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Properties

Label 27.9.d.a.17.3
Level $27$
Weight $9$
Character 27.17
Analytic conductor $10.999$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,Mod(8,27)] 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("27.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{30} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.3
Root \(-2.00397 + 3.47098i\) of defining polynomial
Character \(\chi\) \(=\) 27.17
Dual form 27.9.d.a.8.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-6.01192 + 3.47098i) q^{2} +(-103.905 + 179.968i) q^{4} +(-331.396 - 191.331i) q^{5} +(467.516 + 809.762i) q^{7} -3219.75i q^{8} +2656.43 q^{10} +(-4890.10 + 2823.30i) q^{11} +(17663.1 - 30593.3i) q^{13} +(-5621.34 - 3245.48i) q^{14} +(-15423.9 - 26714.9i) q^{16} -152914. i q^{17} +191248. q^{19} +(68867.1 - 39760.4i) q^{20} +(19599.3 - 33947.0i) q^{22} +(-132313. - 76390.7i) q^{23} +(-122097. - 211478. i) q^{25} +245233. i q^{26} -194308. q^{28} +(-401612. + 231871. i) q^{29} +(-393956. + 682351. i) q^{31} +(899280. + 519200. i) q^{32} +(530763. + 919308. i) q^{34} -357802. i q^{35} -1.10561e6 q^{37} +(-1.14977e6 + 663818. i) q^{38} +(-616039. + 1.06701e6i) q^{40} +(3.14915e6 + 1.81816e6i) q^{41} +(-1.51318e6 - 2.62091e6i) q^{43} -1.17342e6i q^{44} +1.06060e6 q^{46} +(-4.75308e6 + 2.74419e6i) q^{47} +(2.44526e6 - 4.23531e6i) q^{49} +(1.46808e6 + 847594. i) q^{50} +(3.67054e6 + 6.35757e6i) q^{52} -1.41381e7i q^{53} +2.16075e6 q^{55} +(2.60723e6 - 1.50528e6i) q^{56} +(1.60964e6 - 2.78797e6i) q^{58} +(-7.58082e6 - 4.37679e6i) q^{59} +(-3.47102e6 - 6.01198e6i) q^{61} -5.46965e6i q^{62} +688484. q^{64} +(-1.17069e7 + 6.75900e6i) q^{65} +(7.08899e6 - 1.22785e7i) q^{67} +(2.75197e7 + 1.58885e7i) q^{68} +(1.24193e6 + 2.15108e6i) q^{70} +7.97888e6i q^{71} -4.61414e6 q^{73} +(6.64682e6 - 3.83754e6i) q^{74} +(-1.98715e7 + 3.44184e7i) q^{76} +(-4.57241e6 - 2.63988e6i) q^{77} +(-1.37621e6 - 2.38366e6i) q^{79} +1.18043e7i q^{80} -2.52433e7 q^{82} +(-3.52291e7 + 2.03395e7i) q^{83} +(-2.92573e7 + 5.06751e7i) q^{85} +(1.81943e7 + 1.05045e7i) q^{86} +(9.09033e6 + 1.57449e7i) q^{88} +2.90621e7i q^{89} +3.30311e7 q^{91} +(2.74958e7 - 1.58747e7i) q^{92} +(1.90501e7 - 3.29957e7i) q^{94} +(-6.33787e7 - 3.65917e7i) q^{95} +(-4.58061e7 - 7.93386e7i) q^{97} +3.39498e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{2} + 767 q^{4} - 438 q^{5} + 922 q^{7} - 516 q^{10} + 28677 q^{11} + 1684 q^{13} - 120966 q^{14} - 65281 q^{16} - 269630 q^{19} - 539454 q^{20} + 61311 q^{22} + 1000452 q^{23} + 65177 q^{25}+ \cdots + 127049161 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.01192 + 3.47098i −0.375745 + 0.216937i −0.675965 − 0.736933i \(-0.736273\pi\)
0.300220 + 0.953870i \(0.402940\pi\)
\(3\) 0 0
\(4\) −103.905 + 179.968i −0.405877 + 0.703000i
\(5\) −331.396 − 191.331i −0.530233 − 0.306130i 0.210878 − 0.977512i \(-0.432368\pi\)
−0.741111 + 0.671382i \(0.765701\pi\)
\(6\) 0 0
\(7\) 467.516 + 809.762i 0.194717 + 0.337260i 0.946808 − 0.321800i \(-0.104288\pi\)
−0.752091 + 0.659060i \(0.770954\pi\)
\(8\) − 3219.75i − 0.786071i
\(9\) 0 0
\(10\) 2656.43 0.265643
\(11\) −4890.10 + 2823.30i −0.334001 + 0.192835i −0.657616 − 0.753353i \(-0.728435\pi\)
0.323615 + 0.946189i \(0.395102\pi\)
\(12\) 0 0
\(13\) 17663.1 − 30593.3i 0.618433 − 1.07116i −0.371339 − 0.928497i \(-0.621101\pi\)
0.989772 − 0.142660i \(-0.0455654\pi\)
\(14\) −5621.34 − 3245.48i −0.146328 − 0.0844826i
\(15\) 0 0
\(16\) −15423.9 − 26714.9i −0.235350 − 0.407637i
\(17\) − 152914.i − 1.83085i −0.402491 − 0.915424i \(-0.631856\pi\)
0.402491 − 0.915424i \(-0.368144\pi\)
\(18\) 0 0
\(19\) 191248. 1.46751 0.733756 − 0.679413i \(-0.237766\pi\)
0.733756 + 0.679413i \(0.237766\pi\)
\(20\) 68867.1 − 39760.4i 0.430419 − 0.248503i
\(21\) 0 0
\(22\) 19599.3 − 33947.0i 0.0836661 − 0.144914i
\(23\) −132313. − 76390.7i −0.472814 − 0.272979i 0.244603 − 0.969623i \(-0.421342\pi\)
−0.717417 + 0.696644i \(0.754676\pi\)
\(24\) 0 0
\(25\) −122097. − 211478.i −0.312568 − 0.541384i
\(26\) 245233.i 0.536643i
\(27\) 0 0
\(28\) −194308. −0.316125
\(29\) −401612. + 231871.i −0.567825 + 0.327834i −0.756280 − 0.654248i \(-0.772985\pi\)
0.188455 + 0.982082i \(0.439652\pi\)
\(30\) 0 0
\(31\) −393956. + 682351.i −0.426580 + 0.738858i −0.996567 − 0.0827958i \(-0.973615\pi\)
0.569987 + 0.821654i \(0.306948\pi\)
\(32\) 899280. + 519200.i 0.857621 + 0.495147i
\(33\) 0 0
\(34\) 530763. + 919308.i 0.397178 + 0.687932i
\(35\) − 357802.i − 0.238435i
\(36\) 0 0
\(37\) −1.10561e6 −0.589921 −0.294960 − 0.955509i \(-0.595306\pi\)
−0.294960 + 0.955509i \(0.595306\pi\)
\(38\) −1.14977e6 + 663818.i −0.551410 + 0.318357i
\(39\) 0 0
\(40\) −616039. + 1.06701e6i −0.240640 + 0.416801i
\(41\) 3.14915e6 + 1.81816e6i 1.11444 + 0.643425i 0.939977 − 0.341238i \(-0.110846\pi\)
0.174468 + 0.984663i \(0.444180\pi\)
\(42\) 0 0
\(43\) −1.51318e6 − 2.62091e6i −0.442607 − 0.766617i 0.555275 − 0.831667i \(-0.312613\pi\)
−0.997882 + 0.0650494i \(0.979280\pi\)
\(44\) − 1.17342e6i − 0.313070i
\(45\) 0 0
\(46\) 1.06060e6 0.236876
\(47\) −4.75308e6 + 2.74419e6i −0.974055 + 0.562371i −0.900470 − 0.434918i \(-0.856777\pi\)
−0.0735846 + 0.997289i \(0.523444\pi\)
\(48\) 0 0
\(49\) 2.44526e6 − 4.23531e6i 0.424170 − 0.734685i
\(50\) 1.46808e6 + 847594.i 0.234892 + 0.135615i
\(51\) 0 0
\(52\) 3.67054e6 + 6.35757e6i 0.502015 + 0.869516i
\(53\) − 1.41381e7i − 1.79179i −0.444269 − 0.895894i \(-0.646536\pi\)
0.444269 − 0.895894i \(-0.353464\pi\)
\(54\) 0 0
\(55\) 2.16075e6 0.236131
\(56\) 2.60723e6 − 1.50528e6i 0.265111 − 0.153062i
\(57\) 0 0
\(58\) 1.60964e6 − 2.78797e6i 0.142238 − 0.246364i
\(59\) −7.58082e6 − 4.37679e6i −0.625617 − 0.361200i 0.153436 − 0.988159i \(-0.450966\pi\)
−0.779052 + 0.626959i \(0.784299\pi\)
\(60\) 0 0
\(61\) −3.47102e6 − 6.01198e6i −0.250690 − 0.434209i 0.713026 − 0.701138i \(-0.247324\pi\)
−0.963716 + 0.266929i \(0.913991\pi\)
\(62\) − 5.46965e6i − 0.370163i
\(63\) 0 0
\(64\) 688484. 0.0410368
\(65\) −1.17069e7 + 6.75900e6i −0.655827 + 0.378642i
\(66\) 0 0
\(67\) 7.08899e6 − 1.22785e7i 0.351791 − 0.609321i −0.634772 − 0.772700i \(-0.718906\pi\)
0.986563 + 0.163379i \(0.0522393\pi\)
\(68\) 2.75197e7 + 1.58885e7i 1.28709 + 0.743099i
\(69\) 0 0
\(70\) 1.24193e6 + 2.15108e6i 0.0517254 + 0.0895909i
\(71\) 7.97888e6i 0.313985i 0.987600 + 0.156992i \(0.0501798\pi\)
−0.987600 + 0.156992i \(0.949820\pi\)
\(72\) 0 0
\(73\) −4.61414e6 −0.162480 −0.0812399 − 0.996695i \(-0.525888\pi\)
−0.0812399 + 0.996695i \(0.525888\pi\)
\(74\) 6.64682e6 − 3.83754e6i 0.221660 − 0.127975i
\(75\) 0 0
\(76\) −1.98715e7 + 3.44184e7i −0.595630 + 1.03166i
\(77\) −4.57241e6 − 2.63988e6i −0.130071 − 0.0750968i
\(78\) 0 0
\(79\) −1.37621e6 − 2.38366e6i −0.0353326 − 0.0611979i 0.847818 − 0.530287i \(-0.177916\pi\)
−0.883151 + 0.469089i \(0.844582\pi\)
\(80\) 1.18043e7i 0.288191i
\(81\) 0 0
\(82\) −2.52433e7 −0.558329
\(83\) −3.52291e7 + 2.03395e7i −0.742316 + 0.428577i −0.822911 − 0.568170i \(-0.807651\pi\)
0.0805945 + 0.996747i \(0.474318\pi\)
\(84\) 0 0
\(85\) −2.92573e7 + 5.06751e7i −0.560478 + 0.970776i
\(86\) 1.81943e7 + 1.05045e7i 0.332614 + 0.192035i
\(87\) 0 0
\(88\) 9.09033e6 + 1.57449e7i 0.151582 + 0.262548i
\(89\) 2.90621e7i 0.463199i 0.972811 + 0.231599i \(0.0743958\pi\)
−0.972811 + 0.231599i \(0.925604\pi\)
\(90\) 0 0
\(91\) 3.30311e7 0.481678
\(92\) 2.74958e7 − 1.58747e7i 0.383808 − 0.221592i
\(93\) 0 0
\(94\) 1.90501e7 − 3.29957e7i 0.243997 − 0.422616i
\(95\) −6.33787e7 − 3.65917e7i −0.778124 − 0.449250i
\(96\) 0 0
\(97\) −4.58061e7 − 7.93386e7i −0.517412 − 0.896185i −0.999795 − 0.0202242i \(-0.993562\pi\)
0.482383 − 0.875960i \(-0.339771\pi\)
\(98\) 3.39498e7i 0.368072i
\(99\) 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 27.9.d.a.17.3 14
3.2 odd 2 9.9.d.a.5.5 yes 14
4.3 odd 2 432.9.q.a.17.3 14
9.2 odd 6 inner 27.9.d.a.8.3 14
9.4 even 3 81.9.b.a.80.6 14
9.5 odd 6 81.9.b.a.80.9 14
9.7 even 3 9.9.d.a.2.5 ✓ 14
12.11 even 2 144.9.q.a.113.4 14
36.7 odd 6 144.9.q.a.65.4 14
36.11 even 6 432.9.q.a.305.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.5 ✓ 14 9.7 even 3
9.9.d.a.5.5 yes 14 3.2 odd 2
27.9.d.a.8.3 14 9.2 odd 6 inner
27.9.d.a.17.3 14 1.1 even 1 trivial
81.9.b.a.80.6 14 9.4 even 3
81.9.b.a.80.9 14 9.5 odd 6
144.9.q.a.65.4 14 36.7 odd 6
144.9.q.a.113.4 14 12.11 even 2
432.9.q.a.17.3 14 4.3 odd 2
432.9.q.a.305.3 14 36.11 even 6