Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4046,2,Mod(1,4046)] 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("4046.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4046, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4046 = 2 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4046.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,0,1,2,0,1,-1,-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: \(32.3074726578\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 238)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4046.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} +8.00000 q^{23} -1.00000 q^{25} +1.00000 q^{28} -8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{35} -3.00000 q^{36} +4.00000 q^{37} +2.00000 q^{38} -2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} +2.00000 q^{44} -6.00000 q^{45} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{53} +4.00000 q^{55} -1.00000 q^{56} +10.0000 q^{59} -10.0000 q^{61} +8.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +8.00000 q^{67} -2.00000 q^{70} -4.00000 q^{71} +3.00000 q^{72} +10.0000 q^{73} -4.00000 q^{74} -2.00000 q^{76} +2.00000 q^{77} +4.00000 q^{79} +2.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{86} -2.00000 q^{88} -6.00000 q^{89} +6.00000 q^{90} +8.00000 q^{92} -8.00000 q^{94} -4.00000 q^{95} +14.0000 q^{97} -1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) −2.00000 −0.632456
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 3.00000 0.707107
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) −3.00000 −0.500000
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) −6.00000 −0.894427
\(46\) −8.00000 −1.17954
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000 1.01600
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 3.00000 0.353553
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 2.00000 0.223607
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 6.00000 0.632456
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −1.00000 −0.101015
\(99\) −6.00000 −0.603023
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 4046.2.a.d.1.1 1
17.16 even 2 238.2.a.a.1.1 ✓ 1
51.50 odd 2 2142.2.a.s.1.1 1
68.67 odd 2 1904.2.a.c.1.1 1
85.84 even 2 5950.2.a.o.1.1 1
119.118 odd 2 1666.2.a.e.1.1 1
136.67 odd 2 7616.2.a.h.1.1 1
136.101 even 2 7616.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
238.2.a.a.1.1 ✓ 1 17.16 even 2
1666.2.a.e.1.1 1 119.118 odd 2
1904.2.a.c.1.1 1 68.67 odd 2
2142.2.a.s.1.1 1 51.50 odd 2
4046.2.a.d.1.1 1 1.1 even 1 trivial
5950.2.a.o.1.1 1 85.84 even 2
7616.2.a.g.1.1 1 136.101 even 2
7616.2.a.h.1.1 1 136.67 odd 2