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Properties

Label 888.2.q.e.121.1
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,-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 121.1
Root \(-1.05745 - 1.83156i\) of defining polynomial
Character \(\chi\) \(=\) 888.121
Dual form 888.2.q.e.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-2.05745 - 3.56361i) q^{5} +(-2.29387 - 3.97310i) q^{7} +(-0.500000 + 0.866025i) q^{9} +1.83076 q^{11} +(0.0846199 + 0.146566i) q^{13} +(-2.05745 + 3.56361i) q^{15} +(0.500000 - 0.866025i) q^{17} +(-2.03029 - 3.51656i) q^{19} +(-2.29387 + 3.97310i) q^{21} +0.527166 q^{23} +(-5.96623 + 10.3338i) q^{25} +1.00000 q^{27} +5.22982 q^{29} +6.94567 q^{31} +(-0.915380 - 1.58548i) q^{33} +(-9.43907 + 16.3489i) q^{35} +(-5.84472 + 1.68500i) q^{37} +(0.0846199 - 0.146566i) q^{39} +(4.63547 + 8.02887i) q^{41} -10.6289 q^{43} +4.11491 q^{45} +1.64207 q^{47} +(-7.02369 + 12.1654i) q^{49} -1.00000 q^{51} +(1.82104 - 3.15413i) q^{53} +(-3.76670 - 6.52412i) q^{55} +(-2.03029 + 3.51656i) q^{57} +(5.50312 - 9.53169i) q^{59} +(0.963110 + 1.66816i) q^{61} +4.58774 q^{63} +(0.348203 - 0.603105i) q^{65} +(-0.851324 - 1.47454i) q^{67} +(-0.263583 - 0.456539i) q^{69} +(-3.30359 - 5.72199i) q^{71} -3.89134 q^{73} +11.9325 q^{75} +(-4.19953 - 7.27379i) q^{77} +(-6.49340 - 11.2469i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(2.94567 - 5.10205i) q^{83} -4.11491 q^{85} +(-2.61491 - 4.52915i) q^{87} +(-5.62463 + 9.74215i) q^{89} +(0.388214 - 0.672406i) q^{91} +(-3.47283 - 6.01512i) q^{93} +(-8.35445 + 14.4703i) q^{95} +12.8308 q^{97} +(-0.915380 + 1.58548i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} - 2 q^{7} - 3 q^{9} + 2 q^{11} + 5 q^{13} - 6 q^{15} + 3 q^{17} + 5 q^{19} - 2 q^{21} + 14 q^{23} - 5 q^{25} + 6 q^{27} + 6 q^{29} + 20 q^{31} - q^{33} - 15 q^{35} + 3 q^{37} + 5 q^{39}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −2.05745 − 3.56361i −0.920121 − 1.59370i −0.799225 − 0.601032i \(-0.794757\pi\)
−0.120896 − 0.992665i \(-0.538577\pi\)
\(6\) 0 0
\(7\) −2.29387 − 3.97310i −0.867002 − 1.50169i −0.865046 − 0.501693i \(-0.832711\pi\)
−0.00195578 − 0.999998i \(-0.500623\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.83076 0.551995 0.275997 − 0.961158i \(-0.410992\pi\)
0.275997 + 0.961158i \(0.410992\pi\)
\(12\) 0 0
\(13\) 0.0846199 + 0.146566i 0.0234693 + 0.0406501i 0.877522 − 0.479537i \(-0.159195\pi\)
−0.854052 + 0.520187i \(0.825862\pi\)
\(14\) 0 0
\(15\) −2.05745 + 3.56361i −0.531232 + 0.920121i
\(16\) 0 0
\(17\) 0.500000 − 0.866025i 0.121268 − 0.210042i −0.799000 − 0.601331i \(-0.794637\pi\)
0.920268 + 0.391289i \(0.127971\pi\)
\(18\) 0 0
\(19\) −2.03029 − 3.51656i −0.465780 − 0.806755i 0.533456 − 0.845828i \(-0.320893\pi\)
−0.999236 + 0.0390730i \(0.987559\pi\)
\(20\) 0 0
\(21\) −2.29387 + 3.97310i −0.500564 + 0.867002i
\(22\) 0 0
\(23\) 0.527166 0.109922 0.0549609 − 0.998489i \(-0.482497\pi\)
0.0549609 + 0.998489i \(0.482497\pi\)
\(24\) 0 0
\(25\) −5.96623 + 10.3338i −1.19325 + 2.06676i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.22982 0.971152 0.485576 − 0.874194i \(-0.338610\pi\)
0.485576 + 0.874194i \(0.338610\pi\)
\(30\) 0 0
\(31\) 6.94567 1.24748 0.623739 − 0.781632i \(-0.285613\pi\)
0.623739 + 0.781632i \(0.285613\pi\)
\(32\) 0 0
\(33\) −0.915380 − 1.58548i −0.159347 − 0.275997i
\(34\) 0 0
\(35\) −9.43907 + 16.3489i −1.59549 + 2.76348i
\(36\) 0 0
\(37\) −5.84472 + 1.68500i −0.960866 + 0.277012i
\(38\) 0 0
\(39\) 0.0846199 − 0.146566i 0.0135500 − 0.0234693i
\(40\) 0 0
\(41\) 4.63547 + 8.02887i 0.723939 + 1.25390i 0.959409 + 0.282018i \(0.0910037\pi\)
−0.235470 + 0.971882i \(0.575663\pi\)
\(42\) 0 0
\(43\) −10.6289 −1.62089 −0.810444 − 0.585816i \(-0.800774\pi\)
−0.810444 + 0.585816i \(0.800774\pi\)
\(44\) 0 0
\(45\) 4.11491 0.613414
\(46\) 0 0
\(47\) 1.64207 0.239521 0.119761 − 0.992803i \(-0.461787\pi\)
0.119761 + 0.992803i \(0.461787\pi\)
\(48\) 0 0
\(49\) −7.02369 + 12.1654i −1.00338 + 1.73791i
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 1.82104 − 3.15413i 0.250139 − 0.433253i −0.713425 − 0.700731i \(-0.752857\pi\)
0.963564 + 0.267479i \(0.0861905\pi\)
\(54\) 0 0
\(55\) −3.76670 − 6.52412i −0.507902 − 0.879713i
\(56\) 0 0
\(57\) −2.03029 + 3.51656i −0.268918 + 0.465780i
\(58\) 0 0
\(59\) 5.50312 − 9.53169i 0.716445 − 1.24092i −0.245954 − 0.969282i \(-0.579101\pi\)
0.962399 − 0.271638i \(-0.0875655\pi\)
\(60\) 0 0
\(61\) 0.963110 + 1.66816i 0.123314 + 0.213586i 0.921073 − 0.389391i \(-0.127315\pi\)
−0.797759 + 0.602977i \(0.793981\pi\)
\(62\) 0 0
\(63\) 4.58774 0.578001
\(64\) 0 0
\(65\) 0.348203 − 0.603105i 0.0431893 − 0.0748060i
\(66\) 0 0
\(67\) −0.851324 − 1.47454i −0.104006 − 0.180143i 0.809326 − 0.587360i \(-0.199833\pi\)
−0.913332 + 0.407217i \(0.866499\pi\)
\(68\) 0 0
\(69\) −0.263583 − 0.456539i −0.0317317 − 0.0549609i
\(70\) 0 0
\(71\) −3.30359 − 5.72199i −0.392064 − 0.679076i 0.600657 − 0.799507i \(-0.294906\pi\)
−0.992722 + 0.120431i \(0.961572\pi\)
\(72\) 0 0
\(73\) −3.89134 −0.455446 −0.227723 − 0.973726i \(-0.573128\pi\)
−0.227723 + 0.973726i \(0.573128\pi\)
\(74\) 0 0
\(75\) 11.9325 1.37784
\(76\) 0 0
\(77\) −4.19953 − 7.27379i −0.478581 − 0.828926i
\(78\) 0 0
\(79\) −6.49340 − 11.2469i −0.730564 − 1.26537i −0.956642 − 0.291265i \(-0.905924\pi\)
0.226078 − 0.974109i \(-0.427410\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 2.94567 − 5.10205i 0.323329 − 0.560022i −0.657844 − 0.753154i \(-0.728531\pi\)
0.981173 + 0.193132i \(0.0618646\pi\)
\(84\) 0 0
\(85\) −4.11491 −0.446324
\(86\) 0 0
\(87\) −2.61491 − 4.52915i −0.280348 − 0.485576i
\(88\) 0 0
\(89\) −5.62463 + 9.74215i −0.596210 + 1.03267i 0.397165 + 0.917747i \(0.369994\pi\)
−0.993375 + 0.114918i \(0.963339\pi\)
\(90\) 0 0
\(91\) 0.388214 − 0.672406i 0.0406959 − 0.0704873i
\(92\) 0 0
\(93\) −3.47283 − 6.01512i −0.360116 − 0.623739i
\(94\) 0 0
\(95\) −8.35445 + 14.4703i −0.857148 + 1.48462i
\(96\) 0 0
\(97\) 12.8308 1.30277 0.651383 − 0.758749i \(-0.274189\pi\)
0.651383 + 0.758749i \(0.274189\pi\)
\(98\) 0 0
\(99\) −0.915380 + 1.58548i −0.0919992 + 0.159347i
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.121.1 ✓ 6
3.2 odd 2 2664.2.r.l.1009.3 6
4.3 odd 2 1776.2.q.n.1009.1 6
37.26 even 3 inner 888.2.q.e.433.1 yes 6
111.26 odd 6 2664.2.r.l.433.3 6
148.63 odd 6 1776.2.q.n.433.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.e.121.1 ✓ 6 1.1 even 1 trivial
888.2.q.e.433.1 yes 6 37.26 even 3 inner
1776.2.q.n.433.1 6 148.63 odd 6
1776.2.q.n.1009.1 6 4.3 odd 2
2664.2.r.l.433.3 6 111.26 odd 6
2664.2.r.l.1009.3 6 3.2 odd 2