Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,2,Mod(1,392)] 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("392.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-2,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.13013575923\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 392.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{5} -3.00000 q^{9} -4.00000 q^{11} -2.00000 q^{13} +6.00000 q^{17} -8.00000 q^{19} -1.00000 q^{25} +6.00000 q^{29} -8.00000 q^{31} -2.00000 q^{37} -2.00000 q^{41} -4.00000 q^{43} +6.00000 q^{45} +8.00000 q^{47} +6.00000 q^{53} +8.00000 q^{55} +6.00000 q^{61} +4.00000 q^{65} -4.00000 q^{67} -8.00000 q^{71} -10.0000 q^{73} +16.0000 q^{79} +9.00000 q^{81} -8.00000 q^{83} -12.0000 q^{85} +6.00000 q^{89} +16.0000 q^{95} +6.00000 q^{97} +12.0000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.0000 1.64157
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 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 392.2.a.d.1.1 1
3.2 odd 2 3528.2.a.x.1.1 1
4.3 odd 2 784.2.a.e.1.1 1
5.4 even 2 9800.2.a.u.1.1 1
7.2 even 3 392.2.i.d.361.1 2
7.3 odd 6 392.2.i.c.177.1 2
7.4 even 3 392.2.i.d.177.1 2
7.5 odd 6 392.2.i.c.361.1 2
7.6 odd 2 56.2.a.a.1.1 ✓ 1
8.3 odd 2 3136.2.a.p.1.1 1
8.5 even 2 3136.2.a.q.1.1 1
12.11 even 2 7056.2.a.bo.1.1 1
21.2 odd 6 3528.2.s.e.361.1 2
21.5 even 6 3528.2.s.t.361.1 2
21.11 odd 6 3528.2.s.e.3313.1 2
21.17 even 6 3528.2.s.t.3313.1 2
21.20 even 2 504.2.a.c.1.1 1
28.3 even 6 784.2.i.e.177.1 2
28.11 odd 6 784.2.i.g.177.1 2
28.19 even 6 784.2.i.e.753.1 2
28.23 odd 6 784.2.i.g.753.1 2
28.27 even 2 112.2.a.b.1.1 1
35.13 even 4 1400.2.g.g.449.2 2
35.27 even 4 1400.2.g.g.449.1 2
35.34 odd 2 1400.2.a.g.1.1 1
56.13 odd 2 448.2.a.d.1.1 1
56.27 even 2 448.2.a.e.1.1 1
77.76 even 2 6776.2.a.g.1.1 1
84.83 odd 2 1008.2.a.d.1.1 1
91.90 odd 2 9464.2.a.c.1.1 1
112.13 odd 4 1792.2.b.i.897.1 2
112.27 even 4 1792.2.b.d.897.2 2
112.69 odd 4 1792.2.b.i.897.2 2
112.83 even 4 1792.2.b.d.897.1 2
140.27 odd 4 2800.2.g.p.449.2 2
140.83 odd 4 2800.2.g.p.449.1 2
140.139 even 2 2800.2.a.p.1.1 1
168.83 odd 2 4032.2.a.bk.1.1 1
168.125 even 2 4032.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.a.1.1 ✓ 1 7.6 odd 2
112.2.a.b.1.1 1 28.27 even 2
392.2.a.d.1.1 1 1.1 even 1 trivial
392.2.i.c.177.1 2 7.3 odd 6
392.2.i.c.361.1 2 7.5 odd 6
392.2.i.d.177.1 2 7.4 even 3
392.2.i.d.361.1 2 7.2 even 3
448.2.a.d.1.1 1 56.13 odd 2
448.2.a.e.1.1 1 56.27 even 2
504.2.a.c.1.1 1 21.20 even 2
784.2.a.e.1.1 1 4.3 odd 2
784.2.i.e.177.1 2 28.3 even 6
784.2.i.e.753.1 2 28.19 even 6
784.2.i.g.177.1 2 28.11 odd 6
784.2.i.g.753.1 2 28.23 odd 6
1008.2.a.d.1.1 1 84.83 odd 2
1400.2.a.g.1.1 1 35.34 odd 2
1400.2.g.g.449.1 2 35.27 even 4
1400.2.g.g.449.2 2 35.13 even 4
1792.2.b.d.897.1 2 112.83 even 4
1792.2.b.d.897.2 2 112.27 even 4
1792.2.b.i.897.1 2 112.13 odd 4
1792.2.b.i.897.2 2 112.69 odd 4
2800.2.a.p.1.1 1 140.139 even 2
2800.2.g.p.449.1 2 140.83 odd 4
2800.2.g.p.449.2 2 140.27 odd 4
3136.2.a.p.1.1 1 8.3 odd 2
3136.2.a.q.1.1 1 8.5 even 2
3528.2.a.x.1.1 1 3.2 odd 2
3528.2.s.e.361.1 2 21.2 odd 6
3528.2.s.e.3313.1 2 21.11 odd 6
3528.2.s.t.361.1 2 21.5 even 6
3528.2.s.t.3313.1 2 21.17 even 6
4032.2.a.bb.1.1 1 168.125 even 2
4032.2.a.bk.1.1 1 168.83 odd 2
6776.2.a.g.1.1 1 77.76 even 2
7056.2.a.bo.1.1 1 12.11 even 2
9464.2.a.c.1.1 1 91.90 odd 2
9800.2.a.u.1.1 1 5.4 even 2