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Properties

Label 567.2.a.c.1.3
Level $567$
Weight $2$
Character 567.1
Self dual yes
Analytic conductor $4.528$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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 = [3,-3,0,3,-3] 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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.87939\) 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)
 
\(f(q)\) \(=\) \(q+0.879385 q^{2} -1.22668 q^{4} -1.34730 q^{5} +1.00000 q^{7} -2.83750 q^{8} -1.18479 q^{10} -1.65270 q^{11} -3.36959 q^{13} +0.879385 q^{14} -0.0418891 q^{16} -0.467911 q^{17} -3.22668 q^{19} +1.65270 q^{20} -1.45336 q^{22} -8.94356 q^{23} -3.18479 q^{25} -2.96316 q^{26} -1.22668 q^{28} -6.26857 q^{29} +9.23442 q^{31} +5.63816 q^{32} -0.411474 q^{34} -1.34730 q^{35} +9.23442 q^{37} -2.83750 q^{38} +3.82295 q^{40} -3.41147 q^{41} -4.41147 q^{43} +2.02734 q^{44} -7.86484 q^{46} -9.35504 q^{47} +1.00000 q^{49} -2.80066 q^{50} +4.13341 q^{52} +0.573978 q^{53} +2.22668 q^{55} -2.83750 q^{56} -5.51249 q^{58} +10.3969 q^{59} +7.63816 q^{61} +8.12061 q^{62} +5.04189 q^{64} +4.53983 q^{65} +0.596267 q^{67} +0.573978 q^{68} -1.18479 q^{70} +0.554378 q^{71} +2.04963 q^{73} +8.12061 q^{74} +3.95811 q^{76} -1.65270 q^{77} -2.40373 q^{79} +0.0564370 q^{80} -3.00000 q^{82} +15.0496 q^{83} +0.630415 q^{85} -3.87939 q^{86} +4.68954 q^{88} -9.08647 q^{89} -3.36959 q^{91} +10.9709 q^{92} -8.22668 q^{94} +4.34730 q^{95} -1.89899 q^{97} +0.879385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} + 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} + 3 q^{26} + 3 q^{28} - 9 q^{29} - 3 q^{31}+ \cdots - 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.879385 0.621819 0.310910 − 0.950439i \(-0.399366\pi\)
0.310910 + 0.950439i \(0.399366\pi\)
\(3\) 0 0
\(4\) −1.22668 −0.613341
\(5\) −1.34730 −0.602529 −0.301265 − 0.953541i \(-0.597409\pi\)
−0.301265 + 0.953541i \(0.597409\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.83750 −1.00321
\(9\) 0 0
\(10\) −1.18479 −0.374664
\(11\) −1.65270 −0.498309 −0.249154 − 0.968464i \(-0.580153\pi\)
−0.249154 + 0.968464i \(0.580153\pi\)
\(12\) 0 0
\(13\) −3.36959 −0.934555 −0.467277 − 0.884111i \(-0.654765\pi\)
−0.467277 + 0.884111i \(0.654765\pi\)
\(14\) 0.879385 0.235026
\(15\) 0 0
\(16\) −0.0418891 −0.0104723
\(17\) −0.467911 −0.113485 −0.0567426 − 0.998389i \(-0.518071\pi\)
−0.0567426 + 0.998389i \(0.518071\pi\)
\(18\) 0 0
\(19\) −3.22668 −0.740252 −0.370126 − 0.928982i \(-0.620685\pi\)
−0.370126 + 0.928982i \(0.620685\pi\)
\(20\) 1.65270 0.369556
\(21\) 0 0
\(22\) −1.45336 −0.309858
\(23\) −8.94356 −1.86486 −0.932431 − 0.361348i \(-0.882317\pi\)
−0.932431 + 0.361348i \(0.882317\pi\)
\(24\) 0 0
\(25\) −3.18479 −0.636959
\(26\) −2.96316 −0.581124
\(27\) 0 0
\(28\) −1.22668 −0.231821
\(29\) −6.26857 −1.16404 −0.582022 − 0.813173i \(-0.697738\pi\)
−0.582022 + 0.813173i \(0.697738\pi\)
\(30\) 0 0
\(31\) 9.23442 1.65855 0.829276 − 0.558840i \(-0.188753\pi\)
0.829276 + 0.558840i \(0.188753\pi\)
\(32\) 5.63816 0.996695
\(33\) 0 0
\(34\) −0.411474 −0.0705672
\(35\) −1.34730 −0.227735
\(36\) 0 0
\(37\) 9.23442 1.51813 0.759065 − 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) −2.83750 −0.460303
\(39\) 0 0
\(40\) 3.82295 0.604461
\(41\) −3.41147 −0.532783 −0.266391 − 0.963865i \(-0.585831\pi\)
−0.266391 + 0.963865i \(0.585831\pi\)
\(42\) 0 0
\(43\) −4.41147 −0.672743 −0.336372 − 0.941729i \(-0.609200\pi\)
−0.336372 + 0.941729i \(0.609200\pi\)
\(44\) 2.02734 0.305633
\(45\) 0 0
\(46\) −7.86484 −1.15961
\(47\) −9.35504 −1.36457 −0.682286 − 0.731085i \(-0.739014\pi\)
−0.682286 + 0.731085i \(0.739014\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.80066 −0.396073
\(51\) 0 0
\(52\) 4.13341 0.573201
\(53\) 0.573978 0.0788419 0.0394210 − 0.999223i \(-0.487449\pi\)
0.0394210 + 0.999223i \(0.487449\pi\)
\(54\) 0 0
\(55\) 2.22668 0.300246
\(56\) −2.83750 −0.379176
\(57\) 0 0
\(58\) −5.51249 −0.723825
\(59\) 10.3969 1.35356 0.676782 − 0.736183i \(-0.263374\pi\)
0.676782 + 0.736183i \(0.263374\pi\)
\(60\) 0 0
\(61\) 7.63816 0.977966 0.488983 − 0.872293i \(-0.337368\pi\)
0.488983 + 0.872293i \(0.337368\pi\)
\(62\) 8.12061 1.03132
\(63\) 0 0
\(64\) 5.04189 0.630236
\(65\) 4.53983 0.563097
\(66\) 0 0
\(67\) 0.596267 0.0728456 0.0364228 − 0.999336i \(-0.488404\pi\)
0.0364228 + 0.999336i \(0.488404\pi\)
\(68\) 0.573978 0.0696051
\(69\) 0 0
\(70\) −1.18479 −0.141610
\(71\) 0.554378 0.0657925 0.0328963 − 0.999459i \(-0.489527\pi\)
0.0328963 + 0.999459i \(0.489527\pi\)
\(72\) 0 0
\(73\) 2.04963 0.239891 0.119946 − 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) 8.12061 0.944002
\(75\) 0 0
\(76\) 3.95811 0.454026
\(77\) −1.65270 −0.188343
\(78\) 0 0
\(79\) −2.40373 −0.270441 −0.135221 − 0.990816i \(-0.543174\pi\)
−0.135221 + 0.990816i \(0.543174\pi\)
\(80\) 0.0564370 0.00630985
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 15.0496 1.65191 0.825956 − 0.563735i \(-0.190636\pi\)
0.825956 + 0.563735i \(0.190636\pi\)
\(84\) 0 0
\(85\) 0.630415 0.0683781
\(86\) −3.87939 −0.418325
\(87\) 0 0
\(88\) 4.68954 0.499907
\(89\) −9.08647 −0.963164 −0.481582 − 0.876401i \(-0.659938\pi\)
−0.481582 + 0.876401i \(0.659938\pi\)
\(90\) 0 0
\(91\) −3.36959 −0.353228
\(92\) 10.9709 1.14380
\(93\) 0 0
\(94\) −8.22668 −0.848517
\(95\) 4.34730 0.446023
\(96\) 0 0
\(97\) −1.89899 −0.192813 −0.0964064 − 0.995342i \(-0.530735\pi\)
−0.0964064 + 0.995342i \(0.530735\pi\)
\(98\) 0.879385 0.0888313
\(99\) 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.c.1.3 3
3.2 odd 2 567.2.a.h.1.1 3
4.3 odd 2 9072.2.a.bs.1.2 3
7.6 odd 2 3969.2.a.l.1.3 3
9.2 odd 6 63.2.f.a.22.3 ✓ 6
9.4 even 3 189.2.f.b.127.1 6
9.5 odd 6 63.2.f.a.43.3 yes 6
9.7 even 3 189.2.f.b.64.1 6
12.11 even 2 9072.2.a.ca.1.2 3
21.20 even 2 3969.2.a.q.1.1 3
36.7 odd 6 3024.2.r.k.1009.2 6
36.11 even 6 1008.2.r.h.337.2 6
36.23 even 6 1008.2.r.h.673.2 6
36.31 odd 6 3024.2.r.k.2017.2 6
63.2 odd 6 441.2.g.c.67.3 6
63.4 even 3 1323.2.g.d.667.1 6
63.5 even 6 441.2.h.e.214.1 6
63.11 odd 6 441.2.h.d.373.1 6
63.13 odd 6 1323.2.f.d.883.1 6
63.16 even 3 1323.2.g.d.361.1 6
63.20 even 6 441.2.f.c.148.3 6
63.23 odd 6 441.2.h.d.214.1 6
63.25 even 3 1323.2.h.c.226.3 6
63.31 odd 6 1323.2.g.e.667.1 6
63.32 odd 6 441.2.g.c.79.3 6
63.34 odd 6 1323.2.f.d.442.1 6
63.38 even 6 441.2.h.e.373.1 6
63.40 odd 6 1323.2.h.b.802.3 6
63.41 even 6 441.2.f.c.295.3 6
63.47 even 6 441.2.g.b.67.3 6
63.52 odd 6 1323.2.h.b.226.3 6
63.58 even 3 1323.2.h.c.802.3 6
63.59 even 6 441.2.g.b.79.3 6
63.61 odd 6 1323.2.g.e.361.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 9.2 odd 6
63.2.f.a.43.3 yes 6 9.5 odd 6
189.2.f.b.64.1 6 9.7 even 3
189.2.f.b.127.1 6 9.4 even 3
441.2.f.c.148.3 6 63.20 even 6
441.2.f.c.295.3 6 63.41 even 6
441.2.g.b.67.3 6 63.47 even 6
441.2.g.b.79.3 6 63.59 even 6
441.2.g.c.67.3 6 63.2 odd 6
441.2.g.c.79.3 6 63.32 odd 6
441.2.h.d.214.1 6 63.23 odd 6
441.2.h.d.373.1 6 63.11 odd 6
441.2.h.e.214.1 6 63.5 even 6
441.2.h.e.373.1 6 63.38 even 6
567.2.a.c.1.3 3 1.1 even 1 trivial
567.2.a.h.1.1 3 3.2 odd 2
1008.2.r.h.337.2 6 36.11 even 6
1008.2.r.h.673.2 6 36.23 even 6
1323.2.f.d.442.1 6 63.34 odd 6
1323.2.f.d.883.1 6 63.13 odd 6
1323.2.g.d.361.1 6 63.16 even 3
1323.2.g.d.667.1 6 63.4 even 3
1323.2.g.e.361.1 6 63.61 odd 6
1323.2.g.e.667.1 6 63.31 odd 6
1323.2.h.b.226.3 6 63.52 odd 6
1323.2.h.b.802.3 6 63.40 odd 6
1323.2.h.c.226.3 6 63.25 even 3
1323.2.h.c.802.3 6 63.58 even 3
3024.2.r.k.1009.2 6 36.7 odd 6
3024.2.r.k.2017.2 6 36.31 odd 6
3969.2.a.l.1.3 3 7.6 odd 2
3969.2.a.q.1.1 3 21.20 even 2
9072.2.a.bs.1.2 3 4.3 odd 2
9072.2.a.ca.1.2 3 12.11 even 2