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Properties

Label 8280.2.a.bf.1.1
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 1.1
Root \(2.56155\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -2.56155 q^{7} +2.00000 q^{11} -3.56155 q^{13} -1.43845 q^{17} -2.00000 q^{19} +1.00000 q^{23} +1.00000 q^{25} +8.12311 q^{29} +0.123106 q^{31} -2.56155 q^{35} +0.561553 q^{37} -4.12311 q^{41} +6.24621 q^{43} +8.68466 q^{47} -0.438447 q^{49} -8.56155 q^{53} +2.00000 q^{55} +1.43845 q^{59} -0.876894 q^{61} -3.56155 q^{65} -7.43845 q^{67} +5.00000 q^{71} +7.56155 q^{73} -5.12311 q^{77} -0.876894 q^{79} -7.68466 q^{83} -1.43845 q^{85} -8.00000 q^{89} +9.12311 q^{91} -2.00000 q^{95} +5.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7} + 4 q^{11} - 3 q^{13} - 7 q^{17} - 4 q^{19} + 2 q^{23} + 2 q^{25} + 8 q^{29} - 8 q^{31} - q^{35} - 3 q^{37} - 4 q^{43} + 5 q^{47} - 5 q^{49} - 13 q^{53} + 4 q^{55} + 7 q^{59} - 10 q^{61}+ \cdots + 2 q^{97}+O(q^{100}) \) 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
\(7\) −2.56155 −0.968176 −0.484088 − 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(8\) 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
\(13\) −3.56155 −0.987797 −0.493899 − 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.43845 −0.348875 −0.174437 − 0.984668i \(-0.555811\pi\)
−0.174437 + 0.984668i \(0.555811\pi\)
\(18\) 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
\(29\) 8.12311 1.50842 0.754211 − 0.656632i \(-0.228019\pi\)
0.754211 + 0.656632i \(0.228019\pi\)
\(30\) 0 0
\(31\) 0.123106 0.0221104 0.0110552 − 0.999939i \(-0.496481\pi\)
0.0110552 + 0.999939i \(0.496481\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.56155 −0.432981
\(36\) 0 0
\(37\) 0.561553 0.0923187 0.0461594 − 0.998934i \(-0.485302\pi\)
0.0461594 + 0.998934i \(0.485302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.12311 −0.643921 −0.321960 − 0.946753i \(-0.604342\pi\)
−0.321960 + 0.946753i \(0.604342\pi\)
\(42\) 0 0
\(43\) 6.24621 0.952538 0.476269 − 0.879300i \(-0.341989\pi\)
0.476269 + 0.879300i \(0.341989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.68466 1.26679 0.633394 − 0.773830i \(-0.281661\pi\)
0.633394 + 0.773830i \(0.281661\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.56155 −1.17602 −0.588010 − 0.808854i \(-0.700088\pi\)
−0.588010 + 0.808854i \(0.700088\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.43845 0.187270 0.0936349 − 0.995607i \(-0.470151\pi\)
0.0936349 + 0.995607i \(0.470151\pi\)
\(60\) 0 0
\(61\) −0.876894 −0.112275 −0.0561374 − 0.998423i \(-0.517878\pi\)
−0.0561374 + 0.998423i \(0.517878\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.56155 −0.441756
\(66\) 0 0
\(67\) −7.43845 −0.908751 −0.454375 − 0.890810i \(-0.650138\pi\)
−0.454375 + 0.890810i \(0.650138\pi\)
\(68\) 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
\(73\) 7.56155 0.885013 0.442506 − 0.896765i \(-0.354089\pi\)
0.442506 + 0.896765i \(0.354089\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −0.876894 −0.0986583 −0.0493292 − 0.998783i \(-0.515708\pi\)
−0.0493292 + 0.998783i \(0.515708\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.68466 −0.843501 −0.421750 − 0.906712i \(-0.638584\pi\)
−0.421750 + 0.906712i \(0.638584\pi\)
\(84\) 0 0
\(85\) −1.43845 −0.156022
\(86\) 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
\(91\) 9.12311 0.956361
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 5.12311 0.520173 0.260086 − 0.965585i \(-0.416249\pi\)
0.260086 + 0.965585i \(0.416249\pi\)
\(98\) 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.1.1 2
3.2 odd 2 920.2.a.e.1.2 ✓ 2
12.11 even 2 1840.2.a.o.1.1 2
15.2 even 4 4600.2.e.n.4049.2 4
15.8 even 4 4600.2.e.n.4049.3 4
15.14 odd 2 4600.2.a.t.1.1 2
24.5 odd 2 7360.2.a.bp.1.1 2
24.11 even 2 7360.2.a.bl.1.2 2
60.59 even 2 9200.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.e.1.2 ✓ 2 3.2 odd 2
1840.2.a.o.1.1 2 12.11 even 2
4600.2.a.t.1.1 2 15.14 odd 2
4600.2.e.n.4049.2 4 15.2 even 4
4600.2.e.n.4049.3 4 15.8 even 4
7360.2.a.bl.1.2 2 24.11 even 2
7360.2.a.bp.1.1 2 24.5 odd 2
8280.2.a.bf.1.1 2 1.1 even 1 trivial
9200.2.a.bq.1.2 2 60.59 even 2