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 1.2
Root \(-0.231361\) 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)
 
\(f(q)\) \(=\) \(q-0.231361 q^{2} +3.32225 q^{3} -1.94647 q^{4} -2.23136 q^{5} -0.768639 q^{6} +0.913059 q^{8} +8.03736 q^{9} +0.516249 q^{10} -3.32225 q^{11} -6.46667 q^{12} -7.41314 q^{15} +3.68170 q^{16} +1.37578 q^{17} -1.85953 q^{18} +3.23531 q^{19} +4.34328 q^{20} +0.768639 q^{22} +0.838502 q^{23} +3.03341 q^{24} -0.0210289 q^{25} +16.7354 q^{27} -0.607142 q^{29} +1.71511 q^{30} +1.71511 q^{31} -2.67792 q^{32} -11.0374 q^{33} -0.318302 q^{34} -15.6445 q^{36} -1.55361 q^{37} -0.748524 q^{38} -2.03736 q^{40} -9.17783 q^{41} +1.23136 q^{43} +6.46667 q^{44} -17.9343 q^{45} -0.193997 q^{46} -1.62817 q^{47} +12.2315 q^{48} +0.00486525 q^{50} +4.57069 q^{51} +8.39607 q^{53} -3.87192 q^{54} +7.41314 q^{55} +10.7485 q^{57} +0.140469 q^{58} +8.82234 q^{59} +14.4295 q^{60} -5.46667 q^{61} -0.396810 q^{62} -6.74383 q^{64} +2.55361 q^{66} +10.1857 q^{67} -2.67792 q^{68} +2.78572 q^{69} +5.21428 q^{71} +7.33859 q^{72} -3.96355 q^{73} +0.359445 q^{74} -0.0698632 q^{75} -6.29744 q^{76} +6.45051 q^{79} -8.21520 q^{80} +31.4871 q^{81} +2.12339 q^{82} -4.64055 q^{83} -3.06986 q^{85} -0.284889 q^{86} -2.01708 q^{87} -3.03341 q^{88} +9.12826 q^{89} +4.14929 q^{90} -1.63212 q^{92} +5.69803 q^{93} +0.376695 q^{94} -7.21915 q^{95} -8.89672 q^{96} -15.3589 q^{97} -26.7022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} + 5 q^{4} - 7 q^{5} - 5 q^{6} + 6 q^{8} + 7 q^{9} + 11 q^{10} + q^{11} - 12 q^{12} - 3 q^{15} + 19 q^{16} + 4 q^{17} + 3 q^{18} + q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - 3 q^{24}+ \cdots - 23 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.231361 −0.163597 −0.0817984 − 0.996649i \(-0.526066\pi\)
−0.0817984 + 0.996649i \(0.526066\pi\)
\(3\) 3.32225 1.91810 0.959052 − 0.283231i \(-0.0914062\pi\)
0.959052 + 0.283231i \(0.0914062\pi\)
\(4\) −1.94647 −0.973236
\(5\) −2.23136 −0.997895 −0.498947 − 0.866632i \(-0.666280\pi\)
−0.498947 + 0.866632i \(0.666280\pi\)
\(6\) −0.768639 −0.313796
\(7\) 0 0
\(8\) 0.913059 0.322815
\(9\) 8.03736 2.67912
\(10\) 0.516249 0.163252
\(11\) −3.32225 −1.00170 −0.500848 − 0.865535i \(-0.666979\pi\)
−0.500848 + 0.865535i \(0.666979\pi\)
\(12\) −6.46667 −1.86677
\(13\) 0 0
\(14\) 0 0
\(15\) −7.41314 −1.91407
\(16\) 3.68170 0.920425
\(17\) 1.37578 0.333676 0.166838 − 0.985984i \(-0.446644\pi\)
0.166838 + 0.985984i \(0.446644\pi\)
\(18\) −1.85953 −0.438296
\(19\) 3.23531 0.742231 0.371116 − 0.928587i \(-0.378975\pi\)
0.371116 + 0.928587i \(0.378975\pi\)
\(20\) 4.34328 0.971187
\(21\) 0 0
\(22\) 0.768639 0.163874
\(23\) 0.838502 0.174840 0.0874199 − 0.996172i \(-0.472138\pi\)
0.0874199 + 0.996172i \(0.472138\pi\)
\(24\) 3.03341 0.619193
\(25\) −0.0210289 −0.00420577
\(26\) 0 0
\(27\) 16.7354 3.22073
\(28\) 0 0
\(29\) −0.607142 −0.112743 −0.0563717 − 0.998410i \(-0.517953\pi\)
−0.0563717 + 0.998410i \(0.517953\pi\)
\(30\) 1.71511 0.313135
\(31\) 1.71511 0.308043 0.154022 − 0.988067i \(-0.450777\pi\)
0.154022 + 0.988067i \(0.450777\pi\)
\(32\) −2.67792 −0.473394
\(33\) −11.0374 −1.92136
\(34\) −0.318302 −0.0545883
\(35\) 0 0
\(36\) −15.6445 −2.60742
\(37\) −1.55361 −0.255413 −0.127706 − 0.991812i \(-0.540761\pi\)
−0.127706 + 0.991812i \(0.540761\pi\)
\(38\) −0.748524 −0.121427
\(39\) 0 0
\(40\) −2.03736 −0.322136
\(41\) −9.17783 −1.43334 −0.716668 − 0.697414i \(-0.754334\pi\)
−0.716668 + 0.697414i \(0.754334\pi\)
\(42\) 0 0
\(43\) 1.23136 0.187781 0.0938904 − 0.995583i \(-0.470070\pi\)
0.0938904 + 0.995583i \(0.470070\pi\)
\(44\) 6.46667 0.974888
\(45\) −17.9343 −2.67348
\(46\) −0.193997 −0.0286032
\(47\) −1.62817 −0.237493 −0.118747 − 0.992925i \(-0.537888\pi\)
−0.118747 + 0.992925i \(0.537888\pi\)
\(48\) 12.2315 1.76547
\(49\) 0 0
\(50\) 0.00486525 0.000688051 0
\(51\) 4.57069 0.640025
\(52\) 0 0
\(53\) 8.39607 1.15329 0.576644 − 0.816995i \(-0.304362\pi\)
0.576644 + 0.816995i \(0.304362\pi\)
\(54\) −3.87192 −0.526901
\(55\) 7.41314 0.999588
\(56\) 0 0
\(57\) 10.7485 1.42368
\(58\) 0.140469 0.0184445
\(59\) 8.82234 1.14857 0.574285 − 0.818655i \(-0.305280\pi\)
0.574285 + 0.818655i \(0.305280\pi\)
\(60\) 14.4295 1.86284
\(61\) −5.46667 −0.699936 −0.349968 − 0.936762i \(-0.613807\pi\)
−0.349968 + 0.936762i \(0.613807\pi\)
\(62\) −0.396810 −0.0503949
\(63\) 0 0
\(64\) −6.74383 −0.842979
\(65\) 0 0
\(66\) 2.55361 0.314328
\(67\) 10.1857 1.24439 0.622193 − 0.782864i \(-0.286242\pi\)
0.622193 + 0.782864i \(0.286242\pi\)
\(68\) −2.67792 −0.324745
\(69\) 2.78572 0.335361
\(70\) 0 0
\(71\) 5.21428 0.618822 0.309411 − 0.950928i \(-0.399868\pi\)
0.309411 + 0.950928i \(0.399868\pi\)
\(72\) 7.33859 0.864861
\(73\) −3.96355 −0.463898 −0.231949 − 0.972728i \(-0.574510\pi\)
−0.231949 + 0.972728i \(0.574510\pi\)
\(74\) 0.359445 0.0417847
\(75\) −0.0698632 −0.00806711
\(76\) −6.29744 −0.722366
\(77\) 0 0
\(78\) 0 0
\(79\) 6.45051 0.725739 0.362869 − 0.931840i \(-0.381797\pi\)
0.362869 + 0.931840i \(0.381797\pi\)
\(80\) −8.21520 −0.918487
\(81\) 31.4871 3.49857
\(82\) 2.12339 0.234489
\(83\) −4.64055 −0.509367 −0.254684 − 0.967024i \(-0.581971\pi\)
−0.254684 + 0.967024i \(0.581971\pi\)
\(84\) 0 0
\(85\) −3.06986 −0.332973
\(86\) −0.284889 −0.0307203
\(87\) −2.01708 −0.216253
\(88\) −3.03341 −0.323363
\(89\) 9.12826 0.967593 0.483797 − 0.875180i \(-0.339257\pi\)
0.483797 + 0.875180i \(0.339257\pi\)
\(90\) 4.14929 0.437373
\(91\) 0 0
\(92\) −1.63212 −0.170160
\(93\) 5.69803 0.590859
\(94\) 0.376695 0.0388531
\(95\) −7.21915 −0.740669
\(96\) −8.89672 −0.908018
\(97\) −15.3589 −1.55946 −0.779729 − 0.626117i \(-0.784643\pi\)
−0.779729 + 0.626117i \(0.784643\pi\)
\(98\) 0 0
\(99\) −26.7022 −2.68367
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.1.2 4
7.6 odd 2 1183.2.a.l.1.2 4
13.4 even 6 637.2.f.i.393.2 8
13.10 even 6 637.2.f.i.295.2 8
13.12 even 2 8281.2.a.bp.1.3 4
91.4 even 6 637.2.h.i.471.3 8
91.10 odd 6 637.2.g.k.373.2 8
91.17 odd 6 637.2.h.h.471.3 8
91.23 even 6 637.2.h.i.165.3 8
91.30 even 6 637.2.g.j.263.2 8
91.34 even 4 1183.2.c.g.337.4 8
91.62 odd 6 91.2.f.c.22.2 ✓ 8
91.69 odd 6 91.2.f.c.29.2 yes 8
91.75 odd 6 637.2.h.h.165.3 8
91.82 odd 6 637.2.g.k.263.2 8
91.83 even 4 1183.2.c.g.337.5 8
91.88 even 6 637.2.g.j.373.2 8
91.90 odd 2 1183.2.a.k.1.3 4
273.62 even 6 819.2.o.h.568.3 8
273.251 even 6 819.2.o.h.757.3 8
364.251 even 6 1456.2.s.q.1121.1 8
364.335 even 6 1456.2.s.q.113.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.2 ✓ 8 91.62 odd 6
91.2.f.c.29.2 yes 8 91.69 odd 6
637.2.f.i.295.2 8 13.10 even 6
637.2.f.i.393.2 8 13.4 even 6
637.2.g.j.263.2 8 91.30 even 6
637.2.g.j.373.2 8 91.88 even 6
637.2.g.k.263.2 8 91.82 odd 6
637.2.g.k.373.2 8 91.10 odd 6
637.2.h.h.165.3 8 91.75 odd 6
637.2.h.h.471.3 8 91.17 odd 6
637.2.h.i.165.3 8 91.23 even 6
637.2.h.i.471.3 8 91.4 even 6
819.2.o.h.568.3 8 273.62 even 6
819.2.o.h.757.3 8 273.251 even 6
1183.2.a.k.1.3 4 91.90 odd 2
1183.2.a.l.1.2 4 7.6 odd 2
1183.2.c.g.337.4 8 91.34 even 4
1183.2.c.g.337.5 8 91.83 even 4
1456.2.s.q.113.1 8 364.335 even 6
1456.2.s.q.1121.1 8 364.251 even 6
8281.2.a.bp.1.3 4 13.12 even 2
8281.2.a.bt.1.2 4 1.1 even 1 trivial