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 1189.3
Root \(-0.392182 - 0.226426i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.c.2629.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.82467 + 1.29251i) q^{5} -2.54366i q^{7} -2.22377 q^{11} +(-6.08116 + 3.51096i) q^{13} +(2.21492 + 1.27878i) q^{17} +(2.70498 - 3.41805i) q^{19} +(-6.95328 + 4.01448i) q^{23} +(1.65885 - 4.71680i) q^{25} +(0.941734 + 1.63113i) q^{29} +5.98111 q^{31} +(3.28770 + 4.64135i) q^{35} +2.86105i q^{37} +(3.67524 - 6.36571i) q^{41} +(-3.19919 - 1.84706i) q^{43} +(4.09540 - 2.36448i) q^{47} +0.529782 q^{49} +(-8.91226 + 5.14549i) q^{53} +(4.05764 - 2.87424i) q^{55} +(3.73666 - 6.47208i) q^{59} +(4.17839 + 7.23719i) q^{61} +(6.55817 - 14.2663i) q^{65} +(10.7040 - 6.17997i) q^{67} +(4.13931 - 7.16950i) q^{71} +(10.9489 + 6.32134i) q^{73} +5.65651i q^{77} +(-2.13067 + 3.69043i) q^{79} -14.7613i q^{83} +(-5.69434 + 0.529441i) q^{85} +(7.19403 + 12.4604i) q^{89} +(8.93069 + 15.4684i) q^{91} +(-0.517834 + 9.73303i) q^{95} +(-5.04871 - 2.91488i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) 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
\(5\) −1.82467 + 1.29251i −0.816018 + 0.578027i
\(6\) 0 0
\(7\) − 2.54366i − 0.961414i −0.876881 − 0.480707i \(-0.840380\pi\)
0.876881 − 0.480707i \(-0.159620\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.22377 −0.670491 −0.335246 − 0.942131i \(-0.608819\pi\)
−0.335246 + 0.942131i \(0.608819\pi\)
\(12\) 0 0
\(13\) −6.08116 + 3.51096i −1.68661 + 0.973765i −0.729526 + 0.683953i \(0.760259\pi\)
−0.957084 + 0.289812i \(0.906407\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.21492 + 1.27878i 0.537197 + 0.310151i 0.743942 − 0.668244i \(-0.232954\pi\)
−0.206745 + 0.978395i \(0.566287\pi\)
\(18\) 0 0
\(19\) 2.70498 − 3.41805i 0.620565 − 0.784155i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.95328 + 4.01448i −1.44986 + 0.837076i −0.998472 − 0.0552521i \(-0.982404\pi\)
−0.451387 + 0.892329i \(0.649070\pi\)
\(24\) 0 0
\(25\) 1.65885 − 4.71680i 0.331769 − 0.943361i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.941734 + 1.63113i 0.174876 + 0.302893i 0.940118 − 0.340849i \(-0.110714\pi\)
−0.765243 + 0.643742i \(0.777381\pi\)
\(30\) 0 0
\(31\) 5.98111 1.07424 0.537120 − 0.843506i \(-0.319512\pi\)
0.537120 + 0.843506i \(0.319512\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.28770 + 4.64135i 0.555723 + 0.784531i
\(36\) 0 0
\(37\) 2.86105i 0.470353i 0.971953 + 0.235177i \(0.0755669\pi\)
−0.971953 + 0.235177i \(0.924433\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.67524 − 6.36571i 0.573977 − 0.994157i −0.422175 − 0.906514i \(-0.638733\pi\)
0.996152 − 0.0876426i \(-0.0279333\pi\)
\(42\) 0 0
\(43\) −3.19919 − 1.84706i −0.487873 − 0.281673i 0.235819 − 0.971797i \(-0.424223\pi\)
−0.723691 + 0.690124i \(0.757556\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.09540 − 2.36448i 0.597376 − 0.344895i −0.170633 − 0.985335i \(-0.554581\pi\)
0.768008 + 0.640440i \(0.221248\pi\)
\(48\) 0 0
\(49\) 0.529782 0.0756832
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.91226 + 5.14549i −1.22419 + 0.706788i −0.965809 − 0.259255i \(-0.916523\pi\)
−0.258383 + 0.966042i \(0.583190\pi\)
\(54\) 0 0
\(55\) 4.05764 − 2.87424i 0.547133 − 0.387562i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.73666 − 6.47208i 0.486472 − 0.842593i −0.513408 − 0.858145i \(-0.671617\pi\)
0.999879 + 0.0155515i \(0.00495040\pi\)
\(60\) 0 0
\(61\) 4.17839 + 7.23719i 0.534988 + 0.926627i 0.999164 + 0.0408838i \(0.0130174\pi\)
−0.464176 + 0.885743i \(0.653649\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.55817 − 14.2663i 0.813441 − 1.76952i
\(66\) 0 0
\(67\) 10.7040 − 6.17997i 1.30771 − 0.755004i 0.325993 − 0.945372i \(-0.394302\pi\)
0.981713 + 0.190368i \(0.0609682\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.13931 − 7.16950i 0.491246 − 0.850863i −0.508703 − 0.860942i \(-0.669875\pi\)
0.999949 + 0.0100790i \(0.00320829\pi\)
\(72\) 0 0
\(73\) 10.9489 + 6.32134i 1.28147 + 0.739857i 0.977117 − 0.212703i \(-0.0682266\pi\)
0.304352 + 0.952559i \(0.401560\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.65651i 0.644620i
\(78\) 0 0
\(79\) −2.13067 + 3.69043i −0.239719 + 0.415206i −0.960634 − 0.277818i \(-0.910389\pi\)
0.720914 + 0.693024i \(0.243722\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 14.7613i − 1.62026i −0.586248 − 0.810132i \(-0.699396\pi\)
0.586248 − 0.810132i \(-0.300604\pi\)
\(84\) 0 0
\(85\) −5.69434 + 0.529441i −0.617638 + 0.0574259i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.19403 + 12.4604i 0.762566 + 1.32080i 0.941524 + 0.336946i \(0.109394\pi\)
−0.178958 + 0.983857i \(0.557273\pi\)
\(90\) 0 0
\(91\) 8.93069 + 15.4684i 0.936191 + 1.62153i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.517834 + 9.73303i −0.0531286 + 0.998588i
\(96\) 0 0
\(97\) −5.04871 − 2.91488i −0.512619 − 0.295961i 0.221291 − 0.975208i \(-0.428973\pi\)
−0.733910 + 0.679247i \(0.762306\pi\)
\(98\) 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.1189.3 20
3.2 odd 2 380.2.r.a.49.5 ✓ 20
5.4 even 2 inner 3420.2.bj.c.1189.5 20
15.2 even 4 1900.2.i.g.201.5 20
15.8 even 4 1900.2.i.g.201.6 20
15.14 odd 2 380.2.r.a.49.6 yes 20
19.7 even 3 inner 3420.2.bj.c.2629.5 20
57.26 odd 6 380.2.r.a.349.6 yes 20
95.64 even 6 inner 3420.2.bj.c.2629.3 20
285.83 even 12 1900.2.i.g.501.6 20
285.197 even 12 1900.2.i.g.501.5 20
285.254 odd 6 380.2.r.a.349.5 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.5 ✓ 20 3.2 odd 2
380.2.r.a.49.6 yes 20 15.14 odd 2
380.2.r.a.349.5 yes 20 285.254 odd 6
380.2.r.a.349.6 yes 20 57.26 odd 6
1900.2.i.g.201.5 20 15.2 even 4
1900.2.i.g.201.6 20 15.8 even 4
1900.2.i.g.501.5 20 285.197 even 12
1900.2.i.g.501.6 20 285.83 even 12
3420.2.bj.c.1189.3 20 1.1 even 1 trivial
3420.2.bj.c.1189.5 20 5.4 even 2 inner
3420.2.bj.c.2629.3 20 95.64 even 6 inner
3420.2.bj.c.2629.5 20 19.7 even 3 inner