Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,16,Mod(1,144)] 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("144.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-69660,0,-2491504] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(205.478647344\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{8017}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2004 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-44.2689\) of defining polynomial
Character \(\chi\) \(=\) 144.1

$q$-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)
 
\(f(q)\) \(=\) \(q+223039. q^{5} -3.56657e6 q^{7} -8.12384e7 q^{11} +3.44718e8 q^{13} -2.43697e9 q^{17} -3.23300e9 q^{19} +1.26437e10 q^{23} +1.92286e10 q^{25} -9.41624e10 q^{29} -7.80665e10 q^{31} -7.95483e11 q^{35} -4.06319e11 q^{37} +2.98195e11 q^{41} +1.91876e12 q^{43} +5.51593e11 q^{47} +7.97285e12 q^{49} +1.20870e13 q^{53} -1.81193e13 q^{55} +6.58404e12 q^{59} -4.21447e12 q^{61} +7.68854e13 q^{65} +5.45476e13 q^{67} -1.17707e14 q^{71} +1.31548e14 q^{73} +2.89742e14 q^{77} -2.57200e14 q^{79} +3.45415e14 q^{83} -5.43537e14 q^{85} -3.21904e14 q^{89} -1.22946e15 q^{91} -7.21083e14 q^{95} -1.01652e15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 69660 q^{5} - 2491504 q^{7} - 14975928 q^{11} + 11757580 q^{13} - 4256077284 q^{17} - 9241689400 q^{19} + 24254822736 q^{23} + 74383511150 q^{25} - 147290258412 q^{29} + 76677530432 q^{31} - 1110152602080 q^{35}+ \cdots - 18\!\cdots\!52 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 223039. 1.27675 0.638374 − 0.769727i \(-0.279607\pi\)
0.638374 + 0.769727i \(0.279607\pi\)
\(6\) 0 0
\(7\) −3.56657e6 −1.63687 −0.818437 − 0.574596i \(-0.805159\pi\)
−0.818437 + 0.574596i \(0.805159\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.12384e7 −1.25694 −0.628472 − 0.777832i \(-0.716319\pi\)
−0.628472 + 0.777832i \(0.716319\pi\)
\(12\) 0 0
\(13\) 3.44718e8 1.52366 0.761832 − 0.647775i \(-0.224300\pi\)
0.761832 + 0.647775i \(0.224300\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.43697e9 −1.44040 −0.720199 − 0.693768i \(-0.755949\pi\)
−0.720199 + 0.693768i \(0.755949\pi\)
\(18\) 0 0
\(19\) −3.23300e9 −0.829761 −0.414881 − 0.909876i \(-0.636177\pi\)
−0.414881 + 0.909876i \(0.636177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.26437e10 0.774309 0.387154 − 0.922015i \(-0.373458\pi\)
0.387154 + 0.922015i \(0.373458\pi\)
\(24\) 0 0
\(25\) 1.92286e10 0.630084
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.41624e10 −1.01366 −0.506830 − 0.862046i \(-0.669183\pi\)
−0.506830 + 0.862046i \(0.669183\pi\)
\(30\) 0 0
\(31\) −7.80665e10 −0.509626 −0.254813 − 0.966990i \(-0.582014\pi\)
−0.254813 + 0.966990i \(0.582014\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.95483e11 −2.08988
\(36\) 0 0
\(37\) −4.06319e11 −0.703646 −0.351823 − 0.936067i \(-0.614438\pi\)
−0.351823 + 0.936067i \(0.614438\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.98195e11 0.239123 0.119562 − 0.992827i \(-0.461851\pi\)
0.119562 + 0.992827i \(0.461851\pi\)
\(42\) 0 0
\(43\) 1.91876e12 1.07648 0.538241 − 0.842791i \(-0.319089\pi\)
0.538241 + 0.842791i \(0.319089\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.51593e11 0.158813 0.0794063 − 0.996842i \(-0.474698\pi\)
0.0794063 + 0.996842i \(0.474698\pi\)
\(48\) 0 0
\(49\) 7.97285e12 1.67936
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.20870e13 1.41335 0.706676 − 0.707538i \(-0.250194\pi\)
0.706676 + 0.707538i \(0.250194\pi\)
\(54\) 0 0
\(55\) −1.81193e13 −1.60480
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.58404e12 0.344431 0.172216 − 0.985059i \(-0.444907\pi\)
0.172216 + 0.985059i \(0.444907\pi\)
\(60\) 0 0
\(61\) −4.21447e12 −0.171700 −0.0858498 − 0.996308i \(-0.527361\pi\)
−0.0858498 + 0.996308i \(0.527361\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.68854e13 1.94533
\(66\) 0 0
\(67\) 5.45476e13 1.09955 0.549774 − 0.835313i \(-0.314714\pi\)
0.549774 + 0.835313i \(0.314714\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.17707e14 −1.53591 −0.767956 − 0.640502i \(-0.778726\pi\)
−0.767956 + 0.640502i \(0.778726\pi\)
\(72\) 0 0
\(73\) 1.31548e14 1.39368 0.696840 − 0.717226i \(-0.254589\pi\)
0.696840 + 0.717226i \(0.254589\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.89742e14 2.05746
\(78\) 0 0
\(79\) −2.57200e14 −1.50684 −0.753422 − 0.657537i \(-0.771598\pi\)
−0.753422 + 0.657537i \(0.771598\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.45415e14 1.39719 0.698595 − 0.715518i \(-0.253809\pi\)
0.698595 + 0.715518i \(0.253809\pi\)
\(84\) 0 0
\(85\) −5.43537e14 −1.83902
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.21904e14 −0.771438 −0.385719 − 0.922616i \(-0.626047\pi\)
−0.385719 + 0.922616i \(0.626047\pi\)
\(90\) 0 0
\(91\) −1.22946e15 −2.49405
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.21083e14 −1.05940
\(96\) 0 0
\(97\) −1.01652e15 −1.27740 −0.638699 − 0.769457i \(-0.720527\pi\)
−0.638699 + 0.769457i \(0.720527\pi\)
\(98\) 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 144.16.a.r.1.2 2
3.2 odd 2 48.16.a.i.1.1 2
4.3 odd 2 36.16.a.c.1.2 2
12.11 even 2 12.16.a.b.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.16.a.b.1.1 ✓ 2 12.11 even 2
36.16.a.c.1.2 2 4.3 odd 2
48.16.a.i.1.1 2 3.2 odd 2
144.16.a.r.1.2 2 1.1 even 1 trivial