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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,Mod(49,50)] 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("50.49"); 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([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8388608,0,198172672] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 631153566 x^{6} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{4}\cdot 5^{14}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.8
Root \(-18808.5i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.h.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1024.00i q^{2} +163894. i q^{3} -1.04858e6 q^{4} -1.67827e8 q^{6} +1.27262e8i q^{7} -1.07374e9i q^{8} -1.64008e10 q^{9} +1.26472e10 q^{11} -1.71855e11i q^{12} +3.89950e11i q^{13} -1.30317e11 q^{14} +1.09951e12 q^{16} -6.14944e12i q^{17} -1.67944e13i q^{18} -4.71776e13 q^{19} -2.08575e13 q^{21} +1.29507e13i q^{22} -2.22845e14i q^{23} +1.75980e14 q^{24} -3.99308e14 q^{26} -9.73600e14i q^{27} -1.33444e14i q^{28} +1.52745e15 q^{29} -5.97515e15 q^{31} +1.12590e15i q^{32} +2.07279e15i q^{33} +6.29702e15 q^{34} +1.71975e16 q^{36} -4.33170e16i q^{37} -4.83098e16i q^{38} -6.39103e16 q^{39} +7.52194e16 q^{41} -2.13581e16i q^{42} +1.62856e17i q^{43} -1.32615e16 q^{44} +2.28194e17 q^{46} -2.41564e17i q^{47} +1.80203e17i q^{48} +5.42350e17 q^{49} +1.00785e18 q^{51} -4.08892e17i q^{52} -4.40493e17i q^{53} +9.96966e17 q^{54} +1.36647e17 q^{56} -7.73211e18i q^{57} +1.56411e18i q^{58} +5.30056e18 q^{59} +5.54333e18 q^{61} -6.11856e18i q^{62} -2.08720e18i q^{63} -1.15292e18 q^{64} -2.12254e18 q^{66} +4.60716e18i q^{67} +6.44815e18i q^{68} +3.65230e19 q^{69} +2.90933e19 q^{71} +1.76102e19i q^{72} +2.14785e19i q^{73} +4.43566e19 q^{74} +4.94693e19 q^{76} +1.60951e18i q^{77} -6.54441e19i q^{78} -1.49183e20 q^{79} -1.19912e19 q^{81} +7.70247e19i q^{82} -2.68194e20i q^{83} +2.18707e19 q^{84} -1.66765e20 q^{86} +2.50340e20i q^{87} -1.35798e19i q^{88} -1.29561e20 q^{89} -4.96259e19 q^{91} +2.33670e20i q^{92} -9.79290e20i q^{93} +2.47361e20 q^{94} -1.84528e20 q^{96} +7.45555e20i q^{97} +5.55367e20i q^{98} -2.07424e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8388608 q^{4} + 198172672 q^{6} - 47229523424 q^{9} + 175118508216 q^{11} - 333333561344 q^{14} + 8796093022208 q^{16} - 167026213839880 q^{19} - 8313944227904 q^{21} - 207799107715072 q^{24} - 200905406660608 q^{26}+ \cdots + 56\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1024.00i 0.707107i
\(3\) 163894.i 1.60247i 0.598352 + 0.801233i \(0.295822\pi\)
−0.598352 + 0.801233i \(0.704178\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) −1.67827e8 −1.13312
\(7\) 1.27262e8i 0.170283i 0.996369 + 0.0851414i \(0.0271342\pi\)
−0.996369 + 0.0851414i \(0.972866\pi\)
\(8\) − 1.07374e9i − 0.353553i
\(9\) −1.64008e10 −1.56790
\(10\) 0 0
\(11\) 1.26472e10 0.147018 0.0735090 − 0.997295i \(-0.476580\pi\)
0.0735090 + 0.997295i \(0.476580\pi\)
\(12\) − 1.71855e11i − 0.801233i
\(13\) 3.89950e11i 0.784519i 0.919855 + 0.392259i \(0.128306\pi\)
−0.919855 + 0.392259i \(0.871694\pi\)
\(14\) −1.30317e11 −0.120408
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 6.14944e12i − 0.739813i −0.929069 − 0.369906i \(-0.879390\pi\)
0.929069 − 0.369906i \(-0.120610\pi\)
\(18\) − 1.67944e13i − 1.10867i
\(19\) −4.71776e13 −1.76532 −0.882659 − 0.470015i \(-0.844249\pi\)
−0.882659 + 0.470015i \(0.844249\pi\)
\(20\) 0 0
\(21\) −2.08575e13 −0.272872
\(22\) 1.29507e13i 0.103957i
\(23\) − 2.22845e14i − 1.12166i −0.827931 − 0.560830i \(-0.810482\pi\)
0.827931 − 0.560830i \(-0.189518\pi\)
\(24\) 1.75980e14 0.566558
\(25\) 0 0
\(26\) −3.99308e14 −0.554739
\(27\) − 9.73600e14i − 0.910041i
\(28\) − 1.33444e14i − 0.0851414i
\(29\) 1.52745e15 0.674201 0.337100 − 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(30\) 0 0
\(31\) −5.97515e15 −1.30934 −0.654668 − 0.755916i \(-0.727192\pi\)
−0.654668 + 0.755916i \(0.727192\pi\)
\(32\) 1.12590e15i 0.176777i
\(33\) 2.07279e15i 0.235592i
\(34\) 6.29702e15 0.523126
\(35\) 0 0
\(36\) 1.71975e16 0.783950
\(37\) − 4.33170e16i − 1.48095i −0.672085 − 0.740474i \(-0.734601\pi\)
0.672085 − 0.740474i \(-0.265399\pi\)
\(38\) − 4.83098e16i − 1.24827i
\(39\) −6.39103e16 −1.25717
\(40\) 0 0
\(41\) 7.52194e16 0.875184 0.437592 − 0.899174i \(-0.355831\pi\)
0.437592 + 0.899174i \(0.355831\pi\)
\(42\) − 2.13581e16i − 0.192950i
\(43\) 1.62856e17i 1.14917i 0.818444 + 0.574587i \(0.194837\pi\)
−0.818444 + 0.574587i \(0.805163\pi\)
\(44\) −1.32615e16 −0.0735090
\(45\) 0 0
\(46\) 2.28194e17 0.793134
\(47\) − 2.41564e17i − 0.669891i −0.942237 − 0.334946i \(-0.891282\pi\)
0.942237 − 0.334946i \(-0.108718\pi\)
\(48\) 1.80203e17i 0.400617i
\(49\) 5.42350e17 0.971004
\(50\) 0 0
\(51\) 1.00785e18 1.18553
\(52\) − 4.08892e17i − 0.392259i
\(53\) − 4.40493e17i − 0.345973i −0.984924 − 0.172987i \(-0.944658\pi\)
0.984924 − 0.172987i \(-0.0553418\pi\)
\(54\) 9.96966e17 0.643496
\(55\) 0 0
\(56\) 1.36647e17 0.0602040
\(57\) − 7.73211e18i − 2.82886i
\(58\) 1.56411e18i 0.476732i
\(59\) 5.30056e18 1.35013 0.675064 − 0.737759i \(-0.264116\pi\)
0.675064 + 0.737759i \(0.264116\pi\)
\(60\) 0 0
\(61\) 5.54333e18 0.994965 0.497482 − 0.867474i \(-0.334258\pi\)
0.497482 + 0.867474i \(0.334258\pi\)
\(62\) − 6.11856e18i − 0.925841i
\(63\) − 2.08720e18i − 0.266986i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −2.12254e18 −0.166588
\(67\) 4.60716e18i 0.308779i 0.988010 + 0.154390i \(0.0493411\pi\)
−0.988010 + 0.154390i \(0.950659\pi\)
\(68\) 6.44815e18i 0.369906i
\(69\) 3.65230e19 1.79742
\(70\) 0 0
\(71\) 2.90933e19 1.06067 0.530336 − 0.847788i \(-0.322066\pi\)
0.530336 + 0.847788i \(0.322066\pi\)
\(72\) 1.76102e19i 0.554336i
\(73\) 2.14785e19i 0.584943i 0.956274 + 0.292471i \(0.0944776\pi\)
−0.956274 + 0.292471i \(0.905522\pi\)
\(74\) 4.43566e19 1.04719
\(75\) 0 0
\(76\) 4.94693e19 0.882659
\(77\) 1.60951e18i 0.0250346i
\(78\) − 6.54441e19i − 0.888950i
\(79\) −1.49183e20 −1.77269 −0.886347 − 0.463021i \(-0.846765\pi\)
−0.886347 + 0.463021i \(0.846765\pi\)
\(80\) 0 0
\(81\) −1.19912e19 −0.109590
\(82\) 7.70247e19i 0.618849i
\(83\) − 2.68194e20i − 1.89727i −0.316368 − 0.948637i \(-0.602463\pi\)
0.316368 − 0.948637i \(-0.397537\pi\)
\(84\) 2.18707e19 0.136436
\(85\) 0 0
\(86\) −1.66765e20 −0.812589
\(87\) 2.50340e20i 1.08038i
\(88\) − 1.35798e19i − 0.0519787i
\(89\) −1.29561e20 −0.440431 −0.220215 − 0.975451i \(-0.570676\pi\)
−0.220215 + 0.975451i \(0.570676\pi\)
\(90\) 0 0
\(91\) −4.96259e19 −0.133590
\(92\) 2.33670e20i 0.560830i
\(93\) − 9.79290e20i − 2.09817i
\(94\) 2.47361e20 0.473685
\(95\) 0 0
\(96\) −1.84528e20 −0.283279
\(97\) 7.45555e20i 1.02654i 0.858227 + 0.513271i \(0.171566\pi\)
−0.858227 + 0.513271i \(0.828434\pi\)
\(98\) 5.55367e20i 0.686603i
\(99\) −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.b.h.49.8 8
5.2 odd 4 50.22.a.i.1.4 ✓ 4
5.3 odd 4 50.22.a.j.1.1 yes 4
5.4 even 2 inner 50.22.b.h.49.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.4 ✓ 4 5.2 odd 4
50.22.a.j.1.1 yes 4 5.3 odd 4
50.22.b.h.49.1 8 5.4 even 2 inner
50.22.b.h.49.8 8 1.1 even 1 trivial