Properties

Label 888.2.a.j.1.3
Level $888$
Weight $2$
Character 888.1
Self dual yes
Analytic conductor $7.091$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
Root \(3.12489\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.64002 q^{5} -0.484862 q^{7} +1.00000 q^{9} +2.48486 q^{11} +0.484862 q^{13} +2.64002 q^{15} -7.37466 q^{17} +7.76491 q^{19} -0.484862 q^{21} +7.12489 q^{23} +1.96972 q^{25} +1.00000 q^{27} -2.64002 q^{29} -6.24977 q^{31} +2.48486 q^{33} -1.28005 q^{35} -1.00000 q^{37} +0.484862 q^{39} +2.00000 q^{41} +7.21949 q^{43} +2.64002 q^{45} -7.52982 q^{47} -6.76491 q^{49} -7.37466 q^{51} +12.9844 q^{53} +6.56009 q^{55} +7.76491 q^{57} +1.11021 q^{59} -6.00000 q^{61} -0.484862 q^{63} +1.28005 q^{65} -5.21949 q^{67} +7.12489 q^{69} +10.5601 q^{71} -9.70436 q^{73} +1.96972 q^{75} -1.20482 q^{77} -11.5298 q^{79} +1.00000 q^{81} +13.0450 q^{83} -19.4693 q^{85} -2.64002 q^{87} -9.43521 q^{89} -0.235091 q^{91} -6.24977 q^{93} +20.4995 q^{95} +15.7796 q^{97} +2.48486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{7} + 3 q^{9} + 7 q^{11} + q^{13} + 3 q^{17} + 7 q^{19} - q^{21} + 13 q^{23} + 5 q^{25} + 3 q^{27} - 2 q^{31} + 7 q^{33} + 12 q^{35} - 3 q^{37} + q^{39} + 6 q^{41} + 4 q^{43} + 10 q^{47}+ \cdots + 7 q^{99}+O(q^{100}) \) 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
\(5\) 2.64002 1.18065 0.590327 − 0.807164i \(-0.298999\pi\)
0.590327 + 0.807164i \(0.298999\pi\)
\(6\) 0 0
\(7\) −0.484862 −0.183261 −0.0916303 − 0.995793i \(-0.529208\pi\)
−0.0916303 + 0.995793i \(0.529208\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.48486 0.749214 0.374607 − 0.927184i \(-0.377778\pi\)
0.374607 + 0.927184i \(0.377778\pi\)
\(12\) 0 0
\(13\) 0.484862 0.134477 0.0672383 − 0.997737i \(-0.478581\pi\)
0.0672383 + 0.997737i \(0.478581\pi\)
\(14\) 0 0
\(15\) 2.64002 0.681651
\(16\) 0 0
\(17\) −7.37466 −1.78862 −0.894308 − 0.447451i \(-0.852332\pi\)
−0.894308 + 0.447451i \(0.852332\pi\)
\(18\) 0 0
\(19\) 7.76491 1.78139 0.890696 − 0.454599i \(-0.150217\pi\)
0.890696 + 0.454599i \(0.150217\pi\)
\(20\) 0 0
\(21\) −0.484862 −0.105806
\(22\) 0 0
\(23\) 7.12489 1.48564 0.742821 − 0.669490i \(-0.233488\pi\)
0.742821 + 0.669490i \(0.233488\pi\)
\(24\) 0 0
\(25\) 1.96972 0.393945
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.64002 −0.490240 −0.245120 − 0.969493i \(-0.578827\pi\)
−0.245120 + 0.969493i \(0.578827\pi\)
\(30\) 0 0
\(31\) −6.24977 −1.12249 −0.561246 − 0.827649i \(-0.689678\pi\)
−0.561246 + 0.827649i \(0.689678\pi\)
\(32\) 0 0
\(33\) 2.48486 0.432559
\(34\) 0 0
\(35\) −1.28005 −0.216367
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0.484862 0.0776400
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 7.21949 1.10096 0.550481 − 0.834847i \(-0.314444\pi\)
0.550481 + 0.834847i \(0.314444\pi\)
\(44\) 0 0
\(45\) 2.64002 0.393551
\(46\) 0 0
\(47\) −7.52982 −1.09834 −0.549168 − 0.835712i \(-0.685055\pi\)
−0.549168 + 0.835712i \(0.685055\pi\)
\(48\) 0 0
\(49\) −6.76491 −0.966416
\(50\) 0 0
\(51\) −7.37466 −1.03266
\(52\) 0 0
\(53\) 12.9844 1.78355 0.891773 − 0.452484i \(-0.149462\pi\)
0.891773 + 0.452484i \(0.149462\pi\)
\(54\) 0 0
\(55\) 6.56009 0.884563
\(56\) 0 0
\(57\) 7.76491 1.02849
\(58\) 0 0
\(59\) 1.11021 0.144536 0.0722682 − 0.997385i \(-0.476976\pi\)
0.0722682 + 0.997385i \(0.476976\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −0.484862 −0.0610869
\(64\) 0 0
\(65\) 1.28005 0.158770
\(66\) 0 0
\(67\) −5.21949 −0.637663 −0.318831 − 0.947811i \(-0.603290\pi\)
−0.318831 + 0.947811i \(0.603290\pi\)
\(68\) 0 0
\(69\) 7.12489 0.857735
\(70\) 0 0
\(71\) 10.5601 1.25325 0.626626 − 0.779320i \(-0.284435\pi\)
0.626626 + 0.779320i \(0.284435\pi\)
\(72\) 0 0
\(73\) −9.70436 −1.13581 −0.567905 − 0.823094i \(-0.692246\pi\)
−0.567905 + 0.823094i \(0.692246\pi\)
\(74\) 0 0
\(75\) 1.96972 0.227444
\(76\) 0 0
\(77\) −1.20482 −0.137301
\(78\) 0 0
\(79\) −11.5298 −1.29721 −0.648603 − 0.761127i \(-0.724646\pi\)
−0.648603 + 0.761127i \(0.724646\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.0450 1.43187 0.715935 − 0.698167i \(-0.246001\pi\)
0.715935 + 0.698167i \(0.246001\pi\)
\(84\) 0 0
\(85\) −19.4693 −2.11174
\(86\) 0 0
\(87\) −2.64002 −0.283040
\(88\) 0 0
\(89\) −9.43521 −1.00013 −0.500065 − 0.865988i \(-0.666691\pi\)
−0.500065 + 0.865988i \(0.666691\pi\)
\(90\) 0 0
\(91\) −0.235091 −0.0246442
\(92\) 0 0
\(93\) −6.24977 −0.648071
\(94\) 0 0
\(95\) 20.4995 2.10321
\(96\) 0 0
\(97\) 15.7796 1.60217 0.801087 − 0.598548i \(-0.204255\pi\)
0.801087 + 0.598548i \(0.204255\pi\)
\(98\) 0 0
\(99\) 2.48486 0.249738
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.1.3 ✓ 3
3.2 odd 2 2664.2.a.o.1.1 3
4.3 odd 2 1776.2.a.r.1.3 3
8.3 odd 2 7104.2.a.bx.1.1 3
8.5 even 2 7104.2.a.br.1.1 3
12.11 even 2 5328.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.j.1.3 ✓ 3 1.1 even 1 trivial
1776.2.a.r.1.3 3 4.3 odd 2
2664.2.a.o.1.1 3 3.2 odd 2
5328.2.a.bm.1.1 3 12.11 even 2
7104.2.a.br.1.1 3 8.5 even 2
7104.2.a.bx.1.1 3 8.3 odd 2