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Properties

Label 567.2.a.c.1.2
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 1.2
Root \(-0.347296\) 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)
 
\(f(q)\) \(=\) \(q-1.34730 q^{2} -0.184793 q^{4} -2.53209 q^{5} +1.00000 q^{7} +2.94356 q^{8} +3.41147 q^{10} -0.467911 q^{11} +5.82295 q^{13} -1.34730 q^{14} -3.59627 q^{16} -3.87939 q^{17} -2.18479 q^{19} +0.467911 q^{20} +0.630415 q^{22} +0.106067 q^{23} +1.41147 q^{25} -7.84524 q^{26} -0.184793 q^{28} -8.78106 q^{29} -7.68004 q^{31} -1.04189 q^{32} +5.22668 q^{34} -2.53209 q^{35} -7.68004 q^{37} +2.94356 q^{38} -7.45336 q^{40} +2.22668 q^{41} +1.22668 q^{43} +0.0864665 q^{44} -0.142903 q^{46} +5.33275 q^{47} +1.00000 q^{49} -1.90167 q^{50} -1.07604 q^{52} +0.716881 q^{53} +1.18479 q^{55} +2.94356 q^{56} +11.8307 q^{58} -0.736482 q^{59} +0.958111 q^{61} +10.3473 q^{62} +8.59627 q^{64} -14.7442 q^{65} -9.63816 q^{67} +0.716881 q^{68} +3.41147 q^{70} -13.2344 q^{71} -10.2686 q^{73} +10.3473 q^{74} +0.403733 q^{76} -0.467911 q^{77} -12.6382 q^{79} +9.10607 q^{80} -3.00000 q^{82} +2.73143 q^{83} +9.82295 q^{85} -1.65270 q^{86} -1.37733 q^{88} +8.11381 q^{89} +5.82295 q^{91} -0.0196004 q^{92} -7.18479 q^{94} +5.53209 q^{95} -13.6040 q^{97} -1.34730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} + 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} + 3 q^{26} + 3 q^{28} - 9 q^{29} - 3 q^{31}+ \cdots - 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34730 −0.952682 −0.476341 − 0.879261i \(-0.658037\pi\)
−0.476341 + 0.879261i \(0.658037\pi\)
\(3\) 0 0
\(4\) −0.184793 −0.0923963
\(5\) −2.53209 −1.13238 −0.566192 − 0.824273i \(-0.691584\pi\)
−0.566192 + 0.824273i \(0.691584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.94356 1.04071
\(9\) 0 0
\(10\) 3.41147 1.07880
\(11\) −0.467911 −0.141081 −0.0705403 − 0.997509i \(-0.522472\pi\)
−0.0705403 + 0.997509i \(0.522472\pi\)
\(12\) 0 0
\(13\) 5.82295 1.61500 0.807498 − 0.589871i \(-0.200821\pi\)
0.807498 + 0.589871i \(0.200821\pi\)
\(14\) −1.34730 −0.360080
\(15\) 0 0
\(16\) −3.59627 −0.899067
\(17\) −3.87939 −0.940889 −0.470445 − 0.882430i \(-0.655906\pi\)
−0.470445 + 0.882430i \(0.655906\pi\)
\(18\) 0 0
\(19\) −2.18479 −0.501226 −0.250613 − 0.968087i \(-0.580632\pi\)
−0.250613 + 0.968087i \(0.580632\pi\)
\(20\) 0.467911 0.104628
\(21\) 0 0
\(22\) 0.630415 0.134405
\(23\) 0.106067 0.0221165 0.0110582 − 0.999939i \(-0.496480\pi\)
0.0110582 + 0.999939i \(0.496480\pi\)
\(24\) 0 0
\(25\) 1.41147 0.282295
\(26\) −7.84524 −1.53858
\(27\) 0 0
\(28\) −0.184793 −0.0349225
\(29\) −8.78106 −1.63060 −0.815301 − 0.579038i \(-0.803428\pi\)
−0.815301 + 0.579038i \(0.803428\pi\)
\(30\) 0 0
\(31\) −7.68004 −1.37938 −0.689688 − 0.724106i \(-0.742252\pi\)
−0.689688 + 0.724106i \(0.742252\pi\)
\(32\) −1.04189 −0.184182
\(33\) 0 0
\(34\) 5.22668 0.896368
\(35\) −2.53209 −0.428001
\(36\) 0 0
\(37\) −7.68004 −1.26259 −0.631296 − 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 2.94356 0.477509
\(39\) 0 0
\(40\) −7.45336 −1.17848
\(41\) 2.22668 0.347749 0.173875 − 0.984768i \(-0.444371\pi\)
0.173875 + 0.984768i \(0.444371\pi\)
\(42\) 0 0
\(43\) 1.22668 0.187067 0.0935336 − 0.995616i \(-0.470184\pi\)
0.0935336 + 0.995616i \(0.470184\pi\)
\(44\) 0.0864665 0.0130353
\(45\) 0 0
\(46\) −0.142903 −0.0210700
\(47\) 5.33275 0.777861 0.388931 − 0.921267i \(-0.372845\pi\)
0.388931 + 0.921267i \(0.372845\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.90167 −0.268937
\(51\) 0 0
\(52\) −1.07604 −0.149220
\(53\) 0.716881 0.0984712 0.0492356 − 0.998787i \(-0.484321\pi\)
0.0492356 + 0.998787i \(0.484321\pi\)
\(54\) 0 0
\(55\) 1.18479 0.159757
\(56\) 2.94356 0.393350
\(57\) 0 0
\(58\) 11.8307 1.55345
\(59\) −0.736482 −0.0958818 −0.0479409 − 0.998850i \(-0.515266\pi\)
−0.0479409 + 0.998850i \(0.515266\pi\)
\(60\) 0 0
\(61\) 0.958111 0.122674 0.0613368 − 0.998117i \(-0.480464\pi\)
0.0613368 + 0.998117i \(0.480464\pi\)
\(62\) 10.3473 1.31411
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) −14.7442 −1.82880
\(66\) 0 0
\(67\) −9.63816 −1.17749 −0.588744 − 0.808320i \(-0.700377\pi\)
−0.588744 + 0.808320i \(0.700377\pi\)
\(68\) 0.716881 0.0869346
\(69\) 0 0
\(70\) 3.41147 0.407749
\(71\) −13.2344 −1.57064 −0.785318 − 0.619092i \(-0.787501\pi\)
−0.785318 + 0.619092i \(0.787501\pi\)
\(72\) 0 0
\(73\) −10.2686 −1.20185 −0.600923 − 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(74\) 10.3473 1.20285
\(75\) 0 0
\(76\) 0.403733 0.0463114
\(77\) −0.467911 −0.0533234
\(78\) 0 0
\(79\) −12.6382 −1.42190 −0.710952 − 0.703241i \(-0.751736\pi\)
−0.710952 + 0.703241i \(0.751736\pi\)
\(80\) 9.10607 1.01809
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 2.73143 0.299813 0.149907 − 0.988700i \(-0.452103\pi\)
0.149907 + 0.988700i \(0.452103\pi\)
\(84\) 0 0
\(85\) 9.82295 1.06545
\(86\) −1.65270 −0.178216
\(87\) 0 0
\(88\) −1.37733 −0.146823
\(89\) 8.11381 0.860062 0.430031 − 0.902814i \(-0.358503\pi\)
0.430031 + 0.902814i \(0.358503\pi\)
\(90\) 0 0
\(91\) 5.82295 0.610411
\(92\) −0.0196004 −0.00204348
\(93\) 0 0
\(94\) −7.18479 −0.741055
\(95\) 5.53209 0.567580
\(96\) 0 0
\(97\) −13.6040 −1.38128 −0.690639 − 0.723200i \(-0.742671\pi\)
−0.690639 + 0.723200i \(0.742671\pi\)
\(98\) −1.34730 −0.136097
\(99\) 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.1.2 3
3.2 odd 2 567.2.a.h.1.2 3
4.3 odd 2 9072.2.a.bs.1.1 3
7.6 odd 2 3969.2.a.l.1.2 3
9.2 odd 6 63.2.f.a.22.2 ✓ 6
9.4 even 3 189.2.f.b.127.2 6
9.5 odd 6 63.2.f.a.43.2 yes 6
9.7 even 3 189.2.f.b.64.2 6
12.11 even 2 9072.2.a.ca.1.3 3
21.20 even 2 3969.2.a.q.1.2 3
36.7 odd 6 3024.2.r.k.1009.3 6
36.11 even 6 1008.2.r.h.337.1 6
36.23 even 6 1008.2.r.h.673.1 6
36.31 odd 6 3024.2.r.k.2017.3 6
63.2 odd 6 441.2.g.c.67.2 6
63.4 even 3 1323.2.g.d.667.2 6
63.5 even 6 441.2.h.e.214.2 6
63.11 odd 6 441.2.h.d.373.2 6
63.13 odd 6 1323.2.f.d.883.2 6
63.16 even 3 1323.2.g.d.361.2 6
63.20 even 6 441.2.f.c.148.2 6
63.23 odd 6 441.2.h.d.214.2 6
63.25 even 3 1323.2.h.c.226.2 6
63.31 odd 6 1323.2.g.e.667.2 6
63.32 odd 6 441.2.g.c.79.2 6
63.34 odd 6 1323.2.f.d.442.2 6
63.38 even 6 441.2.h.e.373.2 6
63.40 odd 6 1323.2.h.b.802.2 6
63.41 even 6 441.2.f.c.295.2 6
63.47 even 6 441.2.g.b.67.2 6
63.52 odd 6 1323.2.h.b.226.2 6
63.58 even 3 1323.2.h.c.802.2 6
63.59 even 6 441.2.g.b.79.2 6
63.61 odd 6 1323.2.g.e.361.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 9.2 odd 6
63.2.f.a.43.2 yes 6 9.5 odd 6
189.2.f.b.64.2 6 9.7 even 3
189.2.f.b.127.2 6 9.4 even 3
441.2.f.c.148.2 6 63.20 even 6
441.2.f.c.295.2 6 63.41 even 6
441.2.g.b.67.2 6 63.47 even 6
441.2.g.b.79.2 6 63.59 even 6
441.2.g.c.67.2 6 63.2 odd 6
441.2.g.c.79.2 6 63.32 odd 6
441.2.h.d.214.2 6 63.23 odd 6
441.2.h.d.373.2 6 63.11 odd 6
441.2.h.e.214.2 6 63.5 even 6
441.2.h.e.373.2 6 63.38 even 6
567.2.a.c.1.2 3 1.1 even 1 trivial
567.2.a.h.1.2 3 3.2 odd 2
1008.2.r.h.337.1 6 36.11 even 6
1008.2.r.h.673.1 6 36.23 even 6
1323.2.f.d.442.2 6 63.34 odd 6
1323.2.f.d.883.2 6 63.13 odd 6
1323.2.g.d.361.2 6 63.16 even 3
1323.2.g.d.667.2 6 63.4 even 3
1323.2.g.e.361.2 6 63.61 odd 6
1323.2.g.e.667.2 6 63.31 odd 6
1323.2.h.b.226.2 6 63.52 odd 6
1323.2.h.b.802.2 6 63.40 odd 6
1323.2.h.c.226.2 6 63.25 even 3
1323.2.h.c.802.2 6 63.58 even 3
3024.2.r.k.1009.3 6 36.7 odd 6
3024.2.r.k.2017.3 6 36.31 odd 6
3969.2.a.l.1.2 3 7.6 odd 2
3969.2.a.q.1.2 3 21.20 even 2
9072.2.a.bs.1.1 3 4.3 odd 2
9072.2.a.ca.1.3 3 12.11 even 2