Properties

Label 50.22.a.i.1.4
Level $50$
Weight $22$
Character 50.1
Self dual yes
Analytic conductor $139.739$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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 1.4
Root \(18808.5\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +163894. q^{3} +1.04858e6 q^{4} -1.67827e8 q^{6} -1.27262e8 q^{7} -1.07374e9 q^{8} +1.64008e10 q^{9} +1.26472e10 q^{11} +1.71855e11 q^{12} +3.89950e11 q^{13} +1.30317e11 q^{14} +1.09951e12 q^{16} +6.14944e12 q^{17} -1.67944e13 q^{18} +4.71776e13 q^{19} -2.08575e13 q^{21} -1.29507e13 q^{22} -2.22845e14 q^{23} -1.75980e14 q^{24} -3.99308e14 q^{26} +9.73600e14 q^{27} -1.33444e14 q^{28} -1.52745e15 q^{29} -5.97515e15 q^{31} -1.12590e15 q^{32} +2.07279e15 q^{33} -6.29702e15 q^{34} +1.71975e16 q^{36} +4.33170e16 q^{37} -4.83098e16 q^{38} +6.39103e16 q^{39} +7.52194e16 q^{41} +2.13581e16 q^{42} +1.62856e17 q^{43} +1.32615e16 q^{44} +2.28194e17 q^{46} +2.41564e17 q^{47} +1.80203e17 q^{48} -5.42350e17 q^{49} +1.00785e18 q^{51} +4.08892e17 q^{52} -4.40493e17 q^{53} -9.96966e17 q^{54} +1.36647e17 q^{56} +7.73211e18 q^{57} +1.56411e18 q^{58} -5.30056e18 q^{59} +5.54333e18 q^{61} +6.11856e18 q^{62} -2.08720e18 q^{63} +1.15292e18 q^{64} -2.12254e18 q^{66} -4.60716e18 q^{67} +6.44815e18 q^{68} -3.65230e19 q^{69} +2.90933e19 q^{71} -1.76102e19 q^{72} +2.14785e19 q^{73} -4.43566e19 q^{74} +4.94693e19 q^{76} -1.60951e18 q^{77} -6.54441e19 q^{78} +1.49183e20 q^{79} -1.19912e19 q^{81} -7.70247e19 q^{82} -2.68194e20 q^{83} -2.18707e19 q^{84} -1.66765e20 q^{86} -2.50340e20 q^{87} -1.35798e19 q^{88} +1.29561e20 q^{89} -4.96259e19 q^{91} -2.33670e20 q^{92} -9.79290e20 q^{93} -2.47361e20 q^{94} -1.84528e20 q^{96} -7.45555e20 q^{97} +5.55367e20 q^{98} +2.07424e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4096 q^{2} - 96764 q^{3} + 4194304 q^{4} + 99086336 q^{6} - 162760528 q^{7} - 4294967296 q^{8} + 23614761712 q^{9} + 87559254108 q^{11} - 101464408064 q^{12} + 98098343096 q^{13} + 166666780672 q^{14}+ \cdots - 28\!\cdots\!76 q^{99}+O(q^{100}) \) 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
\(3\) 163894. 1.60247 0.801233 − 0.598352i \(-0.204178\pi\)
0.801233 + 0.598352i \(0.204178\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) −1.67827e8 −1.13312
\(7\) −1.27262e8 −0.170283 −0.0851414 − 0.996369i \(-0.527134\pi\)
−0.0851414 + 0.996369i \(0.527134\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 1.64008e10 1.56790
\(10\) 0 0
\(11\) 1.26472e10 0.147018 0.0735090 − 0.997295i \(-0.476580\pi\)
0.0735090 + 0.997295i \(0.476580\pi\)
\(12\) 1.71855e11 0.801233
\(13\) 3.89950e11 0.784519 0.392259 − 0.919855i \(-0.371694\pi\)
0.392259 + 0.919855i \(0.371694\pi\)
\(14\) 1.30317e11 0.120408
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 6.14944e12 0.739813 0.369906 − 0.929069i \(-0.379390\pi\)
0.369906 + 0.929069i \(0.379390\pi\)
\(18\) −1.67944e13 −1.10867
\(19\) 4.71776e13 1.76532 0.882659 − 0.470015i \(-0.155751\pi\)
0.882659 + 0.470015i \(0.155751\pi\)
\(20\) 0 0
\(21\) −2.08575e13 −0.272872
\(22\) −1.29507e13 −0.103957
\(23\) −2.22845e14 −1.12166 −0.560830 − 0.827931i \(-0.689518\pi\)
−0.560830 + 0.827931i \(0.689518\pi\)
\(24\) −1.75980e14 −0.566558
\(25\) 0 0
\(26\) −3.99308e14 −0.554739
\(27\) 9.73600e14 0.910041
\(28\) −1.33444e14 −0.0851414
\(29\) −1.52745e15 −0.674201 −0.337100 − 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(30\) 0 0
\(31\) −5.97515e15 −1.30934 −0.654668 − 0.755916i \(-0.727192\pi\)
−0.654668 + 0.755916i \(0.727192\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 2.07279e15 0.235592
\(34\) −6.29702e15 −0.523126
\(35\) 0 0
\(36\) 1.71975e16 0.783950
\(37\) 4.33170e16 1.48095 0.740474 − 0.672085i \(-0.234601\pi\)
0.740474 + 0.672085i \(0.234601\pi\)
\(38\) −4.83098e16 −1.24827
\(39\) 6.39103e16 1.25717
\(40\) 0 0
\(41\) 7.52194e16 0.875184 0.437592 − 0.899174i \(-0.355831\pi\)
0.437592 + 0.899174i \(0.355831\pi\)
\(42\) 2.13581e16 0.192950
\(43\) 1.62856e17 1.14917 0.574587 − 0.818444i \(-0.305163\pi\)
0.574587 + 0.818444i \(0.305163\pi\)
\(44\) 1.32615e16 0.0735090
\(45\) 0 0
\(46\) 2.28194e17 0.793134
\(47\) 2.41564e17 0.669891 0.334946 − 0.942237i \(-0.391282\pi\)
0.334946 + 0.942237i \(0.391282\pi\)
\(48\) 1.80203e17 0.400617
\(49\) −5.42350e17 −0.971004
\(50\) 0 0
\(51\) 1.00785e18 1.18553
\(52\) 4.08892e17 0.392259
\(53\) −4.40493e17 −0.345973 −0.172987 − 0.984924i \(-0.555342\pi\)
−0.172987 + 0.984924i \(0.555342\pi\)
\(54\) −9.96966e17 −0.643496
\(55\) 0 0
\(56\) 1.36647e17 0.0602040
\(57\) 7.73211e18 2.82886
\(58\) 1.56411e18 0.476732
\(59\) −5.30056e18 −1.35013 −0.675064 − 0.737759i \(-0.735884\pi\)
−0.675064 + 0.737759i \(0.735884\pi\)
\(60\) 0 0
\(61\) 5.54333e18 0.994965 0.497482 − 0.867474i \(-0.334258\pi\)
0.497482 + 0.867474i \(0.334258\pi\)
\(62\) 6.11856e18 0.925841
\(63\) −2.08720e18 −0.266986
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) −2.12254e18 −0.166588
\(67\) −4.60716e18 −0.308779 −0.154390 − 0.988010i \(-0.549341\pi\)
−0.154390 + 0.988010i \(0.549341\pi\)
\(68\) 6.44815e18 0.369906
\(69\) −3.65230e19 −1.79742
\(70\) 0 0
\(71\) 2.90933e19 1.06067 0.530336 − 0.847788i \(-0.322066\pi\)
0.530336 + 0.847788i \(0.322066\pi\)
\(72\) −1.76102e19 −0.554336
\(73\) 2.14785e19 0.584943 0.292471 − 0.956274i \(-0.405522\pi\)
0.292471 + 0.956274i \(0.405522\pi\)
\(74\) −4.43566e19 −1.04719
\(75\) 0 0
\(76\) 4.94693e19 0.882659
\(77\) −1.60951e18 −0.0250346
\(78\) −6.54441e19 −0.888950
\(79\) 1.49183e20 1.77269 0.886347 − 0.463021i \(-0.153235\pi\)
0.886347 + 0.463021i \(0.153235\pi\)
\(80\) 0 0
\(81\) −1.19912e19 −0.109590
\(82\) −7.70247e19 −0.618849
\(83\) −2.68194e20 −1.89727 −0.948637 − 0.316368i \(-0.897537\pi\)
−0.948637 + 0.316368i \(0.897537\pi\)
\(84\) −2.18707e19 −0.136436
\(85\) 0 0
\(86\) −1.66765e20 −0.812589
\(87\) −2.50340e20 −1.08038
\(88\) −1.35798e19 −0.0519787
\(89\) 1.29561e20 0.440431 0.220215 − 0.975451i \(-0.429324\pi\)
0.220215 + 0.975451i \(0.429324\pi\)
\(90\) 0 0
\(91\) −4.96259e19 −0.133590
\(92\) −2.33670e20 −0.560830
\(93\) −9.79290e20 −2.09817
\(94\) −2.47361e20 −0.473685
\(95\) 0 0
\(96\) −1.84528e20 −0.283279
\(97\) −7.45555e20 −1.02654 −0.513271 − 0.858227i \(-0.671566\pi\)
−0.513271 + 0.858227i \(0.671566\pi\)
\(98\) 5.55367e20 0.686603
\(99\) 2.07424e20 0.230510
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.i.1.4 ✓ 4
5.2 odd 4 50.22.b.h.49.1 8
5.3 odd 4 50.22.b.h.49.8 8
5.4 even 2 50.22.a.j.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.4 ✓ 4 1.1 even 1 trivial
50.22.a.j.1.1 yes 4 5.4 even 2
50.22.b.h.49.1 8 5.2 odd 4
50.22.b.h.49.8 8 5.3 odd 4