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Properties

Label 162.2.e.a.127.1
Level $162$
Weight $2$
Character 162.127
Analytic conductor $1.294$
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] = [162,2,Mod(19,162)] 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("162.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.29357651274\)
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 127.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 162.127
Dual form 162.2.e.a.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 + 0.642788i) q^{2} +(0.173648 + 0.984808i) q^{4} +(1.26604 + 0.460802i) q^{5} +(-0.0209445 + 0.118782i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(0.673648 + 1.16679i) q^{10} +(3.49273 - 1.27125i) q^{11} +(-4.64543 + 3.89798i) q^{13} +(-0.0923963 + 0.0775297i) q^{14} +(-0.939693 + 0.342020i) q^{16} +(-2.58512 - 4.47756i) q^{17} +(2.96064 - 5.12797i) q^{19} +(-0.233956 + 1.32683i) q^{20} +(3.49273 + 1.27125i) q^{22} +(0.826352 + 4.68647i) q^{23} +(-2.43969 - 2.04715i) q^{25} -6.06418 q^{26} -0.120615 q^{28} +(-4.55303 - 3.82045i) q^{29} +(-0.875515 - 4.96529i) q^{31} +(-0.939693 - 0.342020i) q^{32} +(0.897804 - 5.09170i) q^{34} +(-0.0812519 + 0.140732i) q^{35} +(-0.145430 - 0.251892i) q^{37} +(5.56418 - 2.02520i) q^{38} +(-1.03209 + 0.866025i) q^{40} +(-4.44356 + 3.72859i) q^{41} +(-0.426022 + 0.155059i) q^{43} +(1.85844 + 3.21891i) q^{44} +(-2.37939 + 4.12122i) q^{46} +(0.134285 - 0.761570i) q^{47} +(6.56418 + 2.38917i) q^{49} +(-0.553033 - 3.13641i) q^{50} +(-4.64543 - 3.89798i) q^{52} +7.29086 q^{53} +5.00774 q^{55} +(-0.0923963 - 0.0775297i) q^{56} +(-1.03209 - 5.85327i) q^{58} +(-1.40033 - 0.509678i) q^{59} +(-0.656574 + 3.72362i) q^{61} +(2.52094 - 4.36640i) q^{62} +(-0.500000 - 0.866025i) q^{64} +(-7.67752 + 2.79439i) q^{65} +(-5.08512 + 4.26692i) q^{67} +(3.96064 - 3.32337i) q^{68} +(-0.152704 + 0.0555796i) q^{70} +(2.87211 + 4.97464i) q^{71} +(-5.20961 + 9.02330i) q^{73} +(0.0505072 - 0.286441i) q^{74} +(5.56418 + 2.02520i) q^{76} +(0.0778483 + 0.441500i) q^{77} +(10.7194 + 8.99465i) q^{79} -1.34730 q^{80} -5.80066 q^{82} +(1.81521 + 1.52314i) q^{83} +(-1.20961 - 6.86002i) q^{85} +(-0.426022 - 0.155059i) q^{86} +(-0.645430 + 3.66041i) q^{88} +(-1.08512 + 1.87949i) q^{89} +(-0.365715 - 0.633436i) q^{91} +(-4.47178 + 1.62760i) q^{92} +(0.592396 - 0.497079i) q^{94} +(6.11128 - 5.12797i) q^{95} +(3.21941 - 1.17177i) q^{97} +(3.49273 + 6.04958i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{10} + 3 q^{11} - 12 q^{13} + 3 q^{14} + 6 q^{17} + 9 q^{19} - 6 q^{20} + 3 q^{22} + 6 q^{23} - 9 q^{25} - 18 q^{26} - 12 q^{28} - 15 q^{29} - 18 q^{31} + 6 q^{34}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.766044 + 0.642788i 0.541675 + 0.454519i
\(3\) 0 0
\(4\) 0.173648 + 0.984808i 0.0868241 + 0.492404i
\(5\) 1.26604 + 0.460802i 0.566192 + 0.206077i 0.609226 − 0.792996i \(-0.291480\pi\)
−0.0430339 + 0.999074i \(0.513702\pi\)
\(6\) 0 0
\(7\) −0.0209445 + 0.118782i −0.00791629 + 0.0448955i −0.988510 − 0.151155i \(-0.951701\pi\)
0.980594 + 0.196051i \(0.0628118\pi\)
\(8\) −0.500000 + 0.866025i −0.176777 + 0.306186i
\(9\) 0 0
\(10\) 0.673648 + 1.16679i 0.213026 + 0.368972i
\(11\) 3.49273 − 1.27125i 1.05310 − 0.383296i 0.243266 − 0.969960i \(-0.421781\pi\)
0.809831 + 0.586664i \(0.199559\pi\)
\(12\) 0 0
\(13\) −4.64543 + 3.89798i −1.28841 + 1.08110i −0.296385 + 0.955069i \(0.595781\pi\)
−0.992026 + 0.126036i \(0.959775\pi\)
\(14\) −0.0923963 + 0.0775297i −0.0246939 + 0.0207207i
\(15\) 0 0
\(16\) −0.939693 + 0.342020i −0.234923 + 0.0855050i
\(17\) −2.58512 − 4.47756i −0.626984 − 1.08597i −0.988154 − 0.153468i \(-0.950956\pi\)
0.361169 − 0.932500i \(-0.382378\pi\)
\(18\) 0 0
\(19\) 2.96064 − 5.12797i 0.679217 − 1.17644i −0.296000 − 0.955188i \(-0.595653\pi\)
0.975217 − 0.221250i \(-0.0710137\pi\)
\(20\) −0.233956 + 1.32683i −0.0523141 + 0.296688i
\(21\) 0 0
\(22\) 3.49273 + 1.27125i 0.744652 + 0.271031i
\(23\) 0.826352 + 4.68647i 0.172306 + 0.977197i 0.941207 + 0.337830i \(0.109693\pi\)
−0.768901 + 0.639368i \(0.779196\pi\)
\(24\) 0 0
\(25\) −2.43969 − 2.04715i −0.487939 − 0.409429i
\(26\) −6.06418 −1.18928
\(27\) 0 0
\(28\) −0.120615 −0.0227940
\(29\) −4.55303 − 3.82045i −0.845477 − 0.709440i 0.113312 − 0.993559i \(-0.463854\pi\)
−0.958789 + 0.284120i \(0.908299\pi\)
\(30\) 0 0
\(31\) −0.875515 − 4.96529i −0.157247 − 0.891793i −0.956703 − 0.291067i \(-0.905990\pi\)
0.799455 − 0.600725i \(-0.205121\pi\)
\(32\) −0.939693 − 0.342020i −0.166116 − 0.0604612i
\(33\) 0 0
\(34\) 0.897804 − 5.09170i 0.153972 − 0.873219i
\(35\) −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 \(0.840944\pi\)
\(38\) 5.56418 − 2.02520i 0.902629 − 0.328530i
\(39\) 0 0
\(40\) −1.03209 + 0.866025i −0.163188 + 0.136931i
\(41\) −4.44356 + 3.72859i −0.693968 + 0.582308i −0.920050 − 0.391800i \(-0.871853\pi\)
0.226082 + 0.974108i \(0.427408\pi\)
\(42\) 0 0
\(43\) −0.426022 + 0.155059i −0.0649678 + 0.0236463i −0.374300 − 0.927308i \(-0.622117\pi\)
0.309332 + 0.950954i \(0.399895\pi\)
\(44\) 1.85844 + 3.21891i 0.280170 + 0.485270i
\(45\) 0 0
\(46\) −2.37939 + 4.12122i −0.350821 + 0.607640i
\(47\) 0.134285 − 0.761570i 0.0195875 − 0.111086i −0.973446 − 0.228915i \(-0.926482\pi\)
0.993034 + 0.117829i \(0.0375933\pi\)
\(48\) 0 0
\(49\) 6.56418 + 2.38917i 0.937740 + 0.341309i
\(50\) −0.553033 − 3.13641i −0.0782107 − 0.443555i
\(51\) 0 0
\(52\) −4.64543 − 3.89798i −0.644205 − 0.540552i
\(53\) 7.29086 1.00148 0.500738 − 0.865599i \(-0.333062\pi\)
0.500738 + 0.865599i \(0.333062\pi\)
\(54\) 0 0
\(55\) 5.00774 0.675244
\(56\) −0.0923963 − 0.0775297i −0.0123470 − 0.0103603i
\(57\) 0 0
\(58\) −1.03209 − 5.85327i −0.135520 − 0.768572i
\(59\) −1.40033 − 0.509678i −0.182307 − 0.0663545i 0.249253 − 0.968438i \(-0.419815\pi\)
−0.431561 + 0.902084i \(0.642037\pi\)
\(60\) 0 0
\(61\) −0.656574 + 3.72362i −0.0840657 + 0.476760i 0.913489 + 0.406864i \(0.133378\pi\)
−0.997554 + 0.0698959i \(0.977733\pi\)
\(62\) 2.52094 − 4.36640i 0.320160 − 0.554534i
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −7.67752 + 2.79439i −0.952279 + 0.346601i
\(66\) 0 0
\(67\) −5.08512 + 4.26692i −0.621247 + 0.521288i −0.898195 − 0.439597i \(-0.855121\pi\)
0.276949 + 0.960885i \(0.410677\pi\)
\(68\) 3.96064 − 3.32337i 0.480298 − 0.403018i
\(69\) 0 0
\(70\) −0.152704 + 0.0555796i −0.0182516 + 0.00664303i
\(71\) 2.87211 + 4.97464i 0.340857 + 0.590381i 0.984592 − 0.174867i \(-0.0559495\pi\)
−0.643735 + 0.765248i \(0.722616\pi\)
\(72\) 0 0
\(73\) −5.20961 + 9.02330i −0.609738 + 1.05610i 0.381545 + 0.924350i \(0.375392\pi\)
−0.991283 + 0.131748i \(0.957941\pi\)
\(74\) 0.0505072 − 0.286441i 0.00587134 − 0.0332980i
\(75\) 0 0
\(76\) 5.56418 + 2.02520i 0.638255 + 0.232306i
\(77\) 0.0778483 + 0.441500i 0.00887164 + 0.0503136i
\(78\) 0 0
\(79\) 10.7194 + 8.99465i 1.20603 + 1.01198i 0.999437 + 0.0335498i \(0.0106812\pi\)
0.206591 + 0.978427i \(0.433763\pi\)
\(80\) −1.34730 −0.150632
\(81\) 0 0
\(82\) −5.80066 −0.640576
\(83\) 1.81521 + 1.52314i 0.199245 + 0.167186i 0.736951 − 0.675946i \(-0.236265\pi\)
−0.537706 + 0.843132i \(0.680709\pi\)
\(84\) 0 0
\(85\) −1.20961 − 6.86002i −0.131200 − 0.744074i
\(86\) −0.426022 − 0.155059i −0.0459391 − 0.0167205i
\(87\) 0 0
\(88\) −0.645430 + 3.66041i −0.0688030 + 0.390201i
\(89\) −1.08512 + 1.87949i −0.115023 + 0.199225i −0.917789 − 0.397069i \(-0.870027\pi\)
0.802766 + 0.596294i \(0.203361\pi\)
\(90\) 0 0
\(91\) −0.365715 − 0.633436i −0.0383373 − 0.0664022i
\(92\) −4.47178 + 1.62760i −0.466215 + 0.169689i
\(93\) 0 0
\(94\) 0.592396 − 0.497079i 0.0611010 − 0.0512698i
\(95\) 6.11128 − 5.12797i 0.627004 − 0.526119i
\(96\) 0 0
\(97\) 3.21941 − 1.17177i 0.326881 − 0.118975i −0.173366 − 0.984857i \(-0.555464\pi\)
0.500248 + 0.865882i \(0.333242\pi\)
\(98\) 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 162.2.e.a.127.1 6
3.2 odd 2 54.2.e.a.43.1 ✓ 6
9.2 odd 6 486.2.e.b.217.1 6
9.4 even 3 486.2.e.a.55.1 6
9.5 odd 6 486.2.e.d.55.1 6
9.7 even 3 486.2.e.c.217.1 6
12.11 even 2 432.2.u.a.97.1 6
27.2 odd 18 1458.2.c.d.973.2 6
27.4 even 9 486.2.e.c.271.1 6
27.5 odd 18 54.2.e.a.49.1 yes 6
27.7 even 9 1458.2.a.d.1.2 3
27.11 odd 18 1458.2.c.d.487.2 6
27.13 even 9 486.2.e.a.433.1 6
27.14 odd 18 486.2.e.d.433.1 6
27.16 even 9 1458.2.c.a.487.2 6
27.20 odd 18 1458.2.a.a.1.2 3
27.22 even 9 inner 162.2.e.a.37.1 6
27.23 odd 18 486.2.e.b.271.1 6
27.25 even 9 1458.2.c.a.973.2 6
108.59 even 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 3.2 odd 2
54.2.e.a.49.1 yes 6 27.5 odd 18
162.2.e.a.37.1 6 27.22 even 9 inner
162.2.e.a.127.1 6 1.1 even 1 trivial
432.2.u.a.49.1 6 108.59 even 18
432.2.u.a.97.1 6 12.11 even 2
486.2.e.a.55.1 6 9.4 even 3
486.2.e.a.433.1 6 27.13 even 9
486.2.e.b.217.1 6 9.2 odd 6
486.2.e.b.271.1 6 27.23 odd 18
486.2.e.c.217.1 6 9.7 even 3
486.2.e.c.271.1 6 27.4 even 9
486.2.e.d.55.1 6 9.5 odd 6
486.2.e.d.433.1 6 27.14 odd 18
1458.2.a.a.1.2 3 27.20 odd 18
1458.2.a.d.1.2 3 27.7 even 9
1458.2.c.a.487.2 6 27.16 even 9
1458.2.c.a.973.2 6 27.25 even 9
1458.2.c.d.487.2 6 27.11 odd 18
1458.2.c.d.973.2 6 27.2 odd 18