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 = [32,0,0,0,0] 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: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1189.5
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.d.2629.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.18735 - 1.89478i) q^{5} +0.635261i q^{7} -2.41745 q^{11} +(3.34250 - 1.92979i) q^{13} +(3.50975 + 2.02636i) q^{17} +(-2.29113 + 3.70819i) q^{19} +(-5.15400 + 2.97567i) q^{23} +(-2.18040 + 4.49954i) q^{25} +(-2.75814 - 4.77724i) q^{29} +1.63802 q^{31} +(1.20368 - 0.754277i) q^{35} +8.46276i q^{37} +(5.23276 - 9.06341i) q^{41} +(-7.87799 - 4.54836i) q^{43} +(-9.77942 + 5.64615i) q^{47} +6.59644 q^{49} +(-5.31255 + 3.06720i) q^{53} +(2.87036 + 4.58055i) q^{55} +(1.09083 - 1.88938i) q^{59} +(7.47972 + 12.9553i) q^{61} +(-7.62525 - 4.04197i) q^{65} +(-3.63700 + 2.09982i) q^{67} +(4.09245 - 7.08832i) q^{71} +(9.07329 + 5.23847i) q^{73} -1.53571i q^{77} +(7.87141 - 13.6337i) q^{79} +13.5453i q^{83} +(-0.327797 - 9.05621i) q^{85} +(4.68019 + 8.10632i) q^{89} +(1.22592 + 2.12336i) q^{91} +(9.74660 - 0.0617195i) q^{95} +(8.90415 + 5.14082i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{11} + 12 q^{19} - 12 q^{25} - 22 q^{29} - 56 q^{31} + 2 q^{35} + 16 q^{41} - 88 q^{49} - 14 q^{55} + 86 q^{59} + 6 q^{61} - 36 q^{65} - 10 q^{71} + 50 q^{79} + 30 q^{85} - 34 q^{89} + 4 q^{91}+ \cdots - 46 q^{95}+O(q^{100}) \) 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
\(5\) −1.18735 − 1.89478i −0.530999 − 0.847373i
\(6\) 0 0
\(7\) 0.635261i 0.240106i 0.992767 + 0.120053i \(0.0383065\pi\)
−0.992767 + 0.120053i \(0.961694\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.41745 −0.728889 −0.364445 − 0.931225i \(-0.618741\pi\)
−0.364445 + 0.931225i \(0.618741\pi\)
\(12\) 0 0
\(13\) 3.34250 − 1.92979i 0.927042 − 0.535228i 0.0411673 − 0.999152i \(-0.486892\pi\)
0.885875 + 0.463924i \(0.153559\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.50975 + 2.02636i 0.851240 + 0.491464i 0.861069 − 0.508488i \(-0.169795\pi\)
−0.00982904 + 0.999952i \(0.503129\pi\)
\(18\) 0 0
\(19\) −2.29113 + 3.70819i −0.525622 + 0.850718i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.15400 + 2.97567i −1.07468 + 0.620469i −0.929458 − 0.368929i \(-0.879725\pi\)
−0.145227 + 0.989398i \(0.546391\pi\)
\(24\) 0 0
\(25\) −2.18040 + 4.49954i −0.436081 + 0.899908i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.75814 − 4.77724i −0.512174 − 0.887112i −0.999900 − 0.0141152i \(-0.995507\pi\)
0.487726 − 0.872997i \(-0.337827\pi\)
\(30\) 0 0
\(31\) 1.63802 0.294198 0.147099 − 0.989122i \(-0.453006\pi\)
0.147099 + 0.989122i \(0.453006\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.20368 − 0.754277i 0.203459 − 0.127496i
\(36\) 0 0
\(37\) 8.46276i 1.39127i 0.718396 + 0.695635i \(0.244877\pi\)
−0.718396 + 0.695635i \(0.755123\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.23276 − 9.06341i 0.817220 − 1.41547i −0.0905027 − 0.995896i \(-0.528847\pi\)
0.907723 − 0.419570i \(-0.137819\pi\)
\(42\) 0 0
\(43\) −7.87799 − 4.54836i −1.20138 − 0.693619i −0.240520 − 0.970644i \(-0.577318\pi\)
−0.960863 + 0.277026i \(0.910651\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.77942 + 5.64615i −1.42647 + 0.823575i −0.996841 − 0.0794279i \(-0.974691\pi\)
−0.429634 + 0.903003i \(0.641357\pi\)
\(48\) 0 0
\(49\) 6.59644 0.942349
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.31255 + 3.06720i −0.729735 + 0.421313i −0.818325 − 0.574755i \(-0.805097\pi\)
0.0885900 + 0.996068i \(0.471764\pi\)
\(54\) 0 0
\(55\) 2.87036 + 4.58055i 0.387039 + 0.617641i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.09083 − 1.88938i 0.142014 − 0.245976i −0.786241 − 0.617920i \(-0.787975\pi\)
0.928255 + 0.371944i \(0.121309\pi\)
\(60\) 0 0
\(61\) 7.47972 + 12.9553i 0.957680 + 1.65875i 0.728113 + 0.685458i \(0.240398\pi\)
0.229567 + 0.973293i \(0.426269\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.62525 − 4.04197i −0.945796 − 0.501345i
\(66\) 0 0
\(67\) −3.63700 + 2.09982i −0.444330 + 0.256534i −0.705433 − 0.708777i \(-0.749247\pi\)
0.261103 + 0.965311i \(0.415914\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.09245 − 7.08832i 0.485684 − 0.841229i −0.514181 − 0.857682i \(-0.671904\pi\)
0.999865 + 0.0164525i \(0.00523724\pi\)
\(72\) 0 0
\(73\) 9.07329 + 5.23847i 1.06195 + 0.613116i 0.925971 − 0.377596i \(-0.123249\pi\)
0.135978 + 0.990712i \(0.456582\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.53571i − 0.175011i
\(78\) 0 0
\(79\) 7.87141 − 13.6337i 0.885603 − 1.53391i 0.0405818 − 0.999176i \(-0.487079\pi\)
0.845021 − 0.534733i \(-0.179588\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.5453i 1.48679i 0.668854 + 0.743394i \(0.266785\pi\)
−0.668854 + 0.743394i \(0.733215\pi\)
\(84\) 0 0
\(85\) −0.327797 − 9.05621i −0.0355546 − 0.982284i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.68019 + 8.10632i 0.496099 + 0.859268i 0.999990 − 0.00449879i \(-0.00143201\pi\)
−0.503891 + 0.863767i \(0.668099\pi\)
\(90\) 0 0
\(91\) 1.22592 + 2.12336i 0.128512 + 0.222589i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.74660 − 0.0617195i 0.999980 − 0.00633229i
\(96\) 0 0
\(97\) 8.90415 + 5.14082i 0.904080 + 0.521971i 0.878522 − 0.477703i \(-0.158530\pi\)
0.0255583 + 0.999673i \(0.491864\pi\)
\(98\) 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.d.1189.5 32
3.2 odd 2 1140.2.bg.c.49.4 ✓ 32
5.4 even 2 inner 3420.2.bj.d.1189.16 32
15.14 odd 2 1140.2.bg.c.49.13 yes 32
19.7 even 3 inner 3420.2.bj.d.2629.16 32
57.26 odd 6 1140.2.bg.c.349.14 yes 32
95.64 even 6 inner 3420.2.bj.d.2629.5 32
285.254 odd 6 1140.2.bg.c.349.3 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.c.49.4 ✓ 32 3.2 odd 2
1140.2.bg.c.49.13 yes 32 15.14 odd 2
1140.2.bg.c.349.3 yes 32 285.254 odd 6
1140.2.bg.c.349.14 yes 32 57.26 odd 6
3420.2.bj.d.1189.5 32 1.1 even 1 trivial
3420.2.bj.d.1189.16 32 5.4 even 2 inner
3420.2.bj.d.2629.5 32 95.64 even 6 inner
3420.2.bj.d.2629.16 32 19.7 even 3 inner