Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,26,Mod(1,50)] 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("50.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8192,-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: \(197.998389976\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 50.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} -1.75788e6 q^{3} +1.67772e7 q^{4} +7.20027e9 q^{6} +3.04153e10 q^{7} -6.87195e10 q^{8} +2.24285e12 q^{9} +2.58704e12 q^{11} -2.94923e13 q^{12} +9.57327e13 q^{13} -1.24581e14 q^{14} +2.81475e14 q^{16} +1.64685e15 q^{17} -9.18671e15 q^{18} +4.95030e15 q^{19} -5.34664e16 q^{21} -1.05965e16 q^{22} +1.07650e16 q^{23} +1.20801e17 q^{24} -3.92121e17 q^{26} -2.45323e18 q^{27} +5.10284e17 q^{28} -1.36741e18 q^{29} -4.42000e18 q^{31} -1.15292e18 q^{32} -4.54771e18 q^{33} -6.74549e18 q^{34} +3.76288e19 q^{36} -1.01944e19 q^{37} -2.02764e19 q^{38} -1.68287e20 q^{39} +1.58687e20 q^{41} +2.18999e20 q^{42} -1.83575e20 q^{43} +4.34034e19 q^{44} -4.40934e19 q^{46} -1.40203e21 q^{47} -4.94799e20 q^{48} -4.15978e20 q^{49} -2.89496e21 q^{51} +1.60613e21 q^{52} +1.99903e21 q^{53} +1.00484e22 q^{54} -2.09012e21 q^{56} -8.70202e21 q^{57} +5.60091e21 q^{58} -4.16691e21 q^{59} +3.42128e22 q^{61} +1.81043e22 q^{62} +6.82170e22 q^{63} +4.72237e21 q^{64} +1.86274e22 q^{66} -8.67051e22 q^{67} +2.76295e22 q^{68} -1.89236e22 q^{69} -5.13159e22 q^{71} -1.54128e23 q^{72} -3.49147e22 q^{73} +4.17563e22 q^{74} +8.30522e22 q^{76} +7.86857e22 q^{77} +6.89302e23 q^{78} +2.91588e23 q^{79} +2.41214e24 q^{81} -6.49981e23 q^{82} +1.64916e24 q^{83} -8.97018e23 q^{84} +7.51925e23 q^{86} +2.40374e24 q^{87} -1.77780e23 q^{88} +8.74435e23 q^{89} +2.91174e24 q^{91} +1.80607e23 q^{92} +7.76982e24 q^{93} +5.74273e24 q^{94} +2.02670e24 q^{96} -1.00608e25 q^{97} +1.70384e24 q^{98} +5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} - 379848 q^{3} + 33554432 q^{4} + 1555857408 q^{6} + 376536944 q^{7} - 137438953472 q^{8} + 3294531432666 q^{9} + 8323034610264 q^{11} - 6372791943168 q^{12} + 106467053152292 q^{13} - 1542295322624 q^{14}+ \cdots + 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) −1.75788e6 −1.90974 −0.954868 − 0.297031i \(-0.904004\pi\)
−0.954868 + 0.297031i \(0.904004\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) 7.20027e9 1.35039
\(7\) 3.04153e10 0.830552 0.415276 − 0.909696i \(-0.363685\pi\)
0.415276 + 0.909696i \(0.363685\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 2.24285e12 2.64709
\(10\) 0 0
\(11\) 2.58704e12 0.248539 0.124270 − 0.992248i \(-0.460341\pi\)
0.124270 + 0.992248i \(0.460341\pi\)
\(12\) −2.94923e13 −0.954868
\(13\) 9.57327e13 1.13964 0.569821 − 0.821769i \(-0.307012\pi\)
0.569821 + 0.821769i \(0.307012\pi\)
\(14\) −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 \(0.388632\pi\)
\(18\) −9.18671e15 −1.87178
\(19\) 4.95030e15 0.513111 0.256555 − 0.966530i \(-0.417412\pi\)
0.256555 + 0.966530i \(0.417412\pi\)
\(20\) 0 0
\(21\) −5.34664e16 −1.58613
\(22\) −1.05965e16 −0.175744
\(23\) 1.07650e16 0.102427 0.0512136 − 0.998688i \(-0.483691\pi\)
0.0512136 + 0.998688i \(0.483691\pi\)
\(24\) 1.20801e17 0.675194
\(25\) 0 0
\(26\) −3.92121e17 −0.805849
\(27\) −2.45323e18 −3.14551
\(28\) 5.10284e17 0.415276
\(29\) −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 \(-0.668113\pi\)
−0.503931 + 0.863744i \(0.668113\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) −4.54771e18 −0.474645
\(34\) −6.74549e18 −0.484760
\(35\) 0 0
\(36\) 3.76288e19 1.32355
\(37\) −1.01944e19 −0.254590 −0.127295 − 0.991865i \(-0.540629\pi\)
−0.127295 + 0.991865i \(0.540629\pi\)
\(38\) −2.02764e19 −0.362824
\(39\) −1.68287e20 −2.17642
\(40\) 0 0
\(41\) 1.58687e20 1.09835 0.549177 − 0.835706i \(-0.314941\pi\)
0.549177 + 0.835706i \(0.314941\pi\)
\(42\) 2.18999e20 1.12157
\(43\) −1.83575e20 −0.700582 −0.350291 − 0.936641i \(-0.613917\pi\)
−0.350291 + 0.936641i \(0.613917\pi\)
\(44\) 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 \(0.842488\pi\)
\(48\) −4.94799e20 −0.477434
\(49\) −4.15978e20 −0.310184
\(50\) 0 0
\(51\) −2.89496e21 −1.30923
\(52\) 1.60613e21 0.569821
\(53\) 1.99903e21 0.558946 0.279473 − 0.960154i \(-0.409840\pi\)
0.279473 + 0.960154i \(0.409840\pi\)
\(54\) 1.00484e22 2.22421
\(55\) 0 0
\(56\) −2.09012e21 −0.293644
\(57\) −8.70202e21 −0.979906
\(58\) 5.60091e21 0.507468
\(59\) −4.16691e21 −0.304905 −0.152452 − 0.988311i \(-0.548717\pi\)
−0.152452 + 0.988311i \(0.548717\pi\)
\(60\) 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
\(63\) 6.82170e22 2.19855
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) 1.86274e22 0.335624
\(67\) −8.67051e22 −1.29452 −0.647261 − 0.762268i \(-0.724086\pi\)
−0.647261 + 0.762268i \(0.724086\pi\)
\(68\) 2.76295e22 0.342777
\(69\) −1.89236e22 −0.195609
\(70\) 0 0
\(71\) −5.13159e22 −0.371127 −0.185564 − 0.982632i \(-0.559411\pi\)
−0.185564 + 0.982632i \(0.559411\pi\)
\(72\) −1.54128e23 −0.935888
\(73\) −3.49147e22 −0.178432 −0.0892159 − 0.996012i \(-0.528436\pi\)
−0.0892159 + 0.996012i \(0.528436\pi\)
\(74\) 4.17563e22 0.180022
\(75\) 0 0
\(76\) 8.30522e22 0.256555
\(77\) 7.86857e22 0.206425
\(78\) 6.89302e23 1.53896
\(79\) 2.91588e23 0.555176 0.277588 − 0.960700i \(-0.410465\pi\)
0.277588 + 0.960700i \(0.410465\pi\)
\(80\) 0 0
\(81\) 2.41214e24 3.36000
\(82\) −6.49981e23 −0.776654
\(83\) 1.64916e24 1.69351 0.846753 − 0.531986i \(-0.178554\pi\)
0.846753 + 0.531986i \(0.178554\pi\)
\(84\) −8.97018e23 −0.793067
\(85\) 0 0
\(86\) 7.51925e23 0.495386
\(87\) 2.40374e24 1.37056
\(88\) −1.77780e23 −0.0878720
\(89\) 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
\(92\) 1.80607e23 0.0512136
\(93\) 7.76982e24 1.92475
\(94\) 5.74273e24 1.24457
\(95\) 0 0
\(96\) 2.02670e24 0.337597
\(97\) −1.00608e25 −1.47227 −0.736134 − 0.676835i \(-0.763351\pi\)
−0.736134 + 0.676835i \(0.763351\pi\)
\(98\) 1.70384e24 0.219333
\(99\) 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 50.26.a.c.1.1 2
5.2 odd 4 50.26.b.e.49.2 4
5.3 odd 4 50.26.b.e.49.3 4
5.4 even 2 2.26.a.b.1.2 ✓ 2
15.14 odd 2 18.26.a.e.1.2 2
20.19 odd 2 16.26.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.2 ✓ 2 5.4 even 2
16.26.a.c.1.1 2 20.19 odd 2
18.26.a.e.1.2 2 15.14 odd 2
50.26.a.c.1.1 2 1.1 even 1 trivial
50.26.b.e.49.2 4 5.2 odd 4
50.26.b.e.49.3 4 5.3 odd 4