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 28.2
Root \(2.26556 + 3.92407i\) of defining polynomial
Character \(\chi\) \(=\) 81.28
Dual form 81.8.c.h.55.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(8.29669 + 14.3703i) q^{2} +(-73.6702 + 127.601i) q^{4} +(57.0934 - 98.8886i) q^{5} +(-719.202 - 1245.70i) q^{7} -320.924 q^{8} +1894.75 q^{10} +(2964.37 + 5134.44i) q^{11} +(5723.62 - 9913.60i) q^{13} +(11934.0 - 20670.3i) q^{14} +(6767.18 + 11721.1i) q^{16} +20235.6 q^{17} -6354.94 q^{19} +(8412.17 + 14570.3i) q^{20} +(-49188.9 + 85197.7i) q^{22} +(37922.8 - 65684.2i) q^{23} +(32543.2 + 56366.5i) q^{25} +189948. q^{26} +211935. q^{28} +(-37392.1 - 64765.1i) q^{29} +(94681.4 - 163993. i) q^{31} +(-132830. + 230068. i) q^{32} +(167889. + 290792. i) q^{34} -164247. q^{35} -33407.2 q^{37} +(-52725.0 - 91322.4i) q^{38} +(-18322.6 + 31735.7i) q^{40} +(70622.4 - 122322. i) q^{41} +(123099. + 213213. i) q^{43} -873543. q^{44} +1.25854e6 q^{46} +(167566. + 290234. i) q^{47} +(-622733. + 1.07860e6i) q^{49} +(-540002. + 935310. i) q^{50} +(843321. + 1.46067e6i) q^{52} -1.65156e6 q^{53} +676983. q^{55} +(230809. + 399774. i) q^{56} +(620462. - 1.07467e6i) q^{58} +(1.02411e6 - 1.77382e6i) q^{59} +(295235. + 511361. i) q^{61} +3.14217e6 q^{62} -2.67579e6 q^{64} +(-653562. - 1.13200e6i) q^{65} +(-26787.7 + 46397.7i) q^{67} +(-1.49076e6 + 2.58208e6i) q^{68} +(-1.36271e6 - 2.36027e6i) q^{70} +4.95678e6 q^{71} +817542. q^{73} +(-277169. - 480071. i) q^{74} +(468170. - 810895. i) q^{76} +(4.26396e6 - 7.38540e6i) q^{77} +(-3.78629e6 - 6.55804e6i) q^{79} +1.54545e6 q^{80} +2.34373e6 q^{82} +(-509455. - 882402. i) q^{83} +(1.15532e6 - 2.00108e6i) q^{85} +(-2.04262e6 + 3.53792e6i) q^{86} +(-951337. - 1.64776e6i) q^{88} -1.37281e6 q^{89} -1.64658e7 q^{91} +(5.58756e6 + 9.67794e6i) q^{92} +(-2.78049e6 + 4.81596e6i) q^{94} +(-362825. + 628432. i) q^{95} +(5.30114e6 + 9.18185e6i) q^{97} -2.06665e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{2} - 77 q^{4} + 180 q^{5} - 700 q^{7} - 3654 q^{8} + 2790 q^{10} + 10890 q^{11} + 5480 q^{13} + 29475 q^{14} + 15967 q^{16} - 32832 q^{17} + 32048 q^{19} + 12195 q^{20} - 60705 q^{22} + 24372 q^{23}+ \cdots - 45559476 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.29669 + 14.3703i 0.733331 + 1.27017i 0.955452 + 0.295147i \(0.0953688\pi\)
−0.222121 + 0.975019i \(0.571298\pi\)
\(3\) 0 0
\(4\) −73.6702 + 127.601i −0.575549 + 0.996880i
\(5\) 57.0934 − 98.8886i 0.204264 − 0.353795i −0.745634 − 0.666355i \(-0.767853\pi\)
0.949898 + 0.312561i \(0.101187\pi\)
\(6\) 0 0
\(7\) −719.202 − 1245.70i −0.792516 − 1.37268i −0.924405 − 0.381413i \(-0.875437\pi\)
0.131889 − 0.991265i \(-0.457896\pi\)
\(8\) −320.924 −0.221609
\(9\) 0 0
\(10\) 1894.75 0.599171
\(11\) 2964.37 + 5134.44i 0.671518 + 1.16310i 0.977474 + 0.211058i \(0.0676908\pi\)
−0.305956 + 0.952046i \(0.598976\pi\)
\(12\) 0 0
\(13\) 5723.62 − 9913.60i 0.722552 − 1.25150i −0.237422 − 0.971407i \(-0.576302\pi\)
0.959974 − 0.280090i \(-0.0903643\pi\)
\(14\) 11934.0 − 20670.3i 1.16235 − 2.01325i
\(15\) 0 0
\(16\) 6767.18 + 11721.1i 0.413036 + 0.715399i
\(17\) 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 \(0.533893\pi\)
\(20\) 8412.17 + 14570.3i 0.235127 + 0.407252i
\(21\) 0 0
\(22\) −49188.9 + 85197.7i −0.984890 + 1.70588i
\(23\) 37922.8 − 65684.2i 0.649910 − 1.12568i −0.333235 − 0.942844i \(-0.608140\pi\)
0.983144 − 0.182832i \(-0.0585266\pi\)
\(24\) 0 0
\(25\) 32543.2 + 56366.5i 0.416553 + 0.721491i
\(26\) 189948. 2.11948
\(27\) 0 0
\(28\) 211935. 1.82453
\(29\) −37392.1 − 64765.1i −0.284700 − 0.493115i 0.687836 − 0.725866i \(-0.258561\pi\)
−0.972536 + 0.232751i \(0.925227\pi\)
\(30\) 0 0
\(31\) 94681.4 − 163993.i 0.570820 − 0.988688i −0.425663 − 0.904882i \(-0.639959\pi\)
0.996482 − 0.0838064i \(-0.0267077\pi\)
\(32\) −132830. + 230068.i −0.716589 + 1.24117i
\(33\) 0 0
\(34\) 167889. + 290792.i 0.732565 + 1.26884i
\(35\) −164247. −0.647528
\(36\) 0 0
\(37\) −33407.2 −0.108426 −0.0542130 − 0.998529i \(-0.517265\pi\)
−0.0542130 + 0.998529i \(0.517265\pi\)
\(38\) −52725.0 − 91322.4i −0.155874 − 0.269982i
\(39\) 0 0
\(40\) −18322.6 + 31735.7i −0.0452666 + 0.0784041i
\(41\) 70622.4 − 122322.i 0.160029 − 0.277179i −0.774850 − 0.632145i \(-0.782174\pi\)
0.934879 + 0.354967i \(0.115508\pi\)
\(42\) 0 0
\(43\) 123099. + 213213.i 0.236109 + 0.408954i 0.959595 − 0.281386i \(-0.0907943\pi\)
−0.723485 + 0.690340i \(0.757461\pi\)
\(44\) −873543. −1.54597
\(45\) 0 0
\(46\) 1.25854e6 1.90640
\(47\) 167566. + 290234.i 0.235421 + 0.407761i 0.959395 − 0.282067i \(-0.0910199\pi\)
−0.723974 + 0.689827i \(0.757687\pi\)
\(48\) 0 0
\(49\) −622733. + 1.07860e6i −0.756163 + 1.30971i
\(50\) −540002. + 935310.i −0.610942 + 1.05818i
\(51\) 0 0
\(52\) 843321. + 1.46067e6i 0.831728 + 1.44059i
\(53\) −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
\(56\) 230809. + 399774.i 0.175629 + 0.304198i
\(57\) 0 0
\(58\) 620462. − 1.07467e6i 0.417559 − 0.723233i
\(59\) 1.02411e6 − 1.77382e6i 0.649182 − 1.12442i −0.334136 − 0.942525i \(-0.608445\pi\)
0.983319 − 0.181892i \(-0.0582220\pi\)
\(60\) 0 0
\(61\) 295235. + 511361.i 0.166538 + 0.288452i 0.937200 − 0.348792i \(-0.113408\pi\)
−0.770663 + 0.637243i \(0.780075\pi\)
\(62\) 3.14217e6 1.67440
\(63\) 0 0
\(64\) −2.67579e6 −1.27591
\(65\) −653562. − 1.13200e6i −0.295182 − 0.511270i
\(66\) 0 0
\(67\) −26787.7 + 46397.7i −0.0108811 + 0.0188467i −0.871415 − 0.490547i \(-0.836797\pi\)
0.860534 + 0.509394i \(0.170130\pi\)
\(68\) −1.49076e6 + 2.58208e6i −0.574947 + 0.995838i
\(69\) 0 0
\(70\) −1.36271e6 − 2.36027e6i −0.474853 − 0.822469i
\(71\) 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 \(0.460753\pi\)
\(74\) −277169. − 480071.i −0.0795121 − 0.137719i
\(75\) 0 0
\(76\) 468170. − 810895.i 0.122337 − 0.211893i
\(77\) 4.26396e6 − 7.38540e6i 1.06438 − 1.84356i
\(78\) 0 0
\(79\) −3.78629e6 − 6.55804e6i −0.864009 − 1.49651i −0.868027 − 0.496516i \(-0.834612\pi\)
0.00401783 − 0.999992i \(-0.498721\pi\)
\(80\) 1.54545e6 0.337473
\(81\) 0 0
\(82\) 2.34373e6 0.469417
\(83\) −509455. − 882402.i −0.0977986 − 0.169392i 0.812975 − 0.582299i \(-0.197847\pi\)
−0.910773 + 0.412907i \(0.864513\pi\)
\(84\) 0 0
\(85\) 1.15532e6 − 2.00108e6i 0.204050 − 0.353425i
\(86\) −2.04262e6 + 3.53792e6i −0.346293 + 0.599797i
\(87\) 0 0
\(88\) −951337. − 1.64776e6i −0.148814 − 0.257754i
\(89\) −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
\(92\) 5.58756e6 + 9.67794e6i 0.748109 + 1.29576i
\(93\) 0 0
\(94\) −2.78049e6 + 4.81596e6i −0.345283 + 0.598047i
\(95\) −362825. + 628432.i −0.0434175 + 0.0752013i
\(96\) 0 0
\(97\) 5.30114e6 + 9.18185e6i 0.589750 + 1.02148i 0.994265 + 0.106946i \(0.0341072\pi\)
−0.404514 + 0.914532i \(0.632559\pi\)
\(98\) −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 81.8.c.h.28.2 4
3.2 odd 2 81.8.c.d.28.1 4
9.2 odd 6 81.8.c.d.55.1 4
9.4 even 3 27.8.a.b.1.1 ✓ 2
9.5 odd 6 27.8.a.e.1.2 yes 2
9.7 even 3 inner 81.8.c.h.55.2 4
36.23 even 6 432.8.a.q.1.2 2
36.31 odd 6 432.8.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.1 ✓ 2 9.4 even 3
27.8.a.e.1.2 yes 2 9.5 odd 6
81.8.c.d.28.1 4 3.2 odd 2
81.8.c.d.55.1 4 9.2 odd 6
81.8.c.h.28.2 4 1.1 even 1 trivial
81.8.c.h.55.2 4 9.7 even 3 inner
432.8.a.j.1.1 2 36.31 odd 6
432.8.a.q.1.2 2 36.23 even 6