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: \(1\)
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 1.1
Root \(4.53113\) 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 \(-0.565480\pi\)
−0.204264 + 0.978916i \(0.565480\pi\)
\(6\) 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 \(-0.265643\pi\)
0.671518 + 0.740988i \(0.265643\pi\)
\(12\) 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 \(-0.333525\pi\)
0.499477 + 0.866327i \(0.333525\pi\)
\(18\) 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 \(-0.274807\pi\)
0.649910 + 0.760012i \(0.274807\pi\)
\(24\) 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 \(-0.408106\pi\)
0.284700 + 0.958617i \(0.408106\pi\)
\(30\) 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 \(-0.551159\pi\)
−0.160029 + 0.987112i \(0.551159\pi\)
\(42\) 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 \(-0.424353\pi\)
0.235421 + 0.971894i \(0.424353\pi\)
\(48\) 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 \(-0.775735\pi\)
−0.761904 + 0.647691i \(0.775735\pi\)
\(54\) 0 0
\(55\) −676983. −0.548667
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.04823e6 1.29836 0.649182 − 0.760633i \(-0.275111\pi\)
0.649182 + 0.760633i \(0.275111\pi\)
\(60\) 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 \(-0.807029\pi\)
−0.821798 + 0.569779i \(0.807029\pi\)
\(72\) 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 \(-0.531180\pi\)
−0.0977986 + 0.995206i \(0.531180\pi\)
\(84\) 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 \(-0.532911\pi\)
−0.103209 + 0.994660i \(0.532911\pi\)
\(90\) 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.j.1.1 2
3.2 odd 2 432.8.a.q.1.2 2
4.3 odd 2 27.8.a.b.1.1 ✓ 2
12.11 even 2 27.8.a.e.1.2 yes 2
36.7 odd 6 81.8.c.h.28.2 4
36.11 even 6 81.8.c.d.28.1 4
36.23 even 6 81.8.c.d.55.1 4
36.31 odd 6 81.8.c.h.55.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.1 ✓ 2 4.3 odd 2
27.8.a.e.1.2 yes 2 12.11 even 2
81.8.c.d.28.1 4 36.11 even 6
81.8.c.d.55.1 4 36.23 even 6
81.8.c.h.28.2 4 36.7 odd 6
81.8.c.h.55.2 4 36.31 odd 6
432.8.a.j.1.1 2 1.1 even 1 trivial
432.8.a.q.1.2 2 3.2 odd 2