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 1.4
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 \(-0.0793115\pi\)
0.969119 + 0.246594i \(0.0793115\pi\)
\(3\) −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 \(-0.447008\pi\)
0.165711 + 0.986174i \(0.447008\pi\)
\(6\) −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 \(-0.434035\pi\)
0.205754 + 0.978604i \(0.434035\pi\)
\(12\) −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 \(-0.186349\pi\)
0.833474 + 0.552559i \(0.186349\pi\)
\(20\) 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 \(-0.580093\pi\)
−0.248973 + 0.968510i \(0.580093\pi\)
\(32\) 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 \(-0.332634\pi\)
0.501902 + 0.864924i \(0.332634\pi\)
\(38\) 19.9169 3.23094
\(39\) 0 0
\(40\) 7.13727 1.12850
\(41\) 1.25461 0.195938 0.0979688 − 0.995189i \(-0.468765\pi\)
0.0979688 + 0.995189i \(0.468765\pi\)
\(42\) 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 \(-0.640525\pi\)
−0.427270 + 0.904124i \(0.640525\pi\)
\(48\) −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 \(-0.753580\pi\)
−0.715014 + 0.699110i \(0.753580\pi\)
\(60\) −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 \(-0.182258\pi\)
0.840505 + 0.541804i \(0.182258\pi\)
\(68\) 22.8740 2.77389
\(69\) 3.18474 0.383398
\(70\) 0 0
\(71\) 4.81526 0.571466 0.285733 − 0.958309i \(-0.407763\pi\)
0.285733 + 0.958309i \(0.407763\pi\)
\(72\) −10.9529 −1.29081
\(73\) 6.06987 0.710425 0.355212 − 0.934786i \(-0.384409\pi\)
0.355212 + 0.934786i \(0.384409\pi\)
\(74\) 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 \(-0.277216\pi\)
0.644139 + 0.764908i \(0.277216\pi\)
\(84\) 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 \(-0.529752\pi\)
−0.0933331 + 0.995635i \(0.529752\pi\)
\(90\) −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 \(-0.339181\pi\)
0.484006 + 0.875065i \(0.339181\pi\)
\(98\) 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.bt.1.4 4
7.6 odd 2 1183.2.a.l.1.4 4
13.4 even 6 637.2.f.i.393.4 8
13.10 even 6 637.2.f.i.295.4 8
13.12 even 2 8281.2.a.bp.1.1 4
91.4 even 6 637.2.h.i.471.1 8
91.10 odd 6 637.2.g.k.373.4 8
91.17 odd 6 637.2.h.h.471.1 8
91.23 even 6 637.2.h.i.165.1 8
91.30 even 6 637.2.g.j.263.4 8
91.34 even 4 1183.2.c.g.337.8 8
91.62 odd 6 91.2.f.c.22.4 ✓ 8
91.69 odd 6 91.2.f.c.29.4 yes 8
91.75 odd 6 637.2.h.h.165.1 8
91.82 odd 6 637.2.g.k.263.4 8
91.83 even 4 1183.2.c.g.337.1 8
91.88 even 6 637.2.g.j.373.4 8
91.90 odd 2 1183.2.a.k.1.1 4
273.62 even 6 819.2.o.h.568.1 8
273.251 even 6 819.2.o.h.757.1 8
364.251 even 6 1456.2.s.q.1121.3 8
364.335 even 6 1456.2.s.q.113.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.4 ✓ 8 91.62 odd 6
91.2.f.c.29.4 yes 8 91.69 odd 6
637.2.f.i.295.4 8 13.10 even 6
637.2.f.i.393.4 8 13.4 even 6
637.2.g.j.263.4 8 91.30 even 6
637.2.g.j.373.4 8 91.88 even 6
637.2.g.k.263.4 8 91.82 odd 6
637.2.g.k.373.4 8 91.10 odd 6
637.2.h.h.165.1 8 91.75 odd 6
637.2.h.h.471.1 8 91.17 odd 6
637.2.h.i.165.1 8 91.23 even 6
637.2.h.i.471.1 8 91.4 even 6
819.2.o.h.568.1 8 273.62 even 6
819.2.o.h.757.1 8 273.251 even 6
1183.2.a.k.1.1 4 91.90 odd 2
1183.2.a.l.1.4 4 7.6 odd 2
1183.2.c.g.337.1 8 91.83 even 4
1183.2.c.g.337.8 8 91.34 even 4
1456.2.s.q.113.3 8 364.335 even 6
1456.2.s.q.1121.3 8 364.251 even 6
8281.2.a.bp.1.1 4 13.12 even 2
8281.2.a.bt.1.4 4 1.1 even 1 trivial