Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3969,2,Mod(1,3969)] 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("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3969, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,0,0,0,0,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 567)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.456850\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.18890 q^{2} +2.79129 q^{4} +0.913701 q^{5} -1.73205 q^{8} -2.00000 q^{10} -2.64575 q^{11} -4.00000 q^{13} -1.79129 q^{16} +3.46410 q^{17} -5.58258 q^{19} +2.55040 q^{20} +5.79129 q^{22} +3.46410 q^{23} -4.16515 q^{25} +8.75560 q^{26} +8.75560 q^{29} +9.16515 q^{31} +7.38505 q^{32} -7.58258 q^{34} +3.00000 q^{37} +12.2197 q^{38} -1.58258 q^{40} -0.913701 q^{41} +0.582576 q^{43} -7.38505 q^{44} -7.58258 q^{46} -13.1334 q^{47} +9.11710 q^{50} -11.1652 q^{52} +8.66025 q^{53} -2.41742 q^{55} -19.1652 q^{58} +3.46410 q^{59} -11.5826 q^{61} -20.0616 q^{62} -12.5826 q^{64} -3.65480 q^{65} +8.58258 q^{67} +9.66930 q^{68} +4.47315 q^{71} -15.1652 q^{73} -6.56670 q^{74} -15.5826 q^{76} +0.582576 q^{79} -1.63670 q^{80} +2.00000 q^{82} -9.66930 q^{83} +3.16515 q^{85} -1.27520 q^{86} +4.58258 q^{88} -1.82740 q^{89} +9.66930 q^{92} +28.7477 q^{94} -5.10080 q^{95} +1.58258 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 8 q^{10} - 16 q^{13} + 2 q^{16} - 4 q^{19} + 14 q^{22} + 20 q^{25} - 12 q^{34} + 12 q^{37} + 12 q^{40} - 16 q^{43} - 12 q^{46} - 8 q^{52} - 28 q^{55} - 40 q^{58} - 28 q^{61} - 32 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18890 −1.54779 −0.773893 − 0.633316i \(-0.781693\pi\)
−0.773893 + 0.633316i \(0.781693\pi\)
\(3\) 0 0
\(4\) 2.79129 1.39564
\(5\) 0.913701 0.408619 0.204310 − 0.978906i \(-0.434505\pi\)
0.204310 + 0.978906i \(0.434505\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −2.64575 −0.797724 −0.398862 − 0.917011i \(-0.630595\pi\)
−0.398862 + 0.917011i \(0.630595\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.79129 −0.447822
\(17\) 3.46410 0.840168 0.420084 − 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −5.58258 −1.28073 −0.640365 − 0.768070i \(-0.721217\pi\)
−0.640365 + 0.768070i \(0.721217\pi\)
\(20\) 2.55040 0.570287
\(21\) 0 0
\(22\) 5.79129 1.23471
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) −4.16515 −0.833030
\(26\) 8.75560 1.71712
\(27\) 0 0
\(28\) 0 0
\(29\) 8.75560 1.62587 0.812937 − 0.582351i \(-0.197867\pi\)
0.812937 + 0.582351i \(0.197867\pi\)
\(30\) 0 0
\(31\) 9.16515 1.64611 0.823055 − 0.567962i \(-0.192268\pi\)
0.823055 + 0.567962i \(0.192268\pi\)
\(32\) 7.38505 1.30551
\(33\) 0 0
\(34\) −7.58258 −1.30040
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 12.2197 1.98230
\(39\) 0 0
\(40\) −1.58258 −0.250227
\(41\) −0.913701 −0.142696 −0.0713480 − 0.997451i \(-0.522730\pi\)
−0.0713480 + 0.997451i \(0.522730\pi\)
\(42\) 0 0
\(43\) 0.582576 0.0888420 0.0444210 − 0.999013i \(-0.485856\pi\)
0.0444210 + 0.999013i \(0.485856\pi\)
\(44\) −7.38505 −1.11334
\(45\) 0 0
\(46\) −7.58258 −1.11799
\(47\) −13.1334 −1.91570 −0.957852 − 0.287262i \(-0.907255\pi\)
−0.957852 + 0.287262i \(0.907255\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 9.11710 1.28935
\(51\) 0 0
\(52\) −11.1652 −1.54833
\(53\) 8.66025 1.18958 0.594789 − 0.803882i \(-0.297236\pi\)
0.594789 + 0.803882i \(0.297236\pi\)
\(54\) 0 0
\(55\) −2.41742 −0.325965
\(56\) 0 0
\(57\) 0 0
\(58\) −19.1652 −2.51651
\(59\) 3.46410 0.450988 0.225494 − 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) −11.5826 −1.48300 −0.741498 − 0.670955i \(-0.765885\pi\)
−0.741498 + 0.670955i \(0.765885\pi\)
\(62\) −20.0616 −2.54783
\(63\) 0 0
\(64\) −12.5826 −1.57282
\(65\) −3.65480 −0.453322
\(66\) 0 0
\(67\) 8.58258 1.04853 0.524264 − 0.851556i \(-0.324340\pi\)
0.524264 + 0.851556i \(0.324340\pi\)
\(68\) 9.66930 1.17258
\(69\) 0 0
\(70\) 0 0
\(71\) 4.47315 0.530866 0.265433 − 0.964129i \(-0.414485\pi\)
0.265433 + 0.964129i \(0.414485\pi\)
\(72\) 0 0
\(73\) −15.1652 −1.77495 −0.887473 − 0.460859i \(-0.847541\pi\)
−0.887473 + 0.460859i \(0.847541\pi\)
\(74\) −6.56670 −0.763364
\(75\) 0 0
\(76\) −15.5826 −1.78744
\(77\) 0 0
\(78\) 0 0
\(79\) 0.582576 0.0655449 0.0327724 − 0.999463i \(-0.489566\pi\)
0.0327724 + 0.999463i \(0.489566\pi\)
\(80\) −1.63670 −0.182989
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −9.66930 −1.06134 −0.530672 − 0.847577i \(-0.678060\pi\)
−0.530672 + 0.847577i \(0.678060\pi\)
\(84\) 0 0
\(85\) 3.16515 0.343309
\(86\) −1.27520 −0.137508
\(87\) 0 0
\(88\) 4.58258 0.488504
\(89\) −1.82740 −0.193704 −0.0968521 − 0.995299i \(-0.530877\pi\)
−0.0968521 + 0.995299i \(0.530877\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.66930 1.00809
\(93\) 0 0
\(94\) 28.7477 2.96510
\(95\) −5.10080 −0.523331
\(96\) 0 0
\(97\) 1.58258 0.160686 0.0803431 − 0.996767i \(-0.474398\pi\)
0.0803431 + 0.996767i \(0.474398\pi\)
\(98\) 0 0
\(99\) 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 3969.2.a.u.1.1 4
3.2 odd 2 inner 3969.2.a.u.1.4 4
7.6 odd 2 567.2.a.i.1.1 ✓ 4
21.20 even 2 567.2.a.i.1.4 yes 4
28.27 even 2 9072.2.a.ci.1.2 4
63.13 odd 6 567.2.f.n.379.4 8
63.20 even 6 567.2.f.n.190.1 8
63.34 odd 6 567.2.f.n.190.4 8
63.41 even 6 567.2.f.n.379.1 8
84.83 odd 2 9072.2.a.ci.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.1 ✓ 4 7.6 odd 2
567.2.a.i.1.4 yes 4 21.20 even 2
567.2.f.n.190.1 8 63.20 even 6
567.2.f.n.190.4 8 63.34 odd 6
567.2.f.n.379.1 8 63.41 even 6
567.2.f.n.379.4 8 63.13 odd 6
3969.2.a.u.1.1 4 1.1 even 1 trivial
3969.2.a.u.1.4 4 3.2 odd 2 inner
9072.2.a.ci.1.2 4 28.27 even 2
9072.2.a.ci.1.3 4 84.83 odd 2