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Properties

Label 9680.2.a.bt.1.2
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
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] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,2,0,1,0,11,0,0,0,-4,0,1,0,3] 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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.37228 q^{3} +1.00000 q^{5} +3.37228 q^{7} +8.37228 q^{9} -2.00000 q^{13} +3.37228 q^{15} -1.37228 q^{17} +0.627719 q^{19} +11.3723 q^{21} -2.74456 q^{23} +1.00000 q^{25} +18.1168 q^{27} -1.37228 q^{29} -3.37228 q^{31} +3.37228 q^{35} +9.37228 q^{37} -6.74456 q^{39} +11.4891 q^{41} -4.00000 q^{43} +8.37228 q^{45} -2.74456 q^{47} +4.37228 q^{49} -4.62772 q^{51} -4.11684 q^{53} +2.11684 q^{57} +2.74456 q^{59} +5.37228 q^{61} +28.2337 q^{63} -2.00000 q^{65} -8.00000 q^{67} -9.25544 q^{69} -10.1168 q^{71} +15.4891 q^{73} +3.37228 q^{75} -1.25544 q^{79} +35.9783 q^{81} -2.74456 q^{83} -1.37228 q^{85} -4.62772 q^{87} -1.37228 q^{89} -6.74456 q^{91} -11.3723 q^{93} +0.627719 q^{95} -12.7446 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 11 q^{9} - 4 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} + 19 q^{27} + 3 q^{29} - q^{31} + q^{35} + 13 q^{37} - 2 q^{39} - 8 q^{43} + 11 q^{45}+ \cdots - 14 q^{97}+O(q^{100}) \) 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
\(3\) 3.37228 1.94699 0.973494 − 0.228714i \(-0.0734519\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.37228 1.27460 0.637301 − 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 0 0
\(9\) 8.37228 2.79076
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 3.37228 0.870719
\(16\) 0 0
\(17\) −1.37228 −0.332827 −0.166414 − 0.986056i \(-0.553219\pi\)
−0.166414 + 0.986056i \(0.553219\pi\)
\(18\) 0 0
\(19\) 0.627719 0.144009 0.0720043 − 0.997404i \(-0.477060\pi\)
0.0720043 + 0.997404i \(0.477060\pi\)
\(20\) 0 0
\(21\) 11.3723 2.48164
\(22\) 0 0
\(23\) −2.74456 −0.572281 −0.286140 − 0.958188i \(-0.592372\pi\)
−0.286140 + 0.958188i \(0.592372\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 18.1168 3.48659
\(28\) 0 0
\(29\) −1.37228 −0.254826 −0.127413 − 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 − 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.37228 0.570020
\(36\) 0 0
\(37\) 9.37228 1.54079 0.770397 − 0.637565i \(-0.220058\pi\)
0.770397 + 0.637565i \(0.220058\pi\)
\(38\) 0 0
\(39\) −6.74456 −1.07999
\(40\) 0 0
\(41\) 11.4891 1.79430 0.897150 − 0.441726i \(-0.145634\pi\)
0.897150 + 0.441726i \(0.145634\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 8.37228 1.24807
\(46\) 0 0
\(47\) −2.74456 −0.400336 −0.200168 − 0.979762i \(-0.564149\pi\)
−0.200168 + 0.979762i \(0.564149\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) −4.62772 −0.648010
\(52\) 0 0
\(53\) −4.11684 −0.565492 −0.282746 − 0.959195i \(-0.591245\pi\)
−0.282746 + 0.959195i \(0.591245\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.11684 0.280383
\(58\) 0 0
\(59\) 2.74456 0.357312 0.178656 − 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 0 0
\(61\) 5.37228 0.687850 0.343925 − 0.938997i \(-0.388243\pi\)
0.343925 + 0.938997i \(0.388243\pi\)
\(62\) 0 0
\(63\) 28.2337 3.55711
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −9.25544 −1.11422
\(70\) 0 0
\(71\) −10.1168 −1.20065 −0.600324 − 0.799757i \(-0.704962\pi\)
−0.600324 + 0.799757i \(0.704962\pi\)
\(72\) 0 0
\(73\) 15.4891 1.81286 0.906432 − 0.422351i \(-0.138795\pi\)
0.906432 + 0.422351i \(0.138795\pi\)
\(74\) 0 0
\(75\) 3.37228 0.389398
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.25544 −0.141248 −0.0706239 − 0.997503i \(-0.522499\pi\)
−0.0706239 + 0.997503i \(0.522499\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) 0 0
\(83\) −2.74456 −0.301255 −0.150627 − 0.988591i \(-0.548129\pi\)
−0.150627 + 0.988591i \(0.548129\pi\)
\(84\) 0 0
\(85\) −1.37228 −0.148845
\(86\) 0 0
\(87\) −4.62772 −0.496144
\(88\) 0 0
\(89\) −1.37228 −0.145462 −0.0727308 − 0.997352i \(-0.523171\pi\)
−0.0727308 + 0.997352i \(0.523171\pi\)
\(90\) 0 0
\(91\) −6.74456 −0.707022
\(92\) 0 0
\(93\) −11.3723 −1.17925
\(94\) 0 0
\(95\) 0.627719 0.0644026
\(96\) 0 0
\(97\) −12.7446 −1.29401 −0.647007 − 0.762484i \(-0.723980\pi\)
−0.647007 + 0.762484i \(0.723980\pi\)
\(98\) 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 9680.2.a.bt.1.2 2
4.3 odd 2 1210.2.a.r.1.1 2
11.10 odd 2 880.2.a.n.1.2 2
20.19 odd 2 6050.2.a.cb.1.2 2
33.32 even 2 7920.2.a.bq.1.1 2
44.43 even 2 110.2.a.d.1.1 ✓ 2
55.32 even 4 4400.2.b.p.4049.1 4
55.43 even 4 4400.2.b.p.4049.4 4
55.54 odd 2 4400.2.a.bl.1.1 2
88.21 odd 2 3520.2.a.bj.1.1 2
88.43 even 2 3520.2.a.bq.1.2 2
132.131 odd 2 990.2.a.m.1.2 2
220.43 odd 4 550.2.b.f.199.3 4
220.87 odd 4 550.2.b.f.199.2 4
220.219 even 2 550.2.a.n.1.2 2
308.307 odd 2 5390.2.a.bp.1.2 2
660.263 even 4 4950.2.c.bc.199.1 4
660.527 even 4 4950.2.c.bc.199.4 4
660.659 odd 2 4950.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 ✓ 2 44.43 even 2
550.2.a.n.1.2 2 220.219 even 2
550.2.b.f.199.2 4 220.87 odd 4
550.2.b.f.199.3 4 220.43 odd 4
880.2.a.n.1.2 2 11.10 odd 2
990.2.a.m.1.2 2 132.131 odd 2
1210.2.a.r.1.1 2 4.3 odd 2
3520.2.a.bj.1.1 2 88.21 odd 2
3520.2.a.bq.1.2 2 88.43 even 2
4400.2.a.bl.1.1 2 55.54 odd 2
4400.2.b.p.4049.1 4 55.32 even 4
4400.2.b.p.4049.4 4 55.43 even 4
4950.2.a.bw.1.1 2 660.659 odd 2
4950.2.c.bc.199.1 4 660.263 even 4
4950.2.c.bc.199.4 4 660.527 even 4
5390.2.a.bp.1.2 2 308.307 odd 2
6050.2.a.cb.1.2 2 20.19 odd 2
7920.2.a.bq.1.1 2 33.32 even 2
9680.2.a.bt.1.2 2 1.1 even 1 trivial