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(10,27)] 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("27.10"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.c (of order \(3\), 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: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 19.3
Root \(0.500000 - 1.48508i\) of defining polynomial
Character \(\chi\) \(=\) 27.19
Dual form 27.8.c.a.10.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.536120 + 0.928588i) q^{2} +(63.4251 + 109.856i) q^{4} +(-47.9866 - 83.1153i) q^{5} +(-189.000 + 327.358i) q^{7} -273.261 q^{8} +102.906 q^{10} +(-3436.63 + 5952.41i) q^{11} +(4826.64 + 8359.99i) q^{13} +(-202.654 - 351.007i) q^{14} +(-7971.92 + 13807.8i) q^{16} -21431.3 q^{17} +5518.94 q^{19} +(6087.12 - 10543.2i) q^{20} +(-3684.89 - 6382.42i) q^{22} +(31486.4 + 54536.1i) q^{23} +(34457.1 - 59681.4i) q^{25} -10350.6 q^{26} -47949.5 q^{28} +(111113. - 192454. i) q^{29} +(-57729.1 - 99989.7i) q^{31} +(-26036.5 - 45096.6i) q^{32} +(11489.7 - 19900.8i) q^{34} +36277.9 q^{35} +81737.7 q^{37} +(-2958.82 + 5124.82i) q^{38} +(13112.9 + 22712.2i) q^{40} +(298773. + 517491. i) q^{41} +(-33874.2 + 58671.8i) q^{43} -871875. q^{44} -67522.1 q^{46} +(151740. - 262822. i) q^{47} +(340329. + 589468. i) q^{49} +(36946.3 + 63992.8i) q^{50} +(-612261. + 1.06047e6i) q^{52} -846755. q^{53} +659649. q^{55} +(51646.4 - 89454.2i) q^{56} +(119140. + 206357. i) q^{58} +(793119. + 1.37372e6i) q^{59} +(1.12706e6 - 1.95213e6i) q^{61} +123799. q^{62} -1.98498e6 q^{64} +(463228. - 802335. i) q^{65} +(-1.51172e6 - 2.61838e6i) q^{67} +(-1.35928e6 - 2.35434e6i) q^{68} +(-19449.3 + 33687.3i) q^{70} +4.41675e6 q^{71} +2.21484e6 q^{73} +(-43821.3 + 75900.6i) q^{74} +(350040. + 606287. i) q^{76} +(-1.29905e6 - 2.25002e6i) q^{77} +(-153821. + 266426. i) q^{79} +1.53018e6 q^{80} -640714. q^{82} +(-1.57735e6 + 2.73204e6i) q^{83} +(1.02841e6 + 1.78127e6i) q^{85} +(-36321.3 - 62910.3i) q^{86} +(939096. - 1.62656e6i) q^{88} -1.93441e6 q^{89} -3.64895e6 q^{91} +(-3.99406e6 + 6.91792e6i) q^{92} +(162702. + 281808. i) q^{94} +(-264836. - 458709. i) q^{95} +(-4.94528e6 + 8.56548e6i) q^{97} -729830. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{2} - 321 q^{4} + 180 q^{5} - 84 q^{7} - 5922 q^{8} + 252 q^{10} + 8460 q^{11} - 1848 q^{13} + 16272 q^{14} - 12417 q^{16} - 30564 q^{17} + 24432 q^{19} + 40788 q^{20} - 35001 q^{22} + 51588 q^{23}+ \cdots + 95833314 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.536120 + 0.928588i −0.0473868 + 0.0820763i −0.888746 − 0.458400i \(-0.848423\pi\)
0.841359 + 0.540476i \(0.181756\pi\)
\(3\) 0 0
\(4\) 63.4251 + 109.856i 0.495509 + 0.858247i
\(5\) −47.9866 − 83.1153i −0.171682 − 0.297362i 0.767326 − 0.641257i \(-0.221587\pi\)
−0.939008 + 0.343895i \(0.888254\pi\)
\(6\) 0 0
\(7\) −189.000 + 327.358i −0.208266 + 0.360728i −0.951169 − 0.308672i \(-0.900115\pi\)
0.742902 + 0.669400i \(0.233449\pi\)
\(8\) −273.261 −0.188696
\(9\) 0 0
\(10\) 102.906 0.0325419
\(11\) −3436.63 + 5952.41i −0.778499 + 1.34840i 0.154308 + 0.988023i \(0.450685\pi\)
−0.932807 + 0.360377i \(0.882648\pi\)
\(12\) 0 0
\(13\) 4826.64 + 8359.99i 0.609317 + 1.05537i 0.991353 + 0.131220i \(0.0418896\pi\)
−0.382036 + 0.924147i \(0.624777\pi\)
\(14\) −202.654 − 351.007i −0.0197382 − 0.0341875i
\(15\) 0 0
\(16\) −7971.92 + 13807.8i −0.486567 + 0.842759i
\(17\) −21431.3 −1.05798 −0.528989 − 0.848629i \(-0.677429\pi\)
−0.528989 + 0.848629i \(0.677429\pi\)
\(18\) 0 0
\(19\) 5518.94 0.184594 0.0922972 − 0.995732i \(-0.470579\pi\)
0.0922972 + 0.995732i \(0.470579\pi\)
\(20\) 6087.12 − 10543.2i 0.170140 − 0.294691i
\(21\) 0 0
\(22\) −3684.89 − 6382.42i −0.0737812 − 0.127793i
\(23\) 31486.4 + 54536.1i 0.539605 + 0.934623i 0.998925 + 0.0463526i \(0.0147598\pi\)
−0.459320 + 0.888271i \(0.651907\pi\)
\(24\) 0 0
\(25\) 34457.1 − 59681.4i 0.441050 − 0.763922i
\(26\) −10350.6 −0.115494
\(27\) 0 0
\(28\) −47949.5 −0.412792
\(29\) 111113. − 192454.i 0.846004 − 1.46532i −0.0387428 − 0.999249i \(-0.512335\pi\)
0.884747 − 0.466072i \(-0.154331\pi\)
\(30\) 0 0
\(31\) −57729.1 − 99989.7i −0.348040 − 0.602822i 0.637862 − 0.770151i \(-0.279819\pi\)
−0.985901 + 0.167329i \(0.946486\pi\)
\(32\) −26036.5 − 45096.6i −0.140462 − 0.243287i
\(33\) 0 0
\(34\) 11489.7 − 19900.8i 0.0501342 − 0.0868350i
\(35\) 36277.9 0.143023
\(36\) 0 0
\(37\) 81737.7 0.265287 0.132644 − 0.991164i \(-0.457653\pi\)
0.132644 + 0.991164i \(0.457653\pi\)
\(38\) −2958.82 + 5124.82i −0.00874734 + 0.0151508i
\(39\) 0 0
\(40\) 13112.9 + 22712.2i 0.0323957 + 0.0561111i
\(41\) 298773. + 517491.i 0.677015 + 1.17262i 0.975875 + 0.218328i \(0.0700602\pi\)
−0.298860 + 0.954297i \(0.596606\pi\)
\(42\) 0 0
\(43\) −33874.2 + 58671.8i −0.0649725 + 0.112536i −0.896682 − 0.442676i \(-0.854029\pi\)
0.831709 + 0.555211i \(0.187363\pi\)
\(44\) −871875. −1.54301
\(45\) 0 0
\(46\) −67522.1 −0.102281
\(47\) 151740. − 262822.i 0.213186 − 0.369249i −0.739524 − 0.673130i \(-0.764949\pi\)
0.952710 + 0.303881i \(0.0982827\pi\)
\(48\) 0 0
\(49\) 340329. + 589468.i 0.413250 + 0.715770i
\(50\) 36946.3 + 63992.8i 0.0417999 + 0.0723996i
\(51\) 0 0
\(52\) −612261. + 1.06047e6i −0.603844 + 1.04589i
\(53\) −846755. −0.781254 −0.390627 − 0.920549i \(-0.627742\pi\)
−0.390627 + 0.920549i \(0.627742\pi\)
\(54\) 0 0
\(55\) 659649. 0.534618
\(56\) 51646.4 − 89454.2i 0.0392990 − 0.0680679i
\(57\) 0 0
\(58\) 119140. + 206357.i 0.0801788 + 0.138874i
\(59\) 793119. + 1.37372e6i 0.502755 + 0.870797i 0.999995 + 0.00318395i \(0.00101348\pi\)
−0.497240 + 0.867613i \(0.665653\pi\)
\(60\) 0 0
\(61\) 1.12706e6 − 1.95213e6i 0.635760 − 1.10117i −0.350593 − 0.936528i \(-0.614020\pi\)
0.986353 − 0.164641i \(-0.0526467\pi\)
\(62\) 123799. 0.0659699
\(63\) 0 0
\(64\) −1.98498e6 −0.946510
\(65\) 463228. − 802335.i 0.209218 − 0.362376i
\(66\) 0 0
\(67\) −1.51172e6 − 2.61838e6i −0.614060 − 1.06358i −0.990549 − 0.137162i \(-0.956202\pi\)
0.376489 − 0.926421i \(-0.377131\pi\)
\(68\) −1.35928e6 − 2.35434e6i −0.524238 − 0.908006i
\(69\) 0 0
\(70\) −19449.3 + 33687.3i −0.00677738 + 0.0117388i
\(71\) 4.41675e6 1.46453 0.732266 − 0.681018i \(-0.238463\pi\)
0.732266 + 0.681018i \(0.238463\pi\)
\(72\) 0 0
\(73\) 2.21484e6 0.666366 0.333183 − 0.942862i \(-0.391877\pi\)
0.333183 + 0.942862i \(0.391877\pi\)
\(74\) −43821.3 + 75900.6i −0.0125711 + 0.0217738i
\(75\) 0 0
\(76\) 350040. + 606287.i 0.0914682 + 0.158428i
\(77\) −1.29905e6 − 2.25002e6i −0.324271 − 0.561653i
\(78\) 0 0
\(79\) −153821. + 266426.i −0.0351011 + 0.0607969i −0.883042 − 0.469293i \(-0.844509\pi\)
0.847941 + 0.530090i \(0.177842\pi\)
\(80\) 1.53018e6 0.334140
\(81\) 0 0
\(82\) −640714. −0.128326
\(83\) −1.57735e6 + 2.73204e6i −0.302798 + 0.524462i −0.976769 − 0.214296i \(-0.931254\pi\)
0.673970 + 0.738758i \(0.264588\pi\)
\(84\) 0 0
\(85\) 1.02841e6 + 1.78127e6i 0.181636 + 0.314603i
\(86\) −36321.3 − 62910.3i −0.00615768 − 0.0106654i
\(87\) 0 0
\(88\) 939096. − 1.62656e6i 0.146900 − 0.254438i
\(89\) −1.93441e6 −0.290859 −0.145430 − 0.989369i \(-0.546456\pi\)
−0.145430 + 0.989369i \(0.546456\pi\)
\(90\) 0 0
\(91\) −3.64895e6 −0.507601
\(92\) −3.99406e6 + 6.91792e6i −0.534758 + 0.926229i
\(93\) 0 0
\(94\) 162702. + 281808.i 0.0202044 + 0.0349950i
\(95\) −264836. − 458709.i −0.0316916 − 0.0548914i
\(96\) 0 0
\(97\) −4.94528e6 + 8.56548e6i −0.550161 + 0.952907i 0.448101 + 0.893983i \(0.352100\pi\)
−0.998262 + 0.0589243i \(0.981233\pi\)
\(98\) −729830. −0.0783304
\(99\) 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.c.a.19.3 12
3.2 odd 2 9.8.c.a.7.4 yes 12
4.3 odd 2 432.8.i.c.289.3 12
9.2 odd 6 81.8.a.e.1.3 6
9.4 even 3 inner 27.8.c.a.10.3 12
9.5 odd 6 9.8.c.a.4.4 ✓ 12
9.7 even 3 81.8.a.c.1.4 6
12.11 even 2 144.8.i.c.97.4 12
36.23 even 6 144.8.i.c.49.4 12
36.31 odd 6 432.8.i.c.145.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.4 ✓ 12 9.5 odd 6
9.8.c.a.7.4 yes 12 3.2 odd 2
27.8.c.a.10.3 12 9.4 even 3 inner
27.8.c.a.19.3 12 1.1 even 1 trivial
81.8.a.c.1.4 6 9.7 even 3
81.8.a.e.1.3 6 9.2 odd 6
144.8.i.c.49.4 12 36.23 even 6
144.8.i.c.97.4 12 12.11 even 2
432.8.i.c.145.3 12 36.31 odd 6
432.8.i.c.289.3 12 4.3 odd 2