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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 = [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 1189.6
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.d.2629.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.842697 - 2.07120i) q^{5} +4.65052i q^{7} -4.12988 q^{11} +(1.79297 - 1.03517i) q^{13} +(-0.415138 - 0.239680i) q^{17} +(-2.33753 - 3.67913i) q^{19} +(-1.32570 + 0.765396i) q^{23} +(-3.57972 + 3.49078i) q^{25} +(2.49088 + 4.31433i) q^{29} -1.44535 q^{31} +(9.63215 - 3.91898i) q^{35} -7.84217i q^{37} +(6.02367 - 10.4333i) q^{41} +(8.86698 + 5.11935i) q^{43} +(6.56042 - 3.78766i) q^{47} -14.6273 q^{49} +(-3.90126 + 2.25239i) q^{53} +(3.48024 + 8.55380i) q^{55} +(5.31226 - 9.20110i) q^{59} +(-4.21205 - 7.29549i) q^{61} +(-3.65498 - 2.84126i) q^{65} +(11.1806 - 6.45514i) q^{67} +(-6.36448 + 11.0236i) q^{71} +(9.92484 + 5.73011i) q^{73} -19.2061i q^{77} +(2.54620 - 4.41014i) q^{79} +13.8472i q^{83} +(-0.146589 + 1.06181i) q^{85} +(-1.66388 - 2.88193i) q^{89} +(4.81409 + 8.33824i) q^{91} +(-5.65037 + 7.94187i) q^{95} +(2.84117 + 1.64035i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{11} + 12 q^{19} - 12 q^{25} - 22 q^{29} - 56 q^{31} + 2 q^{35} + 16 q^{41} - 88 q^{49} - 14 q^{55} + 86 q^{59} + 6 q^{61} - 36 q^{65} - 10 q^{71} + 50 q^{79} + 30 q^{85} - 34 q^{89} + 4 q^{91}+ \cdots - 46 q^{95}+O(q^{100}) \) 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
\(5\) −0.842697 − 2.07120i −0.376865 − 0.926268i
\(6\) 0 0
\(7\) 4.65052i 1.75773i 0.477070 + 0.878866i \(0.341699\pi\)
−0.477070 + 0.878866i \(0.658301\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.12988 −1.24521 −0.622603 − 0.782538i \(-0.713925\pi\)
−0.622603 + 0.782538i \(0.713925\pi\)
\(12\) 0 0
\(13\) 1.79297 − 1.03517i 0.497280 − 0.287105i −0.230309 − 0.973117i \(-0.573974\pi\)
0.727590 + 0.686013i \(0.240640\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.415138 − 0.239680i −0.100686 − 0.0581309i 0.448812 − 0.893626i \(-0.351847\pi\)
−0.549497 + 0.835495i \(0.685181\pi\)
\(18\) 0 0
\(19\) −2.33753 − 3.67913i −0.536265 − 0.844049i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.32570 + 0.765396i −0.276428 + 0.159596i −0.631805 − 0.775127i \(-0.717686\pi\)
0.355377 + 0.934723i \(0.384353\pi\)
\(24\) 0 0
\(25\) −3.57972 + 3.49078i −0.715945 + 0.698157i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.49088 + 4.31433i 0.462545 + 0.801151i 0.999087 − 0.0427222i \(-0.0136030\pi\)
−0.536542 + 0.843874i \(0.680270\pi\)
\(30\) 0 0
\(31\) −1.44535 −0.259592 −0.129796 − 0.991541i \(-0.541432\pi\)
−0.129796 + 0.991541i \(0.541432\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.63215 − 3.91898i 1.62813 − 0.662428i
\(36\) 0 0
\(37\) − 7.84217i − 1.28924i −0.764501 − 0.644622i \(-0.777015\pi\)
0.764501 − 0.644622i \(-0.222985\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.02367 − 10.4333i 0.940739 − 1.62941i 0.176673 − 0.984270i \(-0.443466\pi\)
0.764066 − 0.645138i \(-0.223200\pi\)
\(42\) 0 0
\(43\) 8.86698 + 5.11935i 1.35220 + 0.780694i 0.988558 − 0.150844i \(-0.0481992\pi\)
0.363644 + 0.931538i \(0.381532\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.56042 − 3.78766i 0.956935 − 0.552487i 0.0617066 − 0.998094i \(-0.480346\pi\)
0.895228 + 0.445608i \(0.147012\pi\)
\(48\) 0 0
\(49\) −14.6273 −2.08962
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.90126 + 2.25239i −0.535879 + 0.309390i −0.743407 − 0.668839i \(-0.766792\pi\)
0.207528 + 0.978229i \(0.433458\pi\)
\(54\) 0 0
\(55\) 3.48024 + 8.55380i 0.469275 + 1.15339i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.31226 − 9.20110i 0.691597 − 1.19788i −0.279717 − 0.960082i \(-0.590241\pi\)
0.971314 − 0.237799i \(-0.0764259\pi\)
\(60\) 0 0
\(61\) −4.21205 − 7.29549i −0.539298 − 0.934092i −0.998942 − 0.0459886i \(-0.985356\pi\)
0.459644 − 0.888103i \(-0.347977\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.65498 − 2.84126i −0.453344 − 0.352415i
\(66\) 0 0
\(67\) 11.1806 − 6.45514i 1.36593 − 0.788621i 0.375526 − 0.926812i \(-0.377462\pi\)
0.990406 + 0.138191i \(0.0441289\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.36448 + 11.0236i −0.755325 + 1.30826i 0.189888 + 0.981806i \(0.439187\pi\)
−0.945213 + 0.326455i \(0.894146\pi\)
\(72\) 0 0
\(73\) 9.92484 + 5.73011i 1.16161 + 0.670659i 0.951690 − 0.307059i \(-0.0993449\pi\)
0.209924 + 0.977718i \(0.432678\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 19.2061i − 2.18874i
\(78\) 0 0
\(79\) 2.54620 − 4.41014i 0.286470 − 0.496180i −0.686495 − 0.727135i \(-0.740852\pi\)
0.972964 + 0.230955i \(0.0741849\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.8472i 1.51992i 0.649967 + 0.759962i \(0.274783\pi\)
−0.649967 + 0.759962i \(0.725217\pi\)
\(84\) 0 0
\(85\) −0.146589 + 1.06181i −0.0158998 + 0.115169i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.66388 − 2.88193i −0.176371 − 0.305484i 0.764264 − 0.644904i \(-0.223103\pi\)
−0.940635 + 0.339420i \(0.889769\pi\)
\(90\) 0 0
\(91\) 4.81409 + 8.33824i 0.504653 + 0.874085i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.65037 + 7.94187i −0.579716 + 0.814819i
\(96\) 0 0
\(97\) 2.84117 + 1.64035i 0.288477 + 0.166553i 0.637255 − 0.770653i \(-0.280070\pi\)
−0.348778 + 0.937206i \(0.613403\pi\)
\(98\) 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.1189.6 32
3.2 odd 2 1140.2.bg.c.49.9 yes 32
5.4 even 2 inner 3420.2.bj.d.1189.15 32
15.14 odd 2 1140.2.bg.c.49.8 ✓ 32
19.7 even 3 inner 3420.2.bj.d.2629.15 32
57.26 odd 6 1140.2.bg.c.349.8 yes 32
95.64 even 6 inner 3420.2.bj.d.2629.6 32
285.254 odd 6 1140.2.bg.c.349.9 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.c.49.8 ✓ 32 15.14 odd 2
1140.2.bg.c.49.9 yes 32 3.2 odd 2
1140.2.bg.c.349.8 yes 32 57.26 odd 6
1140.2.bg.c.349.9 yes 32 285.254 odd 6
3420.2.bj.d.1189.6 32 1.1 even 1 trivial
3420.2.bj.d.1189.15 32 5.4 even 2 inner
3420.2.bj.d.2629.6 32 95.64 even 6 inner
3420.2.bj.d.2629.15 32 19.7 even 3 inner