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 55.2
Root \(2.26556 - 3.92407i\) of defining polynomial
Character \(\chi\) \(=\) 81.55
Dual form 81.8.c.h.28.2

$q$-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 \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.29669 − 14.3703i 0.733331 − 1.27017i −0.222121 − 0.975019i \(-0.571298\pi\)
0.955452 − 0.295147i \(-0.0953688\pi\)
\(3\) 0 0
\(4\) −73.6702 − 127.601i −0.575549 − 0.996880i
\(5\) 57.0934 + 98.8886i 0.204264 + 0.353795i 0.949898 − 0.312561i \(-0.101187\pi\)
−0.745634 + 0.666355i \(0.767853\pi\)
\(6\) 0 0
\(7\) −719.202 + 1245.70i −0.792516 + 1.37268i 0.131889 + 0.991265i \(0.457896\pi\)
−0.924405 + 0.381413i \(0.875437\pi\)
\(8\) −320.924 −0.221609
\(9\) 0 0
\(10\) 1894.75 0.599171
\(11\) 2964.37 − 5134.44i 0.671518 − 1.16310i −0.305956 − 0.952046i \(-0.598976\pi\)
0.977474 − 0.211058i \(-0.0676908\pi\)
\(12\) 0 0
\(13\) 5723.62 + 9913.60i 0.722552 + 1.25150i 0.959974 + 0.280090i \(0.0903643\pi\)
−0.237422 + 0.971407i \(0.576302\pi\)
\(14\) 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 0.983144 + 0.182832i \(0.0585266\pi\)
−0.333235 + 0.942844i \(0.608140\pi\)
\(24\) 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 −0.972536 − 0.232751i \(-0.925227\pi\)
0.687836 + 0.725866i \(0.258561\pi\)
\(30\) 0 0
\(31\) 94681.4 + 163993.i 0.570820 + 0.988688i 0.996482 + 0.0838064i \(0.0267077\pi\)
−0.425663 + 0.904882i \(0.639959\pi\)
\(32\) −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 0.934879 − 0.354967i \(-0.115508\pi\)
−0.774850 + 0.632145i \(0.782174\pi\)
\(42\) 0 0
\(43\) 123099. − 213213.i 0.236109 − 0.408954i −0.723485 − 0.690340i \(-0.757461\pi\)
0.959595 + 0.281386i \(0.0907943\pi\)
\(44\) −873543. −1.54597
\(45\) 0 0
\(46\) 1.25854e6 1.90640
\(47\) 167566. − 290234.i 0.235421 − 0.407761i −0.723974 − 0.689827i \(-0.757687\pi\)
0.959395 + 0.282067i \(0.0910199\pi\)
\(48\) 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 0.983319 + 0.181892i \(0.0582220\pi\)
−0.334136 + 0.942525i \(0.608445\pi\)
\(60\) 0 0
\(61\) 295235. − 511361.i 0.166538 − 0.288452i −0.770663 − 0.637243i \(-0.780075\pi\)
0.937200 + 0.348792i \(0.113408\pi\)
\(62\) 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 0.860534 − 0.509394i \(-0.170130\pi\)
−0.871415 + 0.490547i \(0.836797\pi\)
\(68\) −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 0.00401783 + 0.999992i \(0.498721\pi\)
−0.868027 + 0.496516i \(0.834612\pi\)
\(80\) 1.54545e6 0.337473
\(81\) 0 0
\(82\) 2.34373e6 0.469417
\(83\) −509455. + 882402.i −0.0977986 + 0.169392i −0.910773 − 0.412907i \(-0.864513\pi\)
0.812975 + 0.582299i \(0.197847\pi\)
\(84\) 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 −0.404514 − 0.914532i \(-0.632559\pi\)
0.994265 − 0.106946i \(-0.0341072\pi\)
\(98\) −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.55.2 4
3.2 odd 2 81.8.c.d.55.1 4
9.2 odd 6 27.8.a.e.1.2 yes 2
9.4 even 3 inner 81.8.c.h.28.2 4
9.5 odd 6 81.8.c.d.28.1 4
9.7 even 3 27.8.a.b.1.1 ✓ 2
36.7 odd 6 432.8.a.j.1.1 2
36.11 even 6 432.8.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.1 ✓ 2 9.7 even 3
27.8.a.e.1.2 yes 2 9.2 odd 6
81.8.c.d.28.1 4 9.5 odd 6
81.8.c.d.55.1 4 3.2 odd 2
81.8.c.h.28.2 4 9.4 even 3 inner
81.8.c.h.55.2 4 1.1 even 1 trivial
432.8.a.j.1.1 2 36.7 odd 6
432.8.a.q.1.2 2 36.11 even 6