Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,26,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8192,0,33554432,-741953100] 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: yes
Analytic conductor: \(71.2794203914\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-162.829\) of defining polynomial
Character \(\chi\) \(=\) 18.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +1.67772e7 q^{4} +1.37041e8 q^{5} -3.04153e10 q^{7} -6.87195e10 q^{8} -5.61320e11 q^{10} -2.58704e12 q^{11} -9.57327e13 q^{13} +1.24581e14 q^{14} +2.81475e14 q^{16} +1.64685e15 q^{17} +4.95030e15 q^{19} +2.29916e15 q^{20} +1.05965e16 q^{22} +1.07650e16 q^{23} -2.79243e17 q^{25} +3.92121e17 q^{26} -5.10284e17 q^{28} +1.36741e18 q^{29} -4.42000e18 q^{31} -1.15292e18 q^{32} -6.74549e18 q^{34} -4.16814e18 q^{35} +1.01944e19 q^{37} -2.02764e19 q^{38} -9.41738e18 q^{40} -1.58687e20 q^{41} +1.83575e20 q^{43} -4.34034e19 q^{44} -4.40934e19 q^{46} -1.40203e21 q^{47} -4.15978e20 q^{49} +1.14378e21 q^{50} -1.60613e21 q^{52} +1.99903e21 q^{53} -3.54531e20 q^{55} +2.09012e21 q^{56} -5.60091e21 q^{58} +4.16691e21 q^{59} +3.42128e22 q^{61} +1.81043e22 q^{62} +4.72237e21 q^{64} -1.31193e22 q^{65} +8.67051e22 q^{67} +2.76295e22 q^{68} +1.70727e22 q^{70} +5.13159e22 q^{71} +3.49147e22 q^{73} -4.17563e22 q^{74} +8.30522e22 q^{76} +7.86857e22 q^{77} +2.91588e23 q^{79} +3.85736e22 q^{80} +6.49981e23 q^{82} +1.64916e24 q^{83} +2.25686e23 q^{85} -7.51925e23 q^{86} +1.77780e23 q^{88} -8.74435e23 q^{89} +2.91174e24 q^{91} +1.80607e23 q^{92} +5.74273e24 q^{94} +6.78393e23 q^{95} +1.00608e25 q^{97} +1.70384e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} + 33554432 q^{4} - 741953100 q^{5} - 376536944 q^{7} - 137438953472 q^{8} + 3039039897600 q^{10} - 8323034610264 q^{11} - 106467053152292 q^{13} + 1542295322624 q^{14} + 562949953421312 q^{16}+ \cdots + 35\!\cdots\!16 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) 0 0
\(4\) 1.67772e7 0.500000
\(5\) 1.37041e8 0.251030 0.125515 − 0.992092i \(-0.459942\pi\)
0.125515 + 0.992092i \(0.459942\pi\)
\(6\) 0 0
\(7\) −3.04153e10 −0.830552 −0.415276 − 0.909696i \(-0.636315\pi\)
−0.415276 + 0.909696i \(0.636315\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 0 0
\(10\) −5.61320e11 −0.177505
\(11\) −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 \(0.692988\pi\)
\(14\) 1.24581e14 0.587289
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 1.64685e15 0.685555 0.342777 − 0.939417i \(-0.388632\pi\)
0.342777 + 0.939417i \(0.388632\pi\)
\(18\) 0 0
\(19\) 4.95030e15 0.513111 0.256555 − 0.966530i \(-0.417412\pi\)
0.256555 + 0.966530i \(0.417412\pi\)
\(20\) 2.29916e15 0.125515
\(21\) 0 0
\(22\) 1.05965e16 0.175744
\(23\) 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
\(26\) 3.92121e17 0.805849
\(27\) 0 0
\(28\) −5.10284e17 −0.415276
\(29\) 1.36741e18 0.717668 0.358834 − 0.933401i \(-0.383174\pi\)
0.358834 + 0.933401i \(0.383174\pi\)
\(30\) 0 0
\(31\) −4.42000e18 −1.00786 −0.503931 − 0.863744i \(-0.668113\pi\)
−0.503931 + 0.863744i \(0.668113\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 0 0
\(34\) −6.74549e18 −0.484760
\(35\) −4.16814e18 −0.208493
\(36\) 0 0
\(37\) 1.01944e19 0.254590 0.127295 − 0.991865i \(-0.459371\pi\)
0.127295 + 0.991865i \(0.459371\pi\)
\(38\) −2.02764e19 −0.362824
\(39\) 0 0
\(40\) −9.41738e18 −0.0887524
\(41\) −1.58687e20 −1.09835 −0.549177 − 0.835706i \(-0.685059\pi\)
−0.549177 + 0.835706i \(0.685059\pi\)
\(42\) 0 0
\(43\) 1.83575e20 0.700582 0.350291 − 0.936641i \(-0.386083\pi\)
0.350291 + 0.936641i \(0.386083\pi\)
\(44\) −4.34034e19 −0.124270
\(45\) 0 0
\(46\) −4.40934e19 −0.0724270
\(47\) −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
\(50\) 1.14378e21 0.662548
\(51\) 0 0
\(52\) −1.60613e21 −0.569821
\(53\) 1.99903e21 0.558946 0.279473 − 0.960154i \(-0.409840\pi\)
0.279473 + 0.960154i \(0.409840\pi\)
\(54\) 0 0
\(55\) −3.54531e20 −0.0623908
\(56\) 2.09012e21 0.293644
\(57\) 0 0
\(58\) −5.60091e21 −0.507468
\(59\) 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 \(0.191087\pi\)
\(62\) 1.81043e22 0.712666
\(63\) 0 0
\(64\) 4.72237e21 0.125000
\(65\) −1.31193e22 −0.286084
\(66\) 0 0
\(67\) 8.67051e22 1.29452 0.647261 − 0.762268i \(-0.275914\pi\)
0.647261 + 0.762268i \(0.275914\pi\)
\(68\) 2.76295e22 0.342777
\(69\) 0 0
\(70\) 1.70727e22 0.147427
\(71\) 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 \(0.471564\pi\)
\(74\) −4.17563e22 −0.180022
\(75\) 0 0
\(76\) 8.30522e22 0.256555
\(77\) 7.86857e22 0.206425
\(78\) 0 0
\(79\) 2.91588e23 0.555176 0.277588 − 0.960700i \(-0.410465\pi\)
0.277588 + 0.960700i \(0.410465\pi\)
\(80\) 3.85736e22 0.0627574
\(81\) 0 0
\(82\) 6.49981e23 0.776654
\(83\) 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
\(86\) −7.51925e23 −0.495386
\(87\) 0 0
\(88\) 1.77780e23 0.0878720
\(89\) −8.74435e23 −0.375277 −0.187639 − 0.982238i \(-0.560083\pi\)
−0.187639 + 0.982238i \(0.560083\pi\)
\(90\) 0 0
\(91\) 2.91174e24 0.946532
\(92\) 1.80607e23 0.0512136
\(93\) 0 0
\(94\) 5.74273e24 1.24457
\(95\) 6.78393e23 0.128806
\(96\) 0 0
\(97\) 1.00608e25 1.47227 0.736134 − 0.676835i \(-0.236649\pi\)
0.736134 + 0.676835i \(0.236649\pi\)
\(98\) 1.70384e24 0.219333
\(99\) 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 18.26.a.e.1.2 2
3.2 odd 2 2.26.a.b.1.2 ✓ 2
12.11 even 2 16.26.a.c.1.1 2
15.2 even 4 50.26.b.e.49.3 4
15.8 even 4 50.26.b.e.49.2 4
15.14 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 3.2 odd 2
16.26.a.c.1.1 2 12.11 even 2
18.26.a.e.1.2 2 1.1 even 1 trivial
50.26.a.c.1.1 2 15.14 odd 2
50.26.b.e.49.2 4 15.8 even 4
50.26.b.e.49.3 4 15.2 even 4