↑

Properties

Label 888.2.q.e.433.2
Level $888$
Weight $2$
Character 888.433
Analytic conductor $7.091$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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(121,888)] 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("888.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-3,0,-6,0,-2,0,-3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1415907.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 4x^{4} - 2x^{3} + 16x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.2
Root \(0.127051 - 0.220059i\) of defining polynomial
Character \(\chi\) \(=\) 888.433
Dual form 888.2.q.e.121.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(-0.872949 + 1.51199i) q^{5} +(1.09477 - 1.89619i) q^{7} +(-0.500000 - 0.866025i) q^{9} -4.61676 q^{11} +(3.30838 - 5.73028i) q^{13} +(-0.872949 - 1.51199i) q^{15} +(0.500000 + 0.866025i) q^{17} +(3.56248 - 6.17040i) q^{19} +(1.09477 + 1.89619i) q^{21} +4.93543 q^{23} +(0.975920 + 1.69034i) q^{25} +1.00000 q^{27} +0.491797 q^{29} -1.87086 q^{31} +(2.30838 - 3.99823i) q^{33} +(1.91135 + 3.31056i) q^{35} +(1.26231 + 5.95034i) q^{37} +(3.30838 + 5.73028i) q^{39} +(4.14084 - 7.17215i) q^{41} -7.60036 q^{43} +1.74590 q^{45} +3.68133 q^{47} +(1.10297 + 1.91040i) q^{49} -1.00000 q^{51} +(2.84067 + 4.92018i) q^{53} +(4.03020 - 6.98051i) q^{55} +(3.56248 + 6.17040i) q^{57} +(-4.49792 - 7.79062i) q^{59} +(4.02200 - 6.96630i) q^{61} -2.18953 q^{63} +(5.77610 + 10.0045i) q^{65} +(3.72182 - 6.44638i) q^{67} +(-2.46772 + 4.27421i) q^{69} +(7.55220 - 13.0808i) q^{71} +13.7417 q^{73} -1.95184 q^{75} +(-5.05428 + 8.75427i) q^{77} +(-3.95951 + 6.85808i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-5.87086 - 10.1686i) q^{83} -1.74590 q^{85} +(-0.245898 + 0.425908i) q^{87} +(4.21153 + 7.29458i) q^{89} +(-7.24381 - 12.5467i) q^{91} +(0.935432 - 1.62022i) q^{93} +(6.21973 + 10.7729i) q^{95} +6.38324 q^{97} +(2.30838 + 3.99823i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} - 2 q^{7} - 3 q^{9} + 2 q^{11} + 5 q^{13} - 6 q^{15} + 3 q^{17} + 5 q^{19} - 2 q^{21} + 14 q^{23} - 5 q^{25} + 6 q^{27} + 6 q^{29} + 20 q^{31} - q^{33} - 15 q^{35} + 3 q^{37} + 5 q^{39}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) −0.872949 + 1.51199i −0.390395 + 0.676184i −0.992502 − 0.122232i \(-0.960995\pi\)
0.602107 + 0.798416i \(0.294328\pi\)
\(6\) 0 0
\(7\) 1.09477 − 1.89619i 0.413783 − 0.716693i −0.581517 − 0.813534i \(-0.697541\pi\)
0.995300 + 0.0968411i \(0.0308739\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −4.61676 −1.39201 −0.696003 − 0.718039i \(-0.745040\pi\)
−0.696003 + 0.718039i \(0.745040\pi\)
\(12\) 0 0
\(13\) 3.30838 − 5.73028i 0.917580 − 1.58930i 0.114500 − 0.993423i \(-0.463473\pi\)
0.803080 − 0.595872i \(-0.203193\pi\)
\(14\) 0 0
\(15\) −0.872949 − 1.51199i −0.225395 − 0.390395i
\(16\) 0 0
\(17\) 0.500000 + 0.866025i 0.121268 + 0.210042i 0.920268 − 0.391289i \(-0.127971\pi\)
−0.799000 + 0.601331i \(0.794637\pi\)
\(18\) 0 0
\(19\) 3.56248 − 6.17040i 0.817290 − 1.41559i −0.0903824 − 0.995907i \(-0.528809\pi\)
0.907672 − 0.419680i \(-0.137858\pi\)
\(20\) 0 0
\(21\) 1.09477 + 1.89619i 0.238898 + 0.413783i
\(22\) 0 0
\(23\) 4.93543 1.02911 0.514554 − 0.857458i \(-0.327957\pi\)
0.514554 + 0.857458i \(0.327957\pi\)
\(24\) 0 0
\(25\) 0.975920 + 1.69034i 0.195184 + 0.338068i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.491797 0.0913243 0.0456622 − 0.998957i \(-0.485460\pi\)
0.0456622 + 0.998957i \(0.485460\pi\)
\(30\) 0 0
\(31\) −1.87086 −0.336017 −0.168009 − 0.985786i \(-0.553734\pi\)
−0.168009 + 0.985786i \(0.553734\pi\)
\(32\) 0 0
\(33\) 2.30838 − 3.99823i 0.401838 − 0.696003i
\(34\) 0 0
\(35\) 1.91135 + 3.31056i 0.323077 + 0.559587i
\(36\) 0 0
\(37\) 1.26231 + 5.95034i 0.207522 + 0.978230i
\(38\) 0 0
\(39\) 3.30838 + 5.73028i 0.529765 + 0.917580i
\(40\) 0 0
\(41\) 4.14084 − 7.17215i 0.646691 − 1.12010i −0.337217 − 0.941427i \(-0.609486\pi\)
0.983908 − 0.178675i \(-0.0571810\pi\)
\(42\) 0 0
\(43\) −7.60036 −1.15904 −0.579522 − 0.814957i \(-0.696761\pi\)
−0.579522 + 0.814957i \(0.696761\pi\)
\(44\) 0 0
\(45\) 1.74590 0.260263
\(46\) 0 0
\(47\) 3.68133 0.536977 0.268489 − 0.963283i \(-0.413476\pi\)
0.268489 + 0.963283i \(0.413476\pi\)
\(48\) 0 0
\(49\) 1.10297 + 1.91040i 0.157567 + 0.272914i
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 2.84067 + 4.92018i 0.390195 + 0.675838i 0.992475 − 0.122447i \(-0.0390741\pi\)
−0.602280 + 0.798285i \(0.705741\pi\)
\(54\) 0 0
\(55\) 4.03020 − 6.98051i 0.543432 − 0.941252i
\(56\) 0 0
\(57\) 3.56248 + 6.17040i 0.471862 + 0.817290i
\(58\) 0 0
\(59\) −4.49792 − 7.79062i −0.585579 − 1.01425i −0.994803 − 0.101818i \(-0.967534\pi\)
0.409224 − 0.912434i \(-0.365799\pi\)
\(60\) 0 0
\(61\) 4.02200 − 6.96630i 0.514964 − 0.891943i −0.484885 − 0.874578i \(-0.661139\pi\)
0.999849 − 0.0173658i \(-0.00552798\pi\)
\(62\) 0 0
\(63\) −2.18953 −0.275855
\(64\) 0 0
\(65\) 5.77610 + 10.0045i 0.716437 + 1.24090i
\(66\) 0 0
\(67\) 3.72182 − 6.44638i 0.454692 − 0.787550i −0.543978 − 0.839099i \(-0.683083\pi\)
0.998670 + 0.0515491i \(0.0164159\pi\)
\(68\) 0 0
\(69\) −2.46772 + 4.27421i −0.297078 + 0.514554i
\(70\) 0 0
\(71\) 7.55220 − 13.0808i 0.896281 − 1.55240i 0.0640691 − 0.997945i \(-0.479592\pi\)
0.832212 − 0.554458i \(-0.187074\pi\)
\(72\) 0 0
\(73\) 13.7417 1.60835 0.804174 − 0.594394i \(-0.202608\pi\)
0.804174 + 0.594394i \(0.202608\pi\)
\(74\) 0 0
\(75\) −1.95184 −0.225379
\(76\) 0 0
\(77\) −5.05428 + 8.75427i −0.575989 + 0.997642i
\(78\) 0 0
\(79\) −3.95951 + 6.85808i −0.445480 + 0.771594i −0.998086 − 0.0618490i \(-0.980300\pi\)
0.552606 + 0.833443i \(0.313634\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −5.87086 − 10.1686i −0.644411 − 1.11615i −0.984437 − 0.175737i \(-0.943769\pi\)
0.340026 − 0.940416i \(-0.389564\pi\)
\(84\) 0 0
\(85\) −1.74590 −0.189369
\(86\) 0 0
\(87\) −0.245898 + 0.425908i −0.0263631 + 0.0456622i
\(88\) 0 0
\(89\) 4.21153 + 7.29458i 0.446421 + 0.773224i 0.998150 − 0.0607993i \(-0.0193650\pi\)
−0.551729 + 0.834024i \(0.686032\pi\)
\(90\) 0 0
\(91\) −7.24381 − 12.5467i −0.759358 − 1.31525i
\(92\) 0 0
\(93\) 0.935432 − 1.62022i 0.0969998 − 0.168009i
\(94\) 0 0
\(95\) 6.21973 + 10.7729i 0.638131 + 1.10528i
\(96\) 0 0
\(97\) 6.38324 0.648120 0.324060 − 0.946037i \(-0.394952\pi\)
0.324060 + 0.946037i \(0.394952\pi\)
\(98\) 0 0
\(99\) 2.30838 + 3.99823i 0.232001 + 0.401838i
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.q.e.433.2 yes 6
3.2 odd 2 2664.2.r.l.433.2 6
4.3 odd 2 1776.2.q.n.433.2 6
37.10 even 3 inner 888.2.q.e.121.2 ✓ 6
111.47 odd 6 2664.2.r.l.1009.2 6
148.47 odd 6 1776.2.q.n.1009.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.e.121.2 ✓ 6 37.10 even 3 inner
888.2.q.e.433.2 yes 6 1.1 even 1 trivial
1776.2.q.n.433.2 6 4.3 odd 2
1776.2.q.n.1009.2 6 148.47 odd 6
2664.2.r.l.433.2 6 3.2 odd 2
2664.2.r.l.1009.2 6 111.47 odd 6