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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 49.7
Root \(-1195.65i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.h.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1024.00i q^{2} -12234.5i q^{3} -1.04858e6 q^{4} +1.25282e7 q^{6} +9.06115e8i q^{7} -1.07374e9i q^{8} +1.03107e10 q^{9} +8.75575e10 q^{11} +1.28288e10i q^{12} -3.78299e10i q^{13} -9.27862e11 q^{14} +1.09951e12 q^{16} +1.03197e13i q^{17} +1.05581e13i q^{18} +4.47347e13 q^{19} +1.10859e13 q^{21} +8.96589e13i q^{22} -2.93024e14i q^{23} -1.31367e13 q^{24} +3.87378e13 q^{26} -2.54124e14i q^{27} -9.50131e14i q^{28} +4.56584e14 q^{29} +5.31739e15 q^{31} +1.12590e15i q^{32} -1.07123e15i q^{33} -1.05674e16 q^{34} -1.08115e16 q^{36} -2.33624e16i q^{37} +4.58083e16i q^{38} -4.62831e14 q^{39} -7.54341e16 q^{41} +1.13520e16i q^{42} -1.59218e17i q^{43} -9.18107e16 q^{44} +3.00056e17 q^{46} +5.36127e15i q^{47} -1.34520e16i q^{48} -2.62499e17 q^{49} +1.26257e17 q^{51} +3.96675e16i q^{52} -1.47056e18i q^{53} +2.60223e17 q^{54} +9.72934e17 q^{56} -5.47308e17i q^{57} +4.67542e17i q^{58} +2.78753e18 q^{59} -5.95048e18 q^{61} +5.44501e18i q^{62} +9.34266e18i q^{63} -1.15292e18 q^{64} +1.09694e18 q^{66} +3.15077e18i q^{67} -1.08210e19i q^{68} -3.58501e18 q^{69} +3.60686e19 q^{71} -1.10710e19i q^{72} -5.16840e19i q^{73} +2.39231e19 q^{74} -4.69077e19 q^{76} +7.93372e19i q^{77} -4.73939e17i q^{78} +1.21308e20 q^{79} +1.04744e20 q^{81} -7.72445e19i q^{82} -1.00531e20i q^{83} -1.16244e19 q^{84} +1.63039e20 q^{86} -5.58609e18i q^{87} -9.40142e19i q^{88} -2.72642e20 q^{89} +3.42782e19 q^{91} +3.07257e20i q^{92} -6.50559e19i q^{93} -5.48994e18 q^{94} +1.37749e19 q^{96} -5.06240e20i q^{97} -2.68799e20i q^{98} +9.02777e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8388608 q^{4} + 198172672 q^{6} - 47229523424 q^{9} + 175118508216 q^{11} - 333333561344 q^{14} + 8796093022208 q^{16} - 167026213839880 q^{19} - 8313944227904 q^{21} - 207799107715072 q^{24} - 200905406660608 q^{26}+ \cdots + 56\!\cdots\!52 q^{99}+O(q^{100}) \) 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
\(3\) − 12234.5i − 0.119623i −0.998210 − 0.0598115i \(-0.980950\pi\)
0.998210 − 0.0598115i \(-0.0190500\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 1.25282e7 0.0845862
\(7\) 9.06115e8i 1.21242i 0.795304 + 0.606211i \(0.207311\pi\)
−0.795304 + 0.606211i \(0.792689\pi\)
\(8\) − 1.07374e9i − 0.353553i
\(9\) 1.03107e10 0.985690
\(10\) 0 0
\(11\) 8.75575e10 1.01782 0.508909 − 0.860820i \(-0.330049\pi\)
0.508909 + 0.860820i \(0.330049\pi\)
\(12\) 1.28288e10i 0.0598115i
\(13\) − 3.78299e10i − 0.0761079i −0.999276 − 0.0380539i \(-0.987884\pi\)
0.999276 − 0.0380539i \(-0.0121159\pi\)
\(14\) −9.27862e11 −0.857312
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.03197e13i 1.24152i 0.784000 + 0.620761i \(0.213176\pi\)
−0.784000 + 0.620761i \(0.786824\pi\)
\(18\) 1.05581e13i 0.696988i
\(19\) 4.47347e13 1.67391 0.836954 − 0.547273i \(-0.184334\pi\)
0.836954 + 0.547273i \(0.184334\pi\)
\(20\) 0 0
\(21\) 1.10859e13 0.145034
\(22\) 8.96589e13i 0.719706i
\(23\) − 2.93024e14i − 1.47489i −0.675406 − 0.737446i \(-0.736032\pi\)
0.675406 − 0.737446i \(-0.263968\pi\)
\(24\) −1.31367e13 −0.0422931
\(25\) 0 0
\(26\) 3.87378e13 0.0538164
\(27\) − 2.54124e14i − 0.237534i
\(28\) − 9.50131e14i − 0.606211i
\(29\) 4.56584e14 0.201531 0.100765 − 0.994910i \(-0.467871\pi\)
0.100765 + 0.994910i \(0.467871\pi\)
\(30\) 0 0
\(31\) 5.31739e15 1.16520 0.582600 − 0.812759i \(-0.302035\pi\)
0.582600 + 0.812759i \(0.302035\pi\)
\(32\) 1.12590e15i 0.176777i
\(33\) − 1.07123e15i − 0.121754i
\(34\) −1.05674e16 −0.877888
\(35\) 0 0
\(36\) −1.08115e16 −0.492845
\(37\) − 2.33624e16i − 0.798728i −0.916792 − 0.399364i \(-0.869231\pi\)
0.916792 − 0.399364i \(-0.130769\pi\)
\(38\) 4.58083e16i 1.18363i
\(39\) −4.62831e14 −0.00910425
\(40\) 0 0
\(41\) −7.54341e16 −0.877682 −0.438841 − 0.898565i \(-0.644611\pi\)
−0.438841 + 0.898565i \(0.644611\pi\)
\(42\) 1.13520e16i 0.102554i
\(43\) − 1.59218e17i − 1.12350i −0.827307 − 0.561750i \(-0.810128\pi\)
0.827307 − 0.561750i \(-0.189872\pi\)
\(44\) −9.18107e16 −0.508909
\(45\) 0 0
\(46\) 3.00056e17 1.04291
\(47\) 5.36127e15i 0.0148676i 0.999972 + 0.00743378i \(0.00236627\pi\)
−0.999972 + 0.00743378i \(0.997634\pi\)
\(48\) − 1.34520e16i − 0.0299057i
\(49\) −2.62499e17 −0.469969
\(50\) 0 0
\(51\) 1.26257e17 0.148514
\(52\) 3.96675e16i 0.0380539i
\(53\) − 1.47056e18i − 1.15501i −0.816386 − 0.577507i \(-0.804026\pi\)
0.816386 − 0.577507i \(-0.195974\pi\)
\(54\) 2.60223e17 0.167962
\(55\) 0 0
\(56\) 9.72934e17 0.428656
\(57\) − 5.47308e17i − 0.200238i
\(58\) 4.67542e17i 0.142504i
\(59\) 2.78753e18 0.710024 0.355012 − 0.934862i \(-0.384477\pi\)
0.355012 + 0.934862i \(0.384477\pi\)
\(60\) 0 0
\(61\) −5.95048e18 −1.06804 −0.534021 − 0.845471i \(-0.679320\pi\)
−0.534021 + 0.845471i \(0.679320\pi\)
\(62\) 5.44501e18i 0.823922i
\(63\) 9.34266e18i 1.19507i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) 1.09694e18 0.0860934
\(67\) 3.15077e18i 0.211169i 0.994410 + 0.105585i \(0.0336714\pi\)
−0.994410 + 0.105585i \(0.966329\pi\)
\(68\) − 1.08210e19i − 0.620761i
\(69\) −3.58501e18 −0.176431
\(70\) 0 0
\(71\) 3.60686e19 1.31497 0.657485 − 0.753467i \(-0.271620\pi\)
0.657485 + 0.753467i \(0.271620\pi\)
\(72\) − 1.10710e19i − 0.348494i
\(73\) − 5.16840e19i − 1.40756i −0.710420 − 0.703778i \(-0.751495\pi\)
0.710420 − 0.703778i \(-0.248505\pi\)
\(74\) 2.39231e19 0.564786
\(75\) 0 0
\(76\) −4.69077e19 −0.836954
\(77\) 7.93372e19i 1.23403i
\(78\) − 4.73939e17i − 0.00643768i
\(79\) 1.21308e20 1.44146 0.720732 − 0.693214i \(-0.243806\pi\)
0.720732 + 0.693214i \(0.243806\pi\)
\(80\) 0 0
\(81\) 1.04744e20 0.957276
\(82\) − 7.72445e19i − 0.620615i
\(83\) − 1.00531e20i − 0.711180i −0.934642 − 0.355590i \(-0.884280\pi\)
0.934642 − 0.355590i \(-0.115720\pi\)
\(84\) −1.16244e19 −0.0725168
\(85\) 0 0
\(86\) 1.63039e20 0.794434
\(87\) − 5.58609e18i − 0.0241077i
\(88\) − 9.40142e19i − 0.359853i
\(89\) −2.72642e20 −0.926825 −0.463413 − 0.886143i \(-0.653375\pi\)
−0.463413 + 0.886143i \(0.653375\pi\)
\(90\) 0 0
\(91\) 3.42782e19 0.0922749
\(92\) 3.07257e20i 0.737446i
\(93\) − 6.50559e19i − 0.139385i
\(94\) −5.48994e18 −0.0105130
\(95\) 0 0
\(96\) 1.37749e19 0.0211466
\(97\) − 5.06240e20i − 0.697033i −0.937303 − 0.348516i \(-0.886686\pi\)
0.937303 − 0.348516i \(-0.113314\pi\)
\(98\) − 2.68799e20i − 0.332318i
\(99\) 9.02777e20 1.00325
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.49.7 8
5.2 odd 4 50.22.a.i.1.3 ✓ 4
5.3 odd 4 50.22.a.j.1.2 yes 4
5.4 even 2 inner 50.22.b.h.49.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.3 ✓ 4 5.2 odd 4
50.22.a.j.1.2 yes 4 5.3 odd 4
50.22.b.h.49.2 8 5.4 even 2 inner
50.22.b.h.49.7 8 1.1 even 1 trivial