Show commands: Magma / Pari/GP / SageMath

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,-32] 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: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.4
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 288.127
Dual form 288.5.g.d.127.3

$q$-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)
 
\(f(q)\) \(=\) \(q+18.8328 q^{5} +33.6656i q^{7} -43.3313i q^{11} +140.663 q^{13} +90.3344 q^{17} +131.331i q^{19} +771.988i q^{23} -270.325 q^{25} +523.173 q^{29} -754.322i q^{31} +634.019i q^{35} -743.313 q^{37} +2409.65 q^{41} +347.356i q^{43} +2126.69i q^{47} +1267.63 q^{49} +4665.13 q^{53} -816.050i q^{55} +6593.26i q^{59} +749.988 q^{61} +2649.07 q^{65} +7833.24i q^{67} +3056.05i q^{71} +320.675 q^{73} +1458.77 q^{77} -10500.9i q^{79} +7227.28i q^{83} +1701.25 q^{85} +9671.14 q^{89} +4735.49i q^{91} +2473.34i q^{95} -10097.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{5} - 296 q^{13} + 576 q^{17} + 636 q^{25} + 3488 q^{29} + 1320 q^{37} + 6848 q^{41} - 3516 q^{49} + 11040 q^{53} + 424 q^{61} + 25408 q^{65} + 3000 q^{73} + 28160 q^{77} - 10368 q^{85} - 384 q^{89}+ \cdots + 11128 q^{97}+O(q^{100}) \) 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
\(5\) 18.8328 0.753313 0.376656 − 0.926353i \(-0.377074\pi\)
0.376656 + 0.926353i \(0.377074\pi\)
\(6\) 0 0
\(7\) 33.6656i 0.687054i 0.939143 + 0.343527i \(0.111622\pi\)
−0.939143 + 0.343527i \(0.888378\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 43.3313i − 0.358110i −0.983839 − 0.179055i \(-0.942696\pi\)
0.983839 − 0.179055i \(-0.0573039\pi\)
\(12\) 0 0
\(13\) 140.663 0.832323 0.416161 − 0.909291i \(-0.363375\pi\)
0.416161 + 0.909291i \(0.363375\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 90.3344 0.312576 0.156288 − 0.987712i \(-0.450047\pi\)
0.156288 + 0.987712i \(0.450047\pi\)
\(18\) 0 0
\(19\) 131.331i 0.363799i 0.983317 + 0.181899i \(0.0582245\pi\)
−0.983317 + 0.181899i \(0.941776\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 771.988i 1.45933i 0.683803 + 0.729667i \(0.260325\pi\)
−0.683803 + 0.729667i \(0.739675\pi\)
\(24\) 0 0
\(25\) −270.325 −0.432520
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 523.173 0.622085 0.311042 − 0.950396i \(-0.399322\pi\)
0.311042 + 0.950396i \(0.399322\pi\)
\(30\) 0 0
\(31\) − 754.322i − 0.784934i −0.919766 − 0.392467i \(-0.871622\pi\)
0.919766 − 0.392467i \(-0.128378\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 634.019i 0.517566i
\(36\) 0 0
\(37\) −743.313 −0.542960 −0.271480 − 0.962444i \(-0.587513\pi\)
−0.271480 + 0.962444i \(0.587513\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2409.65 1.43346 0.716732 − 0.697349i \(-0.245637\pi\)
0.716732 + 0.697349i \(0.245637\pi\)
\(42\) 0 0
\(43\) 347.356i 0.187862i 0.995579 + 0.0939308i \(0.0299432\pi\)
−0.995579 + 0.0939308i \(0.970057\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2126.69i 0.962738i 0.876518 + 0.481369i \(0.159860\pi\)
−0.876518 + 0.481369i \(0.840140\pi\)
\(48\) 0 0
\(49\) 1267.63 0.527957
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4665.13 1.66078 0.830390 − 0.557183i \(-0.188118\pi\)
0.830390 + 0.557183i \(0.188118\pi\)
\(54\) 0 0
\(55\) − 816.050i − 0.269768i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6593.26i 1.89407i 0.321128 + 0.947036i \(0.395938\pi\)
−0.321128 + 0.947036i \(0.604062\pi\)
\(60\) 0 0
\(61\) 749.988 0.201555 0.100778 − 0.994909i \(-0.467867\pi\)
0.100778 + 0.994909i \(0.467867\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2649.07 0.626999
\(66\) 0 0
\(67\) 7833.24i 1.74499i 0.488627 + 0.872493i \(0.337498\pi\)
−0.488627 + 0.872493i \(0.662502\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3056.05i 0.606239i 0.952953 + 0.303119i \(0.0980281\pi\)
−0.952953 + 0.303119i \(0.901972\pi\)
\(72\) 0 0
\(73\) 320.675 0.0601754 0.0300877 − 0.999547i \(-0.490421\pi\)
0.0300877 + 0.999547i \(0.490421\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1458.77 0.246041
\(78\) 0 0
\(79\) − 10500.9i − 1.68257i −0.540595 − 0.841283i \(-0.681801\pi\)
0.540595 − 0.841283i \(-0.318199\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7227.28i 1.04910i 0.851378 + 0.524552i \(0.175767\pi\)
−0.851378 + 0.524552i \(0.824233\pi\)
\(84\) 0 0
\(85\) 1701.25 0.235467
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9671.14 1.22095 0.610475 − 0.792036i \(-0.290979\pi\)
0.610475 + 0.792036i \(0.290979\pi\)
\(90\) 0 0
\(91\) 4735.49i 0.571850i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2473.34i 0.274054i
\(96\) 0 0
\(97\) −10097.8 −1.07320 −0.536601 − 0.843836i \(-0.680292\pi\)
−0.536601 + 0.843836i \(0.680292\pi\)
\(98\) 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.d.127.4 yes 4
3.2 odd 2 288.5.g.e.127.2 yes 4
4.3 odd 2 inner 288.5.g.d.127.3 ✓ 4
8.3 odd 2 576.5.g.n.127.1 4
8.5 even 2 576.5.g.n.127.2 4
12.11 even 2 288.5.g.e.127.1 yes 4
24.5 odd 2 576.5.g.k.127.4 4
24.11 even 2 576.5.g.k.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.5.g.d.127.3 ✓ 4 4.3 odd 2 inner
288.5.g.d.127.4 yes 4 1.1 even 1 trivial
288.5.g.e.127.1 yes 4 12.11 even 2
288.5.g.e.127.2 yes 4 3.2 odd 2
576.5.g.k.127.3 4 24.11 even 2
576.5.g.k.127.4 4 24.5 odd 2
576.5.g.n.127.1 4 8.3 odd 2
576.5.g.n.127.2 4 8.5 even 2