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Properties

Label 8280.2.a.bf.1.2
Level $8280$
Weight $2$
Character 8280.1
Self dual yes
Analytic conductor $66.116$
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] = [8280,2,Mod(1,8280)] 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("8280.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,-1,0,0,0,4,0,-3,0,0,0,-7] 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: \(66.1161328736\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 8280.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.56155 q^{7} +2.00000 q^{11} +0.561553 q^{13} -5.56155 q^{17} -2.00000 q^{19} +1.00000 q^{23} +1.00000 q^{25} -0.123106 q^{29} -8.12311 q^{31} +1.56155 q^{35} -3.56155 q^{37} +4.12311 q^{41} -10.2462 q^{43} -3.68466 q^{47} -4.56155 q^{49} -4.43845 q^{53} +2.00000 q^{55} +5.56155 q^{59} -9.12311 q^{61} +0.561553 q^{65} -11.5616 q^{67} +5.00000 q^{71} +3.43845 q^{73} +3.12311 q^{77} -9.12311 q^{79} +4.68466 q^{83} -5.56155 q^{85} -8.00000 q^{89} +0.876894 q^{91} -2.00000 q^{95} -3.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7} + 4 q^{11} - 3 q^{13} - 7 q^{17} - 4 q^{19} + 2 q^{23} + 2 q^{25} + 8 q^{29} - 8 q^{31} - q^{35} - 3 q^{37} - 4 q^{43} + 5 q^{47} - 5 q^{49} - 13 q^{53} + 4 q^{55} + 7 q^{59} - 10 q^{61}+ \cdots + 2 q^{97}+O(q^{100}) \) 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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.56155 0.590211 0.295106 − 0.955465i \(-0.404645\pi\)
0.295106 + 0.955465i \(0.404645\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 − 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.56155 −1.34887 −0.674437 − 0.738332i \(-0.735614\pi\)
−0.674437 + 0.738332i \(0.735614\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.123106 −0.0228601 −0.0114301 − 0.999935i \(-0.503638\pi\)
−0.0114301 + 0.999935i \(0.503638\pi\)
\(30\) 0 0
\(31\) −8.12311 −1.45895 −0.729476 − 0.684006i \(-0.760236\pi\)
−0.729476 + 0.684006i \(0.760236\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.56155 0.263951
\(36\) 0 0
\(37\) −3.56155 −0.585516 −0.292758 − 0.956187i \(-0.594573\pi\)
−0.292758 + 0.956187i \(0.594573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.12311 0.643921 0.321960 − 0.946753i \(-0.395658\pi\)
0.321960 + 0.946753i \(0.395658\pi\)
\(42\) 0 0
\(43\) −10.2462 −1.56253 −0.781266 − 0.624198i \(-0.785426\pi\)
−0.781266 + 0.624198i \(0.785426\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.68466 −0.537463 −0.268731 − 0.963215i \(-0.586604\pi\)
−0.268731 + 0.963215i \(0.586604\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.43845 −0.609668 −0.304834 − 0.952406i \(-0.598601\pi\)
−0.304834 + 0.952406i \(0.598601\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.56155 0.724053 0.362026 − 0.932168i \(-0.382085\pi\)
0.362026 + 0.932168i \(0.382085\pi\)
\(60\) 0 0
\(61\) −9.12311 −1.16809 −0.584047 − 0.811720i \(-0.698532\pi\)
−0.584047 + 0.811720i \(0.698532\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.561553 0.0696521
\(66\) 0 0
\(67\) −11.5616 −1.41247 −0.706234 − 0.707978i \(-0.749607\pi\)
−0.706234 + 0.707978i \(0.749607\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.00000 0.593391 0.296695 − 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) 3.43845 0.402440 0.201220 − 0.979546i \(-0.435509\pi\)
0.201220 + 0.979546i \(0.435509\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.12311 0.355911
\(78\) 0 0
\(79\) −9.12311 −1.02643 −0.513215 − 0.858260i \(-0.671546\pi\)
−0.513215 + 0.858260i \(0.671546\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.68466 0.514208 0.257104 − 0.966384i \(-0.417232\pi\)
0.257104 + 0.966384i \(0.417232\pi\)
\(84\) 0 0
\(85\) −5.56155 −0.603235
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0.876894 0.0919235
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −3.12311 −0.317103 −0.158552 − 0.987351i \(-0.550682\pi\)
−0.158552 + 0.987351i \(0.550682\pi\)
\(98\) 0 0
\(99\) 0 0
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 8280.2.a.bf.1.2 2
3.2 odd 2 920.2.a.e.1.1 ✓ 2
12.11 even 2 1840.2.a.o.1.2 2
15.2 even 4 4600.2.e.n.4049.4 4
15.8 even 4 4600.2.e.n.4049.1 4
15.14 odd 2 4600.2.a.t.1.2 2
24.5 odd 2 7360.2.a.bp.1.2 2
24.11 even 2 7360.2.a.bl.1.1 2
60.59 even 2 9200.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.e.1.1 ✓ 2 3.2 odd 2
1840.2.a.o.1.2 2 12.11 even 2
4600.2.a.t.1.2 2 15.14 odd 2
4600.2.e.n.4049.1 4 15.8 even 4
4600.2.e.n.4049.4 4 15.2 even 4
7360.2.a.bl.1.1 2 24.11 even 2
7360.2.a.bp.1.2 2 24.5 odd 2
8280.2.a.bf.1.2 2 1.1 even 1 trivial
9200.2.a.bq.1.1 2 60.59 even 2