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 = [3,3072,46383] 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: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 30959316x - 11291332284 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-366.307\) 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)
 
\(f(q)\) \(=\) \(q+1024.00 q^{2} +26460.2 q^{3} +1.04858e6 q^{4} +2.70953e7 q^{6} +3.30980e8 q^{7} +1.07374e9 q^{8} -9.76021e9 q^{9} +1.32998e9 q^{11} +2.77455e10 q^{12} -1.22401e11 q^{13} +3.38924e11 q^{14} +1.09951e12 q^{16} +1.18162e12 q^{17} -9.99446e12 q^{18} +1.96281e13 q^{19} +8.75781e12 q^{21} +1.36190e12 q^{22} -1.45964e14 q^{23} +2.84114e13 q^{24} -1.25339e14 q^{26} -5.35040e14 q^{27} +3.47058e14 q^{28} -2.54346e15 q^{29} -8.14052e13 q^{31} +1.12590e15 q^{32} +3.51915e13 q^{33} +1.20998e15 q^{34} -1.02343e16 q^{36} -5.96053e15 q^{37} +2.00992e16 q^{38} -3.23876e15 q^{39} +2.16034e16 q^{41} +8.96800e15 q^{42} -5.16519e15 q^{43} +1.39458e15 q^{44} -1.49467e17 q^{46} -4.47352e17 q^{47} +2.90933e16 q^{48} -4.48998e17 q^{49} +3.12660e16 q^{51} -1.28347e17 q^{52} +2.16134e17 q^{53} -5.47881e17 q^{54} +3.55387e17 q^{56} +5.19363e17 q^{57} -2.60450e18 q^{58} +3.65866e18 q^{59} -6.71330e17 q^{61} -8.33589e16 q^{62} -3.23044e18 q^{63} +1.15292e18 q^{64} +3.60361e16 q^{66} -5.53900e18 q^{67} +1.23902e18 q^{68} -3.86223e18 q^{69} +9.93356e18 q^{71} -1.04799e19 q^{72} -4.06075e19 q^{73} -6.10358e18 q^{74} +2.05815e19 q^{76} +4.40197e17 q^{77} -3.31649e18 q^{78} -6.51458e19 q^{79} +8.79380e19 q^{81} +2.21218e19 q^{82} -1.48741e20 q^{83} +9.18323e18 q^{84} -5.28916e18 q^{86} -6.73005e19 q^{87} +1.42805e18 q^{88} -2.72017e20 q^{89} -4.05124e19 q^{91} -1.53054e20 q^{92} -2.15400e18 q^{93} -4.58089e20 q^{94} +2.97916e19 q^{96} -1.44113e19 q^{97} -4.59774e20 q^{98} -1.29809e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3072 q^{2} + 46383 q^{3} + 3145728 q^{4} + 47496192 q^{6} - 911775234 q^{7} + 3221225472 q^{8} + 25062837354 q^{9} - 77565926349 q^{11} + 48636100608 q^{12} + 29305708548 q^{13} - 933657839616 q^{14}+ \cdots - 11\!\cdots\!82 q^{99}+O(q^{100}) \) 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
\(3\) 26460.2 0.258714 0.129357 − 0.991598i \(-0.458709\pi\)
0.129357 + 0.991598i \(0.458709\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) 2.70953e7 0.182939
\(7\) 3.30980e8 0.442866 0.221433 − 0.975176i \(-0.428927\pi\)
0.221433 + 0.975176i \(0.428927\pi\)
\(8\) 1.07374e9 0.353553
\(9\) −9.76021e9 −0.933067
\(10\) 0 0
\(11\) 1.32998e9 0.0154604 0.00773021 − 0.999970i \(-0.497539\pi\)
0.00773021 + 0.999970i \(0.497539\pi\)
\(12\) 2.77455e10 0.129357
\(13\) −1.22401e11 −0.246252 −0.123126 − 0.992391i \(-0.539292\pi\)
−0.123126 + 0.992391i \(0.539292\pi\)
\(14\) 3.38924e11 0.313154
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.18162e12 0.142156 0.0710780 − 0.997471i \(-0.477356\pi\)
0.0710780 + 0.997471i \(0.477356\pi\)
\(18\) −9.99446e12 −0.659778
\(19\) 1.96281e13 0.734455 0.367227 − 0.930131i \(-0.380307\pi\)
0.367227 + 0.930131i \(0.380307\pi\)
\(20\) 0 0
\(21\) 8.75781e12 0.114576
\(22\) 1.36190e12 0.0109322
\(23\) −1.45964e14 −0.734688 −0.367344 − 0.930085i \(-0.619733\pi\)
−0.367344 + 0.930085i \(0.619733\pi\)
\(24\) 2.84114e13 0.0914693
\(25\) 0 0
\(26\) −1.25339e14 −0.174127
\(27\) −5.35040e14 −0.500112
\(28\) 3.47058e14 0.221433
\(29\) −2.54346e15 −1.12265 −0.561327 − 0.827594i \(-0.689709\pi\)
−0.561327 + 0.827594i \(0.689709\pi\)
\(30\) 0 0
\(31\) −8.14052e13 −0.0178383 −0.00891917 − 0.999960i \(-0.502839\pi\)
−0.00891917 + 0.999960i \(0.502839\pi\)
\(32\) 1.12590e15 0.176777
\(33\) 3.51915e13 0.00399983
\(34\) 1.20998e15 0.100519
\(35\) 0 0
\(36\) −1.02343e16 −0.466533
\(37\) −5.96053e15 −0.203782 −0.101891 − 0.994796i \(-0.532489\pi\)
−0.101891 + 0.994796i \(0.532489\pi\)
\(38\) 2.00992e16 0.519338
\(39\) −3.23876e15 −0.0637090
\(40\) 0 0
\(41\) 2.16034e16 0.251357 0.125678 − 0.992071i \(-0.459889\pi\)
0.125678 + 0.992071i \(0.459889\pi\)
\(42\) 8.96800e15 0.0810173
\(43\) −5.16519e15 −0.0364475 −0.0182237 − 0.999834i \(-0.505801\pi\)
−0.0182237 + 0.999834i \(0.505801\pi\)
\(44\) 1.39458e15 0.00773021
\(45\) 0 0
\(46\) −1.49467e17 −0.519503
\(47\) −4.47352e17 −1.24057 −0.620286 − 0.784376i \(-0.712984\pi\)
−0.620286 + 0.784376i \(0.712984\pi\)
\(48\) 2.90933e16 0.0646785
\(49\) −4.48998e17 −0.803869
\(50\) 0 0
\(51\) 3.12660e16 0.0367778
\(52\) −1.28347e17 −0.123126
\(53\) 2.16134e17 0.169756 0.0848782 − 0.996391i \(-0.472950\pi\)
0.0848782 + 0.996391i \(0.472950\pi\)
\(54\) −5.47881e17 −0.353632
\(55\) 0 0
\(56\) 3.55387e17 0.156577
\(57\) 5.19363e17 0.190014
\(58\) −2.60450e18 −0.793836
\(59\) 3.65866e18 0.931913 0.465957 − 0.884808i \(-0.345710\pi\)
0.465957 + 0.884808i \(0.345710\pi\)
\(60\) 0 0
\(61\) −6.71330e17 −0.120496 −0.0602480 − 0.998183i \(-0.519189\pi\)
−0.0602480 + 0.998183i \(0.519189\pi\)
\(62\) −8.33589e16 −0.0126136
\(63\) −3.23044e18 −0.413224
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) 3.60361e16 0.00282831
\(67\) −5.53900e18 −0.371232 −0.185616 − 0.982622i \(-0.559428\pi\)
−0.185616 + 0.982622i \(0.559428\pi\)
\(68\) 1.23902e18 0.0710780
\(69\) −3.86223e18 −0.190074
\(70\) 0 0
\(71\) 9.93356e18 0.362153 0.181077 − 0.983469i \(-0.442042\pi\)
0.181077 + 0.983469i \(0.442042\pi\)
\(72\) −1.04799e19 −0.329889
\(73\) −4.06075e19 −1.10590 −0.552951 − 0.833214i \(-0.686498\pi\)
−0.552951 + 0.833214i \(0.686498\pi\)
\(74\) −6.10358e18 −0.144096
\(75\) 0 0
\(76\) 2.05815e19 0.367227
\(77\) 4.40197e17 0.00684690
\(78\) −3.31649e18 −0.0450491
\(79\) −6.51458e19 −0.774108 −0.387054 − 0.922057i \(-0.626507\pi\)
−0.387054 + 0.922057i \(0.626507\pi\)
\(80\) 0 0
\(81\) 8.79380e19 0.803681
\(82\) 2.21218e19 0.177736
\(83\) −1.48741e20 −1.05223 −0.526114 − 0.850414i \(-0.676351\pi\)
−0.526114 + 0.850414i \(0.676351\pi\)
\(84\) 9.18323e18 0.0572879
\(85\) 0 0
\(86\) −5.28916e18 −0.0257723
\(87\) −6.73005e19 −0.290447
\(88\) 1.42805e18 0.00546609
\(89\) −2.72017e20 −0.924700 −0.462350 − 0.886697i \(-0.652994\pi\)
−0.462350 + 0.886697i \(0.652994\pi\)
\(90\) 0 0
\(91\) −4.05124e19 −0.109057
\(92\) −1.53054e20 −0.367344
\(93\) −2.15400e18 −0.00461503
\(94\) −4.58089e20 −0.877217
\(95\) 0 0
\(96\) 2.97916e19 0.0457346
\(97\) −1.44113e19 −0.0198427 −0.00992134 − 0.999951i \(-0.503158\pi\)
−0.00992134 + 0.999951i \(0.503158\pi\)
\(98\) −4.59774e20 −0.568421
\(99\) −1.29809e19 −0.0144256
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.h.1.2 yes 3
5.2 odd 4 50.22.b.g.49.5 6
5.3 odd 4 50.22.b.g.49.2 6
5.4 even 2 50.22.a.g.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.g.1.2 ✓ 3 5.4 even 2
50.22.a.h.1.2 yes 3 1.1 even 1 trivial
50.22.b.g.49.2 6 5.3 odd 4
50.22.b.g.49.5 6 5.2 odd 4