Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,8,Mod(1,432)] 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("432.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,180,0,-700] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(134.950331009\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 432.1

$q$-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)
 
\(f(q)\) \(=\) \(q+114.187 q^{5} -1438.40 q^{7} -5928.74 q^{11} -11447.2 q^{13} -20235.6 q^{17} +6354.94 q^{19} -75845.6 q^{23} -65086.4 q^{25} -74784.3 q^{29} +189363. q^{31} -164247. q^{35} -33407.2 q^{37} +141245. q^{41} +246197. q^{43} -335133. q^{47} +1.24547e6 q^{49} +1.65156e6 q^{53} -676983. q^{55} -2.04823e6 q^{59} -590469. q^{61} -1.30712e6 q^{65} -53575.5 q^{67} +4.95678e6 q^{71} +817542. q^{73} +8.52792e6 q^{77} -7.57257e6 q^{79} +1.01891e6 q^{83} -2.31064e6 q^{85} +1.37281e6 q^{89} +1.64658e7 q^{91} +725650. q^{95} -1.06023e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 180 q^{5} - 700 q^{7} - 10890 q^{11} - 5480 q^{13} + 16416 q^{17} - 16024 q^{19} - 24372 q^{23} - 138880 q^{25} - 143280 q^{29} + 38708 q^{31} - 115650 q^{35} + 455620 q^{37} + 731880 q^{41} + 1088840 q^{43}+ \cdots + 4098670 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 114.187 0.408527 0.204264 − 0.978916i \(-0.434520\pi\)
0.204264 + 0.978916i \(0.434520\pi\)
\(6\) 0 0
\(7\) −1438.40 −1.58503 −0.792516 − 0.609851i \(-0.791229\pi\)
−0.792516 + 0.609851i \(0.791229\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5928.74 −1.34304 −0.671518 − 0.740988i \(-0.734357\pi\)
−0.671518 + 0.740988i \(0.734357\pi\)
\(12\) 0 0
\(13\) −11447.2 −1.44510 −0.722552 − 0.691317i \(-0.757031\pi\)
−0.722552 + 0.691317i \(0.757031\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −20235.6 −0.998955 −0.499477 − 0.866327i \(-0.666475\pi\)
−0.499477 + 0.866327i \(0.666475\pi\)
\(18\) 0 0
\(19\) 6354.94 0.212556 0.106278 − 0.994336i \(-0.466107\pi\)
0.106278 + 0.994336i \(0.466107\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −75845.6 −1.29982 −0.649910 − 0.760012i \(-0.725193\pi\)
−0.649910 + 0.760012i \(0.725193\pi\)
\(24\) 0 0
\(25\) −65086.4 −0.833106
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −74784.3 −0.569400 −0.284700 − 0.958617i \(-0.591894\pi\)
−0.284700 + 0.958617i \(0.591894\pi\)
\(30\) 0 0
\(31\) 189363. 1.14164 0.570820 − 0.821076i \(-0.306626\pi\)
0.570820 + 0.821076i \(0.306626\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −164247. −0.647528
\(36\) 0 0
\(37\) −33407.2 −0.108426 −0.0542130 − 0.998529i \(-0.517265\pi\)
−0.0542130 + 0.998529i \(0.517265\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 141245. 0.320058 0.160029 − 0.987112i \(-0.448841\pi\)
0.160029 + 0.987112i \(0.448841\pi\)
\(42\) 0 0
\(43\) 246197. 0.472219 0.236109 − 0.971726i \(-0.424128\pi\)
0.236109 + 0.971726i \(0.424128\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −335133. −0.470841 −0.235421 − 0.971894i \(-0.575647\pi\)
−0.235421 + 0.971894i \(0.575647\pi\)
\(48\) 0 0
\(49\) 1.24547e6 1.51233
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.65156e6 1.52381 0.761904 − 0.647691i \(-0.224265\pi\)
0.761904 + 0.647691i \(0.224265\pi\)
\(54\) 0 0
\(55\) −676983. −0.548667
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.04823e6 −1.29836 −0.649182 − 0.760633i \(-0.724889\pi\)
−0.649182 + 0.760633i \(0.724889\pi\)
\(60\) 0 0
\(61\) −590469. −0.333076 −0.166538 − 0.986035i \(-0.553259\pi\)
−0.166538 + 0.986035i \(0.553259\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.30712e6 −0.590364
\(66\) 0 0
\(67\) −53575.5 −0.0217623 −0.0108811 − 0.999941i \(-0.503464\pi\)
−0.0108811 + 0.999941i \(0.503464\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.95678e6 1.64360 0.821798 − 0.569779i \(-0.192971\pi\)
0.821798 + 0.569779i \(0.192971\pi\)
\(72\) 0 0
\(73\) 817542. 0.245969 0.122984 − 0.992409i \(-0.460753\pi\)
0.122984 + 0.992409i \(0.460753\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.52792e6 2.12875
\(78\) 0 0
\(79\) −7.57257e6 −1.72802 −0.864009 − 0.503476i \(-0.832054\pi\)
−0.864009 + 0.503476i \(0.832054\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.01891e6 0.195597 0.0977986 − 0.995206i \(-0.468820\pi\)
0.0977986 + 0.995206i \(0.468820\pi\)
\(84\) 0 0
\(85\) −2.31064e6 −0.408100
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.37281e6 0.206418 0.103209 − 0.994660i \(-0.467089\pi\)
0.103209 + 0.994660i \(0.467089\pi\)
\(90\) 0 0
\(91\) 1.64658e7 2.29054
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 725650. 0.0868350
\(96\) 0 0
\(97\) −1.06023e7 −1.17950 −0.589750 − 0.807586i \(-0.700774\pi\)
−0.589750 + 0.807586i \(0.700774\pi\)
\(98\) 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 432.8.a.q.1.2 2
3.2 odd 2 432.8.a.j.1.1 2
4.3 odd 2 27.8.a.e.1.2 yes 2
12.11 even 2 27.8.a.b.1.1 ✓ 2
36.7 odd 6 81.8.c.d.28.1 4
36.11 even 6 81.8.c.h.28.2 4
36.23 even 6 81.8.c.h.55.2 4
36.31 odd 6 81.8.c.d.55.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.1 ✓ 2 12.11 even 2
27.8.a.e.1.2 yes 2 4.3 odd 2
81.8.c.d.28.1 4 36.7 odd 6
81.8.c.d.55.1 4 36.31 odd 6
81.8.c.h.28.2 4 36.11 even 6
81.8.c.h.55.2 4 36.23 even 6
432.8.a.j.1.1 2 3.2 odd 2
432.8.a.q.1.2 2 1.1 even 1 trivial