.3 odd 2 576.5.g.o.127.4 4
8.5 even 2 576.5.g.o.127.3 4
12.11 even 2 96.5.g.b.31.4 yes 4
24.5 odd 2 192.5.g.c.127.3 4
24.11 even 2 192.5.g.c.127.1 4
48.5 odd 4 768.5.b.a.127.2 4
48.11 even 4 768.5.b.f.127.4 4
48.29 odd 4 768.5.b.f.127.3 4
48.35 even 4 768.5.b.a.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.5.g.b.31.2 ✓ 4 3.2 odd 2
96.5.g.b.31.4 yes 4 12.11 even 2
192.5.g.c.127.1 4 24.11 even 2
192.5.g.c.127.3 4 24.5 odd 2
288.5.g.c.127.1 4 1.1 even 1 trivial
288.5.g.c.127.2 4 4.3 odd 2 inner
576.5.g.o.127.3 4 8.5 even 2
576.5.g.o.127.4 4 8.3 odd 2
768.5.b.a.127.1 4 48.35 even 4
768.5.b.a.127.2 4 48.5 odd 4
768.5.b.f.127.3 4 48.29 odd 4
768.5.b.f.127.4 4 48.11 even 4