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: \(0\)
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 1.3
Root \(-1.41812\) 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)
 
\(f(q)\) \(=\) \(q-1.41812 q^{2} +0.0110547 q^{4} +0.849430 q^{5} +5.06334 q^{7} +2.82056 q^{8} -1.20459 q^{10} +1.47830 q^{11} +1.01815 q^{13} -7.18040 q^{14} -4.02199 q^{16} -5.71934 q^{17} +4.56726 q^{19} +0.00939015 q^{20} -2.09640 q^{22} -4.27847 q^{25} -1.44385 q^{26} +0.0559734 q^{28} +2.87836 q^{29} +5.27820 q^{31} +0.0625339 q^{32} +8.11068 q^{34} +4.30095 q^{35} -0.462189 q^{37} -6.47690 q^{38} +2.39586 q^{40} +6.89693 q^{41} +4.85049 q^{43} +0.0163421 q^{44} -4.11922 q^{47} +18.6374 q^{49} +6.06737 q^{50} +0.0112552 q^{52} -2.82472 q^{53} +1.25571 q^{55} +14.2814 q^{56} -4.08185 q^{58} -2.11728 q^{59} +9.77162 q^{61} -7.48511 q^{62} +7.95529 q^{64} +0.864843 q^{65} -5.48351 q^{67} -0.0632253 q^{68} -6.09924 q^{70} +12.6544 q^{71} +1.75910 q^{73} +0.655438 q^{74} +0.0504895 q^{76} +7.48511 q^{77} -2.34094 q^{79} -3.41639 q^{80} -9.78065 q^{82} -16.4396 q^{83} -4.85817 q^{85} -6.87857 q^{86} +4.16962 q^{88} +10.1854 q^{89} +5.15521 q^{91} +5.84154 q^{94} +3.87956 q^{95} -11.2699 q^{97} -26.4300 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 8 q^{4} - 8 q^{5} + 19 q^{7} + 6 q^{8} + 13 q^{10} - 3 q^{11} - 4 q^{13} - 4 q^{16} - 11 q^{17} + 22 q^{19} - q^{20} + 13 q^{22} - 2 q^{25} - 4 q^{26} + 26 q^{28} + 5 q^{29} - 7 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41812 −1.00276 −0.501380 − 0.865227i \(-0.667174\pi\)
−0.501380 + 0.865227i \(0.667174\pi\)
\(3\) 0 0
\(4\) 0.0110547 0.00552733
\(5\) 0.849430 0.379876 0.189938 − 0.981796i \(-0.439171\pi\)
0.189938 + 0.981796i \(0.439171\pi\)
\(6\) 0 0
\(7\) 5.06334 1.91376 0.956880 − 0.290482i \(-0.0938156\pi\)
0.956880 + 0.290482i \(0.0938156\pi\)
\(8\) 2.82056 0.997217
\(9\) 0 0
\(10\) −1.20459 −0.380925
\(11\) 1.47830 0.445723 0.222862 − 0.974850i \(-0.428460\pi\)
0.222862 + 0.974850i \(0.428460\pi\)
\(12\) 0 0
\(13\) 1.01815 0.282383 0.141191 − 0.989982i \(-0.454907\pi\)
0.141191 + 0.989982i \(0.454907\pi\)
\(14\) −7.18040 −1.91904
\(15\) 0 0
\(16\) −4.02199 −1.00550
\(17\) −5.71934 −1.38714 −0.693571 − 0.720388i \(-0.743964\pi\)
−0.693571 + 0.720388i \(0.743964\pi\)
\(18\) 0 0
\(19\) 4.56726 1.04780 0.523900 − 0.851780i \(-0.324476\pi\)
0.523900 + 0.851780i \(0.324476\pi\)
\(20\) 0.00939015 0.00209970
\(21\) 0 0
\(22\) −2.09640 −0.446953
\(23\) 0 0
\(24\) 0 0
\(25\) −4.27847 −0.855694
\(26\) −1.44385 −0.283162
\(27\) 0 0
\(28\) 0.0559734 0.0105780
\(29\) 2.87836 0.534498 0.267249 − 0.963627i \(-0.413885\pi\)
0.267249 + 0.963627i \(0.413885\pi\)
\(30\) 0 0
\(31\) 5.27820 0.947993 0.473997 − 0.880527i \(-0.342811\pi\)
0.473997 + 0.880527i \(0.342811\pi\)
\(32\) 0.0625339 0.0110545
\(33\) 0 0
\(34\) 8.11068 1.39097
\(35\) 4.30095 0.726993
\(36\) 0 0
\(37\) −0.462189 −0.0759835 −0.0379917 − 0.999278i \(-0.512096\pi\)
−0.0379917 + 0.999278i \(0.512096\pi\)
\(38\) −6.47690 −1.05069
\(39\) 0 0
\(40\) 2.39586 0.378819
\(41\) 6.89693 1.07712 0.538560 − 0.842587i \(-0.318969\pi\)
0.538560 + 0.842587i \(0.318969\pi\)
\(42\) 0 0
\(43\) 4.85049 0.739693 0.369847 − 0.929093i \(-0.379410\pi\)
0.369847 + 0.929093i \(0.379410\pi\)
\(44\) 0.0163421 0.00246366
\(45\) 0 0
\(46\) 0 0
\(47\) −4.11922 −0.600850 −0.300425 − 0.953805i \(-0.597129\pi\)
−0.300425 + 0.953805i \(0.597129\pi\)
\(48\) 0 0
\(49\) 18.6374 2.66248
\(50\) 6.06737 0.858055
\(51\) 0 0
\(52\) 0.0112552 0.00156082
\(53\) −2.82472 −0.388006 −0.194003 − 0.981001i \(-0.562147\pi\)
−0.194003 + 0.981001i \(0.562147\pi\)
\(54\) 0 0
\(55\) 1.25571 0.169320
\(56\) 14.2814 1.90844
\(57\) 0 0
\(58\) −4.08185 −0.535973
\(59\) −2.11728 −0.275646 −0.137823 − 0.990457i \(-0.544011\pi\)
−0.137823 + 0.990457i \(0.544011\pi\)
\(60\) 0 0
\(61\) 9.77162 1.25113 0.625564 − 0.780173i \(-0.284869\pi\)
0.625564 + 0.780173i \(0.284869\pi\)
\(62\) −7.48511 −0.950609
\(63\) 0 0
\(64\) 7.95529 0.994412
\(65\) 0.864843 0.107271
\(66\) 0 0
\(67\) −5.48351 −0.669918 −0.334959 − 0.942233i \(-0.608722\pi\)
−0.334959 + 0.942233i \(0.608722\pi\)
\(68\) −0.0632253 −0.00766719
\(69\) 0 0
\(70\) −6.09924 −0.728999
\(71\) 12.6544 1.50181 0.750903 − 0.660413i \(-0.229619\pi\)
0.750903 + 0.660413i \(0.229619\pi\)
\(72\) 0 0
\(73\) 1.75910 0.205887 0.102943 − 0.994687i \(-0.467174\pi\)
0.102943 + 0.994687i \(0.467174\pi\)
\(74\) 0.655438 0.0761932
\(75\) 0 0
\(76\) 0.0504895 0.00579154
\(77\) 7.48511 0.853008
\(78\) 0 0
\(79\) −2.34094 −0.263377 −0.131688 − 0.991291i \(-0.542040\pi\)
−0.131688 + 0.991291i \(0.542040\pi\)
\(80\) −3.41639 −0.381965
\(81\) 0 0
\(82\) −9.78065 −1.08009
\(83\) −16.4396 −1.80448 −0.902240 − 0.431235i \(-0.858078\pi\)
−0.902240 + 0.431235i \(0.858078\pi\)
\(84\) 0 0
\(85\) −4.85817 −0.526943
\(86\) −6.87857 −0.741735
\(87\) 0 0
\(88\) 4.16962 0.444483
\(89\) 10.1854 1.07965 0.539823 − 0.841778i \(-0.318491\pi\)
0.539823 + 0.841778i \(0.318491\pi\)
\(90\) 0 0
\(91\) 5.15521 0.540413
\(92\) 0 0
\(93\) 0 0
\(94\) 5.84154 0.602508
\(95\) 3.87956 0.398035
\(96\) 0 0
\(97\) −11.2699 −1.14429 −0.572145 − 0.820153i \(-0.693889\pi\)
−0.572145 + 0.820153i \(0.693889\pi\)
\(98\) −26.4300 −2.66983
\(99\) 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.bt.1.3 10
3.2 odd 2 1587.2.a.u.1.8 10
23.9 even 11 207.2.i.d.127.1 20
23.18 even 11 207.2.i.d.163.1 20
23.22 odd 2 4761.2.a.bu.1.3 10
69.32 odd 22 69.2.e.c.58.2 yes 20
69.41 odd 22 69.2.e.c.25.2 ✓ 20
69.68 even 2 1587.2.a.t.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.e.c.25.2 ✓ 20 69.41 odd 22
69.2.e.c.58.2 yes 20 69.32 odd 22
207.2.i.d.127.1 20 23.9 even 11
207.2.i.d.163.1 20 23.18 even 11
1587.2.a.t.1.8 10 69.68 even 2
1587.2.a.u.1.8 10 3.2 odd 2
4761.2.a.bt.1.3 10 1.1 even 1 trivial
4761.2.a.bu.1.3 10 23.22 odd 2