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Properties

Label 243.3.d.d.161.1
Level $243$
Weight $3$
Character 243.161
Analytic conductor $6.621$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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] = [243,3,Mod(80,243)] 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("243.80"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 243.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.62127042396\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 161.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 243.161
Dual form 243.3.d.d.80.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.00000 + 3.46410i) q^{4} +(-1.00000 - 1.73205i) q^{7} +(-11.5000 + 19.9186i) q^{13} +(-8.00000 - 13.8564i) q^{16} -37.0000 q^{19} +(-12.5000 - 21.6506i) q^{25} +8.00000 q^{28} +(-29.5000 + 51.0955i) q^{31} +26.0000 q^{37} +(30.5000 + 52.8275i) q^{43} +(22.5000 - 38.9711i) q^{49} +(-46.0000 - 79.6743i) q^{52} +(60.5000 + 104.789i) q^{61} +64.0000 q^{64} +(54.5000 - 94.3968i) q^{67} -97.0000 q^{73} +(74.0000 - 128.172i) q^{76} +(-5.50000 - 9.52628i) q^{79} +46.0000 q^{91} +(84.5000 + 146.358i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{7} - 23 q^{13} - 16 q^{16} - 74 q^{19} - 25 q^{25} + 16 q^{28} - 59 q^{31} + 52 q^{37} + 61 q^{43} + 45 q^{49} - 92 q^{52} + 121 q^{61} + 128 q^{64} + 109 q^{67} - 194 q^{73} + 148 q^{76}+ \cdots + 169 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 0 0
\(4\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(5\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) −1.00000 − 1.73205i −0.142857 − 0.247436i 0.785714 − 0.618590i \(-0.212296\pi\)
−0.928571 + 0.371154i \(0.878962\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) −11.5000 + 19.9186i −0.884615 + 1.53220i −0.0384615 + 0.999260i \(0.512246\pi\)
−0.846154 + 0.532939i \(0.821088\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.00000 − 13.8564i −0.500000 − 0.866025i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −37.0000 −1.94737 −0.973684 − 0.227901i \(-0.926814\pi\)
−0.973684 + 0.227901i \(0.926814\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −12.5000 − 21.6506i −0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 8.00000 0.285714
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −29.5000 + 51.0955i −0.951613 + 1.64824i −0.209677 + 0.977771i \(0.567241\pi\)
−0.741935 + 0.670471i \(0.766092\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 − 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 30.5000 + 52.8275i 0.709302 + 1.22855i 0.965116 + 0.261822i \(0.0843232\pi\)
−0.255814 + 0.966726i \(0.582343\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0 0
\(49\) 22.5000 − 38.9711i 0.459184 − 0.795329i
\(50\) 0 0
\(51\) 0 0
\(52\) −46.0000 − 79.6743i −0.884615 − 1.53220i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 60.5000 + 104.789i 0.991803 + 1.71785i 0.606557 + 0.795040i \(0.292550\pi\)
0.385246 + 0.922814i \(0.374117\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 54.5000 − 94.3968i 0.813433 − 1.40891i −0.0970149 − 0.995283i \(-0.530929\pi\)
0.910448 − 0.413624i \(-0.135737\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −97.0000 −1.32877 −0.664384 − 0.747392i \(-0.731306\pi\)
−0.664384 + 0.747392i \(0.731306\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 74.0000 − 128.172i 0.973684 − 1.68647i
\(77\) 0 0
\(78\) 0 0
\(79\) −5.50000 − 9.52628i −0.0696203 − 0.120586i 0.829114 − 0.559080i \(-0.188845\pi\)
−0.898734 + 0.438494i \(0.855512\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 46.0000 0.505495
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 84.5000 + 146.358i 0.871134 + 1.50885i 0.860825 + 0.508902i \(0.169948\pi\)
0.0103093 + 0.999947i \(0.496718\pi\)
\(98\) 0 0
\(99\) 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 243.3.d.d.161.1 2
3.2 odd 2 CM 243.3.d.d.161.1 2
9.2 odd 6 inner 243.3.d.d.80.1 2
9.4 even 3 243.3.b.a.242.1 ✓ 1
9.5 odd 6 243.3.b.a.242.1 ✓ 1
9.7 even 3 inner 243.3.d.d.80.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.3.b.a.242.1 ✓ 1 9.4 even 3
243.3.b.a.242.1 ✓ 1 9.5 odd 6
243.3.d.d.80.1 2 9.2 odd 6 inner
243.3.d.d.80.1 2 9.7 even 3 inner
243.3.d.d.161.1 2 1.1 even 1 trivial
243.3.d.d.161.1 2 3.2 odd 2 CM