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Properties

Label 486.2.e.b.217.1
Level $486$
Weight $2$
Character 486.217
Analytic conductor $3.881$
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] = [486,2,Mod(55,486)] 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("486.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(486, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 486 = 2 \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 486.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.88072953823\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 217.1
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 486.217
Dual form 486.2.e.b.271.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.173648 + 0.984808i) q^{2} +(-0.939693 - 0.342020i) q^{4} +(1.03209 - 0.866025i) q^{5} +(0.113341 - 0.0412527i) q^{7} +(0.500000 - 0.866025i) q^{8} +(0.673648 + 1.16679i) q^{10} +(2.84730 + 2.38917i) q^{11} +(-1.05303 - 5.97205i) q^{13} +(0.0209445 + 0.118782i) q^{14} +(0.766044 + 0.642788i) q^{16} +(2.58512 + 4.47756i) q^{17} +(2.96064 - 5.12797i) q^{19} +(-1.26604 + 0.460802i) q^{20} +(-2.84730 + 2.38917i) q^{22} +(4.47178 + 1.62760i) q^{23} +(-0.553033 + 3.13641i) q^{25} +6.06418 q^{26} -0.120615 q^{28} +(1.03209 - 5.85327i) q^{29} +(4.73783 + 1.72443i) q^{31} +(-0.766044 + 0.642788i) q^{32} +(-4.85844 + 1.76833i) q^{34} +(0.0812519 - 0.140732i) q^{35} +(-0.145430 - 0.251892i) q^{37} +(4.53596 + 3.80612i) q^{38} +(-0.233956 - 1.32683i) q^{40} +(1.00727 + 5.71253i) q^{41} +(0.347296 + 0.291416i) q^{43} +(-1.85844 - 3.21891i) q^{44} +(-2.37939 + 4.12122i) q^{46} +(0.726682 - 0.264490i) q^{47} +(-5.35117 + 4.49016i) q^{49} +(-2.99273 - 1.08926i) q^{50} +(-1.05303 + 5.97205i) q^{52} -7.29086 q^{53} +5.00774 q^{55} +(0.0209445 - 0.118782i) q^{56} +(5.58512 + 2.03282i) q^{58} +(-1.14156 + 0.957882i) q^{59} +(3.55303 - 1.29320i) q^{61} +(-2.52094 + 4.36640i) q^{62} +(-0.500000 - 0.866025i) q^{64} +(-6.25877 - 5.25173i) q^{65} +(-1.15270 - 6.53731i) q^{67} +(-0.897804 - 5.09170i) q^{68} +(0.124485 + 0.104455i) q^{70} +(-2.87211 - 4.97464i) q^{71} +(-5.20961 + 9.02330i) q^{73} +(0.273318 - 0.0994798i) q^{74} +(-4.53596 + 3.80612i) q^{76} +(0.421274 + 0.153331i) q^{77} +(2.42989 - 13.7806i) q^{79} +1.34730 q^{80} -5.80066 q^{82} +(-0.411474 + 2.33359i) q^{83} +(6.54576 + 2.38246i) q^{85} +(-0.347296 + 0.291416i) q^{86} +(3.49273 - 1.27125i) q^{88} +(1.08512 - 1.87949i) q^{89} +(-0.365715 - 0.633436i) q^{91} +(-3.64543 - 3.05888i) q^{92} +(0.134285 + 0.761570i) q^{94} +(-1.38532 - 7.85651i) q^{95} +(-2.62449 - 2.20220i) q^{97} +(-3.49273 - 6.04958i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 15 q^{11} + 6 q^{13} - 3 q^{14} - 6 q^{17} + 9 q^{19} - 3 q^{20} - 15 q^{22} + 12 q^{23} + 9 q^{25} + 18 q^{26} - 12 q^{28} - 3 q^{29} + 9 q^{31} - 21 q^{34}+ \cdots - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/486\mathbb{Z}\right)^\times\).

\(n\) \(245\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.173648 + 0.984808i −0.122788 + 0.696364i
\(3\) 0 0
\(4\) −0.939693 − 0.342020i −0.469846 − 0.171010i
\(5\) 1.03209 − 0.866025i 0.461564 − 0.387298i −0.382142 − 0.924104i \(-0.624813\pi\)
0.843706 + 0.536805i \(0.180369\pi\)
\(6\) 0 0
\(7\) 0.113341 − 0.0412527i 0.0428388 − 0.0155920i −0.320512 − 0.947244i \(-0.603855\pi\)
0.363351 + 0.931652i \(0.381633\pi\)
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) 0 0
\(10\) 0.673648 + 1.16679i 0.213026 + 0.368972i
\(11\) 2.84730 + 2.38917i 0.858492 + 0.720360i 0.961643 − 0.274305i \(-0.0884479\pi\)
−0.103151 + 0.994666i \(0.532892\pi\)
\(12\) 0 0
\(13\) −1.05303 − 5.97205i −0.292059 − 1.65635i −0.678921 − 0.734211i \(-0.737552\pi\)
0.386862 − 0.922137i \(-0.373559\pi\)
\(14\) 0.0209445 + 0.118782i 0.00559766 + 0.0317459i
\(15\) 0 0
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) 2.58512 + 4.47756i 0.626984 + 1.08597i 0.988154 + 0.153468i \(0.0490443\pi\)
−0.361169 + 0.932500i \(0.617622\pi\)
\(18\) 0 0
\(19\) 2.96064 − 5.12797i 0.679217 − 1.17644i −0.296000 − 0.955188i \(-0.595653\pi\)
0.975217 − 0.221250i \(-0.0710137\pi\)
\(20\) −1.26604 + 0.460802i −0.283096 + 0.103039i
\(21\) 0 0
\(22\) −2.84730 + 2.38917i −0.607046 + 0.509372i
\(23\) 4.47178 + 1.62760i 0.932431 + 0.339377i 0.763173 − 0.646195i \(-0.223641\pi\)
0.169258 + 0.985572i \(0.445863\pi\)
\(24\) 0 0
\(25\) −0.553033 + 3.13641i −0.110607 + 0.627282i
\(26\) 6.06418 1.18928
\(27\) 0 0
\(28\) −0.120615 −0.0227940
\(29\) 1.03209 − 5.85327i 0.191654 − 1.08692i −0.725449 − 0.688275i \(-0.758368\pi\)
0.917104 − 0.398649i \(-0.130521\pi\)
\(30\) 0 0
\(31\) 4.73783 + 1.72443i 0.850939 + 0.309716i 0.730423 − 0.682995i \(-0.239323\pi\)
0.120516 + 0.992711i \(0.461545\pi\)
\(32\) −0.766044 + 0.642788i −0.135419 + 0.113630i
\(33\) 0 0
\(34\) −4.85844 + 1.76833i −0.833216 + 0.303266i
\(35\) 0.0812519 − 0.140732i 0.0137341 − 0.0237881i
\(36\) 0 0
\(37\) −0.145430 − 0.251892i −0.0239085 − 0.0414107i 0.853824 − 0.520562i \(-0.174278\pi\)
−0.877732 + 0.479152i \(0.840944\pi\)
\(38\) 4.53596 + 3.80612i 0.735830 + 0.617434i
\(39\) 0 0
\(40\) −0.233956 − 1.32683i −0.0369916 − 0.209790i
\(41\) 1.00727 + 5.71253i 0.157310 + 0.892148i 0.956644 + 0.291261i \(0.0940748\pi\)
−0.799334 + 0.600887i \(0.794814\pi\)
\(42\) 0 0
\(43\) 0.347296 + 0.291416i 0.0529622 + 0.0444406i 0.668885 − 0.743366i \(-0.266772\pi\)
−0.615922 + 0.787807i \(0.711216\pi\)
\(44\) −1.85844 − 3.21891i −0.280170 − 0.485270i
\(45\) 0 0
\(46\) −2.37939 + 4.12122i −0.350821 + 0.607640i
\(47\) 0.726682 − 0.264490i 0.105997 − 0.0385799i −0.288477 − 0.957487i \(-0.593149\pi\)
0.394474 + 0.918907i \(0.370927\pi\)
\(48\) 0 0
\(49\) −5.35117 + 4.49016i −0.764452 + 0.641452i
\(50\) −2.99273 − 1.08926i −0.423235 − 0.154045i
\(51\) 0 0
\(52\) −1.05303 + 5.97205i −0.146029 + 0.828174i
\(53\) −7.29086 −1.00148 −0.500738 − 0.865599i \(-0.666938\pi\)
−0.500738 + 0.865599i \(0.666938\pi\)
\(54\) 0 0
\(55\) 5.00774 0.675244
\(56\) 0.0209445 − 0.118782i 0.00279883 − 0.0158730i
\(57\) 0 0
\(58\) 5.58512 + 2.03282i 0.733362 + 0.266922i
\(59\) −1.14156 + 0.957882i −0.148618 + 0.124706i −0.714065 − 0.700079i \(-0.753148\pi\)
0.565447 + 0.824785i \(0.308704\pi\)
\(60\) 0 0
\(61\) 3.55303 − 1.29320i 0.454919 − 0.165577i −0.104389 − 0.994536i \(-0.533289\pi\)
0.559309 + 0.828959i \(0.311067\pi\)
\(62\) −2.52094 + 4.36640i −0.320160 + 0.554534i
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −6.25877 − 5.25173i −0.776305 − 0.651397i
\(66\) 0 0
\(67\) −1.15270 − 6.53731i −0.140825 − 0.798659i −0.970625 − 0.240598i \(-0.922656\pi\)
0.829800 − 0.558061i \(-0.188455\pi\)
\(68\) −0.897804 − 5.09170i −0.108875 − 0.617459i
\(69\) 0 0
\(70\) 0.124485 + 0.104455i 0.0148788 + 0.0124848i
\(71\) −2.87211 − 4.97464i −0.340857 − 0.590381i 0.643735 − 0.765248i \(-0.277384\pi\)
−0.984592 + 0.174867i \(0.944050\pi\)
\(72\) 0 0
\(73\) −5.20961 + 9.02330i −0.609738 + 1.05610i 0.381545 + 0.924350i \(0.375392\pi\)
−0.991283 + 0.131748i \(0.957941\pi\)
\(74\) 0.273318 − 0.0994798i 0.0317726 − 0.0115643i
\(75\) 0 0
\(76\) −4.53596 + 3.80612i −0.520310 + 0.436592i
\(77\) 0.421274 + 0.153331i 0.0480087 + 0.0174737i
\(78\) 0 0
\(79\) 2.42989 − 13.7806i 0.273384 − 1.55044i −0.470664 − 0.882313i \(-0.655985\pi\)
0.744048 − 0.668126i \(-0.232903\pi\)
\(80\) 1.34730 0.150632
\(81\) 0 0
\(82\) −5.80066 −0.640576
\(83\) −0.411474 + 2.33359i −0.0451652 + 0.256144i −0.999027 − 0.0441014i \(-0.985958\pi\)
0.953862 + 0.300246i \(0.0970687\pi\)
\(84\) 0 0
\(85\) 6.54576 + 2.38246i 0.709987 + 0.258414i
\(86\) −0.347296 + 0.291416i −0.0374499 + 0.0314242i
\(87\) 0 0
\(88\) 3.49273 − 1.27125i 0.372326 − 0.135516i
\(89\) 1.08512 − 1.87949i 0.115023 − 0.199225i −0.802766 − 0.596294i \(-0.796639\pi\)
0.917789 + 0.397069i \(0.129973\pi\)
\(90\) 0 0
\(91\) −0.365715 − 0.633436i −0.0383373 − 0.0664022i
\(92\) −3.64543 − 3.05888i −0.380062 − 0.318910i
\(93\) 0 0
\(94\) 0.134285 + 0.761570i 0.0138505 + 0.0785499i
\(95\) −1.38532 − 7.85651i −0.142130 − 0.806061i
\(96\) 0 0
\(97\) −2.62449 − 2.20220i −0.266476 − 0.223600i 0.499752 − 0.866168i \(-0.333424\pi\)
−0.766228 + 0.642568i \(0.777869\pi\)
\(98\) −3.49273 − 6.04958i −0.352819 − 0.611100i
\(99\) 0 0
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 486.2.e.b.217.1 6
3.2 odd 2 486.2.e.c.217.1 6
9.2 odd 6 486.2.e.a.55.1 6
9.4 even 3 54.2.e.a.43.1 ✓ 6
9.5 odd 6 162.2.e.a.127.1 6
9.7 even 3 486.2.e.d.55.1 6
27.2 odd 18 1458.2.c.a.487.2 6
27.4 even 9 54.2.e.a.49.1 yes 6
27.5 odd 18 486.2.e.a.433.1 6
27.7 even 9 1458.2.c.d.973.2 6
27.11 odd 18 1458.2.a.d.1.2 3
27.13 even 9 inner 486.2.e.b.271.1 6
27.14 odd 18 486.2.e.c.271.1 6
27.16 even 9 1458.2.a.a.1.2 3
27.20 odd 18 1458.2.c.a.973.2 6
27.22 even 9 486.2.e.d.433.1 6
27.23 odd 18 162.2.e.a.37.1 6
27.25 even 9 1458.2.c.d.487.2 6
36.31 odd 6 432.2.u.a.97.1 6
108.31 odd 18 432.2.u.a.49.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.e.a.43.1 ✓ 6 9.4 even 3
54.2.e.a.49.1 yes 6 27.4 even 9
162.2.e.a.37.1 6 27.23 odd 18
162.2.e.a.127.1 6 9.5 odd 6
432.2.u.a.49.1 6 108.31 odd 18
432.2.u.a.97.1 6 36.31 odd 6
486.2.e.a.55.1 6 9.2 odd 6
486.2.e.a.433.1 6 27.5 odd 18
486.2.e.b.217.1 6 1.1 even 1 trivial
486.2.e.b.271.1 6 27.13 even 9 inner
486.2.e.c.217.1 6 3.2 odd 2
486.2.e.c.271.1 6 27.14 odd 18
486.2.e.d.55.1 6 9.7 even 3
486.2.e.d.433.1 6 27.22 even 9
1458.2.a.a.1.2 3 27.16 even 9
1458.2.a.d.1.2 3 27.11 odd 18
1458.2.c.a.487.2 6 27.2 odd 18
1458.2.c.a.973.2 6 27.20 odd 18
1458.2.c.d.487.2 6 27.25 even 9
1458.2.c.d.973.2 6 27.7 even 9