Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1323,2,Mod(1,1323)] 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("1323.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,-1,4,0,0,-3,0,4,2,0,-1] 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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1323.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{8} +4.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} -4.00000 q^{20} +2.00000 q^{22} +6.00000 q^{23} +11.0000 q^{25} -1.00000 q^{26} +2.00000 q^{29} -3.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +3.00000 q^{37} -4.00000 q^{38} -12.0000 q^{40} +2.00000 q^{41} -1.00000 q^{43} -2.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} +11.0000 q^{50} +1.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} +2.00000 q^{58} -6.00000 q^{59} +5.00000 q^{61} -3.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} +7.00000 q^{67} -6.00000 q^{68} +6.00000 q^{73} +3.00000 q^{74} +4.00000 q^{76} +11.0000 q^{79} -4.00000 q^{80} +2.00000 q^{82} -6.00000 q^{83} +24.0000 q^{85} -1.00000 q^{86} -6.00000 q^{88} +4.00000 q^{89} -6.00000 q^{92} -6.00000 q^{94} -16.0000 q^{95} -9.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −12.0000 −1.89737
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 11.0000 1.55563
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −3.00000 −0.381000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 − 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 − 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 24.0000 2.60317
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −16.0000 −1.64157
\(96\) 0 0
\(97\) −9.00000 −0.913812 −0.456906 − 0.889515i \(-0.651042\pi\)
−0.456906 + 0.889515i \(0.651042\pi\)
\(98\) 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 1323.2.a.p.1.1 1
3.2 odd 2 1323.2.a.d.1.1 1
7.3 odd 6 189.2.e.a.163.1 yes 2
7.5 odd 6 189.2.e.a.109.1 ✓ 2
7.6 odd 2 1323.2.a.m.1.1 1
21.5 even 6 189.2.e.c.109.1 yes 2
21.17 even 6 189.2.e.c.163.1 yes 2
21.20 even 2 1323.2.a.g.1.1 1
63.5 even 6 567.2.h.b.298.1 2
63.31 odd 6 567.2.g.b.541.1 2
63.38 even 6 567.2.h.b.352.1 2
63.40 odd 6 567.2.h.e.298.1 2
63.47 even 6 567.2.g.e.109.1 2
63.52 odd 6 567.2.h.e.352.1 2
63.59 even 6 567.2.g.e.541.1 2
63.61 odd 6 567.2.g.b.109.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.e.a.109.1 ✓ 2 7.5 odd 6
189.2.e.a.163.1 yes 2 7.3 odd 6
189.2.e.c.109.1 yes 2 21.5 even 6
189.2.e.c.163.1 yes 2 21.17 even 6
567.2.g.b.109.1 2 63.61 odd 6
567.2.g.b.541.1 2 63.31 odd 6
567.2.g.e.109.1 2 63.47 even 6
567.2.g.e.541.1 2 63.59 even 6
567.2.h.b.298.1 2 63.5 even 6
567.2.h.b.352.1 2 63.38 even 6
567.2.h.e.298.1 2 63.40 odd 6
567.2.h.e.352.1 2 63.52 odd 6
1323.2.a.d.1.1 1 3.2 odd 2
1323.2.a.g.1.1 1 21.20 even 2
1323.2.a.m.1.1 1 7.6 odd 2
1323.2.a.p.1.1 1 1.1 even 1 trivial