Properties

Label 10.22.a.b.1.2
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 1.2
Root \(-198.196\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +47924.0 q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} -4.90741e7 q^{6} +6.67863e8 q^{7} -1.07374e9 q^{8} -8.16365e9 q^{9} +1.00000e10 q^{10} -1.26793e11 q^{11} +5.02519e10 q^{12} +9.12750e11 q^{13} -6.83892e11 q^{14} -4.68008e11 q^{15} +1.09951e12 q^{16} +8.60406e12 q^{17} +8.35957e12 q^{18} -6.93702e12 q^{19} -1.02400e13 q^{20} +3.20066e13 q^{21} +1.29836e14 q^{22} -3.30974e14 q^{23} -5.14580e13 q^{24} +9.53674e13 q^{25} -9.34656e14 q^{26} -8.92536e14 q^{27} +7.00305e14 q^{28} -3.90534e15 q^{29} +4.79240e14 q^{30} +3.32185e15 q^{31} -1.12590e15 q^{32} -6.07642e15 q^{33} -8.81056e15 q^{34} -6.52210e15 q^{35} -8.56020e15 q^{36} -4.33804e16 q^{37} +7.10351e15 q^{38} +4.37426e16 q^{39} +1.04858e16 q^{40} -9.56620e16 q^{41} -3.27748e16 q^{42} -7.81706e16 q^{43} -1.32952e17 q^{44} +7.97231e16 q^{45} +3.38917e17 q^{46} -2.03634e17 q^{47} +5.26930e16 q^{48} -1.12505e17 q^{49} -9.76562e16 q^{50} +4.12341e17 q^{51} +9.57088e17 q^{52} +1.42121e18 q^{53} +9.13957e17 q^{54} +1.23821e18 q^{55} -7.17112e17 q^{56} -3.32449e17 q^{57} +3.99907e18 q^{58} +2.28919e18 q^{59} -4.90741e17 q^{60} -5.32798e18 q^{61} -3.40158e18 q^{62} -5.45220e18 q^{63} +1.15292e18 q^{64} -8.91357e18 q^{65} +6.22226e18 q^{66} -1.31401e19 q^{67} +9.02201e18 q^{68} -1.58616e19 q^{69} +6.67863e18 q^{70} -2.98159e19 q^{71} +8.76565e18 q^{72} +1.37094e18 q^{73} +4.44215e19 q^{74} +4.57039e18 q^{75} -7.27399e18 q^{76} -8.46803e19 q^{77} -4.47924e19 q^{78} +1.74281e19 q^{79} -1.07374e19 q^{80} +4.26208e19 q^{81} +9.79579e19 q^{82} +3.75717e19 q^{83} +3.35614e19 q^{84} -8.40240e19 q^{85} +8.00467e19 q^{86} -1.87159e20 q^{87} +1.36143e20 q^{88} +2.43231e20 q^{89} -8.16365e19 q^{90} +6.09592e20 q^{91} -3.47051e20 q^{92} +1.59196e20 q^{93} +2.08521e20 q^{94} +6.77443e19 q^{95} -5.39576e19 q^{96} -1.80740e20 q^{97} +1.15205e20 q^{98} +1.03509e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} - 126692 q^{3} + 2097152 q^{4} - 19531250 q^{5} + 129732608 q^{6} - 292598684 q^{7} - 2147483648 q^{8} + 11866737826 q^{9} + 20000000000 q^{10} - 41326831776 q^{11} - 132846190592 q^{12}+ \cdots + 27\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) 47924.0 0.468576 0.234288 − 0.972167i \(-0.424724\pi\)
0.234288 + 0.972167i \(0.424724\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −9.76562e6 −0.447214
\(6\) −4.90741e7 −0.331333
\(7\) 6.67863e8 0.893631 0.446815 − 0.894626i \(-0.352558\pi\)
0.446815 + 0.894626i \(0.352558\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −8.16365e9 −0.780437
\(10\) 1.00000e10 0.316228
\(11\) −1.26793e11 −1.47391 −0.736957 − 0.675940i \(-0.763738\pi\)
−0.736957 + 0.675940i \(0.763738\pi\)
\(12\) 5.02519e10 0.234288
\(13\) 9.12750e11 1.83631 0.918156 − 0.396219i \(-0.129678\pi\)
0.918156 + 0.396219i \(0.129678\pi\)
\(14\) −6.83892e11 −0.631892
\(15\) −4.68008e11 −0.209553
\(16\) 1.09951e12 0.250000
\(17\) 8.60406e12 1.03512 0.517559 − 0.855648i \(-0.326841\pi\)
0.517559 + 0.855648i \(0.326841\pi\)
\(18\) 8.35957e12 0.551852
\(19\) −6.93702e12 −0.259573 −0.129787 − 0.991542i \(-0.541429\pi\)
−0.129787 + 0.991542i \(0.541429\pi\)
\(20\) −1.02400e13 −0.223607
\(21\) 3.20066e13 0.418733
\(22\) 1.29836e14 1.04221
\(23\) −3.30974e14 −1.66591 −0.832955 − 0.553341i \(-0.813352\pi\)
−0.832955 + 0.553341i \(0.813352\pi\)
\(24\) −5.14580e13 −0.165666
\(25\) 9.53674e13 0.200000
\(26\) −9.34656e14 −1.29847
\(27\) −8.92536e14 −0.834269
\(28\) 7.00305e14 0.446815
\(29\) −3.90534e15 −1.72377 −0.861886 − 0.507103i \(-0.830716\pi\)
−0.861886 + 0.507103i \(0.830716\pi\)
\(30\) 4.79240e14 0.148177
\(31\) 3.32185e15 0.727918 0.363959 − 0.931415i \(-0.381425\pi\)
0.363959 + 0.931415i \(0.381425\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) −6.07642e15 −0.690640
\(34\) −8.81056e15 −0.731939
\(35\) −6.52210e15 −0.399644
\(36\) −8.56020e15 −0.390218
\(37\) −4.33804e16 −1.48312 −0.741558 − 0.670889i \(-0.765913\pi\)
−0.741558 + 0.670889i \(0.765913\pi\)
\(38\) 7.10351e15 0.183546
\(39\) 4.37426e16 0.860451
\(40\) 1.04858e16 0.158114
\(41\) −9.56620e16 −1.11304 −0.556518 − 0.830836i \(-0.687863\pi\)
−0.556518 + 0.830836i \(0.687863\pi\)
\(42\) −3.27748e16 −0.296089
\(43\) −7.81706e16 −0.551601 −0.275800 − 0.961215i \(-0.588943\pi\)
−0.275800 + 0.961215i \(0.588943\pi\)
\(44\) −1.32952e17 −0.736957
\(45\) 7.97231e16 0.349022
\(46\) 3.38917e17 1.17798
\(47\) −2.03634e17 −0.564707 −0.282354 − 0.959310i \(-0.591115\pi\)
−0.282354 + 0.959310i \(0.591115\pi\)
\(48\) 5.26930e16 0.117144
\(49\) −1.12505e17 −0.201425
\(50\) −9.76562e16 −0.141421
\(51\) 4.12341e17 0.485031
\(52\) 9.57088e17 0.918156
\(53\) 1.42121e18 1.11625 0.558127 − 0.829756i \(-0.311520\pi\)
0.558127 + 0.829756i \(0.311520\pi\)
\(54\) 9.13957e17 0.589917
\(55\) 1.23821e18 0.659154
\(56\) −7.17112e17 −0.315946
\(57\) −3.32449e17 −0.121630
\(58\) 3.99907e18 1.21889
\(59\) 2.28919e18 0.583089 0.291545 − 0.956557i \(-0.405831\pi\)
0.291545 + 0.956557i \(0.405831\pi\)
\(60\) −4.90741e17 −0.104777
\(61\) −5.32798e18 −0.956311 −0.478155 − 0.878275i \(-0.658694\pi\)
−0.478155 + 0.878275i \(0.658694\pi\)
\(62\) −3.40158e18 −0.514716
\(63\) −5.45220e18 −0.697422
\(64\) 1.15292e18 0.125000
\(65\) −8.91357e18 −0.821224
\(66\) 6.22226e18 0.488356
\(67\) −1.31401e19 −0.880668 −0.440334 − 0.897834i \(-0.645140\pi\)
−0.440334 + 0.897834i \(0.645140\pi\)
\(68\) 9.02201e18 0.517559
\(69\) −1.58616e19 −0.780604
\(70\) 6.67863e18 0.282591
\(71\) −2.98159e19 −1.08702 −0.543508 − 0.839404i \(-0.682904\pi\)
−0.543508 + 0.839404i \(0.682904\pi\)
\(72\) 8.76565e18 0.275926
\(73\) 1.37094e18 0.0373361 0.0186681 − 0.999826i \(-0.494057\pi\)
0.0186681 + 0.999826i \(0.494057\pi\)
\(74\) 4.44215e19 1.04872
\(75\) 4.57039e18 0.0937151
\(76\) −7.27399e18 −0.129787
\(77\) −8.46803e19 −1.31713
\(78\) −4.47924e19 −0.608431
\(79\) 1.74281e19 0.207093 0.103546 − 0.994625i \(-0.466981\pi\)
0.103546 + 0.994625i \(0.466981\pi\)
\(80\) −1.07374e19 −0.111803
\(81\) 4.26208e19 0.389519
\(82\) 9.79579e19 0.787035
\(83\) 3.75717e19 0.265792 0.132896 − 0.991130i \(-0.457572\pi\)
0.132896 + 0.991130i \(0.457572\pi\)
\(84\) 3.35614e19 0.209367
\(85\) −8.40240e19 −0.462919
\(86\) 8.00467e19 0.390040
\(87\) −1.87159e20 −0.807717
\(88\) 1.36143e20 0.521107
\(89\) 2.43231e20 0.826845 0.413423 − 0.910539i \(-0.364333\pi\)
0.413423 + 0.910539i \(0.364333\pi\)
\(90\) −8.16365e19 −0.246796
\(91\) 6.09592e20 1.64098
\(92\) −3.47051e20 −0.832955
\(93\) 1.59196e20 0.341084
\(94\) 2.08521e20 0.399308
\(95\) 6.77443e19 0.116085
\(96\) −5.39576e19 −0.0828332
\(97\) −1.80740e20 −0.248858 −0.124429 − 0.992229i \(-0.539710\pi\)
−0.124429 + 0.992229i \(0.539710\pi\)
\(98\) 1.15205e20 0.142429
\(99\) 1.03509e21 1.15030
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.1.2 ✓ 2
4.3 odd 2 80.22.a.d.1.1 2
5.2 odd 4 50.22.b.f.49.1 4
5.3 odd 4 50.22.b.f.49.4 4
5.4 even 2 50.22.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.b.1.2 ✓ 2 1.1 even 1 trivial
50.22.a.f.1.1 2 5.4 even 2
50.22.b.f.49.1 4 5.2 odd 4
50.22.b.f.49.4 4 5.3 odd 4
80.22.a.d.1.1 2 4.3 odd 2