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 = [3,0,0,6,-3,0,0,-9,0,-3,6] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) 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 1.2
Root \(-0.523976\) 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)
 
\(f(q)\) \(=\) \(q+0.523976 q^{2} -1.72545 q^{4} +2.20147 q^{5} -1.95205 q^{8} +1.15352 q^{10} +5.20147 q^{11} -3.15352 q^{13} +2.42807 q^{16} +3.24943 q^{17} -7.45090 q^{19} -3.79853 q^{20} +2.72545 q^{22} +4.40294 q^{23} -0.153520 q^{25} -1.65237 q^{26} -1.15352 q^{29} -2.00000 q^{31} +5.17635 q^{32} +1.70262 q^{34} +5.00000 q^{37} -3.90409 q^{38} -4.29738 q^{40} +11.4509 q^{41} +9.29738 q^{43} -8.97487 q^{44} +2.30704 q^{46} +1.04795 q^{47} -0.0804406 q^{50} +5.44124 q^{52} -0.249425 q^{53} +11.4509 q^{55} -0.604417 q^{58} +8.09591 q^{59} -8.60442 q^{61} -1.04795 q^{62} -2.14386 q^{64} -6.94239 q^{65} -7.60442 q^{67} -5.60672 q^{68} +9.60442 q^{71} -0.846480 q^{73} +2.61988 q^{74} +12.8561 q^{76} -7.60442 q^{79} +5.34533 q^{80} +6.00000 q^{82} -11.4509 q^{83} +7.15352 q^{85} +4.87161 q^{86} -10.1535 q^{88} -9.24943 q^{89} -7.59706 q^{92} +0.549103 q^{94} -16.4029 q^{95} +3.45090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{4} - 3 q^{5} - 9 q^{8} - 3 q^{10} + 6 q^{11} - 3 q^{13} + 12 q^{16} - 3 q^{17} - 21 q^{20} - 3 q^{22} - 6 q^{23} + 6 q^{25} + 27 q^{26} + 3 q^{29} - 6 q^{31} - 18 q^{32} + 21 q^{34} + 15 q^{37}+ \cdots - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.523976 0.370507 0.185254 − 0.982691i \(-0.440689\pi\)
0.185254 + 0.982691i \(0.440689\pi\)
\(3\) 0 0
\(4\) −1.72545 −0.862724
\(5\) 2.20147 0.984528 0.492264 − 0.870446i \(-0.336169\pi\)
0.492264 + 0.870446i \(0.336169\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.95205 −0.690153
\(9\) 0 0
\(10\) 1.15352 0.364775
\(11\) 5.20147 1.56830 0.784151 − 0.620569i \(-0.213099\pi\)
0.784151 + 0.620569i \(0.213099\pi\)
\(12\) 0 0
\(13\) −3.15352 −0.874629 −0.437314 − 0.899309i \(-0.644070\pi\)
−0.437314 + 0.899309i \(0.644070\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.42807 0.607018
\(17\) 3.24943 0.788101 0.394051 − 0.919089i \(-0.371073\pi\)
0.394051 + 0.919089i \(0.371073\pi\)
\(18\) 0 0
\(19\) −7.45090 −1.70935 −0.854677 − 0.519161i \(-0.826245\pi\)
−0.854677 + 0.519161i \(0.826245\pi\)
\(20\) −3.79853 −0.849377
\(21\) 0 0
\(22\) 2.72545 0.581068
\(23\) 4.40294 0.918077 0.459039 − 0.888416i \(-0.348194\pi\)
0.459039 + 0.888416i \(0.348194\pi\)
\(24\) 0 0
\(25\) −0.153520 −0.0307039
\(26\) −1.65237 −0.324056
\(27\) 0 0
\(28\) 0 0
\(29\) −1.15352 −0.214203 −0.107102 − 0.994248i \(-0.534157\pi\)
−0.107102 + 0.994248i \(0.534157\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 5.17635 0.915057
\(33\) 0 0
\(34\) 1.70262 0.291997
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) −3.90409 −0.633328
\(39\) 0 0
\(40\) −4.29738 −0.679475
\(41\) 11.4509 1.78833 0.894165 − 0.447738i \(-0.147770\pi\)
0.894165 + 0.447738i \(0.147770\pi\)
\(42\) 0 0
\(43\) 9.29738 1.41784 0.708918 − 0.705290i \(-0.249183\pi\)
0.708918 + 0.705290i \(0.249183\pi\)
\(44\) −8.97487 −1.35301
\(45\) 0 0
\(46\) 2.30704 0.340154
\(47\) 1.04795 0.152860 0.0764298 − 0.997075i \(-0.475648\pi\)
0.0764298 + 0.997075i \(0.475648\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −0.0804406 −0.0113760
\(51\) 0 0
\(52\) 5.44124 0.754564
\(53\) −0.249425 −0.0342612 −0.0171306 − 0.999853i \(-0.505453\pi\)
−0.0171306 + 0.999853i \(0.505453\pi\)
\(54\) 0 0
\(55\) 11.4509 1.54404
\(56\) 0 0
\(57\) 0 0
\(58\) −0.604417 −0.0793638
\(59\) 8.09591 1.05400 0.526999 − 0.849866i \(-0.323317\pi\)
0.526999 + 0.849866i \(0.323317\pi\)
\(60\) 0 0
\(61\) −8.60442 −1.10168 −0.550841 − 0.834610i \(-0.685693\pi\)
−0.550841 + 0.834610i \(0.685693\pi\)
\(62\) −1.04795 −0.133090
\(63\) 0 0
\(64\) −2.14386 −0.267982
\(65\) −6.94239 −0.861097
\(66\) 0 0
\(67\) −7.60442 −0.929027 −0.464514 − 0.885566i \(-0.653771\pi\)
−0.464514 + 0.885566i \(0.653771\pi\)
\(68\) −5.60672 −0.679914
\(69\) 0 0
\(70\) 0 0
\(71\) 9.60442 1.13983 0.569917 − 0.821702i \(-0.306975\pi\)
0.569917 + 0.821702i \(0.306975\pi\)
\(72\) 0 0
\(73\) −0.846480 −0.0990730 −0.0495365 − 0.998772i \(-0.515774\pi\)
−0.0495365 + 0.998772i \(0.515774\pi\)
\(74\) 2.61988 0.304555
\(75\) 0 0
\(76\) 12.8561 1.47470
\(77\) 0 0
\(78\) 0 0
\(79\) −7.60442 −0.855564 −0.427782 − 0.903882i \(-0.640705\pi\)
−0.427782 + 0.903882i \(0.640705\pi\)
\(80\) 5.34533 0.597626
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −11.4509 −1.25690 −0.628450 − 0.777850i \(-0.716310\pi\)
−0.628450 + 0.777850i \(0.716310\pi\)
\(84\) 0 0
\(85\) 7.15352 0.775908
\(86\) 4.87161 0.525319
\(87\) 0 0
\(88\) −10.1535 −1.08237
\(89\) −9.24943 −0.980437 −0.490219 − 0.871600i \(-0.663083\pi\)
−0.490219 + 0.871600i \(0.663083\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.59706 −0.792048
\(93\) 0 0
\(94\) 0.549103 0.0566356
\(95\) −16.4029 −1.68291
\(96\) 0 0
\(97\) 3.45090 0.350386 0.175193 − 0.984534i \(-0.443945\pi\)
0.175193 + 0.984534i \(0.443945\pi\)
\(98\) 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.n.1.2 3
3.2 odd 2 3969.2.a.o.1.2 3
7.6 odd 2 567.2.a.f.1.2 yes 3
21.20 even 2 567.2.a.e.1.2 ✓ 3
28.27 even 2 9072.2.a.cb.1.1 3
63.13 odd 6 567.2.f.l.379.2 6
63.20 even 6 567.2.f.m.190.2 6
63.34 odd 6 567.2.f.l.190.2 6
63.41 even 6 567.2.f.m.379.2 6
84.83 odd 2 9072.2.a.bu.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.e.1.2 ✓ 3 21.20 even 2
567.2.a.f.1.2 yes 3 7.6 odd 2
567.2.f.l.190.2 6 63.34 odd 6
567.2.f.l.379.2 6 63.13 odd 6
567.2.f.m.190.2 6 63.20 even 6
567.2.f.m.379.2 6 63.41 even 6
3969.2.a.n.1.2 3 1.1 even 1 trivial
3969.2.a.o.1.2 3 3.2 odd 2
9072.2.a.bu.1.3 3 84.83 odd 2
9072.2.a.cb.1.1 3 28.27 even 2