Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4096,96764] 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: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 315576783x^{2} - 792388522562x + 1396510961585520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1195.65\) 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)
 
\(f(q)\) \(=\) \(q+1024.00 q^{2} +12234.5 q^{3} +1.04858e6 q^{4} +1.25282e7 q^{6} +9.06115e8 q^{7} +1.07374e9 q^{8} -1.03107e10 q^{9} +8.75575e10 q^{11} +1.28288e10 q^{12} +3.78299e10 q^{13} +9.27862e11 q^{14} +1.09951e12 q^{16} +1.03197e13 q^{17} -1.05581e13 q^{18} -4.47347e13 q^{19} +1.10859e13 q^{21} +8.96589e13 q^{22} +2.93024e14 q^{23} +1.31367e13 q^{24} +3.87378e13 q^{26} -2.54124e14 q^{27} +9.50131e14 q^{28} -4.56584e14 q^{29} +5.31739e15 q^{31} +1.12590e15 q^{32} +1.07123e15 q^{33} +1.05674e16 q^{34} -1.08115e16 q^{36} -2.33624e16 q^{37} -4.58083e16 q^{38} +4.62831e14 q^{39} -7.54341e16 q^{41} +1.13520e16 q^{42} +1.59218e17 q^{43} +9.18107e16 q^{44} +3.00056e17 q^{46} +5.36127e15 q^{47} +1.34520e16 q^{48} +2.62499e17 q^{49} +1.26257e17 q^{51} +3.96675e16 q^{52} +1.47056e18 q^{53} -2.60223e17 q^{54} +9.72934e17 q^{56} -5.47308e17 q^{57} -4.67542e17 q^{58} -2.78753e18 q^{59} -5.95048e18 q^{61} +5.44501e18 q^{62} -9.34266e18 q^{63} +1.15292e18 q^{64} +1.09694e18 q^{66} +3.15077e18 q^{67} +1.08210e19 q^{68} +3.58501e18 q^{69} +3.60686e19 q^{71} -1.10710e19 q^{72} +5.16840e19 q^{73} -2.39231e19 q^{74} -4.69077e19 q^{76} +7.93372e19 q^{77} +4.73939e17 q^{78} -1.21308e20 q^{79} +1.04744e20 q^{81} -7.72445e19 q^{82} +1.00531e20 q^{83} +1.16244e19 q^{84} +1.63039e20 q^{86} -5.58609e18 q^{87} +9.40142e19 q^{88} +2.72642e20 q^{89} +3.42782e19 q^{91} +3.07257e20 q^{92} +6.50559e19 q^{93} +5.48994e18 q^{94} +1.37749e19 q^{96} -5.06240e20 q^{97} +2.68799e20 q^{98} -9.02777e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4096 q^{2} + 96764 q^{3} + 4194304 q^{4} + 99086336 q^{6} + 162760528 q^{7} + 4294967296 q^{8} + 23614761712 q^{9} + 87559254108 q^{11} + 101464408064 q^{12} - 98098343096 q^{13} + 166666780672 q^{14}+ \cdots - 28\!\cdots\!76 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1024.00 0.707107
\(3\) 12234.5 0.119623 0.0598115 − 0.998210i \(-0.480950\pi\)
0.0598115 + 0.998210i \(0.480950\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) 1.25282e7 0.0845862
\(7\) 9.06115e8 1.21242 0.606211 − 0.795304i \(-0.292689\pi\)
0.606211 + 0.795304i \(0.292689\pi\)
\(8\) 1.07374e9 0.353553
\(9\) −1.03107e10 −0.985690
\(10\) 0 0
\(11\) 8.75575e10 1.01782 0.508909 − 0.860820i \(-0.330049\pi\)
0.508909 + 0.860820i \(0.330049\pi\)
\(12\) 1.28288e10 0.0598115
\(13\) 3.78299e10 0.0761079 0.0380539 − 0.999276i \(-0.487884\pi\)
0.0380539 + 0.999276i \(0.487884\pi\)
\(14\) 9.27862e11 0.857312
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.03197e13 1.24152 0.620761 − 0.784000i \(-0.286824\pi\)
0.620761 + 0.784000i \(0.286824\pi\)
\(18\) −1.05581e13 −0.696988
\(19\) −4.47347e13 −1.67391 −0.836954 − 0.547273i \(-0.815666\pi\)
−0.836954 + 0.547273i \(0.815666\pi\)
\(20\) 0 0
\(21\) 1.10859e13 0.145034
\(22\) 8.96589e13 0.719706
\(23\) 2.93024e14 1.47489 0.737446 − 0.675406i \(-0.236032\pi\)
0.737446 + 0.675406i \(0.236032\pi\)
\(24\) 1.31367e13 0.0422931
\(25\) 0 0
\(26\) 3.87378e13 0.0538164
\(27\) −2.54124e14 −0.237534
\(28\) 9.50131e14 0.606211
\(29\) −4.56584e14 −0.201531 −0.100765 − 0.994910i \(-0.532129\pi\)
−0.100765 + 0.994910i \(0.532129\pi\)
\(30\) 0 0
\(31\) 5.31739e15 1.16520 0.582600 − 0.812759i \(-0.302035\pi\)
0.582600 + 0.812759i \(0.302035\pi\)
\(32\) 1.12590e15 0.176777
\(33\) 1.07123e15 0.121754
\(34\) 1.05674e16 0.877888
\(35\) 0 0
\(36\) −1.08115e16 −0.492845
\(37\) −2.33624e16 −0.798728 −0.399364 − 0.916792i \(-0.630769\pi\)
−0.399364 + 0.916792i \(0.630769\pi\)
\(38\) −4.58083e16 −1.18363
\(39\) 4.62831e14 0.00910425
\(40\) 0 0
\(41\) −7.54341e16 −0.877682 −0.438841 − 0.898565i \(-0.644611\pi\)
−0.438841 + 0.898565i \(0.644611\pi\)
\(42\) 1.13520e16 0.102554
\(43\) 1.59218e17 1.12350 0.561750 − 0.827307i \(-0.310128\pi\)
0.561750 + 0.827307i \(0.310128\pi\)
\(44\) 9.18107e16 0.508909
\(45\) 0 0
\(46\) 3.00056e17 1.04291
\(47\) 5.36127e15 0.0148676 0.00743378 − 0.999972i \(-0.497634\pi\)
0.00743378 + 0.999972i \(0.497634\pi\)
\(48\) 1.34520e16 0.0299057
\(49\) 2.62499e17 0.469969
\(50\) 0 0
\(51\) 1.26257e17 0.148514
\(52\) 3.96675e16 0.0380539
\(53\) 1.47056e18 1.15501 0.577507 − 0.816386i \(-0.304026\pi\)
0.577507 + 0.816386i \(0.304026\pi\)
\(54\) −2.60223e17 −0.167962
\(55\) 0 0
\(56\) 9.72934e17 0.428656
\(57\) −5.47308e17 −0.200238
\(58\) −4.67542e17 −0.142504
\(59\) −2.78753e18 −0.710024 −0.355012 − 0.934862i \(-0.615523\pi\)
−0.355012 + 0.934862i \(0.615523\pi\)
\(60\) 0 0
\(61\) −5.95048e18 −1.06804 −0.534021 − 0.845471i \(-0.679320\pi\)
−0.534021 + 0.845471i \(0.679320\pi\)
\(62\) 5.44501e18 0.823922
\(63\) −9.34266e18 −1.19507
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) 1.09694e18 0.0860934
\(67\) 3.15077e18 0.211169 0.105585 − 0.994410i \(-0.466329\pi\)
0.105585 + 0.994410i \(0.466329\pi\)
\(68\) 1.08210e19 0.620761
\(69\) 3.58501e18 0.176431
\(70\) 0 0
\(71\) 3.60686e19 1.31497 0.657485 − 0.753467i \(-0.271620\pi\)
0.657485 + 0.753467i \(0.271620\pi\)
\(72\) −1.10710e19 −0.348494
\(73\) 5.16840e19 1.40756 0.703778 − 0.710420i \(-0.251495\pi\)
0.703778 + 0.710420i \(0.251495\pi\)
\(74\) −2.39231e19 −0.564786
\(75\) 0 0
\(76\) −4.69077e19 −0.836954
\(77\) 7.93372e19 1.23403
\(78\) 4.73939e17 0.00643768
\(79\) −1.21308e20 −1.44146 −0.720732 − 0.693214i \(-0.756194\pi\)
−0.720732 + 0.693214i \(0.756194\pi\)
\(80\) 0 0
\(81\) 1.04744e20 0.957276
\(82\) −7.72445e19 −0.620615
\(83\) 1.00531e20 0.711180 0.355590 − 0.934642i \(-0.384280\pi\)
0.355590 + 0.934642i \(0.384280\pi\)
\(84\) 1.16244e19 0.0725168
\(85\) 0 0
\(86\) 1.63039e20 0.794434
\(87\) −5.58609e18 −0.0241077
\(88\) 9.40142e19 0.359853
\(89\) 2.72642e20 0.926825 0.463413 − 0.886143i \(-0.346625\pi\)
0.463413 + 0.886143i \(0.346625\pi\)
\(90\) 0 0
\(91\) 3.42782e19 0.0922749
\(92\) 3.07257e20 0.737446
\(93\) 6.50559e19 0.139385
\(94\) 5.48994e18 0.0105130
\(95\) 0 0
\(96\) 1.37749e19 0.0211466
\(97\) −5.06240e20 −0.697033 −0.348516 − 0.937303i \(-0.613314\pi\)
−0.348516 + 0.937303i \(0.613314\pi\)
\(98\) 2.68799e20 0.332318
\(99\) −9.02777e20 −1.00325
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 50.22.a.j.1.2 yes 4
5.2 odd 4 50.22.b.h.49.7 8
5.3 odd 4 50.22.b.h.49.2 8
5.4 even 2 50.22.a.i.1.3 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.3 ✓ 4 5.4 even 2
50.22.a.j.1.2 yes 4 1.1 even 1 trivial
50.22.b.h.49.2 8 5.3 odd 4
50.22.b.h.49.7 8 5.2 odd 4