Properties

Label 50.22.a.g.1.1
Level $50$
Weight $22$
Character 50.1
Self dual yes
Analytic conductor $139.739$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.1
Root \(-5371.39\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} -176613. q^{3} +1.04858e6 q^{4} +1.80851e8 q^{6} +1.28981e9 q^{7} -1.07374e9 q^{8} +2.07317e10 q^{9} -8.21479e9 q^{11} -1.85192e11 q^{12} +2.70307e11 q^{13} -1.32077e12 q^{14} +1.09951e12 q^{16} +1.05954e13 q^{17} -2.12292e13 q^{18} -2.60103e13 q^{19} -2.27798e14 q^{21} +8.41194e12 q^{22} -3.27037e14 q^{23} +1.89636e14 q^{24} -2.76795e14 q^{26} -1.81405e15 q^{27} +1.35247e15 q^{28} +2.01684e15 q^{29} -7.82354e15 q^{31} -1.12590e15 q^{32} +1.45084e15 q^{33} -1.08496e16 q^{34} +2.17387e16 q^{36} +2.12834e16 q^{37} +2.66345e16 q^{38} -4.77397e16 q^{39} -9.00182e16 q^{41} +2.33265e17 q^{42} -6.03505e16 q^{43} -8.61383e15 q^{44} +3.34886e17 q^{46} +3.50369e17 q^{47} -1.94188e17 q^{48} +1.10508e18 q^{49} -1.87127e18 q^{51} +2.83438e17 q^{52} +1.24046e18 q^{53} +1.85758e18 q^{54} -1.38493e18 q^{56} +4.59374e18 q^{57} -2.06524e18 q^{58} -5.56450e18 q^{59} -5.81826e18 q^{61} +8.01130e18 q^{62} +2.67400e19 q^{63} +1.15292e18 q^{64} -1.48566e18 q^{66} +2.21752e19 q^{67} +1.11100e19 q^{68} +5.77588e19 q^{69} +3.61226e19 q^{71} -2.22605e19 q^{72} -4.54704e19 q^{73} -2.17942e19 q^{74} -2.72738e19 q^{76} -1.05956e19 q^{77} +4.88854e19 q^{78} -9.81840e19 q^{79} +1.03523e20 q^{81} +9.21786e19 q^{82} -4.19887e19 q^{83} -2.38863e20 q^{84} +6.17989e19 q^{86} -3.56199e20 q^{87} +8.82056e18 q^{88} -3.94595e20 q^{89} +3.48646e20 q^{91} -3.42923e20 q^{92} +1.38174e21 q^{93} -3.58778e20 q^{94} +1.98848e20 q^{96} -6.34667e20 q^{97} -1.13160e21 q^{98} -1.70306e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3072 q^{2} - 46383 q^{3} + 3145728 q^{4} + 47496192 q^{6} + 911775234 q^{7} - 3221225472 q^{8} + 25062837354 q^{9} - 77565926349 q^{11} - 48636100608 q^{12} - 29305708548 q^{13} - 933657839616 q^{14}+ \cdots - 11\!\cdots\!82 q^{99}+O(q^{100}) \) 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
\(3\) −176613. −1.72683 −0.863413 − 0.504497i \(-0.831678\pi\)
−0.863413 + 0.504497i \(0.831678\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) 1.80851e8 1.22105
\(7\) 1.28981e9 1.72583 0.862915 − 0.505349i \(-0.168636\pi\)
0.862915 + 0.505349i \(0.168636\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 2.07317e10 1.98193
\(10\) 0 0
\(11\) −8.21479e9 −0.0954934 −0.0477467 − 0.998859i \(-0.515204\pi\)
−0.0477467 + 0.998859i \(0.515204\pi\)
\(12\) −1.85192e11 −0.863413
\(13\) 2.70307e11 0.543817 0.271908 − 0.962323i \(-0.412345\pi\)
0.271908 + 0.962323i \(0.412345\pi\)
\(14\) −1.32077e12 −1.22035
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.05954e13 1.27468 0.637341 − 0.770582i \(-0.280034\pi\)
0.637341 + 0.770582i \(0.280034\pi\)
\(18\) −2.12292e13 −1.40144
\(19\) −2.60103e13 −0.973267 −0.486634 − 0.873606i \(-0.661775\pi\)
−0.486634 + 0.873606i \(0.661775\pi\)
\(20\) 0 0
\(21\) −2.27798e14 −2.98021
\(22\) 8.41194e12 0.0675240
\(23\) −3.27037e14 −1.64609 −0.823046 − 0.567974i \(-0.807727\pi\)
−0.823046 + 0.567974i \(0.807727\pi\)
\(24\) 1.89636e14 0.610525
\(25\) 0 0
\(26\) −2.76795e14 −0.384536
\(27\) −1.81405e15 −1.69562
\(28\) 1.35247e15 0.862915
\(29\) 2.01684e15 0.890208 0.445104 − 0.895479i \(-0.353167\pi\)
0.445104 + 0.895479i \(0.353167\pi\)
\(30\) 0 0
\(31\) −7.82354e15 −1.71437 −0.857187 − 0.515006i \(-0.827790\pi\)
−0.857187 + 0.515006i \(0.827790\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 1.45084e15 0.164900
\(34\) −1.08496e16 −0.901336
\(35\) 0 0
\(36\) 2.17387e16 0.990965
\(37\) 2.12834e16 0.727649 0.363825 − 0.931467i \(-0.381471\pi\)
0.363825 + 0.931467i \(0.381471\pi\)
\(38\) 2.66345e16 0.688204
\(39\) −4.77397e16 −0.939077
\(40\) 0 0
\(41\) −9.00182e16 −1.04737 −0.523684 − 0.851912i \(-0.675443\pi\)
−0.523684 + 0.851912i \(0.675443\pi\)
\(42\) 2.33265e17 2.10733
\(43\) −6.03505e16 −0.425855 −0.212928 − 0.977068i \(-0.568300\pi\)
−0.212928 + 0.977068i \(0.568300\pi\)
\(44\) −8.61383e15 −0.0477467
\(45\) 0 0
\(46\) 3.34886e17 1.16396
\(47\) 3.50369e17 0.971624 0.485812 − 0.874063i \(-0.338524\pi\)
0.485812 + 0.874063i \(0.338524\pi\)
\(48\) −1.94188e17 −0.431707
\(49\) 1.10508e18 1.97849
\(50\) 0 0
\(51\) −1.87127e18 −2.20115
\(52\) 2.83438e17 0.271908
\(53\) 1.24046e18 0.974284 0.487142 − 0.873323i \(-0.338039\pi\)
0.487142 + 0.873323i \(0.338039\pi\)
\(54\) 1.85758e18 1.19899
\(55\) 0 0
\(56\) −1.38493e18 −0.610173
\(57\) 4.59374e18 1.68066
\(58\) −2.06524e18 −0.629472
\(59\) −5.56450e18 −1.41736 −0.708680 − 0.705530i \(-0.750709\pi\)
−0.708680 + 0.705530i \(0.750709\pi\)
\(60\) 0 0
\(61\) −5.81826e18 −1.04431 −0.522155 − 0.852850i \(-0.674872\pi\)
−0.522155 + 0.852850i \(0.674872\pi\)
\(62\) 8.01130e18 1.21224
\(63\) 2.67400e19 3.42047
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) −1.48566e18 −0.116602
\(67\) 2.21752e19 1.48621 0.743107 − 0.669172i \(-0.233351\pi\)
0.743107 + 0.669172i \(0.233351\pi\)
\(68\) 1.11100e19 0.637341
\(69\) 5.77588e19 2.84252
\(70\) 0 0
\(71\) 3.61226e19 1.31694 0.658471 − 0.752606i \(-0.271204\pi\)
0.658471 + 0.752606i \(0.271204\pi\)
\(72\) −2.22605e19 −0.700718
\(73\) −4.54704e19 −1.23834 −0.619168 − 0.785258i \(-0.712530\pi\)
−0.619168 + 0.785258i \(0.712530\pi\)
\(74\) −2.17942e19 −0.514526
\(75\) 0 0
\(76\) −2.72738e19 −0.486634
\(77\) −1.05956e19 −0.164805
\(78\) 4.88854e19 0.664028
\(79\) −9.81840e19 −1.16669 −0.583346 − 0.812224i \(-0.698257\pi\)
−0.583346 + 0.812224i \(0.698257\pi\)
\(80\) 0 0
\(81\) 1.03523e20 0.946115
\(82\) 9.21786e19 0.740602
\(83\) −4.19887e19 −0.297038 −0.148519 − 0.988910i \(-0.547451\pi\)
−0.148519 + 0.988910i \(0.547451\pi\)
\(84\) −2.38863e20 −1.49010
\(85\) 0 0
\(86\) 6.17989e19 0.301125
\(87\) −3.56199e20 −1.53723
\(88\) 8.82056e18 0.0337620
\(89\) −3.94595e20 −1.34139 −0.670697 − 0.741732i \(-0.734005\pi\)
−0.670697 + 0.741732i \(0.734005\pi\)
\(90\) 0 0
\(91\) 3.48646e20 0.938535
\(92\) −3.42923e20 −0.823046
\(93\) 1.38174e21 2.96042
\(94\) −3.58778e20 −0.687042
\(95\) 0 0
\(96\) 1.98848e20 0.305263
\(97\) −6.34667e20 −0.873862 −0.436931 − 0.899495i \(-0.643935\pi\)
−0.436931 + 0.899495i \(0.643935\pi\)
\(98\) −1.13160e21 −1.39900
\(99\) −1.70306e20 −0.189261
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.g.1.1 ✓ 3
5.2 odd 4 50.22.b.g.49.3 6
5.3 odd 4 50.22.b.g.49.4 6
5.4 even 2 50.22.a.h.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.g.1.1 ✓ 3 1.1 even 1 trivial
50.22.a.h.1.3 yes 3 5.4 even 2
50.22.b.g.49.3 6 5.2 odd 4
50.22.b.g.49.4 6 5.3 odd 4