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Properties

Label 8281.2.a.z.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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 = [2,3,0,3,0,-5,0,6,4,-5,6,-15,0,0,10,13,-6] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q+0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} -1.47214 q^{8} +2.00000 q^{9} +0.854102 q^{10} +3.00000 q^{11} -4.14590 q^{12} +5.00000 q^{15} +3.14590 q^{16} -7.47214 q^{17} +0.763932 q^{18} -3.00000 q^{19} -4.14590 q^{20} +1.14590 q^{22} -3.76393 q^{23} -3.29180 q^{24} -2.23607 q^{27} -4.47214 q^{29} +1.90983 q^{30} -5.00000 q^{31} +4.14590 q^{32} +6.70820 q^{33} -2.85410 q^{34} -3.70820 q^{36} +8.70820 q^{37} -1.14590 q^{38} -3.29180 q^{40} -4.47214 q^{41} -8.00000 q^{43} -5.56231 q^{44} +4.47214 q^{45} -1.43769 q^{46} -1.47214 q^{47} +7.03444 q^{48} -16.7082 q^{51} +1.47214 q^{53} -0.854102 q^{54} +6.70820 q^{55} -6.70820 q^{57} -1.70820 q^{58} -7.47214 q^{59} -9.27051 q^{60} +3.00000 q^{61} -1.90983 q^{62} -4.70820 q^{64} +2.56231 q^{66} +3.00000 q^{67} +13.8541 q^{68} -8.41641 q^{69} -8.94427 q^{71} -2.94427 q^{72} -10.7082 q^{73} +3.32624 q^{74} +5.56231 q^{76} +10.7082 q^{79} +7.03444 q^{80} -11.0000 q^{81} -1.70820 q^{82} -16.7082 q^{85} -3.05573 q^{86} -10.0000 q^{87} -4.41641 q^{88} +2.23607 q^{89} +1.70820 q^{90} +6.97871 q^{92} -11.1803 q^{93} -0.562306 q^{94} -6.70820 q^{95} +9.27051 q^{96} +17.4164 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9} - 5 q^{10} + 6 q^{11} - 15 q^{12} + 10 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} + 15 q^{30}+ \cdots + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.381966 0.270091 0.135045 − 0.990839i \(-0.456882\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) 2.23607 1.29099 0.645497 − 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) −1.85410 −0.927051
\(5\) 2.23607 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0.854102 0.348686
\(7\) 0 0
\(8\) −1.47214 −0.520479
\(9\) 2.00000 0.666667
\(10\) 0.854102 0.270091
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −4.14590 −1.19682
\(13\) 0 0
\(14\) 0 0
\(15\) 5.00000 1.29099
\(16\) 3.14590 0.786475
\(17\) −7.47214 −1.81226 −0.906130 − 0.423000i \(-0.860977\pi\)
−0.906130 + 0.423000i \(0.860977\pi\)
\(18\) 0.763932 0.180061
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −4.14590 −0.927051
\(21\) 0 0
\(22\) 1.14590 0.244306
\(23\) −3.76393 −0.784834 −0.392417 − 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) −3.29180 −0.671935
\(25\) 0 0
\(26\) 0 0
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 1.90983 0.348686
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 4.14590 0.732898
\(33\) 6.70820 1.16775
\(34\) −2.85410 −0.489474
\(35\) 0 0
\(36\) −3.70820 −0.618034
\(37\) 8.70820 1.43162 0.715810 − 0.698295i \(-0.246058\pi\)
0.715810 + 0.698295i \(0.246058\pi\)
\(38\) −1.14590 −0.185889
\(39\) 0 0
\(40\) −3.29180 −0.520479
\(41\) −4.47214 −0.698430 −0.349215 − 0.937043i \(-0.613552\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −5.56231 −0.838549
\(45\) 4.47214 0.666667
\(46\) −1.43769 −0.211976
\(47\) −1.47214 −0.214733 −0.107367 − 0.994220i \(-0.534242\pi\)
−0.107367 + 0.994220i \(0.534242\pi\)
\(48\) 7.03444 1.01533
\(49\) 0 0
\(50\) 0 0
\(51\) −16.7082 −2.33962
\(52\) 0 0
\(53\) 1.47214 0.202213 0.101107 − 0.994876i \(-0.467762\pi\)
0.101107 + 0.994876i \(0.467762\pi\)
\(54\) −0.854102 −0.116229
\(55\) 6.70820 0.904534
\(56\) 0 0
\(57\) −6.70820 −0.888523
\(58\) −1.70820 −0.224298
\(59\) −7.47214 −0.972789 −0.486395 − 0.873739i \(-0.661688\pi\)
−0.486395 + 0.873739i \(0.661688\pi\)
\(60\) −9.27051 −1.19682
\(61\) 3.00000 0.384111 0.192055 − 0.981384i \(-0.438485\pi\)
0.192055 + 0.981384i \(0.438485\pi\)
\(62\) −1.90983 −0.242549
\(63\) 0 0
\(64\) −4.70820 −0.588525
\(65\) 0 0
\(66\) 2.56231 0.315398
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 13.8541 1.68006
\(69\) −8.41641 −1.01322
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 − 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) −2.94427 −0.346986
\(73\) −10.7082 −1.25330 −0.626650 − 0.779301i \(-0.715575\pi\)
−0.626650 + 0.779301i \(0.715575\pi\)
\(74\) 3.32624 0.386667
\(75\) 0 0
\(76\) 5.56231 0.638040
\(77\) 0 0
\(78\) 0 0
\(79\) 10.7082 1.20477 0.602384 − 0.798207i \(-0.294218\pi\)
0.602384 + 0.798207i \(0.294218\pi\)
\(80\) 7.03444 0.786475
\(81\) −11.0000 −1.22222
\(82\) −1.70820 −0.188640
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −16.7082 −1.81226
\(86\) −3.05573 −0.329508
\(87\) −10.0000 −1.07211
\(88\) −4.41641 −0.470791
\(89\) 2.23607 0.237023 0.118511 − 0.992953i \(-0.462188\pi\)
0.118511 + 0.992953i \(0.462188\pi\)
\(90\) 1.70820 0.180061
\(91\) 0 0
\(92\) 6.97871 0.727581
\(93\) −11.1803 −1.15935
\(94\) −0.562306 −0.0579974
\(95\) −6.70820 −0.688247
\(96\) 9.27051 0.946167
\(97\) 17.4164 1.76837 0.884184 − 0.467139i \(-0.154715\pi\)
0.884184 + 0.467139i \(0.154715\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
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.z.1.1 2
7.2 even 3 1183.2.e.d.508.2 4
7.4 even 3 1183.2.e.d.170.2 4
7.6 odd 2 8281.2.a.ba.1.1 2
13.12 even 2 637.2.a.f.1.2 2
39.38 odd 2 5733.2.a.v.1.1 2
91.12 odd 6 637.2.e.h.508.1 4
91.25 even 6 91.2.e.b.79.1 yes 4
91.38 odd 6 637.2.e.h.79.1 4
91.51 even 6 91.2.e.b.53.1 ✓ 4
91.90 odd 2 637.2.a.e.1.2 2
273.116 odd 6 819.2.j.c.352.2 4
273.233 odd 6 819.2.j.c.235.2 4
273.272 even 2 5733.2.a.w.1.1 2
364.51 odd 6 1456.2.r.j.417.2 4
364.207 odd 6 1456.2.r.j.625.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.1 ✓ 4 91.51 even 6
91.2.e.b.79.1 yes 4 91.25 even 6
637.2.a.e.1.2 2 91.90 odd 2
637.2.a.f.1.2 2 13.12 even 2
637.2.e.h.79.1 4 91.38 odd 6
637.2.e.h.508.1 4 91.12 odd 6
819.2.j.c.235.2 4 273.233 odd 6
819.2.j.c.352.2 4 273.116 odd 6
1183.2.e.d.170.2 4 7.4 even 3
1183.2.e.d.508.2 4 7.2 even 3
1456.2.r.j.417.2 4 364.51 odd 6
1456.2.r.j.625.2 4 364.207 odd 6
5733.2.a.v.1.1 2 39.38 odd 2
5733.2.a.w.1.1 2 273.272 even 2
8281.2.a.z.1.1 2 1.1 even 1 trivial
8281.2.a.ba.1.1 2 7.6 odd 2