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(177,1232)] 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("1232.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,1] 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: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1232.177
Dual form 1232.2.q.d.529.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(0.500000 + 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +(-0.500000 - 0.866025i) q^{11} -1.00000 q^{13} +(3.00000 + 5.19615i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-2.00000 + 1.73205i) q^{21} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +5.00000 q^{27} +9.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(0.500000 - 0.866025i) q^{33} +(-1.00000 + 1.73205i) q^{37} +(-0.500000 - 0.866025i) q^{39} -6.00000 q^{41} +4.00000 q^{43} +(-3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-3.00000 + 5.19615i) q^{51} +2.00000 q^{57} +(-1.50000 - 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(5.00000 + 1.73205i) q^{63} +(5.50000 + 9.52628i) q^{67} -6.00000 q^{69} +(-1.00000 - 1.73205i) q^{73} +(-2.50000 + 4.33013i) q^{75} +(2.00000 - 1.73205i) q^{77} +(2.50000 - 4.33013i) q^{79} +(-0.500000 - 0.866025i) q^{81} +6.00000 q^{83} +(4.50000 + 7.79423i) q^{87} +(9.00000 - 15.5885i) q^{89} +(-0.500000 - 2.59808i) q^{91} +(2.00000 - 3.46410i) q^{93} -13.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{7} + 2 q^{9} - q^{11} - 2 q^{13} + 6 q^{17} + 2 q^{19} - 4 q^{21} - 6 q^{23} + 5 q^{25} + 10 q^{27} + 18 q^{29} - 4 q^{31} + q^{33} - 2 q^{37} - q^{39} - 12 q^{41} + 8 q^{43} - 6 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i 0.973494 − 0.228714i \(-0.0734519\pi\)
−0.684819 + 0.728714i \(0.740119\pi\)
\(4\) 0 0
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 1.00000 − 1.73205i 0.333333 − 0.577350i
\(10\) 0 0
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) 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
\(16\) 0 0
\(17\) 3.00000 + 5.19615i 0.727607 + 1.26025i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) 1.00000 − 1.73205i 0.229416 − 0.397360i −0.728219 − 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) −2.00000 + 1.73205i −0.436436 + 0.377964i
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −2.00000 − 3.46410i −0.359211 − 0.622171i 0.628619 − 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0.500000 − 0.866025i 0.0870388 − 0.150756i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 + 1.73205i −0.164399 + 0.284747i −0.936442 − 0.350823i \(-0.885902\pi\)
0.772043 + 0.635571i \(0.219235\pi\)
\(38\) 0 0
\(39\) −0.500000 − 0.866025i −0.0800641 − 0.138675i
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 − 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) 0 0
\(53\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −1.50000 − 2.59808i −0.195283 − 0.338241i 0.751710 − 0.659494i \(-0.229229\pi\)
−0.946993 + 0.321253i \(0.895896\pi\)
\(60\) 0 0
\(61\) −5.50000 + 9.52628i −0.704203 + 1.21972i 0.262776 + 0.964857i \(0.415362\pi\)
−0.966978 + 0.254858i \(0.917971\pi\)
\(62\) 0 0
\(63\) 5.00000 + 1.73205i 0.629941 + 0.218218i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.50000 + 9.52628i 0.671932 + 1.16382i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.00000 − 1.73205i −0.117041 − 0.202721i 0.801553 − 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) −2.50000 + 4.33013i −0.288675 + 0.500000i
\(76\) 0 0
\(77\) 2.00000 − 1.73205i 0.227921 − 0.197386i
\(78\) 0 0
\(79\) 2.50000 − 4.33013i 0.281272 − 0.487177i −0.690426 − 0.723403i \(-0.742577\pi\)
0.971698 + 0.236225i \(0.0759104\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.50000 + 7.79423i 0.482451 + 0.835629i
\(88\) 0 0
\(89\) 9.00000 − 15.5885i 0.953998 − 1.65237i 0.217354 − 0.976093i \(-0.430258\pi\)
0.736644 − 0.676280i \(-0.236409\pi\)
\(90\) 0 0
\(91\) −0.500000 − 2.59808i −0.0524142 − 0.272352i
\(92\) 0 0
\(93\) 2.00000 − 3.46410i 0.207390 − 0.359211i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 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.q.d.177.1 2
4.3 odd 2 154.2.e.c.23.1 ✓ 2
7.2 even 3 8624.2.a.k.1.1 1
7.4 even 3 inner 1232.2.q.d.529.1 2
7.5 odd 6 8624.2.a.u.1.1 1
12.11 even 2 1386.2.k.e.793.1 2
28.3 even 6 1078.2.e.k.67.1 2
28.11 odd 6 154.2.e.c.67.1 yes 2
28.19 even 6 1078.2.a.c.1.1 1
28.23 odd 6 1078.2.a.e.1.1 1
28.27 even 2 1078.2.e.k.177.1 2
84.11 even 6 1386.2.k.e.991.1 2
84.23 even 6 9702.2.a.br.1.1 1
84.47 odd 6 9702.2.a.bs.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.c.23.1 ✓ 2 4.3 odd 2
154.2.e.c.67.1 yes 2 28.11 odd 6
1078.2.a.c.1.1 1 28.19 even 6
1078.2.a.e.1.1 1 28.23 odd 6
1078.2.e.k.67.1 2 28.3 even 6
1078.2.e.k.177.1 2 28.27 even 2
1232.2.q.d.177.1 2 1.1 even 1 trivial
1232.2.q.d.529.1 2 7.4 even 3 inner
1386.2.k.e.793.1 2 12.11 even 2
1386.2.k.e.991.1 2 84.11 even 6
8624.2.a.k.1.1 1 7.2 even 3
8624.2.a.u.1.1 1 7.5 odd 6
9702.2.a.br.1.1 1 84.23 even 6
9702.2.a.bs.1.1 1 84.47 odd 6