Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1078,2,Mod(1,1078)] 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("1078.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1078, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,2,1,2,2,0,1,1,2,1,2,-2,0,4,1,0,1,2,2,0,1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1078.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} +2.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -4.00000 q^{27} +6.00000 q^{29} +4.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -8.00000 q^{41} +12.0000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -12.0000 q^{47} +2.00000 q^{48} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -4.00000 q^{54} +2.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} -10.0000 q^{59} +4.00000 q^{60} +10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +2.00000 q^{66} -12.0000 q^{67} +4.00000 q^{71} +1.00000 q^{72} -12.0000 q^{73} +2.00000 q^{74} -2.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} +2.00000 q^{80} -11.0000 q^{81} -8.00000 q^{82} +18.0000 q^{83} +12.0000 q^{86} +12.0000 q^{87} +1.00000 q^{88} +2.00000 q^{90} -8.00000 q^{93} -12.0000 q^{94} +4.00000 q^{95} +2.00000 q^{96} +12.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 2.00000 0.408248
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 4.00000 0.730297
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 2.00000 0.324443
\(39\) −4.00000 −0.640513
\(40\) 2.00000 0.316228
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 2.00000 0.288675
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −4.00000 −0.544331
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 4.00000 0.516398
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 2.00000 0.246183
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.0000 −1.40449 −0.702247 − 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 2.00000 0.232495
\(75\) −2.00000 −0.230940
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) −11.0000 −1.22222
\(82\) −8.00000 −0.883452
\(83\) 18.0000 1.97576 0.987878 − 0.155230i \(-0.0496119\pi\)
0.987878 + 0.155230i \(0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 12.0000 1.28654
\(88\) 1.00000 0.106600
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) −12.0000 −1.23771
\(95\) 4.00000 0.410391
\(96\) 2.00000 0.204124
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
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 1078.2.a.l.1.1 yes 1
3.2 odd 2 9702.2.a.e.1.1 1
4.3 odd 2 8624.2.a.g.1.1 1
7.2 even 3 1078.2.e.a.67.1 2
7.3 odd 6 1078.2.e.e.177.1 2
7.4 even 3 1078.2.e.a.177.1 2
7.5 odd 6 1078.2.e.e.67.1 2
7.6 odd 2 1078.2.a.h.1.1 ✓ 1
21.20 even 2 9702.2.a.t.1.1 1
28.27 even 2 8624.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.h.1.1 ✓ 1 7.6 odd 2
1078.2.a.l.1.1 yes 1 1.1 even 1 trivial
1078.2.e.a.67.1 2 7.2 even 3
1078.2.e.a.177.1 2 7.4 even 3
1078.2.e.e.67.1 2 7.5 odd 6
1078.2.e.e.177.1 2 7.3 odd 6
8624.2.a.g.1.1 1 4.3 odd 2
8624.2.a.y.1.1 1 28.27 even 2
9702.2.a.e.1.1 1 3.2 odd 2
9702.2.a.t.1.1 1 21.20 even 2