Properties

Label 888.2.a.g.1.1
Level $888$
Weight $2$
Character 888.1
Self dual yes
Analytic conductor $7.091$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(1,888)] 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("888.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.09071569949\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 888.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.41421 q^{5} +0.828427 q^{7} +1.00000 q^{9} -2.82843 q^{11} +0.828427 q^{13} -3.41421 q^{15} -0.585786 q^{17} -5.65685 q^{19} +0.828427 q^{21} -2.58579 q^{23} +6.65685 q^{25} +1.00000 q^{27} -10.2426 q^{29} -1.17157 q^{31} -2.82843 q^{33} -2.82843 q^{35} -1.00000 q^{37} +0.828427 q^{39} -0.343146 q^{41} +2.82843 q^{43} -3.41421 q^{45} -5.65685 q^{47} -6.31371 q^{49} -0.585786 q^{51} -7.17157 q^{53} +9.65685 q^{55} -5.65685 q^{57} +4.24264 q^{59} -9.31371 q^{61} +0.828427 q^{63} -2.82843 q^{65} -0.828427 q^{67} -2.58579 q^{69} +8.00000 q^{71} +8.00000 q^{73} +6.65685 q^{75} -2.34315 q^{77} +2.34315 q^{79} +1.00000 q^{81} +16.4853 q^{83} +2.00000 q^{85} -10.2426 q^{87} +1.07107 q^{89} +0.686292 q^{91} -1.17157 q^{93} +19.3137 q^{95} +16.1421 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{9} - 4 q^{13} - 4 q^{15} - 4 q^{17} - 4 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 8 q^{31} - 2 q^{37} - 4 q^{39} - 12 q^{41} - 4 q^{45} + 10 q^{49} - 4 q^{51}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 − 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) 0.828427 0.313116 0.156558 − 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 − 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) 0.828427 0.229764 0.114882 − 0.993379i \(-0.463351\pi\)
0.114882 + 0.993379i \(0.463351\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) −0.585786 −0.142074 −0.0710370 − 0.997474i \(-0.522631\pi\)
−0.0710370 + 0.997474i \(0.522631\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 − 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) 0.828427 0.180778
\(22\) 0 0
\(23\) −2.58579 −0.539174 −0.269587 − 0.962976i \(-0.586887\pi\)
−0.269587 + 0.962976i \(0.586887\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.2426 −1.90201 −0.951005 − 0.309175i \(-0.899947\pi\)
−0.951005 + 0.309175i \(0.899947\pi\)
\(30\) 0 0
\(31\) −1.17157 −0.210421 −0.105210 − 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0.828427 0.132655
\(40\) 0 0
\(41\) −0.343146 −0.0535904 −0.0267952 − 0.999641i \(-0.508530\pi\)
−0.0267952 + 0.999641i \(0.508530\pi\)
\(42\) 0 0
\(43\) 2.82843 0.431331 0.215666 − 0.976467i \(-0.430808\pi\)
0.215666 + 0.976467i \(0.430808\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 0 0
\(47\) −5.65685 −0.825137 −0.412568 − 0.910927i \(-0.635368\pi\)
−0.412568 + 0.910927i \(0.635368\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) −0.585786 −0.0820265
\(52\) 0 0
\(53\) −7.17157 −0.985091 −0.492546 − 0.870287i \(-0.663934\pi\)
−0.492546 + 0.870287i \(0.663934\pi\)
\(54\) 0 0
\(55\) 9.65685 1.30213
\(56\) 0 0
\(57\) −5.65685 −0.749269
\(58\) 0 0
\(59\) 4.24264 0.552345 0.276172 − 0.961108i \(-0.410934\pi\)
0.276172 + 0.961108i \(0.410934\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 − 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) 0.828427 0.104372
\(64\) 0 0
\(65\) −2.82843 −0.350823
\(66\) 0 0
\(67\) −0.828427 −0.101208 −0.0506042 − 0.998719i \(-0.516115\pi\)
−0.0506042 + 0.998719i \(0.516115\pi\)
\(68\) 0 0
\(69\) −2.58579 −0.311292
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 6.65685 0.768667
\(76\) 0 0
\(77\) −2.34315 −0.267026
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 − 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.4853 1.80949 0.904747 − 0.425949i \(-0.140060\pi\)
0.904747 + 0.425949i \(0.140060\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −10.2426 −1.09813
\(88\) 0 0
\(89\) 1.07107 0.113533 0.0567665 − 0.998387i \(-0.481921\pi\)
0.0567665 + 0.998387i \(0.481921\pi\)
\(90\) 0 0
\(91\) 0.686292 0.0719429
\(92\) 0 0
\(93\) −1.17157 −0.121486
\(94\) 0 0
\(95\) 19.3137 1.98154
\(96\) 0 0
\(97\) 16.1421 1.63899 0.819493 − 0.573090i \(-0.194255\pi\)
0.819493 + 0.573090i \(0.194255\pi\)
\(98\) 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 888.2.a.g.1.1 ✓ 2
3.2 odd 2 2664.2.a.l.1.2 2
4.3 odd 2 1776.2.a.l.1.1 2
8.3 odd 2 7104.2.a.bn.1.2 2
8.5 even 2 7104.2.a.bh.1.2 2
12.11 even 2 5328.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.g.1.1 ✓ 2 1.1 even 1 trivial
1776.2.a.l.1.1 2 4.3 odd 2
2664.2.a.l.1.2 2 3.2 odd 2
5328.2.a.bi.1.2 2 12.11 even 2
7104.2.a.bh.1.2 2 8.5 even 2
7104.2.a.bn.1.2 2 8.3 odd 2