Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,8,Mod(28,81)] 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("81.28"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,9,0,-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: no
Analytic conductor: \(25.3031870642\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{65})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 17x^{2} + 16x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 28.1
Root \(-1.76556 - 3.05805i\) of defining polynomial
Character \(\chi\) \(=\) 81.28
Dual form 81.8.c.h.55.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.79669 - 6.57607i) q^{2} +(35.1702 - 60.9166i) q^{4} +(32.9066 - 56.9959i) q^{5} +(369.202 + 639.477i) q^{7} -1506.08 q^{8} -499.745 q^{10} +(2480.63 + 4296.58i) q^{11} +(-2983.62 + 5167.78i) q^{13} +(2803.50 - 4855.80i) q^{14} +(1216.32 + 2106.72i) q^{16} -36651.6 q^{17} +22378.9 q^{19} +(-2314.67 - 4009.12i) q^{20} +(18836.4 - 32625.6i) q^{22} +(-25736.8 + 44577.4i) q^{23} +(36896.8 + 63907.1i) q^{25} +45311.5 q^{26} +51939.7 q^{28} +(-34247.9 - 59319.0i) q^{29} +(-75327.4 + 130471. i) q^{31} +(-87152.9 + 150953. i) q^{32} +(139155. + 241024. i) q^{34} +48596.8 q^{35} +489027. q^{37} +(-84966.0 - 147165. i) q^{38} +(-49559.9 + 85840.2i) q^{40} +(295318. - 511505. i) q^{41} +(421321. + 729750. i) q^{43} +348978. q^{44} +390859. q^{46} +(613184. + 1.06207e6i) q^{47} +(139151. - 241016. i) q^{49} +(280172. - 485272. i) q^{50} +(209869. + 363504. i) q^{52} -958904. q^{53} +326517. q^{55} +(-556047. - 963101. i) q^{56} +(-260057. + 450432. i) q^{58} +(-158135. + 273897. i) q^{59} +(14861.5 + 25740.8i) q^{61} +1.14398e6 q^{62} +1.63495e6 q^{64} +(196362. + 340108. i) q^{65} +(-146512. + 253767. i) q^{67} +(-1.28905e6 + 2.23270e6i) q^{68} +(-184507. - 319576. i) q^{70} -714537. q^{71} -3.96273e6 q^{73} +(-1.85669e6 - 3.21587e6i) q^{74} +(787073. - 1.36325e6i) q^{76} +(-1.83171e6 + 3.17262e6i) q^{77} +(-1.26902e6 - 2.19801e6i) q^{79} +160100. q^{80} -4.48492e6 q^{82} +(831556. + 1.44030e6i) q^{83} +(-1.20608e6 + 2.08899e6i) q^{85} +(3.19926e6 - 5.54128e6i) q^{86} +(-3.73602e6 - 6.47098e6i) q^{88} -4.64819e6 q^{89} -4.40624e6 q^{91} +(1.81034e6 + 3.13560e6i) q^{92} +(4.65614e6 - 8.06467e6i) q^{94} +(736415. - 1.27551e6i) q^{95} +(-7.35048e6 - 1.27314e7i) q^{97} -2.11325e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{2} - 77 q^{4} + 180 q^{5} - 700 q^{7} - 3654 q^{8} + 2790 q^{10} + 10890 q^{11} + 5480 q^{13} + 29475 q^{14} + 15967 q^{16} - 32832 q^{17} + 32048 q^{19} + 12195 q^{20} - 60705 q^{22} + 24372 q^{23}+ \cdots - 45559476 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.79669 − 6.57607i −0.335583 − 0.581248i 0.648013 − 0.761629i \(-0.275600\pi\)
−0.983597 + 0.180381i \(0.942267\pi\)
\(3\) 0 0
\(4\) 35.1702 − 60.9166i 0.274767 − 0.475911i
\(5\) 32.9066 − 56.9959i 0.117730 − 0.203915i −0.801138 − 0.598480i \(-0.795772\pi\)
0.918868 + 0.394565i \(0.129105\pi\)
\(6\) 0 0
\(7\) 369.202 + 639.477i 0.406838 + 0.704664i 0.994533 − 0.104418i \(-0.0332981\pi\)
−0.587696 + 0.809082i \(0.699965\pi\)
\(8\) −1506.08 −1.04000
\(9\) 0 0
\(10\) −499.745 −0.158033
\(11\) 2480.63 + 4296.58i 0.561937 + 0.973304i 0.997327 + 0.0730623i \(0.0232772\pi\)
−0.435390 + 0.900242i \(0.643389\pi\)
\(12\) 0 0
\(13\) −2983.62 + 5167.78i −0.376653 + 0.652382i −0.990573 − 0.136986i \(-0.956258\pi\)
0.613920 + 0.789368i \(0.289592\pi\)
\(14\) 2803.50 − 4855.80i 0.273056 − 0.472947i
\(15\) 0 0
\(16\) 1216.32 + 2106.72i 0.0742381 + 0.128584i
\(17\) −36651.6 −1.80935 −0.904674 − 0.426104i \(-0.859886\pi\)
−0.904674 + 0.426104i \(0.859886\pi\)
\(18\) 0 0
\(19\) 22378.9 0.748518 0.374259 − 0.927324i \(-0.377897\pi\)
0.374259 + 0.927324i \(0.377897\pi\)
\(20\) −2314.67 − 4009.12i −0.0646969 − 0.112058i
\(21\) 0 0
\(22\) 18836.4 − 32625.6i 0.377154 − 0.653250i
\(23\) −25736.8 + 44577.4i −0.441069 + 0.763955i −0.997769 − 0.0667591i \(-0.978734\pi\)
0.556700 + 0.830714i \(0.312067\pi\)
\(24\) 0 0
\(25\) 36896.8 + 63907.1i 0.472279 + 0.818012i
\(26\) 45311.5 0.505594
\(27\) 0 0
\(28\) 51939.7 0.447143
\(29\) −34247.9 − 59319.0i −0.260760 − 0.451649i 0.705684 − 0.708526i \(-0.250640\pi\)
−0.966444 + 0.256877i \(0.917306\pi\)
\(30\) 0 0
\(31\) −75327.4 + 130471.i −0.454137 + 0.786589i −0.998638 − 0.0521715i \(-0.983386\pi\)
0.544501 + 0.838760i \(0.316719\pi\)
\(32\) −87152.9 + 150953.i −0.470172 + 0.814362i
\(33\) 0 0
\(34\) 139155. + 241024.i 0.607187 + 1.05168i
\(35\) 48596.8 0.191589
\(36\) 0 0
\(37\) 489027. 1.58718 0.793591 − 0.608451i \(-0.208209\pi\)
0.793591 + 0.608451i \(0.208209\pi\)
\(38\) −84966.0 − 147165.i −0.251190 − 0.435074i
\(39\) 0 0
\(40\) −49559.9 + 85840.2i −0.122439 + 0.212071i
\(41\) 295318. − 511505.i 0.669184 − 1.15906i −0.308948 − 0.951079i \(-0.599977\pi\)
0.978133 − 0.207982i \(-0.0666897\pi\)
\(42\) 0 0
\(43\) 421321. + 729750.i 0.808117 + 1.39970i 0.914167 + 0.405338i \(0.132846\pi\)
−0.106050 + 0.994361i \(0.533820\pi\)
\(44\) 348978. 0.617609
\(45\) 0 0
\(46\) 390859. 0.592062
\(47\) 613184. + 1.06207e6i 0.861486 + 1.49214i 0.870495 + 0.492177i \(0.163799\pi\)
−0.00900942 + 0.999959i \(0.502868\pi\)
\(48\) 0 0
\(49\) 139151. − 241016.i 0.168966 − 0.292658i
\(50\) 280172. − 485272.i 0.316978 − 0.549022i
\(51\) 0 0
\(52\) 209869. + 363504.i 0.206984 + 0.358507i
\(53\) −958904. −0.884727 −0.442364 − 0.896836i \(-0.645860\pi\)
−0.442364 + 0.896836i \(0.645860\pi\)
\(54\) 0 0
\(55\) 326517. 0.264628
\(56\) −556047. − 963101.i −0.423110 − 0.732848i
\(57\) 0 0
\(58\) −260057. + 450432.i −0.175013 + 0.303132i
\(59\) −158135. + 273897.i −0.100241 + 0.173622i −0.911784 − 0.410670i \(-0.865295\pi\)
0.811543 + 0.584293i \(0.198628\pi\)
\(60\) 0 0
\(61\) 14861.5 + 25740.8i 0.00838315 + 0.0145200i 0.870187 − 0.492722i \(-0.163998\pi\)
−0.861803 + 0.507242i \(0.830665\pi\)
\(62\) 1.14398e6 0.609604
\(63\) 0 0
\(64\) 1.63495e6 0.779604
\(65\) 196362. + 340108.i 0.0886870 + 0.153610i
\(66\) 0 0
\(67\) −146512. + 253767.i −0.0595130 + 0.103080i −0.894247 − 0.447574i \(-0.852288\pi\)
0.834734 + 0.550654i \(0.185621\pi\)
\(68\) −1.28905e6 + 2.23270e6i −0.497150 + 0.861089i
\(69\) 0 0
\(70\) −184507. − 319576.i −0.0642939 − 0.111360i
\(71\) −714537. −0.236930 −0.118465 − 0.992958i \(-0.537797\pi\)
−0.118465 + 0.992958i \(0.537797\pi\)
\(72\) 0 0
\(73\) −3.96273e6 −1.19224 −0.596121 − 0.802894i \(-0.703292\pi\)
−0.596121 + 0.802894i \(0.703292\pi\)
\(74\) −1.85669e6 − 3.21587e6i −0.532632 − 0.922546i
\(75\) 0 0
\(76\) 787073. − 1.36325e6i 0.205668 − 0.356228i
\(77\) −1.83171e6 + 3.17262e6i −0.457235 + 0.791954i
\(78\) 0 0
\(79\) −1.26902e6 − 2.19801e6i −0.289584 − 0.501574i 0.684126 − 0.729363i \(-0.260184\pi\)
−0.973710 + 0.227789i \(0.926850\pi\)
\(80\) 160100. 0.0349603
\(81\) 0 0
\(82\) −4.48492e6 −0.898269
\(83\) 831556. + 1.44030e6i 0.159631 + 0.276490i 0.934736 − 0.355344i \(-0.115636\pi\)
−0.775104 + 0.631833i \(0.782303\pi\)
\(84\) 0 0
\(85\) −1.20608e6 + 2.08899e6i −0.213015 + 0.368953i
\(86\) 3.19926e6 − 5.54128e6i 0.542381 − 0.939432i
\(87\) 0 0
\(88\) −3.73602e6 − 6.47098e6i −0.584413 − 1.01223i
\(89\) −4.64819e6 −0.698906 −0.349453 − 0.936954i \(-0.613632\pi\)
−0.349453 + 0.936954i \(0.613632\pi\)
\(90\) 0 0
\(91\) −4.40624e6 −0.612947
\(92\) 1.81034e6 + 3.13560e6i 0.242383 + 0.419820i
\(93\) 0 0
\(94\) 4.65614e6 − 8.06467e6i 0.578201 − 1.00147i
\(95\) 736415. − 1.27551e6i 0.0881232 − 0.152634i
\(96\) 0 0
\(97\) −7.35048e6 − 1.27314e7i −0.817738 − 1.41636i −0.907345 − 0.420387i \(-0.861894\pi\)
0.0896064 − 0.995977i \(-0.471439\pi\)
\(98\) −2.11325e6 −0.226809
\(99\) 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 81.8.c.h.28.1 4
3.2 odd 2 81.8.c.d.28.2 4
9.2 odd 6 81.8.c.d.55.2 4
9.4 even 3 27.8.a.b.1.2 ✓ 2
9.5 odd 6 27.8.a.e.1.1 yes 2
9.7 even 3 inner 81.8.c.h.55.1 4
36.23 even 6 432.8.a.q.1.1 2
36.31 odd 6 432.8.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.2 ✓ 2 9.4 even 3
27.8.a.e.1.1 yes 2 9.5 odd 6
81.8.c.d.28.2 4 3.2 odd 2
81.8.c.d.55.2 4 9.2 odd 6
81.8.c.h.28.1 4 1.1 even 1 trivial
81.8.c.h.55.1 4 9.7 even 3 inner
432.8.a.j.1.2 2 36.31 odd 6
432.8.a.q.1.1 2 36.23 even 6