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Properties

Label 567.2.o.f.377.4
Level $567$
Weight $2$
Character 567.377
Analytic conductor $4.528$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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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(188,567)] 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("567.188"); 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, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,8,0,0,0] 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: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 377.4
Root \(2.23256 - 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 567.377
Dual form 567.2.o.f.188.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.23256 - 1.28897i) q^{2} +(2.32288 - 4.02334i) q^{4} +(-1.32288 - 2.29129i) q^{7} -6.82058i q^{8} +(-0.790881 + 0.456615i) q^{11} +(-5.90679 - 3.41029i) q^{14} +(-4.14575 - 7.18065i) q^{16} +(-1.17712 + 2.03884i) q^{22} +(8.13935 + 4.69926i) q^{23} +(2.50000 + 4.33013i) q^{25} -12.2915 q^{28} +(5.25600 - 3.03455i) q^{29} +(-6.69767 - 3.86690i) q^{32} -10.5830 q^{37} +(-2.64575 - 4.58258i) q^{43} +4.24264i q^{44} +24.2288 q^{46} +(-3.50000 + 6.06218i) q^{49} +(11.1628 + 6.44484i) q^{50} +14.5544i q^{53} +(-15.6279 + 9.02277i) q^{56} +(7.82288 - 13.5496i) q^{58} -3.35425 q^{64} +(2.00000 - 3.46410i) q^{67} +7.57205i q^{71} +(-23.6272 + 13.6412i) q^{74} +(2.09247 + 1.20809i) q^{77} +(-4.00000 - 6.92820i) q^{79} +(-11.8136 - 6.82058i) q^{86} +(3.11438 + 5.39426i) q^{88} +(37.8134 - 21.8316i) q^{92} +18.0455i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 12 q^{16} - 20 q^{22} + 20 q^{25} - 56 q^{28} + 88 q^{46} - 28 q^{49} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23256 − 1.28897i 1.57866 − 0.911438i 0.583609 − 0.812035i \(-0.301640\pi\)
0.995047 − 0.0994033i \(-0.0316934\pi\)
\(3\) 0 0
\(4\) 2.32288 − 4.02334i 1.16144 − 2.01167i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −1.32288 − 2.29129i −0.500000 − 0.866025i
\(8\) − 6.82058i − 2.41144i
\(9\) 0 0
\(10\) 0 0
\(11\) −0.790881 + 0.456615i −0.238459 + 0.137675i −0.614468 − 0.788941i \(-0.710630\pi\)
0.376009 + 0.926616i \(0.377296\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) −5.90679 − 3.41029i −1.57866 − 0.911438i
\(15\) 0 0
\(16\) −4.14575 − 7.18065i −1.03644 − 1.79516i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.17712 + 2.03884i −0.250964 + 0.434682i
\(23\) 8.13935 + 4.69926i 1.69717 + 0.979863i 0.948422 + 0.317009i \(0.102679\pi\)
0.748749 + 0.662853i \(0.230655\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) −12.2915 −2.32288
\(29\) 5.25600 − 3.03455i 0.976014 − 0.563502i 0.0749496 − 0.997187i \(-0.476120\pi\)
0.901064 + 0.433685i \(0.142787\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −6.69767 − 3.86690i −1.18399 − 0.683578i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.5830 −1.73984 −0.869918 − 0.493197i \(-0.835828\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 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
\(43\) −2.64575 − 4.58258i −0.403473 − 0.698836i 0.590669 − 0.806914i \(-0.298864\pi\)
−0.994142 + 0.108078i \(0.965531\pi\)
\(44\) 4.24264i 0.639602i
\(45\) 0 0
\(46\) 24.2288 3.57234
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(50\) 11.1628 + 6.44484i 1.57866 + 0.911438i
\(51\) 0 0
\(52\) 0 0
\(53\) 14.5544i 1.99920i 0.0283132 + 0.999599i \(0.490986\pi\)
−0.0283132 + 0.999599i \(0.509014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −15.6279 + 9.02277i −2.08837 + 1.20572i
\(57\) 0 0
\(58\) 7.82288 − 13.5496i 1.02719 − 1.77915i
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −3.35425 −0.419281
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.57205i 0.898637i 0.893372 + 0.449319i \(0.148333\pi\)
−0.893372 + 0.449319i \(0.851667\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −23.6272 + 13.6412i −2.74660 + 1.58575i
\(75\) 0 0
\(76\) 0 0
\(77\) 2.09247 + 1.20809i 0.238459 + 0.137675i
\(78\) 0 0
\(79\) −4.00000 − 6.92820i −0.450035 − 0.779484i 0.548352 − 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 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
\(86\) −11.8136 − 6.82058i −1.27389 − 0.735482i
\(87\) 0 0
\(88\) 3.11438 + 5.39426i 0.331994 + 0.575030i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 37.8134 − 21.8316i 3.94232 − 2.27610i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 18.0455i 1.82288i
\(99\) 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.o.f.377.4 8
3.2 odd 2 inner 567.2.o.f.377.1 8
7.6 odd 2 CM 567.2.o.f.377.4 8
9.2 odd 6 inner 567.2.o.f.188.4 8
9.4 even 3 63.2.c.a.62.4 yes 4
9.5 odd 6 63.2.c.a.62.1 ✓ 4
9.7 even 3 inner 567.2.o.f.188.1 8
21.20 even 2 inner 567.2.o.f.377.1 8
36.23 even 6 1008.2.k.a.881.1 4
36.31 odd 6 1008.2.k.a.881.2 4
45.4 even 6 1575.2.b.a.251.1 4
45.13 odd 12 1575.2.g.d.1574.7 8
45.14 odd 6 1575.2.b.a.251.4 4
45.22 odd 12 1575.2.g.d.1574.2 8
45.23 even 12 1575.2.g.d.1574.1 8
45.32 even 12 1575.2.g.d.1574.8 8
63.4 even 3 441.2.p.b.215.1 8
63.5 even 6 441.2.p.b.80.1 8
63.13 odd 6 63.2.c.a.62.4 yes 4
63.20 even 6 inner 567.2.o.f.188.4 8
63.23 odd 6 441.2.p.b.80.1 8
63.31 odd 6 441.2.p.b.215.1 8
63.32 odd 6 441.2.p.b.215.4 8
63.34 odd 6 inner 567.2.o.f.188.1 8
63.40 odd 6 441.2.p.b.80.4 8
63.41 even 6 63.2.c.a.62.1 ✓ 4
63.58 even 3 441.2.p.b.80.4 8
63.59 even 6 441.2.p.b.215.4 8
72.5 odd 6 4032.2.k.c.3905.3 4
72.13 even 6 4032.2.k.c.3905.4 4
72.59 even 6 4032.2.k.b.3905.2 4
72.67 odd 6 4032.2.k.b.3905.1 4
252.139 even 6 1008.2.k.a.881.2 4
252.167 odd 6 1008.2.k.a.881.1 4
315.13 even 12 1575.2.g.d.1574.7 8
315.104 even 6 1575.2.b.a.251.4 4
315.139 odd 6 1575.2.b.a.251.1 4
315.167 odd 12 1575.2.g.d.1574.8 8
315.202 even 12 1575.2.g.d.1574.2 8
315.293 odd 12 1575.2.g.d.1574.1 8
504.13 odd 6 4032.2.k.c.3905.4 4
504.139 even 6 4032.2.k.b.3905.1 4
504.293 even 6 4032.2.k.c.3905.3 4
504.419 odd 6 4032.2.k.b.3905.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.1 ✓ 4 9.5 odd 6
63.2.c.a.62.1 ✓ 4 63.41 even 6
63.2.c.a.62.4 yes 4 9.4 even 3
63.2.c.a.62.4 yes 4 63.13 odd 6
441.2.p.b.80.1 8 63.5 even 6
441.2.p.b.80.1 8 63.23 odd 6
441.2.p.b.80.4 8 63.40 odd 6
441.2.p.b.80.4 8 63.58 even 3
441.2.p.b.215.1 8 63.4 even 3
441.2.p.b.215.1 8 63.31 odd 6
441.2.p.b.215.4 8 63.32 odd 6
441.2.p.b.215.4 8 63.59 even 6
567.2.o.f.188.1 8 9.7 even 3 inner
567.2.o.f.188.1 8 63.34 odd 6 inner
567.2.o.f.188.4 8 9.2 odd 6 inner
567.2.o.f.188.4 8 63.20 even 6 inner
567.2.o.f.377.1 8 3.2 odd 2 inner
567.2.o.f.377.1 8 21.20 even 2 inner
567.2.o.f.377.4 8 1.1 even 1 trivial
567.2.o.f.377.4 8 7.6 odd 2 CM
1008.2.k.a.881.1 4 36.23 even 6
1008.2.k.a.881.1 4 252.167 odd 6
1008.2.k.a.881.2 4 36.31 odd 6
1008.2.k.a.881.2 4 252.139 even 6
1575.2.b.a.251.1 4 45.4 even 6
1575.2.b.a.251.1 4 315.139 odd 6
1575.2.b.a.251.4 4 45.14 odd 6
1575.2.b.a.251.4 4 315.104 even 6
1575.2.g.d.1574.1 8 45.23 even 12
1575.2.g.d.1574.1 8 315.293 odd 12
1575.2.g.d.1574.2 8 45.22 odd 12
1575.2.g.d.1574.2 8 315.202 even 12
1575.2.g.d.1574.7 8 45.13 odd 12
1575.2.g.d.1574.7 8 315.13 even 12
1575.2.g.d.1574.8 8 45.32 even 12
1575.2.g.d.1574.8 8 315.167 odd 12
4032.2.k.b.3905.1 4 72.67 odd 6
4032.2.k.b.3905.1 4 504.139 even 6
4032.2.k.b.3905.2 4 72.59 even 6
4032.2.k.b.3905.2 4 504.419 odd 6
4032.2.k.c.3905.3 4 72.5 odd 6
4032.2.k.c.3905.3 4 504.293 even 6
4032.2.k.c.3905.4 4 72.13 even 6
4032.2.k.c.3905.4 4 504.13 odd 6