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Properties

Label 567.2.f.n.379.3
Level $567$
Weight $2$
Character 567.379
Analytic conductor $4.528$
Analytic rank $0$
Dimension $8$
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(190,567)] 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("567.190"); 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([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-2,0,0,4,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(i, \sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 379.3
Root \(0.228425 - 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 567.379
Dual form 567.2.f.n.190.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.228425 - 0.395644i) q^{2} +(0.895644 + 1.55130i) q^{4} +(2.18890 + 3.79129i) q^{5} +(0.500000 - 0.866025i) q^{7} +1.73205 q^{8} +2.00000 q^{10} +(1.32288 - 2.29129i) q^{11} +(-2.00000 - 3.46410i) q^{13} +(-0.228425 - 0.395644i) q^{14} +(-1.39564 + 2.41733i) q^{16} +3.46410 q^{17} -3.58258 q^{19} +(-3.92095 + 6.79129i) q^{20} +(-0.604356 - 1.04678i) q^{22} +(1.73205 + 3.00000i) q^{23} +(-7.08258 + 12.2674i) q^{25} -1.82740 q^{26} +1.79129 q^{28} +(-0.913701 + 1.58258i) q^{29} +(-4.58258 - 7.93725i) q^{31} +(2.36965 + 4.10436i) q^{32} +(0.791288 - 1.37055i) q^{34} +4.37780 q^{35} +3.00000 q^{37} +(-0.818350 + 1.41742i) q^{38} +(3.79129 + 6.56670i) q^{40} +(-2.18890 - 3.79129i) q^{41} +(4.29129 - 7.43273i) q^{43} +4.73930 q^{44} +1.58258 q^{46} +(-1.37055 + 2.37386i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(3.23568 + 5.60436i) q^{50} +(3.58258 - 6.20520i) q^{52} -8.66025 q^{53} +11.5826 q^{55} +(0.866025 - 1.50000i) q^{56} +(0.417424 + 0.723000i) q^{58} +(-1.73205 - 3.00000i) q^{59} +(-1.20871 + 2.09355i) q^{61} -4.18710 q^{62} -3.41742 q^{64} +(8.75560 - 15.1652i) q^{65} +(0.291288 + 0.504525i) q^{67} +(3.10260 + 5.37386i) q^{68} +(1.00000 - 1.73205i) q^{70} +11.4014 q^{71} -3.16515 q^{73} +(0.685275 - 1.18693i) q^{74} +(-3.20871 - 5.55765i) q^{76} +(-1.32288 - 2.29129i) q^{77} +(4.29129 - 7.43273i) q^{79} -12.2197 q^{80} -2.00000 q^{82} +(-3.10260 + 5.37386i) q^{83} +(7.58258 + 13.1334i) q^{85} +(-1.96048 - 3.39564i) q^{86} +(2.29129 - 3.96863i) q^{88} +8.75560 q^{89} -4.00000 q^{91} +(-3.10260 + 5.37386i) q^{92} +(0.626136 + 1.08450i) q^{94} +(-7.84190 - 13.5826i) q^{95} +(-3.79129 + 6.56670i) q^{97} -0.456850 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 4 q^{7} + 16 q^{10} - 16 q^{13} - 2 q^{16} + 8 q^{19} - 14 q^{22} - 20 q^{25} - 4 q^{28} - 12 q^{34} + 24 q^{37} + 12 q^{40} + 16 q^{43} - 24 q^{46} - 4 q^{49} - 8 q^{52} + 56 q^{55} + 40 q^{58}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.228425 − 0.395644i 0.161521 − 0.279763i −0.773893 − 0.633316i \(-0.781693\pi\)
0.935414 + 0.353553i \(0.115027\pi\)
\(3\) 0 0
\(4\) 0.895644 + 1.55130i 0.447822 + 0.775650i
\(5\) 2.18890 + 3.79129i 0.978906 + 1.69552i 0.666390 + 0.745603i \(0.267838\pi\)
0.312516 + 0.949913i \(0.398828\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 1.32288 − 2.29129i 0.398862 − 0.690849i −0.594724 − 0.803930i \(-0.702739\pi\)
0.993586 + 0.113081i \(0.0360719\pi\)
\(12\) 0 0
\(13\) −2.00000 − 3.46410i −0.554700 − 0.960769i −0.997927 − 0.0643593i \(-0.979500\pi\)
0.443227 − 0.896410i \(-0.353834\pi\)
\(14\) −0.228425 − 0.395644i −0.0610492 − 0.105740i
\(15\) 0 0
\(16\) −1.39564 + 2.41733i −0.348911 + 0.604332i
\(17\) 3.46410 0.840168 0.420084 − 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −3.58258 −0.821899 −0.410950 − 0.911658i \(-0.634803\pi\)
−0.410950 + 0.911658i \(0.634803\pi\)
\(20\) −3.92095 + 6.79129i −0.876751 + 1.51858i
\(21\) 0 0
\(22\) −0.604356 − 1.04678i −0.128849 − 0.223173i
\(23\) 1.73205 + 3.00000i 0.361158 + 0.625543i 0.988152 − 0.153481i \(-0.0490483\pi\)
−0.626994 + 0.779024i \(0.715715\pi\)
\(24\) 0 0
\(25\) −7.08258 + 12.2674i −1.41652 + 2.45348i
\(26\) −1.82740 −0.358383
\(27\) 0 0
\(28\) 1.79129 0.338522
\(29\) −0.913701 + 1.58258i −0.169670 + 0.293877i −0.938304 − 0.345812i \(-0.887603\pi\)
0.768634 + 0.639689i \(0.220937\pi\)
\(30\) 0 0
\(31\) −4.58258 − 7.93725i −0.823055 − 1.42557i −0.903397 − 0.428806i \(-0.858935\pi\)
0.0803419 − 0.996767i \(-0.474399\pi\)
\(32\) 2.36965 + 4.10436i 0.418899 + 0.725555i
\(33\) 0 0
\(34\) 0.791288 − 1.37055i 0.135705 − 0.235048i
\(35\) 4.37780 0.739984
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −0.818350 + 1.41742i −0.132754 + 0.229937i
\(39\) 0 0
\(40\) 3.79129 + 6.56670i 0.599455 + 1.03829i
\(41\) −2.18890 − 3.79129i −0.341849 − 0.592100i 0.642927 − 0.765927i \(-0.277720\pi\)
−0.984776 + 0.173828i \(0.944386\pi\)
\(42\) 0 0
\(43\) 4.29129 − 7.43273i 0.654415 − 1.13348i −0.327625 − 0.944808i \(-0.606248\pi\)
0.982040 − 0.188673i \(-0.0604185\pi\)
\(44\) 4.73930 0.714477
\(45\) 0 0
\(46\) 1.58258 0.233338
\(47\) −1.37055 + 2.37386i −0.199915 + 0.346264i −0.948501 − 0.316775i \(-0.897400\pi\)
0.748585 + 0.663038i \(0.230733\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 3.23568 + 5.60436i 0.457594 + 0.792576i
\(51\) 0 0
\(52\) 3.58258 − 6.20520i 0.496814 − 0.860507i
\(53\) −8.66025 −1.18958 −0.594789 − 0.803882i \(-0.702764\pi\)
−0.594789 + 0.803882i \(0.702764\pi\)
\(54\) 0 0
\(55\) 11.5826 1.56179
\(56\) 0.866025 − 1.50000i 0.115728 − 0.200446i
\(57\) 0 0
\(58\) 0.417424 + 0.723000i 0.0548105 + 0.0949346i
\(59\) −1.73205 − 3.00000i −0.225494 − 0.390567i 0.730974 − 0.682406i \(-0.239066\pi\)
−0.956467 + 0.291839i \(0.905733\pi\)
\(60\) 0 0
\(61\) −1.20871 + 2.09355i −0.154760 + 0.268052i −0.932972 − 0.359950i \(-0.882794\pi\)
0.778212 + 0.628002i \(0.216127\pi\)
\(62\) −4.18710 −0.531762
\(63\) 0 0
\(64\) −3.41742 −0.427178
\(65\) 8.75560 − 15.1652i 1.08600 − 1.88101i
\(66\) 0 0
\(67\) 0.291288 + 0.504525i 0.0355865 + 0.0616376i 0.883270 − 0.468865i \(-0.155337\pi\)
−0.847684 + 0.530502i \(0.822003\pi\)
\(68\) 3.10260 + 5.37386i 0.376246 + 0.651677i
\(69\) 0 0
\(70\) 1.00000 − 1.73205i 0.119523 − 0.207020i
\(71\) 11.4014 1.35309 0.676546 − 0.736400i \(-0.263476\pi\)
0.676546 + 0.736400i \(0.263476\pi\)
\(72\) 0 0
\(73\) −3.16515 −0.370453 −0.185226 − 0.982696i \(-0.559302\pi\)
−0.185226 + 0.982696i \(0.559302\pi\)
\(74\) 0.685275 − 1.18693i 0.0796616 − 0.137978i
\(75\) 0 0
\(76\) −3.20871 − 5.55765i −0.368065 − 0.637506i
\(77\) −1.32288 − 2.29129i −0.150756 − 0.261116i
\(78\) 0 0
\(79\) 4.29129 − 7.43273i 0.482808 − 0.836247i −0.516998 − 0.855987i \(-0.672950\pi\)
0.999805 + 0.0197396i \(0.00628371\pi\)
\(80\) −12.2197 −1.36620
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −3.10260 + 5.37386i −0.340555 + 0.589858i −0.984536 − 0.175183i \(-0.943948\pi\)
0.643981 + 0.765041i \(0.277282\pi\)
\(84\) 0 0
\(85\) 7.58258 + 13.1334i 0.822446 + 1.42452i
\(86\) −1.96048 − 3.39564i −0.211404 − 0.366162i
\(87\) 0 0
\(88\) 2.29129 − 3.96863i 0.244252 − 0.423057i
\(89\) 8.75560 0.928092 0.464046 − 0.885811i \(-0.346397\pi\)
0.464046 + 0.885811i \(0.346397\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −3.10260 + 5.37386i −0.323469 + 0.560264i
\(93\) 0 0
\(94\) 0.626136 + 1.08450i 0.0645810 + 0.111858i
\(95\) −7.84190 − 13.5826i −0.804562 − 1.39354i
\(96\) 0 0
\(97\) −3.79129 + 6.56670i −0.384947 + 0.666748i −0.991762 − 0.128096i \(-0.959114\pi\)
0.606815 + 0.794843i \(0.292447\pi\)
\(98\) −0.456850 −0.0461488
\(99\) 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.f.n.379.3 8
3.2 odd 2 inner 567.2.f.n.379.2 8
9.2 odd 6 567.2.a.i.1.3 yes 4
9.4 even 3 inner 567.2.f.n.190.3 8
9.5 odd 6 inner 567.2.f.n.190.2 8
9.7 even 3 567.2.a.i.1.2 ✓ 4
36.7 odd 6 9072.2.a.ci.1.1 4
36.11 even 6 9072.2.a.ci.1.4 4
63.20 even 6 3969.2.a.u.1.3 4
63.34 odd 6 3969.2.a.u.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.2 ✓ 4 9.7 even 3
567.2.a.i.1.3 yes 4 9.2 odd 6
567.2.f.n.190.2 8 9.5 odd 6 inner
567.2.f.n.190.3 8 9.4 even 3 inner
567.2.f.n.379.2 8 3.2 odd 2 inner
567.2.f.n.379.3 8 1.1 even 1 trivial
3969.2.a.u.1.2 4 63.34 odd 6
3969.2.a.u.1.3 4 63.20 even 6
9072.2.a.ci.1.1 4 36.7 odd 6
9072.2.a.ci.1.4 4 36.11 even 6