Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1058,4,Mod(1,1058)] 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("1058.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1058.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,3,8,-10,6,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.4240207861\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.77200\) of defining polynomial
Character \(\chi\) \(=\) 1058.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -2.77200 q^{3} +4.00000 q^{4} -13.5440 q^{5} -5.54400 q^{6} -23.0880 q^{7} +8.00000 q^{8} -19.3160 q^{9} -27.0880 q^{10} +6.00000 q^{11} -11.0880 q^{12} -72.4920 q^{13} -46.1760 q^{14} +37.5440 q^{15} +16.0000 q^{16} +58.6320 q^{17} -38.6320 q^{18} -77.4400 q^{19} -54.1760 q^{20} +64.0000 q^{21} +12.0000 q^{22} -22.1760 q^{24} +58.4400 q^{25} -144.984 q^{26} +128.388 q^{27} -92.3520 q^{28} -169.476 q^{29} +75.0880 q^{30} -303.268 q^{31} +32.0000 q^{32} -16.6320 q^{33} +117.264 q^{34} +312.704 q^{35} -77.2640 q^{36} -265.336 q^{37} -154.880 q^{38} +200.948 q^{39} -108.352 q^{40} +310.388 q^{41} +128.000 q^{42} +193.304 q^{43} +24.0000 q^{44} +261.616 q^{45} +322.700 q^{47} -44.3520 q^{48} +190.056 q^{49} +116.880 q^{50} -162.528 q^{51} -289.968 q^{52} +137.336 q^{53} +256.776 q^{54} -81.2640 q^{55} -184.704 q^{56} +214.664 q^{57} -338.952 q^{58} +562.320 q^{59} +150.176 q^{60} +283.224 q^{61} -606.536 q^{62} +445.968 q^{63} +64.0000 q^{64} +981.832 q^{65} -33.2640 q^{66} +182.768 q^{67} +234.528 q^{68} +625.408 q^{70} +109.788 q^{71} -154.528 q^{72} -625.684 q^{73} -530.672 q^{74} -161.996 q^{75} -309.760 q^{76} -138.528 q^{77} +401.896 q^{78} -172.072 q^{79} -216.704 q^{80} +165.640 q^{81} +620.776 q^{82} -994.984 q^{83} +256.000 q^{84} -794.112 q^{85} +386.608 q^{86} +469.788 q^{87} +48.0000 q^{88} -1099.90 q^{89} +523.232 q^{90} +1673.70 q^{91} +840.660 q^{93} +645.400 q^{94} +1048.85 q^{95} -88.7041 q^{96} +1625.53 q^{97} +380.112 q^{98} -115.896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} - 10 q^{5} + 6 q^{6} - 12 q^{7} + 16 q^{8} - 13 q^{9} - 20 q^{10} + 12 q^{11} + 12 q^{12} - 51 q^{13} - 24 q^{14} + 58 q^{15} + 32 q^{16} + 66 q^{17} - 26 q^{18} + 16 q^{19}+ \cdots - 78 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −2.77200 −0.533472 −0.266736 − 0.963770i \(-0.585945\pi\)
−0.266736 + 0.963770i \(0.585945\pi\)
\(4\) 4.00000 0.500000
\(5\) −13.5440 −1.21141 −0.605706 − 0.795688i \(-0.707109\pi\)
−0.605706 + 0.795688i \(0.707109\pi\)
\(6\) −5.54400 −0.377222
\(7\) −23.0880 −1.24664 −0.623318 − 0.781969i \(-0.714216\pi\)
−0.623318 + 0.781969i \(0.714216\pi\)
\(8\) 8.00000 0.353553
\(9\) −19.3160 −0.715408
\(10\) −27.0880 −0.856598
\(11\) 6.00000 0.164461 0.0822304 − 0.996613i \(-0.473796\pi\)
0.0822304 + 0.996613i \(0.473796\pi\)
\(12\) −11.0880 −0.266736
\(13\) −72.4920 −1.54659 −0.773295 − 0.634046i \(-0.781393\pi\)
−0.773295 + 0.634046i \(0.781393\pi\)
\(14\) −46.1760 −0.881504
\(15\) 37.5440 0.646255
\(16\) 16.0000 0.250000
\(17\) 58.6320 0.836491 0.418246 − 0.908334i \(-0.362645\pi\)
0.418246 + 0.908334i \(0.362645\pi\)
\(18\) −38.6320 −0.505870
\(19\) −77.4400 −0.935051 −0.467525 − 0.883980i \(-0.654854\pi\)
−0.467525 + 0.883980i \(0.654854\pi\)
\(20\) −54.1760 −0.605706
\(21\) 64.0000 0.665045
\(22\) 12.0000 0.116291
\(23\) 0 0
\(24\) −22.1760 −0.188611
\(25\) 58.4400 0.467520
\(26\) −144.984 −1.09360
\(27\) 128.388 0.915122
\(28\) −92.3520 −0.623318
\(29\) −169.476 −1.08520 −0.542602 − 0.839990i \(-0.682561\pi\)
−0.542602 + 0.839990i \(0.682561\pi\)
\(30\) 75.0880 0.456971
\(31\) −303.268 −1.75705 −0.878525 − 0.477696i \(-0.841472\pi\)
−0.878525 + 0.477696i \(0.841472\pi\)
\(32\) 32.0000 0.176777
\(33\) −16.6320 −0.0877352
\(34\) 117.264 0.591489
\(35\) 312.704 1.51019
\(36\) −77.2640 −0.357704
\(37\) −265.336 −1.17895 −0.589473 − 0.807788i \(-0.700664\pi\)
−0.589473 + 0.807788i \(0.700664\pi\)
\(38\) −154.880 −0.661181
\(39\) 200.948 0.825062
\(40\) −108.352 −0.428299
\(41\) 310.388 1.18230 0.591152 − 0.806560i \(-0.298673\pi\)
0.591152 + 0.806560i \(0.298673\pi\)
\(42\) 128.000 0.470258
\(43\) 193.304 0.685549 0.342775 − 0.939418i \(-0.388633\pi\)
0.342775 + 0.939418i \(0.388633\pi\)
\(44\) 24.0000 0.0822304
\(45\) 261.616 0.866654
\(46\) 0 0
\(47\) 322.700 1.00150 0.500751 − 0.865591i \(-0.333057\pi\)
0.500751 + 0.865591i \(0.333057\pi\)
\(48\) −44.3520 −0.133368
\(49\) 190.056 0.554099
\(50\) 116.880 0.330587
\(51\) −162.528 −0.446245
\(52\) −289.968 −0.773295
\(53\) 137.336 0.355935 0.177968 − 0.984036i \(-0.443048\pi\)
0.177968 + 0.984036i \(0.443048\pi\)
\(54\) 256.776 0.647089
\(55\) −81.2640 −0.199230
\(56\) −184.704 −0.440752
\(57\) 214.664 0.498823
\(58\) −338.952 −0.767355
\(59\) 562.320 1.24081 0.620405 − 0.784281i \(-0.286968\pi\)
0.620405 + 0.784281i \(0.286968\pi\)
\(60\) 150.176 0.323127
\(61\) 283.224 0.594477 0.297239 − 0.954803i \(-0.403934\pi\)
0.297239 + 0.954803i \(0.403934\pi\)
\(62\) −606.536 −1.24242
\(63\) 445.968 0.891852
\(64\) 64.0000 0.125000
\(65\) 981.832 1.87356
\(66\) −33.2640 −0.0620382
\(67\) 182.768 0.333263 0.166632 − 0.986019i \(-0.446711\pi\)
0.166632 + 0.986019i \(0.446711\pi\)
\(68\) 234.528 0.418246
\(69\) 0 0
\(70\) 625.408 1.06787
\(71\) 109.788 0.183513 0.0917566 − 0.995781i \(-0.470752\pi\)
0.0917566 + 0.995781i \(0.470752\pi\)
\(72\) −154.528 −0.252935
\(73\) −625.684 −1.00316 −0.501581 − 0.865111i \(-0.667248\pi\)
−0.501581 + 0.865111i \(0.667248\pi\)
\(74\) −530.672 −0.833640
\(75\) −161.996 −0.249409
\(76\) −309.760 −0.467525
\(77\) −138.528 −0.205023
\(78\) 401.896 0.583407
\(79\) −172.072 −0.245058 −0.122529 − 0.992465i \(-0.539101\pi\)
−0.122529 + 0.992465i \(0.539101\pi\)
\(80\) −216.704 −0.302853
\(81\) 165.640 0.227216
\(82\) 620.776 0.836015
\(83\) −994.984 −1.31583 −0.657914 − 0.753093i \(-0.728561\pi\)
−0.657914 + 0.753093i \(0.728561\pi\)
\(84\) 256.000 0.332522
\(85\) −794.112 −1.01334
\(86\) 386.608 0.484756
\(87\) 469.788 0.578926
\(88\) 48.0000 0.0581456
\(89\) −1099.90 −1.30999 −0.654993 − 0.755635i \(-0.727328\pi\)
−0.654993 + 0.755635i \(0.727328\pi\)
\(90\) 523.232 0.612817
\(91\) 1673.70 1.92803
\(92\) 0 0
\(93\) 840.660 0.937337
\(94\) 645.400 0.708169
\(95\) 1048.85 1.13273
\(96\) −88.7041 −0.0943054
\(97\) 1625.53 1.70152 0.850760 − 0.525555i \(-0.176142\pi\)
0.850760 + 0.525555i \(0.176142\pi\)
\(98\) 380.112 0.391807
\(99\) −115.896 −0.117656
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 1058.4.a.j.1.1 2
23.22 odd 2 46.4.a.d.1.1 ✓ 2
69.68 even 2 414.4.a.f.1.1 2
92.91 even 2 368.4.a.f.1.2 2
115.22 even 4 1150.4.b.j.599.4 4
115.68 even 4 1150.4.b.j.599.1 4
115.114 odd 2 1150.4.a.j.1.2 2
161.160 even 2 2254.4.a.f.1.2 2
184.45 odd 2 1472.4.a.k.1.2 2
184.91 even 2 1472.4.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.a.d.1.1 ✓ 2 23.22 odd 2
368.4.a.f.1.2 2 92.91 even 2
414.4.a.f.1.1 2 69.68 even 2
1058.4.a.j.1.1 2 1.1 even 1 trivial
1150.4.a.j.1.2 2 115.114 odd 2
1150.4.b.j.599.1 4 115.68 even 4
1150.4.b.j.599.4 4 115.22 even 4
1472.4.a.k.1.2 2 184.45 odd 2
1472.4.a.n.1.1 2 184.91 even 2
2254.4.a.f.1.2 2 161.160 even 2