Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [529,4,Mod(1,529)] 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("529.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(529, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 529 = 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 529.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [25,0,-1,80,-51] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.2120103930\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 529.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.17387 q^{2} +9.69393 q^{3} -6.62203 q^{4} -0.587346 q^{5} +11.3794 q^{6} -24.4606 q^{7} -17.1643 q^{8} +66.9723 q^{9} -0.689467 q^{10} -30.6659 q^{11} -64.1935 q^{12} -27.6225 q^{13} -28.7136 q^{14} -5.69369 q^{15} +32.8276 q^{16} +30.7916 q^{17} +78.6166 q^{18} -92.5849 q^{19} +3.88942 q^{20} -237.120 q^{21} -35.9977 q^{22} -166.390 q^{24} -124.655 q^{25} -32.4252 q^{26} +387.488 q^{27} +161.979 q^{28} -73.6185 q^{29} -6.68364 q^{30} -105.303 q^{31} +175.850 q^{32} -297.273 q^{33} +36.1452 q^{34} +14.3669 q^{35} -443.493 q^{36} -89.7975 q^{37} -108.682 q^{38} -267.771 q^{39} +10.0814 q^{40} -88.7884 q^{41} -278.347 q^{42} -365.622 q^{43} +203.070 q^{44} -39.3359 q^{45} +181.434 q^{47} +318.228 q^{48} +255.323 q^{49} -146.329 q^{50} +298.491 q^{51} +182.917 q^{52} -612.828 q^{53} +454.860 q^{54} +18.0115 q^{55} +419.851 q^{56} -897.511 q^{57} -86.4184 q^{58} +78.0294 q^{59} +37.7038 q^{60} -111.873 q^{61} -123.612 q^{62} -1638.18 q^{63} -56.1959 q^{64} +16.2240 q^{65} -348.959 q^{66} +408.709 q^{67} -203.903 q^{68} +16.8648 q^{70} +449.763 q^{71} -1149.53 q^{72} -93.4293 q^{73} -105.410 q^{74} -1208.40 q^{75} +613.100 q^{76} +750.107 q^{77} -314.328 q^{78} +277.946 q^{79} -19.2812 q^{80} +1948.03 q^{81} -104.226 q^{82} +1111.38 q^{83} +1570.21 q^{84} -18.0853 q^{85} -429.192 q^{86} -713.652 q^{87} +526.360 q^{88} +1272.90 q^{89} -46.1752 q^{90} +675.665 q^{91} -1020.80 q^{93} +212.979 q^{94} +54.3794 q^{95} +1704.68 q^{96} +1087.37 q^{97} +299.716 q^{98} -2053.76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - q^{3} + 80 q^{4} - 51 q^{5} + 86 q^{6} - 73 q^{7} + 3 q^{8} + 166 q^{9} - 139 q^{10} - 221 q^{11} - 191 q^{12} - 27 q^{13} - 372 q^{14} - 310 q^{15} + 152 q^{16} - 365 q^{17} - 538 q^{18} - 405 q^{19}+ \cdots - 7317 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17387 0.415025 0.207513 − 0.978232i \(-0.433463\pi\)
0.207513 + 0.978232i \(0.433463\pi\)
\(3\) 9.69393 1.86560 0.932799 − 0.360398i \(-0.117359\pi\)
0.932799 + 0.360398i \(0.117359\pi\)
\(4\) −6.62203 −0.827754
\(5\) −0.587346 −0.0525338 −0.0262669 − 0.999655i \(-0.508362\pi\)
−0.0262669 + 0.999655i \(0.508362\pi\)
\(6\) 11.3794 0.774270
\(7\) −24.4606 −1.32075 −0.660375 − 0.750936i \(-0.729603\pi\)
−0.660375 + 0.750936i \(0.729603\pi\)
\(8\) −17.1643 −0.758564
\(9\) 66.9723 2.48045
\(10\) −0.689467 −0.0218029
\(11\) −30.6659 −0.840556 −0.420278 − 0.907396i \(-0.638067\pi\)
−0.420278 + 0.907396i \(0.638067\pi\)
\(12\) −64.1935 −1.54426
\(13\) −27.6225 −0.589317 −0.294658 − 0.955603i \(-0.595206\pi\)
−0.294658 + 0.955603i \(0.595206\pi\)
\(14\) −28.7136 −0.548145
\(15\) −5.69369 −0.0980070
\(16\) 32.8276 0.512931
\(17\) 30.7916 0.439297 0.219648 − 0.975579i \(-0.429509\pi\)
0.219648 + 0.975579i \(0.429509\pi\)
\(18\) 78.6166 1.02945
\(19\) −92.5849 −1.11792 −0.558959 − 0.829196i \(-0.688799\pi\)
−0.558959 + 0.829196i \(0.688799\pi\)
\(20\) 3.88942 0.0434851
\(21\) −237.120 −2.46399
\(22\) −35.9977 −0.348852
\(23\) 0 0
\(24\) −166.390 −1.41518
\(25\) −124.655 −0.997240
\(26\) −32.4252 −0.244581
\(27\) 387.488 2.76193
\(28\) 161.979 1.09326
\(29\) −73.6185 −0.471400 −0.235700 − 0.971826i \(-0.575738\pi\)
−0.235700 + 0.971826i \(0.575738\pi\)
\(30\) −6.68364 −0.0406754
\(31\) −105.303 −0.610095 −0.305048 − 0.952337i \(-0.598672\pi\)
−0.305048 + 0.952337i \(0.598672\pi\)
\(32\) 175.850 0.971443
\(33\) −297.273 −1.56814
\(34\) 36.1452 0.182319
\(35\) 14.3669 0.0693841
\(36\) −443.493 −2.05321
\(37\) −89.7975 −0.398989 −0.199495 − 0.979899i \(-0.563930\pi\)
−0.199495 + 0.979899i \(0.563930\pi\)
\(38\) −108.682 −0.463964
\(39\) −267.771 −1.09943
\(40\) 10.0814 0.0398503
\(41\) −88.7884 −0.338205 −0.169103 − 0.985598i \(-0.554087\pi\)
−0.169103 + 0.985598i \(0.554087\pi\)
\(42\) −278.347 −1.02262
\(43\) −365.622 −1.29667 −0.648334 − 0.761356i \(-0.724534\pi\)
−0.648334 + 0.761356i \(0.724534\pi\)
\(44\) 203.070 0.695773
\(45\) −39.3359 −0.130308
\(46\) 0 0
\(47\) 181.434 0.563082 0.281541 − 0.959549i \(-0.409154\pi\)
0.281541 + 0.959549i \(0.409154\pi\)
\(48\) 318.228 0.956923
\(49\) 255.323 0.744382
\(50\) −146.329 −0.413880
\(51\) 298.491 0.819551
\(52\) 182.917 0.487809
\(53\) −612.828 −1.58827 −0.794136 − 0.607740i \(-0.792076\pi\)
−0.794136 + 0.607740i \(0.792076\pi\)
\(54\) 454.860 1.14627
\(55\) 18.0115 0.0441576
\(56\) 419.851 1.00187
\(57\) −897.511 −2.08558
\(58\) −86.4184 −0.195643
\(59\) 78.0294 0.172179 0.0860895 − 0.996287i \(-0.472563\pi\)
0.0860895 + 0.996287i \(0.472563\pi\)
\(60\) 37.7038 0.0811257
\(61\) −111.873 −0.234818 −0.117409 − 0.993084i \(-0.537459\pi\)
−0.117409 + 0.993084i \(0.537459\pi\)
\(62\) −123.612 −0.253205
\(63\) −1638.18 −3.27606
\(64\) −56.1959 −0.109758
\(65\) 16.2240 0.0309591
\(66\) −348.959 −0.650817
\(67\) 408.709 0.745249 0.372625 − 0.927982i \(-0.378458\pi\)
0.372625 + 0.927982i \(0.378458\pi\)
\(68\) −203.903 −0.363630
\(69\) 0 0
\(70\) 16.8648 0.0287961
\(71\) 449.763 0.751790 0.375895 − 0.926662i \(-0.377335\pi\)
0.375895 + 0.926662i \(0.377335\pi\)
\(72\) −1149.53 −1.88158
\(73\) −93.4293 −0.149796 −0.0748978 − 0.997191i \(-0.523863\pi\)
−0.0748978 + 0.997191i \(0.523863\pi\)
\(74\) −105.410 −0.165591
\(75\) −1208.40 −1.86045
\(76\) 613.100 0.925361
\(77\) 750.107 1.11016
\(78\) −314.328 −0.456290
\(79\) 277.946 0.395840 0.197920 − 0.980218i \(-0.436581\pi\)
0.197920 + 0.980218i \(0.436581\pi\)
\(80\) −19.2812 −0.0269462
\(81\) 1948.03 2.67220
\(82\) −104.226 −0.140364
\(83\) 1111.38 1.46976 0.734878 − 0.678199i \(-0.237239\pi\)
0.734878 + 0.678199i \(0.237239\pi\)
\(84\) 1570.21 2.03958
\(85\) −18.0853 −0.0230779
\(86\) −429.192 −0.538150
\(87\) −713.652 −0.879443
\(88\) 526.360 0.637615
\(89\) 1272.90 1.51604 0.758018 − 0.652234i \(-0.226168\pi\)
0.758018 + 0.652234i \(0.226168\pi\)
\(90\) −46.1752 −0.0540810
\(91\) 675.665 0.778340
\(92\) 0 0
\(93\) −1020.80 −1.13819
\(94\) 212.979 0.233693
\(95\) 54.3794 0.0587285
\(96\) 1704.68 1.81232
\(97\) 1087.37 1.13821 0.569103 − 0.822266i \(-0.307290\pi\)
0.569103 + 0.822266i \(0.307290\pi\)
\(98\) 299.716 0.308937
\(99\) −2053.76 −2.08496
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 529.4.a.m.1.16 25
23.15 odd 22 23.4.c.a.18.3 yes 50
23.20 odd 22 23.4.c.a.9.3 ✓ 50
23.22 odd 2 529.4.a.n.1.16 25
69.20 even 22 207.4.i.a.55.3 50
69.38 even 22 207.4.i.a.64.3 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.4.c.a.9.3 ✓ 50 23.20 odd 22
23.4.c.a.18.3 yes 50 23.15 odd 22
207.4.i.a.55.3 50 69.20 even 22
207.4.i.a.64.3 50 69.38 even 22
529.4.a.m.1.16 25 1.1 even 1 trivial
529.4.a.n.1.16 25 23.22 odd 2