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Properties

Label 567.2.a.i.1.3
Level $567$
Weight $2$
Character 567.1
Self dual yes
Analytic conductor $4.528$
Analytic rank $0$
Dimension $4$
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] = [567,2,Mod(1,567)] 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("567.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.18890\) of defining polynomial
Character \(\chi\) \(=\) 567.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.456850 q^{2} -1.79129 q^{4} +4.37780 q^{5} -1.00000 q^{7} -1.73205 q^{8} +2.00000 q^{10} +2.64575 q^{11} +4.00000 q^{13} -0.456850 q^{14} +2.79129 q^{16} -3.46410 q^{17} -3.58258 q^{19} -7.84190 q^{20} +1.20871 q^{22} +3.46410 q^{23} +14.1652 q^{25} +1.82740 q^{26} +1.79129 q^{28} -1.82740 q^{29} +9.16515 q^{31} +4.73930 q^{32} -1.58258 q^{34} -4.37780 q^{35} +3.00000 q^{37} -1.63670 q^{38} -7.58258 q^{40} -4.37780 q^{41} -8.58258 q^{43} -4.73930 q^{44} +1.58258 q^{46} -2.74110 q^{47} +1.00000 q^{49} +6.47135 q^{50} -7.16515 q^{52} +8.66025 q^{53} +11.5826 q^{55} +1.73205 q^{56} -0.834849 q^{58} -3.46410 q^{59} +2.41742 q^{61} +4.18710 q^{62} -3.41742 q^{64} +17.5112 q^{65} -0.582576 q^{67} +6.20520 q^{68} -2.00000 q^{70} -11.4014 q^{71} -3.16515 q^{73} +1.37055 q^{74} +6.41742 q^{76} -2.64575 q^{77} -8.58258 q^{79} +12.2197 q^{80} -2.00000 q^{82} -6.20520 q^{83} -15.1652 q^{85} -3.92095 q^{86} -4.58258 q^{88} -8.75560 q^{89} -4.00000 q^{91} -6.20520 q^{92} -1.25227 q^{94} -15.6838 q^{95} +7.58258 q^{97} +0.456850 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{7} + 8 q^{10} + 16 q^{13} + 2 q^{16} + 4 q^{19} + 14 q^{22} + 20 q^{25} - 2 q^{28} + 12 q^{34} + 12 q^{37} - 12 q^{40} - 16 q^{43} - 12 q^{46} + 4 q^{49} + 8 q^{52} + 28 q^{55} - 40 q^{58}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.456850 0.323042 0.161521 − 0.986869i \(-0.448360\pi\)
0.161521 + 0.986869i \(0.448360\pi\)
\(3\) 0 0
\(4\) −1.79129 −0.895644
\(5\) 4.37780 1.95781 0.978906 − 0.204310i \(-0.0654949\pi\)
0.978906 + 0.204310i \(0.0654949\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 2.64575 0.797724 0.398862 − 0.917011i \(-0.369405\pi\)
0.398862 + 0.917011i \(0.369405\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −0.456850 −0.122098
\(15\) 0 0
\(16\) 2.79129 0.697822
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −3.58258 −0.821899 −0.410950 − 0.911658i \(-0.634803\pi\)
−0.410950 + 0.911658i \(0.634803\pi\)
\(20\) −7.84190 −1.75350
\(21\) 0 0
\(22\) 1.20871 0.257698
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 14.1652 2.83303
\(26\) 1.82740 0.358383
\(27\) 0 0
\(28\) 1.79129 0.338522
\(29\) −1.82740 −0.339340 −0.169670 − 0.985501i \(-0.554270\pi\)
−0.169670 + 0.985501i \(0.554270\pi\)
\(30\) 0 0
\(31\) 9.16515 1.64611 0.823055 − 0.567962i \(-0.192268\pi\)
0.823055 + 0.567962i \(0.192268\pi\)
\(32\) 4.73930 0.837798
\(33\) 0 0
\(34\) −1.58258 −0.271409
\(35\) −4.37780 −0.739984
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −1.63670 −0.265508
\(39\) 0 0
\(40\) −7.58258 −1.19891
\(41\) −4.37780 −0.683698 −0.341849 − 0.939755i \(-0.611053\pi\)
−0.341849 + 0.939755i \(0.611053\pi\)
\(42\) 0 0
\(43\) −8.58258 −1.30883 −0.654415 − 0.756135i \(-0.727085\pi\)
−0.654415 + 0.756135i \(0.727085\pi\)
\(44\) −4.73930 −0.714477
\(45\) 0 0
\(46\) 1.58258 0.233338
\(47\) −2.74110 −0.399831 −0.199915 − 0.979813i \(-0.564067\pi\)
−0.199915 + 0.979813i \(0.564067\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 6.47135 0.915188
\(51\) 0 0
\(52\) −7.16515 −0.993628
\(53\) 8.66025 1.18958 0.594789 − 0.803882i \(-0.297236\pi\)
0.594789 + 0.803882i \(0.297236\pi\)
\(54\) 0 0
\(55\) 11.5826 1.56179
\(56\) 1.73205 0.231455
\(57\) 0 0
\(58\) −0.834849 −0.109621
\(59\) −3.46410 −0.450988 −0.225494 − 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 2.41742 0.309519 0.154760 − 0.987952i \(-0.450540\pi\)
0.154760 + 0.987952i \(0.450540\pi\)
\(62\) 4.18710 0.531762
\(63\) 0 0
\(64\) −3.41742 −0.427178
\(65\) 17.5112 2.17200
\(66\) 0 0
\(67\) −0.582576 −0.0711729 −0.0355865 − 0.999367i \(-0.511330\pi\)
−0.0355865 + 0.999367i \(0.511330\pi\)
\(68\) 6.20520 0.752491
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −11.4014 −1.35309 −0.676546 − 0.736400i \(-0.736524\pi\)
−0.676546 + 0.736400i \(0.736524\pi\)
\(72\) 0 0
\(73\) −3.16515 −0.370453 −0.185226 − 0.982696i \(-0.559302\pi\)
−0.185226 + 0.982696i \(0.559302\pi\)
\(74\) 1.37055 0.159323
\(75\) 0 0
\(76\) 6.41742 0.736129
\(77\) −2.64575 −0.301511
\(78\) 0 0
\(79\) −8.58258 −0.965615 −0.482808 − 0.875726i \(-0.660383\pi\)
−0.482808 + 0.875726i \(0.660383\pi\)
\(80\) 12.2197 1.36620
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −6.20520 −0.681110 −0.340555 − 0.940225i \(-0.610615\pi\)
−0.340555 + 0.940225i \(0.610615\pi\)
\(84\) 0 0
\(85\) −15.1652 −1.64489
\(86\) −3.92095 −0.422807
\(87\) 0 0
\(88\) −4.58258 −0.488504
\(89\) −8.75560 −0.928092 −0.464046 − 0.885811i \(-0.653603\pi\)
−0.464046 + 0.885811i \(0.653603\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −6.20520 −0.646937
\(93\) 0 0
\(94\) −1.25227 −0.129162
\(95\) −15.6838 −1.60912
\(96\) 0 0
\(97\) 7.58258 0.769894 0.384947 − 0.922939i \(-0.374220\pi\)
0.384947 + 0.922939i \(0.374220\pi\)
\(98\) 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.a.i.1.3 yes 4
3.2 odd 2 inner 567.2.a.i.1.2 ✓ 4
4.3 odd 2 9072.2.a.ci.1.4 4
7.6 odd 2 3969.2.a.u.1.3 4
9.2 odd 6 567.2.f.n.190.3 8
9.4 even 3 567.2.f.n.379.2 8
9.5 odd 6 567.2.f.n.379.3 8
9.7 even 3 567.2.f.n.190.2 8
12.11 even 2 9072.2.a.ci.1.1 4
21.20 even 2 3969.2.a.u.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.2 ✓ 4 3.2 odd 2 inner
567.2.a.i.1.3 yes 4 1.1 even 1 trivial
567.2.f.n.190.2 8 9.7 even 3
567.2.f.n.190.3 8 9.2 odd 6
567.2.f.n.379.2 8 9.4 even 3
567.2.f.n.379.3 8 9.5 odd 6
3969.2.a.u.1.2 4 21.20 even 2
3969.2.a.u.1.3 4 7.6 odd 2
9072.2.a.ci.1.1 4 12.11 even 2
9072.2.a.ci.1.4 4 4.3 odd 2