Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1232,2,Mod(1,1232)] 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("1232.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.83756952902\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1232.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} -4.00000 q^{13} -3.00000 q^{15} -6.00000 q^{17} -2.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} -6.00000 q^{29} -5.00000 q^{31} -1.00000 q^{33} -3.00000 q^{35} +11.0000 q^{37} +4.00000 q^{39} +6.00000 q^{41} -8.00000 q^{43} -6.00000 q^{45} +1.00000 q^{49} +6.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +2.00000 q^{57} +9.00000 q^{59} -10.0000 q^{61} +2.00000 q^{63} -12.0000 q^{65} -5.00000 q^{67} +3.00000 q^{69} -9.00000 q^{71} +2.00000 q^{73} -4.00000 q^{75} -1.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -18.0000 q^{85} +6.00000 q^{87} -3.00000 q^{89} +4.00000 q^{91} +5.00000 q^{93} -6.00000 q^{95} -1.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 − 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 − 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 − 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −18.0000 −1.95237
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
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 1232.2.a.d.1.1 1
4.3 odd 2 77.2.a.b.1.1 ✓ 1
7.6 odd 2 8624.2.a.s.1.1 1
8.3 odd 2 4928.2.a.i.1.1 1
8.5 even 2 4928.2.a.x.1.1 1
12.11 even 2 693.2.a.b.1.1 1
20.3 even 4 1925.2.b.g.1849.2 2
20.7 even 4 1925.2.b.g.1849.1 2
20.19 odd 2 1925.2.a.f.1.1 1
28.3 even 6 539.2.e.e.177.1 2
28.11 odd 6 539.2.e.d.177.1 2
28.19 even 6 539.2.e.e.67.1 2
28.23 odd 6 539.2.e.d.67.1 2
28.27 even 2 539.2.a.b.1.1 1
44.3 odd 10 847.2.f.f.372.1 4
44.7 even 10 847.2.f.g.148.1 4
44.15 odd 10 847.2.f.f.148.1 4
44.19 even 10 847.2.f.g.372.1 4
44.27 odd 10 847.2.f.f.729.1 4
44.31 odd 10 847.2.f.f.323.1 4
44.35 even 10 847.2.f.g.323.1 4
44.39 even 10 847.2.f.g.729.1 4
44.43 even 2 847.2.a.c.1.1 1
84.83 odd 2 4851.2.a.k.1.1 1
132.131 odd 2 7623.2.a.i.1.1 1
308.307 odd 2 5929.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.b.1.1 ✓ 1 4.3 odd 2
539.2.a.b.1.1 1 28.27 even 2
539.2.e.d.67.1 2 28.23 odd 6
539.2.e.d.177.1 2 28.11 odd 6
539.2.e.e.67.1 2 28.19 even 6
539.2.e.e.177.1 2 28.3 even 6
693.2.a.b.1.1 1 12.11 even 2
847.2.a.c.1.1 1 44.43 even 2
847.2.f.f.148.1 4 44.15 odd 10
847.2.f.f.323.1 4 44.31 odd 10
847.2.f.f.372.1 4 44.3 odd 10
847.2.f.f.729.1 4 44.27 odd 10
847.2.f.g.148.1 4 44.7 even 10
847.2.f.g.323.1 4 44.35 even 10
847.2.f.g.372.1 4 44.19 even 10
847.2.f.g.729.1 4 44.39 even 10
1232.2.a.d.1.1 1 1.1 even 1 trivial
1925.2.a.f.1.1 1 20.19 odd 2
1925.2.b.g.1849.1 2 20.7 even 4
1925.2.b.g.1849.2 2 20.3 even 4
4851.2.a.k.1.1 1 84.83 odd 2
4928.2.a.i.1.1 1 8.3 odd 2
4928.2.a.x.1.1 1 8.5 even 2
5929.2.a.d.1.1 1 308.307 odd 2
7623.2.a.i.1.1 1 132.131 odd 2
8624.2.a.s.1.1 1 7.6 odd 2