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 = [2,0,0,0,0,0,5182520] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{370}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 370 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-19.2354\) 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)
 
\(f(q)\) \(=\) \(q-160654. q^{5} +2.59126e6 q^{7} +3.77814e7 q^{11} -7.79120e7 q^{13} -7.52193e7 q^{17} +9.76566e8 q^{19} -2.08375e10 q^{23} -4.70789e9 q^{25} +1.13824e11 q^{29} -1.62228e11 q^{31} -4.16296e11 q^{35} -6.06605e11 q^{37} +2.27020e12 q^{41} -1.76483e12 q^{43} +3.83036e12 q^{47} +1.96707e12 q^{49} -5.84453e12 q^{53} -6.06973e12 q^{55} +2.75390e13 q^{59} +3.19783e13 q^{61} +1.25169e13 q^{65} -1.26768e11 q^{67} -1.45175e13 q^{71} +1.01137e14 q^{73} +9.79014e13 q^{77} -1.63798e14 q^{79} -3.09698e14 q^{83} +1.20843e13 q^{85} -6.77864e13 q^{89} -2.01890e14 q^{91} -1.56889e14 q^{95} -1.02585e15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5182520 q^{7} - 155823980 q^{13} + 1953132368 q^{19} - 9415787290 q^{25} - 324455854216 q^{31} - 1213210694660 q^{37} - 3529665379120 q^{43} + 3934133755314 q^{49} - 12139451596800 q^{55} + 63956550008524 q^{61}+ \cdots - 20\!\cdots\!20 q^{97}+O(q^{100}) \) 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
\(5\) −160654. −0.919637 −0.459818 − 0.888013i \(-0.652085\pi\)
−0.459818 + 0.888013i \(0.652085\pi\)
\(6\) 0 0
\(7\) 2.59126e6 1.18926 0.594628 − 0.804001i \(-0.297299\pi\)
0.594628 + 0.804001i \(0.297299\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.77814e7 0.584565 0.292282 − 0.956332i \(-0.405585\pi\)
0.292282 + 0.956332i \(0.405585\pi\)
\(12\) 0 0
\(13\) −7.79120e7 −0.344373 −0.172187 − 0.985064i \(-0.555083\pi\)
−0.172187 + 0.985064i \(0.555083\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.52193e7 −0.0444593 −0.0222296 − 0.999753i \(-0.507076\pi\)
−0.0222296 + 0.999753i \(0.507076\pi\)
\(18\) 0 0
\(19\) 9.76566e8 0.250640 0.125320 − 0.992116i \(-0.460004\pi\)
0.125320 + 0.992116i \(0.460004\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.08375e10 −1.27610 −0.638052 − 0.769993i \(-0.720259\pi\)
−0.638052 + 0.769993i \(0.720259\pi\)
\(24\) 0 0
\(25\) −4.70789e9 −0.154268
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.13824e11 1.22532 0.612660 − 0.790347i \(-0.290100\pi\)
0.612660 + 0.790347i \(0.290100\pi\)
\(30\) 0 0
\(31\) −1.62228e11 −1.05904 −0.529521 − 0.848297i \(-0.677628\pi\)
−0.529521 + 0.848297i \(0.677628\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.16296e11 −1.09368
\(36\) 0 0
\(37\) −6.06605e11 −1.05049 −0.525247 − 0.850950i \(-0.676027\pi\)
−0.525247 + 0.850950i \(0.676027\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.27020e12 1.82048 0.910239 − 0.414082i \(-0.135897\pi\)
0.910239 + 0.414082i \(0.135897\pi\)
\(42\) 0 0
\(43\) −1.76483e12 −0.990125 −0.495063 − 0.868857i \(-0.664855\pi\)
−0.495063 + 0.868857i \(0.664855\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.83036e12 1.10282 0.551412 − 0.834233i \(-0.314089\pi\)
0.551412 + 0.834233i \(0.314089\pi\)
\(48\) 0 0
\(49\) 1.96707e12 0.414332
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.84453e12 −0.683409 −0.341705 − 0.939807i \(-0.611004\pi\)
−0.341705 + 0.939807i \(0.611004\pi\)
\(54\) 0 0
\(55\) −6.06973e12 −0.537587
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.75390e13 1.44065 0.720323 − 0.693639i \(-0.243994\pi\)
0.720323 + 0.693639i \(0.243994\pi\)
\(60\) 0 0
\(61\) 3.19783e13 1.30281 0.651405 − 0.758730i \(-0.274180\pi\)
0.651405 + 0.758730i \(0.274180\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.25169e13 0.316698
\(66\) 0 0
\(67\) −1.26768e11 −0.00255534 −0.00127767 − 0.999999i \(-0.500407\pi\)
−0.00127767 + 0.999999i \(0.500407\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.45175e13 −0.189433 −0.0947164 − 0.995504i \(-0.530194\pi\)
−0.0947164 + 0.995504i \(0.530194\pi\)
\(72\) 0 0
\(73\) 1.01137e14 1.07149 0.535744 − 0.844380i \(-0.320031\pi\)
0.535744 + 0.844380i \(0.320031\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.79014e13 0.695198
\(78\) 0 0
\(79\) −1.63798e14 −0.959633 −0.479817 − 0.877369i \(-0.659297\pi\)
−0.479817 + 0.877369i \(0.659297\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.09698e14 −1.25271 −0.626357 − 0.779536i \(-0.715455\pi\)
−0.626357 + 0.779536i \(0.715455\pi\)
\(84\) 0 0
\(85\) 1.20843e13 0.0408864
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.77864e13 −0.162449 −0.0812246 − 0.996696i \(-0.525883\pi\)
−0.0812246 + 0.996696i \(0.525883\pi\)
\(90\) 0 0
\(91\) −2.01890e14 −0.409548
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.56889e14 −0.230497
\(96\) 0 0
\(97\) −1.02585e15 −1.28913 −0.644563 − 0.764552i \(-0.722960\pi\)
−0.644563 + 0.764552i \(0.722960\pi\)
\(98\) 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.t.1.1 2
3.2 odd 2 inner 144.16.a.t.1.2 2
4.3 odd 2 9.16.a.e.1.1 ✓ 2
12.11 even 2 9.16.a.e.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.16.a.e.1.1 ✓ 2 4.3 odd 2
9.16.a.e.1.2 yes 2 12.11 even 2
144.16.a.t.1.1 2 1.1 even 1 trivial
144.16.a.t.1.2 2 3.2 odd 2 inner