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 = [2,6,0,92] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 55.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 81.55
Dual form 81.8.c.c.28.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.00000 - 5.19615i) q^{2} +(46.0000 + 79.6743i) q^{4} +(195.000 + 337.750i) q^{5} +(32.0000 - 55.4256i) q^{7} +1320.00 q^{8} +2340.00 q^{10} +(-474.000 + 820.992i) q^{11} +(2549.00 + 4415.00i) q^{13} +(-192.000 - 332.554i) q^{14} +(-1928.00 + 3339.39i) q^{16} -28386.0 q^{17} -8620.00 q^{19} +(-17940.0 + 31073.0i) q^{20} +(2844.00 + 4925.95i) q^{22} +(-7644.00 - 13239.8i) q^{23} +(-36987.5 + 64064.2i) q^{25} +30588.0 q^{26} +5888.00 q^{28} +(18255.0 - 31618.6i) q^{29} +(138404. + 239723. i) q^{31} +(96048.0 + 166360. i) q^{32} +(-85158.0 + 147498. i) q^{34} +24960.0 q^{35} +268526. q^{37} +(-25860.0 + 44790.8i) q^{38} +(257400. + 445830. i) q^{40} +(-314859. - 545352. i) q^{41} +(-342886. + 593896. i) q^{43} -87216.0 q^{44} -91728.0 q^{46} +(291648. - 505149. i) q^{47} +(409724. + 709662. i) q^{49} +(221925. + 384385. i) q^{50} +(-234508. + 406180. i) q^{52} +428058. q^{53} -369720. q^{55} +(42240.0 - 73161.8i) q^{56} +(-109530. - 189712. i) q^{58} +(653190. + 1.13136e6i) q^{59} +(-150331. + 260381. i) q^{61} +1.66085e6 q^{62} +659008. q^{64} +(-994110. + 1.72185e6i) q^{65} +(253622. + 439286. i) q^{67} +(-1.30576e6 - 2.26164e6i) q^{68} +(74880.0 - 129696. i) q^{70} -5.56063e6 q^{71} +1.36908e6 q^{73} +(805578. - 1.39530e6i) q^{74} +(-396520. - 686793. i) q^{76} +(30336.0 + 52543.5i) q^{77} +(3.45686e6 - 5.98746e6i) q^{79} -1.50384e6 q^{80} -3.77831e6 q^{82} +(-2.18837e6 + 3.79037e6i) q^{83} +(-5.53527e6 - 9.58737e6i) q^{85} +(2.05732e6 + 3.56338e6i) q^{86} +(-625680. + 1.08371e6i) q^{88} +8.52831e6 q^{89} +326272. q^{91} +(703248. - 1.21806e6i) q^{92} +(-1.74989e6 - 3.03089e6i) q^{94} +(-1.68090e6 - 2.91140e6i) q^{95} +(4.41341e6 - 7.64425e6i) q^{97} +4.91668e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 92 q^{4} + 390 q^{5} + 64 q^{7} + 2640 q^{8} + 4680 q^{10} - 948 q^{11} + 5098 q^{13} - 384 q^{14} - 3856 q^{16} - 56772 q^{17} - 17240 q^{19} - 35880 q^{20} + 5688 q^{22} - 15288 q^{23}+ \cdots + 9833364 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 − 5.19615i 0.265165 − 0.459279i −0.702442 − 0.711741i \(-0.747907\pi\)
0.967607 + 0.252462i \(0.0812402\pi\)
\(3\) 0 0
\(4\) 46.0000 + 79.6743i 0.359375 + 0.622456i
\(5\) 195.000 + 337.750i 0.697653 + 1.20837i 0.969278 + 0.245968i \(0.0791057\pi\)
−0.271625 + 0.962403i \(0.587561\pi\)
\(6\) 0 0
\(7\) 32.0000 − 55.4256i 0.0352620 − 0.0610756i −0.847856 − 0.530227i \(-0.822107\pi\)
0.883118 + 0.469151i \(0.155440\pi\)
\(8\) 1320.00 0.911505
\(9\) 0 0
\(10\) 2340.00 0.739973
\(11\) −474.000 + 820.992i −0.107375 + 0.185979i −0.914706 − 0.404120i \(-0.867578\pi\)
0.807331 + 0.590099i \(0.200911\pi\)
\(12\) 0 0
\(13\) 2549.00 + 4415.00i 0.321787 + 0.557351i 0.980857 − 0.194731i \(-0.0623833\pi\)
−0.659070 + 0.752082i \(0.729050\pi\)
\(14\) −192.000 − 332.554i −0.0187005 − 0.0323902i
\(15\) 0 0
\(16\) −1928.00 + 3339.39i −0.117676 + 0.203820i
\(17\) −28386.0 −1.40131 −0.700653 − 0.713502i \(-0.747108\pi\)
−0.700653 + 0.713502i \(0.747108\pi\)
\(18\) 0 0
\(19\) −8620.00 −0.288317 −0.144158 − 0.989555i \(-0.546047\pi\)
−0.144158 + 0.989555i \(0.546047\pi\)
\(20\) −17940.0 + 31073.0i −0.501438 + 0.868517i
\(21\) 0 0
\(22\) 2844.00 + 4925.95i 0.0569443 + 0.0986304i
\(23\) −7644.00 − 13239.8i −0.131001 − 0.226900i 0.793062 − 0.609141i \(-0.208486\pi\)
−0.924063 + 0.382241i \(0.875152\pi\)
\(24\) 0 0
\(25\) −36987.5 + 64064.2i −0.473440 + 0.820022i
\(26\) 30588.0 0.341306
\(27\) 0 0
\(28\) 5888.00 0.0506891
\(29\) 18255.0 − 31618.6i 0.138992 − 0.240741i −0.788124 − 0.615517i \(-0.788947\pi\)
0.927115 + 0.374776i \(0.122281\pi\)
\(30\) 0 0
\(31\) 138404. + 239723.i 0.834416 + 1.44525i 0.894505 + 0.447058i \(0.147528\pi\)
−0.0600887 + 0.998193i \(0.519138\pi\)
\(32\) 96048.0 + 166360.i 0.518159 + 0.897478i
\(33\) 0 0
\(34\) −85158.0 + 147498.i −0.371577 + 0.643591i
\(35\) 24960.0 0.0984026
\(36\) 0 0
\(37\) 268526. 0.871526 0.435763 − 0.900061i \(-0.356479\pi\)
0.435763 + 0.900061i \(0.356479\pi\)
\(38\) −25860.0 + 44790.8i −0.0764515 + 0.132418i
\(39\) 0 0
\(40\) 257400. + 445830.i 0.635914 + 1.10144i
\(41\) −314859. − 545352.i −0.713465 − 1.23576i −0.963549 − 0.267533i \(-0.913791\pi\)
0.250084 − 0.968224i \(-0.419542\pi\)
\(42\) 0 0
\(43\) −342886. + 593896.i −0.657673 + 1.13912i 0.323543 + 0.946213i \(0.395126\pi\)
−0.981216 + 0.192910i \(0.938207\pi\)
\(44\) −87216.0 −0.154352
\(45\) 0 0
\(46\) −91728.0 −0.138947
\(47\) 291648. − 505149.i 0.409748 − 0.709704i −0.585114 − 0.810951i \(-0.698950\pi\)
0.994861 + 0.101248i \(0.0322834\pi\)
\(48\) 0 0
\(49\) 409724. + 709662.i 0.497513 + 0.861718i
\(50\) 221925. + 384385.i 0.251079 + 0.434882i
\(51\) 0 0
\(52\) −234508. + 406180.i −0.231284 + 0.400596i
\(53\) 428058. 0.394945 0.197473 − 0.980308i \(-0.436727\pi\)
0.197473 + 0.980308i \(0.436727\pi\)
\(54\) 0 0
\(55\) −369720. −0.299643
\(56\) 42240.0 − 73161.8i 0.0321415 − 0.0556707i
\(57\) 0 0
\(58\) −109530. − 189712.i −0.0737115 − 0.127672i
\(59\) 653190. + 1.13136e6i 0.414054 + 0.717163i 0.995329 − 0.0965444i \(-0.0307790\pi\)
−0.581274 + 0.813708i \(0.697446\pi\)
\(60\) 0 0
\(61\) −150331. + 260381.i −0.0847997 + 0.146877i −0.905306 − 0.424760i \(-0.860358\pi\)
0.820506 + 0.571638i \(0.193692\pi\)
\(62\) 1.66085e6 0.885032
\(63\) 0 0
\(64\) 659008. 0.314240
\(65\) −994110. + 1.72185e6i −0.448991 + 0.777675i
\(66\) 0 0
\(67\) 253622. + 439286.i 0.103021 + 0.178437i 0.912928 − 0.408121i \(-0.133816\pi\)
−0.809907 + 0.586558i \(0.800482\pi\)
\(68\) −1.30576e6 − 2.26164e6i −0.503594 − 0.872251i
\(69\) 0 0
\(70\) 74880.0 − 129696.i 0.0260929 − 0.0451943i
\(71\) −5.56063e6 −1.84383 −0.921913 − 0.387397i \(-0.873374\pi\)
−0.921913 + 0.387397i \(0.873374\pi\)
\(72\) 0 0
\(73\) 1.36908e6 0.411907 0.205954 − 0.978562i \(-0.433970\pi\)
0.205954 + 0.978562i \(0.433970\pi\)
\(74\) 805578. − 1.39530e6i 0.231098 − 0.400274i
\(75\) 0 0
\(76\) −396520. − 686793.i −0.103614 − 0.179464i
\(77\) 30336.0 + 52543.5i 0.00757253 + 0.0131160i
\(78\) 0 0
\(79\) 3.45686e6 − 5.98746e6i 0.788836 − 1.36630i −0.137844 − 0.990454i \(-0.544017\pi\)
0.926680 − 0.375851i \(-0.122649\pi\)
\(80\) −1.50384e6 −0.328388
\(81\) 0 0
\(82\) −3.77831e6 −0.756744
\(83\) −2.18837e6 + 3.79037e6i −0.420096 + 0.727627i −0.995948 − 0.0899264i \(-0.971337\pi\)
0.575853 + 0.817553i \(0.304670\pi\)
\(84\) 0 0
\(85\) −5.53527e6 − 9.58737e6i −0.977626 − 1.69330i
\(86\) 2.05732e6 + 3.56338e6i 0.348784 + 0.604111i
\(87\) 0 0
\(88\) −625680. + 1.08371e6i −0.0978730 + 0.169521i
\(89\) 8.52831e6 1.28232 0.641162 − 0.767405i \(-0.278453\pi\)
0.641162 + 0.767405i \(0.278453\pi\)
\(90\) 0 0
\(91\) 326272. 0.0453874
\(92\) 703248. − 1.21806e6i 0.0941567 − 0.163084i
\(93\) 0 0
\(94\) −1.74989e6 − 3.03089e6i −0.217302 − 0.376377i
\(95\) −1.68090e6 − 2.91140e6i −0.201145 − 0.348393i
\(96\) 0 0
\(97\) 4.41341e6 − 7.64425e6i 0.490990 − 0.850420i −0.508956 − 0.860793i \(-0.669968\pi\)
0.999946 + 0.0103725i \(0.00330171\pi\)
\(98\) 4.91668e6 0.527692
\(99\) 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.c.55.1 2
3.2 odd 2 81.8.c.a.55.1 2
9.2 odd 6 3.8.a.a.1.1 ✓ 1
9.4 even 3 inner 81.8.c.c.28.1 2
9.5 odd 6 81.8.c.a.28.1 2
9.7 even 3 9.8.a.a.1.1 1
36.7 odd 6 144.8.a.b.1.1 1
36.11 even 6 48.8.a.g.1.1 1
45.2 even 12 75.8.b.c.49.2 2
45.7 odd 12 225.8.b.f.199.1 2
45.29 odd 6 75.8.a.a.1.1 1
45.34 even 6 225.8.a.i.1.1 1
45.38 even 12 75.8.b.c.49.1 2
45.43 odd 12 225.8.b.f.199.2 2
63.2 odd 6 147.8.e.b.67.1 2
63.11 odd 6 147.8.e.b.79.1 2
63.20 even 6 147.8.a.b.1.1 1
63.34 odd 6 441.8.a.a.1.1 1
63.38 even 6 147.8.e.a.79.1 2
63.47 even 6 147.8.e.a.67.1 2
72.11 even 6 192.8.a.a.1.1 1
72.29 odd 6 192.8.a.i.1.1 1
72.43 odd 6 576.8.a.x.1.1 1
72.61 even 6 576.8.a.w.1.1 1
99.65 even 6 363.8.a.b.1.1 1
117.38 odd 6 507.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.8.a.a.1.1 ✓ 1 9.2 odd 6
9.8.a.a.1.1 1 9.7 even 3
48.8.a.g.1.1 1 36.11 even 6
75.8.a.a.1.1 1 45.29 odd 6
75.8.b.c.49.1 2 45.38 even 12
75.8.b.c.49.2 2 45.2 even 12
81.8.c.a.28.1 2 9.5 odd 6
81.8.c.a.55.1 2 3.2 odd 2
81.8.c.c.28.1 2 9.4 even 3 inner
81.8.c.c.55.1 2 1.1 even 1 trivial
144.8.a.b.1.1 1 36.7 odd 6
147.8.a.b.1.1 1 63.20 even 6
147.8.e.a.67.1 2 63.47 even 6
147.8.e.a.79.1 2 63.38 even 6
147.8.e.b.67.1 2 63.2 odd 6
147.8.e.b.79.1 2 63.11 odd 6
192.8.a.a.1.1 1 72.11 even 6
192.8.a.i.1.1 1 72.29 odd 6
225.8.a.i.1.1 1 45.34 even 6
225.8.b.f.199.1 2 45.7 odd 12
225.8.b.f.199.2 2 45.43 odd 12
363.8.a.b.1.1 1 99.65 even 6
441.8.a.a.1.1 1 63.34 odd 6
507.8.a.a.1.1 1 117.38 odd 6
576.8.a.w.1.1 1 72.61 even 6
576.8.a.x.1.1 1 72.43 odd 6