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Properties

Label 567.2.a.i.1.1
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 1.1
Root \(0.456850\) 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)
 
\(f(q)\) \(=\) \(q-2.18890 q^{2} +2.79129 q^{4} -0.913701 q^{5} -1.00000 q^{7} -1.73205 q^{8} +2.00000 q^{10} -2.64575 q^{11} +4.00000 q^{13} +2.18890 q^{14} -1.79129 q^{16} -3.46410 q^{17} +5.58258 q^{19} -2.55040 q^{20} +5.79129 q^{22} +3.46410 q^{23} -4.16515 q^{25} -8.75560 q^{26} -2.79129 q^{28} +8.75560 q^{29} -9.16515 q^{31} +7.38505 q^{32} +7.58258 q^{34} +0.913701 q^{35} +3.00000 q^{37} -12.2197 q^{38} +1.58258 q^{40} +0.913701 q^{41} +0.582576 q^{43} -7.38505 q^{44} -7.58258 q^{46} +13.1334 q^{47} +1.00000 q^{49} +9.11710 q^{50} +11.1652 q^{52} +8.66025 q^{53} +2.41742 q^{55} +1.73205 q^{56} -19.1652 q^{58} -3.46410 q^{59} +11.5826 q^{61} +20.0616 q^{62} -12.5826 q^{64} -3.65480 q^{65} +8.58258 q^{67} -9.66930 q^{68} -2.00000 q^{70} +4.47315 q^{71} +15.1652 q^{73} -6.56670 q^{74} +15.5826 q^{76} +2.64575 q^{77} +0.582576 q^{79} +1.63670 q^{80} -2.00000 q^{82} +9.66930 q^{83} +3.16515 q^{85} -1.27520 q^{86} +4.58258 q^{88} +1.82740 q^{89} -4.00000 q^{91} +9.66930 q^{92} -28.7477 q^{94} -5.10080 q^{95} -1.58258 q^{97} -2.18890 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{7} + 8 q^{10} + 16 q^{13} + 2 q^{16} + 4 q^{19} + 14 q^{22} + 20 q^{25} - 2 q^{28} + 12 q^{34} + 12 q^{37} - 12 q^{40} - 16 q^{43} - 12 q^{46} + 4 q^{49} + 8 q^{52} + 28 q^{55} - 40 q^{58}+ \cdots + 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18890 −1.54779 −0.773893 − 0.633316i \(-0.781693\pi\)
−0.773893 + 0.633316i \(0.781693\pi\)
\(3\) 0 0
\(4\) 2.79129 1.39564
\(5\) −0.913701 −0.408619 −0.204310 − 0.978906i \(-0.565495\pi\)
−0.204310 + 0.978906i \(0.565495\pi\)
\(6\) 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 \(-0.630595\pi\)
−0.398862 + 0.917011i \(0.630595\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 2.18890 0.585008
\(15\) 0 0
\(16\) −1.79129 −0.447822
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 5.58258 1.28073 0.640365 − 0.768070i \(-0.278783\pi\)
0.640365 + 0.768070i \(0.278783\pi\)
\(20\) −2.55040 −0.570287
\(21\) 0 0
\(22\) 5.79129 1.23471
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) −4.16515 −0.833030
\(26\) −8.75560 −1.71712
\(27\) 0 0
\(28\) −2.79129 −0.527504
\(29\) 8.75560 1.62587 0.812937 − 0.582351i \(-0.197867\pi\)
0.812937 + 0.582351i \(0.197867\pi\)
\(30\) 0 0
\(31\) −9.16515 −1.64611 −0.823055 − 0.567962i \(-0.807732\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) 7.38505 1.30551
\(33\) 0 0
\(34\) 7.58258 1.30040
\(35\) 0.913701 0.154444
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −12.2197 −1.98230
\(39\) 0 0
\(40\) 1.58258 0.250227
\(41\) 0.913701 0.142696 0.0713480 − 0.997451i \(-0.477270\pi\)
0.0713480 + 0.997451i \(0.477270\pi\)
\(42\) 0 0
\(43\) 0.582576 0.0888420 0.0444210 − 0.999013i \(-0.485856\pi\)
0.0444210 + 0.999013i \(0.485856\pi\)
\(44\) −7.38505 −1.11334
\(45\) 0 0
\(46\) −7.58258 −1.11799
\(47\) 13.1334 1.91570 0.957852 − 0.287262i \(-0.0927450\pi\)
0.957852 + 0.287262i \(0.0927450\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 9.11710 1.28935
\(51\) 0 0
\(52\) 11.1652 1.54833
\(53\) 8.66025 1.18958 0.594789 − 0.803882i \(-0.297236\pi\)
0.594789 + 0.803882i \(0.297236\pi\)
\(54\) 0 0
\(55\) 2.41742 0.325965
\(56\) 1.73205 0.231455
\(57\) 0 0
\(58\) −19.1652 −2.51651
\(59\) −3.46410 −0.450988 −0.225494 − 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 11.5826 1.48300 0.741498 − 0.670955i \(-0.234115\pi\)
0.741498 + 0.670955i \(0.234115\pi\)
\(62\) 20.0616 2.54783
\(63\) 0 0
\(64\) −12.5826 −1.57282
\(65\) −3.65480 −0.453322
\(66\) 0 0
\(67\) 8.58258 1.04853 0.524264 − 0.851556i \(-0.324340\pi\)
0.524264 + 0.851556i \(0.324340\pi\)
\(68\) −9.66930 −1.17258
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 4.47315 0.530866 0.265433 − 0.964129i \(-0.414485\pi\)
0.265433 + 0.964129i \(0.414485\pi\)
\(72\) 0 0
\(73\) 15.1652 1.77495 0.887473 − 0.460859i \(-0.152459\pi\)
0.887473 + 0.460859i \(0.152459\pi\)
\(74\) −6.56670 −0.763364
\(75\) 0 0
\(76\) 15.5826 1.78744
\(77\) 2.64575 0.301511
\(78\) 0 0
\(79\) 0.582576 0.0655449 0.0327724 − 0.999463i \(-0.489566\pi\)
0.0327724 + 0.999463i \(0.489566\pi\)
\(80\) 1.63670 0.182989
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) 9.66930 1.06134 0.530672 − 0.847577i \(-0.321940\pi\)
0.530672 + 0.847577i \(0.321940\pi\)
\(84\) 0 0
\(85\) 3.16515 0.343309
\(86\) −1.27520 −0.137508
\(87\) 0 0
\(88\) 4.58258 0.488504
\(89\) 1.82740 0.193704 0.0968521 − 0.995299i \(-0.469123\pi\)
0.0968521 + 0.995299i \(0.469123\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 9.66930 1.00809
\(93\) 0 0
\(94\) −28.7477 −2.96510
\(95\) −5.10080 −0.523331
\(96\) 0 0
\(97\) −1.58258 −0.160686 −0.0803431 − 0.996767i \(-0.525602\pi\)
−0.0803431 + 0.996767i \(0.525602\pi\)
\(98\) −2.18890 −0.221112
\(99\) 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.1.1 ✓ 4
3.2 odd 2 inner 567.2.a.i.1.4 yes 4
4.3 odd 2 9072.2.a.ci.1.2 4
7.6 odd 2 3969.2.a.u.1.1 4
9.2 odd 6 567.2.f.n.190.1 8
9.4 even 3 567.2.f.n.379.4 8
9.5 odd 6 567.2.f.n.379.1 8
9.7 even 3 567.2.f.n.190.4 8
12.11 even 2 9072.2.a.ci.1.3 4
21.20 even 2 3969.2.a.u.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.1 ✓ 4 1.1 even 1 trivial
567.2.a.i.1.4 yes 4 3.2 odd 2 inner
567.2.f.n.190.1 8 9.2 odd 6
567.2.f.n.190.4 8 9.7 even 3
567.2.f.n.379.1 8 9.5 odd 6
567.2.f.n.379.4 8 9.4 even 3
3969.2.a.u.1.1 4 7.6 odd 2
3969.2.a.u.1.4 4 21.20 even 2
9072.2.a.ci.1.2 4 4.3 odd 2
9072.2.a.ci.1.3 4 12.11 even 2