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Properties

Label 81.9.b.a.80.5
Level $81$
Weight $9$
Character 81.80
Analytic conductor $32.998$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,9,Mod(80,81)] 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("81.80"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{63} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 80.5
Root \(4.05115 - 7.01679i\) of defining polynomial
Character \(\chi\) \(=\) 81.80
Dual form 81.9.b.a.80.10

$q$-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)
 
\(f(q)\) \(=\) \(q-14.0336i q^{2} +59.0584 q^{4} -780.826i q^{5} -4337.21 q^{7} -4421.40i q^{8} -10957.8 q^{10} +6474.45i q^{11} +16995.5 q^{13} +60866.7i q^{14} -46929.1 q^{16} -29881.7i q^{17} -91768.1 q^{19} -46114.4i q^{20} +90859.8 q^{22} +122846. i q^{23} -219065. q^{25} -238508. i q^{26} -256149. q^{28} +501795. i q^{29} -1.01186e6 q^{31} -473294. i q^{32} -419347. q^{34} +3.38661e6i q^{35} +190279. q^{37} +1.28784e6i q^{38} -3.45235e6 q^{40} +2.25881e6i q^{41} +3.28994e6 q^{43} +382371. i q^{44} +1.72397e6 q^{46} -127214. i q^{47} +1.30466e7 q^{49} +3.07427e6i q^{50} +1.00373e6 q^{52} -1.40856e6i q^{53} +5.05542e6 q^{55} +1.91766e7i q^{56} +7.04199e6 q^{58} +2.07119e7i q^{59} -1.92431e7 q^{61} +1.42000e7i q^{62} -1.86559e7 q^{64} -1.32705e7i q^{65} -2.34312e7 q^{67} -1.76476e6i q^{68} +4.75263e7 q^{70} -9.12361e6i q^{71} +1.43931e7 q^{73} -2.67030e6i q^{74} -5.41968e6 q^{76} -2.80811e7i q^{77} -5.39607e7 q^{79} +3.66435e7i q^{80} +3.16992e7 q^{82} +3.07329e7i q^{83} -2.33324e7 q^{85} -4.61696e7i q^{86} +2.86261e7 q^{88} +2.32435e7i q^{89} -7.37131e7 q^{91} +7.25509e6i q^{92} -1.78527e6 q^{94} +7.16550e7i q^{95} -4.93189e7 q^{97} -1.83091e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 1534 q^{4} - 1844 q^{7} - 516 q^{10} - 3368 q^{13} + 130562 q^{16} - 269630 q^{19} - 122622 q^{22} - 130354 q^{25} + 1075708 q^{28} + 328264 q^{31} - 1309986 q^{34} - 1671668 q^{37} - 1226652 q^{40}+ \cdots - 254098322 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 14.0336i − 0.877099i −0.898707 − 0.438550i \(-0.855492\pi\)
0.898707 − 0.438550i \(-0.144508\pi\)
\(3\) 0 0
\(4\) 59.0584 0.230697
\(5\) − 780.826i − 1.24932i −0.780896 − 0.624661i \(-0.785237\pi\)
0.780896 − 0.624661i \(-0.214763\pi\)
\(6\) 0 0
\(7\) −4337.21 −1.80642 −0.903210 − 0.429199i \(-0.858796\pi\)
−0.903210 + 0.429199i \(0.858796\pi\)
\(8\) − 4421.40i − 1.07944i
\(9\) 0 0
\(10\) −10957.8 −1.09578
\(11\) 6474.45i 0.442214i 0.975250 + 0.221107i \(0.0709669\pi\)
−0.975250 + 0.221107i \(0.929033\pi\)
\(12\) 0 0
\(13\) 16995.5 0.595060 0.297530 − 0.954713i \(-0.403837\pi\)
0.297530 + 0.954713i \(0.403837\pi\)
\(14\) 60866.7i 1.58441i
\(15\) 0 0
\(16\) −46929.1 −0.716082
\(17\) − 29881.7i − 0.357774i −0.983870 − 0.178887i \(-0.942750\pi\)
0.983870 − 0.178887i \(-0.0572497\pi\)
\(18\) 0 0
\(19\) −91768.1 −0.704170 −0.352085 − 0.935968i \(-0.614527\pi\)
−0.352085 + 0.935968i \(0.614527\pi\)
\(20\) − 46114.4i − 0.288215i
\(21\) 0 0
\(22\) 90859.8 0.387865
\(23\) 122846.i 0.438985i 0.975614 + 0.219492i \(0.0704401\pi\)
−0.975614 + 0.219492i \(0.929560\pi\)
\(24\) 0 0
\(25\) −219065. −0.560806
\(26\) − 238508.i − 0.521926i
\(27\) 0 0
\(28\) −256149. −0.416736
\(29\) 501795.i 0.709471i 0.934967 + 0.354736i \(0.115429\pi\)
−0.934967 + 0.354736i \(0.884571\pi\)
\(30\) 0 0
\(31\) −1.01186e6 −1.09566 −0.547828 − 0.836591i \(-0.684545\pi\)
−0.547828 + 0.836591i \(0.684545\pi\)
\(32\) − 473294.i − 0.451369i
\(33\) 0 0
\(34\) −419347. −0.313803
\(35\) 3.38661e6i 2.25680i
\(36\) 0 0
\(37\) 190279. 0.101528 0.0507638 − 0.998711i \(-0.483834\pi\)
0.0507638 + 0.998711i \(0.483834\pi\)
\(38\) 1.28784e6i 0.617627i
\(39\) 0 0
\(40\) −3.45235e6 −1.34857
\(41\) 2.25881e6i 0.799364i 0.916654 + 0.399682i \(0.130879\pi\)
−0.916654 + 0.399682i \(0.869121\pi\)
\(42\) 0 0
\(43\) 3.28994e6 0.962308 0.481154 − 0.876636i \(-0.340218\pi\)
0.481154 + 0.876636i \(0.340218\pi\)
\(44\) 382371.i 0.102017i
\(45\) 0 0
\(46\) 1.72397e6 0.385033
\(47\) − 127214.i − 0.0260702i −0.999915 − 0.0130351i \(-0.995851\pi\)
0.999915 − 0.0130351i \(-0.00414932\pi\)
\(48\) 0 0
\(49\) 1.30466e7 2.26315
\(50\) 3.07427e6i 0.491883i
\(51\) 0 0
\(52\) 1.00373e6 0.137279
\(53\) − 1.40856e6i − 0.178514i −0.996009 − 0.0892571i \(-0.971551\pi\)
0.996009 − 0.0892571i \(-0.0284493\pi\)
\(54\) 0 0
\(55\) 5.05542e6 0.552467
\(56\) 1.91766e7i 1.94993i
\(57\) 0 0
\(58\) 7.04199e6 0.622276
\(59\) 2.07119e7i 1.70928i 0.519224 + 0.854638i \(0.326221\pi\)
−0.519224 + 0.854638i \(0.673779\pi\)
\(60\) 0 0
\(61\) −1.92431e7 −1.38981 −0.694907 − 0.719099i \(-0.744555\pi\)
−0.694907 + 0.719099i \(0.744555\pi\)
\(62\) 1.42000e7i 0.960999i
\(63\) 0 0
\(64\) −1.86559e7 −1.11198
\(65\) − 1.32705e7i − 0.743421i
\(66\) 0 0
\(67\) −2.34312e7 −1.16277 −0.581386 − 0.813628i \(-0.697489\pi\)
−0.581386 + 0.813628i \(0.697489\pi\)
\(68\) − 1.76476e6i − 0.0825374i
\(69\) 0 0
\(70\) 4.75263e7 1.97944
\(71\) − 9.12361e6i − 0.359032i −0.983755 − 0.179516i \(-0.942547\pi\)
0.983755 − 0.179516i \(-0.0574532\pi\)
\(72\) 0 0
\(73\) 1.43931e7 0.506832 0.253416 − 0.967357i \(-0.418446\pi\)
0.253416 + 0.967357i \(0.418446\pi\)
\(74\) − 2.67030e6i − 0.0890498i
\(75\) 0 0
\(76\) −5.41968e6 −0.162450
\(77\) − 2.80811e7i − 0.798824i
\(78\) 0 0
\(79\) −5.39607e7 −1.38538 −0.692691 − 0.721235i \(-0.743575\pi\)
−0.692691 + 0.721235i \(0.743575\pi\)
\(80\) 3.66435e7i 0.894617i
\(81\) 0 0
\(82\) 3.16992e7 0.701121
\(83\) 3.07329e7i 0.647577i 0.946130 + 0.323788i \(0.104957\pi\)
−0.946130 + 0.323788i \(0.895043\pi\)
\(84\) 0 0
\(85\) −2.33324e7 −0.446975
\(86\) − 4.61696e7i − 0.844039i
\(87\) 0 0
\(88\) 2.86261e7 0.477345
\(89\) 2.32435e7i 0.370460i 0.982695 + 0.185230i \(0.0593030\pi\)
−0.982695 + 0.185230i \(0.940697\pi\)
\(90\) 0 0
\(91\) −7.37131e7 −1.07493
\(92\) 7.25509e6i 0.101273i
\(93\) 0 0
\(94\) −1.78527e6 −0.0228662
\(95\) 7.16550e7i 0.879735i
\(96\) 0 0
\(97\) −4.93189e7 −0.557091 −0.278546 − 0.960423i \(-0.589852\pi\)
−0.278546 + 0.960423i \(0.589852\pi\)
\(98\) − 1.83091e8i − 1.98501i
\(99\) 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 81.9.b.a.80.5 14
3.2 odd 2 inner 81.9.b.a.80.10 14
9.2 odd 6 27.9.d.a.17.5 14
9.4 even 3 27.9.d.a.8.5 14
9.5 odd 6 9.9.d.a.2.3 ✓ 14
9.7 even 3 9.9.d.a.5.3 yes 14
36.7 odd 6 144.9.q.a.113.2 14
36.11 even 6 432.9.q.a.17.1 14
36.23 even 6 144.9.q.a.65.2 14
36.31 odd 6 432.9.q.a.305.1 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.3 ✓ 14 9.5 odd 6
9.9.d.a.5.3 yes 14 9.7 even 3
27.9.d.a.8.5 14 9.4 even 3
27.9.d.a.17.5 14 9.2 odd 6
81.9.b.a.80.5 14 1.1 even 1 trivial
81.9.b.a.80.10 14 3.2 odd 2 inner
144.9.q.a.65.2 14 36.23 even 6
144.9.q.a.113.2 14 36.7 odd 6
432.9.q.a.17.1 14 36.11 even 6
432.9.q.a.305.1 14 36.31 odd 6