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 1.2
Root \(4.77200\) 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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +5.77200 q^{3} +4.00000 q^{4} +3.54400 q^{5} +11.5440 q^{6} +11.0880 q^{7} +8.00000 q^{8} +6.31601 q^{9} +7.08801 q^{10} +6.00000 q^{11} +23.0880 q^{12} +21.4920 q^{13} +22.1760 q^{14} +20.4560 q^{15} +16.0000 q^{16} +7.36799 q^{17} +12.6320 q^{18} +93.4400 q^{19} +14.1760 q^{20} +64.0000 q^{21} +12.0000 q^{22} +46.1760 q^{24} -112.440 q^{25} +42.9840 q^{26} -119.388 q^{27} +44.3520 q^{28} +112.476 q^{29} +40.9120 q^{30} +286.268 q^{31} +32.0000 q^{32} +34.6320 q^{33} +14.7360 q^{34} +39.2959 q^{35} +25.2640 q^{36} +59.3361 q^{37} +186.880 q^{38} +124.052 q^{39} +28.3520 q^{40} +62.6119 q^{41} +128.000 q^{42} -507.304 q^{43} +24.0000 q^{44} +22.3839 q^{45} +536.300 q^{47} +92.3520 q^{48} -220.056 q^{49} -224.880 q^{50} +42.5280 q^{51} +85.9681 q^{52} -187.336 q^{53} -238.776 q^{54} +21.2640 q^{55} +88.7041 q^{56} +539.336 q^{57} +224.952 q^{58} +49.6799 q^{59} +81.8240 q^{60} +778.776 q^{61} +572.536 q^{62} +70.0319 q^{63} +64.0000 q^{64} +76.1678 q^{65} +69.2640 q^{66} +661.232 q^{67} +29.4720 q^{68} +78.5919 q^{70} +289.212 q^{71} +50.5280 q^{72} -651.316 q^{73} +118.672 q^{74} -649.004 q^{75} +373.760 q^{76} +66.5280 q^{77} +248.104 q^{78} +50.0720 q^{79} +56.7041 q^{80} -859.640 q^{81} +125.224 q^{82} -807.016 q^{83} +256.000 q^{84} +26.1122 q^{85} -1014.61 q^{86} +649.212 q^{87} +48.0000 q^{88} -946.104 q^{89} +44.7679 q^{90} +238.304 q^{91} +1652.34 q^{93} +1072.60 q^{94} +331.152 q^{95} +184.704 q^{96} -715.529 q^{97} -440.112 q^{98} +37.8960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} - 10 q^{5} + 6 q^{6} - 12 q^{7} + 16 q^{8} - 13 q^{9} - 20 q^{10} + 12 q^{11} + 12 q^{12} - 51 q^{13} - 24 q^{14} + 58 q^{15} + 32 q^{16} + 66 q^{17} - 26 q^{18} + 16 q^{19}+ \cdots - 78 q^{99}+O(q^{100}) \) 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
\(3\) 5.77200 1.11082 0.555411 − 0.831576i \(-0.312561\pi\)
0.555411 + 0.831576i \(0.312561\pi\)
\(4\) 4.00000 0.500000
\(5\) 3.54400 0.316985 0.158493 − 0.987360i \(-0.449337\pi\)
0.158493 + 0.987360i \(0.449337\pi\)
\(6\) 11.5440 0.785470
\(7\) 11.0880 0.598696 0.299348 − 0.954144i \(-0.403231\pi\)
0.299348 + 0.954144i \(0.403231\pi\)
\(8\) 8.00000 0.353553
\(9\) 6.31601 0.233926
\(10\) 7.08801 0.224142
\(11\) 6.00000 0.164461 0.0822304 − 0.996613i \(-0.473796\pi\)
0.0822304 + 0.996613i \(0.473796\pi\)
\(12\) 23.0880 0.555411
\(13\) 21.4920 0.458524 0.229262 − 0.973365i \(-0.426369\pi\)
0.229262 + 0.973365i \(0.426369\pi\)
\(14\) 22.1760 0.423342
\(15\) 20.4560 0.352114
\(16\) 16.0000 0.250000
\(17\) 7.36799 0.105118 0.0525588 − 0.998618i \(-0.483262\pi\)
0.0525588 + 0.998618i \(0.483262\pi\)
\(18\) 12.6320 0.165411
\(19\) 93.4400 1.12824 0.564121 − 0.825692i \(-0.309215\pi\)
0.564121 + 0.825692i \(0.309215\pi\)
\(20\) 14.1760 0.158493
\(21\) 64.0000 0.665045
\(22\) 12.0000 0.116291
\(23\) 0 0
\(24\) 46.1760 0.392735
\(25\) −112.440 −0.899520
\(26\) 42.9840 0.324226
\(27\) −119.388 −0.850972
\(28\) 44.3520 0.299348
\(29\) 112.476 0.720217 0.360108 − 0.932911i \(-0.382740\pi\)
0.360108 + 0.932911i \(0.382740\pi\)
\(30\) 40.9120 0.248982
\(31\) 286.268 1.65856 0.829279 − 0.558835i \(-0.188752\pi\)
0.829279 + 0.558835i \(0.188752\pi\)
\(32\) 32.0000 0.176777
\(33\) 34.6320 0.182687
\(34\) 14.7360 0.0743294
\(35\) 39.2959 0.189778
\(36\) 25.2640 0.116963
\(37\) 59.3361 0.263643 0.131821 − 0.991273i \(-0.457917\pi\)
0.131821 + 0.991273i \(0.457917\pi\)
\(38\) 186.880 0.797788
\(39\) 124.052 0.509339
\(40\) 28.3520 0.112071
\(41\) 62.6119 0.238496 0.119248 − 0.992864i \(-0.461952\pi\)
0.119248 + 0.992864i \(0.461952\pi\)
\(42\) 128.000 0.470258
\(43\) −507.304 −1.79914 −0.899572 − 0.436773i \(-0.856121\pi\)
−0.899572 + 0.436773i \(0.856121\pi\)
\(44\) 24.0000 0.0822304
\(45\) 22.3839 0.0741512
\(46\) 0 0
\(47\) 536.300 1.66441 0.832206 − 0.554466i \(-0.187077\pi\)
0.832206 + 0.554466i \(0.187077\pi\)
\(48\) 92.3520 0.277706
\(49\) −220.056 −0.641563
\(50\) −224.880 −0.636057
\(51\) 42.5280 0.116767
\(52\) 85.9681 0.229262
\(53\) −187.336 −0.485521 −0.242760 − 0.970086i \(-0.578053\pi\)
−0.242760 + 0.970086i \(0.578053\pi\)
\(54\) −238.776 −0.601728
\(55\) 21.2640 0.0521316
\(56\) 88.7041 0.211671
\(57\) 539.336 1.25328
\(58\) 224.952 0.509270
\(59\) 49.6799 0.109623 0.0548116 − 0.998497i \(-0.482544\pi\)
0.0548116 + 0.998497i \(0.482544\pi\)
\(60\) 81.8240 0.176057
\(61\) 778.776 1.63462 0.817312 − 0.576195i \(-0.195463\pi\)
0.817312 + 0.576195i \(0.195463\pi\)
\(62\) 572.536 1.17278
\(63\) 70.0319 0.140051
\(64\) 64.0000 0.125000
\(65\) 76.1678 0.145345
\(66\) 69.2640 0.129179
\(67\) 661.232 1.20571 0.602853 − 0.797852i \(-0.294030\pi\)
0.602853 + 0.797852i \(0.294030\pi\)
\(68\) 29.4720 0.0525588
\(69\) 0 0
\(70\) 78.5919 0.134193
\(71\) 289.212 0.483425 0.241712 − 0.970348i \(-0.422291\pi\)
0.241712 + 0.970348i \(0.422291\pi\)
\(72\) 50.5280 0.0827054
\(73\) −651.316 −1.04426 −0.522129 − 0.852867i \(-0.674862\pi\)
−0.522129 + 0.852867i \(0.674862\pi\)
\(74\) 118.672 0.186424
\(75\) −649.004 −0.999207
\(76\) 373.760 0.564121
\(77\) 66.5280 0.0984620
\(78\) 248.104 0.360157
\(79\) 50.0720 0.0713107 0.0356554 − 0.999364i \(-0.488648\pi\)
0.0356554 + 0.999364i \(0.488648\pi\)
\(80\) 56.7041 0.0792463
\(81\) −859.640 −1.17920
\(82\) 125.224 0.168642
\(83\) −807.016 −1.06725 −0.533624 − 0.845722i \(-0.679170\pi\)
−0.533624 + 0.845722i \(0.679170\pi\)
\(84\) 256.000 0.332522
\(85\) 26.1122 0.0333207
\(86\) −1014.61 −1.27219
\(87\) 649.212 0.800033
\(88\) 48.0000 0.0581456
\(89\) −946.104 −1.12682 −0.563409 − 0.826178i \(-0.690510\pi\)
−0.563409 + 0.826178i \(0.690510\pi\)
\(90\) 44.7679 0.0524328
\(91\) 238.304 0.274517
\(92\) 0 0
\(93\) 1652.34 1.84236
\(94\) 1072.60 1.17692
\(95\) 331.152 0.357636
\(96\) 184.704 0.196367
\(97\) −715.529 −0.748978 −0.374489 − 0.927231i \(-0.622182\pi\)
−0.374489 + 0.927231i \(0.622182\pi\)
\(98\) −440.112 −0.453654
\(99\) 37.8960 0.0384717
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.1.2 2
23.22 odd 2 46.4.a.d.1.2 ✓ 2
69.68 even 2 414.4.a.f.1.2 2
92.91 even 2 368.4.a.f.1.1 2
115.22 even 4 1150.4.b.j.599.3 4
115.68 even 4 1150.4.b.j.599.2 4
115.114 odd 2 1150.4.a.j.1.1 2
161.160 even 2 2254.4.a.f.1.1 2
184.45 odd 2 1472.4.a.k.1.1 2
184.91 even 2 1472.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.a.d.1.2 ✓ 2 23.22 odd 2
368.4.a.f.1.1 2 92.91 even 2
414.4.a.f.1.2 2 69.68 even 2
1058.4.a.j.1.2 2 1.1 even 1 trivial
1150.4.a.j.1.1 2 115.114 odd 2
1150.4.b.j.599.2 4 115.68 even 4
1150.4.b.j.599.3 4 115.22 even 4
1472.4.a.k.1.1 2 184.45 odd 2
1472.4.a.n.1.2 2 184.91 even 2
2254.4.a.f.1.1 2 161.160 even 2