↑

Properties

Label 58.22.a.b.1.5
Level $58$
Weight $22$
Character 58.1
Self dual yes
Analytic conductor $162.097$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,22,Mod(1,58)] 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("58.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 58.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12288] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(162.096859686\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 85606746065 x^{10} + 168391612240800 x^{9} + \cdots - 43\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{32}\cdot 3^{8}\cdot 5^{3}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-88134.1\) of defining polynomial
Character \(\chi\) \(=\) 58.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} -84712.1 q^{3} +1.04858e6 q^{4} +3.01193e7 q^{5} +8.67452e7 q^{6} +1.46527e9 q^{7} -1.07374e9 q^{8} -3.28421e9 q^{9} -3.08421e10 q^{10} -7.02903e10 q^{11} -8.88271e10 q^{12} +7.71605e11 q^{13} -1.50043e12 q^{14} -2.55147e12 q^{15} +1.09951e12 q^{16} +3.47245e12 q^{17} +3.36303e12 q^{18} -3.50307e13 q^{19} +3.15823e13 q^{20} -1.24126e14 q^{21} +7.19773e13 q^{22} +1.33677e14 q^{23} +9.09590e13 q^{24} +4.30333e14 q^{25} -7.90123e14 q^{26} +1.16433e15 q^{27} +1.53644e15 q^{28} +4.20707e14 q^{29} +2.61270e15 q^{30} +8.16961e15 q^{31} -1.12590e15 q^{32} +5.95444e15 q^{33} -3.55579e15 q^{34} +4.41327e16 q^{35} -3.44374e15 q^{36} +3.56944e16 q^{37} +3.58714e16 q^{38} -6.53643e16 q^{39} -3.23403e16 q^{40} -4.06761e16 q^{41} +1.27105e17 q^{42} -1.96188e17 q^{43} -7.37047e16 q^{44} -9.89179e16 q^{45} -1.36885e17 q^{46} +4.45509e17 q^{47} -9.31420e16 q^{48} +1.58846e18 q^{49} -4.40661e17 q^{50} -2.94159e17 q^{51} +8.09086e17 q^{52} -9.01948e17 q^{53} -1.19228e18 q^{54} -2.11709e18 q^{55} -1.57332e18 q^{56} +2.96753e18 q^{57} -4.30804e17 q^{58} +2.88427e18 q^{59} -2.67541e18 q^{60} -4.84386e18 q^{61} -8.36568e18 q^{62} -4.81224e18 q^{63} +1.15292e18 q^{64} +2.32402e19 q^{65} -6.09735e18 q^{66} -3.46593e18 q^{67} +3.64113e18 q^{68} -1.13240e19 q^{69} -4.51919e19 q^{70} +2.83123e19 q^{71} +3.52639e18 q^{72} -3.84779e19 q^{73} -3.65511e19 q^{74} -3.64544e19 q^{75} -3.67324e19 q^{76} -1.02994e20 q^{77} +6.69330e19 q^{78} -4.21052e19 q^{79} +3.31165e19 q^{80} -6.42790e19 q^{81} +4.16524e19 q^{82} +6.29393e19 q^{83} -1.30155e20 q^{84} +1.04588e20 q^{85} +2.00896e20 q^{86} -3.56390e19 q^{87} +7.54736e19 q^{88} -2.66870e19 q^{89} +1.01292e20 q^{90} +1.13061e21 q^{91} +1.40170e20 q^{92} -6.92065e20 q^{93} -4.56201e20 q^{94} -1.05510e21 q^{95} +9.53774e19 q^{96} -4.78492e18 q^{97} -1.62658e21 q^{98} +2.30848e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12288 q^{2} + 41066 q^{3} + 12582912 q^{4} + 32000498 q^{5} - 42051584 q^{6} + 442380608 q^{7} - 12884901888 q^{8} + 45829788394 q^{9} - 32768509952 q^{10} - 75612722022 q^{11} + 43060822016 q^{12}+ \cdots - 77\!\cdots\!40 q^{99}+O(q^{100}) \) 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
\(3\) −84712.1 −0.828271 −0.414136 − 0.910215i \(-0.635916\pi\)
−0.414136 + 0.910215i \(0.635916\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 3.01193e7 1.37930 0.689651 − 0.724142i \(-0.257764\pi\)
0.689651 + 0.724142i \(0.257764\pi\)
\(6\) 8.67452e7 0.585676
\(7\) 1.46527e9 1.96059 0.980296 − 0.197534i \(-0.0632933\pi\)
0.980296 + 0.197534i \(0.0632933\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −3.28421e9 −0.313967
\(10\) −3.08421e10 −0.975314
\(11\) −7.02903e10 −0.817094 −0.408547 − 0.912737i \(-0.633964\pi\)
−0.408547 + 0.912737i \(0.633964\pi\)
\(12\) −8.88271e10 −0.414136
\(13\) 7.71605e11 1.55235 0.776175 − 0.630517i \(-0.217157\pi\)
0.776175 + 0.630517i \(0.217157\pi\)
\(14\) −1.50043e12 −1.38635
\(15\) −2.55147e12 −1.14244
\(16\) 1.09951e12 0.250000
\(17\) 3.47245e12 0.417756 0.208878 − 0.977942i \(-0.433019\pi\)
0.208878 + 0.977942i \(0.433019\pi\)
\(18\) 3.36303e12 0.222008
\(19\) −3.50307e13 −1.31080 −0.655400 − 0.755282i \(-0.727500\pi\)
−0.655400 + 0.755282i \(0.727500\pi\)
\(20\) 3.15823e13 0.689651
\(21\) −1.24126e14 −1.62390
\(22\) 7.19773e13 0.577773
\(23\) 1.33677e14 0.672842 0.336421 − 0.941712i \(-0.390783\pi\)
0.336421 + 0.941712i \(0.390783\pi\)
\(24\) 9.09590e13 0.292838
\(25\) 4.30333e14 0.902473
\(26\) −7.90123e14 −1.09768
\(27\) 1.16433e15 1.08832
\(28\) 1.53644e15 0.980296
\(29\) 4.20707e14 0.185695
\(30\) 2.61270e15 0.807824
\(31\) 8.16961e15 1.79021 0.895104 − 0.445858i \(-0.147101\pi\)
0.895104 + 0.445858i \(0.147101\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 5.95444e15 0.676776
\(34\) −3.55579e15 −0.295398
\(35\) 4.41327e16 2.70425
\(36\) −3.44374e15 −0.156984
\(37\) 3.56944e16 1.22034 0.610172 − 0.792269i \(-0.291101\pi\)
0.610172 + 0.792269i \(0.291101\pi\)
\(38\) 3.58714e16 0.926875
\(39\) −6.53643e16 −1.28577
\(40\) −3.23403e16 −0.487657
\(41\) −4.06761e16 −0.473270 −0.236635 − 0.971599i \(-0.576045\pi\)
−0.236635 + 0.971599i \(0.576045\pi\)
\(42\) 1.27105e17 1.14827
\(43\) −1.96188e17 −1.38437 −0.692186 − 0.721719i \(-0.743352\pi\)
−0.692186 + 0.721719i \(0.743352\pi\)
\(44\) −7.37047e16 −0.408547
\(45\) −9.89179e16 −0.433055
\(46\) −1.36885e17 −0.475771
\(47\) 4.45509e17 1.23546 0.617731 − 0.786390i \(-0.288052\pi\)
0.617731 + 0.786390i \(0.288052\pi\)
\(48\) −9.31420e16 −0.207068
\(49\) 1.58846e18 2.84392
\(50\) −4.40661e17 −0.638145
\(51\) −2.94159e17 −0.346015
\(52\) 8.09086e17 0.776175
\(53\) −9.01948e17 −0.708411 −0.354205 − 0.935168i \(-0.615249\pi\)
−0.354205 + 0.935168i \(0.615249\pi\)
\(54\) −1.19228e18 −0.769559
\(55\) −2.11709e18 −1.12702
\(56\) −1.57332e18 −0.693174
\(57\) 2.96753e18 1.08570
\(58\) −4.30804e17 −0.131306
\(59\) 2.88427e18 0.734666 0.367333 − 0.930089i \(-0.380271\pi\)
0.367333 + 0.930089i \(0.380271\pi\)
\(60\) −2.67541e18 −0.571218
\(61\) −4.84386e18 −0.869417 −0.434708 − 0.900571i \(-0.643149\pi\)
−0.434708 + 0.900571i \(0.643149\pi\)
\(62\) −8.36568e18 −1.26587
\(63\) −4.81224e18 −0.615561
\(64\) 1.15292e18 0.125000
\(65\) 2.32402e19 2.14116
\(66\) −6.09735e18 −0.478553
\(67\) −3.46593e18 −0.232292 −0.116146 − 0.993232i \(-0.537054\pi\)
−0.116146 + 0.993232i \(0.537054\pi\)
\(68\) 3.64113e18 0.208878
\(69\) −1.13240e19 −0.557296
\(70\) −4.51919e19 −1.91219
\(71\) 2.83123e19 1.03220 0.516098 − 0.856530i \(-0.327384\pi\)
0.516098 + 0.856530i \(0.327384\pi\)
\(72\) 3.52639e18 0.111004
\(73\) −3.84779e19 −1.04790 −0.523951 − 0.851748i \(-0.675543\pi\)
−0.523951 + 0.851748i \(0.675543\pi\)
\(74\) −3.65511e19 −0.862913
\(75\) −3.64544e19 −0.747492
\(76\) −3.67324e19 −0.655400
\(77\) −1.02994e20 −1.60199
\(78\) 6.69330e19 0.909174
\(79\) −4.21052e19 −0.500324 −0.250162 − 0.968204i \(-0.580484\pi\)
−0.250162 + 0.968204i \(0.580484\pi\)
\(80\) 3.31165e19 0.344825
\(81\) −6.42790e19 −0.587458
\(82\) 4.16524e19 0.334653
\(83\) 6.29393e19 0.445248 0.222624 − 0.974904i \(-0.428538\pi\)
0.222624 + 0.974904i \(0.428538\pi\)
\(84\) −1.30155e20 −0.811951
\(85\) 1.04588e20 0.576212
\(86\) 2.00896e20 0.978899
\(87\) −3.56390e19 −0.153806
\(88\) 7.54736e19 0.288887
\(89\) −2.66870e19 −0.0907203 −0.0453602 − 0.998971i \(-0.514444\pi\)
−0.0453602 + 0.998971i \(0.514444\pi\)
\(90\) 1.01292e20 0.306216
\(91\) 1.13061e21 3.04353
\(92\) 1.40170e20 0.336421
\(93\) −6.92065e20 −1.48278
\(94\) −4.56201e20 −0.873603
\(95\) −1.05510e21 −1.80799
\(96\) 9.53774e19 0.146419
\(97\) −4.78492e18 −0.00658827 −0.00329414 − 0.999995i \(-0.501049\pi\)
−0.00329414 + 0.999995i \(0.501049\pi\)
\(98\) −1.62658e21 −2.01096
\(99\) 2.30848e20 0.256541
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 58.22.a.b.1.5 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.22.a.b.1.5 ✓ 12 1.1 even 1 trivial