Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [16,26,Mod(1,16)] 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("16.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-379848] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3594847924\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(163.829\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.75788e6 q^{3} -1.37041e8 q^{5} +3.04153e10 q^{7} +2.24285e12 q^{9} -2.58704e12 q^{11} -9.57327e13 q^{13} +2.40901e14 q^{15} -1.64685e15 q^{17} -4.95030e15 q^{19} -5.34664e16 q^{21} +1.07650e16 q^{23} -2.79243e17 q^{25} -2.45323e18 q^{27} -1.36741e18 q^{29} +4.42000e18 q^{31} +4.54771e18 q^{33} -4.16814e18 q^{35} +1.01944e19 q^{37} +1.68287e20 q^{39} +1.58687e20 q^{41} -1.83575e20 q^{43} -3.07362e20 q^{45} -1.40203e21 q^{47} -4.15978e20 q^{49} +2.89496e21 q^{51} -1.99903e21 q^{53} +3.54531e20 q^{55} +8.70202e21 q^{57} +4.16691e21 q^{59} +3.42128e22 q^{61} +6.82170e22 q^{63} +1.31193e22 q^{65} -8.67051e22 q^{67} -1.89236e22 q^{69} +5.13159e22 q^{71} +3.49147e22 q^{73} +4.90875e23 q^{75} -7.86857e22 q^{77} -2.91588e23 q^{79} +2.41214e24 q^{81} +1.64916e24 q^{83} +2.25686e23 q^{85} +2.40374e24 q^{87} +8.74435e23 q^{89} -2.91174e24 q^{91} -7.76982e24 q^{93} +6.78393e23 q^{95} +1.00608e25 q^{97} -5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 379848 q^{3} + 741953100 q^{5} + 376536944 q^{7} + 3294531432666 q^{9} - 8323034610264 q^{11} - 106467053152292 q^{13} + 14\!\cdots\!00 q^{15} + 13\!\cdots\!56 q^{17} + 477079242949400 q^{19} - 94\!\cdots\!56 q^{21}+ \cdots - 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) −1.75788e6 −1.90974 −0.954868 − 0.297031i \(-0.904004\pi\)
−0.954868 + 0.297031i \(0.904004\pi\)
\(4\) 0 0
\(5\) −1.37041e8 −0.251030 −0.125515 − 0.992092i \(-0.540058\pi\)
−0.125515 + 0.992092i \(0.540058\pi\)
\(6\) 0 0
\(7\) 3.04153e10 0.830552 0.415276 − 0.909696i \(-0.363685\pi\)
0.415276 + 0.909696i \(0.363685\pi\)
\(8\) 0 0
\(9\) 2.24285e12 2.64709
\(10\) 0 0
\(11\) −2.58704e12 −0.248539 −0.124270 − 0.992248i \(-0.539659\pi\)
−0.124270 + 0.992248i \(0.539659\pi\)
\(12\) 0 0
\(13\) −9.57327e13 −1.13964 −0.569821 − 0.821769i \(-0.692988\pi\)
−0.569821 + 0.821769i \(0.692988\pi\)
\(14\) 0 0
\(15\) 2.40901e14 0.479400
\(16\) 0 0
\(17\) −1.64685e15 −0.685555 −0.342777 − 0.939417i \(-0.611368\pi\)
−0.342777 + 0.939417i \(0.611368\pi\)
\(18\) 0 0
\(19\) −4.95030e15 −0.513111 −0.256555 − 0.966530i \(-0.582588\pi\)
−0.256555 + 0.966530i \(0.582588\pi\)
\(20\) 0 0
\(21\) −5.34664e16 −1.58613
\(22\) 0 0
\(23\) 1.07650e16 0.102427 0.0512136 − 0.998688i \(-0.483691\pi\)
0.0512136 + 0.998688i \(0.483691\pi\)
\(24\) 0 0
\(25\) −2.79243e17 −0.936984
\(26\) 0 0
\(27\) −2.45323e18 −3.14551
\(28\) 0 0
\(29\) −1.36741e18 −0.717668 −0.358834 − 0.933401i \(-0.616826\pi\)
−0.358834 + 0.933401i \(0.616826\pi\)
\(30\) 0 0
\(31\) 4.42000e18 1.00786 0.503931 − 0.863744i \(-0.331887\pi\)
0.503931 + 0.863744i \(0.331887\pi\)
\(32\) 0 0
\(33\) 4.54771e18 0.474645
\(34\) 0 0
\(35\) −4.16814e18 −0.208493
\(36\) 0 0
\(37\) 1.01944e19 0.254590 0.127295 − 0.991865i \(-0.459371\pi\)
0.127295 + 0.991865i \(0.459371\pi\)
\(38\) 0 0
\(39\) 1.68287e20 2.17642
\(40\) 0 0
\(41\) 1.58687e20 1.09835 0.549177 − 0.835706i \(-0.314941\pi\)
0.549177 + 0.835706i \(0.314941\pi\)
\(42\) 0 0
\(43\) −1.83575e20 −0.700582 −0.350291 − 0.936641i \(-0.613917\pi\)
−0.350291 + 0.936641i \(0.613917\pi\)
\(44\) 0 0
\(45\) −3.07362e20 −0.664499
\(46\) 0 0
\(47\) −1.40203e21 −1.76009 −0.880045 − 0.474890i \(-0.842488\pi\)
−0.880045 + 0.474890i \(0.842488\pi\)
\(48\) 0 0
\(49\) −4.15978e20 −0.310184
\(50\) 0 0
\(51\) 2.89496e21 1.30923
\(52\) 0 0
\(53\) −1.99903e21 −0.558946 −0.279473 − 0.960154i \(-0.590160\pi\)
−0.279473 + 0.960154i \(0.590160\pi\)
\(54\) 0 0
\(55\) 3.54531e20 0.0623908
\(56\) 0 0
\(57\) 8.70202e21 0.979906
\(58\) 0 0
\(59\) 4.16691e21 0.304905 0.152452 − 0.988311i \(-0.451283\pi\)
0.152452 + 0.988311i \(0.451283\pi\)
\(60\) 0 0
\(61\) 3.42128e22 1.65031 0.825156 − 0.564904i \(-0.191087\pi\)
0.825156 + 0.564904i \(0.191087\pi\)
\(62\) 0 0
\(63\) 6.82170e22 2.19855
\(64\) 0 0
\(65\) 1.31193e22 0.286084
\(66\) 0 0
\(67\) −8.67051e22 −1.29452 −0.647261 − 0.762268i \(-0.724086\pi\)
−0.647261 + 0.762268i \(0.724086\pi\)
\(68\) 0 0
\(69\) −1.89236e22 −0.195609
\(70\) 0 0
\(71\) 5.13159e22 0.371127 0.185564 − 0.982632i \(-0.440589\pi\)
0.185564 + 0.982632i \(0.440589\pi\)
\(72\) 0 0
\(73\) 3.49147e22 0.178432 0.0892159 − 0.996012i \(-0.471564\pi\)
0.0892159 + 0.996012i \(0.471564\pi\)
\(74\) 0 0
\(75\) 4.90875e23 1.78939
\(76\) 0 0
\(77\) −7.86857e22 −0.206425
\(78\) 0 0
\(79\) −2.91588e23 −0.555176 −0.277588 − 0.960700i \(-0.589535\pi\)
−0.277588 + 0.960700i \(0.589535\pi\)
\(80\) 0 0
\(81\) 2.41214e24 3.36000
\(82\) 0 0
\(83\) 1.64916e24 1.69351 0.846753 − 0.531986i \(-0.178554\pi\)
0.846753 + 0.531986i \(0.178554\pi\)
\(84\) 0 0
\(85\) 2.25686e23 0.172095
\(86\) 0 0
\(87\) 2.40374e24 1.37056
\(88\) 0 0
\(89\) 8.74435e23 0.375277 0.187639 − 0.982238i \(-0.439917\pi\)
0.187639 + 0.982238i \(0.439917\pi\)
\(90\) 0 0
\(91\) −2.91174e24 −0.946532
\(92\) 0 0
\(93\) −7.76982e24 −1.92475
\(94\) 0 0
\(95\) 6.78393e23 0.128806
\(96\) 0 0
\(97\) 1.00608e25 1.47227 0.736134 − 0.676835i \(-0.236649\pi\)
0.736134 + 0.676835i \(0.236649\pi\)
\(98\) 0 0
\(99\) −5.80235e24 −0.657907
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 16.26.a.c.1.1 2
4.3 odd 2 2.26.a.b.1.2 ✓ 2
12.11 even 2 18.26.a.e.1.2 2
20.3 even 4 50.26.b.e.49.2 4
20.7 even 4 50.26.b.e.49.3 4
20.19 odd 2 50.26.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.2 ✓ 2 4.3 odd 2
16.26.a.c.1.1 2 1.1 even 1 trivial
18.26.a.e.1.2 2 12.11 even 2
50.26.a.c.1.1 2 20.19 odd 2
50.26.b.e.49.2 4 20.3 even 4
50.26.b.e.49.3 4 20.7 even 4