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Properties

Label 50.22.b.d.49.2
Level $50$
Weight $22$
Character 50.49
Analytic conductor $139.739$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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(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 49.2
Root \(-343.930i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.d.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} +94506.7i q^{3} -1.04858e6 q^{4} +9.67749e7 q^{6} +3.65434e8i q^{7} +1.07374e9i q^{8} +1.52883e9 q^{9} -3.92190e10 q^{11} -9.90975e10i q^{12} +4.64598e11i q^{13} +3.74204e11 q^{14} +1.09951e12 q^{16} +2.43124e11i q^{17} -1.56552e12i q^{18} +6.74889e12 q^{19} -3.45360e13 q^{21} +4.01602e13i q^{22} +1.66102e14i q^{23} -1.01476e14 q^{24} +4.75749e14 q^{26} +1.13306e15i q^{27} -3.83185e14i q^{28} -2.85150e15 q^{29} +3.48202e15 q^{31} -1.12590e15i q^{32} -3.70646e15i q^{33} +2.48959e14 q^{34} -1.60309e15 q^{36} +3.87028e15i q^{37} -6.91086e15i q^{38} -4.39077e16 q^{39} +1.53913e17 q^{41} +3.53648e16i q^{42} +6.75910e15i q^{43} +4.11241e16 q^{44} +1.70089e17 q^{46} -1.11128e17i q^{47} +1.03911e17i q^{48} +4.25004e17 q^{49} -2.29769e16 q^{51} -4.87167e17i q^{52} +2.06153e18i q^{53} +1.16025e18 q^{54} -3.92382e17 q^{56} +6.37815e17i q^{57} +2.91994e18i q^{58} -5.84178e18 q^{59} +4.90762e18 q^{61} -3.56559e18i q^{62} +5.58686e17i q^{63} -1.15292e18 q^{64} -3.79541e18 q^{66} +1.35893e19i q^{67} -2.54934e17i q^{68} -1.56978e19 q^{69} +3.28855e19 q^{71} +1.64157e18i q^{72} -3.25004e19i q^{73} +3.96317e18 q^{74} -7.07672e18 q^{76} -1.43319e19i q^{77} +4.49615e19i q^{78} -5.11527e19 q^{79} -9.10896e19 q^{81} -1.57607e20i q^{82} -2.02190e20i q^{83} +3.62136e19 q^{84} +6.92131e18 q^{86} -2.69486e20i q^{87} -4.21111e19i q^{88} -2.17189e20 q^{89} -1.69780e20 q^{91} -1.74171e20i q^{92} +3.29075e20i q^{93} -1.13795e20 q^{94} +1.06405e20 q^{96} -1.03286e21i q^{97} -4.35204e20i q^{98} -5.99592e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4194304 q^{4} - 205430784 q^{6} - 51927197652 q^{9} + 43738388448 q^{11} + 2721577607168 q^{14} + 4398046511104 q^{16} + 104145722023120 q^{19} + 306321025902528 q^{21} + 215409789763584 q^{24} + 843899164205056 q^{26}+ \cdots - 34\!\cdots\!24 q^{99}+O(q^{100}) \) 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
\(3\) 94506.7i 0.924037i 0.886870 + 0.462019i \(0.152875\pi\)
−0.886870 + 0.462019i \(0.847125\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 9.67749e7 0.653393
\(7\) 3.65434e8i 0.488967i 0.969653 + 0.244483i \(0.0786184\pi\)
−0.969653 + 0.244483i \(0.921382\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) 1.52883e9 0.146155
\(10\) 0 0
\(11\) −3.92190e10 −0.455904 −0.227952 − 0.973672i \(-0.573203\pi\)
−0.227952 + 0.973672i \(0.573203\pi\)
\(12\) − 9.90975e10i − 0.462019i
\(13\) 4.64598e11i 0.934701i 0.884072 + 0.467350i \(0.154791\pi\)
−0.884072 + 0.467350i \(0.845209\pi\)
\(14\) 3.74204e11 0.345752
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 2.43124e11i 0.0292492i 0.999893 + 0.0146246i \(0.00465533\pi\)
−0.999893 + 0.0146246i \(0.995345\pi\)
\(18\) − 1.56552e12i − 0.103347i
\(19\) 6.74889e12 0.252534 0.126267 − 0.991996i \(-0.459700\pi\)
0.126267 + 0.991996i \(0.459700\pi\)
\(20\) 0 0
\(21\) −3.45360e13 −0.451824
\(22\) 4.01602e13i 0.322373i
\(23\) 1.66102e14i 0.836052i 0.908435 + 0.418026i \(0.137278\pi\)
−0.908435 + 0.418026i \(0.862722\pi\)
\(24\) −1.01476e14 −0.326697
\(25\) 0 0
\(26\) 4.75749e14 0.660933
\(27\) 1.13306e15i 1.05909i
\(28\) − 3.83185e14i − 0.244483i
\(29\) −2.85150e15 −1.25862 −0.629310 − 0.777155i \(-0.716662\pi\)
−0.629310 + 0.777155i \(0.716662\pi\)
\(30\) 0 0
\(31\) 3.48202e15 0.763017 0.381508 − 0.924365i \(-0.375405\pi\)
0.381508 + 0.924365i \(0.375405\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) − 3.70646e15i − 0.421272i
\(34\) 2.48959e14 0.0206823
\(35\) 0 0
\(36\) −1.60309e15 −0.0730774
\(37\) 3.87028e15i 0.132320i 0.997809 + 0.0661598i \(0.0210747\pi\)
−0.997809 + 0.0661598i \(0.978925\pi\)
\(38\) − 6.91086e15i − 0.178568i
\(39\) −4.39077e16 −0.863698
\(40\) 0 0
\(41\) 1.53913e17 1.79079 0.895396 − 0.445272i \(-0.146893\pi\)
0.895396 + 0.445272i \(0.146893\pi\)
\(42\) 3.53648e16i 0.319488i
\(43\) 6.75910e15i 0.0476947i 0.999716 + 0.0238473i \(0.00759156\pi\)
−0.999716 + 0.0238473i \(0.992408\pi\)
\(44\) 4.11241e16 0.227952
\(45\) 0 0
\(46\) 1.70089e17 0.591178
\(47\) − 1.11128e17i − 0.308174i −0.988057 − 0.154087i \(-0.950756\pi\)
0.988057 − 0.154087i \(-0.0492436\pi\)
\(48\) 1.03911e17i 0.231009i
\(49\) 4.25004e17 0.760911
\(50\) 0 0
\(51\) −2.29769e16 −0.0270274
\(52\) − 4.87167e17i − 0.467350i
\(53\) 2.06153e18i 1.61917i 0.587003 + 0.809584i \(0.300307\pi\)
−0.587003 + 0.809584i \(0.699693\pi\)
\(54\) 1.16025e18 0.748890
\(55\) 0 0
\(56\) −3.92382e17 −0.172876
\(57\) 6.37815e17i 0.233351i
\(58\) 2.91994e18i 0.889978i
\(59\) −5.84178e18 −1.48799 −0.743993 − 0.668188i \(-0.767070\pi\)
−0.743993 + 0.668188i \(0.767070\pi\)
\(60\) 0 0
\(61\) 4.90762e18 0.880861 0.440431 − 0.897787i \(-0.354826\pi\)
0.440431 + 0.897787i \(0.354826\pi\)
\(62\) − 3.56559e18i − 0.539534i
\(63\) 5.58686e17i 0.0714648i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −3.79541e18 −0.297884
\(67\) 1.35893e19i 0.910776i 0.890293 + 0.455388i \(0.150500\pi\)
−0.890293 + 0.455388i \(0.849500\pi\)
\(68\) − 2.54934e17i − 0.0146246i
\(69\) −1.56978e19 −0.772543
\(70\) 0 0
\(71\) 3.28855e19 1.19893 0.599463 − 0.800403i \(-0.295381\pi\)
0.599463 + 0.800403i \(0.295381\pi\)
\(72\) 1.64157e18i 0.0516735i
\(73\) − 3.25004e19i − 0.885112i −0.896741 − 0.442556i \(-0.854072\pi\)
0.896741 − 0.442556i \(-0.145928\pi\)
\(74\) 3.96317e18 0.0935641
\(75\) 0 0
\(76\) −7.07672e18 −0.126267
\(77\) − 1.43319e19i − 0.222922i
\(78\) 4.49615e19i 0.610727i
\(79\) −5.11527e19 −0.607832 −0.303916 − 0.952699i \(-0.598294\pi\)
−0.303916 + 0.952699i \(0.598294\pi\)
\(80\) 0 0
\(81\) −9.10896e19 −0.832484
\(82\) − 1.57607e20i − 1.26628i
\(83\) − 2.02190e20i − 1.43034i −0.698949 − 0.715171i \(-0.746349\pi\)
0.698949 − 0.715171i \(-0.253651\pi\)
\(84\) 3.62136e19 0.225912
\(85\) 0 0
\(86\) 6.92131e18 0.0337252
\(87\) − 2.69486e20i − 1.16301i
\(88\) − 4.21111e19i − 0.161186i
\(89\) −2.17189e20 −0.738316 −0.369158 − 0.929367i \(-0.620354\pi\)
−0.369158 + 0.929367i \(0.620354\pi\)
\(90\) 0 0
\(91\) −1.69780e20 −0.457038
\(92\) − 1.74171e20i − 0.418026i
\(93\) 3.29075e20i 0.705056i
\(94\) −1.13795e20 −0.217912
\(95\) 0 0
\(96\) 1.06405e20 0.163348
\(97\) − 1.03286e21i − 1.42213i −0.703129 − 0.711063i \(-0.748214\pi\)
0.703129 − 0.711063i \(-0.251786\pi\)
\(98\) − 4.35204e20i − 0.538046i
\(99\) −5.99592e19 −0.0666325
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.49.2 4
5.2 odd 4 50.22.a.e.1.2 2
5.3 odd 4 10.22.a.c.1.1 ✓ 2
5.4 even 2 inner 50.22.b.d.49.3 4
20.3 even 4 80.22.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.c.1.1 ✓ 2 5.3 odd 4
50.22.a.e.1.2 2 5.2 odd 4
50.22.b.d.49.2 4 1.1 even 1 trivial
50.22.b.d.49.3 4 5.4 even 2 inner
80.22.a.b.1.2 2 20.3 even 4