Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,2,Mod(1,126)] 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("126.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 126.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{19} -5.00000 q^{25} -4.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +2.00000 q^{37} +2.00000 q^{38} -6.00000 q^{41} +8.00000 q^{43} +12.0000 q^{47} +1.00000 q^{49} -5.00000 q^{50} -4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{56} +6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{67} -6.00000 q^{68} +2.00000 q^{73} +2.00000 q^{74} +2.00000 q^{76} +8.00000 q^{79} -6.00000 q^{82} +6.00000 q^{83} +8.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} +12.0000 q^{94} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −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 \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 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
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −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
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) 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 126.2.a.b.1.1 1
3.2 odd 2 14.2.a.a.1.1 ✓ 1
4.3 odd 2 1008.2.a.h.1.1 1
5.2 odd 4 3150.2.g.j.2899.2 2
5.3 odd 4 3150.2.g.j.2899.1 2
5.4 even 2 3150.2.a.i.1.1 1
7.2 even 3 882.2.g.c.361.1 2
7.3 odd 6 882.2.g.d.667.1 2
7.4 even 3 882.2.g.c.667.1 2
7.5 odd 6 882.2.g.d.361.1 2
7.6 odd 2 882.2.a.i.1.1 1
8.3 odd 2 4032.2.a.r.1.1 1
8.5 even 2 4032.2.a.w.1.1 1
9.2 odd 6 1134.2.f.l.757.1 2
9.4 even 3 1134.2.f.f.379.1 2
9.5 odd 6 1134.2.f.l.379.1 2
9.7 even 3 1134.2.f.f.757.1 2
12.11 even 2 112.2.a.c.1.1 1
15.2 even 4 350.2.c.d.99.1 2
15.8 even 4 350.2.c.d.99.2 2
15.14 odd 2 350.2.a.f.1.1 1
21.2 odd 6 98.2.c.b.67.1 2
21.5 even 6 98.2.c.a.67.1 2
21.11 odd 6 98.2.c.b.79.1 2
21.17 even 6 98.2.c.a.79.1 2
21.20 even 2 98.2.a.a.1.1 1
24.5 odd 2 448.2.a.g.1.1 1
24.11 even 2 448.2.a.a.1.1 1
28.27 even 2 7056.2.a.bd.1.1 1
33.32 even 2 1694.2.a.e.1.1 1
39.5 even 4 2366.2.d.b.337.2 2
39.8 even 4 2366.2.d.b.337.1 2
39.38 odd 2 2366.2.a.j.1.1 1
48.5 odd 4 1792.2.b.c.897.2 2
48.11 even 4 1792.2.b.g.897.1 2
48.29 odd 4 1792.2.b.c.897.1 2
48.35 even 4 1792.2.b.g.897.2 2
51.50 odd 2 4046.2.a.f.1.1 1
57.56 even 2 5054.2.a.c.1.1 1
60.23 odd 4 2800.2.g.h.449.2 2
60.47 odd 4 2800.2.g.h.449.1 2
60.59 even 2 2800.2.a.g.1.1 1
69.68 even 2 7406.2.a.a.1.1 1
84.11 even 6 784.2.i.c.177.1 2
84.23 even 6 784.2.i.c.753.1 2
84.47 odd 6 784.2.i.i.753.1 2
84.59 odd 6 784.2.i.i.177.1 2
84.83 odd 2 784.2.a.b.1.1 1
105.62 odd 4 2450.2.c.c.99.1 2
105.83 odd 4 2450.2.c.c.99.2 2
105.104 even 2 2450.2.a.t.1.1 1
168.83 odd 2 3136.2.a.z.1.1 1
168.125 even 2 3136.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 ✓ 1 3.2 odd 2
98.2.a.a.1.1 1 21.20 even 2
98.2.c.a.67.1 2 21.5 even 6
98.2.c.a.79.1 2 21.17 even 6
98.2.c.b.67.1 2 21.2 odd 6
98.2.c.b.79.1 2 21.11 odd 6
112.2.a.c.1.1 1 12.11 even 2
126.2.a.b.1.1 1 1.1 even 1 trivial
350.2.a.f.1.1 1 15.14 odd 2
350.2.c.d.99.1 2 15.2 even 4
350.2.c.d.99.2 2 15.8 even 4
448.2.a.a.1.1 1 24.11 even 2
448.2.a.g.1.1 1 24.5 odd 2
784.2.a.b.1.1 1 84.83 odd 2
784.2.i.c.177.1 2 84.11 even 6
784.2.i.c.753.1 2 84.23 even 6
784.2.i.i.177.1 2 84.59 odd 6
784.2.i.i.753.1 2 84.47 odd 6
882.2.a.i.1.1 1 7.6 odd 2
882.2.g.c.361.1 2 7.2 even 3
882.2.g.c.667.1 2 7.4 even 3
882.2.g.d.361.1 2 7.5 odd 6
882.2.g.d.667.1 2 7.3 odd 6
1008.2.a.h.1.1 1 4.3 odd 2
1134.2.f.f.379.1 2 9.4 even 3
1134.2.f.f.757.1 2 9.7 even 3
1134.2.f.l.379.1 2 9.5 odd 6
1134.2.f.l.757.1 2 9.2 odd 6
1694.2.a.e.1.1 1 33.32 even 2
1792.2.b.c.897.1 2 48.29 odd 4
1792.2.b.c.897.2 2 48.5 odd 4
1792.2.b.g.897.1 2 48.11 even 4
1792.2.b.g.897.2 2 48.35 even 4
2366.2.a.j.1.1 1 39.38 odd 2
2366.2.d.b.337.1 2 39.8 even 4
2366.2.d.b.337.2 2 39.5 even 4
2450.2.a.t.1.1 1 105.104 even 2
2450.2.c.c.99.1 2 105.62 odd 4
2450.2.c.c.99.2 2 105.83 odd 4
2800.2.a.g.1.1 1 60.59 even 2
2800.2.g.h.449.1 2 60.47 odd 4
2800.2.g.h.449.2 2 60.23 odd 4
3136.2.a.e.1.1 1 168.125 even 2
3136.2.a.z.1.1 1 168.83 odd 2
3150.2.a.i.1.1 1 5.4 even 2
3150.2.g.j.2899.1 2 5.3 odd 4
3150.2.g.j.2899.2 2 5.2 odd 4
4032.2.a.r.1.1 1 8.3 odd 2
4032.2.a.w.1.1 1 8.5 even 2
4046.2.a.f.1.1 1 51.50 odd 2
5054.2.a.c.1.1 1 57.56 even 2
7056.2.a.bd.1.1 1 28.27 even 2
7406.2.a.a.1.1 1 69.68 even 2