Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4761,2,Mod(1,4761)] 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("4761.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4761, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4761 = 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4761.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,2,0,8,8,0,-19,6,0,-13,3,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.0167764023\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: 10.10.5791333887977.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(2.24419\) of defining polynomial
Character \(\chi\) \(=\) 4761.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.24419 q^{2} +3.03638 q^{4} +2.78323 q^{5} -3.39142 q^{7} +2.32582 q^{8} +6.24609 q^{10} -5.13420 q^{11} -2.18086 q^{13} -7.61098 q^{14} -0.853169 q^{16} -2.58806 q^{17} -2.31236 q^{19} +8.45093 q^{20} -11.5221 q^{22} +2.74636 q^{25} -4.89426 q^{26} -10.2976 q^{28} -5.34289 q^{29} +8.94893 q^{31} -6.56632 q^{32} -5.80809 q^{34} -9.43909 q^{35} -2.19455 q^{37} -5.18936 q^{38} +6.47330 q^{40} -0.305385 q^{41} -9.17975 q^{43} -15.5894 q^{44} +4.99153 q^{47} +4.50171 q^{49} +6.16335 q^{50} -6.62191 q^{52} +8.19642 q^{53} -14.2897 q^{55} -7.88784 q^{56} -11.9905 q^{58} -4.78368 q^{59} -5.85406 q^{61} +20.0831 q^{62} -13.0297 q^{64} -6.06983 q^{65} +0.915436 q^{67} -7.85832 q^{68} -21.1831 q^{70} +1.32972 q^{71} +10.1834 q^{73} -4.92498 q^{74} -7.02119 q^{76} +17.4122 q^{77} -14.2581 q^{79} -2.37456 q^{80} -0.685341 q^{82} +2.68661 q^{83} -7.20316 q^{85} -20.6011 q^{86} -11.9413 q^{88} -0.780082 q^{89} +7.39621 q^{91} +11.2019 q^{94} -6.43582 q^{95} +3.71181 q^{97} +10.1027 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24419 1.58688 0.793440 − 0.608648i \(-0.208288\pi\)
0.793440 + 0.608648i \(0.208288\pi\)
\(3\) 0 0
\(4\) 3.03638 1.51819
\(5\) 2.78323 1.24470 0.622349 − 0.782740i \(-0.286179\pi\)
0.622349 + 0.782740i \(0.286179\pi\)
\(6\) 0 0
\(7\) −3.39142 −1.28184 −0.640918 − 0.767610i \(-0.721446\pi\)
−0.640918 + 0.767610i \(0.721446\pi\)
\(8\) 2.32582 0.822303
\(9\) 0 0
\(10\) 6.24609 1.97519
\(11\) −5.13420 −1.54802 −0.774010 − 0.633173i \(-0.781752\pi\)
−0.774010 + 0.633173i \(0.781752\pi\)
\(12\) 0 0
\(13\) −2.18086 −0.604862 −0.302431 − 0.953171i \(-0.597798\pi\)
−0.302431 + 0.953171i \(0.597798\pi\)
\(14\) −7.61098 −2.03412
\(15\) 0 0
\(16\) −0.853169 −0.213292
\(17\) −2.58806 −0.627697 −0.313848 − 0.949473i \(-0.601618\pi\)
−0.313848 + 0.949473i \(0.601618\pi\)
\(18\) 0 0
\(19\) −2.31236 −0.530491 −0.265246 − 0.964181i \(-0.585453\pi\)
−0.265246 + 0.964181i \(0.585453\pi\)
\(20\) 8.45093 1.88969
\(21\) 0 0
\(22\) −11.5221 −2.45652
\(23\) 0 0
\(24\) 0 0
\(25\) 2.74636 0.549273
\(26\) −4.89426 −0.959843
\(27\) 0 0
\(28\) −10.2976 −1.94607
\(29\) −5.34289 −0.992150 −0.496075 − 0.868280i \(-0.665226\pi\)
−0.496075 + 0.868280i \(0.665226\pi\)
\(30\) 0 0
\(31\) 8.94893 1.60728 0.803638 − 0.595118i \(-0.202895\pi\)
0.803638 + 0.595118i \(0.202895\pi\)
\(32\) −6.56632 −1.16077
\(33\) 0 0
\(34\) −5.80809 −0.996079
\(35\) −9.43909 −1.59550
\(36\) 0 0
\(37\) −2.19455 −0.360782 −0.180391 − 0.983595i \(-0.557736\pi\)
−0.180391 + 0.983595i \(0.557736\pi\)
\(38\) −5.18936 −0.841826
\(39\) 0 0
\(40\) 6.47330 1.02352
\(41\) −0.305385 −0.0476931 −0.0238466 − 0.999716i \(-0.507591\pi\)
−0.0238466 + 0.999716i \(0.507591\pi\)
\(42\) 0 0
\(43\) −9.17975 −1.39990 −0.699949 − 0.714192i \(-0.746794\pi\)
−0.699949 + 0.714192i \(0.746794\pi\)
\(44\) −15.5894 −2.35019
\(45\) 0 0
\(46\) 0 0
\(47\) 4.99153 0.728090 0.364045 − 0.931381i \(-0.381395\pi\)
0.364045 + 0.931381i \(0.381395\pi\)
\(48\) 0 0
\(49\) 4.50171 0.643102
\(50\) 6.16335 0.871630
\(51\) 0 0
\(52\) −6.62191 −0.918294
\(53\) 8.19642 1.12587 0.562933 − 0.826503i \(-0.309673\pi\)
0.562933 + 0.826503i \(0.309673\pi\)
\(54\) 0 0
\(55\) −14.2897 −1.92682
\(56\) −7.88784 −1.05406
\(57\) 0 0
\(58\) −11.9905 −1.57442
\(59\) −4.78368 −0.622782 −0.311391 − 0.950282i \(-0.600795\pi\)
−0.311391 + 0.950282i \(0.600795\pi\)
\(60\) 0 0
\(61\) −5.85406 −0.749535 −0.374768 − 0.927119i \(-0.622278\pi\)
−0.374768 + 0.927119i \(0.622278\pi\)
\(62\) 20.0831 2.55055
\(63\) 0 0
\(64\) −13.0297 −1.62871
\(65\) −6.06983 −0.752870
\(66\) 0 0
\(67\) 0.915436 0.111838 0.0559191 − 0.998435i \(-0.482191\pi\)
0.0559191 + 0.998435i \(0.482191\pi\)
\(68\) −7.85832 −0.952962
\(69\) 0 0
\(70\) −21.1831 −2.53186
\(71\) 1.32972 0.157808 0.0789042 − 0.996882i \(-0.474858\pi\)
0.0789042 + 0.996882i \(0.474858\pi\)
\(72\) 0 0
\(73\) 10.1834 1.19187 0.595936 − 0.803032i \(-0.296781\pi\)
0.595936 + 0.803032i \(0.296781\pi\)
\(74\) −4.92498 −0.572517
\(75\) 0 0
\(76\) −7.02119 −0.805386
\(77\) 17.4122 1.98431
\(78\) 0 0
\(79\) −14.2581 −1.60416 −0.802080 − 0.597216i \(-0.796273\pi\)
−0.802080 + 0.597216i \(0.796273\pi\)
\(80\) −2.37456 −0.265484
\(81\) 0 0
\(82\) −0.685341 −0.0756833
\(83\) 2.68661 0.294894 0.147447 − 0.989070i \(-0.452894\pi\)
0.147447 + 0.989070i \(0.452894\pi\)
\(84\) 0 0
\(85\) −7.20316 −0.781293
\(86\) −20.6011 −2.22147
\(87\) 0 0
\(88\) −11.9413 −1.27294
\(89\) −0.780082 −0.0826885 −0.0413443 − 0.999145i \(-0.513164\pi\)
−0.0413443 + 0.999145i \(0.513164\pi\)
\(90\) 0 0
\(91\) 7.39621 0.775333
\(92\) 0 0
\(93\) 0 0
\(94\) 11.2019 1.15539
\(95\) −6.43582 −0.660301
\(96\) 0 0
\(97\) 3.71181 0.376877 0.188439 − 0.982085i \(-0.439657\pi\)
0.188439 + 0.982085i \(0.439657\pi\)
\(98\) 10.1027 1.02053
\(99\) 0 0
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 4761.2.a.bu.1.9 10
3.2 odd 2 1587.2.a.t.1.2 10
23.7 odd 22 207.2.i.d.118.1 20
23.10 odd 22 207.2.i.d.100.1 20
23.22 odd 2 4761.2.a.bt.1.9 10
69.53 even 22 69.2.e.c.49.2 yes 20
69.56 even 22 69.2.e.c.31.2 ✓ 20
69.68 even 2 1587.2.a.u.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.e.c.31.2 ✓ 20 69.56 even 22
69.2.e.c.49.2 yes 20 69.53 even 22
207.2.i.d.100.1 20 23.10 odd 22
207.2.i.d.118.1 20 23.7 odd 22
1587.2.a.t.1.2 10 3.2 odd 2
1587.2.a.u.1.2 10 69.68 even 2
4761.2.a.bt.1.9 10 23.22 odd 2
4761.2.a.bu.1.9 10 1.1 even 1 trivial