Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1216,2,Mod(1,1216)] 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("1216.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,0,0,1] 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: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1216.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{7} -2.00000 q^{9} -6.00000 q^{11} -5.00000 q^{13} +3.00000 q^{17} +1.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} -5.00000 q^{25} -5.00000 q^{27} -9.00000 q^{29} +4.00000 q^{31} -6.00000 q^{33} -2.00000 q^{37} -5.00000 q^{39} +8.00000 q^{43} -6.00000 q^{49} +3.00000 q^{51} +3.00000 q^{53} +1.00000 q^{57} +9.00000 q^{59} +10.0000 q^{61} -2.00000 q^{63} +5.00000 q^{67} -3.00000 q^{69} +6.00000 q^{71} -7.00000 q^{73} -5.00000 q^{75} -6.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -9.00000 q^{87} -12.0000 q^{89} -5.00000 q^{91} +4.00000 q^{93} -10.0000 q^{97} +12.0000 q^{99} +O(q^{100})\)

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 \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 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
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 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 \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −5.00000 −0.577350
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 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
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
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 1216.2.a.m.1.1 1
4.3 odd 2 1216.2.a.e.1.1 1
8.3 odd 2 38.2.a.a.1.1 ✓ 1
8.5 even 2 304.2.a.c.1.1 1
24.5 odd 2 2736.2.a.n.1.1 1
24.11 even 2 342.2.a.e.1.1 1
40.3 even 4 950.2.b.b.799.2 2
40.19 odd 2 950.2.a.d.1.1 1
40.27 even 4 950.2.b.b.799.1 2
40.29 even 2 7600.2.a.n.1.1 1
56.27 even 2 1862.2.a.b.1.1 1
88.43 even 2 4598.2.a.p.1.1 1
104.51 odd 2 6422.2.a.h.1.1 1
120.59 even 2 8550.2.a.m.1.1 1
152.3 even 18 722.2.e.e.389.1 6
152.11 odd 6 722.2.c.e.653.1 2
152.27 even 6 722.2.c.c.653.1 2
152.35 odd 18 722.2.e.f.389.1 6
152.37 odd 2 5776.2.a.m.1.1 1
152.43 odd 18 722.2.e.f.595.1 6
152.51 even 18 722.2.e.e.245.1 6
152.59 even 18 722.2.e.e.99.1 6
152.67 even 18 722.2.e.e.423.1 6
152.75 even 2 722.2.a.e.1.1 1
152.83 odd 6 722.2.c.e.429.1 2
152.91 even 18 722.2.e.e.415.1 6
152.99 odd 18 722.2.e.f.415.1 6
152.107 even 6 722.2.c.c.429.1 2
152.123 odd 18 722.2.e.f.423.1 6
152.131 odd 18 722.2.e.f.99.1 6
152.139 odd 18 722.2.e.f.245.1 6
152.147 even 18 722.2.e.e.595.1 6
456.227 odd 2 6498.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.a.1.1 ✓ 1 8.3 odd 2
304.2.a.c.1.1 1 8.5 even 2
342.2.a.e.1.1 1 24.11 even 2
722.2.a.e.1.1 1 152.75 even 2
722.2.c.c.429.1 2 152.107 even 6
722.2.c.c.653.1 2 152.27 even 6
722.2.c.e.429.1 2 152.83 odd 6
722.2.c.e.653.1 2 152.11 odd 6
722.2.e.e.99.1 6 152.59 even 18
722.2.e.e.245.1 6 152.51 even 18
722.2.e.e.389.1 6 152.3 even 18
722.2.e.e.415.1 6 152.91 even 18
722.2.e.e.423.1 6 152.67 even 18
722.2.e.e.595.1 6 152.147 even 18
722.2.e.f.99.1 6 152.131 odd 18
722.2.e.f.245.1 6 152.139 odd 18
722.2.e.f.389.1 6 152.35 odd 18
722.2.e.f.415.1 6 152.99 odd 18
722.2.e.f.423.1 6 152.123 odd 18
722.2.e.f.595.1 6 152.43 odd 18
950.2.a.d.1.1 1 40.19 odd 2
950.2.b.b.799.1 2 40.27 even 4
950.2.b.b.799.2 2 40.3 even 4
1216.2.a.e.1.1 1 4.3 odd 2
1216.2.a.m.1.1 1 1.1 even 1 trivial
1862.2.a.b.1.1 1 56.27 even 2
2736.2.a.n.1.1 1 24.5 odd 2
4598.2.a.p.1.1 1 88.43 even 2
5776.2.a.m.1.1 1 152.37 odd 2
6422.2.a.h.1.1 1 104.51 odd 2
6498.2.a.f.1.1 1 456.227 odd 2
7600.2.a.n.1.1 1 40.29 even 2
8550.2.a.m.1.1 1 120.59 even 2