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Properties

Label 888.2.q.e.121.3
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 121.3
Root \(0.930403 + 1.61151i\) of defining polynomial
Character \(\chi\) \(=\) 888.121
Dual form 888.2.q.e.433.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-0.0695971 - 0.120546i) q^{5} +(0.199104 + 0.344858i) q^{7} +(-0.500000 + 0.866025i) q^{9} +3.78600 q^{11} +(-0.893001 - 1.54672i) q^{13} +(-0.0695971 + 0.120546i) q^{15} +(0.500000 - 0.866025i) q^{17} +(0.967805 + 1.67629i) q^{19} +(0.199104 - 0.344858i) q^{21} +1.53740 q^{23} +(2.49031 - 4.31335i) q^{25} +1.00000 q^{27} -2.72161 q^{29} +4.92520 q^{31} +(-1.89300 - 3.27877i) q^{33} +(0.0277141 - 0.0480022i) q^{35} +(6.08242 - 0.0647814i) q^{37} +(-0.893001 + 1.54672i) q^{39} +(-5.77632 - 10.0049i) q^{41} +7.22923 q^{43} +0.139194 q^{45} -1.32340 q^{47} +(3.42072 - 5.92485i) q^{49} -1.00000 q^{51} +(0.338298 - 0.585949i) q^{53} +(-0.263495 - 0.456386i) q^{55} +(0.967805 - 1.67629i) q^{57} +(1.49479 - 2.58906i) q^{59} +(-3.48511 - 6.03638i) q^{61} -0.398207 q^{63} +(-0.124301 + 0.215295i) q^{65} +(3.62951 + 6.28649i) q^{67} +(-0.768701 - 1.33143i) q^{69} +(-4.24860 - 7.35879i) q^{71} +0.149606 q^{73} -4.98062 q^{75} +(0.753807 + 1.30563i) q^{77} +(0.952911 + 1.65049i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(0.925197 - 1.60249i) q^{83} -0.139194 q^{85} +(1.36081 + 2.35698i) q^{87} +(-5.08690 + 8.81077i) q^{89} +(0.355600 - 0.615917i) q^{91} +(-2.46260 - 4.26535i) q^{93} +(0.134713 - 0.233329i) q^{95} +14.7860 q^{97} +(-1.89300 + 3.27877i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} - 2 q^{7} - 3 q^{9} + 2 q^{11} + 5 q^{13} - 6 q^{15} + 3 q^{17} + 5 q^{19} - 2 q^{21} + 14 q^{23} - 5 q^{25} + 6 q^{27} + 6 q^{29} + 20 q^{31} - q^{33} - 15 q^{35} + 3 q^{37} + 5 q^{39}+ \cdots - q^{99}+O(q^{100}) \) 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
\(5\) −0.0695971 − 0.120546i −0.0311248 − 0.0539097i 0.850043 − 0.526713i \(-0.176576\pi\)
−0.881168 + 0.472803i \(0.843242\pi\)
\(6\) 0 0
\(7\) 0.199104 + 0.344858i 0.0752541 + 0.130344i 0.901197 − 0.433410i \(-0.142690\pi\)
−0.825943 + 0.563754i \(0.809357\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 3.78600 1.14152 0.570761 − 0.821116i \(-0.306648\pi\)
0.570761 + 0.821116i \(0.306648\pi\)
\(12\) 0 0
\(13\) −0.893001 − 1.54672i −0.247674 − 0.428984i 0.715206 − 0.698914i \(-0.246333\pi\)
−0.962880 + 0.269930i \(0.913000\pi\)
\(14\) 0 0
\(15\) −0.0695971 + 0.120546i −0.0179699 + 0.0311248i
\(16\) 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
\(19\) 0.967805 + 1.67629i 0.222030 + 0.384567i 0.955424 − 0.295237i \(-0.0953985\pi\)
−0.733395 + 0.679803i \(0.762065\pi\)
\(20\) 0 0
\(21\) 0.199104 − 0.344858i 0.0434480 − 0.0752541i
\(22\) 0 0
\(23\) 1.53740 0.320570 0.160285 − 0.987071i \(-0.448759\pi\)
0.160285 + 0.987071i \(0.448759\pi\)
\(24\) 0 0
\(25\) 2.49031 − 4.31335i 0.498062 − 0.862670i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.72161 −0.505391 −0.252695 − 0.967546i \(-0.581317\pi\)
−0.252695 + 0.967546i \(0.581317\pi\)
\(30\) 0 0
\(31\) 4.92520 0.884591 0.442296 − 0.896869i \(-0.354164\pi\)
0.442296 + 0.896869i \(0.354164\pi\)
\(32\) 0 0
\(33\) −1.89300 − 3.27877i −0.329529 − 0.570761i
\(34\) 0 0
\(35\) 0.0277141 − 0.0480022i 0.00468453 − 0.00811385i
\(36\) 0 0
\(37\) 6.08242 − 0.0647814i 0.999943 − 0.0106500i
\(38\) 0 0
\(39\) −0.893001 + 1.54672i −0.142995 + 0.247674i
\(40\) 0 0
\(41\) −5.77632 − 10.0049i −0.902109 − 1.56250i −0.824752 − 0.565495i \(-0.808685\pi\)
−0.0773571 − 0.997003i \(-0.524648\pi\)
\(42\) 0 0
\(43\) 7.22923 1.10245 0.551223 − 0.834358i \(-0.314161\pi\)
0.551223 + 0.834358i \(0.314161\pi\)
\(44\) 0 0
\(45\) 0.139194 0.0207498
\(46\) 0 0
\(47\) −1.32340 −0.193038 −0.0965192 − 0.995331i \(-0.530771\pi\)
−0.0965192 + 0.995331i \(0.530771\pi\)
\(48\) 0 0
\(49\) 3.42072 − 5.92485i 0.488674 − 0.846408i
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 0.338298 − 0.585949i 0.0464688 − 0.0804863i −0.841855 − 0.539703i \(-0.818537\pi\)
0.888324 + 0.459217i \(0.151870\pi\)
\(54\) 0 0
\(55\) −0.263495 − 0.456386i −0.0355296 − 0.0615391i
\(56\) 0 0
\(57\) 0.967805 − 1.67629i 0.128189 − 0.222030i
\(58\) 0 0
\(59\) 1.49479 − 2.58906i 0.194606 − 0.337067i −0.752166 − 0.658974i \(-0.770991\pi\)
0.946771 + 0.321907i \(0.104324\pi\)
\(60\) 0 0
\(61\) −3.48511 − 6.03638i −0.446222 − 0.772879i 0.551914 − 0.833901i \(-0.313898\pi\)
−0.998136 + 0.0610214i \(0.980564\pi\)
\(62\) 0 0
\(63\) −0.398207 −0.0501694
\(64\) 0 0
\(65\) −0.124301 + 0.215295i −0.0154176 + 0.0267040i
\(66\) 0 0
\(67\) 3.62951 + 6.28649i 0.443415 + 0.768017i 0.997940 − 0.0641498i \(-0.0204335\pi\)
−0.554525 + 0.832167i \(0.687100\pi\)
\(68\) 0 0
\(69\) −0.768701 − 1.33143i −0.0925407 − 0.160285i
\(70\) 0 0
\(71\) −4.24860 − 7.35879i −0.504216 − 0.873328i −0.999988 − 0.00487530i \(-0.998448\pi\)
0.495772 − 0.868453i \(-0.334885\pi\)
\(72\) 0 0
\(73\) 0.149606 0.0175101 0.00875505 − 0.999962i \(-0.497213\pi\)
0.00875505 + 0.999962i \(0.497213\pi\)
\(74\) 0 0
\(75\) −4.98062 −0.575113
\(76\) 0 0
\(77\) 0.753807 + 1.30563i 0.0859043 + 0.148791i
\(78\) 0 0
\(79\) 0.952911 + 1.65049i 0.107211 + 0.185695i 0.914639 − 0.404271i \(-0.132475\pi\)
−0.807428 + 0.589965i \(0.799141\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 0.925197 − 1.60249i 0.101554 − 0.175896i −0.810771 − 0.585363i \(-0.800952\pi\)
0.912325 + 0.409467i \(0.134285\pi\)
\(84\) 0 0
\(85\) −0.139194 −0.0150977
\(86\) 0 0
\(87\) 1.36081 + 2.35698i 0.145894 + 0.252695i
\(88\) 0 0
\(89\) −5.08690 + 8.81077i −0.539210 + 0.933939i 0.459737 + 0.888055i \(0.347944\pi\)
−0.998947 + 0.0458841i \(0.985389\pi\)
\(90\) 0 0
\(91\) 0.355600 − 0.615917i 0.0372770 − 0.0645656i
\(92\) 0 0
\(93\) −2.46260 − 4.26535i −0.255360 − 0.442296i
\(94\) 0 0
\(95\) 0.134713 − 0.233329i 0.0138212 − 0.0239391i
\(96\) 0 0
\(97\) 14.7860 1.50129 0.750646 − 0.660705i \(-0.229743\pi\)
0.750646 + 0.660705i \(0.229743\pi\)
\(98\) 0 0
\(99\) −1.89300 + 3.27877i −0.190254 + 0.329529i
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.121.3 ✓ 6
3.2 odd 2 2664.2.r.l.1009.1 6
4.3 odd 2 1776.2.q.n.1009.3 6
37.26 even 3 inner 888.2.q.e.433.3 yes 6
111.26 odd 6 2664.2.r.l.433.1 6
148.63 odd 6 1776.2.q.n.433.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.e.121.3 ✓ 6 1.1 even 1 trivial
888.2.q.e.433.3 yes 6 37.26 even 3 inner
1776.2.q.n.433.3 6 148.63 odd 6
1776.2.q.n.1009.3 6 4.3 odd 2
2664.2.r.l.433.1 6 111.26 odd 6
2664.2.r.l.1009.1 6 3.2 odd 2