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 337.1
Root \(-2.74108i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.g.337.8

$q$-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)
 
\(f(q)\) \(=\) \(q-2.74108i q^{2} +1.36482 q^{3} -5.51353 q^{4} +0.741082i q^{5} -3.74108i q^{6} +1.00000i q^{7} +9.63087i q^{8} -1.13727 q^{9} +2.03137 q^{10} +1.36482i q^{11} -7.52497 q^{12} +2.74108 q^{14} +1.01144i q^{15} +15.3720 q^{16} +4.14871 q^{17} +3.11734i q^{18} +7.26606i q^{19} -4.08598i q^{20} +1.36482i q^{21} +3.74108 q^{22} +2.33345 q^{23} +13.1444i q^{24} +4.45080 q^{25} -5.64662 q^{27} -5.51353i q^{28} -0.407629 q^{29} +2.77245 q^{30} -2.77245i q^{31} -22.8740i q^{32} +1.86273i q^{33} -11.3720i q^{34} -0.741082 q^{35} +6.27036 q^{36} +6.10590i q^{37} +19.9169 q^{38} -7.13727 q^{40} +1.25461i q^{41} +3.74108 q^{42} +1.74108 q^{43} -7.52497i q^{44} -0.842809i q^{45} -6.39619i q^{46} +5.85843i q^{47} +20.9799 q^{48} -1.00000 q^{49} -12.2000i q^{50} +5.66224 q^{51} +4.56778 q^{53} +15.4779i q^{54} -1.01144 q^{55} -9.63087 q^{56} +9.91685i q^{57} +1.11734i q^{58} +10.9843i q^{59} -5.57662i q^{60} +6.52497 q^{61} -7.59951 q^{62} -1.13727i q^{63} -31.9557 q^{64} +5.10590 q^{66} -13.7597i q^{67} -22.8740 q^{68} +3.18474 q^{69} +2.03137i q^{70} -4.81526i q^{71} -10.9529i q^{72} -6.06987i q^{73} +16.7368 q^{74} +6.07453 q^{75} -40.0616i q^{76} -1.36482 q^{77} -9.12582 q^{79} +11.3919i q^{80} -4.29482 q^{81} +3.43900 q^{82} +11.7368i q^{83} -7.52497i q^{84} +3.07453i q^{85} -4.77245i q^{86} -0.556340 q^{87} -13.1444 q^{88} +1.76101i q^{89} -2.31021 q^{90} -12.8656 q^{92} -3.78389i q^{93} +16.0584 q^{94} -5.38474 q^{95} -31.2189i q^{96} +9.53381i q^{97} +2.74108i q^{98} -1.55217i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9} + 22 q^{10} - 24 q^{12} + 2 q^{14} + 38 q^{16} + 8 q^{17} + 10 q^{22} + 4 q^{23} - 10 q^{25} - 52 q^{27} + 2 q^{29} + 8 q^{30} + 14 q^{35} + 68 q^{36} + 46 q^{38} - 34 q^{40}+ \cdots - 42 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.74108i − 1.93824i −0.246594 − 0.969119i \(-0.579311\pi\)
0.246594 − 0.969119i \(-0.420689\pi\)
\(3\) 1.36482 0.787979 0.393989 − 0.919115i \(-0.371095\pi\)
0.393989 + 0.919115i \(0.371095\pi\)
\(4\) −5.51353 −2.75677
\(5\) 0.741082i 0.331422i 0.986174 + 0.165711i \(0.0529919\pi\)
−0.986174 + 0.165711i \(0.947008\pi\)
\(6\) − 3.74108i − 1.52729i
\(7\) 1.00000i 0.377964i
\(8\) 9.63087i 3.40503i
\(9\) −1.13727 −0.379089
\(10\) 2.03137 0.642374
\(11\) 1.36482i 0.411509i 0.978604 + 0.205754i \(0.0659648\pi\)
−0.978604 + 0.205754i \(0.934035\pi\)
\(12\) −7.52497 −2.17227
\(13\) 0 0
\(14\) 2.74108 0.732585
\(15\) 1.01144i 0.261153i
\(16\) 15.3720 3.84299
\(17\) 4.14871 1.00621 0.503105 − 0.864225i \(-0.332191\pi\)
0.503105 + 0.864225i \(0.332191\pi\)
\(18\) 3.11734i 0.734765i
\(19\) 7.26606i 1.66695i 0.552559 + 0.833474i \(0.313651\pi\)
−0.552559 + 0.833474i \(0.686349\pi\)
\(20\) − 4.08598i − 0.913652i
\(21\) 1.36482i 0.297828i
\(22\) 3.74108 0.797601
\(23\) 2.33345 0.486559 0.243279 − 0.969956i \(-0.421777\pi\)
0.243279 + 0.969956i \(0.421777\pi\)
\(24\) 13.1444i 2.68309i
\(25\) 4.45080 0.890159
\(26\) 0 0
\(27\) −5.64662 −1.08669
\(28\) − 5.51353i − 1.04196i
\(29\) −0.407629 −0.0756948 −0.0378474 − 0.999284i \(-0.512050\pi\)
−0.0378474 + 0.999284i \(0.512050\pi\)
\(30\) 2.77245 0.506178
\(31\) − 2.77245i − 0.497946i −0.968510 − 0.248973i \(-0.919907\pi\)
0.968510 − 0.248973i \(-0.0800931\pi\)
\(32\) − 22.8740i − 4.04360i
\(33\) 1.86273i 0.324260i
\(34\) − 11.3720i − 1.95027i
\(35\) −0.741082 −0.125266
\(36\) 6.27036 1.04506
\(37\) 6.10590i 1.00380i 0.864924 + 0.501902i \(0.167366\pi\)
−0.864924 + 0.501902i \(0.832634\pi\)
\(38\) 19.9169 3.23094
\(39\) 0 0
\(40\) −7.13727 −1.12850
\(41\) 1.25461i 0.195938i 0.995189 + 0.0979688i \(0.0312345\pi\)
−0.995189 + 0.0979688i \(0.968765\pi\)
\(42\) 3.74108 0.577261
\(43\) 1.74108 0.265513 0.132756 − 0.991149i \(-0.457617\pi\)
0.132756 + 0.991149i \(0.457617\pi\)
\(44\) − 7.52497i − 1.13443i
\(45\) − 0.842809i − 0.125639i
\(46\) − 6.39619i − 0.943066i
\(47\) 5.85843i 0.854539i 0.904124 + 0.427270i \(0.140525\pi\)
−0.904124 + 0.427270i \(0.859475\pi\)
\(48\) 20.9799 3.02819
\(49\) −1.00000 −0.142857
\(50\) − 12.2000i − 1.72534i
\(51\) 5.66224 0.792872
\(52\) 0 0
\(53\) 4.56778 0.627433 0.313717 − 0.949517i \(-0.398426\pi\)
0.313717 + 0.949517i \(0.398426\pi\)
\(54\) 15.4779i 2.10627i
\(55\) −1.01144 −0.136383
\(56\) −9.63087 −1.28698
\(57\) 9.91685i 1.31352i
\(58\) 1.11734i 0.146715i
\(59\) 10.9843i 1.43003i 0.699110 + 0.715014i \(0.253580\pi\)
−0.699110 + 0.715014i \(0.746420\pi\)
\(60\) − 5.57662i − 0.719939i
\(61\) 6.52497 0.835437 0.417719 − 0.908576i \(-0.362830\pi\)
0.417719 + 0.908576i \(0.362830\pi\)
\(62\) −7.59951 −0.965139
\(63\) − 1.13727i − 0.143282i
\(64\) −31.9557 −3.99446
\(65\) 0 0
\(66\) 5.10590 0.628493
\(67\) − 13.7597i − 1.68101i −0.541804 − 0.840505i \(-0.682258\pi\)
0.541804 − 0.840505i \(-0.317742\pi\)
\(68\) −22.8740 −2.77389
\(69\) 3.18474 0.383398
\(70\) 2.03137i 0.242795i
\(71\) − 4.81526i − 0.571466i −0.958309 − 0.285733i \(-0.907763\pi\)
0.958309 − 0.285733i \(-0.0922370\pi\)
\(72\) − 10.9529i − 1.29081i
\(73\) − 6.06987i − 0.710425i −0.934786 − 0.355212i \(-0.884409\pi\)
0.934786 − 0.355212i \(-0.115591\pi\)
\(74\) 16.7368 1.94561
\(75\) 6.07453 0.701427
\(76\) − 40.0616i − 4.59538i
\(77\) −1.36482 −0.155536
\(78\) 0 0
\(79\) −9.12582 −1.02674 −0.513368 − 0.858169i \(-0.671602\pi\)
−0.513368 + 0.858169i \(0.671602\pi\)
\(80\) 11.3919i 1.27365i
\(81\) −4.29482 −0.477202
\(82\) 3.43900 0.379774
\(83\) 11.7368i 1.28828i 0.764908 + 0.644139i \(0.222784\pi\)
−0.764908 + 0.644139i \(0.777216\pi\)
\(84\) − 7.52497i − 0.821042i
\(85\) 3.07453i 0.333480i
\(86\) − 4.77245i − 0.514626i
\(87\) −0.556340 −0.0596459
\(88\) −13.1444 −1.40120
\(89\) 1.76101i 0.186666i 0.995635 + 0.0933331i \(0.0297522\pi\)
−0.995635 + 0.0933331i \(0.970248\pi\)
\(90\) −2.31021 −0.243517
\(91\) 0 0
\(92\) −12.8656 −1.34133
\(93\) − 3.78389i − 0.392371i
\(94\) 16.0584 1.65630
\(95\) −5.38474 −0.552463
\(96\) − 31.2189i − 3.18627i
\(97\) 9.53381i 0.968012i 0.875065 + 0.484006i \(0.160819\pi\)
−0.875065 + 0.484006i \(0.839181\pi\)
\(98\) 2.74108i 0.276891i
\(99\) − 1.55217i − 0.155998i
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.337.1 8
13.2 odd 12 91.2.f.c.22.4 ✓ 8
13.5 odd 4 1183.2.a.k.1.1 4
13.6 odd 12 91.2.f.c.29.4 yes 8
13.8 odd 4 1183.2.a.l.1.4 4
13.12 even 2 inner 1183.2.c.g.337.8 8
39.2 even 12 819.2.o.h.568.1 8
39.32 even 12 819.2.o.h.757.1 8
52.15 even 12 1456.2.s.q.113.3 8
52.19 even 12 1456.2.s.q.1121.3 8
91.2 odd 12 637.2.h.h.165.1 8
91.6 even 12 637.2.f.i.393.4 8
91.19 even 12 637.2.g.j.263.4 8
91.32 odd 12 637.2.h.h.471.1 8
91.34 even 4 8281.2.a.bt.1.4 4
91.41 even 12 637.2.f.i.295.4 8
91.45 even 12 637.2.h.i.471.1 8
91.54 even 12 637.2.h.i.165.1 8
91.58 odd 12 637.2.g.k.263.4 8
91.67 odd 12 637.2.g.k.373.4 8
91.80 even 12 637.2.g.j.373.4 8
91.83 even 4 8281.2.a.bp.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.4 ✓ 8 13.2 odd 12
91.2.f.c.29.4 yes 8 13.6 odd 12
637.2.f.i.295.4 8 91.41 even 12
637.2.f.i.393.4 8 91.6 even 12
637.2.g.j.263.4 8 91.19 even 12
637.2.g.j.373.4 8 91.80 even 12
637.2.g.k.263.4 8 91.58 odd 12
637.2.g.k.373.4 8 91.67 odd 12
637.2.h.h.165.1 8 91.2 odd 12
637.2.h.h.471.1 8 91.32 odd 12
637.2.h.i.165.1 8 91.54 even 12
637.2.h.i.471.1 8 91.45 even 12
819.2.o.h.568.1 8 39.2 even 12
819.2.o.h.757.1 8 39.32 even 12
1183.2.a.k.1.1 4 13.5 odd 4
1183.2.a.l.1.4 4 13.8 odd 4
1183.2.c.g.337.1 8 1.1 even 1 trivial
1183.2.c.g.337.8 8 13.12 even 2 inner
1456.2.s.q.113.3 8 52.15 even 12
1456.2.s.q.1121.3 8 52.19 even 12
8281.2.a.bp.1.1 4 91.83 even 4
8281.2.a.bt.1.4 4 91.34 even 4