Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1183,2,Mod(337,1183)] 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("1183.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1183, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.11667456256.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(-0.231361i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.g.337.5

$q$-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)
 
\(f(q)\) \(=\) \(q-0.231361i q^{2} -3.32225 q^{3} +1.94647 q^{4} +2.23136i q^{5} +0.768639i q^{6} -1.00000i q^{7} -0.913059i q^{8} +8.03736 q^{9} +0.516249 q^{10} +3.32225i q^{11} -6.46667 q^{12} -0.231361 q^{14} -7.41314i q^{15} +3.68170 q^{16} +1.37578 q^{17} -1.85953i q^{18} -3.23531i q^{19} +4.34328i q^{20} +3.32225i q^{21} +0.768639 q^{22} -0.838502 q^{23} +3.03341i q^{24} +0.0210289 q^{25} -16.7354 q^{27} -1.94647i q^{28} -0.607142 q^{29} -1.71511 q^{30} -1.71511i q^{31} -2.67792i q^{32} -11.0374i q^{33} -0.318302i q^{34} +2.23136 q^{35} +15.6445 q^{36} +1.55361i q^{37} -0.748524 q^{38} +2.03736 q^{40} +9.17783i q^{41} +0.768639 q^{42} -1.23136 q^{43} +6.46667i q^{44} +17.9343i q^{45} +0.193997i q^{46} -1.62817i q^{47} -12.2315 q^{48} -1.00000 q^{49} -0.00486525i q^{50} -4.57069 q^{51} +8.39607 q^{53} +3.87192i q^{54} -7.41314 q^{55} -0.913059 q^{56} +10.7485i q^{57} +0.140469i q^{58} +8.82234i q^{59} -14.4295i q^{60} +5.46667 q^{61} -0.396810 q^{62} -8.03736i q^{63} +6.74383 q^{64} -2.55361 q^{66} +10.1857i q^{67} +2.67792 q^{68} +2.78572 q^{69} -0.516249i q^{70} +5.21428i q^{71} -7.33859i q^{72} -3.96355i q^{73} +0.359445 q^{74} -0.0698632 q^{75} -6.29744i q^{76} +3.32225 q^{77} +6.45051 q^{79} +8.21520i q^{80} +31.4871 q^{81} +2.12339 q^{82} +4.64055i q^{83} +6.46667i q^{84} +3.06986i q^{85} +0.284889i q^{86} +2.01708 q^{87} +3.03341 q^{88} +9.12826i q^{89} +4.14929 q^{90} -1.63212 q^{92} +5.69803i q^{93} -0.376695 q^{94} +7.21915 q^{95} +8.89672i q^{96} +15.3589i q^{97} +0.231361i q^{98} +26.7022i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9} + 22 q^{10} - 24 q^{12} + 2 q^{14} + 38 q^{16} + 8 q^{17} + 10 q^{22} + 4 q^{23} - 10 q^{25} - 52 q^{27} + 2 q^{29} + 8 q^{30} + 14 q^{35} + 68 q^{36} + 46 q^{38} - 34 q^{40}+ \cdots - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.231361i − 0.163597i −0.996649 − 0.0817984i \(-0.973934\pi\)
0.996649 − 0.0817984i \(-0.0260664\pi\)
\(3\) −3.32225 −1.91810 −0.959052 − 0.283231i \(-0.908594\pi\)
−0.959052 + 0.283231i \(0.908594\pi\)
\(4\) 1.94647 0.973236
\(5\) 2.23136i 0.997895i 0.866632 + 0.498947i \(0.166280\pi\)
−0.866632 + 0.498947i \(0.833720\pi\)
\(6\) 0.768639i 0.313796i
\(7\) − 1.00000i − 0.377964i
\(8\) − 0.913059i − 0.322815i
\(9\) 8.03736 2.67912
\(10\) 0.516249 0.163252
\(11\) 3.32225i 1.00170i 0.865535 + 0.500848i \(0.166979\pi\)
−0.865535 + 0.500848i \(0.833021\pi\)
\(12\) −6.46667 −1.86677
\(13\) 0 0
\(14\) −0.231361 −0.0618338
\(15\) − 7.41314i − 1.91407i
\(16\) 3.68170 0.920425
\(17\) 1.37578 0.333676 0.166838 − 0.985984i \(-0.446644\pi\)
0.166838 + 0.985984i \(0.446644\pi\)
\(18\) − 1.85953i − 0.438296i
\(19\) − 3.23531i − 0.742231i −0.928587 − 0.371116i \(-0.878975\pi\)
0.928587 − 0.371116i \(-0.121025\pi\)
\(20\) 4.34328i 0.971187i
\(21\) 3.32225i 0.724975i
\(22\) 0.768639 0.163874
\(23\) −0.838502 −0.174840 −0.0874199 − 0.996172i \(-0.527862\pi\)
−0.0874199 + 0.996172i \(0.527862\pi\)
\(24\) 3.03341i 0.619193i
\(25\) 0.0210289 0.00420577
\(26\) 0 0
\(27\) −16.7354 −3.22073
\(28\) − 1.94647i − 0.367849i
\(29\) −0.607142 −0.112743 −0.0563717 − 0.998410i \(-0.517953\pi\)
−0.0563717 + 0.998410i \(0.517953\pi\)
\(30\) −1.71511 −0.313135
\(31\) − 1.71511i − 0.308043i −0.988067 − 0.154022i \(-0.950777\pi\)
0.988067 − 0.154022i \(-0.0492225\pi\)
\(32\) − 2.67792i − 0.473394i
\(33\) − 11.0374i − 1.92136i
\(34\) − 0.318302i − 0.0545883i
\(35\) 2.23136 0.377169
\(36\) 15.6445 2.60742
\(37\) 1.55361i 0.255413i 0.991812 + 0.127706i \(0.0407615\pi\)
−0.991812 + 0.127706i \(0.959239\pi\)
\(38\) −0.748524 −0.121427
\(39\) 0 0
\(40\) 2.03736 0.322136
\(41\) 9.17783i 1.43334i 0.697414 + 0.716668i \(0.254334\pi\)
−0.697414 + 0.716668i \(0.745666\pi\)
\(42\) 0.768639 0.118604
\(43\) −1.23136 −0.187781 −0.0938904 − 0.995583i \(-0.529930\pi\)
−0.0938904 + 0.995583i \(0.529930\pi\)
\(44\) 6.46667i 0.974888i
\(45\) 17.9343i 2.67348i
\(46\) 0.193997i 0.0286032i
\(47\) − 1.62817i − 0.237493i −0.992925 − 0.118747i \(-0.962112\pi\)
0.992925 − 0.118747i \(-0.0378876\pi\)
\(48\) −12.2315 −1.76547
\(49\) −1.00000 −0.142857
\(50\) − 0.00486525i 0 0.000688051i
\(51\) −4.57069 −0.640025
\(52\) 0 0
\(53\) 8.39607 1.15329 0.576644 − 0.816995i \(-0.304362\pi\)
0.576644 + 0.816995i \(0.304362\pi\)
\(54\) 3.87192i 0.526901i
\(55\) −7.41314 −0.999588
\(56\) −0.913059 −0.122013
\(57\) 10.7485i 1.42368i
\(58\) 0.140469i 0.0184445i
\(59\) 8.82234i 1.14857i 0.818655 + 0.574285i \(0.194720\pi\)
−0.818655 + 0.574285i \(0.805280\pi\)
\(60\) − 14.4295i − 1.86284i
\(61\) 5.46667 0.699936 0.349968 − 0.936762i \(-0.386193\pi\)
0.349968 + 0.936762i \(0.386193\pi\)
\(62\) −0.396810 −0.0503949
\(63\) − 8.03736i − 1.01261i
\(64\) 6.74383 0.842979
\(65\) 0 0
\(66\) −2.55361 −0.314328
\(67\) 10.1857i 1.24439i 0.782864 + 0.622193i \(0.213758\pi\)
−0.782864 + 0.622193i \(0.786242\pi\)
\(68\) 2.67792 0.324745
\(69\) 2.78572 0.335361
\(70\) − 0.516249i − 0.0617036i
\(71\) 5.21428i 0.618822i 0.950928 + 0.309411i \(0.100132\pi\)
−0.950928 + 0.309411i \(0.899868\pi\)
\(72\) − 7.33859i − 0.864861i
\(73\) − 3.96355i − 0.463898i −0.972728 − 0.231949i \(-0.925490\pi\)
0.972728 − 0.231949i \(-0.0745103\pi\)
\(74\) 0.359445 0.0417847
\(75\) −0.0698632 −0.00806711
\(76\) − 6.29744i − 0.722366i
\(77\) 3.32225 0.378606
\(78\) 0 0
\(79\) 6.45051 0.725739 0.362869 − 0.931840i \(-0.381797\pi\)
0.362869 + 0.931840i \(0.381797\pi\)
\(80\) 8.21520i 0.918487i
\(81\) 31.4871 3.49857
\(82\) 2.12339 0.234489
\(83\) 4.64055i 0.509367i 0.967024 + 0.254684i \(0.0819713\pi\)
−0.967024 + 0.254684i \(0.918029\pi\)
\(84\) 6.46667i 0.705572i
\(85\) 3.06986i 0.332973i
\(86\) 0.284889i 0.0307203i
\(87\) 2.01708 0.216253
\(88\) 3.03341 0.323363
\(89\) 9.12826i 0.967593i 0.875180 + 0.483797i \(0.160743\pi\)
−0.875180 + 0.483797i \(0.839257\pi\)
\(90\) 4.14929 0.437373
\(91\) 0 0
\(92\) −1.63212 −0.170160
\(93\) 5.69803i 0.590859i
\(94\) −0.376695 −0.0388531
\(95\) 7.21915 0.740669
\(96\) 8.89672i 0.908018i
\(97\) 15.3589i 1.55946i 0.626117 + 0.779729i \(0.284643\pi\)
−0.626117 + 0.779729i \(0.715357\pi\)
\(98\) 0.231361i 0.0233710i
\(99\) 26.7022i 2.68367i
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 1183.2.c.g.337.4 8
13.5 odd 4 1183.2.a.l.1.2 4
13.7 odd 12 91.2.f.c.29.2 yes 8
13.8 odd 4 1183.2.a.k.1.3 4
13.11 odd 12 91.2.f.c.22.2 ✓ 8
13.12 even 2 inner 1183.2.c.g.337.5 8
39.11 even 12 819.2.o.h.568.3 8
39.20 even 12 819.2.o.h.757.3 8
52.7 even 12 1456.2.s.q.1121.1 8
52.11 even 12 1456.2.s.q.113.1 8
91.11 odd 12 637.2.g.k.373.2 8
91.20 even 12 637.2.f.i.393.2 8
91.24 even 12 637.2.g.j.373.2 8
91.33 even 12 637.2.g.j.263.2 8
91.34 even 4 8281.2.a.bp.1.3 4
91.37 odd 12 637.2.h.h.165.3 8
91.46 odd 12 637.2.h.h.471.3 8
91.59 even 12 637.2.h.i.471.3 8
91.72 odd 12 637.2.g.k.263.2 8
91.76 even 12 637.2.f.i.295.2 8
91.83 even 4 8281.2.a.bt.1.2 4
91.89 even 12 637.2.h.i.165.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.2 ✓ 8 13.11 odd 12
91.2.f.c.29.2 yes 8 13.7 odd 12
637.2.f.i.295.2 8 91.76 even 12
637.2.f.i.393.2 8 91.20 even 12
637.2.g.j.263.2 8 91.33 even 12
637.2.g.j.373.2 8 91.24 even 12
637.2.g.k.263.2 8 91.72 odd 12
637.2.g.k.373.2 8 91.11 odd 12
637.2.h.h.165.3 8 91.37 odd 12
637.2.h.h.471.3 8 91.46 odd 12
637.2.h.i.165.3 8 91.89 even 12
637.2.h.i.471.3 8 91.59 even 12
819.2.o.h.568.3 8 39.11 even 12
819.2.o.h.757.3 8 39.20 even 12
1183.2.a.k.1.3 4 13.8 odd 4
1183.2.a.l.1.2 4 13.5 odd 4
1183.2.c.g.337.4 8 1.1 even 1 trivial
1183.2.c.g.337.5 8 13.12 even 2 inner
1456.2.s.q.113.1 8 52.11 even 12
1456.2.s.q.1121.1 8 52.7 even 12
8281.2.a.bp.1.3 4 91.34 even 4
8281.2.a.bt.1.2 4 91.83 even 4