Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [54,6,Mod(19,54)] 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("54.19"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 54.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.66072626990\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.2
Root \(0.500000 - 8.40123i\) of defining polynomial
Character \(\chi\) \(=\) 54.37
Dual form 54.6.c.b.19.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.00000 + 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(20.8014 - 36.0292i) q^{5} +(101.661 + 176.082i) q^{7} -64.0000 q^{8} +166.412 q^{10} +(235.168 + 407.323i) q^{11} +(-241.506 + 418.300i) q^{13} +(-406.643 + 704.326i) q^{14} +(-128.000 - 221.703i) q^{16} -1259.86 q^{17} +1978.94 q^{19} +(332.823 + 576.466i) q^{20} +(-940.672 + 1629.29i) q^{22} +(239.119 - 414.166i) q^{23} +(697.100 + 1207.41i) q^{25} -1932.04 q^{26} -3253.14 q^{28} +(-580.249 - 1005.02i) q^{29} +(1186.50 - 2055.07i) q^{31} +(512.000 - 886.810i) q^{32} +(-2519.72 - 4364.28i) q^{34} +8458.76 q^{35} +8185.10 q^{37} +(3957.89 + 6855.26i) q^{38} +(-1331.29 + 2305.87i) q^{40} +(8758.87 - 15170.8i) q^{41} +(-11435.0 - 19806.1i) q^{43} -7525.37 q^{44} +1912.95 q^{46} +(-8685.43 - 15043.6i) q^{47} +(-12266.3 + 21245.9i) q^{49} +(-2788.40 + 4829.65i) q^{50} +(-3864.09 - 6692.80i) q^{52} +5390.58 q^{53} +19567.3 q^{55} +(-6506.29 - 11269.2i) q^{56} +(2321.00 - 4020.09i) q^{58} +(-22281.8 + 38593.2i) q^{59} +(-2084.17 - 3609.88i) q^{61} +9491.97 q^{62} +4096.00 q^{64} +(10047.3 + 17402.5i) q^{65} +(-1228.46 + 2127.76i) q^{67} +(10078.9 - 17457.1i) q^{68} +(16917.5 + 29302.0i) q^{70} -2184.37 q^{71} +3037.34 q^{73} +(16370.2 + 28354.0i) q^{74} +(-15831.5 + 27421.0i) q^{76} +(-47814.7 + 82817.5i) q^{77} +(25266.2 + 43762.3i) q^{79} -10650.3 q^{80} +70070.9 q^{82} +(-25913.9 - 44884.2i) q^{83} +(-26206.9 + 45391.7i) q^{85} +(45740.1 - 79224.3i) q^{86} +(-15050.7 - 26068.7i) q^{88} +20154.7 q^{89} -98206.5 q^{91} +(3825.91 + 6626.66i) q^{92} +(34741.7 - 60174.4i) q^{94} +(41164.9 - 71299.6i) q^{95} +(-40214.4 - 69653.3i) q^{97} -98130.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 48 q^{4} + 54 q^{5} - 132 q^{7} - 384 q^{8} + 432 q^{10} + 315 q^{11} - 744 q^{13} + 528 q^{14} - 768 q^{16} - 2898 q^{17} + 2262 q^{19} + 864 q^{20} - 1260 q^{22} + 3168 q^{23} - 2883 q^{25}+ \cdots - 422856 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 + 3.46410i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −8.00000 + 13.8564i −0.250000 + 0.433013i
\(5\) 20.8014 − 36.0292i 0.372108 − 0.644509i −0.617782 − 0.786349i \(-0.711969\pi\)
0.989890 + 0.141840i \(0.0453019\pi\)
\(6\) 0 0
\(7\) 101.661 + 176.082i 0.784166 + 1.35822i 0.929496 + 0.368832i \(0.120242\pi\)
−0.145330 + 0.989383i \(0.546424\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 166.412 0.526239
\(11\) 235.168 + 407.323i 0.585998 + 1.01498i 0.994750 + 0.102333i \(0.0326308\pi\)
−0.408752 + 0.912646i \(0.634036\pi\)
\(12\) 0 0
\(13\) −241.506 + 418.300i −0.396341 + 0.686482i −0.993271 − 0.115811i \(-0.963053\pi\)
0.596931 + 0.802293i \(0.296387\pi\)
\(14\) −406.643 + 704.326i −0.554489 + 0.960403i
\(15\) 0 0
\(16\) −128.000 − 221.703i −0.125000 − 0.216506i
\(17\) −1259.86 −1.05730 −0.528652 − 0.848839i \(-0.677302\pi\)
−0.528652 + 0.848839i \(0.677302\pi\)
\(18\) 0 0
\(19\) 1978.94 1.25762 0.628810 − 0.777559i \(-0.283542\pi\)
0.628810 + 0.777559i \(0.283542\pi\)
\(20\) 332.823 + 576.466i 0.186054 + 0.322255i
\(21\) 0 0
\(22\) −940.672 + 1629.29i −0.414363 + 0.717698i
\(23\) 239.119 − 414.166i 0.0942529 − 0.163251i −0.815044 − 0.579400i \(-0.803287\pi\)
0.909297 + 0.416149i \(0.136620\pi\)
\(24\) 0 0
\(25\) 697.100 + 1207.41i 0.223072 + 0.386372i
\(26\) −1932.04 −0.560511
\(27\) 0 0
\(28\) −3253.14 −0.784166
\(29\) −580.249 − 1005.02i −0.128121 − 0.221912i 0.794828 − 0.606835i \(-0.207561\pi\)
−0.922949 + 0.384923i \(0.874228\pi\)
\(30\) 0 0
\(31\) 1186.50 − 2055.07i 0.221749 − 0.384081i −0.733590 − 0.679592i \(-0.762157\pi\)
0.955339 + 0.295511i \(0.0954901\pi\)
\(32\) 512.000 − 886.810i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2519.72 − 4364.28i −0.373813 − 0.647464i
\(35\) 8458.76 1.16718
\(36\) 0 0
\(37\) 8185.10 0.982923 0.491462 − 0.870899i \(-0.336463\pi\)
0.491462 + 0.870899i \(0.336463\pi\)
\(38\) 3957.89 + 6855.26i 0.444636 + 0.770132i
\(39\) 0 0
\(40\) −1331.29 + 2305.87i −0.131560 + 0.227868i
\(41\) 8758.87 − 15170.8i 0.813745 − 1.40945i −0.0964809 − 0.995335i \(-0.530759\pi\)
0.910226 − 0.414113i \(-0.135908\pi\)
\(42\) 0 0
\(43\) −11435.0 − 19806.1i −0.943119 − 1.63353i −0.759475 − 0.650537i \(-0.774544\pi\)
−0.183644 − 0.982993i \(-0.558789\pi\)
\(44\) −7525.37 −0.585998
\(45\) 0 0
\(46\) 1912.95 0.133294
\(47\) −8685.43 − 15043.6i −0.573518 − 0.993362i −0.996201 − 0.0870843i \(-0.972245\pi\)
0.422683 − 0.906277i \(-0.361088\pi\)
\(48\) 0 0
\(49\) −12266.3 + 21245.9i −0.729833 + 1.26411i
\(50\) −2788.40 + 4829.65i −0.157736 + 0.273206i
\(51\) 0 0
\(52\) −3864.09 − 6692.80i −0.198170 − 0.343241i
\(53\) 5390.58 0.263600 0.131800 − 0.991276i \(-0.457924\pi\)
0.131800 + 0.991276i \(0.457924\pi\)
\(54\) 0 0
\(55\) 19567.3 0.872218
\(56\) −6506.29 − 11269.2i −0.277245 − 0.480202i
\(57\) 0 0
\(58\) 2321.00 − 4020.09i 0.0905951 − 0.156915i
\(59\) −22281.8 + 38593.2i −0.833335 + 1.44338i 0.0620431 + 0.998073i \(0.480238\pi\)
−0.895379 + 0.445306i \(0.853095\pi\)
\(60\) 0 0
\(61\) −2084.17 − 3609.88i −0.0717147 − 0.124213i 0.827938 − 0.560819i \(-0.189514\pi\)
−0.899653 + 0.436606i \(0.856180\pi\)
\(62\) 9491.97 0.313601
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 10047.3 + 17402.5i 0.294963 + 0.510891i
\(66\) 0 0
\(67\) −1228.46 + 2127.76i −0.0334330 + 0.0579076i −0.882258 − 0.470767i \(-0.843977\pi\)
0.848825 + 0.528674i \(0.177311\pi\)
\(68\) 10078.9 − 17457.1i 0.264326 − 0.457826i
\(69\) 0 0
\(70\) 16917.5 + 29302.0i 0.412659 + 0.714747i
\(71\) −2184.37 −0.0514256 −0.0257128 − 0.999669i \(-0.508186\pi\)
−0.0257128 + 0.999669i \(0.508186\pi\)
\(72\) 0 0
\(73\) 3037.34 0.0667093 0.0333546 − 0.999444i \(-0.489381\pi\)
0.0333546 + 0.999444i \(0.489381\pi\)
\(74\) 16370.2 + 28354.0i 0.347516 + 0.601915i
\(75\) 0 0
\(76\) −15831.5 + 27421.0i −0.314405 + 0.544565i
\(77\) −47814.7 + 82817.5i −0.919040 + 1.59182i
\(78\) 0 0
\(79\) 25266.2 + 43762.3i 0.455482 + 0.788918i 0.998716 − 0.0506633i \(-0.0161335\pi\)
−0.543234 + 0.839582i \(0.682800\pi\)
\(80\) −10650.3 −0.186054
\(81\) 0 0
\(82\) 70070.9 1.15081
\(83\) −25913.9 − 44884.2i −0.412893 − 0.715152i 0.582311 − 0.812966i \(-0.302148\pi\)
−0.995205 + 0.0978135i \(0.968815\pi\)
\(84\) 0 0
\(85\) −26206.9 + 45391.7i −0.393431 + 0.681442i
\(86\) 45740.1 − 79224.3i 0.666886 − 1.15508i
\(87\) 0 0
\(88\) −15050.7 − 26068.7i −0.207182 − 0.358849i
\(89\) 20154.7 0.269713 0.134857 − 0.990865i \(-0.456943\pi\)
0.134857 + 0.990865i \(0.456943\pi\)
\(90\) 0 0
\(91\) −98206.5 −1.24319
\(92\) 3825.91 + 6626.66i 0.0471264 + 0.0816254i
\(93\) 0 0
\(94\) 34741.7 − 60174.4i 0.405538 − 0.702413i
\(95\) 41164.9 − 71299.6i 0.467970 − 0.810547i
\(96\) 0 0
\(97\) −40214.4 − 69653.3i −0.433962 − 0.751644i 0.563248 − 0.826288i \(-0.309552\pi\)
−0.997210 + 0.0746433i \(0.976218\pi\)
\(98\) −98130.4 −1.03214
\(99\) 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 54.6.c.b.37.2 6
3.2 odd 2 18.6.c.b.13.2 yes 6
4.3 odd 2 432.6.i.b.145.2 6
9.2 odd 6 18.6.c.b.7.2 ✓ 6
9.4 even 3 162.6.a.i.1.2 3
9.5 odd 6 162.6.a.j.1.2 3
9.7 even 3 inner 54.6.c.b.19.2 6
12.11 even 2 144.6.i.b.49.2 6
36.7 odd 6 432.6.i.b.289.2 6
36.11 even 6 144.6.i.b.97.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.2 ✓ 6 9.2 odd 6
18.6.c.b.13.2 yes 6 3.2 odd 2
54.6.c.b.19.2 6 9.7 even 3 inner
54.6.c.b.37.2 6 1.1 even 1 trivial
144.6.i.b.49.2 6 12.11 even 2
144.6.i.b.97.2 6 36.11 even 6
162.6.a.i.1.2 3 9.4 even 3
162.6.a.j.1.2 3 9.5 odd 6
432.6.i.b.145.2 6 4.3 odd 2
432.6.i.b.289.2 6 36.7 odd 6