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 2629.6
Root \(1.74361 - 1.00667i\) of defining polynomial
Character \(\chi\) \(=\) 3420.2629
Dual form 3420.2.bj.c.1189.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0408382 + 2.23570i) q^{5} +1.34403i q^{7} -5.25594 q^{11} +(-2.10918 - 1.21773i) q^{13} +(-1.17765 + 0.679914i) q^{17} +(2.89815 - 3.25587i) q^{19} +(7.05514 + 4.07329i) q^{23} +(-4.99666 - 0.182603i) q^{25} +(1.03597 - 1.79435i) q^{29} -0.513207 q^{31} +(-3.00483 - 0.0548876i) q^{35} +5.57175i q^{37} +(-2.70353 - 4.68265i) q^{41} +(-11.0197 + 6.36221i) q^{43} +(-2.82785 - 1.63266i) q^{47} +5.19359 q^{49} +(-10.1892 - 5.88276i) q^{53} +(0.214643 - 11.7507i) q^{55} +(-0.0175979 - 0.0304805i) q^{59} +(0.518372 - 0.897846i) q^{61} +(2.80861 - 4.66574i) q^{65} +(0.664028 + 0.383377i) q^{67} +(-5.68450 - 9.84583i) q^{71} +(-1.86429 + 1.07635i) q^{73} -7.06413i q^{77} +(-6.48576 - 11.2337i) q^{79} -4.20304i q^{83} +(-1.47199 - 2.66062i) q^{85} +(3.65426 - 6.32937i) q^{89} +(1.63667 - 2.83479i) q^{91} +(7.16079 + 6.61235i) q^{95} +(0.721716 - 0.416683i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) 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
\(5\) −0.0408382 + 2.23570i −0.0182634 + 0.999833i
\(6\) 0 0
\(7\) 1.34403i 0.507994i 0.967205 + 0.253997i \(0.0817454\pi\)
−0.967205 + 0.253997i \(0.918255\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.25594 −1.58473 −0.792363 − 0.610050i \(-0.791149\pi\)
−0.792363 + 0.610050i \(0.791149\pi\)
\(12\) 0 0
\(13\) −2.10918 − 1.21773i −0.584980 − 0.337738i 0.178130 − 0.984007i \(-0.442995\pi\)
−0.763110 + 0.646269i \(0.776329\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.17765 + 0.679914i −0.285621 + 0.164903i −0.635965 − 0.771718i \(-0.719398\pi\)
0.350344 + 0.936621i \(0.386065\pi\)
\(18\) 0 0
\(19\) 2.89815 − 3.25587i 0.664882 − 0.746949i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.05514 + 4.07329i 1.47110 + 0.849339i 0.999473 − 0.0324603i \(-0.0103342\pi\)
0.471625 + 0.881799i \(0.343668\pi\)
\(24\) 0 0
\(25\) −4.99666 − 0.182603i −0.999333 − 0.0365207i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.03597 − 1.79435i 0.192375 − 0.333203i −0.753662 − 0.657262i \(-0.771714\pi\)
0.946037 + 0.324059i \(0.105048\pi\)
\(30\) 0 0
\(31\) −0.513207 −0.0921747 −0.0460873 − 0.998937i \(-0.514675\pi\)
−0.0460873 + 0.998937i \(0.514675\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00483 − 0.0548876i −0.507910 − 0.00927770i
\(36\) 0 0
\(37\) 5.57175i 0.915991i 0.888955 + 0.457995i \(0.151432\pi\)
−0.888955 + 0.457995i \(0.848568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.70353 − 4.68265i −0.422220 − 0.731307i 0.573936 − 0.818900i \(-0.305416\pi\)
−0.996156 + 0.0875933i \(0.972082\pi\)
\(42\) 0 0
\(43\) −11.0197 + 6.36221i −1.68048 + 0.970228i −0.719146 + 0.694859i \(0.755467\pi\)
−0.961339 + 0.275369i \(0.911200\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82785 − 1.63266i −0.412485 − 0.238148i 0.279372 − 0.960183i \(-0.409874\pi\)
−0.691857 + 0.722035i \(0.743207\pi\)
\(48\) 0 0
\(49\) 5.19359 0.741942
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.1892 − 5.88276i −1.39960 − 0.808060i −0.405250 − 0.914206i \(-0.632815\pi\)
−0.994351 + 0.106146i \(0.966149\pi\)
\(54\) 0 0
\(55\) 0.214643 − 11.7507i 0.0289425 − 1.58446i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.0175979 − 0.0304805i −0.00229105 − 0.00396822i 0.864878 − 0.501983i \(-0.167396\pi\)
−0.867169 + 0.498015i \(0.834063\pi\)
\(60\) 0 0
\(61\) 0.518372 − 0.897846i 0.0663707 − 0.114957i −0.830930 − 0.556376i \(-0.812191\pi\)
0.897301 + 0.441419i \(0.145525\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.80861 − 4.66574i 0.348366 − 0.578714i
\(66\) 0 0
\(67\) 0.664028 + 0.383377i 0.0811239 + 0.0468369i 0.540013 − 0.841657i \(-0.318419\pi\)
−0.458889 + 0.888493i \(0.651753\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.68450 − 9.84583i −0.674625 − 1.16849i −0.976578 − 0.215163i \(-0.930972\pi\)
0.301953 − 0.953323i \(-0.402362\pi\)
\(72\) 0 0
\(73\) −1.86429 + 1.07635i −0.218199 + 0.125977i −0.605116 − 0.796137i \(-0.706873\pi\)
0.386917 + 0.922115i \(0.373540\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 7.06413i − 0.805032i
\(78\) 0 0
\(79\) −6.48576 − 11.2337i −0.729705 − 1.26389i −0.957008 − 0.290062i \(-0.906324\pi\)
0.227302 − 0.973824i \(-0.427009\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 4.20304i − 0.461343i −0.973032 − 0.230672i \(-0.925908\pi\)
0.973032 − 0.230672i \(-0.0740923\pi\)
\(84\) 0 0
\(85\) −1.47199 − 2.66062i −0.159660 − 0.288585i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.65426 − 6.32937i 0.387351 − 0.670912i −0.604741 − 0.796422i \(-0.706723\pi\)
0.992092 + 0.125510i \(0.0400568\pi\)
\(90\) 0 0
\(91\) 1.63667 − 2.83479i 0.171569 − 0.297166i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.16079 + 6.61235i 0.734681 + 0.678413i
\(96\) 0 0
\(97\) 0.721716 − 0.416683i 0.0732791 − 0.0423077i −0.462913 − 0.886404i \(-0.653196\pi\)
0.536192 + 0.844096i \(0.319862\pi\)
\(98\) 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.2629.6 20
3.2 odd 2 380.2.r.a.349.8 yes 20
5.4 even 2 inner 3420.2.bj.c.2629.8 20
15.2 even 4 1900.2.i.g.501.3 20
15.8 even 4 1900.2.i.g.501.8 20
15.14 odd 2 380.2.r.a.349.3 yes 20
19.11 even 3 inner 3420.2.bj.c.1189.8 20
57.11 odd 6 380.2.r.a.49.3 ✓ 20
95.49 even 6 inner 3420.2.bj.c.1189.6 20
285.68 even 12 1900.2.i.g.201.8 20
285.182 even 12 1900.2.i.g.201.3 20
285.239 odd 6 380.2.r.a.49.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.3 ✓ 20 57.11 odd 6
380.2.r.a.49.8 yes 20 285.239 odd 6
380.2.r.a.349.3 yes 20 15.14 odd 2
380.2.r.a.349.8 yes 20 3.2 odd 2
1900.2.i.g.201.3 20 285.182 even 12
1900.2.i.g.201.8 20 285.68 even 12
1900.2.i.g.501.3 20 15.2 even 4
1900.2.i.g.501.8 20 15.8 even 4
3420.2.bj.c.1189.6 20 95.49 even 6 inner
3420.2.bj.c.1189.8 20 19.11 even 3 inner
3420.2.bj.c.2629.6 20 1.1 even 1 trivial
3420.2.bj.c.2629.8 20 5.4 even 2 inner