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Properties

Label 8624.2.a.bg.1.2
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
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] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,0,0,0,0,-2,0,-2,0,4,0,-4,0,8,0,0,0,-8,0,-6,0,-2, 0,2,0,8,0,2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 616)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.414214 q^{3} +1.41421 q^{5} -2.82843 q^{9} -1.00000 q^{11} +1.82843 q^{13} +0.585786 q^{15} -2.00000 q^{17} +2.58579 q^{19} -5.41421 q^{23} -3.00000 q^{25} -2.41421 q^{27} +1.00000 q^{29} +9.65685 q^{31} -0.414214 q^{33} -9.07107 q^{37} +0.757359 q^{39} +6.24264 q^{41} +8.00000 q^{43} -4.00000 q^{45} -4.82843 q^{47} -0.828427 q^{51} +2.58579 q^{53} -1.41421 q^{55} +1.07107 q^{57} -4.41421 q^{59} -8.17157 q^{61} +2.58579 q^{65} -8.07107 q^{67} -2.24264 q^{69} +5.75736 q^{71} -3.41421 q^{73} -1.24264 q^{75} -8.07107 q^{79} +7.48528 q^{81} +10.4853 q^{83} -2.82843 q^{85} +0.414214 q^{87} -9.17157 q^{89} +4.00000 q^{93} +3.65685 q^{95} -15.8284 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{11} - 2 q^{13} + 4 q^{15} - 4 q^{17} + 8 q^{19} - 8 q^{23} - 6 q^{25} - 2 q^{27} + 2 q^{29} + 8 q^{31} + 2 q^{33} - 4 q^{37} + 10 q^{39} + 4 q^{41} + 16 q^{43} - 8 q^{45} - 4 q^{47}+ \cdots - 26 q^{97}+O(q^{100}) \) 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
\(3\) 0.414214 0.239146 0.119573 − 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(4\) 0 0
\(5\) 1.41421 0.632456 0.316228 − 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 − 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 2.58579 0.593220 0.296610 − 0.954999i \(-0.404144\pi\)
0.296610 + 0.954999i \(0.404144\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.41421 −1.12894 −0.564471 − 0.825453i \(-0.690920\pi\)
−0.564471 + 0.825453i \(0.690920\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 − 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 9.65685 1.73442 0.867211 − 0.497941i \(-0.165910\pi\)
0.867211 + 0.497941i \(0.165910\pi\)
\(32\) 0 0
\(33\) −0.414214 −0.0721053
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.07107 −1.49127 −0.745637 − 0.666352i \(-0.767855\pi\)
−0.745637 + 0.666352i \(0.767855\pi\)
\(38\) 0 0
\(39\) 0.757359 0.121275
\(40\) 0 0
\(41\) 6.24264 0.974937 0.487468 − 0.873141i \(-0.337920\pi\)
0.487468 + 0.873141i \(0.337920\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −4.00000 −0.596285
\(46\) 0 0
\(47\) −4.82843 −0.704298 −0.352149 − 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.828427 −0.116003
\(52\) 0 0
\(53\) 2.58579 0.355185 0.177593 − 0.984104i \(-0.443169\pi\)
0.177593 + 0.984104i \(0.443169\pi\)
\(54\) 0 0
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) 1.07107 0.141866
\(58\) 0 0
\(59\) −4.41421 −0.574682 −0.287341 − 0.957828i \(-0.592771\pi\)
−0.287341 + 0.957828i \(0.592771\pi\)
\(60\) 0 0
\(61\) −8.17157 −1.04626 −0.523131 − 0.852252i \(-0.675236\pi\)
−0.523131 + 0.852252i \(0.675236\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.58579 0.320727
\(66\) 0 0
\(67\) −8.07107 −0.986038 −0.493019 − 0.870019i \(-0.664107\pi\)
−0.493019 + 0.870019i \(0.664107\pi\)
\(68\) 0 0
\(69\) −2.24264 −0.269982
\(70\) 0 0
\(71\) 5.75736 0.683273 0.341636 − 0.939832i \(-0.389019\pi\)
0.341636 + 0.939832i \(0.389019\pi\)
\(72\) 0 0
\(73\) −3.41421 −0.399603 −0.199802 − 0.979836i \(-0.564030\pi\)
−0.199802 + 0.979836i \(0.564030\pi\)
\(74\) 0 0
\(75\) −1.24264 −0.143488
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.07107 −0.908066 −0.454033 − 0.890985i \(-0.650015\pi\)
−0.454033 + 0.890985i \(0.650015\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) 10.4853 1.15091 0.575455 − 0.817834i \(-0.304825\pi\)
0.575455 + 0.817834i \(0.304825\pi\)
\(84\) 0 0
\(85\) −2.82843 −0.306786
\(86\) 0 0
\(87\) 0.414214 0.0444084
\(88\) 0 0
\(89\) −9.17157 −0.972185 −0.486092 − 0.873907i \(-0.661578\pi\)
−0.486092 + 0.873907i \(0.661578\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 3.65685 0.375185
\(96\) 0 0
\(97\) −15.8284 −1.60713 −0.803567 − 0.595215i \(-0.797067\pi\)
−0.803567 + 0.595215i \(0.797067\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
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 8624.2.a.bg.1.2 2
4.3 odd 2 4312.2.a.u.1.1 2
7.2 even 3 1232.2.q.h.529.1 4
7.4 even 3 1232.2.q.h.177.1 4
7.6 odd 2 8624.2.a.cd.1.1 2
28.11 odd 6 616.2.q.b.177.2 ✓ 4
28.23 odd 6 616.2.q.b.529.2 yes 4
28.27 even 2 4312.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.q.b.177.2 ✓ 4 28.11 odd 6
616.2.q.b.529.2 yes 4 28.23 odd 6
1232.2.q.h.177.1 4 7.4 even 3
1232.2.q.h.529.1 4 7.2 even 3
4312.2.a.m.1.2 2 28.27 even 2
4312.2.a.u.1.1 2 4.3 odd 2
8624.2.a.bg.1.2 2 1.1 even 1 trivial
8624.2.a.cd.1.1 2 7.6 odd 2