Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,6,Mod(145,432)] 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("432.145"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.i (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: \(69.2858101592\)
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 145.2
Root \(0.500000 - 8.40123i\) of defining polynomial
Character \(\chi\) \(=\) 432.145
Dual form 432.6.i.b.289.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(20.8014 - 36.0292i) q^{5} +(-101.661 - 176.082i) q^{7} +(-235.168 - 407.323i) q^{11} +(-241.506 + 418.300i) q^{13} -1259.86 q^{17} -1978.94 q^{19} +(-239.119 + 414.166i) q^{23} +(697.100 + 1207.41i) q^{25} +(-580.249 - 1005.02i) q^{29} +(-1186.50 + 2055.07i) q^{31} -8458.76 q^{35} +8185.10 q^{37} +(8758.87 - 15170.8i) q^{41} +(11435.0 + 19806.1i) q^{43} +(8685.43 + 15043.6i) q^{47} +(-12266.3 + 21245.9i) q^{49} +5390.58 q^{53} -19567.3 q^{55} +(22281.8 - 38593.2i) q^{59} +(-2084.17 - 3609.88i) q^{61} +(10047.3 + 17402.5i) q^{65} +(1228.46 - 2127.76i) q^{67} +2184.37 q^{71} +3037.34 q^{73} +(-47814.7 + 82817.5i) q^{77} +(-25266.2 - 43762.3i) q^{79} +(25913.9 + 44884.2i) q^{83} +(-26206.9 + 45391.7i) q^{85} +20154.7 q^{89} +98206.5 q^{91} +(-41164.9 + 71299.6i) q^{95} +(-40214.4 - 69653.3i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47}+ \cdots + 46245 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 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 \(-0.879758\pi\)
0.145330 − 0.989383i \(-0.453576\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −235.168 − 407.323i −0.585998 − 1.01498i −0.994750 − 0.102333i \(-0.967369\pi\)
0.408752 − 0.912646i \(-0.365964\pi\)
\(12\) 0 0
\(13\) −241.506 + 418.300i −0.396341 + 0.686482i −0.993271 − 0.115811i \(-0.963053\pi\)
0.596931 + 0.802293i \(0.296387\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −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 \(-0.716458\pi\)
−0.628810 + 0.777559i \(0.716458\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −239.119 + 414.166i −0.0942529 + 0.163251i −0.909297 − 0.416149i \(-0.863380\pi\)
0.815044 + 0.579400i \(0.196713\pi\)
\(24\) 0 0
\(25\) 697.100 + 1207.41i 0.223072 + 0.386372i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −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 −0.955339 − 0.295511i \(-0.904510\pi\)
0.733590 + 0.679592i \(0.237843\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8458.76 −1.16718
\(36\) 0 0
\(37\) 8185.10 0.982923 0.491462 − 0.870899i \(-0.336463\pi\)
0.491462 + 0.870899i \(0.336463\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(0.225456\pi\)
0.183644 + 0.982993i \(0.441211\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8685.43 + 15043.6i 0.573518 + 0.993362i 0.996201 + 0.0870843i \(0.0277550\pi\)
−0.422683 + 0.906277i \(0.638912\pi\)
\(48\) 0 0
\(49\) −12266.3 + 21245.9i −0.729833 + 1.26411i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 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
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 22281.8 − 38593.2i 0.833335 − 1.44338i −0.0620431 − 0.998073i \(-0.519762\pi\)
0.895379 − 0.445306i \(-0.146905\pi\)
\(60\) 0 0
\(61\) −2084.17 − 3609.88i −0.0717147 − 0.124213i 0.827938 − 0.560819i \(-0.189514\pi\)
−0.899653 + 0.436606i \(0.856180\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10047.3 + 17402.5i 0.294963 + 0.510891i
\(66\) 0 0
\(67\) 1228.46 − 2127.76i 0.0334330 − 0.0579076i −0.848825 − 0.528674i \(-0.822689\pi\)
0.882258 + 0.470767i \(0.156023\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2184.37 0.0514256 0.0257128 − 0.999669i \(-0.491814\pi\)
0.0257128 + 0.999669i \(0.491814\pi\)
\(72\) 0 0
\(73\) 3037.34 0.0667093 0.0333546 − 0.999444i \(-0.489381\pi\)
0.0333546 + 0.999444i \(0.489381\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −47814.7 + 82817.5i −0.919040 + 1.59182i
\(78\) 0 0
\(79\) −25266.2 − 43762.3i −0.455482 − 0.788918i 0.543234 − 0.839582i \(-0.317200\pi\)
−0.998716 + 0.0506633i \(0.983866\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 25913.9 + 44884.2i 0.412893 + 0.715152i 0.995205 − 0.0978135i \(-0.0311849\pi\)
−0.582311 + 0.812966i \(0.697852\pi\)
\(84\) 0 0
\(85\) −26206.9 + 45391.7i −0.393431 + 0.681442i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 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
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −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 \(0.976218\pi\)
\(98\) 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 432.6.i.b.145.2 6
3.2 odd 2 144.6.i.b.49.2 6
4.3 odd 2 54.6.c.b.37.2 6
9.2 odd 6 144.6.i.b.97.2 6
9.7 even 3 inner 432.6.i.b.289.2 6
12.11 even 2 18.6.c.b.13.2 yes 6
36.7 odd 6 54.6.c.b.19.2 6
36.11 even 6 18.6.c.b.7.2 ✓ 6
36.23 even 6 162.6.a.j.1.2 3
36.31 odd 6 162.6.a.i.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.2 ✓ 6 36.11 even 6
18.6.c.b.13.2 yes 6 12.11 even 2
54.6.c.b.19.2 6 36.7 odd 6
54.6.c.b.37.2 6 4.3 odd 2
144.6.i.b.49.2 6 3.2 odd 2
144.6.i.b.97.2 6 9.2 odd 6
162.6.a.i.1.2 3 36.31 odd 6
162.6.a.j.1.2 3 36.23 even 6
432.6.i.b.145.2 6 1.1 even 1 trivial
432.6.i.b.289.2 6 9.7 even 3 inner