Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,8,Mod(1,27)] 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("27.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,9,0,77] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.43439568807\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.53113\) of defining polynomial
Character \(\chi\) \(=\) 27.1

$q$-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)
 
\(f(q)\) \(=\) \(q+16.5934 q^{2} +147.340 q^{4} +114.187 q^{5} +1438.40 q^{7} +320.924 q^{8} +1894.75 q^{10} +5928.74 q^{11} -11447.2 q^{13} +23868.0 q^{14} -13534.4 q^{16} -20235.6 q^{17} -6354.94 q^{19} +16824.3 q^{20} +98377.8 q^{22} +75845.6 q^{23} -65086.4 q^{25} -189948. q^{26} +211935. q^{28} -74784.3 q^{29} -189363. q^{31} -265659. q^{32} -335778. q^{34} +164247. q^{35} -33407.2 q^{37} -105450. q^{38} +36645.3 q^{40} +141245. q^{41} -246197. q^{43} +873543. q^{44} +1.25854e6 q^{46} +335133. q^{47} +1.24547e6 q^{49} -1.08000e6 q^{50} -1.68664e6 q^{52} +1.65156e6 q^{53} +676983. q^{55} +461619. q^{56} -1.24092e6 q^{58} +2.04823e6 q^{59} -590469. q^{61} -3.14217e6 q^{62} -2.67579e6 q^{64} -1.30712e6 q^{65} +53575.5 q^{67} -2.98153e6 q^{68} +2.72541e6 q^{70} -4.95678e6 q^{71} +817542. q^{73} -554338. q^{74} -936340. q^{76} +8.52792e6 q^{77} +7.57257e6 q^{79} -1.54545e6 q^{80} +2.34373e6 q^{82} -1.01891e6 q^{83} -2.31064e6 q^{85} -4.08524e6 q^{86} +1.90267e6 q^{88} +1.37281e6 q^{89} -1.64658e7 q^{91} +1.11751e7 q^{92} +5.56099e6 q^{94} -725650. q^{95} -1.06023e7 q^{97} +2.06665e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{2} + 77 q^{4} + 180 q^{5} + 700 q^{7} + 1827 q^{8} + 1395 q^{10} + 10890 q^{11} - 5480 q^{13} + 29475 q^{14} - 15967 q^{16} + 16416 q^{17} + 16024 q^{19} + 12195 q^{20} + 60705 q^{22} + 24372 q^{23}+ \cdots + 22779738 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.5934 1.46666 0.733331 − 0.679872i \(-0.237965\pi\)
0.733331 + 0.679872i \(0.237965\pi\)
\(3\) 0 0
\(4\) 147.340 1.15110
\(5\) 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 \(-0.208771\pi\)
0.792516 + 0.609851i \(0.208771\pi\)
\(8\) 320.924 0.221609
\(9\) 0 0
\(10\) 1894.75 0.599171
\(11\) 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 \(0.757031\pi\)
\(14\) 23868.0 2.32471
\(15\) 0 0
\(16\) −13534.4 −0.826072
\(17\) −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 \(-0.533893\pi\)
−0.106278 + 0.994336i \(0.533893\pi\)
\(20\) 16824.3 0.470254
\(21\) 0 0
\(22\) 98377.8 1.96978
\(23\) 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
\(26\) −189948. −2.11948
\(27\) 0 0
\(28\) 211935. 1.82453
\(29\) −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 \(-0.693374\pi\)
−0.570820 + 0.821076i \(0.693374\pi\)
\(32\) −265659. −1.43318
\(33\) 0 0
\(34\) −335778. −1.46513
\(35\) 164247. 0.647528
\(36\) 0 0
\(37\) −33407.2 −0.108426 −0.0542130 − 0.998529i \(-0.517265\pi\)
−0.0542130 + 0.998529i \(0.517265\pi\)
\(38\) −105450. −0.311748
\(39\) 0 0
\(40\) 36645.3 0.0905332
\(41\) 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 \(-0.575872\pi\)
−0.236109 + 0.971726i \(0.575872\pi\)
\(44\) 873543. 1.54597
\(45\) 0 0
\(46\) 1.25854e6 1.90640
\(47\) 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
\(50\) −1.08000e6 −1.22188
\(51\) 0 0
\(52\) −1.68664e6 −1.66346
\(53\) 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
\(56\) 461619. 0.351257
\(57\) 0 0
\(58\) −1.24092e6 −0.835117
\(59\) 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 \(0.553259\pi\)
\(62\) −3.14217e6 −1.67440
\(63\) 0 0
\(64\) −2.67579e6 −1.27591
\(65\) −1.30712e6 −0.590364
\(66\) 0 0
\(67\) 53575.5 0.0217623 0.0108811 − 0.999941i \(-0.496536\pi\)
0.0108811 + 0.999941i \(0.496536\pi\)
\(68\) −2.98153e6 −1.14989
\(69\) 0 0
\(70\) 2.72541e6 0.949705
\(71\) −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 \(0.460753\pi\)
\(74\) −554338. −0.159024
\(75\) 0 0
\(76\) −936340. −0.244673
\(77\) 8.52792e6 2.12875
\(78\) 0 0
\(79\) 7.57257e6 1.72802 0.864009 − 0.503476i \(-0.167946\pi\)
0.864009 + 0.503476i \(0.167946\pi\)
\(80\) −1.54545e6 −0.337473
\(81\) 0 0
\(82\) 2.34373e6 0.469417
\(83\) −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
\(86\) −4.08524e6 −0.692585
\(87\) 0 0
\(88\) 1.90267e6 0.297629
\(89\) 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
\(92\) 1.11751e7 1.49622
\(93\) 0 0
\(94\) 5.56099e6 0.690565
\(95\) −725650. −0.0868350
\(96\) 0 0
\(97\) −1.06023e7 −1.17950 −0.589750 − 0.807586i \(-0.700774\pi\)
−0.589750 + 0.807586i \(0.700774\pi\)
\(98\) 2.06665e7 2.21807
\(99\) 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 27.8.a.e.1.2 yes 2
3.2 odd 2 27.8.a.b.1.1 ✓ 2
4.3 odd 2 432.8.a.q.1.2 2
9.2 odd 6 81.8.c.h.28.2 4
9.4 even 3 81.8.c.d.55.1 4
9.5 odd 6 81.8.c.h.55.2 4
9.7 even 3 81.8.c.d.28.1 4
12.11 even 2 432.8.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.1 ✓ 2 3.2 odd 2
27.8.a.e.1.2 yes 2 1.1 even 1 trivial
81.8.c.d.28.1 4 9.7 even 3
81.8.c.d.55.1 4 9.4 even 3
81.8.c.h.28.2 4 9.2 odd 6
81.8.c.h.55.2 4 9.5 odd 6
432.8.a.j.1.1 2 12.11 even 2
432.8.a.q.1.2 2 4.3 odd 2