Show commands: Magma / Pari/GP / SageMath

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 1189.7
Root \(-2.10552 - 1.21562i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.c.2629.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.896156 + 2.04863i) q^{5} -0.663818i q^{7} +1.80905 q^{11} +(-1.99526 + 1.15197i) q^{13} +(-3.77643 - 2.18033i) q^{17} +(-4.21168 + 1.12329i) q^{19} +(1.81374 - 1.04716i) q^{23} +(-3.39381 + 3.67179i) q^{25} +(-0.974621 - 1.68809i) q^{29} -9.52527 q^{31} +(1.35992 - 0.594885i) q^{35} -2.97461i q^{37} +(0.247657 - 0.428954i) q^{41} +(-6.81715 - 3.93588i) q^{43} +(-5.69449 + 3.28772i) q^{47} +6.55935 q^{49} +(1.99575 - 1.15225i) q^{53} +(1.62119 + 3.70609i) q^{55} +(-3.88559 + 6.73003i) q^{59} +(-5.36021 - 9.28415i) q^{61} +(-4.14802 - 3.05522i) q^{65} +(-3.96984 + 2.29199i) q^{67} +(2.95914 - 5.12538i) q^{71} +(4.86313 + 2.80773i) q^{73} -1.20088i q^{77} +(-2.99810 + 5.19286i) q^{79} +6.20090i q^{83} +(1.08242 - 9.69045i) q^{85} +(6.65028 + 11.5186i) q^{89} +(0.764696 + 1.32449i) q^{91} +(-6.07552 - 7.62155i) q^{95} +(-8.80695 - 5.08470i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 0.896156 + 2.04863i 0.400773 + 0.916177i
\(6\) 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 \(-0.770181\pi\)
0.197100 + 0.980383i \(0.436847\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.77643 − 2.18033i −0.915920 − 0.528807i −0.0335887 − 0.999436i \(-0.510694\pi\)
−0.882331 + 0.470629i \(0.844027\pi\)
\(18\) 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 \(-0.596600\pi\)
0.677031 + 0.735955i \(0.263266\pi\)
\(24\) 0 0
\(25\) −3.39381 + 3.67179i −0.678762 + 0.734359i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.974621 − 1.68809i −0.180983 − 0.313471i 0.761233 − 0.648479i \(-0.224594\pi\)
−0.942215 + 0.335008i \(0.891261\pi\)
\(30\) 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
\(35\) 1.35992 − 0.594885i 0.229869 − 0.100554i
\(36\) 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 −0.846039 − 0.533122i \(-0.821019\pi\)
0.884716 + 0.466130i \(0.154352\pi\)
\(42\) 0 0
\(43\) −6.81715 − 3.93588i −1.03960 − 0.600216i −0.119884 − 0.992788i \(-0.538252\pi\)
−0.919721 + 0.392572i \(0.871585\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.69449 + 3.28772i −0.830627 + 0.479563i −0.854067 − 0.520163i \(-0.825871\pi\)
0.0234403 + 0.999725i \(0.492538\pi\)
\(48\) 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 \(-0.616074\pi\)
0.630766 + 0.775973i \(0.282741\pi\)
\(54\) 0 0
\(55\) 1.62119 + 3.70609i 0.218602 + 0.499729i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.88559 + 6.73003i −0.505860 + 0.876176i 0.494117 + 0.869396i \(0.335492\pi\)
−0.999977 + 0.00678007i \(0.997842\pi\)
\(60\) 0 0
\(61\) −5.36021 − 9.28415i −0.686304 − 1.18871i −0.973025 − 0.230700i \(-0.925899\pi\)
0.286721 − 0.958014i \(-0.407435\pi\)
\(62\) 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 \(-0.757005\pi\)
0.237502 + 0.971387i \(0.423671\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.95914 − 5.12538i 0.351185 − 0.608270i −0.635272 − 0.772288i \(-0.719112\pi\)
0.986457 + 0.164018i \(0.0524455\pi\)
\(72\) 0 0
\(73\) 4.86313 + 2.80773i 0.569187 + 0.328620i 0.756824 − 0.653618i \(-0.226750\pi\)
−0.187638 + 0.982238i \(0.560083\pi\)
\(74\) 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 −0.983926 − 0.178575i \(-0.942851\pi\)
0.646614 + 0.762817i \(0.276185\pi\)
\(80\) 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
\(85\) 1.08242 − 9.69045i 0.117404 − 1.05108i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.65028 + 11.5186i 0.704928 + 1.22097i 0.966717 + 0.255847i \(0.0823542\pi\)
−0.261789 + 0.965125i \(0.584312\pi\)
\(90\) 0 0
\(91\) 0.764696 + 1.32449i 0.0801619 + 0.138844i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.07552 − 7.62155i −0.623335 − 0.781955i
\(96\) 0 0
\(97\) −8.80695 − 5.08470i −0.894211 − 0.516273i −0.0188932 − 0.999822i \(-0.506014\pi\)
−0.875317 + 0.483549i \(0.839348\pi\)
\(98\) 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.1189.7 20
3.2 odd 2 380.2.r.a.49.2 ✓ 20
5.4 even 2 inner 3420.2.bj.c.1189.1 20
15.2 even 4 1900.2.i.g.201.2 20
15.8 even 4 1900.2.i.g.201.9 20
15.14 odd 2 380.2.r.a.49.9 yes 20
19.7 even 3 inner 3420.2.bj.c.2629.1 20
57.26 odd 6 380.2.r.a.349.9 yes 20
95.64 even 6 inner 3420.2.bj.c.2629.7 20
285.83 even 12 1900.2.i.g.501.9 20
285.197 even 12 1900.2.i.g.501.2 20
285.254 odd 6 380.2.r.a.349.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.2 ✓ 20 3.2 odd 2
380.2.r.a.49.9 yes 20 15.14 odd 2
380.2.r.a.349.2 yes 20 285.254 odd 6
380.2.r.a.349.9 yes 20 57.26 odd 6
1900.2.i.g.201.2 20 15.2 even 4
1900.2.i.g.201.9 20 15.8 even 4
1900.2.i.g.501.2 20 285.197 even 12
1900.2.i.g.501.9 20 285.83 even 12
3420.2.bj.c.1189.1 20 5.4 even 2 inner
3420.2.bj.c.1189.7 20 1.1 even 1 trivial
3420.2.bj.c.2629.1 20 19.7 even 3 inner
3420.2.bj.c.2629.7 20 95.64 even 6 inner