Properties

Label 10.22.a.b.1.1
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
Analytic rank $1$
Dimension $2$
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] = [10,22,Mod(1,10)] 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("10.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2048,-126692] 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: \(27.9477344287\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{157921}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 39480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(199.196\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} -174616. q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} +1.78807e8 q^{6} -9.60462e8 q^{7} -1.07374e9 q^{8} +2.00304e10 q^{9} +1.00000e10 q^{10} +8.54661e10 q^{11} -1.83098e11 q^{12} +9.74302e11 q^{13} +9.83513e11 q^{14} +1.70523e12 q^{15} +1.09951e12 q^{16} -1.17045e13 q^{17} -2.05111e13 q^{18} +1.41128e13 q^{19} -1.02400e13 q^{20} +1.67712e14 q^{21} -8.75173e13 q^{22} -2.65862e13 q^{23} +1.87492e14 q^{24} +9.53674e13 q^{25} -9.97685e14 q^{26} -1.67108e15 q^{27} -1.00712e15 q^{28} +1.45075e15 q^{29} -1.74616e15 q^{30} -7.63253e15 q^{31} -1.12590e15 q^{32} -1.49237e16 q^{33} +1.19854e16 q^{34} +9.37951e15 q^{35} +2.10034e16 q^{36} +1.09912e16 q^{37} -1.44515e16 q^{38} -1.70129e17 q^{39} +1.04858e16 q^{40} +7.93639e16 q^{41} -1.71737e17 q^{42} -8.36751e16 q^{43} +8.96177e16 q^{44} -1.95609e17 q^{45} +2.72243e16 q^{46} +3.57127e17 q^{47} -1.91992e17 q^{48} +3.63941e17 q^{49} -9.76562e16 q^{50} +2.04379e18 q^{51} +1.02163e18 q^{52} -8.17016e17 q^{53} +1.71119e18 q^{54} -8.34630e17 q^{55} +1.03129e18 q^{56} -2.46432e18 q^{57} -1.48557e18 q^{58} +8.21011e17 q^{59} +1.78807e18 q^{60} +4.53683e18 q^{61} +7.81571e18 q^{62} -1.92384e19 q^{63} +1.15292e18 q^{64} -9.51466e18 q^{65} +1.52819e19 q^{66} +8.02826e18 q^{67} -1.22730e19 q^{68} +4.64238e18 q^{69} -9.60462e18 q^{70} -5.25728e19 q^{71} -2.15075e19 q^{72} +9.28698e18 q^{73} -1.12549e19 q^{74} -1.66527e19 q^{75} +1.47984e19 q^{76} -8.20869e19 q^{77} +1.74212e20 q^{78} +1.08861e19 q^{79} -1.07374e19 q^{80} +8.22724e19 q^{81} -8.12687e19 q^{82} -4.79994e19 q^{83} +1.75859e20 q^{84} +1.14302e20 q^{85} +8.56833e19 q^{86} -2.53325e20 q^{87} -9.17685e19 q^{88} +2.35270e20 q^{89} +2.00304e20 q^{90} -9.35779e20 q^{91} -2.78777e19 q^{92} +1.33276e21 q^{93} -3.65698e20 q^{94} -1.37820e20 q^{95} +1.96600e20 q^{96} +7.25967e20 q^{97} -3.72675e20 q^{98} +1.71192e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} - 126692 q^{3} + 2097152 q^{4} - 19531250 q^{5} + 129732608 q^{6} - 292598684 q^{7} - 2147483648 q^{8} + 11866737826 q^{9} + 20000000000 q^{10} - 41326831776 q^{11} - 132846190592 q^{12}+ \cdots + 27\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) −174616. −1.70730 −0.853652 − 0.520844i \(-0.825617\pi\)
−0.853652 + 0.520844i \(0.825617\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −9.76562e6 −0.447214
\(6\) 1.78807e8 1.20725
\(7\) −9.60462e8 −1.28514 −0.642570 − 0.766227i \(-0.722132\pi\)
−0.642570 + 0.766227i \(0.722132\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 2.00304e10 1.91489
\(10\) 1.00000e10 0.316228
\(11\) 8.54661e10 0.993507 0.496753 − 0.867892i \(-0.334525\pi\)
0.496753 + 0.867892i \(0.334525\pi\)
\(12\) −1.83098e11 −0.853652
\(13\) 9.74302e11 1.96015 0.980073 − 0.198640i \(-0.0636523\pi\)
0.980073 + 0.198640i \(0.0636523\pi\)
\(14\) 9.83513e11 0.908732
\(15\) 1.70523e12 0.763529
\(16\) 1.09951e12 0.250000
\(17\) −1.17045e13 −1.40812 −0.704058 − 0.710143i \(-0.748630\pi\)
−0.704058 + 0.710143i \(0.748630\pi\)
\(18\) −2.05111e13 −1.35403
\(19\) 1.41128e13 0.528081 0.264041 − 0.964512i \(-0.414945\pi\)
0.264041 + 0.964512i \(0.414945\pi\)
\(20\) −1.02400e13 −0.223607
\(21\) 1.67712e14 2.19413
\(22\) −8.75173e13 −0.702515
\(23\) −2.65862e13 −0.133818 −0.0669090 − 0.997759i \(-0.521314\pi\)
−0.0669090 + 0.997759i \(0.521314\pi\)
\(24\) 1.87492e14 0.603623
\(25\) 9.53674e13 0.200000
\(26\) −9.97685e14 −1.38603
\(27\) −1.67108e15 −1.56199
\(28\) −1.00712e15 −0.642570
\(29\) 1.45075e15 0.640346 0.320173 − 0.947359i \(-0.396259\pi\)
0.320173 + 0.947359i \(0.396259\pi\)
\(30\) −1.74616e15 −0.539897
\(31\) −7.63253e15 −1.67252 −0.836259 − 0.548335i \(-0.815262\pi\)
−0.836259 + 0.548335i \(0.815262\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) −1.49237e16 −1.69622
\(34\) 1.19854e16 0.995688
\(35\) 9.37951e15 0.574732
\(36\) 2.10034e16 0.957443
\(37\) 1.09912e16 0.375773 0.187886 − 0.982191i \(-0.439836\pi\)
0.187886 + 0.982191i \(0.439836\pi\)
\(38\) −1.44515e16 −0.373410
\(39\) −1.70129e17 −3.34656
\(40\) 1.04858e16 0.158114
\(41\) 7.93639e16 0.923406 0.461703 − 0.887035i \(-0.347239\pi\)
0.461703 + 0.887035i \(0.347239\pi\)
\(42\) −1.71737e17 −1.55148
\(43\) −8.36751e16 −0.590442 −0.295221 − 0.955429i \(-0.595393\pi\)
−0.295221 + 0.955429i \(0.595393\pi\)
\(44\) 8.96177e16 0.496753
\(45\) −1.95609e17 −0.856363
\(46\) 2.72243e16 0.0946236
\(47\) 3.57127e17 0.990365 0.495183 − 0.868789i \(-0.335101\pi\)
0.495183 + 0.868789i \(0.335101\pi\)
\(48\) −1.91992e17 −0.426826
\(49\) 3.63941e17 0.651586
\(50\) −9.76562e16 −0.141421
\(51\) 2.04379e18 2.40408
\(52\) 1.02163e18 0.980073
\(53\) −8.17016e17 −0.641702 −0.320851 − 0.947130i \(-0.603969\pi\)
−0.320851 + 0.947130i \(0.603969\pi\)
\(54\) 1.71119e18 1.10449
\(55\) −8.34630e17 −0.444310
\(56\) 1.03129e18 0.454366
\(57\) −2.46432e18 −0.901595
\(58\) −1.48557e18 −0.452793
\(59\) 8.21011e17 0.209124 0.104562 − 0.994518i \(-0.466656\pi\)
0.104562 + 0.994518i \(0.466656\pi\)
\(60\) 1.78807e18 0.381765
\(61\) 4.53683e18 0.814310 0.407155 − 0.913359i \(-0.366521\pi\)
0.407155 + 0.913359i \(0.366521\pi\)
\(62\) 7.81571e18 1.18265
\(63\) −1.92384e19 −2.46090
\(64\) 1.15292e18 0.125000
\(65\) −9.51466e18 −0.876604
\(66\) 1.52819e19 1.19941
\(67\) 8.02826e18 0.538066 0.269033 − 0.963131i \(-0.413296\pi\)
0.269033 + 0.963131i \(0.413296\pi\)
\(68\) −1.22730e19 −0.704058
\(69\) 4.64238e18 0.228468
\(70\) −9.60462e18 −0.406397
\(71\) −5.25728e19 −1.91668 −0.958338 − 0.285638i \(-0.907794\pi\)
−0.958338 + 0.285638i \(0.907794\pi\)
\(72\) −2.15075e19 −0.677014
\(73\) 9.28698e18 0.252921 0.126460 − 0.991972i \(-0.459638\pi\)
0.126460 + 0.991972i \(0.459638\pi\)
\(74\) −1.12549e19 −0.265711
\(75\) −1.66527e19 −0.341461
\(76\) 1.47984e19 0.264041
\(77\) −8.20869e19 −1.27680
\(78\) 1.74212e20 2.36638
\(79\) 1.08861e19 0.129356 0.0646781 − 0.997906i \(-0.479398\pi\)
0.0646781 + 0.997906i \(0.479398\pi\)
\(80\) −1.07374e19 −0.111803
\(81\) 8.22724e19 0.751902
\(82\) −8.12687e19 −0.652947
\(83\) −4.79994e19 −0.339560 −0.169780 − 0.985482i \(-0.554306\pi\)
−0.169780 + 0.985482i \(0.554306\pi\)
\(84\) 1.75859e20 1.09706
\(85\) 1.14302e20 0.629728
\(86\) 8.56833e19 0.417506
\(87\) −2.53325e20 −1.09327
\(88\) −9.17685e19 −0.351258
\(89\) 2.35270e20 0.799783 0.399891 − 0.916563i \(-0.369048\pi\)
0.399891 + 0.916563i \(0.369048\pi\)
\(90\) 2.00304e20 0.605540
\(91\) −9.35779e20 −2.51906
\(92\) −2.78777e19 −0.0669090
\(93\) 1.33276e21 2.85549
\(94\) −3.65698e20 −0.700294
\(95\) −1.37820e20 −0.236165
\(96\) 1.96600e20 0.301812
\(97\) 7.25967e20 0.999571 0.499786 − 0.866149i \(-0.333412\pi\)
0.499786 + 0.866149i \(0.333412\pi\)
\(98\) −3.72675e20 −0.460741
\(99\) 1.71192e21 1.90245
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 10.22.a.b.1.1 ✓ 2
4.3 odd 2 80.22.a.d.1.2 2
5.2 odd 4 50.22.b.f.49.2 4
5.3 odd 4 50.22.b.f.49.3 4
5.4 even 2 50.22.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.b.1.1 ✓ 2 1.1 even 1 trivial
50.22.a.f.1.2 2 5.4 even 2
50.22.b.f.49.2 4 5.2 odd 4
50.22.b.f.49.3 4 5.3 odd 4
80.22.a.d.1.2 2 4.3 odd 2