Properties

Label 888.2.a.j.1.1
Level $888$
Weight $2$
Character 888.1
Self dual yes
Analytic conductor $7.091$
Analytic rank $0$
Dimension $3$
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] = [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 = [3,0,3,0,0,0,-1] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.363328\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.50466 q^{5} -3.14134 q^{7} +1.00000 q^{9} +5.14134 q^{11} +3.14134 q^{13} -3.50466 q^{15} +3.08998 q^{17} -1.86799 q^{19} -3.14134 q^{21} +3.63667 q^{23} +7.28267 q^{25} +1.00000 q^{27} +3.50466 q^{29} +0.726656 q^{31} +5.14134 q^{33} +11.0093 q^{35} -1.00000 q^{37} +3.14134 q^{39} +2.00000 q^{41} +5.55602 q^{43} -3.50466 q^{45} +11.7360 q^{47} +2.86799 q^{49} +3.08998 q^{51} +1.68802 q^{53} -18.0187 q^{55} -1.86799 q^{57} +14.2313 q^{59} -6.00000 q^{61} -3.14134 q^{63} -11.0093 q^{65} -3.55602 q^{67} +3.63667 q^{69} -14.0187 q^{71} -10.6974 q^{73} +7.28267 q^{75} -16.1507 q^{77} +7.73599 q^{79} +1.00000 q^{81} -8.87732 q^{83} -10.8294 q^{85} +3.50466 q^{87} +11.6553 q^{89} -9.86799 q^{91} +0.726656 q^{93} +6.54669 q^{95} -10.4626 q^{97} +5.14134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{7} + 3 q^{9} + 7 q^{11} + q^{13} + 3 q^{17} + 7 q^{19} - q^{21} + 13 q^{23} + 5 q^{25} + 3 q^{27} - 2 q^{31} + 7 q^{33} + 12 q^{35} - 3 q^{37} + q^{39} + 6 q^{41} + 4 q^{43} + 10 q^{47}+ \cdots + 7 q^{99}+O(q^{100}) \) 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
\(5\) −3.50466 −1.56733 −0.783667 − 0.621181i \(-0.786653\pi\)
−0.783667 + 0.621181i \(0.786653\pi\)
\(6\) 0 0
\(7\) −3.14134 −1.18731 −0.593657 − 0.804718i \(-0.702316\pi\)
−0.593657 + 0.804718i \(0.702316\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.14134 1.55017 0.775086 − 0.631856i \(-0.217707\pi\)
0.775086 + 0.631856i \(0.217707\pi\)
\(12\) 0 0
\(13\) 3.14134 0.871250 0.435625 − 0.900128i \(-0.356527\pi\)
0.435625 + 0.900128i \(0.356527\pi\)
\(14\) 0 0
\(15\) −3.50466 −0.904900
\(16\) 0 0
\(17\) 3.08998 0.749431 0.374716 − 0.927140i \(-0.377740\pi\)
0.374716 + 0.927140i \(0.377740\pi\)
\(18\) 0 0
\(19\) −1.86799 −0.428547 −0.214273 − 0.976774i \(-0.568738\pi\)
−0.214273 + 0.976774i \(0.568738\pi\)
\(20\) 0 0
\(21\) −3.14134 −0.685496
\(22\) 0 0
\(23\) 3.63667 0.758298 0.379149 − 0.925336i \(-0.376217\pi\)
0.379149 + 0.925336i \(0.376217\pi\)
\(24\) 0 0
\(25\) 7.28267 1.45653
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.50466 0.650800 0.325400 − 0.945576i \(-0.394501\pi\)
0.325400 + 0.945576i \(0.394501\pi\)
\(30\) 0 0
\(31\) 0.726656 0.130511 0.0652557 − 0.997869i \(-0.479214\pi\)
0.0652557 + 0.997869i \(0.479214\pi\)
\(32\) 0 0
\(33\) 5.14134 0.894992
\(34\) 0 0
\(35\) 11.0093 1.86092
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 3.14134 0.503016
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 5.55602 0.847284 0.423642 − 0.905830i \(-0.360751\pi\)
0.423642 + 0.905830i \(0.360751\pi\)
\(44\) 0 0
\(45\) −3.50466 −0.522445
\(46\) 0 0
\(47\) 11.7360 1.71187 0.855935 − 0.517084i \(-0.172983\pi\)
0.855935 + 0.517084i \(0.172983\pi\)
\(48\) 0 0
\(49\) 2.86799 0.409713
\(50\) 0 0
\(51\) 3.08998 0.432684
\(52\) 0 0
\(53\) 1.68802 0.231868 0.115934 − 0.993257i \(-0.463014\pi\)
0.115934 + 0.993257i \(0.463014\pi\)
\(54\) 0 0
\(55\) −18.0187 −2.42964
\(56\) 0 0
\(57\) −1.86799 −0.247422
\(58\) 0 0
\(59\) 14.2313 1.85276 0.926380 − 0.376590i \(-0.122903\pi\)
0.926380 + 0.376590i \(0.122903\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −3.14134 −0.395771
\(64\) 0 0
\(65\) −11.0093 −1.36554
\(66\) 0 0
\(67\) −3.55602 −0.434436 −0.217218 − 0.976123i \(-0.569698\pi\)
−0.217218 + 0.976123i \(0.569698\pi\)
\(68\) 0 0
\(69\) 3.63667 0.437804
\(70\) 0 0
\(71\) −14.0187 −1.66371 −0.831854 − 0.554994i \(-0.812721\pi\)
−0.831854 + 0.554994i \(0.812721\pi\)
\(72\) 0 0
\(73\) −10.6974 −1.25203 −0.626015 − 0.779811i \(-0.715315\pi\)
−0.626015 + 0.779811i \(0.715315\pi\)
\(74\) 0 0
\(75\) 7.28267 0.840931
\(76\) 0 0
\(77\) −16.1507 −1.84054
\(78\) 0 0
\(79\) 7.73599 0.870366 0.435183 − 0.900342i \(-0.356684\pi\)
0.435183 + 0.900342i \(0.356684\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.87732 −0.974413 −0.487206 − 0.873287i \(-0.661984\pi\)
−0.487206 + 0.873287i \(0.661984\pi\)
\(84\) 0 0
\(85\) −10.8294 −1.17461
\(86\) 0 0
\(87\) 3.50466 0.375739
\(88\) 0 0
\(89\) 11.6553 1.23546 0.617731 − 0.786389i \(-0.288052\pi\)
0.617731 + 0.786389i \(0.288052\pi\)
\(90\) 0 0
\(91\) −9.86799 −1.03445
\(92\) 0 0
\(93\) 0.726656 0.0753508
\(94\) 0 0
\(95\) 6.54669 0.671676
\(96\) 0 0
\(97\) −10.4626 −1.06232 −0.531160 − 0.847271i \(-0.678244\pi\)
−0.531160 + 0.847271i \(0.678244\pi\)
\(98\) 0 0
\(99\) 5.14134 0.516724
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.j.1.1 ✓ 3
3.2 odd 2 2664.2.a.o.1.3 3
4.3 odd 2 1776.2.a.r.1.1 3
8.3 odd 2 7104.2.a.bx.1.3 3
8.5 even 2 7104.2.a.br.1.3 3
12.11 even 2 5328.2.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.j.1.1 ✓ 3 1.1 even 1 trivial
1776.2.a.r.1.1 3 4.3 odd 2
2664.2.a.o.1.3 3 3.2 odd 2
5328.2.a.bm.1.3 3 12.11 even 2
7104.2.a.br.1.3 3 8.5 even 2
7104.2.a.bx.1.3 3 8.3 odd 2