Properties

Label 10.22.a.c.1.1
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
Analytic rank $0$
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,-2048,100308] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{474529}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 118632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(344.930\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} -94506.7 q^{3} +1.04858e6 q^{4} +9.76562e6 q^{5} +9.67749e7 q^{6} +3.65434e8 q^{7} -1.07374e9 q^{8} -1.52883e9 q^{9} -1.00000e10 q^{10} -3.92190e10 q^{11} -9.90975e10 q^{12} -4.64598e11 q^{13} -3.74204e11 q^{14} -9.22917e11 q^{15} +1.09951e12 q^{16} +2.43124e11 q^{17} +1.56552e12 q^{18} -6.74889e12 q^{19} +1.02400e13 q^{20} -3.45360e13 q^{21} +4.01602e13 q^{22} -1.66102e14 q^{23} +1.01476e14 q^{24} +9.53674e13 q^{25} +4.75749e14 q^{26} +1.13306e15 q^{27} +3.83185e14 q^{28} +2.85150e15 q^{29} +9.45067e14 q^{30} +3.48202e15 q^{31} -1.12590e15 q^{32} +3.70646e15 q^{33} -2.48959e14 q^{34} +3.56869e15 q^{35} -1.60309e15 q^{36} +3.87028e15 q^{37} +6.91086e15 q^{38} +4.39077e16 q^{39} -1.04858e16 q^{40} +1.53913e17 q^{41} +3.53648e16 q^{42} -6.75910e15 q^{43} -4.11241e16 q^{44} -1.49300e16 q^{45} +1.70089e17 q^{46} -1.11128e17 q^{47} -1.03911e17 q^{48} -4.25004e17 q^{49} -9.76562e16 q^{50} -2.29769e16 q^{51} -4.87167e17 q^{52} -2.06153e18 q^{53} -1.16025e18 q^{54} -3.82998e17 q^{55} -3.92382e17 q^{56} +6.37815e17 q^{57} -2.91994e18 q^{58} +5.84178e18 q^{59} -9.67749e17 q^{60} +4.90762e18 q^{61} -3.56559e18 q^{62} -5.58686e17 q^{63} +1.15292e18 q^{64} -4.53709e18 q^{65} -3.79541e18 q^{66} +1.35893e19 q^{67} +2.54934e17 q^{68} +1.56978e19 q^{69} -3.65434e18 q^{70} +3.28855e19 q^{71} +1.64157e18 q^{72} +3.25004e19 q^{73} -3.96317e18 q^{74} -9.01286e18 q^{75} -7.07672e18 q^{76} -1.43319e19 q^{77} -4.49615e19 q^{78} +5.11527e19 q^{79} +1.07374e19 q^{80} -9.10896e19 q^{81} -1.57607e20 q^{82} +2.02190e20 q^{83} -3.62136e19 q^{84} +2.37426e18 q^{85} +6.92131e18 q^{86} -2.69486e20 q^{87} +4.21111e19 q^{88} +2.17189e20 q^{89} +1.52883e19 q^{90} -1.69780e20 q^{91} -1.74171e20 q^{92} -3.29075e20 q^{93} +1.13795e20 q^{94} -6.59071e19 q^{95} +1.06405e20 q^{96} -1.03286e21 q^{97} +4.35204e20 q^{98} +5.99592e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} + 100308 q^{3} + 2097152 q^{4} + 19531250 q^{5} - 102715392 q^{6} + 1328895316 q^{7} - 2147483648 q^{8} + 25963598826 q^{9} - 20000000000 q^{10} + 21869194224 q^{11} + 105180561408 q^{12}+ \cdots + 17\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) −94506.7 −0.924037 −0.462019 − 0.886870i \(-0.652875\pi\)
−0.462019 + 0.886870i \(0.652875\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 9.76562e6 0.447214
\(6\) 9.67749e7 0.653393
\(7\) 3.65434e8 0.488967 0.244483 − 0.969653i \(-0.421382\pi\)
0.244483 + 0.969653i \(0.421382\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −1.52883e9 −0.146155
\(10\) −1.00000e10 −0.316228
\(11\) −3.92190e10 −0.455904 −0.227952 − 0.973672i \(-0.573203\pi\)
−0.227952 + 0.973672i \(0.573203\pi\)
\(12\) −9.90975e10 −0.462019
\(13\) −4.64598e11 −0.934701 −0.467350 − 0.884072i \(-0.654791\pi\)
−0.467350 + 0.884072i \(0.654791\pi\)
\(14\) −3.74204e11 −0.345752
\(15\) −9.22917e11 −0.413242
\(16\) 1.09951e12 0.250000
\(17\) 2.43124e11 0.0292492 0.0146246 − 0.999893i \(-0.495345\pi\)
0.0146246 + 0.999893i \(0.495345\pi\)
\(18\) 1.56552e12 0.103347
\(19\) −6.74889e12 −0.252534 −0.126267 − 0.991996i \(-0.540300\pi\)
−0.126267 + 0.991996i \(0.540300\pi\)
\(20\) 1.02400e13 0.223607
\(21\) −3.45360e13 −0.451824
\(22\) 4.01602e13 0.322373
\(23\) −1.66102e14 −0.836052 −0.418026 − 0.908435i \(-0.637278\pi\)
−0.418026 + 0.908435i \(0.637278\pi\)
\(24\) 1.01476e14 0.326697
\(25\) 9.53674e13 0.200000
\(26\) 4.75749e14 0.660933
\(27\) 1.13306e15 1.05909
\(28\) 3.83185e14 0.244483
\(29\) 2.85150e15 1.25862 0.629310 − 0.777155i \(-0.283338\pi\)
0.629310 + 0.777155i \(0.283338\pi\)
\(30\) 9.45067e14 0.292206
\(31\) 3.48202e15 0.763017 0.381508 − 0.924365i \(-0.375405\pi\)
0.381508 + 0.924365i \(0.375405\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 3.70646e15 0.421272
\(34\) −2.48959e14 −0.0206823
\(35\) 3.56869e15 0.218673
\(36\) −1.60309e15 −0.0730774
\(37\) 3.87028e15 0.132320 0.0661598 − 0.997809i \(-0.478925\pi\)
0.0661598 + 0.997809i \(0.478925\pi\)
\(38\) 6.91086e15 0.178568
\(39\) 4.39077e16 0.863698
\(40\) −1.04858e16 −0.158114
\(41\) 1.53913e17 1.79079 0.895396 − 0.445272i \(-0.146893\pi\)
0.895396 + 0.445272i \(0.146893\pi\)
\(42\) 3.53648e16 0.319488
\(43\) −6.75910e15 −0.0476947 −0.0238473 − 0.999716i \(-0.507592\pi\)
−0.0238473 + 0.999716i \(0.507592\pi\)
\(44\) −4.11241e16 −0.227952
\(45\) −1.49300e16 −0.0653624
\(46\) 1.70089e17 0.591178
\(47\) −1.11128e17 −0.308174 −0.154087 − 0.988057i \(-0.549244\pi\)
−0.154087 + 0.988057i \(0.549244\pi\)
\(48\) −1.03911e17 −0.231009
\(49\) −4.25004e17 −0.760911
\(50\) −9.76562e16 −0.141421
\(51\) −2.29769e16 −0.0270274
\(52\) −4.87167e17 −0.467350
\(53\) −2.06153e18 −1.61917 −0.809584 − 0.587003i \(-0.800307\pi\)
−0.809584 + 0.587003i \(0.800307\pi\)
\(54\) −1.16025e18 −0.748890
\(55\) −3.82998e17 −0.203886
\(56\) −3.92382e17 −0.172876
\(57\) 6.37815e17 0.233351
\(58\) −2.91994e18 −0.889978
\(59\) 5.84178e18 1.48799 0.743993 − 0.668188i \(-0.232930\pi\)
0.743993 + 0.668188i \(0.232930\pi\)
\(60\) −9.67749e17 −0.206621
\(61\) 4.90762e18 0.880861 0.440431 − 0.897787i \(-0.354826\pi\)
0.440431 + 0.897787i \(0.354826\pi\)
\(62\) −3.56559e18 −0.539534
\(63\) −5.58686e17 −0.0714648
\(64\) 1.15292e18 0.125000
\(65\) −4.53709e18 −0.418011
\(66\) −3.79541e18 −0.297884
\(67\) 1.35893e19 0.910776 0.455388 − 0.890293i \(-0.349500\pi\)
0.455388 + 0.890293i \(0.349500\pi\)
\(68\) 2.54934e17 0.0146246
\(69\) 1.56978e19 0.772543
\(70\) −3.65434e18 −0.154625
\(71\) 3.28855e19 1.19893 0.599463 − 0.800403i \(-0.295381\pi\)
0.599463 + 0.800403i \(0.295381\pi\)
\(72\) 1.64157e18 0.0516735
\(73\) 3.25004e19 0.885112 0.442556 − 0.896741i \(-0.354072\pi\)
0.442556 + 0.896741i \(0.354072\pi\)
\(74\) −3.96317e18 −0.0935641
\(75\) −9.01286e18 −0.184807
\(76\) −7.07672e18 −0.126267
\(77\) −1.43319e19 −0.222922
\(78\) −4.49615e19 −0.610727
\(79\) 5.11527e19 0.607832 0.303916 − 0.952699i \(-0.401706\pi\)
0.303916 + 0.952699i \(0.401706\pi\)
\(80\) 1.07374e19 0.111803
\(81\) −9.10896e19 −0.832484
\(82\) −1.57607e20 −1.26628
\(83\) 2.02190e20 1.43034 0.715171 − 0.698949i \(-0.246349\pi\)
0.715171 + 0.698949i \(0.246349\pi\)
\(84\) −3.62136e19 −0.225912
\(85\) 2.37426e18 0.0130807
\(86\) 6.92131e18 0.0337252
\(87\) −2.69486e20 −1.16301
\(88\) 4.21111e19 0.161186
\(89\) 2.17189e20 0.738316 0.369158 − 0.929367i \(-0.379646\pi\)
0.369158 + 0.929367i \(0.379646\pi\)
\(90\) 1.52883e19 0.0462182
\(91\) −1.69780e20 −0.457038
\(92\) −1.74171e20 −0.418026
\(93\) −3.29075e20 −0.705056
\(94\) 1.13795e20 0.217912
\(95\) −6.59071e19 −0.112937
\(96\) 1.06405e20 0.163348
\(97\) −1.03286e21 −1.42213 −0.711063 − 0.703129i \(-0.751786\pi\)
−0.711063 + 0.703129i \(0.751786\pi\)
\(98\) 4.35204e20 0.538046
\(99\) 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 10.22.a.c.1.1 ✓ 2
4.3 odd 2 80.22.a.b.1.2 2
5.2 odd 4 50.22.b.d.49.2 4
5.3 odd 4 50.22.b.d.49.3 4
5.4 even 2 50.22.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.c.1.1 ✓ 2 1.1 even 1 trivial
50.22.a.e.1.2 2 5.4 even 2
50.22.b.d.49.2 4 5.2 odd 4
50.22.b.d.49.3 4 5.3 odd 4
80.22.a.b.1.2 2 4.3 odd 2