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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 = [4,0,0,-4194304,0,-205430784] 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: \(4\)
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: \( x^{4} + 237265x^{2} + 14073551424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(344.930i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.d.49.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} -194815. i q^{3} -1.04858e6 q^{4} -1.99490e8 q^{6} +9.63461e8i q^{7} +1.07374e9i q^{8} -2.74924e10 q^{9} +6.10882e10 q^{11} +2.04278e11i q^{12} -5.25383e10i q^{13} +9.86584e11 q^{14} +1.09951e12 q^{16} -1.13179e13i q^{17} +2.81522e13i q^{18} +4.53240e13 q^{19} +1.87696e14 q^{21} -6.25543e13i q^{22} -2.47091e14i q^{23} +2.09181e14 q^{24} -5.37992e13 q^{26} +3.31810e15i q^{27} -1.01026e15i q^{28} +2.58718e15 q^{29} -1.35601e15 q^{31} -1.12590e15i q^{32} -1.19009e16i q^{33} -1.15895e16 q^{34} +2.88279e16 q^{36} -1.35920e16i q^{37} -4.64117e16i q^{38} -1.02352e16 q^{39} +6.72480e16 q^{41} -1.92201e17i q^{42} +3.70641e16i q^{43} -6.40556e16 q^{44} -2.53021e17 q^{46} +3.17182e17i q^{47} -2.14201e17i q^{48} -3.69712e17 q^{49} -2.20489e18 q^{51} +5.50904e16i q^{52} -1.44909e18i q^{53} +3.39773e18 q^{54} -1.03451e18 q^{56} -8.82978e18i q^{57} -2.64928e18i q^{58} +1.08957e18 q^{59} -1.59498e17 q^{61} +1.38855e18i q^{62} -2.64879e19i q^{63} -1.15292e18 q^{64} -1.21865e19 q^{66} +8.71262e18i q^{67} +1.18677e19i q^{68} -4.81370e19 q^{69} -3.06845e19 q^{71} -2.95198e19i q^{72} -2.36491e19i q^{73} -1.39182e19 q^{74} -4.75256e19 q^{76} +5.88561e19i q^{77} +1.04809e19i q^{78} +9.35354e19 q^{79} +3.58834e20 q^{81} -6.88619e19i q^{82} +2.00855e20i q^{83} -1.96814e20 q^{84} +3.79536e19 q^{86} -5.04022e20i q^{87} +6.55929e19i q^{88} -3.85510e19 q^{89} +5.06186e19 q^{91} +2.59094e20i q^{92} +2.64170e20i q^{93} +3.24795e20 q^{94} -2.19342e20 q^{96} -5.46564e20i q^{97} +3.78585e20i q^{98} -1.67946e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4194304 q^{4} - 205430784 q^{6} - 51927197652 q^{9} + 43738388448 q^{11} + 2721577607168 q^{14} + 4398046511104 q^{16} + 104145722023120 q^{19} + 306321025902528 q^{21} + 215409789763584 q^{24} + 843899164205056 q^{26}+ \cdots - 34\!\cdots\!24 q^{99}+O(q^{100}) \) 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
\(3\) − 194815.i − 1.90480i −0.304856 − 0.952398i \(-0.598608\pi\)
0.304856 − 0.952398i \(-0.401392\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) −1.99490e8 −1.34689
\(7\) 9.63461e8i 1.28915i 0.764539 + 0.644577i \(0.222967\pi\)
−0.764539 + 0.644577i \(0.777033\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) −2.74924e10 −2.62825
\(10\) 0 0
\(11\) 6.10882e10 0.710124 0.355062 − 0.934843i \(-0.384460\pi\)
0.355062 + 0.934843i \(0.384460\pi\)
\(12\) 2.04278e11i 0.952398i
\(13\) − 5.25383e10i − 0.105699i −0.998602 − 0.0528495i \(-0.983170\pi\)
0.998602 − 0.0528495i \(-0.0168303\pi\)
\(14\) 9.86584e11 0.911570
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 1.13179e13i − 1.36161i −0.732466 − 0.680803i \(-0.761631\pi\)
0.732466 − 0.680803i \(-0.238369\pi\)
\(18\) 2.81522e13i 1.85845i
\(19\) 4.53240e13 1.69596 0.847979 − 0.530030i \(-0.177819\pi\)
0.847979 + 0.530030i \(0.177819\pi\)
\(20\) 0 0
\(21\) 1.87696e14 2.45558
\(22\) − 6.25543e13i − 0.502133i
\(23\) − 2.47091e14i − 1.24370i −0.783137 − 0.621849i \(-0.786382\pi\)
0.783137 − 0.621849i \(-0.213618\pi\)
\(24\) 2.09181e14 0.673447
\(25\) 0 0
\(26\) −5.37992e13 −0.0747404
\(27\) 3.31810e15i 3.10149i
\(28\) − 1.01026e15i − 0.644577i
\(29\) 2.58718e15 1.14195 0.570977 − 0.820966i \(-0.306565\pi\)
0.570977 + 0.820966i \(0.306565\pi\)
\(30\) 0 0
\(31\) −1.35601e15 −0.297142 −0.148571 − 0.988902i \(-0.547467\pi\)
−0.148571 + 0.988902i \(0.547467\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) − 1.19009e16i − 1.35264i
\(34\) −1.15895e16 −0.962801
\(35\) 0 0
\(36\) 2.88279e16 1.31413
\(37\) − 1.35920e16i − 0.464692i −0.972633 − 0.232346i \(-0.925360\pi\)
0.972633 − 0.232346i \(-0.0746401\pi\)
\(38\) − 4.64117e16i − 1.19922i
\(39\) −1.02352e16 −0.201335
\(40\) 0 0
\(41\) 6.72480e16 0.782436 0.391218 − 0.920298i \(-0.372054\pi\)
0.391218 + 0.920298i \(0.372054\pi\)
\(42\) − 1.92201e17i − 1.73636i
\(43\) 3.70641e16i 0.261538i 0.991413 + 0.130769i \(0.0417446\pi\)
−0.991413 + 0.130769i \(0.958255\pi\)
\(44\) −6.40556e16 −0.355062
\(45\) 0 0
\(46\) −2.53021e17 −0.879427
\(47\) 3.17182e17i 0.879592i 0.898098 + 0.439796i \(0.144949\pi\)
−0.898098 + 0.439796i \(0.855051\pi\)
\(48\) − 2.14201e17i − 0.476199i
\(49\) −3.69712e17 −0.661919
\(50\) 0 0
\(51\) −2.20489e18 −2.59358
\(52\) 5.50904e16i 0.0528495i
\(53\) − 1.44909e18i − 1.13815i −0.822286 − 0.569074i \(-0.807302\pi\)
0.822286 − 0.569074i \(-0.192698\pi\)
\(54\) 3.39773e18 2.19308
\(55\) 0 0
\(56\) −1.03451e18 −0.455785
\(57\) − 8.82978e18i − 3.23046i
\(58\) − 2.64928e18i − 0.807483i
\(59\) 1.08957e18 0.277529 0.138764 − 0.990325i \(-0.455687\pi\)
0.138764 + 0.990325i \(0.455687\pi\)
\(60\) 0 0
\(61\) −1.59498e17 −0.0286281 −0.0143140 − 0.999898i \(-0.504556\pi\)
−0.0143140 + 0.999898i \(0.504556\pi\)
\(62\) 1.38855e18i 0.210111i
\(63\) − 2.64879e19i − 3.38822i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −1.21865e19 −0.956462
\(67\) 8.71262e18i 0.583934i 0.956428 + 0.291967i \(0.0943097\pi\)
−0.956428 + 0.291967i \(0.905690\pi\)
\(68\) 1.18677e19i 0.680803i
\(69\) −4.81370e19 −2.36899
\(70\) 0 0
\(71\) −3.06845e19 −1.11868 −0.559340 − 0.828938i \(-0.688946\pi\)
−0.559340 + 0.828938i \(0.688946\pi\)
\(72\) − 2.95198e19i − 0.929227i
\(73\) − 2.36491e19i − 0.644059i −0.946730 − 0.322029i \(-0.895635\pi\)
0.946730 − 0.322029i \(-0.104365\pi\)
\(74\) −1.39182e19 −0.328587
\(75\) 0 0
\(76\) −4.75256e19 −0.847979
\(77\) 5.88561e19i 0.915459i
\(78\) 1.04809e19i 0.142365i
\(79\) 9.35354e19 1.11145 0.555727 − 0.831365i \(-0.312440\pi\)
0.555727 + 0.831365i \(0.312440\pi\)
\(80\) 0 0
\(81\) 3.58834e20 3.27945
\(82\) − 6.88619e19i − 0.553266i
\(83\) 2.00855e20i 1.42090i 0.703750 + 0.710448i \(0.251508\pi\)
−0.703750 + 0.710448i \(0.748492\pi\)
\(84\) −1.96814e20 −1.22779
\(85\) 0 0
\(86\) 3.79536e19 0.184935
\(87\) − 5.04022e20i − 2.17519i
\(88\) 6.55929e19i 0.251067i
\(89\) −3.85510e19 −0.131051 −0.0655255 − 0.997851i \(-0.520872\pi\)
−0.0655255 + 0.997851i \(0.520872\pi\)
\(90\) 0 0
\(91\) 5.06186e19 0.136262
\(92\) 2.59094e20i 0.621849i
\(93\) 2.64170e20i 0.565996i
\(94\) 3.24795e20 0.621965
\(95\) 0 0
\(96\) −2.19342e20 −0.336724
\(97\) − 5.46564e20i − 0.752554i −0.926507 − 0.376277i \(-0.877204\pi\)
0.926507 − 0.376277i \(-0.122796\pi\)
\(98\) 3.78585e20i 0.468047i
\(99\) −1.67946e21 −1.86638
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.d.49.1 4
5.2 odd 4 50.22.a.e.1.1 2
5.3 odd 4 10.22.a.c.1.2 ✓ 2
5.4 even 2 inner 50.22.b.d.49.4 4
20.3 even 4 80.22.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.c.1.2 ✓ 2 5.3 odd 4
50.22.a.e.1.1 2 5.2 odd 4
50.22.b.d.49.1 4 1.1 even 1 trivial
50.22.b.d.49.4 4 5.4 even 2 inner
80.22.a.b.1.1 2 20.3 even 4