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Properties

Label 888.2.q.h.121.2
Level $888$
Weight $2$
Character 888.121
Analytic conductor $7.091$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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(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 = [6,0,3,0,3,0,-3,0,-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: 6.0.50898483.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \) 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 121.2
Root \(0.330560 + 0.572547i\) of defining polynomial
Character \(\chi\) \(=\) 888.121
Dual form 888.2.q.h.433.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(0.169440 + 0.293478i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +4.56292 q^{11} +(0.669440 + 1.15950i) q^{13} +(-0.169440 + 0.293478i) q^{15} +(-0.450900 + 0.780981i) q^{17} +(2.62034 + 4.53856i) q^{19} +(0.500000 - 0.866025i) q^{21} -8.14248 q^{23} +(2.44258 - 4.23067i) q^{25} -1.00000 q^{27} +10.2240 q^{29} +3.90180 q^{31} +(2.28146 + 3.95160i) q^{33} +(0.169440 - 0.293478i) q^{35} +(1.12866 + 5.97713i) q^{37} +(-0.669440 + 1.15950i) q^{39} +(6.18326 + 10.7097i) q^{41} +0.661120 q^{43} -0.338880 q^{45} -8.22404 q^{47} +(3.00000 - 5.19615i) q^{49} -0.901800 q^{51} +(-3.78978 + 6.56409i) q^{53} +(0.773140 + 1.33912i) q^{55} +(-2.62034 + 4.53856i) q^{57} +(-1.94258 + 3.36465i) q^{59} +(-5.56292 - 9.63526i) q^{61} +1.00000 q^{63} +(-0.226860 + 0.392932i) q^{65} +(-3.44258 - 5.96272i) q^{67} +(-4.07124 - 7.05159i) q^{69} +(4.32224 + 7.48634i) q^{71} +13.0276 q^{73} +4.88516 q^{75} +(-2.28146 - 3.95160i) q^{77} +(-5.02214 - 8.69860i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-4.57956 + 7.93203i) q^{83} -0.305602 q^{85} +(5.11202 + 8.85428i) q^{87} +(-3.24068 + 5.61302i) q^{89} +(0.669440 - 1.15950i) q^{91} +(1.95090 + 3.37906i) q^{93} +(-0.887980 + 1.53803i) q^{95} -6.20740 q^{97} +(-2.28146 + 3.95160i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 3 q^{5} - 3 q^{7} - 3 q^{9} - 2 q^{11} + 6 q^{13} - 3 q^{15} + 10 q^{17} + 5 q^{19} + 3 q^{21} - 2 q^{23} - 4 q^{25} - 6 q^{27} + 28 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} + 2 q^{37} - 6 q^{39}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) 0.169440 + 0.293478i 0.0757758 + 0.131248i 0.901423 − 0.432939i \(-0.142523\pi\)
−0.825648 + 0.564186i \(0.809190\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i 0.755929 − 0.654654i \(-0.227186\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 4.56292 1.37577 0.687886 − 0.725819i \(-0.258539\pi\)
0.687886 + 0.725819i \(0.258539\pi\)
\(12\) 0 0
\(13\) 0.669440 + 1.15950i 0.185669 + 0.321588i 0.943802 − 0.330512i \(-0.107221\pi\)
−0.758133 + 0.652100i \(0.773888\pi\)
\(14\) 0 0
\(15\) −0.169440 + 0.293478i −0.0437492 + 0.0757758i
\(16\) 0 0
\(17\) −0.450900 + 0.780981i −0.109359 + 0.189416i −0.915511 − 0.402293i \(-0.868213\pi\)
0.806152 + 0.591709i \(0.201547\pi\)
\(18\) 0 0
\(19\) 2.62034 + 4.53856i 0.601147 + 1.04122i 0.992648 + 0.121040i \(0.0386228\pi\)
−0.391501 + 0.920178i \(0.628044\pi\)
\(20\) 0 0
\(21\) 0.500000 − 0.866025i 0.109109 − 0.188982i
\(22\) 0 0
\(23\) −8.14248 −1.69782 −0.848912 − 0.528534i \(-0.822742\pi\)
−0.848912 + 0.528534i \(0.822742\pi\)
\(24\) 0 0
\(25\) 2.44258 − 4.23067i 0.488516 − 0.846135i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.2240 1.89856 0.949278 − 0.314437i \(-0.101816\pi\)
0.949278 + 0.314437i \(0.101816\pi\)
\(30\) 0 0
\(31\) 3.90180 0.700784 0.350392 − 0.936603i \(-0.386048\pi\)
0.350392 + 0.936603i \(0.386048\pi\)
\(32\) 0 0
\(33\) 2.28146 + 3.95160i 0.397151 + 0.687886i
\(34\) 0 0
\(35\) 0.169440 − 0.293478i 0.0286406 − 0.0496069i
\(36\) 0 0
\(37\) 1.12866 + 5.97713i 0.185550 + 0.982635i
\(38\) 0 0
\(39\) −0.669440 + 1.15950i −0.107196 + 0.185669i
\(40\) 0 0
\(41\) 6.18326 + 10.7097i 0.965663 + 1.67258i 0.707823 + 0.706389i \(0.249677\pi\)
0.257840 + 0.966188i \(0.416989\pi\)
\(42\) 0 0
\(43\) 0.661120 0.100820 0.0504100 − 0.998729i \(-0.483947\pi\)
0.0504100 + 0.998729i \(0.483947\pi\)
\(44\) 0 0
\(45\) −0.338880 −0.0505172
\(46\) 0 0
\(47\) −8.22404 −1.19960 −0.599800 − 0.800150i \(-0.704753\pi\)
−0.599800 + 0.800150i \(0.704753\pi\)
\(48\) 0 0
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) −0.901800 −0.126277
\(52\) 0 0
\(53\) −3.78978 + 6.56409i −0.520566 + 0.901647i 0.479148 + 0.877734i \(0.340946\pi\)
−0.999714 + 0.0239130i \(0.992388\pi\)
\(54\) 0 0
\(55\) 0.773140 + 1.33912i 0.104250 + 0.180567i
\(56\) 0 0
\(57\) −2.62034 + 4.53856i −0.347072 + 0.601147i
\(58\) 0 0
\(59\) −1.94258 + 3.36465i −0.252902 + 0.438040i −0.964324 − 0.264726i \(-0.914718\pi\)
0.711421 + 0.702766i \(0.248052\pi\)
\(60\) 0 0
\(61\) −5.56292 − 9.63526i −0.712259 − 1.23367i −0.964007 − 0.265876i \(-0.914339\pi\)
0.251748 − 0.967793i \(-0.418994\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −0.226860 + 0.392932i −0.0281385 + 0.0487373i
\(66\) 0 0
\(67\) −3.44258 − 5.96272i −0.420578 − 0.728463i 0.575418 − 0.817860i \(-0.304839\pi\)
−0.995996 + 0.0893968i \(0.971506\pi\)
\(68\) 0 0
\(69\) −4.07124 − 7.05159i −0.490120 − 0.848912i
\(70\) 0 0
\(71\) 4.32224 + 7.48634i 0.512956 + 0.888465i 0.999887 + 0.0150250i \(0.00478279\pi\)
−0.486932 + 0.873440i \(0.661884\pi\)
\(72\) 0 0
\(73\) 13.0276 1.52477 0.762385 − 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(74\) 0 0
\(75\) 4.88516 0.564090
\(76\) 0 0
\(77\) −2.28146 − 3.95160i −0.259996 − 0.450327i
\(78\) 0 0
\(79\) −5.02214 − 8.69860i −0.565035 − 0.978669i −0.997046 − 0.0768010i \(-0.975529\pi\)
0.432012 − 0.901868i \(-0.357804\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −4.57956 + 7.93203i −0.502672 + 0.870653i 0.497323 + 0.867565i \(0.334316\pi\)
−0.999995 + 0.00308800i \(0.999017\pi\)
\(84\) 0 0
\(85\) −0.305602 −0.0331471
\(86\) 0 0
\(87\) 5.11202 + 8.85428i 0.548066 + 0.949278i
\(88\) 0 0
\(89\) −3.24068 + 5.61302i −0.343511 + 0.594979i −0.985082 − 0.172085i \(-0.944950\pi\)
0.641571 + 0.767064i \(0.278283\pi\)
\(90\) 0 0
\(91\) 0.669440 − 1.15950i 0.0701764 − 0.121549i
\(92\) 0 0
\(93\) 1.95090 + 3.37906i 0.202299 + 0.350392i
\(94\) 0 0
\(95\) −0.887980 + 1.53803i −0.0911048 + 0.157798i
\(96\) 0 0
\(97\) −6.20740 −0.630266 −0.315133 − 0.949048i \(-0.602049\pi\)
−0.315133 + 0.949048i \(0.602049\pi\)
\(98\) 0 0
\(99\) −2.28146 + 3.95160i −0.229295 + 0.397151i
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.h.121.2 ✓ 6
3.2 odd 2 2664.2.r.h.1009.2 6
4.3 odd 2 1776.2.q.m.1009.2 6
37.26 even 3 inner 888.2.q.h.433.2 yes 6
111.26 odd 6 2664.2.r.h.433.2 6
148.63 odd 6 1776.2.q.m.433.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.h.121.2 ✓ 6 1.1 even 1 trivial
888.2.q.h.433.2 yes 6 37.26 even 3 inner
1776.2.q.m.433.2 6 148.63 odd 6
1776.2.q.m.1009.2 6 4.3 odd 2
2664.2.r.h.433.2 6 111.26 odd 6
2664.2.r.h.1009.2 6 3.2 odd 2