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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3420,2,Mod(1189,3420)] 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("3420.1189"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,-1] 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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2629.1
Root \(2.10552 - 1.21562i\) of defining polynomial
Character \(\chi\) \(=\) 3420.2629
Dual form 3420.2.bj.c.1189.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.22225 - 0.248224i) q^{5} -0.663818i q^{7} +1.80905 q^{11} +(1.99526 + 1.15197i) q^{13} +(3.77643 - 2.18033i) q^{17} +(-4.21168 - 1.12329i) q^{19} +(-1.81374 - 1.04716i) q^{23} +(4.87677 + 1.10323i) q^{25} +(-0.974621 + 1.68809i) q^{29} -9.52527 q^{31} +(-0.164775 + 1.47517i) q^{35} -2.97461i q^{37} +(0.247657 + 0.428954i) q^{41} +(6.81715 - 3.93588i) q^{43} +(5.69449 + 3.28772i) q^{47} +6.55935 q^{49} +(-1.99575 - 1.15225i) q^{53} +(-4.02016 - 0.449050i) q^{55} +(-3.88559 - 6.73003i) q^{59} +(-5.36021 + 9.28415i) q^{61} +(-4.14802 - 3.05522i) q^{65} +(3.96984 + 2.29199i) q^{67} +(2.95914 + 5.12538i) q^{71} +(-4.86313 + 2.80773i) q^{73} -1.20088i q^{77} +(-2.99810 - 5.19286i) q^{79} +6.20090i q^{83} +(-8.93338 + 3.90782i) q^{85} +(6.65028 - 11.5186i) q^{89} +(0.764696 - 1.32449i) q^{91} +(9.08057 + 3.54166i) q^{95} +(8.80695 - 5.08470i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.22225 − 0.248224i −0.993819 − 0.111009i
\(6\) 0 0
\(7\) − 0.663818i − 0.250900i −0.992100 − 0.125450i \(-0.959963\pi\)
0.992100 − 0.125450i \(-0.0400374\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.80905 0.545450 0.272725 − 0.962092i \(-0.412075\pi\)
0.272725 + 0.962092i \(0.412075\pi\)
\(12\) 0 0
\(13\) 1.99526 + 1.15197i 0.553386 + 0.319498i 0.750487 − 0.660886i \(-0.229819\pi\)
−0.197100 + 0.980383i \(0.563153\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.77643 − 2.18033i 0.915920 − 0.528807i 0.0335887 − 0.999436i \(-0.489306\pi\)
0.882331 + 0.470629i \(0.155973\pi\)
\(18\) 0 0
\(19\) −4.21168 − 1.12329i −0.966225 − 0.257699i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.81374 − 1.04716i −0.378191 − 0.218349i 0.298840 − 0.954303i \(-0.403400\pi\)
−0.677031 + 0.735955i \(0.736734\pi\)
\(24\) 0 0
\(25\) 4.87677 + 1.10323i 0.975354 + 0.220646i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.974621 + 1.68809i −0.180983 + 0.313471i −0.942215 − 0.335008i \(-0.891261\pi\)
0.761233 + 0.648479i \(0.224594\pi\)
\(30\) 0 0
\(31\) −9.52527 −1.71079 −0.855394 − 0.517977i \(-0.826685\pi\)
−0.855394 + 0.517977i \(0.826685\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.164775 + 1.47517i −0.0278521 + 0.249349i
\(36\) 0 0
\(37\) − 2.97461i − 0.489023i −0.969646 − 0.244511i \(-0.921372\pi\)
0.969646 − 0.244511i \(-0.0786276\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.247657 + 0.428954i 0.0386775 + 0.0669914i 0.884716 − 0.466130i \(-0.154352\pi\)
−0.846039 + 0.533122i \(0.821019\pi\)
\(42\) 0 0
\(43\) 6.81715 − 3.93588i 1.03960 − 0.600216i 0.119884 − 0.992788i \(-0.461748\pi\)
0.919721 + 0.392572i \(0.128415\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.69449 + 3.28772i 0.830627 + 0.479563i 0.854067 − 0.520163i \(-0.174129\pi\)
−0.0234403 + 0.999725i \(0.507462\pi\)
\(48\) 0 0
\(49\) 6.55935 0.937049
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.99575 − 1.15225i −0.274137 − 0.158273i 0.356629 − 0.934246i \(-0.383926\pi\)
−0.630766 + 0.775973i \(0.717259\pi\)
\(54\) 0 0
\(55\) −4.02016 − 0.449050i −0.542079 − 0.0605498i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.88559 − 6.73003i −0.505860 − 0.876176i −0.999977 − 0.00678007i \(-0.997842\pi\)
0.494117 − 0.869396i \(-0.335492\pi\)
\(60\) 0 0
\(61\) −5.36021 + 9.28415i −0.686304 + 1.18871i 0.286721 + 0.958014i \(0.407435\pi\)
−0.973025 + 0.230700i \(0.925899\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.14802 − 3.05522i −0.514499 − 0.378954i
\(66\) 0 0
\(67\) 3.96984 + 2.29199i 0.484993 + 0.280011i 0.722495 − 0.691376i \(-0.242995\pi\)
−0.237502 + 0.971387i \(0.576329\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.95914 + 5.12538i 0.351185 + 0.608270i 0.986457 − 0.164018i \(-0.0524455\pi\)
−0.635272 + 0.772288i \(0.719112\pi\)
\(72\) 0 0
\(73\) −4.86313 + 2.80773i −0.569187 + 0.328620i −0.756824 − 0.653618i \(-0.773250\pi\)
0.187638 + 0.982238i \(0.439917\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.20088i − 0.136853i
\(78\) 0 0
\(79\) −2.99810 − 5.19286i −0.337312 − 0.584242i 0.646614 − 0.762817i \(-0.276185\pi\)
−0.983926 + 0.178575i \(0.942851\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.20090i 0.680638i 0.940310 + 0.340319i \(0.110535\pi\)
−0.940310 + 0.340319i \(0.889465\pi\)
\(84\) 0 0
\(85\) −8.93338 + 3.90782i −0.968961 + 0.423863i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.65028 − 11.5186i 0.704928 − 1.22097i −0.261789 − 0.965125i \(-0.584312\pi\)
0.966717 − 0.255847i \(-0.0823542\pi\)
\(90\) 0 0
\(91\) 0.764696 − 1.32449i 0.0801619 − 0.138844i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.08057 + 3.54166i 0.931646 + 0.363366i
\(96\) 0 0
\(97\) 8.80695 − 5.08470i 0.894211 − 0.516273i 0.0188932 − 0.999822i \(-0.493986\pi\)
0.875317 + 0.483549i \(0.160652\pi\)
\(98\) 0 0
\(99\) 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 3420.2.bj.c.2629.1 20
3.2 odd 2 380.2.r.a.349.9 yes 20
5.4 even 2 inner 3420.2.bj.c.2629.7 20
15.2 even 4 1900.2.i.g.501.2 20
15.8 even 4 1900.2.i.g.501.9 20
15.14 odd 2 380.2.r.a.349.2 yes 20
19.11 even 3 inner 3420.2.bj.c.1189.7 20
57.11 odd 6 380.2.r.a.49.2 ✓ 20
95.49 even 6 inner 3420.2.bj.c.1189.1 20
285.68 even 12 1900.2.i.g.201.9 20
285.182 even 12 1900.2.i.g.201.2 20
285.239 odd 6 380.2.r.a.49.9 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.2 ✓ 20 57.11 odd 6
380.2.r.a.49.9 yes 20 285.239 odd 6
380.2.r.a.349.2 yes 20 15.14 odd 2
380.2.r.a.349.9 yes 20 3.2 odd 2
1900.2.i.g.201.2 20 285.182 even 12
1900.2.i.g.201.9 20 285.68 even 12
1900.2.i.g.501.2 20 15.2 even 4
1900.2.i.g.501.9 20 15.8 even 4
3420.2.bj.c.1189.1 20 95.49 even 6 inner
3420.2.bj.c.1189.7 20 19.11 even 3 inner
3420.2.bj.c.2629.1 20 1.1 even 1 trivial
3420.2.bj.c.2629.7 20 5.4 even 2 inner