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Properties

Label 567.2.s.a.458.1
Level $567$
Weight $2$
Character 567.458
Analytic conductor $4.528$
Analytic rank $1$
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] = [567,2,Mod(26,567)] 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("567.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,0,-5] 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: \(4.52751779461\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 458.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 567.458
Dual form 567.2.s.a.26.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(-1.50000 - 0.866025i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-4.50000 + 2.59808i) q^{19} -5.00000 q^{25} +(4.00000 + 3.46410i) q^{28} +(-7.50000 + 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{37} +(2.50000 + 4.33013i) q^{43} +(5.50000 - 4.33013i) q^{49} +3.46410i q^{52} +(-6.00000 - 3.46410i) q^{61} +8.00000 q^{64} +(-5.50000 - 9.52628i) q^{67} +(-13.5000 - 7.79423i) q^{73} +(9.00000 + 5.19615i) q^{76} +(6.50000 - 11.2583i) q^{79} +(4.50000 + 0.866025i) q^{91} +(12.0000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 5 q^{7} - 3 q^{13} - 4 q^{16} - 9 q^{19} - 10 q^{25} + 8 q^{28} - 15 q^{31} - q^{37} + 5 q^{43} + 11 q^{49} - 12 q^{61} + 16 q^{64} - 11 q^{67} - 27 q^{73} + 18 q^{76} + 13 q^{79} + 9 q^{91}+ \cdots + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) −1.00000 − 1.73205i −0.500000 − 0.866025i
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.50000 + 0.866025i −0.944911 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.50000 − 0.866025i −0.416025 − 0.240192i 0.277350 − 0.960769i \(-0.410544\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −4.50000 + 2.59808i −1.03237 + 0.596040i −0.917663 − 0.397360i \(-0.869927\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 4.00000 + 3.46410i 0.755929 + 0.654654i
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −7.50000 + 4.33013i −1.34704 + 0.777714i −0.987829 − 0.155543i \(-0.950287\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.500000 − 0.866025i −0.0821995 − 0.142374i 0.821995 − 0.569495i \(-0.192861\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 2.50000 + 4.33013i 0.381246 + 0.660338i 0.991241 − 0.132068i \(-0.0421616\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 5.50000 − 4.33013i 0.785714 − 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.46410i 0.480384i
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −6.00000 − 3.46410i −0.768221 − 0.443533i 0.0640184 − 0.997949i \(-0.479608\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.50000 − 9.52628i −0.671932 − 1.16382i −0.977356 − 0.211604i \(-0.932131\pi\)
0.305424 − 0.952217i \(-0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −13.5000 − 7.79423i −1.58006 − 0.912245i −0.994850 − 0.101361i \(-0.967680\pi\)
−0.585206 − 0.810885i \(-0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 9.00000 + 5.19615i 1.03237 + 0.596040i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 − 11.2583i 0.731307 − 1.26666i −0.225018 − 0.974355i \(-0.572244\pi\)
0.956325 − 0.292306i \(-0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 4.50000 + 0.866025i 0.471728 + 0.0907841i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 − 6.92820i 1.21842 − 0.703452i 0.253837 − 0.967247i \(-0.418307\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 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 567.2.s.a.458.1 2
3.2 odd 2 CM 567.2.s.a.458.1 2
7.5 odd 6 567.2.i.b.215.1 2
9.2 odd 6 567.2.i.b.269.1 2
9.4 even 3 21.2.g.a.17.1 yes 2
9.5 odd 6 21.2.g.a.17.1 yes 2
9.7 even 3 567.2.i.b.269.1 2
21.5 even 6 567.2.i.b.215.1 2
36.23 even 6 336.2.bc.c.17.1 2
36.31 odd 6 336.2.bc.c.17.1 2
45.4 even 6 525.2.t.c.101.1 2
45.13 odd 12 525.2.q.d.374.2 4
45.14 odd 6 525.2.t.c.101.1 2
45.22 odd 12 525.2.q.d.374.1 4
45.23 even 12 525.2.q.d.374.2 4
45.32 even 12 525.2.q.d.374.1 4
63.4 even 3 147.2.c.a.146.2 2
63.5 even 6 21.2.g.a.5.1 ✓ 2
63.13 odd 6 147.2.g.a.80.1 2
63.23 odd 6 147.2.g.a.68.1 2
63.31 odd 6 147.2.c.a.146.1 2
63.32 odd 6 147.2.c.a.146.2 2
63.40 odd 6 21.2.g.a.5.1 ✓ 2
63.41 even 6 147.2.g.a.80.1 2
63.47 even 6 inner 567.2.s.a.26.1 2
63.58 even 3 147.2.g.a.68.1 2
63.59 even 6 147.2.c.a.146.1 2
63.61 odd 6 inner 567.2.s.a.26.1 2
252.31 even 6 2352.2.k.c.881.2 2
252.59 odd 6 2352.2.k.c.881.2 2
252.67 odd 6 2352.2.k.c.881.1 2
252.95 even 6 2352.2.k.c.881.1 2
252.103 even 6 336.2.bc.c.257.1 2
252.131 odd 6 336.2.bc.c.257.1 2
315.68 odd 12 525.2.q.d.299.1 4
315.103 even 12 525.2.q.d.299.1 4
315.194 even 6 525.2.t.c.26.1 2
315.229 odd 6 525.2.t.c.26.1 2
315.257 odd 12 525.2.q.d.299.2 4
315.292 even 12 525.2.q.d.299.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.g.a.5.1 ✓ 2 63.5 even 6
21.2.g.a.5.1 ✓ 2 63.40 odd 6
21.2.g.a.17.1 yes 2 9.4 even 3
21.2.g.a.17.1 yes 2 9.5 odd 6
147.2.c.a.146.1 2 63.31 odd 6
147.2.c.a.146.1 2 63.59 even 6
147.2.c.a.146.2 2 63.4 even 3
147.2.c.a.146.2 2 63.32 odd 6
147.2.g.a.68.1 2 63.23 odd 6
147.2.g.a.68.1 2 63.58 even 3
147.2.g.a.80.1 2 63.13 odd 6
147.2.g.a.80.1 2 63.41 even 6
336.2.bc.c.17.1 2 36.23 even 6
336.2.bc.c.17.1 2 36.31 odd 6
336.2.bc.c.257.1 2 252.103 even 6
336.2.bc.c.257.1 2 252.131 odd 6
525.2.q.d.299.1 4 315.68 odd 12
525.2.q.d.299.1 4 315.103 even 12
525.2.q.d.299.2 4 315.257 odd 12
525.2.q.d.299.2 4 315.292 even 12
525.2.q.d.374.1 4 45.22 odd 12
525.2.q.d.374.1 4 45.32 even 12
525.2.q.d.374.2 4 45.13 odd 12
525.2.q.d.374.2 4 45.23 even 12
525.2.t.c.26.1 2 315.194 even 6
525.2.t.c.26.1 2 315.229 odd 6
525.2.t.c.101.1 2 45.4 even 6
525.2.t.c.101.1 2 45.14 odd 6
567.2.i.b.215.1 2 7.5 odd 6
567.2.i.b.215.1 2 21.5 even 6
567.2.i.b.269.1 2 9.2 odd 6
567.2.i.b.269.1 2 9.7 even 3
567.2.s.a.26.1 2 63.47 even 6 inner
567.2.s.a.26.1 2 63.61 odd 6 inner
567.2.s.a.458.1 2 1.1 even 1 trivial
567.2.s.a.458.1 2 3.2 odd 2 CM
2352.2.k.c.881.1 2 252.67 odd 6
2352.2.k.c.881.1 2 252.95 even 6
2352.2.k.c.881.2 2 252.31 even 6
2352.2.k.c.881.2 2 252.59 odd 6