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 = [12] 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 91.2
Root \(0.500000 - 2.42499i\) of defining polynomial
Character \(\chi\) \(=\) 162.91
Dual form 162.2.e.b.73.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.173648 - 0.984808i) q^{2} +(-0.939693 + 0.342020i) q^{4} +(3.10057 + 2.60168i) q^{5} +(0.144365 + 0.0525446i) q^{7} +(0.500000 + 0.866025i) q^{8} +(2.02375 - 3.50524i) q^{10} +(-0.169211 + 0.141985i) q^{11} +(0.103202 - 0.585289i) q^{13} +(0.0266776 - 0.151296i) q^{14} +(0.766044 - 0.642788i) q^{16} +(2.78255 - 4.81952i) q^{17} +(-1.91041 - 3.30893i) q^{19} +(-3.80341 - 1.38433i) q^{20} +(0.169211 + 0.141985i) q^{22} +(-5.50570 + 2.00391i) q^{23} +(1.97651 + 11.2093i) q^{25} -0.594318 q^{26} -0.153630 q^{28} +(-0.129880 - 0.736585i) q^{29} +(-4.77702 + 1.73869i) q^{31} +(-0.766044 - 0.642788i) q^{32} +(-5.22949 - 1.90338i) q^{34} +(0.310909 + 0.538510i) q^{35} +(-1.87388 + 3.24566i) q^{37} +(-2.92692 + 2.45598i) q^{38} +(-0.702841 + 3.98601i) q^{40} +(-0.690156 + 3.91407i) q^{41} +(7.81896 - 6.56088i) q^{43} +(0.110445 - 0.191296i) q^{44} +(2.92952 + 5.07408i) q^{46} +(-0.447025 - 0.162704i) q^{47} +(-5.34423 - 4.48434i) q^{49} +(10.6958 - 3.89296i) q^{50} +(0.103202 + 0.585289i) q^{52} -3.29955 q^{53} -0.894051 q^{55} +(0.0266776 + 0.151296i) q^{56} +(-0.702841 + 0.255813i) q^{58} +(-5.57221 - 4.67564i) q^{59} +(-3.16654 - 1.15253i) q^{61} +(2.54180 + 4.40252i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(1.84272 - 1.54623i) q^{65} +(1.29509 - 7.34481i) q^{67} +(-0.966370 + 5.48056i) q^{68} +(0.476340 - 0.399697i) q^{70} +(1.42889 - 2.47490i) q^{71} +(0.638922 + 1.10665i) q^{73} +(3.52175 + 1.28181i) q^{74} +(2.92692 + 2.45598i) q^{76} +(-0.0318887 + 0.0116066i) q^{77} +(-0.574268 - 3.25684i) q^{79} +4.04750 q^{80} +3.97445 q^{82} +(1.43627 + 8.14551i) q^{83} +(21.1664 - 7.70392i) q^{85} +(-7.81896 - 6.56088i) q^{86} +(-0.207568 - 0.0755487i) q^{88} +(-2.47882 - 4.29345i) q^{89} +(0.0456525 - 0.0790725i) q^{91} +(4.48829 - 3.76612i) q^{92} +(-0.0826070 + 0.468487i) q^{94} +(2.68543 - 15.2298i) q^{95} +(4.33127 - 3.63437i) q^{97} +(-3.48820 + 6.04174i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{5} - 3 q^{7} + 6 q^{8} - 3 q^{10} + 12 q^{11} + 12 q^{13} + 3 q^{14} + 6 q^{17} - 9 q^{19} - 6 q^{20} - 12 q^{22} - 30 q^{23} - 9 q^{25} - 18 q^{26} + 12 q^{28} - 15 q^{29} - 15 q^{34} - 3 q^{35}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\)

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
\(5\) 3.10057 + 2.60168i 1.38661 + 1.16351i 0.966690 + 0.255949i \(0.0823880\pi\)
0.419925 + 0.907559i \(0.362056\pi\)
\(6\) 0 0
\(7\) 0.144365 + 0.0525446i 0.0545648 + 0.0198600i 0.369158 − 0.929366i \(-0.379646\pi\)
−0.314594 + 0.949226i \(0.601868\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) 0 0
\(10\) 2.02375 − 3.50524i 0.639966 − 1.10845i
\(11\) −0.169211 + 0.141985i −0.0510191 + 0.0428101i −0.667941 − 0.744215i \(-0.732824\pi\)
0.616922 + 0.787025i \(0.288380\pi\)
\(12\) 0 0
\(13\) 0.103202 − 0.585289i 0.0286231 − 0.162330i −0.967146 − 0.254222i \(-0.918180\pi\)
0.995769 + 0.0918925i \(0.0292916\pi\)
\(14\) 0.0266776 − 0.151296i 0.00712988 − 0.0404356i
\(15\) 0 0
\(16\) 0.766044 − 0.642788i 0.191511 − 0.160697i
\(17\) 2.78255 − 4.81952i 0.674868 − 1.16891i −0.301639 − 0.953422i \(-0.597534\pi\)
0.976507 − 0.215484i \(-0.0691328\pi\)
\(18\) 0 0
\(19\) −1.91041 − 3.30893i −0.438278 − 0.759120i 0.559279 − 0.828980i \(-0.311078\pi\)
−0.997557 + 0.0698599i \(0.977745\pi\)
\(20\) −3.80341 − 1.38433i −0.850468 − 0.309545i
\(21\) 0 0
\(22\) 0.169211 + 0.141985i 0.0360760 + 0.0302713i
\(23\) −5.50570 + 2.00391i −1.14802 + 0.417844i −0.844803 − 0.535078i \(-0.820282\pi\)
−0.303215 + 0.952922i \(0.598060\pi\)
\(24\) 0 0
\(25\) 1.97651 + 11.2093i 0.395302 + 2.24187i
\(26\) −0.594318 −0.116555
\(27\) 0 0
\(28\) −0.153630 −0.0290333
\(29\) −0.129880 − 0.736585i −0.0241181 − 0.136780i 0.970371 − 0.241619i \(-0.0776785\pi\)
−0.994489 + 0.104839i \(0.966567\pi\)
\(30\) 0 0
\(31\) −4.77702 + 1.73869i −0.857978 + 0.312278i −0.733289 − 0.679917i \(-0.762016\pi\)
−0.124689 + 0.992196i \(0.539793\pi\)
\(32\) −0.766044 − 0.642788i −0.135419 − 0.113630i
\(33\) 0 0
\(34\) −5.22949 − 1.90338i −0.896850 − 0.326427i
\(35\) 0.310909 + 0.538510i 0.0525532 + 0.0910248i
\(36\) 0 0
\(37\) −1.87388 + 3.24566i −0.308065 + 0.533584i −0.977939 − 0.208891i \(-0.933015\pi\)
0.669874 + 0.742474i \(0.266348\pi\)
\(38\) −2.92692 + 2.45598i −0.474809 + 0.398412i
\(39\) 0 0
\(40\) −0.702841 + 3.98601i −0.111129 + 0.630244i
\(41\) −0.690156 + 3.91407i −0.107784 + 0.611275i 0.882287 + 0.470711i \(0.156003\pi\)
−0.990072 + 0.140564i \(0.955109\pi\)
\(42\) 0 0
\(43\) 7.81896 − 6.56088i 1.19238 − 1.00053i 0.192565 − 0.981284i \(-0.438319\pi\)
0.999815 − 0.0192411i \(-0.00612500\pi\)
\(44\) 0.110445 − 0.191296i 0.0166502 − 0.0288390i
\(45\) 0 0
\(46\) 2.92952 + 5.07408i 0.431935 + 0.748133i
\(47\) −0.447025 − 0.162704i −0.0652054 − 0.0237328i 0.309212 − 0.950993i \(-0.399935\pi\)
−0.374417 + 0.927260i \(0.622157\pi\)
\(48\) 0 0
\(49\) −5.34423 − 4.48434i −0.763462 − 0.640620i
\(50\) 10.6958 − 3.89296i 1.51262 − 0.550548i
\(51\) 0 0
\(52\) 0.103202 + 0.585289i 0.0143116 + 0.0811650i
\(53\) −3.29955 −0.453228 −0.226614 − 0.973985i \(-0.572766\pi\)
−0.226614 + 0.973985i \(0.572766\pi\)
\(54\) 0 0
\(55\) −0.894051 −0.120554
\(56\) 0.0266776 + 0.151296i 0.00356494 + 0.0202178i
\(57\) 0 0
\(58\) −0.702841 + 0.255813i −0.0922876 + 0.0335899i
\(59\) −5.57221 − 4.67564i −0.725440 − 0.608716i 0.203444 − 0.979086i \(-0.434786\pi\)
−0.928884 + 0.370370i \(0.879231\pi\)
\(60\) 0 0
\(61\) −3.16654 − 1.15253i −0.405434 − 0.147566i 0.131250 − 0.991349i \(-0.458101\pi\)
−0.536684 + 0.843783i \(0.680323\pi\)
\(62\) 2.54180 + 4.40252i 0.322809 + 0.559121i
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 1.84272 − 1.54623i 0.228561 − 0.191786i
\(66\) 0 0
\(67\) 1.29509 − 7.34481i 0.158220 − 0.897312i −0.797562 − 0.603237i \(-0.793877\pi\)
0.955782 − 0.294075i \(-0.0950115\pi\)
\(68\) −0.966370 + 5.48056i −0.117190 + 0.664615i
\(69\) 0 0
\(70\) 0.476340 − 0.399697i 0.0569335 − 0.0477729i
\(71\) 1.42889 − 2.47490i 0.169578 − 0.293717i −0.768694 − 0.639617i \(-0.779093\pi\)
0.938271 + 0.345900i \(0.112426\pi\)
\(72\) 0 0
\(73\) 0.638922 + 1.10665i 0.0747802 + 0.129523i 0.900991 − 0.433838i \(-0.142841\pi\)
−0.826210 + 0.563362i \(0.809508\pi\)
\(74\) 3.52175 + 1.28181i 0.409395 + 0.149008i
\(75\) 0 0
\(76\) 2.92692 + 2.45598i 0.335740 + 0.281720i
\(77\) −0.0318887 + 0.0116066i −0.00363406 + 0.00132269i
\(78\) 0 0
\(79\) −0.574268 − 3.25684i −0.0646102 − 0.366423i −0.999921 − 0.0125947i \(-0.995991\pi\)
0.935310 − 0.353828i \(-0.115120\pi\)
\(80\) 4.04750 0.452524
\(81\) 0 0
\(82\) 3.97445 0.438904
\(83\) 1.43627 + 8.14551i 0.157651 + 0.894085i 0.956322 + 0.292316i \(0.0944261\pi\)
−0.798670 + 0.601769i \(0.794463\pi\)
\(84\) 0 0
\(85\) 21.1664 − 7.70392i 2.29581 − 0.835608i
\(86\) −7.81896 − 6.56088i −0.843140 − 0.707478i
\(87\) 0 0
\(88\) −0.207568 − 0.0755487i −0.0221269 − 0.00805352i
\(89\) −2.47882 − 4.29345i −0.262755 − 0.455105i 0.704218 − 0.709984i \(-0.251298\pi\)
−0.966973 + 0.254879i \(0.917964\pi\)
\(90\) 0 0
\(91\) 0.0456525 − 0.0790725i 0.00478569 − 0.00828905i
\(92\) 4.48829 − 3.76612i 0.467936 − 0.392645i
\(93\) 0 0
\(94\) −0.0826070 + 0.468487i −0.00852026 + 0.0483208i
\(95\) 2.68543 − 15.2298i 0.275519 − 1.56255i
\(96\) 0 0
\(97\) 4.33127 − 3.63437i 0.439774 − 0.369014i −0.395851 − 0.918315i \(-0.629550\pi\)
0.835625 + 0.549301i \(0.185106\pi\)
\(98\) −3.48820 + 6.04174i −0.352361 + 0.610308i
\(99\) 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.b.91.2 12
3.2 odd 2 54.2.e.b.13.2 ✓ 12
9.2 odd 6 486.2.e.h.109.2 12
9.4 even 3 486.2.e.g.433.2 12
9.5 odd 6 486.2.e.f.433.1 12
9.7 even 3 486.2.e.e.109.1 12
12.11 even 2 432.2.u.b.337.1 12
27.2 odd 18 54.2.e.b.25.2 yes 12
27.4 even 9 1458.2.c.g.487.1 12
27.5 odd 18 1458.2.a.g.1.1 6
27.7 even 9 486.2.e.e.379.1 12
27.11 odd 18 486.2.e.f.55.1 12
27.13 even 9 1458.2.c.g.973.1 12
27.14 odd 18 1458.2.c.f.973.6 12
27.16 even 9 486.2.e.g.55.2 12
27.20 odd 18 486.2.e.h.379.2 12
27.22 even 9 1458.2.a.f.1.6 6
27.23 odd 18 1458.2.c.f.487.6 12
27.25 even 9 inner 162.2.e.b.73.2 12
108.83 even 18 432.2.u.b.241.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.e.b.13.2 ✓ 12 3.2 odd 2
54.2.e.b.25.2 yes 12 27.2 odd 18
162.2.e.b.73.2 12 27.25 even 9 inner
162.2.e.b.91.2 12 1.1 even 1 trivial
432.2.u.b.241.1 12 108.83 even 18
432.2.u.b.337.1 12 12.11 even 2
486.2.e.e.109.1 12 9.7 even 3
486.2.e.e.379.1 12 27.7 even 9
486.2.e.f.55.1 12 27.11 odd 18
486.2.e.f.433.1 12 9.5 odd 6
486.2.e.g.55.2 12 27.16 even 9
486.2.e.g.433.2 12 9.4 even 3
486.2.e.h.109.2 12 9.2 odd 6
486.2.e.h.379.2 12 27.20 odd 18
1458.2.a.f.1.6 6 27.22 even 9
1458.2.a.g.1.1 6 27.5 odd 18
1458.2.c.f.487.6 12 27.23 odd 18
1458.2.c.f.973.6 12 27.14 odd 18
1458.2.c.g.487.1 12 27.4 even 9
1458.2.c.g.973.1 12 27.13 even 9