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(177,392)] 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("392.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-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: no
Analytic conductor: \(3.13013575923\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 392.361
Dual form 392.2.i.c.177.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{5} +(1.50000 + 2.59808i) q^{9} +(2.00000 - 3.46410i) q^{11} +2.00000 q^{13} +(3.00000 - 5.19615i) q^{17} +(-4.00000 - 6.92820i) q^{19} +(0.500000 - 0.866025i) q^{25} +6.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(1.00000 + 1.73205i) q^{37} +2.00000 q^{41} -4.00000 q^{43} +(3.00000 - 5.19615i) q^{45} +(4.00000 + 6.92820i) q^{47} +(-3.00000 + 5.19615i) q^{53} -8.00000 q^{55} +(3.00000 + 5.19615i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} -8.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(-8.00000 - 13.8564i) q^{79} +(-4.50000 + 7.79423i) q^{81} +8.00000 q^{83} -12.0000 q^{85} +(3.00000 + 5.19615i) q^{89} +(-8.00000 + 13.8564i) q^{95} -6.00000 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 3 q^{9} + 4 q^{11} + 4 q^{13} + 6 q^{17} - 8 q^{19} + q^{25} + 12 q^{29} - 8 q^{31} + 2 q^{37} + 4 q^{41} - 8 q^{43} + 6 q^{45} + 8 q^{47} - 6 q^{53} - 16 q^{55} + 6 q^{61} - 4 q^{65} + 4 q^{67}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) −1.00000 − 1.73205i −0.447214 − 0.774597i 0.550990 − 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 2.00000 − 3.46410i 0.603023 − 1.04447i −0.389338 − 0.921095i \(-0.627296\pi\)
0.992361 − 0.123371i \(-0.0393705\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 − 5.19615i 0.727607 − 1.26025i −0.230285 − 0.973123i \(-0.573966\pi\)
0.957892 − 0.287129i \(-0.0927008\pi\)
\(18\) 0 0
\(19\) −4.00000 − 6.92820i −0.917663 − 1.58944i −0.802955 − 0.596040i \(-0.796740\pi\)
−0.114708 − 0.993399i \(-0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 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
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 − 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 3.00000 − 5.19615i 0.447214 − 0.774597i
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 3.00000 + 5.19615i 0.384111 + 0.665299i 0.991645 − 0.128994i \(-0.0411748\pi\)
−0.607535 + 0.794293i \(0.707841\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 − 3.46410i −0.248069 − 0.429669i
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 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
\(73\) −5.00000 + 8.66025i −0.585206 + 1.01361i 0.409644 + 0.912245i \(0.365653\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 − 13.8564i −0.900070 − 1.55897i −0.827401 − 0.561611i \(-0.810182\pi\)
−0.0726692 − 0.997356i \(-0.523152\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 − 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 + 13.8564i −0.820783 + 1.42164i
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 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.i.c.361.1 2
3.2 odd 2 3528.2.s.t.361.1 2
4.3 odd 2 784.2.i.e.753.1 2
7.2 even 3 inner 392.2.i.c.177.1 2
7.3 odd 6 392.2.a.d.1.1 1
7.4 even 3 56.2.a.a.1.1 ✓ 1
7.5 odd 6 392.2.i.d.177.1 2
7.6 odd 2 392.2.i.d.361.1 2
21.2 odd 6 3528.2.s.t.3313.1 2
21.5 even 6 3528.2.s.e.3313.1 2
21.11 odd 6 504.2.a.c.1.1 1
21.17 even 6 3528.2.a.x.1.1 1
21.20 even 2 3528.2.s.e.361.1 2
28.3 even 6 784.2.a.e.1.1 1
28.11 odd 6 112.2.a.b.1.1 1
28.19 even 6 784.2.i.g.177.1 2
28.23 odd 6 784.2.i.e.177.1 2
28.27 even 2 784.2.i.g.753.1 2
35.4 even 6 1400.2.a.g.1.1 1
35.18 odd 12 1400.2.g.g.449.2 2
35.24 odd 6 9800.2.a.u.1.1 1
35.32 odd 12 1400.2.g.g.449.1 2
56.3 even 6 3136.2.a.p.1.1 1
56.11 odd 6 448.2.a.e.1.1 1
56.45 odd 6 3136.2.a.q.1.1 1
56.53 even 6 448.2.a.d.1.1 1
77.32 odd 6 6776.2.a.g.1.1 1
84.11 even 6 1008.2.a.d.1.1 1
84.59 odd 6 7056.2.a.bo.1.1 1
91.25 even 6 9464.2.a.c.1.1 1
112.11 odd 12 1792.2.b.d.897.2 2
112.53 even 12 1792.2.b.i.897.2 2
112.67 odd 12 1792.2.b.d.897.1 2
112.109 even 12 1792.2.b.i.897.1 2
140.39 odd 6 2800.2.a.p.1.1 1
140.67 even 12 2800.2.g.p.449.2 2
140.123 even 12 2800.2.g.p.449.1 2
168.11 even 6 4032.2.a.bk.1.1 1
168.53 odd 6 4032.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.a.1.1 ✓ 1 7.4 even 3
112.2.a.b.1.1 1 28.11 odd 6
392.2.a.d.1.1 1 7.3 odd 6
392.2.i.c.177.1 2 7.2 even 3 inner
392.2.i.c.361.1 2 1.1 even 1 trivial
392.2.i.d.177.1 2 7.5 odd 6
392.2.i.d.361.1 2 7.6 odd 2
448.2.a.d.1.1 1 56.53 even 6
448.2.a.e.1.1 1 56.11 odd 6
504.2.a.c.1.1 1 21.11 odd 6
784.2.a.e.1.1 1 28.3 even 6
784.2.i.e.177.1 2 28.23 odd 6
784.2.i.e.753.1 2 4.3 odd 2
784.2.i.g.177.1 2 28.19 even 6
784.2.i.g.753.1 2 28.27 even 2
1008.2.a.d.1.1 1 84.11 even 6
1400.2.a.g.1.1 1 35.4 even 6
1400.2.g.g.449.1 2 35.32 odd 12
1400.2.g.g.449.2 2 35.18 odd 12
1792.2.b.d.897.1 2 112.67 odd 12
1792.2.b.d.897.2 2 112.11 odd 12
1792.2.b.i.897.1 2 112.109 even 12
1792.2.b.i.897.2 2 112.53 even 12
2800.2.a.p.1.1 1 140.39 odd 6
2800.2.g.p.449.1 2 140.123 even 12
2800.2.g.p.449.2 2 140.67 even 12
3136.2.a.p.1.1 1 56.3 even 6
3136.2.a.q.1.1 1 56.45 odd 6
3528.2.a.x.1.1 1 21.17 even 6
3528.2.s.e.361.1 2 21.20 even 2
3528.2.s.e.3313.1 2 21.5 even 6
3528.2.s.t.361.1 2 3.2 odd 2
3528.2.s.t.3313.1 2 21.2 odd 6
4032.2.a.bb.1.1 1 168.53 odd 6
4032.2.a.bk.1.1 1 168.11 even 6
6776.2.a.g.1.1 1 77.32 odd 6
7056.2.a.bo.1.1 1 84.59 odd 6
9464.2.a.c.1.1 1 91.25 even 6
9800.2.a.u.1.1 1 35.24 odd 6