Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,2,Mod(10,81)] 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("81.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 81.e (of order \(9\), degree \(6\), 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: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 64.1
Root \(0.500000 - 0.0126039i\) of defining polynomial
Character \(\chi\) \(=\) 81.64
Dual form 81.2.e.a.19.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.753189 - 0.274138i) q^{2} +(-1.03995 - 0.872619i) q^{4} +(0.477505 - 2.70806i) q^{5} +(1.82076 - 1.52780i) q^{7} +(1.34559 + 2.33062i) q^{8} +(-1.10204 + 1.90878i) q^{10} +(0.0434396 + 0.246358i) q^{11} +(-2.45446 + 0.893351i) q^{13} +(-1.79020 + 0.651581i) q^{14} +(0.0969067 + 0.549585i) q^{16} +(-0.146688 + 0.254072i) q^{17} +(1.39237 + 2.41166i) q^{19} +(-2.85969 + 2.39956i) q^{20} +(0.0348180 - 0.197463i) q^{22} +(5.12472 + 4.30015i) q^{23} +(-2.40714 - 0.876128i) q^{25} +2.09357 q^{26} -3.22668 q^{28} +(-0.333645 - 0.121437i) q^{29} +(2.11847 + 1.77761i) q^{31} +(1.01231 - 5.74108i) q^{32} +(0.180135 - 0.151151i) q^{34} +(-3.26796 - 5.66027i) q^{35} +(3.49619 - 6.05558i) q^{37} +(-0.387591 - 2.19814i) q^{38} +(6.95400 - 2.53105i) q^{40} +(-9.13156 + 3.32362i) q^{41} +(0.0452712 + 0.256746i) q^{43} +(0.169802 - 0.294106i) q^{44} +(-2.68104 - 4.64370i) q^{46} +(8.75249 - 7.34421i) q^{47} +(-0.234540 + 1.33014i) q^{49} +(1.57285 + 1.31978i) q^{50} +(3.33207 + 1.21277i) q^{52} -5.43137 q^{53} +0.687897 q^{55} +(6.01071 + 2.18772i) q^{56} +(0.218007 + 0.182930i) q^{58} +(-1.03788 + 5.88612i) q^{59} +(-9.07515 + 7.61495i) q^{61} +(-1.10830 - 1.91963i) q^{62} +(-1.77824 + 3.08001i) q^{64} +(1.24723 + 7.07342i) q^{65} +(-1.70113 + 0.619160i) q^{67} +(0.374256 - 0.136218i) q^{68} +(0.909693 + 5.15912i) q^{70} +(-0.185255 + 0.320871i) q^{71} +(-2.51339 - 4.35333i) q^{73} +(-4.29336 + 3.60255i) q^{74} +(0.656467 - 3.72301i) q^{76} +(0.455479 + 0.382193i) q^{77} +(-0.754406 - 0.274581i) q^{79} +1.53459 q^{80} +7.78892 q^{82} +(-2.58947 - 0.942488i) q^{83} +(0.617998 + 0.518562i) q^{85} +(0.0362861 - 0.205789i) q^{86} +(-0.515717 + 0.432738i) q^{88} +(5.22533 + 9.05054i) q^{89} +(-3.10412 + 5.37650i) q^{91} +(-1.57704 - 8.94385i) q^{92} +(-8.60560 + 3.13218i) q^{94} +(7.19580 - 2.61906i) q^{95} +(-2.57600 - 14.6092i) q^{97} +(0.541296 - 0.937552i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 6 q^{4} + 3 q^{5} - 6 q^{7} - 6 q^{8} - 3 q^{10} - 3 q^{11} - 6 q^{13} - 15 q^{14} - 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 12 q^{23} + 3 q^{25} + 30 q^{26} - 12 q^{28} + 6 q^{29}+ \cdots + 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.753189 − 0.274138i −0.532585 − 0.193845i 0.0617072 − 0.998094i \(-0.480346\pi\)
−0.594292 + 0.804249i \(0.702568\pi\)
\(3\) 0 0
\(4\) −1.03995 − 0.872619i −0.519974 − 0.436310i
\(5\) 0.477505 − 2.70806i 0.213547 − 1.21108i −0.669864 − 0.742484i \(-0.733648\pi\)
0.883411 − 0.468600i \(-0.155241\pi\)
\(6\) 0 0
\(7\) 1.82076 − 1.52780i 0.688183 − 0.577454i −0.230202 − 0.973143i \(-0.573939\pi\)
0.918385 + 0.395689i \(0.129494\pi\)
\(8\) 1.34559 + 2.33062i 0.475736 + 0.823999i
\(9\) 0 0
\(10\) −1.10204 + 1.90878i −0.348494 + 0.603610i
\(11\) 0.0434396 + 0.246358i 0.0130975 + 0.0742798i 0.990656 − 0.136385i \(-0.0435483\pi\)
−0.977558 + 0.210665i \(0.932437\pi\)
\(12\) 0 0
\(13\) −2.45446 + 0.893351i −0.680745 + 0.247771i −0.659167 − 0.751996i \(-0.729091\pi\)
−0.0215777 + 0.999767i \(0.506869\pi\)
\(14\) −1.79020 + 0.651581i −0.478452 + 0.174142i
\(15\) 0 0
\(16\) 0.0969067 + 0.549585i 0.0242267 + 0.137396i
\(17\) −0.146688 + 0.254072i −0.0355772 + 0.0616215i −0.883266 − 0.468873i \(-0.844660\pi\)
0.847689 + 0.530494i \(0.177994\pi\)
\(18\) 0 0
\(19\) 1.39237 + 2.41166i 0.319432 + 0.553273i 0.980370 − 0.197168i \(-0.0631745\pi\)
−0.660937 + 0.750441i \(0.729841\pi\)
\(20\) −2.85969 + 2.39956i −0.639446 + 0.536559i
\(21\) 0 0
\(22\) 0.0348180 − 0.197463i 0.00742323 − 0.0420992i
\(23\) 5.12472 + 4.30015i 1.06858 + 0.896643i 0.994923 − 0.100641i \(-0.0320894\pi\)
0.0736543 + 0.997284i \(0.476534\pi\)
\(24\) 0 0
\(25\) −2.40714 − 0.876128i −0.481428 − 0.175226i
\(26\) 2.09357 0.410584
\(27\) 0 0
\(28\) −3.22668 −0.609786
\(29\) −0.333645 − 0.121437i −0.0619562 − 0.0225502i 0.310856 − 0.950457i \(-0.399384\pi\)
−0.372812 + 0.927907i \(0.621606\pi\)
\(30\) 0 0
\(31\) 2.11847 + 1.77761i 0.380488 + 0.319268i 0.812894 − 0.582411i \(-0.197891\pi\)
−0.432406 + 0.901679i \(0.642335\pi\)
\(32\) 1.01231 − 5.74108i 0.178952 − 1.01489i
\(33\) 0 0
\(34\) 0.180135 − 0.151151i 0.0308929 − 0.0259222i
\(35\) −3.26796 − 5.66027i −0.552386 − 0.956760i
\(36\) 0 0
\(37\) 3.49619 − 6.05558i 0.574770 − 0.995531i −0.421297 − 0.906923i \(-0.638425\pi\)
0.996067 − 0.0886080i \(-0.0282418\pi\)
\(38\) −0.387591 − 2.19814i −0.0628756 − 0.356585i
\(39\) 0 0
\(40\) 6.95400 − 2.53105i 1.09952 − 0.400194i
\(41\) −9.13156 + 3.32362i −1.42611 + 0.519062i −0.935814 − 0.352494i \(-0.885334\pi\)
−0.490296 + 0.871556i \(0.663112\pi\)
\(42\) 0 0
\(43\) 0.0452712 + 0.256746i 0.00690379 + 0.0391534i 0.988065 − 0.154037i \(-0.0492276\pi\)
−0.981161 + 0.193191i \(0.938116\pi\)
\(44\) 0.169802 − 0.294106i 0.0255986 − 0.0443381i
\(45\) 0 0
\(46\) −2.68104 − 4.64370i −0.395298 − 0.684677i
\(47\) 8.75249 − 7.34421i 1.27668 − 1.07126i 0.282989 − 0.959123i \(-0.408674\pi\)
0.993692 − 0.112140i \(-0.0357704\pi\)
\(48\) 0 0
\(49\) −0.234540 + 1.33014i −0.0335057 + 0.190020i
\(50\) 1.57285 + 1.31978i 0.222435 + 0.186645i
\(51\) 0 0
\(52\) 3.33207 + 1.21277i 0.462074 + 0.168181i
\(53\) −5.43137 −0.746056 −0.373028 − 0.927820i \(-0.621680\pi\)
−0.373028 + 0.927820i \(0.621680\pi\)
\(54\) 0 0
\(55\) 0.687897 0.0927560
\(56\) 6.01071 + 2.18772i 0.803215 + 0.292346i
\(57\) 0 0
\(58\) 0.218007 + 0.182930i 0.0286257 + 0.0240198i
\(59\) −1.03788 + 5.88612i −0.135121 + 0.766308i 0.839655 + 0.543121i \(0.182757\pi\)
−0.974776 + 0.223188i \(0.928354\pi\)
\(60\) 0 0
\(61\) −9.07515 + 7.61495i −1.16195 + 0.974995i −0.999930 − 0.0117924i \(-0.996246\pi\)
−0.162023 + 0.986787i \(0.551802\pi\)
\(62\) −1.10830 − 1.91963i −0.140754 − 0.243793i
\(63\) 0 0
\(64\) −1.77824 + 3.08001i −0.222281 + 0.385001i
\(65\) 1.24723 + 7.07342i 0.154700 + 0.877350i
\(66\) 0 0
\(67\) −1.70113 + 0.619160i −0.207826 + 0.0756424i −0.443835 − 0.896108i \(-0.646383\pi\)
0.236010 + 0.971751i \(0.424160\pi\)
\(68\) 0.374256 − 0.136218i 0.0453852 − 0.0165189i
\(69\) 0 0
\(70\) 0.909693 + 5.15912i 0.108729 + 0.616633i
\(71\) −0.185255 + 0.320871i −0.0219857 + 0.0380804i −0.876809 − 0.480839i \(-0.840332\pi\)
0.854823 + 0.518919i \(0.173666\pi\)
\(72\) 0 0
\(73\) −2.51339 − 4.35333i −0.294171 − 0.509518i 0.680621 − 0.732636i \(-0.261710\pi\)
−0.974792 + 0.223117i \(0.928377\pi\)
\(74\) −4.29336 + 3.60255i −0.499093 + 0.418789i
\(75\) 0 0
\(76\) 0.656467 − 3.72301i 0.0753019 − 0.427059i
\(77\) 0.455479 + 0.382193i 0.0519067 + 0.0435549i
\(78\) 0 0
\(79\) −0.754406 − 0.274581i −0.0848773 − 0.0308928i 0.299233 − 0.954180i \(-0.403269\pi\)
−0.384110 + 0.923287i \(0.625492\pi\)
\(80\) 1.53459 0.171572
\(81\) 0 0
\(82\) 7.78892 0.860143
\(83\) −2.58947 − 0.942488i −0.284231 − 0.103452i 0.195971 − 0.980610i \(-0.437214\pi\)
−0.480201 + 0.877158i \(0.659436\pi\)
\(84\) 0 0
\(85\) 0.617998 + 0.518562i 0.0670313 + 0.0562460i
\(86\) 0.0362861 − 0.205789i 0.00391283 − 0.0221908i
\(87\) 0 0
\(88\) −0.515717 + 0.432738i −0.0549756 + 0.0461300i
\(89\) 5.22533 + 9.05054i 0.553884 + 0.959356i 0.997989 + 0.0633809i \(0.0201883\pi\)
−0.444105 + 0.895975i \(0.646478\pi\)
\(90\) 0 0
\(91\) −3.10412 + 5.37650i −0.325401 + 0.563611i
\(92\) −1.57704 − 8.94385i −0.164418 − 0.932461i
\(93\) 0 0
\(94\) −8.60560 + 3.13218i −0.887600 + 0.323060i
\(95\) 7.19580 − 2.61906i 0.738273 − 0.268709i
\(96\) 0 0
\(97\) −2.57600 − 14.6092i −0.261553 − 1.48334i −0.778673 − 0.627430i \(-0.784107\pi\)
0.517120 − 0.855913i \(-0.327004\pi\)
\(98\) 0.541296 − 0.937552i 0.0546791 − 0.0947070i
\(99\) 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 81.2.e.a.64.1 12
3.2 odd 2 27.2.e.a.4.2 ✓ 12
9.2 odd 6 243.2.e.c.28.1 12
9.4 even 3 243.2.e.a.109.2 12
9.5 odd 6 243.2.e.d.109.1 12
9.7 even 3 243.2.e.b.28.2 12
12.11 even 2 432.2.u.c.193.2 12
15.2 even 4 675.2.u.b.274.2 24
15.8 even 4 675.2.u.b.274.3 24
15.14 odd 2 675.2.l.c.301.1 12
27.2 odd 18 243.2.e.d.136.1 12
27.4 even 9 729.2.c.b.244.3 12
27.5 odd 18 729.2.c.e.487.4 12
27.7 even 9 inner 81.2.e.a.19.1 12
27.11 odd 18 243.2.e.c.217.1 12
27.13 even 9 729.2.a.d.1.4 6
27.14 odd 18 729.2.a.a.1.3 6
27.16 even 9 243.2.e.b.217.2 12
27.20 odd 18 27.2.e.a.7.2 yes 12
27.22 even 9 729.2.c.b.487.3 12
27.23 odd 18 729.2.c.e.244.4 12
27.25 even 9 243.2.e.a.136.2 12
108.47 even 18 432.2.u.c.385.2 12
135.47 even 36 675.2.u.b.574.3 24
135.74 odd 18 675.2.l.c.601.1 12
135.128 even 36 675.2.u.b.574.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.2 ✓ 12 3.2 odd 2
27.2.e.a.7.2 yes 12 27.20 odd 18
81.2.e.a.19.1 12 27.7 even 9 inner
81.2.e.a.64.1 12 1.1 even 1 trivial
243.2.e.a.109.2 12 9.4 even 3
243.2.e.a.136.2 12 27.25 even 9
243.2.e.b.28.2 12 9.7 even 3
243.2.e.b.217.2 12 27.16 even 9
243.2.e.c.28.1 12 9.2 odd 6
243.2.e.c.217.1 12 27.11 odd 18
243.2.e.d.109.1 12 9.5 odd 6
243.2.e.d.136.1 12 27.2 odd 18
432.2.u.c.193.2 12 12.11 even 2
432.2.u.c.385.2 12 108.47 even 18
675.2.l.c.301.1 12 15.14 odd 2
675.2.l.c.601.1 12 135.74 odd 18
675.2.u.b.274.2 24 15.2 even 4
675.2.u.b.274.3 24 15.8 even 4
675.2.u.b.574.2 24 135.128 even 36
675.2.u.b.574.3 24 135.47 even 36
729.2.a.a.1.3 6 27.14 odd 18
729.2.a.d.1.4 6 27.13 even 9
729.2.c.b.244.3 12 27.4 even 9
729.2.c.b.487.3 12 27.22 even 9
729.2.c.e.244.4 12 27.23 odd 18
729.2.c.e.487.4 12 27.5 odd 18