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 1.3
Root \(2.66908\) 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)
 
\(f(q)\) \(=\) \(q+2.66908 q^{2} +5.12398 q^{4} +1.45490 q^{5} +8.33816 q^{8} +3.88325 q^{10} -1.54510 q^{11} -5.88325 q^{13} +12.0072 q^{16} +6.79306 q^{17} +6.24797 q^{19} +7.45490 q^{20} -4.12398 q^{22} +2.90981 q^{23} -2.88325 q^{25} -15.7029 q^{26} +3.88325 q^{29} -2.00000 q^{31} +15.3719 q^{32} +18.1312 q^{34} +5.00000 q^{37} +16.6763 q^{38} +12.1312 q^{40} +2.24797 q^{41} -7.13122 q^{43} -7.91705 q^{44} +7.76651 q^{46} +5.33816 q^{47} -7.69563 q^{50} -30.1457 q^{52} -9.79306 q^{53} -2.24797 q^{55} +10.3647 q^{58} +4.67632 q^{59} +2.36471 q^{61} -5.33816 q^{62} +17.0145 q^{64} -8.55957 q^{65} +3.36471 q^{67} +34.8075 q^{68} +1.36471 q^{71} +1.88325 q^{73} +13.3454 q^{74} +32.0145 q^{76} +3.36471 q^{79} +17.4694 q^{80} +6.00000 q^{82} -2.24797 q^{83} +9.88325 q^{85} -19.0338 q^{86} -12.8833 q^{88} -0.793062 q^{89} +14.9098 q^{92} +14.2480 q^{94} +9.09019 q^{95} -10.2480 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{4} + 3 q^{5} + 9 q^{8} - 3 q^{10} - 6 q^{11} - 3 q^{13} + 12 q^{16} + 3 q^{17} + 21 q^{20} - 3 q^{22} + 6 q^{23} + 6 q^{25} - 27 q^{26} - 3 q^{29} - 6 q^{31} + 18 q^{32} + 21 q^{34} + 15 q^{37}+ \cdots - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66908 1.88732 0.943662 − 0.330911i \(-0.107356\pi\)
0.943662 + 0.330911i \(0.107356\pi\)
\(3\) 0 0
\(4\) 5.12398 2.56199
\(5\) 1.45490 0.650653 0.325326 − 0.945602i \(-0.394526\pi\)
0.325326 + 0.945602i \(0.394526\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.33816 2.94798
\(9\) 0 0
\(10\) 3.88325 1.22799
\(11\) −1.54510 −0.465864 −0.232932 − 0.972493i \(-0.574832\pi\)
−0.232932 + 0.972493i \(0.574832\pi\)
\(12\) 0 0
\(13\) −5.88325 −1.63172 −0.815861 − 0.578249i \(-0.803736\pi\)
−0.815861 + 0.578249i \(0.803736\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 12.0072 3.00181
\(17\) 6.79306 1.64756 0.823780 − 0.566910i \(-0.191861\pi\)
0.823780 + 0.566910i \(0.191861\pi\)
\(18\) 0 0
\(19\) 6.24797 1.43338 0.716691 − 0.697391i \(-0.245656\pi\)
0.716691 + 0.697391i \(0.245656\pi\)
\(20\) 7.45490 1.66697
\(21\) 0 0
\(22\) −4.12398 −0.879236
\(23\) 2.90981 0.606737 0.303368 − 0.952873i \(-0.401889\pi\)
0.303368 + 0.952873i \(0.401889\pi\)
\(24\) 0 0
\(25\) −2.88325 −0.576651
\(26\) −15.7029 −3.07959
\(27\) 0 0
\(28\) 0 0
\(29\) 3.88325 0.721102 0.360551 − 0.932739i \(-0.382589\pi\)
0.360551 + 0.932739i \(0.382589\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 15.3719 2.71740
\(33\) 0 0
\(34\) 18.1312 3.10948
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 16.6763 2.70526
\(39\) 0 0
\(40\) 12.1312 1.91811
\(41\) 2.24797 0.351073 0.175537 − 0.984473i \(-0.443834\pi\)
0.175537 + 0.984473i \(0.443834\pi\)
\(42\) 0 0
\(43\) −7.13122 −1.08750 −0.543750 − 0.839247i \(-0.682996\pi\)
−0.543750 + 0.839247i \(0.682996\pi\)
\(44\) −7.91705 −1.19354
\(45\) 0 0
\(46\) 7.76651 1.14511
\(47\) 5.33816 0.778650 0.389325 − 0.921100i \(-0.372708\pi\)
0.389325 + 0.921100i \(0.372708\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −7.69563 −1.08833
\(51\) 0 0
\(52\) −30.1457 −4.18046
\(53\) −9.79306 −1.34518 −0.672590 − 0.740015i \(-0.734818\pi\)
−0.672590 + 0.740015i \(0.734818\pi\)
\(54\) 0 0
\(55\) −2.24797 −0.303116
\(56\) 0 0
\(57\) 0 0
\(58\) 10.3647 1.36095
\(59\) 4.67632 0.608805 0.304402 − 0.952544i \(-0.401543\pi\)
0.304402 + 0.952544i \(0.401543\pi\)
\(60\) 0 0
\(61\) 2.36471 0.302770 0.151385 − 0.988475i \(-0.451627\pi\)
0.151385 + 0.988475i \(0.451627\pi\)
\(62\) −5.33816 −0.677947
\(63\) 0 0
\(64\) 17.0145 2.12681
\(65\) −8.55957 −1.06168
\(66\) 0 0
\(67\) 3.36471 0.411065 0.205533 − 0.978650i \(-0.434107\pi\)
0.205533 + 0.978650i \(0.434107\pi\)
\(68\) 34.8075 4.22103
\(69\) 0 0
\(70\) 0 0
\(71\) 1.36471 0.161962 0.0809808 − 0.996716i \(-0.474195\pi\)
0.0809808 + 0.996716i \(0.474195\pi\)
\(72\) 0 0
\(73\) 1.88325 0.220418 0.110209 − 0.993908i \(-0.464848\pi\)
0.110209 + 0.993908i \(0.464848\pi\)
\(74\) 13.3454 1.55137
\(75\) 0 0
\(76\) 32.0145 3.67231
\(77\) 0 0
\(78\) 0 0
\(79\) 3.36471 0.378560 0.189280 − 0.981923i \(-0.439385\pi\)
0.189280 + 0.981923i \(0.439385\pi\)
\(80\) 17.4694 1.95314
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −2.24797 −0.246746 −0.123373 − 0.992360i \(-0.539371\pi\)
−0.123373 + 0.992360i \(0.539371\pi\)
\(84\) 0 0
\(85\) 9.88325 1.07199
\(86\) −19.0338 −2.05247
\(87\) 0 0
\(88\) −12.8833 −1.37336
\(89\) −0.793062 −0.0840644 −0.0420322 − 0.999116i \(-0.513383\pi\)
−0.0420322 + 0.999116i \(0.513383\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 14.9098 1.55445
\(93\) 0 0
\(94\) 14.2480 1.46957
\(95\) 9.09019 0.932634
\(96\) 0 0
\(97\) −10.2480 −1.04052 −0.520262 − 0.854007i \(-0.674166\pi\)
−0.520262 + 0.854007i \(0.674166\pi\)
\(98\) 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.o.1.3 3
3.2 odd 2 3969.2.a.n.1.1 3
7.6 odd 2 567.2.a.e.1.3 ✓ 3
21.20 even 2 567.2.a.f.1.1 yes 3
28.27 even 2 9072.2.a.bu.1.2 3
63.13 odd 6 567.2.f.m.379.1 6
63.20 even 6 567.2.f.l.190.3 6
63.34 odd 6 567.2.f.m.190.1 6
63.41 even 6 567.2.f.l.379.3 6
84.83 odd 2 9072.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.e.1.3 ✓ 3 7.6 odd 2
567.2.a.f.1.1 yes 3 21.20 even 2
567.2.f.l.190.3 6 63.20 even 6
567.2.f.l.379.3 6 63.41 even 6
567.2.f.m.190.1 6 63.34 odd 6
567.2.f.m.379.1 6 63.13 odd 6
3969.2.a.n.1.1 3 3.2 odd 2
3969.2.a.o.1.3 3 1.1 even 1 trivial
9072.2.a.bu.1.2 3 28.27 even 2
9072.2.a.cb.1.2 3 84.83 odd 2