Properties

Label 27.8.a.b.1.1
Level $27$
Weight $8$
Character 27.1
Self dual yes
Analytic conductor $8.434$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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 \(-0.762035\pi\)
−0.733331 + 0.679872i \(0.762035\pi\)
\(3\) 0 0
\(4\) 147.340 1.15110
\(5\) −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 \(-0.208771\pi\)
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 \(-0.734357\pi\)
−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 \(0.757031\pi\)
\(14\) −23868.0 −2.32471
\(15\) 0 0
\(16\) −13534.4 −0.826072
\(17\) 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 \(-0.533893\pi\)
−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 \(-0.725193\pi\)
−0.649910 + 0.760012i \(0.725193\pi\)
\(24\) 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 \(-0.408106\pi\)
0.284700 + 0.958617i \(0.408106\pi\)
\(30\) 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 \(-0.551159\pi\)
−0.160029 + 0.987112i \(0.551159\pi\)
\(42\) 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 \(-0.575647\pi\)
−0.235421 + 0.971894i \(0.575647\pi\)
\(48\) 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 \(-0.775735\pi\)
−0.761904 + 0.647691i \(0.775735\pi\)
\(54\) 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 \(-0.724889\pi\)
−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 \(0.553259\pi\)
\(62\) 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 \(-0.192971\pi\)
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 \(0.460753\pi\)
\(74\) 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 \(-0.468820\pi\)
0.0977986 + 0.995206i \(0.468820\pi\)
\(84\) 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 \(-0.532911\pi\)
−0.103209 + 0.994660i \(0.532911\pi\)
\(90\) 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.b.1.1 ✓ 2
3.2 odd 2 27.8.a.e.1.2 yes 2
4.3 odd 2 432.8.a.j.1.1 2
9.2 odd 6 81.8.c.d.28.1 4
9.4 even 3 81.8.c.h.55.2 4
9.5 odd 6 81.8.c.d.55.1 4
9.7 even 3 81.8.c.h.28.2 4
12.11 even 2 432.8.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.1 ✓ 2 1.1 even 1 trivial
27.8.a.e.1.2 yes 2 3.2 odd 2
81.8.c.d.28.1 4 9.2 odd 6
81.8.c.d.55.1 4 9.5 odd 6
81.8.c.h.28.2 4 9.7 even 3
81.8.c.h.55.2 4 9.4 even 3
432.8.a.j.1.1 2 4.3 odd 2
432.8.a.q.1.2 2 12.11 even 2