Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8281,2,Mod(1,8281)] 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("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-1,-1,5,7,5,0,-6,7,11,-1,-12,0,0,3,19,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.27004.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.74108\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.74108 q^{2} -1.36482 q^{3} +5.51353 q^{4} -0.741082 q^{5} +3.74108 q^{6} -9.63087 q^{8} -1.13727 q^{9} +2.03137 q^{10} -1.36482 q^{11} -7.52497 q^{12} +1.01144 q^{15} +15.3720 q^{16} +4.14871 q^{17} +3.11734 q^{18} -7.26606 q^{19} -4.08598 q^{20} +3.74108 q^{22} -2.33345 q^{23} +13.1444 q^{24} -4.45080 q^{25} +5.64662 q^{27} -0.407629 q^{29} -2.77245 q^{30} +2.77245 q^{31} -22.8740 q^{32} +1.86273 q^{33} -11.3720 q^{34} -6.27036 q^{36} -6.10590 q^{37} +19.9169 q^{38} +7.13727 q^{40} -1.25461 q^{41} -1.74108 q^{43} -7.52497 q^{44} +0.842809 q^{45} +6.39619 q^{46} +5.85843 q^{47} -20.9799 q^{48} +12.2000 q^{50} -5.66224 q^{51} +4.56778 q^{53} -15.4779 q^{54} +1.01144 q^{55} +9.91685 q^{57} +1.11734 q^{58} +10.9843 q^{59} +5.57662 q^{60} -6.52497 q^{61} -7.59951 q^{62} +31.9557 q^{64} -5.10590 q^{66} -13.7597 q^{67} +22.8740 q^{68} +3.18474 q^{69} -4.81526 q^{71} +10.9529 q^{72} -6.06987 q^{73} +16.7368 q^{74} +6.07453 q^{75} -40.0616 q^{76} -9.12582 q^{79} -11.3919 q^{80} -4.29482 q^{81} +3.43900 q^{82} -11.7368 q^{83} -3.07453 q^{85} +4.77245 q^{86} +0.556340 q^{87} +13.1444 q^{88} +1.76101 q^{89} -2.31021 q^{90} -12.8656 q^{92} -3.78389 q^{93} -16.0584 q^{94} +5.38474 q^{95} +31.2189 q^{96} -9.53381 q^{97} +1.55217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{3} + 5 q^{4} + 7 q^{5} + 5 q^{6} - 6 q^{8} + 7 q^{9} + 11 q^{10} - q^{11} - 12 q^{12} + 3 q^{15} + 19 q^{16} + 4 q^{17} - 3 q^{18} - q^{19} + 2 q^{20} + 5 q^{22} - 2 q^{23} + 3 q^{24}+ \cdots + 23 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74108 −1.93824 −0.969119 − 0.246594i \(-0.920689\pi\)
−0.969119 + 0.246594i \(0.920689\pi\)
\(3\) −1.36482 −0.787979 −0.393989 − 0.919115i \(-0.628905\pi\)
−0.393989 + 0.919115i \(0.628905\pi\)
\(4\) 5.51353 2.75677
\(5\) −0.741082 −0.331422 −0.165711 − 0.986174i \(-0.552992\pi\)
−0.165711 + 0.986174i \(0.552992\pi\)
\(6\) 3.74108 1.52729
\(7\) 0 0
\(8\) −9.63087 −3.40503
\(9\) −1.13727 −0.379089
\(10\) 2.03137 0.642374
\(11\) −1.36482 −0.411509 −0.205754 − 0.978604i \(-0.565965\pi\)
−0.205754 + 0.978604i \(0.565965\pi\)
\(12\) −7.52497 −2.17227
\(13\) 0 0
\(14\) 0 0
\(15\) 1.01144 0.261153
\(16\) 15.3720 3.84299
\(17\) 4.14871 1.00621 0.503105 − 0.864225i \(-0.332191\pi\)
0.503105 + 0.864225i \(0.332191\pi\)
\(18\) 3.11734 0.734765
\(19\) −7.26606 −1.66695 −0.833474 − 0.552559i \(-0.813651\pi\)
−0.833474 + 0.552559i \(0.813651\pi\)
\(20\) −4.08598 −0.913652
\(21\) 0 0
\(22\) 3.74108 0.797601
\(23\) −2.33345 −0.486559 −0.243279 − 0.969956i \(-0.578223\pi\)
−0.243279 + 0.969956i \(0.578223\pi\)
\(24\) 13.1444 2.68309
\(25\) −4.45080 −0.890159
\(26\) 0 0
\(27\) 5.64662 1.08669
\(28\) 0 0
\(29\) −0.407629 −0.0756948 −0.0378474 − 0.999284i \(-0.512050\pi\)
−0.0378474 + 0.999284i \(0.512050\pi\)
\(30\) −2.77245 −0.506178
\(31\) 2.77245 0.497946 0.248973 − 0.968510i \(-0.419907\pi\)
0.248973 + 0.968510i \(0.419907\pi\)
\(32\) −22.8740 −4.04360
\(33\) 1.86273 0.324260
\(34\) −11.3720 −1.95027
\(35\) 0 0
\(36\) −6.27036 −1.04506
\(37\) −6.10590 −1.00380 −0.501902 − 0.864924i \(-0.667366\pi\)
−0.501902 + 0.864924i \(0.667366\pi\)
\(38\) 19.9169 3.23094
\(39\) 0 0
\(40\) 7.13727 1.12850
\(41\) −1.25461 −0.195938 −0.0979688 − 0.995189i \(-0.531235\pi\)
−0.0979688 + 0.995189i \(0.531235\pi\)
\(42\) 0 0
\(43\) −1.74108 −0.265513 −0.132756 − 0.991149i \(-0.542383\pi\)
−0.132756 + 0.991149i \(0.542383\pi\)
\(44\) −7.52497 −1.13443
\(45\) 0.842809 0.125639
\(46\) 6.39619 0.943066
\(47\) 5.85843 0.854539 0.427270 − 0.904124i \(-0.359475\pi\)
0.427270 + 0.904124i \(0.359475\pi\)
\(48\) −20.9799 −3.02819
\(49\) 0 0
\(50\) 12.2000 1.72534
\(51\) −5.66224 −0.792872
\(52\) 0 0
\(53\) 4.56778 0.627433 0.313717 − 0.949517i \(-0.398426\pi\)
0.313717 + 0.949517i \(0.398426\pi\)
\(54\) −15.4779 −2.10627
\(55\) 1.01144 0.136383
\(56\) 0 0
\(57\) 9.91685 1.31352
\(58\) 1.11734 0.146715
\(59\) 10.9843 1.43003 0.715014 − 0.699110i \(-0.246420\pi\)
0.715014 + 0.699110i \(0.246420\pi\)
\(60\) 5.57662 0.719939
\(61\) −6.52497 −0.835437 −0.417719 − 0.908576i \(-0.637170\pi\)
−0.417719 + 0.908576i \(0.637170\pi\)
\(62\) −7.59951 −0.965139
\(63\) 0 0
\(64\) 31.9557 3.99446
\(65\) 0 0
\(66\) −5.10590 −0.628493
\(67\) −13.7597 −1.68101 −0.840505 − 0.541804i \(-0.817742\pi\)
−0.840505 + 0.541804i \(0.817742\pi\)
\(68\) 22.8740 2.77389
\(69\) 3.18474 0.383398
\(70\) 0 0
\(71\) −4.81526 −0.571466 −0.285733 − 0.958309i \(-0.592237\pi\)
−0.285733 + 0.958309i \(0.592237\pi\)
\(72\) 10.9529 1.29081
\(73\) −6.06987 −0.710425 −0.355212 − 0.934786i \(-0.615591\pi\)
−0.355212 + 0.934786i \(0.615591\pi\)
\(74\) 16.7368 1.94561
\(75\) 6.07453 0.701427
\(76\) −40.0616 −4.59538
\(77\) 0 0
\(78\) 0 0
\(79\) −9.12582 −1.02674 −0.513368 − 0.858169i \(-0.671602\pi\)
−0.513368 + 0.858169i \(0.671602\pi\)
\(80\) −11.3919 −1.27365
\(81\) −4.29482 −0.477202
\(82\) 3.43900 0.379774
\(83\) −11.7368 −1.28828 −0.644139 − 0.764908i \(-0.722784\pi\)
−0.644139 + 0.764908i \(0.722784\pi\)
\(84\) 0 0
\(85\) −3.07453 −0.333480
\(86\) 4.77245 0.514626
\(87\) 0.556340 0.0596459
\(88\) 13.1444 1.40120
\(89\) 1.76101 0.186666 0.0933331 − 0.995635i \(-0.470248\pi\)
0.0933331 + 0.995635i \(0.470248\pi\)
\(90\) −2.31021 −0.243517
\(91\) 0 0
\(92\) −12.8656 −1.34133
\(93\) −3.78389 −0.392371
\(94\) −16.0584 −1.65630
\(95\) 5.38474 0.552463
\(96\) 31.2189 3.18627
\(97\) −9.53381 −0.968012 −0.484006 − 0.875065i \(-0.660819\pi\)
−0.484006 + 0.875065i \(0.660819\pi\)
\(98\) 0 0
\(99\) 1.55217 0.155998
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 8281.2.a.bp.1.1 4
7.6 odd 2 1183.2.a.k.1.1 4
13.3 even 3 637.2.f.i.295.4 8
13.9 even 3 637.2.f.i.393.4 8
13.12 even 2 8281.2.a.bt.1.4 4
91.3 odd 6 637.2.g.k.373.4 8
91.9 even 3 637.2.g.j.263.4 8
91.16 even 3 637.2.h.i.165.1 8
91.34 even 4 1183.2.c.g.337.1 8
91.48 odd 6 91.2.f.c.29.4 yes 8
91.55 odd 6 91.2.f.c.22.4 ✓ 8
91.61 odd 6 637.2.g.k.263.4 8
91.68 odd 6 637.2.h.h.165.1 8
91.74 even 3 637.2.h.i.471.1 8
91.81 even 3 637.2.g.j.373.4 8
91.83 even 4 1183.2.c.g.337.8 8
91.87 odd 6 637.2.h.h.471.1 8
91.90 odd 2 1183.2.a.l.1.4 4
273.146 even 6 819.2.o.h.568.1 8
273.230 even 6 819.2.o.h.757.1 8
364.55 even 6 1456.2.s.q.113.3 8
364.139 even 6 1456.2.s.q.1121.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.4 ✓ 8 91.55 odd 6
91.2.f.c.29.4 yes 8 91.48 odd 6
637.2.f.i.295.4 8 13.3 even 3
637.2.f.i.393.4 8 13.9 even 3
637.2.g.j.263.4 8 91.9 even 3
637.2.g.j.373.4 8 91.81 even 3
637.2.g.k.263.4 8 91.61 odd 6
637.2.g.k.373.4 8 91.3 odd 6
637.2.h.h.165.1 8 91.68 odd 6
637.2.h.h.471.1 8 91.87 odd 6
637.2.h.i.165.1 8 91.16 even 3
637.2.h.i.471.1 8 91.74 even 3
819.2.o.h.568.1 8 273.146 even 6
819.2.o.h.757.1 8 273.230 even 6
1183.2.a.k.1.1 4 7.6 odd 2
1183.2.a.l.1.4 4 91.90 odd 2
1183.2.c.g.337.1 8 91.34 even 4
1183.2.c.g.337.8 8 91.83 even 4
1456.2.s.q.113.3 8 364.55 even 6
1456.2.s.q.1121.3 8 364.139 even 6
8281.2.a.bp.1.1 4 1.1 even 1 trivial
8281.2.a.bt.1.4 4 13.12 even 2